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THE MATH classroom is, and probably always will be, a center of controversy. Teachers, mathematicians, and researchers may never come to agreement on exactly how their beloved subject is to be represented in school. That said, aren't we all making some obvious mistakes? I contend that there are some practices, common to nearly all math classrooms, that we can all agree simply must be done away with. Here are my four prime candidates.
1. Forty problems a night. Most of my mathematician friends and I are only able to solve about two problems a year - if we're lucky! Tell a mathematician you've solved even five problems in a single day, and the first thing she will think is, "They must not have been very interesting problems." Outside of mathematics, does anyone you know ever get 40 things done in a day? [ed.: Everyone I know gets 40 things done in a day.]
2. The third-person czars of math problems. A strange, anonymous set of people are constantly referred to in math classrooms. We frequently hear teachers and students ask such questions as "What do you think they mean in problem number 4?"...Who are these people?...If we decide, as we often do, that our classrooms are going to be guided by the mathematics of the past, then let's at least talk about real people - not mythical ones. [ed.: Perhaps math textbooks could include a photograph of Skip Fennell.]Here is our situation, demographically speaking. Ed schools stopped teaching the methods of direct instruction in the second half of the 1980s. Teachers who earned Masters degrees in education in 1980 are now in their mid-50s which is retirement age in my district (and, I gather elsewhere, too). They are leaving the profession in droves.3. Teachers give problems; students give answers. If only mathematics were that easy! A mathematician would arrive at her desk to find that her problems were all there, waiting for her in a list. The reality is that the largest challenge in mathematics is finding a good problem to solve or theorem to prove - a single conjecture that is both interesting and approachable...[Y]ou would be hard pressed to find a classroom where the students regularly face the challenge of finding a good problem.
4. Suppose a student still "doesn't get it" by the end of a math class, and the teacher decides not to set him straight for the time being. Many people would label such a decision as "immoral," fearing that it would endanger the student's academic future. Because of the high stakes we attach to learning math in school, we seem to lose our perspective on these matters. Do we really think that mathematical learning is that simple and straightforward? Might a student have a richer mathematical experience if he is allowed to fumble around with a misconception for a few days than if he is steered promptly to the "truth"? [answer: Having now watched my own child fumble around with a misconception concerning the distributive property for the last 2 1/2 years, I would prefer America's math teachers stick to the business of clearing up misconceptions not fostering them. Alternatively, if fumbling around is to be the goal, let's knock off giving students grades of C, D, and F on tests due to fumbled answers, shall we?]
We must deemphasize answers and correctness as the only worthy goals in mathematics. Sure, "right answers" are an important part of math, but they aren't always the bottom line. Instead of always asking, "What's the right answer?" we should also wonder, "What's the right question?" and "What's the most interesting way to the answer?" Mathematics is about bold, adventuresome ideas, and the history of the subject is therefore fraught with mistakes, contusion, and invalid convictions. Let's make the classroom a bit more like the discipline and allow our students to revel in the "wrong" while they pursue the "right." [ed.: Reveling in the wrong is incompatible with being graded on a curve. You'd think a person majoring in math education would know this.]
source:
Four Practices That Math Classrooms Could Do Without
Nick Fiori
Phi Delta Kappan May 2007 Vol. 88, Issue 9, p. 695
author bio: NICK FIORI is a doctoral candidate in mathematics education at Stanford University, Stanford, Calif.
Nearly all certified teachers under the age of, say, 45 have been taught constructivism and constructivism alone. And while Robert Slavin claims that a new back-to-basics movement is brewing inside ed schools, I'm skeptical. (If others have seen a shift, let me know.) The fact that a doctoral candidate in Stanford's school of education would assume that "we can all agree" on these four propositions tells me that as yet there has been no challenge to the orthodoxy.
This means that ed schools are still producing constructivist teachers who will teach for 25 years before retirement benefits kick in. Many if not most of these new teachers will not be mentored and overseen by baby-boom era teachers trained in direct instruction as earlier cohorts of constructivist-trained teachers were.
They will be mentored and overseen by 40-year old constructivists.
38 comments:
" Forty problems a night. Most of my mathematician friends and I are only able to solve about two problems a year - if we're lucky!"
Yes because they are MATHEMATICIANS, they know math, their job is to expand humanities understanding of the world. They are not math STUDENTS. These mathematician friends of hers have learned the math and have it down cold and can use what they know to explore what they do not know.
"The reality is that the largest challenge in mathematics is finding a good problem to solve or theorem to prove - a single conjecture that is both interesting and approachable"
Yes and again the people she speaks of are MATHEMATICIANS and that is their job, after they have learned the mechanics. They are not going to tackle these problems if they do not know 2 + 2 = 4.
"Mathematics is about bold, adventuresome ideas, and the history of the subject is therefore fraught with mistakes, contusion, and invalid convictions."
Yes it is, if you are a Ph.D. in Mathematics like my cousin and doing mathematical research or if you are using it to do engineering work like I do or doing theoretical phyisics like a Richard Feynman, but if you are LEARNING math you must learn the basics before you can tackle the big ideas.
"NICK FIORI is a doctoral candidate in mathematics education "
Yep, that pretty much explains it.
For a somewhat different view, you may wish to check out dy/dan the blog of Dan Meyer who teaches high school math in northern California. He just came back from the California Mathematics Council professional development conference -- here's the index post to his posts on all the sessions he attended.. Here's the post where he talks about the presentations he judged too light on content.
Have to agree with Jeff.
And this puzzles me: "...fraught with mistakes, contusion, and invalid convictions."
Would that contusion come from hitting their heads against them nasty 40 mathematical problems?
Might a student have a richer mathematical experience if he is allowed to fumble around with a misconception for a few days than if he is steered promptly to the "truth" Sure, if they actually fumbled with the problem, but they don't. Most of them offer up some token gesture scratch work to placate their teacher and get a participation grade but they really don't find the problem intrinsically interesting enough to think about once they leave the classroom.
And I guess this is where the math ed person comes in and says, "but this is why they should choose their own problems." But again, they will only choose a problem because it's required of them, you still can't guarantee that they find this problem interesting. If a student REALLY finds math interesting and fun they would do it on their own without any prompting from the teacher in much the same way they eat candy bars and play video games without adult coercion.
I wish I could find the GH Hardy quote where he says that as a boy he never liked math--he was just better at it than the others.
"I cannot remember ever having wanted to be anything but a mathematician. I suppose that it was always clear that my specific abilities lay that way, and it never occurred to me to question the verdict of my elders. I do not remember having felt, as a boy, any passion for mathematics, and such notions as I may have had of the career of a mathematician were far from noble. I thought of mathematics in terms of examinations and scholarships: I wanted to beat other boys, and this seemed to be the way in which I could do so most decisively."
GH Hardy
A Mathematician's Apology
http://www.math.ualberta.ca/~mss/misc/A%20Mathematician's%20Apology.pdf
The shift isn't going to come until upper middle class white adults are losing out their cushy jobs to every South Korean, Singaporean, Indian, Czech, etc. person. Then maybe someone might be willing to stop with this nonsense. But not before then. If the upper middle class schools keep pushing this crap and the parents nod along while all hiring Kumon on the side, it may never change.
1. My son now has at least 25 math problems a night, thank goodness. Fortunately, I'm around to help out when the teacher and book don't do their jobs. His math homework take little time compared to the hours and hours of art homework masquerading as learning.
2. I'm at a loss for words. Wait. How about vacuous nonsense.
3. "Teachers give problems; students give answers. If only mathematics were that easy!"
It IS that easy for grade school, and they can't get that part right. Apparently, there is some other kind of math that make it OK to not know your times table.
4. "Many people would label such a decision as 'immoral,' fearing that it would endanger the student's academic future."
Immoral. That's the right word.
"Might a student have a richer mathematical experience if he is allowed to fumble around with a misconception for a few days than if he is steered promptly to the 'truth'? "
Since you asked, the answer is no. Fumbling is neither necessary or sufficient. One kid will achieve the lightbulb effect (or be taught by a parent) and then directly teach it (poorly) to the other kids. There is absolutely no advantage to fumbling in grade school. I have had teachers who carefully lead us students down the garden path, so to speak, but never any who just tossed us into the deep end of the pool on purpose.
"We must deemphasize answers and correctness as the only worthy goals in mathematics."
"Only?" OK, you get the "only" part done and then we'll talk about other "worthy" goals. I won't hold my breath.
You better go start your own school and see how your students like "to revel in the 'wrong'". i.e. bad teaching.
Yes because they are MATHEMATICIANS, they know math, their job is to expand humanities understanding of the world. They are not math STUDENTS. These mathematician friends of hers have learned the math and have it down cold and can use what they know to explore what they do not know.
well, of course that's the profound misconception upon which the entire essay rests: beginner = expert
Ed schools, it seems to me, have systematically erased all concept of sequence from development and from curriculum.
The way to teach math is to have 7-year olds acting like mathematicians.
Which, as Myrtle et al have pointed out, is something no self-respecting 7 year old will do.
Actually, it's not even this sensible.
The workshop approach to writing is based in the belief that beginning writers should act like professional writers.
What they mean by that is that beginning writers should not act like professional writers. Beginning writers should act like Lucy Calkins' notion of professional writers.
Which happens to be wrong.
Liz - that looks great!
Everyone - go take a look at the list of conference titles - fabulous.
I agree with his picks. Every last one.
Actually, if you have the time, root around the Dy/Dan blog. It's got some really fantastic entries. I find his perspecttive on many issues refreshing.
By the way, folks, go vote for us over here:
http://edublogawards.com/2007/best-group-edublog-2007/
SusanS
I give homework every night but before I assign it, I do it myself. If it takes me more than 10 minutes (5 for a slow class), I shorten it. A good kid will do all the homework, but, if it is too much, nothing worthwhile will get done.
My motto--keep it sensible and it will get done.
I am a scientist-biophysics (faculty at University of Chicago - yes, the place where everyday math comes from. I had nothing to do with it, and I don't get royalties). I have chewed through a lot of math education, but it was never my favorite subject. It was boring, dry, and repetitive. I was good at getting answers, but completely uninterested in the process. Obviously, I had the old-style curriculum. I struggled with the algorithms, because my teachers just taught them as rote processes - no theory. Do it because it works. Now do it again. I remember asking a simple question about remainders, "Why is it the remainder over the denominator." I was put into a remedial math group (to do it again some more). I felt stupid and bad at math. You can see where I am going with this.
But before you condemn me, consider why we are teaching math in school. What's the point? To make them 1/10th the speed of a calculator? What a pointless waste. I use math all the time and never need to do long division. I haven't seen paper and pencil long division being done since high school. have heard that some people do it by hand. I think they must enjoy it. Normal people use calculators. University level courses require calculators for tests now. A huge improvement, since the problems can be so much richer and more complex. If we only teach process level mathematics, then once the students become adults (and use calculators) they will not understand what is going on in the calculator. They are left with the impression that all that is going on is a whole bunch of the only algorithm that they know.
Now, estimation is something I do all the time. "That must be around 10^-8." and if 6oz is $3.49 then 11 oz must be less than $7. (I do use math for 'normal' activities also). My kids' math curriculum has a big section on this kind of estimation. I never got that.
My kids still have to memorize single digit multiplication tables, and that is necessary. You can't even estimate without those facts and the only way to get them is memorization. Likewise, the terminology of mathematics needs to be learned. But why should it be connected to a process, rather than connected to a conceptual understanding.
I suspect that the dismal crash in student math scores has more to do with teachers not understanding math well enough, on top of all the other ills (My children now have classes of more than 30 with one teacher, no aide) Many people here are noticing a correlation with teaching method and assigning causality. That is not a demonstration of logical thinking. There are other correlations that could be responsible for the change.
I also guess that people react badly to newer curriculums simply from the boot camp effect. "It worked for me, so they should have to suffer like I did." Having to learn a new way of thinking about a math problem is hard for us grownups, so the homework seems strange and inefficient.
Going through my kids' math curriculum has been wonderful for me. I see all of the interesting facets and connections that I had to work out for myself. Yes, my kids can't solve multi-digit problems as fast as I could when I was their age. If that is your definition of "Mastery" then you are going to be skeptical. But master violinists aren't simply more technically gifted, and composers don't even have to be able to play all the instruments. Many technical champions fail to be masters of their endeavor. It is simply easier to test speed than comprehension, and we do care so much about testing.
Are these new math curricula perfect? That is a ludicrous question. We have to constantly work to make things better, and we will never reach perfect. I oppose the idea that we should just go back to the old way. Give up in the attempt to make things better. Boring, dry, and repetitive was far from perfect in my education.
I don't want boot camp for my kids. They don't need boring, dry, and repetitive either. They need Singapore Math.
No matter how you slice it, Everyday Math just doesn't cut it. Singapore Math does.
Adam, I strongly recommend that you have your children complete the assessment for Singapore Math. It's free, it's not timed, and it will open your eyes to what your children could be learning... and what the top notch students in the world already know.
Your children will thank you someday (or not) and you'll sleep better at night knowing you did a really good thing.
I feel sort of bad for Adam. My own math education was not boring, dry, and repetitive. If that's what you think KTM is advocating, I suggest you peruse the archives a bit longer.
The billion dollar question (inflation) is, does Everyday Math prepare kids to succeed in Algebra? Can they do it by 8th grade? Because that is the international standard. Algebra (in any grade) is critical to college success. This we know from research (not suspicions or guesses).
I don't think you use suspicious or guesses in biophysics, do you? Don't lower your standards when you approach math education.
EM is woefully inadequate at preparing kids for algebra. Once the train falls off the tracks, it is very hard to get it back on.
Adam, you should spend some time at KTM. Some of the threads go off on big tangents, but most of the issues you raise have been covered in great detail.
concernedCTparent says it best. This is not about old versus new. Do the Singapore Math assessment.
I will, however, touch on a few of your comments.
It's a strawman to think that people simply want to go back to the way things were. There are two big issues; curriculum and teacher quality. It's not a simple case of bad traditional curricula versus poor teaching of modern curricula.
"I use math all the time and never need to do long division."
This is just one small part of the issue, and even Everyday Math teaches long division by hand. The argument has never been about speed of doing long division by hand, although it's a very good way to to develop one's estimating ability.
"University level courses require calculators for tests now. A huge improvement, since the problems can be so much richer and more complex."
I was in college when they changed from sliderules to calculators. It WAS a big improvement and the problems that could be done were much more complex and interesting, BUT, we already knew basic arithmetic. Nowadays, reform math curricula use calculators as avoidance tools. They are not used to tackle more complex problems.
"They are left with the impression that all that is going on is a whole bunch of the only algorithm that they know."
You are confusing bad teaching with curriculum. Like modern math reformers, you are trying to break the link between mastery and understanding. Mastery is NOT just speed.
"But why should it be connected to a process, rather than connected to a conceptual understanding."
Understanding without process is math appreciation. It's not real understanding.
"Many people here are noticing a correlation with teaching method and assigning causality. That is not a demonstration of logical thinking."
There are many reasons for poor results, but that doesn't mean that the effects of curricula cannot be isolated. It's easy enough to compare content and whether or not it leads to a rigorous course in algebra in 8th grade and the AP calculus track in high school. Our public school used to offer only CMP in middle school. There was a clear gap in content between 8th and 9th grades. This kind of analysis is quite logical.
"I also guess that people react badly to newer curriculums simply from the boot camp effect. "It worked for me, so they should have to suffer like I did." Having to learn a new way of thinking about a math problem is hard for us grownups, so the homework seems strange and inefficient."
Oh please. This is so old and trite that it marks you as a real newcomer to the debate.
"But master violinists aren't simply more technically gifted, and composers don't even have to be able to play all the instruments."
You are again trying to unlink mastery with understanding. Mastery is not just speed. A concert violinist could be technically proficient and still be lacking in musicality, but you won't find a single concert violinist who has musicality and lacks technical ability. It can't be done. Also, composers darn well better know something about all instruments, even if they can't play them. They require mastery of a different sort.
"Many technical champions fail to be masters of their endeavor. It is simply easier to test speed than comprehension, and we do care so much about testing."
The math testing that's done is so trivially easy that the big question is what's so important that it's OK to flunk these simple tests. Good schools laugh at standardized tests.
"Are these new math curricula perfect?"
Strawman.
"Boring, dry, and repetitive was far from perfect in my education."
Yes, but you have to go in the right direction.
Adam,
I graduated from MIT with a math degree. My first job out of college was a program that tried to teach high school math teachers some better math skills to help their students be able to go to college. I was placed in the worst school in the Oakland, CA school district. I worked with kids in the honors pre calc class.
your comments about calculators and estimation reminded me of them. They had all had tons of instruction on estimation. and they all used calculators every day.
They had NO IDEA if they had typed something into the calculator wrong.
If you asked them "what's 1000 times 10000" they'd go to their calculator. If they got .000001, they'd tell you that was the answer. Why? Because teaching estimation and calculator usage WITHOUT NUMBER SENSE meant they knew NOTHING.
You know how to estimate because you did enough rote arithmetic to know if .0001 times .000001 is smaller than one or larger than one. You know enough to be able to ask yourself "is this reasonable?" The kids learning estimation are learning a process that never teaches them enough facts to assess if they applied the process correctly. But once you are past multiplication tables, the only math facts at your fingertips ARE the processes.
These kids were going to fail out of college. All of them. They thought they knew math. They were sharp enough to know more than their teachers knew, and they were never asked to do enough rote arithmetic so they never got good enough to work quickly. So instead, they can't do it in their heads, can't check their work against reality. For them ,it had to be connected to a process so they could have confidence they were doing it right, because they aren't yet sophisticated enough for the conceptual understanding.
Your kids' curriculum is wonderful FOR YOU BECAUSE OF YOUR BASE KNOWLEDGE. It's BECAUSE you already learned so much that you can go back and say "aha! that's WHY that was true!"
but that's not where your kids are. That's where you are because the more you learn, and the more you think, the more you brain is able to understand things it couldn't before. Sure in biophysics you've noticed you're better at understanding thermo than you were the first time you took it? That enthalpy MEANS something to you now that it didnt' at 19? That doesn't mean that the curriculum appropriate for a 35 yr old is appropriate for the 19 yr old.
do you remember Feynman's Lectures? have you read them? Have you thought about why all of the freshmen at Caltech when he taught them dropped the course? Because his physics lectures were DISASTERS for incoming freshman. They needed to DO PROBLEMS. They needed to learn F = ma, and how to draw a free body diagram. Only professors got something out of it.
When you talk of conceptual understanding, you are basically saying that the kid should be taught what the professor understands, even though he spent years doing the process before he got that understanding himself.
Finally,
yes, many technical champions fail to be masters. But no master of their endeavor fails to be a technical champion. By preventing kids from getting to that stage, you prevent anyone from ever becoming that master.
Thank you for your reasoned comments, but I disagree with many of them. I suspect that a conversation with any of you would be enjoyable and would increase my understanding of the subject. Please, consider the possibility that you talk to each other too much. You have reached a conclusion that is not unassailable, but by repeating it you become dogmatic. It happens in academia all the time and web communities are also prone to it. When someone who disagrees with you stops posting it is not always because you have convinced the person.
Adam,
You're right. Being dogmatic is a very poor strategy. Even worse when you're talking about the education of the most important young people in your life. Perhaps the results so many of us have had are anecdotal, but enough anecdotal results really start to mean something when it all comes together.
If you're willing to explore a little, consider comparing Everyday Math and Singapore Math in an open, objective manner. Having your children take the assessment is a really good way to do that.
Here's the link to the placement tests if you're willing to consider another point of view:
http://www.singaporemath.com/Placement_Test_s/86.htm
You may find that you still prefer Everyday Mathematics for your own children, and if so, your feelings will at least be backed with some of your own independent research. Conversely, you just may find that Singapore Math isn't the way you were taught math. You might just find that it's way, way better. The only way to know is to weigh the outcomes objectively.
So, don't stop posting because we don't agree on everything. It's important to consider your experience with this and compare it to our own, at a very minimum as a good faith effort to avoid dogmatism.
Along the way, you just might learn something new. That's almost always a good thing.
"You have reached a conclusion that is not unassailable, but by repeating it you become dogmatic."
This blog is all about an earnest discussion of assumptions, facts, and opinions. Feel free to argue your case, but you better be willing to back it up.
I've been studying this problem for over 6 years while my son has had to endure MathLand and Everyday Math. I've been waiting for anyone to show me what I'm missing. It hasn't happened. Dogmatic? Feel free to justify this opinion.
This wouldn't be such a big issue if there was full school choice. There isn't. Schools select math curricula based on whatever criteria they want. If parents complain, they have an open house (our school is having one in January about Everyday Math) where parents hear the same sorts of things you are saying. There is no discussion. They don't want to defend their positions either.
"When someone who disagrees with you stops posting it is not always because you have convinced the person."
Perhaps these people can't be convinced. Perhaps they have different assumptions, (low) expectations and goals. If some parents want to select fuzzy math for their kids, I won't get in their way. However, as long as the public school monopoly foists their ed school opinions on me, I will fight them. The onus is on the schools to justify their curricula, not the other way around.
People go away because they can't defend their position.
You have reached a conclusion that is not unassailable
Apparently it is unassailable.
I will stay tuned.
I haven't seen paper and pencil long division being done since high school. have heard that some people do it by hand. I think they must enjoy it. Normal people use calculators.
Okay, I love my calculator. Sharp EL-5120. It's on my desk at the moment. It's not much to look at, but its functionality means that it rocks my world. In terms of calculator-adoration I am probably in the top 1% of the world's population. My calculator has literally travelled around the world with me (there's no way I'd trust it to any removal company). I'm not a poet, but if I was I would write love poems to my calculator. The only reason I do not sleep with my calculator is that I fear it will disappear down the end of the bed and I will never see it again. When it comes to using calculators, I strongly suspect I am not normal. However, despite my deep and undying affection for my calculator I am sometimes without it, and on those occasions it is useful to be able to do basic arithmetic such as long division with pencil and paper or in my head. This may not be normal, but why should we educate kids merely to be normal people anyway?
Going through my kids' math curriculum has been wonderful for me. I see all of the interesting facets and connections that I had to work out for myself.
I had the same experience at high school when I was tutoring my fellow students one to four years behind me in the coursework.
At my uni, the professor who taught the Econ 101 course once said "It's amazing, I'll be up there explaining something for the tenth year running, and suddenly I'll understand it myself". Understanding things better the second time around is common regardless of whether the course outline has changed. You are noticing a correlation and assigning causality. That is not a demonstration of logical thinking. There are other correlations that could be responsible for the change.
Well, Adam could have had a stinko math education, but why assume that it was the norm? He could also have had a few stinko teachers, but why assume that it was a problem with the curriculum? My math education in the 1960's left me wanting more (teaching and curriculum-wise) for my son in 2001, but my reaction was quite different than Adam's. One look at MathLand, and then Everyday Math, convinced me that they were going in the wrong direction. The problem is that you have to get from point A (counting in Kindergarten) to point B (a rigorous course in algebra in eighth grade). Modern reform math does not do that. Many (myself included) made it to calculus in high school without any tutoring or help from parents. You can't do that nowadays.
Modern reform math blames the problems in the old days on curriculum, but claims that modern reform math only needs better teacher preparation. Either the curriculum matters or it doesn't. None of them want to consider that the the best solution is a rigorous modern curriculum like Singapore Math AND better teacher preparation. That's because they think that Singapore Math is too difficult.
In our public schools, there is a very strong teacher bias against any sort of rigor or expectations of mastery in the lower grades. Education is all about full-inclusion, child-centered, low expectation learning. Everyday Math is based on the idea that there is no timetable for mastery. This might be appealing to overly earnest parents who feel that it's their responsibility to help their kids master the times table or to really figure out how to divide fractions, but this directly hurts kids who have to do it on their own. All of the talk of understanding only hides low expectations and poor teaching.
"I also guess that people react badly to newer curriculums simply from the boot camp effect. "It worked for me, so they should have to suffer like I did." Having to learn a new way of thinking about a math problem is hard for us grownups, so the homework seems strange and inefficient."
I was thinking about this some more. This is the big lie. This is what schools tell parents over and over and over, but they KNOW that this is not the issue.
So, perhaps Adam will be quite earnest and do what has to be done to keep his kids up to the proper level. Maybe he will even think that this is what all parents should do. I make sure my son is kept up to the proper level, but I know that there are a lot of other kids getting screwed. And they will probably end up thinking that it's their own fault.
"Many people here are noticing a correlation with teaching method and assigning causality."
I know that stating that correlation does not mean causation immediately marks one as highly sophisticated. But amid all that happy chatter about correlation, the notion that some things really cause other things is easily lost. For example, if a student has difficulty grasping a concept or solving a problem and the teacher provides a good explanation that leads to student understanding, one can confidently state that the teacher caused the student to understand.
The rote recitation of correlation reminds me of the reflexive invocation of "stereotype" in some smug quarters when taboo subjects are brought up. The invocation of "stereotype" then serves as a tool to deny painful reality.
[But before you condemn me, consider why we are teaching math in school. What's the point? To make them 1/10th the speed of a calculator? What a pointless waste. I use math all the time and never need to do long division. I haven't seen paper and pencil long division being done since high school.]
One of the fallacies afflicting constructivists is the failure to distinguish between a child who is taking baby steps and a full-blown adult sprinter. What an adult does with math, say in engineering, is in no way comparable to a child who is gradually acquiring math knowledge. Being able to do computations manually not only makes the child self-reliant (not a calculator-dependent math cripple), but has the great advantage of teaching NUMBER SENSE. By requiring the student to estimate how many times one number goes into another number, long division, in particular, is a great teacher of number sense, besides tapping into a host of other skills. Number sense is the great casualty of today's calculator-raised math cripples.
I see that lack of number sense all the time. It's pathetic.
>>And while Robert Slavin claims that a new back-to-basics movement is brewing inside ed schools, I'm skeptical. (If others have seen a shift, let me know.)<<
From my personal experience, it comes a bit later. I'm noticing the colleagues I lunch with are all between 7-10 years experience and, while they were feeling discontent before they hit year 7, it wasn't until then that they began to find ways to sort, analyze, and define that discontent. For one of our colleagues, who is on year 4, I've noticed lunching with us has been enlightening - she's asking more questions and making more concrete observations than when she joined us two years ago. She's talking less about her frustration being a result of student behavior than a result of poor teaching methods she learned in ed school.
Redkudu, what happens when there are no more teachers like you and your lunch colleagues for the newer teachers to exchange experiences with? What happens when everyone's experience is the typical ed school fare and that's all there is, period.
Most districts are only bringing on brand spanking new teachers and the core of experienced, qualified, effective teachers is shrinking. Those new teachers will have few if any mentors worth their grain of salt.
Poor methods learned in ed school is quickly becoming the norm, I'm afraid.
Poor methods learned in ed schools are quickly becoming the norm.
(Clearly, multitasking on the PC with a five year old on your lap has its downside.)
Just discovered this thread & haven't read yet....but I remember reading a Boston mom's comments on TERC. I think she may have attended MIT (unless I'm hallucinating)....in any event, I remember thinking of her as a "math brain" or "math type."
She liked TERC.
I remember, at the time, having the perception that TERC was a completely different kettle of fish for her than it would be for me.
I've got to post my Farm girl goes to Wellesley story.
My math curriculum was fun.
I don't think it was fun for other kids but it was fun for me.
That's probably because it moved so slowly I learned everything to mastery and back and then to mastery and back again.
I've only recently discovered that although I took courses called "Algebra 1" and "Algebra 2," the material we covered had nothing to do with Algebra 2 in any conventional sense of the term.
well...I STILL have only skimmed (short attention span theater - cow chapter may kill me)...but I noticed this line:
yes, many technical champions fail to be masters. But no master of their endeavor fails to be a technical champion. By preventing kids from getting to that stage, you prevent anyone from ever becoming that master.
ditto!
ditto, ditto, ditto!
Wasn't it Carolyn who used to say you don't find any mathematicians who can't do long division?
This is one of my lines on the relationship of knowledge to expertise.
You never find an expert in any field who does not possess massive amounts of knowledge -- knowledge including Brute Facts -- in long-term memory.
It simply doesn't happen.
And, btw, I know this because we've taken our two autistic kids to some of the best physicians and researchers in the world. (I'm pretty sure our current physician has spent quite a bit of time seeing the autistic children in the Saudi royal family -- I'm talking about international-level expertise.)
These people have what used to be called encyclopedic memories. It's unreal, the material they can pull out of memory 1 second after you mention a problem or symptom your child is having.
Experts possess so much knowledge, stored in long-term memory not on Google, that psychologists have studied the phenomena. For a time there they were hypothesizing that experts possess a different form of memory, which I believe they were calling something like "long-term working memory"....
Take that with a grain of salt; it's not fact-checked.
As I understand the research, they've abandoned that idea for the concept that experts build a schema inside their memories in which everything-activates-everything, i.e.: one memory awakens the entire web.
Carolyn once said that math was "a seamless whole" inside her head, which made it hard for her to teach her son because she couldn't deconstruct the whole.
This is one of the fallacies of constructivism.
Constructivists have probably correctly divined the (apparent) fact that the knowledge of experts is a seamless whole.
They have failed to examine the process by which experts acquired these seamless wholes.
[Carolyn once said that math was "a seamless whole" inside her head,...]
I don't know if this ties in with the idea of a seamless whole, but it has occurred to me that discrete skills are needed first before one can appreciate the connectedness of math. Without these concrete skills, math is more like a seamless black hole.
This became apparent to me again when teaching a group of seventh and eighth graders brought up on EM and currently using CMP who are a tabula rasa when it comes to the simplest bits of math knowledge. They can't do any operations with fractions (e.g. change mixed numbers to improper fractions let alone addition and division), can't divide decimals, don't have knowledge of even rudimentary geometry... One wonders what they have been doing for seven and eight years.
The seventh graders are currently in the CMP stretching and shrinking stage. Their homework consisted of finding the scale factor of two rectangles the width of which goes from 1.5 cm to 3 cm. So the idea was to divide 3 by 1.5 (they can't do it because they can't divide decimals). When I tried to show an alternative way of division using fractions to demonstrate the connectedness of math (seamless whole), I ran into trouble, too. They don't have the discrete skills of seeing 1.5 as 1 1/2, then changing this mixed number to 3/2 and dividing 3 by 3/2 (they absolutely can't divide fractions and moreover don't see 3 as 3/1. It would have been spectacular to make them experience with understanding that the more complicated decimal division problem 3/1.5 virtually solves itself when you divide the respective fractions (3 divided by 3/2). Invert and multiply but they have never heard of reciprocals and how they work. The 3 cancels and 2 is left standing without much ado!
So the upshot is: they use Connected Mathematics but can't see the connectedness of math because they don't have discrete skills (skills they could have learned through drill and kill but haven't). So to them, math is a seamless black hole from which not even light can escape.
[T]hey absolutely can't divide fractions and moreover don't see 3 as 3/1
Yes! They come out of EM not knowing that the bar in a fraction means division. They cannot convert fractions to decimals because of this blind spot. They only see fractions graphically, as pieces of a pie.
My son's charter school uses Impact Mathematics in middle school, which I think is better than CMP, and is perhaps supplemented? In the 7th grade class for the first couple of months they got tons of review sheets for homework that made sure they were fluent with fractions and decimals (including multiplying, dividing and converting).
I couldn't agree with you more about the interconnectedness coming from mastery of the components. This is the typical "Aha" moment. I've had many of them in math. I had many more when studing for my written prelims in biochemistry.
The complex, conceptual work done in EM is mostly smoke and mirrors--designed mostly to look impressive and make for impressive talk in this age of emphasis on math and science.
As an aside, I'm frustrated on a larger scale these days. Our local neighborhood public school is slated for closure unless the community comes up with a plan to improve achievement and attract back local kids. (Now in addition to doing the teachers' work, we're doing the adminstration's work: they announced closure with no warning, and in the face of the outcry gave the neighborhood 2 months to come up with an alternative...).
The school is 89% ESL. No wonder it's having problems under NCLB. And they are the sweetest, hardest working kids you'll ever see....
In any event, the school I would build would be the anti-school. Raise achievement? I think we all know how. In a nutshell, whatever the district school are doing, I would do the opposite. However, the curriculum, including EM, is mandated by the district. We can't put Singapore or Saxon in there. Imagine how the scores of these ESL student would increase if we could. What a frustration!!
the school I would build would be the anti-school
Boy, you and me both.
Though I must say, I am STILL having problems with that damn division bar, in spite of my years with Saxon.
[However, the curriculum, including EM, is mandated by the district.]
This makes my skin crawl. It is another example of how an ed school-indoctrinated cadre in power is able to wreak havoc. It puts the lie to Jay Mathews' insouciant claim that ed schools may be the butt of jokes but have no influence.
See my post http://instructivist.blogspot.com/2007/02/buzzword-education.html
Buzzword Education
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