It was 1986, folks (or perhaps 1929), for those keeping score at home. Twenty-three (or eighty) years later and the same arguments are going on, the same mistakes are being made, as if nothing at all has been said like what Louis P. Benezet or Hassler Whitney offered. As if Constance Kamii's work has never been done or published.
My thanks to G. S. Chandy for pointing me to this article. It was published while I was in the process of taking undergraduate mathematics courses in NYC and slowly gravitating towards changing fields from literature to mathematics education. I'd never heard of NCTM or any of the other organizations involved in mathematics teaching and research, the Math Wars hadn't officially started yet, and had I read this piece at the time, I would have naively wondered how anyone could be on the other side from people like Benezet and Whitney. Having suffered through a K-12 mathematics education that was about as inspiring as a dead fish in the gutter, it is remarkable to me even today that I took it upon myself in my thirties to go back to school just to prove to myself that I could indeed learn more mathematics, from calculus to where I was led. Didn't plan to go into the field, and it was fortune, more than anything, the attention of one of my instructors, that led me to start teaching remedial mathematics and, eventually, to do graduate work in math education at the University of Michigan in the 1990s.
Yes, there has been some progress since then, but the entrenchment of traditionalists is fiercer than ever. The lies, distortions, selective quotations, meaningless and carefully culled data, shifting criteria for what "counts" when it comes to evaluating programs, teachers, schools, kids, materials, etc., and many other shady tactics continue unabated, fueled by a hatred for innovation and purveyed by politically-motivated, educationally conservative and reactionary pundits, think-tanks, and foundations, all fiercely determined to see to it that mathematics teaching and learning in this country remain in the hands of a smug, patronizing elite. As long as they are successful in reducing us to a standardized-test crazy culture, as Whitney so accurately describes below, the country as a whole and millions of children will suffer unnecessary torment and boredom when it comes to mathematics. And a populace that is mathematically ignorant is a populace that is far easier to lead by the nose.
Are Benezet's and Whitney's ideas really just those of a couple of isolated cranks, as the anti-progressives from groups like Mathematically Correct would have us believe? Consider for a moment the following anecdote about the great mathematician, Augustin Louis Cauchy:
A mathematical friend of Cauchy's father, Lagrange, recognized the young boy's precocious talent and commented to a contemporary, 'You see that little young man? Well! He will supplant all of us in so far as we are mathematicians.' But he had interesting advice for Cauchy's father. 'Do not let him touch a mathematical book till he is seventeen.' Instead, he suggested stimulating the boy's literary skills so that when eventually he returned to mathematics he would be able to write with his own mathematical voice and not one he had picked up from the books of the day.
It proved to be sound advice. Cauchy developed a new voice that was irrepressible once the floodgates protecting Cauchy from the outside world had been reopened." (Marcus du Sautoy's THE MUSIC OF THE PRIMES pp. 65-66)
No, the analogy to Benezet's experiment is not perfect by any means. But it does suggest that there has long been an awareness that some aspects of formal instruction as they become institutionalized can be stifling to creativity and originality.
Does this mean I am advocating for the destruction of public (or any) formal education and schooling? Not quite. What I am advocating for is a coming to sanity on the part of educators in this country when it comes to mathematics teaching (if nothing else). We are destroying our children, en masse, with the most stultifying approaches imaginable to learning and doing real mathematics, substituting instead a phony "school mathematics" that serves no one truly well, and from which only a small minority emerges able to actually do mathematics, in spite of, rather than because of, the way the subject is taught in most instances.
Few American K-12 teachers have the smallest idea what mathematics is, what it means to do mathematics, or what it means to be a professional mathematician. And what these teachers wind up doing, consciously or not, is to guarantee that very few students will ever find out.
Every time I bump into a piece like the one about Whitney and Benezet below, I am both amazed and sickened: amazed that I hadn't seen this before (though I've been aware of Benezet for over a decade now); sickened that the same lies rise like a foul smokescreen every time Benezet's name or any idea that sounds even vaguely like his, is presented. How much longer do our children have to be tormented by meaningless mathematics education? When will it be time for real mathematics to be taught and learned, in a manner suitable to how children are? If not now, when? If not us, who?
By FRED M. HECHINGERPublished: June 10, 1986
SCHOOL reformers, business executives and politicians are demanding more mathematics for American children. Schools are responding, at least in terms of the hours given to math. Not all mathematicians are cheering. They worry that pressures for more hours of mathematics may hurt rather than help, unless mathematics is taught differently.
Dr. Hassler Whitney, a distinguished mathematician at the Institute for Advanced Study in Princeton, says that for several decades mathematics teaching has largely failed. He predicts that the current round of tougher standards and longer hours threatens to ''throw great numbers, already with great math anxiety, into severe crisis.''
Dr. Whitney has spent many years in classrooms, both teaching mathematics and observing how it is taught, and he calls for an end to what he considers wrongheaded ways.
Long before school, he says, very young children ''learn in manifold ways, at a rate that will never be equaled in later life, and with no formal teaching.'' For example, they learn to speak and communicate, and to deal with their environment. Yet the same children find much simpler things far more difficult as soon as they are formally taught in school.
Learning mathematics, Dr. Whitney says, should mean ''finding one's way through problems of new sorts, and taking responsiblity for the results.''
''This has been completely forgotten'' in most schools, he finds. ''The pressure is now to pass standardized tests. This means simply to remember the rules for a certain number of standard exercises at the moment of the test and thus 'show achievement.' This is the lowest form of learning, of no use in the outside world.''
Dr. Whitney, in a recent report in The Journal of Mathematical Behaviour, recalled an experiment begun in 1929 by L. P. Benezet, then superintendent of schools in Manchester, N.H. Mr. Benezet was distressed over eighth graders' poor command of English and their inability to communicate ideas.
''In the fall of 1929,'' he wrote in 1935, ''I made up my mind to try an experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrate instead on teaching the children to read, to reason and to recite'' by reporting on books they had read and on incidents they had seen. The children were no longer made to struggle with long-division. ''For some years,'' Mr. Benezet went on, ''I had noticed that the effect of the early introduction of arithmetic had been to dull and almost chloroform the child's reasoning faculties.''
Over the years numbers crept into the children's experience, Mr. Benezet said. They learned to deal with ''halves'' and ''doubles,'' with estimates of size, with a natural development of multiplication tables and slowly, with formal arithmetic.
Mr. Benezet concluded that children who had not been dragged into early but only dimly understood mathematics eventually outdistanced those who had. Literacy in English and a capacity to think independently and to speak and write clearly helped many to do well in mathematics, too.
Dr. Whitney points to that experiment as he looks at today's mathematics teaching. He cites the responses to a problem on a recent test given by the National Assessment of Educational Progress: John and Lewis are planning a rectangular garden 10 feet long and 6 feet wide, and they want to put a fence around it. Ignoring such real matters as the need for a gate, the question was simply how many feet of fencing was needed.
Of the 9-year-olds who took the test, 9 percent chose 32 feet; 59 percent, 16 feet; 14 percent, 60 feet, and the remaining 18 percent gave other answers. Of the 13-year-olds taking the test, 31 percent said 32 feet; 38 percent, 16 feet; and 21 per cent, 60 feet, with 10 percent giving other answers that apparently did not use any arithmetical formulas.
''Why did not all the children get the correct answer?'' Dr. Whitney asks. ''If they were involved in it as a real problem they could have drawn a picture or made it real in some way, and looked to find the answer.'' Instead, he said, they did it ''the school way,'' guessing at what kind of ''operation'' to use - multiplying or adding the numbers.
Numbers, Dr. Whitney says, become a tool when you use them for a purpose. In a class of 6-year-olds, he recalls, the teacher explained how to find the sum of 3 plus 5 by drawing ducks on the board, not noticing a boy in the back of the room saying to another, ''Yesterday I gave you 10 cards; now you gave me 7, so you still owe me 3.''
In the traditional school climate, Dr. Whitney writes, children's natural thinking ''becomes gradually replaced by attempts at rote learning, with disaster as a result.'' In high school, students increasingly say, ''Just tell me which formula to use,'' a way of saying, ''Don't ask me to think.''
Because teachers must ''cover the material,'' Dr. Whitney adds, there is less time to think. When students are called on, they must answer instantly. Wrong answers are not discussed.
''Students and teachers are all victims'' as national commissions clamor for more mathematics without realizing, Dr. Whitney warns, that they may create less knowledge and more anxiety. He says it is crucial to stop just learning the rules.
Dr. Whitney's views are controversial, as were Mr. Benezet's in 1935. Some mathematics teachers and other experts may denounce them as soft on mathematics, but others may welcome relief from demands that turn youngsters off mathematics. Of course, some teachers, ignoring the demands of the moment, actually do teach in the Benezet-Whitney fashion.
However controversial his views, Dr. Whitney deserves a hearing. Present attitudes, he writes, ''lead to the lowest of goals, passing standardized tests,'' instead of encouraging the kind of thinking ''essential for true progress in science, techology and elsewhere.''
The mathematics teaching Dr. Whitney talks about makes children want to know the answers in situations that are real to them. It makes mathematics come alive for them as they do their own thinking and take control over their work, not for tests but for themselves.
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12 comments:
I believe that the true root of the "math wars" is not inherent in math itself, but in the current system of education. The very of idea of a standard public education demands standard tests. Our society explicitly demands that any person of age 16 should be able to show competency in a many areas (what we call curriculum). Those who understand education and brain development realize that is unreasonable to expect all 16 year olds to have a similar grasp of mathematics.
In past centuries (I think primarily of the Golden Age of mathematics in the late 18th and early 19th century and the Classical Greeks) the idea that all people learn mathematics was absurd. When would a farmer ever need to consider the intricacies of projectile motion? Those who had opportunity to consider ideas such as this were brought up in households where education was emphasized, and tutors were readily available to stimulate at most a handful of young minds. Their minds were sharpened because their class afforded them that possibility.
I teach mathematics in a rural school in Saskatchewan and I see first hand how these contrasting situations can play out today. I do not worry about the quality of mathematics instruction my daughter will receive, because I know that, like Cauchy's father, I can guide her through what is important and what is not. I know that grades do not necessarily mean understanding. But I also teach numbers of students and understand the difficulty of giving students a grade that I believe accurately reflects their level of understanding of a curriculum set out by our society. I don't believe that I am given the freedom to evaluate a students "mathematical ability" whatever that is. I would prefer to be given more flexibility in what I am allowed to teach, but that is not the mandate set out for me by society.
How do you feel about the content and pedagogy of Singapore Math?
If more U.S. schools were to use Singapore Math instead of Everyday Math, Trailblazers, or Investigations, do you believe that would lead to an improvement, or a further decline, in U.S. math education?
My blog (http://oilf.blogspot.com/) gives regular comparison problems between Singapore Math, French Math, and Reform Math.
Lefty: first, I didn't publish your post earlier because I could not access my blog for moderating comments until just now: there was no need to resubmit your comment.
Second, I'm not in possession of a copy of any Singapore Math materials and hence will not offer a critical view - positive, negative, or neutral. As long as the American publishers refuse to offer examination copies free of charge, I am unlikely to form an adequate opinion to comment. I have familiarity to a very limited extent with some of what is taught and how, but absent the opportunity to see student, teacher, and assessment materials in full, I must take the principled position of reserving comment.
I'll look at what you've got on-line, but I am skeptical that there is a single solution to the various challenges we face here to raise the level of mathematics teaching and learning for a significant, large percentage of students. I am intrigued by the use of Everyday Math as the primary teaching materials and Singapore as the supplementary materials in the Seattle Public Schools, an approach which began in the fall of 2007. I do not know what results have emerged thus far, though less than two years into the process, I would believe them to be premature.
I am a high school math teacher. I love math very much, which makes me an exception among math teachers. I feel a bit torn between battling camps over math education. Everything stated in the article is true. I often don't feel like I'm really teaching my kids math (the creative, artistic aspects--the more fundamental aspects).
I think that we must decide if we want to "make students good at mathematical procedures" or "get students to LIKE math and be good at thinking". I think that's the hang up.
After a year of being in my Calculus class, my students can DO all sorts of things. And they're prepared for all sorts of math-science tracks in college. For instance, without this type of formal Calculus class, the task of someone teaching Calculus-based Physics would be impossible. So in this case, formalism in education is a GOOD thing.
Taking a formal course in Calculus doesn't make my students good at math, however. Many don't see the beauty of it; and many don't get the joy of discovery, the hallmark of mathematical thought. But if we want to accomplish these objectives, we would structure the class like an art class (think Paul Lockhart's Mathematician's Lament). We risk not having them learn a set amount of material, but it might be worth it.
But I don't think it's an easy battle. I think formalism is still valuable in meeting some objectives. I certainly think we could use a little more space in our curriculum for creativity, as this article suggests. I think it's time to do away with testing and life-sucking curriculum. Grades are a pesky thing. I'd love to just have a math-art class. Just let me know when I can teach real math.
John Chase
@Richard Montgomery High School
Montgomery County, MD
Michael, I think you are ignoring the most important factor—the teacher. Yes, I believe the curriculum is important and puts constraints on the teacher, but the teacher is even more important. The best curriculum won’t get good results with a poor teacher. And a bad curriculum will make it hard on a good teacher, but I would bet that a lot of learning takes place anyway.
Did you read the Teaching Gap by Stigler and Hiebert? It describes the results of a video study of classrooms in three countries: the United States, Japan, and Germany. It was part of the Third International Mathematics and Science Study (TIMSS). What struck the team of researchers was that each country had a theme, or cultural approach to teaching. The image of teaching in Germany was “developing advanced procedures;” in Japan it was “structured problem solving;” in the United States it was “learning terms and practicing procedures.” One math educator on the team said that in Japan, the students engage in mathematics and the teacher mediates; in Germany, the teacher owns the mathematics and parcels it out to students, giving facts and explanations at the right time; in the United States, there is interaction between teacher and students, but he couldn’t find the mathematics. He didn’t see any real math in memorizing terms and procedures.
We have a cultural mindset that thinks of math in terms of procedures and calculating an answer and not in understanding mathematical concepts and relationships. As Stigler and Hiebert point out, we aren’t even aware of this mindset. It was in studying classroom teaching in different countries that they realized this cultural tendency existed.
The TIMSS video study also showed how difficult it is to break the pattern. About 70% of the teachers said they were implementing reforms such as those published by the NCTM, and they pointed to places in the video where they were doing so. When the researchers looked at the video, they found only surface changes; the lessons were still consistent with the image of memorizing definitions and practicing procedures. In fact, in many cases the teachers’ actions were worse than what they might have done otherwise.
Did you read Liping Ma’s book, Knowing and Teaching Elementary Mathematics? She compared the mathematical knowledge of Chinese and American teachers. The American teachers’ knowledge of concepts was poor. The Chinese teachers’ knowledge was very good. If you haven’t read the book I strongly recommend it. The Chinese teachers use a standard curriculum (as do the Japanese) with desks in rows facing the teacher. It is very much the teacher-centered approach that you have criticized. But I think you would like the interactions and the real thinking and learning that goes on there. If teachers do not understand mathematical concepts, then they can’t teach conceptually. If they understand the math, then they can teach in any approach that they have learned to use. They need to be comfortable with the methods and genuine in their interaction with the children.
The reform that is needed is not in the curriculum so much as in changing the cultural mindset and in teaching the teachers (especially elementary school generalists) math.
Wow, Burt, thanks for that comment. I'm going to make sure to read The Teaching Gap now.
I teach at a community college, and my students really push me to "show them the steps". No matter how I conceive of the math, they want to keep it in the realm of procedures. It is hard to change, and your analysis (or the authors') helps me see it more clearly.
Burt,
Interesting that you think I am ignoring teachers in the equation. Have to assume from that and some of the other things you ask (like whether I know the Liping Ma book) that you aren't familiar with me or this blog.
Teachers are key, but for them to be effective, they need to be free to teach real mathematics from a broad range of perspectives. Our attempts here to limit K-12 mathematics to the same boring nonsense we've been trying, mostly unsuccessfully, to shove down kids' throats for over a century is just utterly inadequate for the needs of a 21st century democracy. There is SO much wrong with how restricted and narrow our vision is that it would take years of blogging by me and many others to touch on it all.
For a look at just a few of the possibilities, look at the Computer Science Unplugged Web Site, the This Is MegaMathematics! site, and google a book called DIGITAL MATHEMATICS FOR THE DIGITAL AGE and PROGRAMMING IN PYTHON. These aren't all that's out there, but they begin to touch on alternatives few teachers in regular K-12 are looking at or exploring. Some would, if they could, but almost no one is in a position to do so.
Why not? Because the move towards increased standardization and the pressure to make crappy high-stakes assessments the tune to which everyone must dance are killing creativity and individual thought amongst teachers and students alike. It's a tragedy that can't be addressed by looking at only the content/curriculum side or the teacher/pedagogy side. It drives me berserk to hear from the conservatives that it's all about content and that pedagogy is a non-issue, and equally crazy to hear from a few others that it's all or mostly about pedagogy. Not suggesting you believe either extreme. Please don't think that I do, either.
Good points, Michael. I'm going to add those books and websites to my reading list.
Michael,
I am sorry I assumed your were more extreme in your views than you actually are. You are right that I only read two of your blog entries before I posted my comment. I did not see a balance in those postings. After reading your last post in May, and your comments to Stilger’s presentation, I think you and I agree more than we disagree. I agree strongly that the high stakes testing is pushing schools and teaching in the wrong direction.
You have much more experience in math ed and broader knowledge than I do. I am more focused. I still believe all attempts at reform will fail if we do not address the mathematical knowledge of elementary school teachers. Do you think that the “above average” teachers in Liping Ma’s study could conduct a lesson the way that the Japanese teachers do? I don’t think that either of us believes they could, especially given their poor responses to the open ended item on Liping Ma’s interview. You include as a need “d. Provide teachers with opportunities to learn the knowledge, skills, and judgment they will need for continuous improvement: stable settings to work with colleagues to learn what works better and share what is learned with the profession.” I just put more emphasis—a lot more emphasis--on this than you do. I think it gets underemphasized beside all the other issues, but it needs to be priority number 1.
Professor of mathematics Patricia Kenschaft wrote an article titled “Racial Equality Requires Teaching Elementary School Teachers More Mathematics” in the February 2005 issue of the Notices of the American Mathematical Society. For years she visited elementary schools on her own time because of “the great need to teach mathematics to elementary school teachers. Like most Americans, I found it difficult to believe how poorly prepared mathematically they are. They are well chosen. They are kind, diligent, and smart, qualities that nobody can teach. They have been failed mathematically by our system. They need to be taught. I have found them eager and quick to learn—and appallingly ignorant of the most basic mathematics.” Teachers who do not understand the concepts can only teach the procedures in a somewhat rigid way. They cannot facilitate discussions, they cannot entertain students’ novel ideas, they cannot deviate from the narrow procedural knowledge that they possess, and they cannot explain why the procedures work. What they do teach is that math is memorizing definitions and procedures.
Burt,
I'm not sure what the #1 priority is any more (if I ever was). I think it's clear that too many K-5 teachers in this country don't know very much mathematics or how to teach what they know very effectively. I'm also sure from my experiences and reading and observations that too many teachers at higher levels who do know mathematics aren't effective at teaching it to a wide-enough spectrum of kids.
Further, I'm sure that too many kids come to school ill-prepared by their home environments to learn much of anything: as I wrote somewhere today, factors from poverty (including poverty of intellectual engagement by care-givers) bring kids to school who aren't able to behave or learn in the ways teachers expect (even when those expectations are reasonable).
So is the problem mostly the mathematical ignorance in K-5, the pedagogical weaknesses among teachers in 6-12, the textbooks, the curricula, the high-stakes tests,or the impact of poverty and horrid environments?
I certainly think it's all of the above and more. Singling out any one of those is, on my view, a mistake. Focusing one's efforts on any one of them a given person feels well-positioned to address, however, is only realistic and sensible.
I have my fingers in a lot of pies (probably far too many) and, unfortunately, far less influence or leverage than I wish I had. I am neither part of the educational establishment nor of the organize right wing attacks on it. I have to operate as best I can from the margins.
What I find very frustrating is how much energy is poured into trying to hold back positive, progressive reform efforts by the educational right wing (and, of course, by the foot-dragging of entrenched teachers and administrators who fear change and the hard work it demands). The ability of anti-progressives to stir up fear in parents is also highly problematic.
While I never am inclined to give up, I do find it to be a horrid waste of energy to have to keeping fighting the same fights over and over each time positive efforts are attacked or rejected in advance. And I also despair at times at how silly people who would likely be seen (incorrectly) as representing practices and viewpoints I support can be in how they implement reform. I'm hardly alone in this. Deborah Ball's piece from way back about the use of manipulatives in elementary mathematics instruction as if they were panaceas (and as if the mathematics were in the models and tools such that merely using them guaranteed understanding) is one such exploration of good ideas gone badly awry through naive good intentions. But I think I could do better with teachers misusing good ideas than "correct" use of truly bad ideas. Maybe I'm deluding myself.
Sue, if your students “want to keep it in the realm of procedures,” have you considered changing the incentives? Your students may think the tasks in math involve using procedures to compute an answer, but what if you changed some questions so no computation was required, that a small number of questions on your quizzes and exams would be concept questions? Then they would have to pay attention to, and think about the concepts. You might even focus on common misconceptions. On way of thinking about this is to compare different solution methods. Another way of approaching this is adapting formative assessment techniques. A book that might help is Uncovering Student Thinking in Mathematics: 25 Formative Assessment Probes.
I think it would be interesting to look at how autonomously educated children come to terms with mathematical concepts.
This is a bit of a new area for me, though with regard to literacy skills, I have already observed over and over again, that autonomously educated children don't usually learn to read before the age of 8 through to 10, but then rapidly progress to become very competent and avid readers.
I am now discovering that the same unusual approach applies with maths concepts. We haven't done any formal maths with son, though of course he encounters maths all the time one way or another..games he plays, shopping etc.
He has just turned 12, and one way or another wanted to see how he was getting on. We brought him those workbooks from WHSmith and has whizzed through most of KS2 stuff as well as other sorts of stuff in about 6 hours study!
Another autonomously educated young person we know well who last year received the top first class degree in maths and computing at Imperial College only ever had one short furious maths lesson with his mother which ended up on a stand-up row between the two of them. The mother never tried to inflict such a lesson on the boy again and reports that she's still not convinced that he knows his times tables!
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