## Not all learning is equal

How would you define knowledge?

What is learning?

Before going any further, I would like to start by defining knowledge as everything that we know (or can do) which is stored in our long-term memory. This allows us to define learning as a permanent change in the content of long-term memory; an accumulation, or a re-organising, of stored knowledge. It should follow then, that learning should be as simple as encountering and remembering knowledge about the to-be-learned topic. Unfortunately, it is not that simple! Some knowledge is acquired more easily than other knowledge and hence some learning comes easily while some learning is hard. For example, consider learning to speak versus learning to write. Our ability to speak has been evolving since the days of the first humans. We are so programmed to learn to speak that even children of profoundly deaf couples turn out to be fluent speakers (Harris, 1998). Writing, on the other hand, is a relatively new cultural invention that has only existed for the past few millennia; “far too short a time to be influenced by biological evolution” (Sweller, Ayres & Kalyuga, 2011). Humans are yet to develop the biological mechanisms required to learn to write without instruction.

## Primary or secondary

Knowledge can be classified as either biologically primary or biologically secondary (Geary, 1995). Biologically primary knowledge is knowledge which we have evolved to acquire over many generations – often for reasons of survival. Biologically primary knowledge does not usually need to be taught and is acquired with relative ease and speed. Examples include learning to speak, learning to read faces, learning basic social interactions and general cognitive skills such as basic problem-solving or self-regulation (Tricot & Sweller, 2014). In contrast, biologically secondary knowledge is the knowledge we have developed for cultural reasons and tends to be acquired relatively slowly and with conscious effort. Biologically secondary knowledge has emerged relatively recently in human existence and – as we have not evolved to acquire this type of knowledge – explicit instruction is required for us to learn it. Examples of biologically secondary knowledge include learning to read and write. In fact, almost everything that is taught in schools can be considered biologically secondary knowledge.

## Learning mathematics is hard

Mathematics may be the most interdependent and hierarchical body of knowledge we expect students to learn. Early experience with counting and numerosity occurs pre-school with most students going on to study mathematics in some form until they are at least 16 years old. That is a lot of content! If we accept that students develop new ideas and understanding by reference to ideas they already know, then the importance of building robust and complex knowledge structures (schema) in long-term memory is critical to successful learning. Poorly formed schema, lack of relational understanding, misconceptions, or simply missing knowledge, all contribute to a lack of progress when learning mathematics.

We must not forget that practice is essential for successful learning in mathematics. However, time spent practising must result in maximum gains. Practice should take advantage of research-based approaches like retrieval, spacing and interleaving. Practice should also be purposeful and effortful on the part of the student.

Learning mathematics is hard because it involves concept formation. This is different from learning facts and requires students to develop – for themselves – an understanding of how or why a concept must be so. This cannot be ‘told’, rather it requires students to use their own awareness to come to ‘see’ the concept for themselves. Learning mathematics is hard because it involves a progressive change in what a student knows (or can do) and relies on a robust understanding of subordinate concepts. Figure 1 is my attempt at exploring the concept of “Adding Fractions”. You can see the subordinate concepts that connect directly to “Adding Fractions” also have their own subordinate concepts. The concepts around the outside of the diagram will also have their own subordinate concepts, highlighting the deeply structured and interrelated nature of even a relatively simple superordinate concept like adding fractions. I feel this is a worthwhile exercise to carry out when planning a series of lessons as it highlights the key prior knowledge that should be present before the new learning is encountered.

Figure 1

I have met many students who believe mathematics to be a series of unrelated procedures that are to be memorised. Sometimes, that is because they have been taught that way. Perhaps, this is due to predictable and formulaic assessments leading to the assessment tail wagging the pedagogical dog. I suspect that some of our most successful students are – in fact – just very good at remembering processes and recognising when to apply them. Might it be possible that this apparent fluency masks a lack of conceptual or relational understanding? These students may well achieve a top grade come exam time, but are they really mastering the subject?

Learning mathematics is hard because so much of it exists in the abstract. We can touch, hold and count seven objects, but what about x objects? We can see and feel what it means to add three but it is not so easy to ‘see’ what happens when we subtract negative three. Learning mathematics involves extracting information and making sense of experience. We cannot control what students think but we can try to influence what they think about. Mathematical thought requires ‘awareness’ such that we might form concepts for ourselves before using them in new situations (Wheeler, 2001). I believe that students’ awareness is a powerful and often under-used resource. Indeed, Guy Claxton claims that “the learning mechanism is fuelled by awareness” (Claxton, 1984). The notions of awareness and of teachers ‘forcing’ awareness stem from the fascinating work done by Caleb Gattegno in the 70s and 80s. Gattegno tells us that, for students, learning or building knowledge does not happen as the teacher narrates information but rather as students make a conscious effort to focus their attention on developing or “educating” their own awareness (Gattegno, 1987). A student can educate their awareness by observing what happens in a situation. Here we have an example of two tasks relating to place value.

Figure 2 shows my representation of a standard textbook exercise on place value. The problem with this type of practice is that the student either gets it correct, i.e. they already know how to do it and probably don’t need the practice, or the student gets it wrong. If they get it wrong, there is no opportunity for them to educate their awareness. Contrast this with Figure 3 which shows an adapted version of a task found in the wonderful “Practising Mathematics – Developing the Mathematician as well as the Mathematics” available from the Association of Teachers of Mathematics. This task offers some practice of working with place value but with the added benefit of allowing students to observe and make sense of experience. A wrong answer here is useful as students have the opportunity to “see” the effect of their attempt and to take corrective action. In this way, this task not only offers practice but is also a learning opportunity.

Figure 3

## Necessary and arbitrary

To help us create learning experiences which require students to use their own awareness, we can consider the mathematics curriculum in terms of “arbitrary” or “necessary” knowledge (Hewitt, 1999). Arbitrary knowledge cannot be worked out. It exists in the realm of memory and is either remembered or forgotten. Examples include words like “factor”, “prime” or “hypotenuse”; definitions such as 360 degrees in a full turn; notation such as f(x) for functions; or x before y in a coordinate. I acknowledge that there may be cultural or etymological reasons why we use particular words or conventions but these are not things that can be worked out in a mathematical sense. Students need to be ‘told’ this arbitrary knowledge. In contrast, necessary knowledge can be worked out. Examples include the concept of a factor or the concept of prime-ness; results such as ‘the angles in triangles sum to 180 degrees’ or ‘dividing by a fraction is the same as multiplying by its reciprocal’. Necessary knowledge lies in the realm of awareness and the teacher’s job is to use tasks that allow students to use their own awareness to come to know what is necessary. As Hewitt tells us, mathematics lies in the necessary – not the arbitrary.

## Received Wisdom

Teaching is more than the telling of facts and the demonstration of processes. As Elliot Eisner writes, “The aim of the education process… is not to cover the curriculum, but to uncover it.”. While it is true that some “telling” must take place to pass on arbitrary knowledge, we must beware of teaching necessary results as if they were arbitrary. By that I mean we should avoid ‘telling’ knowledge and demonstrating results that students can (with our guidance) work out for themselves. If we teach necessary as arbitrary, this knowledge becomes “received wisdom”. Of course, some students will have the awareness to see the ‘why’ behind the result, i.e. they will use their awareness to convert this received wisdom into a necessary result. However, not all students will have the awareness (or even the ‘bothered-ness’) to do so. Too often, students don’t make sense of the ‘why’ and if necessary results are taught as arbitrary then the received wisdom becomes another unrelated fact to be remembered, or forgotten.

Imagine a student who has never heard of a “factor”. The learning experience might proceed as follows:

Teacher: “Today we are learning about factors.”

Teacher: “A factor is… insert teacher’s definition of a factor…”

Teacher: “Here are some examples of numbers and their factors”

Teacher: “Now you find all the factors for these numbers”

By ‘telling’ students a definition of a factor, I think we have missed an opportunity to educate their awareness. Yes, the word “factor” needs to be told (arbitrary), but the meaning or concept can be seen by students. Consider Figure 4. Having said nothing about factors, I would reveal this diagram line by line.

Teacher: “What do you notice?”

Teacher: “Why are some numbers circled?”

As more lines are revealed it is entirely possible for students to come to realise that the numbers circled are all the whole numbers that divide exactly into the “end” number. The students can provide the ‘meaning’ of “factor”. Leaving the teacher to say: “Yes, and we call these numbers ‘factors’”.

Figure 4

In Figure 5 we see a subset of the pattern in Figure 4.

Teacher: What do you notice?

Teacher: “Compare these rows to Figure 4”

Teacher: Why have these rows remained and the others have gone?

Again, as more lines are revealed it is entirely possible for students to come to see “prime-ness”. Leaving the teacher to say: “Yes, and we call these numbers ‘prime numbers’”.

Figure 5

In Figure 6, we see the same approach being used to explore square numbers.

Teacher: “What is special about these rows?”

Figure 6

Developing this use of awareness even further, figure 7 shows an interesting subset of the original set of rows.

Teacher: “Why have these rows been kept?”

Figure 7

As you will notice, Figure 7 focuses awareness on the numbers with exactly four factors. This can lead to an investigation into the number of numbers between 1 and 100 with exactly four factors.

## Nix the tricks

It can be tempting to show students shortcuts or tricks – especially the lower-attaining ones. Tricks such as: “KFC” (Keep, Flip, Change) – used when dividing by a fraction; FOIL (First, Outer, Inner, Last) for expanding brackets; adding zeros when multiplying by 10, cross multiplication, etc. Consider what has been said here about arbitrary and necessary. Tricks are arbitrary and may be forgotten or confused. Have you ever seen something along the lines of (-7) + (-2) = 9 and been told that “two negatives make a positive”? This is learning without awareness; received wisdom becomes a misconception. Tricks do nothing to educate students’ awareness and only fuel the belief that mathematics is a series of procedures to be memorised. See www.nixthetricks.com for a guide to avoiding the shortcuts that cut out mathematical concept development.

## Conclusion

Many students find learning mathematics hard but there is much we can do to support them. Consider the hierarchical and interrelated nature of mathematics when writing courses. Test for and teach (where required) the subordinate concepts before introducing anything new. Beware of teaching necessary as arbitrary. Spend time with colleagues discussing where the necessary content lies in your courses. We cannot control what students think but we can guide what they think about. Through careful task design, through the use of multiple representations, and by offering a variety of experiences to ‘uncover’ concepts and results for themselves, we should aspire to provide opportunities for students to use their own awareness to come to see what is necessary. Anyone who has ever figured out the ‘why’ of a seemingly mystifying mathematical concept will tell you that educating one’s own awareness is an experience in itself and that it is a satisfying one. I believe our job is to give students the chance to experience this satisfaction for themselves. Learning mathematics is hard, but learning mathematics deeply is a truly rewarding experience.

References

Claxton, G. (1984). Live and Learn. London: Harper and Row.

Eisner, E.W. (2002). The Arts and the Creation of the Mind. New Haven, Connecticut: Yale University Press, p.90.

Gattegno, C. (1987). The Science of Education – Part 1: Theoretical Considerations. New York: Educational Solutions Worldwide.

Geary, D. (1995). Reflections of evolution and culture in children’s cognition: Implications for mathematical development and instruction. American Psychologist, 50(1), pp.24-37.

Harris, JR. (1998). The Nurture Assumption: Why children turn out the way they do. USA: The Free Press.

Hewitt, D. (1999). Arbitrary and Necessary – Part 1: A way of viewing the mathematics curriculum. For the Learning of Mathematics, 19(3), pp.2-9.

Sweller, J., Ayres, P., & Kalyuga, S. (2011). Cognitive Load Theory. New York: Springer.

Tricot, A. & Sweller, J. (2014). Domain-specific knowledge and why teaching generic skills does not work. Educational Psychology Review, 26, pp.265-283.

Welsh, S. (2018). Thinking about learning mathematics. Scottish Mathematical Council, Journal 48, pp.32-49.

Wheeler, D. (2001). Mathematisation as a pedagogical tool. For the Learning of Mathematics, 21, pp.50-53.

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