The great comments about Algebra 2 have got me thinking about relationships, descriptions of “having to do with”.

Every pair of people that have anything to do with each other is a different case, yet each relationship is pretty unambiguously father-son, or business partners, or best friends, or lovers, or whatever it is. We all relate to each other in a small number of very common ways. The examples of each category contain a dazzling, unending variety–but in spite of that, every variation on grandmother-granddaughter is still an example of grandmother-granddaughter. The grandmother and the granddaughter have something to do with one another, and it’s a specific something.

Variables–anything we can count or measure; any quantity that changes, or could–can be related, or not. They can have something to do with each other, or not. The period and length of a pendulum have something to do with each other. The height of a projectile and time have something to do with each other. We need not be applied, we can stay pure: x^2, the product of a number with itself, certainly has something to do with how big x is. 2^x also has something to do with how big x is, but it’s a different something. There are a few, very common ways things can be related (along with lots more less common ways). Is the goal of Algebra 2 to find out what those common ways are?

What are the different kinds of relationships among variables? What can we infer about one if we know something about the other? If Mike is my father, then I must be Mike’s … what? If y=10^x, then x must be … what? What is the only number that has the following complicated relationship to 3? What makes a relationship quadratic? If two people are seen at the movies kissing, what can you infer about their relationship? What can’t you? If two variables take on values together in the following pairs, what kind of relationship is it? What does that tell you about the other ways those variables will be seen–excuse me, observed–together?

This might work.

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Dan, I’m a bit confused. I’m not sure I understand how this is different from saying “Here’s what a function is, and now let’s catalog a bunch of functions.”

Is it bad manners to free-style on an idea in someone else’s comments? If so, all apologies.

What makes big, course-spanning questions so great is not that they motivate students with a tantalizing question. No question that I ask is going to be able to motivate students 5 months after I ask it. Any moment of curiosity will have passed. For motivation and engagement, we need daily questions, curiosities and (for +10 Meyer points!) perplexities.

I think that the big unit/course spanning questions are wonderful because they provide meaning to the curriculum. It’s harder to ask the question “Why are we learning this?” when what we’re learning is clearly situated in a larger, obviously meaningful framework. For instance, the question “Is there life on other planets?” naturally leads to the questions “What are the conditions for life?”, “How hot are other stars?”, “How far away are planets from stars?” and “What’s in the atmospheres of alien planets?” Bam, there’s your calculus-based intro Astronomy course. And while students in a different class might wonder, “What good are absorption lines?” my bet is that students in this class will (a) be more likely to situate them correctly as helpful in determining temperatures of distant stars or atmospheric content of exoplanets and (b) won’t think that astronomy is useless and boring. So that’s what we’re going for here, I think.

And now, a problem. Let’s partition the world of course-spanning questions into the purely mathematical and applied mathematical questions. Let’s take an applied mathematical question such as “Can we predict the motion of a basketball?” or “How do electronics work?” or “Can we beat the stock market?” If we really and honestly pursue these questions, we’re going to have to go beyond our mathematics, since we’re going to need to use the tools of physics, or economics, or engineering. In other words, doggedly pursuing non-mathematical questions quickly leads us out of the mathematical domain.

On the other hand, rich mathematical questions don’t typically do the work of being obviously meaningful to students. The best that I can think of is “What’s a number?” which I imagine as a narrative arc spanning the first bits of a first year of Algebra.

This is a long-winded way of saying that I think we’re either looking for mathematical questions that are big and basic enough to motivate this month-long investigation, or applied mathematical questions that are closed under honest inquiry.

> I’m not sure I understand how this is different from saying “Here’s what a function is, and now let’s catalog a bunch of functions.”

I guess it isn’t, except that it draws an analogy to a different abstraction that teenagers already comprehend and are interested in. I imagine if you tell students we’re cataloging functions, then ask them what we’re doing and how today’s skill fits into it, they won’t be able to answer. If you keep connecting that vocabulary to the more familiar idea,

> For instance, the question “Is there life on other planets?” naturally leads to the questions [….] So that’s what we’re going for here, I think.

Me too. The relationship/function metaphor at least gives a solid answer to the question: what is this course about? But it doesn’t in itself lead to questions like “How can I make a parabola move 3 units left?”

I don’t know what the etiquette is but I’m grateful for the freestyling!

I agree with you, Dan. But it’s not just that your (excellent) metaphor doesn’t just motivate “How can I make a parabola move 3 units left?” It’s that I don’t think it really motivates “What’s a quadratic relationship?” Why should we study quadratic relationships instead of quintic relationships?

Ok, I’m with you. So: what’s the next move? (Anyone?)

What are some good “big” questions that can drive a course, or even a few months of math learning, on any level? I feel like I need to get some practice before taking on Alg2.

I’ll start a list of things that I’ve encountered, a few “S’s” that yielded mathematical “R’s”.

* What are Godel’s incompleteness proofs?

* What is a number?

* Who invented Algebra?

What are some other questions that can drive months of mathematical learning?

I love the relationships metaphor!

I wish I had a clear motivating question to share but none is at the top of my head right now.

However: Are most of your Algebra 2 students also taking a particular science class together at the same time? Which one?

At my engineering school, our freshman physics and calculus professors worked together to line up the curriculum so that new calculus techniques were being taught at roughly the same time as their motivating physics-problem counterparts. I imagine it would be possible (though admittedly a lot of work!) to do the same with high school Physics and Algebra classes. It might be more difficult, but still possible, to match up the Algebra sequence with Biology or Chemistry instead.

Your posts also reminded me of a loosely-related idea on statistician Andrew Gelman’s blog:

http://www.stat.columbia.edu/~cook/movabletype/archives/2010/02/genres_and_anti.html

See his points 1 and 2 midway through the post. The point is that mathematicians aren’t born with superpowers like Superman. Rather, learning math is like putting on an Iron Man suit that lets you solve problems which are impossible without it.

Admittedly, Iron Man also has a clear motivating question, i.e. how best to fight crime. But even just “cataloging a bunch of functions” is valuable: even Iron Man has to practice and get familiar with using the bunch of functions that he’s built into his suit.

Maybe there’s a way to spin this into a motivating question. The typical word problem about “how few buses can it take to transport 50 people” is a terrible motivation for systems of equations because you can easily solve it in your head, by guess and check, etc. But if you motivate it with a much bigger problem that *requires* many equations, math comes to the rescue and lets you solve something you couldn’t do by intuitive guess and check. Not only that, but math reduces many dissimilar-on-the-surface problems into something with a common solution. So… Algebra class is where you learn to use part of this superhero-toolbelt and recognize commonalities among these different problems?

Hmmm.

a) Does anyone know an expression synonymous with “there’s lots of ways to skin a cat” that’s not disgusting?

b) I’m having a very (fill in answer to above question) feeling about this conversation, MBP. Dan’s question seems to be, what’s a coherent and compelling theme for Alg. 2? Everyone in the room seems to agree that this question is pedagogically valuable and more or less why. (A coherent and compelling theme helps kids situate the knowledge in broader context, and makes it more relevant.) The proposed theme is, what are the kinds of relationships numbers can have? (Guiding metaphor: how is this like relationships between people?) If this theme gets at all or even most of the content of the course without too much stretching, and feels compelling at least to Dan, then what’s the problem? Try it and see how it plays. Why wouldn’t it motivate quadratic relationships? They are a) a kind of relationship between numbers and b) in many contexts an important kind. Dan can call on any of these contexts at any time within the broader theme of the course to motivate a particular kind of relationship.

a) There are really only two ways to skin a cat: starting from the head or starting from the tail. Any other way gets messy very, very quickly. There are, however, lots of ways to kill a cat…

b) …anyway, the thing that you said that I agree the most with is “try it and see how it plays.” If it works, it works. And Dan’s idea is well-thought out, and resonates with him. Still, I’ll try to defend my perspective.

A broader question is supposed to do two things. First, it’s supposed to help students situate knowledge. Second, it’s supposed to make the content more meaningful to students. How does a question have this effect?

My (totally made up) analysis is that we’re trying to bootstrap our Alg2 content onto a question that students can quickly recognize as meaningful, and an approach to answering the question that students can quickly recognize as natural. We’re hoping that they care about the question (giving our Alg2 content value) and that they’ll remember the natural approach to answering the question (so that they can associate our Alg2 content the approach).

A question such as “What simple functions are there?” is no help to students because (a) they’re not interested in the answer and (b) they don’t have any idea how to go about answering it. As a consequence, the question (a) is unable to make Alg2 more meaningful to students and (b) unable to provide students with a framework for their knowledge.

So, Dan’s question is “What kinds of relationships can numbers have?” My issue with the question (and I really don’t mean to be hard on you, Dan, I do think that it’s really an excellent metaphor) is that (a) I don’t think my kids will think that it’s worth answering and (b) I can’t think of a natural way to go about answering the question.

My last few posts on this theme have all been asking “What’s a meaningful compelling question / unifying framework for algebra 2”? This discussion is helping me to think more clearly about what “/” means in that question. If the course can be described as understanding the properties and applications of a short list of important functions, then the relationship metaphor will (I hope) help make that unifying framework meaningful to students. And, I agree with MBP that it’s not in itself a meaningful compelling question.

At least not the way it’s stated. The Kaplans talk about human “architectual instinct”. I think there is a human desire to understand by discerning structure. I’ve seen this most vividly in an exercise like the handshake problem (if everyone in the room shakes hands with everyone else, how many handshakes will occur? If 100 people are in the room? If n people are in the room?”). That question has never failed (in 6 attempts) to engage 90% of students for 100% of the period. And it’s not about life on other planets (I don’t mean that disparagingly — that’s an AWESOME example of what we’re going for), it’s not about fitting in or being honorable or having courage or determining your destiny or any of the other preoccupations of adolescence. It’s just a compelling question about structure.

The fact that so much mathematics and so much of the universe shares connections described by this small set of functions is, to me, a very deep miracle. And I agree with MBP that we haven’t yet stumbled on an essential question that motivates investigation of that structure.

So: the relationship metaphor is in for next year and I don’t think anything I’ve read here is being hard on that. The quest is still on for the question. Or in other words, how do I generalize the handshake problem to Algebra 2?

Thinking about Algebra II in the context of moving towards calculus, I think a lot of the “describing a non-linear world” stuff makes sense… If we’re thinking about graphs, rates of change, transformations of graphs, families of functions, it’s very calculus oriented. I got a lot of bang (accidentally) out of a project of having kids use http://www.linerider.org/play.php to design roller coaster tracks and then describe them mathematically. What I wished for was a technology where they could input their mathematical descriptions and have the track sketched and sledded for them. It’s appealing because drawing by hand is less precise.

I wonder if meeting families of functions through the shape of their graph would then make it possible to apply families of functions to relevant data sets and then eventually to say with some clarity what the mathematical connection is between exponential growth and, say, bacteria. I had students do a lot of learning from a task in which they had to defend in terms of rate of change why a quadratic model made sense for vertical motion, a logistic curve for the spread of a virus, and an exponential for population growth.

I also had success with this data (you need to create a free account to access it: http://mathforum.org/pows/submission-flow.do?publicationId=3729) just asking, “what can you say about the data? What predictions can you make? How would you best describe it mathematically?”

But I’m not really that into calculus. I’m into algebra. I like to think about sets and operations and inverses and identities and mappings. The idea of quantities and relationships can be taken in a calculus direction (asking about rate of change of x^2 as x changes, for example) but it can also be taken in an algebraic direction. What operations happened to x? Do all those operations have inverses? Can you map from x^2 back to x? If we restrict the domain of x, what happens to x^2? If the domain of x is all real numbers, what’s the range? What’s the domain of the inverse?

Is it plausible to argue that up to Algebra I, we worked on sets of numbers and operations on those numbers, expanding the sets we worked with as the operations required. Then in Algebra I we used properties of those sets and operations to express relationships between unknown quantities, but it was still mainly numbers that we worked with. Could Algebra II actually be where we begin to thing of relationships (functions) as a set with operations, inverses, identities, etc?

If “what is a number?” and “why do we need more kinds of numbers?” and “what numbers are special?” were big questions in prior math class, then could the big question of Algebra 2 be, “do functions behave like numbers? Are there kinds of functions? Special functions? Do all functions have inverses? Are there operations on functions? Do those make new kinds of functions or keep them the same?” etc.?

Dan,

In response to your work, I sat down and tried to think this through. I came up with the question, “How can we predict the future?” and I tried to rejigger my Algebra 2 standards to work this out. I’d like to know your thoughts: http://bit.ly/pMSC5n.

Michael

@Michael – I like the predictions theme a lot. It has room for all kinds of cool models and most of your year fits under that umbrella. I think any sequence will have pros and cons but fwiw your isn’t too different from what I did last year and it seemed fine.

@jerzysblog – “But if you motivate it with a much bigger problem … math comes to the rescue ….” I agree, and finding a much bigger problem that students can still (eventually) solve for themselves, or at least make satisfying partial progress on, is a good challenge. I was at the Math Circle institute last week and Bob Kaplan suggested “what would we have to know to launch a rocket from the parking lot and get it to the moon?” I don’t know yet if this can be made tractable and compelling at the same time.

@Max – I like algebraic questions, too. I don’t think “what is a number” and related have been big questions of previous courses. But I’m hopeful that by the end of the coming year my Alg2 students could sensibly discuss some of the year’s material from an algebraic point of view. I guess that’s what I’m really going for: inducing some sense of functions as objects from the specific models that arise in class.

[…] rich and engaging “pure” problems. MBP discussed part of the reason for this in his comment (“Let’s partition the world of course-spanning questions into the purely mathematical and […]

[…] II? The comments on this post are excellent, and Dan followed it up an excellent first stab: Relationships. Another interesting theme is MBP’s, How can we predict the […]

[…] My colleagues and I are re-writing our Algebra 2 course over the summer. At the moment we’re going with Michael’s excellent suggestion of “predictions” as the main idea […]