In school mathematics, we spend a lot of time making math very formal, very sophisticated, and very unreachable for most people because it doesn’t feel real. Perhaps more time should be spent playing with math, exploring math, and making math real for everyone.
In ancient times, people often did very sophisticated math problems, but they did so without algebra and many of the other high-powered symbols we use.
Here is an example of such a problem. It is from ancient Babylon about 8000 years ago:
I have a square with area 1000. It is equal in area to the total of two other squares, such that the side of one of the squares is ten less than 2/3 the side of the other. Find the lengths of the sides of the other two squares.
Solving this problem using algebraic means is somewhat involved. But what about doing it without algebra? I started “playing around” with square numbers, and building a small table:
(20)2 = 400
(25)2 = 625
(30)2 = 900
Then I thought…
⅔ of 30 is 20 (⅔ of a side…)
20 – 10 is 10 (ten less than ⅔ of that side…)
I had my solution. 30 and 10 are the sides of the two squares, whose total area (900 + 100) is 1000.
While algebra is a powerful tool which adds to our ability to solve problems, it has more often become a gatekeeper in society, limiting the future options for those who cannot master this tool. For them, math is a barrier. It is something to avoid, However, I found the solution by “playing” with square numbers which made it very real for me. It was pretty quick and easy. This solution is within the reach of most people.
Reify means to make real. School mathematics is not real for students, hence, “When are we ever going to use this?” My grandson is in the fourth grade. In a grocery store, I posed a question to him that was certainly within his ability to solve: “If this box of cookies costs $4, how much would 7 boxes cost?” He was perplexed because it was math in real life, rather than in his classroom. After he got over the initial shock, he was able to do the problem. He saw this as a “school problem” in a non-school context. The situation made sense to him – he knew that he wanted $4 seven times (or 4×7). Why shouldn’t we be as comfortable doing basic math as we are, for instance, reading? Reading the box, the labels, the signs in the store felt “real” and not like a “school problem.” How can we do more with math in real life so that the use of math outside of school becomes “real,” aka reified?
In our research, we’ve found that a lot of cognitive development must take place before true understanding is possible. Adding ‘reifying days’ in with the regular curriculum would help make math real and understandable.
“The Gains and Pitfalls of Reification – The Case of Algebra”
Anna Sfard and Liora Linchevski, 1994