As I discussed previously, I am often in agreement with Alfie Kohn in his writings about education.
One big disagreement is over the issue of Chicago Math, aka "constructivist math". Kohn is a proponent of this approach, and he has written with contempt about the parents' groups that oppose it (see "Only for My Kid".) I've done a lot of reading about Chicago Math (on sites like kitchen table math and Out in Left Field), and my daughter suffered through two years of Trailblazers at her private school. I have come to the conclusion that on this issue, Kohn is just plain wrong.
The Chicago Math curricula are really terrible. Why? Because they contain hardly any math, and a great deal of pointless wankery, such as "write an essay about your favorite number". (True!) Kids can have years of this stuff, and still be unable to do even the simplest computations. When my daughter had Trailblazers, I went out of my way to tutor her at home with Singapore Math (I recommend it). Several of her classmates are now taking remedial math classes.
So, much as I want to like progressive ed, I also genuinely care about content. I want my kids to be capable and confident with math, and Trailblazers won't get them there. I want my kids to learn stuff, and I also want them to enjoy their childhood and not be overburdened with homework, pressure and competition.
In a similar vein, I really wanted to like this essay, "Turning Schools Into Robot Factories". And I did like it, right up to the point that the writer complained that upper elementary students "were also taught the formula for dividing by fractions (as if anyone ever does such a thing)". I want my kids to be able to divide by a fraction. It's part of the logical system known as arithmetic, and I think it's basic for any educated person. Is that too much to ask?
We used Singapore Maths and loved it. One of the big complaints about it is that it can't easily be used to teach children of varying abilities that WILL be in your standard classroom. Heaven forbid one child be more proficient, or get ahead of, another child.
ReplyDeleteMore and more I'm convinced that NCLB really is going to mean less REAL teaching for white middle-class children. Upper-class parents won't put up with this not teaching their children properly stuff.
PS. My nine-year-old can divide by a fraction. It involves multiplying by the inverse. When you're done with Singapore Maths, give Teaching Textbooks a try. Only take the placement quizzes first so you buy the right one... it is behind Singapore and you will probably start into seventh grade after fifth grade Singapore as we did. (Hope that makes sense.)
FedUpMom,
ReplyDeleteAlthough I consider myself a "progressive," when it comes to educational issues, I have to agree with you on this one. What irks me most is the way math education is framed as a liberal/conservative, left/right issue. The ongoing "math wars"--as conceived and reported on in the media (more so in the States than here, where the war has been more of a minor skirmish followed by complete surrender)--reinforce this counterproductive, politicized way of looking at the issue.
The problem I have with "constructivist" math is simple: it frustrates and confuses kids, even--perhaps especially--those kids who have a knack or gift for math. It does this by eschewing incrementalism (always putting the cart before the horse--eg, giving kids algebra problems without first teaching the tools of algebra), and by expecting kids to reinvent the wheel when it comes to algorithms that have taken mathematicians 2000 or so years to perfect. Another issue I have with this type of math instruction (I have so many that I'm planning to write my own blog piece on the subject--but it's daunting because I have so much to say) is that it is patronizing in its emphasis on concrete materials and group work. Both my daughters enjoy math much more when it is abstract: numbers on a page that they can play around with. As opposed to a million pictures and silly blocks. My husband and I were the same when we were young: the more abstract the math became, the better we performed. My husband went on to become a very gifted math student; today he uses high-level math in his job every day, and he hates the kids' math program--thinks it's "anti-math." It's interesting to me that when I look at the authors of my daughters' textbook (called Math Makes Sense, referred to in our household as Math Makes Nonsense), there is not one actual mathematician listed. All the authors and consultants are education professors or elementary teachers. I think that speaks volumes.
One more thing: the constructivists' main argument against teaching traditional algorithms is that they don't allow kids to learn the "concept" behind them. Only a non-mathematician would be unable to see the obvious truth that all algorithms contain within them "the concept," (which becomes evident with use); otherwise they wouldn't work!
northTOmom: however, it is possible to teach math from concept to algorithm without a so-called constructivist approach that you are mentioning. Singapore Math is brilliant at this. While some students will figure out the concept behind the algorithm, for most, learning the algorithm first (and only) is frustrating and does not lead to deep understanding. That's a big reason so many modern adults fear math.
ReplyDeleteIf you teach from concept to algorithm, or concrete to pictorial to abstract (and then to mental math), you cover all the bases. And for the students who don't understand a concept, you just keep backing up a step until they do get it, and you can help them build understanding to abstract (algorithm) from there. It leads to faster advancement and mastery.
I'm just chiming in to repeat myself and agree with all the posters so far; Singapore Math is the bomb. The little drawings are very helpful, and then they go right to the equations and algorithms. northTOmom, you really might consider getting a few workbooks and going through them with your kids.
ReplyDeleteI've seen 2 different kinds of bad math instruction so far. One was the Trailblazers kind, and the other was an accelerated math class that my daughter got stuck in, which accelerated us right out of the public schools. The accelerated math class was like a caricature of traditionalist teaching; the kids were just being marched through a lot of algorithms without any conceptual understanding.
I feel another post coming on ...
I am too uninformed on these different approaches to math instruction to say much about them. But what northTOmom says rings true to my experience of watching my kids deal with math in school. (Our school uses "Everyday Math," whatever that is.) For the kids, the expectations are very unclear, and the experience is frustrating and confidence-undermining. It's as if there's a taboo against actually teaching the kids any real math facts, when in fact doing some of that early on would just end up being empowering.
ReplyDeleteThat said, I have my doubts about the whole project of making kids learn a set of predetermined math subjects on a predetermined schedule in some predetermined way, especially in the lower grades. In that way, maybe I'm even farther out than Alfie. I think we spend years teaching concepts that, if we just waited until the kids were a little older, they could learn in a matter of a few months and with much less anguish. (I guess this where the unschooler wanna-be comes out in me.) In the meantime, the younger kids are bound to learn some of it anyway just out of curiosity, and maybe then they'd think of it as something interesting rather than as something aversive.
I agree with FedUpMom that some math concepts are very helpful to know and I wouldn't want to completely omit them from my kids' education. I'd include dividing by fractions. (And, as Mrs. C points out, dividing by fractions is really easy!) I also think a certain amount of algebra can come in handy in life even if you're not in a mathy career. Some knowledge of probability and statistics also seems useful for just about anybody to have.
But advanced algebra and trigonometry? Sines and cosines? The quadratic equation? I'm mystified about why we not only insist that everyone learn these subjects, but we even use them as a basis for sorting people out at college application time. Seriously, is there an adult in America who knows trigonometry? People don't know those subjects because there really isn't much use for them outside of very specialized fields. At that point, I'd say: leave those subjects to the kids who are interested in pursuing them or to the college students who decide to go into fields that make use of them.
Chris, if you can, BORROW THE TEACHER'S MANUALS for Everyday Math. Just a perusal at the section on what students should know will seriously blow you out of the water. They COVER tons of concepts under this "spiral" program, but the children are not meant to be "secure" (as in, know their stuff) until much later.
ReplyDeleteI have nothing against spiral curriculum per se, but I *do* have a problem with parents being expected to do homework etc. with their children on concepts they are really not meant to master until much later. Meanwhile, the curriculum *appears* to be teaching at a higher level than it is.
Hard to express myself properly in this... but suffice to say... LOOK at the teacher's manuals. OH! And check this out:
http://www.youtube.com/watch?v=Tr1qee-bTZI
Thanks for the tip, Mrs. C. Does anyone tell the students that they aren't expected to be "secure"? Seems pretty close to saying that we don't really expect you to understand what we're teaching. If I were a kid, I'd want to know what they *do* expect.
ReplyDeleteSusan,
ReplyDeleteYou're right that it's entirely possible (and desirable, in my opinion) to teach math from concept to algorithm in a non-constructivist way. I guess what I was trying to say is that the algorithm does embody the concept, so one way to reinforce understanding of the concept would be to pick apart the algorithm (so to speak), and show kids how and why it works. This is how I taught my own kids long division at a time when the teachers were not yet teaching the algorithm for it in school, yet were assigning problems that clearly demanded it.
FedUpMom,
Yes, I am familiar with Singapore Math, and used it for a few years when the girls were young. I agree that it is excellent. But I have to put in a plug for a Canadian program which I really like: JUMP Math. It was started by a guy called John Mighton, who was actually a playwright who went back to university later in life and earned a PhD in math. He is Canada's only vocal critic of constructivist math, and his programs have been met with resistance by the math teaching establishment here; recently, though, some public schools have adopted his program and have had great success with it. The workbooks are deliberately incremental, easy for kids to work on independently, and free of fancy graphics and pictures. Mighton has written two interesting books, one called the Myth of Ability (mostly about teaching math), and another called The End of Ignorance. What I love about Mighton is that he is an educational progressive who does not endorse constructivist teaching in math. He's really the only one I know of who has criticized "fuzzy" math from a progressive point of view. (If anyone's interested, JUMP has a website with more info: http://jumpmath.org/)
Chris,
ReplyDeleteEveryday Math is one of the more (in)famous of the constructivist programs, as the YouTube link Mrs. C provided attests. Because it is an American program, it is not used here in Canada except (ironically) at some of the most elite, traditional private schools.
I sort of agree with you about early math instruction. I know in my own case, I struggled with math early on, when basic concepts were being introduced, then suddenly became an excellent math student in grade 7 or 8 when we started what I considered to be the fun stuff (eg, algebra). Actually, Peter Gray (to whose writing you introduced me--thank you!) wrote an interesting article on this very subject, which supports your unschooling instincts. See: http://www.psychologytoday.com/blog/freedom-learn/201003/when-less-is-more-the-case-teaching-less-math-in-schools
I just read through that Alfie Kohn article you linked to. I'm sympathetic to some of the points he makes -- for example, about education becoming more about credentialing than about learning, and about the value of de-emphasizing grades. But I agree with you, FedUpMom, that he is too ready to dismiss as mean-spirited or selfish genuine concerns about whether the detracking and conceptual learning that he describes will be effective in practice.
ReplyDeleteParents like the ones he describe may exist, but I haven't encountered many of them. I really don't think parents who complain about detracking are worried that other kids will "catch up" to theirs. They're worried that their own kids' education -- and not just their relative test scores -- will suffer. (At least Kohn admits that "heterogeneity is hard to do well." So shouldn't the parents be concerned?) Ditto with the more conceptual approaches to math.
In any event, I think Kohn would be smart to focus not so much on the supposedly "selfish" parents who want other people's kids to fail, but on the parents who are understandably skeptical about the broad implementation of practices that are "hard to do well."
NorthTOmom -- Yes, that Peter Gray article is really interesting. That experiment he describes was eighty years ago. Despite all the talk of "evidence-based" educational techniques, I doubt we'll be seeing any studies testing that hypothesis any time soon. But I bet we won't have to wait long before someone studies whether we can introduce all these math concepts at even younger ages . . .
ReplyDeleteTo echo something Chris said:
ReplyDeleteBy starting more and more advanced levels of math sooner, we are forcing kids into irrelevant levels of math to meet graduation requirements.
If you follow the G/T track in math in my county you end up with two years of calculus at the end. For my oldest son it may be relevant because he is considering going into engineering, but for 99% of the population...who needs two years of calculus (or one for that mater)???
Sorry..."matter" not "mater."
ReplyDeleteMatthew, if 99% of your population is in the G/T track, there is something very wrong with your definition of the track. G/T tracks are intended for the top 1-5% of the students, a large fraction of whom will need that much calculus.
ReplyDelete@gasstationwithoutpumps: I think I didn't phrase my statement well, but I don't really agree with what you're trying to say.
ReplyDeleteI'd guess that about 25-30% of students are in G/T math at some point, which is probably a bit higher than it should be, but a class focused on 1-5% of the student population is never going to happen. You'd need a grade with 500-2,000 students in it to have enough to fill a G/T class by your definition.
There are plenty of careers for smart people that don't need 2 years of calculus. I work in IT, was in G/T math in school (with good cause I believe) and have never used my one year of calculus in my life. My wife, also in G/T math I believe, is in a health field and has never used calculus professionally.
What bothers me is not that the schools want all their students to take four years of math (which is what leads to 2 years of calculus for kids in the G/T program, 1 year for honors students), but that they are so rigid about what you have to take. There are other valid subjects that could be fulfill the goal--programming, accounting & finance, statistics, etc.--and be more useful to a lot of kids.
Matthew, I'd love to see a good programming course offered in high school. That's a really useful skill. Finance would be a good idea, too -- even on the scale of "how to handle your money".
ReplyDeleteA personal finance class should really be made part of every school system, but they have to do it well (and we know how that usually goes...).
ReplyDeleteI remember when one of my kids was in either elementary or early middle school and they had a section on finance. Much of the teaching was based on material saying "if you put your money into a savings account and earn 10% interest, in a few years you'll have..." Meanwhile, I was looking at a bank statement saying I had earned .05% (yes, .05%, and no, I don't keep much money in that account) interest that month.
No mention of loss of value due to inflation, taxes on interest, etc. No mention of avoiding fees. Basically, the curriculum hadn't been updated since 1980 despite all the changes in the economy.
PsychMom chimes in....
ReplyDeleteIn the 70's our high school had a cool math teacher named Mr. Sportun, who did the classes for the weaker students and for the very advanced kids. He was a special teacher. His advanced Grade 10 kids did things like tax returns.....it was very cool. Everyone loved Mr. Sportun. My sister still credits him for teaching her how to fill out a tax return.
Practical things like that should be a part of our kids' growing up years...it should be available at the community level as well.
The New York Times has an article today on Singapore Math here.
ReplyDelete