# Is it convenient to classify problem by level of difficulty?

When leaving homework to students, should we state that there is an excercise with a special level of difficulty?

I have seen this done in a lot of physics books, and some books on mathematics have their section of "advanced excercises". Is it convenient al all to use this in classroom?

To make the question more specific, I'm talking about from the point of view of the student, do they benefit from it?

For example: I think that it may make students to feel challenged and more satisfied if they solve it.

Papers on the subject are welcomed.

As a counterpoint to Joe's answer, I would say that classifying problems can be very helpful.

Without this kind of gradation, the student does not know your expectations: if they struggle on some problems but not others, they may assume that the things they are struggling with are "challenge problems", rather than something which is supposed to be considered basic for the course. Then, when this comes up on a test, or in later material, they are lost. Also the reverse problem can occur: a student may think that their lack of success with a challenging problem indicates they are not doing well, which could be discouraging.

If you clearly state which problems are basic and which are challenging, then they know that to do well in the course they must at least understand all of the "basic" problems. Also, the students who are doing well will know not to feel too bad when the challenging problems are, well, challenging.

To summarize, if you do not classify the problems, the students will make their own classification, and they will probably classify them incorrectly. This leads to you and your students having a different idea of what success in the course means.

• +1 for a well reasoned answer. I still wonder how much of a time sink such a process would be. Apr 22 '14 at 20:45
• thanks for your answer, seems reasonable enough. @JoeTaxPayer I really don't think that it may take much time. If you have a system, such as the one StasK explained, it should be quick to determine which problems are "challenging". Apr 23 '14 at 11:25

What I was doing in my (undergraduate statistics) class was to release the rubrics to the students, explaining what they are expected to know at the "C" proficiency level (usually, mechanics), and what they are expected to know at the "A" proficiency level (understanding where a given technique applies). With that, I would

1. give a list of problems from the text book to solve, and ask them to rate their difficulty level;
2. since most of the textbook problems where C level (mechanics), anyway, on some homeworks I would give the list of C problems from the textbook, and give my own A problems.
3. I would also ask them to compose their own problem, and rate its level of difficulty. In the beginning of the next class, I would have them trade problems with the neighbor for an (ungraded) D-I-Y quiz. "A"-level students were sometimes able to come up with A-difficulty problems that I used on the tests :).

If we are talking about derivative of trig functions, say, then

• Find $\frac{d}{dx} \sin^3 5x$

is obviously a C-level problem: mechanics, boring, not that much to do other than applying the rules. But something like

• True fact: $\frac{d}{dx} \sin x = a \sin x + b \cos x$ for some $a$ and $b$. (1) Using the standard trig identities between sine and cosine, and the chain rule, find $\frac{d}{dx} \cos x$ in terms of $\sin x$, $\cos x$, $a$ and $b$; (2) using the chain rule for $\sin 2x$ and $\cos 2x$, as well as derivatives of products and squares, find $a$ and $b$.

-- well, that's a noble, if not upscale, "A" problem (which incidentally also reminds/tests solving a system of two linear equations with two unknowns, with coefficients being symbols rather than numbers).

If you cannot solve "A" problems on the homeworks, it is fairly unlikely you'd do one on the test (which also followed the difficulty levels outlined in rubrics, and problems were explicitly listed as "C", "B" or "A" difficulty).

• Welcome to the site! I really like this answer, but I think there is something missing from your "True fact," since it is a pretty trivial true fact right now (a = 0, b = 1)! What did you mean here? Apr 23 '14 at 1:23

No, I don't think it's convenient, and worse, I think it may be a waste of time. I think there are better things to do with your time than classify questions like this.

The difficulty level can be counterproductive, setting up a student to give up on a difficult problem, or to feel stupid for not solving one that's deemed easy. Last, a given problem might not strike every student the same, e.g. some happen to be better at geometry, so for them a hard geometry problem might be far easier than for student who is otherwise at the same overall level.

I'd be curious to understand the benefits of such classification.

• "setting up a student to give up on a difficult problem" I was thinking the same thing, some students may give up on a problem too early because "it's labeled as hard, therefore I can't do it". But isn't that the case anyway? Some students stops trying when they feel stuck, no matter the "level" of the problem. "Or to feel stupid for not solving one that's deemed easy." no problem should ever be labeled as easy because of that. I was talking about those particular problems that are "harder than others". Apr 22 '14 at 16:42
• If you mean to just have a couple/few extra that are labeled challenge or bonus, that will separate out the harder workers a bit. Apr 22 '14 at 16:46
• Is this an answer? The OP: Is it convenient to classify problem by level of difficulty? In response, you conclude: I'd be curious to understand the benefits of such classification. I think this should probably be a comment... Apr 22 '14 at 17:23
• I agree that this feels close to not being an answer, but one can also read it as a 'no you should not do this' and a somewhat rhetorical add-on. I do not yet convert it, in the hope that OP will expand/clarify.
– quid
Apr 22 '14 at 17:49
• I offered no and potential pitfall. The way the question is phrased invites yes/no answer, and hopefully a bit of reason behind it. I'm on thin ground, I understand, no offense if converted to comment after a time. Apr 22 '14 at 18:04

Either add a challenging problem for "bonus points," or just asign points according to difficulty.

Average students will just do enough to get a passing grade anyway.

• This is sad, and I will agree... Apr 23 '14 at 2:56