Hidden Treasure in Math Class

A course design can facilitate learning, and a course design can hinder learning.  I suspect that we get so focused on the details of our math courses that we may not notice whether our course is facilitating or hindering.

In our Math Literacy class, we have been working on algebraic reasoning.  On the surface, the class looks like we are not ‘covering’ very much because we don’t include some typical algebraic (developmental) topics.  We found some hidden treasure this week in class.

As we often do, part of class is based on groups figuring out problems with some guidance and reflection.  Today this meant that we had each group do an equation ‘tag-team’ style — each student could either do the next step, or erase the last step.  Students had a little trouble playing by the rules, and wanted to switch to ‘their’ method to solve the equation.  The payoff came when we talked about the different choices, as more students figured out that they have options for linear equations.

The hidden treasure came next, not that students saw it as totally good.  We looked at how we could solve equations of a type never seen before, starting with a simple rational equation (namely, 5 = 200/x).  Students could see the solution (40) though not always obtained formally, so we talked about doing ‘opposite’ operations to solve.  We followed this with a radical equation (the pendulum model), which is not normally seen in this level of math course.  To solve for the length inside the radical, we listed the calculation steps if we knew the length and wanted to calculate the period.  Then, we reversed — the opposite operations in the reverse order.

To me, the hidden treasure in this is that students get to think about both types of skills that we use in mathematics — we have routine procedures (often based on properties) and we have reasoning about statements (often based on relationships unique to the problem).  Wouldn’t it be wonderful if students developed both strategies, instead of just using routine procedures (often memorized)?

It’s clear that my hidden treasure was not perfectly clear to students; after this discussion, we had a worksheet which included an equation of related design.  They generally understood the reverse order idea, but thought they should do them in a different order — a choice which requires applying properties of expressions.  Our conversation was more satisfying than normal because we had used the reasoning approach, and talked about choices.

Students may still ‘want’ a recipe for solving equations and simplifying expressions.  Giving students a recipe hides the math treasure; emphasizing choices and reasoning allows for the possibility of students finding our hidden treasure.

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