I had the privilege of being in Tom Kendall’s classroom recently, and saw something wonderful.

A child in Year 7 said something that showed he’d understood something essential to arithmetic development, or advanced conceptualisation of arithmetic, or an important threshold concept… I’ve never been sure how to categorise it, but everyone who’s mathematically competent recognises the ability to do this: take an expression like:

**9 + 6 – 7 + 1 – 2**

And see it simultaneously as ‘nine add six subtract seven add one subtract two’ **and** ‘positive nine **add** positive six **add** negative seven **add** positive one **add** negative two.’

David Chart introduced me to the idea of procepts – ideas in mathematics that can simultaneously be thought of procedures or concepts, leading to children who manifest as ‘better mathematicians’ i.e. often observed as being able to perform more calculation in the classroom, more quickly, more easily and more accurately, as actually undertaking an *easier* form of calculation in their heads due to this ‘proceputal understanding.’

I think I can articulate what this is now better than I’ve ever been able to before:

**Case 1**

This is how the vast majority of people, and all children at first, will see the expression:

(+9) + (+6) **–** (**+7**) + (+1) **–** (**+2**)

**Case 2**

But people who are mathematically trained also see it as:

(+9) + (+6) **+** (**-7**) + (+1) **+** (**-2**)

The result is the same in both cases, and the ability to switch between the two at will means you can make choices that simplify arithmetic calculations you have to perform in your head.

There’s a beautiful symmetry here; the symbols **+** and **–** unfortunately represent two concepts each: the **operations **of addition and subtraction and the **position **of a number relative to zero (many other ways of conceptualising it, but suffice it to say, ‘sign,’) so each case either:

**(1) Holds constant the sign, and allows the operation to vary**

or

**(2) Holds constant the operation, and allows the sign to vary**

Either you (1) see the expression as the addition and subtraction of positive numbers only, or you see it as the addition only of positive and negative numbers.

**Method**

I’ve always articulated this idea to pupils, but never had any sense of whether any of them were seeing arithmetic expressions this way, and never really known how to properly communicate the idea.

Perhaps it’s because it’s an idea that sits within an uncomfortable space which cannot be explicitly or directly taught/communicated, but what should we do about this? I’m wondering whether Tom has the answer.

As Michael Fordham aptly points out here, people differ in their thoughts. Curiosity is another good example of something that cannot be directly taught; while some believe that we can directly teach curiosity by having ‘lessons in curiosity’ and measuring pupils progress in ‘being curious,’ most believe that we must facilitate curiosity, but differ in how. Some feel that the solution is to ask a big interesting question, and then leave children alone with books and computers to go answer it, sating their *natural *curiosity. Others believe we need to emerge them in fantasy, a la the Mantle of the Expert approach. Others believe that these approaches limit children to their current narrow experience, and so the solution is to do lots of good teaching of the stuff we *can* teach explicitly, believing we first need to know something about something before we can express curiosity – and the more we know about it the more curious we are likely to become.

This same problem strikes at the heart of mathematical problem solving, approaches to which I’ve felt were lacking since forever. The typical approach is to say ‘we need to teach problem solving,’ then turn to Polya or his anaemic derivatives and create step by step approaches for children to follow. For the longest time I’ve had little faith in this approach, partly because I see it everywhere yet see it making no difference, but mostly because the words of Daisy Christodoulou still ring in my head **“We underestimate our own knowledge, and overestimate theirs;”** consequently whenever I’ve approached a problem solving task I’ve viewed it not just in terms of ‘what steps am I undertaking,’ but ‘what mathematical knowledge am I bringing to bear,’ and ‘what previous problem solutions am I adapting here,’ as suggested by Willingham. in other words, if we aren’t teaching the mathematical content well, and we’re not showing pupils solutions to a wide variety of problem types with overlapping features, then we aren’t going to develop their problem solving ability.

Further to this, I suspect that the ability to problem solve is rooted in the idea of ‘noticing’ expressed here.

As White (1967: 69) puts it:

We can ask someone how he [sic] `would’ discover or cure, but not how he

`would’ notice, although it is as legitimate to ask how he `did’ notice as it is to ask how he `did’ discover or cure. For the former `how’ question asks for the method, but the latter for the opportunity. Although appropriate schooling and practice can put us in a condition to notice what we used to miss, people cannot be taught nor can they learn how to notice, as they can be taught or can learn how to detect. Noticing, unlike solving, is not the exercise of a skill.

When we ask people how they solved a problem, they will at some point tell us ‘the things they tried,’ and ** what **they eventually noticed following all this trying, but will not mention

*they noticed it.*

**how**In this sense, problem solving might be an **emergent property **of other teachings, in the same way that Fordham suspects ‘curiosity’ is an emergent property of our existing knowledge, not a skill to be honed or developed in isolation.

**Back to arithmetic**

Tom’s lesson was about how to evaluate multi-term expressions, and I couldn’t help but wonder if his approach to this had increased the probability that any given child would **notice** these two ways of seeing the expression, as described above (whether or not the teacher also attempts to point to it verbally.)

So, we have:

**9 + 6 – 7 + 1 – 2**

And two ways of evaluating it.

**Method 1 – **Go from left to right.

**9 + 6** – 7 + 1 – 2

**15 – 7** + 1 – 2

**8 + 1** – 2

** 9 – 2**

**7**

This method focuses the mind on the early concept of ‘adding and subtracting positives.’ (I also like that there’s no = sign here either)

**Method 2** – Sum the positives and negatives independently, then resolve

(Tom used column addition for all stages of this, but that’s a little harder to set out on a computer)

(+ves) 9 + 6 + 1 = **16**

(-ves) 7 + 2 = **9**

+ves subtract -ves:

**16** – **9** = **7**

That second method hints at Case 2 described above; the fact that you can treat all those subtractions as adding together negative numbers, and that you can treat addition of negatives as subtraction.

I don’t know yet if this is helping those children notice this important concept, or if the child who spoke up was just ‘one of the bright ones’ who learns more in spite of us than because of us. I also haven’t yet had time to think up a way, or search to see if someone else has, of determining whether someone can see the expressions each way. But just logically analysing the methodology… there’s certainly something here. As I come to the end I wonder if maybe it needs a further activity to be introduced, just literally asking pupils for ‘the other way’ of writing:

(+9) + (+6) – (+7) + (+1) – (+2)

Or ‘the other way’ of writing:

(+9) + (+6) + (-7) + (+1) + (-2)

I also wonder, on reflection, if that *would* be an explicit/direct method of teaching the idea…

Either way, we’re not yet getting enough pupil to this realisation/conceptualisation. This is something we must do.

In my school we are in our first year using the ark ‘Mathematics Mastery’ scheme. There is a whole lot of this kind of stuff in there! I think they have taken some of it from Singapore, where a lot of this stuff (that we often assume is intuitive but that most students don’t automatically just ‘get’) is explicitly planned for in the curriculum.