Students generating examples
In the current (15 November 2016) posting of the MAA Teaching Tidbits Blog, Dana Ernst, Northern Arizona University, writes, “Who generates the examples?” Dana is suggesting that students can demonstrate their knowledge very well by creating examples and problems. He says,
“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 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.”
By creating problems and examples one demonstrates good knowledge of the material – backwards and forwards as it were. Indeed, we often say to ourselves, “I really learn material when I teach it.” The same applies to the subset of activities called creating examples.
Thus we agree with Dana and encourage you to let your students make examples, create modeling situations, modify situations, and build their own question sets.
Sydney Welton, when he was teaching my geometry class in 1963 – over 50 years ago and I still remember this – at my undergraduate school, Wagner College on Staten Island NY, asked us to create questions for the final exam and we would construct the final exam from submitted questions. Wow! We get to make up the exam! Great! SO I made up five questions and one of them I though was really cool, but I could not answer. Lo and behold three of my questions were on the exam including the one I could not do. Was this Sydney’s revenge for me being a smart alec during class? No, he could not answer it either, for it was a type of open question and he liked that fact to see what we were all thinking so he put it on the exam. This was an early sign I was destined to be a professor, I always thought. He had given credibility to my thought process, to my inquisitiveness, and he had sanctified my curiosity as being worthy.
I have done this activity from time to time in my career, indeed I have had students make up short quizzes and bring them to class, have them pass it to their classmate on the left, whereupon that receiving student takes the short quiz and returns it to the originator to grade and I enter a grade for the quiz AND the nature and level of the problem. Always works!
So let your student generate examples and occasionally “steal” them for your exams and homework sets and give them credit for originating materials and doing the examples.