I got the periodic/wave/Hertz thing. But I tried to see “Google” in it and failed. So I tuned out, feeling annoyed.

Then Frank Noschese, high school physics (etc.) teacher and all around smart person stepped in:

Is anyone else upset that the Hertz Google doodle isn't actually sinusoidal in shape?

—
Frank Noschese (@fnoschese) February 23, 2012

Conversation ensued. Basically, Chris Lusto (high school math teacher and all around smart person) and Frank went back and forth on what exactly constitutes a sinusoidal curve. 

On Twitter.

Now I was interested. Lusto’s question prompted me to respond:

@Trianglemancsd @fnoschese So the question is, do we care about finite vs. infinite wave combinations? bit.ly/zzFSca

—
Mr. Lusto (@Lustomatical) February 23, 2012

And then this question got inside my head. Is it still sinusoidal if it’s composed of an infinite series of sinusoids?

I didn’t care even a little bit what the formal definition would be. I was struggling with the spirit of things. Does it make sense to call this thing sinusoidal?

And then walking to the parking lot at 9:00 at night, it struck me.

@Lustomatical @fnoschese That doodle is sinusoidal like e^x is polynomial.

—
Christopher (@Trianglemancsd) February 23, 2012

On Twitter.

They are working on a short-term intervention curriculum to improve seventh-grade students’ proportional reasoning. The project is funded by the Institute of Education Sciences in the federal Department of Education.

They are looking for 100 middle school math teachers in Minnesota. Here’s their blurb:

Project MARS (Mathematical Reasoning Strategies)  is  a collaborative seventh-grade math study between the University of Minnesota and Harvard University. Project MARS is seeking to recruit 100 seventh-grade math teachers throughout the state of Minnesota in addition to 100 seventh-grade math teachers throughout the east coast.  The 6-week intervention with seventh-graders and their teachers will span two consecutive academic years beginning January of 2013.  Specifically,  Project MARS targets students’ learning of ratio, proportion and percents through word problems, critically important concepts in middle school mathematics instruction and challenging for many students.

Our project website is located at www.cehd.umn.edu/EdPsych/MARS/ which will provide you with more detailed information. It is important to note that Project MARS is an  Efficacy and Replication study that builds upon the previous success of several smaller studies conducted over the last five years.  Our previous smaller scale studies have been conducted in South Washington County,  Bloomington, and Osseo School districts. To date we have worked with nearly 2000 students and have yielded very positive results.

There is compensation; it’s professional development; you’re helping the world to better understand the learning of middle school kids. It’s win-win-win. See their site for details.

I’ll spare you her lengthy plot summary. But if you haven’t seen the movie, the plot surrounds a boy born into a human family, but who is, in fact, a Muppet. Fish out of water. Happy ending.

Here's the guy.

So I ask Tabitha (who, recall is 4 years old and will soon be 5) a question.

Me: Is it possible that you’re a Muppet?

Tabitha: Dad! No! I don’t look like a cartoon.

This is great, right? All Muppets look like cartoons. I don’t look like a cartoon. Therefore I am not a Muppet.

In formal logic,

Ever the teacher, I want to assess.

Me: Nice. Are all cartoons Muppets?

Tabitha: Sheep in WordWorld isn’t a Muppet. Clifford’s not. 

Sheep from WordWorld (on PBS)

Me: One more question. If you’re not a Muppet, does that mean you don’t look like a cartoon?

I’m asking about whether:

She thinks for a while. Finally.

Tabitha: Dad! Clifford’s not a Muppet, but he looks like a cartoon!

She also seems frustrated by the obviousness of my question, given that:

Let me tell you a secret about teaching.

If you give a kid a cougar paw (or a reward) the first time they do something, don’t do it the next time. They’re probably just doing it to get the reward.

Then it won’t count as a reward because they’re trying to get it. They’re trying to trick the teacher into giving it to them.

This will help you be a better teacher.

Basically, they’re mini McNuggets.

McNuggets have a lot of fat. Most of it (as with all fried foods with the possible exception of cheese curds) is in the outer coating.

So what happens when we shrink the McNugget?

We could model the chicken by volume and the outer coating by surface area. In this case, shrinking the McNugget by a factor 1/2 would yield a doubling of the ratio of coating to chicken. As a result we predict a substantial increase in fat as a percentage of weight.

But that’s a crude model-the coating isn’t infinitely thin.

A more realistic model might treat the coating as having a thickness, and thus a volume. When we shrink the McNugget to make a McBite, do we shrink the thickness of the coating proportionately? This seems unlikely. But even if we do, the ratio of coating to chicken increases.

More likely is that the coating is of constant thickness. In other words, McBites are probably smaller chicken bits covered with the same coating as McNuggets. Once again, we increase the ratio of coating to chicken, and we predict a substantial increase in fat as a percentage of weight.

Heading over to McDonald’s online Nutrition Information, we learn that McNuggets have about 12 grams of fat for a 65 gram serving. That’s about 18% fat by weight. This figure is consistent across serving sizes.

Mc Bites? 19 grams of fat for an 85 gram serving. That’s about 22% fat by weight. Also consistent across serving sizes.

How different is 22% fat from 18% fat, you ask? Very different. They’re both bad news, but 22% fat is (coincidentally) 22% more fat than 18% is. So take a serving of McNuggets, which has a lot of fat. And put almost a quarter more fat in. That’s McBites. Oh, and also increase the portion size from 97 grams (6 piece McNuggets) to 128 grams (Regular size McBites).

Middle school math will get you far, kids.

Now if we can get an estimate of the McNugget to McBite scale factor, we can set up a system of equations and figure out just how much of that fat is in the coating.

Tabitha: What is an umbrella bird?

Me: I don’t know. I’ve never heard of one before. Did you hear about umbrella birds somewhere?

Tabitha: No.

Me: Did you make it up?

Tabitha: (casually) No.

Four-year olds don’t concern themselves with the law of the excluded middle.

One of my favorite parts of my hard copy of the Sunday New York Times each week is the Business section. Ordinarily this has nothing to do with my job. It’s just a brief trip into a fantasy world in which I earn real money; a chance to bump elbows with people who do.

But this week there were three articles about badges.

Exhibit A: Natasha Singer’s “Slipstream” column. This covers the online trend of virtual rewards, referred to as gamification and reminds us that there may be danger in allowing ourselves to be manipulated. There is discussion of the Twitter badge on Samsung’s site, a company called Badgeville that designs game-based sites, and the difference between virtual rewards (badges) and real ones (free airplane trips).

Exhibit B: Randall Stross’s “Digital Domain” column. This covers a medical social networking site that allows patients to ask question and doctors to post and discuss answers to those questions. There are the Paramedic Award, the Good Samaritan Award, the Louis Pasteur Award and many others.

Exhibit C: A review by Nancy Koehn of Strings Attached by Ruth Grant. From the  review:

[Grant] says that paying children to elicit certain behavior may have destructive consequences in developing character, potentially nurturing self-interest at the expense of kinder motives. “Where students work in an environment that values only extrinsic rewards for learning,” she writes, “cheating goes up.”

Right. When the rewards are extrinsic (can you say “badges” or “test scores”?), those who buy in become motivated for the reward.

I don’t work for badges. Or grades. Or external evaluations of any kind, really. Of course this meant a lot of B’s and C’s in high school and much teacher and parental hand-wringing about “underachievement”. And it meant quitting the Boy Scouts after my first year because I was uninterested in earning literal badges.

Now I’m supposed to be interested in your virtual badges?

Please.

So teachers, let’s be careful about this stuff. Best case scenario in the long term seems to be annoying kids with this stuff. Worst case scenario reduces their interest in what we’re trying to motivate them to do.

And it’s time to vote.

That’s right, I’m running for office. Not against Rick Santorum (did he really just win the Minnesota Republican caucus?) And not for president.

Nope, I’m running for Vice-President of Mathematics for MCTM. Election starts today. So head on over to mctm.org and cast your vote. I did. It takes like 10 seconds. Seriously.

Thanks.

Paid for by the Committee to Overthink Teaching.

TI SmartView is a lovely piece of software… (see embedded video):

If I am teaching Advanced Mathematical Button Pushing, I am going to need this software. But honestly, that’s about it. It costs more than the calculator it is emulating and requires a fully functioning computer to run on.

It cripples the teacher with the same crummy graphics (94 pixels, horizontally) that students have in their hands. And it records every single button push. All of them, down to the last 5 up arrows.

You know how annoying it is when you get directions from Google Maps and the first five turns tell you how to get out of your own neighborhood? Same deal here. There is truly no intelligence built in.

Oh, and licensing? Crazy restrictive. My institution pays per license, and it’s not the “running on x computers at a time” sort of license. It’s the “installed on x computers at a time” sort of license.

Putting my money where my mouth is tonight.

I have had some saucy things to say about our adopted Calc text. In particular, that it has not been written from a perspective that is empathetic with students’ states of mind.

So tonight we’re proving the area of a circle formula as an introduction to trigonometric substitution. There’s a rabbit-out-of-the-hat moment in which the substitution in question gets introduced:

x=r•cos (theta)

I anticipate that this will be a challenging point for students. So we’ll stop there and collect responses to the question, What does this make you wonder, question or think about?

Here’s what I anticipate they’ll say:

• Why r?
• Why cosine (instead of, say, sine)?
• Why theta?
• Is this like u-substitution?

As always, I am prepared to be schooled.

I know they’ll have ideas I never thought of. Can’t wait to see what they are.

Post mortem

Sue writes in the comments:

I don’t start with the strange substitution. I start with a triangle. For me, the triangle I draw explains the substitution I decide to use. Then it’s all about reasoning, instead of memorizing (?) 3 very strange substitutions.

This is spot on.

Students asked about the theta, they asked about the r. They didn’t make any connection to u-substitution. Lots of them worked on triangles. Students thought about vectors in physics and they thought about the unit circle.

In contrast with Sue’s suggestion, and in contrast with where my students’ minds were, our text talks about this:

In general, we can make a substitution of the form x=g(t) by using the Substitution Rule in reverse. To make our calculations simpler, we assume that g has an inverse function; that is, g is one-to-one…This kind of substitution is called inverse substitution.

And the triangle Sue suggests? That comes in at the end of the technique as a tool for changing variables back from theta to x. So we finish with where students’ minds are, but we start with an abstract, backwards version of something they’re not even thinking about to begin with.

And we wonder why students don’t become better problem solvers in their college courses, and why they don’t develop the kinds of critical thinking skills we would like them to. And why students don’t read the textbook. They don’t read the textbook because it’s not speaking to them.