In my first, brief report on the forum sponsored by Where's the Math? I described the basic argument made there, that our current math curriculum is a poor one, and that we could do better by moving from what is commonly called "constructivist" mathematics to a more traditional curriculum. In this post, I would like to give you a brief description of the two sides in this controversy.
On one side is what former Education Secretary William Bennett called the "blob", which is short for "bloated bureaucracy". If we want to more precise, though it slightly spoils the blob metaphor, we would say that advocates of the status quo are in a set of bloated bureaucracies, in our public schools, in our state departments of education, and, most of all, in the National Council of Teachers of Mathematics. (If you want to correct the metaphor, imagine a set of blobs, side by side, blocking a road.) In 1989, the NCTM proposed a math curriculum that was widely adopted, in part because of federal support for textbook purchases that follow the NCTM standard. This standard has been, research shows, a disaster, seriously damaging mathematics education in the United States. It has been worst for the children of parents who have fewer resources, less money and less education, because those parents find it harder to make up for the defects of the math education in most public schools.
On the other side from the blob are moms and mathematicians, in a wonderful ad hoc coalition. Moms noticed that their kids were not learning mathematics, and, a little later, mathematicians (or to be more precise, mathematically adept professors) noticed that students were coming to them weak in basic mathematics. There were many dads at that meeting in Bellevue, but I would judge that the moms outnumbered them by two to one, and were more passionate about how their kids were being cheated by the blob — as moms usually are when they believe that their kids are not getting what they should. And the mathematicians were rightly distressed that they had to spend so much time on remedial mathematics, even in supposedly competitive colleges and universities.
Both groups at first reacted with individual solutions. The moms tutored their kids, enrolled them in private tutoring organizations, or even started home schooling them in a few cases. The mathematicians changed their courses so that less was assumed about their students' knowledge of mathematics. As time went by, some moms and some mathematicians began to think that these individual solutions were not enough; those moms began to organize, and those mathematicians began to research the problem. The result is what I saw at that Bellevue meeting, a grassroots organization with strong scientific support for its cause.
It is, as I said, a wonderful ad hoc coalition. And the six member panel at the forum showed the quality of the people in the coalition. (Unfortunately, I have barely adequate pictures of just five of the six. The lighting conditions were poor, my little Olympus C-765 has a modest flash, and I am a very amateur photographer.) I mentioned the first member of the panel, Shalimar Backman, in my first post, saying that I thought her a remarkable woman. The other five were also remarkable, though in different ways. The first picture shows Bill Hook and Niki Hayes. (Scroll to the bottom for brief sketch of her career.)
(You can find a sample of his work here, and samples of her work here and here.)
Hook presented research on California schools showing that schools that used traditional textbooks outperformed schools using constructivist textbooks. Hayes, who has been a principal, and is now a math teacher, sparkled as she gave us lessons from her experience.
The second picture shows, left to right, Craig Parsley, a fifth grade teacher in Seattle, Elizabeth Carson, the executive director of New York City HOLD, and University of Washington Atmospheric Sciences Professor Clifford Mass.
Parsley, like Hayes, gave us the benefit of his own experience. When he said that his goal was to produce students who "dominated" in the sixth grade, I knew that he understands boys that age. (I suspect girls that age just tolerate that kind of talk.) And he also made a general point of great importance: Those who hope that we can improve math teaching by better training of math teachers are forgetting the brutally high turnover among teachers. Carson gave us a fine overview of the problem. (And her organization's web site has many useful links.) Mass, as I mentioned in the previous post, gave us his experience as a father, and as a professor teaching a mathematically demanding subject.
All of the panel members were willing to learn from the nations that outperform the United States in teaching mathematics. In that, they differ sharply from their opponents in the blob. I have long thought that, in education, we should look at what works and copy it. We don't have the same theoretical understanding of the psychology of learning that we do of, for instance, physics, to put it mildly. The NCTM began with a theory and built a proposed curriculum around it; they would have done better to study nations that succeed at teaching math, such as Singapore, and copy their methods. And their textbooks. Where's the Math? had exhibits outside the forum; one of the most interesting was the math textbooks from Singapore, which were plainer, cheaper, and much better than the comparable American textbooks. (The Singapore textbooks were, as far as I could tell, much like the Japanese textbooks that I discussed here.) And the extra material in the American textbooks was not just distracting, but, in some cases, incorrect. (Mass found a silly meteorological error in one textbook; a diagram showed a cirrus cloud, which is made of ice crystals, below the freezing line.)
So how should this coalition confront the blob? I have never seen the original movie, so I am not sure what strategic advice we could gain from it. But often the best way to defeat bureaucracies is to go around them.
Cross posted at Jim Miller on Politics.
Correction: I misread my notes, and misspelled Shalimar's last name, both in this post and the previous post. It's "Backman", not "Buckman". I've corrected the text in both posts. My apologies to Ms. Backman.
Posted by Jim Miller at May 18, 2006 07:03 PM | Email ThisNow, are there drawbacks? Absolutely. I don't think that the curriculum provides enough practice in each new concept so that the kids can obtain some level of automaticity. We also have to leave out some things that might have otherwise been taught as it does take longer to teach this way. A concept that might have been a day in a traditional curriculum might be two or three in this system. However, I have been pleasantly surprised by the level of sophistication of the math that we are presenting and that the kids appear to be understanding.
My suspicion is that the struggles that students encounter in college has less to do with a lack of knowledge and much to do with a lack of understanding of how the system works. Being unprepared to sit through a lecture and take notes; or performing poorly on an exam that provides out of context problems and asks for application of memorized formulae; neither of these actually proves that a curriculum is flawed.
So, would I reject a return to a traditional curriculum? Not necessarily. One of the challenges during student teaching in a traditional math program was making the lessons interesting. It was really easy to get stuck in the sound of my own voice as I patiently explained to the kids what they needed to know. Some of them got it, some of them memorized the formula and what the clues were that told them when to apply it; and some of them zoned out.
Now, we explore the mathematics. Some of them get it, some of them wait for the formula to be presented so that they can try and memorize it; and some of them zone out. Has anything changed? Well, it might be different kids succeeding. The kids that were good at recognizing situations and memorization are probably still able to do so, but with less success since problems are presented "in context". Kids who are more tactile learners are probably doing better than when I lectured them. The kids who are failing, well, they probably were failing under the old system.
Are there "bad" integrated curricula? Probably. Are there "bad" traditional textbooks and curricula? Probably. Are there "bad" teachers teaching either system? No doubt.
In summary, there are drawbacks to each system, and there are benefits to each system. I will probably stay in the school system I'm in with the curriculum that we have. I will also probably develop more worksheets to allow them to develop automaticity in more skills.
Comparisons to Japan, Germany, etc? Who really cares. We are kicking their asses in everything else that matters. Our productiviy is higher, our unemployment is lower, our GDP is higher, our growth is higher, in virtually every measure that matters we are better. I am less than concerned about where we stand in test scores on some standardized test that maybe tests how well they learn under a traditional curriculum I would be interested in having their kids come here and take the WASL and see how they compared. Lets make the standardized testing playing field level before we start dumping on our curriculum.
Posted by: Calvin A on May 18, 2006 09:04 PMThis is a hot topic in Bellevue. Jim's outline is a good summation of those who want to implement Singapore math as a response to the blob. Your summation is a different path which I think gets us where we should be going. It isn't the blob, which is a problem, but it isn't a return to drilling and memorizing which is the core of Singapore math.
The goal of math should be to learn a set of principles and theories that allow students to solve problems.
Posted by: Janet S on May 18, 2006 10:36 PMIn the hands of a person who is a good teacher and has a good understanding of math, I am quite confident that the constructivist approach is superior. Both of those ingredients are necessary, because the teacher must first herself have a clear understanding of the concepts and then figure out how to familiarize her students with that same conceptual landscape. The result is a student who is both conceptually and computationally proficient.
The trouble is, the vast majority of primary teachers are missing at least one of those qualifications. (Some are even missing both.) In the hands of such teachers, the traditional curriculum will at least yield a student who is computationally (though not conceptually) proficient; the constructivist approach will yield a student who is neither computationally nor conceptually proficient. Therein lies the rub.
We are paying teachers to teach our kids how to perform computational arithmetic. Not to have them sit around in circles and try to do what Al Kwarizme (a genius) did: Invent computational arithmetic.
What we engineers call "Reinventing the wheel"
Not that there is no place for asking our children to brainstorm solutions to problems, but I would like them to be able to perform on an engineering midterm when their calculator battery goes blooey halfway through the test!
I would suggest the teacher set up a problem on the board, say how many feet are there in 29 miles.
5280
__29
First, we take the 0 at the right hand part of the number of feet in a mile and multiply it by the 9. Answer? 0. 0 multiplied by anything is 0! That we put in the "ones" place in the answer.
Then we take the 8 and multiply it by that same 9. What is that, class? You there, sneaking a peek into your PeeChee at the multiplication table, you are not allowed to answer. But I'll let you grin and laugh! Someone else have an answer? 72? That's right! 1/8 of a fathom is 9 inches and 1/9 of a fathom is 8 inches! Well, it IS! So, we put the 2 into the "tens" place in the answer. But, what do we do with the 7? Well, we'll just stick it up here for safekeeping, we'll be using it in a second or two.
_7
5280
__29
__20
2 times 9 is? 18. Correcto! But we don't just stick the 8 in the "hundreds" place in the answer. No, no, no, uh-uh! Remember that 7 we got hanging up there above the 2? Right! We add that to the 18. 25 is the result, and we put the 5 in the "hundreds" place. The 2 we store up above for safekeeping.
27
5280
__29
_520
5 times 9? Anyone? No, not 54. Wrong answer. Anyone else? 45! Right, Colt 45! But there is that 2 up there, what do we do with it? We add it to the 45 and come up with 47. But now that we have run out of numbers in the number of feet in a mile, we can just place the 7 in the "thousands" place and the 4 in the "ten thousands" place:
_5280
___29
47520
Before we go further, can anyone tell me what this number, 47520, is? That's right! It is the number of feet in 9 miles! But we still have another 20 miles worth of feet to account for here. So we take the 0 and multiply it by the 2, and get 0. But where do we put it? Why we put it right under the 2 below, in the "tens" place of the second answer.
_5280
___29
47520
___0
And we just do with these new numbers what we did before:
___1
__5280
____29
_47520
10560
That is right folks! 8 time 2 is 16, stick the 6 down there and the 1 up there, 2 times 2 is 4 add the 1 to get 5 and, of course, 5 times 2 is 10 and we write that in. Now we got two answers: The number of feet in 9 miles and the number of feet in 20 miles! What is 20 plus 9? 29. If there are 105600 feet in 20 miles and 47520 feet in 9 miles, it follows that the number of feet in 29 miles is the 105600 feet added to the 47520 feet:
__5280
____29
_47520
10560
153120
153120 is the number of feet in 29 miles!
And they said the Metric System is easier! Only if you don't know how to do computational arithmetic! And if you don't know arithmetic, you will flub the Metric System too!
If our teachers want to succeed and actually do the job we pay them to, I would suggest they just teach the kids how to do the math and save the brainstorming sessions for when they are older and already know the basic stuff!
Oh, Calvin...
Of course it matters. Are we succeeding because of our educational system, or in spite the system?
Posted by: South County on May 19, 2006 05:26 AMIn my business, all the fluffy constructivist junk ain't going to make me money. My success is based upon simple adding and subtracting and making math dance to my music. I don't need that other stuff.
I've recently been asked by some teenagers what good differential equations and calculus are in the real world. I tell them it does not do any good, but also that it trains their mind.
But you have to have the basics. I remember talking to a number of the teachers (disgruntled at the quality of basic math skills in high school) a few years ago at a curriculum forum with business people (I was a rep for the business community). Their beef is that there were too many kids graduating that couldn't balance a checkbook. In other words, they, too, were having trouble with 8+6=.
Posted by: swatter on May 19, 2006 06:19 AMYour first year, you feel overwhelmed by the "system," but generally not the subject matter. Your second year, when you know the "system," you are surprised by what you didn't know about the content/instructional process. The third year, you begin to see how your instruction and the system finally mesh together. AND, you say, "Oh, my goodness, when you learn that those teachers who receive your kids (and witness your students' knowledge/skills) are the greatest critics of your work (besides parents who may not realize until much later the cumulative effect of all the teaching).
Thus, while I appreciate your enjoyment of reform math approaches, I recommend you keep a copy of your message and review it at the end of your third year of teaching. (I know whereof I speak, because this same recommendation was made of me as a first year teacher in 1961.)
Posted by: Niki Hayes on May 19, 2006 07:42 AMIt has taken 6 years to "start" this discussion from the time I found out about a "fuzzy math" curriculum adopted by the Sumner School District (one of the 6 districts who implemented education reform in the state BEFORE everyone else).
The elementary math curriculum was called "Mathland" by Creative Publications (which is no longer a company).
Needless to say, I started supplementing our 3rd grade daughter in traditional math at home during that year and PULLED HER OUT to HOMESCHOOL her the next!
Mathland didn't succeed in California and it's not succeeding here in WA State.
A Sumner SD board member recently communicated that he sees Mathland as a success by referring to ITBS scores in 3rd grade. What this ignorant board member with blinders on doesn't realize is that the reason the ITBS scores are high is because of companies like Sylvan, Kmon and math tutors (traditional math tutors)... and parents supplementing their children with traditional math! Duh!
My children were NOT going to be the district's guinea pigs.
Again, thank you Shalimar for bringing this to the forefront. Let's keep this in the forefront until the pendulum swings back to offering traditional math from Kindergarten on up!
Posted by: Jean Ward on May 19, 2006 10:13 AMOh, Calvin. Where do I start? First off, did it occur to you that we simly may appear that we are kicking buts of these nations? Our productivitity is higher than what in comparisone? Could it be that Japan may have a higher relative per capita productivity than US? US may higher per capita GDP, what about the relative GDP adjusted to each country's living standard? Is US's higher in that sense? I highly doubt it.
Second, even if what you stated about higher productivity, GDP, etc. is true, could it be possible that that's on the back of imported competent workers or outsourcing?
I could go on and on, but I am you get my drift. And, even if we are enjoying higher level of productivity, GDP, etc., it is not because our education system is superior by any means but in spite of the quality of it.
As for math education, I don't doubt that there are obvious benefits to constructivist, but the fact remains, at least to me, that if you don't understand, simply memorize how it works (formulas, patterns, etc,) and there will come a time when you suddenly get it. The modern school system simply fails because those who don't get it will eventually stop getting it altogether because they just pass through without even an attempt to at least memorize what they didn't get in the first place.
Posted by: C. Oh on May 19, 2006 10:18 AMI'd say that all depends on what you do in the real world.
Posted by: RBW on May 19, 2006 01:58 PMMy profession as well as all that practice the sciences are based upon differntial equations and calculus.
But please, how many times have I used them over thirty years? Not many and even then it was just to plug in the right numbers. I didn't have to know the math, I just need to know what to plug into those equations.
So, why not teach me how to plug numbers in and not the theory? Well, it is the discipline and problem solving that is learned.
But again, what use is it for even the average student other than discipline?
Posted by: swatter on May 19, 2006 03:00 PMPosted by: Shalimar on May 20, 2006 07:54 AM
"I've recently been asked by some teenagers what good differential equations and calculus are in the real world. I tell them it does not do any good, but also that it trains their mind."
Well, as has been stated, it really depends what you do for a living. I might mention at first that the average student is not taking differential equations or calculus, merely the top performing students headed for a technical degree.
As far as some examples go, I just got home from a calculus conference and we listened to a speaker who does biometric analysis and uses calculus on a regular basis. We listed to an engineer who was called in to redesign dumpsters because they were buckling due to poor design. Calculus also. I have a degree in Aerospace Engineering and, although I have never worked as an engineer, I can tell you that calculus was and is extensively used in the design of both aircraft and spacecraft. And for the fellow with the idea that you can just plug the numbers in and use the results without knowing what the process is doing is hooey. I certainly would never ride in anything you designed. How do you know that you are using the right equation if you all you need to know is where to plug in the numbers? How do you know if you got the right answer if you have no understanding of the concept behind the process?
A few more that come to mind.
Analysis of Gini Coefficients which are used to make comparisons of various statistical data is also more efficiently processed using calculus.
Heat diffusion, wave equations, pressure distributions, coriolis acceleration. I'm sure there are more, those are just the ones that come to mind.
Posted by: Calvin A on May 20, 2006 07:27 PMI also see the difficulty choosing between concept and computation. Many students can function quite well on a computational level and not on a conceptual level. It is like the student who can form good sentences in his writing but could veer diagram that sentence. One does not need a thorough understanding of the parts of speach to be able to write clear and gramatically correct compositions. In math, one needs the abilty to "know" their addition and multimplication to a certain level like they "know" the words they intend to use when they speak. But unlike english, which we use every day in huge volumes, math is something we do a few hours a week. Math that focuses too much on concept and not enough of computation will prevent a student from "knowing" their math and using it as effectively as they do languange.
Now, if a student were to aspire to move into a career that requires mathematical proficiencey, having the abiltiy to "derive" formulas and to conceptualize math will be important. However, IF they ARE to move on in math, I would bet they ALREADY kow how to do that without being taught.
So, my off the cuff conclusion is that for students to perform daily "life" math, they need to have computational math drilled into them so they can be nimble in computations. With that ability, they will be less likely to struggle in algebra and geometry because when they don't stumble and stutter on basic addition, subtraction, multiplication and fractions, they will focus on the new material and not get so frustrated with how long it takes to do problems and how often they get the answer wrong (even IF they have the concept correct.)
Let me say that concepts in math ARE important in algebra and above, and a student who has been trained effectively in it before hand will do better, but concept is worthless if the basics are not first as "intutive" as speaking.
Therefore, my conclusion is that grades through the 5th should just be focused on getting them to know their numbers and then from pre-algebra on, you start helping them to understand the patterns of math and to learn to see how things relate and how they were derived. True Constructivism probably should be reserved for the "advanced" class math students who would be bored with "basics" anyway.
Posted by: Eyago on May 21, 2006 06:21 AMSo, no, I don't think you can say they are using the high math. But they are doing problem solving using the discipline of learning high level math.
So, you have proved my point. I also said above that there are a lot of higher formulas that are used in all professions, but you are plugging and chugging; you are not doing the higher math. It is a matter of adding and subtracting.
That is why more emphasis on the basics are important than the fancy math.
We live & work in what is the "Information Age", regardless of whether or not we recognize it. We want & get information instantly in nearly every aspect of our lives. We simply have the ability to communicate information much more effectively at this point in time than any other in history. We use google to find the equation we learned in HS to solve a problem once every five years etc.
Problem solving is what math, and higher level math attempts to teach. Not knowing the solution or how to find the solution is completely different than not even knowing where to look for the solution. I'm more concerned that students today, and my players, can solve the problem (with or without memorization) when needed. If they were at a loss, and cannot move forward, I can be disappointed. To do this, a basic understanding of Algebra is required (calc is not).
Posted by: Kyle on May 22, 2006 03:32 PMInstead of talking too much about my country's education system, i would like to show you the Singapore's education system.
Website: www.the-teamwork.com/?BV10011lis
Click on Quick Demo, and you will know what our students learn.
You can then judge for yourselves if this is more useful for American students.
Posted by: Elisabeth Wong on June 8, 2006 11:42 AMhttp://www.dailypress.com/news/local/dp-37219sy0jun08,0,1902654.story?coll=dp-features-wmsbg
Posted by: Elisabeth Wong on June 8, 2006 11:49 AM