Dear Dr. Shaw,
Thank you for your talk "The Mozart Effect," which I heard last week at MSRI.
I was intrigued to hear you say that proportional reasoning is hard to teach. As soon as I began thinking about it, I got a sense of the difficulty. There are a lot of pieces to proportion! In your lemonade example, we start with four quantities,
sugar1, lemon1, sugar2, lemon2.
We are then asked to consider the sweetness that results from mixing them
and determine what the appropriate relation is:
sweetness1 [<, =, or >] sweetness2
Already, we are getting dangerously close to the "magic number" 7 (the number of elements that it's possible to keep in one's mind at a time, see Miller, 1956) -- and we haven't even enumerated the intermediate relations we have to keep track of:
sugar1/lemon1 = sweetness1
sugar2/lemon2 = sweetness2
If we are sophisticated enough to conceive of sweetness as a single quantity (a fraction), and ask the question "what is the sweetness of the first mixture and what is the sweetness of the second mixture?" then we can group the original elements and forget about them as distinct:
(sugar1, lemon1) :: sweetness1
(sugar2, lemon2) :: sweetness2
which gets us down to a more manageable number.
But this is only possible if we're sophisticated at using fractions; what if we aren't?
In some cases, there are tricks which help. In your lemonade example, sugar2 was twice as much as sugar1. "Twice as much" can be thought of as the fraction 2/1, of course, but it can also be understood without fractions, as simply 2. The relation between lemon1 and lemon2 was not so easy to characterize, though it was easy to see that lemon2 was not as much as twice lemon1. But then, what are you supposed to do with that?
sugar2/sugar1 = 2
lemon2/lemon1 < 2
(<2) < (2)
therefore ... ?
To get from there to the answer, we need to consider yet more relations.
Compared to other mathematical operations we learn to do (comparing two quantities, finding the difference between two quantities, etc.), proportional reasoning is very complicated!
The usual tack in dealing with complexity is to decompose it. How can a proportion be decomposed? If we have the sophistication to turn each proportion into a quantity (a fraction), then we can simplify the problem into comparing two quantities. But what if we have had no experience (or inadequate experience) with fractions?
One approach would be to replace "ratio" or "fraction" with "slope." Slope is a quantity which can be directly perceived, compared, without resort to fractions.
Here, we can see
as lemon increases, sweetness decreases
as sugar increases, sweetness increases
We can also see
that if both lemon and sugar double, the sweetness stays the same. Using this tool, we can view the solution to the lemonade problem directly:
When I first drew this figure, I was stunned; I was expecting something like this:
I had been wrong in my assumption about which mixture was sweeter! I had been thinking something along the lines of "there's LESS _____ in the second case ... so it's LESS sweet." Now that I put words on it, it's obviously wrong, but at the time, I was not verbalizing these thoughts.
The utility of the visualization extends beyond merely being able to give an answer to the question given; it's possible to see other related answers
"How much sugar would have to be added in the first case to make it as sweet as in the second case?" (a)
"How much less sugar would have been needed in the second case to make it as sweet as the first case?" (b)
Also, if these drawings are done on graph paper, other interesting points can be considered:
A toy for playing with slope could be fashioned with a grid pegboard, pegs, and string (or, less tangible and thus less effective: made into a computer program).
BTW, my first slope experiment almost led me to abandon the idea. I picked the two proportions
and drew the diagram:
I thought: "this is no good; you can hardly see that they are different!"
Only when I did the math
3:5 :: 24:40
5:8 :: 25:40
did I realize that they WERE almost the same.
Would the slope idea help people with proportional reasoning? That would depend on how well they could apply it to problems appearing in various forms. It's easy to recognize when we are being asked to compare two quantities; it is less obvious to recognize when we are being asked to compare two proportions/ratios/fractions/slopes.
What we need to be able to do is transform one view of the problem into another.
There's a spatial reasoning test that involves identifying a certain shape when it has been rotated 90 or 180 degrees. Presumably, people who have trouble with this would have less trouble if the shape had been rotated only 5 or 10 degrees. Could they increase their range of mental rotation a few degrees at a time?
Similarly, are there closer and further analogues of slope? Yes: angle is very closely related; fractions are less close. Perhaps exercises which increase the stretch of analogy gradually could be used to build familiarity and facility with proportional reasoning through the idea of slope.