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.

******************************************
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.


I Don’t Do Homework

Our school had its second Back to School Night. Attendance was spotty. I don’t judge. As a parent, I rarely attended.

But boy oh boy, could four sets of parents generate some excitement. I had a genuine culture clash.

It all began when I was going through my brief dog and pony show for my second trig class.

“Student grades are 80% tests and quizzes, 20% classwork. But I don’t grade classwork. Students get a B or A- just for showing up and working, which bumps their grade slightly.”

Until recently, I weighted homework for 10% and classwork for 15%–but not really. More accurately, if a student did most of his homework in a relatively timely manner, he’d get a little more of a boost. He couldn’t get the boost by “making up” missed homework; nor could he get the boost for just a couple homework completions. But if he didn’t do the homework at all, no harm no foul.

A few of my students got the boost, and they came from all points on the ability spectrum. I always remembered to assign homework through the first semester, then I’d fall off. For my first five years of teaching, homework had always completely stopped at some point in the third quarter.

“But last term, I suddenly realized that the end of the first semester was weeks away, and I hadn’t been assigning homework for a very long time.”

Remember my mentioning it had been a busy first term? Well, yeah.

“Most of my kids don’t do homework. So this realization just reinforced my awareness that I was only engaging in the homework ritual because I didn’t want to stray too far off the beaten path in comparison to my colleagues. But once I’d given up homework by accident, it seemed natural to make it official.”

The fact that I got that glorious tenure email and didn’t have to worry too much if my colleagues complained may have played a teensy, tiny part.

“So if you’ve got one of those kids who gets an A on tests but pulls his grade down by ignoring all homework, he–and it’s a usually a he–has probably mentioned it by now, and worships at my feet. I accept Starbucks cards or sixpacks of Diet Coke in tribute.”

One parent raises his hand.

“But don’t you find that homework ensures the students will get more practice? They need practice, just as we did when we were kids. I think it’s best for students to genuinely learn the math with practice.”

Uh oh. I take a deep breath.

“My students have always been graded overwhelmingly by what they do in class and the learning they demonstrate on tests. Homework was always optional, and I didn’t assign enough of it for students to practice fluency.”

“But I want my son to have practice material.”

“Well, I use the book pretty regularly, and there’s plenty of relevant practice material in there.”

“But do you think that’s how we all learned math?”

“Well, we weren’t all required to take advanced math. Look, I want to be clear: my method is the ultimate in hippy dippy squish.” Two parents laughed.

“I’m not trying to pretend that it’s normal for a math teacher to abandon homework. The whole homework ediscussion is basically a religious issue–and I don’t mean Muslim, Christian, and Jewish. People have strong ideological beliefs about the best way to achieve academically. However, the research on the intellectual impact of homework is very weak. But no research has shown that doing homework is the cause of comprehension.”

Another parent spoke up with a, er, very pointed tone. “I am so happy that you grade based on their work in class. So much better than to have them confused with nothing more than busy work after school. They can’t ask questions, they feel lost, and then they get discouraged.” Another parent nodded.

Original parent: “But the confusion is part of learning. Then they can come in the next day and ask for help.”

“They learn in class. If I take the bulk of one class to explain something, then they spend the next day working on that concept. I ensure students demonstrate their understanding, to the best of their ability. They won’t be able to copy the work from someone else; if I spot them not working, I work with them until I can see them understand it. If they’re talking or goofing around, they move to a different seat. My kids work math while they’re here. And ninety minutes of working or thinking about math is plenty.”

“But shouldn’t the students be practicing at home? Couldn’t you go through the course much quicker if they did?” the original parent is not to be discouraged.

“Again, they are welcome to work additional problems of their choice. But in my experience, students forget a lot of what they ‘go through’. My goal is to ensure that if they do forget material in this course, at least they really did understand at the time, rather than just follow through on some algorithms.”

“Exactly. I want them to understand the math.” said another parent.

“One last thing: I follow my students’ progress in subsequent classes. For the most part, they are keeping up and doing fine. I teach some of those subsequent classes, and so am able to compare my students to those given a more traditional course, and they’re doing fine. Many of my students go to junior college or local public universities, and I track their placement results as well. They, too, are ending up just as I’d expect. The weakest ones need some small amount of remediation, but most are placing in college credit courses. Meanwhile, they have far more accurate GPAs and weren’t forced to retake courses and slow down their progress simply because they didn’t do homework.”

And….the bell rang. Saved!

The original parent came up to me and asked, “You will assign my son additional homework?”

I smiled at the dad and the son. “All he has to do is ask.”

(He hasn’t.)

I decided to describe my policy change thusly because, well, the story happened and it was fun. All parents were respectful; I did not feel insulted or bothered by the first parent’s concerns. If I have in any way seemed contemptuous of the parents involved it’s unintentional. That said, ethnic stereotypes will prove helpful in deciphering the anecdote. The reason for the change is as described—I was busy, suddenly realized I had stopped assigning homework, decided it was time to cut the cord.

I usually just pick holes in everyone else’s arguments, but math homework is a teaching issue I have strong feelings about. Grading homework compliance is hurting a lot of kids, and all it does for those who comply is give them higher grades, not better academic skills.

Administrators understand this more than most, as they’re the ones putting additional math sections on their master schedule to accommodate all the kids with reasonable test scores who nonetheless flunked for not doing their homework. That’s the impetus behind all those stories you read of a district limiting homework’s percentage on the grade.

So as I wave goodbye to homework, let me take this opportunity to urge my compatriots to consider a similar policy, particularly if their classes look something like this:

The class opens with a warmup, designed to either review the previous material or introduce a new concept. Teacher reviews the warmup problem, then lectures or holds a class discussion on a new concept, works a few problems, has the class work a few problems, assigns a problem set, and those problems are called “homework”. Your basic I tell, I do, we do, you do.

The kids have the rest of the period to work on the problems, while the teacher is available to answer questions. If they finish in class, no “homework”! If they don’t work in class or do work for some other teacher, no big deal. It’s just time-shifting. They’ll turn in the work tomorrow, maybe do it with their tutors, maybe just copy it from friends who did it with their tutor.

Or they won’t do the problem set, either because they don’t understand, can’t be bothered, or just forget. The teacher will encourage them to come in and ask for help, or go to after school tutoring. Some of them will. Many of them won’t show up. Then they’ll get a zero, or turn it in late for a reduced grade, or stop doing homework altogether until they flunk. Or maybe their parents will call a conference and the teacher will be persuaded to accept a bunch of late homework to help the student pass the class.

How many high school math classrooms does this describe, with the occasional variation? A whole lot.

Notice that it’s only “homework” for those who can’t finish the work in class. The kids who don’t understand the material have to struggle at home. The students who really understand the material and could use more challenge get the night off.

High school teachers borrowed this method from colleges fifty years ago or more, a method designed for highly ambitious 20-somethings with demonstrated ability and interest. Today, our well-meaning education policy forces everyone into three years or more of advanced math, regardless of their demonstrated ability and interest. The college model is unlikely to work well with many students.

So go ahead and sneer at me for being a softie who skips homework, but understand that my students work to the bell. More often than not, my introduction is 10-20 minutes or even less, so the students are working the entire class period, taking on problems of increasing challenge. On those occasions where I have to explain something complicated, they focus on the relevant concepts for another day or more. But all my students are getting 60-90 minutes each day actively thinking and working about math, and my student engagement level has always been high. Strong students who finish early just do more problems. The student who treats my class as a study hall for her other homework because she has a tutor will experience teacher disapproval, often for the first time, and I’m a cranky cuss. She rarely makes the mistake twice.

When I did assign homework, I didn’t just continue from the same classwork problems, but created or selected much easier problems, designed for students to determined if they understood the basics of that particular concept.

Most education debates are tediously binary and thus wholly inaccurate. And so the math homework debate becomes “teachers who want to challenge their kids assign demanding homework” vs. “teachers who want to coddle their kids neglect their responsibility to prepare kids for college.”

In my classroom, kids are working pretty much non-stop, usually much harder on average than in the classrooms where kids are left to their own devices to finish their work. But somehow I’m the squish because I don’t engage in the great morality play known as homework. Are there teachers who don’t assign homework and also allow their kids to discover their pagh? Sure. That’s why the binary debate is a waste of time. The reality of classroom activity requires many additional points on a compass–not a bi-directional spectrum.

Finally, none of this really has anything to do with the actual teacher quality. Many teachers are doing a great job explaining math in those I do, etc lessons. Nor would any observer consider me hippy dippy or squish, which is why the comment always gets a laugh.

I was going to end with a joke about being a Unitarian in a Calvinist world. But hell, that plays right into the wrong sterotype.


Troubling Students

My classes are easy to pass, hard to do really well in. I’m a pushover for a D, but think three or four times about giving out an A. I didn’t fail a single kid last year. Save for Year Two, All Algebra All the Time, I’ve failed fewer than six kids a year, and even Year Two I had the second lowest fail rate of the math teachers.

I teach mostly math at a comprehensive high school, and the previous paragraph is very near heresy. Some math teachers cheer me on as a brave, admirable soul, but I spot them making the Mano Pantea while they walk away, just in case the Overlord is Watching. Others think I’m What’s Wrong With Education Today. These teachers hold as gospel that math standards could be upheld if we teachers were just willing to fail 60-70% of our students. In contrast to Checker Finn, who thinks teachers like me are spreading out two years of math content over three years of instruction because we can’t be bothered, these folks don’t think I’m lazy. They think I’m soft. They think I’m damaging their ability to cover all the course content they could get through if there weren’t all these kids who shouldn’t be there.

I became a lot less conflicted about my high pass rate–not that I ever lost sleep over it–after teaching precalc and discovering that a third of the kids had forgotten how to graph a linear equation and half couldn’t graph a parabola. These were kids that those other teachers had, teachers who had covered everything. Meanwhile, my kids do well in subsequent classes, so I’m not doing any harm.

But I digress. The students who trouble me aren’t the strugglers. I can take a kid who hates math, doesn’t want to be in class, and get him (it’s usually a him) to try. I can get that kid to attack a projectile motion problem and, even while making multiple small mistakes, beam with pride because by god, he kind of gets this and who ever would have thought? Kids like that, I can pass with nary a qualm.

The worrisome ones pretend they understand, but don’t have a clue. They cheat whenever they can, and not just on tests. They copy classwork in the guise of “working together” or “getting help”, and do their best to sit next to strong students. I group students by ability and, unless they can cheat on my assessment test, they are outed and placed up front, where I can keep an eye on then. They will then ask if they can sit next to John, or Sally, or Patel, their friend, because “they explain it so well”. I say no.

But if they cheated on the test, they can sometimes escape notice for a while. I circle constantly, watching kids work, changing seating when I see too much “consulting” with little discussion. Still others are more clever, and it takes a while before I realize they’ve been cheating not only in classwork, but on the tests–even when I create multiple tests. As a new teacher, I would sometimes miss these kids through the first semester. My success rate at pegging them early has improved.

This isn’t a big group, thank god. I might run into one or two a year. They have a telltale bipolar profile: for example, failing English entirely one year, and passing it the next year with Bs. Passing algebra with straight As, failing geometry completely–and failing the mostly pre-algebra and algebra state graduation test with a spectacularly low score. They aren’t fooling all of the teachers all of the time.

These kids are not your Stuyvesant cheaters, conspiring with others to satisfy demanding parents and create a fraudulent resume to get into a good school. Nor are these the low achievers who just want to get a passing grade in these time units called classes organized into a larger time period called school that others apparently view as a place of learning but they see as little more than a community network in which they have invested considerable social capital.

In fact, they’re almost worse than identified low incentive low achievers, cheating or otherwise. These kids almost seem incapable of learning. I can’t get them to slow down. They often resist help from me. Typical conversation:

Me, stopping by: “Okay, let’s start this again. You’ve plotted these points….”

Student: “Oh, yeah, I see.” Frantically erases.

Me: “Well, hang on, I want to be sure…”

Student: “I got it I got it I got it.” Starts to plot a point, then pauses.

I realize the student is waiting for me to say where to plot it in order to say “Yes, I know, I know.” So I wait. The student takes a deep breath and plots the point then lifts his pencil. “No, that’s not right, duh…”

Me: “You aren’t sure how to plot points.”

Student: “Yes, I am.”

Me: “Great. Plot (7,-7).”

Student plots (-7, -7).

Me: “Stop there.” I go grab a handout I have specifically for these situations, a simple handout that explains plotting points with some amusing activities to drive the point home.

Student: “I don’t need this. I know how to do it!”

Me: “Great. Then it should just take you a few minutes.”

At this point, I get a variety of reactions. Some students become furious. Others get sulky. Still others do the handout, making many mistakes, all the while assuring me that this is easy. I obligingly correct the mistakes, make them do it correctly. The ones that get furious, I shrug and let them continue.

Regardless, within a day, they are making the same mistakes. Nothing sinks in. Don’t get overly focused on plotting points; the problem could be anything–factoring, solving multi-step equations, working with negatives, exponential properties, fractions, whatever. Or a new concept. They have absolutely no clue, and can’t do much of anything.

Yet they don’t have the profile of a low ability student. Test scores, yes. Profile, no. They often have As, win praise from teachers for their teamwork and effort. They are heavily invested in appearing “normal”. Serious control freaks. Sometimes, but not always, with parents who expect success. More often, but not always, Asian. All races. Both genders.

I haven’t taught freshmen since oh, lord, fall 2012.1 I teach relatively few sophomores these days, running into them only in Algebra 2.

That matters because when I taught freshmen and sophomores, I would go full-scale intervention. I might talk to a counselor to see if they should be assessed for a learning disability. I would insist that they stop lying to me and themselves. I had no small success at getting some of them to acknowledge their desperate attempts at fraud, get them to work at their actual level, deal with the discomfort. They didn’t make much progress, but it was real progress, and they had skills to move forward. I ran into some of them again the next year, and we could start on an honest basis and make additional progress. Those who didn’t acknowledge their issues were among the few students I failed.

But that’s a lot harder to do when dealing with juniors taking trigonometry or, god forbid, precalc. Should I fail them? They will probably do better in a class with teachers who give “practice tests”, study guides that have exactly the same questions as the eventual real test but with different numbers. They will definitely do better with teachers who actually grade homework and count it as 25% of the overall.

A small problem. This approach turns my grading policy into: work hard and honestly acknowledge your ignorance and I’ll pass you. Lie and do your best to cheat with similar ignorance and I’ll fail you. I’m comfortable with holistic grading at the bottom of the scale, but I don’t like morality plays.

Then I remember that kids who honestly acknowledge their inability in a trig or pre-calc class are usually seniors, off to junior college and a placement test that will accurately put them in remedial math. I’m only ensuring they are learning as much as possible for free before paying. If they are juniors, I always have a talk with them about their next steps, telling them not to take the next course in the sequence but maybe stats or something else that will keep them working math, but not out of their league.

The kids who cheat and fake it in trig and precalc are usually juniors, and they will be going onto another course. They will not listen to me when I tell them under no circumstances should they continue into pre-calc or, god forbid, calculus. I might be teaching that course, which just gives me the same problem again. Or they’ll be cheating their way through with another teacher–or, that teacher will do what I should have done and flunked them.

This quandary doesn’t make any sense unless you realize that in my view, these kids are pathologically terrified of facing reality, the sort of thing that some of them, forced to face up, might not survive in good form. These aren’t blithe liars gaming the system to look good. Then I remind myself that they’ve been caught before, they’ve flunked other classes, they’ll survive. But I still don’t like the quandary, because these are kids who literally can’t learn. (And remember, I’ve seen them in my non-math classes, too). By junior year, given their denial and fear, does it do any good to make them aware of this? They’re going to be able to point to any number of teachers who disagree with my assessment, and have all sorts of excuses for why they got those Fs. Besides, they just don’t test well. It’s always been a problem.

At times like this, I envy my colleagues who never notice the cheating, or who focus purely on achievement and aren’t interested in the distinctions I’m making.

But these are the students who trouble me.

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1Holy Crap. That’s an amazing realization. New math teachers doing your time in the algebra/geometry trenches, take heed. If you want variety, it will come.


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