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My Disagreements Shot Down

Brian D. Rude 2004

It was on February 19, 2004 that I sent my article, “Some Disagreements With The Standards”^{1} to The Mathematics Teacher for consideration for publication. On May 27 I wrote to withdraw it from consideration for publication, as I had heard nothing in the previous three months other than a routine letter that they had received it.. I withdrew it also because I decided to put it on my new web site. I had learned enough by that time to know that it was very unlikely that it would be published. There are “math wars” going on, unfortunately, and the NCTM is stuck on one side. In spite of my withdrawal, which was acknowledged in a June 2 letter, I got a rejection letter on Aug 7. They were also kind enough to explain to me the error of my ways. In addition to the letter saying my article “is not appropriate for the Mathematics Teacher as written”, there was a page of comments on my article. It is primarily these comments that I will analyze here. First I will reproduce this page of comments in full, as the form, as well as the content, is the subject to some speculation. Then I will try to return the favor by explaining to the NCTM the error of their ways. Hopefully in the process I will say something worthwhile about teaching math.

Here is the complete page of comments:

- description of standards is superficial and inaccurate

- presents a deficit view of students

- misses the point that the standards advocate for connections and not discrete topics which are mastered, or not, in a day

- students portrayed in the standards are not supposed to be geniuses, they’re people with natural curiosity and ways of viewing the world. They’re not supposed to be uber-problem solvers

- author implies that he is teaching in a standards-based way but his description belies that

- author states that ‘assumption of fluency’ is common in the standards but does not provide evidence of this

- there is no ‘principle of practice’ in the Standards because students are doing mathematics throughout class; rote drills do not constitute an authentic mathematical practice

- criticism is really the implementation of the Standards, not the Standards document itself

The author claims to make an argument about the Standards document when he is in fact making an argument about superficial interpretations of the Standards (including his own). The author claims that implementing the Standards results in students doing a series of unrelated activities, indicating that he has not had any experience with curricula designed on the principles expressed in the Standards. These curricula are perhaps the best manifestations of the vision of math instruction in the Standards - because these curricula are based on carefully sequenced and related tasks.

I am guessing that this is a compilation of comments made by different reviewers. I sent five copies of my article, according to directions, which I assume means that at least five reviewers are involved. I can imagine that each reviewer made notes in the margins, and these notes were then compiled. There is no explanation why the first eight comments are put in a list form, while the last comment is in a paragraph form. I wonder if the first comments were marginal notes, and then someone summarized everything in the final paragraph. I have checked carefully to make sure that all spelling and punctuation of these comments are accurately reproduced.

Comment One, “description of standards is superficial and inaccurate”

I suppose I’ll have to plead at least partially guilty on this one. The Standards is a long document. I read it on the internet over a period of a number of months. I don’t claim to have read every word. I have no intention of ever reading every word. It didn’t take too much reading for me to realize that the perspective and premises presented are not ones that I can relate to and support. I did a lot more reading to try to figure out just what my disagreements were, so I am satisfied that I read enough to have an understanding and to form my opinions .

Comment Two: “presents a deficit view of students”

I’m not sure what a “deficit view” is. I’ll interpret it as “negative view”. This is important. One of my main disagreements with the Standards is the portrayal of students. I’ll stick to my argument that the Standards portrays students as having an IQ of 130 or more. Of course there is plenty of mention of students needing special accommodations or extra help. But this seems mostly rhetorical. And the portrayal of the motivations and attitudes of students do not seem real. Or course it can be argued that if we teach as recommended, then students will respond as portrayed. But that is too much for me to accept on faith. I don’t consider my criticism of the Standards as negative to students in any way. Portraying students idealistically may seem positive, but there is a downside to that. If we’re not realistic we risk doing the wrong thing. Motives are always complex. Students will respond with interest to a subject well taught, but at any given time they are not primarily interested in what we want them to learn at that particular time. The phrase “students have a natural desire to learn” needs much investigation. Taken uncritically it can only lead us astray.

Comment Three: “misses the point that the standards advocate for connections and not discrete topics which are mastered, or not, in a day”

This seems to be saying that the standards try to promote high level thinking skills. Of course they do! Who would not? But I disagree that the methods promoted by the Standards are effective to that end. I’m glad to give the Standards credit for good intentions. But that is not saying much. Good intentions without effective means are worth very little. The term “discrete topics” does bring up an idea that I think needs some careful thought. Is learning math more like the unfolding of a flower or like the laying up of bricks? I think that a lot of learning math is more like the laying up of bricks, and practice is an important ingredient in the mortar. Many topics are discrete, and though they are not normally mastered in a day, they are typically presented in a day. Typically there is an initial understanding on that day. Of course this will be followed by forgetting if nothing more is done. “Mastering” a topic, or building a structure of mathematical knowledge over the long term, may indeed be more like the unfolding of a flower, and that is very important. But the flower won’t unfold without the bricks being laid down, and cemented in place with practice. Every detail is a brick, and it must be set in place carefully, and if not reinforced with practice it will come unglued and fall out of place. If the bricks fall out of place then the flower can never unfold. So practice is important. That is the whole idea of my argument. Practice must be planned for and managed by the teacher.

Comment Four: “students portrayed in the standards are not supposed to be geniuses, they’re people with natural curiosity and ways of viewing the world. They’re not supposed to be uber-problems solvers”

I’ll stick to my contention that students are not portrayed realistically in the Standards.

Comment Five: “author implies that he is teaching in a standards-based way but his description belies that”

I don’t claim to teach in a “standards-based” way. I don’t know what part of my article would lead anyone to think that. The whole point of the article is that I disagree with the standards in some very important ways.

Comment Six: “author states that ‘assumption of fluency’ is common in the standards but does not provide evidence of this.

The evidence is that there is very little discussion of appropriate kinds and amounts of practice for various topics, and there is no principle of practice, and there is not much recognition of the different levels of fluency. This leads to another important topic which I have given some thought to, the idea of incidental learning^{2}. The “project method” and the “activity method”, which date back at least eighty years, rely on incidental learning. But there are limitations to incidental learning.

Comment Seven: “there is no ‘principle of practice’ in the Standards because students are doing mathematics throughout class; rote drills do not constitute an authentic mathematical practice”

I think the distinction between “doing mathematics” and “learning mathematics” is an important one. The Standards seem to equate them, or to assume that the former is the obvious, and only, path to the latter. I would argue that they differ in various ways, and the differences are great enough to assert that the former is necessary condition for the latter, but definitely not a sufficient condition. I think this needs to be investigated much more extensively. Progressive educators like to say “We learn by doing”. Obviously there is a lot of truth to this. But there is also much more to be said about it.^{3}

I think it is unfortunate that “drill” has become a bad word in education. We may quibble about whether or not to call it a “mathematical” practice, but drill, including rote drill, certainly is an authentic learning practice. I think it is extremely shortsighted to dismiss it.

Comment Eight: “criticism is really the implementation of the Standards, not the Standards document itself”

This is a little unclear. It must mean that my criticism is about how the Standards are implemented. That is not the case. I don’t have much of an idea how the Standards are implemented by people who really believe in them.. I question that the ideas in the standards can be implemented. I suspect they are not realistic enough to be implemented. No, my criticism is about the Standards. I don’t agree with the Standards.

Paragraph comment: “The author claims . . . . .” (see above)

I am accused of making “superficial interpretations” of the Standards. This is like comment one. I don’t claim to have read every word of the standards, but I will stick to the interpretations I make. The standards do not give any guidance about practice (see comment seven above). The standards do not advocate or emphasize direct instruction. The Standards do not portray realistic student behavior and capabilities.

I am accused of not having experience with curricula aligned with the standards. This is mostly true, for which I am grateful. I do have experience with one curriculum which gives some evidence of being aligned with the standards, and that has not been a good experience.^{4} But all that misses the point. The Standards should be able to stand by itself, to make sense even if implementation is variable and imperfect. It is the standards I disagree with, not any particular implementation.

And what’s this about “carefully sequenced and related tasks”? What are “these curricula” that we are talking about. It seems to me that “carefully sequenced and related tasks” is one thing that the Standards definitely are not. I’m not sure they should be. We don’t want a catalog of recipes. We want general principles that can be applied, but not so general as to be meaningless. And I think some benchmarks, some content goals to strive for, such as the goal that students should be comfortable with the four operations on fractions by the end of the sixth grade, would be beneficial. As I stated before, the Standards seem primarily to be a collection of unrealistic ideals.

And the curricula that claim to be aligned with the Standards seem to be controversial. That’s what the current “math wars” are all about. “These curricula are perhaps the best manifestations of the vision of math instruction.” That’s possibly true. But I don’t know. I don’t think the vision is realistic, and I don’t think the vision is helpful. So it is not surprising that programs based on the vision are frustrating and much criticized

Now I want to present a few thoughts about disagreements and how they are handled. People may certainly disagree about things. I am quite accustomed to people disagreeing with me. Disagreements are frustrating, but that’s life. Sometimes a discussion can help resolve those differences. Sometimes a discussion merely causes bad feelings and no benefit. When discussion doesn’t come easily or natural, one can certainly look at more than just the content of the disagreement. Each side can also look at how the other side handles that disagreement. One might also look at how one’s own side handles the disagreement, but that is a bit harder.

I am disappointed that the NCTM does not view my article as a disagreement worth pursuing in The Mathematics Teacher. I think it is the logical place for discussion to take place, and obviously I think the Standards need a lot of discussion. When a newspaper publishes an editorial, it expects to publish letters from readers critical to their editorial as well as supportive. Newspapers are sometimes accused of delaying publication of critical letters, but generally they do not reject them entirely. Does The Mathematics Teacher publish any criticisms of the Standards and the NCTM’s ideals?

The first letter I received acknowledging receipt of my article, back in February, mentions my article as being submitted for the “Sound Off Department”. I never liked the sound of that. No, I didn’t submit my article for the “Sound Off Department”. I meant my article to be a serious discussion of issues of teaching and learning mathematics, not anything less.

I’ve never been an expert on educational trends, but all my life I’ve been at least a little bit aware of them, and I’ve also been aware all my life that educational trends are usually irrelevant to educational practice. This irrelevancy has some implications, it seems, to me. The educational establishment ought not to be irrelevant to educational practice. I will first explain why I make this claim of irrelevancy.

For a history of American education in the twentieth century I think there is no better book than “Left Back” by Diane Ravitch.^{5} Again and again in this book she reminds us that for whatever trend or idea is being touted by education professors at a particular time in history, we seldom have much evidence that these ideas are really put into practice by most teachers. The ideas of progressive education were virtually unchallenged during much of the thirties and forties, but there is evidence that a lot of educational practice was determined by community and traditional expectations, not progressive ideas. The PEA, The Progressive Education Association, disbanded in the fifties - not a good indication that the ideas had value or permanence. Teachers in any historical period usually have some awareness of the latest educational ideas. Some accept the new ideas. Some reject the new ideas. Some try hard to implement them. Some would implement them only over strenuous objection. There is some evidence that sometimes teachers think they are implementing them, but they really are not. But many teachers have always simply ignored the new ideas. They close the classroom door and carry on as best they can, mainly relying on their common sense as a guide.

I have my personal evidence of the irrelevancy and impermanence of trends in math education, and this evidence covers many years. I was in high school in 1960, when “modern math” was making news. Everyone was supposed to learn set theory. But we didn’t use set theory in the geometry and trigonometry that I took in the eleventh grade. And we didn’t use set theory in “senior math” that I took the next year. Our teachers were aware of the promise of set theory and modern math, and set up some sort of extra curricular opportunity for interested students to learn a bit of it. My brother, a year older than I, took advantage of that opportunity, but did not think too well of the experience. I didn’t try it myself. I don’t remember why, I think it was not an active rejection of the idea so much as just the press of other things that I was interested in doing.

Next year in college I went straight into calculus, which gave us no set theory, and so I didn’t give modern math much more thought. However I was acquainted with people who took college algebra. The consensus of opinion seemed to be that it was rather difficult course. “Why do we have to mess around with set theory?” seemed a common complaint. I didn’t have much of an idea what ought to be in a course called college algebra, but my impression was that at the University of Missouri in the early 60’s it was a collection of topics, some algebraic and some not, that probably didn’t add up to coherent whole. There was an attempt, apparently, to incorporate “modern math”.

I graduated with a major in math, and taught for several years - math, music, and eventually science - and then drifted into other things. I didn’t learn anything more about math education until our youngest daughter was in the seventh grade in 1995 and I looked at her math book. That piqued my interest and over the next year I wrote my article “Chicago Math”.^{6}

I wrote it with very little idea of the larger context of math education. I thought probably the Chicago Math program was something unique, the labors of a group of writers who went off the deep end in one direction. I never did get much of a picture of how my daughter’s teacher used this book. I felt a little sorry for the teacher, for I decided it would not be a very good book to teach from. But any teacher must sometimes make do with less than ideal textbooks. It’s no big thing.

It was only in 1997 that, for various reasons, I thought about getting back into math. In 1998 I began work on a masters degree at South Dakota State University. I got a graduate teaching assistantship, which involved teaching two sections of college algebra each semester. What, I wondered, will be the content of college algebra? Will it be a review of high school algebra? Will it extend high school algebra? Will it have set theory? Will it be “modern math”? Will I have to learn whole new subjects that didn’t exist when I was young? Will it be a poorly selected collection of topics that is hard for teachers to teach and hard for the students to learn?

When I finally got the book I would use, about a month before school began, I was pleasantly surprised to see it would be a review and extension of the high school math that I had almost 40 years before. It was not identical, to be sure. I don’t think I ever had functions, not to mention composite and inverse functions, in high school.

So what happened to all that modern math that was going to revolutionize the world? All that remained, so far as I could tell, was that the language of sets was used a little. Perhaps the topics of functions is a legacy of modern math. I’m not sure. But primarily college algebra was pretty sensible. “Modern math”, by 1998, was history. I can’t say it vanished without a trace, but it seems safe to say that it was largely irrelevant to what actual teachers did in their classrooms.

None of the professors at SDSU that I had any contact with seemed to be concerned with what the NCTM, or any other part of the educational establishment, was saying. So far as I could tell everyone went about their business of teaching their courses pretty much as college math courses have always been taught. As part of my graduate teaching assistantship I had to take a course called Teaching College Mathematics. It was only a one hour course, and canceled a few times for one reason or another, but it was pretty good. We dealt with the practical concerns of teaching college algebra - homework, tests, grading, and so on. Our teacher, as I recall, had a masters degree in math education as well as a doctorate in math. But apparently she saw no need to delve into the latest educational ideas. I don’t recall that the NCTM was mentioned even once.

After graduating with a masters in math in 2001 I taught for two years at St. Cloud State University in Minnesota. Again the math department seemed not much concerned with what the NCTM had to say. There were several professors in the math department who taught math ed courses. It was quite apparent that they knew what was going on in math ed. They looked to the NCTM for guidance and tried to apply in their own classes current ideas that they felt were of value. I don’t know all the details of what they tried to do. I knew that they assigned group projects. I also decided, as time went by, that I didn’t want to assign group projects in my classes. But different teachers do things differently. That has always been the case. My point is that what the educational establishment touts at any given time is mostly irrelevant to what most teachers actually do.

So in over forty years of my experience educational trends have always been on the sidelines, always of interest to some teachers, but seldom entering mainstream practice. Their irrelevancy may not be total, but is substantial.

The most important implication of this irrelevancy, it seems to me, are that the ideas promoted do not have much value. One can certainly argue that there is some value, and that value is simply incorporated into standard teaching practice. I grant this, to a certain extent, but if the ideas are not important enough to retain their name, then how important can they be? The “activity method”, for example, has some value, but not as a organizing principle. Lots of activities can be profitably used in the classroom, but so can lots of things that don’t quite fit the label “activity”. Currently, apparently, problem based learning (PBL) is given considerable attention. But problems used in the service of learning are nothing new, and the principle that ideas will be approached only through problems is going too far. It will be scrapped in due time, or perhaps it will morph into a related idea with a new name.

A second implication of the irrelevancy, or perhaps the result, is that the NCTM is not accustomed to criticism. It is insulated from the reality of everyday life in the classroom. Of course this is a common criticism of anyone not actually in the classroom everyday. It’s inevitable to at least some extent for every principal or superintendent. If you’re not in the classroom everyday it’s hard to stay connected. But this disconnection should be challenged by teachers. As it is, apparently, no one expects the NCTM to address the real issues of teaching and learning mathematics, so they don’t. Criticism is always painful. It ought to be very carefully considered before it is given. But it does no one any good to just keep quiet. When there are honest disagreements, everyone profits, in the long run, by being aware of them. I think the irrelevancy of the NCTM, indeed of the entire professional educational establishment, is something that should not be allowed to continue generation after generation.

The NCTM shouldn’t be irrelevant. It should be, among other things perhaps, a clearinghouse of ideas about teaching math. It should not be only a mouthpiece for one minority segment of opinion and practice. I don’t mind if the NCTM advocates for that narrow segment (though I surely don’t understand why they want to), but they ought to acknowledge, present, and discuss other segments of opinion.

They should try to figure out why they are mostly irrelevant, and do something about it. I think the answer is pretty clear. They are irrelevant because they are not realistic. Their ideals don’t fit the real world. They claim the students portrayed in the Standards are typical students. I claim they are figments of imagination. They claim “rote drills do not constitute authentic mathematical practice”. I claim rote drills are valuable, even indispensable, in learning some topics in mathematics. They claim, apparently, that “doing math” is sufficient to learn math. I claim it is not sufficient.

They don’t care what I think, of course, but I would argue that there is much to be gained if they would open their ideas up to discussion - and what better place for discussion of the NCTM Standards than the pages of The Mathematics Teacher?

It seems to me that the NCTM has staked out an ideological position, not a pragmatic one. An ideological position does not start with observations of reality that lead to ideas that are subject to criticism and disagreement. An ideological position starts with ideas in which there is an emotional investment. Observations of reality must then be rationalized to fit with the ideological position. Ideologies are intolerant of criticism. I think the ideology of the NCTM, and of the educational establishment in general, can still be called “progressivism”, although perhaps that term is rejected as out of style. There does seem to be a solid thread of connection between the progressive educational ideas early in the twentieth century and NCTM’s ideas today. .

No publication ought to be pressured into publishing what they believe is not of value. It’s their magazine, they can publish what they choose to publish. Similarly I should not be pressured into putting on my web site anything that I don’t think belongs on my web site. But I certainly can criticize the NCTM, just as they can criticize me. I can say that I think the NCTM would be wise to open up to discussion.

If The Mathematics Teacher does not wish to publish anything critical of the Standards or the NCTM, then where will such a discussion take place? I think that other critics of “fuzzy math” would do well to call on the NCTM to be open to discussion of their ideas. It seems to me that one article of disagreement in every issue is not asking too much. It would be in everyone’s interest, including the NCTM’s.

In my internet investigations about the current “math wars” I have discovered two articles that I think are especially relevant. One is “Romancing the Child” by E. D. Hirsch^{7}. The other is “Developmentalism: an Obscure But Pervasive Restriction On Educational Improvement” by J. E. Stone^{8}. They are not short articles, and not particularly easy to read. I haven’t figured them all out yet. Both address, in one form or another, the problem of substituting ideology for reason.

Notes:

1. Click here for “Some Disagreements With The Standards”, an article which I have had on my web site since May.

2 Click here for “The Case Against Incidental Learning”.

3. There is obviously some truth to the statement, “We learn by doing”, but it is often stretched much too far. It is not true that we learn one thing by doing another. We do not learn A by doing B. However doing B may help to learn A if there is some connection between A and B. It is true that we learn to manipulate mental concepts by manipulating mental concepts. For more thoughts along this line click here for Chapter Eight in my proposed book on teaching and learning, and/or click here for The Rationale For Laboratory Exercises In The Teaching Of Science.

4. My experience at NDSU is what I am talking about here. Click here for Thoughts On My Teaching At NDSU. Click here for “NDSU Math” in which I try to analyze not my experience, but the program, to much greater depth.

5. “Left Back”, Diane Ravitch, 2000 ISBN 0-7432-0326-7

6. Click here for Chicago Math.

7. Click here for E. D. Hirsch, "Romancing The Child" (http://www.educationnext.org/2001sp/34.html)

8. Click here for J. E. Stone, "Developmentalism: an Obscure But Pervasive Restriction On Educational Improvement" (http://epaa.asu.edu/v4n8.html)