In my post on the Alberta Math Dialogue, in which a group of Alberta university professors got together and offered their critique of our current curriculum, I mentioned that I heard some things that really offended me. Two of them aren’t even worth elaborating on (and for the record, were not uttered by a university math professor). Two of them had to do with math and I addressed one in a previous post, in which I discussed how I sometimes wonder if university math professors truly understand who we teach in K-12 math. It has some relevance here. It’s ridiculously long. Perhaps you should read it first. The second comment that offended me had to do with the use of concrete (hands on materials) and pictorial (drawing) representations.
In her critique of the Junior High curriculum, Christina Anton from Grant MacEwan University, talked about visiting a junior high math classroom and seeing the students colouring and using fabrics. From the context she described (polynomials), I suspect she saw a frugal teacher who had made algebra tiles out of old fabric, rather than spending sparse school money on a commercial set. Because she got a good laugh out of this, Christina kept coming back to it, and it became the running joke of the day. The Edmonton Journal even published the joke.
It may come as a surprise to you, as it did to me, but Grade 9 students here are required to use sticks, tiles, swatches of cloth and colouring to do complex math operations such as multiplying polynomials with monomials.
Here’s the thing, though. It’s not funny. After her session, I offered to show her how algebra tiles connect to base 10 blocks and make a nice bridge to symbolic algebra in grade 10. Christina dismissed me, and stated emphatically that concrete and pictorial representations are not real mathematics and have no place in the junior high curriculum. Only symbolic representations (the x’s and y’s and so on) are real mathematics and they are the only things that should be taught.
Such statements show the true naiveté of (some, not all) mathematics professors about who we teach in K-12 schools, and how those students learn. Concrete and pictorial representations help students make the jump to symbolic. For many students, they help form a critical bridge to understanding.
It is true that many of our students can make the jump to symbolic representations fairly quickly. But even those students still benefit from the bridge that concrete and pictorial representations make to that symbolic notation. We could probably even leave out the concrete and pictorial for our strongest students and they would be able to replicate the algebra without too much difficulty. The manipulatives will deepen their understanding, though.
For our visual and tactile learners, though, these concrete and pictorial representations are absolutely critical pieces. That’s no joke.
Would I force a student who can do it symbolically to draw it for me on an assignment or test? No. Would I let a student who can’t do it symbolically show me concretely or pictorially instead? Certainly. Would I expect a student bound for university calculus to be able to do it symbolically? Absolutely.
Do we still like what Singapore is doing? To those who speak derisively about concrete and pictorial representations, I leave you with the Singapore Bar Model. (Sorry, that was the best video I could find quickly with a google search.) The Singapore Bar Model creates lovely pictorial representations that help students make the bridge to symbolic notation. These representations work, even for high school algebra.