Month: November 2011

Taking a Moment… in Calculus

In calculus, I’ve historically asked kids to take the derivative of:

f(x)=\frac{2x^2+\sqrt{x}}{\sqrt{x}}

and students will immediately go to the quotient rule. OBVIOUSLY! There’s a numerator and denominator. Duh. So go at it!

Unfortunately, this is VERY UNWISE because it leads to a lot more work. And I was sick of my kids not taking a moment to think: what are my options, and what might be the best option available? Also, kids generally found it hard to deal when we started mixing the derivative rules up!

So I came up with a sheet to address this and paired kids to work on it.

(I’ve also had kids think they can do some crazy algebra with g(x)=\frac{x^2+1}{x+1}. This sheet also helped me talk with kids individually about that.)

For a little context, my kids have only learned the power rule, the product rule, the quotient rule, and that the derivative of e^x is e^x. They have not yet been formally exposed to the chain rule.

Without further ado…

[.pdf, .doc]

Believe it or not… a log question which was briefly stumping us

Hi all,

A teacher approached me with the following question.

The function \ln(x^{-2}) has a graph that looks like:

It makes sense that the function exists for all negative x values, because when you raise a negative number to the -2 power, you’re going to get a positive number. And you can take the natural log of a positive number.

Then the teacher said to consider the following function: -2 \ln(x), and the graph looks like:

Notice that you can’t input negative x values, because the domain of natural log doesn’t allow for it.

Here’s the question.

According to the log rules/properties, we know that:

\ln(a^b)=b\ln(a) (obviously).

So \ln(x^{-2})=-2\ln(x). But the graphs are different.

We went a little crazy trying to figure out what’s going on… For about 3 minutes, we were having a great conversation. But we quickly converged on the little text that accompanies the log rules in any textbook… and this text says that these rules work but are only valid for a>0.

I kinda love this as an in-class exercise (I’ll probably forget this when I get to logarithms, but maybe posting it here will prevent me from forgetting it). Because it will force kids to (a) be confuzzled, (b) talk through ideas, (c) go back to the definition and qualifications for the log rules, and (d) see that these rules are indeed valid (we didn’t break math), but they are a bit more restrictive that we might have thought.

What I love is that \ln(x^{-2})=-2\ln(x) isn’t actually an identity. But we are so used to using the rules blindly, robotically, that we never think about it. But for it to be a good mathematical statement, you need to qualify it! You need to say this is only an equivalence for x>0. This was a good reminder for us.

Review Activity for Rational Equations

Last year, I did a review game that I got from Sue Van Hattum. I wrote that:

[this game] forces students to ask themselves: what do I know and how confident am I in what I know? (It’s meta-cognitive like that).

I set kids up in pre-chosen pairs, and they are asked to work together. In fact, I gave kids their new seats for the quarter, so this was their introduction to their new seat partner! They then are given a booklet with problems — and each pair is asked to work only on ONE problem at a time. (For those who finish a problem before others, I have alternative problems for them to work on.) When I see almost all pairs are done, I’ll give a one minute warning… Then I ask all students to put their pencils down and pick up a pen. We go over each problem, kids correct their own work, and using the honor system, they figure out how many points they have. (Scoring below.)

You can see three sample questions from our review game below…

[The .pdf and .doc file of the 6 questions are linked.]

I explained in my last post how scoring worked…

Each group started with 100 points to wager — and they lost the points if they got the question wrong, and the gained the points if they got the question right.

Some possible game trajectories:

100 –> 150 –> 250 –> 490 etc.

100 –> 10 –> 15 –> 30 etc.

Anyway, what was great was that the game really got students engaged and talking. Each student tended to work on the problem individually, and then when they were done, they would compare with their partner.

(If you try this, you have to make sure that students know NOT to skip ahead… everyone is working on one problem at a time. Then you go over the problem, and THEN everyone starts the next problem.)

So there you go… I don’t do reviews a lot, but for rational expressions, rational equations, and circuit problems, I figured we’d need a day to tie things up. And since this is one review I think works amazingly, I figure I’d share it a second time! Thanks Sue!

Digressions! Hints of the Chain Rule via the Power Rule.

In calculus today, I went off the beaten path a bit and it was a lovely digression. I think this works so well with my kids not only because they’re awesome, but because quite a few of them like to notice patterns and explore.

So far we’ve learned and proved the power rule for derivatives, and we’ve been practicing using it. So if students are given y=\sqrt{x}(1+x) and asked to find y', they know that they have to distribute and then take the derivative. [We don’t know the product rule yet.]

So… for their work due today, they were asked to find the derivative of y=(1+x)^3. And my kids wanted to go over this together in class. So when we worked it all out, we got y'=3x^2+6x+3. And then someone noticed that was the same as y'=3(1+x)^2 which looked like the power rule! Like if we had y=x^3, the derivative would be y'=3x^2, so similarly since we have y=(1+x)^3, it’s makes sense that the derivative was y=3(1+x)^2.

At this point, I decided I wanted to capitalize on this. So I said: okay, neat observation. Does it always hold?

And I threw this up…

and had students — using the rule they observed — make a conjecture as to what the derivative would be (without calculating things out). They got (working in pairs, and then sharing as a class):

And then they checked…

… and saw it was wrong. So based on this, I had them revise their conjecture, and take a stab at:

which they did… and they came up with (and worked out):

So they believed they had something that always worked… so I had them prove it. Which they did.

And it worked out!!!

So now we had proved something about the derivative of y=(1+ax)^2, and I asked them: would it work to the third power? would it work to the nth power? And I left it as an exercise for their home enjoyment (our corny term for homework). I’m really curious to see who gets how far on this!

It’s cool. They’re getting whiffs of the chain rule. I’m not going to give it to ’em or do anything else with this. We’ll wait a while. But I really like how this digression took 15 minutes, but it capitalized on something they were curious about. And we’ll see the connection later.

I felt strongly enough about how this worked out that I engineered this discussion to happen in my second calculus class. I treated it like a big surprise. What a strange observation. Instead of forging forward in class, let’s take a digression. I loved that it worked a second time too.

Another “How To Fix Math Education” Article

One of my students sent me a Slate article, yet another piece of tripe with an attention-grabbing, gag-inducing headline: “How To Fix Math Education in High School and College.” Barf.

And the article is short and doesn’t really say how to fix math education in high school and college. So there ya go. But my student asked me for my thoughts. And I gave myself 20 minutes to compose a response. I had to give myself a time limit because I know myself. I’d obsess, second guess, and then think: well, that’s not precisely right, and then get diverted to go into this or that tangent, and never actually send it. And if I did, I wouldn’t be happy with it and it would be maybe 5 pages of things I wouldn’t be happy with.

So I did it under time constraints. And I figured I’d share it here. It is not precisely what I believe, and it is a lot of broad strokes. And it certainly is choppy (because I didn’t having time to proof). But here you go…

Hi [Stu],

I think this article brings up a lot of good points, and I know at all the math conferences I attend and all the conversations I have with math teachers (at Packer and around the country), these are the discussions we are having.

When it gets down to it, there are two claims that I think are worth discussing.

First, that our kids are being pushed on a “calculus” track, while the real action and usefulness is elsewhere. I do think that there is this standardized curriculum in high schools, where kids are being put on a track where calculus is the pinnacle of their math studies. It’s not just Packer, but everywhere in the US. And I think that is not always the appropriate track — and we could come up with alternatives. We could have multiple tracks, culminating is statistics, discrete math and number theory, alternative geometries, or something interdisciplinary. Of course there are about a zillion things in the way, including staffing (who would teach these courses, how would they get paid, when would they have time to write the curriculum which would have to be something untraditional) and colleges (which look for calculus on a transcript, or so I’ve been told… I don’t really know much about that world). But I think most math teachers would say that calculus is just one possible, and not always the best, ending to a high school math career (depending on who the kid is and what the kid’s interests are in math). Very deep-seeded cultural, social, institutional, and even political barriers get in the way of revolutionizing what math is taught and how it is taught. On the other hand, I disagree with the argument that calculus should not be pushed because it doesn’t have as much “practical” “applied” use to most people. If we only cared about pushing the things that would be useful for students in the real world, why teach Shakespeare and Pynchon and hydrogen bonds and what makes a rainbow — if most students aren’t going to be working in a lab or becoming writers or critics? I think there’s a value to calculus for the sake of it being calculus, for it showing (for many, the very first time) the abstractness and beauty that a few simple ideas can bring to the table — and how these simple ideas can be stretched in crazy and amazing ways. (Given that a student has the algebra tools to accomplish it.) But to be clear, I honestly believe most math curricula in high school aren’t solely bent on helping kids understand calculus. If that were the case, I could come up with a curriculum where we elminiate geometry, and combine Algebra I and Algebra II into a 1.5 year course… and students would have the background to do calculus afterwards. That’s not the goal. The goal is building up ways of thinking, putting tools in your mathematical toolbelt, and leading up to abstraction and reasoning… with the hope that the structure, logic, and incredible beauty and creativity of it all come tumbling out. Now whether that actually happens… let’s just say it’s not easy to accomplish. We teachers don’t get students as blank slates, and we aren’t always perfect at executing our vision under the constraints we have.

Second, there is the claim that ” that schools should focus less on teaching facts—which can be easily ascertained from Google—and more on teaching them how to think.” I think most teachers would agree with that. But then the article goes on to claim: “mathematical education will be less about computation—we’ve got calculators for that!—and more conceptual, like ‘understanding when you need to do integrals, when you need to do a square root.’   This is a much bigger issue and it can’t be simplified into these two sentences. There is a large discussion going on in the math education community about the use of graphing calculators, and if they can be the panacea for math education. That students who struggle with basic algebra can still explore and discover using their calculators. I half-agree with that. Pattern-finding is great. It invites creativity and expression, this sort-of calculator-based discovery-learning. But if the calculator is used as a black-box, and we don’t know what it’s calculating for us, or how we could calculate what it’s doing (but just much slower, and possibly with different algorithms), we’re in trouble. If you can find patterns in pascal’s triangle, but you can’t prove them or at least have some plausible argument as to why they exist, then you’re just finding patterns. It’s cool, but has very little depth. If you let a calculator factor for you (the new ones can! like wolfram alpha!), but you don’t know what it’s doing, then I fear math can easily turn into magic, where the magician is the calculuator. And that’s one thing I big thing I worry about as a teacher: math being seen as a bag of magic tricks, where there is no logic or structure to it. And if the calculator is the magician, and the student is the audience, the student might marvel at the trick, be excited by whatever pattern is found, but never really understand what makes it all hang together. That’s why you hear me harping on understanding so much. And why when you found the power rule pattern, you did the first step, but the real learning came when you went off to prove it. It stretched your mind, and you spent a long while working it out. You wanted to understand the pattern, the logic, the conjecture. When technology helps with understanding, I LOVE IT. When technology helps generate questions, I LOVE IT. But when it replaces understanding, I’m a bit more wary.

So there are my very quickly typed two cents. They might not make a whole lot of sense, but they just sort of poured out. My thoughts change in subtle ways on these issues all the time, so ask me again in a few months and I might have switched some of my thinking.

Best,
Mr. Shah

To be honest, I’m posting this as part of my desire to archive my evolution as a teacher. You’re welcome to comment, and have discussions, if you so wish, but I probably won’t engage too much. I’m tired.

In other news, explaining why I’m so tired, I spent the last week and half writing narrative comments on all my students. I think they are better this year than in years past (each year I try to improve a tiny bit), so maybe if I have the time and desire, I’ll post about my process. But who knows, school is like a train and time just keeps whooshing by. I can’t believe a quarter is already done. It feels like we just started, and I barely have scratched the surface of my kids.  (Right at this moment, that is. You know, by Thursday or Friday it’s going to have felt like this year is turning into a piece of taffy that keeps getting stretched out, the end getting further and further away while my grip on reality is getting as delicate as the taffy is getting thin.)

PS. On the views of math:

Compiling A List Of Posts

Hi all,

I need some help, if you have a few minutes. I am looking for some quality blog posts and/or websites which offer the following:

Stories from the Front: On the ground experiences of teachers teaching problem solving in the math… the good, the bad, the ugly

War Strategies:  Different ways teachers actually do problem solving in the classroom, and maybe some hints/tips/technqiues (e.g. whiteboarding, Moore Method, Harkness Table, problem sets, grouping ideas, hint tokens, etc.)

Weapons:  Good websites (or books) which contain good math problem solving problems (e.g. Exeter problem sets, AMC questions, etc.). My personal thought on questions is that they don’t need to be hard to be problem solving… In fact, the harder the problems are, the less accessible and fun the problem solving will be, and the more my kids will be turned off.

What I’m not really looking for is Polya’s How To Solve It, which is great reading but lacks in the day-to-day practicality and concreteness I’m looking for. I don’t need to know what problem solving is (like Potter Stewart, I know it when I see it), or read philosophical exhortations about how important it is in promoting meaningful and deep learning. I want practicality. Stories, resources, tips, etc.

If I get some responses in the comments, I will compile them into either a comprehensive post, or if there are a lot, I’ll make a new page (a la the Virtual Filing Cabinet) for it.

The reason behind this is selfish, but I’m hoping the output could be collectively useful. My department is thinking seriously about how to integrate problem solving into our curricula… and I wanted to show them: “hey, there are a ton of good ideas from teachers who do it!”

So if you could help a teacher out…

PS. Not to make you jealous, but yesterday I designed and ordered these buttons! (You have to recognize I don’t know what I’m doing with Photoshop, so the pictures aren’t all that great. And these buttons have a large bleed area, so the text will actually be just near the outer rim of the button instead of with all that blank space between the text and the outside of the pin.)