Functions vs. Equations: f(x) is y and more

I wanted to talk about function algebra, which naturally would include a reference to function notation.

So here’s the frustrating thing about writing this blog. I try to include links to other sites that explain a concept, so that I don’t have to reinvent the wheel for my reading audience. But a google gives me these results: useless links that do little more than say “f(x) is the same as y”. That’s not math. That’s test prep. And there’s nothing wrong with test prep, but every one of these sites purport to be math teaching sites, and hey, I’m not a mathematician, but shouldn’t we be explaining what f(x) means?

Someone somewhere is saying “See, this is why we need teachers to be math majors, instead of English majors who get 800 on the GRE quant section. You can’t substitute math understanding that comes with the study of these important principles.” That someone somewhere is wrong. I used to think that in my early days, until I had too many conversations like this:

Me, to AP Calculus teacher WHO MAJORED IN MATH: Hey, what do you tell your kids about function notation?

AP Calculus teacher WHO MAJORED IN MATH: f(x) is the same as y.

Me, nonplused: Well. Yeah. But I mean about why we developed function notation, what it serves that can’t be served by….

AP Calculus teacher WHO MAJORED IN MATH: It’s just notation. Don’t be confused.

Me: I’m not confused. But they serve different purposes, and I’m just trying to be sure I accurately capture…

AP Calculus teacher WHO MAJORED IN MATH: They don’t serve different purposes. It’s just notation f(x) is the same as y.

Me: Ok.

In my experience, very few math teachers WHO ACTUALLY MAJORED IN MATH care about these things either. My beer drinking buddy is an exception (and he’s now department head), and he’s the only math teacher I’ve found so far who was interested in my work on this subject.

Textbooks? McDougall Litell, CPM has a lot of those function machines. But no explanation. Holt does a little better but I didn’t understand that until I understood what I was looking for.

So I spend more time looking for a good link. Otherwise, I have to spend a lot of time figuring out how to explain function notation accurately, or at least inoffensively, so that people reading this blog don’t make me remind them that, for chrissakes, I’m an English major not a mathematician! That takes time. It’s not time I wanted to spend. I don’t want to tell you what function notation is, in a way that will pass expert muster. I want to tell how I build on function notation to teach function algebra. But I can’t do that well without explaining function notation, which I didn’t set out to do. This leads to many blog entries taking much more time than they should. The original intent for my function algebra post was to be just a quick little throwaway.

I began writing this post nearly a month ago, and got stalled looking for a way to characterize the explanation. You may be wondering why I would explain something I don’t understand—but that’s not it, really. I just don’t know what to call it. And that’s fine for teaching, not so much for writing, and so I spend hours trying to figure out the correct query. Which took me, literally, up until today.

Just fifteen minutes ago (as I write this sentence) I finally found the kernel in this discussion on function notation before Euler, in which someone writes:

but [Newton] refers to these as equations, not functions, and admittedly (written the way they are) that is exactly what they are. It seems anything that we would today write as a function, Newton described in words, such as:

HA. I learned something I hadn’t quite understood completely before–a function and an equation are not the same thing. Googling “what is the difference between an equation and a function” led me to the right websites. I realize now that I wasn’t just looking for an explanation of function notation, but rather why and when we use functions vs. equations.

Here’s an explanation that covers what I was trying to say.

So my research paid off. In practice, what I’ve been doing in this lesson is introducing function operations and function notation as a way to overcome a constraint in using equations.

******************************************************************

Sami needs $15 more to buy the new hoodie that he wants. But if Sami skips the hoodie, he needs just three more dollars to buy a ticket to the pizza feed on Friday. If Sami has x dollars, how much money, in terms of x, does Sami need if he wants both the hoodie and the ticket to the pizza feed?

The first thing the kids think is that Sami needs $18 more.

I say okay, Sami has $20. How much does the hoodie cost? $35. How much does the pizza feed cost? $23. How much ….oh. Huh, say the kids. He needs a lot more than $18.

Depending on how goofy I feel, I might get out some fake money. I count out $20, give it to a quiet student. How much more for the hoodie? Count out another $15. Now how about the…Right about then, a student gets it: you need the $20 twice.

So then we go to the board and model the two different equations for each purchase.

y=x+15
y=x+3

So if we are getting both things, what are we doing? Adding, the class choruses.

Ah, now there’s a new wrinkle. The kids have been adding equations for a while now, in systems. So I say, let’s try to add these equations.

2y=2x+18.

Is that right? We test it with $20 and the kids realize that the right side “works” (that is, we get $68) but the left side says we still need to divide by 2, which would be…wrong.

“So what’s happening is that we are running into the limits of an equation. An equation tells us that two expressions occupy the same point on a number line–that is, after all, what “equal” means.”

“But when we use multiple variables in equations, then the equation becomes a relationship between two variables, an if-then. If y=x + 15, then the point (3, 18) is a solution because setting x=3 and y=18 creates an equation that has both sides occupying the same point on the number line. If 3x + 2y=12, then (2,3) is a solution because setting x=2 and y=3, etc.”

But in an equation, the variables are values. So in the Sami case, we can’t treat y as a collection point. We can’t keep track of the dependent variable because it varies, obviously. The y in the first equation has a different value from the y in the second equation. If we wanted to keep them separate, we could use two different variables, like z = x + 15 and y = x + 3. Or we could number the ys: y1 = x+15, y2 = x+3.

“Using the language of functions makes a lot of these constraints disappear.”

“First, logically. Functions are different in a key way from equations: a function is an output. An equation is a relationship between variables. Yes, y=x+3 and f(x)= x+3 yield the same results, which is why we teachers always tell you to remember that ‘y and f(x) are the same thing’. However f(x) isn’t a variable, but an output. So when we add two functions, we’re adding outputs. Remember, too, that a function doesn’t even have to be an equation, like in the cell phone code example.

Then there’s function notation, invented by Euler. Function notation enables unique names, usually a single letter. But it doesn’t have to be. You can get creative with the letter names and the input values.”

“Function notation is just more elegant and efficient, too. Instead of saying ‘if x=7′ you can just say f(3). Once you define the function named ‘f’, anything can be input, even another expression, like f(a+7). And then, instead of saying ‘y=’ and solving for x, write f(x)= 3.”

“So let’s call Sammy’s cash on hand c, and then create a function h for hoodie, and p for pizza feed.

h(c) = c+15
p(c) = c+3

In both cases, c represents the money Sami has, so the input value is the same. But the output value varies based on the function used.”

“Now, this is a small difference. But how many have you been told that f(x) is the same as y?” Bunch of hands raised.

“Yep. And in a lot of ways, it is. But you have to be wondering why, if they’re the same thing, we bother teaching you about function notation.” Lots of nods.

“So as you move on into advanced math, you’ll start to learn other reasons why we sometimes use functions and other times use equations. For now, it’s enough to know that function notation allows us to keep track of our different outcomes.

“Once we can do this, we can actually create an entire math with functions. They can be added, subtracted, multiplied. They have inverse operations.”

“But then why do we use equations?”

“Well, for one thing, functions don’t do systems well. Remember, when we solve systems, we are expecting both the x and the y (and any other variables) to be equal. Functions don’t handle that well. So you’ll see that we switch back and forth between equations and functions as needed.”

When you need to add expressions, functions are great. So now we can add h(c) and g(c).

h(c) + p(c) = (x + 15) + (x + 3) = 2x + 18

“Because we are adding outcomes, and have a unique way of tracking each outcome, we can add them properly. Remember, too, that since a function doesn’t need to be an equation, I can add or subtract outcomes without even having an equation. If a(x) = 9 and b(y) = 17, then b(y) – a(x) is 8, and I don’t have to care if a(x) and b(y) are generated by an expression or a rule or a code or a random happenstance—provided, of course, that random happenstance is only one per input.”

******************************************************************************

I know. You’re wondering why I don’t just follow the AP Calculus teacher’s “f(x) is the same as y”. Well, it turns out that function operations are a big part of pre-calc, so they’ll use this later.

In the meantime, I give them some practice with function notation (I stole this at random). Not enough. Kids don’t really know it later. But at least they’re exposed to it.

Then I go on to linear function addition and subtraction. I usually just put problems on the board.

Sample quiz:

functionquiz

Here’s a test question:

functionma

And from here I go on to linear function multiplication (aka quadratics) and, eventually, rational expressions (linear function division).

Like teaching congruence with isometries, I can’t argue that using functions to further our work in linear and quadratic equations is better. I find it more…elegant, maybe?

But the execution isn’t quite there. This is the first year I’ve really taught this whole sequence: introducing functions, function addition/subtraction/notation, function multiplication, inverse functions, rational expressions. Writing it up has revealed an obvious improvement. Up to now, my function illustration has been a quick standalone lesson. Then later I introduce the notion of function addition and in doing so, bring up function notation.

This is goofy, now that I look at it. In the future, I’ll introduce functions and then go into function notation. I can spend a day or two on that, quiz that early. Then I can go back into linear equations or inequalities (the placement is flexible) and then bring up function addition and subtraction, with function notation already covered.

You know what’s irritating? The huge effort described at the beginning of this post to figure out how to describe what I was teaching led me to this. The huge effort underwent solely in order to write this post. Which I was griping about. In learning how to describe function notation for my readers, I learned that the proper way to characterize my work is as a difference between functions and equations, and that led to an idea for better sequencing.

This is kind of a placeholder post. Obviously, I’m in flux about this right now. My linear equations unit has been in good shape for a while. This gives me plenty of room to add flourishes, introduce more complicated topics onto a subject the students know well. Meanwhile, linear function multiplication has proven to be a great introduction to quadratics. So now I’m involved in putting it all together.

Next up in this sequence: the post that I really wanted to write, on my quadratics introduction.

Sorry for the slow rate of posts lately. I did five in April, then got lazy.


The Day of Three Miracles

I often hook illustrative anecdotes into essays making a larger point. But this anecdote has so many applications that I’m just going to put it out there in its pure form.

A colleague who I’ll call Chuck is pushing the math department to set a department goal. Chuck is in the process of upgrading our algebra 1 classes, and his efforts were really improving outcomes for mid to high ability levels, although the failure rates were a tad terrifying. He has been worried for a while that the successful algebra kids would be let down by subsequent math teachers who would hold his kids to lower standards.

“If we set ourselves the goal of getting one kid from freshman algebra all the way through to pass AP Calculus, we’ll improve instruction for everyone.” (Note: while the usual school year doesn’t allow enough time, our “4×4 full-metal block” schedule makes it possible for a dedicated kid to take a double year of math if he chooses).

Chuck isn’t pushing this goal for the sake of that one kid, as he pointed out in a recent meeting. “If we are all thinking about the kid who might make it to calculus, we’ll all be focused on keeping standards high, on making sure that we are teaching the class that will prepare that kid–if he exists–to pass AP Calculus.”

I debated internally, then spoke up. “I think the best way to evaluate your proposal is by considering a second, incompatible objective. Instead of trying to prepare every kid who starts out behind as if he can get to calculus, we could try to improve the math outcomes for the maximum number of students.”

“What do you mean?”

“We could look at our historical math completion patterns for entering freshmen algebra students, and try to improve on those outcomes. Suppose that a quarter of our freshmen take algebra. Of those students, 10% make it to pre-calc or higher. 30% make it to trigonometry, 50% make it to algebra 2, and the other 10% make it to geometry or less. And we set ourselves the goal of reducing the percentages of students who get no further than geometry or even, ideally, algebra 2, while increasing the percentages of kids who make it into trigonometry and pre-calc by senior year.”

“That’s what will happen with my proposal, too.”

“No. You want us to set standards higher, to ensure that kids getting through each course are only those qualified enough to go to Calculus and pass the AP test. That’s a small group anyway, and while you’re more sanguine than I am about the efficacy of instruction on academic outcomes, I think you’ll agree that a large chunk of kids simply won’t be the right combination of interested and capable to go all the way through.”

“Yes, exactly. But we can teach our classes as if they are.”

“Which means we’ll lose a whole bunch of kids who might be convinced to try harder to pass advanced math classes that weren’t taught as if the only objective was to pass calculus. Thus those kids won’t try, and our overall failure rate will increase. This will lower math completion outcomes.”

Chuck waved this away. “I don’t think you understand what I’m saying. There’s nothing incompatible about increasing math completion and setting standards high enough to get kids from algebra to calculus. We can do both.”

I opened my mouth…and decided against further discussion. I’d made my point. Half the department probably agreed with me. So I decided not to argue. No, really. It was, like, a miracle.

Chuck asked us all to think about committing to this instruction model.

Later that day, I ran into Chuck in the copyroom, and lo, a second miracle took place.

“Hey,” he said. “I just realized you were right. We can’t have both. If we get the lowest ability kids motivated just to try, we have to have a C to offer them, and that lowers the standard for a C, which ripples on up. We can’t keep kids working for the highest quality of A if we lower the standards for failure.”

Both copiers were working. That’s three.

**************************************************************

I do not discuss my colleagues to trash them, and if this story in any way reflects negatively on Chuck it’s not intentional. Quite the contrary, in fact. Chuck took less than a day to grasp my point and realized his goal was impossible. We couldn’t enforce higher standards in advanced math without dooming far more kids to failure, which would never be tolerated.

Thus the two of us collapsed a typical reform cycle to six hours from the ten years our country normally takes to abandon a well-meant but impossible chimera.

Many of my readers will understand the larger point implicitly. For those wondering why I chose to tell this story now, I offer up Marc Tucker, whose twopart epic on American education’s purported failures illustrates everything that’s wrong with educational thinking today. I would have normally gone into greater detail enumerating the flaws in reasoning, facts, and ambition but that’s a lot of work and this is a damn good anecdote.

Some other work of mine that strikes me as related:

I think I’ve written about my suggested solution somewhere, but where…(rummages)….oh, yes. Here it is: Philip Dick, Preschool and Schrödinger’s Cat–the last few paragraphs.

“Reality is that which, when you stop believing in it, doesn’t go away.”

When everyone finally accepts reality, we can start crafting an educational policy that will actually improve on our current system, which does a much better job than most people understand.

But that’s a miracle for another day.


Evaluating the New PSAT: Math

Well, after the high drama of writing, the math section is pretty tame. Except the whole oh, my god, are they serious? part. Caveat: I’m assuming that the SAT is still a harder version of the PSAT, and that this is a representative test.

Metric Old SAT Old PSAT ACT New PSAT
Questions
 
54 
44 MC, 10 grid
38 
28 MC, 10 grid
60 MC 
 
48 
40 MC, 8 grid

Sections
 
 
1: 20 q, 25 m 
2: 18 q, 25 m 
3: 16 q, 20 m
1: 20 q, 25 m 
2: 18 q, 25 m
 
1: 60 q, 60 m 
 
 
NC: 17 q, 25 m 
Calc: 31 q, 45 m
 
MPQ
 
 
1: 1.25 mpq 
2: 1.38 mpq
3: 1.25 mpq
1: 1.25 mpq 
2: 1.38 mpq
 
1 mpq 
 
 
NC: 1.47 mpq 
Calc: 1.45 mpq
 
Category 
 
 
 
 
 
 

Number Operations 
Algebra & Functions
Geometry & Measurement
Data & Statistics
 
 
 

Same  
 
 
 
 
 
 

Pre-algebra 
Algebra
elem & intermed.
Geometry
coord & plane
Trigonometry
 
 
1) Heart of Algebra 
2) Passport to
Advanced Math
3) Probability &
4) Data Analysis
Additional Topics
in math
 

It’s going to take me a while to fully process the math section. For my first go-round, I thought I’d point out the instant takeaways, and then discuss the math questions that are going to make any SAT expert sit up and take notice.

Format
The SAT and PSAT always gave an average of 1.25 minutes for multiple choice question sections. On the 18 question section that has 10 grid-ins, giving 1.25 minutes for the 8 multiple choice questions leaves 1.5 minutes for each grid in.

That same conversion doesn’t work on the new PSAT. However, both sections have exactly 4 grid-ins, which makes a nifty linear system. Here you go, boys and girls, check my work.

The math section that doesn’t allow a calculator has 13 multiple choice questions and 4 grid-ins, and a time limit of 25 minutes. The calculator math section has 27 multiple choice questions and 4 grid-ins, and a time limit of 45 minutes.

13x + 4y = 1500
27x + 4y = 2700

Flip them around and subtract for
14x = 1200
x = 85.714 seconds, or 1.42857 minutes. Let’s round it up to 14.3
y = 96.428 seconds, or 1.607 minutes, which I shall round down to 1.6 minutes.

If–and this is a big if–the test is using a fixed average time for multiple choice and another for grid-ins, then each multiple choice question is getting a 14.4% boost in time, and each grid-in a 7% boost. But the test may be using an entirely different parameter.

Question Organization

In the old SAT and ACT, the questions move from easier to more difficult. The SAT and PSAT difficulty level resets for the grid-in questions. The new PSAT does not organize the problems by difficulty. Easy problems (there are only 4) are more likely to be at the beginning, but they are interlaced with medium difficulty problems. I saw only two Hard problems in the non-calculator section, both near but not at the end. The Hard problems in the calculator section are tossed throughout the second half, with the first one showing up at 15. However, the coding is inexplicable, as I’ll discuss later.

As nearly everyone has mentioned, any evaluation of the questions in the new test doesn’t lead to an easy distinction between “no calc” and “calc”. I didn’t use a calculator more than two or three times at any point in the test. However, the College Board may have knowledge about what questions kids can game with a good calculator. I know that the SAT Math 2c test is a fifteen minute endeavor if you get a series of TI-84 programs. (Note: Not a 15 minute endeavor to get the programs, but a 15 minute endeavor to take the test. And get an 800. Which is my theory as to why the results are so skewed towards 800.) So there may be a good organizing principle behind this breakdown.

That said, I’m doubtful. The only trig question on the test is categorized as “hard”. But the question is simplicity itself if the student knows any right triangle trigonometry, which is taught in geometry. But for students who don’t know any trigonometry, will a calculator help? If the answer is “no”, then why is it in this section? Worse, what if the answer is “yes”? Do not underestimate the ability of people who turned the Math 2c into a 15 minute plug and play to come up with programs to automate checks for this sort of thing.

Categories

Geometry has disappeared. Not just from the categories, either. The geometry formula box has been expanded considerably.

There are only three plane geometry questions on the test. One was actually an algebra question using the perimeter formula Another is a variation question using a trapezoid’s area. Interestingly, neither rectangle perimeter nor trapezoid formula were provided. (To reinforce an earlier point, both of these questions were in the calculator section. I don’t know why; they’re both pure algebra.)

The last geometry question really involves ratios; I simply picked the multiple choice answer that had 7 as a factor.

I could only find one coordinate geometry question, barely. Most of the other xy plane questions were analytic geometry, rather than the basic skills that you usually see regarding midpoint and distance–both of which were completely absent. Nothing on the Pythagorean Theorem, either. Freaky deaky weird.

When I wrote about the Common Core math standards, I mentioned that most of geometry had been pushed down into seventh and eighth grade. In theory, anyway. Apparently the College Board thinks that testing geometry will be too basic for a test on college-level math? Don’t know.

Don’t you love the categories? You can see which ones the makers cared about. Heart of Algebra. Passport to Advanced Math! Meanwhile, geometry and the one trig question are stuck under “Additional Topic in Math”. As opposed to the “Additional Topic in History”, I guess.

Degree of Difficulty;

I worked the new PSAT test while sitting at a Starbucks. Missed three on the no-calculator section, but two of them were careless errors due to clatter and haste. In one case I flipped a negative in a problem I didn’t even bother to write down, in the other I missed a unit conversion (have I mentioned before how measurement issues are the obsessions of petty little minds?)

The one I actually missed was a function notation problem. I’m not fully versed in function algebra and I hadn’t really thought this one through. I think I’ve seen it before on the SAT Math 2c test, which I haven’t looked at in years. Takeaway— if I’m weak on that, so are a lot of kids. I didn’t miss any on the calculator section, and I rarely used a calculator.

But oh, my lord, the problems. They aren’t just difficult. The original, pre-2005 SAT had a lot of tough questions. But those questions relied on logic and intelligence—that is, they sought out aptitude. So a classic “diamond in the rough” who hadn’t had access to advanced math could still score quite well. Meanwhile, on both the pre and post 2005 tests, kids who weren’t terribly advanced in either ability or transcript faced a test that had plenty of familiar material, with or without coaching, because the bulk of the test is arithmetic, algebra I, and geometry.

The new PSAT and, presumably, the SAT, is impossible to do unless the student has taken and understood two years of algebra. Some will push back and say oh, don’t be silly, all the linear systems work is covered in algebra I. Yeah, but kids don’t really get it then. Not even many of the top students. You need two years of algebra even as a strong student, to be able to work these problems with the speed and confidence needed to get most of these answers in the time required.

And this is the PSAT, a test that students take at the beginning of their junior year (or sophomore, in many schools), so the College Board has created a test with material that most students won’t have covered by the time they are expected to take the test. As I mentioned earlier, California alone has nearly a quarter of a million sophomores and juniors in algebra and geometry. Will the new PSAT or the SAT be able to accurately assess their actual math knowledge?

Key point: The SAT and the ACT’s ability to reflect a full range of abilities is an unacknowledged attribute of these tests. Many colleges use these tests as placement proxies, including many, if not most or all, of the public university systems.

The difficulty level I see in this new PSAT makes me wonder what the hell the organization is up to. How can the test will reveal anything meaningful about kids who a) haven’t yet taken algebra 2 or b) have taken algebra 2 but didn’t really understand it? And if David Coleman’s answer is “Those testers aren’t ready for college so they shouldn’t be taking the test” then I have deep doubts that David Coleman understands the market for college admissions tests.

Of course, it’s also possible that the SAT will yield the same range of scores and abilities despite being considerably harder. I don’t do psychometrics.

Examples:

newpsatmath10

Here’s the function question I missed. I think I get it now. I don’t generally cover this degree of complexity in Precalc, much less algebra 2. I suspect this type of question will be the sort covered in new SAT test prep courses.

mathnocalcquads

These two are fairly complicated quadratic questions. The question on the left reveals that the SAT is moving into new territory; previously, SAT never expected testers to factor a quadratic unless a=1. Notice too how it uses the term “divisible by x” rather than the more common term, “x is a factor”. While all students know that “2 is a factor of 6″ is the same as “6 is divisible by 2″, it’s not a completely intuitive leap to think of variable factors in the same way. That’s why we cover the concept–usually in late algebra 2, but much more likely in pre-calc. That’s when synthetic division/substitution is covered–as I write in that piece, I’m considered unusual for introducing “division” of this form so early in the math cycle.

The question on the right is a harder version of an SAT classic misdirection. The test question doesn’t appear to give enough information, until you realize it’s not asking you to identify the equation and solve for a, b, and c–just plug in the point and yield a new relationship between the variables. But these questions always used to show up in linear equations, not quadratics.

That’s the big news: the new PSAT is pushing quadratic fluency in a big way.

Here, the student is expected to find the factors of 1890:

newpsatperimeter

This is a quadratic system. I don’t usually teach these until Pre-Calc, but then my algebra 2 classes are basically algebra one on steroids. I’m not alone in this.

No doubt there’s a way to game this problem with the answer choices that I’m missing, but to solve this in the forward fashion you either have to use the quadratic formula or, as I said, find all the factors of 1890, which is exactly what the answer document suggests. I know of no standardized test that requires knowledge of the quadratic formula. The old school GRE never did; the new one might (I don’t coach it anymore). The GMAT does not require knowledge of the quadratic formula. It’s possible that the CATs push a quadratic formula question to differentiate at the 800 level, but I’ve never heard of it. The ACT has not ever required knowledge of the quadratic formula. I’ve taught for Kaplan and other test prep companies, and the quadratic formula is not covered in most test prep curricula.

Here’s one of the inexplicable difficulty codings I mentioned–this is coded as of Medium difficulty.

As big a deal as that is, this one’s even more of a shock: a quadratic and linear system.

newpsatsystemlineparabola

The answer document suggests putting the quadratic into vertex form, then plugging in the point and solving for a. I solved it with a linear system. Either way, after solving the quadratic you find the equation of the line and set them equal to each other to solve. I am….stunned. Notice it’s not a multiple choice question, so no plug and play.

Then, a negative 16 problem–except it uses meters, not feet. That’s just plain mean.
newpsatmathneg16

Notice that the problem gives three complicated equations. However, those who know the basic algorithm (h(t)=-4.9t2 + v0 + s0) can completely ignore the equations and solve a fairly easy problem. Those who don’t know the basic algorithm will have to figure out how to coordinate the equations to solve the problem, which is much more difficult. So this problem represents dramatically different levels of difficulty based on whether or not the student has been taught the algorithm. And in that case, the problem is quite straightforward, so should be coded as of Medium difficulty. But no, it’s tagged as Hard. As is this extremely simple graph interpretation problem. I’m confused.

Recall: if the College Board keeps the traditional practice, the SAT will be more difficult.

So this piece is long enough. I have some thoughts–rather, questions–on what on earth the College Board’s intentions are, but that’s for another test.

tl;dr Testers will get a little more time to work much harder problems. Geometry has disappeared almost entirely. Quadratics beefed up to the point of requiring a steroids test. Inexplicable “calc/no calc” categorization. College Board didn’t rip off the ACT math section. If the new PSAT is any indication, I do not see how the SAT can be used by the same population for the same purpose unless the CB does very clever things with the grading scale.


Evaluating the New PSAT: Reading and Writing

The College Board has released a new practice PSAT, which gives us a lot of info on the new SAT. This essay focuses on the reading and writing sections.

As I predicted in my essay on the SAT’s competitive advantage, the College Board has released a test that has much in common with the ACT. I did not predict that the homage would go so far as test plagiarism.

This is a pretty technical piece, but not in the psychometric sense. I’m writing this as a long-time coach of the SAT and, more importantly, the ACT, trying to convey the changes as I see them from that viewpoint.

For comparison, I used these two sample ACT, this practice SAT (old version), and this old PSAT.

Reading

The old SAT had a reading word count of about 2800 words, broken up into eight passages. Four passages were very short, just 100 words each. The longest was 800 words. The PSAT reading count was around 2000 words in six passages. This word count is reading passages only; the SAT has 19 sentence completions to the PSAT’s 13.

So SAT testers had 70 minutes to complete 19 sentence completions and 47 questions over eight passages of 2800 words total. PSAT testers had 50 minutes to complete 13 sentence and 27 questions over six passages of 2000 words total.

The ACT has always had 4 passages averaging 750 words, giving the tester 35 minutes to complete 40 questions (ten for each passage). No sentence completions.

Comparisons are difficult, but if you figure about 45 seconds per sentence completion, you can deduct that from the total time and come up with two rough metrics comparing reading passages only: minutes per question and words per question (on average, how many words is the tester reading to answer the questions).

Metric Old SAT Old PSAT ACT New PSAT
Word Count 2800 2000 3000 3200
Passage Count 8 6 4 5
Passage Length 100-850 100-850 750 500-800
MPQ 1.18 1.49 1.14 1.27
WPQ 59.57 74.07 75 69.21

I’ve read a lot of assertions that the new SAT reading text is more complex, but my brief Lexile analysis on random passages in the same category (humanities, science) showed the same range of difficulty and sentence lengths for old SAT, current ACT, and old and new PSAT. Someone with more time and tools than I have should do an indepth analysis.

Question types are much the same as the old format: inference, function, vocabulary in context, main idea. The new PSAT requires the occasional figure analysis, which the College Board will undoubtedly flaunt as unprecedented. However, the College Board doesn’t have an entire Science section, which is where the ACT assesses a reader’s ability to evaluate data and text.

Sentence completions are gone, completely. In passage length and overall reading demands, the new PSAT is remarkably similar in structure and word length to the ACT. This suggests that the SAT is going to be even longer? I don’t see how, given the time constraints.

tl;dr: The new PSAT reading section looks very similar to the current ACT reading test in structure and reading demands. The paired passage and the questions types are the only holdover from the old SAT/PSAT structure. The only new feature is actually a cobbled up homage to the ACT science test in the form of occasional table or graph analysis.

Writing

I am so flummoxed by the overt plagiarism in this section that I seriously wonder if the test I have isn’t a fake, designed to flush out leaks within the College Board. This can’t be serious.

The old PSAT/SAT format consisted of three question types: Sentence Improvements, Identifying Sentence Error, and Paragraph Improvements. The first two question types presented a single sentence. In the first case, the student would identify a correct (or improved) version or say that the given version was best (option A). In the ISEs, the student had to read the sentence cold with no alternatives and indicate which if any underlined word or phrase was erroneous (much, much more difficult, option E was no change). In Paragraph Improvements, the reader had to answer grammar or rhetoric questions about a given passage. All questions had five options.

The ACT English section is five passages running down the left hand side of the page, with underlined words or phrases. As the tester goes along, he or she stops at each underlined section and looks to the right for a question. Some questions are simple grammar checks. Others ask about logic or writing choices—is the right transition used, is the passage redundant, what would provide the most relevant detail. Each passage has 15 questions, for a total of 75 questions in 45 minutes (9 minutes per passage, or 36 seconds per question). The tester has four choices and the “No Change” option is always A.

The new PSAT/SAT Writing/Language section is four passages running down the left hand side of the page, with underlined words or phrases. As the tester goes along, he or she stops at each underlined section and looks to the right for a question. Some questions are simple grammar checks. Others ask about logic or writing choices—is the right transition used, is the passage redundant, what would provide the most relevant detail. Each passage has 11 questions, for a total of 44 questions in 35 minutes (about 8.75 minutes per passage or 47 seconds a question). The tester has four choices and the “No Change” option is always A.

Oh, did I forget? Sometimes the tester has to analyze a graph.

The College Board appears to have simply stolen not only the structure, but various common question types that the ACT has used for years—as long as I’ve been coaching the test, which is coming on for twelve years this May.

I’ll give some samples, but this isn’t a random thing. The entire look and feel of the ACT English test has been copied wholesale—I’ll add “in my opinion” but don’t know how anyone could see this differently.

Writing Objective:

Style and Logic:

Grammar/Punctuation:

tl;dr: The College Board ripped off the ACT English test. I don’t really understand copyright law, much less plagiarism. But if the American College Test company is not considering legal action, I’d love to know why.

The PSAT reading and writing sections don’t ramp up dramatically in difficulty. Timing, yes. But the vocabulary load appears to be similar.

The College Board and the poorly informed reporters will make much of the data analysis questions, but I hope to see any such claims addressed in the context of the ACT’s considerably more challenging data analysis section. The ACT should change the name; the “Science” section only uses science contexts to test data analysis. All the College Board has done is add a few questions and figures. Weak tea compared to the ACT.

As I predicted, The College Board has definitely chosen to make the test more difficult for gaming. I’ve been slowly untangling the process by which someone who can barely speak English is able to get a high SAT verbal and writing score, and what little I know suggests that all the current methods will have to be tossed. Moving to longer passages with less time will reward strong readers, not people who are deciphering every word and comparing it to a memory bank. And the sentence completions, which I quite liked, were likely being gamed by non-English speakers.

In writing, leaving the plagiarism issue aside for more knowledgeable folk, the move to passage-based writing tests will reward English speakers with lower ability levels and should hurt anyone with no English skills trying to game the test. That can only be a good thing.

Of course, that brings up my larger business question that I addressed in the competitive advantage piece: given that Asians show a strong preference for the SAT over the ACT, why would Coleman decide to kill the golden goose? But I’ll put big picture considerations aside for now.

Here’s my evaluation of the math section.


Designing Multiple Answer Math Tests

I got the idea for Multiple Answer Tests originally because I wanted to prepare my kids for Common Core Tests. (I’d rather people not use that post as the primary link, as I have done a lot more work since then.)

About six months later (a little over a year ago), I gave an update, which goes extensively into the grading of these tests, if you’re curious. At that time, I was teaching Pre-Calc and Algebra 2/Trig. This past year, I’ve been teaching Trigonometry and Algebra II. I’d never taught trig before, so all my work was new. In contrast, I have a lot of Algebra 2 tests, so I often rework a multiple choice question into a multiple answer.

I thought I’d go into the work of designing a multiple answer test, as well as give some examples of my newer work.

I design my questions almost in an ad hoc basis. Some questions I really like and keep practically intact; others get tweaked each time. I build tests from a mental question database, pulling them in from tests. So when I start a new test, I take the previous unit test, evaluate it, see if I’ve covered the same information, create new questions as needed, pull in questions I didn’t use on an earlier test, whatever. I don’t know how teachers can use the same test time and again. I’d get bored.

I recently realized my questions have a typology. Realizing this has helped me construct questions more clearly, sometimes adding a free response activity just to get the students started down the right path.

The first type of question requires modeling and/or solving one equation completely. The answer choices all involve that one process.

Trigonometry:

matrig1

I’m very proud of this question. My kids had learned how to graph the functions, but we hadn’t yet turned to modeling applications. So they got this cold, and did really well with it. (In the first class, anyway. We’ll see how the next group does in a month or so.) I had to design it in such a way to really telegraph the question’s simplicity, to convince the students to give it a shot.

Algebra II:
maratsimp

The rational expression question is incredibly flexible. I’m probably teaching pre-calc again next year and am really looking forward to beefing this question up with analysis.

Other questions are a situation or graph that can be addressed from multiple aspects. The student ends up working 2 or 3 actual calculations per question. I realized the questions look the same as the previous type, but they represent much more work and I need to start making that clear.

Trigonometry:

mypythruler

Algebra II:
mafurnquest

I love the Pythagorean Ruler question, which could be used purely for plane geometry questions, or right triangle trig. Or both. The furniture question is an early draft; I needed an inverse question and wanted some linear modeling review, so I threw together something that gave me both.

I can also use this format to test fluency on basic functions very efficiently. Instead of wasting one whole question on a trig identity, I can test four or five identities at once.

matrigalg

Or this one, also trig, where I toss in some simplification (re-expression) coupled with an understanding of the actual ratios (cosine and secant), even though they haven’t yet done any graphing. So even if they have graphing calculators (most don’t), they wouldn’t know what to look for.

matrigvals

I’m not much for “math can be used in the real world” lectures, but trigonometry is the one class where I can be all, “in your FACE!” when kids complain that they’d never see this in real life.

maisuzu

I stole the above concept from a trig book and converted to multiple answer, but the one below I came up with all by myself, and there’s all sorts of ways to take it. (and yes, as Mark Roulo points out, it should be “the B29’s circumference is blah blah blah.” Fixed in the source.)

mapropspeed

Some other questions for Algebra II, although they can easily be beefed up for pre-calc.

maparlinesys

maparabolaeq

One of the last things I do in creating a test is consider the weight I give each question. Sometimes I realize that I’ve created a really tough question with only five answer choices (my minimum). So I’ll add some easier answer choices to give kids credit for knowledge, even if they aren’t up to the toughest concepts yet.

That’s something I’ve really liked about the format. I can push the kids at different levels with the same question, and create more answer choices to give more weight to important concepts.

The kids mostly hate the tests, but readily admit that the hatred is for all the right reasons. Many kids used to As in math are flummoxed by the format, which forces them to realize they don’t really know the math as well as they think they do. They’ve really trained their brains to spot the correct answer in a multiple choice format–or identify the wrong ones. (These are the same kids who have memorized certain freeform response questions, but are flattened by unusual situations that don’t fit directly into the algorithms.)

Other strong students do exceptionally well, often spotting question interpretations I didn’t think of, or asking excellent clarifications that I incorporate into later tests. This tells me that I’m on the right track, exposing and differentiating ability levels.

At the lower ability levels, students actually do pretty well, once I convince them not to randomly circle answers. So, for example, on a rational expression question, they might screw up the final answer, but they can identify factors in common. Or they might make a mistake in calculating linear velocity, but they correctly calculate the circumference, and can answer questions about it.

I’ve already written about the frustrations, as when the kids have correctly worked problems but didn’t identify the verbal description of their process. But that, too, is useful, as they can plainly see the evidence. It forces them to (ahem) attend to precision.

Of course, I’m less than precise myself, and one thing I really love about these tests is my ability to hide this personality flaw. But if you spot any ambiguities, please let me know.


Ian Malcolm on Eva Moskowitz

malcolmquote1

Another good piece documenting the lack of “there” at the Success Academy schools, this one by Kate Taylor at the Times.

Pretend that Judge Patrice Lessner is interrupting me every four words for this next bit:

Success Academies’ “success” will eventually be revealed as a chimera. Certainly they are skimming on a massive scale, and their attrition rates over time are pretty telling. Despite Moskowitz’s constant denials,the kids spend a shocking amount of time in test prep—one witness even saw an early slam the exam class.

But skimming, test prep, and attrition don’t explain enough. If Carol Burris is providing correct information here, then 45% of whites were proficient in math, and 31% in ELA. According to Robert Pondiscio, the numbers for the overwhelmingly low income black and Hispanic Success Academies were over 90% and 68%, respectively. That suggests the schools are doing more than cherrypicking.

I don’t know how. Unlikely to be anything as obvious as fixing the tests later or telling the kids the answers, or we’d hear about it. Possibly they are engaging in the Chinese variety of test prep.

But if low income black and Hispanic proficiency rates are twice that of whites, then the dinosaurs have escaped.

Paul Bruno is more careful, less intuitive (in his writing) and far more data-driven than, say, me. So maybe everyone doesn’t read his explication of everything we don’t know about Success Academy as howlingly skeptical, but nor would anyone see the piece as a ringing endorsement. More surprisingly, Robert Pondiscio asks “what the hell is going on at Success Academy? in a way that doesn’t sound very flattering.

In no way are Bruno or Pondiscio going out on the ledge with me. Not for them the wise words of Ian Malcolm. I’m just saying that their articles signal considerable skepticism to me, a frequent reader of both.

I haven’t seen many respectable reformers touting Success Academy, either. Take that as you will.

Here’s a story idea for some enterprising reporter:

Contact Success Academy and ask to see score progressions for their early students. Presumably, all the students didn’t come in scoring at the top level (don’t laugh, skeptics!). So Eva and her minions should be able to provide initial scores for students–they are testing them constantly, yes?–and connect these scores to their actual state exam scores. By year. Then that enterprising reporter should track down Success Academy alumni and get their scores year by year since they’ve left. In a year, that could include SAT/ACT scores.

This would provide actual data to answer the following questions:

  1. Are the weakest students leaving the schools?
  2. Are specific students improving their demonstrated abilities during their tenure at the schools?
  3. Are alumni still doing well after they leave school?

Those questions would eliminate or at least reduce the charges of skimming, attrition, and prepping-to-the-extent-of-cheating.

I note that Kate Taylor or the Times is looking for students or parents to “share their stories”. Less stories. More data. Get test scores over time per student, stat!

If I’m wrong, nothing happens! No one gets fired. I’m just an amateur. It’s not like I’m claiming a frat party instigated a gang rape, or anything. And oh, yeah, the achievement gap that has plagued our education efforts for over fifty years has finally been beaten.

So if I’m wrong, someone should go look for Isla Nublar to see if the T-Rex has eaten all the velociraptors.


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:
functionoriginalexample

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.

functionactivity

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?

alphabetgraph

cellphonegraph

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.
functionalgebra1version

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.
cpmfunct

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):

functionvendingmachine

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.

kirshappendix

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.


Education: No Iron Triangle

I came from the corporate world, which invented the project management triangle. (“Fast, Good, Cheap: Pick Two.”)

Education has no triangle.

Money, of course, doesn’t work. Just ask Kansas City. Or Roland Fryer, who learned that kids would read more books for money but couldn’t seem to produce higher test scores for cash. Increased teacher salaries, merit pay, reduced class size are all suggestions that either don’t have any impact or have a limited impact….sometimes. Maybe. But not in any linear, scalable pattern.

“Good”? Don’t make me laugh. We don’t have a consensus on what it means. Most education reformers use the word “quality” exclusively to mean higher test scores. Teachers do not. Nor do parents, as Rahm Emanuel, Cami Anderson, Adrian Fenty and Michelle Rhee have learned. Common Core supporters have had similar moments of revelation.

So until we agree on what “good” is, what a “high quality education” means, we can’t even pretend that quality is a vertex of education’s triangle, even if it existed. We could save a whole lot of wasted dollars if people could just grasp that fact.

Time is an odd one. We never use the word directly, but clearly, politicians, many parents, and education reformers of all stripes believe we can educate “faster”. Until sixty years ago, calculus was an upper level college course. Once the high school movement began, fewer than 3% of students nationwide took trigonometry, between 10-20% took geometry, and the high point for algebra was 57%–over one hundred years ago–then declining to 25%. (Cite.) One of the little noted achievements of the New Math movement was to alter the math curriculum and make high school calculus a possibility. At first, just kids with interest and ability took that path. Then someone noticed that success in algebra I predicted college readiness and everyone got all cargo cult about it. By the turn of the century, if not earlier, more of our kids were taking advanced math in high school than at any point in our history.

And that was before kids started taking algebra in seventh grade. Sophomores take now take honors pre-calculus so they can get a second year of AP calculus in before graduation. Common Core has gone further and pushed algebra 2 down into algebra I.

Yet 17 year old NAEP scores have been basically stagnant for the same amount of time our high school students have been first encouraged, then required, to take three or more years of advanced math.

Not only do we try to educate kids faster, we measure their gain or loss by time. Poor kids of uneducated parents lose two months learning over the summer. CREDO, source of all those charter studies, refers to additional days of learning. Everyone comparing our results to Singapore always mentions the calendar, how much earlier their kids start working with advanced math. These same people also point out that Singapore has a longer school year. Longer school years don’t appear to work reliably either.

Except maybe KIPP, whose success is mostly likely due to extended school hours. KIPP focuses on middle school and has not really been scrutinized at the high school level. Scrutiny would reveal that the program doesn’t turn out stellar candidates, and while more KIPP alumni complete college than the average low income black or Hispanic student, the numbers are reasonable but not extraordinary when compared against motivated students in the same category who attended traditional schools. Particularly given the additional support and instruction hours the KIPP kids get.

So KIPP’s “success” actually adds weight to the NAEP scores as evidence that time–like money and quality–doesn’t respond to the project management constraints.

Kids learn what they have the capacity to learn. Spending more instruction hours will–well, may–help kids learn more of what they are capable of learning in fewer school years. But the NAEP scores and all sorts of other evidence says that learning more early doesn’t lead to increased capacity later. And so, we’ve moved 1979 first grader readiness rules to preschool with considerable success, but that success hasn’t given us any traction in increasing college readiness at the other end of childhood. Quite the contrary.

I probably don’t have much of a point. I was actually thinking about the increasing graduation rates. It’ll be a while until part 2. I’m swamped at work, moving again, writing some longer pieces, and really would like to post some math curriculum rather than detangle my mullings.

But the triangle thing is important. Really.

Take note: under 1000 words. Hey, I have to do it every year or so.


Group Work vs. Working In Groups

I sit my kids in groups. But I don’t like “group work”.

No, that’s not a paradox. Sitting in groups isn’t “group work”.

Group work is an activity that falls under the larger rubric of “collaborative learning”, an organizing bubble to collect techniques and strategies like “Think Pair Share”, jigsawing, peer tutoring, and the like. The most fully-realized collaborative learning pedagogy is probably complex instruction, which was developed by Elizabeth Cohen. (That’s CI, not CISC.) To illustrate, CPM curriculum is based on complex instruction, whereas Everyday Math is not.

Complex Instruction had been in development for over 20 years, but really caught on during the early 90s, when detracking was all the rage, thanks to the Demon Goddess Jeannie Oakes and her book Keeping Track, a synthesis of the arguments against tracking developed since the late 60s, when the feds and the Supreme Court decided by god, they were serious about this integration business, enforcing busing and other means of insuring that no schools were too white or too brown.

In Keeping Track, Oakes accused parents and schools of racial discrimination, and a good chunk of the 90s was wasted as districts and states desperately tried to win her approval. Fortunately, they all ultimately learned it was easier to disappoint her.1

Complex Instruction was also developed by tracking opponents, but opponents who nonetheless cared about learning. It’s explicitly designed to give schools a tool for the havoc that results when kids with a 3 to 8 year range in abilities are put in the same room, and thus was grabbed at by many schools back in the early 90s. Many CI concepts are also found in “reform math”—Jo Boaler’s Railside study on San Lorenzo High School was all about Complex Instruction. Carlos Cabana and Estelle Woodbury, who just co-authored Mathematics for Equity, a book on teaching math with Complex Instruction, both worked at San Lorenzo High School during Boaler’s study.

So start with the theory, articulated here by Rachel Lotan, the late Cohen’s key associate. You should watch this, or at least fast forward through parts, because Lotan clearly articulates the admirable goals of complex instruction minus the castigation, blame, and fuming ideology. Or, Complex Instruction’s major components in written form:

ci3components

Both Lotan and the writeup offer much that is problematic. Reducing the ability range: not good. Creating busywork tasks (writing down questions, getting supplies) to let everyone feel “smart”: not good.

The write up mentions “status problems”. Lotan gives a great account of an absurdly pretentious term, “mitigating status” that is something every teacher in every classroom–no matter how they are seated—should take seriously. Lotan does a better job of explaining it, but since many won’t listen to the video, here’s a written version:

CI targets equity and, in particular, three ideas: first, that all students are smart; second, that issues of status—who is perceived as smart and who is not—interfere with students’ participation and learning; and third, that it is teachers’ responsibility to provide all students with opportunities to reveal how they are smart and develop/recognize new ways of being smart. The complex instruction model aims to “disrupt typical hierarchies of who is ‘smart’ and who is not” (Sapon-Shevin, 2004) by promoting equal status interactions amongst students so that they engage with tasks that have high cognitive demand within a cooperative learning environment.

(emphasis mine)

Ed schools wanting to hammer home how putting kids in groups doesn’t by itself address status usually show this video, but brace yourself. I tell myself that the ignored kid is probably a pest all the time, that everyone in the class is tired of his nonsense, that we’re just seeing a carefully culled selection to maximize the impact of exclusion and of course, race. It doesn’t matter. It’s still hard to watch.

And the video also reinforces the practical message that CI advocates are pushing, as opposed to the theory. In theory, status can be unearned by anyone of any gender or color. In practice, most CI advocates expect teachers to shut down white males. In theory, kids learn that everyone is smart. In practice, kids still know who’s “smart” and who’s not.

But then, CI advocates have their own frustrations. In theory, they’d put teachers in PD designed to indoctrinate them into realizing the error of their racist ways. In practice, teachers who haven’t already drunk the Koolaid either politely fake it until they can find an exit or get really annoyed when they’re called racists, as an excerpt for Mathematics for Equity makes clear:

CIPurpose
Cite: Mathematics for Equity1

Complex Instruction done well is pretty interesting and often thought-provoking. Cathy Humphreys is a long-time advocate of “reform math” and complex instruction. She was at the center of one of those “rich educated parents” meltdowns that you saw over reform math back in the 90s. Humphreys represented the reform side, of course, and further endeared herself to parents by proposing to get rid of tracking at a Palo Alto, CA middle school. That went over like a water balloon down a balcony, she quit, worked as a math coach for a while, and then taught for years at a diverse high school in the Bay Area that had ended tracking. She also teaches at Stanford’s education program, according to her bio. Carlos Cabana, one of the co-authors of Mathematics for Equity, has also been teaching complex instruction for a long time; he’s one of the teachers at Railside, Jo Boaler’s pseudonym for San Lorenzo High School.

You can see both Humphreys and Cabana here at a website put together by the Noyce Foundation to promote the 8 essential practices. (Notice the link between “reform math” and supporting “common core”? As Tom Loveless says, Common Core is a “dog whistle” for reform math. Humphreys and Cabana are teaching high school math in the videos. You can also see Humphreys teaching at what I assume is the middle school that melted down. Humphreys and Cabana are much better demonstrations of complex instruction than the absurdly flashy promos that Jo Boaler puts out.

When I began teaching, I thought sitting kids in groups was absurd. I remember being pleased one of my mentoring teachers put kids in rows. But my primary student teaching assignment required me to sit kids in groups, as we were using CPM, a reform text that requires groups. I adjusted and liked it much more than I thought I would, especially when I took over the class and could group by ability. But my first year out, I happily put my desks in rows, thinking that groups were good, but now I could finally run my class the way I wanted.

Four weeks later, I put the kids in groups. It just….felt better. Year 2, I was teaching all-algebra, all the time, and thought rows would make more sense. The rows lasted 2 weeks and since around September of 2010, the only time my kids sit in rows is for tests.

I have….mixed feelings about CI. When promoted by the fanatic adherents, it’s both Orwellian and despicable. Teachers have to squelch kids who know the answer, force kids who understand the concept to explain, endlessly, to the kids who don’t, and then grade the kids who know the answer not on their demonstrated knowledge but on the success of their explanation and their willingness to do so. Teachers have to pretend to their students that asking a good question or taking notes is just as important as understanding the math (no, say the fanatic adherents, teachers aren’t pretending. These tasks are just as important!).

But while no student is ever going to believe that everyone is smart, “issues of status” do absolutely impact a students’ willingness to participate. Let the “smart kids” talk, everyone thinks, and sits back and zones out.

However, in my opinion and experience, CI methods often achieve exactly what they are defined to avoid, precisely because of their Orwellian insistence on ignoring reality. Kids know who is smart. They shut down if the smart kid is in their group, and go through the motions when the teacher walks by.

Ironically, I “mitigate status” by violating Complex Instruction’s most sacred tenet. Complex Instruction holds that student groups must be heterogeneous. Organization can’t be based on the rigid, academic version of “smart”. But I group my kids by ability as the most effective way of “mitigating status”.

I don’t want the weakest students in my class feeling as if any success short of an “A” is irrelevant. I also don’t want to try and convince them they’re just as “smart” as students who don’t struggle with the same material. That way, my students know that they can talk about math, what they need to know, what questions they have, knowing that other students probably have similar issues.

I don’t want to make it sound as if “mitigating status” is the only reason I sit kids in groups. Groups allow me to differentiate tasks slightly (or extensively) and enables me to quickly give help or new tasks. Groups allow kids to work together, discussing math, developing at their own speed with peers who have similar abilities.

But whether it’s status or some other curricular reason, when I sit them in groups, they start working and talking about math. They discover they are working with peers who won’t make them feel stupid, and they start to have discussions. Should we do this or this? They call me over to adjudicate. They try things. They check their notes, engage in all those excellent student behaviors. Not always, of course. But many times. They are less likely to sit passively and wait until I come by to personally tutor them through problems.

Moreover, because they are working with students of their own ability, they don’t feel alone or stupid. They work to improve. Maybe not great, maybe not good. But better.

Sitting kids in groups is not group work. But sitting kids in groups based on ability and giving them achievable tasks makes them more likely to work, and as math teachers often know, that’s no small thing.

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1 I was thinking crap, I don’t want to have to look up the whole history of the ebb and flow of tracking and then went hey, Tom Loveless has to have something on this and by golly he does: The Resurgence of Ability Grouping and Persistence of Tracking covers the whole era, Oakes included. I would only quibble slightly with this sentence: Although the call to detrack was not accompanied by conventional incentives—the big budgets, regulatory regimes, and rewards and sanctions that draw the attention of policy analysts—detracking was, in a field famous for ignored or subverted policies, adopted by a large number of schools.

Loveless appears to forget the biggest incentive of all: lawsuit avoidance. Detracking lawsuits were the rage in this time period. Unlike new curriculum or teaching styles, detracking is achieved by executive fiat by district superintendents. No training, no carrots needed. Shazam! But leaving aside that minor quibble, a great piece documenting the move to and then the move away from heterogeneous classrooms (de-tracked).


Teaching: My Retrospective

Okay, I’m rolling along on my task of drawing clear lines of demarcation between my particular brand of squish and traditional progressive education (heh–traditional progressive. Get it?). First up was my new no homework policy.

I then decided to take on sitting my kids in groups (as opposed to group work), which led me to look back at some old post, which forced me to look back at my practice over the years, and that’s been a trip. So much of a trip that I decided to do the retrospective first.

The introspection kicked off when I reread one of the first posts I ever wrote on this site, over 3 years ago, halfway through my third year of teaching. Some key observations:

  1. I focused almost entirely on classwork, even then. The essay doesn’t even mention homework which, at that time, I assigned in much the way I describe in my last essay.
  2. At that time, the school I worked at used a traditional schedule of 60 minute classes, so the 3 day span per lesson is about two days at my current school. Additional evidence I was focused primarily on what kids learned in class, although as I said, my original homework policy goes back even further than this post.
  3. Here’s a real change. Me on low ability students three years ago: lowabilstds3yrs
    I’m so cheered to realize how much I’ve improved. I had good student engagement back then, but in rereading this I can remember how many students I had to nudge endlessly, how I had to constantly pick up pencils and hand them to kids to get them to work. Recall I was teaching algebra and geometry, and had just begun what is now my bread and butter class of Algebra 2. So my experience at the time of writing those words was with a lower level of math class, which will always mean lower engagement. Nonetheless, that simple paragraphs reminds me of the struggles I had to get total engagement. I’ve come a long way. Yay, me.

  4. Interesting to see my off-hand mention of EDI. No one seeing my teaching would think of me as using the direct instruction mode, but in fact I always, at some point, give kids specific, explicit instructions on the concept at hand.
  5. While I talked about differentiation and my need to challenge top students, I have actually moved away from different assessments for different students. At that time, I was just three months of teaching out from year two, all-algebra I-all-the-time, and I basically taught 4 different classes. I’d tentatively planned on continuing this approach, but learned that year (and confirmed in later years) that this wouldn’t work for any class but algebra I.

I wrote this post on January 8, 2012, at almost exactly the same time I began an experiment that utterly transformed my teaching. I speak, of course, of Modeling Linear Equations, which I’m amazed to realize I wrote just one week after the “How I Teach” post. So shortly after I began this blog and described my teaching method, I started on a path that took my existing teaching approach–which was pretty good, I think–and gave it a form and shape that has allowed me to grow and progress even further.

I haven’t really read this post in over two years—I tend to link in Modeling Linear Equations, Part 3, written a year later (two years ago today!), when I’d realized how much my teaching had changed. So reading the original is instructive. I talk about the Christmas Mull, something that stands very large in my memory but don’t remember quite as described here:

modelingchristmasmull

The part that’s consistent with my memory: Christmas 2011, I was depressed by the dismal finals in my three algebra II classes. In the first semester, I had gone through all of linear and quadratic equations, including complex numbers, at a rate considerably slower than two colleagues also teaching the course. Yet the kids remembered next to nothing. Every single person failed the multiple choice test–the top students had around half right. I had experienced knowledge fall-offs in algebra and geometry, but nothing that had so sublimely illustrated how much time I’d wasted in three months. So I came out of the Christmas break determined to reteach linear and quadratic equations, because to continue on teaching more advanced topics with these numbers was purely insane. And I wasn’t just going to reteach, but come up with an entirely different, less structured approach that allowed my students to use their own understanding of real-life situations.

What I hadn’t remembered until reading this closely was my rationale for ignoring the regular curriculum requrements. At the time, Algebra 2 was considered a “terminal” class; students weren’t expected to take another course in the college-prep sequence. This has changed, of course–these days, algebra 2/trig is, if anything, experiencing a fall-off in favor of a full year of each course. But at the time, I justified my decision to go off-curriculum based on the student needs. These students’ primary concern, whether they knew it or not, was what happened to them in college. How much remediation were they going to need? Could the best of them escape any remedial work and go straight onto credit bearing courses? This, of course, still remains my priority–I’d just forgotten how linked it was to my initial decision to try something new.

Also interesting that I described this approach by the specific method I used for linear equations–using “inherent math ability”. That’s not how I describe my approach these days, but I can see the germination of the idea. At the time I wrote this, I had no idea I would go beyond linear equations and use this approach consistently throughout my instruction.

I think the best description I’ve come up with for my approach is modified instructivist, which comes in one of two forms: “highly structured instructivist discovery, and classroom discussions with lots of student involvement”.

As for the latter: I don’t lecture, with or without powerpoints. When I do explanations, they are classroom discussions, and you can see this demonstrated in all my pedagogy posts. However, I am constantly migrating my classroom discussions to structured discovery.

What’s structured discovery? Imagine a teacher and students on a cliff, with a beach below. There’s a path, but it’s not visible.

In a traditional lecture or classroom discussion, the teacher shows them the path and leads them down to the beach.

In a discovery class, the teacher doesn’t even tell them there’s a path or even a beach. In fact, to the discovery/reform teacher, it doesn’t matter whether there’s a path or not—the kids will all find their own way down. Or maybe they’ll just find some really cool flowers and stop to examine their biology. Or maybe they’ll just kick back and have a picnic. It’s all good, in reform math. (sez the skeptic)

In what I call structured discovery, the kids are given a series of tasks that use their existing knowledge base and find the path themselves. They may not yet know there’s a beach. They may not know what the path means. But they will find the path and recognize it as a consistent finding that makes them go “hmm”. In some cases, an interesting finding. In other cases, just something they can see and understand.

Sometimes the path they’ve found is the concept–for example, modeling linear equations or exponential functions, or finding gravity in projectile motion problems.

In other cases, the model just introduces an inevitable observation that leads to the new concept. For example, I teach my kids about function operations when we do linear equations–adding and subtracting are good models for simple profit and loss applications.

So I kick off quadratics by asking my students to multiply linear functions, which they can see clearly as an extension of adding and subtracting them. This is an activity they can start off cold, with no intro (I haven’t written it up yet). I designed this because parabolas just don’t have a natural “real life” model other than area, which gets kind of boring. Plus, I need to cover function operations anyway, so hey, synergy. In any event, the kids are seeing an extension of a concept they already know (function operations) and seeing a new graph form consistently emerge. Then we can talk about factors (the zeros) and realize that we are looking at products of two lines. Could a parabola exist without being a product of two lines? Well, this is algebra 2 so they are fully aware that parabolas don’t have to have zeros. But what does that mean in terms of multiplying lines being factors of parabolas? Well, they must not have factors. So are all parabolas the product of two lines? And we go from there.

Understand that my classes still have lots of practice time where kids just factor equations and graph parabolas, learn about the different forms, and so on. But rather than just saying “now we’ll do this new thing called a parabola”, I give them a task that builds on their existing work and leads them into the new equation type. I don’t define the path. But nor do I let them go off on their own. I give them something to do that looks kind of random, but is in fact a path.

And all of this came from the results of the Great Christmas Mull. The previous Christmas had been productive, too–it’s when I came up with differentiated instruction for my algebra class.

So what can I say about my teaching, 5.5 years in? What’s consistent, what’s changed?

  1. I never lectured. I always explained, with increasing emphasis on classroom discussion.

  2. I have always been focused on student work during class, emphasizing demonstrated test ability above everything, and minimizing (or now eliminating) homework.
  3. I have always tried to move the student needle at all ability levels, from the no-hopers to the strugglers to the average achievers to the top-tier thinkers. I’m not always successful, but that’s consistently my stated priority.
  4. I have always designed my own curriculum and assessments.

  5. My teaching was transformed Christmas of 2011, when I realized I could introduce and teach topics using existing knowledge, forcing students to engage immediately with the material and start “doing” right away, increasing engagement and understanding. I have evolved from a teacher who mostly explains first to a teacher who only occasionally explains first. And that is a huge change that takes a lot of work.
  6. The observer might think that this change makes my classes student-centered, but I disagree. My classes are definitely teacher-centered, and let’s be clear, I’m the star of my teaching movie.
  7. Thanks also to the Great Christmas Mull, I’ve become far less concerned about curriculum coverage than I was in my first two years of teaching.
  8. I have always been a teacher who values explanation. It’s the heart of my teaching. I’ll explain through discussion or demonstration, but I’m not a reformer letting kids “construct” the meaning of math. I’m there to tell them what it all means.

I have plenty of development areas ahead. I’m working on tossing in the occasional open-ended instruction, just to see if I can come up with ideas that don’t waste hours and have some interesting learning objectives. I still have many concepts waiting to be converted to a “path to the beach”. And I’m now teaching something other than math, which gives me new challenges and more opportunities to see how to construct those paths without running off the cliff.


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