Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.

I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?

At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):

And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.

On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.

Developing Curriculum

Ed schools, particularly elite schools, preach the need for teachers to create their own curriculum. We don’t get any lessons on how to take a few pages of a textbook and break it down into explanation and practice, or how to select a good range of problems from a textbook to assign for classwork. Many ed schools use the Wiggins and McTighe text Understanding by Design (and its follow up on differentiating instruction), which argues that adherence to textbooks lead to the sin of coverage. No, not only shouldn’t we follow the book, said my ed school (and many others) but we should throw the book out and work backwards from out learning goals.

I thought this an idiotic idea. Open the book, give some examples with a good explanation, and have them work some problems. I explain things well, and any decent textbook has a wide range of problems to assign.

I had good instructors, particularly in C&I, and I expressed this opinion frequently and openly. I was not shot down—ed schools are doctrinaire at the administrative level, but at the instruction level, I found all my professors to be open to challenge. In fact, when I look back, I’m struck by how often my instructors reiterated what Wiggins himself says time and again: Understanding by Design is a framework, not a philosophy.

But it was impossible for me to believe that because the examples in the readings, and the openly progressive politics of ed school always sold curriculum design as a way to indoctrinate. The UBD books use, as an example, a history teacher who created an elaborate project for students to design their own constitution, reflecting the needs and interests of everyone in the community, not like those racist Founding Fathers. Or the instructor might describe a math teacher with an equally elaborate projects for students to “discover” transformations of functions when none of them have the skills to solve the functions in the first place.

Or there was the time we had to listen to an absolute idiot of an English teacher at an inner city charter school yammer on ignorantly about the “culturally whitewashed curriculum” that gives urban kids Robert Frost, who (to paraphrase) didn’t know trouble, didn’t know suffering, and wrote peaceful rural poems this teacher’s unfortunate inner city students couldn’t relate to, instead of the “real” poetry of Gwendolyn Brooks. There I was, paying huge chunks of money to listen to a brain dead jackwit present Frost as an out of touch white guy who wrote pretty poems about happiness and peace. Surely there’s a teacher out there who has considered a lesson for inner city kids helping them to see Frost’s hidden bleakness by contrasting it with Brooks’ open despair, and surely that would be the teacher invited to lecture at an elite university instead of this buffoonish hack? But I digress.

Ironically, I learned that textbooks could be a problem when, my first year out, I worked with a famously progressive, constructivist text known as CPM. I’ve used CPM to teach geometry and both years of algebra, and all the books had a few moments of interesting brilliance, way way WAY too much text, not enough practice problems, insufficient respect for the real priorities of any subject and an ordering approach that drove me crazy. I hated it, just like the good little progressive teachers hate their cold, formula-laden traditional text, and so I learned to ignore the book and develop my own curriculum.

Three years out, I have a very different view of textbooks. Good ones are great tools for ideas and problem sets that I can dip into as needed. But all textbooks fall short in some ways, many of which aren’t their fault.

The big problem: I often teach kids who won’t use them.They certainly won’t take them home and back to school (most of them just leave the books at home). They won’t use them as a resource. In fact, many kids actively prefer a worksheet to a textbook, as they get a sense of completion from finishing a page of problems. A textbook never ends.

Another problem is, of course, the size. At more than one of my schools, the kids don’t all get lockers. So the books are either going to stay home or stay at school. Last year, (2011-2012), I asked my kids to keep the books in the classroom, so I could use them on and off as needed. The textbook supervisor at the school got very upset at this, for good reason, but it worked. I was able to pull in the books as needed, particularly for my strongest kids, and ignore them the rest of the time. (Update: this year, at a different school, all my kids have lockers. It’s very convenient–I just write BOOKS in big letters on the board, and the kids go back to their lockers to get the books on the days I need them.)

Another problem is that textbooks are designed for one audience. Progressive texts, like CPM, are primarily designed for low ability kids in a constructivist classroom. Not enough problems, very few challenging problems, too much text, usually strained efforts to connect math to “real-life”, way too much indoctrination, and an exhaustive preference for explanations over answers. Spare me.

So when I got to my current school, using Prentice for algebra and Holt for Geometry, I was happy to shake the CPM dust off and use textbooks daily. Alas, I realized that these books were the Papa Bear to CPM’s baby—a fire hose for all but a fraction of my kids. Most textbooks are designed for students who are actually ready for the material, with low ability kids an afterthought. These textbooks cover material far too quickly. In my current geometry book, three pages in one section covers both 30-60-90 and 45-45-90 triangles. Sure, because kids pick it up just like that. These books often include different worksheets on a CD for lower ability kids, but at that point, you’re not using the textbook anymore.

I also find books are too limiting. They rarely provide teachers with useful illustrating activities–sometimes the book will sketch out an interesting possibility, but leave the details to the teacher. For example, the Holt Algebra II text introduces complex numbers as if the topic is little more than a walk to the drugstore. I mean, the numbers are imaginary, for chrissakes, and the text just spells it out in a sentence and moves on. For as huge as math books have gotten, publishers still haven’t used any of that space to lay out an explanation that works for low ability kids. Of course, the kids wouldn’t use it anyway, since they’d avoid the textbook, but at least it would give me something to copy so I didn’t have to create my own or steal a good start off the Internet.

Then, the book ordering is often insane. The year I taught all Algebra I, the Holt book introduced rate problems and work formula problems in Chapter 2. I laughed. My kids are still shaky on subtraction, and I’m going to cover high-complexity word problems in the third week. Sure. Only my top students got these problems, and then only at the end of the year.

When I realized how advanced the book was, I checked with a senior teacher, and she snorted. “Oh, I don’t use the books.” At our school, in algebra and geometry, relatively few teachers use the books on a regular basis. They develop their own lessons, their own tests, they borrow worksheets, and cobble together a curriculum that, in their view, meets their students’ needs.

I know my experiences aren’t universal. I know many teachers teach from the book, and many teachers work collaboratively to produce a class taught over 13-15 sections by multiple teachers. I student taught at a school that planned course-alikes collaboratively, were faithful both to the (CPM) text and a common schedule for all classes. The next three schools I’ve taught at gave me a solid grounding in “teacher as island”; everyone does their own thing. Given my druthers, I’d rather the latter. While I do wish I worked in departments that did more course-alike planning, I’m becoming increasingly sympathetic to the teachers who resist lockstep synchronicity. (Update: Right now, I’m dealing with a math teacher who insists that everyone teach trinomial factoring in exactly the same way. Um. No. Unless y’all want to use my way.)

What spun off this post was a review of The Tyranny of Textbooks, a purported expose of the textbook selection committee, with proposals to change and improve the process.

But that makes me laugh. Improve the process? Tons of teachers don’t even use the books! Why waste billions on textbooks that go home to serve as doorstops? Pick a few approved texts. Buy a few sets of each. Let the teachers who want to use them get a class set, or (in the case of advanced classes) check them out to the students. In all but a few cases, schools could save money by using class sets—except, of course, many states legally require districts to give every kid a book. Taxdollars in action, baby.

Here’s the really funny thing: I’ve described what I do, and what many high school teachers do—develop our own curriculum to cover the standards. But it’s clear from even a cursory overview of the education debate, that a million teachers planning their own curriculum is not what eduformers foresee as the future of education in this country. It’s also clear, however, that most education reforms never make it to the classroom and don’t have a clue how teaching actually happens, particularly in high school. (Eduformers believe that once elementary school is fixed, all will be well. They’re wrong.)

So if you don’t like teachers overriding local and national priorities by developing their own curriculum, and using the books, too bad. First off, progressives own ed schools and they’re always going to be pushing teacher curriculum development.

But more to the point, the range of student abilities, and the expectation that low ability students are to be taught a college-prep curriculum, pretty much mandates curriculum development at the district, school, or even classroom level. You want lockstep classroom curriculum? Bring back tracking and develop different texts for different ability levels. Let’s all laugh at that idea.

And now, a mea culpa: given how much misery I caused my ed school, I feel it only fair to acknowledge that my disdain for the progressive agenda and my dislike of constructivism was drowning out my instructors’ message about Understanding By Design: Here’s a framework for building your own. Take what works and toss what doesn’t. While they might approve of a particular agenda, the framework is ideologically neutral.

My last three plus years of teaching have done much to increase my approval of progressives. Yes, their agenda is still overtly political and yes, they still ignore ability in much the same way that eduformers do. But progressives in ed schools know far more about teaching than eduformers will ever know, and buried underneath their squishy curricular nonsense is a core of useful knowledge that I tap into quite often.

(Note: I first wrote this while at my last school; the updates were made in mid-September at my third school.)