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[Apologies for the repost. The original (reportedly) defected to Canada leaving me to reconstruct it from pieces. I would have let the post expire gracefully but the comments were — and I'm not kidding about this — eye-blisteringly incredible. Check them out.]

I teach Algebra 1 and Remedial Algebra 1, a schedule which offers me interesting contrasts and case studies daily. The remedial population, as you might expect, features more behavior problems, lower rates of attendance, higher mobility, higher incidence of poverty, weaker student skills, more individualized education plans on file with the district, and those students are more likely to have disliked math (or their math teacher) in the past. After three years of trial and error, I have found intermittently successful ways to remediate most of these issues.

The feature of this group that confounds me and defies my remediation is this: they are far less likely find our daily show and tell interesting than are their contemporaries in non-remedial Algebra.

What I’m saying is that, when I play, for example, this fantastic loop of time lapse photography, my Algebra 1 students sit a few millimeters closer to the edges of their seats and lean a few degrees closer to the screen than do my Remedial Algebra students. They call out observations and deconstruct the movie in ways the remedial classes do not anticipate. In general, they seem eager to engage the unknown whereas my Remedial Algebra students seem to prefer that the unknown stay unknown, that life’s unturned rocks stay unturned.

Pixel’s Revenge timelapse showreel from David Coiffier on Vimeo.

This bears out even between sections of the same course. The length of a class’ discussion of show and tell media correlates positively to the class’ average grade.

No pithy conclusion. I have no idea what I can do with this.

81 Responses to “But How Do I Remediate THAT?”

  1. on 02 Jun 2009 at 5:46 pmJen

    Crack this nut and you’ll be a rich, rich man.

  2. on 02 Jun 2009 at 5:47 pmMichael

    Is this a social problem or an education problem. One can’t be fixed only in the classroom, the other can.

    Are your remedial students waiting for you to give the answer or the “hand out?” Are these students curious about anything and perhaps that time lapse photography is just not interesting to them?

    Many of these students don’t have some basic needs met therefore may not care about their intellectual needs. Perhaps with this crowd is to show them how knowing math you can prevent people from cheating you. Being cheated is something they all should be somewhat familiar with.

    These are just some ideas. Where to go from here . . . I too am still working on that.

    Perhaps you can get some great ideas from Ruby Payne’s Understanding Poverty. Maybe there is a definite cultural clash between student experience and the videos you show in class.

  3. on 02 Jun 2009 at 5:47 pmsylvia martinez

    I’m wondering something else. Perhaps these groups aren’t breaking down into curious vs. not curious, but how they show their interest. Good students are tuned to teacher (and cultural) expectations of how to react. They are on the whole compliant, interested, “easy”, and do better in school. Other students are either defiant or have been trained through endless disappointment to not bother to show interest.

    What if students in your former group are actually less interested than you think, or students in your later group are actually more interested than it seems. They’ve just both been “schooled” in how they are supposed to react.

  4. on 02 Jun 2009 at 5:48 pmSue

    Ooh, I like both those thoughts! (I’ve added Understanding Poverty to my reading list.)

    I’m thinking part of the problem is the tracking (but no easy way to remedy that this far along). In What’s Math Got to Do With It?, Jo Boaler devotes a whole chapter, Stuck in the Slow Lane, to the problem. She discusses why Japanese schools do not track (Japanese students excel in math), and considers the problems tracking brings even to students in the highest groups, who feel too much pressure and end up liking math less than students who were in mixed-ability classes.

    Here’s a bit about what’s needed to make mixed-ability classes work:

    For mixed-ability classes to work well, two critical conditions need to be met. First, the students must be given open work that can be accessed at different levels and taken to different levels. Teachers have to provide problems that people will find challenging in different ways, not small problems targeting a small, specific piece of content. These are also the most interesting problems in mathematics so they carry the additional advantage of being more engaging. … In addition to open, multilevel problems, the second critical condition for mixed-ability classes to work is that students are taught to work respectfully with each other. (page 118)

    I should have a review of this book up on my blog in the next few weeks.

  5. on 02 Jun 2009 at 5:48 pmJason Dyer

    Yep. I’ve observed the same.

    On very rare occasion I have been able to get strong reaction from a remedial crowd. I have yet to find a definite pattern to when this happens.

  6. on 02 Jun 2009 at 5:49 pmBen S.

    I recently read an essay on writing great essays: http://paulgraham.com/essay.html. In it, the author argues that what makes essays exciting are surprises. These surprises can be anything, but in general they are the little things in life that so many people pass over.

    The pertinent quote:
    “Collecting surprises is a [learning] process. The more anomalies you’ve seen, the more easily you’ll notice new ones. Which means, oddly enough, that as you grow older, life should become more and more surprising. When I was a kid, I used to think adults had it all figured out. I had it backwards. Kids are the ones who have it all figured out. They’re just mistaken.”

    In my mind, this is the root of the problem. The remedial kids haven’t been collecting surprises. The ones I teach are pretty shut off to the world for one reason or another. Maybe whenever they asked why something happened, they were told to not ask questions because their parent was embarrassed for not knowing the answer? I can imagine hundreds of scenarios like this one for a child in poverty. This means that surprises whiz by their heads left and right.

    The higher kids, on the other hand, have been practicing noticing these surprises in life from a very young age. Moreover, they were probably rewarded by excitement from their parents for their curiosity. Rewards beget further curiosity, and the cycle repeats.

    My point is this: your kids probably are on a 1st grade level when it comes to recognizing surprises. This doesn’t mean they have to stay there. It does mean you have to meet them where they are. Work them up to more complex situations and bigger problems. They need to see this skill modeled many, many times before it becomes their own.

    That being said, I struggle with executing what I describe above. To me, the interesting problems are the complex ones. It is always tough to find things simple enough to analyze, yet interesting enough to keep everyone’s attention. If you know where to start, please let me know.

  7. on 02 Jun 2009 at 5:49 pmEric

    Reminds me of the “16 Habits of Mind”: http://www.ascd.org/publications/books/108008/chapters/Describing_the_Habits_of_Mind.aspx

    —which doesn’t help a lot, I know.

    Many years ago I ‘taught’ boys in court school, who were supposed to be working on their GED. Soon discovered their main problem: the narrowness of their world. Girls, cars, heavy rock music, cigarettes, alcohol, drugs—that was about it. If your world is that small, no wonder you lack curiosity, inspiration, ambition, etc.

  8. on 02 Jun 2009 at 5:50 pmMichael Paul Goldenberg

    Very thought-provoking post and comments. There’s clearly no one simple answer to the question. Keep in mind that what you say can apply to affluent kids, too. In fact, it can apply to highly-educated adults in certain contexts. It’s a matter of knowing how to look intelligently at any given phenomenon (and willingness to try). I’ve seen a bunch of educators and cognitive psych professors and graduate students look at an amazing video of a 3rd grade lesson taught by Deborah Ball (now the dean of the School of Education at the University of Michigan). Most of the folks at this session seemed completely focused on answering a single question for themselves which boiled down to: “Did the teacher make the correct decision in letting students pursue a particular question that arose during the less?” Once they settled that for themselves one way or the other, there was nothing more for them to observe, or so it seemed. Very disconcerting, given the incredible richness of the video.

    That said, obviously what you’re hoping to uncover here is how to engage kids who seem disinclined to be engaged. That’s a non-trivial question, and it’s good that you don’t expect to find the answer easily or quickly. One potentially useful source, despite the difference in ages focused upon, is Watson & Ecken’s book, LEARNING TO TRUST, which looks at two years of Ecken’s teaching of a combined grade 1/2 classroom in inner-city Louisville, KY. She comes to realize the crucial need to model and teach many basic interpersonal and personal skills such as how to listen, how to have a conversation with one other person without having it turn into a brawl (literally or otherwise), etc. We teachers who come from well-behaved backgrounds and schools that mostly consisted of kids who bought into school forget the danger of taking all this stuff for granted as what kids walk in the door with. But if students don’t bring this with them in Kindergarten, there’s a very real possibility they still aren’t bringing in the door with them in high school, especially if they are not academically successful and have been shunted into remedial classrooms with other low-achieving and turned-off kids. It’s a downward spiral from which escape is difficult if not impossible.

    Assuming that stuff that catches the attention of any group of kids will work with most kids is probably naive. My recent guest teaching a lesson at a charter school for at-risk kids in grades 7 to 10 in Saginaw, MI, a really impoverished city, is a case in point. I taught the same basic lesson to five classes. It really clicked in some, fell flat in others. It’s way too complex to write about the variables here, but I can’t say that I was surprised (only pleasantly so that according to the classroom teacher, the lesson DID engage some kids who hadn’t previously shown the slightest interest in anything the whole year). I tend to be highly skeptical of magic bullets, miracles, and so forth when I hear about them in the educational “literature” on-line. And I certainly don’t have any. But that doesn’t mean we shouldn’t continue to try to understand why some things do click unexpectedly or how to increase the percentage of kids we engage.

  9. on 02 Jun 2009 at 5:50 pmNick

    Sylvia nailed it. I don’t know if one group is less interested than you’d expect. Maybe I’m just mincing words, but it seems to me that kids in remedial class are schooled into being reluctant about their own ideas. That it’s harder for teachers to find the stuff in a wrong answer that is partially correct, and therefore the reaction they get for most of their participation is negative. They’re more reluctant to engage, they’re more anxious/unsure and yeah, they don’t like it.

    My Theory: Kids in remedial classes know it. Also, when they take risks in class and offer their ideas, they get negative feedback. The effect is that they’re less confident in their ideas and outwardly less engaged. Their ideas are muffled by reluctance.

    Kids in the non-remedial classes are more often engaged with their ideas, teachers find ways to illuminate the good ideas in their mistakes more often, they receive a lot of positive responses to their ideas. Higher confidence in their ideas manifests itself in less restraint, and so they participate.

  10. on 02 Jun 2009 at 5:51 pmNick

    I’m gonna throw out an answer, that I think there’s a strong reason to reject, but it seems like a straight forward solution.

    Remediate it like this: increase your tolerance for imprecision/incorrect responses in remedial classes. Allow other students to weigh in. Let conversations go south longer. Maybe even let them conclude on wrong notes. Practice the absolute poker face/posturing so as not to condition kids to look for the tell that in-fact they’re wayyy off. Wait to give your feedback until its less personal, wait until you understand the ingredients in a students wrong answer, and then gently work them towards more solid ground.

    An objection: This takes time. A lot of time. In fact, it might be more effective to simply give a textbook correction. Illuminate all the ways in which an idea is wrong, and power on. Taking the time to pursue incorrect answers, sometimes leaves sound bites bouncing around that lead others down the wrong path as well.

  11. on 02 Jun 2009 at 5:52 pmMichael Paul Goldenberg

    Nick: Your first comment about kids being conditioned to not take risks because they are generally wrong and their teachers have little ability to piece out what’s valuable in student errors has a good deal of merit. I have argued many times that in fact we condition most kids in math classes to stop taking risks (not answering unless they are: A) sure what the teacher is asking for; and B) sure that s/he (the student) knows that answer) by the time they are in about 3rd grade. From then on, there may be a few bright kids who always seem to know the answer, but when any challenging question is posed (if indeed the teacher is capable of framing one), everyone plays it safe, knowing from experience/conditioning that the teacher will be unable to tolerate silence for more than 3 seconds and will answer the question him/herself. For those on the low end of achievement, there is even LESS reason to take risks, as you’ve suggested, because that puts the student in a position to be ridiculed by the class, the teacher, or both.

    But I think your second post misses the mark in that kids who have not been pushed to take risks, even bright ones, are disinclined to do so when some “progressive” teacher who understands the importance of wait time, student-centered questions, etc., comes in with a lesson that would promote more student participation. In remedial classes, the kids can generally out-wait any teacher. I’m sure many teachers can try to tease out answers if they can provoke any meaningful responses whatsoever, but getting the ball rolling in the first place is no easy task.

    Having been in front of many at-risk/remedial classrooms, I can tell you that the kids HATE me for trying to “force” them to think and answer. They find it highly-disturbing when I won’t just give them the solutions immediately. And they quickly tell me that what I’m doing “isn’t mathematics” and “isn’t teaching.” They swear that what they want is “book work.” And by that they mean work sheets. Just hand them a set of mindless calculation problems to do and they are content.

    Of course, they don’t ENJOY such work. They don’t really generally put much effort into it. But it is what they are used to. It “looks like math” to them, so failing at it is safe, familiar, no new humiliation. They can just “not be good at math” in the usual ways.

    Trying to get them to actually speak, to attempt to explain their answers, to think about each other’s reasoning, is asking them to engage in risk-taking and meta-cognition. Do I think that sort of teaching is vitally important? You bet! Do I think getting kids to engage in it, especially at-risk and remedial kids, is a simple process? I know from a lot of experience and observation of teachers in similar classrooms that it is not.

  12. on 02 Jun 2009 at 5:52 pmBonnie

    Does it make me a “remedial student” if I’m left speechless from that video?

    I very much enjoy reading the discussion here. Interesting thoughts.

  13. on 02 Jun 2009 at 5:53 pmNick

    Michael so I guess the idea is that both the students and teacher are conditioned by the “remedial” class. Seems as if your position is that this is inescapable or nearly inescapable. I don’t know if I’d mudsling with “progressive” teacher, when the connotation is a teacher that is willing to wait when waiting is not the effective thing to do. So, yea, I agree, I think the objection may overpower my suggested solution, so should we settle for feeding them what they expect?

  14. on 02 Jun 2009 at 5:54 pmMichael Paul Goldenberg

    Nick: No mudslinging. I am a progressive reform advocate in all my work. I simply was being ironic in suggesting that it’s some radically progressive insight to attend to wait time, try to get students at the center of the class instead of lecturing to (or at) them, etc. (Not all progressive teachers are very good at doing this, and not everyone who does this would consider him/herself progressive. It’s just a word I use as shorthand for a host of practices and attitudes).

    And no, we don’t settle. We HAVE to move kids out of their comfort zones (and for me, a math coach and teacher educator as well as a classroom teacher, I have to help move other teachers and myself out of comfort zones of theirs and my own). But to do so without a sense of the difficulties is to set ourselves up for a great deal of misery. My message isn’t that it shouldn’t be done, but that it isn’t at all easy, and it requires persistence, support from administrators and colleagues, and a knack for really connecting with students as individuals.

    Considering the accurate comments from others about the “face” many remedial kids are committed to putting out – disinterest, disengagement, too cool for school, etc., and the narrow range of interests many have, as was also pointed out by someone else – it takes more than neat lessons. That’s not to denigrate such lessons, but only to suggest that it takes more to get some kids to take the risk of participating in such lessons, and probably a hell of a lot of time and patience to help bring them to the point where they can succeed and will be motivated to try things they don’t already feel comfortable with on their own. (Of course, that’s true for lots of teachers, too). ;^)

  15. on 02 Jun 2009 at 5:55 pmDavid Cox

    Sean
    What curriculum is your school using for your integrated Geometry/Algebra approach?

  16. on 02 Jun 2009 at 5:55 pmNick

    Does it require magic or potions to work properly on your sub-group 1 students?

  17. on 02 Jun 2009 at 5:56 pmJackie Ballarini

    What else is going on in the lives of the students who “just don’t care”?

    Is it possible that in their world, there are so many other big problems for which they don’t have answers, that the WCYDWT problems are irrelevant?

  18. on 02 Jun 2009 at 5:56 pmsylvia martinez

    @michael thanks for jumping in here, it gave me a chance to rediscover your excellent blog, Rational Math Education. http://rationalmathed.blogspot.com/

    @dan not to add to the expansion of the problem well beyond your control… (but, I’ll do it anyway)… math curriculum has to be acknowledged as part of the problem. School math has been decoupled from the physical reality of the world in an most unfortunate way. It’s irrelevant, not only to most K-12 students, but to most real mathematicians, scientists and engineers.

    Then we compound it with rigid one-dimensional assessment; we do not test them on interest or curiosity.

    These kids know that your content (like this video) are onramps to stuff that will not be as interesting or relevant. So, an attitude of “let’s cut to the chase” seems nearly rational. You are asking them to ignore 10 years of their first-hand experience of how school math works. You are asking them to have an interest in an “unknown thing” when they have correctly come to the conclusion that this thing will not be on the test, and therefore doesn’t matter. There will be no clouds, raindrops, or videos of anything on the test. School has taught them well, it seems.

    I think what you are doing with your WCYDWT graphics and videos are great. For me, that’s real math. it’s what happens after that that should be tossed out.

  19. on 02 Jun 2009 at 5:57 pmSean Morris

    In response to David, we looked at various articulated curricula, mostly IMP out of Berkeley, but my high school(Albany HS in CA) had implemented an Integrated Math Curriculum across all grades and courses about 7 years ago and it failed for various reasons but particularly it failed on the “remedial” group. The community wasn’t going to go for it.

    So we looked at California State Standards for Integrated Math and went from there. It is essentially our own integration without using a textbook and just giving the students problems to practice with. We have a solid foundation but need to add in the WCYDWT-type approach to fill it out more.

    No magic here. It is pretty straight-forward typical math instruction. Hence, I have been spending the last four or five months trying to figure how to get better engagement.

    As it is 50% passing is pretty horrible, it just feels a lot better than it used too.

    I would avoid the blame the curriculum, blame math instruction in general, blame the school system, blame the “too many other issues”, blame middle school, blame the student kind of thinking. It isn’t that they aren’t all true in some way for every student it is just that they lack a solution to help these students now.

    I go on the assumption that every student wants to be successful, wants feedback that will help them suceed, and in general wants your approval. The question I ask is if I could teach this student one-on-one for however many weeks could I teach them the material. Very rarely is the answer “no.” So then the only question is what does it take and how to make it happen.

    There is much more to say here but I’ll spare everyone until it picks it up again. Have you read Paul Tough, “Whatever it Takes”?

    Sean

  20. on 02 Jun 2009 at 5:57 pmMichael

    Are we fighting social/media Frankenstein that have been pieced together with government handouts, advertisements that remind these kids and their parents that their lives are not complete without the newest gadget, television programs and movies that discourage people from reading by giving away all the imaginative stuff all sewn together by feelings of entitlement?

    Ok, so I am exaggerating a bit, but there is truth to the social and media influences that acculturate our students much longer than our 55 minutes in the classroom.

    Apathy is a social problem. It starts at home and reinforced by TV, movies, games and advertising. Sure, we teachers can create lessons like action packed movies, but unless the sequel lesson is bigger and badder than the first, it will not be a blockbuster, and then we will loose our audience. Kids don’t know how to endure boredom and many lack curiosity.

    This is a social problem and no where do I see society or media saying “GET AN EDUCATION. IT IS IMPORTANT FOR YOUR FUTURE.”

    It is true that many of our students will never have use of the Quadratic Formula or Quadratic Equations in their lives. But the process of being able to understand something so abstract is extremely important when learning the abstract elements of a student’s career after high school.

    So, what do we educators do with this, when our hands are partially tied by state standards and high stakes testing?

    We clearly can’t change society. We clearly can’t overhaul the education system overnight.

    What can we do, now, in our classrooms given all our obstacles, in and out of the classroom, and our wealth of curricular, technological, and collaborative resources? Do we succumb to the advertising model or the TV/video game model of education? Do we keep the status quo? Is there something education has been too slow to catch onto?

    If every math teacher had Sean’s statistics of 50% failure rate, we will have half of our nation’s future voters and leaders undereducated for their future. With this deficit, our country will quickly become less of a world leader than we are now.

    The line must be drawn here. We must forsake the factory model of education expecting every student to learn the same way in the same time learning the same thing.

    Perhaps we need to teach those 50% failures something else. Perhaps we are not fulfilling a certain need that they have, because we have failed to identify it. Lets stop treating the symptoms and focus on finding the true cause(s) and start addressing those that we as educators can do something about.

    There are educators out there that are successful with dealing with apathy. We need to track them down, identify what they do and say, and try to implement that into our classrooms.

    Check out Power Teaching http://powerteachers.net/. I haven’t used it, but I plan to next school year. Maybe it will help, maybe it won’t. It can’t do any worse than what I am currently doing. Maybe it will work for you.

  21. on 02 Jun 2009 at 6:03 pmDan Meyer

    It’s like I need to take a personal day to do justice to the commentary here. Brief remarks follow:

    Sylvia: You are asking them to have an interest in an “unknown thing” when they have correctly come to the conclusion that this thing will not be on the test, and therefore doesn’t matter. There will be no clouds, raindrops, or videos of anything on the test.

    One of the steepest climbs of my (short) career has been an effort at blurring the lines between a) our miscellaneous opener questions, photosets, videos and b) our mathematics. We treat each with as much curiosity as we can muster, treating disagreement and agreement neutrally, always insisting on justification, holding correctness and speed in a loose hand, prizing intuition alongside calculation.

    We are trying to narrow the distance between the bread and the circuses, I guess. We don’t treat show and tell as a respite from serious, abstract thinking. It is the serious, abstract thinking.

    It’s pretty easy for me to criticize standards-based math curriculum but this year has been the most fun my classes and I have had trudging through all this algebraic abstraction. You have any worthy alternatives I should put an eye on, Sylvia?

    I dig Michael’s description here:

    Trying to get them to actually speak, to attempt to explain their answers, to think about each other’s reasoning, is asking them to engage in risk-taking and meta-cognition. Do I think that sort of teaching is vitally important? You bet! Do I think getting kids to engage in it, especially at-risk and remedial kids, is a simple process? I know from a lot of experience and observation of teachers in similar classrooms that it is not.

    While WCYDWT? media is often hit-or-miss, it’s fundamental to the process that these units scaffold from a visceral, initially-unanswerable question, the kind that begs an answer, an answer that no one will mock because everyone is working from the same limited set of data.

    My students chattered over, “Will the ball hit the can?” for example. Students went on record for “hit” or “miss” without fear because it was, at best, an intuitive, educated guess. They invested themselves. Then, with their implicit permission, we layered mathematical structures slowly on top of their investment.

  22. on 03 Jun 2009 at 3:11 amJen

    Sean said: The question I ask is if I could teach this student one-on-one for however many weeks could I teach them the material. Very rarely is the answer “no.” So then the only question is what does it take and how to make it happen.

    That’s a great question for any teacher. Though I agree that figuring out the implementation of it is the kicker. Any insights you’ve had so far?

  23. on 03 Jun 2009 at 6:18 ammacsilver

    Someone pointed out this blog to me and I find it fascinating to say the least. I gather that Dan feels there is a correlation between a lack of interest in his time lapse photography and a lack of mathematical ability. I find the video curious from a totally different point of view that I am sure Dan does. When an idea pops into a child’s head the idea pops out in some form or anther over and over and over. Very curious.
    But back to THE MATH PROBLEM, it is understanding the nature of the problem is what will lead to a solution. The problem is not with the teachers, nor the students, nor society, nor parents, it is actually in the mathematics itself. It is the smart people who can’t learn the math because it doesn’t make any sense and they refuse to try and learn something that doesn’t make sense. It is possible once you understand the problem to actually solve the problem which we have done. We have developed a program called cognitive instruction in mathematical modeling (CIMM). We are able to get 99.99% of students to LOVE mathematics and excel in it. The program (even to me) is just stunning but it required rewriting the mathematics from the ground up. Once done properly, the difficulties with fractions, decimals, percents, place value, negative numbers, subtraction, etc. all, fade away evaporate, disappear, as in your time lapse video.
    Curious, very curious indeed

  24. on 03 Jun 2009 at 7:39 amSue

    (I haven’t figured out how to get comments to dy/dan into my google reader, so I got behind on this thread, and I’m catching up now.)

    Wow! This is such a great discussion! I’m going through all the references right now, and wanted to comment on one.

    Michael, I went to that Power Teaching site, and read until I got distressed. It feels so false and manipulative. How can you have a real relationship with the students if you’re yanking their chains so thoroughly? It was the no guff page that finally put me over the edge.

    I did like the bull’s eye game, and a few other bits. But I would have absolutely HATED to be a student in a class like this. It is not acceptable to me to be forced to do silly things.

    On the other hand, thanks for the Summerhil anecdote. I think I’ll read that book again now.

  25. on 03 Jun 2009 at 7:39 ammrc

    You just need to love them more.

  26. on 03 Jun 2009 at 7:51 amMichael Paul Goldenberg

    I am always skeptical, to put it mildly, when I read about some new miracle cure for an educational illness that has plagued the nation (and/or world) for decades or much longer. So when I read the post above about CIMM, I have to be honest: my crap detector started beeping “May Day!” at a record-setting pace.

    My fears were not allayed when I found the relevant web-site and read the following in “The Math Problem, ” by Robert MacDuff, Ph.D., President, InfoDynamics Applications Ltd.:

    “To solve a problem, the first requirement is to understand what the problem is. The reason for lack of progress is that the math problem is poorly defined. Most mathematics education researchers have targeted teacher knowledge, both content and pedagogical, as the problem. Others assume that it is caused by a lack of parental support, by effects of change or breakdown in society or unwillingness of students to learn. All math approaches point to one or more of these issues as the crux of the problem.

    “An alternative possible source of the math problem, simply stated, is that it lies in the mathematics itself. Could mathematics itself be flawed? Frege2, Russell3, Cassirer4, 5, Kline6, and Hart7 and others have pointed out difficulties in the foundations of mathematics.

    “If the teachers, despite their considerable exposure to current mathematical content, cannot master it, then we must consider the possibility that the problem is with the content itself, rather than with the teachers. Certainly, if teachers have not been able to master this content, we cannot expect that any redesign of teaching methods, re-ordering of topic sequences, raising of standards or programs of high-stakes testing will result in their students being able to do so!”

    Okay, I’m all for being clear about the difficulties in effective mathematics education, but the notion that “problems in the foundations of mathematics” translates into “mathematics itself [is] flawed” is so stupid and dishonest that it’s a little stunning.

    Philosophical problems of the foundations of mathematics had nothing to do with the idea that mathematics is flawed, and anyone who is familiar with the philosophy of mathematics, the efforts of various logicians and mathematicians to ground mathematics in something more fundamental (e.g., logic), and the destruction of those efforts and the fond wishes that spawned them by Godel’s work is not likely to be worrying that mathematics is flawed, at least not in the ways that would require that we redo mathematics from the ground up.

    The fact is, from my viewpoint, is that foundationalism is always an enormous waste of time. And the problems that kids have with learning various aspects of mathematics has nothing to do with the issues that the people cited above by Doc McDuff were grappling with.

    Perhaps McDuff is just a little out of his depth on this one point, and the rest of his work is super solid. I haven’t read too far into his proposed miracle cure, but to be honest, I’m not optimistic that it’s quite the panacea suggested by macsilver or the glowing rhetoric on the site. Indeed, I’m expecting a lot of hand-waving and cog-psych jargon intended to hide that there’s just another naked emperor walking past.

    I’ve been around long enough to think I can recognize hogwash bottled as an elixir. Unless CIMM’s advocates can lay off on the typical rhetorical quick-steps and get down to how it would address the real problems of a 15 y.o. high school kid who can’t do 3rd grade arithmetic and frankly would like to be just about anywhere but in math class, and then multiply such results in ways that would work in real classrooms like the ones where I’ve taught and consulted, I think I’ll save my $0.50 for another barker.

  27. on 03 Jun 2009 at 9:03 amsylvia martinez

    Dan, I think you answered your own question to me when you said, “…these units scaffold from a visceral, initially-unanswerable question, the kind that begs an answer…”

    To me, that’s it. Finding those questions that lead to bigger questions and a need to answer them in precise ways. I’m not questioning the need for abstraction, but rather, the enormous amount of time spent devolving everything into algorithms (tricks).

    I don’t see much difference between the teaching tricks advocated by “Power Teaching” (yuck, by the way) and solution tricks like FOIL. It all skirts the issue of how to address big ideas and solve real problems. and as you point out, finding those onramps of interest to big ideas.

    As I read this post again, I’m wondering about the “show and tell” structure. Do students bring stuff in?

  28. on 03 Jun 2009 at 5:55 pmmacsilver

    I feel I must respond to Goldenberg. Here is a letter from an elementary school principal.
    To Whom It May Concern:
    This past fall, a third grade teacher, Sandy xxxxxx, began to take classes in Cognition Ignition Math. Sandy began to plan lessons in math using the CI approach. After just a couple of lessons, Sandy came to me to share how excited the students were in math and that they were doing CI math during recess!!! Sandy asked me to observe a CI math lessons and the students were not only actively engaged in the lesson, there was a buzz of excitement about their ability to solve difficult problems. One day, Sandy even called me to the room because the students had solved a very difficult problem and she wasn’t sure how to write the equation!!!
    Mid-year, Sandy’s third graders took a district math test to monitor their progress in math and Sandy’s students showed the most improvement of any third grade in the district. She has several students who have learning difficulties and several students who are in a class for Second English Language and yet each student showed significant improvement across the board. The students feel so confident about their math skills that they are offering to tutor the upper grade students at lunch and during the after school programs!
    Using CI math has not only affected their math scores. Our school had all students in third grade through sixth grade participate in a movie making project on computers. Sandy’s class did an exceptional job creating their movies and both teachers working on the project stated that this class was the most successful in the project because they are thinking outside of the box. Our students in grades three through six have just completed their state mandated tests and we are anxious to receive our scores in July to see just exactly what this third grade has accomplished. The students are confident that they have aced the tests, as they were bragging to their peers at lunch about how easy the math test was for them to do.
    Perhaps the most supportive statement from CI is a statement from one of Sandy’s students, “My sister is in middle school (7th/8th grade) and I was able to help her with her homework!” At xxxxxx Hills, we are thrilled that our district supports the Cognitive Ignition math approach and are training more teachers to implement this program next year.
    Sincerely xxxxx Principal

    Let me explain a little about this group. The schools grade three students were grouped into low, middle and enriched. The low group (20% special ed) out performed the enriched kids on the districts tests. The teacher initially disliked math and took our math course to upgrade her skills to be able to teach grade three, now all she wants to do is teach math.

    Goldenberg, a mathematics educator??, is clearly out of his depth on this one, but to be fair, if I hadn’t seen it I wouldn’t believe it either.

  29. on 03 Jun 2009 at 6:30 pmMichael Paul Goldenberg

    I’m not going to get into an insulting exchange with you, especially given the tone of your response. Questioning my profession, something you have no grounds for doing, is hardly collegial. (Of course, you don’t mention what it is that YOU do).

    I didn’t make ANY personal comment about you whatsoever, so what you offer above is indicative of the level of conversation you’d like to have. I note that you completely ignored one of my main points: that the material I quoted from Prof. McDuff was at best a questionable attempt to twist the debates about foundationalism in the philosophy of mathematics in the late 19th and early 20th centuries into some nonsense about there being something “wrong” with mathematics itself. Do you defend McDuff’s use of this in the way he did? I suspect not, as you avoided dealing with it. And you were wise to have done so.

    Further, I note that you don’t want to address the fact that this approaching is being touted as a cure-all. Naturally, your response is to cite some miraculous results. Maybe they’re real; maybe they’re not quite as advertised. As you’ve removed any identifying information, there’s really no way to check.

    But that’s not the point. What IS the point is that the style of promotion just reeks of commercial hucksterism. If it’s really as marvelous as is being suggested, why the thick coating of salesmanship and jargon? In my experience, whenever this sort of rhetoric appears surrounding some book, tool, technology, software, etc., the operant words are “caveat emptor.” And your response didn’t exactly alleviate my suspicions. On the contrary, they heightened them dramatically.

    p.s.: I personally find it rude when people who don’t know me address me as “Goldenberg.” Of course, since you’re posting under an alias, I can’t respond in kind were I inclined to do so. As I’m not a professor, feel free to refer to me as “Mr. Goldenberg”; “MPG” is fine, too.

  30. on 03 Jun 2009 at 6:34 pmMichael Paul Goldenberg

    There is apparent confusion on the part of a couple people here between “Michael” and “Michael Paul Goldenberg.” I am the latter. Don’t know who the former is, but it would be good if he’d provide a last name so as to avoid the already-burgeoning confusion. I was the one Dan quoted, though he referred to me only as “Michael.” And I was the one who mentioned the Summerhill anecdote to which Sue referred. However, I was most decidedly NOT the one who posted about PowerTeaching, and my response to what I saw there was precisely the same as Sue’s reaction. It’s a bit ironic to be thought of as someone who would advocate that sort of gimmickry. Indeed, given my posts today about another highly-touted panacea, it might have been obvious that I wouldn’t have been promoting a similarly-slick approach or web-site.

  31. [...] But How Do I Remediate THAT? May 31st, 2009 | Category: Uncategorized | Leave a comment [...]

  32. on 04 Jun 2009 at 3:46 ammacsilver

    To Michael Paul Goldenberg: If I insulted you by calling you Goldenberg, I must apologize as I didn’t realize this. If I had intended to insult you I would of called you Michael Goldenboy.
    I will not address the issue of the problems of mathematics as by your own admission you lack the qualifications to do so.
    However I would like to address your issue of ….”how it would address the real problems of a 15 y.o. high school kid who can’t do 3rd grade arithmetic…”. I gather that you feel that you and your colleagues have destroyed this individuals self-esteem to the point where EVEN OUR program can’t remediate the damage. And you are perfectly correct, the anger, frustration, anxiety, and fear have become so deeply ingrained that it is impossible to remove all of it. However, we can get the most to pass algebra I and geometry and get enough math credits to graduate (we have plenty of evidence documenting this). Will this student ever become an engineer, unlikely, you have done your job too well.
    Mr. Goldenberg, mathematics educator, think of it this way, approximately 85% of all students by the time they leave school are mathephobic (many A students fall into this category). Just suppose that what was wanted was a program that would create mathephobes, would the programs you advocate be a good choice?
    I would like to make a request, if you teach mathematics or consult in the teaching of mathematics, …….. I wish you wouldn’t. Why would you want to punish the 15 y.o. further for knowing that what you are attempting to teach doesn’t make any sense.

  33. on 04 Jun 2009 at 5:43 amMichael Paul Goldenberg

    Okay, got it: you’ve got nothing to say but offering up groundless insults to anyone who dares to question your miracle cure. Seen your kind come and go. After you rip off enough people, they’ll look for another miracle cure. And it will be just as big of a “miracle” as this one.

    As you know nothing about my work, I will not take personally your attempts to denigrate it. However, it’s very dirty pool you’re playing, and you do it hiding behind a pseudonym: that’s cowardice. When you post here using your real name, e-mail address, and details of what you do professionally, I might consider taking anything further you serve up seriously. I don’t indulge anonymous trolls for long, and you’ve used up your fifteen minutes.

  34. on 04 Jun 2009 at 6:42 amSue

    (Yep, I did have the two Michael’s confused. Also, thanks to Sarah, I’ve got the comments in my reader now.)

  35. on 04 Jun 2009 at 8:17 amMichael

    Mr. Goldenberg you are perhaps correct that I should give my last name. I choose not to share my last name for my own reasons which I do not need to defend here in this forum. It is however incumbent upon individuals who quote others or respond to others to ensure correctness in their references.

    As for my Power Teaching comment, I have tried many different ways of teaching my students. I would LOVE to follow Dan’s example in using slides and other media to provide a different perspective and application of mathematics in the classroom. Currently, however, I do not technology to use such a seemingly successful strategy. So I turn to something like Power Teaching to help me in other ways, such as to encourage student summarization of information in a way that is fun and engaging.

    I have never used Power Teaching, but I am willing to try it. Should I not be able to use it effectively in my classroom, I will try something else. I will be looking up “cognitive instruction in mathematical modeling” and evaluate it for myself and perhaps apply it in my classroom. I don’t want to be one of those individuals who keeps doing the same thing in my classroom expecting different results. If something works, use it.

    Finally, I agree with you, Mr. Goldenberg, that there is nothing wrong with the mathematics. But is there something wrong with how that mathematics is taught given all the antecedent conditions our students arrive with in our classrooms?

    Dan’s question is a very deep question. I appreciate all the comments with respects to attempting to come up with a collaborative diagnosis of the underlying issues and perhaps some reasonable solutions to a complex problem.

    I am grateful for Dan’s blog and his willingness to be so reflective about his teaching in such a public forum. Dan, you are very inquisitive, creative, and provide a realistic perspective to teaching math. Thank you.

  36. on 04 Jun 2009 at 9:18 amDon Cooke

    Perhaps give up on showing them cool videos and trying to teach them algebra in a conventional class, and instead give them digital cameras and have them make their own time-lapse videos.

  37. on 04 Jun 2009 at 5:27 pmMichael Paul Goldenberg

    Michael: how about using your first and middle name. Or first name and last initial? I’m not exactly asking that you put your address and phone number down. It’s just a matter of making some clear distinctions when you happen to have the most common male first name in the United states for the past 75 years or so.

    To be sure, there’s a good deal wrong with how mathematics has been taught in this and many other countries. The reason people like Dan Meyer do what they do is precisely to try to do better.

    Some of the comments here, and I need not name names at this point, as it will be obvious from the content of such posts, seem to be saying, unless I’m very confused, that Dan is just wasting kids’ time. Indeed, someone on another list (math-teach@mathforum.org) to which I forwarded the post that started this thread said that THAT is why these kids Dan mentions are apathetic: that they know he’s wasting their precious time by showing time-lapse videos. Of course, this same person hasn’t bothered to visit this blog to find out what Dan means by WCYMOT, etc.: she just KNOWS that it’s a waste and that those apathetic kids are expressing the same negative judgments of innovative teaching ideas that she has herself. Pretty amazing mind-reading, don’t you think?).

    Then we have the idea that some magic wand wavers have hit on a solution to all the ills that plague mathematics education. Just use Program X or Software Y (or Singapore Math, or Direct Instruction, or Saxon, or. . . ) and kids will suddenly become deep thinkers (well, maybe not with Saxon) and start really becoming engaged in mathematics.

    Actually, that’s not usually the claim. The claim if examined carefully is more like “will get better test scores,” the unquestioned assumption being, of course, that the tests they have in mind are deeply meaningful and that performing well on them is a valid measure of mathematical achievement. To which I generally must say, in my most polite tones, “Bull Shit.” For in fact, such tests generally can’t even be shown to reliably measure what they claim to measure, and they most definitely do not measure mathematical thinking beyond the most cursory definition of that term.

    But never mind. In the land of constant miracle cures (“the KIPP miracle,” “the Jaime Escalante miracle,” “the Houston miracle,” “the Texas miracle,” even the Arne Duncan “Chicago miracle”) there’s no shortage of supernatural amazement. Except if you actually try to find out what the kids who are supposed to be the living proof of these miracles can DO mathematically. Then it ain’t quite so miraculous.

    I’m hardly suggesting, of course, that what Dan is doing is a miracle cure, either. But it IS interesting and it does appear to have a good deal of potential. And unlike the charlatans, he is puzzled by a very real question: why isn’t it working with a particular group of kids and kind of kid?

    And that honest question is worth more than all the phony miracles the hucksters care to shill. Your mileage may vary, and most likely does (and I don’t mean Michael, in particular, I hope is clear).

  38. on 05 Jun 2009 at 9:07 ammacsilver

    I would like to pick up on something Michael C. (Mr. Goldenberg’s suggestion) said:
    “Finally, I agree with you, Mr. Goldenberg, that there is nothing wrong with the mathematics. But is there something wrong with how that mathematics is taught given all the antecedent conditions our students arrive with in our classrooms?”
    But before I do this I would like to add to the idea that approximately 85% of people in the US are mathephobic. I just had a long talk with executives of a large educational service provider, they feel the number is much higher. Is this not the exact opposite of what “math educators” are attempting to achieve? What could be more insane? Doing nothing has to be better, right?
    Some of you have guessed correctly (emails) that I do research in mathematics education. I also work with teachers and students in classrooms on a daly basis. In addition, I organize and run workshops for both teachers and students and author textbooks.
    Now, suppose that we accept Mr. Goldenberg and Michael assumptions that there is nothing wrong with the mathematics itself, then the only conclusion is the one that Michael has articulated. Research shows that over the past 60 years there has been no significant change in math scores. Mr. Goldenberg can attest I am sure to the fact that there have been numerous miracle cures that have come and gone. This actually is one of the largest difficulties impeding educational reform. But you have to consider this, there are thousands and thousands of very smart people who are working on or have worked on this problem from the pedagogical and content knowledge perspective.
    So you really have to ask the question why? I can attest to the issue of whether or not it is an intelligence issue and I can assure you that it is not. Don’t just accept my word on this also consider the word of a principal (earlier post) who witnessed a group of low achieving and special ed students out perform a group of enriched students and it only took nine weeks for this to happen. They just didn’t out perform in math, they out performed them in other areas as well.
    My point is that if all you attempt to change is the pedagogy as in “Power Teaching” (there is another program called project seed which is very similar) your chances of success are slim.
    Cognitive Instruction in Mathematical Modeling (CIMM) has a strong neuroscience component as the name suggests.

  39. on 05 Jun 2009 at 9:16 amMichael

    Macsilver, is there another forum which you and I can discuss further CIMM. I would be interested in learning more. Thanks.

  40. on 05 Jun 2009 at 9:48 amMichael Paul Goldenberg

    I doubt highly that you’d find anyone in this country willing to try to defend the position that on the whole, American mathematics education is effective (or at least that it couldn’t stand a lot of improvement). Mathematics educators have been leading efforts to reform and improve how mathematics is taught and what mathematics is taught in K-12, well before cog-sci people decided that they had all the answers (though not everyone in that field is quite as arrogant as the few I’ve dealt with personally and on-line, I presume).

    The figures being bandied about are a bit extreme (85% of the US population as math phobic sounds hyperbolic, unless what is meant by “math phobic” is operationalized much more clearly. Reminds me of claims that some ludicrously high number of women have been sexually assaulted: once the definition of “sexual assault” is made clear, the numbers aren’t quite as impressive). It’s too bad, in that there’s no need for such hyperbole (a sure sign that someone is selling something). It suffices to demonstrate that we’re not doing justice to many of our citizens in how and what we teach in math classes in K-12 (and probably in the first couple of years of college), and then to start to intelligently explore potential improvements.

    In case the cog-sci folks have failed to check the huge body of literature in the field of mathematics education, they really aren’t opening anyone’s eyes (except maybe their own) to a very well-known concern. But in the past 20 years, at least, it’s been difficult to make much progress because, in no small part, of the enormous resistance to change in the fields of mathematics (at the professional/university level) and within K-12 teaching itself. Researchers in the area of teacher practice can state unequivocally that trying to get experienced teachers to revise even small percentages of their practice (20% in a year would be a HUGE amount) is a very difficult task, and that’s looking at teachers who are consciously interested in improving their teaching. In the many cases where the teachers are openly resistant to change, it’s next to impossible.

    One of the single most significant factors in how K-12 mathematics is taught is how mathematics from K-16 was taught to the teachers themselves. So we’re looking at a complex “system” that is for the most part self-perpetuating. (As I write this, I’m trying not to chuckle too much at the suggestions by macsilver that I’m somehow engaged in trying to maintain the status quo and that I shouldn’t be teaching, all without having a single clue about my work and what I’ve stood for as a mathematics teacher and teacher-educator.)

    When I bump into a truly reflective and innovative teacher like Dan, I am always excited. He’s clearly not only doing something outside the box (outside of several boxes, in fact), but he’s self-critical (something I’d love to see evidence of from those who tout these miracle cures, be the cure “Direct Instruction,” from the University of Oregon, or some new magic cog-sci program from Pittsburgh, St. Louis, or wherever macsilver’s wizardry is being concocted).

    To me, as I’m sure I’ve made clear, there are no miracle cures, no panaceas, no magic bullets, and as soon as someone starts promoting a new one, I know that it’s 99.9% certain that it’s a shuck. Not that everyone who comes up with one of these deals realizes that it’s nonsense, but simply that nothing works in the real world quite the way it’s advertised, and the complexities of teaching and learning a subject like mathematics are far too great to be solved definitively and across the board by any one tool, method, book, curriculum, etc.

    What’s needed far more than magic is self-reflective, honest teachers who know mathematics well and are able to communicate what they know effectively to a broad spectrum of students, many of whom think and feel about mathematics quite differently from the teacher and from each other. I’ve been teaching since 1973, and have been teaching mathematics in various contexts since the early 1980s. And while it would be lovely to claim that I’ve got it all (or even a lot of it) figured out, I don’t and don’t expect to. Part of being a decent teacher is realizing that one will NEVER know enough mathematics, pedagogy, psychology, applications, technology, etc. to be able to connect with every student or to engage every student or to get every concept across to every student.

    When someone suggests they’ve got The Answer, I know one thing for sure: s/he doesn’t. And is far less likely to provide anything useful than will those teachers who actually try things and reflect about what is working, what isn’t working, and how they can do better.

  41. on 05 Jun 2009 at 10:06 amSue

    Hmm, 85% didn’t sound like hyperbole to me. I’ve always guessed it at about 80%. ‘It’ being the percent of the population who don’t like math and/or are uncomfortable with it.

    What would you guess that percent is?

    Can you imagine 80% (or whatever percent you give) of people not liking art, or music, or being uncomfortable with cooking? Yikes!

  42. on 05 Jun 2009 at 3:48 pmJason Dyer

    The difficulty I have with “find out what a student likes and tweak your curriculum to it” is that it doesn’t scale. I have done things like note a particular student complaining that a certain math concept was useless (logarithms) find out what they were interested in (psychology) and come back a few days later with a integrated lesson specifically on that topic.

    Sometimes it works brilliantly.

    However, a.) not every math topic intersects with every interest, so sometimes the customization is impossible and b.) with a class of 32 disaffected students there’s a limit to how much one can do; furthermore, getting in on one student’s interest will sometimes be entirely outside the interest of another classroom chunk.

    Here’s a story –

    Last year a taught a class to our seniors who haven’t passed our standardized test to graduate. Students at the end of their rope.

    After they had take said standardized test, I tried a statistics project where they researched any topic at all they liked and presented something about it, with the only requirement that use mathematics in some way (to make an argument, to present a story, etc.)

    Even when students got to pick any topic they wanted, were using technology for a project based assignment, and I used pretty much did every “reform” teaching trick in the book, a good chunk of students had the same apathy they always had.

    In the end, I have the say the experiment was a failure; the ratio of time spent to learning just wasn’t there in a way that would allow a sustained class.

    I’ve got some other experiments up my sleeve to try next year, but I sometimes wonder just how much I can do.

  43. on 05 Jun 2009 at 7:59 pmMichael Paul Goldenberg

    Sue, to answer your question, I will repeat my previous comment: “The figures being bandied about are a bit extreme (85% of the US population as math phobic sounds hyperbolic, unless what is meant by “math phobic” is operationalized much more clearly. Reminds me of claims that some ludicrously high number of women have been sexually assaulted: once the definition of “sexual assault” is made clear, the numbers aren’t quite as impressive).”

    Does that clarify my point? First, there is no real definition being offered of what comprises “math phobic”; second, no information about how that figure was arrived at; finally, if you keep things loose enough, by adding in your phrase “uncomfortable with math,” then it could mean just about anything.

    I have a master’s in mathematics education from an excellent university. Am I math phobic? Depends. Am I “uncomfortable with math?” Again, it depends. What math? In what context?

    Absent a context, and given certain cultural norms, the phrase really has very little meaning. You might also care to note that research by the late Harold Stevenson, co-author of the learning gap, indicated that a much higher percentage of students in the United States than in Japan consider themselves to be “good at math.” So we’re dealing with subjective information that may not mean a whole lot.

    You’ll note, too, I hope, that my previous post said that there was no need for any hyperbole at all. And my reaction is grounded in the lack of support for the claim, the fact that it sounds like it’s exaggerating in order to make a sale, and my overall impression of how this whole deal is being pitched from what I saw on that web site. When someone has to completely distort a well-known (to those of us IN the field) bit of history in order to sell a fraudulent notion (that there’s “something wrong with mathematics”) I frankly am going to deeply mistrust anything else that person has to say, especially when they’re trying to sell me on something that they need to push with an obvious falsehood.

    Did you visit that site? If so, what’s your impression of it, and the main spokesperson, and what they’re offering up?

  44. on 05 Jun 2009 at 8:20 pmMichael Paul Goldenberg

    Jason, a couple of comments: first, the notion of finding out what kids “like” and then adjusting one’s curriculum to that is clearly unworkable, as you’ve described it, but that’s not what I, for one, advocate. Rather, I advocated finding out what is going on in kids’ lives and then pulling things out from the curriculum that could be tweaked for individual kids to connect in some (hopefully meaningful way) to such things. When you find such things for any give kid(s), great. When you don’t, you don’t. So it’s not a matter of needing to make each lesson magical and deeply relevant, but increasing your awareness of kids’ lives and taking a shot when you can.

    Second, what you describe as a failure may in fact have been a very predictable failure, because given the kids (as you describe them, they are ‘disaffected’), you don’t generally such students know how to do original projects. And they’re not motivated to think of something. So freedom in this case is a BAD thing. They need guidance, they need scaffolding (about coming up with ideas, developing them, following through), and they need direction. I’ve had the identical experience you describe in a host of teaching situations, from teaching literature classes at the University of Florida in the 1970s to teaching mathematics in various high school situations over the past decade or so: even motivated kids may need some degree of structure, but your less-motivated, less successful ones definitely do.

    Not that you intended it to go that way, but this almost smacks of self-fulfilling prophecy: to some significant extent, we fail kids as teachers when we don’t provide the minimum requisite structure for the students given where they are as students of mathematics and students in a broader sense. Not blaming you. Made the same errors quite a number of times.

  45. on 05 Jun 2009 at 8:53 pmJason Dyer

    Michael, thanks for the comments.

    While there was some sort of learning, it was the ratio of time spent to learning that was off. If I spent the entire year in the same way at the same I’d only get through a handful of topics, and only scratch superficially at the mathematics.

    There was a school in a district near ours that back in the — I want to say 80s, although my memory is foggy — had math classes with an all-project-based curriculum and lots of interaction between levels. Ed research people observed, cooed over it. End result: “Calculus” students couldn’t even do basic algebra. The project approach led to such a scattershot understanding of mathematics they didn’t know it at all.

    My experiment reminded me of that — even with more involved students, the learning just wouldn’t be in enough a connected or unified framework to be considered a math class at all.

    A different scaffolding should help, as you say, which was one of the aforementioned new experiments. But it makes for (in my mind) an entirely different project.

  46. on 06 Jun 2009 at 4:46 ammacsilver

    I like Sue’s summary of the mathephobia issue: Yikes!
    I have been doing a lot of work with elementary school students lately and I would have to say that at least a third of that population is already turned off of mathematics. In their words it is hard and boring. As this cohort moves through the system it gets larger and larger. Roughly a third of the students entering high school have grade four math skills. I have only one word for this and it is “horrific (Yikes with a different spelling).” And it is even more horrific when you realize that a LOW group of grade three students can easily do grade 6 math along with a significant dose of algebra. We are massively underestimating the ability of kids, WHY?
    Consider the amount of psychological damage that you can do to a child by inflecting seven years of failure on them. Now I don’t know anyone that is eager to fail again after failing so often. Again my point is that longer, louder, and harder is not going to get the job done.

  47. on 06 Jun 2009 at 6:40 ammacsilver

    I would like to bring up Jason’s question but phrase it in a different way. Can we motivate students to engage in mathematical activities by making the problems more meaningful? (Jason is this roughly in the ball park of what you are attempting to get at?)
    When we learn anything we are able to see the world in a different light. If you have not learned it you can’t see it! What I mean is that the teacher can see the math but to the students it is invisible. Learning math by attempting to see the math in some context is a slow and laborious process. This is the main issue underlying the current math wars. Context rich and content poor versus content rich and context poor.
    Using meaningful contexts is good when the mathematical concepts are being applied. Attempting to learn math in a rich context means the understanding is context dependent with little or no transfer to other contexts. Dan’s original premise was what happens if we make the context really real. My point is that we need to have the student engage in different types of tasks in the learning process for those in the application process.

  48. on 07 Jun 2009 at 6:48 ammacsilver

    I would be remiss if I didn’t attempt to illustrate some ways in which mathematics is flawed.
    The problem: One-quarter plus one-quarter is ? (the one-quarter refers to segment of a circle, something we can easily visualize). Since Mr. Goldenberg is a mathematics educator and hopefully he will aid us in providing an experts answer.

    Others can provide answers as well, hopefully we can get some consensus.

  49. on 07 Jun 2009 at 9:23 amSue

    macsilver, i’m curious why you talk about your program with such certainty but don’t provide links to it? (or did i miss that? i’ve tried to run through this whole long thread to find a link.) also, i’m curious why you avoid stating who you are. (a quick google on ‘Cognitive Instruction in Mathematical Modeling’ makes your identity obvious of course.)

    i don’t know if you’ve ever visited math forum, but it’s full of exchanges like the ones here between you and michael paul goldenberg. mostly, these math blogs have more collegial conversations. mostly people identify who they are, and say things like “in my experience…”. it makes for a much more useful exchange, in my opinion.

    regards,
    sue
    mathmamawrites.blogspot.com

  50. on 07 Jun 2009 at 12:01 pmmacsilver

    Sue, first of all I am new to blogging and secondly names don’t really mean a lot. A couple of people have contacted me looking for more information. I sort of thought that those who really wanted to contact me could (macduff@asu.edu).
    As far as the program is concerned it is very new, we have been pilot testing the program for three years now. A school district with approximately 34000 students has adopted it. There is a difficulty with the program in that it you have to take a course in it. There is just no way for even someone with an extensive background in math to understand it. In fact the more math background the more difficult it is. As a mathematician, among other things it was extremely difficult for me to develop this program simply because it violated many of the precepts (called enculturation).
    The program tends to split people down the middle with some declaring it is not math and others who say, this is the essence of mathematics. It is not just the math that we have changed, but the pedagogy, the underlying theories of what does it mean to learn, what are the elements students use to construct knowledge, how does meaning arise, what types of reasoning does the mind do naturally, etc.
    From your experience, what percentage of students entering college take remedial math courses? And do you not think that is excessive? It seems to me that most of the people here feel that they could do better, a lot better if they new how. Are you one of them and could you elaborate?

  51. on 07 Jun 2009 at 1:18 pmMichael Paul Goldenberg

    I’ve read all the available downloads on Prof. McDuff’s site. I’m underwhelmed, to put it politely. The claim is that this is some radical reworking of mathematics and mathematical ideas appears without merit.

    Unless we’re all as gullible as Prof. McDuff would like us to be, and the entire mathematics education profession for the past century really comprises idiots and charlatans, there’s little, if anything, new in what he’s offering.

    Except perhaps for the vast ignorance displayed there of the history of mathematics education research for the past 60 years or so, serious reform efforts for at least the last quarter century, and a host of materials, books, articles, talks, etc. from many American and non-American mathematics educators. To name but one, does Professor McDuff really think he’s onto things that would be news to, say, Hans Freudenthal and the many researchers at his institute?

    I’m particularly amazed by his comment here that the more mathematics one knows, the less one can understand his ideas. Hmm. Does that strike anyone as both incredible and/or remarkably like the sorts of things we hear repeatedly in the history of pseudo-science and charlatanry? Yes, I know: you have to be “Clear” in order to read an E-meter, too.

    But really, this is just too rich: “There is just no way for even someone with an extensive background in math to understand it. In fact the more math background the more difficult it is.”

    Think about that. Prof. McDuff previous attacked me personally as not being qualified to judge the program, and presumed with no evidence whatsoever that I supported traditional mathematics teaching. He suggested that I was harming students and should leave the profession. That’s the sort of outrageous libel routinely offered up by – surprise- some of the folks from Mathematically Correct and NYC-HOLD towards those who challenge traditional mathematics education. I’ve had similar things said to me, quite directly and personally, by some of the more vocal proponents of the “grand” tradition of US math teaching.

    Now, Prof. McDuff comes along and, assuming I suppose that I’m a traditionalist, attacks me pretty much along the same lines. Just an odd coincidence? I think not. But it gets better, for Prof. McDuff has us all coming and going. I’m not qualified because I don’t know enough mathematics. And I’m not qualified because I do know a lot of mathematics. Brilliant. So everyone but he and a handful he’s trained are hence unqualified to understand or judge his fabulous “new” methods. It doesn’t get much sweeter than that, does it?

    Unless, of course, we start to ask for facts instead of rhetoric and hyperbole. Controlled studies? Not a one I can find. Anecdotal evidence? Well, there are some lovely testimonials from a teacher here, a kid there. But few districts and no states are going to commit to something based on endorsements alone.

    And where, I keep wondering as I read through the little information provided on the site, are the amazing breakthroughs? The really different concepts? And, for that matter, where are the truly deep mathematical ideas that this method would help kids understand so that they “loved” mathematics so much, as he says?

    I have a funny feeling that what he means is that that some kids enjoy doing elementary arithmetical computation via this approach. Nothing very new there. There are so many programs and gimmicks on the market that get similar responses from kids. And homeschooling parents. And teachers. Yet not one has been demonstrated in any serious comparative controlled studies to be superior to many other approaches and ideas and tools already out there.

    Nothing ultimately wrong with coming up with another. Of course, dots on paper vs. colored plastic counters? Hmm. Radically, deeply different, eh? I guess I either know too much mathematics or not enough mathematics or both to see the brilliant innovation.

    What has bothered me from the beginning about Professor McDuff, of course, has been, as Sue touches upon, the absolute certainty that he’s not only right, but Right. Not that he merely has something that helps, but that he has The Nuts when it comes to mathematics education and that everyone else is holding rags. It’s evident in his posts here. It’s evident on his web site. There’s only one thing that seems to be missing: real evidence.

    I will cite just one example of the sort of things I found at the site that puzzle and bother me. In a paper called “Enter The Dots,” Prof. McDuff’s co-author, Richard Hewko states in the opening sentence, “It has been estimated that 80% of the high school graduates in the U.S. have some form of math anxiety (anonymous grant reviewer with NSF). ”

    Hmm. So this vague notion, “some form of math anxiety” is backed by what research? None. Just the alleged statement of an ANONYMOUS NSF grant reviewer. Color me skeptical, to put it mildly.

    For one thing, I was a grant reviewer for the NSF in 2003. I am intimately aware of what it takes to work on such panels and how diverse the backgrounds are of those who serve. Lots of people who do this sort of thing are not trained researchers. Some are not educators. Some are not involved in either research or education. This is not to suggest that these folks are stupid or unqualified or anything of the kind. But they may well not be skilled at assessing the validity of research. So this anonymous endorsement, absence any information about training and experience, is effectively meaningless. Heck, *I* did this work, and as we know from Prof. McDuff, *I* am not qualified to do much of anything.

    Perhaps others here have or will visit the site and be able to offer a contrast to my impressions. I’m not claiming to have the definitive take on all this. But I am giving my honest impressions. I went to the site skeptical (as I believe we all should be when being shown miracles) but hoping to learn something useful and new. I have yet to find anything of the kind, but then, I’m not qualified. I suspect that anyone else who goes and isn’t entranced will also turn out to be not qualified. Funny how that works, don’t you think?

  52. on 07 Jun 2009 at 7:05 pmMichael

    There are many here who have commented to this blog entry who are way more experienced in teaching Mathematics than I am. I am grateful for their insight and perspective.

    I think we are all in agreement that there are some fundamental issues with regards to teaching Mathematics here in America, and perhaps worldwide. I think we are also in agreement that the solution is not simple, for if it were, we would know it and we would be using it in our classrooms. And, I think we are also in agreement that there have been methods that clearly don’t work, somewhat do work given certain conditions, and others that do work given certain conditions. We are all looking for a silver bullet to teaching mathematics, that may or may not exist.

    However, when discussing alternative methods of teaching, lets us stick to the evidence, as Mr. Goldenberg has mentioned, in determining whether a method works or does not work. Let’s not discount a method unless we have a counter example or other evidence to show it does not work, at least for a certain subset of students. Likewise, let’s either give the evidence to show a method works, at least for a certain subset of students. If we can do neither of these, then can we really say that a method does not work or does work? Furthermore, can we generalize that if it doesn’t work in our classroom that it doesn’t or won’t work in other classrooms or all classrooms?

    I suspect that we will need multiple methods to successfully ensure that “no child is left behind” (to quote a federally coined phrase that is directing modern education) in Math education.

    As a high school Math teacher, I feel the urgency in finding this solution quickly. Many of the students who enter my classroom are not ready for Algebra 1 and many of these student suffer through this same course for for multiple years in high school because of their misunderstandings and holes in their fundamental knowledge, habits of mind, and the ability to abstract and synthesize their knowledge to make new knowledge. And, as a result, they career choices are perhaps limited as a result, let alone their success in graduating high school.

    Given these antecedent conditions, a high school Math teacher’s job is certainly not an easy one, if it is at all possible, to ensure that “no child is left behind.” I can not imagine the challenges that primary education teachers have to address in their classrooms in meeting this same goal.

    Therefore, it is incumbent upon us to find a solution or solutions to this fundamental issue that Dan has posted.

    Let us focus on solving the problem. Everything else is a waste of time and energy.

  53. on 08 Jun 2009 at 4:52 ammacsilver

    On the question of a silver bullet. If we shoot them (the kids we see in high school who don’t get math) in elementary school the answer is a definite yes. If we wait until high school the answer is no, unless getting them to graduate is a silver bullet. Is it the extensive damage? or is it puberty? or both? I will say a lot more about the mental damage later.
    This program was not originally designed to be a silver bullet. It was designed to solve a common complaint by physics teachers that the math the students learn in math class doesn’t transfer over to the math needed in a science class. (The law of unintended consequences.)
    Michael since Mr. Goldenberg does not seem interested in solving the problem would you mind proving your answer. I can assure you that you will find the ensuing discussion fascinating.

  54. on 08 Jun 2009 at 5:19 amDan Meyer

    I am under-read and under-experienced here so forgive me. My question for Sylvia, Michael Paul Goldenberg, MacDuff, and anyone else who has considered math reform longer than I have is this:

    What do we do with rational expressions?

    These creatures serve no purpose in Algebra 1 except to prepare students for their re-appearance in Algebra II and, subsequently, in calculus where students will graph them, but nothing more than that. If they’re integral to certain engineering formulas I am unaware of it.

    Do we abandon concepts that defy placement in a real-world context? If you had to write your ideal curriculum, would you include rationals?

  55. on 08 Jun 2009 at 5:35 amMichael Paul Goldenberg

    Prof. McDuff, it’s not a “silver bullet” and we’re not shooting kids. It’s a “magic bullet” and what is supposed to be shot is the problems of mathematics education. Let’s at least keep our metaphors aligned, focused, and sensible, if we can’t agree upon anything else.

    Of course, the notion of “waiting” until high school to try to help students with mathematics learning is silly. But there are kids of high school age who need help, and those are the ones for the most part I’ve been asked to teach (had I the chance to go back to square one, I’d have gotten my degrees in elementary mathematics education rather than secondary, but it seemed like the “sensible” move at the time). Dan, too, is teaching older kids, as is the other Michael here. Sue teaches community college, which in my experience with that population is very similar to teaching high school students: similar problems and weaknesses, but a few to a lot of years older).

    Surely, you’re not suggesting that we abandon such folks simply because they’ve not had good teachers or experiences with mathematics previously? And surely your method should work with them, too. I know that I use many of the same tools, models, etc., with remedial students that I use when I work with elementary teachers and their students, as well as with would-be elementary school teachers, and the effect is generally positive (though of course not universally so).

    As for the the emotional (or do you mean something different by “mental”) damage: again, if you believe you’re onto something new, guess again. Or that there aren’t many serious, dedicated, knowledgeable professionals who are aware of, deeply concerned by, and working on ways to effectively address (correct AND prevent) such harm, again, you need to do more homework. I get the distinct impression time and again from what you write here that you don’t know the literature of mathematics education very well or if it. I don’t profess to know cog-sci; why do you seem to believe you know mathematics education?

    Those physics profs have been complaining for a long time that kids can’t do the math they need for science classes, but it’s not an issue of transfer. It’s an issue of approach. The problem has to do with doing applied mathematics in context, reading questions and calling upon appropriate mathematical and problem-solving skills to combine with scientific knowledge.

    Many programs developed in the past two decades have tried to address this weakness in traditional approaches and textbooks. But while some have been used more widely in K-8, few are used on a broad scale in high school? Why not? Because of the self-replicating system of traditional school mathematics that is at its most entrenched at the secondary and post-secondary levels. High school teachers are inclined even more than elementary teachers to teach precisely as they were taught. And as most students were ill-served by the limitations of such instruction when it comes to REAL problem solving, be the context physics, business, or what-have-you, INCLUDING authentic mathematical problem solving (the kind where you’re not just plugging and chugging, doing tiny variants from problems already covered in detail in class and in the textbook). This isn’t simply a mathematics difficulty: it’s a READING difficulty and a difficulty with the failure to help students develop effective habits of mind that scientists, mathematicians, and other theoretical and applied problem solvers MUST have if they are going to be successful.

    If you’ve got real solutions we’ve not seen before, Prof. McDuff, and if there’s evidence to support them beyond some nice endorsements, I have no doubt the world will beat a path to your door. But thus far, I don’t believe you do.

    Finally, don’t play silly games with me, because I will not indulge you. You clearly have something to say about the sum of 1/4 and 1/4 as sections of a circle. Just say it and we can discuss your insights. If you need a patsy so you can wax smug and feel brilliant, you’ve got the wrong fella.

  56. on 08 Jun 2009 at 8:06 ammacsilver

    Wow, Mr. Goldenberg, it is very clear that you are not certain of the answer. Hmmmmmmm not really sure how to respond. Have you not set yourself up as the expert?

  57. on 08 Jun 2009 at 8:13 amsylvia martinez

    Dan – thanks for an invite back into the conversation. I have to admit that I’ve been sitting on the sidelines. As Sue pointed out, this blog (and others like them) tend to be more collegial and respectful. I’ve found incredible value in Dan’s (and others) sharing their experience and willingness to think about what could be done better for his kids, and by extension, all kids.

    MacDuff/macsilver, if you are as you say, “new to blogs” I would think you would wait a bit to understand the culture here before launching so much personal attack against another person. It’s not pleasant and prevents those of us with less backbone and experience from chiming in.

    Because at this point, I’m not comfortable sharing. I don’t want to engage in what this has become. I hate doing this because I’m not a “let’s make rules” person, and I do respect people who speak their minds. But to me, macsilver/macduff, you consistently cross the line into personal attack, ridicule, and top it off with a sales pitch.

    I respectfully ask you, macsilver to take a step back and let others share their ideas without ridicule.

  58. on 08 Jun 2009 at 8:39 amDan Meyer

    I try to take a libertarian’s stance toward comments here. I don’t delete comments without fair warning first so consider this that. All parties here need to keep a safe distance from the line of ad hominem attack. Bring some evidence. Criticize but don’t insult. Cite some research. Keep your tones cool and your voices low.

  59. on 08 Jun 2009 at 8:40 ammacsilver

    Dan I am not sure I am following you. Are you asking what would constitute a “useful or meaningful” curriculum? And would rational expressions fall into it? I must admit that rational expressions do not fall into my list of high priority items. However they do provide an opportunity for playing around with and manipulating expressions. Learning to play and understanding the ramifications of that play are important skills for high levels of mathematical thinking.
    Students struggling with math have not developed an adequate understanding of number sense and this is where the focus should be.

  60. on 08 Jun 2009 at 9:21 amJason Dyer

    I would take rational expressions out of Algebra I. There’s enough mess to deal with as it is the curriculum needs the elbow room.

    They do indeed crop up quite a bit later. Possibly the easiest “early” application I can think of is the harmonic mean.

    Here is a writeup on relativity using a rational expression and the Pythagorean theorem I really like.

  61. on 08 Jun 2009 at 9:30 amMichael Paul Goldenberg

    Professor McDuff: I certainly haven’t proposed that I’m an expert, but you’ve repeatedly (and without foundation) proposed quite the opposite while suggesting from your first post that you have a miraculous cure-all. What you failed to do, however, was immediately identify yourself as the developer and salesman for that product. A shill is a shill is a shill, and you have repeatedly done a host of things that call your integrity into question. I won’t bother to list them again or the things available at your site that are evidence of gross misrepresentation of the history of 20th century mathematics and its philosophy, the weak citations you and your co-author use, etc.

    And now you really think you are going to trap me or anyone here with something? As I stated: if you have a unique insight to offer, by all means do so. Baiting me won’t work, because I’ve been down this path dozens of times with other people. I believe the onus is on you to produce an insight of any sort that is news to me or others with backgrounds in mathematics education history, research, methods, etc.

    But as I pointed out already, you have everyone “covered” with that lovely preemptive strike you launched earlier: anyone who knows math will find it very difficult to understand your work. Well, at the risk of being befuddled, I simply request that you dazzle me and other list members who also know mathematics. I’m sure I’m not the most knowledgeable member when it comes to mathematics, so maybe I’ll only suffer moderate levels of confusion and be mildly awe-struck. It’s a risk I’m very willing to take. All you need do is produce. Can you do so?

  62. on 08 Jun 2009 at 10:20 amMichael Paul Goldenberg

    Dan,

    Regarding the “place” of rational algebraic expressions:

    I rarely have any entrenched views of what must or must not be included in math curricula. I try to take a more organic/holistic view. Certainly there are some very basic things kids need to know, and with those things must come habits of mind about mathematics and solving problems that generally are ignored or given lip-service at best in traditionalists conversations about what mathematics to teach and how to teach it.

    So when someone asks if a particular topic is needed or where it is appropriate, absent some overwhelming evidence for or against it, I tend to think that it all depends. Who is the particular student in question given the particular mathematical fact, topic, technique, concept, etc.? Without that very individual context of person and content, it’s pretty much meaningless (for me, at least) to try to definitively prescribe or proscribe something like “rational algebraic expressions.” Of course, your state DOE and district academic subject coordinator may have a very different view. ;^)

    That said, I like a lot of things that arise in the context of RAEs. Asymptotes, for example, though of course they arise in other contexts (e.g., the tangent function). But the graphs can get awfully interesting when you enter into the world of RAEs, and it can be a lot of fun to play there.

    Another thing they bring into play that I have found heuristic for students is adding and subtracting RAEs. Because of my own experience with them (this was the first topic I hit when I started taking mathematics courses again in my mid-30s to learn calculus and beyond) that I didn’t instantly remember. It took me a bit to realize that to add or subtract RAEs, I needed to see them as (Doh!) fractions. Which meant common denominators. And then the rest was clear. (When we got to multiplying and dividing RAEs, the form of the polynomials was the hint that factoring and canceling factors would come into play: in that regard, just as with fractions, RAEs are easier to multiply and divide (from a mechanical perspective, anyway) than they are to add and subtract (from a conceptual perspective).

    So years later, when I saw some high school kids sitting looking pretty clueless over a worksheet of RAEs they were supposed to add (I was supervising a student teacher at a local high school for the University of Michigan), I suspected that they, too, were stuck on the same thing that had hung me up: they didn’t see the fractions, just a lot of algebraic symbols that they weren’t sure what to do with. Asking some questions got them to where they needed to be, and they were then able to move on.

    Later my student teacher asked me, “How do you do that?” And when I asked him what he meant, he replied, “How do you take something that is so obvious and break it down for the students?” Of course, this is central to a difficulty many high school teachers have, and particularly, in my experience, those coming from stronger universities where they’ve done pretty well as math majors and teacher education students: the things that confuse their kids are just “too easy” for them (because in general, folks who choose to study math as undergrads at places like U of M and who want to teach high school math are NOT kids who struggled with math themselves much in K-12). Such would-be teachers struggle more with empathy than with the mathematics. It’s difficult for them to put themselves in the place (and minds) of kids who don’t find the math “obvious” or “easy.” And everyone suffers.

    So I just told this fellow that I start with the assumption that nothing in mathematics is easy or obvious and go from there. Naturally, we subsequently spoke back at the university in the weekly practicum about this and related issues, and how to break down specifics concepts into smaller, more manageable pieces. But the first step is for the teacher to try to think more like an average or below average student, and to feel what such kids feel when they are struggling and can’t even formulate a decent question. (If I wanted to instantly give them such experiences today, I’d simply have invited some of my mathematics professors or graduate student friends in to lecture on something slightly above the heads of these student teachers. That can be a useful lesson for EVERYONE involved. ;^)

    Don’t know if this helps at all Dan. But my sense is that like most topics, this one can be “useful” in ways that may have absolutely nothing to do with whether you connect it with real-world applications. Never hurts to have some to pull out for those who are interested, of course. But I’ve gone that route, too, and discovered what I’m sure you and many others already realize: for some students, applications are at least as intimidating as are abstractions in mathematics, and if they’re at least committed enough to “school mathematics” to want to pass your class, they would prefer you “just give the equation and show how to do it” and stop bugging them with all this “extra” stuff. :^)

    Other kids really respond to applied situations. Some (probably not many in your remedial classes) actually can appreciate some of the aesthetic aspects of math and don’t care about applications. And then, there are those, as you’ve already mentioned, who seem to respond to nothing whatsoever that you try to show. And therein lies one of several dilemmas we face.

  63. [...] problem involving a rational expression Posted on June 8, 2009 by Jason Dyer Dan Meyer wants to know what to do with rational expressions. Here is an [...]

  64. on 08 Jun 2009 at 9:29 pmMichael

    Jason, what a succinct example showing the need of rational expressions. I had forgotten what the harmonic mean was and how to use it. Your example was very clear, for me. Thank you.

    However, what is the point of knowing all the arithmetic of rational expressions and equations at the Algebra 1 level. What does all of such solutions and simplification mean other than “that is what the book tells us what to do”?

    Could more valuable time be spent focusing on other fundamental elements of Algebra 1 and save such abstract concepts for a more rigorous examination in Advanced Algebra?

    In the mean time, I will be figuring ways of applying Jason’s problem in my teaching and playing with this idea of harmonic mean and its other uses. Thanks for a wonderful example.

  65. on 08 Jun 2009 at 9:35 pmmacsilver

    I would like to share one more thing with the group and that is a miracle. At least that is what the mother reports.

    “My daughter xxxx was in Mrs. yyyy’s third grade class this year. xxxxx is very creatilve. She is a visual learner and has severe language delay. I feel she is very smart, but she has not been able to function well in a traditional classroom.
    However, this year the classroom has been very unorthodox. This creative environment has shared a love of learning all the way around. She was very motivated to learn and her self-esteem grew tremendously. The dot program has helped her use all the parts of her brain. She had one part of her brain that was not functioning due to toxic encephalopathy.
    Last week she stood up at our garden science dinner and recited nine verses of Tennyson’s Lady of Shallot to ten people, just because she felt like it. This level of articulation and confidence was something we did not expect her to be able to do in her lifetime.
    At the beginning of the year she was struggling with simple addition, now she can do multiplication, division, and fractions. She is very worried that next year her teacher will not do the dots and she will loose her security. She is afraid to return to a traditional classroom.
    It is my hope that all children will learn this method starting in kindergarten. This will prevent a fear of math and enable all students to succeed regardless of learning style and ability.”

    This story is only one of many. The reason I am sharing this with you is that there are alternative ways to go about teaching mathematics (this kid can do way more that those skills listed above). But it requires an alternate viewpoint of what it means to know, learn and understand mathematics. Good luck with your endeavors. If anyone would like to get in touch with me send me a private email.

  66. on 08 Jun 2009 at 10:50 pmjosh g.

    So first off I will admit I am still trying to wrap my head around various chunks of algebra curriculum; I still know most of it as student rather than teacher, and as a student the lines blurred.

    That said, Dan, when you asked whether rational expressions matter, I was confused – surely they’re a given? Then I followed up and realized that the curriculum chunks you’re referring to include techniques like adding and subtracting them. My blurry experience from taking (computer) engineering is that rational expressions are all over the place, but I don’t specifically recall having to do addition or subtraction tricks; usually you could shuffle things around an equation, divide / cancel stuff, etc instead. I do recall coming across dividing rational expressions, I think.

    Just to make sure I’m not insane in thinking that these were common, I poked around and found this quote from Wikipedia (for whatever that’s worth):

    Rational functions are used to approximate or model more complex equations in science and engineering including (i) fields and forces in physics, (ii) spectroscopy in analytical chemistry, (iii) enzyme kinetics in biochemistry, (iv) electronic circuitry, (v) aerodynamics, (vi) medicine concentrations in vivo, (vii) wave functions for atoms and molecules, (viii) optics and photography to improve image resolution, and (ix) acoustics and sound.

    I don’t know if any of those would give you a good hands-on application; maybe a hook into relevancy, but Jason’s idea is probably a better approach anyway.

  67. on 09 Jun 2009 at 5:35 amJason Dyer

    However, what is the point of knowing all the arithmetic of rational expressions and equations at the Algebra 1 level. What does all of such solutions and simplification mean other than “that is what the book tells us what to do”?

    I don’t see one. I have never thought of them as a part of Algebra 1.

    Having said that, I think my own school’s Algebra 1 book doesn’t do the topic at all, so others have had the same thought.

  68. on 09 Jun 2009 at 8:46 amMichael Paul Goldenberg

    Is the real concern here WHERE in the curriculum or textbooks RAEs are placed? Seems rather passive to me. I guess I thought folks here were of a somewhat different mind-set. :(

  69. on 09 Jun 2009 at 9:15 amJason Dyer

    She is very worried that next year her teacher will not do the dots and she will loose her security.

    This is part of what worries me. That is –

    a.) Just because a teacher later decides to use numbers shouldn’t cause prior curriculum clash; curriculum design needs to be forward compatible.

    I can’t assume a student will next year be in the same city, state, or even country.

    For instance, a student who uses Singapore Math models and then switches to a class without them isn’t going to be suddenly unable to do math.

    b.) I don’t see how this system is forward compatible with negative numbers, or even multi-digit arithmetic.

    For instance, letting students count on their hands in class will help (and sometimes get reactions as above) but at some point they have to leap to the next step and be able to do 452 + 231. (Unless you want to teach Chisenbop, which isn’t necessarily a bad idea for certain students.)

    However, I realize I’m only seeing the promotional materials on the web and not the full package. macsilver, if you’re willing to send me a review copy (email jason dot dyer, which is at tusd1 dot org) I will do my best to give a fair assessment.

  70. on 09 Jun 2009 at 9:19 amJason Dyer

    Michael G., we are big thinkers *and* pragmatists here. :)

    After all, at some point we have to get dirty and teach it. If Dan has to worry about a California standard saying rational expressions should be in Algebra 1 (is that the case?), then exact placement in the curriculum becomes relevant.

  71. on 09 Jun 2009 at 9:27 amMichael Paul Goldenberg

    Oh, I understand, Jason. And of course, the prospect of even more restrictive NATIONAL standards (drafted mostly by the most clueless educational liberals and test-mad conservatives in the country, I’m sure), will soon help me sell my new textbook series: MATHEMATICS FOR THE PRE-BORN.

    I plan on many volumes, but one of them will surely be EQUATIONS FOR EMBRYOS. Other potential titles include FRACTIONS FOR FETUSES, BINOMIALS FOR BABIES, TRINOMIALS FOR THE FIRST TRIMESTER, ODEs FOR OVA, and DIFFERENTIATING IN DIAPERS.

  72. on 09 Jun 2009 at 9:47 amDan Meyer

    Sorry to have been such a passive contributor to my own thread. Finals and all.

    Though much of my current work involves context-rich mathematics, not all mathematics pins itself to a neat context. Jason’s example notwithstanding (and, if I’m not mistaken, “work problems” also involve the harmonic mean) RAEs resist context like few concepts I have encountered in Algebra.

    MPG entered this thread endorsing individual contact and real-world context — the wheelchair-bound grandparent, the real cost of leaving a lightbulb on. I won’t contradict him on individual contact, but I’m not sure that real-world context should be a primary motivator of our math curriculum. I see value, I guess, in teaching the abstractions of math, just as I see value in teaching the abstractions of philosophy, though they both may lack immediate real-world context.

    MPG seems to be saying, “Take every student separately. Some need more context. Some need more abstraction.” I won’t contradict that either. But class sizes larger than three necessitate a formalized, adopted math curriculum. Which is kind of a drag but, yeah, I’m speaking pragmatically here. Whither rational expressions?

  73. on 09 Jun 2009 at 10:35 amMichael

    Thinking pragmatically, one can weigh the importance of such topics on the standardized tests and make an argument in favor of perhaps forsaking rational expressions and equations completely to give more time for other topics like basic skills at the beginning of the year or on polynomials, factoring, and quadratics because it meets the needs of the students more.

    By meeting the needs of the students and giving them more time on task, could such sacrifices in curriculum pay out greater dividends later? What are such expected dividends?

    What is the loss of forsaking rational expressions and equations if such students never enter higher level mathematics greater than Geometry? Would we be teaching students that enough is enough and just getting by is sufficient and that it is ok to avoid difficult learning?

    Arguments can be made in both directions. Both have a sound basis and are valid. But as professionals, how do we decide? What is best?

  74. on 09 Jun 2009 at 10:40 amMichael Paul Goldenberg

    Dan, just to be crystal clear, I was speaking very specifically to your question about reaching those kids you are having difficulty motivating. It’s a process that needs to start on Day One of the school year by gathering as much information as you can in as non-threatening a manner as possible (lots of ice-breaking games make this sort of thing relatively doable, and it’s good practice for nearly ANY situation involving groups of people who will be working together in classes, business, etc.)

    How and when you use that information (which you try to update periodically) is up to you. You many never find anything useful for a given student; for some you may find useful things that somehow the right situation for never seems to arise. But the point is to try to get these resources as you can and think about them periodically: if opportunity presents itself, great.

    Further, I never suggested or would suggest that it’s only about applications. I don’t believe that and doubt that anyone would get far with me trying to make that point. I’m speaking to a particular dilemma you face and one alternative to what you are doing (not instead of, but in addition to).

    I have no definitive answer to your last question. If you are forced to do something prematurely, you have to scaffold that topic as best you can. And probably give it as minimal a treatment as the state assessment tools indicate you can. Unless, of course, you can find a way to connect that topic to what you’ve already done. I tried to make some suggestions along those lines in a previous post regarding issues of asymptotes, graphs, and ideas about rational numbers extended to algebraic expressions. But as you ask WHITHER RAES, I can’t help you there, at least in that my opinion is as good or bad as yours in that regard: if you HAVE to teach them, then you have to.

  75. on 11 Jun 2009 at 7:42 amSusan Socha

    I once asked one of my remedial students who worked at a local drugstore if she didn’t find her job boring since she had been working there for three years. I was surprised by her answer. She said, not at all. She said I like that I know exactly what I am supposed to do every day, and that there are no surprises. I like that I can suceed at this because there is nothing that happens that I can’t handle. It was apparent that she wasn’t “in” to surprises, and actually didn’t like them. We always assume that this is a bad thing. This girl was perfectly content with her job, and her success in that job. Are we assuming that all kids need to be curious and excited by new things to be happy?

  76. on 11 Jun 2009 at 10:17 amMichael

    What a great perspective that we teachers often don’t get to see after students leave our classroom. Thank you for sharing Susan Socha!

    When students learn a new concept or idea about once every day or two, there is this sense of “surprise” on a daily basis. Students may feel that they don’t have enough experience on the first idea before we move onto the next one, because that is what the district pacing guide says we are supposed to do. Couple this with holes in their foundational knowledge, it is no wonder students feel so frustrated and often give up.

    I will have to keep this idea in my head when planning for next school year.

  77. on 13 Jun 2009 at 5:27 pmstevehar

    Sitting here in San Jose Costa Rica, I’ve read through the 1st 79 posts and for the most part this is a pretty terrific conversation! Please don’t stop!

    Here is my take so far…

    I read from the point of view of a guy who in a earlier career taught engineers in factories to use math [applied descriptive statistics] to find solutions to problems embedded in real concrete “stuff” with cause and effect analysis, frequency distributions and average and range charts.

    Every time I started a class, to a person, each new engineer hated going. They didn’t want to go to another math [factory statistics] class.

    They would say: “Steve, we had statistics in engineering class, I got a great set of class notes.”

    It stumped me for a long time – how to engage them.

    One day I retorted: “let me ask you this, if you were at a Dentist getting a root canal and just before the laughing gas you asked the Doc how many root canals you’ve done and he said – well none but I got a great set of class notes. Would you get a root canal done from that Dentist?

    No? Well, what we teach in class is factory root canals.

    Let me see your last set of notes, how you addressed the problem, distinguished cause & effect how you made a testable hypothesis, how you collected data, let’s see your calculations and charts. They’d say – well we didn’t have to do THAT, we just had to use the formulas and crank the software program.

    The point of that story is it is about handling reluctant learners and inviting them into the game.

    Although they were skilled practitioners in engineering school – the engineers are novices in a factory that applies math to raise quality and reduce cost for the company.

    So…I noticed that in new learning environments novices are mostly “Tell Oriented”; but in a familiar learning environment practitioners are almost always “Question Oriented.”

    So?

    First, it seems to me “remedial” is a little harsh – maybe reluctant would be more helpful. These learners seem to me – as you describe them – to be spectators watching the math game; they are not IN your math game, and some have never ever been in a math game, period. How do you know they are “remedial”?

    Second, most of this conversation has been a rather lively Shared Inquiry among practitioners who are “in the math game”.

    A portion of this conversation has been an Advocacy Dispute involving displays of credentials and positions, challenges and back and forth verbal debate + sometimes a little snarky sharpshooting.

    To my view none of these pairs:
    - Inquiry or Advocacy
    -Ask or Tell orientation
    -Novice or Practitioner
    are good or bad strategies.

    Where the heat comes from and what lowers the quality of the conversation is missing the signals in the game. People are listening for inquiry instead they get advocacy or visa versa. They miss the signal calls. They run terrible plays.

    This is a little like suiting up for football and showing up for a baseball game.

    It is the basic root question about learners: how do you get people to suit up for the right game and start playing by the rules?

    In the factory my goal was to teach engineers to make terrific “root canals”…

    …but one time when my younger daughter Margot was in a 3rd grade math class the class made “stone” [chicken] soup, did taste testing, did frequency distributions of the ingredients and colored the frequency charts posted next to the taste tests…but that is another story

    Please don’t stop the conversation now, seems like you were just getting started…

  78. on 14 Jun 2009 at 3:19 pmMaria Droujkova

    Coming late to the party… I want to make two closely related points.

    1. Intellectual consumption may not be the universal good.

    2. Beauty/interest/learning connection is a huge can of worms, too.

    “How do I remediate that?” assumes “that” is a bad thing. It may be, but let’s be clear that a value judgment and an assumption have been made right there. Ditto about high algebra grades.

    Narrowly, the goal of a math teacher is to help students learn math, as measured by the people who hired the teacher. Broadly, as measured by the teachers’ understanding of what math is about. It is easy to assume that reaching goals toward which we work is good in itself, especially if goals are so darn hard to reach. We got to examine the goals, though, to determine their own value…

    This understanding of what math is about is cultural. For example, in a very bitter article “A Russian teacher in America,” Toom says: “It is a most important duty of a teacher of humans to teach them to be humans, that is, to behave reasonably in unusual situations.” http://www.de.ufpe.br/~toom/articles/engeduc/ARUSSIAN.PDF Then there is a girl Susan observed, who said, “I once asked one of my remedial students who worked at a local drugstore if she didn’t find her job boring since she had been working there for three years. I was surprised by her answer. She said, not at all. She said I like that I know exactly what I am supposed to do every day, and that there are no surprises.”

    According to information processing theory, human sense of beauty is based on seeing similarities (pattern, fractal, repetitions and so on) that reduce our information processing load. We are rather weak on memory and some other information processing capacities, so anything that reduces the load is highly valued. “Interest,” in this theory, is rather poetically described as the first derivative of beauty, equivalent to learning. Something is interesting if it has a high potential to be compressed by the beauty algorithms (i.e. learning). http://en.wikipedia.org/wiki/Mathematical_beauty#Beauty_and_mathematical_information_theory

    And that leads to Stevehar’s point about “spectators” – except I think these learners REFUSE to be spectators. They don’t want to consume/connoisseur mathematical beauty – similarity, fractality or pattern. They don’t want interesting/surprising things, those onramps to learning, because the compression process (learning) is the derivative of the unwanted beauty consumption process.

    Maybe instead of trying to offer different content for intellectual consumption, we can offer activities that aren’t consumption. Then the two cultures can meet and make friends during co-production.

  79. on 16 Jun 2009 at 12:52 pmDan Meyer
    “Interest,” in this theory, is rather poetically described as the first derivative of beauty, equivalent to learning.

    I have no idea what that means or if it has any practical implications, even, but “the first derivative of beauty” is a helluva thing to ponder. Thanks.

  80. on 16 Jun 2009 at 1:43 pmstevehar

    Left in wonder too about 1st derivative of beauty too…and the 2nd…N?
    Wow.

    Regarding spectators vs Maria’s REFUSEnicks.

    Seems like my engineers spectate from ignorance, whereas Maria’s learners are much more caught-up in an aversion to math or maybe an aversion to the whole enterprise.

    Possibly Maria’s refuseniks are headed elsewhere, not “fast-math” but “slow-food” & “slow-math” and craft; which might be a thing of beauty too…[see the New Yorker reference]

    My some-time taxi driver in San Jose talks excitedly of his son’s enthusiasm for math and something like tool design and getting his hands dirty in practice work sites, and what’s going to happen when he graduates into his craft.

    Father & son both love the open road: Pura Vida is what they say here over and over again, which translates poorly into English as… Pure Life.

    See
    Shop Class as Soulcraft: An Inquiry Into the Value of Work which is reviewed in the current New Yorker: http://www.newyorker.com/arts/critics/atlarge/2009/06/22/090622crat_atlarge_sanneh?currentPage=all

  81. on 17 Jun 2009 at 2:24 pmMaria Droujkova

    Stevehar, what a lovely article. It pulls in quite a few ideas and books I love. As a female and a multiple-time immigrant, I don’t feel like Crawford’s target audience, but it looks like he’s full of Pura Vida, which is cool.

    Introducing two more relevant references, both having similar cultural following to the works mentioned in the article:

    “Red Mars” trilogy with the idea of science-driven, purposeful socioeconomic change

    and

    “Here comes everybody” with the idea of cognitive surplus, measured in Wikipedias (100 million hours of volunteer work) http://www.herecomeseverybody.org/2008/04/looking-for-the-mouse.html ” When a TV reporter interviewed the author about social media, he writes, “she shook her head and said, “Where do people find the time?” That was her question. And I just kind of snapped. And I said, “No one who works in TV gets to ask that question. You know where the time comes from. It comes from the cognitive surplus you’ve been masking for 50 years.”

    Studies show watching TV (or YouTube videos) does take a lot of brain energy. The theory where “learning is the derivative of beauty” comes from may explain why. “Refuseniks” (funny word) who don’t spend their cognitive surplus in this manner aren’t necessarily to be helped out of their state. Maybe we can guide them toward some open road they vaguely long to take.

    Here is an interesting piece of data. Despite lack of fast internet, software and good computers at home, poor kids are more likely than rich kids to author and share content online (page 10 of this report http://www.pewinternet.org/~/media/Files/Reports/2005/PIP_Teens_Content_Creation.pdf.pdf). What can we make of it?