In response to yesterday’s post, Jason Dyer offers us a useful thought experiment:

- Is there a problem that could be completely explained using symbolic notation alone?
- Is there a problem that symbolic notation cannot sufficiently explain?

I vote yes for both and have added my examples in the comments.

**BTW.**

I realized that *the headline* from the Garelick & Beals article doesn’t match their *argument*.

The headline: “Explaining Your Math: Unnecessary at Best, Encumbering at Worst”

Their article: “At best, verbal explanations beyond ‘showing the work’ may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.”

I can see why *The Atlantic* would want to sharpen their writing for the headline. They qualify themselves *twice* in the article (“at best” and “may be”) *barely* making a claim.

So if they think symbols are *always* sufficient explanations, let’s offer questions in the comments for which they *aren’t*. If they think verbal explanations are *sometimes* necessary, let’s let them articulate *when*.

**Featured Comment**:

There’s a place in instruction (somewhere between ages 3 and 8) where each of the symbols “3” and “+” and “4” and “=” and “7” each need explanation, which might look like

… + …. = … …. = …….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).

## 19 Comments

## Dan Meyer

November 12, 2015 - 3:28 pm -1. I think it’s possible to fully explain the factorization of x

^{2}+ 5x + 6 into (x+3)(x+2) through symbols alone. I don’t think a verbal explanation adds much, if anything.2. The Pool Border Problem is impossible to fully explain through symbols alone. There are too many possible equivalent expressions for the number of tiles in the pool border. And a full explanation requires the student to map the coefficients and operations in each expression to features of the pool border itself.

## Dan Meyer

November 12, 2015 - 3:50 pm -2. Additionally, any problem that asks a student to

interpretan algebraic expression in context is really tough to answer using symbols. That’s itspoint.## Kenneth Tilton

November 12, 2015 - 4:28 pm -I think the headline writer did their job, excising the namby-pamby qualifiers we all use to avoid over-committing ourselves into an unsustainable position, producing instead a strong statement the full article backs up well.

That full article does not knock the assessment value of requiring an explanation per se, it says that assessment is an needlessly higher bar than executing the right algorithm in the right circumstances.

The best way to learn something is to teach it, precisely because explaining something requires a deeper mastery than does executing it. It also requires mastery from a different field: rhetoric. This why a good math student is not always a good math teacher.

Your quest for counterexamples should still produce some interesting maths. Look forward to it.

## Dylan Kane

November 12, 2015 - 4:41 pm -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.

## l hodge

November 12, 2015 - 6:12 pm -First, I do think it is a good idea to ask for explanations. But not because they are a good way to check for understanding. Because they often are not.

Does a diagram count as mathematical notation in your eyes? If so, I believe mathematical notation is sufficient (and preferable) for the pool problem. Draw a square with a thin path around it. Split up into squares and/or rectangles. Label relevant lengths for a square of an arbitrary size (“97” x “97”) or use a variable like “x”. Color code the diagram & the equation to connect the coefficients in the equation with the parts of the path. Multiple diagrams, obviously, for the multiple answers.

## l hodge

November 12, 2015 - 6:19 pm -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 …”.

## Denis

November 12, 2015 - 7:28 pm -I am really glad Dan got this conversation going. It is one of those arguments where a valid point was made in a sloppy way, but it led to this conversation, and lots of clarifying thoughts… so it is OK.

My take away from the article was that Smarter Balance and PARCC test makers need to take care in designing prompts that are “explanation-worthy tasks” to use the term coined nicely by Dylan in the comment above. And further, teachers need to do the same. Teachers need to be intentional about teaching when, why and how to explain work beyond a symbolic representation. Pretty much any profession that uses math will require these skills for legal reasons, team work reasons, institutional memory reason… and there are pedagogical benefits in the process too.

## John Golden

November 12, 2015 - 7:43 pm -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?

## Kate Nowak

November 13, 2015 - 5:30 am -I thought about sixth grade content since that’s where my head is these days. Here’s my nomination for a task that requires some words. I imagine that someone might come back with an assertion that kids should just memorize a bunch of different area formulas and practice until they could use them correctly, and we wouldn’t have a very productive discussion. Then there are problems where the solver has to make some decisions about constraints, and it seems like explaining those decisions necessitates using some words. Many good modeling tasks have examples of this idea, but part (c) of this task is a straightforward example.

Here’s a problem whose solution I think could be communicated only with symbols, possibly including diagrams if a student wasn’t ready for an algebraic approach.

There’s also a discussion to be had over the differences between problems appropriate for instruction-only, assessment-only, and problems that could be used for both. The authors of the original article seem to take issue with how problems are posed on assessments, and the painful procedures they observed some teachers devising to train students to maximize their scores on those assessment items, and they don’t seem to understand that problems can be written for different purposes.

## Timteachesmath

November 13, 2015 - 6:15 am -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.

## Timteachesmath

November 13, 2015 - 6:20 am -One thing that might be more unique to math: if there isn’t a good answer to “Why do I have to explain?” then there are probably many students saying, “The teacher just doesn’t explain . . .”

## Stan

November 13, 2015 - 7:43 am -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.

## winnie

November 13, 2015 - 10:23 am -Preservice teacher here! Dan, I agree with you! Yes to both questions, and my reasons are below.

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.

## Rob

November 13, 2015 - 12:07 pm -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.

## Karen

November 13, 2015 - 2:28 pm -Taking the following problem as an example:

There are 300 students at a high school and 2/3 of the students are bilingual. How many students are bilingual?

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.

## Chester Draws

November 14, 2015 - 3:10 pm -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 |

## Chester Draws

November 14, 2015 - 3:18 pm -To me the factoring problem is an exercise, a symbolic exercise, which is why symbols suffice.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?

## Ernest Gunn

November 16, 2015 - 7:56 am -… + …. = … …. = …….

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).

## Dan Meyer

November 16, 2015 - 4:17 pm -@Ernest, I loaded all of that up in the main post. Thanks for the comment.