The Maths ability pyramid

Pyramid

The Maths ability pyramid is a communication tool I have created in order to explain more easily to students and parents what we do in Maths and why we do it.

The initial idea behind the Maths ability pyramid is not only to access a better understanding of what learning Maths is, but also to get rid of this highly misleading fiction: the one monolithic so-called ‘Maths ability’ (you know, when parents or students tell you: ‘Sir/Miss, I’m not good at Maths anyway’…).

This is already pretty obvious if you consider the broad range of skills that come under the ‘Maths’ heading. For example, I have always been at ease with algebra, but not so much with geometry. In other words, I do not have one Maths ability, but several Maths abilities.

But even this multi-dimensional idea of our ‘Maths ability’ is not enough. The reality is more complex, and more interesting too. There are at least 3 layers, and far-reaching implications for Maths activities, as well as beyond Maths and in real life.

The first layer is what I call fundamental abilities. There is little or no Maths involved here, except for one of the fundamental human activities: counting and performing basic mental operations. Fundamental abilities are about cultivating your own mental space, becoming more aware of mental objects and processes such as order, numbers, causes and consequences.

Why are fundamental abilities important ? We should always remember that, although there is such a thing as Maths intuition or instant vision, most of the ‘shop-floor’ Maths we ask our students to do relies on slow and analytical mental processes. These processes cannot take place without focus. Fundamental abilities are about increasing the capacity to focus. They ‘muscle up’ this capacity, so to speak. Without focus, no Maths is possible.

In a future post, I will explain how these fundamental abilities can be developed in the classroom, and how they actually make the conceptual understanding of Maths much easier and more natural.

The second layer is the Skills level. This is what most people usually refer to as Maths. There is a lot of learning happening there; it’s about conceptual understanding of Maths skills, but also about technique and systematic practice. This level is particularly important for kids who wonder why they are inflicted all this Maths (left-hand side of the pyramid) and do not see that it can ever be useful to them in real life (right-hand side of the pyramid). Going from right to left also tells an interesting story: students (and parents) actually do a lot more Maths than they think on a daily basis. When you say that an event is more or less likely to happen, you are doing some Maths (feel for probabilities). When you want to buy something and estimate how long you need to economize for it, you are also doing some Maths (feel for numbers).

At the top of the pyramid is problem-solving. Ultimately, the whole point of Maths is problem-solving. This is often reflected in the structure of a lesson or a succession of activities. As a teacher, you may start with a problem, show that you need a specific skill to solve it, teach that skill, get students to practise it, and then solve other related puzzles or problems.

Problem-solving is multidimensional. One aspect of that is problem-solving skills: yes, solving problems has to be taught as well. Another aspect is the intellectual drive that students will develop through problem-solving: curiosity, challenge, strategy, trial and error, etc. all of which will be useful to students, whatever they do later in life. Many future posts in this blog will be about problem-solving, which is one my great passions.

To parents, the pyramid provides an explanation of why simplistic notions such as ‘I’m good / I’m bad at Maths’ are just not good enough. The same goes for students, who may get to understand that there is much more hope than they think: for example, if a student gets a sense of failure when struggling with probabilities, then the teacher can show the pyramid and point to all the areas where the student is performing well, thus avoiding a misleading and damaging sense of failure.

For students, a quick look at the pyramid will help them understand the supreme importance of problem-solving and the purpose of basic practice to help them reach the required proficiency in various skills. By becoming more familiar with that global picture, students get to understand what they are doing and why. Ultimately, it also helps them optimize their way through an exam paper by immediately sorting out lower-order questions from more complex ones.

For teachers, it is important to point out that the content and structure of the pyramid are perceived very differently across age groups, which explains why Maths teachers face different challenges when teaching to different KS. In the early stages of learning, the levels and skills outlined in the pyramid do not exist as such for pupils. For them, it’s all Maths and it has to be playful, it is too early to be explicit about domains and levels. In a way, when you teach younger pupils, you have to re-invent Maths for them, you have to use special ‘sparks’ (stories, pictures, etc.) and ‘driving forces’ (games, manipulation of various objects, competition, etc.) — more on that in future posts. You must also get your conceptual thinking absolutely right, because you are dealing with the fundamental building blocks of numeracy. It requires both rigour and creativity, which is why teaching younger children is always a (fascinating) challenge.

At a later stage, as students go through secondary school, the structure of the pyramid becomes clearer. Students gradually find their way around the concepts. Each student’s natural inclination will typically make him/her favour one of the three layers (fundamentals, skills, problem-solving); which, as a teacher, will be your anchoring point to teach that student. However, that pre-GCSE stage is tricky for a Maths teacher, because that is when some teenagers tend to lose interest, with a risk of dropping out and accumulating gaps. They also forget more easily, which is why it is so important to focus on the main point of each lesson and design the right initial spark to support that main point. At this stage of the curriculum, the key teaching qualities are focus, energy and versatility.

Beyond GCSE, when you get to A level and Further Maths, it’s not so much the story-telling, playfulness and originality that matter, although these continue to exist in different forms. By then, students have more capacity for independent and conceptual thinking, and therefore it is more about clear lesson structure and immaculate logical progression.

Finally, the pyramid is a diagnosis tool for teachers, especially if you couple it with other key learning dimensions such as Confidence. For example, one of my students was asking for help with fractions a few days ago: ‘Sir, I could add the first fractions and thought I could do it, then my mind went blank’. This is typically not a skills problem (the technique has been understood in some way), but a mixed problem of Confidence + Fundamentals (attention and focus). By working on fundamental abilities on the one hand, and providing the right help and practice to increase confidence on the other hand, you know exactly what you’re doing and why you’re doing it.

There is another reading of the pyramid, which operates at quite another level. To explain this, I will need to write about vision, about structure and emotional healing, about Roger Penrose and consciousness. But the moment has not come for that yet. Tons of notes are awaiting to be developed and shared before moving on to this fascinating dimension…

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