Here Is Some Great Slow Motion Of A Cat Falling

We're starting a unit on average/instantaneous rates of change in Precalculus. Roughly, my idea is to give kids lots of scenarios where they can find speeds at various intervals and moments and use that to give them time to develop ways to represent rates of change. Once they have a firm grasp on using secant and tangent lines to represent rates of change we'll start measuring the slopes of tangent lines along curves, and then we'll notice some cool things.

A sticking point is how to explain those cool things. For instance, we'll notice that the slopes of tangent lines along y = x^2 vary as y = 2x. How do we make sense of that observation? Some curricula explain this using polynomial division, but my kids don't have that skill and I don't want to give it to them. I could do it like Newton and use infinitesimals -- so that we're looking at (x+o)^2 - x^2 over o, where o is ridiculously small -- but I'm a little woozy about just being like "here's some magic children!" (Though, I guess it's sort of OK since that magic turns out to be Calculus.)

I've got graphingstories.com and the Shell Centre's Functions and Graphs book. Anyone have advice or resources as I embark on this unit?

Thanks in advance!

The Point of Context is to Provoke Modeling

Q: What's the purpose of context in math class? 

A: To provoke modeling.

Q: What's the problem with the hot air balloon problem?

A: It offers a context without provoking modeling. 

"There were 58 geese and 37 ducks in the marsh. How many birds were in the marsh?"

Q: So is the above bird problem pseudocontext?

A: No, because it's provoking modeling.

In my book, a context doesn't need to be realistic if it serves a purpose, and the purpose of context is to give kids a chance to think about something interesting. If a context provokes modeling? Great! What else can you ask of a context? 

(You could ask that it be "real world," but that's a desire that I don't share. And neither should you.)

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I was checking out Dan's latest thing, and I came across this comment:

I dunno, this one made me wish the follow-up showed him just throwing the paint away and starting over. There’s just not enough investment in materials & time to make me think past. Plus if 6 tablespoons was enough paint to do the job then 30 is just a waste of paint far in excess of throwing away 6.

Which if the class brings it up organically would be a big win; then you can talk about why to choose one path or the other and it’s some good critical thinking. Otherwise it smells like pseudocontext to me.

Pseudocontext, eh? I say, nah, it doesn't really matter if it's realistic. What matters is whether he can do something with his scenario for kids. Context only becomes problematic when it's purposeless.

Update: This is what I mean by "purposeless context."

Everyday of My Life Feels Like This

Too fast.

Too slow.

Too fast.

Too slow.

She's bored.

He's lost.

Too fast.

Too slow...

"You just leave them alone, and it'll be OK."

Patrick Honner interviews Steven Strogatz in the latest Math Horizons. It's great! But this line rubbed me the wrong way:

There's a lot of fun in math. Do we really have to teach such dead material? If we could get a cadre of people who love math and who get it the way you get it or the way I get it -- people who know what math is about -- you don't need to tell them how to teach. You just leave them alone, and it'll be OK.

1. Teaching is hard. You do need to teach them how to teach. It won't be OK.
2. His idea is that the main obstacle to excited students is the content, but kids encounter content through pedagogy. Content is not enough, and for that reason content-knowledgeable people is not enough.

Update: Patrick Honner clarifies Strogatz's position in the comments:

Maybe it's not clear from the context, but the "you just leave them alone, and it'll be ok" line refers to the administrative and curricular micromanaging that many teachers face nowadays. It isn't a claim that content-knowledgeable people are automatically good teachers.
In particular, Strogatz was talking about the freedom his Calculus teacher, Mr. Jofffray (the subject of The Calculus of Friendship) had in the way he taught the subject. The school trusted him. He could "put his own stamp" on the course--that is, teach it the way it made sense to him, and make his own personal pedagogic decisions.
It's also worth noting that, in the book, Strogatz points out that Mr. Joffray was not the most knowledgeable mathematician, even at the school where he taught. As a student, Strogatz was initially skeptical, but learned that a lot more than content went into making a great teacher.

Complex Numbers Are Two-Part Numbers

I have a claim, and I'm trying to figure out whether I believe it or not.

When learning fractions, it's tempting to treat it as a two-part number, but that's a mistake. Really, a fraction is a single number. It's exactly the opposite case with complex numbers. Really, they're two-part numbers, though it's tempting to see it as a single number.

Is this true? What sort of evidence would support or contradict this claim?

Setting A Trap

What sorts of activities do you do with kids to help them avoid common mistakes?

There's obviously no easy fix for common mathematical mistakes. They're common for a reason. Helping kids avoid common math mistakes is synonymous with effective math teaching, and there's no recipe for teaching good.

OK, fine. But here's a move that I often make when I know that there's an especially nasty math mistake coming up.

Example: cos(a+b) = cos(a) + cos(b)

1. I ask kids what they think cos(a + b) is.
2. We get the wrong answer on the table, and we give it a name. In this case, I name it "The Distributive Property for Cosine."
3. I get kids to argue about the wrong answer until it's been disproved.
4. We give language to the truth: "The Distributive Property Isn't True for Cosine," and "Cosine Isn't Linear."
5. We practice the true thing a bunch in the next few days.

Trying to provoke interesting conversations amongst the childrens is a pretty big part of what I try to do. I also think it's important not to tip kids to the fact that you're setting a trap for them, so I try to hide my hand. (In this case, cos(a+b) = cos(a) + cos(b) was included in a long list of Always, Sometimes, Never problems.) I think that giving kids language to talk about this debate is absolutely crucial to them remembering it, so it needs to happen.

This is a go-to pedagogical format for me.

(New Blogging Rule? I only blog if it'll take less than 10 minutes or more than 2 hours to write the post?)

The Part of "Lockhart's Lament" That Gets Quoted Less Often

Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.

In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

It’s perfectly simple. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them!

I'm sure that Lockhart would agree that teachers get better over time. So where does that improvement come from for Lockhart? An increased ability to be real?

[Lockhart's Lament]

Things I've Read Lately

Negroes Are Anti-Semitic Because They're Anti-White by James Baldwin (1967)

"We hated many of our teachers at school because they so clearly despised us and treated us like dirty, ignorant savages. Not all of these teachers were Jewish. Some of them, alas, were black. I used to carry my father's union dues downtown for him sometimes. I hated everyone in that den of thieves, especially the man who took the envelope from me, the envelope which contained my father's hard-earned money, that envelope which contained bread for his children. "Thieves," I thought, "every one of you!" And I know I was right about that, and I have not changed my mind. But whether or not all these people were Jewish, I really do not know."

Learning Modern Algebra From Early Attempts to Solve Fermat's Last Theorem

"But complex numbers arose in the middle of calculations, eventually producing real numbers. To understand this phenomenon, mathematicians were forced to investigate the meaning of number; are complex numbers bona fide numbers? Are negative numbers bona fide numbers?" 

Children's Mathematics: Cognitively Guided Instruction
Extending Children's Mathematics: Fractions & Decimals 

"We found that teachers have a great deal of intuitive knowledge about children's mathematical thinking; however, because that knowledge was fragmented it generally did not play an important role in most teachers' decision making."

Keys to a Rubbled Kingdom

"There's no earthly reason I should be upset that the city is again in disarray. I didn't go because I thought we were going to solve the problems of the Iraqi people. I sure as hell didn't go to defend my country. (Rusty mortars only fly so far.) I went because that's what you do for the dead. You keep the things they give you."

What I Spend My Time Worrying About

The author of my favorite comic, Joe Hill, has been trying to radically reduce the time he spends online:

A couple weeks ago, I tried out an experiment in self-control. I decided to set some new parameters for my internet usage.

My rules for myself were modest: I would not visit any of the social parts of the internet until I finished my daily writing (about 2,500 words), my daily reading (a minimum of 40 pages), and my daily exercise (10,000 steps tracked by my UPband). No exceptions: I would not even send tweets from within apps.

Furthermore, I decided the rest of the internet was off-limits, unless I was looking at it while I was on the treadmill. At least until I get those first 10,000 steps, everything from Wikipedia to Huffington Post would only be available if I was moving. I allowed myself the indulgence of Words with Friends, but again, only on the treadmill.

So how’d that all turn out? Um - really well. I never would’ve guessed how well.

I'm not exactly sure what my "2,500 words" is, but I'd like to try this.