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01 October 2016

Teaching the qualitative-quantitative translation: Why our students use common sense instead of calculation.

Today's question: Planet X has three times the free-fall acceleration of Earth.

(a)          A ball is thrown vertically upward with the same initial velocity on Earth and on X.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?

(b)          Next, a ball is thrown vertically upward on X with three times the initial velocity of an identical ball on earth.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?

The newbie physics student generally doesn't want to learn how to approach a physics problem.  He or she wants to get to an answer.  Usually the unsuccessful path to the answer takes one of two forms:

(1) Common sense.  "Obviously, more gravity means smaller max height, and three times more gravity means three times less height.  And in (b), three times more speed cancels out the three times more gravity, so the same height for each."

(2) Trying to find the one weird trick. "In class, I remember you asked a question like this.  Since the equation has a square in it, three times the gravity means nine times less height.  And in (b), the square terms cancel to give the same height."

(If you've been teaching for more than a month, you've seen these sorts of answers.  Let me know if you haven't seen one, and I'll buy a beer for you in anticipation of when you do.)

Why do our students say these things rather than just do the calculation?

In many of our students' minds, good, smart boys and girls know the answer.  The thought of "figuring out" the answer from first principles isn't part of their skill set.  You don't "figure out" the 3rd person plural present active indicative of cupio; you remember that -io verbs take -iunt in this form.  And if you don't remember, you should, 'cause you've been taught that.  So take a guess, knowing that -nt is a typical 3rd person ending.  You'll at least get close.

There's your common sense approach in (1) above.  Since of course smart students should know the answer, they take a reasonable guess based on their instincts and previous experience.  Those instincts have been good in previous classes, especially math class; so guesses like this should work in physics, too.

The next step for students is to try to mimic what they see in class, what they read in the textbook.  When they recognize that common sense approaches don't work, they despair -- "oh, physics is impossible, every question has a trick to it."  So find the trick.  Note the language used in response (2) above, referring to "the equation".  WHAT EQUATION?  I want to shout.  Shouting is useless, of course...

The student looking for one weird trick doesn't remember that in class I showed how to determine the correct relevant equation, then to solve it for the height of the ball in terms of the other variables.

The student does remember that I got the answer.

The student assumes that I already knew the answer, and only went through the calculations for form's sake, like a prosecutor painstakingly presenting evidence that Jack Ruby shot Lee Harvey Oswald.

The student takes away, then, that HE is supposed to know the answer, too.  He shows his work only because I insist on it for silly teachery reasons.

How do we train students to use quantitative reasoning when answering these sorts of questions?

Understand that we're fighting a war of attrition.  There's no one weird trick for teachers that will suddenly cause enlightenment.  Chip away at the class, getting one student at a time to use quantitative reasoning properly.

Begin the battle by modeling good qualitative-quantitative translation skills.  Model an organized approach, in which you solve for the desired quantity in variables.  Use numbers too, not only variables -- early in the year, most of your class will not yet be comfortable looking at variables with squares and square roots, but they're easily able to compare 9 meters to 3 meters.  As the year goes on, you can help everyone transition to using variables only.

Next, do an experiment to demonstrate your results.  Okay, you can't travel to planet X for the problem I've posed here.  But you can show that doubling a block's initial speed will quadruple its stopping distance on a track.  You can show that dropping from twice the height does not double the time a ball is in the air.  Emphasize that we're making physical predictions, not merely doing abstract mathematics.

And finally, demand to see a quantitative approach.  Look how I've rephrased these same questions below, to emphasize that we're looking for a multiplicative factor.

For the students who still don't do the calculations -- yes, that means about 1/4 of my class -- make them do the problems again with even more explicit instruction, like "Pretend the initial speed is 10 m/s.  Use kinematics to calculate the height on X and on Earth."  Eventually, they'll get it.  Just be patient yet persistent.

The question, rephrased: Planet X has three times the free-fall acceleration of Earth.

(a)          A ball is thrown vertically upward with the same initial velocity on Earth and on X.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?  Justify your answer with both words and kinematics calculations.  Then your answer should state “The ball goes ____ times higher on X.”

(b)          Now, a ball is thrown vertically upward on X with three times the initial velocity of an identical ball on earth.  How does the maximum height reached by the ball on X compare to the maximum height on Earth?  Justify your answer with both words and kinematics calculations.  Then your answer should state “The ball goes ____ times higher on X.”

1. Funny you should post this, Greg. I just assigned this very problem last week (I'm sure I stole it from you). I remember assigning it last year and getting all sorts of answers. This time around, I used the class period before the assignment was due as an opportunity to introduce what I was taught as the "factor-of-change method." Take the equation (vf^2 = v0^2 + 2ad), get rid of stuff that's equal to zero, solve for the thing you want (d = v0^2/2a), and then plug in the factors by which those quantities changed. If something doesn't change, plug in a 1. So, if the initial velocity triples, the equation becomes d = (3)^2/[(1)(1)] = 9, so the new distance traveled is nine times as much. When I assigned the free-fall question the same day as I taught that method, nearly all of the homework was perfect. From now on, we exclusively use this method to solve these sorts of problems quantitatively. The next bit is the qual/quant translation, which I find is made much easier by the use of colored pencils. For example, if you have a statement written in red pencil, use red pencil to write an equation that justifies that statement. Use a different color for different statements. I find that this method helps students form direct links between words and equations.

2. I like that "factor of change method" very much. I'll probably use the term now. Thanks!

3. Greg, you can purchase custom self-inking stamp that says WHAT EQUATION!
In any color you wish. I've considered it.

"The student assumes that I already knew the answer, and only went through the calculations for form's sake..." YES YES and YES!

They think we know the Magic Formula (tm) for every one of the thousands of problems we can solve. Nope. I know a few ideas and a process.

Thanks, Mike V. I teach that method in conceptual classes but had never heard a name for it. That is a great name. And I like that you have an algorithm to eliminate numerical constants along with un-varying variables. And I should probably teach it in every class. Students never learn this kind of equation relationship thinking in their math classes because they are never asked that kind of question in a math class. (Which is why they never had their instincts challenged in a math class.)

I also like that color trick.