Wednesday, June 27, 2018

Why is subtraction so hard?

Subtraction presents various problems to learners of mathematics, not least with the mechanics of hand calculating the result of subtracting one multi-digit number from another. But that's not the kind of difficulty I'm interested in: I'm more interested here in the difficulties that arise when computations involving several additions and subtractions of integers are involved, or at the slightly more advanced level where algebraic computations involving several additions and subtractions are required. I'm going to take a look at what I think lies underneath the difficulty.

The procedure required to deal with subtraction starts off deceptively simple.

Addition and subtraction are of the same priority, and so a calculation involving a string of additions and subtractions is calculated from right to left: \[ 1-2+3-4-5=-1+3-4-5=2-4-5=-2-5=-7 \] Slightly more complicated, we may have things inside parentheses which consequently have to be evaluated before the result is combined with the rest: \[ 2-(3-4) = 2-(-1) = 3 \] But already we find lots of mistakes being made. It isn't uncommon for the first calculation to turn into some variation on \[ 1-2+3-4-5=1-2+3-(-1)=-1+2=1 \] or for the second to go down the route of \[ 2-(3-4)=2-3-4=-1-4=-5 \]

The root of the problem

In one sense, the root of the problem is combinatorial. There are lots of different ways of combining the various numbers in the calculations, but they don't all give the same answer. There are more ways of getting the wrong answer than the right one.

But that doesn't really get to the bottom of it. Why do learners combine things in the wrong way when they've been given careful explanation and instruction of the right way? (And it's not just novices: errors along these lines persist with undergraduates!)

I think that the real problem is that subtraction is a horrible operation, in many ways.

The horrors of subtraction

  • First, we learn to add together positive integers, or natural numbers. The result of adding together two natural numbers is always a natural number: addition is a well defined binary operation on natural numbers.

    But subtraction isn't.

    When we subtract one natural number from another, the result can be negative. Suddenly a whole new can of worms is opened up, because we need to deal with the arithmetic of negative numbers, which, as we all know, is a paradise of opportunity for sign errors.

  • Next, subtraction isn't commutative. This doesn't seem to be subtle, but let's take a quick look: \[ 2-1=1 \qquad \text{ but } \qquad 1-2 =-1 \] The answers are different, but only by being of opposite signs; and to rub salt in the wound, the second may well be done by subtracting \(1\) from \(2\) and changing the sign. Losing a minus sign is easy for all of us.
  • Even more horrible, subtraction is not associative: \[ (1-2)-3 \neq 1-(2-3) \]

    But we get very used to using associativity and commutativity without even noticing it when adding. It's much harder to remember not to do it when it's inappropriate, once you've become thoroughly accustomed to doing it.

  • And lastly, to cope with the problem we end up with a plethora of rules about combinations of signs. More to remember, often just regarded as more rules to learn. I cannot express in words the horror I feel when I hear students muttering (or even writing out) "FOIL" when doing algebraic calculations; I haven't yet heard "KFC" from my students when working with fractions, but I've seen it on the interwebs often enough to know it's only a matter of time.
Now, there are many rules in algebra and arithmetic, but a relatively small number of principles (axioms) from which they all follow.

Unfortunately, you can't really teach arithmetic to five-year-olds by starting off with the axioms of a field.

But maybe there comes a time to revisit the collection of rules and see where they come from. At least, I always find it illuminating to go back to elementary stuff and see what new insight I might have based on further study.

Avoiding the issue

So here's my approach to a better (I have many opinions, and very few of them are humble) view of the arithmetic and algebra of subtraction.

Uninvent subtraction.

OK, so I haven't gone back in time and relived my education without the notion. What I mean is that I look back and say that after all this time I now see that there's a better way to think of subtraction.

First, any number has \(a\) an additive inverse \(-a\): a number we add to it to get \(0\). Then if \(a\) and \(b\) are two numbers, I say that \[ a-b \] is a (somewhat undesirable, but unavoidable) notation for \[ a+(-b) \] which is the sum of \(a\) and \(-b\), the additive inverse of \(b\).

And at this point I really, really wish that there were a conventional typographical distinction between \(-\), the binary subtraction operator, and \(-\), the unary additive inverse operator. Oh well.

This has the immediate consequence that we can write our sums slightly differently: \[ 1-2+3-4-5 = 1+ (-2) + 3 +(-4) + (-5) \] and these quantities can now be rearranged at will, since addition is associative and commutative.

In the second example, we have \[ 2-(3-4) = 2 + (-(3-4)) \] And what do we add to \(3-4\) to get \(0\)? we add \(4-3=1\), so \(-(3-4)=1\) and \[ 2-(3-4) = 2 + 1 = 3 \]

Of course, it's still necessary to be able to add together combinations of positive and negative numbers: there is no free lunch here. But it's a way of thinking about the computation that reduces the temptation/opportunity to make mistakes, so maybe it's a slightly cheaper lunch.

One consequence is that if I see \[ x+5=7 \] I don't think 'subtract \(5\) from each side', I think 'add \(-5\) to each side'.

I find it a useful way to think about what's going on.

I try to stress this with my students, but with mixed success. And just about everybody who's teaching early undergraduate material will surely be doing something like this in the students' first encounter with abstract algebra and axiomatic systems.

My general impression is that as long as they're doing a set problem of the 'show from the axioms of a field' type, most students can be persuaded to work this way.

But I find that as soon as that context is gone and they're back in the old familiar territory of doing algebra or arithmetic, for most it's also back to the old familiar way of proceeding. And for quite a few this has just too many opportunities for the old familiar way of going wrong.

But also

I have very similar thoughts about division, the reconstruction of which I leave as an exercise for the sufficiently motivated reader.

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