Students strictly follow the steps and notations in sample problems without understanding them

It's just an observation, but I'll be highly appreciated if anyone with experience in teaching (or TA-ing) lower-division calculus can explain this phenomenon in detail.

I'm one of a TAs who's responsible to write up a section quiz last week. According to the course instructor's email one of the main topics in lectures was partial fraction. He went over examples that involve repeated root in the denominator.

I composed the problem $\int\frac{2x^2+7x+6}{x^3+x^2}\mathrm{d}x$ then realized it might be too challenged for a short quiz. To make it easier, I broke the problem into two parts, and partially gave the answer.

1(a) $$\frac{2x^2+7x+6}{x^3+x^2}=\frac{1}{x}+\frac{c}{x^2}+\frac{1}{x+1}$$ Compute $c$.

and the next part was evaluating the integral.

The interesting thing I observed when I graded was, about a quarter of students wrote down something similar to,

$$\frac{2x^2+7x+6}{x^3+x^2}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$$

Some of them messed up the calculation after that and some of them forgot the "$B$" they wrote was the "$c$" they had to compute.

The thing I don't understand is why people want to make an easy problem harder, so I come up with the explanation I write in the title of this post. Because most partial fraction problems in textbook/lecture set up that way, they want to do all the related problems using identical notations.

Any insight to this phenomenon is welcome.

• Although there is of course the problem of wanting to follow the formula, use their customary notation, etc., part of the problem is you didn't explain what you were doing. It's natural even for students that understand the concepts to be suspicious of a drastically simplified problem in a test situation. This is why if you ask an interviewee a super simple question like "What is 1 + 1 ?" you will get hesitation. You should have explicitly explained that you solved for part of the problem already and they just need to find the remaining part. – Chan-Ho Suh Feb 15 '16 at 21:00

The 25% of students did not think of the problem as easier. It was harder because it did not exactly follow the form of the example problems. As such, it required some (little tiny bit) of creative thought and understanding.

This was an accident. I recommend learning from this, and asking more such questions. The students who are actually thinking will have an easier time on these questions, and the students who are not will have a harder time.

If you want to actually test understanding of the material, this is ideal.

I have a colleague who is fond of asking people this question:

As you might guess, a lot of people make the same notational mistake with the Pythagorean theorem, in that the $c$ in the customary formula does not map to the $c$ in this problem.

The issue of free and bound variables is legitimately tricky. I might prefer to discuss this complication separately, and not have this kind of collision come up by accident when I'm otherwise trying to quiz on some new skill. In fact, I would want to support students being in the habit of writing down a general and shared formula for the kind of problem they're working on. So: If asking this again in the future I'd be prone to change the $c$ to $r$ or something

That's a separate issue from whether some students failed to recognize that half of the work in the partial fraction decomposition process had already been done for them. If so, then your quiz question is a legitimate instrument for diagnosing those who simply failed to understand why the process works, and providing them with appropriate feedback for fixing that.

• This can sometimes border on the ridiculous for some students. A math teacher I know told the story that she always marked triangles as ABC. Once, for a change, she marked it as MNP and there were students who were baffled and struggled more with the problem, simply because of this. – vsz Feb 17 '16 at 7:20

You're probably familiar with muscle memory, where a certain type of motion becomes instinctive after enough repetition, right? The same thing can happen with problem-solving procedures. If you run through a particular procedure enough times in practice, your brain "optimizes" that procedure, and you can do it almost without thinking.

But in all the repetition, you learn that some things vary from case to case and some things are always the same, and if you run into a situation where one of the things that was always the same in practice is now different, the neural pathways don't work anymore. It's no longer instinctive; now you actually have to think, and that's harder and more error-prone.

As you can imagine, students taking a quiz want to take the quickest, easiest, and most reliable route they can find to the solution. If their brain has "optimized" a procedure that involves writing fractions with $A$, $B$, $C$, etc., then they're going to look for a way to cast any partial fraction problem in terms of that procedure, because it is the quickest, easiest, and most reliable route to the solution for them. The extra effort involved in translating the problem is more than offset by the ease of using the procedure they know well.

Of course, one would hope that eventually the students are able to generalize these optimized procedures into a flexible mental model that allows them to do all sorts of algebraic manipulations. As a TA, you probably did this a long time ago, but most of your students likely have not studied this material long enough to do the same. So it's not surprising that they will fall back on stored procedures.

All this suggests that the students have probably been using the $A$, $B$, $C$ decomposition method a lot in their practice exercises, and have internalized it. If you want to break them of that habit, try encouraging them to look at partial fraction problems in a different way, e.g. as plain old algebra.

• In other words, if you want to encourage thinking, the training set of problems should be much more diverse. Then when the novel testing problems come out, they are not looking for mindless symbol mashing. – Steven Gubkin Feb 16 '16 at 20:43
• Indeed, or at least if the students aren't ready to go beyond mindless symbol mashing, it will become apparent well before the test. – David Z Feb 17 '16 at 4:48

When I was a grad student, I asked my advisor if there was such a thing as Zen mathematics to explore, and I honestly didn't appreciate that I was very bright, that that entailed thinking very differently from how students thought and various things that Please Understand Me II might, if I was lucky, have sensitized me to.

I functioned as something of a mathematical Zen master, intentionally breaking not exactly the mind of reason but the mind of mindless symbol manipulation, as an attempt to share with students what was the beauty of mathematics, what poetry most low level practical math classes utterly fail to convey of what draws mathematicians to mathematics.

The one problem I remember was where we had covered matrix multiplication, and I had a 1x12 and 12x1 matrix multiplied together, and besides stating "Credit will not be awarded for taking long time to do unnecessary work," specifically said of the matrices, "Without writing out the result, what do you get when you multiply these two of the above matrices?" and in some cases got an answer of "The times table" and in other cases got the times table listed out. One student acted shocked and betrayed when I docked points for writing out the multiplication table.

I got reamed in the end-of-class reviews, and I believe students' words about "Now it's payback time" were justified.

• If I'd taken that test, I would have said the result of multiplying the two matrices would either be the multiplication table or the scalar sum of squares, because "when you multiply these two" does not specify the order of multiplication. If I'd graded that test, any answer that didn't cover both possibilities would have received partial credit at best. – Monty Harder Feb 15 '16 at 22:21
• The test's wording was more precise than my summary; it wasn't ambiguous as to ordering of requested matrix multiplication. – Christos Hayward Feb 17 '16 at 13:29

That's the reduce-to-the-previous-case syndrome: A mathematician walks into the break room to make a cup of tea. He takes the kettle off the table, boils water, grabs a teabag and leaves, forgetting the kettle on the stove. A second mathematician wants tea too, he goes to the break room, moves the kettle back to the table and declares the problem trivial. See comment by rudistroyer666 at https://www.reddit.com/r/math/comments/1ose5f/favorite_math_jokes/

• Perhaps it's the Einstellung effect at work! – Vandermonde Feb 16 '16 at 9:28
• I heard it with "1st mathematician fills kettle with cold water", "2nd mathematician puts kettle with boiling water into fridge until water is cold". – gnasher729 Feb 16 '16 at 11:18
• @gnasher729, I heard he pours out it so as to reduce to the previous problem. It may be interesting to find the source for this story. – Mikhail Katz Feb 16 '16 at 13:31
• I've heard it, "Engineer is walking along the road and sees a house with flames coming out of an open upstairs window. He spots a nearby hose, plugs it into a fire hydrant, fetches a ladder, goes up it, and puts out the fire. Some time later, a passing mathematician spots that the ground floor window is also open. So he throws the ladder away, unplugs the hose, sets the downstairs room on fire, digs a hole in front of the house and declares the problem solved". – Steve Jessop Feb 16 '16 at 17:05

In many respects this problem reminds me of the sort of exercises that came out of the calculus reform movement of the 1990s. The problem is "conceptual" in the sense that it cleverly does not rely on what David Z describes as "muscle memory."

I think this partial fraction problem is better suited for short group quiz with formative assessment in mind, rather than in the form of an individual summative assessment.

I'd suggest you vary the problems asked. You could teach them to use a computer algebra system (they will do it anyway, if they are the kind of smart students you want to have) to cut down on routine tasks, freeing up time to teach (and review, and train) understanding.