How does the brain learn abstract maths?

How does the brain learn abstract maths?

I have found quite a bit of information of how, on a neurological level, we learn the most basic forms of maths. We seem to be hardwired from the get go, to deal with manageable quantities, can intuitively decide whether something is more or less and even add or subtract small integers together. There have even been tests that six-month-old children already have an intuitive sense of small numbers. Basically, this covers basic arithmetic on natural numbers.

However, I had a hard time finding anything about how we learn to deal with more abstract, higher level maths. What happens in our brain when we deal with functions and variables? Will they still be processed in the same region that is used for counting? - Especially when it comes to functions that do not even take numbers as inputs or outputs.


I am asking about math that can not be solved with simple arithmetic, and I wonder how an intuition of a generic mathematical concept would be represented in our brains. Has any research on this already been done? - Like something which isn't a numeric algorithm but rather symbolic manipulation.

The parietal area and prefrontal cortex from the neocortex are the source of ones ability to perform algebra and most other logic and analytic intensive tasks.

In a brain imaging study of children learning algebra, it is shown that the same regions are active in children solving equations as are active in experienced adults solving equations. As with adults, practice in symbol manipulation produces a reduced activation in prefrontal cortex area. However, unlike adults, practice seems also to produce a decrease in a parietal area that is holding an image of the equation. This finding suggests that adolescents' brain responses are more plastic and change more with practice. These results are integrated in a cognitive model that predicts both the behavioral and brain imaging results.

-The change of the brain activation patterns as children learn algebra equation solving

The Adolescent Brain:

The period from adolescence through young adulthood is one of great promise and vulnerability. As teenagers approach maturity, they must develop and apply the skills and habits necessary to navigate adulthood and compete in an ever more technological and globalized world. But as parents and researchers have long known, there is a crucial dichotomy between adolescents' cognitive competence and their frequent inability to utilize that competence in everyday decision-making.

This volume brings together an interdisciplinary group of leading scientists to examine how the adolescent brain develops, and how this development impacts various aspects of reasoning and decision-making, from the use and function of memory and representation, to judgment, mathematical problem-solving, and the construction of meaning.

The contributors ask questions that seek to uncover the basic mechanisms underlying brain development in adolescence, such as:

  • How do the concepts of proof and reasoning emerge?
  • What is the relationship between cognitive and procedural understanding in problem-solving?
  • How can researchers build assessments to capture and describe learning over time?

The Adolescent Brain raises questions relevant to young people's educational and health outcomes, as well as to neuroscience research.

Introduction to The Adolescent Brain: Learning, Reasoning, and Decision Making
Valerie F. Reyna, Sandra B. Chapman, Michael R. Dougherty, and Jere Confrey

I. Foundations

  1. Anatomic Magnetic Resonance Imaging of the Developing Child and Adolescent Brain
    Jay N. Giedd, Michael Stockman, Catherine Weddle, Maria Liverpool, Gregory L. Wallace, Nancy Raitano Lee, Francois Lalonde, and Rhoshel K. Lenroot

II. Memory, Meaning, and Representation

  1. Semantic and Associative Relations in Adolescents and Young Adults: Examining a Tenuous Dichotomy
    Ken McRae, Saman Khalkhali, and Mary Hare
  2. Representation and Transfer of Abstract Mathematical Concepts in Adolescence and Young Adulthood
    Jennifer A. Kaminski and Vladimir M. Sloutsky
  3. A Value of Concrete Learning Materials in Adolescence
    Kristen P. Blair and Daniel L. Schwartz
  4. Higher-Order Strategic Gist Reasoning in Adolescence
    Sandra B. Chapman, Jacquelyn F. Gamino, and Raksha Anand Mudar

III. Learning, Reasoning, and Problem Solving

  1. Better Measurement of Higher Cognitive Processes Through Learning Trajectories and Diagnostic Assessments in Mathematics: The Challenge in Adolescence
    Jere Confrey
  2. Adolescent Reasoning in Mathematical and Nonmathematical Domains: Exploring the Paradox
    Eric Knuth, Charles Kalish, Amy Ellis, Caroline Williams, and Mathew D. Felton
  3. Training the Adolescent Brain: Neural Plasticity and the Acquisition of Cognitive Abilities
    Sharona M. Atkins, Michael F. Bunting, Donald J. Bolger, and Michael R. Dougherty
  4. Higher Cognition is Altered by Noncognitive Factors: How Affect Enhances and Disrupts Mathematics Performance in Adolescence and Young Adulthood
    Mark H. Ashcraft and Nathan O. Rudig

IV. Judgment and Decision Making

  1. Risky Behavior in Adolescents: The Role of the Developing Brain
    Adrianna Galván
  2. Affective Motivators and Experience in Adolescents' Development of Health-Related Behavior Patterns
    Sandra L. Schneider and Christine M. Caffray
  3. Judgment and Decision Making in Adolescence: Separating Intelligence From Rationality
    Keith E. Stanovich, Richard F. West, and Maggie E. Toplak
  4. A Fuzzy Trace Theory of Adolescent Risk Taking: Beyond Self-Control and Sensation Seeking
    Christina F. Chick and Valerie F. Reyna
  1. Paradoxes in the Adolescent Brain in Cognition, Emotion, and Rationality
    Valerie F. Reyna and Michael R. Dougherty

Valerie F. Reyna, PhD, is codirector, Center for Behavioral Economics and Decision Research, and professor of human development, psychology, cognitive science, and neuroscience (IMAGINE Program), at Cornell University.

Dr. Reyna is a developer of fuzzy trace theory, an influential model of memory and decision making that has been widely applied in law, medicine, and public health, as well as in neuroscience. A leader in using memory principles and mathematical models to explain judgment and decision making, she helped initiate what is now a burgeoning area of research on developmental differences in judgment and decision making.

Her recent work has focused on neuroeconomics aging, neurocognitive impairment, and genetic risk factors (e.g., in Alzheimer's disease) rationality and risky decision making, particularly risk taking in adolescence and neuroimaging models of framing and decision making. She has also extended fuzzy trace theory to risk perception, numeracy, and medical decision making by both physicians and patients.

She is the past president of the Society for Judgment and Decision Making, and she currently serves on scientific panels of the National Academy of Sciences, National Institutes of Health, and National Science Foundation. Dr. Reyna is an elected Fellow of the American Association for the Advancement of Science, the Association for Psychological Science, and four divisions of APA.

Sandra B. Chapman, PhD, is the founder and chief director of the Center for BrainHealth, Behavioral and Brain Sciences, and Dee Wyly Distinguished Professor, The University of Texas at Dallas.

Dr. Chapman's research as a cognitive neuroscientist, spanning 25 years, is devoted to better understanding how to maximize higher order reasoning, critical thinking, and innovation across the lifespan, and how to protect and heal cognitive brain function from brain injuries and diseases.

Dr. Chapman is actively involved in public policy to address brain health and to discover ways to maximize cognitive brain function from youth through adulthood. Her research reveals that the middle school years represent a pivotal window for developing reasoning skills. Her team has developed ways to measure and advance these skills to address the growing decline in teen reasoning capacity through administering evidenced-based cognitive training programs.

Dr. Chapman coined the term Brainomics © to represent the economics of brain power — our greatest economic asset and cost burden — developed at school and in the workplace, for better or worse. With more than 125 publications and 30 funded research grants, her research spans the age spectrum from studies that establish ways to advance teen reasoning to protocols that enhance cognitive brain function into late life.

Michael R. Dougherty, PhD, is an associate professor of psychology, Program in Neuroscience and Cognitive Science, University of Maryland.

Dr. Dougherty's work focuses on the fundamental bases of judgment and decision making, cognitive plasticity, and the emergence of cognitive ability, and how these capacities interrelate. His research spans such topics as human factors, limitations of attention and working memory, memory search and retrieval, and hypothesis generation and probability judgment. This research involves an integrative approach that implements mathematical and computational modeling, behavioral experiments, and eye-tracking methodologies.

His recent work applies neuroimaging techniques to understanding cognitive adaptation and retraining, collaborating with members of the Neuroscience and Cognitive Science Program. Dr. Dougherty also collaborates with researchers at the University of Maryland Center for Advanced Study of Language on projects related to improving language comprehension and cognitive ability.

He has received numerous scientific awards, including the Hillel Einhorn New Investigator Award from the Society for Judgment and Decision Making, and the early investigator CAREER award from the National Science Foundation.

Jere Confrey, PhD, is the Joseph D. Moore Distinguished Professor of Mathematics Education at North Carolina State University and a senior scholar at the William and Ida Friday Institute for Educational Innovation.

Dr. Confrey is building diagnostic assessments of rational number reasoning using a learning trajectories approach. She is a member of the Validation Committee for the Common Core State Standards, and was vice chairman of the Mathematics Sciences Education Board, National Academy of Sciences (1998–2004).

She chaired the National Research Council (NRC) Committee that produced On Evaluating Curricular Effectiveness, and she was a coauthor of the NRC's Scientific Research in Education. She was also a cofounder of the UTEACH Program at the University of Texas in Austin, the largest secondary teacher education program for mathematics and science teachers at a research university. She was the founder of the SummerMath program for young women at Mount Holyoke College and cofounder of SummerMath for Teachers Program.

She coauthored the software Function Probe, Graph N Glyphs and sets of interactive diagrams. She has served as vice-president of the International Group for the Psychology of Mathematics Education chair of the Special Interest Group — Research in Mathematics Education on the editorial boards of the Journal for Research in Mathematics Education, International Journal for Computers in Mathematics Learning, and Cognition and Instruction and on the Research Committee of the National Council of Teachers of Mathematics.

Dr. Confrey has taught school at the elementary, secondary, and postsecondary levels.

One Final Note

When we wrote to Dr. Alan Castel for permission to use his stimuli in this article, he not only consented, but he also sent us his data and copies of all of his stimuli. He sent copies of research by a variety of people, some research that has supported his work with David McCabe and some that has not. He even included a copy of the 10-experiment paper that you just read about, the one that failed to replicate the McCabe and Castel study.

The goal is to find the truth, not to insist that everything you publish is the last word on the topic. In fact, if it is the last word, then you are probably studying something so boring that no one else really cares.

Scientists disagree with one another all the time. But the disagreements are (usually) not personal. The evidence is not always neat and tidy, and the best interpretation of complex results is seldom obvious. At its best, it is possible for scientists to disagree passionately about theory and evidence, and later to relax over a cool drink, laugh and talk about friends or sports or life and love.

The impact of stress on the structure of the adolescent brain: Implications for adolescent mental health

Adolescent development is associated with major changes in emotional and cognitive functions, as well as a rise in stress-related psychological disorders such as anxiety and depression. It is also a time of significant maturation of the brain, marked by structural alterations in many limbic and cortical regions. Though many elegant human neuroimaging studies have described the adolescent-related changes in these regions, relatively little is known about these changes in non-human animals. Moreover, both human and non-human data are lacking on how exposure to chronic stress may disrupt this structural maturation. Given the fundamental structure-function relationship in the nervous system, it will be important to understand how these normative and stress-induced structural alterations during adolescence influence psychological function, which in turn can modify future neural development. The purpose of this brief review is to describe the impact of stress on the structure of brain regions that continue to show structural maturation during adolescence and are highly sensitive to the effects of chronic stress exposure. Specifically, this review will focus on the amygdala, hippocampal formation, and prefrontal cortex, particularly from a morphological perspective. As many unanswered questions remain in this area of investigation, potential future lines of research are also discussed. A deeper appreciation of how stress affects adolescent brain development will be needed if we are to gain a better understanding of the mechanisms that mediate the increase in stress-related psychological dysfunctions often observed during this stage of development. This article is part of a Special Issue entitled SI: Adolescent plasticity.

Keywords: Adolescence Amygdala Hippocampus Prefrontal Cortex Puberty Stress.

Why We Did the Study

To answer the question “How is building Lego models related to math skills?” we first need to figure out which brain processes are involved in learning math. We do this by testing if certain skills and abilities like general intelligence and memory are related to how well children do in math. These skills and abilities are called variables because they change based on their relationship with each other. For example, we expect children with higher levels of general intelligence and memory to do better at math, than children with lower levels. The variables we tested were different types of memory (please see Box 1) and general intelligence in 7-year-old children, and we compared their relationship to math and reading.

Box 1. Different Types of Memory

Memory is a very large topic, and researchers divide the topic of memory to make it easier to study. There is long-term memory, short-term memory, and working memory. Long-term memory allows you to remember information for some time𠅏or a few minutes, or for your whole life. For example, if you are divided into groups and are given a number, you will remember your group number for the entire activity. Or, you will remember your name for a lifetime. Short-term memory stores immediate information and is erased in less than a minute. For example, if someone tells you a phone number and you dial it immediately and then forget it. It is more difficult to dial a phone number if a person tells you the whole phone number at once, as opposed to telling you the number in bits and pieces. When you work with short-term memory, it is called working memory. For example, when you read a sentence, you need to remember the first words in the sentence until you finish reading the end of the sentence. Or else, the sentence will not make any sense!

Some researchers divide short-term memory into verbal memory and visuospatial memory [ 4 ]. Verbal memory stores information that is in words and is heard. Visuospatial memory stores information on shapes, colors, and locations and can be seen. When you work with verbal or visuospatial short-term memory, it is called verbal working memory (for example: remembering an entire sentence) or visuospatial working memory (for example: remembering directions given on a map).

After discovering the brain processes involved in learning math, we need to find out whether building Lego models is related to learning math and how. Building Lego models, using wooden blocks, sand, and other such toys and equipment to create something is called construction play. While previous studies have found a relationship between construction play and math, these studies do not tell us the brain processes involved in how construction play and math are related. We tested whether the variables of memory and intelligence are the brain processes involved in the relationship between math and construction play.

The first section of this article explains what researchers already know about math and construction play. The second section explains how we tested math, reading, memory, and intelligence. The third section explains what we found, and finally we discuss what our findings mean to children, students, and teachers of math.

What Researchers Already Know

Previous studies have tested the relationship between construction play and math skills. One study found that teenagers who built taller block models had better math skills than teenagers whose block models tumbled at a shorter height [ 1 ]. Another study found that 3-year-old children who could correctly build a block model by following instructions had better math skills [ 2 ]. One more study found that children who built complicated models when they were in preschool (3- to 4-year olds) had better math scores in grade 7 (as 12-year olds) [ 3 ]. These studies tell us that building block models by following instructions is related to math skills. However, these studies do not tell us the brain processes involved in the relationship between construction play and math skills. Memory is an important brain function for math skills, and so we decided to focus on different types of memory, to test how memory is related to math skills and construction play.

There is one study that tested the role of verbal short-term memory and spatial skills in the relationship between construction play and math skills. They tested spatial skills by asking participants to guess what shape would be made if paper was folded in a certain way. While they found that construction play was related to math skills through spatial skills, verbal short-term memory was not involved [ 5 ]. We thought it was more important to test visuospatial memory with construction play, because to build Lego models you need to see and place the bricks in the correct positions. We also tested reading skills and intelligence to see if the results we found were specific to math skills, or if their general intelligence explained why some children were better at all variables (math, reading, and construction play) and not only at math. To understand the brain processes involved in the relationship between construction play and math skills, we tested verbal short-term memory and working memory, visuospatial short-term memory and working memory, general intelligence, math skills, and reading skills.

APA Format Abstract Basics

The abstract is the second page of a lab report or APA-format paper and should immediately follow the title page. Think of an abstract as a highly condensed summary of your entire paper.

The purpose of your abstract is to provide a brief yet thorough overview of your paper. It should function much like your title page—it should allow the person reading it to quickly determine what your paper is all about. Your abstract is the first thing that most people will read, and it is usually what informs their decision to read the rest of your paper.

The abstract is the single most important paragraph in your entire paper, according to the APA Publication Manual.

A good abstract lets the reader know that your paper is worth reading. According to the official guidelines of the American Psychological Association, an abstract should be brief, but packed with information. Each sentence must be written with maximum impact in mind. To keep your abstract short, focus on including just four or five of the essential points, concepts, or findings.

An abstract must also be objective and accurate. The abstract's purpose is to report rather than provide commentary. It should accurately reflect what your paper is about. Only include information that is also included in the body of your paper.

Mindfulness for Tics

Introducing Mindfulness

Mindfulness is an abstract concept and is best introduced experientially. A common way to introduce mindfulness is through the raisin exercise. In the raisin exercise, participants are provided with a raisin, although any other small item of food could serve as a substitute. When presenting the small food item, the clinician does not name the object. Instead, the clinician asks the individual to investigate the object (the raisin) as if it is the first time that he or she is encountering it. For example, the clinician might say:

In a moment, I will give you an object. I want you to imagine that you have just landed on earth from a far away land and you have never encountered this object before. Explore the object as if it is the first time you are seeing it.

In mindfulness, this act of approaching something without expectations or preexisting knowledge is referred to as beginner’s mind. The individual is then given the raisin and guided in silently exploring the raisin with all of their senses: sight, touch, smell, hearing, and finally taste. Throughout, the participant is encouraged to notice any thoughts or emotions that may arise in response to the object. After eating the raisin, the participant is asked, “What did you notice?” Often, in doing this exercise, participants notice a great deal. They notice the way in which the light bounces off the raisin. They notice the variations in color and texture. They notice the juiciness of the raisin and the changing flavors as it enters their mouth. They may also notice thoughts that arise about the past (memories of past encounters with raisins), the present (commentary about this experience, “What am I doing? What is the point of this?”), or the future (e.g., “What will I say about this experience?”). They may also notice emotions or behavioral urges that arise in response to the raisin—liking it or not liking it, wanting or not wanting to eat it. They also often notice how much this encounter with a raisin differed from their previous encounters with raisins. In noticing all of this, the individual is being mindful. They are noticing the stream of perceptions, thoughts, and emotions that are occurring in the present moment. And they are also beginning to appreciate how mindful attention differs from our ordinary way of paying attention. This can be a very rich experience for participants in that they may start to question what they are not noticing in their daily lives. Some may even begin to understand that our views of the world and what is possible are constrained by what we notice. After completing this exercise, participants generally have a much better understanding of what we mean when we say mindfulness. However, if the clinician suspects that the participant is having difficulty connecting the exercise to the concept of mindfulness, the clinician might explicitly connect it. For instance, a clinician could say something like, “What you just did, noticing the details of the raisin, the thoughts, emotions, and bodily sensations that you were having, that was mindfulness.” Most people, however, do not struggle with making this connection. Indeed, this exercise generally serves as a very concrete, nonthreatening, and accessible introduction to mindfulness.

Talent and Creativity

Talent Search Model

In the 1970s, Julian Stanley began to give the Scholastic Aptitude Test to moderate and bright middle schoolers. The highest scorers were viewed as having the highest potential for mathematical achievement , and were given special coursework and accelerated through the lower schools to permit them to enter college early and to pursue a Ph.D. He founded a program called the Study of Mathematically Precocious Youth (SMPY). The model was expanded in the 1980s to include verbally talented middle schoolers as well. The high scorers have been followed since then by Stanley and Benbow, and over 40 years of data have resulted. Their findings have indicated that specific paths of education and mentoring are necessary for the development of talent. These include advanced courses and special programs. Talent Search centers exist at Northwestern University (Midwest Talent Search), Johns Hopkins University Center for Talented Youth (CTY), which recruits from the east and abroad, and the Talent Identification Program (TIP) at Duke University, among others.

Cognition, Learning & Plasticity

We conduct research on the psychological and biological factors that shape learning and cognitive achievement, with focus on mathematical cognition and executive functions. Our unique multidisciplinary research has been recognised by a number of awards for originality, creativity, and theoretical contribution in the fields of psychology, neuroscience, and education.

It is not a science fiction, it is already real and it will be a critical part of our future.

- Stephen Hawking, discussing our research

In our work, we combine cognitive theories and methods with neuroscientific tools. Such an integrative investigation is an endeavour with broad implications for basic science and can greatly inform the development of evidence-based methods for improving human cognition and brain functions.

Aside from its theoretical implications, our research has: 1) a clear translational impact, as evidenced by the national and international patents and our work with industrial partners and 2) societal and ethical implications which we are exploring with neuroethicists.

What is The Concrete Pictorial Abstract (CPA) Approach And How To Use It In Maths

The Concrete Pictorial Abstract approach is now an essential tool in teaching maths at KS1 and KS2, so here we explain what it is, why its use is so widespread, what misconceptions there may be around using concrete resources throughout a child’s primary maths education, and how best to use the CPA approach yourself in your KS1 and KS2 maths lessons.

The maths curriculum is far too broad to cover in one blog, so the focus here will be on specifically how the CPA approach can be used to support the teaching and learning of the four written calculation methods.

What is the Concrete Pictorial Abstract in Maths?

The Concrete Pictorial Abstract (CPA) approach is a system of learning that uses physical and visual aids to build a child’s understanding of abstract topics.

Pupils are introduced to a new mathematical concept through the use of concrete resources (e.g. fruit, Dienes blocks etc). When they are comfortable solving problems with physical aids, they are given problems with pictures – usually pictorial representations of the concrete objects they were using.

Then they are asked to solve problems where they only have the abstract i.e. numbers or other symbols. Building these steps across a lesson can help pupils better understand the relationship between numbers and the real world, and therefore helps secure their understanding of the mathematical concept they are learning.

Origins of Concrete Pictorial Abstract Approach

Anyone working in primary mathematics education can’t fail to have noticed that the word ‘maths’ is rarely heard these days without a mention of the term ‘mastery’ alongside it.

This is no surprise, with ‘mastery’ being the Government’s flagship policy for improving mathematics and with millions of pounds being injected into the Teaching for Mastery programme a programme involving thousands of schools across the country.

Prior to 2015, the term ‘mastery’ was rarely used. With the constant references to high achieving Asian-style Maths from East Asian countries including Singapore and Shanghai (and the much publicised Shanghai Teacher Exchange Programme), a teacher could be forgiven for believing ‘teaching for mastery’ to be something which was imported directly from these countries..

The fact that the CPA approach is a key component in maths teaching in these countries only added to the misconception.

The CPA maths model in Teaching for Mastery

To find the origins of the mastery maths approach, we need to go much further back in time and look much closer to home.

The concept of ‘mastery’ was first proposed in 1968 by Benjamin Bloom. At this time the phrase ‘learning for mastery’ was used instead. Bloom believed students must achieve mastery in prerequisite knowledge before moving forward to learn subsequent information.

Bloom suggested that if learners don’t get something the first time, then they should be taught again and in different ways until they do.

Jerome Bruner and Concrete Pictorial Abstract

Looking more specifically at the origins of the CPA approach, we again need to go back to the teaching methods of the 1960s, when American psychologist Jerome Bruner proposed this approach as a means of scaffolding learning.

He believed the abstract nature of learning (which is especially true in maths) to be a ‘mystery’ to many children. It therefore needs to be scaffolded by the use of effective representations and maths manipulatives.

He found that when pupils used the CPA approach as part of their mathematics education, they were able to build on each stage towards a greater mathematical understanding of the concepts being learned, which in turn led to information and knowledge being internalised to a greater degree.

Many teachers mistakenly believe mastery, and specifically the CPA approach, to have been a method imported from Singapore.

In actual fact, the Singapore Maths curriculum has been heavily influenced by a combination of Bruner’s ideas about learning and recommendations from the 1982 Cockcroft Report (a report by the HMI in England, which suggested that computational skills should be related to practical situations and applied to problems).

To get a better handle on the concept of maths mastery as a whole, take a look at our Ultimate Maths Mastery guide.

Why use the Concrete Pictorial Abstract approach in Maths?

Pupils achieve a much deeper understanding if they don’t have to resort to rote learning and are able to solve problems without having to memorise.

When teaching reading to young children, we accept that children need to have seen what the word is to understand it. Putting together the letters c- a- t would be meaningless and abstract if children had no idea what a cat was or had never seen a picture.

People often don’t think of this when it comes to maths, but to children many mathematical concepts can be equally meaningless without a concrete resource or picture to go with it. This applies equally to mathematics teaching at KS1 or at KS2.

What is a ‘Concrete’ representation in the CPA approach?

As part of the CPA approach, new concepts are introduced through the use of physical objects or practical equipment. These can be physically handled, enabling children to explore different mathematical concepts. These are sometimes referred to as maths manipulatives and can include ordinary household items such as straws or dice, or specific mathematical resources such as dienes or numicon.

The Ultimate Guide to Maths Manipulatives

Unsure of what sort of materials you might use for the CPA approach? Download our ultimate guide to manipulatives to get some ideas.

The abstract nature of maths can be confusing for children, but through the use of concrete materials they are able to ‘see’ and make sense of what is actually happening.

Previously, there has been the misconception that concrete resources are only for learners who find maths difficult. In fact concrete resources can be used in a great variety of ways at every level. All children, regardless of ability, benefit from the use of practical resources in ensuring understanding goes beyond the learning of a procedure.

Practical resources promote reasoning and discussion, enabling children to articulate and explain a concept. Teachers are also able to observe the children to gain a greater understanding of where misconceptions lie and to establish the depth of their understanding.

What is a ‘Pictorial’ representation in the CPA approach?

Once children are confident with a concept using concrete resources, they progress to drawing pictorial representations or quick sketches of the objects. By doing this, they are no longer manipulating the physical resources, but still benefit from the visual support the resources provide.

Some teachers choose to leave this stage out, but pictorial recording is key to ensuring that children can make the link between a concrete resource and abstract notation. Without it, children can find actually visualising a problem difficult.

One of the most common methods of representing the pictorial stage is through the bar model which is often used in more complex multi step problem solving.

What is an ‘Abstract’ representation in the CPA approach?

Once children have a secure understanding of the concept through the use of concrete resources and visual images, they are then able to move on to the abstract stage. Here, children are using abstract symbols to model problems – usually numerals. To be able to access this stage effectively, children need access to the previous two stages alongside it.

For the most effective learning to take place, children need to constantly go back and forth between each of the stages. This ensures concepts are reinforced and understood.

How to teach using the Concrete Pictorial Abstract method at primary school

A common misconception with this CPA model is that you teach the concrete, then the pictorial and finally the abstract. But all stages should be taught simultaneously whenever a new concept is introduced and when the teacher wants to build further on the concept.

When concrete resources, pictorial representations and abstract recordings are all used within the same activity, it ensures pupils are able to make strong links between each stage.

If you’re concerned about differentiating effectively using the CPA approach, have a look at our differentiation strategies guide for ideas to get you started. Or if you’re short on time, our White Rose Maths aligned lesson slides incorporate the CPA approach into them and some are free to download and use.

Using ‘The Four Operations’ to model how CPA works

In the following section I will be looking at the ‘four operations’ and how the CPA approach can be used at different stages of teaching them. In particular, I will examine how the 3 parts of the CPA approach should be intertwined rather than taught as 3 separate things.

As this blog is to share ideas rather than say how the calculation methods should be taught, I am only going to cover the four operations briefly.

It’s important to take your school’s Calculation Policy into account when determining how the CPA approach can work best for you.

Teaching addition using the Concrete Pictorial Abstract approach

In the early stages of learning column addition, it is helpful for children to use familiar objects. For example, straws or lollipop sticks can be bundled into groups of ten and used individually to represent the tens and ones.

Once children are familiar making 2-digit numbers using these resources, they can set the resources out on a baseboard to represent the two numbers in a column addition calculation.

Initially children complete calculations where the units do not add to more than 9, before progressing to calculations involving exchanging/ regrouping. Alongside the concrete resources, children can annotate the baseboard to show the digits being used, which helps to build a link towards the abstract formal method.

Once children are confident using the concrete resources they can then record them pictorially, again recording the digits alongside to ensure links are constantly being made between the concrete, pictorial and abstract stages.

The next step is for children to progress to using more formal mathematical equipment. Dienes base ten should be introduced alongside the straws, to enable children to see what is the same and what is different. As confidence grows using the Dienes, children can be introduced to the hundreds column for column addition, adding together 3-digit and 2-digit numbers

A basic example of pictorial stage of column addition with dienes.

Alongside the concrete resources children should be recording the numbers on the baseboard, and again have the opportunity to record pictorial representations.

Once children are completely secure with the value of digits and the base ten nature of our number system, Dienes equipment can be replaced with place value counters. These help children as they progress towards the abstract, as unlike the dienes they are all the same size.

These should be introduced in the same way as the other resources, with children making use of a baseboard without regrouping initially, then progressing to calculations which do involve regrouping.

As with the other equipment, children should have the opportunity to record the digits alongside the concrete resources and to progress to recording pictorially once they are secure. The place value counters can be used to introduce children to larger numbers, calculating column addition involving the thousands and then the ten thousands column.

Teaching subtraction using the Concrete Pictorial Abstract

The method for teaching column subtraction is very similar to the method for column addition. Children should start by using familiar objects (such as straws) to make the 2-digit numbers, set out on a baseboard as column subtraction.

To begin with, ensure the ones being subtracted don’t exceed those in the first number. Once children are confident with this concept, they can progress to calculations which require exchanging. As with addition, the digits should be recorded alongside the concrete resources to ensure links are being built between the concrete and abstract.

Once secure with using the concrete resources, children should have the opportunity to record pictorially, again recording the digits alongside.

As with addition, children should eventually progress to using formal mathematical equipment, such as Dienes.

These should be introduced alongside the straws so pupils will make the link between the two resource types. Digits are noted down alongside the concrete resources and once secure in their understanding children can record the Dienes pictorially, to ensure links are built between the concrete and abstract.

Once secure with the value of the digits using Dienes, children progress to using place value counters.

Again, the counters enable children to work concretely with larger numbers, as well as bridging the gap from the use of Dienes to the abstract. Along with the counters, children should be recording the digits and they should have the opportunity to record pictorially once confident with the method using concrete resources.

Teaching multiplication using the Concrete Pictorial Abstract approach

The grid method is an important step in the teaching of multiplication, as it helps children to understand the concept of partitioning to multiply each digit separately.

Pupils can begin by drawing out the grid and representing the number being multiplied concretely. For example, 23 x 3 can be shown using straws, setting out 2 tens and 3 ones three times. This way, children can actually see what is happening when they multiply the tens and the ones.

As with addition and subtraction, children should be recording the digits alongside the concrete apparatus, and recording pictorially once they are confident with the concrete resources.

Children are then able to progress to representing the numbers in a grid, using place value counters.

The video above is a great example of how this might be done.

Once children are confident using the counters, they can again record them pictorially, ensuring they are writing the digits alongside both the concrete apparatus and the visual representations.

Progressing to the expanded method and then the short method of column multiplication is much easier for children if these are introduced alongside the grid method, to enable them to see the link.

Teaching division using the Concrete Pictorial Abstract (CPA) Approach

As children work towards the formal written method for division, it is important they understand what is meant by both division as grouping and division as sharing.

Concrete resources are invaluable for representing this concept. Pupils need to understand how numbers can be partitioned and that each digit can be divided by both grouping and sharing.

In this example, the straws will be shared out between the stuffed toys.

Once confident using concrete resources (such bundles of ten and individual straws, or Dienes blocks), children can record them pictorially, before progressing to more formal short division.

As children work towards understanding short division (also known as the bus stop method), concrete resources can be used to help them understand that 2-digit numbers can be partitioned and divided by both sharing and grouping. This can be through the use of bundles of ten straws and individual straws or dienes blocks to represent the tens and ones.

As with the other operations, it’s important that children are recording the digits alongside the concrete resources and are having the opportunity to draw visual representations. As children grow in confidence and once they are ready to progress to larger numbers, place value counters can replace the dienes.

The Concrete Pictorial Abstract Approach is here to stay

It may have taken many years for CPA to reach the level of popularity it has today, but it is definitely here to stay. I’m not one to jump on the bandwagon when it comes to the latest teaching ‘fad’, however this has been one I’ve been happy to jump on.

I have seen first-hand how successful it can be when children have the opportunity to work in this way and I love the fact that children are now starting to have the conceptual understanding in maths that I never had as a child.

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