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What Is Important In School Mathematics released by MSSG The Mathematical Association of America's Finding Common Ground in K-12 Mathematics Education project has released a consensus report "What Is Important In School Mathematics." I'm not sure what to make of this yet:
Near the end of July 2004, the Park City Mathematics Institute, under the direction of Herbert Clemens, hosted two workshops on states' K-12 mathematics standards. Both workshops were supported by the National Science Foundation (NSF). The first, July 21-24, was organized by Johnny Lott and was a meeting of the Association of State Supervisors of Mathematics (ASSM), the National Council of Teachers of Mathematics (NCTM), and some research mathematicians with an interest in K-12 mathematics education. The second workshop, July 25-28, the Mathematics Standards Study Group (MSSG) organized by Roger Howe, was a group of 12 mathematicians, many of whom had just been to the first workshop. During the discussion at the end of the first workshop a representative of the ASSM stood and asked the mathematicians: “What is important?” This essay is an attempt by the MSSG to begin an answer to that question. Other papers in the MSSG proceedings give answers to this question in terms of appropriate problem sets for state standards and in terms of two specific topics, place value arithmetic and proportions.Here are the consensus points:
1. Whole number arithmetic and the place value system are the foundation for school mathematics with most other mathematical strands evolving from this foundation. This foundation should be the subject of most instruction in early grades. 2. In every grade, the mathematics curriculum needs to be carefully focussed on a small number of topics. Almost all mathematics instruction should be devoted to developing deeper mastery of core topics through computation, problem-solving and logical reasoning. 3. Instruction needs to be mathematically rigorous in a grade-appropriate fashion. All terms should be defined with language that is mathematically accurate. Key theorems and formulas ought to be proved, whenever possible. 4. Disciplined, mathematical reasoning is one of the most important goals of a school education. Although it is difficult to assess on standardized tests, it must permeate all mathematical instruction.Nothing seems all that objectionable considering who was involved. Which is why I'm suspicious. I wonder if this is what NCTM's Seeley was referring to last week in her discussion? Could this be the first sign that NCTM is coming to its senses? I doubt it. the devil is always in the details. I also spotted Reaching for Common Ground in K-12 Mathematics Education which has even more sensible points.
A. Automatic recall of basic facts: Certain procedures and algorithms in mathematics are so basic and have such wide application that they should be practiced to the point of automaticity. Computational fluency in whole number arithmetic is vital. B. Calculators: Calculators can have a useful role even in the lower grades, but they must be used carefully, so as not to impede the acquisition of fluency with basic facts and computational procedures. C. Learning algorithms: Students should be able to use the basic algorithms of whole number arithmetic fluently, and they should understand how and why the algorithms work. Fluent use and understanding ought to be developed concurrently. These basic algorithms were a major intellectual accomplishment. Because they embody the structure of the base-ten number system, studying them can reinforce students' understanding of the place value system. D. Fractions: Understanding the number meaning of fractions is critical. E. Teaching mathematics in "real world" contexts: It can be helpful to motivate and introduce mathematical ideas through applied problems. However, this approach should not be elevated to a general principle. F. Instructional methods: ... For example, mathematical conventions and definitions should not be taught by pure discovery. Correct mathematical understanding and conclusions are the responsibility of the teacher. Making good decisions about the appropriate pedagogy to use depends on teachers having solid knowledge of the subject. G. Teacher knowledge: Teaching mathematics effectively depends on a solid understanding of the material. Teachers must be able to do the mathematics they are teaching, but that is not sufficient knowledge for teaching. Effective teaching requires an understanding of the underlying meaning and justifications for the ideas and procedures to be taught, and the ability to make connections among topics.Read the whole thing. -- KDeRosa - 02 Nov 2005 KtmGuest (password: guest) when prompted.
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Whoa, that looks way too sensible. It must be a trick. -- SusanS - 02 Nov 2005
It is for real. A group of mathematicians has attempted to develop a set of common principles that could be agreed upon by the mathematicians, NSF, and NCTM. The mathematicians include Roger Howe, Jim Milgram and Stanley Ocken. There are mixed feelings about this. Some believe it is a step in the right direction. Others are skeptical that it will lead to anything, and suspect that NSF and NCTM will attempt to co-opt the position of the mathematicians toward some fuzziness. I tend to think that neither side will be co-opted. NSF and NCTM will go with the money. If the money suddenly supported good solid mathematics standards, the world would suddenly be quite different. The market for textbooks like Investigations and Trailblazers and EM would dry up overnight. But I don't see anything happening in that quarter. -- BarryGarelick - 02 Nov 2005
It seems like the mathematicians may be winning so far. We need a follow the money diagram to show how the money flows from each organization. -- KDeRosa - 02 Nov 2005
Probably a lot easier to define what projects get money for now, than to trace the path. Right now, fuzzy math projects get the grants and contracts. These may be math/science partnerships with universities and school districts that ostensibly help the district with "professional development" which translates to: help them teach EM, or Investigations or Trailblazers, or Watch me Retch, or whatever the program is. They may also be "implementation centers" which function the same way that the MSPs do. The funding to develop new math texts seems to have died, thank goodness. When a bunch of us met on the Hill last April, we learned that no one on the Hill is going to cut funding to any NSF program. Education and Human Resources (EHR) is the part of NSF that funds a lot of the fuzzy stuff. A politician proposing to cut money from any part of NSF is tantamount to a politican proposing to cut social security. It's political suicide. So things have to proceed behind the scenes. But not much has been going on. In truth, even if not a penny more were spent by NSF, a lot of the damage has been done. The bad math texts have been written, grants have been given to school districts to implement the bad texts, etc etc. What is needed is a flood of money to produce good texts, model standards and frameworks that states can use in writing their standards so that NCTM is not the only game in town. Putting in a knowledgable math-savvy director of EHR might help but it would take more than that. SOme things are moving. A fellow math warrior from the Hill has been appointed to a White House office that is involved with domestic science and technology policy, so she will be an "insider" helping to push a better math ed agenda. And there may be some other changes happening that I can't talk about right now. Stay tuned. -- BarryGarelick - 03 Nov 2005
Isn't this the same document one of the middle school bloggers was reading in order to fend off hostile parent questions on Back to School night? -- CatherineJohnson - 03 Nov 2005
hmm Let me find the one I saw... -- CatherineJohnson - 03 Nov 2005
Here's the article I found (somehow I lost the middle school blog link, which I wanted to post...) Jim Milgram was part of this one: Reaching for Common Ground in K–12 Mathematics Education (pdf file) -- CatherineJohnson - 03 Nov 2005
Are these the same report?? I need to spend some time reading through these things... -- CatherineJohnson - 03 Nov 2005
oh jeez I see you HAVE LINKED TO Reaching for Common Ground..... Time to go do Saxon. -- CatherineJohnson - 03 Nov 2005
yes they are the same. -- KDeRosa - 03 Nov 2005
These mathematicians that are heading this effort up, Clemens and Howe, are solid. Bernie and Herb Clemens were colleagues at the University of Utah for three years while Bernie was a postdoc there. He's both mathematically and politically strong. My mathematical specialty, harmonic analysis, is the same as Roger Howe's -- he is a 'mathematical uncle' of mine. He's at Yale -- absolutely at the top of our field. I had one mathematics paper in the American Journal of Mathematics -- a result I was really proud of. I found out a few years later that Roger had proven the result before I did, and thought it not worth publishing -- it was tucked away in his drawer somewhere. He was kind enough not to mention it! He has good sense and a good heart. I trust his judgment absolutely. It's very exciting to me that these heavy hitters have waded into the fray. I hope they hang in through the 9th inning. -- CarolynJohnston - 03 Nov 2005
To mix my metaphors. -- CarolynJohnston - 03 Nov 2005
Roger will never in a million years go fuzzy. Never, never, never. Bernie is not sure about Herb. -- CarolynJohnston - 03 Nov 2005
I've met both Milgram and Howe and do not believe that either will go fuzzy. I'm just reporting that there are some people who believe that even talking to the other side is a bad thing. In my opinion that's just nonsense. There are risks involved with anything, I imagine, but given the power that NSF and NCTM have, it doesn't seem that ignoring them is going to serve anyone's purpose. -- BarryGarelick - 03 Nov 2005
Those who want better, more rigorous math taught in K-8 generally agree on what needs to be taught. When it comes to how to get there, that is another question. It is both a mathematical and a political problem. The problem with working within the system (ed schools, NCTM, & NSF) is that there are issues that have nothing to do specifically with math. In our town, it is the concept of full-inclusion and no pull-out. This means low expectations. The school has a fundamental conflict; how to teach all common-age kids in a child-centered, full-inclusion environment and still give all students what they need. This means that the advanced (and even average) kids get enrichment rather than curriculum acceleration. The enrichment is usually homework, is optional, and depends entirely on the willingness of the teacher. These problems have to do with fundamental assumptions of education, not technical questions about mathematics. I don't see how our town could possibly use Saxon or Singapore math without changing these fundamental assumptions. It cannot or will not be done. Many of the problems with "fuzzy" math are not so much because they are fuzzy, but that they cover a lot less material and don't emphasize mastery. I prefer emphasizing charter schools and full vouchers, but these provide no guarantee. You cannot get a charter in our state if the charter has to do with setting higher, and more specific grade-by-grade standards. I have said before that if our state opened up the restrictions on charter schools, then I know very many parent groups who would have a school up and going by next fall. Many won't even care what NSF or NCTM are doing. -- SteveH - 03 Nov 2005
The NCTM is not pure evil. "Knowing basic number combinations -- the single-digit addition and multiplication pairs and their counterparts for subtraction and division -- is essential. Equally essential is computationsal fluency -- having and using efficient and accurate methods for computing." (NCTM Principles and Standards for School Mathematics, 2000, page 32) "The point is that students must become fluent in arithmetic computation -- they must have efficient and accurate methods that are supported by an understanding of numbers and operations. 'Standard' algorithms for arithmetic are one means of achieving this fluency." (page 35) I own a copy (actually two: printed and electronic) of PSSM, and I have read the whole thing. Most of the things annoying things that have come out of PSSM seem to be from people who have misinterpreted the more idealistic aspects of it. Overall, though, a curriculum taught to the intents of PSSM (as opposed to radical interpretations of parts of PSSM) would be, in my opinion, a good thing. -- RudbeckiaHirta - 03 Nov 2005
In our town, it is the concept of full-inclusion and no pull-out. This means low expectations. The school has a fundamental conflictOddly enough, I believe in Singapore that there is no ability grouping until algebra in 7th grade; all students learn the same curriculum at the same pace. I was a little surprised by this myself. -- KDeRosa - 03 Nov 2005
The NCTM is not pure evil.I think it was the endorsement of the fuzzy curricula that kicked them into the evil category and belied any good intentions they may have had. The constructivist curricula is the antithesis of mastery learning the way it is implemented today. -- KDeRosa - 03 Nov 2005
"Oddly enough, I believe in Singapore that there is no ability grouping until algebra in 7th grade; all students learn the same curriculum at the same pace. I was a little surprised by this myself" This is the way it should be. This is how they did it when I was growing up. (not Singapore Math) However, with the assumption of full-inclusion, this cannot be done. They have to slow down the pace and lower expectations. They don't believe in a full course in algebra in 8th grade. That is why there is a math content gap between our 8th grade and high school college prep math. Average students who could do the work never get there - it's all over. If a curriculum like Singapore is too rigorous for some, then you can track them into watered-down or slower curricula or hold them back. Schools do not like to hold kids back and they don't like to track, so they slow down the pace, make the subject more fuzzy, and spiral the curriculum for all. As long as they can teach something that sounds like algebra in 8th grade, then everything is OK. In our public schools, full-inclusion and no tracking or pull-out is the governing equation. -- SteveH - 03 Nov 2005
They don't believe in a full course in algebra in 8th grade. That is why there is a math content gap between our 8th grade and high school college prep math. Average students who could do the work never get there - it's all over. Steve has definitely hit the nail right on the head. -- KDeRosa - 03 Nov 2005
I call this mathematics socialism. Since they have been unable to raise achievement of the low performers, they settle on pulling everyone down to the lowest common denominator. -- KDeRosa - 03 Nov 2005
I see evidence of tracking making a comeback. My guess is that it is due to NCLB-enforced realities, but I have no evidence to support that guess. -- DanK - 03 Nov 2005
"A group of mathematicians has attempted to develop a set of common principles that could be agreed upon by the mathematicians, NSF, and NCTM." Reform through common principles? What a novel idea. : ) -- JdFisher - 03 Nov 2005
"Those who want better, more rigorous math taught in K-8 generally agree on what needs to be taught. When it comes to how to get there, that is another question." Agreed, wholeheartedly. I also said it was a good thing to talk with the other side. I didn't mean to imply that such talk be all niceties. I'd like to see them talk turkey. If the people from NCTM and NSF and the ed schools (that would include Deborah Ball) agree with these principles, then let them justify/explain how Investigations, Trailblazers, Core Plus, IMP, etc. meet these goals. And let the mathematicians explain how they do not. Let NSF, NCTM and ed school people explain why curricula represented by Saxon Math and Singapore do not meet these goals. It isn't such a simple matter of "Oh we've always agreed with one another all along, this has all been some terrible misunderstanding." They won't tell why Singapore and Saxon do not meet the goals, because they can't. They'll change the subject. Steve hit it on the head. It's because there is no longer a lowered bar. Some kids will get it quicker than others. In the world of the educationists, everyone is a winner. But in that situation, when everyone wins, everyone loses. -- BarryGarelick - 03 Nov 2005
This means that the advanced (and even average) kids get enrichment rather than curriculum acceleration. This is wrong, wrong, wrong. I noticed this myself two years ago, before I knew Thing One about math ed. That was the reason I wasn't concerned about Christopher's placement. I figured if 'advanced math' meant 'regular math with Math Olympiads problems your parents do' there was no point. I was right. -- CatherineJohnson - 03 Nov 2005
I got a letter from our Assistant Superintendent in charge of curriculum saying the Phase 4 kids 'need to be challenged.' I disagree. They need to be taught to mastery, at as fast a pace as they can manage. -- CatherineJohnson - 03 Nov 2005
My understanding of the Singapore situation is that they don't track until around junior high (though I've seen one report saying otherwise). I don't know what Steve means by full inclusion, but my understanding of Singapore is that they have full inclusion. Everyone learns the same material. Somewhere around 6th grade they separate the faster kids from the slower, and they give the slower kids an extra half hour a day (a huge increase in time) and the best teachers. Those kids continue to learn the same material the faster kids learn. I read one terrific article on tracking that I bet I won't be able to find...it said that the one good tracking situation they'd found was at a Catholic School where every kid in the school knew his track, the tracks were spoken about openly, etc.....and the overriding goal of the lower tracks was to get the kids up to the top track by the time they graduated. Which they did! By the time the kids graduated from high school the kids in the lowest track had managed somehow to catch up to the top track. (I assume this doesn't mean the kids at the bottom were doing AP calculus by the end of high school. BUT they were certainly mastering algebra 1 & algebra 2 & maybe even pre-calc.) -- CatherineJohnson - 03 Nov 2005
Wayne Wickelgren had an interseting proposal for tracking..... I'll have to look it up. It was completely individualized. Basically all kids move through the grade levels as quickly as possible. If some kids move faster, they bump up to the next grade & join that class. -- CatherineJohnson - 03 Nov 2005
If the people from NCTM and NSF and the ed schools (that would include Deborah Ball) agree with these principles, then let them justify/explain how Investigations, Trailblazers, Core Plus, IMP, etc. meet these goals. And let the mathematicians explain how they do not. Let NSF, NCTM and ed school people explain why curricula represented by Saxon Math and Singapore do not meet these goals. It isn't such a simple matter of "Oh we've always agreed with one another all along, this has all been some terrible misunderstanding."Barry has given us a good example of the difference between a dialogue/communication and a debate. They want the former, we want the latter. -- KDeRosa - 03 Nov 2005
My understanding of the post 6th grade Singapore curriculum is that the kids are broken-up into three tracks: the math brains who get accelerated, the normal academic track, and the trade track that learns pretty much the same thing as the academic track with less proofs. This is all from memory and I'm not exactly sure how reliable the source was. -- KDeRosa - 03 Nov 2005
I read one terrific article on tracking that I bet I won't be able to find...it said that the one good tracking situation they'd found was at a Catholic School where every kid in the school knew his track, the tracks were spoken about openly, etc.....and the overriding goal of the lower tracks was to get the kids up to the top track by the time they graduated. I went to a catholic high school in the 80s. We were definitely tracked in all our classes. Movement between the tracks was rare and, if anything, movement was downward, especially in math. -- KDeRosa - 03 Nov 2005 Back to: Main Page.