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I am currently tutoring a few students in an entry level physics course and had some trouble recently when it comes to helping them with problem solving. The students I am helping don't have many issues when the problems are very straightforward. They understand what formulas to use and what each variable in the formulas represents but they seem to struggle when a problem isn't completely straight forward. I'll give a small example as to exactly what I'm refering to.

The students have the following formula for displacement: $$\ d = v_i t + \frac{1}{2}a t^2.$$ The students had no trouble with any of cookie cutter the hw problems that would ask them to find the displacement after some time $t$ of an object that started from rest ($v_i =0$) and had some acceleration $a$.

Inevitably on their test the students received a problem similar to: Two cars $A$ and $B$ start from rest exactly $88 \text{ m}$ away from each other and accelerate towards each other with accelerations of $A_a$ and $B_a$ respectively. How far has car $A$ travelled when it collides with car $B$?

The students all found this problem nearly impossible to solve and really didn't even know where to begin. Most tried something like $$ 88 \text{ m} = 0t_1 + \frac{1}{2}A_a t_1^2 ,$$ and $$88 \text{ m} = 0t_2 + \frac{1}{2} B_a t_2^2$$ but then didn't know how to go any further.

The trick here is that when they collide the total distance covered is $88 \text{ m}$ and the cars have both been traveling for the exact same amount of time so the equation that needs to be solved is $$\ [0t + \frac{1}{2}A_a t^2] + [0t + \frac{1}{2} B_a t^2] = 88 \text{ m}.$$ For me this was easy to see but when I tried to explain this to the students they really did not understand the reasoning behind it.

This is just one example but the same is true in many other cases. How do I teach them the critical thinking skills necessary to solve questions and not just apply formulas to specific situations?

Here is the link ot the other forum https://matheducators.stackexchange.com/questions/4522/teaching-critical-thinking-skills

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    $\begingroup$ It seems like this question belongs at matheducators.stackexchange.com $\endgroup$ – rschwieb Oct 8 '14 at 15:25
  • $\begingroup$ thanks, I will add a version of the same question there. But I would also like to leave it here as well. $\endgroup$ – KBusc Oct 8 '14 at 15:30
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In my experience as a tutor there was never any one true way to teach critical thinking. The best tip I can give is to solve every problem with them in as many different ways as the student can understand. Graph it, figure it out via unit analysis, blindly plug into equations, rephrase the question, demonstrate logical tricks that can make it easier (like arbitrarily saying one car is stationary), use calculus or diff eq even though it's not necessary, you get the idea.

Then once you've done that, tweak the problem and see if they can modify the approaches properly. They'll likely do a better job with some methods than others, and that can give you some insight into how they're thinking about the problem. It can be boring to solve the same problem over and over again but I think it's one of the best ways to learn.

Also, if you don't already, encourage everybody to draw pictures. Especially for people in introductory physics classes, good diagrams are the single easiest shortcut to solving a problem.

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Do like a sport commentator: just say what's happening. As a sport commentator is talking to the public, whatever they are looking to the live-event or listening at the radio, they will understand just because he is saying exactly what's going on and nothing else. I think if you explain to your students, you're probably telling them your interpretation of the facts, so they have first to interpret your interpretation, then maybe understand. Teach as simple as you can!

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  • $\begingroup$ Give me please the link of this post but in the other forum? Thank you! $\endgroup$ – MadMathFourierRoad Oct 8 '14 at 18:52
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I think (in this particular example) this is less to do with "critical thinking" than applying the appropriate tool for the job. Following on from genisage comment above. In the motion of two bodies on a straight line use velocity-time graphs (or at and st graphs if useful) not suvat/kinematics equations to understand more clearly the motion of the objects relative to each other. Yes, of course, the algebraic relationships "underneath" are still suvat but vt graphs will be more accessible and handle changes in acceleration, different starting points, and different starting times much more easily than relating the reference frames of the objects to a single coordinate system in an attempt to apply suvat to motion of separate objects.

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