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Big Ideas Learning’s Mathematics Teaching Practices Series: Implementing Tasks That Promote Reasoning and Problem Solving

Sophie Murphy
August 25 2020

maths problem solving learning intention

In this series, we will explore the eight research-based essential Mathematics Teaching Practices found in NCTM’s Principles to Actions. Today’s blog focuses on the second essential Mathematics Teaching Practices: implementing tasks that promote reasoning and problem-solving.

Today’s blog focuses on the second of NCTM’s eight research-based essential Mathematics Teaching Practices: implementing tasks that promote reasoning and problem solving.

Mathematics Teaching Practice #2 - Implementing Tasks That Promote Reasoning and Problem Solving

The second Mathematics Teaching Practice is an essential stepping-stone for understanding and transferring mathematical concepts and understanding. For students to successfully engage in deep-level tasks that allow for reasoning and problem solving, their mathematics classrooms cannot limit their thinking to pure memorization and carrying out computations with little or no understanding. Students need to explore and investigate mathematical reasoning with confidence and understanding.

In order to implement tasks that promote reasoning and problem solving, teachers and students must aim to do the following.

Teachers must:

Provide opportunities for students to engage in deep-level learning.
Motivate students’ learning of mathematics through opportunities for exploring and solving problems that build on and extend their current mathematical understanding.
Choose and develop tasks that provide multiple entry points for problems to be solved.
Pose tasks on a regular basis that require a high level of cognitive demand.
Know their students to find the “Goldilocks Zone” – not too easy, not too hard (or boring!)
Support students in exploring tasks that connect to real-world mathematics.
Encourage students to use procedures and strategies to make connections which can then be applied to solving tasks.
Value procedural and conceptual thinking. The goal is flexible and transferable thinking from one concept to another.

Students must:

Understand that learning can be difficult but embrace the challenge by persevering in solving problems.
Move into deep-level learning at the right time (after moving through sufficient surface level) to reason through tasks.
Use goals to understand where they are, where they need to go, and how they are going to get there through goal setting and clarity of task/understanding.
Draw upon and make connections with prior understanding and ideas when provided with challenging and deeper-level tasks.
Use tools and representations as needed to support thinking and problem solving.
Use a variety of solution approaches to solve problems.
Move from surface understandings to deeper understandings by defining, describing, discussing, and justifying to one another.
Work to make sense out of tasks and persevere in solving problems, and if the problems are too challenging, be able to return to the surface-level knowledge to gain further understanding and skills in solving the tasks.

Research Behind Deep-Level Learning - Reasoning and Problem Solving in Mathematics

Deep learning is hard. It requires a cognitive demand that can be challenging. It often requires prior learning and the ability to compare and contrast, apply previous knowledge, and make connections. Unless the right conditions are in place to learn how to reason, many students are left at the surface and procedural level of mathematical understanding that relies primarily on using memory alone. Or, on the flip side, students are not ready to go deep, as they have limited skills and or knowledge to move into deeper levels of understanding, reasoning, and problem solving.

If students get into deep-level learning too quickly, they can lose confidence and self-efficacy about their mathematical ability. For this reason, we need to support all students to become engaged in high-level thinking and ensure that all students have the opportunity to experience reasoning and problem solving throughout a sequence of learning, not just those who are considered to be high achievers in mathematics.

When multiple procedures are required, there is a need to understand the surface level, yet find the right balance to move from surface to deep so that students can solve and understand more complex mathematical problems. Hattie (2012) defines this as the “Goldilocks Zone,” where the learning is not too easy and not too hard (or tedious) – it is just right for all learners!

Stein and Lane (1996) suggest that student learning has the greatest impact in classrooms where the learning tasks consistently encourage high-level student thinking and reasoning and the least impact in classrooms where the tasks are routinely procedural in nature. It is well documented that deeper-level mathematical tasks are often perceived to be more complex and difficult to deliver in the classroom and that tasks that are procedural with fewer cognitive demands are prioritized in place of deeper-level instruction (Boaler, 1997).

The Importance of Using a Taxonomy

A taxonomy that is used effectively can encourage reasoning and access to the mathematics through multiple entry points that have supported students through the important learning and understandings being developed prior to deeper-level learning. This includes the use of different representations and tools that foster problem solving through a variety of solution strategies that move through different levels of learning. Furthermore, teachers can use a learning taxonomy to support the appropriate use and development of discourse and thinking.

A taxonomy can be used to support success criteria that moves from surface to deep level as the sequence of learning progresses. For example, using ‘I can,’ success criteria statements move students through the learning as the cognition progresses.

When effectively using a taxonomy, mathematical tasks are viewed as placing higher-level cognitive demands on students as they are encouraged to engage in active inquiry and exploration or to use procedures in ways that are meaningfully connected with concepts or understanding. Tasks that encourage students to use procedures, formulas, or algorithms in ways that are not actively linked to meaning, or that consist primarily of memorization or the reproduction of previously memorized facts, are viewed as placing lower-level cognitive demands on students.

A taxonomy should not be perceived as a checklist, but instead as a framework and chance to move from procedural to conceptual understandings. This gives students the opportunity to transfer their understandings from one context to another as teachers move from placing lower-level cognitive demands on students to higher cognitive demands that require reasoning and problem solving.

The SOLO Taxonomy

The Structure of Observed Learning Outcomes (SOLO) Taxonomy was developed by Biggs & Collis in 1982.

The SOLO taxonomy consists of two major categories with increasingly complex stages:

Surface (uni-structural and multi-structural responses)
Deep (relational and extended abstract responses).

Within the two categories, the SOLO taxonomy consists of four levels (Biggs & Collis, 1982):

Multiple ideas
Relating the ideas
Extending the ideas

Biggs and Collis argue that the purpose of the Structured Observed Learning Outcome (SOLO) taxonomy is to balance deep knowledge with surface knowledge when preparing students for deep-level understandings and transferring these to new contexts.

Each level of the SOLO taxonomy increases the demand on the amount of working memory or attention span. At the surface levels, a student only needs to encode the given information and can use a recall strategy to provide an answer. At the deep levels, a student needs to think not only about more things at once, but also how those objects interrelate.

The SOLO taxonomy is used extensively throughout the Big Ideas Math Series to inform the progression of learning and design goals through the consistent implementation of learning targets and success criteria .

How Can Teachers Effectively Promote Reasoning and Problem Solving?

As outlined in this blog and the prior blog, the use of learning intentions and success criteria are vital in providing instructional practice that offers students clear and purposeful goals and a foundation for what success looks like. The learning intentions should plainly explain what students need to understand and what they should be able to do. In addition, this also helps teachers plan learning activities.

Professor John Hattie’s research demonstrates that having clear learning intentions and success criteria coupled with the support of a taxonomy/framework that moves the students from procedural understandings to more complex deeper understandings help students self-regulate. When teachers set and communicate clear lesson goals to help students understand the success criteria, students know where they are, where they need to go, and how they are going to get there. They move from the procedural through to the conceptual. They cannot move to conceptual without the surface-level procedural.

You can find effective examples of learning intentions (also known as learning targets) and success criteria in the Big Ideas Math Series . These align to both the learning within the series and the teacher notes, providing teachers with an excellent starting point for their teaching strategies.

Key Takeaways

When promoting reasoning and problem solving, we suggest that teachers should consider using a taxonomy. Using a framework such as the SOLO Taxonomy can help teachers to ensure that tasks include lower and higher-level understanding, skills, and knowledge and that students are encouraged to engage in higher-level, problem-solving tasks that build up from lower-level cognitive skills. In order for students to learn mathematics with understanding, they must have opportunities to regularly engage in tasks that allow for reasoning and problem solving and make possible multiple entry points and varied solution strategies.

Remember, deep-level learning is hard, but we embrace challenge. Surface-level understandings are just as important to helping students to reason and problem solve as deep-level understandings.

Together with Big Ideas Learning and National Geographic Learning, I will continue to provide you with support as we navigate the complexities of 2020, whether this is online and remotely or back in the classroom. Despite the challenges, we are committed to supporting teachers in moving forward with high impact teaching and learning, starting with NCTM’s Mathematics Teaching Practices.

We will continue to share practical examples and connections to the NCTM evidence-based Mathematics

Teaching Strategies that will impact all classroom environments when used explicitly and intentionally.

Keep in touch, ask any questions or comment via twitter ( @_sophie_murphy_ ) and through Big Ideas Learning and National Geographic Learning . Stay safe and well. I look forward to connecting with you all again soon.

Biggs , J., & Collis , K. ( 1982 ). Evaluating the quality of learning The SOLO taxonomy . New York Academic Press.

Hattie, J. (2012). Visible learning for teachers: Maximizing impact on learning . Routledge.

Henningsen, M., & Stein, M. K. (1997). Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for research in mathematics education , 524-549.

Stein, M. K., & Smith, M. (1993). Practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics .

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A Year’s Worth of Learning Intentions & Success Criteria for Number Talks
Visible Learning

A Year’s Worth of Learning Intentions & Success Criteria for Number Talks

July 7, 2020 |
Math+Literacy, LLC |
Routines , Visible Learning |

Sometimes the idea of creating and communicating meaningful learning intentions and success criteria for every part of our instructional day is overwhelming.

From the moment learners enter our classroom door, they are engaged in intentional learning experiences. Some of these experiences are part of purposefully short, frequent tasks or routines that provide structure to meaningful, spaced practice, such as Number Talks, Quick Images, Which One Doesn’t Belong?, Counting Jars, Visual Estimation, and Notice and Wonder.

Creating and communicating meaningful learning intentions and success criteria for these routines can be daunting. It feels as though the learning intentions and success criteria could take up more time than the brief task itself. And planning 180 learning intentions and success criteria for short routines would be incredibly time-consuming.

On the other hand, we know communicating teacher clarity is pivotal to effective teaching (effect size = 0.75). We want learners to articulate what they are learning , why they are learning it , and how they will know when they are successful . An effective way to communicate teacher clarity is through learning intentions and success criteria. Learning intentions describe what we want our students to learn (effect size = 0.68) and success criteria specify the evidence learners will produce to show their progress toward mastering the learning intention (effect size 1.13). (Almarode, Fisher, Thunder, Hattie, & Fry, 2019)

So how do we efficiently and effectively create learning intentions and success criteria for math routines?

Our answer: Think big picture. Create year-long learning intentions and success criteria.

Rather than think about our learning goals for each day, week or unit’s Number Talks, we think about our learning goals for the whole year. We ask ourselves: By engaging in Number Talks multiple times each week, what are students learning? Why are they learning this? And how will they know when they are successful?

With the whole year in mind, we create three sets of learning intentions: content, language, and social learning intentions.

Content Learning Intentions communicate the mathematical content of the learning or the mathematical “what”.
Language Learning Intentions communicate the mathematical vocabulary and phrases students need to know to master the mathematical content.
Social Learning Intentions communicate the social skills that foster effective collaboration and communication. Together, Language and Social Learning Intentions communicate the mathematical “why” or the mathematical practices and processes for learning the content.

To create our Content Learning Intentions, we focus on the four main purposes of our Number Talks throughout the year:

growing fluency with part-part-whole relationships (We are learning to see combinations of __.) ,
growing fluency with one/two/ten more or less (We are learning to see __ more and __ less.) ,
subitizing (We are learning to see the quantity fast.) , and
developing number sense through benchmarks of five and ten (We are learning to see the largest group.) .

To create our Language Learning Intentions, we examine the mathematical vocabulary learners need to transfer from Number Talks to other mathematical problem solving tasks. We ask ourselves, “What words are important for talking and thinking about the mathematical content?”

To create our Social Learning Intentions, we consider why we value this significant, yet brief, whole group instruction so much. We ask ourselves, “Why is it valuable for all of our students to work, think, and talk together as a math community in a Number Talk?” Again, we examine the social skills for collaboration and communication that learners need to transfer from Number Talks to other learning tasks.

Here are our learning intentions for a year of Number Talks in PreK and Kindergarten:

Now, we can select one content, language, and social learning intention for each Number Talk and quickly create aligned success criteria. We vary our wording but keep the heart of the learning intentions and success criteria the same over time to help learners’ develop meaning through familiarity. We adjust the numbers, representations, tools, questions, and language frames to create novelty and engagement. And we keep it brief when we communicate the learning intentions and success criteria to learners so the learning task remains central.

Here are two examples:

Alisha’s Kindergarten class:

Alisha planned three Number Talks over the course of one week. The learning intention for all three Number Talks was, “We are learning to see combinations of 7.” The success criteria for the first Number Talk using ten-frames was, “I can share how my friend saw seven.” The second Number Talk used domino dot patterns and the success criteria was, “I can compare how our thoughts are the same and different.” And the final Number Talk of the week’s success criteria was, “I can use a friend’s strategy,” using rekenreks.

Kateri’s PreK class:

Kateri planned two Number Talks over the course of one week. The learning intention for both Number Talks was, “We are learning to identify how many dots by noticing how many are empty,” and “We are learning to listen to friends’ ideas.” The success criteria for both Number Talks was, “I can name the total dots by first looking at the empty spaces,” and “I can notice if I saw the same empty spaces as my friends.” For the first Number Talk, they used ten-frames with 0 and 1 empty spaces in different locations. For the second Number Talk, they again used ten-frames but with 0, 1, and 2 empty spaces in different locations.

As we begin thinking about how to use what we know about math routines in new learning settings – whether in-person, virtual, or hybrid, we know that communicating learning intentions and success criteria will remain a significant instructional practice. We believe relying on year-long learning intentions and success criteria for math routines will create a sense of continuity and purpose that will be imperative for us, our students, and their families.

For more about Math Routines and our work to prepare for teaching and learning in-person, virtually, and hybrid: Register for the 2020 Math+Literacy Summer Series Institute – a series of 4 webinars.

For more about communicating clarity in the early childhood mathematics classroom: Check out our guest blog for Corwin .