In this series, we have been looking at little machines that do things like “add 2”, and we’ve been chaining such machines together by connecting the output of one to the input of the next. We’ve been exploring the behavior of such chains of machines. One of my goals has been to see if material about simple machines can be a useful curriculum element in introducing kids to algebra. The shift from arithmetic to algebra is typically fraught with difficulty for kids, and I think it useful to identify where the traditional approach of introducing algebra makes these difficulties worse unnecessarily.
My focus is on middle school math, where I am assuming a fairly traditional elementary school treatment. Other people, for good reasons, are looking at ways to articulate algebraic thinking in elementary school, underlining it and remarking on it when it shows up naturally in the lower grades. For example, kids may have been asked to look for pairs of numbers that add up to 20, and some may have noticed that if they have such a pair, and make one of the numbers in the pair one higher, and the other number in the pair one lower, they have a new pair. This business of encouraging students in the lower grades to make generalizations, and then think about those generalizations (i.e. think of them as objects you can hold up to the light), seems important to me.
The approach in this blog series is different, but addresses a similar goal: strengthen a students’ ability to look at the arithmetic operations as something not necessarily to be done right now, but something to be done later, once other numbers are available too. And in the mean time, those arithmetic operations can be held up to the light and looked at and examined in terms of their relationship to other arithmetic operations. Notions like equivalence, simplification, generalization, distinguishing between patterns that sometimes hold and patterns that always hold, those notions can be pursued and played with and strengthened independently of the normal language of expressions and order of operations and parentheses. We can introduce the thinking involved in solving equations without even having to introduce variables.
In the prior post, we looked at some (still simple) operators that go beyond our initial set of “add a particular number”, “subtract a particular number”, “multiply by a particular number” and “divide by a particular number”. One of these, “remainder after dividing by 2”, had us become aware of the scope of each operator: something about the kind of numbers it accepts on its input, and something about the kind of numbers it produces on its output. The latter is important in so far as its output is connected to the input of another operator – but then, this is exactly what we have been doing. Our earlier notion of railroad cars that will always fit together is a bit too broad. So part of the determination whether two operators can be chained together is looking at the “shape” of the connectors at each end of the box and making sure they are compatible. For example, a box that will always produce a whole number on its output will connect fine with a box that can handle any number, whole or or not, positive or not, on its input. Conversely, a box that produces an arbitrary (real) number on its output may not fit well with another box that requires a whole number on its input. This, by the way, will also serve as an incentive to construct some operators (e.g. some kind of rounding operator, an absolute-value operator) that can act as adaptors between two other boxes that otherwise wouldn’t fit together.
In that last post, we also introduced an operator “100 minus’, which takes the number coming in and subtracts it from 100. We then asked if you could figure out what the operation is that undoes the effect. It turns out that this operator is its own inverse.
In the figure above, on the left is shown two copies of the “100 -” operators, with the output of one feeding the input of the other. If 20 goes in, 80 comes out; and reversely, if 80 goes in, 20 comes out. The situation is entirely symmetric. A similar situation applies for the “60÷” operator: if 20 goes in, 3 comes out; and if 3 goes in, 20 comes out. And these are not because “20” is somehow special, somehow lucky. Instead, this represents a general symmetry between the two situations: if you think of the “100-” operator as computing the change from 100 based on the price, it can equally be seen as an operator that reconstructs the price from the change. Price plus change together make 100.
In the figure on the right, we grapple with the inverse operator of “multiply by itself”. For now, let’s restrict ourselves to positive numbers at the input of the “multiply by itself” box. At this point, that may seem like a rather arbitrary restriction. The key thing for us here is that the behavior of the black box currently labeled “undo multiply by itself” is fully known – even if we didn’t have the slightest clue as to how to build one of these. But that’s a black box for ya, when you know the behavior without being certain as to what is inside.
My daughter, when she was around 11, would steadfastly deflect any of my questions of “what do you think is inside that might explain this behavior?” with “there is a really smart elf inside, really fast, too, and invisible!” A fast elf, for sure, could make some guesses and then check those guesses with the original “multiply by itself” box, see how close it got, and then modify the guess accordingly and repeat till satisfied.
Even if we were never able to decide how to build a box that undoes “multiply by itself”, we could still do something that mathematicians have done for ages, and that is: bluff. “We don’t know what goes in the box? Well, let’s give it a name, and a symbol, anyway, and then go on from there.” A nice name, alone, can do a lot to tame the beast. When students are at the receiving end of this kind of bluffing, they usually don’t know that this is what is going on, and they feel intimidated. “Ah, square root, I’m somehow supposed to know what square root is.” It would feel a whole lot more accessible if the teacher simply said: “people made up a name for the undo-operator of ‘multiply by itself’, and they called it ‘square root’. Giving it a name doesn’t really solve anything or help us in any direct way, but at least it is concise.
See if, from a purely psychological standpoint, the figure on the right doesn’t strike you as more authoritative, showing you more of a solution, is somehow confronting you more with something you are supposed to already know – even though the figure on the right does nothing more than throw a technical-looking vocabulary at the situation shown on the left. And introducing vocabulary and notation for something we don’t know much about, that sounds much more sophisticated than talking about fast elves.