Mapping the Montessori mathematics curriculum to Dehaene’s four pillars of learning

Abstract

Abstraction is a crucial characteristic of human learning, and a fundamental question for education concerns how teachers can support learners to acquire abstract knowledge. We tackle this question through one particular case: the mathematics curriculum devised for 3 to 6-year-olds by Maria Montessori. We map this curriculum to Dehaene’s Four Pillars of Learning (attention, active engagement, error feedback, consolidation), and we argue that the Montessori mathematics curriculum is aligned with well-established cognitive principles known to support learning.

Introduction

Despite mathematical abilities being important for academic attainment, everyday living, and employment^1,2, children’s maths development often lags behind reading³ and many students lack maths confidence⁴. Models of early mathematical development vary in the breadth of skills they emphasise. Clements and Sarama’s learning trajectories⁵ and Purpura and Lonigan’s early numeracy model⁶ focus primarily on the development of number concepts, including counting, numerical magnitude, and early arithmetic such as addition and subtraction, reflecting a relatively narrow conception of mathematics. In contrast, broader models proposed by Milburn et al.⁷ and Braeuning et al.⁸ conceptualise maths as a multi-domain construct, incorporating not only numeracy but also patterning, measurement (e.g., comparing size, length, or weight), and geometry (e.g., shape recognition and spatial relations).

Importantly, children need to gain abstract knowledge because mathematical concepts are not directly tied to physical objects or tangible experiences. Mathematics frequently relies on symbols and mathematical notation to represent concepts. These symbols do not have inherent meaning on their own but gain significance within the context of mathematical operations and relationships. Similarly, mathematics seeks to identify patterns, structures, and relationships that hold true across different contexts and domains. This process of generalization involves abstracting away from specific instances to identify broader principles: simply put, mathematical learning is a process of moving from concrete experiences to abstract concepts.

Our focus is how maths is taught to 3- to 6-year-olds in one particular educational method, namely that devised by Maria Montessori, chosen because it aims to support abstraction of mathematical knowledge: abstract concepts are ‘materialised’ (i.e., made concrete) in a set of sequentially-presented didactic materials that allow for cumulative learning. We map the Montessori maths curriculum to the ‘Four Pillars of Learning’ identified by cognitive neuroscientist Stanislas Dehaene. In his 2020 book ‘How we learn’ ⁹, Dehaene synthesises the neurocognitive research on learning and identifies four evidence-based ‘pillars’ that must be in place for the learning process to be successful. These are Attention (i.e., focusing on relevant information), Active Engagement (effortful involvement in the learning process), Error Feedback (learning from mistakes) and Consolidation (repeated practice and reflection). He also (but only briefly) discusses the role of abstraction in learning, recognising it as “characteristic of the human species” ⁹.

Although Dehaene focuses on learning rather than pedagogy, he suggests that teachers can rely on these pillars to maximise learning. He specifically mentions the effectiveness of Montessori pedagogy: “Most schools inspired by Montessori … propose a whole series of rational and hierarchical activities, whose purpose is first carefully demonstrated by teachers before being carried out independently by children. Active engagement, pleasure, and autonomy, under the guidance of an explicit teaching method and with stimulating pedagogical materials: these are the ingredients for a winning recipe whose effectiveness has been repeatedly demonstrated” ⁹.

Our aim is to flesh out the details of abstract learning in one domain – mathematics – in the Montessori curriculum. We do this by mapping the Montessori maths curriculum to Dehaene’s pillars, placing particular focus on how it has been designed to enable the child to learn abstract concepts. Hence, we aim to bridge a piece of the gap from cognitive neuroscience to education.

We argue that such a mapping exercise is valuable for four reasons. Firstly, evaluations of Montessori education’s effectiveness are relatively few and small in scale, and randomised control trials (RCTs) are even scarcer. A recent systematic review¹⁰ synthesised 12 high-quality studies on maths where the Montessori and control groups had baseline equivalent scores. Montessori-educated children outperformed those in traditional schools with an effect size of 0.22, which the authors argued was medium-large considering that the studies were carried out in real-world school contexts. Several more recent rigorous studies (including RCTs) have also found that Montessori education boosts maths performance^11–14. Montessori education is not always implemented with high fidelity, but the more faithful the school’s adherence to Montessori principles, the stronger the positive impact on maths¹¹. Furthermore, advantages for maths might emerge after several years rather than being evident early on¹³. Thus, there is a need to understand what it is about Montessori maths education that makes it effective. We therefore take a complementary tack to evaluation studies: by mapping the Montessori approach to evidence within the science of learning, we aim to bring further insight into its effectiveness.

Secondly, in an educational landscape that changes rapidly and where accountability is increasingly demanded¹⁵, practitioners need to understand the rationale for their pedagogy, and this is just as true for Montessorians as it is for those trained in other approaches. Such an understanding arguably enables teachers to better justify what they teach and how they teach it, as well as enabling them to more effectively adapt their teaching as needed.

Thirdly, with a growing number of students not liking maths and not reaching a basic level of competence in the subject⁴, and with teachers lacking self-efficacy in maths and prioritising reading over maths¹⁶, questions have been raised around the current maths curriculum and its teaching¹⁷. If the Montessori approach to teaching maths is effective, it should be considered by other education systems. Again, successful adaptation requires an understanding of which aspects of the Montessori approach are effective and why.

Finally, and more broadly, mathematics is just one domain where abstract knowledge is required. Examining it in detail allows researchers and educators to think how the Montessori principle of ‘materialised abstraction’ (i.e. making abstract concepts concrete through tangible materials) might work for other domains too.

A background to Montessori education

Maria Montessori (1870–1952) originally trained as a physician. She started developing her pedagogy for 3- to 6-year-olds (and subsequently for other ages) over 100 years ago, through painstakingly observing children’s development and noting how they responded to her classroom organisation and teaching materials¹⁸. Key and unexpected discoveries were that even young children show intrinsic motivation and have long attention spans for freely-chosen activities, and that they are interested in learning things that might be judged too ‘difficult’ for them¹⁸.

Two unique facets of Montessori’s approach are the learning materials and the way in which children engage with those materials¹⁹. Children learn by engaging hands-on with the materials individually, in pairs or in small groups, during a 3-h ‘work cycle’. Within the limits of the class rules, children are free to choose what they work on, who they work with, and how long they spend on any particular activity. There is no system of extrinsic rewards and no competition between children. Each class includes 3- to 6-year-olds, and learning from peers is common. Children can practise with the materials as often as they like and gain mastery over what they are learning, moving on to the next challenge only when they are ready. Pedagogical features such as these, which prompt the child to self-correct and problem-solve independently, support the development of executive functions²⁰, and executive functions are positively correlated over time with emerging mathematical skills²¹.

The learning materials themselves are made of durable materials such as wood and are designed to be appealing. They are also designed so that the child’s attention is drawn to just one new thing at a time. For example, each set of materials in the sensorial curriculum isolates a single sensory quality (e.g., length, colour, or sound). Most materials have an in-built ‘control of error’ that allows children to check for themselves whether they have completed the activity correctly.

The Montessori mathematics curriculum

Montessori coined the term ‘psychoarithmetic’ for her approach, writing: “It is certain that mathematics organises the abstract path of the mind, so we must offer it at an early age, in a clear and very accessible manner, as a stimulus to the child whose mind is yet to be organised.”²². Her approach prepares children for mathematics long before they encounter the first piece of specially-designed material. Children need to come to the materials with hands that can make controlled and coordinated movements, eyes that can make distinctions between similar percepts, and the self-discipline to complete a cycle of learning activity even when what is being learnt is challenging. Both the practical life curriculum (also called ‘daily living’ and involving everyday activities for the care of the self and of the environment) and the sensorial curriculum (activities to help refine sensory perception of shape, colour, dimension, sound, etc.) offer children plenty of opportunity to hone these skills. The sensorial curriculum has a further, and very critical, role to play: by providing opportunities for pairing, sorting into sets, and sequencing, it supports the development of discrimination and mathematical reasoning abilities.

Children experience mathematical concepts in physical form before experiencing them in increasingly abstract form. Sets of materials form a learning sequence, with an emphasis on preparing secure foundations for later, more advanced, learning. Later materials within a sequence present concepts more abstractly than previously encountered materials (a feature termed ‘concreteness fading’²³). Because abstraction is the result of individual experience, children vary in how long it takes them to achieve it; hence the importance of children being free to take their time using the materials.

We now explain the design of the Montessori maths curriculum for 3- to 6-year-olds, before describing in more detail the first group of learning materials and how they help the child acquire abstract mathematical concepts. For reasons of length, we focus just on number and arithmetic, leaving out geometry. We follow the curriculum of the Association Montessori Internationale (AMI), which organises the materials into six groups²⁴. Group 1 is completed first and then the child progresses through the other groups in parallel. In this way, mathematical understanding is built on a solid foundation of an understanding of “ten”, and further learning is reinforced because similar concepts are explored from different perspectives, using different materials.

Group 1 [see Fig. 1] introduces the quantities one to ten and the symbols that represent them. Zero is introduced as the absence of quantity. The concepts of odd and even numbers and of the divisibility of numbers by 2 are also introduced, along with sensorial representations of the number bonds up to 10. This knowledge forms the foundation for all further work, because decimality is a fundamental feature of the Hindu-Arabic number system.

Fig. 1. The Group 1 materials.

a Number rods. b Sandpaper numerals. c Number rods and cards. d Spindle box. e Numbers and counters. f Memory game.

The Group 2 materials present a key feature of the decimal system: units, tens, hundreds, and thousands. There are also games that introduce the four operations (addition, subtraction, multiplication, division). Group 3 offers further experience of the decimal system, by introducing the names and quantities of the numbers from 11 to 1000 and by providing opportunities for counting these numbers. Children also experience counting in twos, in threes, etc., which prepares them for the multiplication tables. Group 4 focuses on number facts and operations. Children have opportunities to memorise basic arithmetic facts for the four operations, including the commutative law (i.e., when numbers are added or multiplied, the answer is the same regardless of the order of those numbers) and the linking of multiplication and division. Group 5 provides what is termed the ‘pathway to abstraction’: it helps children to know what they know and to step away from the learning materials to do operations with only pencil and paper and the knowledge they have acquired. Finally, Group 6 focuses on fractions, introducing the quantities, names and symbols, and providing opportunities to manipulate fractions using the four operations (readers who would like more detail can consult Montessori²⁴).

Whichever group of materials children are working with, the same basic sequence is followed. First, the quantity is introduced concretely, and children engage in activities that allow them to become familiar with that quantity and absorb an impression of it. Then they are presented with the symbolic representation (name and digit) of that quantity, and some activities that help them to take in an impression of these symbols. When children know both the quantity and the symbols, they are given some activities that help them to put these together. Montessori devised these activities to help children to clarify concepts and aid the abstraction process. Finally, children extend their experience of the concept through further activities.

To illustrate this ‘materialised abstraction’ and ‘concreteness fading’, we take a closer look at the Group 1 materials. The first set of materials, the Number Rods, is a concrete representation of the quantities from one to ten: a total of ten rods are divided into equally-sized, alternating red and blue sections, meaning that units are presented together but distinct. The length of the rods progresses linearly, with the second rod (i.e., the number two) being twice the length of the number one rod (20 cm versus 10 cm), the third rod (the number three) being three times the length of the number one rod (30 cm versus 10 cm), and so on. One to ten are therefore introduced as whole quantities, in sequence. Children are introduced to the abstract representation of these quantities through the Sandpaper Numerals (the symbols 0–9 cut from sandpaper and mounted on boards), which allow them to gain a tactile and visual impression of the symbols and to recognise and learn to trace them. In order to then link quantities with symbols, Number Cards are combined with the Number Rods, and children use this knowledge to explore number bonds (e.g., the one and nine rods placed end-to-end form the length of the ten rod). The symbol then remains linked to something concrete as the children work through the materials in this group, but the quantities take different forms and become less concrete. The Spindle Box has compartments numbered 0–9 into which children count the corresponding number of wooden spindles. This material teaches the concept of zero, develops early counting skills, and allows children to practice the association between quantity and the symbols 0–9. Importantly, with respect to abstraction, the spindles are distinct and mobile: they are still a concrete representation of each quantity when they are put together, but they are not that quantity until they are together. Finally come two activities that deal with progressively less concrete representations and require children to recall what they know. In the Numbers and Counters, the quantities are represented by counters. Now children have to create the sequence for themselves by laying out the symbols in order from 1 to 10. By allocating the corresponding number of counters under each symbol, the activity confirms that children know the quantities and the symbols from 1 to 10 and can link them and build the sequence for themselves. Finally comes the Memory Game with Zero, a multi-player game where the numbers 0–10 are written on slips of folded paper in a basket. Each child picks a slip and reads the number to themself. They then have to collect that number of objects in the classroom (e.g. 2 beads, 3 pencils, 4 sponges). This game gives children a further opportunity to apply their abstract understanding of the quantities and symbols and reinforces the meaning of zero.

Montessori’s approach depends on some key principles: 1) Concrete-to-Abstract. Concrete representations of mathematical concepts are always the starting point, with subsequent materials becoming increasingly less concrete as children abstract the mental concept from their interaction with the materials and store it in memory. 2) Simple-to-Complex. The sequences are logical, with each new challenge being isolated, so children can explore and understand something simple before it is elaborated upon with something more complex. 3) Repetition is offered in many ways – in the various experiences offered with the material itself or with a parallel material presenting the same concept, or with a consecutive, less concrete material. In this way, children can ‘deep-mine’ their understanding of a concept according to their own individual need for repetition. 4) Control of Error. Most materials are designed so that children can find out for themselves when they make a mistake and address this through as much repetition as they need. The pathway is taken at the child’s pace so that new information is always built on a solid foundation.

Mapping the Montessori maths curriculum to Dehaene’s four pillars of learning

We now interweave the key principles introduced in the previous paragraph with Dehaene’s pillars of learning⁹, using the Group 1 materials for illustration.

Attention is the mechanism by which the brain selects, amplifies, and channels information for processing and encoding as memory⁹. Attention relies on being alerted and oriented to something of interest, and in the Montessori classroom children are guided to choose the activities the teacher has observed they are interested in and have the capacity for. Attention also relies on the capacity to sustain focus and filter out distractions long enough for learning to happen. The Montessori materials support children’s Attention by each isolating a single quality. With just one feature highlighted, each set of materials allows children to focus on one new thing at a time. For example, in the sensorial curriculum children will have worked with the Red Rods (see Fig. 2), where the quality of length is isolated, and so they will have a concept of length. Now, when they encounter the Number Rods, the concept of quantity is isolated because children’s attention is drawn to the alternating red and blue sections. Through counting these sections, quantity now becomes the focus.

Fig. 2. The Red Rods (sensorial curriculum) and the Number Rods (maths curriculum).

a Red Rods. b Number Rods.

Active Engagement requires the learner not to just passively receive information but engage with it so that the experience enables them to form a mental model. The brain can compare this new model to the one it predicts from the information already held in memory. When there is a disparity between what is experienced and what the brain predicts, neurons fire so that information can be processed, and new models built. Without this Active Engagement, the opportunity for building new models of understanding, i.e., learning, is reduced. All Montessori materials encourage Active Engagement. For example, after children have manipulated the number rods, their experience is extended by an activity in which they are being asked to fetch a particular Number Rod and match it to a numeral from across the room. This requires holding the visual image of that quantity and its symbolic representation ‘in their head’. Through this hands-on activity they learn to hold information for longer in working memory, strengthening the connections in the brain related to this experience and enabling Consolidation.

Error Feedback is a natural consequence of Active Engagement, with both relying on Attention. The brain relies on routines and structures to automate processes and ensure that attention is focused, thus allowing it to make predictions. Learning takes place when the predicted expectation is violated, and error feedback takes place. The learner needs feedback for correcting misconceptions and for reinforcement (i.e. positive feedback to strengthen connections). The brain picks up the error and tries to reduce the disparity between what is perceived and what is predicted so that it can form a more accurate mental model of what is now known. Activities that give the learner immediate and explicit feedback of error prompt Attention and Active Engagement in correcting the error. Error Feedback is an important feature of the mathematics materials. For example, the Spindle Box (see Fig. 1) has exactly 55 spindles. If children reach the end and there are not enough spindles, or some are left over, they notice this error, Attention is piqued, and repetition leads to correction. The compartment for 0 remains surprisingly empty, thus bringing Attention to the idea that zero has no quantity but it is important enough to be given a place in the number system, an idea that is picked up later in the Group 2 materials, when there will be an opportunity for repetition and Consolidation of this idea.

Consolidation is a “shift from slow, conscious, and effortful processing to fast, unconscious, and automatic expertise”⁹. It is an important part of the process of learning since it enables the learner to make the transition from Attention and Active Engagement into a state where knowledge is stored and used without conscious direction of attention, thus freeing up attentional resources for deeper and higher-order learning. In the Montessori maths curriculum, Consolidation is achieved through the materials forming a cumulative, learning sequence; there is emphasis on preparing secure foundations for later, more advanced, learning and automatisation of knowledge. For example, after children have explored the Number Rods and Spindle Box, the journey from concrete to abstract continues with the Numbers and Counters. The new challenge here is that the children create the sequence of numbers for themselves, which offers Repetition so that knowledge that is not yet fully acquired can be consolidated. Again, error is controlled by the fact that there are only sufficient counters for the numbers 1 to 10. The counters offer something concrete to manipulate, but the ‘concreteness’ of the experience is reduced compared with the Number Rods, where children embody the differences in length and the difference in weight of the quantities. The counters are also less concrete than the spindles, which they grasped in their hands, thereby receiving a haptic response each time a spindle was added.

In summary, we have shown how Montessori’s maths curriculum aligns well with Dehaene’s Four Pillars of Learning. Indeed, it could be argued that Montessori intuited these pillars from her own extensive observations of how children learn and developed her pedagogical approach around them.

Conclusions

As discussed at the outset, abstraction is key to developing good mathematical abilities. We have argued that the design of the Montessori maths materials and the way they are used support abstraction by guiding the child’s attention and providing opportunities for active engagement, error feedback and consolidation of knowledge. Mapping the Montessori maths curriculum to Dehaene’s Four Pillars of Learning demonstrates that it is strongly supported by the current neurocognitive evidence base regarding how children learn. It is therefore not surprising that studies have shown the Montessori approach to be superior to traditional approaches to teaching maths^10–14. Nor is it surprising that Dehaene acknowledges the effectiveness of Montessori pedagogy⁹.

We conclude by revisiting our motivations for writing this paper. The first relates to supplementing direct evaluations of Montessori education. The mapping exercise shows that Montessori’s maths curriculum links convincingly to the current state of knowledge in the science of learning, adding to our understanding of its effectiveness. In relation to our second motivation, our mapping provides a clear rationale for how and why the curriculum is structured the way it is, and why following a particular sequence of materials to build cumulative knowledge is important. That the maths curriculum is aligned with well-established cognitive principles known to support learning provides practitioners with a stronger position to justify what and how they teach, and to make decisions about how to adapt their teaching. Indeed, much of what we have described for the maths curriculum is relevant to other areas of the Montessori curriculum too. Our third motivation relates to the proposal that understanding what makes the Montessori approach effective will facilitate its wider adoption/adaptation, beyond Montessori schools. Although the evidence currently suggests that adapting the Montessori maths curriculum to non-Montessori settings does not give as good results as implementing it in high-fidelity, classic Montessori settings^11,25, this should not be taken as a reason for not adapting it at all – a better understanding of the critical components might enable greater success. Our final motivation was to provide an example of how a curriculum can be mapped to evidence from the science of learning, and we hope we inspire others to follow suit.

Author contributions

Conceptualisation: C.M., L.L., and J.V.H.; Visualisation: C.M. and L.L.; Writing – Original Draft Preparation: C.M., L.L. and J.V.H.; Writing – Reviewing and Editing: C.M., L.L. and J.V.H.

Data availability

No datasets were generated or analysed during the current study.

Competing interests

J.V.H. was previously an Associate Editor, and C.M. is an editorial board member, of the journal, but neither was involved in the journal’s review of, or decisions related to, this manuscript. L.L. declares no competing interests.