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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 $88\,\mathrm{m}$ 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 $$ 88\,\mathrm{m} = 0t_1 + \frac{1}{2}A_a t_1^2 ,$$ and $$88\,\mathrm{m} = 0t_2 + \frac{1}{2} B_a t_2^2$$ 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 $$\left[0t + \frac{1}{2}A_a t^2\right] + \left[0t + \frac{1}{2} B_a t^2\right] = 88\,\mathrm{m}.$$ 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?

Also feel free to add the appropriate tags. I found none that seemd to fit.

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    $\begingroup$ You might be interested in Brent & Felder's article: www4.ncsu.edu/unity/lockers/users/f/felder/public/Columns/… $\endgroup$ – J W Oct 8 '14 at 16:36
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    $\begingroup$ I took the liberty of adding spacing, fixing your LaTeX, adding the proposed tag and fixing grammar. $\endgroup$ – Mark Fantini Oct 8 '14 at 17:52
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    $\begingroup$ @Jared By stating "I don't think this can be taught" you undermine all attempts to do so. I also disagree with "physics/math problems are essentially pattern matching." Last, but not least, it's entirely reasonable to expect most students to apply critical thinking. They just need guidance on how to do so, and that is the purpose of the class. $\endgroup$ – Mark Fantini Oct 9 '14 at 12:23
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    $\begingroup$ @Jared I may have used this method before but I really don't think so. My thought process was: 1) How long until the collide? I didnt know but i realized that the time was going to be the same for both objects. Then 2) I drew a picture in my head of what a drawing of this scenario would look like and at that point it was obvious to me. This thought process is what I am trying to teach I do not think any specific problem from my past was the reason I knew how to do this one. I cannot possbly remember every problem ive done though so I suppose its possible. $\endgroup$ – KBusc Oct 9 '14 at 17:31
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    $\begingroup$ @Jared Those are bold claims, i.e., "you don't understand how you can solve math/physics problems" and "you don't understand why you are able to solve them." Moreover, they are not true. We can detail our thought processes and investigate how our knowledge was brought together and deployed in order to successfully engage and solve a particular problem. This is related to metacognition, which has been touched upon many times in answers here in MESE. Solutions aren't selected, they are crafted. $\endgroup$ – Mark Fantini Oct 9 '14 at 18:16
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I know I shouldn't add another answer, but I think my other answer went off on a tangent that didn't really address your question. I did not initially read it carefully enough to realize you were tutoring students, not teaching a class. I really don't think the label of "Teaching Critical Thinking Skills" is either relevant or pertinent--teaching is teaching, if we want to label "teaching" as "teaching critical thinking skills" then I think we are defining want teaching is--if you actually want to "teach" critical thinking, then this is a far more broad question which is not at all specific to mathematical instruction.

When tutoring students, you have a very good opportunity to give them deep and meaningful instruction (much more so than in a classroom setting). It's easy to just give them the solution, but obviously not very helpful (and unlikely to further the students learning). Having said that, the end goal must be to have the student leave with a solution, preferably one they arrive at as close to on their own as possible. Different students will come to you with different abilities and for some students, you may have to end up explaining the entire solution (but this should be a last resort).

Asking questions is vital. You need to nudge the student but not push them over a cliff (i.e. give them the entire solution). I will use your problem as an example:

Step 1: Ask the student whether or not they drew a picture. If no, then tell them to draw one, if yes, have them show you their picture.

Step 2: Analyze their picture. Is the picture correct? Perhaps they drew the cars going in the same direction (as opposed to driving towards each other). You want to avoid cuing the student; so do not automatically accept a correct picture (assume it's wrong no matter what they draw). Ask them to explain their picture. Does the picture agree with the setup of the question? Why did they draw car $A$ on the left and car $B$ on the right (or vice versa)? Is that correct? Does the problem specify this? Does it matter?

Step 3: Label the picture. What is given in the question? Can we introduce variables (unknowns) that we may or may not be able to solve for? Assuming the picture is now correct, hopefully the student will, at a minimum, label the initial distance between the cars as $88$ m. If this is all they label, then point out the distance between the starting point of one of the cars and the collision point. Ask them if they can find this distance. No? Then let's just label it was an unknown. If you do that for one car, hopefully they will see you can do that for both cars. If not, you will have to point that one out too.

Step 4: Point out relationships we can deduce from the picture. Is there anything we can say about the two distances? Do we know anything about them? If they are baffled, ask does the initial distance of $88$ m change? They may say yes, it changes as the cars get closer together. If they go this route, then let them continue. Explain that they now need a second picture with the cars closer together. Do we have more or less information in this second picture? Do we know how far apart cars are now? Can we figure it out? How much time as elapsed from the first picture to the second?

Step 5: Start solving the problem. With any luck, you stopped around the 2nd or 3rd step and the student did the rest, but you may still have to go this far. If the student is still confused, you will need to start asking about time. OK, we have the two distances labeled, how long did this take? Can we come up with a relationship between the distance and time? If that doesn't work, you may have to ask what if I told you that car A traveled its distance in 2 seconds? Could you find the distance then? If car A traveled its distance in 2 seconds, then how long should it take car B to travel its distance? Hopefully the student will answer 2 seconds. So what can we say about the times it takes car A to travel its distance and the time it takes car B to travel its distance?

If the student is still baffled, you may have to just tell them the two times must be the same. Now ask, what if the time was 2 seconds? What would the two distances be? Is that correct? How do you know it's not correct? Hopefully the student will realize that the two distances (using $t = 2$ seconds) do not add up to the full $88$ meters or they add up to more (and hopefully $t = 2$ seconds is not coincidentally the correct time--if so choose a time you know won't be correct). Can we find the correct time?

If they are still unsure, ask them again, how you knew $t = 2$ seconds was incorrect. Have them write out the equations without evaluating (i.e. $d_A = \frac{1}{2}A_a(2)^2$ and $d_B = \frac{1}{2}B_a(2)^2$). Ask why these two distances are incorrect (hopefully they already answered and thus have the answer ready). Because they do not add up to the full distance. So if $2$ seconds was the correct time, what equation would you expect to be true. If they are unsure, tell them to write it out in full, you may have to simply tell them to write down $\frac{1}{2}A_a(2)^2 + \frac{1}{2}B_a(2)^2$. Ask them what this equation should equal. They now should essentially have the equation. Can you find the time now? If they are still baffled, ask them what do we do when we have an unknown? We create a variable for it. At this point, if they are still confused, you may have to flat out tell them to substitute $t$ where they placed $2$ (remind them that $2$ was the time we guessed).

Now you have two problems: 1) we need to solve for time, so you are now no longer a physics tutor, but an Algebra I tutor and 2) you need to ask them, after all of this work, what was the question asking for? Did it ask for the time? No? It asked for the distance car A traveled? If we know the time, can we find this value? Hopefully they realize they can, but if not, remind them that we have an expression for $d_A$ already, we just need to plug in our value of $t$ into it.

Conclusion

I know that this explanation was very specific to your example problem, but the strategy is essentially the same: 1) Draw a picture, 2) analyze the picture, 3) label the picture, 4) find relationships, and 5) solve the problem. The absolute hardest part is trying to avoid cuing the students' responses. For instance, don't ask: "does [insert kinematic equation] help us?" If you do have to resort to this, then you should go through them all and have the student explain why each is either useful or not applicable. Don't automatically accept correct answers and don't automatically reject incorrect answers. Let them continue with the wrong way until you hit a point where you can no longer solve, then help the student backtrack to find where the mistake is. Your goal as a tutor, should be to do as little as possible. The more time you spend with a particular student, the more you will learn what they need from you (duh).

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    $\begingroup$ I think this is an extremely good answer. I have seen a huge number of students who have developed methods of "learning" that rely entirely on watching for the teacher/tutor's cues and responding to them, and your advice could prevent such unhelpful strategies from ever developing. $\endgroup$ – Chris Cunningham Oct 12 '14 at 15:09
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There is quite a bit of evidence that critical thinking can be taught (which I found surprising). College students' gains in critical thinking skills over the course of their education are correlated with high standards set by instructors and greater time spent studying. So I think the basic answer is that you have to assign students tasks that require critical thinking, and grade them on those tasks. It's not something you can do overnight, and the gains are not large, but the gains do exist.

Some references:

Academically Adrift: Limited Learning on College Campuses, Arum and Roksa, 2011, summarized in http://www.newyorker.com/arts/critics/atlarge/2011/06/06/110606crat_atlarge_menand

Pascarella and Terenzini, "How Robust Are the Findings of 'Academically Adrift?'" http://www.education.uiowa.edu/docs/default-source/crue-publications/AcadAdriftChangeArticleFINAL.pdf

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    $\begingroup$ The second link doesn't work, Ben. $\endgroup$ – Mark Fantini Oct 9 '14 at 17:50
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    $\begingroup$ @MarkFantini: Thanks, fixed. $\endgroup$ – Ben Crowell Oct 10 '14 at 0:15
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    $\begingroup$ @ChrisCunningham OK, I did so. Also, this answer's references do not support their [the answerer's] conclusions--in fact, the 2nd reference seems to refute the conclusions. Perhaps I didn't read the reference with fine enough detail--I don't see anything about where gains were particularly high (only that they are generally fairly low). $\endgroup$ – Jared Oct 11 '14 at 1:54
  • $\begingroup$ @Jared: I didn't claim that the gains were large, only that they existed. (I just find it surprising that they're nonzero.) Why do you say that the second paper refutes the conclusions? $\endgroup$ – Ben Crowell Oct 15 '14 at 0:01
  • $\begingroup$ @BenCrowell First, I want to avoid being adversarial here (which I did not do initially). Your conclusions seem to conflict with the tone of the second reference (I didn't really read the first one since it's a magazine article). The second reference wasn't saying, look how great college is, it produces marginal increases (sometimes) in critical thinking skills (based on a critical thinking test). The second reference was basically saying, look how bad college is at teaching critical thinking skills--the gains are marginal at best and none at worst. $\endgroup$ – Jared Oct 15 '14 at 4:29
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"Do like a sports 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 happening 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!"

EXPLANATION: by "do like a sports commentator" I mean: tell to the public (your students) step by step what is happening, so that the public can understand everything from the event. After the event, an amount of sports experts meet each other in a tv-show and discuss about the sport match. They do comments about it, they suppose how could it be in a different way, they say which were the mistakes.

Now transfer this situation into your class. The problem has to be solved. The teacher or tutor, explains to the students what is the problem asking for. Then the theacher gives his solution, and says step by step how he achieved it.

After that, he asks to the students to give their solutions. At this point, begins a discussion about the problem and its different ways to solve it. Of course every sentence should be motivated and explained as easy as possible, to avoid any misunderstandings caused by a to personal interpretation.

I think this process could encourage in the students a sort of "critical thinking".

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    $\begingroup$ I have downvoted because this answer is not useful. This is merely an opinion. $\endgroup$ – Mark Fantini Oct 10 '14 at 11:09
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    $\begingroup$ @MarkFantini tell me why is it an opinion? $\endgroup$ – Fabio Costa Oct 10 '14 at 11:18
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    $\begingroup$ Where's your research? $\endgroup$ – Mark Fantini Oct 10 '14 at 11:42
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    $\begingroup$ @MarkFantini experience with students and observe sense. it's true i did no research about, and that's why you can't say with sureness mine is an "merely opinion". If you are so sure, where's your research, which certifices your statement? $\endgroup$ – Fabio Costa Oct 10 '14 at 11:50
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    $\begingroup$ I down-voted for a few reasons. One reason is that I don't understand how one applies the following to teaching critical thinking: "saying exactly what's going on and nothing else." (I'm not even sure what this means.) I also don't know how you can help someone's teaching practice by declaring: "Teach as simple as you can!" (Again: I don't quite know what this means.) And, as an objection less directly relevant to mathematics education (but still related due to its use as an analogy here), I don't think you have accurately characterized what a good sports commentator does. $\endgroup$ – Benjamin Dickman Oct 11 '14 at 3:47

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