Tag Archives: fawn nguyen

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.


Reform Math: An Isolationist’s View

For my sins, I periodically peruse the Method Math teacher blogs. I call them Method Math teachers because, much like those self-important thespians in the Actors Studio can’t just act, these guys can’t just teach. Not for them the order of a structured curriculum; no, they want “meaningful math”. They don’t want their kids to do well unless it’s the right kind of doing well. Do they love math? Do they have the proper respect and curiosity for math? What’s the student’s motivation?

They are correctly described as reform math teachers. In math, “reform” refers to the “progressive” side of the debate, in which math is not so much a field of study as it is an ideological value system. Discovery and complex instruction are the guiding lights of their lesson planning. However, since they are teachers, and most teachers don’t really care about education policy in any coherent sense, many teachers who embrace the tenets may not be aware of the ideological underpinnings of their chosen Method. They Like or Don’t Like, without much sense of anything beyond their classroom. (and in that, they are like most teachers).

Reform math is all about social justice, enabling blacks, Hispanics, and girls to “feel successful” about learning math. Actually being successful at learning math is a whole different thing; certainly these demographic categories are successful with their teachers, but when it comes to outside assessments, not so much—which is why reformers don’t much care for standardized tests. But in the classroom, constructivists and discovery-based lessons can accept multiple methods, which means no one method is wrong. And explaining! Explaining is vital. “Explaining your process” is the way that the “procedurally competent” kids (only in reform math is this a bad thing) can be flunked or at least marked down for not explaining their work, while other kids can find “other ways to be smart”. Convenient for grading, this value system allows teachers to dream up all sorts of ways for top kids to fail with the right answer, while tolerating all sorts of other ways for low ability kids to succeed with the wrong one.

(Ironically, these same people who focus on the importance of explaining the “why” are always insisting that teachers reduce the literacy demand of word problems, for kids who can’t read. That’s because the explaining aspect is meant to assist white girls weak on math but strong on literacy, whereas literacy reduction is all about making the problem set up easier for blacks and Hispanics to interpret.)

Reform math practitioners enthuse about this “open-ended discourse”, which avoids calculations and algorithms and, you know, answers. At least definitively right answers. Which teachers don’t give. Teacher explanations = failure. Hence Dan Meyer, the Lee Strasberg of the math blogosphere, famous for his Ted talk, has a blog that proudly bears the label “less helpful”.

Open-ended discourse requires curiosity and ability, which some might deem a feature, but the knowledgeable understand is a bug. Inquiry teaching deliberately eschews algorithms or process or anything resembling a structured approach (while allowing that “blind memorization” might occasionally be useful). Few reform teachers understand the underlying rationale for this method, which lies in the hope that open-ended problems will narrow the achievement gap—not by improving achievement of the lower half, but by narrowing all achievement into a much thinner band. Hence the importance of grading down the top students, and slowing (well, they call it deepening) instruction to be sure that no one is pulling too far ahead.

Ed schools ferociously pretend that all but a few racist fuddy-duddies teach using constructivist methods, but out in the real world, reform math is mostly fringe. The greatest penetration is at the suburban elementary school level, which has a teaching population disproportionately comprised of cheery young women who care more about their students’ interpersonal skills than intellectual development (a feature, not a bug). Complex instruction requires students to “share ideas and knowledge” and the strongest students are responsible for the weakest students’ learning, entrancing elementary school teachers with the delusion that math lessons can enhance social justice. Besides, elementary school teachers aren’t terribly strong at math to begin with, so a method that de-emphasizes algorithms reinforces their own preferences.

Since elementary school teachers rarely have the math chops to develop their own lessons, most reform curriculum development is found in middle school, where kids don’t stay long enough for the parents to complain and the teachers are knowledgeable—and teaching the subjects most attended to by reformers (pre-algebra, algebra, and geometry). So there’s a big support group and lots of material to build on.

Few high school math teachers embrace reform; those who are committed to the Method don’t have a long shelf-life. Most give it up after a few years, the rest show up at grad school where they can pretend that constructivism and complex instruction are valid, proven methods. They get a Phd and demand conformity from prospective teachers in ed school, successfully selling their dogma to a few eager apostles. These converts, alas, ultimately abandon the method or return to grad school where the cycle begins again. Thus Dan Meyer is no longer teaching math but getting his PhD at Stanford with Jo Boaler, Queen Mother of Reform Math. (Understand, however, that reformers do not practice what they preach in ed school. There aren’t multiple ways and many right answers when training new teachers. Heavens, no.)

To the extent reform math survives for any length of time, it does so in white, suburban elementary schools, although not without a struggle. Elementary teachers’ support is counterbalanced by well-educated parents who generally despise it. Parental protests have killed reform math programs at all levels for decades throughout the country. Districts have to balance happy teachers with howlingly angry parents. The high school battles ended over a decade ago, but elementary school parents have to deal with teachers who actually like the program.

But reform math wars are mostly a tale of suburban woes, as parents push back on well-meaning districts hoping to close the achievement gap of their bottom 10-20% by depressing their top performers. It stresses the parents out, but the kids will catch up. For all that reform math propagandists want to change the world for black and Hispanic kids, the techniques are abandoned quickly in high poverty, low ability schools, particularly at the high school level. The story goes like this: a complex instruction curriculum is introduced with great fanfare, math teachers complain, the complainers that can’t be fired are transferred, cue the fawning news coverage with much noise about “equity” and “access”, a few beaming parents who barely speak English talking about their children’s newfound love of math, clips of young black teens and Hispanic girls talking about how they like this math sooooo much better than “just being told what to do”…..and then the dismal state test scores come rolling in and all the canny zealots who once exhorted the grunts to be guides standing to the side are now publicly championing sage on the stage. Back comes explicit direct instruction and the cycle begins again.

The Jo Boaler brouhaha contains one such example, as James Milgram points out:

Indeed, a high official in the district where Railside is located called and updated me on the situation there in May, 2010. One of that person’s remarks is especially relevant. It was stated that as bad as [our work] indicated the situation was at Railside, the school district’s internal data actually showed it was even worse. Consequently, they had to step in and change the math curriculum at Railside to a more traditional approach. Changing the curriculum seems to have had some effect. This year (2012) there was a very large (27 point) increase in Railside’s API score and an even larger (28 point) increase for socioeconomically disadvantaged students, where the target had been 7 points in each case.

Railside High is San Lorenzo High School, in California. As Milgram says, its 2010 scores are dismal, while 2012 scores are improved. Not substantially—ain’t no getting around basic cognitive ability coupled with absurdly unrealistic expectations. But improved.

So I began this post to explain the tiny twitter tempest I began last week, and I’m not there yet. And to get there would take the post into specifics when thus far it’s been general. Sigh. For some reason, I’m writing very slowly this summer. But I didn’t have any clear description of reform math that I could link to in order to explain reform math as I see it, which is not quite as most critics see it.

If the Method teachers out there in the blogosphere do read this, they may confuse me for a traditionalist and, uh, no. My ed school is committed to complex instruction and inquiry-based learning, and I am very fond of my ed school. It’s not fond of me, of course, but then who is?

At that ed school, my all-discovery, all-inquiry, all-complex-instruction master teacher provided me with the best learning experience of my life, adopting effortlessly to my strengths and skepticisms to give me fantastic advice that I hark back to this day. I am, to put it mildly, Not Easy to Teach. That I got six months of valuable education counts for a lot. Thanks to that teacher’s willingness to focus on goals, not methods, I learned to do the same. I can find a lot of good in reform objectives, and steal interesting concepts in their lessons. I might think reform methods are awful but, like progressive educators in general, reformers are thinking about how to teach math, which as it happens is a subject much on my mind.

I am not and never will be a member of the Method group. I am Switzerland, or the US between world wars. Ignore the fact that my first year out I put my students in rows for three weeks until I couldn’t stand it anymore and put them in groups. Second year out I lasted 10 days. Third year out and beyond, I gave into the inevitable and just put the kids in groups from the start. I use manipulatives, introduce units not with facts but with activities that illustrate facts, minimize my use of algorithms, and always remind students that I’ll take a good estimate in lieu of a calculation (and give most of the credit). Anyone evaluating my teaching practice would conclude I have much more in common with the Method crew than I have with traditionalists.

Still, they are profoundly wrong, and their nonsense grates on me much more than the many ways in which the traditionalists err. That’s what led to my tweet, which yes, I still haven’t explained. But I’m ready to start explaining now, so that’s a step up.

Edited later to add:

A couple points. First, I welcome comments on this much because it will help me determine whether I’m getting the right ideas across. I know I can be tough on commenters who misinterpret me, much as I try not to, but I will really try not to if you want to complain about something you’ve misinterpreted on this particular post.

Second, I paint in broad brushes. Keep that in mind!

I’m going to try hard to get the second part of this up faster than normal, for me. I’m hoping for a couple days, but if I fail, know that I tried.