Blog post: Developing a taxonomy of problems: Not all problems are implemented equally
Response:
I absolutely love Geoff’s idea of creating a taxonomy of problems and I think that this is a really great idea to tie into all of the things that we learned during our assessment class with Peter Liljedahl. I think that this definitely goes hand in hand with tracking the progress of the students by skill and classifying our questions and this taxonomy helps us to create the problems that we can use to get to that point where we can track their progress. I absolutely love the idea of problem based learning and think that it gives the students a great sense of ownership of the learning because ultimately they are the ones coming up with the ideas and that’s an empowering feeling! I also think that more often than not, math educators forget that there is a difference between problems for learning and problems for confirmation and that they let the lines blur between what those really mean. I think that we need to watch that there are questions that help students to practice skills, but ultimately we need to see that they can think their way through a problem and apply the skills correctly. I think that by using these ideas to guide us towards conceptional understanding questions for confirmation and assessment questions for confirmation then we are giving our students the skills that they need and assessing them for the right reasons. Just recently I was helping a cousin study for his foundations 11 final exam and he was given booklet upon booklet of multiple choice review questions. While talking him through a lot of the content in those questions, it was clear that he was just able to take the multiple choice answers, plug them into a great conceptionally worded question, and get the correct answer without understanding any of the concepts that he was actually testing. So when, as teachers, we think about the problems we are assigning and where they fall on this taxonomy, we are giving ourselves a more accurate reading of the students skill set as well planning well rounded lessons that lead to the students further developing their critical thinking skills.
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Yep – learning through problems is empowering. John Van de Walle phrases it as problems for learning give students “mathematical power”. What you described about your cousin is a great (but unfortunate) example that’s all too common. From an assessment point of view, do correct answers to multiple choice questions really tell us that a student understands? I know you know the answer to that question. But a problem, even just one problem, can tell us so much – and can do so much for the learner. I love that Geoff used a single problem for a final exam.
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