I’m still wrestling with many of the questions I posed as a student teacher in this blog post from four plus years ago…

Reflections of a Second-career Math Teacher

The options are many, the outcomes are uncertain, the needs are tremendous, and the urgency is high. Are you with me? If you follow my blog, and its purpose, perhaps. If not, I speak of the current plight of secondary math education in the U.S. As presently defined, the math competencies of our nation’s youth, on average, are falling off the proverbially cliff; they started on this downward trajectory a few decades ago – why exactly is not known, or if so, not widely shared. Nothing has been able to slow its decline, and it is not clear if it is even possible, regardless of claims from both ends of the educational spectrum.

Conservative stalwarts to progressive reformers, public officials to private foundations, seasoned teachers to fresh-out Teach for America volunteers, all offer their laundry list of “do’s & don’ts” that will reverse the national slide. However, none have shown…

View original post 683 more words

Considering your example for your pedagogical pickle, when do you remember realizing you could do things like divide the 54 by 13 instead of multiplying 54 and 6 then dividing by 13? It is so obvious to us adults, who have repeatedly used that technique when solving for a variable, that it is difficult to remember when we actually knew we could do that. I don’t remember the exact grade, but I think I was told I was allowed to use that trick in grade school. I know I used it my freshman year and beyond.

Perhaps consider an alternate view. What if it was 53 instead of 54 and students were only shown examples were division was possible. So z = (53*6)/13. 13 divides neither 53 nor 6. The question is would students be able to think, “I can’t divide 53 or 6 by 13. Guess I should multiply 53 and 6 and write my answer as a fraction.” This may seem like a silly example because most students were taught how to multiple two numbers and write fractions. These facts are not the issue. The issue is will students be able to divert their thinking enough from a current procedure to apply previous knowledge when a problem fails to fit the examples shown by the teacher.

Despite all the talk about students being lazy or lacking ability, I find that most students like to follow rules. When they are given problems that don’t follow the “rules,” students become frustrated.

So perhaps the new trick is to show them that both techniques follow the rules. The cross-multiplication rule is the most general rule. It works for all sets of numbers. But if there is no remainder, then the denominator divides at least one of the numbers in the numerator. Your more efficient technique can then be used.

Consider the type of thinking that students must evolve through to reach your goal of knowing they can divide instead of multiple. Ask yourself, “Do the rules change for naturals, integers (an integral domain), rationals (a field), reals (a field) or complex numbers (a field)?”

Or perhaps consider how the application of the order of operations (OoO) may have contributed to your example and students’ mathematical inflexibility. In the world according the OoO, your act of dividing first is a bit scandalous. But you know your method works. Why? Luckily for you and me, mathematical correctness is determined by axioms and proofs and not OoO. When you think about it, OoO is kind of fail safe – do these things in exactly this order and it will work out every time (given you made no mistakes). But OoO really leaves out the idea of how specific sets of numbers behave. Interestingly, axioms and proofs always discuss the behavior of a specific set. Odd, isn’t it – that they would leave this notion of sets and their behavior out of OoO?

Think of how confusing it must be for students when they go from a teacher that treats OoO like a king, marking correct answers wrong because a student divided first, to a teacher that wants them to see all these connections they were told were wrong by the other teacher(s). If teaching your classes ever felt like you were dragging a stubborn, immovable dog by its collar, maybe now you know at least one of the reasons why.

LikeLike