I take these speaking jobs for three reasons.

- To maintain the tuna-casserole lifestyle to which I have become accustomed, even though I’m only bringing in the part-time research assistant money these days.
- To compel me to find better structures, metaphors, visuals, and exercises for communicating good curriculum design.
- For the helpful feedback and criticism the attendees offer.

These groups of grownups are my classroom for the foreseeable future. It’d be a waste of a blog if I didn’t share what I learned last weekend.

**Mathematical notation isn’t a prerequisite for mathematical exploration**. Mathematical notation can even*deter*mathematical exploration. When the textbook asks a student to “find the area of the annulus” in part (a) of the problem, there are at least two possible points of failure. One, the student doesn’t know what an “annulus” is. (Hand goes in the air.) Two, the student knows the term “annulus” but can’t connect it to its area formula. (Hand goes in the air.) ¶ That’s the outcome of teaching the formula, notation, and vocabulary first: the sense that math is something to be remembered or forgotten but not created. ¶ Meanwhile, let’s not kid ourselves. The area of an annulus isn’t difficult to derive. Let the student subtract the small circle from the big circle.*Then*mention, “by the way, this shape which you now feel like you own, mathematists call it an ‘annulus.’ Tuck that away.” ¶ Similarly, if I give you this pattern, I*know*you can draw the next three pictures in the sequence. That’ll get old so I’ll ask you to describe the pattern in words. You’ll write out, “you add two tiles to the last picture every time to get the next picture.” I’ll show you how much easier it is to write out the recursive formula A_{n+1}= A_{n}+ 2. ¶ I’ll ask you to tell me how many tiles I’ll find on the 100^{th}picture. You’ll get tired of adding two every time, and we’ll develop the*explicit*formula A = 2n + 3, which makes that task so much easier. ¶ Terms like “explicit” and “recursive” and “annulus” can do one of two things to*the exact same student*: make the kid feel like a moron or make the kid feel like the master of the universe.**“Talk to someone who actually**What kind of math does he have to do to*makes*ticket rolls.*make*the thing,” said Russ Campbell, a community college adjunct instructor at least twice my age. Great idea, Russ. Speaking of which, pursuant to some harebrained WCYDWT idea, I spent twenty minutes on the phone with my local and state Departments of Transportation last week and it was almost too much fun to handle, peppering questions at engineers who were all too delighted that anyone gave a damn about how they calculated recommended speeds for curved roads. More of this.

## 16 Comments

## Ashli

September 14, 2010 - 3:04 pmThanks for making this point today. I ended up in conversations with my precalc classes earlier about notation and how it can make maths unnecessarily hard (exhibit a: vectors in math books and physics books often use different letters for the same things).

Will put more thought into the definition/work before the word itself as we head into the wild fields of trigonometry.

And now I’m going to spend the evening thinking about forces around a curve. Cheers :)

## Bryan Cook

September 14, 2010 - 4:01 pmI just read “Traffic: Why We Drive the Way We Do” by Tom Vanderbilt. I’m now unacceptably excited by problems dealing with recommended speeds on curved roads.

## Z. Shiner

September 14, 2010 - 4:37 pmI remember being in an upper division math class and being ‘introduced’ to ordered pairs. The book defines an ordered pair as follows: “Let S be a set and let a and b be members of S. The ordered pair (a, b) is the set {{a},{a, b}}. The element a is called the first term of (a, b), and the element b is called the second term of (a, b).”

I had no idea what this book was talking about; perhaps the proper schema was not activated. If someone mentioned a Cartesian Coordinate System or even said “x’s and y’s” then I think I would have saved a lot of time and energy.

In the same sense, while I’m sure all students have been introduced to annulus in various forms, I doubt many know the mathematical name or formal definition (if they did they wouldn’t need a teacher). Our challenge becomes to help them understand the mathematics behind the shape and give them the tools/language to express that understanding without conveying the idea that mathematics is anything other than natural.

## Tony

September 14, 2010 - 5:16 pmIt’s like you read my mind…or were following me on the bike path. I started cycling again recently and noticed that one section of our city’s 20+mile bike path has a posted speed limit around a curve. I took a pic and called the folks at our local Parks and Rec. Still awaiting a response (only 3 days now…not too bad) too the same question you posed the engineers. I’m waiting for someone to tell me…”We had the same poor guy take the turn at various speeds starting at 20 mph and had him decrease his speed by 1mph each time until he stopped falling.”

## Elizabeth

September 14, 2010 - 5:43 pmOh, you and your tuna casserole lifestyle, making those DOT engineers’ whole year, I’ll bet. :-)

## Gilbert Bernstein

September 15, 2010 - 1:01 amI’m right there with you on #2. I’ve been having a very similar experience with my research (not education related). Just going and talking to people who use the ideas/technology I work on improving has been tremendously useful. I wish more people would emphasize this point.

## Kara

September 15, 2010 - 5:12 amDan – the time you spent with us here in Indiana was awesome – faculty are still buzzing about the day! Thanks so much.

## Mike Mathews

September 15, 2010 - 6:34 amI have to say I’m a little disappointed that I didn’t know you were in Indiana until you weren’t. This probably wasn’t an open session, but I would have tried to crash the party. Thanks for what you do, keep blogging about it.

## Rebecca

September 15, 2010 - 7:53 amWhat’s exciting about this post+comments is the enthusiasm for real-world application of things that cause most high school (and junior high) students to roll their eyes. It’s not just telling or showing them how they COULD use it, but allowing them to interact with people who DO

We have, for example, a rather (actually, it’s very) steep driveway that needs to be paved. In my research on road construction techniques (once a student, always a student), I found clear explanations, wonderful diagrams, and (wait for it…) the same geometric equations my high school sophomore is doing for homework. Instant Teaching Moment…

## Derrick Bowman

September 15, 2010 - 9:47 am@Mike Mathews

I’m in the same predicament. I found out about his presentation on the 12th. What’s worse is that it was a presentation for all math faculty at Ivy Tech, the university where I’m employed.

What’s worse is that I’m missing Bill Nye on the 25th because i elected to go to the 25th conference instead of the 11th. Two of my best friends got to meet Dan. I’m jealous!

@Kara

Which campus do you teach?

## Terri

September 15, 2010 - 11:03 amI have forwarded several of your posts to a friend, and she just sent me a message asking if you have (or plan to create) a book of these concrete, real-world examples.

Based on the comments I see here, you could ask for examples from your readers and put them together as a collection.

Terri

## Karim

September 15, 2010 - 9:06 pmCongrats on what was certainly a cool session. Hope you’re enjoying the tuna, and that you’ve got some good driving tunes. “Free Bird” cassette, anyone?

## Matthew Handy

September 16, 2010 - 10:21 amI agree. I try to use as little Greek in my classes as possible. I think Greek letters can be very off-putting and they’re often unnecessary. What’s wrong with sin(A+B) rather than sin(alpha+beta)?

Similarly, I don’t like Latin plurals. Is it so wrong to say “formulas” rather than “formulae”?

Check out my blog :P

## gasstationwithoutpumps

September 17, 2010 - 7:03 am“Formulas” is a perfectly acceptable plural in English, but the students should be told (at least once) about the plurals that they may encounter in other courses.

The one I see students having the most problem with is “vertex” and “vertices”. Many students in college end up doing the back formation “vertice”, which makes them look embarrassingly uneducated.

## Tim Erickson

October 6, 2010 - 7:58 amI once did an inservicey thing in Beaverton, OR in which I assigned teachers to go out to stoplights with a partner, a watch, and a notepad. they were to observe and collect data about the vague question, “How do you tell if a light is timed well?” (Or something like that; at any rate, the timing of lights is something I had churned about for years without actually taking data, so I thought, ¿que oportunidad?)

Before the debriefing the next day, besides taking data of my own, I thought I’d call up a traffic engineer. It took a while to figure out what they were called, but (as you suggest) when I got hold of Portland’s chief guy, he was thrilled to talk to someone, anyone! who cared about what he had devoted his life to.

And a big takeaway: is there a control room where they can control the lights? Yes, for downtown only. Under what circumstances so they alter the timing? Answer: for emergency vehicles. Not for traffic. Their experience was (at that time) that any attempt to do that in real time just made things worse.