A Response To Critics

Let me wrap up this summer’s series by offering some time at the microphone to two groups of critics.

You can’t be in the business of creating headaches and offering the aspirin.

That’s a conflict of interest and a moral hazard, claims Maya Quinn, one of the most interesting commenters to stop by my blog this summer. You can choose one or the other but choosing both seems a bit like a fireman starting fires just to give the fire department something to do.

But “creating headaches” was perhaps always a misnomer because the headaches exist whether or not we create them. New mathematical techniques were developed to resolve the limitations of old ones. Putting students in the way of those limitations, even briefly, results in those headaches. The teacher’s job isn’t to create the headaches, exactly, but to make sure students don’t miss them.

To briefly review, those headaches serve two purposes.

One, they satisfy cognitive psychologist Daniel Willingham’s observation that interesting lessons are often organized around conflict, specifically conflicts that are central to the discipline itself. (Harel identified those conflicts as needs for certainty, causality, computation, communication, and connection.)

Two, by tying our lessons to those five headaches we create several strong schemas for new learning. For example, many skills of secondary math were developed for the sake of efficiency in computation and communication. That is a theme that can be emphasized and strengthened by repeatedly putting students in a position to experience inefficiency, however briefly. If we instead begin every day by simply stating the new skill we intend to teach students, we will create lots and lots of weak schemas.

So creating these headaches is both useful for motivation and useful for learning.

Which brings me to my other critics.

This One Weird Trick To Motivate All Of Your Students That THEY Don’t Want You To Know About

There is a particular crowd on the internet who think the problem of motivation is overblown and my solutions are incorrect.

Some of them would like to dismiss concerns of motivation altogether. They are visibly and oddly celebratory when PISA revealed that students in many high-performing countries don’t look forward to their math lessons. They hypothesize that learning and motivation trade against each other, that we can choose one or the other but not both. Others even suggest that motivation accelerates inequity. They argue that we shouldn’t motivate students because their professors in college won’t be motivating.

I don’t doubt their sincerity. I believe they sincerely see motivation as a slippery slope to confusing group projects in which students spend too much time learning too much about birdhouses and not enough about the math behind the birdhouses. I share those concerns. Motivation, interest, and curiosity may assist learning but they don’t cause it. In the name of motivation, we have seen some of the worst innovations in education. (Though also some of the best.)

But there are also those who do care about motivation. They just think my solutions are overcomplicated and wrong. They have a competing theory that I don’t understand at all: just get students good at math. It’s that easy, they say, and anybody who tells you it’s any harder is selling something.

Success in a skill is self-motivating.”

Many forget that there’s intrinsic motivation to simply perform well in a subject.”

And, yeah, I’m sorry, friends, but I do have a hard time accepting such a simple premise. And I’m not alone. 62% of our nation’s Algebra teachers told the National Mathematics Advisory Panel that their biggest problem was “working with unmotivated students.”

I see two possibilities here. Either the majority of the nation’s Algebra teachers have never considered the option of simply speaking clearly about mathematics and assigning spiraled practice sets, or they’ve tried that pedagogy (perhaps even twice!) and they and their students have found it wanting.

Tell me that first possibility isn’t as crazy as it sounds to me. Tell me there’s another possibility I’m missing. If you can’t, I think we’re dealing with a failure of empathy.

I mean imagine it.

Imagine that an alien culture scrambles your brain and abducts you. You wake from your stupor and you’re sitting in a room where the aliens introduce you to their cryptic alphabet and symbology. They tell you the names they have for those symbols and show you lots of different ways to manipulate those symbols and how several symbols can be written more compactly as a single symbol. They ask you questions about all of this and you’re lousy at their manipulations at first but they give you feedback and you eventually understand those symbols and their basic manipulations. You’re competent!

I agree that in this situation competence is preferable to incompetence but how is competence preferable to not being abducted in the first place?

If that exercise in empathy strikes you as nonsensical or irrelevant then I don’t think you’ve spent enough time with students who have failed math repeatedly and are still required to take it. If you have put in that time and still disagree, then at least we’ve identified the bedrock of our disagreement.

But just imagine how well these competing theories of motivation would hold up if math were an elective. Imagine what would happen if every student everywhere could suddenly opt out of their math education. If your theory of motivation suddenly starts to shrink and pale in your imagination, then you were never really thinking about motivation at all. You were thinking about coercion.

Previously

About 

I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

31 Comments

  1. For me, the bugaboo is “Fun”, or “This isn’t fun”, or “They need to be more engaged.”

    Passing on the joke that if a few of them ARE engaged, and she IS pregnant …

    I often tell my students that “Math isn’t FUN.” (shock and whispers). “It’s satisfying.”

    “It’s a puzzle, it’s a problem to be figured out. The result is that warm glow of accomplishment at a solution, of taking something you didn’t know and learning it, of taking something complex and making it simple and clear, of finding the ‘A-Ha!’ moment. ”

    “Learning is difficult. If it were easy, you’d get nowhere; you’d remain at the level you were when you were playing with Tonka trucks in the sandbox behind the elementary school. You have to reprogram your brain, make new pathways, learn how the world works and what fascinates you, turn the 12 year old Frozen fan into an 18 year old adult ready for work or college.”

  2. Couldn’t agree more Dan! The only critics I had to this post (http://leahkstewart.com/examseason/) were talking about the importance of exams and everyone must have X knowledge by age 16. My response; Who says that if everything was optional students wouldn’t choose it? Intellectual and personal development (towards the good individuals care about creating) is something humans seek out. I like to think we’re all here to help each other to that end. If someone stands up as a teacher by offering something they know will help others and no one is picking it up, that’s a call for self development, not to make your thing compulsory in order to have a full classroom. No teacher outside of the school system (and there are many) can do that or would want to do that because it would rob them of the joy of working only with the people who need their lessons the most.

  3. Quip I came across recently, which sounds a lot like the self-motivating crowd:

    “The focus of K-8 instruction should be to develop the basic math skills needed for high school and college. It is good to understand concepts, but the goal should be rapid and accurate computation,” the report said. “There are simply too many unidentified prerequisite skills to be covered in the allotted class time. The obvious ripple effect is less critical thinking and a weaker command of the basic skills needed to succeed.”

    http://www.charlotteobserver.com/news/state/north-carolina/article31302599.html#storylink=cpy

  4. I guess maybe some people actually don’t believe motivation matters. And, sure, there are math war type zealots with unrealistic ideas. But, addressing this crowd seems like a bit of a deflection. The best response to critics is to produce the goods.

    Creating a conflict or headache aspirin is difficult to do well. So is “speaking clearly about mathematics” and assigning appropriate “spiraled practice sets”. Your examples are not particularly compelling for headache/aspirin nor are the usual textbook, purple math, etc. particularly compelling for “speaking clearly…”.

  5. This is definitely an interesting and provocative post. I like curmudgeon’s quote that math isn’t fun, it’s satisfying, and I also love the Alien analogy. I know we math teachers like to say that math is the language of the universe or truth but to many high schoolers its an arbitrary set of rules and symbols that adults force them to learn. I agree that this idea of motivation needs to be at the forefront of our thoughts as teachers.

    And I’m also in agreement with l hodge that doing it well is very difficult. Personally, I’ve found that the surest way to create a headache with a particular skill is to embed it in a larger type of a puzzle. In my opinion, algebrafunsheets.com creates lots of great headaches simply by turning skill practice into a joke or a riddle. For my students who struggle, I’ve found that creating worksheets similar to this to be the most effective way to motivate the students who hate math. I’m wondering are some headaches better than others?

  6. I was reading a book by Eric Jensen about cognitive neuroscience and it said that when emotions are tied to an event, that the memory is stored in the episodic memory. The episodic memories are much stronger and easier to recall. It makes sense to me that using motivation while learning would increase student’s retention, let alone their willingness to participate.

  7. Michelle Przybylek

    August 20, 2015 - 2:51 pm -

    This summer I taught math at camp. I had small classes of 8 students. I was able to realize that some “unmotivated” students left problems and homework completely blank because they literally had no point of entry into the problem. They couldn’t use the Pythagorean Theorem to find sides of a triangle because they couldn’t identify which side was the hypotenuse (or didn’t know what a hypotenuse is). Once I could shed some light on the math for them, they were able to start at least attempting the problem. I don’t think their motivation changed but in a bigger class, a problem with content could easily be misidentified as an unwillingness to try.

  8. I kinda think that there are two components for classroom motivation: the motivation of the students and the motivation behind the concept or idea that the lesson is structured around.

    I think what Dan’s trying to say is that if we can show rather than tell (borrowing from our English colleagues) the motivation for the lesson we can add a little bit of motivation for the students.

    Obviously a teacher needs to have the skills to foster motivation and what not in the classroom in an interpersonal sense, but to add the extra structure of making sure the lessons are “motivated” properly you’re going to get even more mileage. And if you can make that motivation genuinely mathematical and not boring, even better.

    I think what was great about these “Headache” pieces that you did was that you showed we can do that type of thing without a bunch of extra bells and whistles, and that there’s not much math content that you can’t do it with, even stuff that seems really dry.

  9. I think you’re undervaluing competence as a motivator. To pick another example, consider weightlifting. Is the “headache” being useless when you show up to help move furniture?[1] Not conforming to the Hollywood/Marvel body image? Lacking a general level of fitness without supportive body strength?

    All are reasonable motivators, but if you look at the most active participants, where the proposed headaches are mere dots in the rearview mirror, you’ll find competence itself is the biggest driving force.

    There’s certainly a gap between the neophytes and the competitors. Of I think back to high school–where I was competitively driven in math and avoided lifting a single weight[2]–I believe that glimpsing a picture of my own competence in weightlifting is the one thing that may have inspired me to do it. Finally this did happen after high school, when I was back in town and saw my same weightlifting-avoidant friends start lifting, and hitting bigger and bigger sets. For no discernible reason other than the challenge of it.

    1: I failed to find a gif or even accurate quote from a Community episode where Buzz Hickey makes fun of Jeff for spending all that time in the gym but reluctant to help move furniture.

    2: Weightlifting was an elective, so it had the advantage of a clean slate reputation when I came into on good terms.

  10. I feel like the aspirin-headache setup was great for the first case or two. Then, the headaches were too close to pure math to have the same effect for me.

    The thought I always circle back to is “what about an innate desire to solve problems?” I had it in school and have it today, and now I’m a non-math teacher who reads math teacher blogs at lunch and works out problems on scratch paper because my brain demands I resolve them.

    And I believe that most very young children have a natural sense of inquisitiveness, and that people in general are pleased — on a deep, core, animal brain level — by resolution of problems and conflicts.

    …so many people seem to not have that. Aspirin-headache feels like a great real-world tool, like an interim crutch that helps the kids without the “innate desire” tune in for today’s math lesson. But it feels like there’s a bigger picture that we’re just on the cusp of breaking through to.

    That innate desire…where does it go between birth and adulthood? Can we foster it, or bring it back? Would our entire schools have to adjust their approach? Would we be better for it? Am I on a wild goose chase that’s too big and complicated to actually ever be solved? We’re so, so close, I think, maybe, or not.

  11. Dan,
    You didn’t need to make up (strawman anyone) what other people say. You could have used this forum for a discussion or provided actual quotes from twitter or comments.

    That you chose instead to imagine what others think says a lot about your approach to debating the merits of your views.

  12. Chester Draws

    August 21, 2015 - 6:11 pm -

    Of the nay-sayers, some aren’t active teachers. I’m suspicious of those that talk about what motivates kids to do things, but don’t have to face the consequences of their theories. I was taught some very useful stuff at Teachers College, and some utter tripe too.

    A couple of others have (deliberately?) confused boredom and motivation. In terms of Maths teaching the two concepts are only slightly related.

    Student perceptions are also a dangerous topic in terms of how much motivation there is. I know students who think they are motivated to learn — they aren’t bored in class and they do some work — but for me as their teacher l don’t think they have the motivation required. They don’t listen carefully, they don’t do their homework properly and, most importantly, they don’t really commit to doing things properly.

  13. Curmudgeon:

    Learning is difficult.

    Difficult, but satisfying. Difficult, but purposeful. Difficult, but interesting.

    None of these are contradictions in terms. Learning is difficult, but that doesn’t mean it has to feel dull or arbitrary. Your emphasis on “satisfaction” works for me just fine.

    l hodge:

    The best response to critics is to produce the goods.

    These critics think the goods are unnecessary, no matter how good they are. That’s my point.

    Ryan:

    I think you’re undervaluing competence as a motivator. To pick another example, consider weightlifting. Is the “headache” being useless when you show up to help move furniture?

    The headache is merely a means to establishing purpose, or a “need” in Harel’s formulation. The “need” for exercise – weightlifting, cardio, whatever – seems pretty well established, especially relative to simplifying rational expressions.

    Stan:

    You didn’t need to make up (strawman anyone) what other people say. You could have used this forum for a discussion or provided actual quotes from twitter or comments.

    Stan, I encourage you to upgrade your browser as soon as possible, until it reveals the hyperlinks I provided in the post to actual quotes from Twitter.

  14. Dan,
    My browser is just fine. But after quoting people instead of engaging them to fill in any blanks in what they mean you head of with an imaginary scenario to paint their view as ridiculous. That is a classic straw-man argument.

    Take the quote “Success in skill is self motivating”. Do you disagree with that statement? Is it true but ignoring that even more motivation is possible or required? Perhaps you agree with it but believe initial success requires some motivation from a different source.

    My point is if you are going to engage with someone who disagrees with you and quote them, either respond to the quote with your argument or with questions for the author of the quote.

    If you don’t their response can be simply that is not what I meant and everything in your piece after “… imagine it” is irrelevant to the discussion.

    A worthwhile piece of writing on this would address the strongest argument posed against your position and not waste time on the weakest or imaginary scenarios.

    If this issue is important please take the time to engage with the best of those who don’t agree with you. For those of us reading both sides it would be far more useful and interesting than just watching echo chambers producing words.

    You could start by clearing up my questions on your opinion on the quote “Success in skill is self motivating”.

    Regards,
    Stan

  15. Thank you for the “time at the microphone.” (And the Harel C framework, which I quite enjoyed.) I will perhaps try and use this time just to mention two items:

    1. The post that you linked to is not actually a criticism of the headache/aspirin setup. Its predecessor-post served this purpose more directly:

    https://mathwater.wordpress.com/2015/07/03/situating-need-in-the-world-vertical-integration-with-headaches-aspirin-ha/

    The link that you have provided is to a couple of excerpts from the writing of Neil Postman, as clarified in the comments, along with the remark that:

    “I do not mean to suggest that I agree with the writing [of Postman]; I only deposit the excerpts as a pre-existing example of the sickness-reliever / mathematics educator analogy that you can find linked to in more recent math-ed weblogs.”

    2. I cannot say that I share in the criticisms espoused by the latter group(s) described: specifically, those believing motivation is “overblown.” I believe that motivation is crucial, and that teachers who describe students as “unmotivated” may (knowingly or not) mean: unmotivated to do the given [math] assignments. (Are these hypothetical teachers motivated to identify why their assignments are not responded to positively? How can teachers carry out such an investigation?)

    As to whether the gamification approach (turning mathematical learning into puzzle-playing, as some others suggest) is fruitful: I would add that, even if achieved, the recreational, satisfactory component of engaging with puzzles must be tied back to the empowering sense of agency that comes with a better understanding of mathematics (and the coeval better understanding of their world).

    MQ

  16. Dan,
    In answer to your point about what might work with students who have repeatedly failed at math I ask have you examined the work of John Mighton at Jumpmath.org?

    Mighton addresses exactly the category of student you describe and claims to have lots of success both at students performance in math and motivation for further study.

    Looking at his 2003 book The Myth of Ability on page 27 of my paperback edition he describes an anecdote that is exactly the sort of issue you want to solve. A student won’t engage because they believe they can’t do division. His solution is not an attempt to motivate a justification for learning. Instead he teaches math as a “mechanical procedure” (p31) that result in the student overcoming the barrier to learning.

    The book describes many similar anecdotes and the jumpmath.org site describes several studies of the Jump method that show positive results over the years since this book was published.

    Mighton offers some on ideas why the seemingly obvious approach is not used universally. For one he doesn’t claim it is the only or best approach just that it works reliably. But he also points out that it is not always obvious to someone comfortable with math what barriers a struggling student faces and how much the steps need to be broken down for the student to be able to cope.

    Mighton is not claiming motivation is unimportant. He clearly wants motivated students. His claim is that you can use a direct instruction like teaching method to get competency and motivation will follow. He also does not claim that students who are not struggling won’t sometimes benefit from the sort of approach you endorse.

    This is what your worthwhile critics are saying too.

  17. Reasonable folks can and did argue that you did not produce the goods with the factoring lesson. The rational expression lesson is basically the same concept. Headache/aspirin doesn’t seem to be readily applicable with these prototypical “skill” lessons.

  18. Stan Blakey:

    But after quoting people instead of engaging them to fill in any blanks in what they mean you head of with an imaginary scenario to paint their view as ridiculous. That is a classic straw-man argument.

    I am not imagining those tweets. Their meaning is plain and my response is direct. They’re welcome to correct the record, but none has.

    This is the second time you’ve applied this standard of discourse to me, and, interestingly, I’ve never seen you apply it to your sympathizers.

    A student won’t engage because they believe they can’t do division. His solution is not an attempt to motivate a justification for learning. Instead he teaches math as a “mechanical procedure” (p31) that result in the student overcoming the barrier to learning.

    This is an identical argument as the one made by the people I quote in the body of my post and my response is identical. (Start from “I mean imagine it.” above.)

  19. I thought the series was thought-provoking and enjoyed reading it. You picked excellent examples to prove the point you wanted to make (a point I disagree with, but still).

    Math isn’t aspirin. You simply can’t convince all kids that they have a headache.

    The trick, to me, is to get kids to think about math and engage with problems even if they don’t care. To do more than working procedures endlessly without thought–or worse, nothing at all. That’s a significant difference in aim, because teachers don’t have to rely on the students to achieve their goals.

  20. Dan,
    You are mistaken that others are not responding. Greg has his own blog and a post that responds to your writing in this blog post https://gregashman.wordpress.com/2015/08/21/wheres-your-evidence-dan/

    It also seems you are mistaken in your quote – I couldn’t find anywhere in the link where the words you ascribe to him appear. Am I missing it or are you mistaken there?

    Why apply this “standard of discourse” to you and not to Greg? I found John Mighton’s writing compelling when I read it ten years ago and JumpMath has continued to grow in popularity and publish reports of its success. If Mighton is missing something I won’t discover it by a dialogue in an echo chamber of people saying basically the same thing as him.

    Engaging with someone who disagrees with him is a more effective way to discover if there is something else to learn.

    So I continue.

    I think you should be concerned that an imaginary anecdote is not a good counter to actual anecdotes and studies of actual people. But you are apparently not worried by that so I’ll address the problem with your metaphor.

    I’ll agree, for the sake of argument, that forcing a child for several hours a day to be away from their parents and home and in a class under the authority of a strange set of adults is the same as being abducted by aliens.

    Now I have never heard you suggest that we should not do that. That parents should wait until their child has sufficient motivation to want to place themselves in a classroom for a day and that at any time the child feels unmotivated they should stop going to school until someone motivates them again.

    So none of us it seems have a problem with the temporary abduction part of your metaphor.

    It is the scrambling their brains part that is problematic. Direct instruction and John Mighton’s methods exactly avoid doing that. Mighton repeats over and over that small steps that are understood by the student are required and if things are not working then even smaller steps must be mastered first.

    You seem to be saying through your metaphor that any motivation that is not based on intrinsic curiosity is either wrong or inferior to a teacher inspired intrinsic curiosity. But Mighton is just taking the coercion that got the reluctant student sitting in front of the teacher one step further and using the charm and authority of the teacher to get the student to try something. A goal is for the student to learn to experience the intrinsic motivation that comes from being capable and he doesn’t mind using extrinsic means such as his charm and authority to kick start the process.

    I am using Mighton here rather than Greg or Robert as I think they basically agree with him and Mighton’s ideas are very well documented and explained both in his books and the jumpmath.org site.

    You seem to fundamentally disagree with what Mighton does to get things started with an unmotivated student. Correct me if I am wrong here.

    I wonder how do you get unmotivated students to appear in front of a teacher and attend to a particular headache aspirin scenario?

  21. Having taken many, many, math classes at various points in my career and early education, I have to say that the single greatest pain I’ve felt with math has been:
    Math Teachers.

    My firm belief is that the extensive study of math literally reconfigures your brains so that you are no longer capable of appreciating anybody else’ solutions, methods or modes of thought, and are, coincidentally, rendered incapable of clearly expressing your own.

    Nearly every math professor I’ve had was impatient, arrogant, demeaning and condescending… Right up to the point that they graded my tests.
    I have had no such problems with ANY of my other professors, and I have noticed PRECISELY the same behaviours from my own childrens’ teachers.

    I see similar qualities in the participants of the MTT2k series of videos. Indeed, the videos demonstrate EXACTLY what I’ve learned to loathe about stepping into a mathematics class of any sort. Namely, there is a strange propensity to consider methods that are not yours to be “wrong”, regardless of the quality of results.

    I love the subject. I can not STAND the “professionals” who teach it.

    Note: Below this box into which I am typing is a series of checkboxes. The first two say the same thing. Why are there two of them?

  22. Hear what you’re saying Brian; school gave me the maths prize and an idea of maths that I’ve since found to be only a shadow of the real thing. Fortunately I stumbled upon this book in the town library after graduation “The Man Who Loved Only Numbers” on Paul Erdos and this MOOC https://www.coursera.org/course/maththink by Keith Devlin. They got me back on track. We can’t (and I wouldn’t want to) prevent anyone from teaching what they feel called to teach. All we can do is teach our own thing, and do it so well that people tell other people about it… like the author of that book, and like Mr Devlin.

  23. Chester Draws

    August 24, 2015 - 1:33 am -

    Well I’d just like to point out that not everyone’s Maths teachers were even remotely like Brian’s.

    For a start a non-patient Maths teacher is going to die early of stress. You are daily given to watch people do slowly and badly what you find easy. The very first lesson I learned about being a good teacher is the ability to wait for someone slow do something at their own pace.

    I’ve also as a teacher learned many different ways of doing the same problem. (As a trick to keep me occupied while students do hard calculus problems, I often solve the same problems the easy way without calculus.) If anything it is students who want to focus on the “one right way”. It’s the teachers who try to get the students to think graphically, numerically and algebraically to solve the same problem.

    It’s true that I stop students doing things by a method that isn’t correct, even though the student thinks it is acceptable (most often “solving” by arithmetic methods). I also shun complicated techniques that are difficult to learn and easy to get wrong, which can annoy the more formula focused students. But a correct method that works is always OK.

    So, actually, I call BS on Brian. It seems much more like a personal animus than an actual statement of truth. I’ve known people say how all PE teachers were bullies and idiots, and that’s not true either.

  24. First, thank you for engaging in some tough questions. Idealism is a difficult topic to tackle while maintaining positivity; haters love to sweep the legs of idealists (see comments above).
    It’s the difference between, “Wouldn’t it be great if…?” and “That’d never work in my classroom!”

    Second, a blog lends itself well to discuss one topic at a time, so it makes sense that some readers may feel disjointed. Motivation–like every other challenge to learning–requires a variety of strategies, none of which work unilaterally for every teacher nor every student. This discussion is an edge-piece of a large puzzle.

    Having heard you speak, I’m familiar with the urgent tone you use for a room full of math teachers, and I hear it in your writing, too. It’s good.

  25. I have a critique from a different direction than the two you specify. I think a major reason students lose motivation is that we teach them procedures that seem arbitrary. Your focus recently has been how to get kids to realize the need for what they’re about to learn, which is great (thank you for exposing me to Harel). But then you seem basically fine with just lecture & notes, once the kids are hooked.

    I know I could use better hooks, so again, I appreciate this summer series, but I’m mostly concerned that students (especially those with big gaps in prior knowledge) can only understand the conclusions of the lecture that follows but don’t really understand the concepts leading to those conclusions. Like a student who never realizes that a graph shows the locus of points that are solutions to an equation, but remembers the procedure for graphing from y=mx+b. In the last couple of years, your work seems not to have addressed this issue as much as I would have liked…or when it did, I felt it overloaded kids’ brains in a way I fear could lead them to the correct conclusion without the time, practice, and discussion needed to consolidate that conclusion and the *reasons* for it in their long-term memories. So although you know I think cognitive load theory is a good guide for instruction, I’m actually criticizing your recent work for being too much like traditional instruction, not for being too crazy or creative.

    How do you get kids to understand that in 2x/5x, you can simplify the x’s, but in (2+x)/(5x), you can’t? And how do you structure a lesson that addresses that kind of thing without being dreadfully boring? Your recent approach would help me find a flashy hook to show kids they need to know when you can and can’t simplify the x, but how do you get them to stay tuned to a long debate about what it really means to simplify a fraction, especially when they have already given up on math meaning anything?

    Finally, I just think you misinterpret cognitive load theory as saying that we should only show kids worked examples and drill problems. There’s no reason you couldn’t do a Schwartz/Harel-inspired hook and then use worked examples in the lecture & notes that follow. And the worked examples could demonstrate both correct examples and common mistakes, and could ask students to explain the mistakes conceptually. All of that would be consistent with cognitive load theory, and as far as I can tell, would also be consistent with you like to do. The one thing cognitive load theory tells you not to do, which has been a big revelation for me, is to lead kids through a sequence of discoveries without giving them time to consolidate each discovery in long-term memory before moving on to the next one. That’s what I was criticizing here and then here.

    It turns out that the process of encoding something in long-term memory actually consumes working memory. So if a student makes a discovery (“Aha!”) and it’s bouncing around in their working memory, they can lose it the next minute unless we give their working memory a break from thinking about new things and give it a task that reprocesses that thought a few times until it gets encoded in long-term storage. It’s simply not true, despite what many teachers think, that we tend to remember the things we discovered for ourselves. Instead, as Daniel Willingham says, memory is the residue of thought, so we tend to remember whatever we spent the most time thinking about. If we spent most of our time thinking something wrong, and only a little bit of time realizing that it was a mistake, we might remember the misconception more strongly that the fact that it’s not right! So we have to be really careful that the moment of correct discovery is followed by a decrease in cognitive load, with appropriate repetition and rehearsal, so that the discovery actually gets remembered. Is that really objectionable?

  26. Kevin:

    I’m actually criticizing your recent work for being too much like traditional instruction, not for being too crazy or creative.

    Explicit instruction is the world’s easiest pedagogical act for the majority of teachers, I’m convinced. So when explicit instruction isn’t incompatible with a technique I’m describing, I try not to foreclose it, or foreclose the teachers who find it easy.

    I don’t love the idea that a teacher would just yammer instructions at her students in tight little IRE loops. But just because I believe there are better ways to help students learn, I don’t want to send her away from this series emptyhanded.

    Finally, I just think you misinterpret cognitive load theory as saying that we should only show kids worked examples and drill problems.

    I use the terms “speaking clearly about mathematics and assigning spiraled practice sets” as a level-best attempt at being fair to my critics. These are their words – and they adhere tightly to CLT – and one of them is a grad student of Sweller.

    So we have to be really careful that the moment of correct discovery is followed by a decrease in cognitive load, with appropriate repetition and rehearsal, so that the discovery actually gets remembered. Is that really objectionable?

    Definitely not. I find CLT very useful. It shares an interesting parallel with constructivism, though, in that its most extreme adherents extract from a useful theory of learning a useless theory of teaching.

  27. Okay, that makes sense. I think we’re on the same page on the big issues, though we’d probably diverge again somewhat if we were to consider a specific lesson plan.

  28. Huh, my most motivating teachers in college were also the ones who got the teaching awards. And the ones where I remember the most now from the classes after years of not using it.

  29. Sometimes I think that much of what teachers think is good for students is clouded by “_______ worked for me and look how good I turned out. So it should work for my students” Not recognizing that most of us who turned out to be math teachers were already either self motivated or teacher pleasers or parent pleasers and no matter what method we were exposed to, we likely would have succeeded.

    And for the most part that probably could have worked with no pushback up until maybe 15 years ago. But the fact is that now students have more available to them in the way of information, entertainment, and savviness, that we cannot give them what they have always got and expect them to just take it.

  30. This is a great topic. I struggle with it for my students and my own children. The default answer to the “motivation doesn’t matter” position is really “the motivation is because you have to.”

    I agree with the comment that math is satisfying but most often it is satisfying for math teachers and this goes to the empathy position. Students often don’t feel the lack of resolution that we feel that compels us to keep thinking about the problem. When we can get them to care about the answer in a way that leaves them unsatisfied until they solve it, then we have really addressed the portion of resilience and persistence that will lead them to hate the blank response.