# Schemas and learning: Perspectives on mathematics teaching

As math teachers we want our learners not just to be able to follow a series of steps in order to obtain a correct solution to a math problem, but to develop mental models of how numbers or numerical representations work and to be able to use these models to solve problems. These mental models cannot be imposed on the child, but need to be built up through thinking (cognitive effort) and those “aha!” moments (understanding). Mathematically, the first step in this process may be when a child goes from being able to touch pebbles one by one and say “One, two, three” to actually being able to look at a group of objects of any type and recognise that there are three objects in the group.

### Building a schema

As learning progresses, more and more concepts are linked together to build an increasingly complex model of how numbers work. This type of mental model or cognitive construct is called a “schema”, a term originally used by Sir Frederic Bartlett in 1932. Schemas do not only apply to mathematics, but to everything we know. For example, you have a “dog” schema, which will include knowledge such as the fact that a dog is a furry domesticated mammal, the feeling of what it is like to pet a dog, the sounds a dog makes, particular instances of dogs you have known and so on.

Schemas seem to be fundamental to the way our minds work and are closely linked to the processes of consciousness: directing attention to an event, linking that event with previously acquired information and then being able to recall it on demand or as a response to a stimulus. Our minds can be imagined as a lot of different complexly interlinked schemas that are continuously changing and evolving as we experience and learn things. Also, the schemas that you already have will help you cope with all the sensory input you receive and also affect your responses to situations and how you remember experiences. A lot of the big-name guys like Piaget and Vygotsky made use of the concept of schemas when developing their ideas.

### Memory and storage

Just an aside on memory here: what we think of as “long-term memory” is effectively unlimited and can be modelled as schema storage. Recalling information from long-term memory does not take effort; for instance if part of your number-sense schema includes “bonds of five” it is not an effort for you to think of 1+4, 2+3 etc. For learning in this sense the mental effort is required in the understanding: making the links and connecting the ideas. Once that it done, the element becomes something you just know.

When a child is learning to read, a few of the most common words will be taught as sight words, and he learns to recognise those by their shape. When he encounters a word he doesn’t know he has a strategy for reading it: he must break down into the sounds of the letters in order to recognise the word. As his fluency in reading increases, more and more words become sight words. This process of recognising a group of letters as a word is called “chunking” and it allows us to make more efficient use of our (extremely limited) short term memory. Once a word is chunked it is stored in long-term memory as part of a schema, and reading this word becomes effortless.

### Chess masters

The schemas of experts are intricate and allow them to recall complex information in chunks. The example of this that resonates most strongly with me is that of chess masters. Now, in chess I know the places the pieces occupy on the board at the start of a game, and I know the permitted moves for each piece, but I am not terrible good at actually playing chess. I gape with impressed astonishment when I see those masters who play multiple matches at the same time, or who play blindfold – how can they remember where every piece is? They don’t of course – they have stored the relationships between the pieces as “chunks”.

If I were to look at a chess game half way through it would take a me lot of mental effort to store in my memory the position of every piece and even then the chances are I would be unlikely to reproduce the setup without an error or two. The chess master would only need to glance at the board to do the same thing. In the past he put in the effort into developing sophisticated schema, although he didn’t necessarily think of it as developing schema. He may have thought of it as recognising the recurring patterns the pieces make, or recognising certain situations in a game, and so on. But now when he sees these patterns he recognises them immediately – he doesn’t have to remember where each piece is on the board, he remembers the entire configuration as a chunk.  And in every chess match he uses these schemas and his knowledge of how they are related to help him choose the moves that will put him in a more advantageous position.

### Modelling schemas

Schemas are domain specific but not isolated from each other. For instance, I was trying to recall the name for the psychological phenomenon in which we don’t notice change (Change Blindness, in case you were wondering!). Somehow my mind came up with “The Persistence of Memory” – but then instantly produced a mental picture of melting clocks, reminding me that this is the name of Dali’s famous painting. The map of how all the schemas we have developed over the years are interlinked is uniquely individual and hugely complex; it is difficult to imagine how this could be modelled. In fact, it is overwhelming to even try to analyse our own schemas!

But as math teachers our task is (fortunately) less complex. We need a rough idea of the schemas our learners should have in place in the mathematics domain when they enter our grade, and we need to know what must be added to those schemas before they leave our class at the end of the year.

### Schemas and teaching

Of course we don’t generally think of our teaching in those terms – we are thinking “Ok they must be able to factorise quadratic equations”. We know that this requires a whole lot of implicit number skills (adding, multiplying, factorising), previously ac

I can imagine learners filling in the gaps in their learning and linking up ideas as a process like mending a fishing net. It takes time, persistence and effort, but once it is done the learner is ready to move on. (Net mending at Washington County Fair, Rhode Island 2013)

quired algebraic skills (such as multiplying binomials, which in turn depends on being able to apply the Distributive Law) and so on. But actually the learning process in math is the process of developing, improving, “complexifying” and linking these schemas. All these interrelated schemas are what we would call “mathematical theory”. The ideal of mathematical fluency appears as the thinking processes become automated. Schemas that are general or abstract can be applied to a whole lot of different situations and can quickly be accessed during problem solving.

What happens when a learner arrives in the math class at the beginning of the year and he doesn’t have the pre-requisite schemas in place? Some learners are able to proceed by slavishly “following the steps” – they can’t really extend their schemas because where the new ideas would normally be connected there are holes and gaps and missing pieces. Some learners simply give up, decide they have reached their math ceiling and there is no point in even trying any more.

But other learners may be able to fill in the gaps. In solving problems beyond their existing schema they will be slow, using time-consuming means-end analysis or forward search. Over time, by the effort of persistence and focus, cognitive engagement and a will to advance they can fill in the gaps, linking into their schemas concepts they should have learned in previous years and getting to a place where they can start adding in the new ideas.

The only one who can actually develop the schema is the learner – clearly we cannot do this for him. At most we can facilitate the process and as teachers, the better we are at this, the more effective we will be.