As part of NCSU’s TH!NK Quality Enhancement Program, I have spent a lot of time learning and thinking about how to encourage critical and creative thinking in mathematics courses. The course I targeted as TH!NK Faculty is MA 225 (an introduction or transition course for mathematics majors).
mathematical thinking is weird, but not that weird
It’s true that mathematicians view the world in somewhat a different way. We have our jargon (which we use outside its technical scope — I can’t tell you how many times I use the word “modulo” in everyday conversation!), and depending on our specialization we have our pet obsessions.
I believe very strongly that, though the reasoning one does in mathematics may be rarefied, it is a rarefied form of ordinary reasoning, by which I mean the kind of reasoning it takes to successfully navigate the physical, social, and emotional world we live in. To train students in mathematics means to help them learn to harness their physical-collision-avoiding, social-bond-forming, managing-tears-and-laughter apparatus and bring it to bear in certain ways on certain kinds of problems.
Learning to think involves taking control over one’s own thinking process. One of the best ways to do this is to engage students in reflection:
- students should reflect on the content of what they are learning (“How are the topics we’ve discussed this week related to the topics discussed earlier in the semester?”)
- students should reflect on their process of working in mathematics (“When you first approached this problem, what were some questions you had?” “What was the hardest part about this problem?”)
- students should reflect on their work (“What went wrong?” “What went right?”)
- students should reflect on their thinking (“What has changed in how you think about [topic] after doing this assignment?”)
the language of critical and creative thinking
One way to make the connection between mathematics and the rest of life is applications (such as the application of ODE to population biology). But I am interested in connecting the practice of mathematics to the practice of (for example) biology. One of the innovations of the TH!NK initiative is the use of a common set of terms and descriptions for aspects of critical and creative thinking. By using the same language to discuss and reflect on thinking processes in the mathematics classroom as in the biology classroom, we emphasize that it’s the same process, just being applied to different tasks.
the standards, and some ways they apply to undergraduate mathematics
The 13 TH!NK standards are:
- for critical thinking, clarity, accuracy, precision, relevance, depth, breadth, logic, significance, fairness;
- for creative thinking, originality, adaptability, appropriateness, contribution to the domain.
As an assignment in my MA 225 class, I ask students to think of some ways that each of the standards might apply to the process of writing proofs. Students also indicated which standards might apply to the formatting, style, content, and process of a proof. Below are my condensation of their ideas (along with some of my own).
|Clarity||Everything in a proof has to be written clearly. The logical steps should be stated explicitly. Simple formatting elements, such as writing “we will show. . .”, and consistently using the word Proof at the beginning of each proof, help improve clarity.||formatting, style, content, process|
|Accuracy||In a proof (except possibly in a proof by contradiction), all the statements in the proof should be true. Definitions and statements of quoted theorems should be stated correctly.||style, content, process|
|Precision||Precisely stating what you’re doing in a proof is essential. Ambiguity and fuzziness about what a statement or term means makes it difficult to be accurate and clear.||formatting, style, content, process|
|Relevance||Not every fact is relevant to every problem; sometimes problems contain lots of information that doesn’t go into the final proof. Determining what’s relevant and what’s not is an essential skill in the proof-writing process.||content, process|
|Breadth||Sometimes a proof requires facts and techniques that aren’t obvious at first — allowing yourself to think broadly about what tools could be used is essential.||content, process|
|Logic||Mathematical proofs are based on logic: you need to understand the logical structure of the claim you’re proving, and you need to make sure that statements in the proof are arranged in a valid logical order.||style, content, process|
|Fairness||It is very easy to fall into the trap of motivated reasoning when writing a proof. When you know what the goal is, you may overlook a logical step, or make an unsupported (or false!) statement that would get you your goal.
When you are trying to prove an “obvious” fact, it is important to remind yourself to be fair to the statement–to treat as worthy of a proof.
|Adaptability||Often the first (and second, and third) approaches to a problem don’t work. They’re good attempts, but they don’t work. It’s important to be adaptable in your thinking, so you don’t get locked into one approach.||process|
|Contribution to the Domain||content|