The Challenge of Teaching for Understanding

Posted: March 24, 2012 in Teaching and Learning Philosophy
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I am working with my students to make sense of the chain rule in a calculus 1 class. This entails engaging students in activities to first make sense of the composition of functions and then to make sense of the rate of change with a composition of functions.

It is really great to see students working to make sense of and to develop their own thinking about an idea. I was reflecting on this today and about how much easier, in one sense, it would be if I just showed them how to “do it.” You know, “first you take the outside derivative and then multiply it by the inside derivative.” Then I watch them practice this procedure and walk around and help students to see the pattern and to correct any mistakes in their working of the procedure.

But, where is the conceptual power in this? Where is the meaning of derivative as a rate of change? And, what is the point in just knowing how to do the procedure? This kind of learning is so shallow. However, this is what I see when I visit classrooms…students trying to mimic the procedure of doing the chain rule without any attempt to develop understanding, to build upon mathematical meanings (like the meaning of the derivative), or to develop mathematical ideas.

So why is it that the most typical teaching style that I observe is that of just showing students how to do a procedure rather than to push students to make sense of a big idea?  After a couple of lessons with the chain rule material here, I have a guess…it is much more challenging to push students to make sense! Students struggle (as they should…making sense of an idea takes work). Students give up quickly (they hold strongly to the belief that math problems can be solve quickly). Students sometimes rebel (their friends in other classes don’t have to work so hard). So, teachers give up (if they try to teach conceptually).

I refuse to give up. I see the value in providing the opportunity for students to develop their ability to reason, make sense, problem solve, and think. I continue to support students in their efforts and I continue to “sell” the approach so that they might continue to try. Teachers need to carefully balance the pushing, pushing, pushing and then backing off as students learn to persevere in their efforts to make sense.

I encourage everyone to use this community to stay motivated by getting involved in this community…ask questions, share ideas, vent, use resources, tell about successes and challenges.


  1. Andrew says:

    Don’t know if you happen to remember “The Mechanical Universe” – an old set of physics shows. They’re available online. The show on the derivative covers the chain rule quite clearly and carefully. It starts at about minute 18.

    People get all caught up in “teaching for understanding” and seem to forget that automaticity in computation is also important to allow understanding of what’s next. Others get caught up in the computations and forget that one has to understand what one is computing. Both are important.

    In your example, it would be tedious to have to derive the formula for the chain rule every time you want to compute a derivative of a composite function, and computing a derivative is useful. On the other hand, if you want to be able to use what you have learned in some way other than answering the question: “compute the derivative of…”, you have to understand what the derivative represents, composite functions, and how derivatives operate on composites.

    • Scott Adamson says:

      You are correct…it is not only about developing the idea and never encapsulating that idea into a powerful procedure/computation. Nor is it only about providing procedures/computations without developing where they come from and why they make sense! It must involve both.

      I think I practice a form of “guided reinvention.” That is, students realize that there is something problematic about computing the derivative of the composition of two functions. They can try what they think will work, compare it against graphs of the function and its alleged derivative, and see if their idea makes sense. Then, we work together to develop the idea. Then we look for patterns in our work to see if we can simplify the procedure. Eventually, we come to the “shortcut” for the chain rule but it was developed with a focus on problem solving, reasoning, and sense making.


  2. Andrew says:

    My memory fails. They don’t spend much time on the chain rule, but it may still be worth a watch.

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