It’s complicated

I’m not a big TV person, but I can get down with some Wheel of Fortune. It’s pretty fun to watch and Vanna and Pat are seemingly ageless…which blows my mind.

But as much as I like the show, this puzzle from last night’s episode bothered me:

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They didn’t choose English, social studies, or computer science. No, they mindfully chose math (and physics) to associate with “complicated.” This is exactly the sort of damaging groupthink that fosters fear, anxiety, and stereotype threat of mathematics in my students (and society) and makes my job so dang hard. We can be better than this Wheel. C’mon now.

 

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Example analysis from DeltaMath

With so much problem-based learning happening this year, I’ve been mixing in plenty of algebra by example-esque problems. They work really well because they get kids to analyze math work on their own and then use it to solve a similar problem.

I’ve been writing some of these problems from scratch (horribly), but DeltaMath has shown up on the scene and helped out in unexpected ways. At the beginning of the year, I originally intended for DeltaMath to be a review of the problems/topics we learned in class. I assign them one big assignment that’s due the day before the next exam and they do it over time as we explore ideas in class.

That’s happening, yes, But what I’ve found is that the kids are also using the DeltaMath to learn the new ideas by means of the examples, not just review them. They’re independently leaning on their own analysis of DeltaMath examples to learn rather than on me to hand-hold them through examples in class. Independent learners, yay!!

The result is that someone regularly comes to class saying “…on DeltaMath I learned that…,” when presenting a problem we’re discussing in class – even when its an introductory problem on a topic. And, more often than not, this opens the door for a complete student-led class discussion around the problem.

For example, take this “Factor by Grouping Six Terms” problem that I assigned earlier in the year:

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When they click the “Show Example” on the top, a worked-out example appears:

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Students can even filter through different types of examples of the same problem by clicking “Next Example.”

 

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Factoring trinomials by first rewriting them

So back in December, I gave this problem to my students:

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To my surprise, it got a lot of traction with the kiddos. We spent the entire period talking about it. The idea was for them to see how rewriting a trinomial with four terms helps us to factor it. I’ve used this approach, often called the “CAB method” and used with a large “X” to organize the product and sum of the A and C terms, to factor trinomials for the last several years and I really like it for two reasons:

  • It doesn’t matter if a is greater than 1.
  • It naturally integrates factoring by grouping. Traditionally, grouping is learned after factoring trinomials. But with this approach, I teach grouping before we even see trinomials. Yeah:

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So, yeah, this is all great, but as I was explaining this approach to a colleague, she asked me why it works. It was in that moment that I realized that I had no idea.

Well, it turns out that later that day she went ahead and wrote up a proof of the method.

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I read a quote somewhere or heard someone say that the real usefulness of algebra is the ability it affords us to rewrite things in order to help reveal their underlying structure. This method surely epitomizes that idea.

 

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