Math Instruction Philosophies: Instructivist and Constructivist

Harry Webb has been on a tear about discovery vs. traditional explanations. The hubbub has pulled the great god Grant Wiggins, originator of backward design, which is a bible of ed schools as a method for developing curriculum.

Now, let us pause, a brief segue, to reflect on those last two words. Developing curriculum. I’m talking about teachers, yes? Teachers, building their own unit lessons, their own tests, their own worksheets. As I’ve written, teachers develop their own curriculum and, to varying degrees, have intellectual property rights (I would argue) to their material. So when reformers, unions, politicians, or whoever stress the importance of curriculum, textbooks, and professional development in implementing Common Core, there’s a whole bunch of teachers nationwide snorfling at them.

So Wiggins and Jay McTighe wrote Understanding by Design, which describes their framework and approach to curriculum. It is, as I said, a bible of ed schools. I have a copy. It’s good, although you have to look past their irritating examples to figure that out.

(Note: See Grant Wiggins’ response below. I’ve reworded this slightly and separated it to respond to his concerns. Also throughout, I changed “direct instruction” to some other term, usually instructivism.)

The book clearly states that there’s no one correct approach for every situation, that arguing between instructivism and constructivism creates a false dichotomy. So I was jokingly sarcastic before, but my point is real: it’s hard to read Grant Wiggins and not think that, so far as K-12 curriculum goes, he leans heavily towards constructivist. As one example, in a text section that discusses the fact that there’s no one right approach, he includes this table on the activities dominant in each approach. When I look at this table, I see a clear preference for constructivist approaches. I also see it in this highly influential essay and much of his writing. But as Wiggins states in the comments, and in the book, he clearly denies this preference. However, Wiggins’ book is the bible of ed schools for a reason, and it’s not for its categoric embrace of all things instructivist. So put it this way: what he says are his preferences and what any instructivist would take away from his preferences are probably not the same thing. I say this as someone who periodically rereads his work because of the value I find in it once I shift my focus away from the trappings and focus in on the substance. I encourage anyone who agrees with my impression of Wiggins’ preference to read him closely, because he’s done a lot over time to inform my approach to curriculum development.

(end major edits–I put the original text at the bottom)

So Wiggins reads all this hooha, and comes out with this outstanding description of lectures and why they are a problem. I agreed with every word of this post (there are two others), so much so that I tweeted on it. (Note: I agree for math. History’s a different issue.) As I did so, I was vaguely disturbed, because look, while I don’t write a lot about ed school per se (and even defend it, slightly), I spent a lot of time in class naysaying. And if they’d been saying reasonable things like this about lectures, what had I been disagreeing about?

And then Harry comes through brilliantly, answering my question and pointing out a huge hole in Wiggins’ 3-part series:

Wiggins writes of a survey of teachers in order to support his view that different pedagogies are required to achieve different aims. Unsurprisingly, the teachers give the right answers; the ones that they probably learnt at Ed School. However, the survey response that is taken to represent lecturing is called, “DIRECT TEACHING Instruction on the knowledge and skills.” Now, although I do not recognise my practice in Wiggins’ definition of lecturing, I do recognise myself in this definition wholeheartedly. And so I think we are being invited here to see all direct teaching – dare I say direct instruction – as non-interactive lecturing that lasts for most of a period.

Hey. Yeah. That’s right! Wiggins naysays the lecture in his essay, but the overall debate is between instructivism, of which lectures are just a part, and it’s , and it’s instructivism that has a bad name in ed school, not solely lectures. Harry says that he explains in classroom discussion, but rarely lectures. Which may sound like someone else.

Harry scoots right by this, because he’s all obsessed about the fuzzy math and constructivist debate, and it occurred to me that this area needs elucidation, because most people—and reporters, I am looking at you—don’t understand this difference.

So here it is: not all explanation is lecture, and not all discovery is constructivist.

In an effort to not turn all my posts into massive tomes (don’t laugh), I’m going to write about this difference later. Here, I’m just going to show you the difference through different teachers.

Before I start: labels are hard. Roughly, the terms reform, student-centered, constructivist, “facilitative” (Grant Wiggins’ term) all refer to the open-ended investigative approach. Instructivist, teacher-centered, traditionalist, direct instruction are all terms used to describe the approach where the teacher either tells you how to do it or wants you to figure out the way (not a way) to do it. (Note: I left “direct instruction” in here, because I believe it’s still an instructivist approach.)

Very few math teachers are pure constructivist. We’re talking degrees. I have no data on usage rates, but I’d be pretty surprised if 80% of all high school math teachers didn’t use traditional instruction-based approach for 90% of their lessons. I speak to a lot of colleagues who dislike pure lecture and would like to teach a more modified instructivist mode, but they aren’t sure how it works. However, most high school math teachers are instructivists who lecture. Full stop.

Constructivist Approach (aka investigation, reform)

Dan Meyer: Dan Meyer’s 3-Act Meatballs
Fawn Nguyen: Barbie Bungee
Fawn Nguyen Vroom Vroom
Michael Pershan: Triangles and Angles (he calls this investigation. I’d personally characterize it as “in between”, but it’s his call.)
Cathy Humphries: Investigation into Quadrilaterals

This is a partial list. Dan’s blog has links to all his various projects, as well as other bloggers committed to the investigative approach.

(By the way, I am dying to do the Vroom Vroom one, but I’m not enough of a mathematician to understand the math behind it. Neither does Fawn, apparently. The math looks quadratic. Is it?)

I’m not a fan of the open-ended reform approach, but I like all sorts of the activities the constructivists come up with. I just modify them to be more instructivist.

Remember that both Meyer and Nguyen use worksheets, practice skills, and many other elements that are pure instructivist. Pershan rarely does open-ended activities. In contrast, Cathy Humphries is very close to pure constructivist math. Total commitment to reform.

Traditional Instructivism (Lectures)

Much MUCH harder to find traditionalists bloggers. I’ve included two of my “lectures” that have relatively little discussion, just to fill out the list:

Me: Geometry: Starting Off
Me: Binomial Multiplication and Factoring with the Rectangle

Dave at MathEquality is traditionalist, a guy who works hard to explain math conceptually, but does so for the most part in lecture form. However, it’s also clear he keeps the lectures fairly short and gives his students lots of in class time for work.

But he’s the only one I can find. Right on the Left Coast appears to be a traditionalist, but he writes more about policy and his disagreement with traditional union views. (Huh, I should have mentioned him in my teacher blogger writeup of a while back.)

In order to give the uninitiated a good idea of what lecture looks like, three google searches are informative:

factoring trinomials power point

holt math power point

McDougall Litell math power point

Many high school teachers build their own power point explanations. Others just take the ones provided by publishers.

Still others use a document camera or, if they’re extremely old-school, transparencies.

What they look like is mostly this:

Khan Academy: Isosceles Right Triangles

Many teachers are really, really irritated at the fuss over Khan Academy because all he does is lecture his explanations—and not very well at that.

The most vigorous voices for traditional direct instruction comes from people who don’t teach high school math. That’s not a dig, it’s just a fact.

Modified instructivist

I’m not sure what to call it. There’s not just one way to depart from instructivist or constructivist. The examples here generally fall into two categories: highly structured instructivist discovery, and classroom discussions with lots of student involvement.

Me:Modeling Linear Equations
Me: Modeling Exponential Growth/Decay
Michael Pershan: Proof with Little Kids
Michael Pershan: Introducing Polar Coordinates
Michael Pershan: The 10K Chart
Ben Orlin: …999…. and the Debate that Repeats Forever.
Ben Orlin: Permutations and combinations

For a complete list of my work, check out the encyclopedia page on teaching. I likewise recommend Pershan and Orlin’s blogs.

A question for Grant Wiggins, and anyone else interested: what differences do you see in these approaches?

A question for reporters: when you write about reform or traditional math, do you have a clear idea of what the fuss is about? And did these examples help?

Question for Harry Webb: You sucked me into this, dammit. Satisfied?

If you have good examples of math instruction that falls into one of these categories, or want to propose it, tweet or add it to the comments. I’m going to write up my own characterizations of this later. Hopefully not much later.

*******

Here’s what I originally said in the changed paragraph:

So Wiggins and Jay McTighe wrote Understanding by Design, which describes their framework and approach to curriculum. It is, as I said, a bible of ed schools. I have a copy. It’s good, although you have to look past their irritating examples to figure that out. The book clearly states that there’s no one right answer, that arguing between direction instruction or constructivism creates a false dichotomy, but then there’s this table on the activities dominant in each approach. Cough. Okay, no one right answer, but a strong preference for facilitative/constructivist.

The Dark Enlightenment and Duck Dynasty

The Dark Enlightenment has been discovered. Eeeek.

I’ve written about my adoption by the Network and have nothing to change–it’s not something I consider myself part of, per se, but they apparently find my writing helpful. I’m fine with that. I would refer Jamie Bartlett to the above image to reinforce what seems to me to be obvious: the “dark enlightenment” is not characterized by political objectives and has very little unity of purpose.

hbd chick wrote a detailed response to the Jamie Bartlett column which, to the extent I understand it, I agree with. But I would refer someone trying to figure this thing out to read the comments, particularly this one by T. Greer:

The many voices in the ‘dark enlightenment’ do not harmonize. They don’t share the same ideals, aims, or even impulses. They defined by a shared enemy; were this enemy to disappear then so would all talk of a cohesive ‘dark enlightenment.’
…
The major strand that unites the entire community is a willingness to frankly state opinions polite society does not accept (but in many cases once did) and listen to others do the same.

That is, as I say in response, the defining element of the Dark Enlightenment is not political, philosophical, or cultural views, but a shared loathing of “The Cathedral”. Unfortunately, I can’t find one clear definition of the Cathedral that doesn’t involve reading all of Mencius Moldebug, who I don’t really understand and makes me feel Hemingway brusque. I use the term Voldemort View to characterize the most likely reason for the achievement gap; the Cathedral can be thought of as the Canon of Modern Anathema, the official dogma of views that must not be spoken. Some of the views are actual truths, others are opinions. But if they are uttered, the speaker must be cast out into the darkness and, more importantly, economically ruined.

I can think of no common objective the nodes in that diagram share, but we all hate and despise the Cathedral. Our touchstones are not racial purity, male dominance or a derailing of democracy (all objectives I unreservedly oppose) but the expulsion of James Watson, Jason Richwine, and John Derbyshire—whether we agree with them or not. I almost never hate people. But I hate the Cathedral. Probably in part because I trusted it a couple decades ago, and there’s nothing like a reformed ex-smoker. Screw you if you want me to righteously disassociate. Take my ideas on their own merit or don’t, never assume I agree with any idea unless I say so. But if you’re the sort who demands indignant condemnation, it will be my considerable pleasure to deprive you of that satisfaction. In short–but why be short when English has so many words?—I will not disavow on principle.

I suggest that if the “dark enlightenment” is spreading, it does so not because of any distaste for democracy, much less some weird white guy radicalization, but because the general public is slowly becoming deeply tired of the elites getting exercised about exorcising yet another heretic.

And so to Duck Dynasty, a show I vaguely knew of before the fuss. Phil Robertson opines, identity groups cluck, and all the pundits write cynically about the outrage, secure in the knowledge that the machine will roll over and crush Robertson. But then, glory be, the Robertson clan doesn’t just refuse to back down, it refuses to apologize, and for once, the cultural segmentation of American society turns out to be a net positive. Christians everywhere have time to make their displeasure known, and A&E realizes that the money move lies in keeping Phil, leaving GLAAD out in the cold. Truly a great day. And if you can’t understand why an agnostic with no interest in denying the reality of pre-civil rights America would celebrate that outcome, you don’t understand how much I hate the Cathedral.

Patton Oswalt quoted Steve Sailer’s pithy statement “Political correctness is a war on noticing”. A few of his followers disapproved. The resulting twitter fest is very funny, as a couple of Oswalt’s followers try to alert him to the evils of Sailer, and Oswalt remains blithely unconcerned. Money quote, from Oswalt: “I’ve never been scared of ideas. I can hear all kinds & still keep my feet. Think I’ll call this stance ‘diversity'”.

But then you’ve got the earnest, well-meaning Michael Pershan, one of the only actual math bloggers I read. Pershan is Jewish, I think, although he never mentions it on the site (I remember his wedding announcement vaguely), and I mention this only because when I read this twitter mess my first thought was “he’s Jewish, he went to Harvard, he lives in New York City, and he didn’t see this coming?” But I think he’s a particularly observant Jew (not like noticing things, like observing Jewish custom), and until recently taught at a Jewish boys’ high school, so perhaps he doesn’t get out much.

Anyway, he takes gentle issue with a PoC teacher blogger who makes what would normally be called racist statements were he talking about anyone but white folks, and gets “schooled”, literally, in a key plot point: in the identity culture, all whites are the same. Michael Pershan, like many reflexive progressives (the sort who haven’t really thought it through but hey, all their friends are doing it) wants race and gender warriors to accept that there are “good” whites and “bad” whites. He wants to be able to point fingers and shame bad whites, but is troubled that the PoC and women seem to paint all whites and all males as the same. The identity divas will have none of that, and kick him around for a while. Pershan has retired from both the fray and Twitter, which is too bad. Not that I sympathize with his point of view. If you want to walk the identity path, baby, then all whites are equally undeserving of their largesse. You either reject or embrace the identity and entitlement game in its entirely; there are no half measures. The correct response is to deny the identity folk all satisfaction. It’s okay, they mostly enjoy the process, gives them something to complain about.

And just to show the compartmentalization of my ideas: I think many of the people beating down Michael Pershan in that conversation are just fine, as teachers. I often agree with them. Not always. Jason, the PoC blogger who started the sound-off, has a good teaching blog, and I don’t find his writings on identity to be insanely insufferable, which is a compliment.

I want more Duck Dynasty victories. I want the Michael Pershans to laugh at the very idea of seeking approval from identity divas. I want the Cathedral thwarted routinely and eventually dismantled. Not as a blogger, but as a person.

As a blogger, I’ll still write about education policy and education itself from all different angles, including the lamentable determination to ignore cognitive ability.

On that point, I’ve noticed a recurring theme that Razib Khan made in the hbd diva post, also seen here in Rod Dreher’s call for silence on HBD: the notion that most people who “embrace” (their word) racial differences don’t have a clue about the science.

I find this flummoxing. I know that Razib, who has his own node on the Network, is not criticizing the ideas themselves, but rather the people promoting them as ignorant. But who are these people promoting science, good or bad? I’m not sure if he’s talking about me. I’m certain the commenters on Rod’s site, from the “reasonable conservatives” to the “moderate progressives” are criticizing the ideas as wrong and the people promoting them as ignorant.

I don’t read the other sites much, save for Steve Sailer and Razib Khan, so maybe they’re doing all sorts of bad science. For myself, I don’t do science. I barely do math.

I often see reporters refer to “beliefs” or “opinions” about IQ. My “beliefs” about IQ involve the degree to which IQ is inaccurate, missing some aspects of intelligence that might be largely irrelevant to measuring IQ among white populations, but highly relevant in others. Actually, they wouldn’t go so far as “belief” or “opinion” but maybe “wonderings”.

But they aren’t talking about those beliefs, but the “belief” that IQ is meaningful, that IQ is not the same in different populations. That’s not a belief.

Or, as Steven Pinker famously wrote of Malcolm Gladwell’s maunderings on IQ: “What Malcolm Gladwell calls a “lonely ice floe” is what psychologists call ‘the mainstream.'”

When taking down a heretic, Cathedral strategy demands that the heretic be easily expelled with a minimal degree of cognitive dissonance. And so no one takes on Steven Pinker. Many reporters regurgitate what they understand of the Flynn Effect, but no one asks James Flynn if black IQs are, on average, lower than white IQs and whether that might make a difference to academic outcomes or whether the gap can easily be fixed with a more nurturing environment. Only one person asked Harvard’s Christopher Jencks why he blessed Jason Richwine’s doctorate, or why Harvard signed on for it. These people are of the Cathedral and if they challenge the canon, maintaining orthodoxy becomes impossible. So they are left alone, ignored politely when they speak anathema.

I don’t do science. I keep my blog anonymous because of I explore the impact of the Voldemort View, the view that must not be spoken, the view that says the achievement gap between different racial and income groups is primarily caused by differences in cognitive ability, on educational outcomes. I believe that IQ is imperfect as a metric of cognitive ability, although I can’t prove it and my opinion is still inchoate (ooh, Thomas of Convenant!). I accept the mainstream findings that shows a clear and largely unchanging difference in IQs by race and income. If Steven Pinker, James Flynn, or Christopher Jencks have said anything that disagrees with my representation of mainstream research, most fully articulated here, I’m unaware of it. So don’t ask me about IQ and race. Ask them.

Isometries and Coordinate Geometry

Michael Pershan’s post on teaching congruence reminded me that way back in the beginning of summer, I’d been meaning to write up some of my geometry work, which I think is pretty unusual. Still on the list is the lesson sequencing, but here is some thoughts and sample problems on integrating Isometries and coordinate geometry.

To summarize my earlier work, explicated in Teaching Congruence, or Are You Happy, Professor Wu?, I was unhappy with the circular reasoning that geometry books present in congruence sections. Triangle ABC is congruent with Triangle DEF because all their sides and angles are congruent, and congruence is when the shapes have congruent sides and angles. Professor Wu’s writing taught me the link between congruence, similarities, and isometries (aka, transformations, or translations, rotations, and reflections). I’d previously skipped isometries, since the kids don’t need them much and they’re easy to figure out, but this discovery led me to use isometries as an introduction to congruence and similarity.

But all book chapters on isometries are very thin, or they rely on non-coordinate shapes, which is largely a waste of time. Was there any way I could bring back some other concept while working with isometries, particularly with my top students?

Which leads me straight to coordinate geometry. The most immediate tie-in is helping students figure out rotation, the most difficult of the transformations. A 90 degree rotation around a point involves perpendicular lines (“..and class, what is the relationship of perpendicular slopes, again? Class? Waiting!”). Moreover, the kids learn that the slope of the line connecting a point and its reflection must be perpendicular to the line of reflection. Finally, dilations involve all sorts of work with parallel lines. All of these reinforcements are excellent for weaker students, and are yet another reason to introduce transformations, even if only as a prelude to congruence.

But I wanted a meaningful connection for my top students, who usually grasp the basics quickly. What could I give them that would integrate algebra, coordinate geometry, and a better understanding of transformations?

Over the summer, I taught an enrichment geometry class to seventh graders whose parents got mad because I wasn’t assigning enough homework. My boss backed me–thanks, boss!—and the kids did, too—thanks, kids!—and not for the usual reasons (these are not kids who celebrate a lack of homework). The kids all told the boss that they were surprised that they weren’t able to just follow the pattern and churn out 50 problems of increasing difficulty in the same vein. “I have to really think about the problem,” said more than one, in some astonishment.

So, for example:

Homework: Reflect Triangle LMN [L: (-1,4), M: (0,7), N: (-4, 10) over line y=x+2. Prove it.

So we discussed the steps before they left. I actually posed it as a couple of questions.

1. If you sketched this and just estimated points the reflection, what would be the key information you’d need to pin down to go from “estimation” to “actual answer”?
2. Can you think of any coordinate geometry algorithms that might help you find these points?

And working with me, they came up with this procedure for each point:

1. Find the equation of the line perpendicular to the reflection line.
2. Find the solution to the reflection line and the perpendicular line. This solution is also the midpoint between the original point and its reflection.
3. Using the original point and the midpoint, find the reflection point.
4. Prove the reflection is accurate by establishing that the sides of the original triangle and the reflection are congruent.

And here it is, mapped out in Desmos—but honestly, it was much easier to do on graph paper. I just wanted to increase my own Desmos capability.

This is the cleaned up version. Maybe I should put the actual work product here. But I’m not very neat. Next time I’ll take pictures of some of the kids’ work; it’s gorgeous.

When we came in the next day, the kids excitedly told me they’d not only done the work, but “figured out how to do it without the work!” Sure, I said, and we then predicted what would happen with the reflection of y=x+3, y=-x + 4, and so on.

But what about reflecting it over the line y=-2x?

Gleesh. I didn’t have time during summer to investigate why the numbers are so ugly. The kids got tired after doing two points, and I told them to use calculators. But we did get it to work. We could see the fractions begin in the perpendicular line solutions, since we’re always adding .5x to 2x. But would it always be like that?

However, I’ve got one great activity for strong kids done–it reinforces knowledge of reflection, coordinate geometry, systems of equations, and some fairly messy algebra. Whoo and hoo.

Down side–for the first time in two years, I’m not teaching geometry this year!

All the more reason to document. Next up in this sequence is my teaching sequence. But if anyone has ideas about the translation that makes the second reflection have such unfriendly numbers, let me know.

Hey, under 1000!

Teaching Math vs. Doing Math

Justin Reich of EdWeek (not to be confused with Justin Baeder of EdWeek) wrote enthusiastically of a new study, asking What If Your Word Problems Knew What You Liked?:

Last week, Education Week ran an article about a recent study from Southern Methodist University showing that students performed better on algebra word problems when the problems tapped into their interests. …The researchers surveyed a group of students, identified some general categories of students’ interests (sports, music, art, video games, etc.), and then modified the word problems to align with those categories. So a problem about costs of of new home construction ($46.50/square foot) could be modified to be about a football game ($46.50/ticket) or the arts ($46.50/new yearbook). Researchers then randomly divided students into two groups, and they gave one group the regular problems while the other group of students received problems aligned to their interests.

The math was exactly the same, but the results weren’t. Students with personalized problems solved them faster and more accurately (emphasis mine), with the biggest gains going to the students with the most difficulty with the mathematics. The gains from the treatment group of students (those who got the personalized problems) persisted even after the personalization treatment ended, suggesting that students didn’t just do better solving the personalized problems, but they actually learned the math better.

Reich has it wrong. From the study:

Students in the experimental group who received personalization for Unit 6 had significantly higher performance within Unit 6, particularly on the most difficult concept in the unit, writing algebraic expressions (10% performance difference, p<.001). The effect of the treatment on expression-writing was significantly larger (p<.05) for students identified as struggling within the tutoring environment1 (22% performance difference). Performance differences favoring the experimental group for solving result and start unknowns did not reach significance (p=.089). In terms of overall efficiency, students in the experimental group obtained 1.88 correct answers per minute in Unit 6, while students in the control group obtained 1.56 correct answers per minute. Students in the experimental group also spent significantly less time (p<.01) writing algebraic expressions (8.6 second reduction). However, just because personalization made problems in Unit 6 easier for students to solve, does not necessary mean that students learned more from solving the personalized problems.

(bold emphasis mine)

and in the Significance section:

As a perceptual scaffold (Goldstone & Son, 2005), personalization allowed students to grasp the deeper, structural characteristics of story situations and then represent them symbolically, and retain this understanding with the support removed. This was evidenced by the transfer, performance, and efficiency effects being strongest for, or even limited to, algebraic expression-writing (even though other concepts, like solving start unknowns, were not near ceiling).

So the students who got personalized instruction did not demonstrate improved accuracy, at least to the same standard as they demonstrated improved ability to model.

I tweeted this as an observation and got into a mild debate with Michael Pershan, who runs a neat blog on math mistakes. Here’s the result:

I’m like oooh, I got snarked at! My own private definition of math!

But I hate having conversations on Twitter, and I probably should have just written a blog entry anyway.

Here’s my point:

Yes, personalizing the context enabled a greater degree of translation. But when did “translating word problems” become, as Michael Pershan puts it, “math”? Probably about 30 years old, back when we began trying to figure out why some kids weren’t doing as well in math as others were. We started noticing that word problems gave kids more difficulty than straight equations, so we start focusing a lot of time and energy on helping students translate word problems into equations—and once the problems are in equation form, the kids can solve them, no sweat!

Except, in this study, that didn’t happen. The kids did better at translating, but no better at solving. That strikes me as interesting, and clearly, the paper’s author also found it relevant.

Pershan chastised me, a tad snootily, for saying the kids “didn’t do better at math”. Translating math IS math. He cited the Common Core standards showing the importance of data modeling. Well, yeah. Go find a grandma and teach her eggsucking. I teach modeling as a fundamental in my algebra classes. It makes sense that Pershan would do this; he’s very much about the why and the how of math, and not as much about the what. Nothing wrong with this in a math teacher, and lord knows I do it as well.

But we shouldn’t confuse the distinction between teaching math and doing it. So I asked the following hypothetical: Suppose you have two groups of kids given a test on word problems. Group 1 translates each problem impeccably into an equation that is then solved incorrectly. Group 2 doesn’t bother with the equations but gives the correct answer to each problem.

Which group would you say was “better at math”?

I mean, really. Think like a real person, instead of a math teacher.

Many math teachers have forgotten that for most people, the point of math is to get the answer. Getting the answer used to be enough for math teachers, too, until kids stopped getting the answer with any reliability. Then we started pretending that the process was more important than the product. Progressives do this all the time: if you can’t explain how you did it, kid, you didn’t really do it. I know a number of math teachers who will give a higher grade to a student who shows his work and “thinking”, even if the answer is completely inaccurate, and give zero credit to a correct answer by a student who did the work in his head.

Not that any of this matters, really. Reich got it wrong. No big deal. The author of the study did not. She understood the difference between translating a word problem into an equation and getting the correct answer.

But Pershan’s objection—and, for that matter, the Common Core standards themselves—shows how far we’ve gone down the path of explaining failure over the past 30-40 years. We’ve moved from not caring how they defined the problem to grading them on how they defined the problem to creating standards so that now they are evaluated solely on how they define the problem. It’s crazy.

End rant.

Remember, though, we’re talking about the lowest ability kids here. Do they need models, or do they need to know how to find the right answer?