Questions with "round" answers only?

Question

Textbook writers are blessed with only solving problems with neat answers. Numerical coefficients are small integers, many terms cancel, polynomials split into simple factors, angles have trigonometric functions with known values. Pure bliss.

The "real life" is different (as any of us knows).

Giving such questions for homework runs the risk that the student knows she is wrong when some crooked formula/value shows up. On the other hand, checking messy results (unless perhaps directly numerical values) is harder.

What do you think about posing questions which don't have neat derivations/results? I presume the answer could depend on the subject matter, the level of the students, and perhaps on exactly what the question should teach.

Answer 1

When teaching linear algebra, I rely heavily on carefully constructed examples where everything can be done with rational numbers of small denominator. (It is faintly amusing that the construction of such examples often requires mathematics that is far harder than the actual content of the course.) However, I also show them the following slide:

I also tell them explicitly and repeatedly about what is generic, emphasising that repeated eigenvalues are only likely to arise when forced by symmetry, for example.

I think it is OK to stick with nice examples in this context, because the whole point is just to illustrate the conceptual theory. If you actually need complex examples then you should obviously use a computer for the calculations anyway.

Answer 2

I agree that such exercises have a built in risk of giving the students the wrong idea what is the generic situation and what is quite a special case.

However, "the risk that the student knows she is wrong when some crooked formula/value shows up" you mention is more a feature in my opinion. It is not a bad thing to do heuristic checks that suggest one might have made an error and to then double-check.

Also, for some things it is more or less unavoidable to present constructed exercises. A polynomial will not have a multiple root just so, you need to take care it has one.

Or, in linear algebra one has to choose the matrices somewhat carefully if one wants to illustrate certain things. And, it is a real pain to calculate even with moderately sized matrices if they involve radical expressions, and if you go for decimal approximations you 'lose' the clear notion of singularity of a matrix and would actually need to do something more advanced. So, somehow one is stuck with matrices whose eigenvalues are "too nice" (and this even leaving the issue of factoring the polynomial aside.)

Personally, I mostly tend to give exercises where things work out nicely, yet I do talk with students about the fact that this is what happens and that this does not capture what would be a "typical" situation were one to choose things "randomly."

That is, I think it is important to raise awareness of the issue of "artificially nice" problems with students, but as mentioned above consider it as unavoidable to use them in certain cases.

Answer 3

I would encourage to have a significant amount of non-"round" numbers in your homework and also in exams. Some reasons:

• Except when you would teach calculations with numbers, normally the way you do things is important, not actual calculations. Even if the students feel that something is wrong with the numbers: As long as his/her way was correct, it is only a very small part of points not gained due to miscalculation.
• If your students know that the only possibility is that number results are "nice", you add some sort of bias to the result: If the calculation is not too complex, students could try to do basic calculations with the given numbers in order to arrive at a "nice" result and can so perform "reverse engineering" in order to get the right idea only by guessing and trying.
• Even when you construct a "nice" example with "round" numbers, there could be a chance that there are two possible correct ways: One with your constructed nice numbers and one with not so nice ones. (For example, if you perform a QR decomposition via Householder reflection, there a freedom in the sign of one real number (on the wikipedia-link this is called $\alpha$). Depending on the choice of the sign, you get different values. There is a recommendation how to choose the sign, but in most examples this does not play a role and you are really free in the choice of the sign.).
• You can train other skills in finding out if a calculation is completely wrong like perfoming rough calculation and estimating lower and upper bounds for some result. In real life this should be also very useful.
• If you allow a calculator in the exam and there is nothing where you really need it, this could be misleading to the students.
• As it was mentioned by the OP and in other answers, it is misleading that there will be only nice values as a result of a calculation since real world values are not nice.

However, it is nice to have "round" results on many examples (in particular those you perform on the board) because you can save time and not distract students from the actual topic.

Answer 4

Like others I more often than not craft the problems to have nice solutions. Just to keep the students on their toes I occasionally insert something not so nice. As we are largely discussing eigenvalue problems, let me propose the following trick I picked up from a senior colleague.

Use a problem, where the precise eigenvalues are not needed, only their signs. For example you can ask whether a given 3-variable quadratic form is positive definite or not. You probably should warn students that they may need to estimate the zeros of the characteristic polynomial as opposed to find them. Also the students need to recall how to do that - always nice to use stuff from courses with very different topics. Your task as the problem designer is then to make sure that the zeros have distinct integral parts, so that it won't be too arduous a task to locate intervals with integral endpoints each containing a single eigenvalue.

Answer 5

It may be useful to present at least one application that is solvable but does not have a simple answer. As an example, quadratic equations with rational coefficients that cannot be easily solved by factoring and require use of the quadratic formula arise in chemistry.