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

1. Is there a problem that could be completely explained using symbolic notation alone?
2. 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:

Ernest Gunn:

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).

Katharine Beals and Barry Garelick argue that students shouldn’t have to explain their answers in math class:

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers–from multi-digit arithmetic through to multi-variable calculus–doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?

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.

Math answers aren’t math understanding any more than the destination of your car trip indicates the route you took. When five people arrive at the same destination, asking how each arrived tells you vastly more about the city, its traffic patterns, and the drivers, than just knowing they arrived.

Their other exemplar of understanding-without-explaining is strange also. Mathematicians advance the frontiers of their field exactly by explaining their answers — in colloquia, in proofs, in journals. Those proofs are some of the most rigorous and exacting explanations you’ll find in any field.

Those explanations aren’t formulaic, though. Mathematicians don’t restrict their explanations to fragile boxes, columns, and rubrics. Beals and Garelick have a valid point that teachers and schools often constrain the function (understanding) to form (boxes, columns, and rubrics). When students are forced to contort explanations to simple problems into complicated graphic organizers, like the one below from their article, we’ve lost our way.

Understanding is the goal. The answer, and even the algebraic work, only approximate that goal. (Does the student know what “80” means in the problem, for example? I have no idea.) Let’s be inflexible in the goal but flexible about the many developmentally appropriate ways students can meet it.

Featured Comments

Really too many to call out individually, but I’ll try.

Susan:

Yet another important thing about students explaining their reasoning is that there is great self-help in a careful explanation of processes. How often have we had a student explain a problem he/she did incorrectly and, in the explanation, the student realizes the mistake without a word from us? This “out-loud-silently-in-my-head” thinking is such an important thing to help students develop.

David Wees articulates a similar point:

Another reason that we might want to listen or read a student explanation of how they solved a problem is just so, in the process of articulating their solution, students may run into their own inconsistencies in their work. I have noticed, quite often, that students will give an answer that I don’t understand, and then when I ask them to explain what they did, in the middle of their explanation they say something like, “Oh, oops! Yeah that isn’t right. I mean this instead” and revise their thinking.

David Coffey:

Mathematicians use words in describing their discoveries all the time — and have for a long time. That’s why some doctorates in mathematics require a foreign language so that the candidate can read the mathematicians’ writings in the original language.

Michael Pershan poses an interesting thought experiment that no one takes up:

1) Can the traditionalist and progressives find a lesson/activity/short video that they both agree is lovely teaching.
2) Same thing, but they both agree it’s lousy teaching.
3) Can each identify a whatever that they like, but their sparring partner doesn’t. Can they explain why.
4) If the disagreement persists, can they explain why they think it does?

Elizabeth Statmore uses an explanation protocol called Talking Points and brings student voices into the conversation.

Tracy Zager excerpts quotes from mathematicians on the value of explanation in their own work.

There is also an exchange between Brett Gilland and Ze’ev Wurman that lays bare two views on teaching that are completely distinct. It’s devastating.

You’ll find another great exchange between Brett Gilland and Katharine Beals (search their names throughout the comments) which ends rather unconventionally for Internet-based discussions of math education.

2015 Nov 21 Katharine Beals has responded on her blog to the objections raised by commenters here. Parts #1, #2, #3.

2015 Nov 28 Education Realist has posted a response that dives into the difference between elementary math ed (the site of Garelick & Beal’s research) and middle school math ed (the site of Garlelick & Beal’s arguments).

Definitions of “inquiry” typically focus on the student’s inquiry. Fine, but I also appreciate Mylène at Shifting Phases’ shifted focus to teacher inquiry:

My definition of “inquiry” as an educational method: it’s the students’ job to inquire into the material, and while they do that, it’s my job to inquire into their thinking.

So she measures the quality of her tasks and instruction by how how much access they grant into her students’ learning. She also shares an organizational strategy that helps her understand which of her tasks grant her the most access. All great.

Featured Comment

Tracy Zager writes in response to several commenters who think this is all obvious and everybody already does it:

I disagree in a big way. My own children come home from school with endless folders of completely useless products. Useless in that they give the teachers no actionable information, no insight into the children’s thinking.

Imagine you had a single piece of student work and were going to talk about it with colleagues for an hour or two. Which pieces of work would lead to rich discussions about students’ thinking and mathematics, and which wouldn’t? If you’d run out of things to talk about in 5 minutes—if the assignment wouldn’t lead to a productive, insightful discussion among teachers—why are we assigning it?

Yeah, practice. Why else? That can’t be the answer for everything.

Most of what I see assigned in schools yields no insight into students’ thinking for teachers.

So I think this is a big, important idea that goes far beyond common sense.

2015 Oct 31. Mylène posts a follow-up, “Who’s Inquiring About What?” and the the last paragraph is a stick of dynamite:

Want to help me improve? Here’s the help I could really use. If you were one of the people whose first reaction to my original post was “I already know that” – either I already know that to be true, or I already know that to be false… what would have helped you respond with curiosity and perplexity, adding your idea as a valuable one of many? If that was your response, what made it work?

2015 Nov 11. Also make sure you read (at least) the intro to a paper linked by Brian Frank that coins the term Discovery Teaching.

Russell Davies, all the way back in 2006, in a post called How to Be Interesting:

The way to be interesting is to be interested. You’ve got to find what’s interesting in everything, you’ve got to be good at noticing things, you’ve got to be good at listening. If you find people (and things) interesting, they’ll find you interesting.

A teacher emailed me after my workshop at the Alaska State Math & Science Conference:

As I mentioned after your session, I watched your CUE talk and have since worked to cultivate my Feedly account to provide more perplexing math content, inspiration, and lesson ideas. I have followed ed-tech and blended learning resources on Twitter for years, but am looking to expand my resources for engaging and interesting math content.

So I’m going to share a certain set of blogs. I follow these blogs with so much devotion, I’d be surprised if I’ve missed more than a handful of their posts since I first started following. And I’ve been following some of them for close to ten years. Some are written by math teachers but most aren’t. They share two features in common:

1. They link. Much of their content isn’t original, and little of it relates directly to math or pedagogy, but they share links that reliably light up the cluster of my neurons that loves to design lessons for kids. (A tech blogger inspired my Joulies lesson, for instance.)
2. They’re interested. Even more than they’re interesting, they are self-evidently interested people who have cultivated a way of looking at the world and being in the world that I want for myself. They are voracious, omnivorous consumers of their surroundings.

I started to share this set of blogs in an email reply to the teacher, but I’d rather share them with all of you, and then I’d rather all of you share your own set of links in the comments, links that fit the bill I’ve described.

• Best of Metafilter
• Colossal
• Daring Fireball
• Devour
• Futility Closet
• Kottke
• Lonely Sandwich
• Marco.org
• Math Munch
• (Mathhombre) Miscellanea
• Sine of the TImes
• Visualizing Math
• Waxy Links

Here is the original Malcolm Swan task, which I love:

Draw a shape on squared paper and plot a point to show its perimeter and area. Which points on the grid represent squares, rectangles, etc? Draw a shape that may be represented by the point (4, 12) or (12, 4). Find all the “impossible” points.

We could talk about adding a context here, but a change of that magnitude would prevent a precise conversation about pedagogy. It’d be like comparing tigers to penguins. We’d learn some high-level differences between quadripeds and bipeds, but comparing tigers to lions, jaguars, and cheetahs gets us down into the details. That’s where I’d like to be with this discussion.

So look at these four representations of the task. What features of the math do they reveal and conceal? What are their advantages and disadvantages?

Paper & Pencil

You’ve met.

Dan Anderson’s Processing Animation

Hit run on this sketch and watch random rectangles graph themselves.

Scott Farrar’s Geogebra Applet

Students click and drag the corner of a rectangle in this applet and the corresponding point traces on the screen.

Desmos’ Activity

277 people on Twitter responded to my prompt:

Draw three rectangles on paper or imagine them. Choose at least one that you think that no one else will think of. Drag one point onto the graph for each rectangle so that the x-coordinate represents its perimeter and the y-coordinate represents its area.

Resulting in this activity on the overlay:

Again: what features of the math do they reveal and conceal? What are their advantages and disadvantages?