# Constructive pathways

Posted on January 25, 2018

Let’s get back to math!   I finished reading the  article about “adapting perception, action and technology for mathematical reasoning” … Some fun snippets:

“People who know algebra show earlier and longer eye fixations to “×”s than “+”s in the context of math problems”

Your brain is trained to look at that first and harder.   I *know* I do this.   I screen out stuff.  I know students often don’t.   How to “teach” it, though?

We believe this slow cultural evolution of notation forms can be accelerated dramatically by modern computational technology.”

… and then they attempt to address the obvious objection, to wit:

They object, “You shouldn’t teach students that they can just move the 2 of y – 2 = 5 to the right side and change its sign. Students should go through the axiomatically justified steps of adding 2 to both sides of the equation, yielding y – 2 + 2 = 5 + 2, and then simplifying to y = 5 + 2.” To this objection, we respond that the teacher’s preferred solution is one justifiable transformation pathway, but mathematics is rich enough to permit multiple pathways to produce valid mathematical reasoning.

They say, regarding teaching students to move things from one side to the other:

Finally, it is conceptually evocative. As the −2 crosses the equal sign and becomes +2, learners experience viscerally a deep mathematical relation: if Y is equal to a one-to-one function of X, then X is equal to the inverse of that function applied to Y. Euclid’s second axiom and spatial transposition are coupled to different, deep mathematical insights.

What’s  missing is evidence that the concept *is* evoked.   Getting problems right doesn’t mean you’ve got the concept.   Now, I don’t know — maybe their instruction includes the talking part, the way we’re taught why we’re adding the same thing to both sides.

On the other hand, “inverse” is pretty abstract… more abstract than “do same thing to both sides.”    I think that’s pretty important for a bunch of learners.   I wonder if a version could be put together that applies that “let’s teach the visual-kinesthetic part” … in a more structured, concrete-to abstract way?   (It’s free — is it open?)  Something that targets things like knowing what the equals sign *means*?

Making things look and feel right can be applied to “do the same thing to both sides,” too.   The practice of plopping a line down from the equals sign and teaching   correct alignment helps w/ students who are still mucking with “what’s a like term?”

And I’m lurking at a webinar about the ACT CollegeReady program, which uses stuff from NROC, and lots of that stuff is OER.   Maybe *those* folks would be up for designing good open stuff that’s accessible and conceptual.   (More likely the EdReady folks — the open part…)  Imagine mixing geogebra and graspable…