Wrong answers are part of the process too. Time and again in the Japanese classroom you’ll see, “Jen discovered an approach that doesn’t work. Jen, explain your discovery to the class.” Jen explains the discovery to the class. “Does everyone understand Jen’s discovery? Now let’s all figure out why it didn’t work. Jen, did you figure out why it didn’t work? Let’s figure it out.” It’s actually often easier to get to the math figuring out why an approach didn’t work than why an approach did work.

[N.B. Often times a student has correctly answered a different question, and asking “For what question would Jen’s approach work very well?” is generative.]

Apostolos Doxiadis argues for more “paramathematicians”:

If our rationale for teaching a subject is circular â€“ â€œyou must learn it because it is useful, because it has uses, because it is useful, because you will need it later, because it is usefulâ€ â€“ we wonâ€™t go a long way. A developing human being is many things, and chief among them a poet, an adventurer and a problem-solver. Give the poetry, the adventure and the problems, through stories, both small stories of environment and large stories of culture. Grip the heart â€“ and the brain will follow.

As for the mathematicians themselves: donâ€™t expect too much help. Most of them are too far removed in their ivory towers to take up such a challenge. And anyway, they are not competent. After all, they are just mathematicians â€“ what we need is paramathematicians, like you…. It is you who can be the welding force, between mathematics and stories, in order to achieve the synthesis.

David Gessner built a shack for himself but left a gap between the door and the roof. He was rewarded when he didn’t patch that gap. Read about it and then imagine the contents of a blog post entitled, “Leave Some Gaps In Your Tasks.”

**Featured Comment**

We may well ask of any item of information that is taught â€¦ whether it is worth knowing? I can only think of two good criteria and one middling one for deciding such an issue: whether the knowledge gives a sense of delight and whether it bestows the gift of intellectual travel beyond the information given, in the sense of containing within it the basis of generalization. The middling criterion is whether the knowledge is useful. It turns out, on the whole, â€¦ that useful knowledge looks after itself. So I would urge that we as school men let it do so and concentrate on the first two criteria. Delight and travel, then.

## 16 Comments

## David Wees

May 30, 2012 - 8:27 am -“Most of them are too far removed in their ivory towers to take up such a challenge.”

I don’t actually know how true this part of his post is. Anecdotally, I know mathematicians who are certainly part of the problem when it comes to reform of mathematics education, but I also know some mathematicians who are really pushing for pedagogical change. Certainly, Paul Lockhart and Gordon Hamilton spring to mind as somewhat progressive thinking in their views on mathematics education.

As for the “math will be useful” argument, I certainly think that falls flat when the math in fact will not be useful, and only a complete idiot would think that it would be useful. We are not fooling any kids by telling them that finding common denominators for rational polynomial functions is a useful life skill.

## Bowen Kerins

May 30, 2012 - 8:44 am -There’s another, longer talk by Phil Daro about CCSSM and a lot of these examples of answer-getting, along with his opinion on what may and should happen when CCSSM becomes The Law Of The Land real soon now.

http://www.lsri.uic.edu/ccss/

## Jason Dyer

May 30, 2012 - 9:14 am -@David: Apostolos’s response seems to the “math will be useful” argument seems not to be “let’s pretend rational polynomial functions will be a useful life skill” but “let’s incorporate the historical narrative.”

This is something I’ve only had mixed success with; I’ve certainly gone out of my way to include historical narrative when I can, but students can be equally apathetic in History Class (even one focused on narrative) as they are in Math Class. The stories Apostolos shares about readers responding to his book are a very particular subset — more or less intellectual folks who like to read books — and not the “average” student.

Apolostolos also hints about problem solving itself being a narrative, but is frustratingly vague about the details. Dan’s certainly got something better going, but I’d really like to hear more about The Hard Part: Act 2, the meaty part with the math stuff the students are dying to skim over. That’s partly where I’ve been trying to go with applying game design structure and non-linear narrative techniques. I believe a lot of issues in applying traditional story structure to the classroom come from the fact it’s not a linear structure at all, and trying to maintain interest and suspense when students can go in fifty directions requires heavier artillery.

## Jerzy

May 30, 2012 - 9:55 am -@ Jason: I don’t Doxiadis necessarily means tying math to history class. In most classrooms, it’d already be an improvement just to admit that equations don’t spring fully-formed from mathematicians’ heads.

I’m sure you’ve seen this cartoon about how math proofs are written:

http://abstrusegoose.com/230

The Daro and Doxiadis quotes remind me that when I speak of “doing math” I’m thinking of the first 8 panels; but most people are never exposed to anything beyond following the directions in panel 9.

## David Wees

May 30, 2012 - 2:24 pm -@Jerzy

I love, love, the cartoon. I’m sure many of those beautiful proofs I learned in real analysis took years to perfect.

@Jason

Given that one can use free tools (see http://WolframAlpha.com, for example) to find the sum or difference of two rational expressions in seconds perhaps we could go a bit further and suggest that we should take some of the computational topics we teach as skills now, and teach them as historical mathematical techniques. The key to using these tools and not be handicapped, is to understand the limitations of the tools, and be able to predict the output of a tool in advance of using it.

I agree that the second act is the hardest part, and it’s where we make our real money as educators.

## Jason Dyer

May 30, 2012 - 3:59 pm -@Jerzy: Ha! Nope, hadn’t read it.

@David Wees: It certainly happened with square roots. Things don’t seem to have budged since then.

## Bowen Kerins

May 30, 2012 - 5:09 pm -I think that’s also happened with logarithms. Log methods used to be taught in gory detail to support by-hand or by-table approximations. There’s still a lot of good meat in the topic — great example of an inverse function, good application problems that can be solved — but the focus on computation is reduced.

Rational expressions, in my opinion, really needs to be shown the door. In the curriculum it mostly exists for rewriting expressions, notably for calculus integrals like the integral of 1/(1-x^2). That stuff is getting deprecated in two ways: the computation can be performed by technology, and there is no longer a need for an explicit solution. At the high school level, there’s not much really happening with it other than talking about function domain, and giving messier and messier algebra work (and factoring). Bah.

There’s a lot of stuff like this — why do we care about converting the form of an ellipse? — but rational expressions is one of easiest topics to zap cleanly out of the way.

## Julie Reulbach

May 31, 2012 - 6:38 am -Phil Daro’s entry reminds me of the “My Favorite No” warm-up activity (found here https://www.teachingchannel.org/videos/class-warm-up-routine).

I read about it in Kate Nowak’s blog and have used it ever since. Students do one problem at the beginning of class on an index card. I sort them and then anonymously pick my favorite incorrect answer, or answers, (“Favorite No”). Then, we go through the problem as a class to see where the student went wrong. Along the way, we emphasize all of the things the student did correctly as well. The students love it and really seem to learn from it. They are often mistakes that several students have made. The students are also encouraged that they only made one mistake and did many steps correctly. The mistakes that we find on the cards are usually ones that the students become more aware of and thus do not seem to repeat in the future. The included video is a much better explanation of this!

## Elizabeth

May 31, 2012 - 7:14 am -Did not know David Gessner’s writing before. Thank you, THANK YOU for the introduction!

## Barry

May 31, 2012 - 10:53 am -@David: “We are not fooling any kids by telling them that finding common denominators for rational polynomial functions is a useful life skill.” Amen to that. Apostolos Doxiadis used one Jerome Bruner quote, but avoided my favorite:

We may well ask of any item of information that is taught … whether it is worth knowing? I can only think of two good criteria and one middling one for deciding such an issue: whether the knowledge gives a sense of delight and whether it bestows the gift of intellectual travel beyond the information given, in the sense of containing within it the basis of generalization. The middling criterion is whether the knowledge is useful. It turns out, on the whole, … that useful knowledge looks after itself. So I would urge that we as school men let it do so and concentrate on the first two criteria. Delight and travel, then.

Narrative certainly provides both delight and travel (including the basis for generalization through use of allegory and metaphor). And for those few students for whom algebra skills actually will be useful later, perhaps we are now in an era where that knowledge can take care of itself (although experience with current college students suggests to me that a desire to be, say, a physical chemist or an actuary along with pointers to information about basic skills that were forgotten or never learned is not usually sufficient for the knowledge to “take care of itself”.)

@Jerzy — great cartoon!

@Bowen — Logarithms are a wonderful example of where historical narrative is essential for making the topic bearable! Perhaps prosthaphaeresis could do the same for trig identities!

However, I disagree with you about the need for rational expressions, at least for students going on to do calculus. As you indicate, they are the prime example for understanding the difference between an expression and the function it defines (more precisely, the phenomenon of equal expressions defining different functions). This particular misunderstanding seems to me the precise reason why students do not understand why all limits aren’t computed simply by “plugging in the value”, which itself leads to them not having a basic understanding of the definition of derivative (even though they might very well understand the intuitive description of a limit of slopes of secant lines.) Now you can argue whether there is a practical need for students for understand the definition of derivative, but having proceeded this far it is easy to introduce history and narrative by talking about differentials, the long process of developing the definition of limit, early successes in computing derivatives, etc.

## James Key

June 1, 2012 - 10:53 am -â€œGrip the heart â€“ and the brain will follow.â€

Priceless words.

@Jason: “It certainly happened with square roots.” I would argue that it is *extremely important* for students to learn to compute square roots. I’m not sadistic enough to suggest they should do it beyond the tenths place, and a handful of practice problems would be enough to satisfy me.

Knowing how to show that the square root of 2 is about 1.4 (i.e. because 1.4^2 is 1.96) is, to me, a very accessible skill (14^2 = 196!), and a very important one to master. How else will students come to understand square roots *conceptually?* Wait a minute — they don’t! And that’s why it’s so hard to teach them abstract, algebraic rules about square roots.

The student who can compute with square roots will have a very easy time learning that the cube root of x^12 is x^4 — it is simply that thing whose cube is x^12.

Having argued for the necessity of computing with square roots, I admit I can make no such argument for a great many other math skills.

## Jason Dyer

June 1, 2012 - 11:17 am -@James: Students used to figure out square roots to five decimal points, by hand. Your situation is a little different.

## mr bombastic

June 1, 2012 - 12:37 pm -@James, I agree. I also think it is important that they be able to do this sort of estimation mentally. For topics like square roots, logs, and fractions for that matter, I think paper and pencil arithmetic can often be just as big a barrier to conceptual understanding as a calculator.

## rachel (@rdkpickle)

June 1, 2012 - 7:41 pm -“A developing human being is many things, and chief among them a poet, an adventurer and a problem-solver.”

Apostolos grips my heart: a swift kick to remind me what I know in my soul – that the subject I love lights me up as much as a transforming piece of music or a long hike in the mountains. Mathematics speaks to – and improves upon – my humanity.

And my brain will follow: I can see how doing what I consider selfish (spending the summer exploring the math that interests me, without regard to how it will directly impact the very particular details of the courses I am teaching next year) is actually an important thing. I’d like to study math in the context of the “complex, adventurous, brave, struggling human beings” who developed it. I’d like to figure out how to invite students to see, understand, and join me on that quest.

## Sue VanHattum

June 2, 2012 - 7:50 am -Hey Rachel, you might enjoy the Math Circle Institute in July.