Math isn’t Aspirin. Neither is Teaching.

First, congrats to Dan Meyer, who finished his doctorate at Stanford and just hired on as CAO for Desmos, a tremendously useful online graphing calculator. He persisted in the face of threatened failure, and didn’t give up even when he had an easy out into a great job. (Presumably Dan and most of the Math Twitter Blogosphere are still annoyed at my jeremiad about the meaning of his meteoric rise, in which Dan played the part of illustration.)

Dan has asked math teachers for ways to create “headaches” for which math can be considered aspirin:

dmeyeraspirin

And this interested me because the request completely, perfectly, captures the difference between our two philosophies, which I also wrote about a couple years ago:

meyervsme

The comparison is an instructive one, I think. Both of us find it necessary to build our own curriculum, rejecting the one on offer, and both of us, I think, tremendously enjoy the creation process. Both of us reject the typical didactic contract described by Guy Brouseau, setting expectations very different from those of typical math teachers: explain, work a few examples, assign a set. Both of us largely eschew textbooks for instruction, although I consider them completely unnecessary save as reference books that often provide interesting problems I can steal, while Dan dreams of the perfect digital textbook.

And yet we couldn’t differ more in both teaching philosophy and curriculum approach.

Dan’s still selling curiosity and desire for knowledge, assuming capability will follow. I’m still selling capability because I see confidence follow.

Dan still believes that student engagement captures their curiosity which leads to academic success, that the Holy Grail of academic success in math lies in finding the perfect problems that universally stimulate interest in finding answers, which leads to understanding for all. I hold that student engagement leads to their willingness to attempt what they previously thought was impossible but that the Holy Grail doesn’t exist.

Meyer thinks teachers skeptical of his methods are resistant to change and the best interests of their students. I advise teachers and recommend curriculum; if they find my advice helpful, great. I encourage them to modify or even reject my advice, to continue to see an approach that works for them and their students.

Dan wants to be “less helpful”. I want to teach kids to use their own resources, but given a kid who wants to give up, I’m offering help every time.

Meyer’s methods would probably need tremendous readjustment if he worked in a low-income school with a wide range of abilities. I’d probably be much “less helpful” if I taught at a school with a high-achieving, homogenous population obsessed about grades.

Meyer rose quickly in the rarefied world of rock star teachers. I aspire to the role of and indie with cult status.

Dan’s query: “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?

My answer: You can’t.

The commenters, mostly teachers, took the question seriously, understanding that it was another way of looking at the students’ demand, “When will we use this?”. Answering this question clearly troubles most of the commenters—or they have an affirmative answer they’re satisfied with.

My answer to the student demand: “Probably never. But the more willing you are to take on challenging tasks you learn from, the more opportunities you’ll have in life, both professional and personal. Call me crazy, but I see this as a good thing.”

Dan Meyer is wrong, I believe, in looking for the Holy Grail that makes math “aspirin”1. But that’s not the point of my running through the Dan vs. Ed showdown.

Instead, consider the comparison yet another data point in my slowly developing thesis that ed schools need more flexibility and even less prescription. Few people understand the vast scale of values, philosophies, management and curriculum found in the teaching population.

Two teachers developing uncommon curriculum who agree on very little—yet both of us are considered successful teachers. (one has much more success selling his ideas to people with money, I grant you.) Take ten more math teachers likewise who build their own curriculum, have their own takes on philosophy, discipline, and even grading and they’re unlikely to change to suit another model. Take 100 more–ditto. Voila! an expanding population of teachers who have successful teaching approaches and curriculum design that they’ve developed and modified. None of them are going to agree on much. They have come to widely varying conclusions that they will continue to develop and enhance on their timeline as they see fit. No one will have anything approaching a convincing argument that could possibly convince them otherwise.

The point: the current push to “fix” ed schools, a fond delusion of reformers, progressives and union leaders alike. People as diverse as Benjamin Riley, Paul Bruno, Rick Hess and others believe we can find (or already have) a teaching knowledge base that can be passed on to novices.

Teachers are never going to agree.

Agreement or even consensus is impossible. Teachers and students form infinite combinations of interests, values, incentives and unlike reformers, teachers are going to value their experience and unique circumstances over anything ed schools tried to pretend was the only way.

Teaching, like math, isn’t aspirin. It’s not medicine. It’s not a cure. It is an art enhanced by skills appropriate to the situation and medium, that will achieve all outcomes including success and failure based on complex interactions between the teachers and their audience. Treat it as a medicine, mandate a particular course of treatment, and hundreds of thousands of teachers will simply refuse to comply because it won’t cure the challenges and opportunities they face.

So when the status quo has prevailed for the next 30 years, don’t say you weren’t warned.
****************************************************************************

1which isn’t to say I don’t plan on writing up the how and why of my quadratic equations section.

Advertisements

About educationrealist


47 responses to “Math isn’t Aspirin. Neither is Teaching.

  • benespen

    “Agreement or even consensus is impossible. Teachers and students form infinite combinations of interests, values, incentives and unlike reformers, teachers are going to value their experience and unique circumstances over anything ed schools tried to pretend was the only way.”

    Thank goodness for the contrariness of ordinary people.

  • Jim

    Based on Dan’s remarks the main problem in math education seems to be a lack of student motivation. How different is math in this regards from other subjects? What subjects seem to generate the greatest student interest and engagement?

    • Roger Sweeny

      I am tempted to say, “Most high school students don’t have much intrinsic interest in any course except sex ed.”

      Oh, what the hell. It’s true.

    • Roger Sweeny

      What subjects seem to generate the greatest student interest and engagement?

      That’s a damn good question. As far as I know, there’s no good research on it. Does anybody know of any?

      (Tyler Cowen’s Second Law says, “There is a literature on everything.” Kevin Drum doesn’t quite agree, “There’s literature on a lot of things, but with some surprising gaps. Furthermore, in many cases the literature is so contradictory and ambiguous as to be almost useless in practical terms.”

      Perhaps determining “[w]hat subjects seem to generate the greatest student interest and engagement” seems too much like market research. Educators require students to take subjects not because the students want to but because the educators think it is a good thing. Though, of course, once kids have been put into a class, it is important to know how to engage and interest them.)

      In any case, as a science teacher, I used to hear a lot of hopeful, “Mr Sweeny, are we doing a lab today?” However, I suspect that had less to do with a love of lab technique and more a desire to stand and do something physical and interact with peers.

      • educationrealist

        Grant Wiggins did a student survey including A students and gender pulled out. But he left off race, saying the school was “mostly white”. That surprises me, based on the responses. In any event, I’d want a much broader survey, with breakdowns by race, GPA, and student SES.

        But they are interesting to think about.

        Student Survey Part 1

        Student Survey Part 2

      • Roger Sweeny

        Thanks for the cites to the Wiggins survey. The fact that PE always comes out as the favorite subject suggests I am not totally off-base about why students look forward to science labs.

        I have not read Wiggins blog before so perhaps this is unfair but I read his statement, “it’s high time we better understood the problem of the high school. (Hard to believe that after 30 years of reform that started with me working with Ted Sizer in the Coalition of Essential Schools, we still lack clear answers” and thought it was like saying, “We’ve been trying to make a perpetual motion machine for 30 years and it’s hard to believe we still don’t know how to do it.”

        We try to do the impossible, to fill young people with academic skills and knowledge that most of them are not very interested in and that many don’t have the mental firepower for. We cover up the impossibility by pretending that passing “memorize and forget” tests shows acquisition of those skills and knowledge. Then we worry that lots of kids don’t even pass that way.

        In the second post, he is surprised and disappointed that he gets mostly “disagree” responses to the statements, “My teachers provide work that lets me play to my talents and interests” and “My teachers always make the work interesting and relevant to me.” Seriously? Seriously?!?! You have to really be living in a delusional world to expect anything different. (I am tempted to rip off Mark Twain, “And you’re an education professor? But I repeat myself.”)

      • educationrealist

        I don’t know if you read my Grant Wiggins tribute (he died recently), but I thought practically every thing he said about students was incorrect. His work on curriculum and thinking about it is stupendous, and it’s clear he taught a great deal and watched a lot of schools. But I would roll my eyes every time he talked about how awful we were with students and the inhumanity of high schools.

    • Mark Roulo

      “What subjects seem to generate the greatest student interest and engagement?”

      Asking what students choose to do on their own might provide a clue.

      If the list of subjects in question is:
         *) Mathematics
         *) Study literature (instead of just reading fiction)
         *) Science
         *) History
         *) Learn a Foreign Language

      The answer is probably math or history. I’d expect that reading fiction would average higher, but literature classes are about a lot more than just reading the stories. Very few adults do the literature class equivalent of joining a book discussion group compared to the number of adults who just enjoy reading fiction.

      But note that what kids do on their own time tends to not be ANY of these. Some kids actually enjoy learning history and will do it on their own. Just like some kids enjoy reading fiction. But mostly what the kids want to do is play (physical sports, game systems, whatever), be passively entertained (TV, movies, broadcast sports) and hang out with their friends. Just like adults.

      What extension classes are popular with the 22 – 25 crowd (I don’t actually know)? I doubt that the answer is “math.” Things like “wine tasting” wouldn’t surprise me. *OR* things that are relevant to job/career … such as “Microsoft Word 2013.” And what percentage of the population takes these classes? I’m sure that the answer to the second question is low.

  • anonymous

    So the plan is to give someone a headache so that they will appreciate the aspirin you subsequently provide? Interesting plan. I suspect most students will view this “aspirin” as actually being more of a second headache. A quick look through the comments is not encouraging for the success of this headache aspirin plan.

    Am I the only one that wonders if Dan was a particularly good teacher? It is almost comical that he is advocating nearly the same strategy that he mocks (silly “real world” book problems for motivation). Does he really think that if we only had some great motivating problem, that kids would get excited about factoring or whatever?

    • educationrealist

      I mentioned in the previous piece that I thought Dan was probably an excellent teacher for his population: underachieving white kids. I hadn’t noticed that parallel. Good point.

      • Michael

        I think just about the only things low ability kids would have liked about him was his homework policy (I believe he didn’t give much).

  • Anub

    Here is a story of Saudis cheating by buying grades, & the comments mention one was a rapist who all got flown out of the US before any consequences could occur, just like with 911
    http://missoulian.com/news/state-and-regional/saudis-tried-to-shield-students-from-scandal-at-montana-tech/article_c9d7440e-b244-5bde-96a6-2f68f06bdb5c.html

  • Texteach

    Wonderful definition of teaching as “an art enhanced by skills appropriate to the situation and medium.” I work with children who have various LDs most of which are labeled as dyslexia. I can assure you the same method does not work for all children. IQ, personality, memory, spatial issue, and attention are some factors to consider when teaching reading. We don’t expect the best question in the world to make everyone interested in fashion, aerospace, or cooking so why do we expect it in math?

  • Dan

    A couple of thoughts. First, I see much of the reasoning behind Meyer’s motivational technique as sound. I think that it can be easily seen that the only reason that any particular math concept/method exists is that someone found it extraordinarily useful in solving some problem. Therefore, a problem does exist in which said method is needed. It is also my opinion that under certain conditions presenting students with this problem can be a very effective motivational technique.

    However, the farther one proceeds into the depths of upper level math, the more difficult and convoluted the original problems become. Often the entire point of a complex mathematical method is to provide additional power to solve a problem of such complexity that it was impossible to solve with a simpler methodology. Thus, the simplest problem in which the new complex method is useful is more difficult than the most complex problem that is able to be solved by the previous, more simplistic method. The two best examples that I can think of at the moment are tensor/vector notation and function/equation notation.

    Ed demonstrated the need for function notation very well in a previous post. In fact, I am guessing Meyer would greatly approve of your method in introducing said topic.

    As for the other example, it is readily apparent that one does not often need Einstein tensor notation when working with force vectors in Newtonian physics. However, Einstein notation becomes extremely useful if one has the misfortune of working with tensors.
    In both of these examples, it is indeed possible to provide students with “headaches”. Ed demonstrated this with function notation, and simply making students write out a few continuum mechanics equations with vector notation before introducing Einstein notation will cause tears of joy to be shed when they realize that they will no longer have to ice their hands after doing their homework nor use an industrial strength stapler to join the resulting pages together.

    Notice however that in each of these examples it is assumed that the student fully understands both the previously taught, simpler method and more complex example problem that was derived from it to show the limitations of the simpler method. This seems unlikely to be a valid assumption in classrooms of heterogeneous ability. It appears to me that when one prioritizes using “headaches” as motivation there exists a high potential of further discretizing the understanding continuum. That is, partial understanding of a topic becomes even less useful to a student. Those who fully understand vector math will be relieved to learn a shorter method exists and will likely be highly motivated. Those individuals who are still struggling with creating systems of equations and have always been told y is equal to f(x) will likely become even more confused and less motivated by something they see as simply increasing complexity and difficulty.

    The second problem I see with using “headaches” to motivate students is that anyone who uses this motivational technique assumes that all of the students he/she is instructing find solving a problem motivating. It assumes that if only students could see the problem, they would of course want to find a solution to said problem. For students of higher ability I would expect this assumption to be often valid. For lower ability students however I would expect a much different reaction. Instead of insatiable desire to solve the new problem, I would estimate their reaction to be anger and annoyance. I myself often had this reaction throughout high school math classes when I suddenly had to put hard work into understanding math for the first time in my life. Rather than being motivated when told that the method or concept I had worked so hard to understand was insufficient and inadequate, I found myself angry that my teachers had forced me to work so hard on something that they were now telling me was (in my view at the time) worthless.
    Thus while agree that Meyer’s idea has merit, I believe it is a concept that is valid only in a severely limited scope and application.

    • educationrealist

      I missed your response the first time through (I do my comments interface through a browser app, and miss some). I wish I hadn’t, because this is EXACTLY RIGHT, particularly the penultimate paragraph (ooh. Alliteration). Thanks for your input.

  • Jim

    Dan – One might think that the development of mathematics was motivated mostly by utilitarian concerns. But a lot of the actual history of mathematics doesn’t conform to this pattern. Greek mathematics didn’t seem to have much utilitarian value. For example Greek mathematicians found the topic of incommensuarability very interesting but I’m not aware of any practical significance to it. I’m not sure if the Greeks found a single practical application of the entire content of the Conic Sections of Apollonius. Likewise one of Archimedes most famous achievements was his quadrature of the parabola. How the hell were the Greeks able to apply that? Archimedes determined the equilibrium floating positions of all kinds of weird shaped solids far beyond any practical applications.

    Similarily there was an immense amount of research on algebraic solutions of polynomial equations over many centuries. But things like Cardano’s formula were never a source of much practical application.

  • Mark Roulo

    “I think that it can be easily seen that the only reason that any particular math concept/method exists is that someone found it extraordinarily useful in solving some problem.”

    I don’t think Cantor discovered that there are different sizes of infinity because he was trying to solve some specific problem. He was just wondering about infinity.

    Lots of math isn’t applied. Or at least, isn’t applied yet!

    And I’d expect a lot of kids to respond to some/many of these problems with some variation of, “But I’m *never* going to care about that problem.”

    When there *is* a good explanation for who uses a given concept and why I think it makes sense to pass this along to the students (e.g. computer programmers often use a base other than base-10, so bases aren’t *just* something thought up to torment pre-algebra students). But I’d be reluctant to force this.

    • educationrealist

      I almost thought of going into that (I was going to use complex numbers) but since I think the obvious response is “yeah, but *I* don’t care”, as you did, I decided to stay focused on my response. But yes, I think you’re absolutely right. That’ s my philosophy.

    • Dan

      Jim and Mark are certainly right about the areas in higher level math that have, as yet, no application, I had not thought of those. As an engineer that sort of math gives me nightmares and I do all I can to avoid it. That’s what MATLAB is for. Or any undergrad minions you have working for you in the lab.

      However in terms of the topics covered up through high school math I would still submit that just about all of it could be described as “applied”. I can’t think of any math below Diff Eq that couldn’t be considered applied – again, quite possibly reflecting my engineer’s bias. Thus I still believe that my original conclusion (slightly modified due to your quite valid points) remains valid: For teaching math up through the high school level, while one will be able to find a “headache” for just about any topic, while this method can be effectively used for high ability students, the method is far from universally effective in application and quite possibly detrimental to some audiences.

    • Jim

      Canter was motivated at least in part by his research on trigonometric series. Of course a lot of the questions about trig series he was interested in were pretty theoretical but Fourier series in general have a lot of practical applications and one can trace trig series historically back to Bernoulli’s study of vibrating strings.

      Problems involving solving quadratic equations can be found in Babylonian sorces dating to the second millenium BC. The problems were exercises for scribal students and were actually pretty artifical. There’s no indication that such problems were anything other than puzzles.

  • anonymousse

    I think I disagree with you, but I’d like to discuss your viewpoint through the point of view of an analogous (rather than directly related) situation.
    Specifically, I have been thinking of the teaching of foreign languages. I think virtually all Americans study foreign languages in school (when I went through the public school system 35 years ago, I was required to take, I believe, 2 years). So it is odd that the net result of this is that virtually no Americans speak a foreign language (or, at least, virtually no Americans learn a foreign language in school).
    If you stop and think about this, it is simply an odd state to be in. We don’t teach English with the assumption that kids won’t learn English. We don’t teach math even, without the assumption that the kids will learn ‘math’. So it is strange that, with all the work, all the money, all the teachers, all the homework, devoted to the study of foreign language, people don’t in fact learn a foreign language.
    As a parent of a young kid, I actually have a very different desire: I’d like my kids to study a language, and actually learn it. Presumably, that means studying it throughout school (12 years), and practicing it. This may also be impossible-I’m not sure that many schools regularly offer 12 full years of language instruction, with the ultimate goal being fluency.
    That being the case: what’s the point of foreign language education? I realize a few will like it, and pursue it enough to become fluent, and I guess there are a few literacy benefits from studying other languages(recognizing roots and source words, and so on). But ultimately, I suspect, most people would argue that the point is similar to what you are arguing for math education: there is none other than the process itself. Studying and learning a new thing, regardless of whether you get good at it, and regardless of whether you will ever use it, has benefits that make it worth while. Under this argument, it is irrelevant what language is actually studied (French, Spanish, or even Klingon): it is the study itself that matters.
    I don’t think I agree with this. I’m coming at it from a real homeschool/contrarian perspective, but I don’t buy that studying arbitrary things is particularly beneficial. Its not beneficial for me as an adult: I don’t do it for fun, and while I do have to do it at work, I don’t think the purpose of a childhood education should be to get used to the boredom that will come with adult employment. Rather, I’d really like to minimize that boredom in my life. So why make it a mandatory part of childhood?
    So rather than make my kids study Klingon for two years (or Tolkienian Elvish) or even French or Spanish, for the sole purpose of making them learn and do hard/tedious/unpleasant/unwanted stuff: I’d like my kids to study Spanish, the the goal of learning to read and speak Spanish. I.e. I’d like my kids to spend their time learning something that will result in an enhanced ability on their part.
    And so, what’s the purpose of math? To gain the ability to use math to function in society. For most people, that probably ends at around 8th grade-balance checkbook, do taxes, understand statistics well enough to understand a mortgage, and so on. There is plenty of realworld math that could be taught (you could institute a societal revolution if you could genuinely teach the concept of compounding interest, for instance). For a few others, that means more advanced math (engineers and scientists, primarily). Just learning arbitrary logical functions for the sole purpose of learning arbitrary logical functions doesn’t strike me as reasonable.

    I also realize that this calls into question the purpose of all education (why read Shakespeare? Why learn about the Constitution and World War II? And so on)-but that’s the subject of another post.

    joeyjoejoe

    • educationrealist

      Totally agree with you about the utter futility of foreign language study. However, the argument is *not* given as “hey, it’s tough, and you’ll get good at learning.” The argument for foreign language is a hippy dippy you’ll learn how other people live, and knowing more than one language is essential to your professional careers. As to the first, give me a class on Greek cuisine, as to the second, no, it’s not.

      Foreign language has no relationship to cognitive ability. My dad is brilliant at spoken languages but has an IQ of 95 or so. That’s by no means unusual, nor is my high IQ monolinguist status uncommon. So I totally agree that foreign languages should be optional.

      As to your daughter learning fluency–very unlikely, unless she has the proclivity.

      Remember, I’m in favor of teaching kids less math. I think IQ and interest affects how kids learn math. But my rationale is not utility.

      Foreign language is a great compare/congtrast point. Thanks!

      • anonymousse

        ‘daughter’?

        You ARE a graduate of an educational department…

        joeyjoejoe

      • educationrealist

        Huh. I don’t where I saw daughter. It’s what I get for posting at work.

      • Jim

        English speakers are lucky that so much stuff is available in English including translations of nearly all important works in other languages. If your native language is Albanian though you will be pretty handicapped if you don’t learn other languages.

        The United States and Canada are fortunate in that one can travel all over a huge continental area and easily get by with just Standard English. Chinese actually consists of about 8 or 9 mutually incomprehensible languages although the wriiten language is the same since it is largely independent of the spoken languages.

    • Jim

      Some people just find the study of languages interesting just as some people find astronomy interesting.(There’s of course no practical value in knowing anything about stars and galaxies.) But few Americans have much need in daily life to know any foreign languages. Unless of course you live in the Valley and then you probably will learn some Spanish.

  • vijay

    There is a ton of engineering headaches for which math is the asprin, but it would be hard to solve the engineering problems with middle or high school math. All of engineering is one giant headache solved by mtah.

  • anonymousskimmer

    “If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”

    For the vast majority of primary and secondary students, “headaches” will go away if they wait long enough for the semester to end. Even the “swotters” know that their pain is imposed by the system (a hammer to the head), not something within them (a genuine headache, or “how can I use this?”).

    The “pain letter” (http://www.humanworkplace.com/whats-pain-letter/) marketing approach may work for job openings, but getting a job/hiring a person is significantly different from teaching/being taught as a semi-captive audience.

    IMO, the best bet is to open up electives and requirements and let students fall where they want, with the occasional pre-req to blatantly show them how knowledge interrelates and force them to expand their knowledge base. And let them know what courses and background will be required for further education or specific jobs.

  • Teacher

    I totally hear you. However, I fear that because you feel that you and Dan Meyer have so different philosophies, you are to quick to discredit an opportunity he has opened up. His aspirin series is a mere exercise for teachers to contemplate over the summer how they may increase the likelihood of student “buy-in” as well as student understanding of, retention of, and confidence in mathematics.

    I too work in a Title I school, and I have been able to build a positive rapport with many students, in part due to my efforts to seek applications of the math they’re being asked/required to learn. I’m perfectly honest with them about the odds that they will use this skill to alleviate any given headache is unlikely. However, many enjoy seeing algebra in a concrete form rather than merely abstract; indeed this is the second standard for mathematical practice to balance to two forms.

    Moreover, I am greatly concerned that when you say “teachers are never going to agree” you are attempting to justify not trying to find common ground or learn from one another’s thoughts. We don’t have to have the same teaching philosophy – we’re actually expected to write our own in college courses – to see value even when I could never teach like someone else does (even my own “coin someone else’s work. I may be an education optimist, but I’m still about to browse your blog to see if I can take an idea you have and adapt it for my own uses. I think we have different philosophies; for example, I think capability follows confidence and curiosity. Still, I think I can learn from and springboard off an idea you have because as you said yourself, you are considered a successful teacher. So my question is, why does a difference in philosophy demand denial of validity of another’s practice?

  • Zach Coverstone

    I’ve been pondering on some of the things I have been reading on this blog. I empathize with some of it. I do think it is interesting, though, that you and Dan seem to agree on how to teach something. It seems that, however, you and he differ in why you do it. But don’t you have more in common than you think if you do many things that he does? (And if your paths naturally converge without really consulting each other?)

    Case-in-point: Your modeling linear linequalities lesson: https://educationrealist.wordpress.com/2013/03/11/modeling-linear-inequalities/

    His linear inequalities lesson: http://blog.mrmeyer.com/2015/if-graphing-linear-inequalities-are-aspirin-then-how-do-you-create-the-headache/

    In both cases, you create a low affective filter and make sense of the mathematics using similar, but not congruent approaches. Could you point out, using these two specific examples, how you two are different?

    • educationrealist

      “It seems that, however, you and he differ in why you do it.”

      Oh, totally. That’s one of our big areas of difference.

      I am not against Dan’s methods. I think he overstates their effectiveness. Edit: I originally restated a bad synthesis of my Gatekeepers article, and deleted it because it only made sense in reference to Gatekeepers. So rather than restate I’ll just say, at the end of the Gatekeepers article, I restated the problem–which wasn’t Dan.

      I think Dan’s current job is a good fit and I wish him well.

  • savantissimo

    I once helped a cousin with his over-vacation algebra homework, and found a good way to motivate factoring: finding maxima and minima.

    First use position, velocity, acceleration as examples of successive differentiation, move to finding instantaneous slope graphically, note that the slope is zero at maxima and minima, then give the cookbook formula for differentiating polynomials. Now he has a much better reason to care where an equation equals zero. Just to be complete, I also gave him a bit on the fundamental theorem of calculus and multiple integrals so that he could go back from acceleration to position. He understood it all and was a lot happier about doing factoring. The whole thing took less than an hour and a half. (And that included a digression into how to count to 1023 on your fingers.)

    I used the same method successfully with a couple of middle school-age homeschooled brothers.

    No doubt this method horrifies math teachers who think calculus is somehow more advanced than Algebra 1, but math teachers very rarely have any real understanding of math, still more rarely are they any good at teaching. No, it likely wouldn’t work with sub-gifted students of any age, but I despise them.

    • educationrealist

      Like most people utterly ignorant of education, you think that just because you taught your nephew all this in 90 minutes that he’ll remember it a week later. He’s unlikely to in most cases.

      And no, the method doesn’t horrify math teachers. But we’re pretty disgusted that you despise sub-gifted students.

      • Savantissimo

        It would be absurd not to despise them, many are literally closer in intelligence to a bright chimp than they are to me.

        Nevertheless, I’m kinder to them that you are. Better to leave them alone than to teach them in your style.

      • educationrealist

        What an unpleasant person. And ignorant, too. While people with IQs over 90 might not remember all the math, they certainly are learning something. Moreover, you are really ignorant on the comparative intelligence of chimps.

      • anonymousskimmer

        https://www.enneagraminstitute.com/type-3/

        “Want to impress others with their superiority: constantly promoting themselves, making themselves sound better than they really are. Narcissistic, with grandiose, inflated notions about themselves and their talents. Exhibitionistic and seductive, as if saying “Look at me!” Arrogance and contempt for others is a defense against feeling jealous of others and their success.

      • Roger Sweeny

        Savantissimo,

        I think you are factually wrong about stupid people and chimps. A nice summary of a lot of relevant research is Thomas Suddendorf’s interesting The Gap: The Science of What Separates Us from Other Animals (Basic, 2013). Check it out.

      • educationrealist

        Remember that he’s not even talking about “stupid people”, but merely “sub-gifted”. Anyone with an IQ less than 115 is apparently chimplike.

      • savantissimo

        Richard Lynn estimated chimp non-verbal intelligence as being equivalent to a 3.5 year old. IQ is not a measure of intelligence (it’s a rarity measure relative to a given age and reference population), s o using the +/-3 s.d. vs. age plot of the Woodcock-Johnson Rasch-based (equal interval with meaningful zero) W-score for block rotation (which is ~6 points lower than the full test and the equivalent CSS scale on the Stanford-Binet, for neither of which a similar chart is available), then an average 3.5 y.o. is at about 464 with an s.d. of 12, adult average is about 508 with s.d. 8.8, I’m around 535 (+3.5 s.d.). Assuming that the chimp s.d. is 8, a +2 s.d. “reasonably bright” chimp is around 480.

      • Roger Sweeny

        The statement “Richard Lynn estimated chimp non-verbal intelligence as being equivalent to a 3.5 year old” is not equivalent to “human and chimp intelligence can be measured on the same scale (in particular, a scale developed for humans) and when it is, chimps and 3.5 year old humans occupy the same spot.”

        Putting that aside, I gather you are saying that the average high school student (100 IQ equivalent) is closer to a “reasonably bright” chimp (2 standard deviations above the mean, or smarter than 95% of chimps) than that high school student is to you (3.5 standard deviations above the mean, or smarter than 99.984% of human adults).

        To go back to the first paragraph, one of the fairly consistent results in experiments on chimp intelligence is that they just don’t go past what 3.5 year old humans can do. There are no “fairly bright” chimps that manage to go past that point on a human scale. There are developmental milestones in humans that are normal and expected that chimps just never can do. There are qualitative differences between even the brightest chimps and a normal 5 year old, not just “less” on a quantitative scale.

  • anonymousskimmer

    @savantissimo
    “The enneagram is twaddle. If you want a better grade of twaddle, or at least one more accurately describing my personality, see the Myers-Briggs type “INTP”.”

    I linked based on a superficial read of you, so may very well have been wrong.

    I don’t understand what people find attractive about a factor-based personality assessment when cluster approaches (of which the Enneagram and instinctual variant typologies are two examples) seem to better fit and better anchor the fuzzy categories that are personality types.

    Sorry for being off-topic ER.

  • anonymousskimmer

    @savantissimo

    And oh yeah: John Gatto never went around yelling at teachers (that I have seen). He yelled at the system, just as ER is doing.

  • Understanding Math, and the Zombie Problem | educationrealist

    […] with lots of homework. Their tests are difficult but predictable. They value students who wrote the didactic contract with Dolores Umbridge’s nasty pen, etching it into their skin. They diligently memorize the […]

  • Curriculum Development: Not Work for Hire | educationrealist

    […] Teaching and Intellectual Property Grant Wiggins Developing Curriculum Handling Teacher Preps Math isn’t Aspirin. Neither is Teaching. […]

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: