Teaching Transformations

One of the most important new concepts in algebra 2 and beyond is the notion of transformation. That is, given the function f(x), we  can change any function’s position and growth by using the same instructions, much like giving directions from a map.

I’ve just introduced functions at this point in the calendar, so I’ve designed this activity to reinforce f(x) as a rule, that once a mapping is created, the mapping holds for all subsequent calls.

So just create a random table, one that’s simpler than anything I’d do in class. (One of the incredibly irritating things about blogging is that it’s insanely time-consuming to create images for publication that take next to no time at all to do on  a smartboard, but I never think of capturing images while on a smartboard.)

x	f(x)
-3	2
-1	5
1	6
3	3
5	-1

That looks like this:

So then I ask if this is f(x), what would f(x+2) look like? Someone brave will always say “Two to the right”.

At that point, I always say “This is a totally logical guess and one of the most annoying things in math from this point on is that your guess is wrong.” (I originally developed the concept of a parabola as the product of two lines as another way of explaining this confusing relationship. Confusing to normal people. Mathies think it makes sense, but they’re weird.)

I add a column to the table. “We start with x. Then we add 2. Then we make the function call. Note the function call comes after the addition of the value. This is important. Now, we have three columns, but we are starting with our x and that’s still our input value. We graph it against the outer column, the output value for f(x+2).”

x	x+2	f(x+2)
-3	-1	5
-1	1	6
1	3	3
3	5	-1

I’ll ask how we can bring the -3 back in, and after some mulling, they’ll suggest that I add -5 to the table. So I add:

-5

-3

2

to the bottom. But I’ve been plotting points all along, so the kids can see it’s not going as expected.

“Yes, indeed. I’ll be teaching this concept in many ways over the next few months, and I ask you to start wrapping your head around this now. We have many ways of envisioning this. When working with points as opposed to an entire function, it might be helpful to think of it this way: Suppose I’m standing at -3, and I want to add two. This has the effect of me reaching to the right on the number line and pulling the output value back to me–to the left, as it were.”

I go through this several times. Whether or not students remember everything I teach, I always want them to remember that at the time, they understood the concept.

“So if standing on -3 and reaching ahead is addition and move the whole function to the left, how would I move the whole function to the right?”

If I don’t get a ready chorus of “subtract?” I know that I need to try one more addition example, but I usually get a good response.

“Exactly. So let’s try that.”

x	x-2	f(x-2)
-3	-5	NS
-1	-3	2
1	-1	5
3	1	6
5	3	3
7	5	-1

One year, I had a doubter who noticed that I’d made up these numbers. How did we know it’d work on any numbers? I told him I’d show him more later, but for now, imagine if I had a table like this:

x	f(x)
1	1
2	2
3	3

etc.

Then I told him, “Now, imagine I put decimal values in there, fractions, whatever. Imagine that no matter how I change the x, the new value has an entry in the table and thus an output. So imagine I added 50. There’d be a value 50 ahead that I could reach forward or backwards.”

“In fact, we’ll eventually do all this with equations that are functions, instead of randomly generated points. But I start with points so you won’t forget that it works with any series of values that I can commit math on. Which isn’t all functions, of course, but that’s another story.”

“But if adding makes it go left and right, how do we make a function go up and down? Discuss that among yourselves for a minute or so.”

Sometimes a student will see that we’ve been changing x so far. Otherwise I’ll point it out.

“The function call itself is key to understanding this. If you change the value before you make the function call, then you are changing the input to the function. Simpler: you’re changing x before you call the function. But once the value comes out of the function, that is, once it’s no longer the input, it’s the….” I always wait for the class to chime in again–are they paying attention?

“Output!”

“Right. But output is no longer x. Output is”

“f of x!”

At this point, I call on a mid-level student. “So, Sanjana, up to now, we’ve been changing x before making the call to the function. See how the new column is in the middle? What could I do differently?”

And I wait until someone suggests making the column on the right, after the f(x).

x	f(x)	f(x)- 3
-3	2	-1
-1	5	2
1	6	3
3	3	0
5	-1	-3

I’m giving a skeletal version of this. Often the kids have whiteboards and are calculating all this along with me. I’ll give some quick learning checks in terms of moving to the right and left, up and down.

The primary learning objective for is to grasp the meaning of horizontal and vertical translations–soon to be known as h and k. But as an introduction, I define them in terms of function notation.

 

We usually end this activity by combining vertical and horizontal shifts.

What would f(x-2)+ 3 look like? Well, you’d need another column.

x	x-2	f(x-2)	f(x-2)+1
-3	-5	NS
-1	-3	2	5
1	-1	5	8
3	1	6	9
5	3	3	7
7	5	-1	2

I connect them this time just to show that one point is in both the original and the transformation.

Ultimately, this goes to transforming functions, not points. That’s the next unit, transforming parent functions. I have a colleague who teaches transformations entirely by points. I start down that path (not from his example, just because that’s how this works), but the purpose of transformations, pedagogically speaking, is for students to understand that entire equations can be changed at the unit level, without replotting points. At the same time, I want the students to know that the process begins at the point level.

Over time, the students start to understand what I often call inside and outside, or before and after. Changes to the input value affect the x, or the horizontal because they occur before the function is called. Changes to the output value affect the y, or the vertical, because they occur after the function is called. Introducing this on a point by point basis creates a memory for that.

At best, this lesson functions as more than just a graphing exercise, something to introduce vertical and horizontal shift. It should ideally give students an understanding of the algebra behind it. Later on, when they are asked to solve equations like:

Find f(a) = 32 for f(x)=3(x-2)²+5

Weaker students have trouble with understanding order of operations, and a memory of “inside” and “outside” the function can be helpful.

If I were writing algebra 1 curriculum, I’d throw out quadratics, introduce a few parent functions, and teach them function notation and simple transformations. It’s a complicated topic that they’ll see all the way through precalc, at least.

I’ll discuss stretch and its complexities in another post.

Glenn, John, and Philip K. Dick

In the last segment of their recent bloggingheads discussion, John McWhorter told Glenn Loury he was writing a piece suggesting that discussions of an IQ gap be deemed unacceptable for public discourse¹:

I don’t understand what possible benefit there could be. I think it should stay in the journals.

Glenn agreed, and mentioned his famous response to the conservative welcome of The Bell Curve, in which he angrily rejected Murray’s assertions.

My first, instantaneous reaction was hey, great idea.  In fact, it will save us billions. Too bad no one proposed it twenty years ago, before we wasted so much time and energy on No Child Left Behind, forcing states to report by racial demographics. Think of all the schools struggling to meet adequate yearly progress and failing because they couldn’t teach students to perform in perfectly average racial unity. Let’s be sure to tell the College Board; they’ll be happy to stop breaking out test results by race; it’ll save them so much criticism.  This will put NAEP out of business of course, so….

What’s that? You don’t think McWhorter would link banning discourse on race and IQ to ending all scrutiny of race-based academic achievement?

Ya think?

Of course John McWhorter knows that race-based academic achievement is at least tangentially related to discourse on race and IQ.  I also think he understands that race and IQ discussions have been The View That Must Not Be Spoken for forty years. In fact, he even mentions that the only sites engaging in this discusssion are “right wing chat sites” and “some blogs”.

So sarcasm aside, I was a bit puzzled by the proposal, as well as Loury’s endorsement. These are two intellectually honest academics, who are generally fearless on racial topics. Here they are declaring certain topics Voldemortean–and doing so in the event that the link between race and IQ is proven out.

Without even going into the suggestion itself, consider that McWhorter posits one response to an acknowledge racial IQ gap:

…are we going to have an arrangement where we allow that black people have these lower IQs and therefore give extra help to black people, with the presupposition that all the average black people aren’t as bright? I think the black people would take it as an insult.

Fifty years and counting on affirmative action (which McWhorter famously opposes), and McWhorter thinks black people would take extra help as an insult? Seriously?

Even more surprisingly, Glenn Loury doesn’t point out this obvious hole.  Or maybe not so suprisingly, I’m a huge Loury fan, but as I’ve mentioned, he’s got some blind spots on education.

The most notable one I can recall is in this  excellent 2010 discussion Loury hosted with Amy Wax, author of Race, Wrongs, and Remedies:

Wax’s central point both in the discussion and in her book is that black academic underperformance is due to the wholesale collapse of the black culture and black family.

Loury pushed back hard (emphasis mine):

“Well, gee, when I look at education, I think, true, what happens at the home is really important….but I do think that the fundamental problem with urban education is its political economic organization. It’s the unions. It’s the work rules. It’s the efficacy of teachers in the classroom, and so on. It may also be to some degree the resources.

And that’s something that government, with great difficulty, with some consternation, can change….that’s something that ONLY government can change, either by making resources available to parents so they can have outside options and not be reliant on a school system that’s failing, and/or by reorganizing the functioning of those systems that are failing so that the kind of things that reformers want to do, that are known to work, are permitted to be put into place.

But in either case, those are political public large government undertakings that need to be done, and to blame the failure of urban education on the culture of African Americans is, well, maybe just a little offensive….Why wouldn’t we want to think of that as a public problem in public terms rather than [blame] the people who are really the victims of failed public policy?”

So Loury holds on to some of the conservative tenets from his youth.  Like all conservatives, he wants to blame schools: crappy, incompetent teachers, unions, and by golly maybe more money would help, too.

Wax responds that blaming schools makes no sense, because black kids in suburban schools are doing poorly. They do poorly in the same excellent schools that whites do well in, and she argues again that it’s culture.

Somewhat surprisingly, Loury agrees. Suburban black kids are doing terribly, too, but not for the same reason:

“I would not identify underperforming middle class African American performance in good suburban schools, which is a real thing, with the absolute wasteland in terms of the cognitive development of the students who have no alternative but to attend the failed inner city public school system. Both are problems, but the latter is just a huge problem and I say it’s mainly a problem of failed institutions not of inadequate culture.“

So Loury has constructed  an absurd dichotomy: Middle class blacks and urban blacks have weak academic performance and he tacitly agrees with Wax that the problem is cultural. But urban black academic performance is about ignorant teachers and union work rules.

Wax comes back with an answer that any teacher would thank her for (which is why I’m quoting it in full)

“I would really argue, are the teachers really not trying to teach them basic stuff, or is it that the students, for whatever reason, are just so ill-equipped when they come in, so indifferent?….We know, from Roland Fryer…that these students are coming in significantly behind their counterparts, so they’re already behind the eight ball…then to turn around and say it’s the school’s fault that they can’t learn arithmetic, or they can’t learn to read. I really question that. I just am not sure that that is the locus of the problem or even the main problem.“

Of course, then she ruins all that good sense by saying that the locus of the problem is  “the abject failure of the family….just collapse.”

Little evidence for either Wax or Loury’s position exists. Roland Fryer’s research on Harlem Children’s Zone ( the 2010 version) gets a mention by Loury as evidence that charters can do a better job. That’s odd, because Fryer’s research has widely been cited as evidence that HCZ does only an adequate job as a charter school, far less impressive academically than KIPP or the others–and of course, their “improvement” has a whole bunch of caveats. But Wax appropriately observes that the results, back in 2010 were new and fadeout often occurs.  As the medium term results from HCZ show,  lottery losers indeed  “caught up” in many ways.

Loury often swats his discussion partners for “asserting from faith”, but his demonization of  “failing schools”is exactly that.   Without any evidence, and considerable evidence against his position, Loury argued two separate culprits for black underachievement, depending on the SES category. Moreover, he uses a big ol’ group of conservative shibboleths to justify his position. That is not the Loury I know from other conversations.

For entirely understandable reasons, both Loury and McWhorter see any discussion of the IQ gap as a personal affront. They both interpret “racial IQ gap” as “blacks are inferior” and I’m sure that there are people who push the topic who see it that way. But one can angrily reject average group IQ as a sign of inferior or superior status while still acknowledging the hard facts of cognitive ability–Fredrik deBoer does it all the time. Whenever they discuss race and IQ, Glenn and John jokingly  mention how smart they are–which they are! But they don’t ever acknowledge that their tremendous intellects aren’t a rebuttal to the discussion at hand.

Men are, on average, taller than women. Michelle Obama is taller than Robert Reich. Both statements are true. Why more people can’t apply this to IQ is a mystery.

I don’t know all the corners of IQ science history, but I’ll stipulate that many unpleasant people discuss IQ gaps with a disgusting glee. I find it incredibly troubling how many people use “genetically inferior” as an equivalent term for “blacks have lower IQs on average”.  But I spend too much time with students of all abilities, and all races, to consider race as the logical grouping for IQ.  I’m more interested in IQ, or more generally the cognitive ability discussion, as a starting point to correctly frame what is now cast as a public education failure.

Our schools “fail” to educate many students. We tend to focus only on the black and Hispanic students–and not as individuals, just as data points that would push the average up nearer to that of whites.  We use white average academic achievement as the standard for success.  We began comparing racial groups back when it was primarily blacks and whites, in that optimistic era after Jim Crow ended, confident that the data would show blacks catching up to whites. If blacks had just caught up, if we just had the same amount of students “failing” to be educated, we’d have moved on.

But blacks never caught up. Since at the national level, we’d begun with the presumption that the gap was caused by racist oppression, we continued with that assumption as long as possible. Over time, other culprits arose. It’s the parents. It’s the culture. It’s the schools. People who offered cognitive explanations were ostracized or at least subject to a barrage of criticism. It’s always odd to hear Loury talk about the Bell Curve era as a traumatic time, when his peers were coolly discussing racial inferiority, while almost everyone else recalls it as a time when Murray was nearly banished from the public square.

As time went on, despite our failure to close “the gap” or, more accurately, despite our failure to provide a needed education for all students, demands went up. High school transcripts got more impressive, more loaded with “college prep” courses and, in many high schools, more akin to fraudulent documents, all designed to push all kids into college in equal proportions. Colleges have obligingly obliterated requirements. Schools have increasingly come in for blame from the political and policy folk, but all attempts to penalize schools for their failure have, well, failed. The public likes their schools and the public, frankly, is more willing to consider cognitive ability relevant than the political and policy folk are.

I’m always reminding myself that most  people see it in, literally, black and white terms for very good reason. But I only discuss IQ in terms of race because society insists on grouping academic achievement by race. Ultimately, I see IQ discussion as an effort to correctly categorize the “failure” of some students, regardless of race. I see it as a way of evaluating student achievement, to see how to best educate them to the extent of their ability and interest. I am well aware that these questions are fraught with reasonable tension.

But I worry, very much, that we won’t take needed steps both in education and immigration policy (as well as a host of other areas, no doubt) if we don’t stop insistently viewing cognitive issues through the prism of race. That is, as I first wrote  here, we need to consider the possibility that the achievement “gap” is just an artifact of IQ distribution. 

I would be pleased to learn this is not the case, as I wrote then. But if in fact IQ distribution explains the variations in academic achievement we see, then we need to face up to that. This facing up does not mean “well, your people are good in sports and music”. The facing up means asking ourselves regardless of race, how do we create meaningful jobs and educational opportunities for everyone?

 

I hope they change their minds. Because we are putting millions of kids in schools each year, making them feel like failures. Yes, some are black. Others are white, Hispanic, Asian. And we’ve spent no time–none–trying to figure out the best ways to educate them.  We’ve only looked for causes, for the right groups to blame.

Of course,  maybe we could trade: no more talk of the achievement gap in exchange for no more talk of race and IQ. That’s not the best approach, though, because in today’s employment environment, we need to educate everyone, not write off “failures”.

But I guess Glenn and John–along with a lot of other people–are still trying to wish reality away.

Note: The piece  Philip Dick, Preschool, and Schrodinger’s Cat is still canonical Ed on IQ.

 ¹: since I wrote this, McWhorter published the piece in the National Review. I made some additional responses directly on Twitter.