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.

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.

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.

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.

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.

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.

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). ;^)

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

@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.

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.

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?

(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.

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.

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.

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.

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.

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.

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).

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

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!

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?

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.

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.

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

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?

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.

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.

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.

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.

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?

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.

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.

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. :(

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.

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.

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.

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.

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