problem solving reasoning and numeracy

Numeracy for all learners

Numeracy is the knowledge, skills, behaviours and dispositions that students need in order to use mathematics in a wide range of situations. It involves recognising and understanding the role of mathematics in the world and having the dispositions and capacities to use mathematical knowledge and skills purposefully.  (Literacy and numeracy strategy version 2).

Number, measurement and geometry, statistics and probability are common aspects of most people’s mathematical experience in everyday personal, study and work situations. Equally important are the essential roles that algebra, functions and relations, logic, mathematical structure and working mathematically play in people’s understanding of the natural and human worlds, and the interaction between them. 

Why numeracy is important

A child's first years are a time of rapid learning and development. Babies and toddlers can recognise number, patterns, and shapes. They use maths concepts to make sense of their world and connect these concepts with their environment and everyday activities. For example, when playing, children may sort or choose toys according to size, shape, weight or colour.

While much of the teaching of concepts and skills to support numeracy happens in the mathematics learning area, it is strengthened as students take part in activities that connect their learning in the mathematics classroom within the context of other curriculum areas.

As they move through their years of schooling, students are exposed to mathematical:

• understanding
• problem solving

These capabilities allow students to respond to familiar and unfamiliar situations by employing mathematics to make informed decisions and solve problems efficiently (VCAA, 2017).

There is also evidence that other areas of development, such as resilience and perseverance, support achievement in numeracy.

Mathematics gives students access to important mathematical ideas, knowledge and skills. Numeracy connects this learning with their personal and work lives.

Numeracy has an increasingly important role in enabling and sustaining cultural, social, economic and technological advances.

Numeracy development

For an overview of numeracy development see  mapping the numeracy focus areas . Resources in the guide are organised by levels:

• Birth to Level 2
• Levels 3 to 8
• Levels 9 to 10

Numeracy across the curriculum

Being numerate involves more than mastering basic mathematics. Numeracy involves connecting the mathematics that students learn at school with the out-of-school situations that require the skills of problem solving, critical judgement, and sense-making related to applied contexts.

Conceptual framework

Learning activities presented draw upon the conceptual framework of Goos, Geiger, and Dole (2014; also discussed in Goos, Geiger, Dole, Forgasz, and Bennison, 2019). In this framework, numeracy is conceptualised as comprising four elements and an orientation: 

Element 1:  Attention to real-life contexts (citizenship, work, and personal and social life)

Element 2:  Application of mathematical knowledge (problem solving, estimation, concepts, and skills)

Element 3:  Use of tools (representational, physical, and digital)

Element 4:  The promotion of positive dispositions towards the use of mathematics to solve problems encountered in day-to-day life (confidence, flexibility, initiative, and risk)

Orientation:  A critical orientation to interpreting mathematical results and making evidence-based judgements

The resources highlight what numeracy is with respect to each learning area, and outline why it is important to develop students' numeracy capabilities within the learning area. Guidance is provided for teachers on the following:

• how to embed numeracy in their learning area
• how to assess numeracy learning
• how to deal with challenges and dilemmas using strategies recommended by experts.

The activities are described in terms of subject-specific learning intentions and content descriptors. The numeracy content and skills are highlighted and explained, with particular focus on how the numeracy links enhance the learning area's specific concepts. Direct links to the Victorian Curriculum: Mathematics highlight the connections between the activity and the students' previously developed mathematical skills and understandings. The VCAA have detailed information regarding the numeracy demands of the Victorian Curriculum on the Numeracy page of the website.

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Early childhood numeracy and mathematics resource

Mathematics is everywhere.

We all use mathematics to navigate our everyday decisions successfully. Children begin to experience and explore mathematical concepts from birth. With support, they participate in mathematical thinking and use mathematical concepts to organise, record and communicate ideas about the world around them.

Understanding and using mathematical concepts, and being numerate, helps children know and describe the world around them and make meaning of these encounters. It is, therefore, an essential skill for successful daily life. Research and practice evidence suggest that mathematics and numeracy skills will support children to be confident and capable learners as they navigate the increasingly complex global community of the 21st century.

Children who are confident and involved learners have positive dispositions toward learning, experience challenge and success in their learning and are able to contribute positively and effectively to others children’s learning. . . .They develop and use their imagination and curiosity as they build a ‘toolkit’ of skills and processes to support problem solving, hypothesising, experimenting researching and investigating (VEYLDF, 2016)

Families and educators play a critical role in introducing children to mathematics and encouraging them to be curious and enthusiastic about mathematics. From a very young age, adults invite children to use mathematics to understand and participate in their world.

Would you like another piece of toast? We need to find the other shoe – we need one for each foot! How old are you today – three – happy birthday! How many plates do we need? We live at number 36.

Building children’s confidence in understanding and using mathematics to explore and know the world will benefit everyone. Children benefit from many opportunities to generate and discuss ideas, make plans, exercise skills, engage in sustained shared thinking, generate solutions to problems, reflect and give reasons for their choices. Children who are confident and involved learners have positive dispositions toward learning, and experience challenge and success in their learning.

Numeracy in early childhood

Numeracy is the capacity, confidence and disposition to use mathematics in daily life. Children bring new mathematical understandings through engaging in problem-solving. The mathematical ideas with which young children interact must be relevant and meaningful in the context of their current lives. Spatial sense, structure and pattern, number, measurement, data argumentation, connections and exploring the world mathematically are the powerful mathematical ideas children need to become numerate (EYLF p. 38).

When educators consider including mathematics and numeracy in early childhood programs, there is often confusion about the relevance of concepts such as algebra or statistics. Children are active learners, exploring the world and beginning to develop explanations for observed phenomena from a young age. With encouragement, guidance, experience and learning, children further develop their capacity to reflect on their own thinking processes, approaches to learning and using mathematics in their everyday engagement with their world..This resource illustrates the variety of ways that educators, working with children from birth to age five, can support numeracy learning and development.  Presented across three key mathematical concepts; Number and Algebra; Measurement and Geometry; Statistics and Probability (reflective of the Victorian Early Years Learning and Development Framework and the Victorian Curriculum) and organised to consider children's learning from birth to age five; early childhood educators are offered ideas for learning experiences, ways to engage families and opportunities for intentional teaching.

The suggestions included in this resource represent only some recommendations to help educators strengthen and enhance numeracy learning in programs for young children. Educators will have their own ideas that will complement this collection and are encouraged to work with their colleagues, as well as children and families, to expand their ideas and resources. Links to a range of resources are included that offer additional materials for further consideration.

Number and Algebra

Number and Algebra for young children involves exploring mathematical concepts such as patterns, symbols, and relationships. A large part of learning in this area involves using numbers in everyday contexts, counting objects and understanding how the numbers combine and connect to describe the world and help us to make meaning.

Children are engaging with number and algebra when they: 

• use mathematical words to describe the world. E.g. ‘lots of’, ‘more than'
• use numbers to count and refer to objects and people in their lives. E.g. 'I'm three years old, 'I have two trucks at home'
• use numbers to solve problems. E.g. ‘I need another glass for the table’
• begin to count objects in a sequence and recognise the way numbers work. 

Measurement and Geometry

Measurement and Geometry for young children involves exploring mathematical concepts such as the size, shape, position and dimensions of objects. A large part of learning in this area involves becoming familiar with and using numbers and words to describe objects and know the difference between objects.

Children are engaging with measurement and geometry when they:

• feel different shaped items
• sort objects according to their shape
• draw shapes in their art
• describe the world around them using concepts such as ‘I like the circle one’ or ‘I put my hat in the big basket’ or ’the snake was really long.’

Statistics and Probability

Statistics and Probability for young children involves sorting, understanding and presenting information from groups of objects in order to understand what is happening.   

Probability is about understanding the chance of something occurring and making decisions based on that thinking.

Children are engaging with statistics and probability when they:

•  collect and sort ideas or groups of objects into categories
• talk about whether they need to take a coat with them when they go on a walk. E.g. ‘Is it going to rain?’

Early childhood educators' beliefs on mathematical learning

Educators’ own beliefs and attitudes towards mathematics and numeracy have a significant impact on the way these ideas are incorporated into programs for children. Increasing numbers of studies (Anders & Robbach, 2015) (Australian Mathematical Sciences Institute, 2018) have identified that many early childhood educators have had negative mathematics experiences in their schooling and therefore believe they will not be able to support children in this area adequately. It is important for adults to reflect on their anxiety in relation to mathematics and shift their perception towards the potential that mathematics provides to make their lives more meaningful. Many early childhood educators are competent users of mathematical concepts, and their numeracy skills are excellent however, these are not always recognised as a positive and necessary part of their daily lives.

Families play a crucial part in the development of children's mathematics and numeracy learning. As is the case for educators, family members’ own beliefs and attitudes towards mathematics and numeracy influence the way that children feel about engaging with and developing their mathematics and numeracy skills. Since numeracy in the early years is so highly connected to daily life and the way we make meaning of the world, families can provide opportunities to explore mathematics and support children to become confident about their mathematics and numeracy learning.

Educators can encourage families to recognise their role in supporting children’s mathematics and numeracy learning in many ways; from formal communication with families (in a family handbook for example or newsletters) about  how they can support children at home to informal conversations that promote positive attitudes and reinforce responses to children that help build their confidence. When educators maintain a commitment to sharing ideas with families about children’s mathematics and numeracy, learning outcomes are more likely to progress.  

Throughout this resource, learning experiences have been identified that are specifically designed for families to try at home. Educators are encouraged to share these ideas with families in their regular communications.  

• The Victorian Numeracy Learning Progressions - helps schools and teachers, in all learning areas, to support their students to engage with the numeracy demands of the Victorian Curriculum F –10.
• F-10 Victorian Curriculum Mathematics glossary - definitions and examples of mathematical vocabulary.

Human Capital Working Group, Council of Australian Government. (2018). National Numeracy Review Report. Canberra: Commonwealth of Australia.         

Jonas, N. (2018). Numeracy practices and numeracy skills among adults. Paris: Organisation for Economic Co-operation and Development.         

Shomos, A., & Forbes, M. (2014). Literacy and Numeracy Skills and Labour Market Outcomes in Australia. Canberra: Productivity Commision Staff working paper.

Attard, C. (2020, Jan 21). Mathematics education in Australia: New decade, new opportunities? Retrieved from Engaging Maths: https://engagingmaths.com/2020/01/21/mathematics-education-in-australia-new-decade-new-opportunities/

Buckley, S. (2011). Deconstructing maths anxiety: Helping students to develop a positive attitude towards learning maths. Retrieved from ACER: https://www.acer.org/au/occasional-essays/deconstructing-maths-anxiety-helping-students-to-develop-a-positive-attitud

Church, A., Cohrssen, C., Ishimine, K., & Tayler, C. (2013). Playing with maths: Facilitating the learning in play-based learning. Australasian Journal of Early Childhood Volume 38 Number 1 March 2013, 95-99.

Cohrssen, C. (2018, June 6). Assessing children’s understanding during play-based maths activities. Canberra, ACT, Australia. Retrieved from http://thespoke.earlychildhoodaustralia.org.au/assessing-childrens-understanding-during-play-based-maths-activities/

DEEWR. (2009). Belonging, Being and Becoming: The early years learning framework for Australia. Canberra: Commonwealth of Australia.

Department of Education and Training . (2012). Integrated Teaching and Learning Approaches Practice Principle Guide 6 . Melbourne : Department of Education and Early Childhood Development).

Department of Education and Training. (2016). Victorian Early Years Learning and Development Framework . Melbourne: Department of Education and Training.         

Knaus, M. (2016). Maths is All Around You: Developing Mathematical Concepts in Early Years . Blairgowrie: Teaching Solutions .        

NAEYC. (2020). Math Talk with Infants and Toddlers. Washington, USA. Retrieved from https://www.naeyc.org/our-work/families/math-talk-infants-and-toddlers

Vogt, F., Hauser, B., Stebler, R., Rechsteiner, K., & Urech, C. (2018). Learning through play – pedagogy and learning outcomes in. EUROPEAN EARLY CHILDHOOD EDUCATION RESEARCH JOURNAL, 589-603.

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Mathematical Reasoning & Problem Solving

In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.

Previously Covered:

• Over the course of the previous lesson, we reviewed some basics about chance and probability, as well as some basics about sampling, surveys, etc. We also covered some ideas about data sets, how they’re represented, and how to interpret the results.

Approaches to Problem Solving

When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.

To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:

Identify the following four-digit number when presented with the following information:

• One of the four digits is a 1.
• The digit in the hundreds place is three times the digit in the thousands place.
• The digit in the ones place is four times the digit in the ten’s place.
• The sum of all four digits is 13.
• The digit 2 is in the thousands place.

Help your students identify and prioritize the information presented.

In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.

We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.

Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.

So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:

If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.

Recall that the clues’ relevance were identified and prioritized as follows:

• clue #3 and clue #1

By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.

Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:

Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?

Let’s solve this problem by representing it in a visual way , in this case, a diagram:

You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:

b = the base of triangle EFA

h = the height of triangle EFA and the height above the ground at which the ropes intersect

If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.

Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.

Let’s try another one.

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.

How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.

Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:

How many more faces does a cube have than a square pyramid?

Reveal Answer

The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:

From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.

Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:

• Sort and prioritize relevant and irrelevant information.
• Represent a problem in different ways, such as words, symbols, concrete models, and diagrams.
• Generate and use estimations to find solutions to mathematical problems.

Mathematical Mistakes

Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.

Circular Arguments

These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:

• Premise: Only an untrustworthy man would become an insurance salesman; the fact that insurance salesmen cannot be trusted is proof of this.
• Conclusion: Therefore, insurance salesmen cannot be trusted.

While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.

Assuming the Truth of the Converse

Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”

Assuming the Truth of the Inverse

Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:

If you grew up in Minnesota , you’ve seen snow.

Now, notice that the inverse of this statement is not necessarily true:

If you didn’t grow up in Minnesota , you’ve never seen snow.

Faulty Generalizations

This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.

Faulty Analogies

It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:

People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.

False (or tenuous) analogies are often used in persuasive arguments.

Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.

Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .

Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:

If p, then q

An arrow is used to indicate that q is derived from p, like this:

This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.

A direct proof will attempt to lay out the shortest number of steps between p and q.

The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.

Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.

Given: Triangle ABC is isosceles with B marking the vertex

Prove: Angles A and C are congruent.

Now, let’s work through this, matching our statements with our reasons.

• Triangle ABC is isosceles . . . . . . . . . . . . Given
• Angle A is the vertex . . . . . . . . . . . . . . . . Given
• Angles A and C are not congruent . . Indirect proof assumption
• Line AB is equal to line BC . . . . . . . . . . . Legs of an isosceles triangle are congruent
• Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are congruent
• Contradiction . . . . . . . . . . . . . . . . . . . . . . Angles can’t be congruent and incongruent
• Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong
• Therefore, if angles A and C are not incongruent, they are congruent.

“Always, Sometimes, and Never”

Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.

Example: x < x 2 for all real numbers x

We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:

Example: For all primes x ≥ 3, x is odd.

This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.

• Know and be able to identify common mathematical errors, such as circular arguments, assuming the truth of the converse, assuming the truth of the inverse, making faulty generalizations, and faulty use of analogical reasoning.
• Be familiar with direct proofs and indirect proofs (proof by contradiction).
• Be able to work with problems to identify “always,” “sometimes,” and “never” statements.

Reasoning Skills

Developing opportunities and ensuring progression in the development of reasoning skills

Achieving the aims of the new National Curriculum:

Developing opportunities and ensuring progression in the development of reasoning skills.

The aims of the National Curriculum are to develop fluency and the ability to reason mathematically and solve problems. Reasoning is not only important in its own right but impacts on the other two aims. Reasoning about what is already known in order to work out what is unknown will improve fluency; for example if I know what 12 × 12 is, I can apply reasoning to work out 12 × 13. The ability to reason also supports the application of mathematics and an ability to solve problems set in unfamiliar contexts.

Research by Nunes (2009) identified the ability to reason mathematically as the most important factor in a pupil’s success in mathematics. It is therefore crucial that opportunities to develop mathematical reasoning skills are integrated fully into the curriculum. Such skills support deep and sustainable learning and enable pupils to make connections in mathematics.

This resource is designed to highlight opportunities and strategies that develop aspects of reasoning throughout the National Curriculum programmes of study. The intention is to offer suggestions of how to enable pupils to become more proficient at reasoning throughout all of their mathematics learning rather than just at the end of a particular unit or topic.

We take the Progression Map for each of the National Curriculum topics, and augment it with a variety of reasoning activities (shaded sections) underneath the relevant programme of study statements for each year group. The overall aim is to support progression in reasoning skills. The activities also offer the opportunity for children to demonstrate depth of understanding, and you might choose to use them for assessment purposes as well as regular classroom activities.

Place Value Reasoning

Addition and subtraction reasoning, multiplication and division reasoning, fractions reasoning, ratio and proportion reasoning, measurement reasoning, geometry - properties of shapes reasoning, geometry - position direction and movement reasoning, statistics reasoning, algebra reasoning.

The strategies embedded in the activities are easily adaptable and can be integrated into your classroom routines. They have been gathered from a range of sources including real lessons, past questions, children’s work and other classroom practice.

Strategies include:

• Spot the mistake / Which is correct?
• True or false?
• What comes next?
• Do, then explain
• Make up an example / Write more statements / Create a question / Another and another
• Possible answers / Other possibilities
• What do you notice?
• Continue the pattern
• Missing numbers / Missing symbols / Missing information/Connected calculations
• Working backwards / Use the inverse / Undoing / Unpicking
• Hard and easy questions
• What else do you know? / Use a fact
• Fact families
• Convince me / Prove it / Generalising / Explain thinking
• Make an estimate / Size of an answer
• Always, sometimes, never
• Making links / Application
• Can you find?
• What’s the same, what’s different?
• Odd one out
• Complete the pattern / Continue the pattern
• Another and another
• Testing conditions
• The answer is…
• Visualising

These strategies are a very powerful way of developing pupils’ reasoning skills and can be used flexibly. Many are transferable to different areas of mathematics and can be differentiated through the choice of different numbers and examples.

Nunes, T. (2009) Development of maths capabilities and confidence in primary school, Research Report DCSF-RR118 (PDF)

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• Mathematics proficiencies

Introduction

The Australian Curriculum: Mathematics aims to be relevant and applicable to the 21st century. The inclusion of the proficiencies of understanding, fluency, problem-solving and reasoning in the curriculum is to ensure that student learning and student independence are at the centre of the curriculum. The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.

The proficiency strands describe the actions in which students can engage when learning and using the content of the Australian Curriculum: Mathematics.

Understanding

Students build a robust knowledge of adaptable and transferable mathematical concepts. They make connections between related concepts and progressively apply the familiar to develop new ideas. They develop an understanding of the relationship between the ‘why’ and the ‘how’ of mathematics. Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and when they interpret mathematical information

Students develop skills in choosing appropriate procedures; carrying out procedures flexibly, accurately, efficiently and appropriately; and recalling factual knowledge and concepts readily. Students are fluent when they calculate answers efficiently, when they recognise robust ways of answering questions, when they choose appropriate methods and approximations, when they recall definitions and regularly use facts, and when they can manipulate expressions and equations to find solutions.

Problem-Solving

Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false, and when they compare and contrast related ideas and explain their choices.

Useful Links

• Australian Curriculum: Mathematics F–10
• Review by Kaye Stacey of 'Adding it up: helping children learn mathematics' report
• Peter Sullivan presentation: Designing learning experiences to exemplify the proficiencies
• Peter Sullivan presentation: Create your own lessons
• Peter Sullivan paper: Using the proficiencies to enrich mathematics teaching and assessment

Explore Mathematics proficiencies portfolios and illustrations

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Learning and Development

Maths problem-solving – Activities for Early Years settings

• Written By: Judith Dancer
• Subject: Maths

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Critical thinking doesn’t have to be a daunting prospect. There are simple, effective and exciting ways to encourage children’s maths problem-solving skills, says Judith Dancer…

Maths is a subject many adults lack confidence in. Having struggled with it at school they often avoid it, wherever possible, when grown up.

But if maths seems scary for some people, then maths problem-solving can cause even more anxiety. There is no ‘safety net’ of knowing the ‘correct answer’ beforehand. This is because maths problem-solving lends itself to investigation and exploration with lots of possible tangents.

Understandably this is often the area of maths where many practitioners feel least confident. However, young children, who are not restrained by right answers, feel the most enthused and animated.

The non-statutory  Development Matters Guidance , as part of ‘creating and thinking critically’ in the Characteristics of Effective Learning, identifies that practitioners need to observe how a child is learning, noting how a child is:

• thinking of ideas;
• finding ways to solve problems;
• finding new ways to do things;
• making links and noticing patterns in their experience;
• making predictions;
• testing their ideas;
• developing ideas of grouping, sequences, cause and effect;
• planning, making decisions about how to approach a task, solve a problem and reach a goal;
• checking how well their activities are going;
• changing strategy as needed;
• reviewing how well the approach worked.

All of these elements are, at one time or another, part of the problem-identifying and solving process – although not at the same time and in the same problem.

Role of the adult

Maths problem-solving for young children involves them understanding and using two kinds of maths:

• Maths knowledge – learning and applying an aspect of maths such as counting, calculating or measuring.
• Maths thinking skills – reasoning, predicting, talking the problem through, making connections, generalising, identifying patterns and finding solutions.

The best maths problems for children are the ones that they identify themselves. They will be enthused, fascinated and more engaged in these ‘real’, meaningful problems.

Children need opportunities to problem-solve together. As they play, they will often find their own mathematical problems.

One of the key roles of practitioners is to provide time, space and support for children. We need to develop situations and provide opportunities in which children can refine their maths problem-solving skills and apply their mathematical knowledge.

Supporting maths problem-solving

You can effectively support children’s developing maths problem-solving strategies through:

• Modelling maths talk and discussion – language is part of maths learning because talking problems through is vital. Children need to hear specific mathematical vocabulary in context. You can promote discussion through the use of comments, enabling statements and open-ended questions.
• Providing hands-on maths problem-solving activities across all areas of the setting. Children learn maths through all their experiences and need frequent opportunities to take part in creative and engaging experiences. Maths doesn’t just happen in the maths learning zone!
• Identifying potential maths learning indoors and outdoors. Provide rich and diverse open-ended resources that children can use in a number of different ways to support their own learning. It is important to include natural and everyday objects and items that have captured children’s imaginations, including popular culture.

Maths problem-solving possibilities

Spell it out.

This experience gives children lots of opportunities to explore calculating, mark making, categorising and decisions about how to approach a task.

What you need to provide:

• Assorted containers filled with natural materials. This includes leaves, pebbles, gravel, conkers, twigs, shells, fir cones, mud and sand. Include some ‘treasure’ – sequins, gold nuggets, jewels and glitter.
• Bottles and jugs of water, large mixing bowls, cups, a ‘cauldron’, small bottles, spoons and ladles.
• Cloaks and wizard hats.
• Laminated ‘spells’ – e.g. “To make a disappearing spell, mix 2 smooth pebbles, 2 gold nuggets, 4 fir cones, a pinch of sparkle dust, 3 cups of water”.
• Writing frameworks for children’s own spell recipes and a shiny ‘Spell Book’ to stick these in.
• Temporary mark-making opportunities such as chalk on slate.

The important thing with open-ended maths problem-solving experiences like this is to observe, wait and listen. Then, if appropriate, join in as a co-player with children, following their play themes.

So if children are mixing potions, note how children sort or categorise the objects. What strategies do they use to solve problems? What happens if they want eight pebbles and they run out? Observe what they do next.

When supporting children’s maths problem-solving, you need to develop a wide range of strategies and ‘dip into’ these appropriately. Rather than asking questions, it is often more effective to make comments about what you can see. For example, say, “Wow, it looks as though there is too much potion for that bottle”.

Acting as a co-player offers lots of opportunities to model mathematical behaviours. This might include reading recipes for potions and spells out loud, focusing on the numbers – one feather, three shells…

Going, going, gone

We all know that children will engage more fully when involved in experiences that fascinate them. If a particular group has a real passion for cars and trucks , consider introducing maths problem-solving opportunities that extend this interest.

This activity offers opportunities for classifying, sorting, counting, adding and subtracting, among many other things.

• Some unfamiliar trucks and cars and some old favourites. Ensure these include metal, plastic and wooden vehicles that can be sorted in different ways.
• Masking tape and scissors.
• Sticky labels and markers.

Mark out some parking lots on a smooth floor, or huge piece of paper using masking tape. Lining paper is great for this. Line the vehicles up around the edge of the floor area.

Encourage one child to select two vehicles that have something the same about them. Ask the child, “What is the same about them?”.

When the children have agreed on what is the same – e.g. size, materials, colour, lorries or racing cars – the child selects a ‘parking lot’ to put the vehicles in. So this first parking lot could be for ‘red vehicles’.

Another child chooses two more vehicles that have something the same. Do they belong in the same ‘parking lot’, or a different parking lot? E.g. these vehicles could both be racing cars.

What happens when a specific vehicle could belong in both lots? E.g. it could belong in the set of red vehicles and also belongs in the set of racing cars.

Support the children as they discuss the vehicle. Make new ‘parking lots’ with masking tape and create labels for the groups, if you choose.

Observe children’s strategies

It’s really important to observe the strategies the children use. Where appropriate, ask the children to explain what they are doing and why.

If necessary, introduce and model the use of the vocabulary ‘the same as’ and ‘different from’. Follow children’s discussions and interests. If they start talking about registration plates, consider making car number plates for all the wheeled toys outdoors.

Do the children know the format of registration plates? Can you take photos of cars you can see in the local environment?

Camping out

Constructing camps and dens outdoors is a good way to give children the opportunity to be involved in lots of maths problem-solving experiences and construction skills learning. This experience offers opportunities for using the language of position, shape and space, and finding solutions to practical problems.

• Materials to construct a tent or den such as sheets, curtains, poles, clips and string.
• Rucksacks, water bottles, compasses and maps.
• Oven shelf and bricks to build a campfire or barbecue.
• Buckets and bowls and water for washing up.

Encourage the children to explore the resources and decide which materials they need to build the camp. Suggest they source extra resources as they are needed.

Talk with the children about the best place to make a den or erect a tent and barbecue. During the discussion, model the use of positional words and phrases.

Follow children’s play themes. This could include going on a scavenger hunt collecting stones, twigs and leaves and going back to the campsite to sort them out.

Encourage children to try different solutions to the practical problems they identify. Use a running commentary on what is happening without providing the solution to the problem.

Look for opportunities to develop children’s mathematical reasoning skills by making comments such as, “I wonder why Rafit chose that box to go on the top of his den.”

If the children are familiar with traditional tales, you could extend this activity by laying a crumb trail round the outdoor area for children to follow. Make sure that there is something exciting at the end of the trail. It could be a large dinosaur sitting in a puddle, or a bear in a ‘cave’.

Children rarely have opportunities to investigate objects that are really heavy. Sometimes they have two objects and are asked the question, “Which one is heavy?” when both objects are actually light.

This experience gives children the chance to explore really heavy things and measures (weight). They also need to cooperate and find new ways to do things.

• A ‘building site’ in the outdoor area. Include hard hats, builders’ buckets, small buckets, shovels, spades, water, sand, pebbles, gravel, guttering, building blocks, huge cardboard boxes and fabric (this could be on a tarpaulin).
• Some distance away, builders’ buckets filled with damp sand and large gravel.
• Bucket balances and bathroom scales.

With an open-ended activity such as this, it is even more important to observe, wait and listen as the children explore the building site and the buckets full of sand and gravel.

Listen to the discussions the children have about moving the sand and the gravel to the building site. What language do they use?

Note the strategies they use when they can’t lift the large buckets. Who empties some of the sand into smaller buckets? Who works together collaboratively to move the full bucket? Does anyone introduce another strategy, for example, finding a wheelbarrow or pull-along truck?

Where and when appropriate, join in the children’s play as a co-player. You could act in role as a customer or new builder. Ask, “How can I get all this sand into my car?”. “How much sand and gravel do we need to make the cement for the foundations?”.

Extend children’s learning by modelling the language of weight:

• heavy/heavier than/heaviest
• light/lighter than/lightest
• about the same weight as/as heavy as
• balance/weigh

Judith Dancer is an author, consultant and trainer specialising in communication and language and mathematics. She is co-author, with Carole Skinner, of  Foundations of Mathematics – An active approach to number, shape and measures in the Early Years .

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Fluency, Reasoning and Problem Solving: What This Looks Like In Every Maths Lesson

Neil Almond

Fluency reasoning and problem solving have been central to the new maths national curriculum for primary schools introduced in 2014. Here we look at how these three approaches or elements of maths can be interwoven in a child’s maths education through KS1 and KS2. We look at what fluency, reasoning and problem solving are, how to teach them, and how to know how a child is progressing in each – as well as what to do when they’re not, and what to avoid.

The hope is that this blog will help primary school teachers think carefully about their practice and the pedagogical choices they make around the teaching of reasoning and problem solving in particular.

Before we can think about what this would look like in practice however, we need to understand the background tothese terms.

What is fluency in maths?

Fluency in maths is a fairly broad concept. The basics of mathematical fluency – as defined by the KS1 / KS2 National Curriculum for maths – involve knowing key mathematical facts and being able to recall them quickly and accurately.

But true fluency in maths (at least up to Key Stage 2) means being able to apply the same skill to multiple contexts, and being able to choose the most appropriate method for a particular task.

Fluency in maths lessons means we teach the content using a range of representations, to ensure that all pupils understand and have sufficient time to practise what is taught.

Read more: How the best schools develop maths fluency at KS2 .

What is reasoning in maths?

Reasoning in maths is the process of applying logical thinking to a situation to derive the correct problem solving strategy for a given question, and using this method to develop and describe a solution.

Put more simply, mathematical reasoning is the bridge between fluency and problem solving. It allows pupils to use the former to accurately carry out the latter.

Read more: Developing maths reasoning at KS2: the mathematical skills required and how to teach them .

What is problem solving in maths?

It’s sometimes easier to start off with what problem solving is not. Problem solving is not necessarily just about answering word problems in maths. If a child already has a readily available method to solve this sort of problem, problem solving has not occurred. Problem solving in maths is finding a way to apply knowledge and skills you have to answer unfamiliar types of problems.

Read more: Maths problem solving: strategies and resources for primary school teachers .

We are all problem solvers

First off, problem solving should not be seen as something that some pupils can do and some cannot. Every single person is born with an innate level of problem-solving ability.

Early on as a species on this planet, we solved problems like recognising faces we know, protecting ourselves against other species, and as babies the problem of getting food (by crying relentlessly until we were fed).

All these scenarios are a form of what the evolutionary psychologist David Geary (1995) calls biologically primary knowledge. We have been solving these problems for millennia and they are so ingrained in our DNA that we learn them without any specific instruction.

Why then, if we have this innate ability, does actually teaching problem solving seem so hard?

Mathematical problem solving is a  learned skill

As you might have guessed, the domain of mathematics is far from innate. Maths doesn’t just happen to us; we need to learn it. It needs to be passed down from experts that have the knowledge to novices who do not.

This is what Geary calls biologically secondary knowledge. Solving problems (within the domain of maths) is a mixture of both primary and secondary knowledge.

The issue is that problem solving in domains that are classified as biologically secondary knowledge (like maths) can only be improved by practising elements of that domain.

So there is no generic problem-solving skill that can be taught in isolation and transferred to other areas.

This will have important ramifications for pedagogical choices, which I will go into more detail about later on in this blog.

The educationalist Dylan Wiliam had this to say on the matter: ‘for…problem solving, the idea that pupils can learn these skills in one context and apply them in another is essentially wrong.’ (Wiliam, 2018)So what is the best method of teaching problem solving to primary maths pupils?

The answer is that we teach them plenty of domain specific biological secondary knowledge – in this case maths. Our ability to successfully problem solve requires us to have a deep understanding of content and fluency of facts and mathematical procedures.

Here is what cognitive psychologist Daniel Willingham (2010) has to say:

‘Data from the last thirty years lead to a conclusion that is not scientifically challengeable: thinking well requires knowing facts, and that’s true not simply because you need something to think about.

The very processes that teachers care about most—critical thinking processes such as reasoning and problem solving—are intimately intertwined with factual knowledge that is stored in long-term memory (not just found in the environment).’

Colin Foster (2019), a reader in Mathematics Education in the Mathematics Education Centre at Loughborough University, says, ‘I think of fluency and mathematical reasoning, not as ends in themselves, but as means to support pupils in the most important goal of all: solving problems.’

In that paper he produces this pyramid:

This is important for two reasons:

1)    It splits up reasoning skills and problem solving into two different entities

2)    It demonstrates that fluency is not something to be rushed through to get to the ‘problem solving’ stage but is rather the foundation of problem solving.

In my own work I adapt this model and turn it into a cone shape, as education seems to have a problem with pyramids and gross misinterpretation of them (think Bloom’s taxonomy).

Notice how we need plenty of fluency of facts, concepts, procedures and mathematical language.

Having this fluency will help with improving logical reasoning skills, which will then lend themselves to solving mathematical problems – but only if it is truly learnt and there is systematic retrieval of this information carefully planned across the curriculum.

Performance vs learning: what to avoid when teaching fluency, reasoning, and problem solving

I mean to make no sweeping generalisation here; this was my experience both at university when training and from working in schools.

At some point schools become obsessed with the ridiculous notion of ‘accelerated progress’. I have heard it used in all manner of educational contexts while training and being a teacher. ‘You will need to show ‘ accelerated progress in maths ’ in this lesson,’ ‘Ofsted will be looking for ‘accelerated progress’ etc.

I have no doubt that all of this came from a good place and from those wanting the best possible outcomes – but it is misguided.

I remember being told that we needed to get pupils onto the problem solving questions as soon as possible to demonstrate this mystical ‘accelerated progress’.

This makes sense; you have a group of pupils and you have taken them from not knowing something to working out pretty sophisticated 2-step or multi-step word problems within an hour. How is that not ‘accelerated progress?’

This was a frequent feature of my lessons up until last academic year: teach a mathematical procedure; get the pupils to do about 10 of them in their books; mark these and if the majority were correct, model some reasoning/problem solving questions from the same content as the fluency content; set the pupils some reasoning and word problem questions and that was it.

I wondered if I was the only one who had been taught this while at university so I did a quick poll on Twitter and found that was not the case.

I know these numbers won’t be big enough for a representative sample but it still shows that others are familiar with this approach.

The issue with the lesson framework I mentioned above is that it does not take into account ‘performance vs learning.’

What IS performance vs learning’?

The premise is that performance in a lesson is not a good proxy for learning.

Yes, those pupils were performing well after I had modeled a mathematical procedure for them, and managed to get questions correct.

But if problem solving depends on a deep knowledge of mathematics, this approach to lesson structure is going to be very ineffective.

As mentioned earlier, the reasoning and problem solving questions were based on the same maths content as the fluency exercises, making it more likely that pupils would solve problems correctly whether they fully understood them or not.

Chances are that all they’d need to do is find the numbers in the questions and use the same method they used in the fluency section to get their answers – not exactly high level problem solving skills.

Teaching to “cover the curriculum” hinders development of strong problem solving skills.

This is one of my worries with ‘maths mastery schemes’ that block content so that, in some circumstances, it is not looked at again until the following year (and with new objectives).

The pressure for teachers to ‘get through the curriculum’ results in many opportunities to revisit content just not happening in the classroom.

Pupils are unintentionally forced to skip ahead in the fluency, reasoning, problem solving chain without proper consolidation of the earlier processes.

As David Didau (2019) puts it, ‘When novices face a problem for which they do not have a conveniently stored solution, they have to rely on the costlier means-end analysis.

This is likely to lead to cognitive overload because it involves trying to work through and hold in mind multiple possible solutions.

It’s a bit like trying to juggle five objects at once without previous practice. Solving problems is an inefficient way to get better at problem solving.’

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Fluency and reasoning – Best practice in a lesson, a unit, and a term

By now I hope you have realised that when it comes to problem solving, fluency is king. As such we should look to mastery maths based teaching to ensure that the fluency that pupils need is there.

The answer to what fluency looks like will obviously depend on many factors, including the content being taught and the year group you find yourself teaching.

But we should not consider rushing them on to problem solving or logical reasoning in the early stages of this new content as it has not been learnt, only performed.

I would say that in the early stages of learning, content that requires the end goal of being fluent should take up the majority of lesson time – approximately 60%. The rest of the time should be spent rehearsing and retrieving other knowledge that is at risk of being forgotten about.

This blog on mental maths strategies pupils should learn in each year group is a good place to start when thinking about the core aspects of fluency that pupils should achieve.

Little and often is a good mantra when we think about fluency, particularly when revisiting the key mathematical skills of number bond fluency or multiplication fluency. So when it comes to what fluency could look like throughout the day, consider all the opportunities to get pupils practicing.

They could chant multiplications when transitioning. If a lesson in another subject has finished earlier than expected, use that time to quiz pupils on number bonds. Have fluency exercises as part of the morning work.

Read more: How to teach times tables KS1 and KS2 for total recall .

What about best practice over a longer period?

Thinking about what fluency could look like across a unit of work would again depend on the unit itself.

Look at this unit below from a popular scheme of work.

They recommend 20 days to cover 9 objectives. One of these specifically mentions problem solving so I will forget about that one at the moment – so that gives 8 objectives.

I would recommend that the fluency of this unit look something like this:

LY = Last Year

This type of structure is heavily borrowed from Mark McCourt’s phased learning idea from his book ‘Teaching for Mastery.’

This should not be seen as something set in stone; it would greatly depend on the needs of the class in front of you. But it gives an idea of what fluency could look like across a unit of lessons – though not necessarily all maths lessons.

When we think about a term, we can draw on similar ideas to the one above except that your lessons could also pull on content from previous units from that term.

So lesson one may focus 60% on the new unit and 40% on what was learnt in the previous unit.

The structure could then follow a similar pattern to the one above.

Best practice for problem solving in a lesson, a unit, and a term 

When an adult first learns something new, we cannot solve a problem with it straight away. We need to become familiar with the idea and practise before we can make connections, reason and problem solve with it.

The same is true for pupils. Indeed, it could take up to two years ‘between the mathematics a student can use in imitative exercises and that they have sufficiently absorbed and connected to use autonomously in non-routine problem solving.’ (Burkhardt, 2017).

Practise with facts that are secure

So when we plan for reasoning and problem solving, we need to be looking at content from 2 years ago to base these questions on.

Now given that much of the content of the KS2 SATs will come from years 5 and 6 it can be hard to stick to this two-year idea as pupils will need to solve problems with content that can be only weeks old to them.

But certainly in other year groups, the argument could be made that content should come from previous years.

You could get pupils in Year 4 to solve complicated place value problems with the numbers they should know from Year 2 or 3. This would lessen the cognitive load, freeing up valuable working memory so they can actually focus on solving the problems using content they are familiar with.

Read more: Cognitive load theory in the classroom

Increase complexity gradually.

Once they practise solving these types of problems, they can draw on this knowledge later when solving problems with more difficult numbers.

This is what Mark McCourt calls the ‘Behave’ phase. In his book he writes:

‘Many teachers find it an uncomfortable – perhaps even illogical – process to plan the ‘Behave’ phase as one that relates to much earlier learning rather than the new idea, but it is crucial to do so if we want to bring about optimal gains in learning, understanding and long term recall.’  (Mark McCourt, 2019)

This just shows the fallacy of ‘accelerated progress’; in the space of 20 minutes some teachers are taught to move pupils from fluency through to non-routine problem solving, or we are somehow not catering to the needs of the child.

When considering what problem solving lessons could look like, here’s an example structure based on the objectives above.

Fluency, Reasoning and Problem Solving should NOT be taught by rote 

It is important to reiterate that this is not something that should be set in stone. Key to getting the most out of this teaching for mastery approach is ensuring your pupils (across abilities) are interested and engaged in their work.

Depending on the previous attainment and abilities of the children in your class, you may find that a few have come across some of the mathematical ideas you have been teaching, and so they are able to problem solve effectively with these ideas.

Equally likely is encountering pupils on the opposite side of the spectrum, who may not have fully grasped the concept of place value and will need to go further back than 2 years and solve even simpler problems.

In order to have the greatest impact on class performance, you will have to account for these varying experiences in your lessons.

Read more: 

• Maths Mastery Toolkit : A Practical Guide To Mastery Teaching And Learning
• Year 6 Maths Reasoning Questions and Answers
• Get to Grips with Maths Problem Solving KS2
• Mixed Ability Teaching for Mastery: Classroom How To
• 21 Maths Challenges To Really Stretch Your More Able Pupils
• Maths Reasoning and Problem Solving CPD Powerpoint
• Why You Should Be Incorporating Stem Sentences Into Your Primary Maths Teaching

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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Problem Solving, Reasoning and Numeracy in the Early Years Foundation Stage (Practical Guidance in the EYFS) 1st Edition

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The Practical Guidance in the Early Years Foundation Stage series will assist practitioners in the smooth and successful implementation of the Early Years Foundation Stage.

Each book gives clear and detailed explanations of each aspect of Learning and Development and encourages readers to consider each area within its broadest context to expand and develop their own knowledge and good practice.

Practical ideas and activities for all age groups are offered along with a wealth of expertise of how elements from the practice guidance can be implemented within all early years settings. The books include suggestions for the innovative use of everyday ressources, popular books and stories.

This book offers an in-depth understanding of children's thinking skills from a psychological perspective. The book introduces the Learning Tools model, a vital cognitive tool used by children to learn and solve problems, and gives practical ideas on how practitioners can use everyday materials to promote problem solving and early numeracy skills through play.

Readers are encouraged to reflect on their own practice and understanding to help them provide learning opportunities to meet the unique needs of all children in their setting.

• ISBN-10 0415476542
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"...Written by Anita Hughes, a chartered educational psychologist, who offers in-depth understanding of children's thinking skills from a psychological perspective. It introduces the learning tools model, a vital cognitive tool used by children to learn and solve problems. It also offers practical ideas on how to use everyday materials to promote problem-solving and early numeracy skills through play. Readers are encourages to reflect on their own practice and understanding to help them provide learning opportunities to meet the unique needs of all children in their setting." - Early Years Update (Issue 65, February 2009)

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Anita M Hughes is a Chartered Educational Psychologist.

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Problem solving, reasoning and numeracy in the early years foundation stage

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What Do the Numbers Say? The Math and Literacy Link

Posted on May 14, 2024 by hallma

Check out article from NAEYC for more information on making math meaningful for young children:  https://www.naeyc.org/resources/pubs/tyc/oct2014/making-math-meaningful 

Adapting math activities for children with special needs  requires individualized approaches to meet each child’s unique needs. It involves sharing information in multiple ways and using various sensory modalities including  visual aids, manipulatives, or tactile materials. Offering individualized supports like  visual schedules or simplified instructions can also help children fully participate in activities.  As with all curricula, taking the time to ensure that math activities incorporate children’s strengths and interests  can enrich their learning experiences. Finally, partnering with families, special educators and therapists  can help make sure that your math activities are  accessible and meaningful to children with special needs.

Email her at  [email protected] 

https://www.iidc.indiana.edu/ecc/index.html

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