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.

About educationrealist

20 responses to “Math Instruction Philosophies: Instructivist and Constructivist

  • grantwiggins

    I appreciate your attempt to make sense out of all this, and for in general representing my views fairly. It is incorrect, however, to say that I am opposed to ‘instructivist’ teaching and in favor of ‘facilitative’ teaching. On the contrary, there is no other way to learn information in a meaningful way nor is there any other way to be taught skills and given feedback. Our entire point – made very clear in the books and my blog – is that different goals require different approaches. You need to learn stuff from an expert? Get taught. But not all learning can be taught.

    Facilitative teaching is needed for meaning-making and inquiry-driven lessons. That’s why socratic seminar is what it is. It is not a ‘better’ way to ‘teach’ Romeo and Juliet; it is a better way to ensure that students learn to grapple with texts and ideas on their own. And I would think it would be terrible teaching, in any subject, if Seminar were the only method used because, again, it would violate what we know about achieving other goals.

    i think you also unhelpfully bring in the phrase “direct instruction”. I actually plan to blog on this unfortunate term. The technical meaning (as opposed to the informal meaning) is an entire cycle of different pedagogies and techniques used to ensure that the learning of stuff is effective. It includes formative assessments and Q & A as well as application. Hattie’s summary of DI is as good as any or you can just go to wikipedia.

    Thanks for pressing on this. And, PS, I don’t think Harry’s rejoinder to be is on point and I plan to respond soon as to why.

    • educationrealist

      Thanks for your response. I agree that “Direct Instruction” has both a technical and informal definition.

      What I was trying to do in that paragraph was reflect the difference between what you *say* and what someone who wasn’t a constructivist would infer from your writings. I agree that you say all methods are useful, that it depends on the situation. However, it’s hard to come away from your writing without getting a sense that you think teachers should be using more open-ended inquiry. I actually mentioned this in an earlier post–when I was in ed school, it was hard for me to realize that your book had any relevance to my teaching, because your examples and tone were so in favor of constructivism. That table I linked in demonstrated what most people would see as a clear preference for the student activities listed under constructivist methods.

      So I tried to cover both aspects–what you *say*, and what someone like me, or someone even more instructivist, would take away.

      • grantwiggins

        I think you’re missing the emphasis and the context of our book. Very few teachers, in my experience, are “constructivist” teachers (not to be confused with believing in the psychology of constructivism as how the mind works). Most teachers focus on getting kids to learn stuff, as if learning = acquisition of content. That’s how textbooks and almost all curricula are written, and that is how most teachers teach. What we say in UbD is that there needs to be greater attention to the “verbs” of understanding: apply, connect, explain, evaluate. Students have to get better at using content with those verbs because it is key to clear and effective thinking and later success. We do not say those are all that matter; we say they matter for anyone teaching for understanding, but are typically reduced to acquisition in the way teaching and testing unfolds.

        I have been in thousands of classrooms. I see mostly coverage, not mostly “discovery” learning. Nor do I think “discovery” learning works very well, as Hattie’s research shows.

        So, I have to say it: I don’t think you are representing the text and its context accurately. We’re addressing a real problem: content coverage that is superficial and not enduring, with little transfer – as reflected in countless assessment results and weaknesses in college – the same weaknesses that gave rise to standards and the Common Core over the last 20 years.

      • educationrealist

        I totally agree with you that most teachers are not constructivist. In fact, I say that probably 80% of teachers are instructivist and do traditional lecture. Which I am not a big fan of. The problem is, I don’t like constructivism, particularly reform math.

        A pretty major theme of my work is that kids don’t remember what they are taught–regardless of the approach.

        Anyway, I edited it to sound less dismissive of your work. Hope that helps, at least.

      • grantwiggins

        Well, you and I are totally on the same page: kids don’t learn enough of what is taught. Add the misconception literature and it’s a double whammy: even lots of teaching and accurate quizzes can hide serious misunderstandings. This is, to me, the challenge. I believe that we have not yet found the optimal pedagogies for most subjects, in part because we fail to understand how meaning is made and people learn and in part because we have holdover ideas about logical sequence in learning from way outmoded theories. It makes logical sense to the teacher to start geometry with axioms. It makes no sense to novices – and it contradicts both history and cognition. We care more and more about the validity of our axioms the more we try to prove “obvious” things like vertical angle theorems. So, the point is not “instructivism” or “constructivism” at all. It’s: buy what sequences and approaches to people really grasp both the key ideas and their meaning? (Hint: I would study non-Euclidean geometries much earlier than we do in school, e.g. taxicab and projective geometry. That then demands that we look more closely at our givens – just as happened historically.)

  • anonpls

    History! Here’s a growing beef I have from my time in Ed School – Mathematics history was mostly treated like an administrative hassle to check off. But as Mr. Wiggins sums up here, the history (i.e. reality) of how mathematics naturally progressed contrasts starkly with the standard (or typical) curriculum progression… or curriculum standards for that mater.

    As for constructivist vs instructivist vs. modified… I think, besides considering history when laying out content and content when deciding on pedagogy, we may need to focus on student backgrounds. Mr. Realist (if I remember correctly) laid this out clearly for me (somewhere, I think) basically along these lines:

    If a topic/idea is new… you can’t expect anyone to dance too far around the cognitive rigor matrix with it.

    So, the constructivist response to Wiggins:

    “How can content move from short-term to long-term memory if there is always more content to memorize tomorrow?”

    Is to… do a lot of constructivist stuff with new material rather than simply memorize and move on (am I reading this right?),when memorization is maybe a more realistic goal for new material? But its also boring. And piling on is self-defeating?

    So, another option would be to intersperse doing a bunch of constructivist/cognitively rigorous stuff on the material students started memorizing 5-10 years ago?

  • Tort

    I read a few paragraphs of “Everything about curriculum…” linked above.

    I coached for many years, and here’s how it works:

    We drill and kill. The game is the outcome of drill and kill, not the curriculum. Nothing about practice is “constructive.” The outcome is a thing of beauty, a creation, a masterful symphony…or it’s a disaster. The coach who does it through drill and repetition usually beats the other guy. If he can’t outcoach the inferior coach, he finds better players. When I played, the coaches screamed, “Why are we doing this drill?” We learned to respond, “To do the drill.”

    Things haven’t changed much as far as that goes.

    I imagine Copernicus was educated in much the same manner; he wasn’t ripped off by unproven, pie-in-the-sky theories. He most likely worked hard and was smart from the beginning. What he accomplished was the result of an education long before the Romantic Era, and long before Dewey and Kilpatrick–and most certainly, long before educational consulting promised fame and fortune. We fool ourselves into believing the next new curriculum will make more serfs into nobles.

    I’ve also been around long enough to see the promise of new curriculums come and go, mere fads that wind up in the ashes of defeat. I’ll say this: what teachers really need is not a new curriculum, but more leverage.

    Sorry, but I’ll stick with Hirsch, Jr.

    • grantwiggins

      Then you do not know much about high level coaching. What you describe is pretty pedestrian HS coaching, not high-level college or professional coaching in, say soccer, music, writing. Read Sven Nater on John Wooden. What you are describing is both a caricature of coaching. You may stick with Hirsch, but your whole position sounds pretty closed-minded to me. Cynicism is not wisdom.

  • Tort

    Opinions are not always wise, either.

    I’ve coached and my record of success speaks for itself. Prudence says, “Follow the money” in the game you are playing.

    Hirsch tells the truth. All others “tinker” and dance around it. If what you believe is true, you would not be offended by my post. Progressives have no evidence of success.

    As for reading, I suggest you consult the name on the Super Bowl Trophy and read about him. Still, Lombardi and Wooden were both successful, and both very different. The progressive agenda wants us all to be the same, and that is not an American ideal.

    • Tort

      “What you describe is pretty pedestrian HS coaching…”

      Precisely! Which is why we teach and coach the pedestrians this way. We wouldn’t want them run over by a bus, would we?

      Now, there is a difference between being “closed minded” and merely keeping things simple and cheap. If I am at all cynical, it is because I know that when students haven’t been taught the basics in K-6, they have no chance to succeed in high school math. Any assertion contrary to this basic, fundamental principle, is an illusion.

  • Teaching Math a Third Way | educationrealist

    […] I was reading Harry Webb’s advice to a new secondary teacher, describing his usual classroom procedure for “senior maths”, as an addendum to his earlier post on classroom management. And I thought hey, I could use this to fully demonstrate the difference in math instruction philosophies. […]

  • Teaching: My Retrospective | educationrealist

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  • Grant Wiggins | educationrealist

    […] a really terrific analysis of lectures that should be required reading for all teachers. (While I also liked Harry Webb’s rejoinder, I reread them in preparation for this essay and Grant’s is far superior.) He approved of […]

  • George

    This is an oldish post, but I thought I would reply to your question about whether the graph of the Vroom, Vroom experiment would be quadratic. The answer is sort of, but not really. This is because friction is complicated. Here’s a brief explanation.

    First idea: when you pull the car back, you’re storing potential energy in a spring, and the energy stored in a spring is proportional to the square of the distance.

    Second idea: when you release the car, the potential energy in the spring is converted to kinetic energy, with the conversion essentially complete at the starting line of the car. Kinetic energy is proportional to the square of velocity, so by the time the car reaches the original starting line, all the energy in the spring has been released, the velocity at that point will be proportional to the distance the car was pulled back.

    Third idea: after the car passes the original starting line at a velocity proportional to distance pulled back, it will coast along, slowing because of friction, until it slows and stops. The only forces acting on it are frictional ones. Calculating how far it will go requires solving a differential equation, which will require deciding how the frictional term behaves. In general, we would expect the magnitude of the friction to be a function of the current velocity of the car: F(v).

    If F(v) is a constant (i.e., it in fact doesn’t depend on velocity), you can solve the differential equation, and find that the distance traveled is proportional to the square of the initial velocity, and so proportional to the square of the distance pulled back, and the graph is a parabola.

    If F(v) is proportional to v, on the other hand, you can still solve the differential equation, but you’ll now find that the distance traveled is proportional to the initial velocity, and so proportional to the distance pulled back, and the graph is a line.

    A more realistic frictional term would probably be somewhere between these, and so the graph is likely to be approximately a power law (at least for reasonable pull-back distances), between linear and quadratic. Estimating the exponent would be doable from data; it’s a standard statistical regression problem. (The data would lie approximately on a line if you plotted it on a log-log scale, and the slope of the log-log graph would give you the exponent of the power law.)

    It’d be a fun experiment to run, though, with plenty of mathematical and statistical content which could be appropriate for students at lots of levels if handled properly.

    • educationrealist

      That’s such a coincidence; I was rereading this post and wondering about Vroom Vroom. So you’re saying if we did it a bunch of times at varying distances we’d get a predictable function, but…exponential?

  • George

    No, not exponential, but a power function: something of the form y = C x^p, for some constants C and p, where x is the distance pulled back, and y is the distance traveled. The exponent, p, is likely to be a number between 1 and 2. Maybe something like (to make up some numbers) y = 10 x^1.6

    You can estimate C and p from data, but looking at the data using a log-log scale. This is because, if we take logarithms of the equation, we get (using the laws of logarithms)

    log y = log C + p log x

    That is, log y is a linear function of log x. So if we plot the data on a graph where the axes use log scales, the data should lie approximately on a line, where the slope is p, and the intercept is log C.

  • George

    My son and I did this experiment yesterday, twice. We got a pretty nice power law graph. When my son did it, he got an exponent of 1.7, and when I did it, I got 1.6.

    For our experiment, we pulled the car back 1 inch, 2 inches, 3 inches, etc. all the way up to 12 inches.

    My son’s distances traveled (in inches): 1, 7, 12, 18, 25, 36, 48, 57, 60, 68, 85, 82.

    My distances traveled: 1, 11, 14, 15, 26, 35, 44, 51, 68, 64, 75, 85.

    One issue to note: the data is fairly noisy. For each of us, we even had one interval where the data wasn’t increasing. If you’re going to use this experiment, be prepared to deal with this in discussion.

    The data looks convincingly close to linear on a log-log graph. So that was nice. Also, the measurement errors are such that small distances have very large relative errors. You may want to start, not at 1 inch, but at 3 or 4 inches.

    • educationrealist

      I’m thinking to get anything real out of it, maybe a pre-calc class? Unless all you want is the data collection.

      • George

        Yeah, or a good statistics class. Real data is always messy. Although it might be a fun way just to generate some numbers to put on a graph for an early graphing lesson. I find that many students tend to do better with numbers for graphing if they have a concrete meaning attached to them.

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