I am having a few issues with our new resource’s attempt at actualizing the recommendations made by the NCTM and the WNCP in order to philosophically align itself with current theories of learning. I deeply appreciate that research supports the idea that students need to construct their own knowledge for deep understanding to occur. I also firmly believe that this construction needs to be made relevant to the students for them to truly engage with the topic at hand.

However, when both the relevance and the constructing are done in primarily artificial ways, I just don’t see the point. For example, to introduce the idea of polynomials, alge-tiles are highly relied upon. The tiles do provide a nice visual for students, and can be easily manipulated to illustrate concepts. To get students using the tiles, the resource uses “games” to get students attention. Well, I don’t know about you, but “let’s pull a handful of alge-tiles out of a bag and then draw both their pictorial and symbolic representation, whee!!” is about the lamest game I’ve heard of yet. This is not relevance. This isn’t even fun past the first throw of the tiles. (Even then it’s likely only fun because you’re throwing something.)

Now, don’t get me wrong, I have no idea how to introduce like terms in an authentic way that would be both relevant and motivating at the grade 9 level. Not yet anyway. (When I do, I have a sneaking suspicion I might be in the business of making lots of money.) But, this is not it. There is nothing to “discover” when the text outlines what the different tiles represent and how to use them. As of yet I have not encountered an alge-tile in my life outside trying to teach students polynomials and cannot imagine when I would pull them out to pictorially represent a situation. There has been no “need” created here, there are no other options that might call upon some problem solving, there is simply compliance wrapped up in a “fun” package.

I have issues with this because I think it misses all the critical pieces that make new theories of learning and engagement so powerful. It’s false advertising. It trivializes what is meant by “deep learning” for both teachers and students. As someone who is desperately trying to see how math could fundamentally look different in my classroom, this fake fun is doubly dangerous. It is both toxic in the staffroom and borderline insulting for both myself and my students. This gives the critics of “new math” the ammunition they are looking for – that the new math has no standards. It also doesn’t help the supporters of math reform envision math in truly new ways. I could have supplemented my old lectures with the introduction of some alge-tiles(and sometimes I did!). I could have had “game” days on Fridays all along. I would, however, still be teaching stuff.

Dare we ask the question “Why do students need to learn about polynomials?” Perhaps if the math being taught was relevant to student lives’, it would be easier to make it engaging. The only math I do in my life relates to money and renovating my house. Haven’t found the limit of a function with absolute values since University Calculus…

Oh yes we may. But I thought it might get a little out of hand in one post…

What really bothers me about the usual introductions to algebra tiles in reform curricula is the way authors are trying to charm or distract students into using them. It feels patronizing and condescending to me, no matter how solid the research is on the effectiveness of using manipulatives to teach math.

The phony motivations seem doubly (or triply) stupid when you could be capturing their attention by giving them the real stuff — historical context on how these tools were used by leading mathematicians and scholars of earlier times to solve otherwise unsolvable problems.

Give students the real historical context — i.e., here is how al-Khwarizmi used them in his groundbreaking 9th-century work on algebraic methods. He used the precursors to algebra tiles because they gave him a geometrical representation that enabled him to solve quadratic equations. He quite literally “completed the square.”

Seeing this in context is engaging for both students and adults.

And there are some fantastic materials on this available for free on the web. The Center for South Asian and Middle Eastern Studies at the University of Illinois has several amazing lesson plans and supporting materials on this online at http://www.csames.illinois.edu/outreach/mideast/math/ . I adapted their Completing the Square materials (lesson plan + worksheet) to use with my students to give them a grounding in a geometric representation of the process of completing the square. They LOVED the fact that there was a historical basis for algebra tiles and they were excited to learn about it. And that motivated them to want to see how the darned things worked.

Another thing I have noticed is that modern mathematics educators run into problems teaching with algebra tiles because they (the tiles) are not good at handling subtraction (negative values) elegantly. But again, this is a way to use a historical problem/fact as a bridge to modern learning. I did a mini-lecture for my students about how these early Arabic mathematicians had trouble with negative values too, and in fact, they dismissed all the negative solutions to quadratics as being “impossible” or “absurd.” And they weren’t the only ones. In fact, it wasn’t until the Renaissance that Western mathematicians figured out how to deal with negative numbers and with (gasp) square roots of negative numbers.

Meanwhile, Indian mathematicians had worked out ways to deal with negative numbers long before the West ever figured it out.

This limitation gave me an opening to show my students how they could move from the very literal geometric representation of an area model for solving quadratics to a more abstract, generalized form for an area model (CPM calls this “using generic rectangles”) that can better accommodate negative numbers. The limitation in one representation motivates the use of another, more abstract representation. And that takes them deeper and deeper into the algebra.

The point is that the limitations of different representational systems do not have to be a

for our students. They can be opportunities to help students understand that the mathematical work they are doing is not trivial. Quite to the contrary, it took humankind thousands of years’ worth of effort by the best mathematical minds in multiple civilizations, working long and hard and together on these challenges.problemI think the very limitations of certain educational tools can be their own greatest strengths, if we are willing to be honest, open, and direct about those limitations and putting them into context. Then those moments can become opportunities to increase curiosity about and appreciation for mathematics.

Elizabeth (aka @cheesemonkeysf on Twitter)