I was immediately drawn to this article
within the first sentence as the author stated that they had been teaching
grade 5 for the last ten years. While I
haven’t quite taught for ten years yet, I have primarily taught grade 5.
The purpose of the study was to analyze how
grade 5 students problem-solved, with respect to a proof. The particular proof that they were asked to
do was, “Find all the squares (in a four by four grid as a figure). Can you prove you have found them all?” (p.3)
There were also extensions to this question, such as finding how many squares
were in a 5 by 5 grid, or 10 by 10, or even 60 by 60.
The author, Vicky Zack, posed possible
solutions to her students and asked them what they thought of them. What I like about this strategy is that it
encourages students to question a possible solution, and they can do it without
potentially hurting any other students’ feelings. Occasionally, I notice these types of
questions asked of students from mathematical textbooks. They are usually at the very end of a lesson,
as they are thought of as being the “challenge” question. I wonder what would happen if these types of
questions were asked right from the start.
Would students become immediately engaged, trying to disprove or prove a
theory? Or would they be more or less apathetic
towards the problem, with no prior knowledge of the subject matter?
Zack found that my analyzing the students’
counter-arguments she gained a higher understanding of their mathematical thinking. In order to prove or dis-prove a potential
solution, often the students had to extend and apply the proof to test it. In this way, the students began to understand
the problem on a much deeper level. Zack
also noted that her students would display sophisticated reasoning, but would
often not vocalize it. She found it
useful to give an alternative voice to their reasoning by putting it in current
mathematical terminology. In this
attempt, Zack connected the problem-solving abilities of her students with the
mathematical curriculum.
If I were to do this activity with my
class, I would be tempted to perform it at the beginning of a patterns unit,
and again at the end of the unit to see how the students’ thinking changed (or
remained the same) over the period of time.
Vicky Zack's method of posing solutions provides closure to the types of problems we did in the early childhood math education class last semester, in which we had to find the total number of embedded rectangles in a large rectangle divided into (at first glance) 10 rectangles with a child. Actually, I think that the idea of presenting multiple solutions can be added to my personal bank of ideas to use in a classroom. Thank you!
ReplyDeleteIt would be useful if Zack were able to get students to analyze their own counter-arguments for this activity. What's useful about this activity is that students would be able to criticize an unknown person's work without having their own work criticized. The one challenge with this would be for students to be able to assess whether or not a solution is valid without either assuming that the solution is wrong and the teacher is trying to "trick" them, or that a solution is right and they then stop thinking about the proof. One modification I might make in this implementation might be to collect all students' solutions on the boards (perhaps throwing in the correct answer if there are enough solutions) and having the students involved in discussion about the correct answers.
David, if you do this activity with your class, would you please let me know how you find it/how it goes/what things work/roadblocks? I'd be interested to know how you implement it.
First off, I think that implementing this activity in your classroom at the start of unit would be a wonderful idea. It is somewhat sad that a "discovery" type problem is only included at the end of a section and that most students would not attempt such a problem. From a recent Schoenfeld article I read, he mentioned that many students have the belief that mathematics problems can be solved in less than 5 minutes, and that if a problem takes over 15 minutes, it's "impossible". Including discovery type problems, especially ones that include generalizations, might help in alleviating such beliefs.
ReplyDeleteI wonder how Zack worked around the problem about not hurting any student's feelings. I think students can take a great deal of pride in their proofs and when presented with the task of "critiquing" mathematics (which is something that doesn't happen often enough!), how can one make sure that it is a constructive environment? I suppose starting early within the curriculum is one idea and I do like the thought of students giving constructive feedback. They do it in other classes after all!
Alex's mention of teachers trying to "trick" their students is a point I haven't thought about until now. As soon as students encounter a non-standard problem, there seems to be an automatic assumption that some sort of trick is involved. I can say from experience that sometimes it's valuable to be able to pull mathematical tricks, but I've never been one to be good at pulling them. They are abundant in number theory type proofs, which is part of the reason why they are often included in math competitions. The proofs are accessible to high school students with no formal experience in proof, but having a back of tricks helps them get through. I suppose this connects to my article and reading about the content of proofs and the mechanisms that develop within them. Once you see one trick in a number theory proof, you will remember to throw it in your box for another occasion.