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Tuesday, December 8, 2009

Rigorous Math Education

Today I gave my 8th graders a summative exam of the algebra topics we've been covering. This mainly included solving for x as well as solving for y to set up a y=mx+b linear equation.

They have not been terribly attentive, to say the least, to the activities we do in class, so it was with some level of trepidation that I decided to give them the exam. I am not a believer it testing for testing sake, nor for holding grades or exams over my students' heads as threats. But I have to admit that my student culturally go there without me doing anything.

Anyway, many of them showed serious misunderstandings of variables and constants, of fractions and of the algebra tiles we've been using. I saw in front of me the many various misunderstandings of algebra that are interfering with their deeper math education. My admin as well as the parents will quickly tell me that it is my fault: that I am not telling them "how to do it". I will say that it is not the "how" that is holding them up, but the "why". They'll say that I must not be explaining the "why"well enough. I will have to consider this carefully, because every teacher should be humble enough to realize that they are not walking on water when it comes to presenting material. But down inside I know I follow all the best practices associated with conceptual learning, and still my students resist (at times, reject) it.

So it all gets me thinking of the intersection between mass culture and education. If my students and families are hell bent on the "how" (because they learned it that way and because that is the "only" way math should be taught), but I am putting my foot down and insisting on justification and conceptual knowledge, then we find ourselves irreconciably miscommunicating.

It gets me thinking, too, about at what level we, as American educators, can hope to really teaching a rigorously conceptual math program along the level of Singapore Math. This is what I wrote in an earlier post:

When we look at something like math and we don't see what we remember from school, we wonder about its validity. Did it really get "watered down" like all the news reports have been saying. Are modern kids being asked to do anything like the rigorous curriculum we used to work through?   

One precept of modern math instruction is that conceptual understanding not only supports math methods, but also should be the precursor much in the way understanding a cat from a holistic standpoint is the important first step before understanding it from a genetic perspective.

I want to believe that I will be able to teach the "real stuff" and not get easily blamed for the "hard" nature of learning math. This evening, though, I can't help but think it might be a fantasy.

When is the next plane to Singapore leaving?



DWees said...

I'm teaching algebra to my 9th graders and have discovered that I need to keep better track of their progress.

I'm using two websites to do this, one is called and the other is

I actually started using the 2nd one recently and it really seems to help the students understand what they are doing. It's very recent so there are occasionally bugs, but the designers of the system are very responsive as there are not many schools using it yet. Maybe 100 schools total.

What is really useful is that I get way more information about what my students know how to do and what they are struggling with before the test/quiz. I've also been using this to modify my instruction as I go.

Next step for me is to include clickers into the equation (individual whiteboards can do the same for a lot less money) so I get feedback about what the students understand immediately.

Anonymous said...

As someone who's fought the battle of "what they can explain is more important than what they can do" and lost, I can see the whole stack of minds we have to change before tackling the core challenge of changing how we teach.

This is frustrating on so many levels, not the least being when we stumble as teachers the immediate response from all corners is "you're doing it wrong." It's easy to see you're doing something different, and when there's trouble it's easy to throw different under the bus.

Let me offer my voice as a counter balance to all those shouting for "how it's always been done." Focusing on conceptual understanding if worthwhile. One of the challenges is that "simple" things like vocabulary become critical, where on a sheet of 40 identical "graph y =3x - 5" problems, the pattern is all that matters.

That said, this is the diagram/tool that always saved this lesson for me. This version is done in Geogebra, but I've written ones in Scratch or Flash in the past. Just fiddling those numbers around for 5 minutes did more to communicate the core concepts than all my time at the board ever did.

ERKO said...

I appreciate both of these comments.

I checked out the site, but I haven't had time to really understand it. My initial concern would be that I wouldn't be able to see student work leading to their answer, which is of great interest to me.

The second comment, about vocabulary and conceptual understanding vs. graphing a bunch of problems is something I am currently grappling with. I spend so much energy on the conceptual understanding and try to limit the practice to what I consider enough, not overkill. And yet, for some learners, the repeated practice to finally discern the pattern may be the way to reach them more effectively. I am just not as sure about this as I used to. Algebra is about patterns, to be sure, but it is really about a deeper, generalized understanding of these patterns.

THanks for these comments, both of which help me consider where I am going with all of this.