|From Discovering Geometry|
Explain, Why?, Justify and Prove
Grab whatever geometry textbook happens to be nearby and scan the reasoning-and-proving exercises. (You can generally find two them at the end of the section after all the practice problems...) Take careful note of the language that's used in these questions. What exactly is the kid being asked to do when they're asked to defend their answers?
There's a variety of language that can be used for these exercises. Kids are asked to elaborate on their thinking using several different -- but apparently interchangeable -- prompts.
In the exercises above we get a few common directives: "Explain." "Why?" "Justify your answer." If we poke around our nearby geometry text we'll pick up a few other phrases, like "How do you know?" and "Explain your reasoning."
Do all of these prompts sound the same to kids? Should they all? Do we want kids to think of an explanation as being roughly identical to a justification? Is answering "why?" the same thing as offering a justification? And how does all of this relate to that other core prompt, "prove"?
Reasoning Problem Makeover
A few weeks ago I wrote about something called the Hexagon of Proof, and that post was half-joke and half-serious. The half-joke part was the idea of making a catchy image that played off Bloom's Taxonomy. The half-serious part was the idea that we can teach proof more effectively if our classes have a healthy and varied diet of proof-like activities. There are natural bridges to be built between everyday discourse and the unnatural act of mathematical proof.
We can do better than just asking kids to "justify" their thinking. There are lots of ways to provoke kids into expressing their reasoning, and there are some prompts that ought to see wider use. As an exercise, I rewrote one of the above problems in five different ways. As you read each problem, think about the different sort of student responses that each bit of prompting language might yield.
Exhibit A: Debating
Exhibit B: Disagreeing
Exhibit C: Convincing
Exhibit D: Explaining
Exhibit E: Teaching
Exhibit F: Proving
Does justifying have a different meaning to students than proving? I have no idea. Thoughts?
Exercises for the Reader
- Which is your favored version of the problem? Would you use different versions in different situations? Explain your answer.
- Are there other versions of this problem that you can imagine? Construct an example.
- "The language used in presenting reasoning problem significantly impacts the sorts of responses that a teacher can expect to receive." Do you agree with this claim? Disagree? Justify your response.
- Challenge Problem! Samuel Otten (and colleagues) wrote a paper called "Reasoning-And-Proving in Geometry Textbooks." In it they analyze the types of reasoning-and-proving activities assigned in popular geometry texts. How does their analysis compare to the one given in this post? How would Otten respond to this post?