For many in maths education, it is often felt necessary to teach pupils "how" to do something before teaching them "why" what they are doing works.
Broadly the reasons for doing this fall into two trains of thought:
- The "why" is considered significantly harder to understand than the "how", either because the level of maths is increased, or the idea is more intricate than its outcome. A prime example is calculating the area of a circle. The actual calculation is a relatively straightforward application of skills around substitution and squaring. To understand why the area of a circle is that calculation, one needs to look to integration, or at least some pretty tricky algebraic manipulation.
- It is felt that learning the "how" improves a pupil’s ability to understand the "why". Becoming "fluent" in a process or procedure can (it is suggested) offer insight into the underlying structure of an idea that makes it easier to see "why" that process/procedure is doing what it is doing, and make sense of the concept. A good example of this is column addition – pupils can learn to do this, with all its "carrying" (or to use a more correct term, exchange) and then later come to understand how this works with the place values of the relative columns.
Now, I have sympathy with both of these viewpoints. But I don’t subscribe to them.
The 'why' first in maths
I believe it is important to support pupils in making at least some sense of an idea as part of becoming fluent in it. And I also believe it is possible, most – if not all – of the time.
Let us take the example of the area of a circle.
Now I agree, I don’t think it enhances pupils’ knowledge to see the derivation of the formula before using it – I suspect for many pupils it would just appear as unnecessary algebra.
The sense I would want pupils to make here is linked back to an earlier understanding of area. By the point they encounter the area of a circle formula, pupils should have already encountered the following ideas:
- That area is the measure of 2D space contained inside a boundary (a perimeter).
- That squares of unit length are used as the unit of area.
- That, for many shapes, the number of squares is related to the product of two lengths measured from the shape, or the square of one length.
- That these relationships can be written algebraically using symbols.
The understanding, or "why", in this situation that I would hope for pupils to recognise, and that I would want to make explicit, is that this is exactly what is going on with the formula for the area of a circle.
Area of a circle
I would want pupils to understand that we are attempting to measure a 2D space, and that this 2D space is related to the length of the radius. So, a calculation involving the radius can tell us the area – which is irrational for whole number lengths because of the curve of the circle (meaning that no matter how small we make the squares, we will never have an exact number of squares).
This sort of approach needs to be well considered. The knowledge used here is built over years, being constantly reinforced and developed at each stage.
Done well, though, this is exactly how the schema for an idea is constructed. Learners slowly, over time, assimilate connected knowledge into their understanding of a concept, and recognise that the connections are an important part of this assimilation.
Being able to "see" the calculations of the area of a circle as a closely related idea to calculating the area of square should help this process.
As for the second point of view, it is undoubtedly true that an understanding of "why" something works can follow from a strong capacity to be able work with an idea in an inflexible way; having automaticity in "how" something works.
An understanding of "why" a process/procedure/idea works (or "what" it does) requires making sense of the underlying structure of that idea. Becoming very intimately aware of the surface aspects of this structure can provide the necessary "window" to allow a learner to look deeper, and to discern an idea’s more permanent and immutable properties.
Many people who hold this view exalt the approach for its “efficiency”, but for me it is patently inefficient. By definition, it means that each idea will have to be revisited multiple times through the same lens – at least once to get good at “doing” it, and then again later to make sense of it.
I also have a concern that, often, this sense-making is simply not supported at all. This leads to lots of pupils being able to (at least temporarily) perform a calculation or carry out a process, but only some are actually going to make sense of "why" the process actually relates to the underlying idea.
Widening the gap
It strikes me that one way this is definitely efficient is to breed inequity into our education system – those with the right mindset and support outside of lessons get to learn about the depth of the structure of an idea, while the more disadvantaged only get to learn about the surface of an idea and quite possibly never really get around to lasting learning at all.
Taking the example of column addition that I mentioned earlier, this procedure works because of two ideas – addition as collection of like objects, and the exchange of equivalent expressions.
The algorithm treats the value in each column as a separate object, and shows the addition of these objects as collecting them together.
For example, if calculating 342 + 255, the column addition algorithm would treat the ones, tens and hundreds of each number as separate objects, resulting in three hundreds, four tens and two ones in the first number, and two hundreds, five tens and five ones in the second.
Adding it up
The addition then simply adds one number to the next by adding the like objects together, i.e. the ones with the ones, the tens with the tens and the hundreds with the hundreds.
This is the same way of making sense of addition that is needed to understand why 3x + 2x = 5x, but that 3x + 2y cannot be written in a simpler form.
Occasionally, an exchange can be carried out that can make the algorithm run properly, for example if we add 348 and 255, the resulting 13 ones can be exchanged for one ten and three ones, which then gives (when all are collected together) 10 tens which can be exchanged for one hundred.
This is required when we want to the write the number using our standard place value system, as in this system each place can only accommodate one of the digits from 0 to 9.
Exactly the same idea allows 3x + 2y to be written in a simpler form only when we know something about the relationship between y and x. For example, if we know that y = 10x then we can add 3x + 2y, as it can be exchanged for 3x + 20x, which is equal to 23x.
Same but different
My point about this is that this understanding of addition as collecting of like objects, and this idea of exchanging one term for an equivalent term are not unique to column addition, and could easily be taught prior to it.
In fact, I would argue that they should be taught prior to it, as they are precisely the ideas that the process of column addition relies on.
Teaching these ideas explicitly, prior to the teaching of column addition (perhaps many weeks or months before) would allow for the "why" and "how" of column addition to come together; for pupils to appreciate "why" this process is doing the job of addition, while at the same time becoming fluent in "how" it works.
Not only that, but then every time addition comes up in the curriculum, and processes for it, these can be made sense of in exactly the same way. When I add surds, I can only add them if they have the same surd part. When I add fractions, I can only add them if they have the same denominator.
This is entirely because of the same reasoning that underlies the column addition algorithm – the idea of addition is the same in all cases.
Put it first
So for me, it is much more efficient to teach this idea, and give pupils as much support as they need in making sense of it, because once they understand this idea, all of the other "hows" and "whys" related to it become much more straightforward to teach.
Of course, to do this properly, what is needed is a very thought through and well-structured curriculum.
Careful attention needs to be given to the different ideas that appear throughout the maths curriculum, with expert practitioners who have a deep understanding of these concepts being able to carefully plot a journey that weaves them together in a coherent narrative both for other teachers and pupils alike.
Attempts at this have been made over the years, but perhaps it is time to revive this work so that all learners of mathematics are supported in developing a deep understanding of both the "hows" and the "whys" of school-level maths.
Peter Mattock is director of maths and numeracy at Brocklington College in Leicestershire. He tweets @MrMattock