There is a report that came out in 2006 that I just discovered. It evaluates the effectiveness of the LSC program. (It was written by Horizon, Inc., under a grant from NSF). I wish I had found it earlier. It can be found here.
This excerpt taken from page 43 of the report is quite telling:
"Other evaluators cited changes in teachers’ beliefs about who can learn science and mathematics. For example, prevailing attitudes among some teachers before LSC workshops included low expectations and the need for ability grouping. LSC professional development helped change these beliefs. Said these teachers about the impact of LSC professional development:"Before IMP, I felt that there were mathematically unreachable students. I felt that students could not go on to more challenging ideas like algebra, statistics, probability, or trig without basic skills. Fortunately, with my IMP training, I have a different feeling about students. I strongly believe in access to mathematics for all. (Teacher, 6–12 mathematics LSC)"
The above quote from a teacher (in italics) is amazing. Before this teacher started using IMP, he/she felt that basic skills were necessary in order to proceed in mathematics. After IMP, which essentially avoids content whenever possible, he/she saw the light. Yes, wonderful things happen when you pretend that content doesn't matter, and that higher order thinking skills occur just by giving students "authentic" problems without the bother of all those and boring drills and instruction. They are able to reach for the stars. Unfortunately they do so by standing on a two legged stool. But NSF has done its duty and the people who wrote this report have confirmed what NSF always knew: Their reform math programs are an unparalleled success.
The only thing this report lacks is a chorus line kick and the ritual singing of Kumbayaa.
Here's an interesting, though admittedly anecdotal observation to add fuel to the fire.
ReplyDeleteI'm currently enrolled in a Master's degree program, specifically a Master's in Math Education. Our current course is Calculus 2 which is touching the beginnings of integration. The entire program to date has been strictly mathematics. The program finishes with two education courses but it has been strictly math so far.
There are about 15 teachers in the course and their backgrounds range from engineering (myself and two others) to math majors and on to folks who hold ed school degrees and masters in various specialties.
These are all very smart people and I'm not for a minute disparaging them or their abilities but there is a huge range of abilities with respect to mathematics in that cohort. Everyone can do the work (to get a correct answer) but the way it gets done across this group is instructive.
I do problem sets with 5 to 10 lines of equations in five minutes or so. Most of my peers are doing the same problems with enourmous amounts of work, covering several pages at times or at the very least covering one page with unimaginable diagrams, arithmetic, equations that lead nowhere, scribbles, arrows, and graphs, etc., in 20-30 minutes.
The difference is striking. Most times that they ask me for help I really struggle to do so. I just can't follow what they've done and my own solution provides them with inadequate insights. It's like we're speaking different languages. But, for the most part, they do get it done, and correctly so, if you're willing to overlook the lack of coherence to how it got done.
So I'm not surprised at the teacher's reaction in the post. There are lots of ways to get an answer to any problem in mathematics. The difference is whether or not you are trying to become a practitioner in the science of mathematics or just get an answer.
I can get an answer to a differential rate problem with skillful manipulation of Excel but that's not Calculus, it's arithmetic on steroids. I can also sheetrock, tape, and mud a room but you'd be nuts to hire me for such work.
My point is that a lot of these programs (CMP, Everyday, IMP) are developing people who can find answers but can't really do math. At least they can't do math in ways that would enable them to ever do the science of math.
My colleagues, most of whom are young enough to be my children, suffer I suspect, from two challenges. One, they've learned their math from the same fuzzy programs they are now teaching. Two, they don't have enough practical application to think mathematically. I don't propose to know if this has an effect on their teaching but I know it effects how they get the problem solved. They just don't have the practical experience and years of practice to pull off the science.
I also wonder if we, as a society, have figured out what we want from our kids in this regard. Do we want kids who can get answers or are we training everyone to be aspiring mathematicians. I suspect there is a difference in how and what you teach and who you teach it with if you can riddle that.
The teacher in the post is thrilled to discover she can get kids to find answers. I would like to ask her if the answer is provably correct or simple happenstance. The former is mathematics, the latter is 'Everyday'.
Exactly right. Take a kid who hasn't learned to swim and throw him in a swimming pool (deep end) and tell him to get to the other side. If he does it, it's happenstance. Maybe he can do it again, maybe not; he's just glad he's done with that assignment and never wants to do it again.
ReplyDeleteYou ask "Do we want kids who can get answers or are we training everyone to be aspiring mathematicians?" I don't think the fuzzies have figured that out. On the one hand they say they want "authentic problem solving" like real mathematicians do, and on the other, they say math has to be "real world" and "relevant". I say, start with the basics, the routine exercises that are held in such disdain, and students will eventually choose what they want to do, and have the foundation that enables them to succeed.
That is horrifying.
ReplyDeletePaul - is there anyone else in the class who can solve these problems efficiently?
There's about three of us who are 'efficient'. Don't get me wrong here. The inefficient solutions are often amazing. They solve things in ways I never would have dreamed of. The problem, if there is one, is that those solutions aren't written in the language of mathematics.
ReplyDeleteFor me, it's a bit like reading a wonderful story written without any rules of grammar or spelling. I don't fault the training or the people. I just don't think you can create mathematical efficiency on a fuzzy foundation.
I suspect this has very strong analogies in the world of ELA. We don't teach grammar, phonics, or spelling and we don't teach algebraic notation, equations, or logical proof. So should we expect great writers and mathematicians, and is this what we want anyway?
Maybe the clues are hiding in plain sight. The titles of what we're about, speak volumes don't they. English Language Arts and Connected Math don't exactly speak to the hard won specifics that build writers and mathematicians.
The fact is, it would be a lie to say that schools teach "reading, writing, and 'rithemetic" these days. Sounding out words, memorizing 6 X 4 - we're obviously going to lose these kids if we demand that they master these basic skills. This is the kind of thinking that pervades education, and as a new educator, I want to do terrible things to the snake oil salesmen who have peddled this ideology.
ReplyDeleteBut don't lay the fault squarely on the teachers. Many publications often pit parents (who depise EM) against teachers (who unqualifiedly embrace it). But practically every teacher I know (that's a lot) depises the fuzzy math programs to which our schools have subscribed.
I am a new teacher, 23, but I learned mathematics from a traditionalist perspective. I came to my school, as a newly minted educator, and I was horrified by the Impact Math textbook that I was expected to use. It is worse than Everyday Mathematics. It is constructivism on steroids, and I wish more people would talk about this program, because it gets short shrift compared to Everyday Math.
At no point does the textbook spell out any procedure necessary to solve a problem. It is riddled with convoluted "discovery" exercises that theoretically will help students induce the right answer, and the right way (or ways) to get there. There is a dearth of practice problems (i.e. the kind of problems that students will be expected to answer on a standardized test).
So how did I adjust? I simply never used the textbook. It could never work with CTT students who, across the board, were years behind in mathematics. I taught the way I was taught the material. It's a challenge because every night I have to hunt for activities and problems. But I do believe that the students have responded very well. It's a shame, because they don't have a standard reference (textbook) to help them do their homework at night, but even if we used the Impact Textbook, they would not have a reference.
Practically every teacher I know who has had the misfortune of inheriting Impact Mathematics does some variation of what I do. Even my principal is baffled by the textbook, but it was sold as the most widely used textbook out there. The higher-ups (chancellor and his ilk) are gung-ho about constructivist learning, and as a result, teachers, and sometimes even administrators, are slaves to a system they despise.
But hey, it's a change from the old way, and any change is good, right?
I find something else disturbing about the quote Barry put up.
ReplyDeleteThe two (false) poles the teacher has in mind are:
1) students are not learning sophisticated math because they can't learn simple things
or
2) mathematics is for all
or, as the report puts it:
low expectations yields ability grouping! how awful!
or
high expectations in math appreciation yields success without grouping by ability--how egalitarian!
NEVER did the teacher think "these kids have NOT BEEN TAUGHT". She thought that it was the kids' fault that the kids couldn't learn. She thought they were unreachable--they didn't know basic skills, and it NEVER OCCURRED to her that she needed to TEACH THEM, that teachers had failed to teach basic skills.
Now that she does math appreciation, though, she appreciates the students--her feelings about the STUDENTS have changed. They are no longer failures! Now she believes in math access for all!
This false dichotomy is a big problem, one of the several competing mentally unstable dualisms ed schools have.
Teachers often parrot the "all children are bright" line but follow it up with
"teaching doesn't mean everyone will succeed; some just aren't self-motivated"
The above shows that they know instinctively that the "all children are bright" is hogwash, but they feel guilty about it--because if it were true, why can't the teachers teach them, hm? And they fall back on blaming the students for being poor in character. But what relief when a curriculum lets them again believe all students are bright! then they don't have to feel guilty for not teaching!
I thought some more about Paul's example. Just a guess here, but I doubt that the students in his class are products of CMP or EM or Investigations. They're probably too old, so they would have escaped those programs. But not too old to have escaped "reform calculus" which is a constructivist approach to the subject and probably results in the inefficient yet clever ways to do problems. These "clever ways" will eventually fail, just like Ruth Parker's "turkey problem" that she always demonstrates (or used to) on her travelling snake oil show. It's a word problem that I can't precisely recall right now, but lends itself to a pictorial solution that eliminates any need for fractional division that a "traditionally" trained student would use. She pooh poohs the traditional approach as a lot of useless work while the approach she touts makes use of (wait for it) higher order thinking skills.
ReplyDeleteThe problem is that if the parameters of the word problem are changed, the diagram she uses doesn't work, and one is forced to use the traditional method. Learning the basic procedures allows students to generalize to a variety of problems, whereas the "draw a picture" and "guess and check" approach has limited application.
While Paul's classmates' approaches seem clever, so does Ruth Parker's. Once you look beyond the clever approach, you see that it will not take them very far beyond calculus because their foundations are too weak.
I suppose someone will say "Yeah but Singapore does the same thing with its bar models and nice numbers". Not exactly. Their approach is used to teach some fundamentals about part/whole and proportion and Singapore does in fact extend problem solving to those that can't be solved pictorially, and must rely on symbolic representation. The reform calculus and other poor math programs don't have such an end in sight and put their trust that higher order thinking skills will supplant any of the "mere exercises" so despised by the charlatans who have sold thousands of students down the tubes.
Where's the Beef in the Turkey Problem
ReplyDeletehttp://www.intres.com/inpage/pub/turkey.shtml
"But what relief when a curriculum lets them again believe all students are bright! then they don't have to feel guilty for not teaching!"
ReplyDeleteI like this comment.
If they talk about discovery and understanding enough, then they don't have to explain why kids don't know what 6 times 7 is in fifth grade. That's what my son's fifth grade Everyday Math teacher did. She didn't cover 35% of the material, but she declared victory over problem solving and critical thinking. If you don't have to define education in concrete terms, then you can't fail.
K-6 teachers love to be nurturing and they want schools to be a pump, rather than a filter. Backwards math fits the bill. Redefine math into some sort of magical thinking process and you're all set.
Then kids start to get tracked in 7th grade and you really can't keep ignoring mastery and content. But now, as with advisories, everything is the student's responsibility. My son apparently has to define goals and sign an agreement about what he will do to meet his goals. It's now a lot easier to blame kids and get them and their parents to believe it.
In K-6, they redefine education so that there is no such thing as failing. It's all about being developmentally appropriate. If a child doesn't learn, it's because they aren't ready for the material yet. Everyday Math's spiral is built around this idea.
Then, starting in 7th grade, it becomes the student's responsibility. Add in some students who do well, and you have an environment that is unwilling or incapable of seeing fundamental problems.
Redefine math into some sort of magical thinking process and you're all set.
ReplyDeleteRight. And if students don't go on to higher math courses, it isn't because they can't do it--after all, they know the thinking processes that will enable them to succeed--it's because they don't choose to.
right again. and this is important *because*
ReplyDeleteone of the most important messages of school is
"you have been weighed in the balance
and found wanting"... but the student must
be made to believe that this "finding"
is *their own fault*...
For those curious about Ruth Parker's "turkey problem", Ralph Raimi wrote a very good description of her presentation of same
ReplyDeleteThere are two (more?) problems I see with this situation: First, is that what has changed is the teachers' belief about whether students can learn. We don't actually have any evidence here that the children have learned anything. As a teacher, I can believe anything I want, but if my students can't do the work, it's not going to help them in the long run.
ReplyDeleteThe second comes from looking at the reform books--especially the middle school and high school books like IMP--they have interesting activities, and the problems make you think and learn stuff, but almost none of the stuff it makes you learn is stuff that it was hard to teach students the old way. If you pick almost any hard topic out of high school algebra, and look for it in a reform book, you will find that they either don't teach it at all, or they they address (and ask) only the easiest sorts of problems. Not very satisfying. That, by the way, is what makes Singapore integrated math (yes, it is integrated) and US integrated math--in Singapore, they assign hard problems.
(hmm--my word-verification word is klutses. I wonder if that says something about me?)
This comment has been removed by the author.
ReplyDeleteA two-legged stool is more properly referred to as "a ladder," and you can indeed use one to reach new heights.
ReplyDeleteThe comments so far propose an "either-or" response of the variety to which they simultaneously object. That's deliciously ironic.
So, it honestly comes to a very basic question: what are the purposes of schools in general, and math instruction in particular?
Strategies like "guess and check" are used legitimately and effectively by adults of all stripes a million times over on a daily basis. Is it the most efficient way? Of course not. But as we used to say back in the day—often times, it's "close enough for Jazz."
99% of the careers in Western civilization don't even require an understanding of algebra. That certainly raises some questions about the need for advanced math in our public schools, no?
The students who become statisticians and physicists have an innate mathematical aptitude. They will succeed no matter what math series is used in the elementary grades, so long as we continue offering advanced math courses in high school and college.
The rest of us need to calculate square footage with calculators and balance our checkbooks. The selection of lower level math series is likewise irrelevant to securing these skills.
Our time would be better spent demanding our educators leave college with a solid grounding in mathematical understanding (not at all true today) and substantive knowledge of educational best practices (also not at all true today).
L-squared: I disagree that IMP is as you describe it. The content taught in traditional algebra and other math texts is NOT taught in IMP. Sequence and structure are lacking. So, by the way, is content. IMP is where the haybaler problem originates, which has been talked about at length at KTM.
ReplyDeletebitch.y, phd: You say The students who become statisticians and physicists have an innate mathematical aptitude. They will succeed no matter what math series is used in the elementary grades, so long as we continue offering advanced math courses in high school and college.
Mathematics is not innate. It's something you work at to learn. Some students work harder than others to learn it. Vern Williams who is a middle school math teacher and was on the President's National Math Advisory Panel, teaches gifted students. Gifted students need instruction in math as much as others--and good instruction. Vern uses a variety of sources, but relies heavily on Dolciani's algebra text which has good explanations, is sequenced well, and has many good problems. The purpose of education is to open doors not slam them shut. That some students will never use algebra and other math in their careers is no reason to lock others out, or presume that those interested will learn it on their own.
You can solve some fairly complex problems, 2 equations with 2 unknowns for example, using simple bar models and not a stitch of algebra. This is exactly what Singapore math does. When you see this for the first time, you might very well go "Wow, that's cool". The problem is that these methods are limited in their range.
ReplyDeleteThe difference between it and our fuzzies is that Singapore does it as a means to an end. The bar model is a transitional tool to get to equations and they are eventually left behind in Singapore Math. In CMP the guess and check, do it with a table, draw a graph, make a picture, is the end. It never really makes an attempt to use this as a transition to algebra.
If your goal is to produce problem solvers who can reason out problems by whatever works that may be enough. If you're trying to create students who will eventually move on to college level math courses, it's weak and even harmful.
I've got some really bright kids this year. You might even say they are innately suited for math but they are stymied by my attempts to transition them away from guess and check and on to a more formal algebraic solution. They understandably argue that my algebraic solution is harder to understand but they fail to see that their solution is not extensible into the next level of complexity.
It's easier to work with kids in the middle whose number sense fails them on guess and check. They're hungry for a way to solve such problems that will always work.
When kids have had 7 years of problems based on small whole numbers to solve by any way they can discover, they develop bad habits and misconceptions that have to be untaught if they're to move beyond that environment.
Bitch.y.phd, you just made a case for not having schooling at all. How many students go on to use chemistry in their careers? Biology? Spanish? World History? Band?
ReplyDeleteNot very many, so why teach them at all? Because as Barry says, "the purpose of education is to open doors (sic) not slam them shut." Students are exposed to a number of different disciplines in order to 1) make them learned members of a civilization and 2) help them choose a career later on. Kids aren't born with a congenital disposition towards engineering. They become engineers after taking, and presumably, succeeding in math.
If we taught watered-down mathematics that catered to the non-mathematically inclined student, we wouldn't have many engineers at all. You might say that America might even lose its cutting edge. Oh wait, that's exactly what is happening.
The students who become statisticians and physicists have an innate mathematical aptitude. They will succeed no matter what math series is used in the elementary grades, so long as we continue offering advanced math courses in high school and college.
ReplyDeleteI'd like to take issue with this, too.
William Schmidt has a very nice passage about this very issue in his Leading Minds lecture:
My personal anecdotal story … is they [tracked] me in the middle grades. I just got pissed off - I probably shouldn’t say that - I just got angry at them and so I went out and got a degree in mathematics and a Ph.D. in statistics. But you know what? But you know what: in 8th grade they said I wasn’t smart enough. But not enough kid has enough fight to want to do that.
We've lived this reality firsthand.
When people constantly tell a child he's not that bright, or he can't do math, most children (I assume it's most children) will take the grown-up's word for it.
I recognize William Schmidt's personality; I was always a bit like him. If somebody told me I wasn't smart enough to do X, Y, or Z, that just ticked me off.
But that is one kind of personality.
I have watched smart kids languish and lower their sights.
I think I'll spell this out a bit more.
ReplyDeleteWhen C. and I went to our high school here for his placement session with the guidance counselor, she said to him, directly and repeatedly, "Math is a struggle for you."
"You struggle with math."
"Math is a challenge for you."
She said this so many times that I finally said, "Math isn't a struggle for him when I teach it."
That is: she said "struggle" so many times that I actually opened my mouth and said what normally I would only have thought to myself.
We are still dealing with psychological problems about math. Test anxiety, specific to math, and probably other things as well.
Talent isn't bulletproof.
Allison's post about mathematically talented kids needs, which she wrote in response to my saying gifted kids don't need much in the way of teaching (!), is here.
ReplyDeleteAlso, Geoff Colvin's book is quite good on the issue of "talent" versus "deliberate practice." (Have only read the first sections...)
ReplyDeleteAnonymous - if you're still around - (thanks for commenting!) - do you have a sense of how teachers fall out in terms of these curricula?
ReplyDeleteA friend of mine told me this week that her child's teacher said that when she first had to teach Everyday Math she thought it was awful but now she's a fan.
Is that common, do you think?
My son apparently has to define goals and sign an agreement about what he will do to meet his goals.
ReplyDeleteI'm still scandalized by this.
If the teacher or principal were signing a document saying what the school was going to do to help him realize his goals I would feel differently.
Remember when Ken told us that a contract with one signature isn't a contract?
one of the most important messages of school is
ReplyDelete"you have been weighed in the balance
and found wanting"... but the student must
be made to believe that this "finding"
is *their own fault*...
right
right, right, right
Diagnosis Diagnosed
How many students go on to use chemistry in their careers? Biology? Spanish? World History? Band?
ReplyDeleteNot very many, so why teach them at all? Because as Barry says, "the purpose of education is to open doors (sic) not slam them shut."
right!
Also, and I still don't know how to put this well, we do use a great deal of the K-12 knowledge we acquire in making sense of the world.
Try reading the TIMES Science section without having taken a sequence of decent high school science courses.
I think the work on "simple heuristics that make us smart" must relate to the question of why it is we believe in mass education (for those of us who do, of course).
But I'm too tired to tackle that at the moment.
Point is: I have used my K12 knowledge in many settings & on many occasions.
I've used some of it in my job as a writer but I've used nearly all of it in trying to understand what's going on around me and to make decisions.
Barry Garelick:
ReplyDeletestudents. Gifted students need instruction in math as much
as others--and good instruction. Vern uses a variety of
I couldn't agree more. Yet I've heard more than once that all the better students will figure everything out themselves, no matter what, and our mission is teach the lower achieving students.
In education it seems that "all students must succeed" has morphed into "all students must succeed AT EVERYTHING". I have students that will likely be very successful fashion designers/marketers or artists, but they're in my class because they have to be somewhere. Yet I'm supposed to take the greatest care that they succeed at the expense of kids who chose to be in my class because they want to be engineers or scientists. The latter group will figure it out for themselves.
It's like forcing someone like myself into a ballet class, identifying me as the student with the "greatest need", and then insisting that the instructor spend the most time on me even though I'll just squander the effort
Kids with high math aptitude do need good teaching to reach their potential; most cannot/will not do it on their own. But there's more to it than that. We are killing their interest and motivation.
ReplyDeleteMany elementary kids with high math aptitude who suffer through the likes of Investigations and EM are left terminally frustrated with math. How many of us have watched the transformation of our kids from eager first or second graders who loved math, to irritated 5th and 6th graders who no longer can even enjoy it. And since they've checked out for a few years, many are going to find it very difficult to pick up the pieces in high school. They'll have gaps. They'll be behind. And they are not too likely to struggle back to the heights they would otherwise reach.
We are losing a generation or two of potential mathematicians. They will be doing other things.
They will succeed no matter what math series is used in the elementary grades, so long as we continue offering advanced math courses in high school and college.
ReplyDeleteJust for the record, IMP is a four year high school math program. It IS supposedly the advanced math courses that some seem to think motivated kids will automatically take in high school.
And while I have met mathematicians who indeed read books and taught themselves, the point is they read books. Even gifted people need well written textbooks. They didn't get there through the 2-legged stool/ladder that some seem to think is all students need.
I should have thought of this earlier, but the easiest way to see the critical importance of teaching to gifted people is to look at sports.
ReplyDeleteIn sports we all understand that the most brilliant athletes need the most brilliant coaches --- or, at least, they need extremely good coaching.
It's the same principle with math.
I wonder what it would be like to have real coaches for real mathematicians --- you know, the way professional athletes have coaches beyond the point at which they're learning the sport & acquiring their skills....
Which reminds me.
ReplyDeleteI must post Ericsson's thoughts on the nature of "deliberate practice" for scientists...
They didn't get there through the 2-legged stool/ladder that some seem to think is all students need.
ReplyDeleteOK, now you've goaded me into saying this: a two-legged stool is not a ladder.
A two-legged stool is a broken stool lying on its side.
My autism genes got a tiny bit activated there.
ReplyDeleteKids with high math aptitude do need good teaching to reach their potential; most cannot/will not do it on their own. But there's more to it than that. We are killing their interest and motivation.
ReplyDeleteABSOLUTELY.
And this is happening everywhere and always.
How much "interest" and "motivation" does C. have where math is concerned after all he went through here?
His teacher this year has done a fantastic "repair" job, but still.
Before middle school he had (some) natural interest in and enthusiasm for math.
I'm hoping he'll get it back as time goes on & I'm thinking it's possible ----- but that doesn't change the fundamental point.
physics teacher (if you're still around) --- would your situation be different if your classes were ability grouped???
ReplyDeleteThat's what I like about ability grouping: you can have a physics class filled with future marketers (or whatever) and they can learn at a brisk clip without dragging down the kids who are headed to math/science careers.
ReplyDelete"So, it honestly comes to a very basic question: what are the purposes of schools in general, and math instruction in particular?"
ReplyDeleteI'm waiting for the answer ...
... and ... ta da! here it is!
"99% of the careers in Western civilization don't even require an understanding of algebra. That certainly raises some questions about the need for advanced math in our public schools, no?"
Darn. It's still a question. Isn't this kind of like saying something without really saying it.
But OK, I'll go first. I say that "advanced math" is anything above algebra in 8th grade. You get the kids there and then we'll talk about high school.