“The most effective way to develop a coding system is to discover it rather than be told by a teacher.”

– Jerome Bruner

“Please don’t help me,” I said to my husband, as I took the wrapping off a set of flat-pack cardboard shelves that needed to be folded and shaped to be ready to use.

Construction is not my strong point but, embracing a growth mindset, I wanted to see if I could work it out myself, and overcome the moments when I got stuck.

Jerome Bruner called this the constructivist approach — the way in which we construct meaning for ourselves. This approach is around us every day in the classroom; pupils are constantly constructing and reconstructing their understanding of the world as new experiences and interactions occur. One of the challenges we have as teachers is to facilitate this rather than do it for them.

Jerome Bruner (1925 - 2016)

American cognitive and developmental psychologist Jerome Bruner revolutionised maths education, shifting the psychological mindset from behaviourism to put cognition at the centre of modern educational theory and modern educational practices everywhere.¹ Bruner was a researcher at Harvard and Oxford, focusing his attention on children’s cognition in developmental psychology and educational psychology.

Some of Bruner’s influential works include The Process of Education and The Culture of Education. In these books, Bruner introduced his theories: Discovery Learning, the CPA Approach, Scaffolding, and the Spiral Curriculum.

These theories have forever changed maths education and education in general.

Jerome Bruner’s Cognitive Development Theory

Jerome Bruner’s theory of cognitive development states that young children are able to learn complex concepts, given the appropriate educational practices, opportunity, support, and tools to do so.²

Any subject can be taught effectively in some intellectually honest form to any child at any stage of development.

– Jerome Bruner, The Process of Education

Compare and Contrast Theorists

Jerome Bruner

Constructivist Theory of Cognitive Development

Suggested Discovery Learning, created Three Modes of Representation (CPA), coined the Spiral Curriculum for deep learning, and introduced Scaffolding for learner support.

Jean Piaget

Stages of Cognitive Development

Proposed four cognitive stages in learning, believed development driven by biology and environment, and emphasized learning through world interaction and self-discovery.³

Richard Skemp

Relational vs. Instrumental Understanding

Distinguished relational from instrumental understanding, and advocated teaching for deeper mathematical comprehension, not rote learning.

Zoltan Dienes

Six-Stage Theory of Learning Mathematics

Promoted manipulatives in maths, introduced Base 10 blocks for place value, and emphasized engaging, enjoyable maths learning through games, songs, and dance.

Lev Vygotsky

Sociocultural Theory

Introduced Zone of Proximal Development (ZPD), supported learning through social interaction where pupils are challenged beyond current abilities with support from others.⁴

Discovery Learning

I looked carefully at the pictorial instructions for the flat-pack cardboard shelves and began. I looked at them again, realised my mistakes and followed them more carefully. I tried things out even when I wasn’t sure and finally after a number of corrections, I had completed the challenge. I sat back and realised that I could in fact do it. I recognised the logic of the steps, and could see how the tearing, folding, and shaping had to be done in sequence.

My “teacher” (in this case, my husband) had not told me what to do. Instead, I made a series of discoveries on my own. I’m much more likely to have valuable insights and remember them next time — When we make our own discoveries, they become more meaningful and deeply embedded in our minds. This is Discovery Learning.

The essence of creativity is figuring out how to use what you already know in order to go beyond what you already think.

– Jerome Bruner

In Bruner’s pivotal work The Process of Education (1960), he introduced the concept of Discovery Learning, or the constructivist approach. The concept of Discovery Learning implies that pupils are active learners in the learning process, discovering knowledge on their own as they explore and navigate the world.⁵ Because of this, Bruner states that children should not be presented with a concept in its final form or complex ideas, but given the opportunity to piece together bits of information to make their own discovery of the subject.⁶ With the constructivist approach, learners construct their own understanding which allows pupils to build strong foundations of knowledge.⁷

Scaffolding

Last year, my teenage daughter took on the somewhat ambitious task of making her own gingerbread house from scratch during the holiday season. She gathered her ingredients, carefully selected the sweets she would decorate it with, and got to work. Once the dough was made, she began to roll it out using a template to mark, and cut out the rectangles for the sides. Again and again she rolled out the first piece of dough and could not get the template to fit with the dough at the required thickness. I came alongside and asked her if it would make any difference if she rolled the dough from a rectangle shape rather than an oval. She tried it and it worked. What was an obvious solution to me was not yet known to her. What she needed was someone suggesting a step forward.

This in effect is what Bruner, building on Lev Vygotsky’s work, coined as scaffolding. Scaffolding is the process by which a more knowledgeable person, such as a teacher, supports pupils as they learn new concepts and provides them with the space to focus solely on that subject.⁸ Then, over time, as their strength in the subject solidifies, slowly taking away the scaffolding. Scaffolding can look different for each pupil depending on their strength in the subject. Think of scaffolding during a construction project, the idea is much the same where we must provide a supportive learning environment for pupils as their knowledge is being built.⁹ In The Process of Education, Bruner explains that the aim of scaffolding is to give a pupil a task that is not too difficult to bore them, but instead provide a task that challenges while not overwhelming the pupil’s current level of their cognitive growth and development.¹⁰

The CPA Approach

Developed from Jean Piaget’s concept of developmental stages and sprung out of Bruner’s emphasis on the heuristic nature of learning (that is, trial and error and self-discovery) are his three Modes of Representation.¹¹ Bruner identified that learning occurs through enactive representation (doing, which is action-based), iconic representation (seeing, which is visual), and symbolic representation (abstract, which is in the form of ‘codes’ or symbols i.e. language). This Concrete, Pictorial, Abstract (CPA) approach enables a natural and supportive progression for learners as they develop their understanding and skills in maths.

For example, last week I explored equivalent fractions with my pupils. We began with multi-link cubes, progressed to the pictorial problems in the book, and then on to questions with no visuals provided except numbers and the necessary, mathematical symbols for symbolic representation. This resulted in a fully embedded understanding of the concepts throughout the classroom.

The Spiral Curriculum

Recently, whilst watching David Attenborough’s “Blue Planet II” series, I sat in awe at seeing the previously undiscovered creatures living at the bottom of our oceans. “Who would have thought it?” I found myself saying as I realised my understanding of the world had changed. Throughout our lives, we all experience this process of learning, revising, and revisiting as we return to old concepts and make sense of new ones.

It is vitally important that we acknowledge the provisionality of learning; that we might return to a subject and make fresh discoveries, allowing us to change our minds about our previously perceived notions.

Bruner termed this spiralling; the process of coming back to an area of learning to build upon it in light of new discoveries. In maths education, the Spiral Curriculum is effective for building deep understanding of topics. Pupils can learn the basics of one subject and then return to this subject in greater depth later, and so on, learning in a spiral. Each time you return to a subject, the focus is more specific and the depth is deeper. Bruner’s Spiral Curriculum from The Process of Education contrasts Jean Piaget’s more rigid child development theory because Bruner’s allows different types and stages of learning to happen simultaneously and blend into each other.¹²

Bringing Jerome Bruner into the Classroom

There are many ways to implement Bruner’s theories into teaching strategies for the classroom. One of the Bruner’s theories, the CPA approach, is already widely used across maths education and you might be using many others without realising how they follow Bruner's key principles.

CPA in the classroom

Use manipulatives, either physical or digital to introduce a concept, then gradually move to a pictorial representation, then to the mathematical symbolic representation. For example, to teach fractions, you may begin with physical manipulatives that you can divide in half to represent one half of a whole, then move onto a picture of a pie with slices to represent one half of a whole, and then finally introduce the mathematical representation as ½.

Guided practice gives another opportunity to implement Bruner’s theories. Give pupils time to challenge themselves and discover the solution on their own, while providing hints and prompts as needed.

Bruner advocates that “a good teacher will design lessons that help students discover the relationship between bits of information. To do this a teacher must give students the information they need, but without organizing it for them” (Saul McLeod). In effect, we provide the tools, the initial task, the questions, and the opportunity for exploration and discovery, rather than methods and procedures.

Scaffolding and Discovery Learning

Modelling is another way to use Bruner’s theories, such as discovery learning and scaffolding learning. Demonstrate how to solve a step, and then let your pupils mirror you. This can look like slowly counting manipulatives for young learners, or breaking down problems into steps and modelling each step for the classroom.

Another great way to utilize Discovery Learning is with interactive maths games, as Bruner was a big supporter of play-based learning. You can find digital games that provide pupils an opportunity to practice and discover a subject on their own. You can also use physical games such as board games, homemade games, or physically active games that get children out of their seats for discovery learning.

Pairing pupils together can also be used as a form of scaffolding and discovery learning as pupils problem-solve together. This is especially effective with pupils of differing knowledge and confidence levels, as more confident or more knowledgeable pupils can assist the problem-solving skills and language development of others who might be struggling while reinforcing their own understanding.

The Spiral Curriculum

The spiral curriculum may also mean we take steps backwards before progressing forward — something to consider when a pupil is not moving forward as we might anticipate. This is all a part of the process of learning.

Last week, as I reflected on a maths lesson that hadn’t gone so well, I saw what I could have done to make it work better. This was not about the class having to do something differently: it was how I could have taken a different approach. Bruner himself was an advocate of reflective practice. This means turning the situation back on ourselves, looking at our own practice and learning, in order to understand how our pupils might be experiencing and managing their own learning experiences. This is a practice that helps us step away, take a breath, and find perspective. Because when it comes to our profession — we’re learners, too.

Jerome Bruner: a Pillar of Maths Mastery

Jerome Bruner is one of the many pillars of theorists that make up our maths mastery programme. At Maths — No Problem! we strongly believe in building deep understanding of maths, rather than rote learning.

With the framework the Maths — No Problem! approach provides, pupils can build on their learning as they progress through the programme — using Bruner’s spiral curriculum by revisiting familiar concepts and adding another layerof deeper understanding. The CPA approach, scaffolding and discovery learning are also fundamental to the structure of the Maths — No Problem! Primary Series.

With a programme that is built on proven, research methods and provides teachers with everything they need to succeed, you don’t have to worry about how to structure pupils’ learning or how to cover every topic — our programme covers everything necessary in a perfect spiral curriculum which is designed to build strong, confident learners. And, we provide teacher guides on top of that to help you in doing the work in practice.

_{1. Harvard University, “Jerome Bruner,” Department of Psychology, 2016, https://psychology.fas.harvard.edu/people/jerome-bruner.}

_{2. Harvard University, “Jerome Bruner”.}

_{3. Saul McLeod, “Piaget’s Theory and Stages of Cognitive Development,” Simply Psychology, August 5, 2024, https://www.simplypsychology.org/piaget.html.}

_{4. Saul McLeod, “Vygotsky’s Sociocultural Theory of Cognitive Development,” Simply Psychology, August 9, 2024, https://www.simplypsychology.org/vygotsky.html.}

_{5. “How Jerome Bruner Transformed Psychological Science,” Association for Psychological Science - APS, June 14, 2016, https://www.psychologicalscience.org/publications/observer/obsonline/how-jerome-bruner-transformed-psychological-science.html.}

_{6. Saul McLeod, “Jerome Bruner Theory of Cognitive Development & Constructivism,” Simply Psychology, November 3, 2022, https://www.simplypsychology.org/bruner.html.}

_{7. “How Jerome Bruner...,” Association for Psychological Science - APS.}

_{8. TeachThought Staff, “Learning Theories: Jerome Bruner on the Scaffolding of Learning,” TeachThought, December 2, 2014, https://www.teachthought.com/learning/jerome-bruner/.}

_{9. TeachThought Staff, “Learning Theories...”.}

_{10. The Editors of Encyclopedia Britannica, “Jerome Bruner,” Encyclopædia Britannica, September 27, 2018, https://www.britannica.com/biography/Jerome-Bruner.}

_{11. The Editors of Encyclopedia Britannica, “Jerome Bruner”.}

_{12. “How Jerome Bruner...,” Association for Psychological Science - APS.}

Sources

The Editors of Encyclopedia Britannica. “Jerome Bruner.” Encyclopædia Britannica, September 27, 2018. https://www.britannica.com/biography/Jerome-Bruner.

Harvard University. “Jerome Bruner.” Department of Psychology, 2016. https://psychology.fas.harvard.edu/people/jerome-bruner.

“How Jerome Bruner Transformed Psychological Science.” Association for Psychological Science - APS, June 14, 2016. https://www.psychologicalscience.org/publications/observer/obsonline/how-jerome-bruner-transformed-psychological-science.htm

McLeod, Saul. “Jerome Bruner Theory of Cognitive Development & Constructivism.” Simply Psychology, November 3, 2022. https://www.simplypsychology.org/bruner.html.

McLeod, Saul. “Piaget’s Theory and Stages of Cognitive Development.” Simply Psychology, August 5, 2024. https://www.simplypsychology.org/piaget.html.

McLeod, Saul. “Vygotsky’s Sociocultural Theory of Cognitive Development.” Simply Psychology, August 9, 2024. https://www.simplypsychology.org/vygotsky.html.

TeachThought Staff. “Learning Theories: Jerome Bruner on the Scaffolding of Learning.” TeachThought, December 2, 2014. https://www.teachthought.com/learning/jerome-bruner/.