… + …. = … …. = …….

I am pretty sure that Common Core haters dislike the notion that any of that ought to be explained, that they would prefer that this just be one of the 55 addition facts that ought to be memorized, and let’s move on.

At the same time, requiring 3rd or 4th graders to explain why 3 + 4 = 7 seems ridiculous, EVEN THOUGH SOME 3RD GRADERS HAVEN’T MASTERED ADDITION.

I would ask my 8th graders to explain why 3x + 4 != 7x, but I wouldn’t ask for this in a Calculus class, EVEN THOUGH IT HAS NOT BEEN MASTERED BY ALL.

My point is that we use symbols for efficiency, to avoid explanation. Symbols are NEVER enough explanation at the beginning of instruction.

A student’s presence in a given class presumes a level of previous mastery and efficiency WHICH IS NOT ALWAYS THERE, and instruction examples demonstrate the level of explanation that is expected.

To Chester Draws about the quadratic, I would hope for words like “Quadratic => 0, 1, or 2 solutions” in an explanation.

So a really good question is “what level of explanation should students be expected to demonstrate on a national test? (all right, multi-state, but I am in favor of a national curriculum).

Except that we factorise in order to solve, eventually, and I would not regard solving as a “symbolic exercise”.

Solve: 2x^2 — 13x + 15 = 0

Answer: 2x^2 — 13x + 15 = 0

=> (2x — 3) (x — 5) = 0

=> 2x — 3 = 0, x — 5 = 0

=> x = 3/2, x = 5

That’s pretty much exactly how I teach my students to lay out such a routine question.

How would words help explain that better?

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 |

If a student writes “300*2/3=600/3=200 students are bilingual”, this student would probably get the problem right and yay for showing work!

However, does it mean that the student understands what is going on? Let’s say that this was a homework problem and the student just learned about multiplying whole numbers with fractions at school that day, the student could be multiplying the numbers that he/she sees because that is the math concept that aligns with this hw problem. Does that mean the student understands why? Will that student be able to reproduce this in a different chapter? Who knows.

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.

1) Is there a problem that could be explained using symbolic notation alone?

Yes. I have seen a few wordless proofs of the pythagorean theorem. They are beautiful. So much information and meaning is given in a symbol.

Also, the question that the article linked to (show 5*3 as a repeated addition problem) can also be done using symbolic notation alone. Writing out 3+3+3+3+3 shows that there are five groups of 3.

2) Is there a problem that symbolic notation cannot sufficiently explain? I say YES.

In problems that rely heavily on context, I would say that symbolic notation cannot be sufficient. Why? I recently read a study that asked students to think of a real world context for a given mathematical procedure. Overwhelmingly, the stories created relied on a “three component structure”: their stories gave a set up, then some necessary (and maybe some unneccesary information), and then posed a question. In essence, they have learned that “context” means write a word problem as they have seen it in textbooks.

In my personal opinion, it seems like we should go for the least explanation necessary to illustrate a concept. This is something that many mathematicians do certainly agree on, but “explanation” is not created equal. Symbolic notation is sufficient when the symbols used carry meaning that can be understood by people trained in the discipline. In the examples in 1), the meanings are unambiguous. In 2), however, simply writing symbols offers no insight into what the question was asking.

This was a thought provoking question, and the article referenced was a good (though anger inducing :)) read.

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.

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.