Conjecture is Not Enough: The need for proof
Lesson 2 of 9
Objective: SWBAT express stereotypes and generalizations as conjectures and find counterexamples to disprove these conjectures.
Where We've Been: In the previous lesson, we just began to use some entry-level inductive reasoning...making conjectures about linear pairs and vertical angles.
Where We're Going: Soon we'll be making the case for why we need deductive reasoning and formal proof in addition to inductive reasoning. [This reminds me of the need to conduct randomized controlled experiments, when possible, and not just rely on observational studies.]
So in this section, I want students to get a gut-level feel for the power, limitations, and potential dangers of inductive reasoning.
I start by explaining that inductive reasoning is a natural function of the human brain. We start by observing the world around us. We are natural data collectors.
As we collect data through our senses, our brain computer naturally looks for patterns and tries to make meaning of the data.
When our brains are convinced that the patterns in the data are revealing a general truth, we make conjectures which are generally constructed as if-then statements.
We tend to think that a conjecture is true until we find a counterexample to disprove it.
Even if we have a million examples that support a conjecture, one counterexample can prove it false.
I then discuss the power and limitations of inductive reasoning using stereotypes as a context.
I start by setting the tone. I say this is a sensitive topic and will call for all of us to conduct ourselves with emotional maturity and intelligence.
I share with students that I want to have an intellectually honest, non-political, conversation about stereotypes. I assert that, in most cases, if we are honest, we have to acknowledge that while stereotypes are not entirely true in all cases, they do have some basis in reality.
[Note: Our students have a bad reputation in the community as being out of control thugs.]
So I pose the question to students: what are some stereotypes that people hold about students at our school?
As students brainstorm stereotypes, I write them on the board.
Next we formulate one of the richer stereotypes as a conjecture. In the most recent iteration of this exercise, the students chose "Muir students are not good in school". We rephrased this to say "If a student is a Muir student, then he or she is an academically weak student."
Next I ask students to think about how this stereotype came to be. In other words, what specific examples would people cite to support their conjecture?
Next I ask 'Is the conjecture true?'
Of course the answer is technically NO. But how do we prove that it's not true? For that, we need a counterexample.
So we're looking for a counterexample to the conjecture: If a student attends Muir High School, then they are an academically weak student....I model the structure of a counterexample for students: I know 'so and so', a student at Muir High School, and (s)he is an academically strong student.
So now that we know the conjecture is false, does that mean that poor academic performance is no longer a problem at our school? Of course not; we know that we have a problem here. There are many examples to support the conjecture that Muir students are academically weak students, and not so many counterexamples to reject the conjecture. So what can we all do if we want to weaken this stereotype? We can each decide to work on being a counterexample to the conjecture. Then it will be harder and harder for people to inductively reason that every student at our school is a bad student.
[Note: While this may appear here to be a monologue, it is actually an interactive discussion between myself and the students. I just know what major points I want to hit as I'm leading the discussion.]
In this section of the lesson, students complete the Inductive Reasoning and Stereotypes activity. I start by having someone read the lead-in paragraph. Next I model for students how to fill out a row on the table. I use an example from my own racial group in order to model the type of openness required for the exercise.
For example, I fill in the first cell on the table with "Black males are not intelligent". For the next cell on what fuels the stereotype, I write (and explain) 'The Achievement Gap'. I also write (and explain) 'Media Portrayals'. Finally I write (and explain) 'Historical Factors'. Next I provide some counterexamples: Charles Drew was African-American, and he was intelligent; Russell Simmons is African American and he is very intelligent; Christian Harvey (a student in my class) is African-American and he is intelligent.
Once I feel satisfied that the tone is healthy and students know what they're supposed to do, I release them to start completing the tables. I then start to walk around and make sure students are on task. I also check in with students to where there having difficulties. As usual, I try to ask open-ended questions that get students thinking about what they want to say and how they want to say it. After students have had enough time to complete the tables, we reconvene as a whole class.
Whole Class Discussion
Once we reconvene as a class, I ask for volunteers to share their analysis of the stereotypes they identified. As students share, I'm being careful to monitor the tone and intervening as necessary. I'm also phrasing the stereotypes as if-then statements and pointing out the potential dangers of this type of if-then thinking. Finally, I'm working with students on communicating their counterexamples. Specifically, I make sure that they establish that person X satisfies the hypothesis of the stereotype/conjecture, but violates the conclusion.
For closure, I have students reflect on the lesson, writing continuously for five minutes. The prompt is:
Reflect on today's learning activity. What were the goals of the learning activity? How do the goals relate to a Geometry Course? How do the goals relate to real life? How can apply what you learned in the activity?