Abstracts can't be avoided
In the eighth column in his series, Aarnout Brombacher explains the important difference between real-life problems and abstracts.
Throughout my columns this year I have shown how, if we regard problems not only as a reason for studying mathematics but also as a way of introducing children to mathematics (learning mathematics by solving problems), we can help children make greater sense of what they are doing as their confidence grows.
In the previous column I warned that while we should use problems to introduce mathematics, not all the problems will be “real life” in nature.
Addition, subtraction, multiplication and division are quite simply names for perfectly natural actions that can be provoked through a range of real-life problems. In this case, real-life problems mean those occurring in everyday experiences.
However there comes a time when the maths can no longer be introduced and/or made sense of by means of so-called real-life problems. Be aware that, when teachers try to create contrived or nonsensical “real-life” applications, they run the risk of causing confusion rather than understanding.
A very good example of this is the negative number.
There are no negative numbers in the “real world”. Some people may argue that being in overdraft at the bank is to have a negative bank balance and/or that temperatures below zero are negative temperatures. These are not negative numbers. These are simply numbers with a “minus” symbol in front of (or behind them—different banks use different conventions) to indicate a direction (below zero).
Some may remember a time when people had bank books and the teller recorded a customer’s bank balance using either a black pen if the person was in credit or a red pen if the person was in overdraft—leading to the expression “being in the red”.
Bank staff would type statements on typewriters with two-colour ribbons—black for credit and red for overdraft. The point of this story is that the bank of yesteryear used colours to indicate credit or overdraft—they did not use the minus sign.
Only with the advent of the modern computer and its early black-ink-only printers did the bank have to adopt a new convention—some used the minus sign and others parenthesis (R25) to denote R25 in overdraft.
What, then, are negative numbers?
Negative numbers are the invention of mathematicians—an invention made to solve a mathematical problem that they had created. Mathematicians are inquisitive; they investigate situations and ask strange questions. In the case of negative numbers, they asked the question: “If x = 2 solves the equation 5 + x = 7, x = 1 solves the equation 5 + x = 6 and x = 0 solves the equation 5 + x = 5, then what values of x will solve the equations: 5 + x = 4 and 5 + x = 3?” and so on.
Now the everyday person is unconcerned with this question—he/she may have responded with: “Who cares!” But, to the mathematician, this is a serious problem and solve it he/she must.
In the case of this particular series of questions, mathematicians solved the problem by inventing what we call negative numbers and said: 5 + (-1) = 4 and 5 + (-2) = 3, etc.
Two quick observations: one, many other numbers such as irrational and imaginary numbers have similar origins—they are the invention of mathematicians to solve problems that would otherwise have no solutions.
Two, do you notice that even the names given to these sets of numbers by mathematicians suggest that they are not “normal” or “common” or “everyday”—“negative” suggests problematic, “irrational” suggests mad and “imaginary” tells its own story. Even the names of these numbers make it clear that they are not ordinary, everyday objects.
If these numbers are not in the real world of everyday people, we make a mistake if we try to introduce them by means of so-called real-world analogies: in large part because the analogies will always fail.
It is not possible to find an analogy to explain why a negative number multiplied by a negative number gives a positive number as the answer and/or why subtracting a positive number is “the same” as adding the negative of the number. As much as real-world problems are useful to give meaning to and help children make sense of mathematics in the early years, and as much as learning mathematics through problem-solving is more effective than not, there comes a time when the nature of the problem has to change.
When working with increasingly abstract mathematical constructs of the later years such as: negative numbers, roots of equations, stationary points of functions and so on, the nature of the problem needs to change from so-called real-life to being more mathematical in nature.
Let me return to the negative numbers. Having invented negative numbers to solve problems of the form 5 + x = 3, mathematicians now had a new problem: how to calculate with these numbers? In particular, what does -2 + -3 equal? What does -2 x -3 equal? And what does 6 ÷ -3 equal?
Space does not allow a full development of the operations with negative numbers, so I will use one way of thinking about this.
Before inventing negative numbers we already knew that: 3 x 4 = 12.
And, having invented negative numbers, we now know that: 5 + (-2) = 3 and 7 + (-3) = 4.
So it follows that: (5 + (-2)) x (7 + (-3)) = 12
And by the distributive property of operations (also well established before the invention of negative numbers), it follows that:
5 x 7 + 5 x (-3) + (-2) x 7 + (-2) x (-3) = 12 that:
5 x 7 = 35 is well established
5 x (-3) = (-3) + (-3) + (-3) + (-3) + (-3) = (-15)
(-2) x 7 = 7 x (-2) = (-14)
So we have: 35 + (-15) + (-14) + (-2) x (-3) = 12
This simplifies to:
35 + (-29) + (-2) x (-3) = 12
6 + (-2) x (-3) = 12
And, of course, we also know that 6 + 6 = 12 and so, if 3 x 4 = 12 and the invention of negative numbers is not going to change that then 6 + (-2) x (-3) = 12 and it follows that (-2) x (-3) had better equal 6.
From here follows the observation that “multiplying a negative number by a negative number” had better result in a positive number or else all our previously held ideas are in trouble.
The gist of what I have tried to point out is that the answer to the question: “What happens when you multiply two negative numbers together?” is not to be found in the real world but rather in a logical analysis of a mathematical situation. The point is that there comes a time in a child’s mathematical development when the problems used to teach mathematical concepts must change from being real-world in nature to being more abstract or mathematical in nature.
The mistakes we make are both to try to teach early mathematics as abstract ideas when there are perfectly good real-life problems that give rise to the mathematics and then to try to teach more sophisticated mathematics using contrived real-world problems which do nothing more than confuse.
The skill of being a good teacher of mathematics is to know the difference between the different kinds of situations.
Aarnout Brombacher is a private maths consultant. For more information, go to www.brombacher.co.za