Have you ever had a student who could recite a definition or theorem word for word, but didn’t really know what it meant? Students often memorized a snippet of mathematical content without understanding where and how it applies.
According to Bloom's Taxonomy, these students have only reached the the first level in the taxonomy--recalling facts and basic concepts. Ideally, we want our students to reach higher levels in the taxonomy such as using information in new situations or producing new original work. In today’s world we need individuals that are capable of asking and exploring questions in contexts that do not yet exist and to be able to tackle problems they have never encountered.
The question is, can we, the instructors, work toward this?
There is a small change we can make to ensure students are progressing on Bloom’s scale and developing the habits of mind of a mathematician: encourage our students to generate examples and counterexamples. Requiring students to construct examples and counterexamples places them in situations where they must wrestle with definitions, concepts, and notation, which provides them with the opportunity to synthesize and analyze mathematical ideas. Below are a few examples from calculus, and additional examples can be found here.
Implementing these types of questions can be done in a number of different ways, but it is important to make it a regular thing. As a starting place, I encourage asking at least one question that requires each student to produce an example or a counterexample each day in class and on every homework assignment and exam. There is likely a limit, but in general, the more often students are asked these types of questions, the better.
Questions that ask students to generate examples are excellent for think-pair-share and small or large group discussions. My experience has been that the discussion surrounding student-generated examples is fruitful for the students and insightful for the teacher. Especially in the classroom, I encourage fellow instructors to allow the students a little room for making mistakes and to foster an environment where it’s okay to experiment in the hope that everyone can learn something from the conversation that follows.
It’s also important to vary the difficulty of the problems. Try not to give away which ones you think are “easy” versus “hard.”
As I've increased the number of opportunities for students to generate examples and counterexamples, I've witnessed an increase in student understanding of mathematical concepts and their interdependence. In particular, students seem to have a much deeper appreciation for the intricacies of key definitions and theorems.
I've included potential questions/problems in the context of calculus to give you a flavor of what such questions might look like, but we certainly we can employ the same strategy in other subject areas. Do you have a favorite question/problem that asks students to generate an example or counterexample? If so, share it in the comments.
Dahlberg, R.P., Housman, D.L. Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283-299, 1997.
Hazzan, O., Zazkis, R. A perspective on ‘give an example’ tasks as opportunities to construct links among mathematical concepts. Focus on Learning Problems in Mathematics, 21(4), 1-14, 1999.
Katz, B., Thoren, E. Call for Papers for PRIMUS Special Issue on Teaching Inquiry. PRIMUS, 2014.
Watson, A., Mason, J. Student‐generated examples in the learning of mathematics. Canadian Journal of Science, Mathematics and Technology Education 2(2), 2002.