On the first weekend of December, the California Math Council held its annual meeting in Asilomar for the 60th time. (I attended for the 33rd time, and presented roughly the same talk I had presented in 1984.) Over the decades, the “must-attend” presenters have changed. Two of my favorites back in the day were Harold Jacobs and G.D. Chakerian. Both of them had written books* that influenced me mightily as a young teacher, and their talks were always interesting, albeit in different ways. Jacobs always did an amazing slide show of mathematics in everyday life, sharing photos and clippings he had gathered during that year. Chakerian gave lectures on specific mathematical topics I usually knew nothing about, often in geometry. I always walked out of their sessions inspired. Both of them are acknowledged in the front matter, and again obliquely in the text of my Algebra: Themes, Tools, Concepts. (Jacobs in Lesson 5.1, Constant Sums, and Chakerian in Lesson 8.8, Percent Decrease.)
Nowadays, at Asilomar, I usually try to catch Scott Farrand’s presentations. He is a math prof at Sacramento State, and his talks are almost always interesting either for their math content, or for his ideas about pedagogy, or both. This time, he gave a talk on using the question “What solution to this problem would be the coolest?” or put differently, “If you were God, designing the universe, what would you want the solution to be?”, or “What do you hope is true?” He gave the example of the volume of a right prism with a square base, with the top sliced off by a random plane. People pretty much agreed that it would be cool if the volume was the area of the base, times the average of the distance to the base from each vertex of the slice quadrilateral. (A 3-D version of what happens when you cut off part of a rectangle with a random line.)
Of course, that sort of question is not always appropriate. Still, it can be added to one’s repertoire, to be deployed in the right situation, whether working with an individual student, or in class discussion.
(Check out my previous blog posts on Scott Farrand’s Asilomar talks.)
The question of what questions to ask must be in the air. In an online forum, mathematician James Propp recently offered these questions for everyday use in the classroom:
– What is the answer?
– Does this answer make sense?
– Is there another way we could arrive at this answer?
– Does this remind you of something else we’ve done?
– What do these things have in common?
– What question might this lead us to ask?
– Is there a pattern here?
– What mistake did I just make?
– How am I fooling you?
– Is this wrong answer the right answer to a different question?
– Are we using the right definition?
– Have I given you enough information to answer the question? What other information might you need?
– Can we think about this a different way?
– Is that a rigorous argument, or is there a subtle point that we’re glossing over?
– How convinced are you?
– Can someone give a concrete example?
– Can we generalize?
– In plain English, what is this equation telling us?
– What kinds of mistakes do you think people are most prone to make when using this procedure?
(Check out James Propp’s monthly Mathematical Enchantments blog post.)
Jim Tanton adds these:
– Now that we see what the answer is, could we have seen that more swiftly?
– Was that approach to answering the question enjoyable? Could we have avoided doing XX?
– Do you think this solution would be a pleasure for someone else to read? Have we made it easy to read and understand? Is what we have presented inviting to read?
– What don’t you like about the question? Shall we just change that and answer an easier question first?
– What do you wish was also in the picture/equation/…? Can we just make it appear?
– Oh bother. I don’t see what we need in order to proceed. I think we should weep. (To spur the class on to do something with the issue/question at hand)
– Who says you have to do the questions in the order presented to you? (If part e is easier for you, do that part first!)
– Why would anyone want to answer this question?
– What would make this question interesting?
– Is there a three-dimensional version of this?
– What is it exactly that made this question hard?
– What did we actually do here? What is the general thing we’ve established?
(Check out Jim Tanton’s G’Day Math Web site.)
Great lists. I’ll just add a few more:
– We have several answers to the question. They can’t all be right. What should we do?
– Would you like to choose a student to help you answer? (if someone is stuck, or blatantly wrong)
– Can you restate what X said, in your own words?
– (to the whole class) Can you give me the answer on your fingers? by “air graphing”? by pointing? (up / down or left / right, depending on the question) agree? disagree? don’t know? (thumbs up, down, horizontal)
(Check out my Nothing Works article, which has a section on classroom discussion.)
Do you have any questions? Please share in the comments!
Sequel to this post: Handling Wrong Answers
* Harold Jacobs’ Mathematics, A Human Endeavor has many wonderful ideas. His Elementary Algebra and Geometry: Seeing, Doing, Understanding are pedagogically not what I look for in a textbook, but they are filled with great cartoons, photos, and other connections with everyday life.
Chakerian, Stein, and Crabill’s Geometry: A Guided Inquiry had an enormous impact on my teaching, as it pioneered the idea of group work, did not rank problems in order of difficulty, gave many answers right there in the margin, etc. It’s probably not possible to use it as is any more, but it is a fantastic source of great problems. Their Algebra books though not as successful, had many of the same powerful pedagogical features. I have used their approach to trigonometry and to complex numbers ever since I came across it.