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Visual Representation

What is visual representation.

Visual Representation refers to the principles by which markings on a surface are made and interpreted. Designers use representations like typography and illustrations to communicate information, emotions and concepts. Color, imagery, typography and layout are crucial in this communication.

Alan Blackwell, cognition scientist and professor, gives a brief introduction to visual representation:

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We can see visual representation throughout human history, from cave drawings to data visualization :

Art uses visual representation to express emotions and abstract ideas.

Financial forecasting graphs condense data and research into a more straightforward format.

Icons on user interfaces (UI) represent different actions users can take.

The color of a notification indicates its nature and meaning.

Van Gogh's "The Starry Night" uses visuals to evoke deep emotions, representing an abstract, dreamy night sky. It exemplifies how art can communicate complex feelings and ideas.

© Public domain

Importance of Visual Representation in Design

Designers use visual representation for internal and external use throughout the design process . For example:

Storyboards are illustrations that outline users’ actions and where they perform them.

Sitemaps are diagrams that show the hierarchy and navigation structure of a website.

Wireframes are sketches that bring together elements of a user interface's structure.

Usability reports use graphs and charts to communicate data gathered from usability testing.

User interfaces visually represent information contained in applications and computerized devices.

This usability report is straightforward to understand. Yet, the data behind the visualizations could come from thousands of answered surveys.

© Interaction Design Foundation, CC BY-SA 4.0

Visual representation simplifies complex ideas and data and makes them easy to understand. Without these visual aids, designers would struggle to communicate their ideas, findings and products . For example, it would be easier to create a mockup of an e-commerce website interface than to describe it with words.

Visual representation simplifies the communication of designs. Without mockups, it would be difficult for developers to reproduce designs using words alone.

Types of Visual Representation

Below are some of the most common forms of visual representation designers use.

Text and Typography

Text represents language and ideas through written characters and symbols. Readers visually perceive and interpret these characters. Typography turns text into a visual form, influencing its perception and interpretation.

We have developed the conventions of typography over centuries , for example, in documents, newspapers and magazines. These conventions include:

Text arranged on a grid brings clarity and structure. Gridded text makes complex information easier to navigate and understand. Tables, columns and other formats help organize content logically and enhance readability.

Contrasting text sizes create a visual hierarchy and draw attention to critical areas. For example, headings use larger text while body copy uses smaller text. This contrast helps readers distinguish between primary and secondary information.

Adequate spacing and paragraphing improve the readability and appearance of the text. These conventions prevent the content from appearing cluttered. Spacing and paragraphing make it easier for the eye to follow and for the brain to process the information.

Balanced image-to-text ratios create engaging layouts. Images break the monotony of text, provide visual relief and illustrate or emphasize points made in the text. A well-planned ratio ensures neither text nor images overwhelm each other. Effective ratios make designs more effective and appealing.

Designers use these conventions because people are familiar with them and better understand text presented in this manner.

This table of funerals from the plague in London in 1665 uses typographic conventions still used today. For example, the author arranged the information in a table and used contrasting text styling to highlight information in the header.

Illustrations and Drawings

Designers use illustrations and drawings independently or alongside text. An example of illustration used to communicate information is the assembly instructions created by furniture retailer IKEA. If IKEA used text instead of illustrations in their instructions, people would find it harder to assemble the furniture.

IKEA assembly instructions use illustrations to inform customers how to build their furniture. The only text used is numeric to denote step and part numbers. IKEA communicates this information visually to: 1. Enable simple communication, 2. Ensure their instructions are easy to follow, regardless of the customer’s language.

© IKEA, Fair use

Illustrations and drawings can often convey the core message of a visual representation more effectively than a photograph. They focus on the core message , while a photograph might distract a viewer with additional details (such as who this person is, where they are from, etc.)

For example, in IKEA’s case, photographing a person building a piece of furniture might be complicated. Further, photographs may not be easy to understand in a black-and-white print, leading to higher printing costs. To be useful, the pictures would also need to be larger and would occupy more space on a printed manual, further adding to the costs.

But imagine a girl winking—this is something we can easily photograph. 

Ivan Sutherland, creator of the first graphical user interface, used his computer program Sketchpad to draw a winking girl. While not realistic, Sutherland's representation effectively portrays a winking girl. The drawing's abstract, generic elements contrast with the distinct winking eye. The graphical conventions of lines and shapes represent the eyes and mouth. The simplicity of the drawing does not draw attention away from the winking.

A photo might distract from the focused message compared to Sutherland's representation. In the photo, the other aspects of the image (i.e., the particular person) distract the viewer from this message.

© Ivan Sutherland, CC BY-SA 3.0 and Amina Filkins, Pexels License

Information and Data Visualization

Designers and other stakeholders use data and information visualization across many industries.

Data visualization uses charts and graphs to show raw data in a graphic form. Information visualization goes further, including more context and complex data sets. Information visualization often uses interactive elements to share a deeper understanding.

For example, most computerized devices have a battery level indicator. This is a type of data visualization. IV takes this further by allowing you to click on the battery indicator for further insights. These insights may include the apps that use the most battery and the last time you charged your device.

macOS displays a battery icon in the menu bar that visualizes your device’s battery level. This is an example of data visualization. Meanwhile, macOS’s settings tell you battery level over time, screen-on-usage and when you last charged your device. These insights are actionable; users may notice their battery drains at a specific time. This is an example of information visualization.

© Low Battery by Jemis Mali, CC BY-NC-ND 4.0, and Apple, Fair use

Information visualization is not exclusive to numeric data. It encompasses representations like diagrams and maps. For example, Google Maps collates various types of data and information into one interface:

Data Representation: Google Maps transforms complex geographical data into an easily understandable and navigable visual map.

Interactivity: Users can interactively customize views that show traffic, satellite imagery and more in real-time.

Layered Information: Google Maps layers multiple data types (e.g., traffic, weather) over geographical maps for comprehensive visualization.

User-Centered Design : The interface is intuitive and user-friendly, with symbols and colors for straightforward data interpretation.

The volume of data contained in one screenshot of Google Maps is massive. However, this information is presented clearly to the user. Google Maps highlights different terrains with colors and local places and businesses with icons and colors. The panel on the left lists the selected location’s profile, which includes an image, rating and contact information.

© Google, Fair use

Symbolic Correspondence

Symbolic correspondence uses universally recognized symbols and signs to convey specific meanings . This method employs widely recognized visual cues for immediate understanding. Symbolic correspondence removes the need for textual explanation.

For instance, a magnifying glass icon in UI design signifies the search function. Similarly, in environmental design, symbols for restrooms, parking and amenities guide visitors effectively.

The Interaction Design Foundation (IxDF) website uses the universal magnifying glass symbol to signify the search function. Similarly, the play icon draws attention to a link to watch a video.

How Designers Create Visual Representations

Visual language.

Designers use elements like color , shape and texture to create a communicative visual experience. Designers use these 8 principles:

Size – Larger elements tend to capture users' attention readily.

Color – Users are typically drawn to bright colors over muted shades.

Contrast – Colors with stark contrasts catch the eye more effectively.

Alignment – Unaligned elements are more noticeable than those aligned ones.

Repetition – Similar styles repeated imply a relationship in content.

Proximity – Elements placed near each other appear to be connected.

Whitespace – Elements surrounded by ample space attract the eye.

Texture and Style – Users often notice richer textures before flat designs.

The 8 visual design principles.

In web design , visual hierarchy uses color and repetition to direct the user's attention. Color choice is crucial as it creates contrast between different elements. Repetition helps to organize the design—it uses recurring elements to establish consistency and familiarity.

In this video, Alan Dix, Professor and Expert in Human-Computer Interaction, explains how visual alignment affects how we read and absorb information:

Correspondence Techniques

Designers use correspondence techniques to align visual elements with their conceptual meanings. These techniques include color coding, spatial arrangement and specific imagery. In information visualization, different colors can represent various data sets. This correspondence aids users in quickly identifying trends and relationships .

Color coding enables the stakeholder to see the relationship and trend between the two pie charts easily.

In user interface design, correspondence techniques link elements with meaning. An example is color-coding notifications to state their nature. For instance, red for warnings and green for confirmation. These techniques are informative and intuitive and enhance the user experience.

The IxDF website uses blue for call-to-actions (CTAs) and red for warnings. These colors inform the user of the nature of the action of buttons and other interactive elements.

Perception and Interpretation

If visual language is how designers create representations, then visual perception and interpretation are how users receive those representations. Consider a painting—the viewer’s eyes take in colors, shapes and lines, and the brain perceives these visual elements as a painting.

In this video, Alan Dix explains how the interplay of sensation, perception and culture is crucial to understanding visual experiences in design:

Copyright holder: Michael Murphy _ Appearance time: 07:19 - 07:37 _ Link: https://www.youtube.com/watch?v=C67JuZnBBDc

Visual perception principles are essential for creating compelling, engaging visual representations. For example, Gestalt principles explain how we perceive visual information. These rules describe how we group similar items, spot patterns and simplify complex images. Designers apply Gestalt principles to arrange content on websites and other interfaces. This application creates visually appealing and easily understood designs.

In this video, design expert and teacher Mia Cinelli discusses the significance of Gestalt principles in visual design . She introduces fundamental principles, like figure/ground relationships, similarity and proximity.

Interpretation

Everyone's experiences, culture and physical abilities dictate how they interpret visual representations. For this reason, designers carefully consider how users interpret their visual representations. They employ user research and testing to ensure their designs are attractive and functional.

Leonardo da Vinci's "Mona Lisa", is one of the most famous paintings in the world. The piece is renowned for its subject's enigmatic expression. Some interpret her smile as content and serene, while others see it as sad or mischievous. Not everyone interprets this visual representation in the same way.

Color is an excellent example of how one person, compared to another, may interpret a visual element. Take the color red:

In Chinese culture, red symbolizes luck, while in some parts of Africa, it can mean death or illness.

A personal experience may mean a user has a negative or positive connotation with red.

People with protanopia and deuteranopia color blindness cannot distinguish between red and green.

In this video, Joann and Arielle Eckstut, leading color consultants and authors, explain how many factors influence how we perceive and interpret color:

Learn More about Visual Representation

Read Alan Blackwell’s chapter on visual representation from The Encyclopedia of Human-Computer Interaction.

Learn about the F-Shaped Pattern For Reading Web Content from Jakob Nielsen.

Read Smashing Magazine’s article, Visual Design Language: The Building Blocks Of Design .

Take the IxDF’s course, Perception and Memory in HCI and UX .

Questions related to Visual Representation

Some highly cited research on visual representation and related topics includes:

Roland, P. E., & Gulyás, B. (1994). Visual imagery and visual representation. Trends in Neurosciences, 17(7), 281-287. Roland and Gulyás' study explores how the brain creates visual imagination. They look at whether imagining things like objects and scenes uses the same parts of the brain as seeing them does. Their research shows the brain uses certain areas specifically for imagination. These areas are different from the areas used for seeing. This research is essential for understanding how our brain works with vision.

Lurie, N. H., & Mason, C. H. (2007). Visual Representation: Implications for Decision Making. Journal of Marketing, 71(1), 160-177.

This article looks at how visualization tools help in understanding complicated marketing data. It discusses how these tools affect decision-making in marketing. The article gives a detailed method to assess the impact of visuals on the study and combination of vast quantities of marketing data. It explores the benefits and possible biases visuals can bring to marketing choices. These factors make the article an essential resource for researchers and marketing experts. The article suggests using visual tools and detailed analysis together for the best results.

Lohse, G. L., Biolsi, K., Walker, N., & Rueter, H. H. (1994, December). A classification of visual representations. Communications of the ACM, 37(12), 36+.

This publication looks at how visuals help communicate and make information easier to understand. It divides these visuals into six types: graphs, tables, maps, diagrams, networks and icons. The article also looks at different ways these visuals share information effectively.

​​If you’d like to cite content from the IxDF website , click the ‘cite this article’ button near the top of your screen.

Some recommended books on visual representation and related topics include:

Chaplin, E. (1994). Sociology and Visual Representation (1st ed.) . Routledge.

Chaplin's book describes how visual art analysis has changed from ancient times to today. It shows how photography, post-modernism and feminism have changed how we see art. The book combines words and images in its analysis and looks into real-life social sciences studies.

Mitchell, W. J. T. (1994). Picture Theory. The University of Chicago Press.

Mitchell's book explores the important role and meaning of pictures in the late twentieth century. It discusses the change from focusing on language to focusing on images in cultural studies. The book deeply examines the interaction between images and text in different cultural forms like literature, art and media. This detailed study of how we see and read visual representations has become an essential reference for scholars and professionals.

Koffka, K. (1935). Principles of Gestalt Psychology. Harcourt, Brace & World.

"Principles of Gestalt Psychology" by Koffka, released in 1935, is a critical book in its field. It's known as a foundational work in Gestalt psychology, laying out the basic ideas of the theory and how they apply to how we see and think. Koffka's thorough study of Gestalt psychology's principles has profoundly influenced how we understand human perception. This book has been a significant reference in later research and writings.

A visual representation, like an infographic or chart, uses visual elements to show information or data. These types of visuals make complicated information easier to understand and more user-friendly.

Designers harness visual representations in design and communication. Infographics and charts, for instance, distill data for easier audience comprehension and retention.

For an introduction to designing basic information visualizations, take our course, Information Visualization .

Text is a crucial design and communication element, transforming language visually. Designers use font style, size, color and layout to convey emotions and messages effectively.

Designers utilize text for both literal communication and aesthetic enhancement. Their typography choices significantly impact design aesthetics, user experience and readability.

Designers should always consider text's visual impact in their designs. This consideration includes font choice, placement, color and interaction with other design elements.

In this video, design expert and teacher Mia Cinelli teaches how Gestalt principles apply to typography:

Designers use visual elements in projects to convey information, ideas, and messages. Designers use images, colors, shapes and typography for impactful designs.

In UI/UX design, visual representation is vital. Icons, buttons and colors provide contrast for intuitive, user-friendly website and app interfaces.

Graphic design leverages visual representation to create attention-grabbing marketing materials. Careful color, imagery and layout choices create an emotional connection.

Product design relies on visual representation for prototyping and idea presentation. Designers and stakeholders use visual representations to envision functional, aesthetically pleasing products.

Our brains process visuals 60,000 times faster than text. This fact highlights the crucial role of visual representation in design.

Our course, Visual Design: The Ultimate Guide , teaches you how to use visual design elements and principles in your work effectively.

Visual representation, crucial in UX, facilitates interaction, comprehension and emotion. It combines elements like images and typography for better interfaces.

Effective visuals guide users, highlight features and improve navigation. Icons and color schemes communicate functions and set interaction tones.

UX design research shows visual elements significantly impact emotions. 90% of brain-transmitted information is visual.

To create functional, accessible visuals, designers use color contrast and consistent iconography. These elements improve readability and inclusivity.

An excellent example of visual representation in UX is Apple's iOS interface. iOS combines a clean, minimalist design with intuitive navigation. As a result, the operating system is both visually appealing and user-friendly.

Michal Malewicz, Creative Director and CEO at Hype4, explains why visual skills are important in design:

Learn more about UI design from Michal in our Master Class, Beyond Interfaces: The UI Design Skills You Need to Know .

The fundamental principles of effective visual representation are:

Clarity : Designers convey messages clearly, avoiding clutter.

Simplicity : Embrace simple designs for ease and recall.

Emphasis : Designers highlight key elements distinctively.

Balance : Balance ensures design stability and structure.

Alignment : Designers enhance coherence through alignment.

Contrast : Use contrast for dynamic, distinct designs.

Repetition : Repeating elements unify and guide designs.

Designers practice these principles in their projects. They also analyze successful designs and seek feedback to improve their skills.

Read our topic description of Gestalt principles to learn more about creating effective visual designs. The Gestalt principles explain how humans group elements, recognize patterns and simplify object perception.

Color theory is vital in design, helping designers craft visually appealing and compelling works. Designers understand color interactions, psychological impacts and symbolism. These elements help designers enhance communication and guide attention.

Designers use complementary , analogous and triadic colors for contrast, harmony and balance. Understanding color temperature also plays a crucial role in design perception.

Color symbolism is crucial, as different colors can represent specific emotions and messages. For instance, blue can symbolize trust and calmness, while red can indicate energy and urgency.

Cultural variations significantly influence color perception and symbolism. Designers consider these differences to ensure their designs resonate with diverse audiences.

For actionable insights, designers should:

Experiment with color schemes for effective messaging. 

Assess colors' psychological impact on the audience. 

Use color contrast to highlight critical elements. 

Ensure color choices are accessible to all.

In this video, Joann and Arielle Eckstut, leading color consultants and authors, give their six tips for choosing color:

Learn more about color from Joann and Arielle in our Master Class, How To Use Color Theory To Enhance Your Designs .

Typography and font choice are crucial in design, impacting readability and mood. Designers utilize them for effective communication and expression.

Designers' perception of information varies with font type. Serif fonts can imply formality, while sans-serifs can give a more modern look.

Typography choices by designers influence readability and user experience. Well-spaced, distinct fonts enhance readability, whereas decorative fonts may hinder it.

Designers use typography to evoke emotions and set a design's tone. Choices in font size, style and color affect the emotional impact and message clarity.

Designers use typography to direct attention, create hierarchy and establish rhythm. These benefits help with brand recognition and consistency across mediums.

Read our article to learn how web fonts are critical to the online user experience .

Designers create a balance between simplicity and complexity in their work. They focus on the main messages and highlight important parts. Designers use the principles of visual hierarchy, like size, color and spacing. They also use empty space to make their designs clear and understandable.

The Gestalt law of Prägnanz suggests people naturally simplify complex images. This principle aids in making even intricate information accessible and engaging.

Through iteration and feedback, designers refine visuals. They remove extraneous elements and highlight vital information. Testing with the target audience ensures the design resonates and is comprehensible.

Michal Malewicz explains how to master hierarchy in UI design using the Gestalt rule of proximity:

Answer a Short Quiz to Earn a Gift

Why do designers use visual representation?

• To guarantee only a specific audience can understand the information
• To replace the need for any form of written communication
• To simplify complex information and make it understandable

Which type of visual representation helps to compare data?

• Article images
• Line charts
• Text paragraphs

What is the main purpose of visual hierarchy in design?

• To decorate the design with more colors
• To guide the viewer’s attention to the most important elements first
• To provide complex text for high-level readers

How does color impact visual representation?

• It has no impact on the design at all.
• It helps to distinguish different elements and set the mood.
• It makes the design less engaging for a serious mood.

Why is consistency important in visual representation?

• It limits creativity, but allows variation in design.
• It makes sure the visual elements are cohesive and easy to understand.
• It makes the design unpredictable yet interesting.

Better luck next time!

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Literature on Visual Representation

Here’s the entire UX literature on Visual Representation by the Interaction Design Foundation, collated in one place:

Learn more about Visual Representation

Take a deep dive into Visual Representation with our course Perception and Memory in HCI and UX .

How does all of this fit with interaction design and user experience? The simple answer is that most of our understanding of human experience comes from our own experiences and just being ourselves. That might extend to people like us, but it gives us no real grasp of the whole range of human experience and abilities. By considering more closely how humans perceive and interact with our world, we can gain real insights into what designs will work for a broader audience: those younger or older than us, more or less capable, more or less skilled and so on.

“You can design for all the people some of the time, and some of the people all the time, but you cannot design for all the people all the time.“ – William Hudson (with apologies to Abraham Lincoln)

While “design for all of the people all of the time” is an impossible goal, understanding how the human machine operates is essential to getting ever closer. And of course, building solutions for people with a wide range of abilities, including those with accessibility issues, involves knowing how and why some human faculties fail. As our course tutor, Professor Alan Dix, points out, this is not only a moral duty but, in most countries, also a legal obligation.

Portfolio Project

In the “ Build Your Portfolio: Perception and Memory Project ”, you’ll find a series of practical exercises that will give you first-hand experience in applying what we’ll cover. If you want to complete these optional exercises, you’ll create a series of case studies for your portfolio which you can show your future employer or freelance customers.

This in-depth, video-based course is created with the amazing Alan Dix , the co-author of the internationally best-selling textbook  Human-Computer Interaction and a superstar in the field of Human-Computer Interaction . Alan is currently a professor and Director of the Computational Foundry at Swansea University.

Gain an Industry-Recognized UX Course Certificate

Use your industry-recognized Course Certificate on your resume , CV , LinkedIn profile or your website.

All open-source articles on Visual Representation

Data visualization for human perception.

The Key Elements & Principles of Visual Design

• 1.1k shares

Guidelines for Good Visual Information Representations

• 5 years ago

Philosophy of Interaction

Information visualization – an introduction to multivariate analysis.

• 8 years ago

Aesthetic Computing

How to represent linear data visually for information visualization.

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Initial Thoughts

Perspectives & resources, what is high-quality mathematics instruction and why is it important.

• Page 1: The Importance of High-Quality Mathematics Instruction
• Page 2: A Standards-Based Mathematics Curriculum
• Page 3: Evidence-Based Mathematics Practices

What evidence-based mathematics practices can teachers employ?

• Page 4: Explicit, Systematic Instruction

Page 5: Visual Representations

• Page 6: Schema Instruction
• Page 7: Metacognitive Strategies
• Page 8: Effective Classroom Practices
• Page 9: References & Additional Resources
• Page 10: Credits

Research Shows

• Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all. (Boonen, van Wesel, Jolles, & van der Schoot, 2014)
• Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities. (van Garderen, Scheuermann, & Jackson, 2012; van Garderen, Scheuermann, & Poch, 2014)
• Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving. (Krawec, 2014)

Visual representations are flexible; they can be used across grade levels and types of math problems. They can be used by teachers to teach mathematics facts and by students to learn mathematics content. Visual representations can take a number of forms. Click on the links below to view some of the visual representations most commonly used by teachers and students.

How does this practice align?

High-leverage practice (hlp).

• HLP15 : Provide scaffolded supports

CCSSM: Standards for Mathematical Practice

• MP1 : Make sense of problems and persevere in solving them.

Number Lines

Definition : A straight line that shows the order of and the relation between numbers.

Common Uses : addition, subtraction, counting

Strip Diagrams

Definition : A bar divided into rectangles that accurately represent quantities noted in the problem.

Common Uses : addition, fractions, proportions, ratios

Definition : Simple drawings of concrete or real items (e.g., marbles, trucks).

Common Uses : counting, addition, subtraction, multiplication, division

Graphs/Charts

Definition : Drawings that depict information using lines, shapes, and colors.

Common Uses : comparing numbers, statistics, ratios, algebra

Graphic Organizers

Definition : Visual that assists students in remembering and organizing information, as well as depicting the relationships between ideas (e.g., word webs, tables, Venn diagrams).

Common Uses : algebra, geometry

Before they can solve problems, however, students must first know what type of visual representation to create and use for a given mathematics problem. Some students—specifically, high-achieving students, gifted students—do this automatically, whereas others need to be explicitly taught how. This is especially the case for students who struggle with mathematics and those with mathematics learning disabilities. Without explicit, systematic instruction on how to create and use visual representations, these students often create visual representations that are disorganized or contain incorrect or partial information. Consider the examples below.

Elementary Example

Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.

Notice that Talia gets the correct answer. However, because Colby draws his dots in haphazard fashion, he fails to count all of them and consequently arrives at the wrong solution.

High School Example

Mr. Huang asks his students to solve the following word problem:

The flagpole needs to be replaced. The school would like to replace it with the same size pole. When Juan stands 11 feet from the base of the pole, the angle of elevation from Juan’s feet to the top of the pole is 70 degrees. How tall is the pole?

Compare the drawings below created by Brody and Zoe to represent this problem. Notice that Brody drew an accurate representation and applied the correct strategy. In contrast, Zoe drew a picture with partially correct information. The 11 is in the correct place, but the 70° is not. As a result of her inaccurate representation, Zoe is unable to move forward and solve the problem. However, given an accurate representation developed by someone else, Zoe is more likely to solve the problem correctly.

Manipulatives

Some students will not be able to grasp mathematics skills and concepts using only the types of visual representations noted in the table above. Very young children and students who struggle with mathematics often require different types of visual representations known as manipulatives. These concrete, hands-on materials and objects—for example, an abacus or coins—help students to represent the mathematical idea they are trying to learn or the problem they are attempting to solve. Manipulatives can help students develop a conceptual understanding of mathematical topics. (For the purpose of this module, the term concrete objects refers to manipulatives and the term visual representations refers to schematic diagrams.)

It is important that the teacher make explicit the connection between the concrete object and the abstract concept being taught. The goal is for the student to eventually understand the concepts and procedures without the use of manipulatives. For secondary students who struggle with mathematics, teachers should show the abstract along with the concrete or visual representation and explicitly make the connection between them.

A move from concrete objects or visual representations to using abstract equations can be difficult for some students. One strategy teachers can use to help students systematically transition among concrete objects, visual representations, and abstract equations is the Concrete-Representational-Abstract (CRA) framework.

If you would like to learn more about this framework, click here.

Concrete-Representational-Abstract Framework

• Concrete —Students interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
• Representational — Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or students may draw their own representation of the problem.
• Abstract — Students solve problems with numbers, symbols, and words without any concrete or representational assistance.

CRA is effective across all age levels and can assist students in learning concepts, procedures, and applications. When implementing each component, teachers should use explicit, systematic instruction and continually monitor student work to assess their understanding, asking them questions about their thinking and providing clarification as needed. Concrete and representational activities must reflect the actual process of solving the problem so that students are able to generalize the process to solve an abstract equation. The illustration below highlights each of these components.

For Your Information

One promising practice for moving secondary students with mathematics difficulties or disabilities from the use of manipulatives and visual representations to the abstract equation quickly is the CRA-I strategy . In this modified version of CRA, the teacher simultaneously presents the content using concrete objects, visual representations of the concrete objects, and the abstract equation. Studies have shown that this framework is effective for teaching algebra to this population of students (Strickland & Maccini, 2012; Strickland & Maccini, 2013; Strickland, 2017).

Kim Paulsen discusses the benefits of manipulatives and a number of things to keep in mind when using them (time: 2:35).

Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University

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Transcript: Kim Paulsen, EdD

Manipulatives are a great way of helping kids understand conceptually. The use of manipulatives really helps students see that conceptually, and it clicks a little more with them. Some of the things, though, that we need to remember when we’re using manipulatives is that it is important to give students a little bit of free time when you’re using a new manipulative so that they can just explore with them. We need to have specific rules for how to use manipulatives, that they aren’t toys, that they really are learning materials, and how students pick them up, how they put them away, the right time to use them, and making sure that they’re not distracters while we’re actually doing the presentation part of the lesson. One of the important things is that we don’t want students to memorize the algorithm or the procedures while they’re using the manipulatives. It really is just to help them understand conceptually. That doesn’t mean that kids are automatically going to understand conceptually or be able to make that bridge between using the concrete manipulatives into them being able to solve the problems. For some kids, it is difficult to use the manipulatives. That’s not how they learn, and so we don’t want to force kids to have to use manipulatives if it’s not something that is helpful for them. So we have to remember that manipulatives are one way to think about teaching math.

I think part of the reason that some teachers don’t use them is because it takes a lot of time, it takes a lot of organization, and they also feel that students get too reliant on using manipulatives. One way to think about using manipulatives is that you do it a couple of lessons when you’re teaching a new concept, and then take those away so that students are able to do just the computation part of it. It is true we can’t walk around life with manipulatives in our hands. And I think one of the other reasons that a lot of schools or teachers don’t use manipulatives is because they’re very expensive. And so it’s very helpful if all of the teachers in the school can pool resources and have a manipulative room where teachers can go check out manipulatives so that it’s not so expensive. Teachers have to know how to use them, and that takes a lot of practice.

What is visual representation?

In the vast landscape of communication, where words alone may fall short, visual representation emerges as a powerful ally. In a world inundated with information, the ability to convey complex ideas, emotions, and data through visual means is becoming increasingly crucial. But what exactly is visual representation, and why does it hold such sway in our understanding?

Defining Visual Representation:

Visual representation is the act of conveying information, ideas, or concepts through visual elements such as images, charts, graphs, maps, and other graphical forms. It’s a means of translating the abstract into the tangible, providing a visual language that transcends the limitations of words alone.

The Power of Images:

The adage “a picture is worth a thousand words” encapsulates the essence of visual representation. Images have an unparalleled ability to evoke emotions, tell stories, and communicate complex ideas in an instant. Whether it’s a photograph capturing a poignant moment or an infographic distilling intricate data, images possess a unique capacity to resonate with and engage the viewer on a visceral level.

Facilitating Understanding:

One of the primary functions of visual representation is to enhance understanding. Humans are inherently visual creatures, and we often process and retain visual information more effectively than text. Complex concepts that might be challenging to grasp through written explanations can be simplified and clarified through visual aids. This is particularly valuable in fields such as science, where intricate processes and structures can be elucidated through diagrams and illustrations.

Visual representation also plays a crucial role in education. In classrooms around the world, teachers leverage visual aids to facilitate learning, making lessons more engaging and accessible. From simple charts that break down historical timelines to interactive simulations that bring scientific principles to life, visual representation is a cornerstone of effective pedagogy.

Data Visualization:

In an era dominated by big data, the importance of data visualization cannot be overstated. Raw numbers and statistics can be overwhelming and abstract, but when presented visually, they transform into meaningful insights. Graphs, charts, and maps are powerful tools for conveying trends, patterns, and correlations, enabling decision-makers to glean actionable intelligence from vast datasets.

Consider the impact of a well-crafted infographic that distills complex research findings into a visually digestible format. Data visualization not only simplifies information but also allows for more informed decision-making in fields ranging from business and healthcare to social sciences and environmental studies.

Cultural and Artistic Expression:

Visual representation extends beyond the realm of information and education; it is also a potent form of cultural and artistic expression. Paintings, sculptures, photographs, and other visual arts serve as mediums through which individuals can convey their emotions, perspectives, and cultural narratives. Artistic visual representation has the power to transcend language barriers, fostering a shared human experience that resonates universally.

Conclusion:

In a world inundated with information, visual representation stands as a beacon of clarity and understanding. Whether it’s simplifying complex concepts, conveying data-driven insights, or expressing the depth of human emotion, visual elements enrich our communication in ways that words alone cannot. As we navigate an increasingly visual society, recognizing and harnessing the power of visual representation is not just a skill but a necessity for effective communication and comprehension. So, let us embrace the visual language that surrounds us, unlocking a deeper, more nuanced understanding of the world.

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The Epistemology of Visual Thinking in Mathematics

Visual thinking is widespread in mathematical practice, and has diverse cognitive and epistemic purposes. This entry discusses potential roles of visual thinking in proving and in discovering, with some examples, and epistemic difficulties and limitations are considered. Also discussed is the bearing of epistemic uses of visual representations on the application of the a priori–a posteriori distinction to mathematical knowledge. A final section looks briefly at how visual means can aid comprehension and deepen understanding of proofs.

1. Introduction

2. historical background, 3.1 the reliability question, 3.2 visual means in non-formal proving, 3.3 a dispute: diagrams in proofs in analysis., 4.1 propositional discovery, 4.2 discovering a proof strategy, 4.3 discovering properties and kinds, 5. visual thinking and mental arithmetic, 6.1 evidential uses of visual experience, 6.2 an evidential use of visual experience in proving, 6.3 a non-evidential use of visual experience, 7. further uses of visual representations, 8. conclusion, other internet resources, related entries.

Visual thinking is a feature of mathematical practice across many subject areas and at many levels. It is so pervasive that the question naturally arises: does visual thinking in mathematics have any epistemically significant roles? A positive answer begets further questions. Can we rationally arrive at a belief with the generality and necessity characteristic of mathematical theorems by attending to specific diagrams or images? If visual thinking contributes to warrant for believing a mathematical conclusion, must the outcome be an empirical belief? How, if at all can visual thinking contribute to understanding abstract mathematical subject matter?

Visual thinking includes thinking with external visual representations (e.g., diagrams, symbol arrays, kinematic computer images) and thinking with internal visual imagery; often the two are used in combination, as when we are required to visually imagine a certain spatial transformation of an object represented by a diagram on paper or on screen. Almost always (and perhaps always) visual thinking in mathematics is used in conjunction with non-visual thinking. Possible epistemic roles include contributions to evidence, proof, discovery, understanding and grasp of concepts. The kinds and the uses of visual thinking in mathematics are numerous and diverse. This entry will deal with some of the topics in this area that have received attention and omit others. Among the omissions is the possible explanatory role of visual representations in mathematics. The topic of explanation within pure mathematics is tricky and best dealt with separately; for this an excellent starting place is the entry on explanation in mathematics (Mancosu 2011). Two other omissions are the development of logic diagrams (Euler, Venn, Pierce and Shin) and the nature and use of geometric diagrams in Euclid’s Elements , both of which are well treated in the entry diagrams (Shin et al. 2013). The focus here is on visual thinking generally, which includes thinking with symbol arrays as well as with diagrams; there will be no attempt here to formulate a criterion for distinguishing between symbolic and diagrammatic thinking. However, the use of visual thinking in proving and in various kinds of discovery will be covered in what follows. Discussions of some related questions and some studies of historical cases not considered here are to be found in the collection Diagrams in Mathematics: History and Philosophy (Mumma and Panza 2012).

“Mathematics can achieve nothing by concepts alone but hastens at once to intuition” wrote Kant (1781/9: A715/B743), before describing the geometrical construction in Euclid’s proof of the angle sum theorem (Euclid, Book 1, proposition 32). In a review of 1816 Gauss echoes Kant:

anybody who is acquainted with the essence of geometry knows that [the logical principles of identity and contradiction] are able to accomplish nothing by themselves, and that they put forth sterile blossoms unless the fertile living intuition of the object itself prevails everywhere. (Ewald 1996 [Vol. 1]: 300)

The word “intuition” here translates the German “ Anschauung ”, a word which applies to visual imagination and perception, though it also has more general uses.

By the late 19 th century a different view had emerged, at least in foundational areas. In a celebrated text giving the first rigorous axiomatization of projective geometry, Pasch wrote: “the theorem is only truly demonstrated if the proof is completely independent of the figure” (Pasch 1882), a view expressed also by Hilbert in writing on the foundations of geometry (Hilbert 1894). A negative attitude to visual thinking was not confined to geometry. Dedekind, for example, wrote of an overpowering feeling of dissatisfaction with appeal to geometric intuitions in basic infinitesimal analysis (Dedekind 1872, Introduction). The grounds were felt to be uncertain, the concepts employed vague and unclear. When such concepts were replaced by precisely defined alternatives without allusions to space, time or motion, our intuitive expectations turned out to be unreliable (Hahn 1933).

In some quarters this view turned into a general disdain for visual thinking in mathematics: “In the best books” Russell pronounced “there are no figures at all” (Russell 1901). Although this attitude was opposed by a few mathematicians, notably Klein (1893), others took it to heart. Landau, for example, wrote a calculus textbook without a single diagram (Landau 1934). But the predominant view was not so extreme: thinking in terms of figures was valued as a means of facilitating grasp of formulae and linguistic text, but only reasoning expressed by means of formulae and text could bear any epistemological weight.

By the late 20 th century the mood had swung back in favour of visualization: Mancosu (2005) provides an excellent survey. Some books advertise their defiance of anti-visual puritanism in their titles, for example Visual Geometry and Topology (Fomenko 1994) and Visual Complex Analysis (Needham 1997); mathematics educators turn their attention to pedagogical uses of visualization (Zimmerman and Cunningham 1991); the use of computer-generated imagery begins to bear fruit at research level (Hoffman 1987; Palais 1999), and diagrams find their way into research papers in abstract fields: see for example the papers on higher dimensional category theory by Joyal et al. (1996), Leinster (2004) and Lauda (2005, Other Internet Resources). But attitudes to the epistemology of visual thinking remain mixed. The discussion is mostly concerned with the role of diagrams in proofs.

3. Visual thinking and proof

In some cases, it is claimed, a picture alone is a proof (Brown 1999: ch. 3). But that view is rare. Even the editor of Proofs without Words: Exercises in Visual Thinking , writes “Of course, ‘proofs without words’ are not really proofs” (Nelsen 1993: vi). Expressions of the other extreme are rare but can be found:

[the diagram] has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array. (Tennant 1986)

Between the extremes we find the view that, even if no picture alone is a proof, visual representations can have a non-superfluous role in reasoning that constitutes a proof. (This is not to deny that there may be another proof of the same conclusion which does not involve any visual representation.) Geometric diagrams, graphs and maps, all carry information. Taking valid deductive reasoning to be the reliable extraction of information from information already obtained, Barwise and Etchemendy (1996:4) pose the following question: Why cannot the representations composing a proof be visual as well as linguistic? The sole reason for denying this role to visual representations is the thought that, with the possible exception of very restricted cases, visual thinking is unreliable, hence cannot contribute to proof. Is that right?

Our concern here is thinking through the steps in a proof, either for the first time (a first successful attempt to construct a proof) or following a given proof. Clearly we want to distinguish between visual thinking which merely accompanies the process of thinking through the steps in a proof and visual thinking which is essential to the process. This is not always straightforward as a proof can be presented in different ways. How different can distinct presentations be and yet be presentations of the same proof? There is no context-invariant answer to this. Often mathematicians are happy to regard two presentations as presenting the same proof if the central idea is the same in both cases. But if one’s main concern is with what is involved in thinking through a proof, its central idea is not enough to individuate it: the overall structure, the sequence of steps and perhaps some other factors affecting the cognitive processes involved will be relevant.

Once individuation of proofs has been settled, we can distinguish between replaceable thinking and superfluous thinking, where these attributions are understood as relative to a given argument or proof. In the process of thinking through a proof, a given part of the thinking is replaceable if thinking of some other kind could stand in place of the given part in a process that would count as thinking through the same proof. A given part of the thinking is superfluous if its excision without replacement would be a process of thinking through the same proof. Superfluous thinking may be extremely valuable in facilitating grasp of the proof text and in enabling one to understand the idea underlying the proof steps; but it is not necessary for thinking through the proof.

It is uncontentious that the visual thinking involved in symbol manipulations, for example in following the “algebraic” steps of proofs of basic lemmas about groups, can be essential, that is neither superfluous nor replaceable. The worry is about thinking visually with diagrams, where “diagram” is used widely to include all non-symbolic visual representations. Let us agree that there can be superfluous diagrammatic thinking in thinking through a proof. This leaves several possibilities.

• (a) All diagrammatic thinking in a process of thinking through a proof is superfluous.
• (b) Not all diagrammatic thinking in a process of thinking through a proof is superfluous; but if not superfluous it will be replaceable by non-diagrammatic thinking.
• (c) Some diagrammatic thinking in a process of thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking.

The negative view stated earlier that diagrams can have no role in proof entails claim (a). The idea behind (a) is that, because diagrammatic reasoning is unreliable, if a process of thinking through an argument contains some non-superfluous diagrammatic thinking, that process lacks the epistemic security to be a case of thinking through a proof.

This view, claim (a) in particular, is threatened by cases in which the reliability of the diagrammatic thinking is demonstrated non-visually. The clearest kind of example would be provided by a formal system which has diagrams in place of formulas among its syntactic objects, and types of inter-diagram transition for inference rules. Suppose you take in such a formal system and an interpretation of it, and then think through a proof of the system’s soundness with respect to that interpretation; suppose you then inspect a sequence of diagrams, checking along the way that it constitutes a derivation in the system; suppose finally that you recover the interpretation to reach a conclusion. (The order is unimportant: one can go through the derivation first and then follow the soundness proof.) That entire process would constitute thinking through a proof of the conclusion; and the diagrammatic thinking involved would not be superfluous.

Shin et al. (2013) report that formal diagrammatic systems of logic and geometry have been proven to be sound. People have indeed followed proofs in these systems. That is enough to refute claim (a), the claim that all diagrammatic thinking in thinking through a proof is superfluous. For a concrete example, Figure 1 presents a derivation of Euclid’s first theorem, that on any straight line segment an equilateral triangle is constructible, in a formal diagrammatic system of a part of Euclidean geometry (Miller 2001).

What about Tennant’s claim that a proof is “a syntactic object consisting only of sentences” as opposed to diagrams? A proof is never a syntactic object. A formal derivation on its own is a syntactic object but not a proof. Without an interpretation of the language of the formal system the end-formula of the derivation says nothing; and so nothing is proved. Without a demonstration of the system’s soundness with respect to the interpretation, one may lack sufficient reason to believe that all derivable conclusions are true. A formal derivation plus an interpretation and soundness proof can be a proof of the derived conclusion, but that whole package is not a syntactic object. Moreover, the part of the proof which really is a syntactic object, the formal derivation, need not consist solely of sentences; it can consist of diagrams.

With claim (a) disposed of, consider again claim (b) that, while not all diagrammatic thinking in a process of thinking through a proof is superfluous, all non-superfluous diagrammatic thinking will be replaceable by non-diagrammatic thinking in a process of thinking through that same proof. The visual thinking in following the proof of Euclid’s first theorem using Miller’s formal system consists in going through a sequence of diagrams and at each step seeing that the next diagram results from a permitted alteration of the previous diagram. It is clear that in a process that counts as thinking through this proof, the diagrammatic thinking is neither superfluous nor replaceable by non-diagrammatic thinking. That knocks out (b), leaving only (c): some thinking that involves a diagram in thinking through a proof is neither superfluous nor replaceable by non-diagrammatic thinking (without changing the proof).

Mathematical practice almost never proceeds by way of formal systems. Outside the context of formal diagrammatic systems, the use of diagrams is widely felt to be unreliable. A diagram can be unfaithful to the described construction: it may represent something with a property that is ruled out by the description, or without a property that is demanded by the description. This is exemplified by diagrams in the famous argument for the proposition that all triangles are isosceles: the meeting point of an angle bisector and the perpendicular bisector of the opposite side is represented as falling inside the triangle, when it has to be outside (Rouse Ball 1939; Maxwell 1959). Errors of this sort are comparatively rare, usually avoidable with a modicum of care, and not inherent in the nature of diagrams; so they do not warrant a general charge of unreliability.

The major sort of error is unwarranted generalisation. Typically diagrams (and other non-verbal visual representations) do not represent their objects as having a property that is actually ruled out by the intention or specification of the object to be represented. But diagrams very frequently do represent their objects as having properties that, though not ruled out by the specification, are not demanded by it. Verbal descriptions can be discrete, in that they supply no more information than is needed. But visual representations are typically indiscrete, in that they supply too much detail. This is often unavoidable, because for many properties or kinds \(F\), a visual representation cannot represent something as being \(F\) without representing it as being \(F\) in a particular way . Any diagram of a triangle, for instance, must represent it as having three acute angles or as having just two acute angles, even if neither property is required by the specification, as would be the case if the specification were “Let ABC be a triangle”. As a result there is a danger that in using a diagram to reason about an arbitrary instance of class \(K\), we will unwittingly rely on a feature represented in the diagram that is not common to all instances of the class \(K\). Thus the risk of unwarranted generalisation is a danger inherent in the use of many diagrams.

Indiscretion of diagrams is not confined to geometrical figures. The dot or pebble diagrams of ancient mathematics used to convince one of elementary truths of number theory necessarily display particular numbers of dots, though the truths are general. Here is an example, used to justify the formula for the \(n\) th triangular number, i.e., the sum of the first \(n\) positive integers.

The conclusion drawn is that the sum of integers from 1 to \(n\) is \((n \times n+1)/2\) for any positive integer \(n\), but the diagram presents the case for \(n = 6\). We can perhaps avoid representing a particular number of dots when we merely imagine a display of the relevant kind; or if a particular number is represented, our experience may not make us aware of the number—just as, when one imagines the sky on a starry night, for no particular number \(k\) are we aware that exactly \(k\) stars are represented. Even so, there is likely to be some extra specificity. For example, in imagining an array of dots of the form just illustrated, one is unlikely to imagine just two columns of three dots, the rectangular array for \(n = 2\). Typically the subject will be aware of imagining an array with more than two columns. This entails that an image is likely to have unintended exclusions. In this case it would exclude the three-by-two array. An image of a triangle representing all angles as acute would exclude triangles with an obtuse angle or a right angle. The danger is that the visual reasoning will not be valid for the cases that are unintentionally excluded by the visual representation, with the result that the step to the conclusion is an unwarranted generalisation.

What should we make of this? First, let us note that in a few cases the image or diagram will not be over-specific. When in geometry all instances of the relevant class are congruent to one another, for instance all circles or all squares, the image or diagram will not be over-specific for a generalisation about that class; so there will be no unintended exclusions and no danger of unwarranted generalisation. Here then are possibilities for reliable visual thinking in proving.

To get clear about the other cases, where there is a danger of over generalizing, it helps to look at generalisation in ordinary non-visual reasoning. Schematically put, in reasoning about things of kind \(K\), once we have shown that from certain premisses it follows that such-and-such a condition is true of arbitrary instance \(c\), we can validly infer from those same premisses that that condition is true of all \(K\)s, with the proviso that neither the condition nor any premiss mentions \(c\). The proviso is required, because if a premiss or the condition does mention \(c\), the reasoning may depend on a property of \(c\) that is not shared by all other \(K\)s and so the generalisation would be unsafe. For a trivial example consider a step from “\(x = c\)” to “\(\forall x [x = c]\)”.

A question we face is whether, in order to come to know the truth of a conclusion by following an argument involving generalisation on an arbitrary instance (a.k.a. universal generalisation, or universal quantifier introduction), the thinking must include a conscious, explicit check that the proviso is met. It is clearly not enough that the proviso is in fact met. For in that case it might just be the thinker’s good luck that the proviso is met; hence the thinker would not know that the generalisation is valid and so would not have genuinely thought through the proof at that step.

This leaves two options. The strict option is that without a conscious, explicit check one has not really thought through the proof. The relaxed option is that one can properly think through the proof without checking that the proviso is met, but only if one is sensitive to the potential error and would detect it in otherwise similar arguments. For then one is not just lucky that the proviso is met. Being sensitive in this context consists in being alert to dependence on features of the arbitrary instance not shared by all members of the class of generalisation, a state produced by a combination of past experience and current vigilance. Without compelling reason to prefer one of these options, decisions on what is to count as proving or following a proof must be conditional.

How does all this apply to generalizing from visual thinking about an arbitrary instance? Take the example of the visual route to the formula for triangular numbers using the diagram of Figure 2 . The diagram reveals that the formula holds for the 6 th triangular number. The generalisation to all triangular numbers is justified only if the visuo-spatial method used is applicable to the \(n\) th triangular number for all positive integers \(n\), that is, provided that the method used does not depend on a property not shared by all positive integers. A conscious, explicit check that this proviso is met requires making explicit the method exemplified for 6 and proving that the method is applicable for all positive integers in place of 6. (For a similar idea in the context of automating visual arguments, see Jamnik 2001). This is not done in practice when thinking visually, and so if we accept the strict option for thinking through a proof involving generalisation, we would have to accept that the visual route to the formula for triangular numbers does not amount to thinking through a proof of it; and the same would apply to the familiar visual routes to other general positive integer formulas, such as that \(n^2 =\) the sum of the first \(n\) odd numbers.

But what if the strict option for proving by generalisation on an arbitrary instance is too strict, and the relaxed option is right? When arriving at the formula in the visual way indicated, one does not pay attention to the fact that the visual display represents the situation for the 6 th triangular number; it is as if the mind had somehow extracted a general schema of visual reasoning from exposure to the particular case, and had then proceeded to reason schematically, converting a schematic result into a universal proposition. What is required, on the relaxed option, is sensitivity to the possibility that the schema is not applicable to all positive integers; one must be so alert to ways a schema of the given kind can fall short of universal applicability that if one had been presented with a schema that did fall short, one would have detected the failure.

In the example at hand, the schema of visual reasoning involves at the start taking a number \(k\) to be represented by a column of \(k\) dots, thence taking the triangular array of \(n\) columns to represent the sum of the first \(n\) positive integers, thence taking that array combined with an inverted copy to make a rectangular array of \(n\) columns of \(n+1\) dots. For a schema starting this way to be universally applicable, it must be possible, given any positive integer \(n\), for the sum of the first \(n\) positive integers to be represented in the form of a triangular array, so that combined with an inverted copy one gets a rectangular array. This actually fails at the extreme case: \(n = 1\). The formula \((n.(n + 1))/2\) holds for this case; but that is something we know by substituting “1” for the variable in the formula, not by the visual method indicated. That method cannot be applied to \(n = 1\), because a single dot does not form a triangular array, and combined with a copy it does not form a rectangular array. But we can check that the method works for all positive integers after the first, using visual reasoning to assure ourselves that it works for 2 and that if the method works for \(k\) it works for \(k+1\). Together with this reflective thinking, the visual thinking sketched earlier constitutes following a proof of the formula for the \(n\) th triangular number for all integers \(n > 1\), at least if the relaxed view of thinking through a proof is correct. Similar conclusions hold in the case of other “dot” arguments (Giaquinto 1993, 2007: ch. 8). So in some cases when the visual representation carries unwanted detail, the danger of over-generalisation in visual reasoning can be overcome.

But the fact that this is frequently missed by commentators suggests that the required sensitivity is often absent. Missing an untypical case is a common hazard in attempts at visual proving. A well-known example is the proof of Euler’s formula \(V - E + F = 2\) for polyhedra by “removing triangles” of a triangulated planar projection of a polyhedron. One is easily convinced by the thinking, but only because the polyhedra we normally think of are convex, while the exceptions are not convex. But it is also easy to miss a case which is not untypical or extreme when thinking visually. An example is Cauchy’s attempted proof (Cauchy 1813) of the claim that if a convex polygon is transformed into another polygon keeping all but one of the sides constant, then if some or all of the internal angles at the vertices increase, the remaining side increases, while if some or all of the internal angles at the vertices decrease, the remaining side decreases. The argument proceeds by considering what happens when one transforms a polygon by increasing (or decreasing) angles, angle by angle. But in a trapezoid, changing a single angle can turn a convex polygon into a concave polygon, and this invalidates the argument (Lyusternik 1963).

The frequency of such mistakes indicates that visual arguments (other than symbol manipulations) often lack the transparency required for proof. Even when a visual argument is in fact sound, its soundness may not be clear, in which case the argument is not a way of proving the truth of the conclusion, though it may be a way of discovering it. But this is consistent with the claim that visual non-symbolic thinking can be (and often is) part of a way of proving something.

An example from knot theory will substantiate the modal part of this claim. To present the example, we need some background information, which will be given with a minimum of technical detail.

A knot is a tame closed non-self-intersecting curve in Euclidean 3-space.

In other words, knots are just the tame curves in Euclidean 3-space which are homeomorphic to a circle. The word “tame” here stands for a property intended to rule out certain pathological cases, such as curves with infinitely nested knotting. There is more than one way of making this mathematically precise, but we have no need for these details. A knot has a specific geometric shape, size and axis-relative position. Now imagine it to be made of flexible yet unbreakable yarn that is stretchable and shrinkable, so that it can be smoothly transformed into other knots without cutting or gluing. Since our interest in a knot is the nature of its knottedness regardless of shape, size or axis-relative position, the real focus of interest is not just the knot but all its possible transforms. A way to think of this is to imagine a knot transforming continuously, so that every possible transform is realized at some time. Then the thing of central interest would be the object that persists over time in varying forms, with knots strictly so called being the things captured in each particular freeze frame. Mathematically, we represent the relevant entity as an equivalence class of knots.

Two knots are equivalent iff one can be smoothly deformed into the other by stretching, shrinking, twisting, flipping, repositioning or in any other way that does not involve cutting, gluing or passing one strand through another.

The relevant kind of deformation forbids eliminating a knotted part by shrinking it down to a point. Again there are mathematically precise definitions of knot-equivalence. Figure 3 gives diagrams of equivalent knots, instances of a trefoil.

Diagrams like these are not merely illustrations; they also have an operational role in knot theory. But not any picture of a knot will do for this purpose. We need to specify:

A knot diagram is a regular projection of a knot onto a plane which, when there is a crossing, tells us which strand passes over the other.

Regularity here is a combination of conditions. In particular, regularity entails that not more than two points of the strict knot project to the same point on the plane, and that two points of the strict knot project to the same point on the plane only where there is a crossing. For more on diagrams in knot theory see (De Toffoli and Giardino 2014).

A major task of knot theory is to find ways of telling whether two knot diagrams are diagrams of equivalent knots. In particular we will want to know if a given knot diagram represents a knot equivalent to an unknot , that is, a knot representable by a knot diagram without crossings.

One way of showing that a knot diagram represents a knot equivalent to an unknot is to show that the diagram can be transformed into one without crossings by a sequence of atomic moves, known as Reidemeister moves. The relevant background fact is Reidemeister’s theorem, which links the visualizable diagrammatic changes to the mathematically precise definition of knot equivalence: Two knots are equivalent if and only if there is a finite sequence of Reidemeister moves taking a knot diagram of one to a knot diagram of the other. Figure 4 illustrates. Each knot diagram is changed into the adjacent knot diagram by a Reidemeister move; hence the knot represented by the leftmost diagram is equivalent to the unknot.

In contrast to these, the knot presented by the left knot diagram of Figure 3 , a trefoil, may seem impossible to deform into an unknot. And in fact it is. To prove it, we can use a knot invariant known as colourability. An arc in a knot diagram is a maximal part between crossings (or the whole thing if there are no crossings). Colourability is this:

A knot diagram is colourable if and only if each of its arcs can be coloured one of three different colours so that (a) at least two colours are used and (b) at each crossing the three arcs are all coloured the same or all coloured differently.

The reference to colours here is inessential. Colourability is in fact a specific case of a kind of combinatorial property known as mod \(p\) labelling (for \(p\) an odd prime). Colourability is a knot invariant in the sense that if one diagram of a knot is colourable every diagram of that knot and of any equivalent knot is colourable. (By Reidemeister’s theorem this can be proved by showing that each Reidemeister move preserves colourability.) A standard diagram of an unknot, a diagram without crossings, is clearly not colourable because it has only one arc (the whole thing) and so two colours cannot be used. So in order to complete proving that the trefoil is not equivalent to an unknot, we only need prove that our trefoil diagram is colourable. This can be done visually. Colour each arc of the knot diagram one of the three colours red, green or blue so that no two arcs have the same colour (or visualize this). Then do a visual check of each crossing, to see that at each crossing the three meeting arcs are all coloured differently. That visual part of the proof is clearly non-superfluous and non-replaceable (without changing the proof). Moreover, the soundness of the argument is quite transparent. So here is a case of a non-formal, non-symbolic visual way of proving a mathematical truth.

Where notions involving the infinite are in play, such as many involving limits, the use of diagrams is famously risky. For this reason it has been widely thought that, beyond some very simple cases, arguments in real and complex analysis in which diagrams have a non-superfluous role are not genuine proofs. Bolzano [1817] expressed this attitude with regard to the intermediate value theorem for the real numbers (IVT) before giving a purely analytic proof, arguing that spatial thinking could not be used to help justify the IVT. James Robert Brown (1999) takes issue with Bolzano on this point. The IVT is this:

If \(f\) is a real-valued function of a real variable continuous on the closed interval \([a, b]\) and \(f(a) < c < f(b)\), then for some \(x\) in \((a, b), f(x) = c\).

Brown focuses on the special case when \(c = 0\). As the IVT can be deduced easily from this special case using the theorem that the difference of two continuous functions is continuous, there is no loss of generality here. Alluding to a diagram like Figure 5, Brown (1999) writes

We have a continuous line running from below to above the \(x\)-axis. Clearly, it must cross that axis in doing so. (1999: 26)

Later he claims:

Using the picture alone, we can be certain of this result—if we can be certain of anything. (1999: 28)

Bolzano’s diagram-free proof of the IVT is an argument from what later became known as the Dedekind completeness of the real numbers: every non-empty set of reals bounded above (below) has a least upper bound (greatest lower bound). The value of Bolzano’s deduction of the IVT from the Dedekind completeness of the reals, according to Brown, is not that it proves the IVT but that it gives us confirmation of Dedekind completeness, just as an empirical hypothesis in empirical science gets confirmed by deducing some consequence of the hypothesis and observing those consequence to be true. This view assumes that we already know the IVT to be true by observing a diagram relevantly like Figure 5 .

That assumption is challenged by Giaquinto (2011). Once we distinguish graphical concepts from associated analytic concepts, the underlying argument from the diagram is essentially this.

• 1. Any function \(f\) which is \(\varepsilon\textrm{-}\delta\) continuous on \([a, b]\) with \(f (a) < 0 < f (b)\) has a visually continuous graphical curve from below the horizontal line representing the \(x\)-axis to above.
• 2. Any visually continuous graphical curve from below a horizontal line to above it meets the line at a crossing point.
• 3. Any function whose graphical curve meets the line representing the \(x\)-axis at a crossing point has a zero value.
• 4. So, any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < 0< f (b)\) has a zero value.

What is inferred from the diagram is premiss 2. Premisses 1 and 3 are assumptions linking analytical with graphical conditions. These linking assumptions are disputed. With regard to premiss 1 Giaquinto (2011) argues that there are functions on the reals which meet the antecedent condition but do not have graphical curves, such as continuous but nowhere differentiable functions and functions which oscillate with unbounded frequency e.g., \(f(x) = x \cdot\sin(1/x)\) for non-zero \(x\) in \([-1, 1]\) and \(f(0) = 0\).

With regard to premiss 3 it is argued that, under the standard conventions of graphical representation of functions in a Cartesian co-ordinate frame, the graphical curve for \(x^2 - 2\) in the rationals is the same as the graphical curve for \(x^2- 2\) in the reals. This is because every real is a limit point of rationals; so for every point \(P\) with one or both co-ordinates irrational, there are points arbitrarily close to \(P\) with both co-ordinates rational; so no gaps would appear if irrational points were removed from the curve for \(x^2- 2\) in the reals. But for \(x\) in the rational interval [0, 2] the function \(x^2- 2\) has no zero value, even though it has a graphical curve which visually crosses the line representing the \(x\)-axis. So one cannot read off the existence of a zero of \(x^2- 2\) on the reals from the diagram; one needs to appeal to some property of the reals which the rationals lack, such as Dedekind completeness.

This raises some obvious questions. Do any theorems of analysis have proofs in which diagrams have a non-superfluous role? Littlewood (1953: 54–5) thought so and gives an example which is examined in Giaquinto (1994). If so, can we demarcate this class of theorems by some mathematical feature of their content? Another question is whether there is a significantly broad class of functions on the reals for which we could prove an intermediate value theorem (i.e., restricted to that class).

If there are theorems of analysis provable with diagrams we do not yet have a mathematical demarcation criterion for them. A natural place to look would be O-minimal structures on the reals—this was brought to the author’s attention by Ethan Galebach. This is because of some remarkable theorems about such structures which exclude all the pathological (hence vision-defying) functions on the reals (Van den Dries 1998), such as continuous nowhere differentiable functions and “space-filling” curves i.e., continuous surjections \(f:(0, 1)\rightarrow(0, 1)^2\). Is the IVT for functions in an O-minimal structure on the reals provable by visual means? Certainly one objection to the visual argument for the unrestricted IVT does not apply when the restriction is in place. This is the objection that continuous nowhere differentiable functions, having no graphical curve, provide counterexamples to the premiss that any \(\varepsilon\textrm{-}\delta\) continuous function \(f\) on \([a, b]\) with \(f (a) < c < f (b)\) has a visually continuous graphical curve from below the horizontal line representing \(y = c\) to above. But the existence of continuous functions with no graphical curve is not the only objection to the visual argument, contrary to a claim of Azzouni (2013: 327). There are also counterexamples to the premiss that any function that does have a graphical curve which visibly crosses the line representing \(y = c\) takes \(c\) as a value, e.g., the function \(x^2 - 2\) on the rationals with \(c = 0\). So the question of a visual proof of the IVT restricted to functions in an O-minimal structure on the reals is still open at the time of writing.

4. Visual thinking and discovery

Though philosophical discussion of visual thinking in mathematics has concentrated on its role in proof, visual thinking may be more valuable for discovery than proof. Three kinds of discovery important in mathematical practice are these:

• (1) propositional discovery (discovering, of a proposition, that it is true),
• (2) discovering a proof strategy (or more loosely, getting the idea for a proof of a proposition), and
• (3) discovering a property or kind of mathematical entity.

In the following subsections visual discovery of these kinds will be discussed and illustrated.

To discover a truth, as that expression is being used here, is to come to believe it by one’s own lights (as opposed to reading it or being told) in a way that is reliable and involves no violation of epistemic rationality (given one’s epistemic state). One can discover a truth without being the first to discover it (in this context); it is enough that one comes to believe it in an independent, reliable and rational way. The difference between merely discovering a truth and proving it is a matter of transparency: for proving or following a proof the subject must be aware of the way in which the conclusion is reached and the soundness of that way; this is not required for discovery.

Sometimes one discovers something by means of visual thinking using background knowledge, resulting in a cogent argument from which one could construct a proof. A nice example is a visual argument that any knot diagram with a finite number of crossings can be turned into a diagram of an unknot by interchanging the over-strand and under-strand of some of its crossings (Adams 2001: 58–90). That argument is a bit too long to present accessibly here. For a short example, here is a way of discovering that the geometric mean of two positive numbers is less than or equal to their arithmetic mean (Eddy 1985) using Figure 6.

Two circles (with diameters \(a\) and \(b\)) meet at a single point. A line is drawn between their centres through their common point; its length is \((a + b)/2\), the sum of the two radii. This line is the hypotenuse of a right angled triangle with one other side of length \((a - b)/2\), the difference of the radii. Pythagoras’s theorem is used to infer that the remaining side of the right-angled triangle has length \(\sqrt{(ab)}\).Then visualizing what happens to the triangle when the diameter of the smaller circle varies between 0 and the diameter of the larger circle, one infers that \(0 < \sqrt{(ab)} < (a + b)/2\); then verifying symbolically that \(\sqrt{(ab)} = (a + b)/2\) when \(a = b\), one concludes that for positive \(a\) and \(b\), \(\sqrt{(ab)} \le (a + b)/2\).

This thinking does not constitute a case of proving or following a proof of the conclusion, because it involves a step which we cannot clearly tell is valid. This is the step of attempting to visually imagine what would happen when the smaller circle varies in diameter between 0 and the diameter of the larger circle and inferring from the resulting experience that the line joining the centres of the circles will always be longer than the horizontal line from the centre of the smaller circle to the vertical diameter of the larger circle. This step seems sound (does not lead us into error) and may be sound; but its soundness is opaque. If in fact it is sound, the whole thinking process is a reliable way of reaching the conclusion; so in the absence of factors that would make it irrational to trust the thinking, it would be a way of discovering the conclusion to be true.

In some cases visual thinking inclines one to believe something on the basis of assumptions suggested by the visual representation that remain to be justified given the subject’s current knowledge. In such cases there is always the danger that the subject takes the visual representation to show the correctness of the assumptions and ends up with an unwarranted belief. In such a case, even if the belief is true, the subject has not made a discovery, as the means of belief-acquisition is unreliable. Here is an example using Figure 7 (Montuchi and Page 1988).

Using this diagram one can come to think the following about the real numbers. When for a constant \(k\) the positive values of \(x\) and \(y\) are constrained to satisfy the equation \(x \cdot y = k\), the positive values of \(x\) and \(y\) for which \(x + y\) is minimal are \(x = \sqrt{k} = y\). (Let “#” denote this claim.)

Suppose that one knows the conventions for representing functions by graphs in a Cartesian co-ordinate system, knows also that the diagonal represents the function \(y = x\), and that a line segment with gradient –1 from \((0, b)\) to \((b, 0)\) represents the function \(x + y = b\). Then looking at the diagram may incline one to think that for no positive value of \(x\) does the value of \(y\) in the function \(x\cdot y = k\) fall below the value of \(y\) in \(x + y = 2\sqrt{k}\), and that these functions coincide just at the diagonal. From these beliefs the subject may (correctly) infer the conclusion #. But mere attention to the diagram cannot warrant believing that, for a given positive \(x\)-value, the \(y\)-value of \(x\cdot y = k\) never falls below the \(y\)-value of \(x + y = 2\sqrt{k}\) and that the functions coincide just at the diagonal; for the conventions of representation do not rule out that the curve of \(x\cdot y = k\) meets the curve of \(x + y = 2\sqrt{k}\) at two points extremely close to the diagonal, and that the former curve falls under the latter in between those two points. So the visual thinking is not in this case a means of discovering proposition #.

But it is useful because it provides the idea for a proof of the conclusion—one of the major benefits of visual thinking in mathematics. In brief: for each equation \((x\cdot y = k\); \(x + y = 2\sqrt{k})\) if \(x = y\), their common value is \(\sqrt{k}\). So the functions expressed by those equations meet at the diagonal. To show that, for a fixed positive \(x\)-value, the \(y\)-values of \(x\cdot y = k\) never fall below the \(y\)-values of \(x + y = 2\sqrt{k}\), it suffices to show that \(2\sqrt{k} - x \le k/x\). As a geometric mean is less than or equal to the corresponding arithmetic mean, \(\sqrt{[x \cdot (k/x)]} \le [x + (k/x)]/2\). So \(2\sqrt{k} \le x + (k/x)\). So \(2\sqrt{k} - x \le k/x\).

In this example, visual attention to, and reasoning about, the diagram is not part of a way of discovering the conclusion. But if it gave one the idea for the argument just given, it would be part of what led to a way of discovering the conclusion, and that is important.

Can visual thinking lead to discovery of an idea for a proof in more advanced contexts? Yes. Carter (2010) gives an example from free probability theory. The case is about certain permutations (those denoted by “\(p\)” with a circumflex in Carter 2010) on a finite set of natural numbers. Using specific kinds of diagram, easily seen properties of the diagrams lead one naturally to certain properties of the permutations (crossing and non-crossing, having neighbouring pairs), and to a certain operation (cancellation of neighbouring pairs). All of these have algebraic definitions, but the ideas defined were noticed by thinking in terms of the diagrams. For the relevant permutations \(\sigma\), \(\sigma(\sigma(n)) = n\); so a permutation can be represented by a set of lines joining dots. The permutations represented on the left and right in Figure 8 are non-crossing and crossing respectively, the former with neighbouring pairs \(\{2, 3\}\) and \(\{6, 7\}\).

A permutation \(\sigma\) of \(\{1, 2, \ldots, 2p\}\) is defined to have a crossing just when there are \(a\), \(b\), \(c\), \(d\) in \(\{1, 2, \ldots, 2p\}\) such that \(a < b < c < d\) and \(\sigma(a) = c\) and \(\sigma(b) = d\). The focus is on the proof of a theorem which employs this notion. (The theorem is that when a permutation of \(\{1, 2, \ldots, 2p\}\) of the relevant kind is non-crossing, there will be exactly \(p+1\) R-equivalence classes, where \(R\) is a certain equivalence relation on \(\{1, 2, \ldots, 2p\}\) defined in terms of the permutation.) Carter says that the proofs of some lemmas “rely on a visualization of the setup”, in that to grasp the correctness of one or more of the steps one needs to visualize the situation. There is also a nice example of some reasoning in terms of a diagram which gives the idea for a proof (“suggests a proof strategy”) for the lemma that every non-crossing permutation has a neighbouring pair. Reflection on a diagram such as Figure 9 does the work.

The reasoning is this. Suppose that \(\pi\) has no neighbouring pair. Choose \(j\) such that \(\pi(j) - j = a\) is minimal, that is, for all \(k, \pi(j) - j \le \pi(k) - k\). As \(\pi\) has no neighbouring pair, \(\pi(j+1) \ne j\). So either \(\pi(j+1)\) is less than \(j\) and we have a crossing, or by minimality of \(\pi(j) - j\), \(\pi(j+1)\) is greater than \(j+a\) and again we have a crossing. Carter reports that this disjunction was initially believed by thinking in term of the diagram, and the proof of the lemma given in the published paper is a non-diagrammatic “version” of that reasoning. In this case study, visual thinking is shown to contribute to discovery in several ways; in particular, by leading the mathematicians to notice crucial properties—the “definitions are based on the diagrams”—and in giving them the ideas for parts of the overall proof.

In this section I will illustrate and then discuss the use of visual thinking in discovering kinds of mathematical entity, by going through a few of the main steps leading to geometric group theory, a subject which really took off in the 1980s through the work of Mikhail Gromov. The material is set out nicely in greater depth in Starikova (2012).

Sometimes it can be fruitful to think of non-spatial entities, such as algebraic structures, in terms of a spatial representation. An example is the representation of a finitely generated group by a Cayley graph. Let \((G, \cdot)\) be a group and \(S\) a finite subset of \(G\). Let \(S^{-1}\) be the set of inverses of members of \(S\). Then \((G, \cdot)\) is generated by \(S\) if and only if every member of \(G\) is the product (with respect to \(\cdot\)) of members of \(S\cup S^{-1}\). In that case \((G, \cdot, S)\) is said to be a finitely generated group. Here are a couple of examples.

First consider the group \(S_{3}\) of permutations of 3 elements under composition. Letting \(\{a, b, c\}\) be the elements, all six permutations can be generated by \(\rf\) and \(\rr\) where

\(\rf\) (for “flip”) fixes a and swaps \(b\) with \(c\), i.e., it takes to \(\langle a, b, c\rangle\) to \(\langle a, c, b\rangle\), and

\(\rr\) (for “rotate”) takes \(\langle a, b, c\rangle\) to \(\langle c, a, b\rangle\).

The Cayley graph for \((S_{3}, \cdot, \{\rf, \rr\})\) is a graph whose vertices represent the members of \(S_{3}\) and two “colours” of directed edges, representing composition with \(\rf\) and composition with \(\rr\). Figure 10 illustrates: red directed edges represent composition with \(\rr\) and black edges represent composition with \(\rf\). So a red edge from a vertex \(\rv\) representing \(\rs\) in \(S_{3}\) ends at a vertex representing \(\rs\rr\) and a black edge from \(\rv\) ends at a vertex representing \(\rs\rf\). (Notation: “\(\rs\rr\)” abbreviates “\(\rs \cdot \rr\)” which here denotes “\(\rs\) followed by \(\rr\)”; same for “\(\rf\)” in place of “\(\rr\)”.) A black edge has arrowheads both ways because \(\rf\) is its own inverse, that is, flipping and flipping again takes you back to where you started. (Sometimes a pair of edges with arrows in opposite directions is used instead.) The symbol “\(\re\)” denotes the identity.

An example of a finitely generated group of infinite order is \((\mathbb{Z}, +, \{1\})\). We can get any integer by successively adding 1 or its additive inverse \(-1\). Since 3 added to the inverse of 2 is 1, and 2 added to the inverse of 3 is \(-1\), we can get any integer by adding members of \(\{2, 3\}\) and their inverses. Thus both \(\{1\}\) and \(\{2, 3\}\) are generating sets for \((\mathbb{Z}, +)\). Figure 11 illustrates part of the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\). The horizontal directed edges represent +2. The directed edges ascending or descending obliquely represent \(+3\).

Another example of a generated group of infinite order is \(F_2\), the free group generated by a pair of members. The first few iterations of its Cayley graph are shown in Figure 12, where \(\{a, b\}\) is the set of generators and a right horizontal move between adjacent vertices represents composition with \(a\), an upward vertical move represents composition with \(b\), and leftward and downward moves represent composition with the inverse of \(a\) and the inverse of \(b\) respectively. The central vertex represents the identity.

Thinking of generated groups in terms of their Cayley graphs makes it very natural to view them as metric spaces. A path is a sequence of consecutively adjacent edges, regardless of direction. For example in the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) the edges from \(-2\) to 1, from 1 to \(-1\), from \(-1\) to 2 (in that order) constitute a path, representing the action, starting from \(-2\), of adding 3, then adding \(-2\), then adding 3. Taking each edge to have unit length, the metric \(d_S\) for a group \(G\) generated by a finite subset \(S\) of \(G\) is defined: for any \(g\), \(h \in G\), \(d_{S}(g, h) =\) the length of a shortest path from \(g\) to \(h\) in the Caley graph of \((G, \cdot, S)\). This is the word metric for this generated group.

Viewing a finitely generated group as a metric space allows us to consider its growth function \(\gamma(n)\) which is the cardinality of the “ball” of radius \(\le n\) centred on the identity (the number of members of the group whose distance from the identity is not greater than \(n\)). A growth function for a given group depends on the set of generators chosen, but when the group is infinite the asymptotic behaviour as \(n \rightarrow \infty\) of the growth functions is independent of the set of generators.

Noticing the possibility of defining a metric on generated groups did not require first viewing diagrams of their Cayley graphs. This is because a word in the generators is just a finite sequence of symbols for the generators or their inverses (we omit the symbol for the group operation), and so has an obvious length visually suggested by the written form of the word, namely the number of symbols in the sequence; and then it is natural to define the distance between group members \(g\) and \(h\) to be the length of a shortest word that gets one from \(g\) to \(h\) by right multiplication, that is, \(\textrm{min}\{\textrm{length}(w): w = g^{-1}h\}\).

However, viewing generated groups by means of their Cayley graphs was the necessary starting point for geometric group theory, which enables us to view finitely generated groups of infinite order not merely as graphs or metric spaces but as geometric entities. The main steps on this route will be sketched briefly here; for more detail see Starikova (2012) and the references therein. The visual key is to start thinking in terms of the “coarse geometry” of the Cayley graph of the generated group, by zooming out in visual imagination so far that the discrete nature of the graph is transformed into a traditional geometrical object. For example, the Cayley graph of a generated group of finite order such as \((S_{3}, \cdot, \{f, r\})\) illustrated in Figure 11 becomes a dot; the Cayley graph for \((\mathbb{Z}, +, \{2, 3\})\) illustrated in Figure 12 becomes an uninterrupted line infinite in both directions.

The word metric of a generated group is discrete: the values are always in \(N\). How is this visuo-spatial association of a discrete metric space with a continuous geometrical object achieved mathematically? By quasi-isometry. While an isometry from one metric space to another is a distance preserving map, a quasi-isometry is a map which preserves distances to within fixed linear bounds. Precisely put, a map \(f\) from \((S, d)\) to \((S', d')\) is a quasi-isometry iff for some real constants \(L > 0\) and \(K \ge 0\) and all \(x\), \(y\) in \(S\) \[ d(x, y)/L - K \le d'(f(x), f(y)) \le L \cdot d(x, y) + K. \]

The spaces \((S, d)\) and \((S', d')\) are quasi - isometric spaces iff the quasi-isometry \(f\) is also quasi-surjective, in the sense that there is a real constant \(M \ge 0\) such that every point of \(S'\) is no further than \(M\) away from some point in the image of \(f\).

For example, \((\mathbb{Z}, d)\) is quasi-isometric to \((\mathbb{R}, d)\) where \(d(x, y) = |y - x|\), because the inclusion map \(\iota\) from \(\mathbb{Z}\) to \(\mathbb{R}\), \(\iota(n) = n\), is an isometry hence a quasi-isometry with \(L = 1\) and \(K = 0\), and each point in \(\mathbb{R}\) is no further than \(1/2\) away from an integer (in \(\mathbb{R}\)). Also, it is easy to see that for any real number \(x\), if \(g(x) =\) the nearest integer to \(x\) (or the greatest integer less than \(x\) if it is midway between integers) then \(g\) is a quasi-isometry from \(\mathbb{R}\) to \(\mathbb{Z}\) with \(L = 1\) and \(K =\frac{1}{2}\);.

The relation between metric spaces of being quasi-isometric is an equivalence relation. Also, if \(S\) and \(T\) are generating sets of a group \((G, \cdot)\), the Cayley graphs of \((G, \cdot, S)\) and \((G, \cdot, T)\) with their word metrics are quasi-isometric spaces. This means that properties of a generated group which are quasi-isometric invariants will be independent of the choice of generating set, and therefore informative about the group itself.

Moreover, it is easy to show that the Cayley graph of a generated group with word metric is quasi-isometric to a geodesic space. [ 1 ] A triangle with vertices \(x\), \(y\), \(z\) in this space is the union of three geodesic segments, between \(x\) and \(y\), between \(y\) and \(z\), and between \(z\) and \(x\). This is the gateway for the application of Gromov’s insights, some of which can be grasped with the help of visual geometric thinking.

Here are some indications. Recall the Poincaré open disc model of hyperbolic geometry: geodesics are diameters or arcs of circles orthogonal to the boundary, with unit distance represented by ever shorter Euclidean distances as one moves from the centre towards the boundary. (The boundary is not part of the model). All triangles have angle sum \(< \pi\) ( Figure 13, left ), and there is a global constant δ such that all triangles are δ-thin in the following sense:

A triangle \(T\) is δ- thin if and only if any point on one side of \(T\) lies within δ of some point on one of the other two sides.

This condition is equivalent to the condition that each side of \(T\) lies within the union of the δ-neighbourhoods of the other two sides, as illustrated in Figure 13 , right. There is no constant δ such that all triangles in a Euclidean plane are δ-thin, because for any δ there are triangles large enough that the midpoint of a longest side lies further than δ from all points on the other two sides.

Figure 13 [ 2 ]

The definition of thin triangles is sufficiently general to apply to any geodesic space and allows a generalisation of the concept of hyperbolicity beyond its original context:

• A geodesic space is hyperbolic iff for some δ all its triangles are δ-thin.
• A group is hyperbolic iff it has a Cayley graph quasi-isometric to a hyperbolic geodesic space.

The class of hyperbolic groups is large and includes important subkinds, such as finite groups, free groups and the fundamental groups of surfaces of genus \(\ge 2\). Some striking theorems have been proved for them. For example, for every hyperbolic group the word problem is solvable, and every hyperbolic group has a finite presentation. So we can reasonably conclude that the discovery of this mathematical kind, the hyperbolic groups, has been fruitful.

How important was visual thinking to the discoveries leading to geometric group theory? Visual thinking was needed to discover Cayley graphs as a means of representing finitely generated groups. This is not the triviality it might seem: Cayley graphs must be distinguished from the diagrams we use to present them visually. A Cayley graph is a mathematical representation of a generated group, not a visual representation. It consists of the following components: a set \(V\) (“vertices”), a set \(E\) of ordered pairs of members of \(V\) (“directed edges”) and a partition of \(E\) into distinguished subsets, (“colours”, each one for representing right multiplication by a particular generator). The Cayley graph of a generated group of infinite order cannot be fully represented by a diagram given the usual conventions of representation for diagrams of graphs, and distinct diagrams may visually represent the same Cayley graph: both diagrams in Figure 14 can be labelled so that under the usual conventions they represent the Cayley graph of \((S_{3}, \cdot, \{f, r\})\), already illustrated by Figure 10 . So the Cayley graph cannot be a diagram.

Diagrams of Cayley graphs were important in prompting mathematicians to think in terms of the coarse-grained geometry of the graphs, in that this idea arises just when one thinks in terms of “zooming out” visually. Gromov (1993) makes the point in a passage quoted in Starikova (2012:138)

This space [a Cayley graph with the word metric] may appear boring and uneventful to a geometer’s eye since it is discrete and the traditional (e.g., topological and infinitesimal) machinery does not run in [the group] Γ. To regain the geometric perspective one has to change one’s position and move the observation point far away from Γ. Then the metric in Γ seen from the distance \(d\) becomes the original distance divided by \(d\) and for \(d \rightarrow \infty\) the points in Γ coalesce into a connected continuous solid unity which occupies the visual horizon without any gaps and holes and fills our geometer’s heart with joy.

In saying that one has to move the observation point far away from Γ so that the points coalesce into a unity which occupies the visual horizon, he makes clear that visual imagination is involved in a crucial step on the road to geometric group theory. Visual thinking is again involved in discovering hyperbolicity as a property of general geodesic spaces from thinking about the Poincaré disk model of hyperbolic geometry. It is hard to see how this property would have been discovered without the use of visual resources.

While there is no reason to think that mental arithmetic (mental calculation in the integers and rational numbers) typically involves much visual thinking, there is strong evidence of substantial visual processing in the mental arithmetic of highly trained abacus users.

In earlier times an abacus would be a rectangular board or table surface marked with lines or grooves along which pebbles or counters could be moved. The oldest surviving abacus, the Salamis abacus, dated around 300 BCE, is a white marble slab, with markings designed for monetary calculation (Fernandes 2015, Other Internet Resources). These were superseded by rectangular frames within which wires or rods parallel to the short sides are fixed, with moveable holed beads on them. There are several kinds of modern abacus — the Chinese suanpan, the Russian schoty and the Japanese soroban for example — each kind with variations. Evidence for visual processing in mental arithmetic comes from studies with well trained users of the soroban, an example of which is shown in Figure 15.

Each column of beads represents a power of 10, increasing to the left. The horizontal bar, sometimes called the reckoning bar , separates the beads on each column into one bead of value 5 above and four beads of value 1 below. The number represented in a column is determined by the beads which are not separated from the reckoning bar. A column on which all beads are separated by a gap from the bar represents zero. For example, the number 6059 is represented on a portion of a schematic soroban in Figure 16.

On some sorobans there is a mark on the reckoning bar at every third column; if a user chooses one of these as a unit column, the marks will help the user keep track of which columns represent which powers of ten. Calculations are made by using forefinger and thumb to move beads according to procedures for the standard four numerical operations and for extraction of square and cube roots (Bernazzani 2005,Other Internet Resources). Despite the fact that the soroban has a decimal place representation of numbers, the soroban procedures are not ‘translations’ of the procedures normally taught for the standard operations using arabic numerals. For example, multidigit addition on a soroban starts by adding highest powers of ten and proceeds rightwards to lower powers, instead of starting with units thence proceeding leftwards to tens, hundreds and so on.

People trained to use a soroban often learn to do mental arithmetic by visualizing an abacus and imagining moving beads on it in accordance with the procedures learned for arithmetical calculations (Frank and Barner 2012). Mental abacus (MA), as this kind of mental arithmetic is known, compares favourably with other kinds of mental calculation for speed and accuracy (Kojima 1954) and MA users are often found among the medallists in the Mental Calculation World Cup.

Although visual and manual motor imagery is likely to occur, cognitive scientists have probed the question whether the actual processes of MA calculation consist in or involve imagining performing operations on a physical abacus. Brain imaging studies provide one source of evidence bearing on this question. Comparing well-trained abacus calculators with matched controls, evidence has been found that MA involves neural resources of visuospatial working memory with a form of abacus which does not depend on the modality (visual or auditory) of the numerical inputs (Chen et al. 2006). Another imaging study found that, compared to controls without abacus training, subjects with long term MA training from a young age had enhanced brain white matter related to motor and visuospatial processes (Hu et al. 2011).

Behavioural studies provide more evidence. Tests on expert and intermediate level abacus users strongly suggest that MA calculators mentally manipulate an abacus representation so that it passes through the same states that an actual abacus would pass through in solving an addition problem. Without using an actual abacus MA calculators were able to answer correctly questions about intermediates states unique to the abacus-based solution of a problem; moreover, their response times were a monotonic function of the position of the probed state in the sequence of states of the abacus process for solving the problem (Stigler 1984). On top of the ‘intermediate states’ evidence, there is ‘error type’ evidence. Mental addition tests comparing abacus users with American subjects revealed that abacus users made errors of a kind which the Americans did not make, but which were predictable from the distribution of errors in physical abacus addition (Stigler 1984).

Another study found evidence that when a sequence of numbers is presented auditorily (as a verbal whole “three thousand five hundred and forty seven” or as a digit sequence “Three, five, four, seven”) abacus experts encode it into an imaged abacus display, while non-experts encode it verbally (Hishitani 1990).

Further evidence comes from behavioural interference studies. In these studies subjects have to perform mental calculations, with and without a task of some other kind to be performed during the calculation, with the aim of seeing which kinds of task interfere with calculation as measured by differences of reaction time and error rate. An early study found that a linguistic task interfered weakly with MA performance (unless the linguistic task was to answer a mathematical question), while motor and visual tasks interfered relatively strongly. These findings suggested to the paper’s authors that MA representations are not linguistic in nature but rely on visual mechanisms and, for intermediate practitioners, on motor mechanisms as well (Hatano et al. 1977).

These studies provide impressive evidence that MA does involve mental manipulation of a visualized abacus. However, limits of the known capacities for perceiving or representing pluralities of objects seem to pose a problem. We have a parallel individuation system for keeping track of up to four objects simultaneously and an approximate number system (ANS) which allows us to gauge roughly the cardinality of a set of things, with an error which increases with the size of the set. The parallel individuation system has a limit of three or four objects and the ANS represents cardinalities greater than four only approximately. Yet mental abacus users would need to hold in mind with precision abacus representations involving a much larger number of beads than four (and the way in which those beads are distributed on the abacus). For example, the number 439 requires a precise distribution of twelve beads. Frank and Barner (2012) address this problem. In some circumstances we can perceive a plurality of objects as a single entity, a set, and simultaneously perceive those objects as individuals. There is evidence that we can keep track of up to three such sets in parallel and simultaneously make reliable estimates of the cardinalities of the sets (if not more than four). If the sets themselves can be easily perceived as (a) divided into disjoint subsets, e.g. columns of beads on an abacus, and (b) structured in a familiar way, e.g. as a distribution of four beads below a reckoning bar and one above, we have the resources for recognising a three-digit number from its abacus representation. The findings of (Frank and Barner 2012) suggest that this is what happens in MA: a mental abacus is represented in visuospatial working memory by splitting it into a series of columns each of which is stored as a unit with its own detailed substructure.

These cognitive investigations confirm the self-reports of mental abacus users that they calculate mentally by visualizing operating on an abacus as they would operate on a physical abacus. (See the 20-second movie Brief interview with mental abacus user , at the Stanford Language and Cognition Lab, for one such self-report.) There is good evidence that MA often involves processes linked to motor cognition in addition to active visual imagination. Intermediate abacus users often make hand movements, without necessarily attending to those movements during MA calculation, as shown in the second of the three short movies just mentioned. Experiments to test the possible role of motor processes in MA resulted in findings which led the authors to conclude that premotor processes involved in the planning of hand movements were involved in MA (Brooks et al. 2018).

6. A priori and a posteriori roles of visual experience

In coming to know a mathematical truth visual experience can play a merely “enabling” role. For example, visual experience may have been a factor in a person’s getting certain concepts involved in a mathematical proposition, thus enabling her to understand the proposition, without giving her reason to believe it. Or the visual experience of reading an argument in a text book may enable one to find out just what the argument is, without helping her tell that the argument is sound. In earlier sections visual experience has been presented as having roles in proof and propositional discovery that are not merely enabling. On the face of it this raises a puzzle: mathematics, as opposed to its application to natural phenomena, has traditionally been thought to be an a priori science; but if visual experience plays a role in acquiring mathematical knowledge which is not merely enabling, the result would surely be a posteriori knowledge, not a priori knowledge. Setting aside knowledge acquired by testimony (reading or hearing that such-&-such is the case), there remain plenty of cases where sensory experience seems to play an evidential role in coming to know some mathematical fact.

A plausible example of the evidential use of sensory experience is the case of a child coming to know that \(5 + 3 = 8\) by counting on her fingers. While there may be an important \(a\) priori element in the child’s appreciation that she can reliably generalise from the result of her counting experiment, getting that result by counting is an a posteriori route to it. For another example, consider the question: how many vertices does a cube have? With the background knowledge that cubes do not vary in shape and that material cubes do not differ from geometrical cubes in number of vertices (where a “vertex” of a material cube is a corner), one can find the answer by visually inspecting a material cube. Or if one does not have a material cube to hand, one can visually imagine a cube, and by attending to its top and bottom faces extract the information that the vertices of the cube are exactly the vertices of these two quadrangular faces. When one gets the answer by inspecting a material cube, the visual experience contributes to one’s grounds for believing the answer and that contribution is part of what makes the belief state knowledge. So the role of the visual experience is evidential; hence the resulting knowledge is not a priori . When one gets the answer by visually imagining a cube, one is drawing on the accumulated cognitive effects of past experiences of seeing material cubes to bring to mind what a cube looks like; so the experience of visual imagining has an indirectly evidential role in this case.

Do such examples show that mathematics is not an a priori science? Yes, if an a priori science is understood to be one whose knowable truths are all knowable only in an a priori way, without use of sense experience as evidence. No, if an a priori science is one whose knowable truths are all knowable in an a priori way, allowing that some may be knowable also in an a posteriori way.

Many cases of proving something (or following a proof of it) involve making, or imagining making, changes in a symbol array. A standard presentation of the proof of left-cancellation in group theory provides an example. “Left-cancellation” is the claim that for any members \(a\), \(b\), \(c\) of a group with operation \(\cdot\) and identity element \(\mathbf{e}\), if \(a \cdot b = a \cdot c\), then \(b = c\). Here is (the core of) a proof of it:

Suppose that one comes to know left-cancellation by following this sequence of steps. Is this an a priori way of getting this knowledge? Although following a mathematical proof is thought to be a paradigmatically a priori way of getting knowledge, attention to the role of visual experience here throws this into doubt. The case for claiming that the visual experience has an evidential role is as follows.

The visual experience reveals not only what the steps of the argument are but also that they are valid, thereby contributing to our grounds for accepting the argument and believing its conclusion. Consider, for example, the step from the second equation to the third. The relevant background knowledge, apart from the logic of identity, is that a group operation is associative. This fact is usually represented in the form of an equation that simply relocates brackets in an obvious way:

We see that relocating the brackets in accord with this format, the left-hand term of the second equation is transformed into the left-hand term of the third equation, and the same for the right-hand terms. So the visual experience plays an evidential role in our recognising as valid the step from the second equation to the third. Hence this quite standard route to knowledge of left-cancellation turns out to be a posteriori , even though it is a clear case of following a proof.

Against this, one may argue that the description just given of what is going on in following the proof is not strictly correct, as follows. Exactly the same proof can be expressed in natural language, using “the composition of \(x\) with \(y\)” for “\(x \cdot y\)”, but the result would be hard to take in. Or the proof can be presented using a different notational convention, one which forces a quite different expression of associativity. For example, we can use the Polish convention of putting the operation symbol before the operands: instead of “\(x \cdot y\)” we put “\(\cdot x y\)”. In that case associativity would be expressed in the following way, without brackets:

The equations of the proof would then need to be re-symbolised; but what is expressed by each equation after re-symbolisation and the steps from one to the next would be exactly as before. So we would be following the very same proof, step by step. But we would not be using visual experiences involved to notice the relocation of brackets this time. This suggests that the role of the different visual experiences involved in following the argument in its different guises is merely to give us access to the common reasoning: the role of the experience is merely enabling. On this account the visual experience does not strictly and literally enable us to see that any of the steps are valid; rather, recognition of (or sensitivity to) the validity of the steps results from cognitive processing at a more abstract level.

Which of these rival views is correct? Does our visual experience in following the argument presented with brackets (1) reveal to us the validity of some of the steps, given the relevant background knowledge ? Or (2) merely give us access to the argument? The core of the argument against view (1) is this:

Seeing the relocation of brackets is not essential to following the argument.

So seeing merely gives access to the argument; it does not reveal any step to be valid.

The step to this conclusion is faulty. How one follows a proof may, and in this case does, depend on how it is presented, and different ways of following a proof may be different ways of coming to know its conclusion. While seeing the relocation of brackets is not essential to all ways of following this argument, it is essential to the normal way of following the argument when it is symbolically presented with brackets in the way given above.

Associativity, expressed without symbols, is this: When the binary group operation is applied twice in succession on an ordered triple of operands \(\langle a, b, c\rangle\), it makes no difference whether the first application is to the initial two operands or the final two operands. While this is the content of associativity, for ease of processing associativity is almost always expressed as a symbol-manipulation rule. Visual perception is used to tell in particular cases whether the rule thus expressed is correctly implemented, in the context of prior knowledge that the rule is correct. What is going on here is a familiar division of labour in mathematical thinking. We first establish the soundness of a rule of symbol-manipulation (in terms of the governing semantic conventions—in this case the matter is trivial); then we check visually that the rule is correctly implemented. Processing at a more abstract, semantic level is often harder than processing at a purely syntactic level; it is for this reason that we often resort to symbol-manipulation techniques as proxy for reasoning directly with meanings to solve a problem. (What is six eighths divided by three fifths, without using any symbolic technique?) When we do use symbol-manipulation in proving or following a proof, visual experience is required to discern that the moves conform to permitted patterns and thus contributes to our grounds for accepting the argument. Then the way of coming to know the conclusion has an a posteriori element.

Must a use of visual experience in knowledge acquisition be evidential , if the visual experience is not merely enabling? Here is an example which supports a negative answer. Imagine a square or look at a drawing of one. Each of its four sides has a midpoint. Now visualize the “inner” square whose sides run between the midpoints of adjacent sides of the original square (Figure 17, left). By visualizing this figure, it should be clear that the original square is composed precisely of the inner square plus four corner triangles, each side of the inner square being the base of a corner triangle. One can now visualize the corner triangles folding over, with creases along the sides of the inner square. The starting and end states of the imagery transformation can be represented by the left and right diagrams of Figure 17.

Visualizing the folding-over within the remembered frame of the original square results in an image of the original square divided into square quarters, its quadrants, and the sides of the inner square seem to be diagonals of the quadrants. Many people conclude that the corner triangles can be arranged to cover the inner square exactly, without any gap or overlap. Thence they infer that the area of the original square is twice the size of the inner square. Let us assume that the propositions concerned are about Euclidean figures. Our concern is with the visual route to the following:

The parts of a square beyond its inner square (formed by joining midpoints of adjacent sides of the original square) can be arranged to fit the inner square exactly, without overlap or gap, without change of size or shape.

The experience of visualizing the corner triangles folding over can lead one to this belief. But it cannot provide good evidence for it. This is because visual experience (of sight or imagination) has limited acuity and so does not enable us to discriminate between a situation in which the outer triangles fit the inner square exactly and a situation in which they fit inexactly but well enough for the mismatch to escape visual detection. (This contrasts with the case of discovering the number of vertices of a cube by seeing or visualizing one.) Even though visualizing the square, the inner square and then visualizing the corner triangles folding over is constrained by the results of earlier perceptual experience of scenes with relevant similarities, we cannot draw from it reliable information about exact equality of areas, because perception itself is not reliable about exact equalities (or exact proportions) of continuous magnitudes.

Though the visual experience could not provide good evidence for the belief, it is possible that we erroneously use the experience evidentially in reaching the belief. But it is also possible, when reaching the belief in the way described, that we do not take the experience to provide evidence. A non-evidential use is more likely, if when one arrives at the belief in this way one feels fairly certain of it, while aware that visual perception and imagination have limited acuity and so cannot provide evidence for a claim of exact fit.

But what could the role of the visualizing experience possibly be, if it were neither merely enabling nor evidential? One suggestion is that we already have relevant beliefs and belief-forming dispositions, and the visualizing experience could serve to bring to mind the beliefs and to activate the belief-forming dispositions (Giaquinto 2007). These beliefs and dispositions will have resulted from prior possession of cognitive resources, some subject-specific such as concepts of geometrical figures, some subject-general such as symmetry perception about perceptually salient vertical and horizontal axes. A relevant prior belief in this case might be that a square is symmetric about a diagonal. A relevant disposition might be the disposition to believe that the quadrants of a square are congruent squares upon seeing or visualizing a square with a horizontal base plus the vertical and horizontal line segments joining midpoints of its opposite sides. (These dispositions differ from ordinary perceptual dispositions to believe what we see in that they are not cancelled when we mistrust the accuracy of the visual experience.)

The question whether the resulting belief would be knowledge depends on whether the belief-forming dispositions are reliable (truth-conducive) and the pre-existing belief states are states of knowledge. As these conditions can be met without any violation of epistemic rationality, the visualizing route described incompletely here can be a route to knowledge. In that case we would have an example of a use of visual experience which is integral to a way of knowing a truth, which is not merely enabling and yet not evidential. A fuller account and discussion is given in chapters 3 and 4 of Giaquinto (2007).

There are other significant uses of visual representations in mathematics. This final section briefly presents a couple of them.

Although the use of diagrams in arguments in analysis faces special dangers (as noted in 3.3 ), the use of diagrams to illustrate symbolically presented operations can be very helpful. Consider, for example, this pair of operations \(\{ f(x) + k, f(x + k) \}\). Grasping them and the difference between them can be aided by a visual illustration; similarly for the sets \(\{ f(x + k), f(x - k) \}\), \(\{ |f(x)|, f(|x|) \}\), \(\{ f(x)^{-1}, f^{-1}(x), f(x^{-1}) \}\). While generalization on the basis of a visual illustration is unreliable, we can use them as checks against calculation errors and overgeneralization. The same holds for properties. Consider for example, functions for which \(f(-x) = f(x)\), known as even functions, and functions for which \(f(-x) = -f(x)\), the odd functions: it can be helpful to have in mind the images of graphs of \(y = x^2\) and \(y = x^{3}\) as instances of evenness and oddness, to remind one that even functions are symmetrical about the \(y\)-axis and odd functions have rotation symmetry by \(\pi\) about the origin. They can serve as a reminder and check against over-generalisation: any general claim true of all odd functions, for example, must be true of \(y = x^{3}\) in particular.

The utility of visual representations in real and complex analysis is not confined to such simple cases. Visual representations can help us grasp what motivates certain definitions and arguments, and thereby deepen our understanding. Abundant confirmation of this claim can be gathered from working through the text Visual Complex Analysis (Needham 1997). Some mathematical subjects have natural visual representations, which then give rise to a domain of mathematical entities in their own right. This is true of geometry but is also true of subjects which become algebraic in nature very quickly, such as graph theory, knot theory and braid theory. Techniques of computer graphics now enable us to use moving images. For an example of the power of kinematic visual representations to provide and increase understanding of a subject, see the first two “chapters” of the online introduction to braid theory by Ester Dalvit (2012, Other Internet Resources).

With regard to proofs, a minimal kind of understanding consists in understanding each line (proposition or formula) and grasping the validity of each step to a new line from earlier lines. But we can have that stepwise grasp of proof without any idea of why it proceeds by those steps. One has a more advanced (or deeper) kind of understanding when one has the minimal understanding and a grasp of the motivating idea(s) and strategy of the proof. The point is sharply expressed by Weyl (1995 [1932]: 453), quoted in (Tappenden 2005:150)

We are not very pleased when we are forced to accept a mathematical truth by virtue of a complicated chain of formal conclusions and computations, which we traverse blindly, link by link, feeling our way by touch. We want first an overview of the aim and the road; we want to understand the idea of the proof, the deeper context.

Occasionally the author of a proof gives readers the desired understanding by adding commentary. But this is not always needed, as the idea of a proof is sometimes revealed in the presentation of the proof itself. Often this is done by using visual representations. An example is Fisk’s proof of Chvátal’s “art gallery” theorem. This theorem is the answer to a combinatorial problem in geometry. Put concretely, the problem is this. Let the \(n\) walls of a single-floored gallery make a polygon. What is the smallest number of stationary guards needed to ensure that every point of the gallery wall can be seen by a guard? If the polygon is convex (all interior angles < 180°), one guard will suffice, as guards may rotate. But if the polygon is not convex, as in Figure 18, one guard may not be enough.

Chvátal’s theorem gives the answer: for a gallery with \(n\) walls, \(\llcorner n/3\lrcorner\) guards suffice, where \(\llcorner n/3\lrcorner\) is the greatest integer \(\le n/3\). (If this does not sound to you sufficiently like a mathematical theorem, it can be restated as follows: Let \(S\) be a subset of the Euclidean plane. For a subset \(B\) of \(S\) let us say that \(B\) supervises \(S\) iff for each \(x \in S\) there is a \(y \in B\) such that the segment \(xy\) lies within \(S\). Then the smallest number \(f(n)\) such that every set bounded by a simple \(n\)-gon is supervised by a set of \(f(n)\) points is at most \(\llcorner n/3.\lrcorner\)

Here is Steve Fisk’s proof. A short induction shows that every polygon can be triangulated, i.e., non-crossing edges between non-adjacent vertices (“diagonals”) can be added so that the polygon is entirely composed of non-overlapping triangles. So take any \(n\)-sided polygon with a fixed triangulation. Think of it as a graph, a set of vertices and connected edges, as in Figure 19.

The first part of the proof shows that the graph is 3-colourable, i.e., every vertex can be coloured with one of just three colours (red, white and blue, say) so that no edge connects vertices of the same colour.

The argument proceeds by induction on \(n \ge 3\), the number of vertices.

For \(n = 3\) it is trivial. Assume it holds for all \(k\), where \(3 \le k < n\).

Let triangulated polygon \(G\) have \(n\) vertices. Let \(u\) and \(v\) be any two vertices connected by diagonal edge \(uv\). The diagonal \(uv\) splits \(G\) into two smaller graphs, both containing \(uv\). Give \(u\) and \(v\) different colours, say red and white, as in Figure 20.

By the inductive assumption, we may colour each of the smaller graphs with the three colours so that no edge joins vertices of the same colour, keeping fixed the colours of \(u\) and \(v\). Pasting together the two smaller graphs as coloured gives us a 3-colouring of the whole graph.

What remains is to show that \(\llcorner n/3\lrcorner\) or fewer guards can be placed on vertices so that every triangle is in the view of a guard. Let \(b\), \(r\) and \(w\) be the number of vertices coloured blue, red and white respectively. Let \(b\) be minimal in \(\{b, r, w\}\). Then \(b \le r\) and \(b \le w\). Then \(2b \le r + w\). So \(3b \le b + r + w = n\). So \(b \le n/3\) and so \(b \le \llcorner n/3\lrcorner\). Place a guard on each blue vertex. Done.

The central idea of this proof, or the proof strategy, is clear. While the actual diagrams produced here are superfluous to the proof, some visualizing enables us to grasp the central idea.

Thinking which involves the use of seen or visualized images, which may be static or moving, is widespread in mathematical practice. Such visual thinking may constitute a non-superfluous and non-replaceable part of thinking through a specific proof. But there is a real danger of over-generalisation when using images, which we need to guard against, and in some contexts, such as real and complex analysis, the apparent soundness of a diagrammatic inference is liable to be illusory.

Even when visual thinking does not contribute to proving a mathematical truth, it may enable one to discover a truth, where to discover a truth is to come to believe it in an independent, reliable and rational way. Visual thinking can also play a large role in discovering a central idea for a proof or a proof-strategy; and in discovering a kind of mathematical entity or a mathematical property.

The (non-superfluous) use of visual thinking in coming to know a mathematical truth does in some cases introduce an a posteriori element into the way one comes to know it, resulting in a posteriori mathematical knowledge. This is not as revolutionary as it may sound as a truth knowable a posteriori may also be knowable a priori . More interesting is the possibility that one can acquire some mathematical knowledge in a way in which visual thinking is essential but does not contribute evidence; in this case the role of the visual thinking may be to activate one’s prior cognitive resources. This opens the possibility that non-superfluous visual thinking may result in a priori knowledge of a mathematical truth.

Visual thinking may contribute to understanding in more than one way. Visual illustrations may be extremely useful in providing examples and non-examples of analytic concepts, thus helping to sharpen our grasp of those concepts. Also, visual thinking accompanying a proof may deepen our understanding of the proof, giving us an awareness of the direction of the proof so that, as Hermann Weyl put it, we are not forced to traverse the steps blindly, link by link, feeling our way by touch.

• Adams, C., 2001, The Knot Book , Providence, Rhode Island: American Mathematical Society.
• Azzouni, J., 2013, “That we see that some diagrammatic proofs are perfectly rigorous”, Philosophia Mathematica , 21: 323–338.
• Brown, J., 1999, Philosophy of Mathematics: an introduction to the world of proofs and pictures , London: Routledge.
• Barwise, J. and J. Etchemendy, 1996, “Visual information and valid reasoning”, in Logical Reasoning with Diagrams , G. Allwein and J. Barwise (eds) Oxford: Oxford University Press.
• Bolzano, B., 1817, “Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation”, in Ewald 1996: vol. 1, 225–248.
• Brooks, N., Barner, D., Frank, M. and Goldin-Meadow, S., 2018, The role of gesture in supporting mental representations: The case of mental abacus arithmetic. Cognitive Science , 45(2), 554–575.
• Carter, J., 2010, “Diagrams and Proofs in Analysis”, International Studies in the Philosophy of Science , 24: 1–14.
• Cauchy, A., 1813, “Recherche sur les polyèdres—premier mémoire”, Journal de l’Ecole Polytechnique , 9: 66–86.
• Chen, F., Hu, Z., Zhao, X., Wang, R. and Tang, X., 2006, Neural correlates of serial abacus mental calculation in children: a functional MRI study. Neuroscience Letters , 403(1–2), 46‐51.
• Chvátal, V., 1975, “A Combinatorial Theorem in Plane Geometry”, Journal of Combinatorial Theory , series B,18: 39–41, 1975.
• Dedekind, R., 1872, “Continuity and the Irrational Numbers”, in Essays on the Theory of Numbers , W. Beman (trans.) New York: Dover Publications.
• De Toffoli, S. and V. Giardino, 2014, “Forms and Roles of Diagrams in Knot Theory”, Erkenntnis , 79: 829–842.
• Eddy, R., 1985, “Behold! The Arithmetic-Geometric Mean Inequality”, College Mathematics Journal , 16: 208. Reprinted in Nelsen 1993: 51.
• Euclid, Elements , Published as Euclid’s Elements: all thirteen books complete in one volume , T. Heath (trans.), D. Densmore (ed.). Santa Fe: Green Lion Press 2002.
• Ewald, W. (ed.), 1996, From Kant to Hilbert. A Source Book in the Foundations of Mathematics , Volumes 1 and 2. Oxford: Clarendon Press.
• Fisk, S., 1978, “A Short Proof of Chvátal’s Watchman Theorem”, Journal of Combinatorial Theory , series B, 24: 374.
• Fomenko, A., 1994, Visual Geometry and Topology , M. Tsaplina (trans.) New York: Springer.
• Frank, M. and Barner, D., 2012, Representing exact number visually using mental abacus. Journal of Experimental Psychology: General , 141(1), 134–149.
• Giaquinto, M., 1993b, “Visualizing in Arithmetic”, Philosophy and Phenomenological Research , 53: 385–396.
• –––, 1994, “Epistemology of visual thinking in elementary real analysis”, British Journal for the Philosophy of Science , 45: 789–813.
• –––, 2007, Visual Thinking in Mathematics , Oxford: Oxford University Press.
• –––, 2011, “Crossing curves: a limit to the use of diagrams in proofs”, Philosophia Mathematica , 19: 281–307.
• Gromov, M., 1993, “Asymptotic invariants of infinite groups”, in Geometric Group Theory , A. Niblo and M. Roller (eds.), LMS Lecture Note Series, Vol. 182, Cambridge: Cambridge University Press, (vol. 2).
• Hahn, H., 1933, “The crisis in intuition”, Translated in Hans Hahn. Empiricism, Logic and Mathematics: Philosophical Papers , B. McGuiness (ed.) Dordrecht: D. Reidel 1980. First published in Krise und Neuaufbau in den exakten Wissenschaften , Fünf Wiener Vorträge, Leipzig and Vienna 1933.
• Hatano, G., Miyake, Y. and Binks, M., 1977, Performance of expert abacus operators. Cognition , 5, 47–55.
• Hilbert, D., 1894, “Die Grundlagen der Geometrie”, Ch. 2, in David Hilbert’s Lectures on the Foundations of Geometry (1891–1902) , M. Hallett and U. Majer (eds) Berlin: Springer 2004.
• Hishitani, S., 1990, Imagery experts: How do expert abacus operators process imagery? Applied Cognitive Psychology , 4(1), 33–46.
• Hoffman, D., 1987, “The Computer-Aided Discovery of New Embedded Minimal Surfaces”, Mathematical Intelligencer , 9: 8–21.
• Hu, Y., Geng, F., Tao, L., Hu, N., Du, F., Fu, K. and Chen, F., 2011, Enhanced white matter tracts integrity in children with abacus training. Human Brain Mapping . 32, 10–21.
• Jamnik, M., 2001, Mathematical Reasoning with Diagrams: From Intuition to Automation , Stanford, California: CSLI Publications.
• Joyal, A., R. Street, and D. Verity, 1996, “Traced monoidal categories”, Mathematical Proceedings of the Cambridge Philosophical Society , 119(3) 447–468.
• Kant, I., 1781/9, Kritik der reinen Vernunft , P. Guyer and A. Wood (trans. & eds), Cambridge: Cambridge University Press, 1998.
• Klein, F., 1893, “Sixth Evanston Colloquium lecture”, in The Evanston Colloquium Lectures on Mathematics , New York: Macmillan 1911. Partially reprinted in Ewald 1996: vol. 2: 958-65.
• Kojima, T., 1954, The Japanese abacus: Its use and theory . Tokyo, Japan. Tuttle.
• Landau, E., 1934, Differential and Integral Calculus , Hausner and Davis (trans.), New York: Chelsea 1950.
• Leinster, T., 2004, “Operads in Higher-dimensional Category theory”, Theory and Applications of Categories , 12(3): 73–194.
• Littlewood, J., 1953, “Postscript on Pictures”, in Littlewood’ Miscellany , Cambridge: Cambridge University Press 1986.
• Lyusternik, L., 1963, Convex Figures and Polyhedra , T. Smith (trans.), New York: Dover Publications.
• Mancosu, P., 2005, “Visualization in Logic and Mathematics”, in P. Mancosu, K. Jørgensen and S. Pedersen (eds), Visualization, Explanation and Reasoning Styles in Mathematics , Dordrecht: Springer.
• –––, 2011, “Explanation in Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = < https://plato.stanford.edu/archives/sum2011/entries/mathematics-explanation/ >.
• Maxwell, E., 1959, Fallacies in Mathematics , Cambridge University Press.
• Montuchi, P. and W. Page, 1988, “Behold! Two extremum problems (and the arithmetic-geometric mean inequality)”, College Mathematics Journal , 19: 347. Reprinted in Nelsen 1993: 52.
• Miller, N., 2001, A Diagrammatic Formal System for Euclidean Geometry , Ph. D. Thesis, Cornell University.
• Mumma, J. and M. Panza, 2012, “Diagrams in Mathematics: History and Philosophy”, Synthese , 186: Issue 1.
• Needham, T., 1997, Visual Complex Analysis , Oxford: Clarendon Press.
• Nelsen, R., 1993, Proofs Without Words: Exercises in Visual Thinking , Washington DC: The Mathematical Association of America.
• Palais, R., 1999, “The visualization of mathematics: towards a mathematical exploratorium”, Notices of the American Mathematical Society , 46: 647–658.
• Pasch, M., 1882, Vorlesungen über neuere Geometrie , Berlin: Springer 1926, 1976 (with introduction by Max Dehn).
• Rouse Ball, W., 1939, Mathematical Recreations and Essays , Revised by H. Coxeter, 11 th edition. (First published in 1892). New York: Macmillan.
• Russell, B., 1901, “Recent Work on the Principles of Mathematics”, International Monthly , 4: 83–101. Reprinted as “Mathematics and the Metaphysicians” in Mysticism and Logic , London: George Allen and Unwin 1918.
• Shin, Sun-Joo, Oliver Lemon, and John Mumma, 2013, “Diagrams”, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL = < https://plato.stanford.edu/archives/fall2013/entries/diagrams/ >.
• Starikova, I., 2012, “From Practice to New Concepts: Geometric Properties of Groups”, Philosophia Scientiae , 16(1): 129–151.
• Stigler, J., 1984, “Mental Abacus”: The effect of abacus training on Chinese children's mental calculation. Cognitive Psychology , 16, 145–176.
• Tappenden, J., 2005, “Proof style and understanding in mathematics I: visualization, unification and axiom choice”, in Mancosu, P., Jørgensen, K. and Pedersen, S. (eds) Visualization, Explanation and Reasoning Styles in Mathematics , Dordrecht: Springer.
• Tennant, N., 1986, “The Withering Away of Formal Semantics?” Mind and Language , 1(4): 382–318.
• Van den Dries, L., 1998, Tame Topology and O-minimal Structures , LMS Lecture Note Series 248, Cambridge University Press.
• Weyl, H., 1995 [1932], “Topology and abstract algebra as two roads of mathematical comprehension”, American Mathematical Monthly , 435–460 and 646–651. Translated by A. Shenitzer from an article of 1932, Gesammelte Abhandlungen , 3: 348–358.
• Zimmermann W. and S. Cunningham (eds), 1991, Visualization in Teaching and Learning Mathematics , Washington, DC: Mathematical Association of America.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Bernazzani, D., 2005, Soroban Abacus Handbook .
• Dalvit, E., 2012, A Journey through the Mathematical Theory of Braids .
• Fernandes, L. 2015 The Abacus: A brief history. .
• Lauda, A., 2005, Frobenius algebras and planar open string topological field theories , at arXiv.org.

a priori justification and knowledge | Bolzano, Bernard | Dedekind, Richard: contributions to the foundations of mathematics | diagrams | mathematical: explanation | proof theory | quantifiers and quantification | Weyl, Hermann

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Creating visual explanations improves learning

Eliza bobek.

1 University of Massachusetts Lowell, Lowell, MA USA

Barbara Tversky

2 Stanford University, Columbia University Teachers College, New York, NY USA

Associated Data

Many topics in science are notoriously difficult for students to learn. Mechanisms and processes outside student experience present particular challenges. While instruction typically involves visualizations, students usually explain in words. Because visual explanations can show parts and processes of complex systems directly, creating them should have benefits beyond creating verbal explanations. We compared learning from creating visual or verbal explanations for two STEM domains, a mechanical system (bicycle pump) and a chemical system (bonding). Both kinds of explanations were analyzed for content and learning assess by a post-test. For the mechanical system, creating a visual explanation increased understanding particularly for participants of low spatial ability. For the chemical system, creating both visual and verbal explanations improved learning without new teaching. Creating a visual explanation was superior and benefitted participants of both high and low spatial ability. Visual explanations often included crucial yet invisible features. The greater effectiveness of visual explanations appears attributable to the checks they provide for completeness and coherence as well as to their roles as platforms for inference. The benefits should generalize to other domains like the social sciences, history, and archeology where important information can be visualized. Together, the findings provide support for the use of learner-generated visual explanations as a powerful learning tool.

Electronic supplementary material

The online version of this article (doi:10.1186/s41235-016-0031-6) contains supplementary material, which is available to authorized users.

Significance

Uncovering cognitive principles for effective teaching and learning is a central application of cognitive psychology. Here we show: (1) creating explanations of STEM phenomena improves learning without additional teaching; and (2) creating visual explanations is superior to creating verbal ones. There are several notable differences between visual and verbal explanations; visual explanations map thought more directly than words and provide checks for completeness and coherence as well as a platform for inference, notably from structure to process. Extensions of the technique to other domains should be possible. Creating visual explanations is likely to enhance students’ spatial thinking skills, skills that are increasingly needed in the contemporary and future world.

Dynamic systems such as those in science and engineering, but also in history, politics, and other domains, are notoriously difficult to learn (e.g. Chi, DeLeeuw, Chiu, & Lavancher, 1994 ; Hmelo-Silver & Pfeffer, 2004 ; Johnstone, 1991 ; Perkins & Grotzer, 2005 ). Mechanisms, processes, and behavior of complex systems present particular challenges. Learners must master not only the individual components of the system or process (structure) but also the interactions and mechanisms (function), which may be complex and frequently invisible. If the phenomena are macroscopic, sub-microscopic, or abstract, there is an additional level of difficulty. Although the teaching of STEM phenomena typically relies on visualizations, such as pictures, graphs, and diagrams, learning is typically revealed in words, both spoken and written. Visualizations have many advantages over verbal explanations for teaching; can creating visual explanations promote learning?

Learning from visual representations in STEM

Given the inherent challenges in teaching and learning complex or invisible processes in science, educators have developed ways of representing these processes to enable and enhance student understanding. External visual representations, including diagrams, photographs, illustrations, flow charts, and graphs, are often used in science to both illustrate and explain concepts (e.g., Hegarty, Carpenter, & Just, 1990 ; Mayer, 1989 ). Visualizations can directly represent many structural and behavioral properties. They also help to draw inferences (Larkin & Simon, 1987 ), find routes in maps (Levine, 1982 ), spot trends in graphs (Kessell & Tversky, 2011 ; Zacks & Tversky, 1999 ), imagine traffic flow or seasonal changes in light from architectural sketches (e.g. Tversky & Suwa, 2009 ), and determine the consequences of movements of gears and pulleys in mechanical systems (e.g. Hegarty & Just, 1993 ; Hegarty, Kriz, & Cate, 2003 ). The use of visual elements such as arrows is another benefit to learning with visualizations. Arrows are widely produced and comprehended as representing a range of kinds of forces as well as changes over time (e.g. Heiser & Tversky, 2002 ; Tversky, Heiser, MacKenzie, Lozano, & Morrison, 2007 ). Visualizations are thus readily able to depict the parts and configurations of systems; presenting the same content via language may be more difficult. Although words can describe spatial properties, because the correspondences of meaning to language are purely symbolic, comprehension and construction of mental representations from descriptions is far more effortful and error prone (e.g. Glenberg & Langston, 1992 ; Hegarty & Just, 1993 ; Larkin & Simon, 1987 ; Mayer, 1989 ). Given the differences in how visual and verbal information is processed, how learners draw inferences and construct understanding in these two modes warrants further investigation.

Benefits of generating explanations

Learner-generated explanations of scientific phenomena may be an important learning strategy to consider beyond the utility of learning from a provided external visualization. Explanations convey information about concepts or processes with the goal of making clear and comprehensible an idea or set of ideas. Explanations may involve a variety of elements, such as the use of examples and analogies (Roscoe & Chi, 2007 ). When explaining something new, learners may have to think carefully about the relationships between elements in the process and prioritize the multitude of information available to them. Generating explanations may require learners to reorganize their mental models by allowing them to make and refine connections between and among elements and concepts. Explaining may also help learners metacognitively address their own knowledge gaps and misconceptions.

Many studies have shown that learning is enhanced when students are actively engaged in creative, generative activities (e.g. Chi, 2009 ; Hall, Bailey, & Tillman, 1997 ). Generative activities have been shown to benefit comprehension of domains involving invisible components, including electric circuits (Johnson & Mayer, 2010 ) and the chemistry of detergents (Schwamborn, Mayer, Thillmann, Leopold, & Leutner, 2010 ). Wittrock’s ( 1990 ) generative theory stresses the importance of learners actively constructing and developing relationships. Generative activities require learners to select information and choose how to integrate and represent the information in a unified way. When learners make connections between pieces of information, knowledge, and experience, by generating headings, summaries, pictures, and analogies, deeper understanding develops.

The information learners draw upon to construct their explanations is likely important. For example, Ainsworth and Loizou ( 2003 ) found that asking participants to self-explain with a diagram resulted in greater learning than self-explaining from text. How might learners explain with physical mechanisms or materials with multi-modal information?

Generating visual explanations

Learner-generated visualizations have been explored in several domains. Gobert and Clement ( 1999 ) investigated the effectiveness of student-generated diagrams versus student-generated summaries on understanding plate tectonics after reading an expository text. Students who generated diagrams scored significantly higher on a post-test measuring spatial and causal/dynamic content, even though the diagrams contained less domain-related information. Hall et al. ( 1997 ) showed that learners who generated their own illustrations from text performed equally as well as learners provided with text and illustrations. Both groups outperformed learners only provided with text. In a study concerning the law of conservation of energy, participants who generated drawings scored higher on a post-test than participants who wrote their own narrative of the process (Edens & Potter, 2003 ). In addition, the quality and number of concept units present in the drawing/science log correlated with performance on the post-test. Van Meter ( 2001 ) found that drawing while reading a text about Newton’s Laws was more effective than answering prompts in writing.

One aspect to explore is whether visual and verbal productions contain different types of information. Learning advantages for the generation of visualizations could be attributed to learners’ translating across modalities, from a verbal format into a visual format. Translating verbal information from the text into a visual explanation may promote deeper processing of the material and more complete and comprehensive mental models (Craik & Lockhart, 1972 ). Ainsworth and Iacovides ( 2005 ) addressed this issue by asking two groups of learners to self-explain while learning about the circulatory system of the human body. Learners given diagrams were asked to self-explain in writing and learners given text were asked to explain using a diagram. The results showed no overall differences in learning outcomes, however the learners provided text included significantly more information in their diagrams than the other group. Aleven and Koedinger ( 2002 ) argue that explanations are most helpful if they can integrate visual and verbal information. Translating across modalities may serve this purpose, although translating is not necessarily an easy task (Ainsworth, Bibby, & Wood, 2002 ).

It is important to remember that not all studies have found advantages to generating explanations. Wilkin ( 1997 ) found that directions to self-explain using a diagram hindered understanding in examples in physical motion when students were presented with text and instructed to draw a diagram. She argues that the diagrams encouraged learners to connect familiar but unrelated knowledge. In particular, “low benefit learners” in her study inappropriately used spatial adjacency and location to connect parts of diagrams, instead of the particular properties of those parts. Wilkin argues that these learners are novices and that experts may not make the same mistake since they have the skills to analyze features of a diagram according to their relevant properties. She also argues that the benefits of self-explaining are highest when the learning activity is constrained so that learners are limited in their possible interpretations. Other studies that have not found a learning advantage from generating drawings have in common an absence of support for the learner (Alesandrini, 1981 ; Leutner, Leopold, & Sumfleth, 2009 ). Another mediating factor may be the learner’s spatial ability.

The role of spatial ability

Spatial thinking involves objects, their size, location, shape, their relation to one another, and how and where they move through space. How then, might learners with different levels of spatial ability gain structural and functional understanding in science and how might this ability affect the utility of learner-generated visual explanations? Several lines of research have sought to explore the role of spatial ability in learning science. Kozhevnikov, Hegarty, and Mayer ( 2002 ) found that low spatial ability participants interpreted graphs as pictures, whereas high spatial ability participants were able to construct more schematic images and manipulate them spatially. Hegarty and Just ( 1993 ) found that the ability to mentally animate mechanical systems correlated with spatial ability, but not verbal ability. In their study, low spatial ability participants made more errors in movement verification tasks. Leutner et al. ( 2009 ) found no effect of spatial ability on the effectiveness of drawing compared to mentally imagining text content. Mayer and Sims ( 1994 ) found that spatial ability played a role in participants’ ability to integrate visual and verbal information presented in an animation. The authors argue that their results can be interpreted within the context of dual-coding theory. They suggest that low spatial ability participants must devote large amounts of cognitive effort into building a visual representation of the system. High spatial ability participants, on the other hand, are more able to allocate sufficient cognitive resources to building referential connections between visual and verbal information.

Benefits of testing

Although not presented that way, creating an explanation could be regarded as a form of testing. Considerable research has documented positive effects of testing on learning. Presumably taking a test requires retrieving and sometimes integrating the learned material and those processes can augment learning without additional teaching or study (e.g. Roediger & Karpicke, 2006 ; Roediger, Putnam, & Smith, 2011 ; Wheeler & Roediger, 1992 ). Hausmann and Vanlehn ( 2007 ) addressed the possibility that generating explanations is beneficial because learners merely spend more time with the content material than learners who are not required to generate an explanation. In their study, they compared the effects of using instructions to self-explain with instructions to merely paraphrase physics (electrodynamics) material. Attending to provided explanations by paraphrasing was not as effective as generating explanations as evidenced by retention scores on an exam 29 days after the experiment and transfer scores within and across domains. Their study concludes, “the important variable for learning was the process of producing an explanation” (p. 423). Thus, we expect benefits from creating either kind of explanation but for the reasons outlined previously, we expect larger benefits from creating visual explanations.

Present experiments

This study set out to answer a number of related questions about the role of learner-generated explanations in learning and understanding of invisible processes. (1) Do students learn more when they generate visual or verbal explanations? We anticipate that learning will be greater with the creation of visual explanations, as they encourage completeness and the integration of structure and function. (2) Does the inclusion of structural and functional information correlate with learning as measured by a post-test? We predict that including greater counts of information, particularly invisible and functional information, will positively correlate with higher post-test scores. (3) Does spatial ability predict the inclusion of structural and functional information in explanations, and does spatial ability predict post-test scores? We predict that high spatial ability participants will include more information in their explanations, and will score higher on post-tests.

Experiment 1

The first experiment examines the effects of creating visual or verbal explanations on the comprehension of a bicycle tire pump’s operation in participants with low and high spatial ability. Although the pump itself is not invisible, the components crucial to its function, notably the inlet and outlet valves, and the movement of air, are located inside the pump. It was predicted that visual explanations would include more information than verbal explanations, particularly structural information, since their construction encourages completeness and the production of a whole mechanical system. It was also predicted that functional information would be biased towards a verbal format, since much of the function of the pump is hidden and difficult to express in pictures. Finally, it was predicted that high spatial ability participants would be able to produce more complete explanations and would thus also demonstrate better performance on the post-test. Explanations were coded for structural and functional content, essential features, invisible features, arrows, and multiple steps.

Participants

Participants were 127 (59 female) seventh and eighth grade students, aged 12–14 years, enrolled in an independent school in New York City. The school’s student body is 70% white, 30% other ethnicities. Approximately 25% of the student body receives financial aid. The sample consisted of three class sections of seventh grade students and three class sections of eighth grade students. Both seventh and eighth grade classes were integrated science (earth, life, and physical sciences) and students were not grouped according to ability in any section. Written parental consent was obtained by means of signed informed consent forms. Each participant was randomly assigned to one of two conditions within each class. There were 64 participants in the visual condition explained the bicycle pump’s function by drawing and 63 participants explained the pump’s function by writing.

The materials consisted of a 12-inch Spalding bicycle pump, a blank 8.5 × 11 in. sheet of paper, and a post-test (Additional file 1 ). The pump’s chamber and hose were made of clear plastic; the handle and piston were black plastic. The parts of the pump (e.g. inlet valve, piston) were labeled.

Spatial ability was assessed using the Vandenberg and Kuse ( 1978 ) mental rotation test (MRT). The MRT is a 20-item test in which two-dimensional drawings of three-dimensional objects are compared. Each item consists of one “target” drawing and four drawings that are to be compared to the target. Two of the four drawings are rotated versions of the target drawing and the other two are not. The task is to identify the two rotated versions of the target. A score was determined by assigning one point to each question if both of the correct rotated versions were chosen. The maximum score was 20 points.

The post-test consisted of 16 true/false questions printed on a single sheet of paper measuring 8.5 × 11 in. Half of the questions related to the structure of the pump and the other half related to its function. The questions were adapted from Heiser and Tversky ( 2002 ) in order to be clear and comprehensible for this age group.

The experiment was conducted over the course of two non-consecutive days during the normal school day and during regularly scheduled class time. On the first day, participants completed the MRT as a whole-class activity. After completing an untimed practice test, they were given 3 min for each of the two parts of the MRT. On the second day, occurring between two and four days after completing the MRT, participants were individually asked to study an actual bicycle tire pump and were then asked to generate explanations of its function. The participants were tested individually in a quiet room away from the rest of the class. In addition to the pump, each participant was one instruction sheet and one blank sheet of paper for their explanations. The post-test was given upon completion of the explanation. The instruction sheet was read aloud to participants and they were instructed to read along. The first set of instructions was as follows: “A bicycle pump is a mechanical device that pumps air into bicycle tires. First, take this bicycle pump and try to understand how it works. Spend as much time as you need to understand the pump.” The next set of instructions differed for participants in each condition. The instructions for the visual condition were as follows: “Then, we would like you to draw your own diagram or set of diagrams that explain how the bike pump works. Draw your explanation so that someone else who has not seen the pump could understand the bike pump from your explanation. Don’t worry about the artistic quality of the diagrams; in fact, if something is hard for you to draw, you can explain what you would draw. What’s important is that the explanation should be primarily visual, in a diagram or diagrams.” The instructions for the verbal condition were as follows: “Then, we would like you to write an explanation of how the bike pump works. Write your explanation so that someone else who has not seen the pump could understand the bike pump from your explanation.” All participants then received these instructions: “You may not use the pump while you create your explanations. Please return it to me when you are ready to begin your explanation. When you are finished with the explanation, you will hand in your explanation to me and I will then give you 16 true/false questions about the bike pump. You will not be able to look at your explanation while you complete the questions.” Study and test were untimed. All students finished within the 45-min class period.

Spatial ability

The mean score on the MRT was 10.56, with a median of 11. Boys scored significantly higher (M = 13.5, SD = 4.4) than girls (M = 8.8, SD = 4.5), F(1, 126) = 19.07, p  < 0.01, a typical finding (Voyer, Voyer, & Bryden, 1995 ). Participants were split into high or low spatial ability by the median. Low and high spatial ability participants were equally distributed in the visual and verbal groups.

Learning outcomes

It was predicted that high spatial ability participants would be better able to mentally animate the bicycle pump system and therefore score higher on the post-test and that post-test scores would be higher for those who created visual explanations. Table  1 shows the scores on the post-test by condition and spatial ability. A two-way factorial ANOVA revealed marginally significant main effect of spatial ability F(1, 124) = 3.680, p  = 0.06, with high spatial ability participants scoring higher on the post-test. There was also a significant interaction between spatial ability and explanation type F(1, 124) = 4.094, p  < 0.01, see Fig.  1 . Creating a visual explanation of the bicycle pump selectively helped low spatial participants.

Post-test scores, by explanation type and spatial ability

Scores on the post-test by condition and spatial ability

Coding explanations

Explanations (see Fig.  2 ) were coded for structural and functional content, essential features, invisible features, arrows, and multiple steps. A subset of the explanations (20%) was coded by the first author and another researcher using the same coding system as a guide. The agreement between scores was above 90% for all measures. Disagreements were resolved through discussion. The first author then scored the remaining explanations.

Examples of visual and verbal explanations of the bicycle pump

Coding for structure and function

A maximum score of 12 points was awarded for the inclusion and labeling of six structural components: chamber, piston, inlet valve, outlet valve, handle, and hose. For the visual explanations, 1 point was given for a component drawn correctly and 1 additional point if the component was labeled correctly. For verbal explanations, sentences were divided into propositions, the smallest unit of meaning in a sentence. Descriptions of structural location e.g. “at the end of the piston is the inlet valve,” or of features of the components, e.g. the shape of a part, counted as structural components. Information was coded as functional if it depicted (typically with an arrow) or described the function/movement of an individual part, or the way multiple parts interact. No explanation contained more than ten functional units.

Visual explanations contained significantly more structural components (M = 6.05, SD = 2.76) than verbal explanations (M = 4.27, SD = 1.54), F(1, 126) = 20.53, p  < 0.05. The number of functional components did not differ between visual and verbal explanations as displayed in Figs.  3 and ​ and4. 4 . Many visual explanations (67%) contained verbal components; the structural and functional information in explanations was coded as depictive or descriptive. Structural and functional information were equally likely to be expressed in words or pictures in visual explanations. It was predicted that explanations created by high spatial participants would include more functional information. However, there were no significant differences found between low spatial (M = 5.15, SD = 2.21) and high spatial (M = 4.62, SD = 2.16) participants in the number of structural units or between low spatial (M = 3.83, SD = 2.51) and high spatial (M = 4.10, SD = 2.13) participants in the number of functional units.

Average number of structural and functional components in visual and verbal explanations

Visual and verbal explanations of chemical bonding

Coding of essential features

To further establish a relationship between the explanations generated and outcomes on the post-test, explanations were also coded for the inclusion of information essential to its function according to a 4-point scale (adapted from Hall et al., 1997 ). One point was given if both the inlet and the outlet valve were clearly present in the drawing or described in writing, 1 point was given if the piston inserted into the chamber was shown or described to be airtight, and 1 point was given for each of the two valves if they were shown or described to be opening/closing in the correct direction.

Visual explanations contained significantly more essential information (M = 1.78, SD = 1.0) than verbal explanations (M = 1.20, SD = 1.21), F(1, 126) = 7.63, p  < 0.05. Inclusion of essential features correlated positively with post-test scores, r = 0.197, p  < 0.05).

Coding arrows and multiple steps

For the visual explanations, three uses of arrows were coded and tallied: labeling a part or action, showing motion, or indicating sequence. Analysis of visual explanations revealed that 87% contained arrows. No significant differences were found between low and high spatial participants’ use of arrows to label and no signification correlations were found between the use of arrows and learning outcomes measured on the post-test.

The explanations were coded for the number of discrete steps used to explain the process of using the bike pump. The number of steps used by participants ranged from one to six. Participants whose explanations, whether verbal or visual, contained multiple steps scored significantly higher (M = 0.76, SD = 0.18) on the post-test than participants whose explanations consisted of a single step (M = 0.67, SD = 0.19), F(1, 126) = 5.02, p  < 0.05.

Coding invisible features

The bicycle tire pump, like many mechanical devices, contains several structural features that are hidden or invisible and must be inferred from the function of the pump. For the bicycle pump the invisible features are the inlet and outlet valves and the three phases of movement of air, entering the pump, moving through the pump, exiting the pump. Each feature received 1 point for a total of 5 possible points.

The mean score for the inclusion of invisible features was 3.26, SD = 1.25. The data were analyzed using linear regression and revealed that the total score for invisible parts significantly predicted scores on the post-test, F(1, 118) = 3.80, p  = 0.05.

In the first experiment, students learned the workings of a bicycle pump from interacting with an actual pump and creating a visual or verbal explanation of its function. Understanding the functionality of a bike pump depends on the actions and consequences of parts that are not visible. Overall, the results provide support for the use of learner-generated visual explanations in developing understanding of a new scientific system. The results show that low spatial ability participants were able to learn as successfully as high spatial ability participants when they first generated an explanation in a visual format.

Visual explanations may have led to greater understanding for a number of reasons. As discussed previously, visual explanations encourage completeness. They force learners to decide on the size, shape, and location of parts/objects. Understanding the “hidden” function of the invisible parts is key to understanding the function of the entire system and requires an understanding of how both the visible and invisible parts interact. The visual format may have been able to elicit components and concepts that are invisible and difficult to integrate into the formation of a mental model. The results show that including more of the essential features and showing multiple steps correlated with superior test performance. Understanding the bicycle pump requires understanding how all of these components are connected through movement, force, and function. Many (67%) of the visual explanations also contained written components to accompany their explanation. Arguably, some types of information may be difficult to depict visually and verbal language has many possibilities that allow for specificity. The inclusion of text as a complement to visual explanations may be key to the success of learner-generated explanations and the development of understanding.

A limitation of this experiment is that participants were not provided with detailed instructions for completing their explanations. In addition, this experiment does not fully clarify the role of spatial ability, since high spatial participants in the visual and verbal groups demonstrated equivalent knowledge of the pump on the post-test. One possibility is that the interaction with the bicycle pump prior to generating explanations was a sufficient learning experience for the high spatial participants. Other researchers (e.g. Flick, 1993 ) have shown that hands-on interactive experiences can be effective learning situations. High spatial ability participants may be better able to imagine the movement and function of a system (e.g. Hegarty, 1992 ).

Experiment 1 examined learning a mechanical system with invisible (hidden) parts. Participants were introduced to the system by being able to interact with an actual bicycle pump. While we did not assess participants’ prior knowledge of the pump with a pre-test, participants were randomly assigned to each condition. The findings have promising implications for teaching. Creating visual explanations should be an effective way to improve performance, especially in low spatial students. Instructors can guide the creation of visual explanations toward the features that augment learning. For example, students can be encouraged to show every step and action and to focus on the essential parts, even if invisible. The coding system shows that visual explanations can be objectively evaluated to provide feedback on students’ understanding. The utility of visual explanations may differ for scientific phenomena that are more abstract, or contain elements that are invisible due to their scale. Experiment 2 addresses this possibility by examining a sub-microscopic area of science: chemical bonding.

Experiment 2

In this experiment, we examine visual and verbal explanations in an area of chemistry: ionic and covalent bonding. Chemistry is often regarded as a difficult subject; one of the essential or inherent features of chemistry which presents difficulty is the interplay between the macroscopic, sub-microscopic, and representational levels (e.g. Bradley & Brand, 1985 ; Johnstone, 1991 ; Taber, 1997 ). In chemical bonding, invisible components engage in complex processes whose scale makes them impossible to observe. Chemists routinely use visual representations to investigate relationships and move between the observable, physical level and the invisible particulate level (Kozma, Chin, Russell, & Marx, 2002 ). Generating explanations in a visual format may be a particularly useful learning tool for this domain.

For this topic, we expect that creating a visual rather than verbal explanation will aid students of both high and low spatial abilities. Visual explanations demand completeness; they were predicted to include more information than verbal explanations, particularly structural information. The inclusion of functional information should lead to better performance on the post-test since understanding how and why atoms bond is crucial to understanding the process. Participants with high spatial ability may be better able to explain function since the sub-microscopic nature of bonding requires mentally imagining invisible particles and how they interact. This experiment also asks whether creating an explanation per se can increase learning in the absence of additional teaching by administering two post-tests of knowledge, one immediately following instruction but before creating an explanation and one after creating an explanation. The scores on this immediate post-test were used to confirm that the visual and verbal groups were equivalent prior to the generation of explanations. Explanations were coded for structural and functional information, arrows, specific examples, and multiple representations. Do the acts of selecting, integrating, and explaining knowledge serve learning even in the absence of further study or teaching?

Participants were 126 (58 female) eighth grade students, aged 13–14 years, with written parental consent and enrolled in the same independent school described in Experiment 1. None of the students previously participated in Experiment 1. As in Experiment 1, randomization occurred within-class, with participants assigned to either the visual or verbal explanation condition.

The materials consisted of the MRT (same as Experiment 1), a video lesson on chemical bonding, two versions of the instructions, the immediate post-test, the delayed post-test, and a blank page for the explanations. All paper materials were typed on 8.5 × 11 in. sheets of paper. Both immediate and delayed post-tests consisted of seven multiple-choice items and three free-response items. The video lesson on chemical bonding consisted of a video that was 13 min 22 s. The video began with a brief review of atoms and their structure and introduced the idea that atoms combine to form molecules. Next, the lesson showed that location in the periodic table reveals the behavior and reactivity of atoms, in particular the gain, loss, or sharing of electrons. Examples of atoms, their valence shell structure, stability, charges, transfer and sharing of electrons, and the formation of ionic, covalent, and polar covalent bonds were discussed. The example of NaCl (table salt) was used to illustrate ionic bonding and the examples of O 2 and H 2 O (water) were used to illustrate covalent bonding. Information was presented verbally, accompanied by drawings, written notes of keywords and terms, and a color-coded periodic table.

On the first of three non-consecutive school days, participants completed the MRT as a whole-class activity. On the second day (occurring between two and three days after completing the MRT), participants viewed the recorded lesson on chemical bonding. They were instructed to pay close attention to the material but were not allowed to take notes. Immediately following the video, participants had 20 min to complete the immediate post-test; all finished within this time frame. On the third day (occurring on the next school day after viewing the video and completing the immediate post-test), the participants were randomly assigned to either the visual or verbal explanation condition. The typed instructions were given to participants along with a blank 8.5 × 11 in. sheet of paper for their explanations. The instructions differed for each condition. For the visual condition, the instructions were as follows: “You have just finished learning about chemical bonding. On the next piece of paper, draw an explanation of how atoms bond and how ionic and covalent bonds differ. Draw your explanation so that another student your age who has never studied this topic will be able to understand it. Be as clear and complete as possible, and remember to use pictures/diagrams only. After you complete your explanation, you will be asked to answer a series of questions about bonding.”

For the verbal condition the instructions were: “You have just finished learning about chemical bonding. On the next piece of paper, write an explanation of how atoms bond and how ionic and covalent bonds differ. Write your explanation so that another student your age who has never studied this topic will be able to understand it. Be as clear and complete as possible. After you complete your explanation, you will be asked to answer a series of questions about bonding.”

Participants were instructed to read the instructions carefully before beginning the task. The participants completed their explanations as a whole-class activity. Participants were given unlimited time to complete their explanations. Upon completion of their explanations, participants were asked to complete the ten-question delayed post-test (comparable to but different from the first) and were given a maximum of 20 min to do so. All participants completed their explanations as well as the post-test during the 45-min class period.

The mean score on the MRT was 10.39, with a median of 11. Boys (M = 12.5, SD = 4.8) scored significantly higher than girls (M = 8.0, SD = 4.0), F(1, 125) = 24.49, p  < 0.01. Participants were split into low and high spatial ability based on the median.

The maximum score for both the immediate and delayed post-test was 10 points. A repeated measures ANOVA showed that the difference between the immediate post-test scores (M = 4.63, SD = 0.469) and delayed post-test scores (M = 7.04, SD = 0.299) was statistically significant F(1, 125) = 18.501, p  < 0.05). Without any further instruction, scores increased following the generation of a visual or verbal explanation. Both groups improved significantly; those who created visual explanations (M = 8.22, SD = 0.208), F(1, 125) = 51.24, p  < 0.01, Cohen’s d  = 1.27 as well as those who created verbal explanations (M = 6.31, SD = 0.273), F(1,125) = 15.796, p  < 0.05, Cohen’s d  = 0.71. As seen in Fig.  5 , participants who generated visual explanations (M = 0.822, SD = 0.208) scored considerably higher on the delayed post-test than participants who generated verbal explanations (M = 0.631, SD = 0.273), F(1, 125) = 19.707, p  < 0.01, Cohen’s d  = 0.88. In addition, high spatial participants (M = 0.824, SD = 0.273) scored significantly higher than low spatial participants (M = 0.636, SD = 0.207), F(1, 125) = 19.94, p  < 0.01, Cohen’s d  = 0.87. The results of the test of the interaction between group and spatial ability was not significant.

Scores on the post-tests by explanation type and spatial ability

Explanations were coded for structural and functional content, arrows, specific examples, and multiple representations. A subset of the explanations (20%) was coded by both the first author and a middle school science teacher with expertise in Chemistry. Both scorers used the same coding system as a guide. The percentage of agreement between scores was above 90 for all measures. The first author then scored the remainder of the explanations. As evident from Fig.  4 , the visual explanations were individual inventions; they neither resembled each other nor those used in teaching. Most contained language, especially labels and symbolic language such as NaCl.

Structure, function, and modality

Visual and verbal explanations were coded for depicting or describing structural and functional components. The structural components included the following: the correct number of valence electrons, the correct charges of atoms, the bonds between non-metals for covalent molecules and between a metal and non-metal for ionic molecules, the crystalline structure of ionic molecules, and that covalent bonds were individual molecules. The functional components included the following: transfer of electrons in ionic bonds, sharing of electrons in covalent bonds, attraction between ions of opposite charge, bonding resulting in atoms with neutral charge and stable electron shell configurations, and outcome of bonding shows molecules with overall neutral charge. The presence of each component was awarded 1 point; the maximum possible points was 5 for structural and 5 for functional information. The modality, visual or verbal, of each component was also coded; if the information was given in both formats, both were coded.

As displayed in Fig.  6 , visual explanations contained a significantly greater number of structural components (M = 2.81, SD = 1.56) than verbal explanations (M = 1.30, SD = 1.54), F(1, 125) = 13.69, p  < 0.05. There were no differences between verbal and visual explanations in the number of functional components. Structural information was more likely to be depicted (M = 3.38, SD = 1.49) than described (M = 0.429, SD = 1.03), F(1, 62) = 21.49, p  < 0.05, but functional information was equally likely to be depicted (M = 1.86, SD = 1.10) or described (M = 1.71, SD = 1.87).

Functional information expressed verbally in the visual explanations significantly predicted scores on the post-test, F(1, 62) = 21.603, p  < 0.01, while functional information in verbal explanations did not. The inclusion of structural information did not significantly predict test scores. As seen Fig.  7 , explanations created by high spatial participants contained significantly more functional components, F(1, 125) = 7.13, p  < 0.05, but there were no ability differences in the amount of structural information created by high spatial participants in either visual or verbal explanations.

Average number of structural and functional components created by low and high spatial ability learners

Ninety-two percent of visual explanations contained arrows. Arrows were used to indicate motion as well as to label. The use of arrows was positively correlated with scores on the post-test, r = 0.293, p  < 0.05. There were no significant differences in the use of arrows between low and high spatial participants.

Specific examples

Explanations were coded for the use of specific examples, such as NaCl, to illustrate ionic bonding and CO 2 and O 2 to illustrate covalent bonding. High spatial participants (M = 1.6, SD = 0.69) used specific examples in their verbal and visual explanations more often than low spatial participants (M = 1.07, SD = 0.79), a marginally significant effect F(1, 125) = 3.65, p  = 0.06. Visual and verbal explanations did not differ in the presence of specific examples. The inclusion of a specific example was positively correlated with delayed test scores, r = 0.555, p  < 0.05.

Use of multiple representations

Many of the explanations (65%) contained multiple representations of bonding. For example, ionic bonding and its properties can be represented at the level of individual atoms or at the level of many atoms bonded together in a crystalline compound. The representations that were coded were as follows: symbolic (e.g. NaCl), atomic (showing structure of atom(s), and macroscopic (visible). Participants who created visual explanations generated significantly more (M =1.79, SD = 1.20) than those who created verbal explanations (M = 1.33, SD = 0.48), F (125) = 6.03, p  < 0.05. However, the use of multiple representations did not significantly correlate with delayed post-test scores on the delayed post-test.

Metaphoric explanations

Although there were too few examples to be included in the statistical analyses, some participants in the visual group created explanations that used metaphors and/or analogies to illustrate the differences between the types of bonding. Figure  4 shows examples of metaphoric explanations. In one example, two stick figures are used to show “transfer” and “sharing” of an object between people. In another, two sharks are used to represent sodium and chlorine, and the transfer of fish instead of electrons.

In the second experiment, students were introduced to chemical bonding, a more abstract and complex set of phenomena than the bicycle pump used in the first experiment. Students were tested immediately after instruction. The following day, half the students created visual explanations and half created verbal explanations. Following creation of the explanations, students were tested again, with different questions. Performance was considerably higher as a consequence of creating either explanation despite the absence of new teaching. Generating an explanation in this way could be regarded as a test of learning. Seen this way, the results echo and amplify previous research showing the advantages of testing over study (e.g. Roediger et al., 2011 ; Roediger & Karpicke, 2006 ; Wheeler & Roediger, 1992 ). Specifically, creating an explanation requires selecting the crucial information, integrating it temporally and causally, and expressing it clearly, processes that seem to augment learning and understanding without additional teaching. Importantly, creating a visual explanation gave an extra boost to learning outcomes over and above the gains provided by creating a verbal explanation. This is most likely due to the directness of mapping complex systems to a visual-spatial format, a format that can also provide a natural check for completeness and coherence as well as a platform for inference. In the case of this more abstract and complex material, generating a visual explanation benefited both low spatial and high spatial participants even if it did not bring low spatial participants up to the level of high spatial participants as for the bicycle pump.

Participants high in spatial ability not only scored better, they also generated better explanations, including more of the information that predicted learning. Their explanations contained more functional information and more specific examples. Their visual explanations also contained more functional information.

As in Experiment 1, qualities of the explanations predicted learning outcomes. Including more arrows, typically used to indicate function, predicted delayed test scores as did articulating more functional information in words in visual explanations. Including more specific examples in both types of explanation also improved learning outcomes. These are all indications of deeper understanding of the processes, primarily expressed in the visual explanations. As before, these findings provide ways that educators can guide students to craft better visual explanations and augment learning.

General discussion

Two experiments examined how learner-generated explanations, particularly visual explanations, can be used to increase understanding in scientific domains, notably those that contain “invisible” components. It was proposed that visual explanations would be more effective than verbal explanations because they encourage completeness and coherence, are more explicit, and are typically multimodal. These two experiments differ meaningfully from previous studies in that the information selected for drawing was not taken from a written text, but from a physical object (bicycle pump) and a class lesson with multiple representations (chemical bonding).

The results show that creating an explanation of a STEM phenomenon benefits learning, even when the explanations are created after learning and in the absence of new instruction. These gains in performance in the absence of teaching bear similarities to recent research showing gains in learning from testing in the absence of new instruction (e.g. Roediger et al., 2011 ; Roediger & Karpicke, 2006 ; Wheeler & Roediger, 1992 ). Many researchers have argued that the retrieval of information required during testing strengthens or enhances the retrieval process itself. Formulating explanations may be an especially effective form of testing for post-instruction learning. Creating an explanation of a complex system requires the retrieval of critical information and then the integration of that information into a coherent and plausible account. Other factors, such as the timing of the creation of the explanations, and whether feedback is provided to students, should help clarify the benefits of generating explanations and how they may be seen as a form of testing. There may even be additional benefits to learners, including increasing their engagement and motivation in school, and increasing their communication and reasoning skills (Ainsworth, Prain, & Tytler, 2011 ). Formulating a visual explanation draws upon students’ creativity and imagination as they actively create their own product.

As in previous research, students with high spatial ability both produced better explanations and performed better on tests of learning (e.g. Uttal et al., 2013 ). The visual explanations of high spatial students contained more information and more of the information that predicts learning outcomes. For the workings of a bicycle pump, creating a visual as opposed to verbal explanation had little impact on students of high spatial ability but brought students of lower spatial ability up to the level of students with high spatial abilities. For the more difficult set of concepts, chemical bonding, creating a visual explanation led to much larger gains than creating a verbal one for students both high and low in spatial ability. It is likely a mistake to assume that how and high spatial learners will remain that way; there is evidence that spatial ability develops with experience (Baenninger & Newcombe, 1989 ). It is possible that low spatial learners need more support in constructing explanations that require imagining the movement and manipulation of objects in space. Students learned the function of the bike pump by examining an actual pump and learned bonding through a video presentation. Future work to investigate methods of presenting material to students may also help to clarify the utility of generating explanations.

Creating visual explanations had greater benefits than those accruing from creating verbal ones. Surely some of the effectiveness of visual explanations is because they represent and communicate more directly than language. Elements of a complex system can be depicted and arrayed spatially to reflect actual or metaphoric spatial configurations of the system parts. They also allow, indeed, encourage, the use of well-honed spatial inferences to substitute for and support abstract inferences (e.g. Larkin & Simon, 1987 ; Tversky, 2011 ). As noted, visual explanations provide checks for completeness and coherence, that is, verification that all the necessary elements of the system are represented and that they work together properly to produce the outcomes of the processes. Visual explanations also provide a concrete reference for making and checking inferences about the behavior, causality, and function of the system. Thus, creating a visual explanation facilitates the selection and integration of information underlying learning even more than creating a verbal explanation.

Creating visual explanations appears to be an underused method of supporting and evaluating students’ understanding of dynamic processes. Two obstacles to using visual explanations in classrooms seem to be developing guidelines for creating visual explanations and developing objective scoring systems for evaluating them. The present findings give insights into both. Creating a complete and coherent visual explanation entails selecting the essential components and linking them by behavior, process, or causality. This structure and organization is familiar from recipes or construction sets: first the ingredients or parts, then the sequence of actions. It is also the ingredients of theater or stories: the players and their actions. In fact, the creation of visual explanations can be practiced on these more familiar cases and then applied to new ones in other domains. Deconstructing and reconstructing knowledge and information in these ways has more generality than visual explanations: these techniques of analysis serve thought and provide skills and tools that underlie creative thought. Next, we have shown that objective scoring systems can be devised, beginning with separating the information into structure and function, then further decomposing the structure into the central parts or actors and the function into the qualities of the sequence of actions and their consequences. Assessing students’ prior knowledge and misconceptions can also easily be accomplished by having students create explanations at different times in a unit of study. Teachers can see how their students’ ideas change and if students can apply their understanding by analyzing visual explanations as a culminating activity.

Creating visual explanations of a range of phenomena should be an effective way to augment students’ spatial thinking skills, thereby increasing the effectiveness of these explanations as spatial ability increases. The proverbial reading, writing, and arithmetic are routinely regarded as the basic curriculum of school learning and teaching. Spatial skills are not typically taught in schools, but should be: these skills can be learned and are essential to functioning in the contemporary and future world (see Uttal et al., 2013 ). In our lives, both daily and professional, we need to understand the maps, charts, diagrams, and graphs that appear in the media and public places, with our apps and appliances, in forms we complete, in equipment we operate. In particular, spatial thinking underlies the skills needed for professional and amateur understanding in STEM fields and knowledge and understanding STEM concepts is increasingly required in what have not been regarded as STEM fields, notably the largest employers, business, and service.

This research has shown that creating visual explanations has clear benefits to students, both specific and potentially general. There are also benefits to teachers, specifically, revealing misunderstandings and gaps in knowledge. Visualizations could be used by teachers as a formative assessment tool to guide further instructional activities and scoring rubrics could allow for the identification of specific misconceptions. The bottom line is clear. Creating a visual explanation is an excellent way to learn and master complex systems.

Additional file

Post-tests. (DOC 44 kb)

Acknowledgments

The authors are indebted to the Varieties of Understanding Project at Fordham University and The John Templeton Foundation and to the following National Science Foundation grants for facilitating the research and/or preparing the manuscript: National Science Foundation NSF CHS-1513841, HHC 0905417, IIS-0725223, IIS-0855995, and REC 0440103. We are grateful to James E. Corter for his helpful suggestions and to Felice Frankel for her inspiration. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the funders. Please address correspondence to Barbara Tversky at the Columbia Teachers College, 525 W. 120th St., New York, NY 10025, USA. Email: [email protected].

Authors’ contributions

This research was part of EB’s doctoral dissertation under the advisement of BT. Both authors contributed to the design, analysis, and drafting of the manuscript. Both authors read and approved the final manuscript.

Competing interests

The author declares that they have no competing interests.

• Ainsworth SE, Bibby PA, Wood DJ. Examining the effects of different multiple representational systems in learning primary mathematics. Journal of the Learning Sciences. 2002; 11 (1):25–62. doi: 10.1207/S15327809JLS1101_2. [ CrossRef ] [ Google Scholar ]
• Ainsworth, S. E., & Iacovides, I. (2005). Learning by constructing self-explanation diagrams. Paper presented at the 11th Biennial Conference of European Association for Resarch on Learning and Instruction, Nicosia, Cyprus.
• Ainsworth SE, Loizou AT. The effects of self-explaining when learning with text or diagrams. Cognitive Science. 2003; 27 (4):669–681. doi: 10.1207/s15516709cog2704_5. [ CrossRef ] [ Google Scholar ]
• Ainsworth S, Prain V, Tytler R. Drawing to learn in science. Science. 2011; 26 :1096–1097. doi: 10.1126/science.1204153. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Alesandrini KL. Pictorial-verbal and analytic-holistic learning strategies in science learning. Journal of Educational Psychology. 1981; 73 :358–368. doi: 10.1037/0022-0663.73.3.358. [ CrossRef ] [ Google Scholar ]
• Aleven, V. & Koedinger, K. R. (2002). An effective metacognitive strategy: learning by doing and explaining with a computer-based cognitive tutor. Cognitive Science , 26 , 147–179.
• Baenninger M, Newcombe N. The role of experience in spatial test performance: A meta-analysis. Sex Roles. 1989; 20 (5–6):327–344. doi: 10.1007/BF00287729. [ CrossRef ] [ Google Scholar ]
• Bradley JD, Brand M. Stamping out misconceptions. Journal of Chemical Education. 1985; 62 (4):318. doi: 10.1021/ed062p318. [ CrossRef ] [ Google Scholar ]
• Chi MT. Active-Constructive-Interactive: A conceptual framework for differentiating learning activities. Topics in Cognitive Science. 2009; 1 :73–105. doi: 10.1111/j.1756-8765.2008.01005.x. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Chi MTH, DeLeeuw N, Chiu M, LaVancher C. Eliciting self-explanations improves understanding. Cognitive Science. 1994; 18 :439–477. [ Google Scholar ]
• Craik F, Lockhart R. Levels of processing: A framework for memory research. Journal of Verbal Learning and Verbal Behavior. 1972; 11 :671–684. doi: 10.1016/S0022-5371(72)80001-X. [ CrossRef ] [ Google Scholar ]
• Edens KM, Potter E. Using descriptive drawings as a conceptual change strategy in elementary science. School Science and Mathematics. 2003; 103 (3):135–144. doi: 10.1111/j.1949-8594.2003.tb18230.x. [ CrossRef ] [ Google Scholar ]
• Flick LB. The meanings of hands-on science. Journal of Science Teacher Education. 1993; 4 :1–8. doi: 10.1007/BF02628851. [ CrossRef ] [ Google Scholar ]
• Glenberg AM, Langston WE. Comprehension of illustrated text: Pictures help to build mental models. Journal of Memory and Language. 1992; 31 :129–151. doi: 10.1016/0749-596X(92)90008-L. [ CrossRef ] [ Google Scholar ]
• Gobert JD, Clement JJ. Effects of student-generated diagrams versus student-generated summaries on conceptual understanding of causal and dynamic knowledge in plate tectonics. Journal of Research in Science Teaching. 1999; 36 :39–53. doi: 10.1002/(SICI)1098-2736(199901)36:1<39::AID-TEA4>3.0.CO;2-I. [ CrossRef ] [ Google Scholar ]
• Hall VC, Bailey J, Tillman C. Can student-generated illustrations be worth ten thousand words? Journal of Educational Psychology. 1997; 89 (4):677–681. doi: 10.1037/0022-0663.89.4.677. [ CrossRef ] [ Google Scholar ]
• Hausmann RGM, Vanlehn K. Explaining self-explaining: A contrast between content and generation. In: Luckin R, Koedinger KR, Greer J, editors. Artificial intelligence in education: Building technology rich learning contexts that work. Amsterdam: Ios Press; 2007. pp. 417–424. [ Google Scholar ]
• Hegarty M. Mental animation: Inferring motion from static displays of mechanical systems. Journal of Experimental Psychology: Learning, Memory & Cognition. 1992; 18 :1084–1102. [ PubMed ] [ Google Scholar ]
• Hegarty M, Carpenter PA, Just MA. Diagrams in the comprehension of scientific text. In: Barr R, Kamil MS, Mosenthal P, Pearson PD, editors. Handbook of reading research. New York: Longman; 1990. pp. 641–669. [ Google Scholar ]
• Hegarty M, Just MA. Constructing mental models of machines from text and diagrams. Journal of Memory and Language. 1993; 32 :717–742. doi: 10.1006/jmla.1993.1036. [ CrossRef ] [ Google Scholar ]
• Hegarty M, Kriz S, Cate C. The roles of mental animations and external animations in understanding mechanical systems. Cognition & Instruction. 2003; 21 (4):325–360. doi: 10.1207/s1532690xci2104_1. [ CrossRef ] [ Google Scholar ]
• Heiser J, Tversky B. Diagrams and descriptions in acquiring complex systems. Proceedings of the Cognitive Science Society. Hillsdale: Erlbaum; 2002. [ Google Scholar ]
• Hmelo-Silver C, Pfeffer MG. Comparing expert and novice understanding of a complex system from the perspective of structures, behaviors, and functions. Cognitive Science. 2004; 28 :127–138. doi: 10.1207/s15516709cog2801_7. [ CrossRef ] [ Google Scholar ]
• Johnson CI, Mayer RE. Applying the self-explanation principle to multimedia learning in a computer-based game-like environment. Computers in Human Behavior. 2010; 26 :1246–1252. doi: 10.1016/j.chb.2010.03.025. [ CrossRef ] [ Google Scholar ]
• Johnstone AH. Why is science difficult to learn? Things are seldom what they seem. Journal of Chemical Education. 1991; 61 (10):847–849. doi: 10.1021/ed061p847. [ CrossRef ] [ Google Scholar ]
• Kessell AM, Tversky B. Visualizing space, time, and agents: Production, performance, and preference. Cognitive Processing. 2011; 12 :43–52. doi: 10.1007/s10339-010-0379-3. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Kozhevnikov M, Hegarty M, Mayer R. Revising the Visualizer–Verbalizer Dimension: Evidence for Two Types of Visualizers. Cognition & Instruction. 2002; 20 :37–77. doi: 10.1207/S1532690XCI2001_3. [ CrossRef ] [ Google Scholar ]
• Kozma R, Chin E, Russell J, Marx N. The roles of representations and tools in the chemistry laboratory and their implication for chemistry learning. Journal of the Learning Sciences. 2002; 9 (2):105–143. doi: 10.1207/s15327809jls0902_1. [ CrossRef ] [ Google Scholar ]
• Larkin J, Simon H. Why a diagram is (sometimes) worth ten thousand words. Cognitive Science. 1987; 11 :65–100. doi: 10.1111/j.1551-6708.1987.tb00863.x. [ CrossRef ] [ Google Scholar ]
• Leutner D, Leopold C, Sumfleth E. Cognitive load and science text comprehension: Effects of drawing and mentally imagining text content. Computers in Human Behavior. 2009; 25 :284–289. doi: 10.1016/j.chb.2008.12.010. [ CrossRef ] [ Google Scholar ]
• Levine M. You-are-here maps: Psychological considerations. Environment and Behavior. 1982; 14 :221–237. doi: 10.1177/0013916584142006. [ CrossRef ] [ Google Scholar ]
• Mayer RE. Systematic thinking fostered by illustrations in scientific text. Journal of Educational Psychology. 1989; 81 :240–246. doi: 10.1037/0022-0663.81.2.240. [ CrossRef ] [ Google Scholar ]
• Mayer RE, Sims VK. For whom is a picture worth a thousand words? Extensions of a dual-coding theory of multimedia learning. Journal of Educational Psychology. 1994; 86 (3):389–401. doi: 10.1037/0022-0663.86.3.389. [ CrossRef ] [ Google Scholar ]
• Perkins DN, Grotzer TA. Dimensions of causal understanding: The role of complex causal models in students’ understanding of science. Studies in Science Education. 2005; 41 :117–166. doi: 10.1080/03057260508560216. [ CrossRef ] [ Google Scholar ]
• Roediger HL, Karpicke JD. Test enhanced learning: Taking memory tests improves long-term retention. Psychological Science. 2006; 17 :249–255. doi: 10.1111/j.1467-9280.2006.01693.x. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Roediger HL, Putnam AL, Smith MA. Ten benefits of testing and their applications to educational practice. In: Ross BH, editor. The psychology of learning and motivation. New York: Elsevier; 2011. pp. 1–36. [ Google Scholar ]
• Roscoe RD, Chi MTH. Understanding tutor learning: Knowledge-building and knowledge-telling in peer tutors’ explanations and questions. Review of Educational Research. 2007; 77 :534–574. doi: 10.3102/0034654307309920. [ CrossRef ] [ Google Scholar ]
• Schwamborn A, Mayer RE, Thillmann H, Leopold C, Leutner D. Drawing as a generative activity and drawing as a prognostic activity. Journal of Educational Psychology. 2010; 102 :872–879. doi: 10.1037/a0019640. [ CrossRef ] [ Google Scholar ]
• Taber KS. Student understanding of ionic bonding: Molecular versus electrostatic framework? School Science Review. 1997; 78 (285):85–95. [ Google Scholar ]
• Tversky B. Visualizing thought. Topics in Cognitive Science. 2011; 3 :499–535. doi: 10.1111/j.1756-8765.2010.01113.x. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Tversky B, Heiser J, MacKenzie R, Lozano S, Morrison JB. Enriching animations. In: Lowe R, Schnotz W, editors. Learning with animation: Research implications for design. New York: Cambridge University Press; 2007. pp. 263–285. [ Google Scholar ]
• Tversky B, Suwa M. Thinking with sketches. In: Markman AB, Wood KL, editors. Tools for innovation. Oxford: Oxford University Press; 2009. pp. 75–84. [ Google Scholar ]
• Uttal DH, Meadow NG, Tipton E, Hand LL, Alden AR, Warren C, et al. The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin. 2013; 139 :352–402. doi: 10.1037/a0028446. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Van Meter P. Drawing construction as a strategy for learning from text. Journal of Educational Psychology. 2001; 93 (1):129–140. doi: 10.1037/0022-0663.93.1.129. [ CrossRef ] [ Google Scholar ]
• Vandenberg SG, Kuse AR. Mental rotations: A group test of three-dimensional spatial visualization. Perceptual Motor Skills. 1978; 47 :599–604. doi: 10.2466/pms.1978.47.2.599. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Voyer D, Voyer S, Bryden MP. Magnitude of sex differences in spatial abilities: A meta-analysis and consideration of critical variables. Psychological Bulletin. 1995; 117 :250–270. doi: 10.1037/0033-2909.117.2.250. [ PubMed ] [ CrossRef ] [ Google Scholar ]
• Wheeler MA, Roediger HL. Disparate effects of repeated testing: Reconciling Ballard’s (1913) and Bartlett’s (1932) results. Psychological Science. 1992; 3 :240–245. doi: 10.1111/j.1467-9280.1992.tb00036.x. [ CrossRef ] [ Google Scholar ]
• Wilkin J. Learning from explanations: Diagrams can “inhibit” the self-explanation effect. In: Anderson M, editor. Reasoning with diagrammatic representations II. Menlo Park: AAAI Press; 1997. [ Google Scholar ]
• Wittrock MC. Generative processes of comprehension. Educational Psychologist. 1990; 24 :345–376. doi: 10.1207/s15326985ep2404_2. [ CrossRef ] [ Google Scholar ]
• Zacks J, Tversky B. Bars and lines: A study of graphic communication. Memory and Cognition. 1999; 27 :1073–1079. doi: 10.3758/BF03201236. [ PubMed ] [ CrossRef ] [ Google Scholar ]

Why is Visualization Important for Learning Mathematics?

By charlene marchese.

How often do we ask students to show their mathematical thinking or explain an answer using words, pictures, or diagrams? If you’re like most of us, the answer to that question is probably “Very often”!

But what does it mean to express mathematical ideas and processes through these modalities? What is our expectation that students’ work contains language that connects to diagrams and pictures? Are representations of thinking created after a solution is found, or are pictures, diagrams, and language part of developing a solution? To explore these questions, let’s first take a close look at four mathematical tasks, each requiring increasingly complex conceptual understandings, and the visual representations and language that can support understanding these tasks.

Examples in Action

Example 1: an early childhood story problem.

Many children are introduced to story problems like this. For young learners, adding one on to a number or collection of objects is a complex concept and tackling a solution using a concrete model is a natural place to start.

If given a container of red and blue bears, a student could line up three red bears, place one blue one at the end of the line, and then count the four bears. To describe how they came up with their solution, a student could say, “I took three red bears and then one blue bear. Now I have 4 bears.”

A semi-concrete representation of a solution could be a drawing of three red circles and one blue circle, and the student then counts the four circles. For older learners, a symbolic or abstract representation shows the solved equation 3 + 1 = 4.

Visual strategies can also be used to support solving more complex problems, as shown in the next several examples. As you read each one, take notice of the images that come to mind.

Example 2 : An Elementary Grade Expression

One way to visualize this problem is to think about jumps on an open number line where a learner employs the critical understanding of decomposing numbers. Starting at 45, the larger number, and moving to the right, one could add 28 by jumping 2 tens (or 20) to land on 65. Next, one could jump 5 and land on the friendly number 70. Finally, one could jump 3 to land on 73. Landing on 73 means the sum of 45 and 28 is 73. In describing their work, a student could say, “I started with the higher number 45, jumped 20, then jumped 5, and then jumped 3, to get to 73. So 45 + 28 is equal to 73.”

Example 3: A Middle School Exploration

Starting with the understanding that the square root of a number is the side length of its square, a learner can visualize the approximate square that is created with a non-perfect square number such as 72.  Since arranging 72 tiles into a square is impossible, one could think about making one square with a number that is less than 72 and one that is greater than 72, such as 64 and 81. The square root of 64 is 8, and the square root of 81 is 9. Since 72 is about halfway between 64 and 81, one can approximate the side length or the square root of 72 to be about 8.5.  A student could describe their work by saying, “I can use the two square numbers that 72 is in between to estimate the square root of 72.”

Example 4: An 8th Grade or High School Exploration

For this question, we explore and apply the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. By building a model of a triangle, either with tiles or by sketching one on graph paper, the student can visualize the right triangle and the squares that can be created by each side length. The leg with a length of 3 units would create a square with an area of 9 square tiles, and the leg with a length of 4 units would create an area of 16 square tiles. The sum of the two squares is 25 square tiles, which would have a side length (or square root) of 5 units. Therefore, a right triangle with leg lengths of 3 units and 4 units would have a hypotenuse of 5 units.

The Pythagorean theorem can be (1) demonstrated concretely using square tiles, (2) shown in a semi-concrete way by sketching and labeling shapes on graph paper, or (3) represented abstractly by the theorem, which can be used to solve for either the missing value of a leg or the hypotenuse of a right triangle.

Connecting both the concrete and semi-concrete models to the Pythagorean theorem deepens students’ understanding of the theorem and its relationship to the physical world. A student making the connection between the concrete and abstract could say, “I built a square for each side of the triangle and determined that the sum of the squares of the legs of the triangle was equal to the square of the hypotenuse.”

There are multiple ways of visualizing mathematics to solve the above questions. Did you use the same visual representations illustrated here, or did you have a different way of visualizing? Were you able to make sense of the visual representations shown here? How do these representations enhance the understanding of the mathematical concept?

The examples illustrate that visualizing mathematics is both a support for students who struggle with a range of math concepts and a critical component of learning mathematics. From young children using concrete objects, like their fingers, to understand counting to college students using representational graphs to grasp calculus, visualization provides an avenue to understand mathematics for all students.

Developing Visualization Skills

How do we help students develop visualization skills, especially with those whose prior math classroom experience may not have included visualizing? In the book Routines for Reasoning (Kelemanik et al., 2016) the authors explore a Recognizing Repetition routine designed to develop this very skill.  Students are asked to build or draw a sequence of figures and then to analyze how the figures are growing by focusing on the repetition, looking specifically at what changes and what stays the same. Students use their analysis to determine the number of tiles needed to build the 10th figure and then determine a mathematical rule that can be applied to determine the number of tiles for any figure number.

Here is an example of a growing pattern that students can explore.

There are multiple ways of visualizing mathematics to solve the above questions. Are you able to make sense of the visual representations shown? Do you use the same ones illustrated here, or do you have a different way of visualizing? How do these representations enhance one’s understanding of the mathematical concept?

Based on this visualization of the growing pattern, students could determine a relationship between each part of the shape and its figure number. For example, each section of pink tiles is one number less than the figure number. To determine how many tiles would be in the 10th figure, students would know that each section of pink tiles would have 9 tiles (one less than the Figure 10). With 2 sets of pink tiles (9 + 9), plus the blue tile that is constant in each figure (1), students could add 9 + 9 + 1 = 19. Making a generalized rule from this line of thinking, a student might say in words, “Take one away from the figure number, double it and add one,” and then connect their words to the rule written symbolically 2( x – 1) + 1, with x representing the figure number.

Another student might have said, “Double the figure number and subtract 1,” or write it symbolically as  “2 x –1.” Can you visualize how they determined this rule from the model? Can you determine a general rule by visualizing the growing figures in a different way?

What about the students who initially cannot visualize how to determine the 10th figure? How can teachers support students who may not be able to initially see how the figure grows and/or how to state a generalization? To support this learner, it is necessary for the teacher to listen to students’ responses and be ready to ask questions based on responses. Questions like the ones below may be helpful to ask:

• Can you tell me how you built the first 4 figures that are displayed? Did you notice anything changing? Was there anything that stayed the same?
• Can you build what you think Figure 5 would look like and explain your thoughts?
• If we skip a few figures, can you use what you noticed about how the figures are growing to build Figure 10?
• Is there a way we can use the figure number to determine how many tiles it took to build Figure 3 and then 4? Could these ideas be used to determine how many tiles are in Figure 10 or Figure 100? Could it be sketched out?
• Could we describe the process to find the number of tiles in any figure?

Notice how the questions refer back to the concrete figures. That is, there is a relationship between the visuals made with materials and the language used . Teacher questioning can also enhance the mathematical practice of “Construct viable arguments and critique the reasoning of others” (Common Core State Standards Initiative, n.d.) and provide a vehicle for student-to-student engagement in mathematical discourse. A teacher could consider asking this series of questions: “Another group came up with the idea that if they doubled the figure number and subtracted 1, they would always get the number of tiles for that figure. Does that work? Will it always work? How do you think they determined this?”

The instructional goal of this routine is focused on students’ process of visualizing how the figures are growing, with an emphasis on what changes and what stays the same. Young children can begin to develop this ability by building and discussing age-appropriate pattern sequences with increasing complexity as they move through the grades. As students move into the upper elementary, middle, and high schools, they build on their ability to identify patterns and to generalize these patterns, which can be expressed in words and then with symbols. Explicitly making the connections between the physical pattern, the visualization of the changes in the pattern, and the symbolic representation of the pattern is a cycle that builds mathematical knowledge with understanding.

Connecting the Concrete, Semi-Concrete, and Abstract

From teacher education courses in college to professional development sessions in the field, the message is that students learn mathematics by starting with the concrete and then moving to the abstract. While this is a powerful message, it is important to take a closer look at the learning process.

There is an important step of semi-concrete or visual representation between the concrete and the abstract. It could be argued that this is the step we are asking students to describe when we ask them to show their work with words, pictures, and diagrams. These three steps—concrete, semi-concrete (visual representations), and abstract or symbolic—are not linear. Rather they repeatedly interact. Throughout a unit of study, students’ experience should be one where they go back and forth among these phases of representation with a focus on making the connection between them. It is the students’ understanding of the connection between the concrete, semi-concrete, and abstract that allows for deep knowledge of the content. We each encounter a whole class of students with different learning styles and ways of thinking. Therefore, all three of these methods always need to be available to all students with a continued focus on their interconnectedness.

A plea to all teachers reading this: With the pressures of test scores, pacing charts, and unfinished learning from the pandemic, many times we feel pulled to emphasize the abstract and only use concrete representations for students who have perceived challenges in learning mathematics. This blog post is making the case to reimagine the idea of moving from the concrete to the abstract to not only include the semi-concrete visual representation but to reconceptualize mathematics instruction as a continual interplay of the concrete, semi-concrete, and abstract representations of mathematical ideas for all students.

Here are a few suggestions:

• Be sure that concrete materials are always present and available for student use.
• Refer to concrete materials and connect them to the visual representations and abstract thinking of all students throughout the unit of study.
• Have an expectation that “show your work” goes beyond the symbolic representation to include pictures, a model, and/or words to explain solutions.
• As teachers, complete the mathematical tasks yourselves, a key component of Math for All, to explore and understand the various physical models, visual representations, and symbolic processes that can be developed in support of the mathematical content.
• Value the concrete models and pictures of the mathematical content in a similar way to how science classes value the physical materials and pictures of the science content.
• Provide opportunities for students to explore mathematics visually.

COMMON CORE STATE STANDARDS FOR MATHEMATICS . (n.d.).

‌Kelemanik, G., Lucenta, A., Janssen Creighton, S., & Lampert, M. (2016). Routines for reasoning: fostering the mathematical practices in all students . Heinemann.

The contents of this blog post were developed under a grant from the Department of Education. However, those contents do not necessarily represent the policy of the Department of Education, and you should not assume endorsement by the Federal Government.

Math for All is a professional development program that brings general and special education teachers together to enhance their skills in planning and adapting mathematics lessons to ensure that all students achieve high-quality learning outcomes in mathematics.

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The Power of Visualization in Math

Creating visual representations for math students can open up understanding. We have resources you can use in class tomorrow.

When do you know it’s time to try something different in your math lesson?

For me, I knew the moment I read this word problem to my fifth-grade summer school students: “On average, the sun’s energy density reaching Earth’s upper atmosphere is 1,350 watts per square meter. Assume the incident, monochromatic light has a wavelength of 800 nanometers (each photon has an energy of 2.48 × 10 -19 joules at this wavelength). How many photons are incident on the Earth’s upper atmosphere in one second?”

My students couldn’t get past the language, the sizes of the different numbers, or the science concepts addressed in the question. In short, I had effectively shut them down, and I needed a new approach to bring them back to their learning. So I started drawing on the whiteboard and created something with a little whimsy, a cartoon photon asking how much energy a photon has.

Immediately, students started yelling out, “2.48 × 10 -19 joules,” and they could even cite the text where they had learned the information. I knew I was on to something, so the next thing I drew was a series of boxes with our friend the photon.

If all of the photons in the image below were to hit in one second, how much energy is represented in the drawing?

Students realized that we were just adding up all the individual energy from each photon and then quickly realized that this was multiplication. And then they knew that the question we were trying to answer was just figuring out the number of photons, and since we knew the total energy in one second, we could compute the number of photons by division.

The point being, we reached a place where my students were able to process the learning. The power of the visual representation made all the difference for these students, and being able to sequence through the problem using the visual supports completely changed the interactions they were having with the problem.

If you’re like me, you’re thinking, “So the visual representations worked with this problem, but what about other types of problems? Surely there isn’t a visual model for every problem!”

The power of this moment, the change in the learning environment, and the excitement of my fifth graders as they could not only understand but explain to others what the problem was about convinced me it was worth the effort to pursue visualization and try to answer these questions: Is there a process to unlock visualizations in math? And are there resources already available to help make mathematics visual?

I realized that the first step in unlocking visualization as a scaffold for students was to change the kind of question I was asking myself. A powerful question to start with is: “How might I represent this learning target in a visual way?” This reframing opens a world of possible representations that we might not otherwise have considered. Thinking about many possible visual representations is the first step in creating a good one for students.

The Progressions published in tandem with the Common Core State Standards for mathematics are one resource for finding specific visual models based on grade level and standard. In my fifth-grade example, what I constructed was a sequenced process to develop a tape diagram—a type of visual model that uses rectangles to represent the parts of a ratio. I didn’t realize it, but to unlock my thinking I had to commit to finding a way to represent the problem in a visual way. Asking yourself a very simple series of questions leads you down a variety of learning paths, and primes you for the next step in the sequence—finding the right resources to complete your visualization journey.

Posing the question of visualization readies your brain to identify the right tool for the desired learning target and your students. That is, you’ll more readily know when you’ve identified the right tool for the job for your students. There are many, many resources available to help make this process even easier, and I’ve created a matrix of clickable tools, articles, and resources .

The process to visualize your math instruction is summarized at the top of my Visualizing Math graphic; below that is a mix of visualization strategies and resources you can use tomorrow in your classroom.

Our job as educators is to set a stage that maximizes the amount of learning done by our students, and teaching students mathematics in this visual way provides a powerful pathway for us to do our job well. The process of visualizing mathematics tests your abilities at first, and you’ll find that it makes both you and your students learn.

What is Visual Representation?

Home >> Neurodiversopedia >> V Terms

Visual representation means using pictures, symbols, or other images to show ideas or things. It helps kids understand things better by showing them instead of just telling them.

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Real World Example of Visual Representation

How does visual representation work, recommended products, related topics, frequently asked question.

Can visual representation help with communication difficulties?

Absolutely! Visual communication tools, like icons and symbols, offer an alternative way for children to express themselves, bridging communication gaps and promoting interaction.

What types of visual tools are commonly used for kids with special needs?

Common visual tools include symbols, pictures, charts, graphs, interactive apps, and social stories that aid comprehension, communication, and learning.

Can visual representation help improve organization and daily routines?

Absolutely, visual schedules and color-coded systems help kids follow routines, stay organized, and anticipate activities, reducing anxiety and promoting independence.

Is visual representation only for young children or can older kids also benefit?

Visual representation is beneficial for children of all ages, including older kids and teenagers. It helps convey complex information, foster independence, and enhance comprehension across various age groups.

Scientific Definition

Visual representation is conveying information, concepts, or data through visual means such as images, charts, graphs, and diagrams. It is crucial in facilitating comprehension and communication, especially for children with special needs . Visual elements make information more accessible and understandable, promoting effective learning and engagement. Unlike solely relying on words, visual representation taps into the brain’s natural ability to process and retain visual information, making it an essential tool in the educational journey of children with diverse learning needs.

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Here’s a look at how visual representation can help kids with special needs. Meet Max, a 7-year-old with ADHD . Max often finds it tough to follow classroom routines. To help him, his teacher uses visual schedules :

• Pictures of daily activities are put on a board.
• Step-by-step images show Max how to start his morning routine.
• Visual cues help Max remember what to do next.

With this system, Max feels more organized and less anxious about what comes next in his day.

Sponsored by - Goally

Visual representation helps make abstract ideas clearer for kids with special needs. Here’s how it’s used:

• Visual schedules display daily routines with pictures.
• Charts and graphs show progress or steps in a task.
• Pictograms break down instructions into easy-to-understand images.
• Social stories use visuals to teach appropriate behaviors.

These tools make learning and following instructions simpler and more engaging.

This post was originally published on 08/26/2023. It was updated on 08/07/2024. 

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Mindfulness

How visualization can benefit your well-being, visualization can help you reach a range of goals..

Updated November 20, 2023 | Reviewed by Gary Drevitch

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Visualization, also called mental imagery , is essentially seeing with the mind’s eye or hearing with the mind’s ear. That is, when visualizing you are having a visual sensory experience without the use of your eyes. In fact, research has shown that visualization recruits the same brain areas that actual seeing does (Pearson et al., 2015).

Humans have evolved to rely heavily on our eyesight, making us highly visually-oriented creatures. Because our brains are adapted to easily process and comprehend visual information, visualization can be a powerful tool for influencing our thoughts, emotions , and behaviors. In fact, research has shown that processing emotions using visualization is more powerful than processing verbally (Blackwell et al., 2019). For example, when research participants listen to descriptions of emotionally valenced situations (i.e., “your boss telling you that they are disappointed with your work”), participants who are instructed to imagine themselves in the situation demonstrate a greater change in mood than those that are instructed only to think about the situation verbally (Blackwell et al., 2019).

There appear to be a number of emotional, cognitive, and behavioral benefits to practicing visualization.

​Emotional. Some forms of visualization have been shown to increase optimism and other positive emotions (Murphy et al., 2015). It has also been shown to be a useful method for regulating negative emotions such as anxiety or overwhelm (Blackwell et al., 2019).

Cognitive. Visualization techniques can be used to facilitate some kinds of decision-making and problem-solving (Blackwell et al., 2019). For example, visualization might be helpful when planning the best route to take on your upcoming road trip. Visualization techniques, such as the mind palace, are also an effective means of improving memory . The mind palace technique involves using a place you are very familiar with, such as your bedroom, and using different locations within that space as mnemonic devices associated with a particular piece of information you are trying to store.

Behavioral. Visualization can also help us achieve our goals by allowing us to determine the appropriate sequences of actions needed to reach our goal and identify any potential obstacles we might encounter as we proceed toward a goal. In other words, we can use visualization as a sort of rough draft for our plans by imagining each step we need to take to reach our goal, what each step might include, what might go wrong, and the ways in which we might need to prepare.

Visualization Tools

Music. Visualization music is music that is specifically intended to facilitate visualization and similar meditative processes. This kind of music can also be described as atmospheric or ambient, as the purpose is not to occupy your attention , but rather to help you focus attention on your visualizations.

Boards. Visualization boards, also called vision boards , are visual representations of your goals , intentions, and desires. Vision boards are typically poster-sized and include a collage-type arrangement of images that symbolize different facets of your goals and intentions. Vision boards are useful for ensuring that your goals remain salient. That is, by creating a visual representation of your goals, you can easily look back at your vision board and remind yourself of the intentions you set. When your intentions are at the forefront of your mind, you are more likely to act in accordance with them.

Guided Imagery

Guided imagery is a visualization exercise in which you engage all of your senses as you imagine yourself in a positive, peaceful environment.

• To begin, find a comfortable position, close your eyes, and begin breathing slowly and deeply as you start to relax.
• Next, visualize a place where you feel calm and content. This can be a place you’ve been before, a place you would like to go, or a place that is wholly the product of your imagination . Engage all of your senses to add depth and detail to the place you are visualizing. Can you feel a soft breeze? Do you hear birds or the sound of water lapping on the shore?
• Reflect on the calm that emerges as you move deeper into your vision.
• As you inhale, imagine peace washing over you and filling your body.
• As you exhale, imagine exhaustion, tension , and stress being washed away.
• Stay in your vision for as long as you like.

Visualization is a simple yet powerful technique that we can use to improve many facets of our lives. We can use visualization to improve our mood, help us remember important information, facilitate problem-solving and decision-making , and boost progress toward our goals. Depending on the purpose, there are many forms of visualization we can practice. For example, if we are trying to regulate our mood we might try visualization meditation , whereas if we are trying to solidify our goals for the new year we might use a vision board or a mind map .

Adapted from a post on visualization published by The Berkeley Well-Being Institute.

Blackwell, S. E. (2019). Mental imagery: From basic research to clinical practice. Journal of Psychotherapy Integration, 29(3), 235.

Murphy, S. E., O’Donoghue, M. C., Drazich, E. H., Blackwell, S. E., Nobre, A. C., & Holmes, E. A. (2015). Imagining a brighter future: the effect of positive imagery training on mood, prospective mental imagery and emotional bias in older adults. Psychiatry Research, 230(1), 36-43.

Pearson, J., Naselaris, T., Holmes, E. A., & Kosslyn, S. M. (2015). Mental imagery: functional mechanisms and clinical applications. Trends in cognitive sciences, 19(10), 590-602.​

Tchiki Davis, Ph.D. , is a consultant, writer, and expert on well-being technology.

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Introducing TeachCatalystAI

TeachCatalystAI is a professional teaching assistant tool designed to help teachers create lesson plan, teaching materials, and many more with ease. Our AI-powered tool will help you streamline your classroom management, making it easier to keep track of students, assignments, and behavior. Our AI-powered tools and templates are great and configured to make you effective in teaching.

19 Benefits of Using Visuals in the Classroom

The use of visuals in the classroom is a great way to engage students and help them learn. Visuals can also be beneficial for teachers, making lessons easier to plan and helping to keep students interested.

In this article, we will explore the 19 important benefits of using visuals in the classroom. From improving student engagement to making lessons more interesting, we will look at how visuals can be used to make learning more effective.

Benefits of Using Visuals in Your Classroom  

1. visuals help to grab students’ attention and maintain their focus, particularly when used at the beginning of a lesson..

Visuals play a crucial role in capturing students’ attention and maintaining their focus throughout the lesson. Teachers can incorporate a variety of visual aids, such as pictures, videos, graphs, and charts to support their lectures. At the beginning of the class session, visuals can be especially effective in piquing students’ curiosity and encouraging them to participate actively. For instance, teachers can show an image or video that relates to the topic being discussed to stimulate student’s interest and inspire questions.

2. They make complex or abstract concepts easier to understand by breaking them down into more digestible parts.

Using visuals in the classroom is an excellent way to make complex or abstract concepts easier for students to understand. By breaking down these ideas into more digestible parts, visual aids like flowcharts, diagrams, and illustrations can help learners grasp difficult concepts with greater ease. This is especially important when topics are particularly technical or scientific, and may be challenging for some students to comprehend through text alone.

Visuals can also help students remember information more easily. When learning new material, our brains use two types of memory: short-term and long-term. Visual aids can help move information from short-term memory into long-term memory by triggering associations that make it easier to recall later on. Additionally, visuals can serve as prompts during exams or assignments and help reinforce key points that might otherwise be forgotten.

3. Graphics and images can provide an emotional connection that enhances learning and retention.

Not only do graphics and images make content more visually appealing, but they also have the power to enhance emotional connection with the material being presented. Research has shown that when learners are emotionally engaged, they are more likely to pay attention and retain information. In fact, studies have found that visuals can increase retention rates by up to 65%.

Moreover, visuals can convey complex ideas and concepts in a more simplified form, making it easier for students to understand and remember them. For instance, diagrams or infographics can help students visualize relationships between different ideas or data sets. Additionally, images can be used as memory cues – associating an image with a piece of information helps the brain retain it better.

4. They help to create a more immersive learning experience, making it easier for students to connect with the subject matter.

Visuals such as images, videos, and infographics are essential in creating an immersive learning experience for students. They enable learners to connect with the subject matter by making it more relatable and understandable. For instance, when studying history, showing images of famous historical events can transport students back in time and allow them to visualize what life was like then. Additionally, visuals can assist in clarifying complex topics that may be difficult to understand through words alone.

Visuals also have an impact on retention rates among learners. According to research studies, people remember 80% of what they see compared to only 20% of what they read. This means that incorporating visuals into classroom instruction is crucial in helping students retain information better. Moreover, using visuals encourages active engagement among students during the learning process since it stimulates their visual senses.

5. Visual aids can help teachers differentiate instruction by presenting information in multiple formats (e.g., text, images, diagrams) that appeal to different types of learners.

Visual aids are a powerful tool for teachers to differentiate instruction and present information in multiple formats that cater to different types of learners. By integrating visual elements such as images, diagrams, and videos into their teaching materials, educators can engage students who learn best through visual cues rather than just text-based lessons. This approach also helps to promote retention by allowing students to visualize concepts more easily.

Moreover, visuals provide a common language between teacher and student that can help bridge communication gaps caused by differing learning styles or language barriers. For example, students who speak English as a second language may find it easier to understand complex concepts when they are presented visually. In addition, using visual aids in the classroom promotes active learning by encouraging students to participate and ask questions about what they see on screen.

6. They facilitate communication between teacher and student by providing a shared reference point for discussion and feedback.

Visuals are an effective way to facilitate communication between teachers and students. By using visuals in the classroom, educators can provide their students with a shared reference point that they can refer back to when discussing lessons or receiving feedback. This not only helps to ensure that everyone is on the same page, but it also enables teachers to better understand how their students are interpreting and processing information.

Visual aids can be particularly useful for learners who struggle with verbal communication, as they offer an alternate way of expressing thoughts and ideas. For instance, visual aids such as diagrams, charts, and maps can help students visualize complex concepts in a more concrete way than words alone would allow. Additionally, visuals can be especially helpful for visual learners who need to see something before they fully understand it.

7. Illustrations and diagrams are particularly useful in science classes because they allow students to visualize complex processes like photosynthesis or cellular respiration.

Illustrations and diagrams are essential tools in science classes as they help students better understand complex processes. Photosynthesis, for example, is a process that can be challenging to grasp without visual aids. However, an illustration showing the chloroplasts and thylakoids within a plant cell can quickly clarify how light energy is converted into chemical energy.

Similarly, cellular respiration is another process that can benefit from illustrations and diagrams. Breaking down glucose into ATP via glycolysis, the Krebs cycle and the electron transport chain may sound like an abstract concept at first glance. But when students have access to well-crafted diagrams of these processes, they gain a deeper understanding of how living organisms produce energy.

8. Using visual aids helps keep pace with today’s technology-dependent world where children have grown up using tablets, smartphones, etc., which makes them adapt better when they see something visually attractive on a screen than just hearing about it.

Visual aids are an essential tool in modern-day classrooms. The use of visual aids helps keep pace with today’s technology-dependent world where children have grown up using tablets, smartphones, etc. This makes them adapt better when they see something visual in the classroom. Visuals make learning more interactive and engaging for students, which leads to better retention of knowledge.

Moreover, visual aids can be used to break down complex ideas into simpler ones, making challenging subjects easier to understand. Teachers can utilize various types of visuals such as diagrams, charts, videos or infographics for this purpose. These tools not only help students grasp abstract concepts but also enhance their understanding of real-world applications of academic material. Overall, the use of visual aids is crucial in creating an effective and inclusive learning environment that caters to all types of learners.

9. Visuals aid memory retention, as people remember pictures better than words; therefore, seeing something visually increases the chances of remembering it long-term.

Visuals are essential tools for educators to increase engagement in the classroom. They help students remember and understand information more effectively. According to research, visuals aid memory retention, as people remember pictures better than words. When students see something visually, it increases the chances of remembering it long-term.

10. Visuals increase motivation levels among students as colorful images break through monotony, making learning fun.

Visuals have the power to increase motivation levels among students, making learning more fun and engaging. Colorful images break through the monotony of reading textbooks or listening to lectures, helping students retain information better. When visuals are used in the classroom, students can visualize concepts and ideas, which makes it easier for them to understand complex topics. This is particularly helpful for subjects that require a lot of memorization, such as history or science.

11. Visuals are a great way to introduce new concepts that may be hard to understand at first but become easier once they’re explained visually.

Visual aids are an essential tool in the classroom as they help to introduce new concepts that may be difficult to understand. For instance, subjects such as science and mathematics rely heavily on diagrams, charts, and other visual aids to explain complex ideas clearly. Students can easily grasp abstract concepts when presented with visual representations of them.

12. Visuals make a topic more relevant, especially for students who have visual impairments or learning disabilities like dyslexia.

Incorporating visuals in the classroom helps to make educational topics more relevant to students, particularly those who may have visual impairments or learning disabilities such as dyslexia. When information is presented through images, videos, and diagrams, students are better able to understand abstract concepts and retain the material being taught. For example, using visual aids can help students with dyslexia overcome challenges in reading comprehension by presenting information in a way that is easier for them to process.

Visuals also improve engagement and participation among all students in the classroom. They help to break up long lectures and add variety to lessons, making them more interesting and enjoyable for learners of all ages. Visual aids can be used for a range of subjects including math equations, scientific concepts, historical events, geographical maps or even art projects. Teachers can use tools such as infographics or interactive whiteboards during presentations which not only enhance learning but also make lessons fun and interactive.

13. Visuals are a great way to show cause-and-effect relationships, and relationships between ideas, concepts, or processes.

Visuals are a powerful tool that can help teachers in the classroom to convey complex concepts, processes or ideas. They make it easy for students to understand abstract or complicated information by breaking them down into simple and easily digestible pieces. Visuals also help to create connections between different ideas, concepts, or processes making it easier for learners to see cause-and-effect relationships.

Visuals can also be used as stimuli during discussions or debates in class. For example, images can be used to introduce topics such as current events or historical events in order to spark discussions among students. Overall, visuals serve as an important supplement to traditional teaching methods, providing an engaging way of presenting information while catering to various learning styles within the classroom.

14. They’re also helpful in explaining elements of a story, especially characters, and their actions.

Visuals are an essential tool to help students understand the elements of a story, including characters and their actions. By using pictures or videos that visually depict each character and their mannerisms, it becomes easier for students to connect with the story’s plot. This engagement leads to a better comprehension of not just the story but also its underlying concepts.

15. Visuals are an easy way to show comparisons and contrasts.

Visuals are an excellent way to show comparisons and contrasts in a classroom setting. This is because visual aids can help to simplify complex information, allowing students to quickly identify similarities and differences between concepts. For example, charts and graphs can be used to compare data sets or demonstrate the relationship between different variables.

16. They work well for summarizing information and main ideas.

Visual aids, such as diagrams, charts, graphs, and tables are effective tools for summarizing information and main ideas. These visual representations help students to condense complex or detailed information into easily understandable formats. Visuals also serve as a memory aid by helping students to retain information better than when presented only through text.

17. Visuals are useful for showing relationships and connections between concepts, events, objects, or ideas.

Visuals are a powerful tool to help students understand the relationships and connections between concepts, events, objects, or ideas. By using diagrams, charts, graphs, and other visual aids, educators can make abstract or complex concepts more concrete and accessible. Visuals also cater to different learning styles; some learners may find it easier to understand information when presented visually rather than through text alone.

Visuals can also be used to help students identify patterns and trends. For example, a line graph can show how data changes over time while a bar chart can highlight differences in quantity between categories. In science class, drawings or diagrams of chemical reactions can illustrate the relationship between reactants and products. In literature classes, concept maps can help students visualize the themes and motifs of a novel.

18. Visuals also help students with visual impairments understand the material better.

Visuals are not only beneficial for students who are able to see, but they also help those with visual impairments understand the material better. Students who have difficulty seeing may rely heavily on text or auditory cues, but incorporating visuals in the classroom can provide a more comprehensive understanding of the material. For example, using tactile graphics can allow visually impaired students to feel and explore physical representations of concepts that may be difficult to visualize through other means.

Overall, incorporating visuals in the classroom is crucial for providing a comprehensive learning experience for all students regardless of their abilities. By utilizing various forms of visual aids and technologies, educators can create an inclusive environment where every student has an equal opportunity to learn and succeed.

19. Visuals can be used to explain abstract concepts.

Using visuals is an effective way of explaining abstract concepts to students in the classroom. These concepts are often difficult to grasp and understand through verbal explanations alone. Visuals such as diagrams, charts, and graphs can help students visualize complex ideas and make connections between them. This enhances their understanding and retention of the material being taught.

In conclusion, using visuals in the classroom is a great way to engage students, make learning more fun, enhance understanding and retention, create an atmosphere of collaboration, aid in critical thinking skills, and help students to see the big picture. Visuals can also be used to differentiate instruction and help to increase student achievement. The use of visuals in the classroom can help students gain knowledge and skills they will need in their future endeavors.

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Step 2: Understanding Visual Representation(s)

• First Online: 28 September 2021

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• Daniel Bühler 2  

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In the second HCD step, the context of use, including the users and the user tasks and goals, needs to be identified, described, and analyzed. This step usually consists in a close examination of real situations in which existing products are used by actual users. Since the context of use in the UIPP project was specified by the Universal Cognitive User Interface project, this chapter discusses universal characteristics of visual representations in HCI. That is, it describes in detail central properties and relations, it discusses existing pictogram systems, and it proposes a taxonomy of visual representations. For example, it argues that always two central properties must be considered: the design and the reference relation. The goal of the chapter is to achieve a general understanding of visual representations that might be the basis for the following steps in the process as much as for other design projects.

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De Souza has argued as early as in 1993 (pp. 771–772) that, in addition to findings in cognitive sciences, semiotics would lead to a better understanding of HCI.

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Bühler, D. (2021). Step 2: Understanding Visual Representation(s). In: Universal, Intuitive, and Permanent Pictograms. Springer, Wiesbaden. https://doi.org/10.1007/978-3-658-32310-3_2

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Problem Solving: Does Visual Representation Truly Help?

• September 30, 2021

Visual Representation in Solving Problems

Problem-solving isn’t always the one we encounter at work when we are tasked to “solve a problem”, it transcends way beyond that into our lives as humans and as professionals. Martinez(1998) defined problem-solving as “the process of moving towards a goal when the path to that goal is uncertain”. Simply put, problem-solving is when we try to achieve a goal without having known ahead of time how to do so.

This means, it could be something as little as “planning a dinner date” or “preparing for a court case” or “constructing a factory on a sloped lot”.  It is the process of looking for a solution. Holyoak(1995)  defined a solution as the sequence of operators that can transform the initial state into the goal state in accordance with the path constraints. Thus, problem-solving requires methods and actions to find a solution. One of such actions or methods is via visual representation. 

Visualization as a problem solving tool

Finke et al(1992 ) maintained that researchers who study the problem-solving process have long recognized visualization as a problem-solving tool. Lloyd(1994) explained that most people find a problem very difficult to solve mentally, and a way to solve problems is to construct a visual representation of the problem’s entry conditions.

Antonietti(1991) in his publication on “ Why does mental visualization facilitate problem-solving?” opined that visual images in problem-solving may result in a notable degree of success. He maintained that figural representations are useful mentally to simulate the situation and the transformations described in the problem. According to Khan(2015), visual aids are tools that help to make an issue or lesson clearer or easier to understand and know (pictures, models, charts, maps, videos, slides, real objects, etc.). 

A picture says 1000 words

A famous Chinese proverb attributed to Confucious says “hearing something a hundred times isn’t better than seeing it once”. This is sometimes presented as being similar to the words of Alan Wilson Watts, a famous English writer and philosopher, saying “one showing is worth a hundred sayings”.

There is another maxim that ”if we hear we forget, if we see we remember, and if we do something we know it”. Napoleon is also quoted as saying “ Un bon croquis vaut mieux qu’un long discours” which means “a good sketch is better than a long speech” in English. All these are not quoted to undermine the extremely valuable importance of words, rather, it’s to explain how problem-solving can be better with well-depicted visual representations and aids. 

Antonietti(1991) further explained that visualization can operate in problem-solving both after and before the problem is given. He opined that “in this way, familiar, but misleading, strategies of reasoning can be substituted with new and productive directions of thinking which avoid the “traps” created by the verbal formulation.”

In conclusion, visual representation can help tackle problems in steps and focus on one part of a whole at separate times. It can help present a problem to others in a way that they can relate to it and form a mental connection and picture of the problem that is to be solved. Simply put, it can help to understand a problem ourselves and then gain consensus with others on it.  An example of this is an attorney in a case using animation to present a visual representation of the medical damage caused to a client. This is done to the end that a decision to “solve the problem” presented before the court is solved in their favor.

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