Archives for category: Algebra II

Compare these 2 sequences:














Will B ever catch up to A?
-after problem 48 from Park Math Book 1 (2011)

It’s obvious, says my student, that B will never catch A, because A starts off so far ahead. I am not convinced—what happens if the table keeps going? She comes back a short while later, with values calculated out to x=24. B is still behind A, which proves her point, she says.

If this exchange were between me and my young niece, I would be delighted. It would be charming. We would continue our debate, and we would be learning, and it would be fun. But since it is in fact between me and an Algebra II student, a young woman who will be a freshman in college next year, that same delight is there, but it is mixed with a discouraged, she-should-know-this, where-do-I-begin kind of feeling that I would not have if she were ten years younger. In short, my emotional response to this situation is far more strongly determined by my ideas about how things should be, than by what the student is actually putting in front of me.

So those ideas about how things should be are not helping. I think that’s why Shawn is so refreshing to read: he’s unwavering in reminding us that learning happens when it happens, not when the curriculum says it will. I have a choice, here, between delight in the learning now, or despair about the learning that hasn’t happened before. My niece proves that it’s possible to delight in a student holding this conviction, so delight in this situation is available. I’ll choose that. It’s not always easy to focus on that, not when it feels like everyone—me, my students, my school, my district, America’s public schools, the whole USA—is being judged by whether or not this eighteen-year-old is ten years more mathematically mature than my eight-year-old niece. But what choice do I have? This can be fun, or this can be miserable. I don’t see how misery will help her learn any more or any faster. What’s the Tom Robbins quote? “There are only two mantras, yum and yuck. Mine is yum.”

Last year I taught one section of an all-writing-and-presenting precalc course. I was spending 5+ minutes reacting to each student paper and students were turning them in at 3-6 per day. So, manageable … but I would get 2-3 days behind, then save them for the weekend, then not get to them on the weekend … pretty soon I was buried.

The superstar ELA coach at our school helped me collect and combine the comments I was writing most often, and the things I wished I were seeing, into a rubric. We then ran off 20 copies and sat down with 20 papers. I had a highlighter and a pen and a stapler. She had a stopwatch. On your marks … get set …

The first batch of 10 took about 10 minutes. The next batch of 10 took 8 minutes. I just highlighted away, with no further annotation except sometimes to elaborate one of the highlighted remarks or to write a quick note that didn’t get covered by the rubric. After a year of practice and tweaking the rubric I now am down near 30 seconds for many papers, and average less than a minute. The feedback for students is much better than when I was writing it all out.

The time savings aren’t just in highlighting vs. writing. It’s also that the rubric focuses my attention. Title. Intro-does it work? Can I follow the explanation? Is it correct? Bam, bam, bam down the page. It also prevents me doing the work for them. “Solution is not correct” is all they need to know. If I write out a correct one, they don’t read it, and if they do, they don’t learn. Better that they find the error, or if they can’t, that they come find me.

The rubric I’m starting with this year is the product of having used and tweaked it for hundreds of papers last year. You can find it around page 4 of this document. But if you want to make your own from scratch, start grading papers freehand and keeping track of what you’re looking for, what you’re always writing. After a few dozen papers you’ll have a good draft of what you need.

This post was inspired by a twitter conversation between @mpershan and @cheesemonkeysf—I know the latter shares my enthusiasm. If we can save even one life, it will have been worth it.

Last year in precalc, everyone presented. (If you solved a problem, you had to present before you could turn in a paper, and the only way to earn a grade was to turn in papers. Publish or perish, baby.) But the audience often sucked. The crowd would wait politely without really trying to follow along, then clap politely and go back to what they were doing. Of course, that didn’t make anyone very eager to present.

So this year in Algebra II and Precalc we’re trying something devised by two of my colleagues and me, based on stuff we learned about feedback from Mylène and Patrick.

Everyone gets a slip of paper to fill out for each presentation:

The questions correspond to the full rubric for presentations (the rubric shows up around page 5).

The goal for the audience is to ask questions of the speaker in order to help the speaker score all YESses—without doing it for the speaker.

Everyone fills out yes/no sheets and turns them in to teacher. Teacher reviews quickly and that day or next gives general feedback to group about their sheets: “Didn’t seem to me that you guys could actually check the steps in Sam’s presentation, but you said you could.” or “Here are 5 samples of comments you wrote for Lianne. Which ones are specific and which are not?” Teacher gives sheets to speaker.

Any troubleshooting you wish to offer prior to launch will be greatly appreciated.

In my district, and many others, Algebra II is a graduation requirement. Given that requirement, I find myself asking what life skills the class can give everyone, whether they continue to practice mathematics or not—and not just from math class in general, but specifically from Algebra II.

For example, an oft-quoted justification for requiring geometry is to teach students to “think logically”. In the 1930’s, Harold Fawcett taught a geometry class in which students learned, and practiced, thinking logically in non-mathematical contexts.

I wonder if a good candidate for a life skill that we hope transfers from Algebra II would be this one from Bowen Kerins, one of the authors of CME:

One thing a great context / question also gives you is the experience of figuring out what information is important and what sort of abstraction is most useful for extracting and using the right information thoughtfully. And that’s a skill a lot more adults will use than factoring …

If that’s to be our transferable skill, then we’ll need to practice it: have lots of non-mathematical examples where “extracting and using the right information thoughtfully” is required. And I have to admit: I’m so unconscious of when and how I am using this skill I’m not at all sure how to begin thinking of examples!

This is not a review. These are my notes from reading Basic Principles of Curriculum and Instruction by Ralph W. Tyler (U. Chicago Press 1949, 128pp.)

How I found it

Brian’s reference led me to this post by Grant Wiggins. Tyler comes in at the bottom, but it turns out Wiggins writes about Tyler a lot. You can get a lot of what follows just from reading those Wiggins links, but it was fun for me to go to the source. Here’s what I found.


The book describes curriculum planning as a process of answering four questions:

  1. What are the right objectives?
  2. What learning experiences are likely to attain those objectives?
  3. How to effectively organize (sequence) those experiences?
  4. How to evaluate those experiences?

He doesn’t answer them directly but outlines very wide-ranging considerations to use when approaching the answers. Every educational philosophy, trend, and approach I can think of sits neatly somewhere in the framework he lays out.

What I’m taking from it

The things that are sticking with me appear early in the book:

A good deal of controversy goes on between essentialists and progressives, between subject specialists and child psychologists, between this group and that school group over the question of the basic source from which objectives can be derived. The progressive emphasizes the importance of studying the child to find out what kinds of interests he has, what problems he encounters, what purposes he has in mind. The progressive sees this information as providing the basic source for selecting objectives. The essentialist, on the other hand, is impressed by the large body of knowledge obtained over many thousands of years, the so-called cultural heritage, and emphasizes this as the primary source for deriving objectives…. Many sociologists and others concerned with the pressing problems of contemporary society see in an analysis of contemporary society the basic information from which objectives can be derived…. On the other hand, the educational philosophers recognize that there are basic values in life, largely transmitted from one generation to another by means of education. They see the school as aiming essentially at the transmission of the basic values…. The point of view taken in this course is that no single source of information is adequate to provide a basis for wise and comprehensive decisions about the objectives of the school. -pp. 4-5

I tend to get identified with one point of view at a time, then suddenly agonize about what I’m not doing. This helped me realize that all these points of view are legitimate and there’s room for all of them in the curriculum, though maybe not all of them at all times. I can relax and think about how to keep them all involved in my teaching overall. Maybe as I get stronger you’ll be able to see more of them in each individual day.

Education is a process of changing the behavior patterns of people. This is using behavior in the broad sense to include thinking and feeling as well as overt action. When education is perceived in this way, it is clear that educational objectives, then, represent the kinds of changes in behavior patterns of the students which the educational institution should seek to produce. -p.5

This quote didn’t really mean something to me until I’d read the rest of the book, but it sticks with me on three levels. On the shallowest level, this helped me move from “skills vs. content?” to “use skills on content.” Tyler recommends a 2D matrix with behaviors on one axis and content on another. So in the Algebra II planning I’m doing for next year with colleagues, we’ve come up with a chart of 26 skills by 5-7 function families. Together these make a year’s worth of daily objectives; it’s what most people think of as “Algebra II”.

The second level is more broad behaviors, or as Wiggins says, “use content well”. For me this is something like “Given some data, make a prediction.” It’s the broad skill of identifying and then using a functional relationship to answer a question. Can you do that and explain it verbally and in writing? This is what our Algebra II course is about. So the matrix is predict, write, present predictions vs. linear, quadratic, exponential etc.

The third level is life skills – what we learn to do in math that helps us everywhere. I took a stab at defining these for myself here, but haven’t made those goals an explicit part of my instruction. Yet. More soon.

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, with functions (using a relationship metaphor) as the main tool (see last year’s discussion about all this). Our evolving google doc is available for anyone who wants to follow along. If you have any input, please chime in. I’ll add to this post as our ideas shape up.

7/16/12: We’ve updated the course standards and grading policy. Next step: design examples & criteria for each standard on what will be accepted as demonstrating proficiency.

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.

Modeling is in the air. Thursday a colleague told me he was looking at Modeling Instruction for next year. Then yesterday, Shawn committed. So I started reading up. I was a full-time modeler before teaching, so I’m biased, but I love it. In Modeling Instruction physics, students organize their year by making and testing hypotheses about observable aspects of a few archetypical physical systems. The hypotheses arise from curiosity: we see the pendulum moving, what are things we could measure? What predicts their values? Here is the world, you are already fluent in it, go make sense of it. What the students construct in response is a solid understanding of the core content of introductory physics.

I could make Algebra 2 into a modeling course. But modeling is using math to describe what you are studying; I would like to be studying the math. I’m looking for a destination such that the study of each of these functions is a step towards something pretty deep and beautiful.

These functions are the structure of everything. Yes, you can model radioactive decay with an exponential. And yes, you can model compound interest with an exponential. But the exponential, the idea of the exponential, is the abstraction of what interest and radioactivity have in common: what you get is proportional to what you have. Yes, the height of a ball over time is quadratic. Yes, as you fence off your rectangular llama pasture, making it squarer and squarer, the area is quadratic. The parabola is the abstraction of what constant acceleration and constrained rectangular area have in common: the unceasing influence of the second difference, patiently turning things its way one bit at a time, until the system is inevitably flying in its direction.

So, what? “Algebra 2: The structure of everything”? It’s grandiose, not compelling. It isn’t even a question. It certainly isn’t “Would You Go to Mars?” (via Dan).

These objects, the stuff of Algebra 2, are too fascinating and too pervasive not to lead to some summit worth attempting.

If you want students to master something–call it R–and R is a means to S, then work on S; R will slip unobtrusively in under the radar….” -Bob and Ellen Kaplan, Out of the Labyrinth, p. 87

I need an overarching theme, question, or mission for Algebra 2 that transcends and motivates the required skills.

The content goals of Algebra 2 are to invert, transform, solve, and apply graphical, tabular, and analytic representations of linear, quadratic, exponential, sinusoidal, and rational equations. That’s R.

What’s S?