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Background: Most textbooks end a section with a set of questions ranked either by topic or by difficulty. A distinction is often made between "exercises", which are for directly practicing a known skill, and "problems", which are generally some kind of application questions where the math expected to be used to answer is not spelled out in the question text. Of course, some books also have sets of questions relating to definitions or big ideas or concepts or challenges, and other texts use the terms "exercises" and "problems" interchangeably -- though the harder questions generally come after the easier ones. So, the student who carefully reads the headers for the end-of-section questions has an idea whether the questions should be easy or hard(er) to answer.

Question: In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a scale of difficulty? Do you have experience or know of any research done on the effects (positive or negative) of including such commentary on questions?

Motivation: I write and assemble questions for an online college math course and tend to put the easier/basic-definition/single-skill questions first, followed by word "problems" and applications. I have been considering breaking this into two separate assignments per topic (or just in sections) with labels indicating whether (I consider) the problems are basic/foundational and should be learned to the point of automaticity, or if they are harder and will require some problem solving. I am wondering if there are any down sides to a plan like this.

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    $\begingroup$ There's an interesting conjuction between (a) this question and (b) it being asked on a website where we insist that all posted questions need context. If we insist on context, I think our students are entitled to as well :) $\endgroup$ Jan 5 at 23:16
  • $\begingroup$ biggest problem for me seems that there seems to be no possible ordering on difficulty: what may seem simple/trivial to some people due to being 'mainly computational' may well be horribly complicated to someone else who can't see/operate below a certain level of abstraction (see: many people who suffer on low-level linear algebra and feel much better with categories and whatnot), and also vice versa, of course $\endgroup$
    – ac15
    Jan 5 at 23:44
  • $\begingroup$ $$ \text{begin quote} $$ is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a scale of difficulty? $$\text{end quote} $$ In one of those courses euphemistically titled "college algebra" an exercise is as follows: $$ \text{exercise} $$ The floating leaf of a water lilly doubles in size each day. If it covers the entire pond after $30$ days, how many days would it take two water lillies to cover the whole pond? $$ \S $$ I told the class it was the easiest exercise they had seen. They insisted it was by far the hardest one. $\endgroup$ Mar 1 at 12:57

4 Answers 4

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In a setting where students aren't working from a book with these labels on questions, is it worthwhile for the instructor to indicate to students where the work they are asking them to do falls on a scale of difficulty?

I would say so. One of the things I learned from experience while teaching was that the better I can communicate to students what an intended learning experience should feel like for them, the smoother it will run, and the quicker we can identify any issues that would otherwise slide under the radar. This is especially true in an online course.

For instance, one time I assigned about 15 exercises to be completed for homework, with each exercise intended to take a minute or two. But I didn't explicitly tell this to the students. While most students completed the work just fine, there was one student who became concerned about the workload of the course. Upon further investigation, it turned out that they had a misconception about what was covered during class and spent hours trying to solve the problems in a weird convoluted way. I asked why they didn't just post on the class forum to get help, and they said they didn't realize they were going about it wrong. From that point onward, if I ever assigned exercises that were meant to be quick, I explicitly told the students "if you find yourself working on any of these exercises for more than 5 minutes, and definitely if it's more than 10 minutes, then you've probably got a misconception that is causing you to go about it wrong and you should ask for help so that you don't waste time grinding away unproductively."

Of course, if you tell students not to spend too long on some exercises, you'll also need to give them a heads up when they are expected to spend a longer time on a problem, otherwise they might misunderstand and ask for help too early.

Basically, it's good to give students an idea of how long they should wrestle with a problem before asking for help. This will prevent them from wasting their time (grinding away unproductively), and also from wasting your time (asking for help too early).

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    $\begingroup$ "if you find yourself working on any of these exercises for more than 5 minutes, and definitely if it's more than 10 minutes, then you've probably got a misconception that is causing you to go about it wrong and you should ask for help so that you don't waste time grinding away unproductively." +100 for this. $\endgroup$ Jan 6 at 12:07
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A Frame Challenge

I think that the premise of the question is slightly flawed: I do not think that the distinction between "exercises" and "problems" is one of difficulty—an exercise is not "easy" and an problem "hard". Instead, the distinction between the two is qualitative: exercises are meant to drill, while problems are meant to provoke thinking. This is even pointed out in the websited linked in the question.

It is entirely appropriate to describe this distinction to students. When a student is presented with an "exercise", they should expect that they should have a good sense of how to solve the problem from the start—the expectation is that an exercise will be an application of skills already learned. This does not mean that the exercise will be "easy". It might involve some tedious computation, or a subtle use of some technique already learned (e.g. I could come up with a very difficult integration by parts exercise by building a function which requires two applications of IBP, with the correct choice of "$u$" and "$v$" not being immediately obvious from the start—but such a problem would still be intended as practice with IBP).

On the other hand, if a student is presented with a "problem", the expectation should be that the difficulty of the problem is not the mathematical concept, but the process of getting to the point where whatever techniques have been learned in the class can actually be applied. Problems often require lateral thinking, translation from English to mathematics (e.g. "application" problems), or the integration of multiple ideas which have been taught in the class. A problem that I often give to my precalculus students, for example, is something like the following:

Let $\ell_1$ be the line given by $y = ax + b$, and $\ell_2$ be the line given by $y = -\frac{1}{a} x + c$; note that these lines are perpendicular. Let $P = (d,e)$ (with $(d,e)$ chosen so that $P$ is on one of the two given lines), and let $O$ denote the point where $\ell_1$ and $\ell_2$ intersect. Find two points $Q$ and $R$ so that $OPQR$ is a square.

In this problem, the constants ($a$, $b$, $c$, $d$, and $e$) are chosen so that the arithmetic is pretty simple. This is not a "hard" problem once you see how to set it up (by this point in the semester, they have seen vectors in plane, so the computation ends up being a little bit of subtraction to obtain the vector $\vec{v} = P-O$, swap the coordinates of $\vec{v}$ and make one of them negative to get $\vec{v}_{\perp}$, then compute $P+\vec{v}_{\perp}$ and $O+\vec{v}_{\perp}$). This is not a hard problem, but it does require some work.

NEVER Call a Problem Easy

Contrary to the other answers, I also think that it is a Bad Idea™ to make an attempt to describe the difficulty level of a problem, particularly if "easy" or "simple" or "routine" or "trivial" or anything like that is one of the possible difficulty levels.

The problem is that you will often have students who get tripped up by problems which you think are simple. Or you think that you have presented the students with a simple problem, and it turns out not to be that simple (for example, after introducing students to the change of variables formula ("$u$-substitution), I asked them to integrate something which involved a trigonometric substitution (in what I thought was a relatively simple and direct way)—I thought it was a very easy problem—but students lacked some of the necessary knowledge about trigonometric functions, and a large portion of the class thought the problem was very, very hard). If you call a problem "easy" and a student struggles with it, they are going to disengage.

I do think it is fair to give some estimate about how long you think it will take to get through a particular set of problems, and make it clear that if a student significantly exceeds that time, then they should talk to you (since it is possible that they are simply approaching the problems incorrectly, or that they are missing some prerequisite knowledge).

My Own Practice

In my own classes, I classify problems into three rough categories, with the categorizations related to Bloom's Taxonomy. From my syllabi:

  • [R] (recall; remember, understand, explain): Demonstrating R-level proficiency typically requires you to recall a definition, complete a short computation, or provide an example.

  • [A] (analyze; apply, evaluate, connect): Demonstrating A-level proficiency typically requires you to complete a more complicated computation, apply a definition or result to solve a problem, or use mathematical tools to solve a “real world” problem.

  • [S] (synthesize; create): Demonstrating S-level proficiency typically requires you to combine several ideas in order to solve a single problem—this could include writing a proof, solving more complicated “real world” problems, or using multiple tools (computers, theorems, etc) in order to generate results.

In the language of "problems" and "exercises", R questions are "exercises", S questions are "problems", and A questions are a mixture of both. But note that these levels are not a about *difficulty—the classifications are based on the kinds of thinking that students are expected to have to utilize in order to finish the problems. In general, R questions should probably be easy, and S questions might be nearly impossible (and I do set the expectation with students that they probably won't be able to solve S problems without help), but these are not hard-and-fast rules.

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  • $\begingroup$ In your last paragraph: "But note that these levels are not a about difficulty—the classifications are based on the kinds of thinking that students are expected to have to utilize in order to finish the problems. In general, R questions should probably be easy, and S questions might be nearly impossible (and I do set the expectation with students that they probably won't be able to solve S problems without help), but these are not hard-and-fast rules." -- Can you help me understand what you mean here? $\endgroup$
    – Nick C
    Jan 4 at 22:41
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    $\begingroup$ @NickC Can you clarify your confusion? The point is that I don't classify problems on the basis of difficulty. I classify them based on where I think they fit with respect to Bloom's Taxonomy. I expect things that are lower on the taxonomy to be easier, but this is not always the case. Sometimes a computation (which is essentially a recall task) can be difficult, and sometimes synthesis can be very easy (if a student quickly recognizes which pieces need to be brought together). $\endgroup$
    – Xander Henderson
    Jan 4 at 22:46
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    $\begingroup$ There is a correlation between "difficulty" and "level on the taxonomy", but it is not a one-to-one relation, and I think that "difficulty" is the wrong way to think about things. $\endgroup$
    – Xander Henderson
    Jan 4 at 22:47
  • $\begingroup$ I guess reading "R questions are exercises" and "R questions should probably be easy", I was seeing an equivalence where you didn't mean to imply one. $\endgroup$
    – Nick C
    Jan 4 at 22:54
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    $\begingroup$ Regarding your "never call a problem easy": I've come to use the category "eventually routine" with my students for those-kinds-of-problems. $\endgroup$ Jan 5 at 23:14
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Donald Knuth certainly thought so: he graded the exercises in his monumental work The Art Of Computer Programming from 00 ‘An extremely easy exercise that can be answered immediately if the material of the text has been understood’ up to 50 ‘A research problem that has not yet been solved satisfactorily’.

To illustrate, the section explaining this has its own set of exercises:

  1. [00] What does the rating “M20” mean?

  2. [10] Of what value can the exercises in a textbook be to the reader?

  3. [14] Prove that 13³ = 2197. Generalize your answer. [This is an example of a horrible kind of problem that the author has tried to avoid.]

  4. [HM45] Prove that when n is an integer, n > 2, the equation xⁿ + yⁿ = zⁿ has no solution in positive integers x, y, z.

(Note that that last exercise hadn't yet been proved when that first volume was originally published…  Interestingly, it has two problems rated even harder than that!)

If you're going to include problems with a range of difficulty even a fraction of that, then I think it's only fair to your students to give them some indication, so they won't give up too easily on foundational exercises, nor waste too much time and get discouraged on really advanced problems.

After all, they won't all be as capable as George Dantzig…  (Who arrived late to a statistics lecture and, not realising that blackboard was showing two famous unsolved problems, took them for a homework assignment and handed in solutions a few days later!)

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    $\begingroup$ I have no idea how to answer 1 and 2 and I don't know how to generalize 3 considering these are specific numbers. $\endgroup$
    – Rusty Core
    Jan 6 at 0:02
  • $\begingroup$ @RustyCore As you might guess, question 1 is trivial if you've read the preceding section in the book, which explains all the ratings.  And if you're visiting this SE site, you ought to be able to think of at least one answer to question 2 :-) $\endgroup$
    – gidds
    Jan 6 at 0:56
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    $\begingroup$ I would expect low-point questions to be easily gradeable, possibly automated. I don't think that 2 can be graded in three seconds by a human just by checking the answer, neither can it be graded automatically unless the machine simply counts fancy words. $\endgroup$
    – Rusty Core
    Jan 6 at 1:01
  • $\begingroup$ @RustyCore The ratings in this case clearly relate to ease of answer, not to ease of marking.  While the two are often correlated, your expectation isn't always justified — as these examples illustrate! $\endgroup$
    – gidds
    Jan 6 at 1:23
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That depends on the student level and on the objective of problem solving. For rookies just learning elementary skills, my answer would be "definitely yes". For graduate students taking a topics course with a view at some potential research in the area later, my answer would be "under no circumstances". In the intermediate cases I would say "to some extent, but not entirely".

The reason is that indicating the difficulty level creates a prejudice in the student mind about his/her ability to solve the problem or, at the very least, about the amount of effort needed for it. Such prejudice by itself is useful and can save one's day when getting exposed to some unfamiliar environment (just like any other prejudice in real life) but becomes a hindrance when you navigate the ground with (justified!) confidence.

Just my two cents :-)

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