Teaching Quantitative Problem-Solving Skills Lies in the Solution

Editor’s Note: One of the themes that emerged from our recent Faculty Focus reader survey was a request for more articles specifically related to teaching in the STEM disciplines. In response, we are pleased to present an article written by true leaders in STEM education and the authors of Teaching and Learning STEM: A Practical Guide (Jossey-Bass, 2016). As its name suggests, the book focuses on the practical application of research-based strategies for designing and teaching STEM courses. It has been called “hands-down the best instruction manual for professors in science, technology, engineering, and mathematics that you can find.” [Barbara Oakley, PhD]

If you teach a course that involves solving quantitative problems, you’ve almost certainly had this experience. You work through a problem in a lecture and ask the students if they have any questions. They don’t. Then you assign a similar problem for homework and collect the solutions, most of which give the impression that the students never saw anything like that problem in their lives. You conclude the students must be incompetent.

A few of them may be in over their heads, but cognitive science suggests that something else is probably going on for the others. Most of the uncountable bits of information perceived by our sense organs are filtered out without our ever being consciously aware of them. The relatively few bits that make it past that filter go to working memory, where we do our conscious processing.

Working memory can hold roughly four chunks of information at any one time. When you lecture straight through a problem solution or derivation in class, you are fire-hosing information at a rate too high for working memory to process—and what you are presenting is only a small fraction of the sights, sounds, and thoughts simultaneously competing for the students’ conscious attention. The result is that most of the content of your problem-solving lecture never gets processed and absorbed. Expecting your students to understand your solution method well enough to solve new problems with it is a recipe for disappointment. Making things worse, even if the students are paying attention, if you’re a good lecturer every step of your solution is likely to seem logical and clear to them. It’s only when they try to do something similar on an assignment or exam do they realize how much of the lecture they didn’t understand at all.

A much more effective way to teach problem-solving is to guide students through complete or partially worked-out problem solutions and derivations using an active-learning structure called Thinking-Aloud Pair Problem Solving (TAPPS) (Felder and Brent, 2016). Here’s how.

  1. Prepare and distribute a class handout containing the problem statement, the solution (possibly with some steps omitted), questions about the solution, and blank spaces (gaps) for the students to insert missing steps and answers to the questions. At the beginning of a class session, tell the students to organize themselves into pairs and to designate one pair member as A and the other B, and have them read the problem statement and ask questions if they need clarifications. They then work through the solution, alternating between the next two steps.
  2. Student A—the explainer—explains a fairly small designated part of the solution, step by step, including why specific formulas and methods were chosen if the reasons are not obvious, and fills in any gaps in that part. Student B—the questioner—asks questions when the explainer says or does anything incorrect or unclear, and gives hints if the explainer doesn’t understand something. Allow a short time for this activity (generally 1-3 minutes), not necessarily enough for every student to finish. The complete solution for that part of the problem will be forthcoming in the next step for any pairs that didn’t get it on their own.
  3. Stop the students when the allotted time has elapsed; randomly call on several of them with questions about the part they just went through. Call for volunteers if you want additional responses to open-ended questions. When students provide correct results that were not in the original handout, write them on the board so everyone can see and copy them, and elaborate when appropriate. If the next part of the handout contains only straightforward calculations and explanations, quickly lecture through it or (better) have the students quickly read it themselves and ask any questions they may have. When you get to the next challenging part, have the pairs reverse roles and repeat Step 2. Proceed in this manner through the entire solution.

After this exercise, most students will understand the solution at a far deeper level than if they merely watched you go through it. Students who had trouble with a difficult step got clarification in minutes, where in a traditional lecture most of them would have gotten lost in that step and stayed lost in whatever followed it.

Consider using TAPPS to guide students through worked-out problem solutions and derivations, ideally after first reviewing some general tips on active learning [Felder and Brent, 2016, Ch. 6] and TAPPS [pp. 121–122] and a narrated video of a TAPPS exercise in an introductory engineering class [Available on YouTube.] It can take an entire class session to work through one problem in this manner so we suggest doing it only three or four times in a semester, mainly for complex problems that tie together much of what you have been teaching for the previous several weeks. If our experience is any indicator, one or two applications of it in your class should be sufficient for you to start noticing significant improvements in your students’ problem-solving skills.

Felder, R.M., and Brent, R. (2016). Teaching and Learning STEM: A Practical Approach. San Francisco: Jossey-Bass.
Felder, R.M. & Brent, R. (nd). Creating Partnerships: Active Learning in an Engineering Class. Posted June 2015. https://www.youtube.com/watch?v=0p7gNXGvcww.

Rebecca Brent is a faculty developer, program evaluator, and president of Education Designs, Inc. Richard M. Felder is the Hoechst Celanese Professor Emeritus of Chemical Engineering at North Carolina State University. For additional information on their latest book, Teaching and Learning STEM: A Practical Approach, visit TeachSTEM.