Life and Musings of Ed

Thinking – With Assumption Focus

In Ed Does Math on 5 March 09 at 12:44 pm

I was reading an interesting post on one of my favorite blogs today.  Two-Pi relates a story about his son’s cub scout meeting.  The leader was talking about “what to do in case of an emergency” and one boy kept escalating the scenario to a more extreme situation; “yeah, but what if…?”.   Two-Pi muses  —

“Hey, he’s thinking like a mathematician!”  He knows the stock answer that is expected, and he’s asking what happens if we change the hypotheses, considering a related problem where the conclusion doesn’t  follow.   HE’S DOING MATH!

— read the full story here:   Thinking Mathematically- by 360

A corollary comes to mind — teaching introductory physics,  I find it useful to frequently remind students to focus on assumptions made when building various “models” .

For example — introductory kinematics equations are generally formed with an assumption of constant acceleration — Newton’s laws are introduced in the context of “point particles” and for velocities << c — torque is introduced on “rigid bodies”, etc.   By stressing the limitations of a particular model,  students often get curious enough to ask the “what if…?” questions.   Of course, it also helps prevent them from misusing/misapplying formulas in situations when these assumptions are violated.

One  common observation as I tutor HS Calculus students,  is that I often find they have not been taught or have not paid attention to the “conditions” which must be met for certain theorems to be valid.  When they are preparing for AP exams, they often flounder because they don’t know the “if” portion of a theorem.   Consider:

Rolle’s Theorem

If   f(x) is continuous over the closed interval  [a, b] ,

if  f(a) = f(b) = 0 ,  and

if  f ‘ (x) exists at least over the open interval  (a, b),

then

there is at least one value (x_1)  in the open interval (a, b) such that the derivative  @  x_1 = 0.

By stressing the importance of wrestling with the assumptions, students develop a deeper understanding of the theorem/concept/model and are trained to ask relevant “what if…?” questions.

So, my suggestion for Two-Pi, who presents the following challenge:

So the challenge, as we prep for our classes:  find a way to ask questions with obvious answers, that will get students motivated to say “yeah yeah, but what about THIS situation?”, and aim to “lead them” (pushing rope comes to mind) toward the actual course content we want to explore.

Try emphasizing the antecedent (or the  conditions which must be met,  the assumptions,  the premise, …) and I think you will find students more naturally asking deep and relevant “what if…?”  questions.

Somehow I am reminded of the framework for the Ten Commandments — “thou shalt not…” — this sets up the boundaries;  you are free to love and work and serve… just do it within these “boundaries” and all will be well.   Carver uses the concept in his Policy Governance model when recommends writing the executive limitations policies in the negative.  In other words, the executive director is free to work toward fulfilling the mission of the organization within certain prescribed limits, which are clearly written into policy statements.  All the other categories of policy are written in the affirmative, as positive statements.

And now a “shout out” to xkcd

What If...

What If...

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