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When reading over the NCTM Standards and Principles over the past year or so, I have often felt dissatisfied, but I was not able to immediately realize why. It took a few months of reflection to begin to realize that there are some things I disagree with in the standards. They are subtle disagreements, hard to pin down, hard to put into words, but I think they are important.

The Standards starts out with a vision. Much of this statement is noncontroversial. It speaks of “high-quality, engaging mathematics instruction”, “ambitious expectations”, “knowledgeable teachers”, “adequate resources”, and so on. I can’t imagine anyone could disagree with these ideas. However then comes this statement, referring to students in an ideal math class:
"They draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers.”
At this point I disagree with the vision. I disagree, first, because I think it paints a not very realistic picture. But, secondly, I disagree with some ideals that are implicit in the vision. Some of these ideals go back many years, and have been taken for granted for many years, as ideals at least. But that does not mean these ideals are not open to question. I believe some of these ideals cannot stand up to careful scrutiny. In this article I will try to verbalize just what these ideals are that I do not accept , and to explain why I disagree.
For several years now I have been teaching freshman college mathematics, and many years ago, in the 60’s, I had several years experience teaching secondary mathematics. My experience, limited though it may be, gives me a picture quite different from what is presented in the vision statement. I will give an example from my personal experience.
Recently, in teaching logarithms in a college algebra course, I put a series of problems like this on a homework assignment:
1. if log_{b} 3 = 1.227, then log_{b} 1/3 = _____;
2. if log_{b} 6 = 1.588, then log_{b} 1/6 = _____;
3. if log_{b} 14 = 1.227, then log_{b} 1/14 = _____;
4. if log_{b} 5 = 1.227, then log_{b} .2 = _____;
I was helping a student in my office with these and similar problems. He seemed to recognize the pattern that this set of problems was supposed to show, but could see no easy answer to the last one. Previously it had not occurred to me that any student in a freshman college math course would not immediately recognize .2 as being 1/5. But this particular student did not. I had to explain to him that 1/5, when changed to a decimal, is .2. Then he understood. I assumed this particular student was an exception to the general rule. However the next semester at this same point in the course I was aware of this possible problem. In the process of helping several students with this type of problem over a period of several days I would ask questions in such a way as to try to ascertain whether or not it was immediately apparent to them that .2 is indeed 1/5. I can’t give statistics, but I did come away with the impression that it is not at all the rare student who doesn’t make this connection. Many of my students do not immediately recognize .2 as the same as 1/5.
So where do we go from here? According to the vision in the standards, as I read it, failure to make this connection between .2 and 1/5 would only be a very temporary stumbling block. The students would puzzle over this problem a bit. They would approach the problem “from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress”. They could do this because, after all, they are “flexible and resourceful problem solvers.”
But that is not what my experience leads me to envision. What I see is students working on their homework with a reasonable amount of diligence, but that reasonable amount of diligence may translate to puzzling over the problem for 15 minutes or so with increasing frustration. They know that there is something missing in their thinking., but they just don’t know what it is. Finally they put the work away. The homework is incomplete. The understanding is incomplete. Perhaps in class the next day I will spend a few minutes on this particular homework problem and the mystery will be cleared up for at least some of the students. Others will still not understand it.
According to the Standards, this isn’t the way it’s supposed to be. My students, it turns out, are not “flexible and resourceful problem solvers.”
Perhaps I am not teaching them right. Perhaps their previous teachers have not taught them right. But here’s another possibility. Perhaps most of the students described in the standards have IQ’s of 130 or better. Though mention is made throughout the standards of students with special needs, students requiring accommodations, students who have difficulty with math, that does not seem to be the picture presented in the various examples. Students described in the standards, especially in the vision, do not seem like students I deal with everyday. I will have a few of those students in my class of course, those “flexible and resourceful problem solvers.” In fact I might have two or three of them in every class. They will make good grades. They will understand the explanations I give. They will turn in well done homework. They may occasionally bring a problem to me for help, but not too often. They will leave the class at the end of the semester with a good grade. These students are not terribly rare, but I will not have a whole class of such students, and I don’t expect anyone else does either, except in very exceptional situations.
I suppose there is the implication that if we all taught according to the ideas in the standards, then our students would be like the students described in the vision quoted. This requires a leap of faith that I am not willing to make. And it would not be a leap of faith so much as a denial of experience and common sense. How do we know that teaching as advocated in the standards will produce “flexible and resourceful problem solvers”? My common sense leads me to suspect quite the opposite. I suspect that my students have been exposed to some teachers who tried to adhere to the ideals advocated by the standards, and I further suspect that that might be part of the problem.
I will continue with the example cited above, about students’ failure to immediately relate .2 to 1/5. How did this situation come about? One possible interpretation jumps quickly to mind. Maybe students don’t know arithmetic very well any more. And maybe this is because fluency is no longer valued. In an attempt to promote understanding, a very desirable goal, perhaps we have down graded the development of fluency. This would be an understandable situation. I have not always kept up with the trends in the teaching of mathematics, but throughout my lifetime I have been aware of rhetoric and ideals concerned with teaching for understanding, and not teaching by rote. In fact there has always, in my lifetime, been a certain amount of rhetoric that dismisses entirely the need for drill. I don’t think most teachers believe we could totally do without drill, but the de-emphasis of drill is unmistakable. I do not disagree with the ideal that we should teach for understanding. I do disagree with the ideal that we should de-emphasize anything that might be interpreted as drill, or anything that cannot be characterized as “higher order thinking skills”.
I have long argued that the number one mistake teachers at all levels make is what I call the assumption of fluency. Simply put, we assume that once we present an idea then the students not only understand the idea, but can fluently use that idea. That this is not always the case is easily demonstrated by reality. A teacher explains something in class, the students nod attentively, and then 20 minutes later they appear confused when the teacher uses that idea in explaining something else.
Assumption of fluency is very common in the standards. The idea of fluency in a given mathematical topic, and the means to gain fluency, are seldom considered. Learning of anything requires practice. What could be a more basic principle of learning than that? At the upper levels of education we leave it to students to find an appropriate amount of practice. Those who do not eventually drop out. But in the lower levels of education we must provide the right amount of practice. The students are in no position to determine that for themselves, and we don’t just let them drop out. We provide the right amount of practice by assigning homework, by taking class time to review, and providing a testing situation in which the results of practice are meaningful and can be rewarded. All this is little mentioned in the Standards. There is no “principle of practice” in Chapter 2. A “principle of practice” would simply state that practice is essential to learning math, and that it is an important part of the teacher’s job to provide for this practice.
There is a very important concept that leads up to the principle of practice. That concept is “levels of fluency.” Fluency is not an all-or-nothing kind of thing. Consider this scenario. A teacher explains to a group of beginning algebra students the idea that adding the same number to each side of an equation results in an equivalent equation. This idea than can be used to either solve the equation, or to at least advance one step toward a solution. She compares an equation with a balance scale, showing that if five ounces are added to each side on the scale the scale remains in balance. She uses examples to explain the idea, and she shows the procedures of how to apply the idea. At the end of a fifteen minute explanation the students in the class seem to understand. Can we assert, then, at this point, that the concept has been taught, that the students have mastered the concept?
It is usually possible, at this point, to elicit evidence from the students that they understand the concept. However I think this is a very limited perspective. If not followed up by practice and application in some way, the concept will slip out of the students’ minds within hours or days. Their fluency in using this concept is at a very low level. In the normal course of teaching, this fifteen minute explanation will be followed up with practice and application. The teacher will assign homework. At the start of the next class period, after having done some homework, the students’ grasp of that concept will have reached a higher level of fluency. Weeks later, after advancing through a few more ideas and applications in which this basic concept of adding the same number to both sides of an equation is used, the students will have yet a higher level of fluency with this concept. After the students take a test on the current chapter and begin a new topic their learning of the concept will reach still another level of fluency. At no particular level of fluency does it seem possible to say that the concept has been “mastered”, but at some level it is possible to say that the students have a level of fluency such that the concept can become a foundation for other mathematical ideas.
In many topics of mathematics initial understanding is not too difficult. But that is not the end of the learning process. An initial flash of understanding can quickly dissipate if not reinforced with practice, if an insufficient level of fluency is attained. It is a very important part of the teacher’s job to provide for the practice that will take the students to appropriate levels of fluency. There can be a great deal of difference of opinion as to what level of fluency is appropriate and what steps will get the students there. But I would argue that the issue itself, the idea of different levels of fluency, is very important. Fluency should not be taken for granted. It should not be assumed prematurely. Therefore there must be a principle of practice, in some form, in the teaching of mathematics.
The Standards do give a limited recognition to the ideas of practice and levels of fluency. It is stated many times in the Standards that year by year the students will become more fluent in math. But I have not found emphasized anywhere in the standards a clear statement that practice and drill are necessary and important activities for gaining fluency in any mathematical topic. Instead there seems to be an idealistic perspective that practice is not needed. This, I believe, is an ideal to disagree with.
I will present several perspectives of teaching and learning that I infer from the standards, perspectives that seem quite different than my perspective. Then I will discuss these perspectives with a view to their effectiveness in promoting learning, and finally will offer what I consider a more realistic perspective on teaching and learning math.
The first picture one can get from the Standards is of the “community of learners”. In fact on page 144 this term is explicitly used.
“In these grades teachers should help students learn to work together as part of building a mathematical community of learners. In such a community, students' ideas are valued and serve as a source of learning, mistakes are seen not as dead ends but rather as potential avenues for learning, and ideas are valued because they are mathematically sound rather than because they are argued strongly or proposed by a particular individual (Hiebert et al. 1997).”
A second perspective is what I will call the “research perspective.” There are many statements in the standards that lead one to the picture of students as mathematical researchers, or mathematical investigators. Of course there is a little bit of truth in this, but only a little bit. Whenever a students listens carefully to the teacher, whenever a student studies the textbook, whenever a student thinks hard about a problem, he or she is indeed a mathematical investigator, but not an independent mathematical investigator.
A third picture I get from the Standards I would call a “seminar perspective”. In this scenario participants work on their individual projects and present their results to the group. Group discussion then follows.
Perhaps other perspectives could be identified. Though different in some ways these perspectives have a lot in common. And they have some limitations in accomplishing our ends.
These perspectives are group oriented, not individualistic. I understand that many teachers put a high value on collaborative effort, and I agree that under the right conditions collaborative effort can be productive. But I would also argue that the individualistic perspective is very important. The basic principle of constuctivism, as I understand it, is that each learner must construct knowledge in his or her own mind. I accept this as a very basic principle of learning. It’s also common sense. And it argues against some kinds of collaborative work in the classroom. Division of labor is valuable in many areas of life. In building a house a plumber can do the plumbing and an electrician can do the electrical work, and the house works. But in learning, such division of labor does not apply. If John learns Chapter One and Jane learns Chapter Two, then neither John nor Jane has done the whole job. We want John to learn all chapters and we want Jane to learn all chapters. Nothing less is acceptable. Any collaborative work should promote this goal, not displace it, not substitute for it, not detract from it in any way. Collaborative work should be encouraged, but if not done right it may simply result in division of labor, and a diminishing of results.
All of these perspectives are open ended. One idea can lead to another. But in real life teaching and learning there is limited time for that, and certainly limited inclination on the part of the students. Yes, there is lots of natural curiosity on the part of students. It’s a part of human nature. But it’s seldom directed at what we need students to learn at the moment. There are lots more interesting things in the life of a child than the math we want him or her to learn. Open ended processes can be very time consuming, and can easily degenerate into unproductive digressions.
These perspectives all have some degree of validity and applicability, of course. They are all variations of what we used to call the “discussion method” of teaching. I would argue that they are too idealistic, and they omit something very important. These perspectives omit the concentration that is needed to really develop mathematical concepts and skills.
Practice is a part of this concentration, as I have already discussed. Another part of that concentration is sticking with a difficult concept until you get it right. This does not come democratically, by working in a group on a project. It comes by being focused, by careful listening to an explanation by an expert, the teacher, and following the directions of that expert in applying the concept.
The concentration needed for good learning does not arise from a seminar perspective. The individual presenting his results to the group does not then assign homework to the members of the group. Nor does he or she give a test to those members. The necessary degree of concentration is not reasonably expected in the “community of learners” perspective either. No specific means of gaining fluency seems to be envisioned by this perspective. Being even less structured than the seminar perspective, it is hard to imagine any real mathematical progress being made.
The researcher perspective perhaps does allow for more concentration. But it requires much more self direction that we expect in elementary school or high school. A true mathematical investigator is very much self directed, has a broad mathematical base to draw from, and has the freedom to take the mathematics wherever it may go. But the typical student is quite different. He or she has a limited mathematical base from which to work. That is unavoidable. The fifth grader trying to figure out fifth grade math typically has only the fourth grade math as his basis. That is indeed a limited base. To expand that base will take years of concentrated effort.
Similar to the “open-endedness” of these perspectives is the idea of mathematical thinking for its own sake, as opposed to building a structure of mathematical knowledge. In many instances, it seems to me, teachers think that if an activity engages students then it must be “educational”. And if a mathematical activity engages students then it must qualify as mathematical learning. But this is a very loose standard. It is another ideal to disagree with. I think it is an important point that one can do a great deal of mathematical thinking without necessarily learning any mathematics. One topic, or activity, may be engaging to a class, and may involve genuine mathematical thinking. However if that activity is followed by an unrelated activity the next day, and yet another unrelated activity the day after, then we may be only engaging in mathematical thinking with no end in view.
Or perhaps mathematical thinking is enough. Where does that mathematical thinking lead? Is the brain a muscle that can be strengthened by exercise? If that is the case then the class seems to be on the right track. However the “brain as muscle” idea has been out of favor for about a century now. Should we return to it?
My view is that mathematical thinking is not enough. Rather we should think in terms of building a structure of mathematical knowledge. That means choosing a series of mathematical topics that fit together, that build on each other to make a useful body of knowledge and competencies. This is the idea of curriculum, of course.
The Standards does have a Curriculum Principle, and I think it is very well stated. But I would argue that once the principle is stated it is largely ignored. It is ignored in order to maintain the idealistic perspectives.
Too much emphasis on these idealistic pictures causes neglect of another picture. This other picture is of students doing assigned homework, studying, and taking quizzes and tests. This is a very mundane picture, not a picture to inspire the idealist. But it is not a bad picture. The vast majority of math that I learned in my lifetime came from taking courses, from doing assigned homework, studying, and taking quizzes and tests.
We need a name for this mundane, non-idealistic, perspective. We perhaps could call it the “traditional” perspective, but that has many connotations that I do not think are desirable. To many people it is not a positive term. For want of a better term I will call this perspective the “student following directions” perspective. The student following directions is other directed. He or she is following the lead of the teacher. That is essential because the teacher knows how to make use of that limited mathematical base when the student possesses, knows how to explain new ideas in terms of that limited base, and knows how not to waste the student’s time. Yes, inner directness is a quality much to be encouraged, but it is not a good basis for the vast majority of students to learn mathematics. And the student following directions will not expect to take the mathematics wherever it may go. The freedom is there of course. If an occasional gifted student takes a mathematical topic in a productive, but unexpected direction, the average teacher will celebrate that accomplishment. But that has nothing to do with making instructional plans for the rest of the class.
Much remains to be discussed about the Standards. By constraints of length I have emphasized the lack of a practice principle and the curriculum principle. I hope this will encourage others to read the Standards carefully and explain their disagreements.