1. Many problems in calculus lend themselves to symbolic answers. There is enormous elegance to graphing a polynomial, noting its x-intercepts, relative extrema, and turning points using the language of calculus — doing so much more concisely than is possible with words. Consider graphing f(x)=x^3-3x^2 as an example.

A “find the missing data point” question, given a mean and an almost-complete data set, can be found and justified very well using notation that reflects the structure of the mean of a data set. That’s a favorite of mine from middle school. Consider “You want your average to be a 90%, and your scores on the first four tests are 81, 93, 88, and 88. Is it possible to reach an average of 90? With what score?” for an example.

2. The locker problem — http://mathforum.org/alejandre/frisbie/student.locker.html — is a great example of a problem that begs explanation. It gets at some great ideas , but using formal notation would obfuscate more than it would help, particular at the middle school level where this problem is accessible and worthwhile.

Another favorite of mine is this one: Would you rather get $0.01 one month, $0.02 the next month, $0.04 the next, and so on, doubling every month for 30 months, or get $10,000 the first month, $20,000 the next month, $30,000 the next, and so on, increasing by $10,000 every month for 30 months. The solution here is not trivial, even with some knowledge of exponential growth, and has an interesting human element beyond the purely mathematical comparison that begs explanation.

There is a large volume of research on the idea of a “group-worthy task” — a task that naturally elicits the kind of thinking and collaboration that makes group work worthwhile. The premise here is that group work supports learning when the group is needed for the work, and putting students in groups when the task isn’t conducive to group work is counterproductive. I wonder if we could put together a comparable framework for “explanation-worthy task”, with the same premise — explanations are valuable when they are necessary, and can get in the way when the task does not naturally elicit a worthwhile explanation.

I wonder what the authors of the original Atlantic article would think of this proposition.

And, continuing with my contrarian nature, I don’t think symbols are sufficient for the the factoring problem. Distributing/factoring could be the posture child for getting a correct answer without understanding. Perhaps I am wrong, but I think the typical explanation would be “because you do this time this and this times this …”.

My students asked today why the volume of a pyramid is 1/3 that of the associated cylinder. Symbols alone, nor words alone would do the trick here without some pictures. It would amaze me if any good problem didn’t benefit from a mix of representations.

To me the factoring problem is an exercise, a symbolic exercise, which is why symbols suffice. The interesting thing is determing the terms, though. How did you determine (2x-9) and (3x+10)? There might be a story there. If there was just a symbolic record, that’s where students’ questions would most likely come.

I spend a lot of time in lessons trying to find what there is for students to discuss. Argue about if I’m lucky.

My best guess:
symbolic – integration technique problem where the symbolic shows the technique and why it applies
verbal – why did you choose that function type to model this context?

Do the examples have to come from math?

Thinking of all the Lockhart’s Lament-like analogies from math to other subjects, is there an analogous article for other disciplines? Would following the authors’ recommendations move math farther away from a student’s other experiences? If I were to, say, tutor a contestant of a spelling bee, I’d discern more from a couple discussions of their process than from a definitive list of words they can spell. With that in mind, maybe this whole conversation just differentiates that it’s easy to get good at something and it’s harder to be great.

Isn’t the pool question impossible to answer using symbols alone because it asks for an answer in terms of a pool.

If we assume it didn’t explicitly ask for an answer in terms of a square pool we could say.

We have an area A = a^2, a in Z+
How much bigger is an area B = (a+2) ^2?
Or in symbols what is B-A where B = (a+2)^2.
The answer would follow as pure algebra and need no explanation unless you are asking for which properties of integers you are applying in any manipulations.

Using symbols the problem easily generalizes to volumes and hyper volumes in 3 or more dimensions.
The algebra is also easily extended to rectangles with area A=a x b and 3d blocks with V = a x b x c or beyond into more dimensions.

Now perhaps you really want to talk about numbers of squares and not integers. But then you are working with a problem about square lattices. That is a problem in Z^2 to give it the accepted mathematical symbol.
That sort of question would generalize to non rectangular shapes on Z^2. Again to be precise about what you are asking accepted mathematical symbols would be a good language to use.

I think it would avoid a pointless debate if people were clear on what they are asking for:

A. Explain your work – which could use mathematical symbols or the symbols of a spoken language that is pure math or pure math and a combination of any of the following:
B. Translate your work from mathematical into a particular spoken language.
C. Provide a graphical description of your work.
D. Relate your mathematical language to a particular real world example.

I’m a physics teacher, and feel like a lot of my effort goes into helping students be able to apply math – not just to be able to solve problems in the text.

From Dan’s original post:
I wouldn’t bet that the student with correct but unexplained answers understands nothing, but I wouldn’t make any confident bets on exactly what that student understands either.

To build on this, I think of students’ wrong answers: without explanation it’s tough to figure out how much work you need to do. Nothing drives me more crazy in grading than a student listing an incorrect number for a problem with no work.

Beyond understanding how to assess learning, though, I think we need to talk about the fact that the skill of being able to talk about math in words is, in and of itself, an important skill. My wife does statistics for the state government, and every day she needs to have productive conversations about what she’s doing with people who aren’t trained in statistics. People who are using math in the world usually have to interact with people who are less trained in math than they are. And that requires language.

Prove | a b | = | a | . | b | where a, b are complex numbers.

Not only would I accept the following answer from students, I would prefer it to ones with words:

| a b | = | (r cis θ)(s cis ϕ) |

= | rs cis (θ + ϕ) |

= rs

= r . s

= | a | . | b |