Archive for October, 2011

In my current position, I spend a lot of time working with high school teachers, assisting them in improving their curriculum and in learning to become more reflective educators. During the course of this work, I am frequently confronted with teachers who struggle to let go of traditional approaches or are loathe to remove topics from their curricula in favor of spending more time developing key ideas, making connections, and working collaboratively to develop genuine problem solving skills.

A long time ago I became highly skeptical of the approach common in math classes around the country, an approach that was so ingrained in the culture of mathematics education that most teachers weren’t consciously aware that teaching in this manner was a choice, and therefore an approach they could choose to abandon.

In classrooms across the country, as documented by research studies such as TIMMS, American students receive a very specific message about the nature of mathematics. With each lesson and each homework assignment, students come to believe that math involves a teacher giving students a new formula or procedure, demonstrating the use of the formula or procedure in a series of examples, culminating in independent student practice involving many similar problems. Students develop this image of mathematics so strongly that, when some of these students grew up and became math teachers, they present mathematics in the same way and using the same approach. In fact, it’s likely the content doesn’t change at all. They drilled and practiced properties of logarithms in school, so their students complete drill practice on properties of logarithms. When they were precalculus students, they learned Descartes’s Law of Signs, so their students learn Descartes’s Law of Sines.

Our community of mathematics educators on the whole seems to lack one very important characteristic: the desire and confidence to challenge the status quo, to step back and say, “Hold it! What am I teaching, and why am I teaching it? What exactly do I want my students to learn, why do I want them to learn this, and what is the best method to help students develop a meaningful, personal understanding of these ideas?”

Consider some anecdotes from my experience pinpointing what I see as troubling characteristics of our mathematics education system. What truly concerns me is not that such events happened, but that they are not the exception, they are the rule.

Anecdote #1: I participated in a small group discussion with high school teachers about inverse functions, with the focus of the discussion on the ideas we presented in our article published in Mathematics Teacher. (If you are unfamiliar with the article, I invite you to follow the link, read it, and consider it.) We got to a point where teachers had accepted the conceptual problems in teaching novice learners to switch x and y to create inverse functions and to graph inverse functions on the same set of axes. One teacher commented that it was an eye-opening experience to think like this and to teach students using our approach. However, two minutes later s/he asked my advice on how to then teach students that inverse functions are reflections across the line y = x, because this is something s/he really wanted students to know. I asked why s/he felt that way – what way of thinking or conceptual understanding was supported in teaching this idea to students. The teacher couldn’t answer my question. Regardless of my beliefs about whether we should be teaching this to students, I was disturbed by the lack of personal reflection. This was an idea the teacher felt was really important, but apparently there was no carefully considered reason why it was important. I suspect the teacher considered it highly important only because s/he had always included it in the curriculum – not because the teacher had reason to believe that it actually led to powerful insights for students.

Anecdote #2: I have worked with many, many teachers who set out to change their skills-driven approach to a more conceptual approach. Over the years, countless teachers have commented that in-depth, conceptual instruction seems to be intended for honors students or future mathematicians, but not for the average student, especially not for students who will finish their mathematics education at precalculus. This seems totally backwards to me. If a student doesn’t plan to take any future math classes, what good are countless lessons focused on drilling skills and procedures? What are they practicing these skills for? I know that if I have a group of learners who are taking their last math class ever, I want them to walk out of that class with a general understanding of rates of change, quantity, variation, and measurement, not the ability to exactly evaluate complicated trigonometric expressions, rationalize denominators, and prove a trigonometric identity. It goes back to a simple question: What am I teaching, and why am I teaching it?

Anecdote #3: I have had former colleagues who have told me that most students hate math, but have to take math classes, so our job as teachers should be to make math classes as straight-forward as possible and that we should focus our efforts on simply helping students “get through” our classes. When I hear this, or a variation of the same comment, I can’t help but carry this forward to the logical implications. This relegates our discipline to pointless busy work, our courses to bureaucratic red tape, and ourselves to little more than babysitters. Our courses need to offer something significant to every learner in the room – or else they shouldn’t exist at all.

When you sit down to plan your next course outline, unit, and even lesson, take a few moments to sit quietly and reflect. Here are some questions to ask yourself.

1) What am I teaching? (Not the topic, which would be obvious, but what are the ideas inherent in the topic you are teaching? For example, if you are teaching about linear functions, an inherent and crucial idea is constant rate of change.)

2) What does it mean to understand these ideas? (For example, what does it mean for someone to have a deep, meaningful, and personal understanding of constant rate of change?)

3) Why are these ideas important? (If you can’t think of a good reason, are they worth teaching?)

4) How can I design a lesson so that students can engage in genuine reasoning about this idea so that they develop a meaningful and personal understanding?

As an example to get you thinking, consider a topic mentioned earlier: properties of logarithms. Most of the properties of logarithms were highly useful before calculators when logarithm tables and slide rules allowed mathematicians to avoid tedious multiplication and division calculations. With the advent of calculators, most of the properties of logarithms, such as the property that states log(AB) = log(A) + log(B), have become little more than mathematical curiosities – statements that are certainly true and make sense with some thought and reflection, but have no real use to the average algebra, precalculus, or calculus student. Yet we teach them, usually in all three courses, and drill students year after year until they “master” these properties even though most of the properties are useless. If you disagree, I recommend you try the following exercise. Come up with two real-world contexts, appropriate to a college algebra classroom, where the property log(AB) = log(A) + log(B) is crucial to understanding the context and/or solving the problem. I find it nearly impossible to complete this task.

I know it can be controversial and uncomfortable to take a stand against teaching obsolete topics or to move away from a focus on procedures to a more conceptual approach, but consider this: if the only defense we can come up with for teaching a topic, or for teaching it the way we do, is that it is something we have always done, what does this say? How can we expect anyone to take us seriously as professionals?