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(Author's note, 9/3/05: I would like to mention two recent articles of mine that I believe are especially relevent to math education - Some Thoughts On Constructivism and The Case Against Incidental Learning . When I wrote this article on Chicago Math I was totally unaware of trends and ideals in math education. At that time, as I recall, I had never heard of constructivism, and I had not thought about the concept of incidental learning. But both ideas are very important and relevant to the current "math wars". Constructivism is a favorite idea of advocates of new math programs, but their concept of constructivism seems very poorly formed. I think the basic idea of constructivism is valuable, but it doesn't lead to group projects. Incidental learning, so far as I am aware, is an idea not considered or analyzed by educators, but basic to much of what has been called progressive education during the past century.)


Chicago Math

Brian D. Rude 1996

      The University of Chicago School Mathematics Project consists of a series of mathematics texts for junior high and high school. When my daughter entered the seventh grade she brought home the text "Transition Mathematics". I generally try to be familiar with the textbooks given my children. Usually at the beginning of the year I thumb through them, find nothing out of the ordinary, and forget about them. However after doing this a few times with "Transition Mathematics" I began to get the idea that this was going to be a special case.

      It was with a certain amount of relish than I pursued this text. It has been said that editors sometimes purposely leave mistakes in books just to give the readers the pleasure of finding them. I doubt if this is generally true, but it certainly is true that we enjoy finding mistakes. At least I do. Usually these are simple typographical errors, or errors not much above that level. Unfortunately that is not the type of thing that interested me in this book. The problems I am concerned with cannot usually be pinned down to page and paragraph number. Rather I think there are some troublesome premises that underlie the writing of this series. These premises are not explicitly stated. They must be inferred from the text by careful reading and reflection. And when these premises are stated they don't really sound too bad. The effects of these premises are often only subtly different from the effects of more sensible premises. Indeed the writers might even deny some of the premises that I identify. Or they might defend the premises. My argument is that the net result of these premises is that learning math becomes more difficult and less satisfying than it should be.

      My analysis of the Chicago program began when I looked for chapters on fractions, decimals, and percents in the seventh grade book. I think of fractions, decimals, and percents as the heart of seventh grade math. Many of the basic ideas of fractions and decimals are introduced in earlier grades, but are not developed to a great extent. Even more importantly, in the earlier grades children do not develop a great deal of fluency in these ideas. The seventh grade and eighth grade, in my experience, is a time when fluency is developed. A sixth grader may understand that four-eighths reduces to one-half, but only to a seventh or eighth grader does this type of understanding become firmly rooted in mind. Sixth graders may be introduced to problems such as finding the price of a $16 dollar item marked with a 15% discount, and they may do a number of such problems. But it is only the seventh or eighth grader to whom it really makes sense. With this perspective then, I expected to find a logical progression of chapters going through fractions, decimals, and percents.

      This I did not find. Topics on fractions, decimals, and percents in a logical progression are not evident in the table of contents. I first became aware of this when my daughter was doing problems in adding fractions. I looked ahead in the book, expecting to find lessons on multiplication of fractions next, and then division of fractions. It appeared that these topics were simply not there. I thought they had to be there. I concluded that the information must be there, but in some form that is not immediately recognizable to me from my perspective. Reflecting on this I decided I should keep an open mind. I had expected to find basically the same approach that I had experienced as a school child myself, and that I had used as a teacher. But this expectation could be a trap. As we grow older it is easy to fall into the habit of thinking that every change from the way things were when we grew up must be a change for the worse. Obviously that is not always the case. I think of my approach to arithmetic as the simple logical approach that has developed over many many years due to both the nature of mathematics and the experience of teachers. But I hold dear the ideal that a little careful thought about something can lead to new and surprising approaches. The old ways are not always the best. There is always room for improvement. And improvements are sometimes a little jarring at first. So I figured the basic information of fractions, decimals, and percents, must be in the book. The book just must start with a different perspective. If I kept at it I would eventually understand this perspective. I might not agree that it is the best perspective, of course, but I would be able to figure it out.

      That was a year and a half ago. This year my daughter has the book of the series titled "Algebra". During this last year and a half I have returned to both books time and again, trying to follow my hypothesis that I just needed to understand the perspective and things will fall into place. I have made some progress. My opinion of the series, however, is not good.

      Keep in mind that as I write this I have little knowledge of how the math teachers actually use this series. How a text is used can be much more important than the merits or problems of the text itself. My daughter is smart. Math does not give her much difficulty. I have helped her on occasion with her math, both last year and this year, but, as is typical of teenagers, she wants to maintain considerable independence from her parents. I will not quiz her every evening about what went on in her math class that day. From what information I do get from her it is apparent that the teachers do follow the book. Apparently they do considerable explaining of the concepts presented in the book, and bring in extra material at times. I also hear of troubles. My daughter will sometimes report that a particular chapter test was failed by a majority of the class. Due to my lack of knowledge I may say many things that would be considered misguided by the teachers, but I hope I will also say some things that will be helpful.

      I have identified five main premises used in writing this series of books that I think are worth very careful attention. These are not the only premises that could be identified, of course. Other premises could be identified with which I would have no disagreement whatsoever. And the premises I identify are certainly not all bad. My argument is simply that these premises give rise to some bad effects. They result in a serious lack of coherence in the text They make it harder for students to learn math. These premises are:

      1. A high level of mathematical logic and language must be used.

      2. Drill need not be used. Understanding will suffice.

      3. Organization of topics will be set by a pattern. The pattern will repeat.

      4. All topics are related to all other topics. The best way to show these relations is to mix everything up together.

      5. The focus should be broad from the beginning.

      Perhaps these premises could be stated better, either as a single premise, or as a smaller number of independent premises. These premises are simply the best formulation that I can come up with to explain what seems to be behind the writing. These five premises are related to each other and all are related to the idea of understanding. Understanding is a very worthy goal. Students should not memorize recipes and parrot them back to the teacher. Teachers should be aware of whether students really understand the material or are just patching together bits and pieces of information that will get them through the next test. I presume understanding is an important goal of the Chicago math program. However I believe they have approached understanding in an idealistic way, a way that may sound good, but which does not result in a solid foundation in mathematics. My thesis is that the Chicago program's approach to understanding is seriously flawed. It is flawed because each of the premises I have identified is faulty in some way, or, perhaps much more importantly, the premises crowd out other considerations that are more important. I will attempt to identify and explain these other considerations that I believe are missing. I will briefly discuss each premise and then analyze one chapter in some detail in terms of these premises.

Premise 1 - Mathematical Logic and Language

      Being against mathematical language and logic in a math class seems like being against motherhood and apple pie. Obviously mathematical language and logic is important, but there can be too much of a good thing, and one good thing may displace another good thing. I will contrast mathematical logic with what I will call "intuitive understanding". I will not attempt to define this with great precision, but will simply say that intuitive understanding is the type of understanding that is best for a given age level, or a given level of learning. Most importantly, perhaps, intuitive understanding includes a level of concreteness versus abstractness that is appropriate for the age. Young children start math in a very concrete way. As they progress through the years their mathematical learning can become more and more complex, and more and more abstract. To attempt to be either too concrete or too abstract at any given level is a mistake. Mathematical logic, in its fullest form, is simply impossible until one has acquired a substantial background in math. I will argue that the Chicago program attempts a level of abstractness that doesn't work, or, perhaps more accurately, it attempts to use a level of mathematical language and logic that is not appropriate for this level of math.

      Mathematical logic may be thought of as a process. It leads to knowledge, which is the product. In higher math the product is impossible without the process. But in lower math, arithmetic, the product is not so much a result of a mathematical logic as it is a result of more concrete common sense. The Chicago program often tries to develop mathematical logic by emphasizing the process rather than the product. This is not necessarily bad, of course, but when overdone it can be detrimental. Mathematical language and logic must develop over a long period of time. It must arise from a concrete beginning and intuitive understanding.

      As an example of intuitive understanding versus mathematical logic consider the concept of area. Chapter Four of the algebra book starts out with the "area model" of multiplication. "Models" as used in this series are not quite the same as definitions, but apparently they almost are. The "model" is set off in a shaded box and reads like a definition. "If a rectangle has length l and width w, then the area of the rectangle is lw." Should this be a definition?

      An intuitive understanding of area, I think, would be along these lines. Area means how many squares it will take to fill up a given space. Area is measured in squares, either square inches, or square miles, etc. One can think in terms of painting an area. It takes a certain amount of paint to fill up a square of a given size, so one might want to know how many squares one needs to paint. When painting a ceiling, for example, it is easy to see that a long narrow ceiling might have the same area, that is the same number of squares, as a short wide ceiling. By this reasoning one can see that inches or feet are not sufficiently specific to measure area, but squares are. To say that the wall is 45 feet long is not enough information to tell you how much paint to buy, but to know that there are 225 square feet is enough information. The simplest way to measure area then would be to mark off the area in squares of equal size and count the squares. Then it is a very short step to the lw formula. A rectangle consists of so many rows of squares with so many squares in each row. Multiplication, at some point, was intuitively defined as a short way of counting. If you have five rows of stars, with eight stars in each row, then you can multiply. Multiplication is just a short way of counting when you have objects in equal-sized groups.

      But the chapter does not use this intuitive definition of multiplication. Rather it simply asserts that in the "area model" of multiplication, A = lw. I would argue that it is much better to start with the intuitive idea of area. The expression A = lw is a result, not a beginning. It is the result of careful thought and intuitive understanding. The reason that A = lw is that the situation fits the intuitive model of multiplication as repeated additions.

      If we start with the Chicago approach, that A = lw, then were do we go from there? And what needs to precede this "model"? Most importantly I would argue that the term "lw" is not very concrete to eighth graders. Eighth graders are not fluent in the language of mathematics. That comes only after considerable experience in mathematics. In my personal experience I would say that the level of fluency so that "A = lw" really makes sense probably came after the first year of algebra in the ninth grade. Until that time the expression would not really convey a relationship. The student may know, on a superficial level at least, that "lw" means to multiply l times w, but that is seen more as a specific recipe to be applied to get a specific result, rather than a general statement about a relationship.

      Consider this example from Chapter Nine. The formula for compound interest is given as T = P(1+i)n. The chapter is on exponents and powering. Certainly compound interest is a valid application of the mathematics. But this example is on the second page of the chapter. It is used less as an application of the topic, and more as an introduction to the topic. The reasoning behind this formula is given, but only in mathematical terms. To someone fluent in the language and logic of math it is a good explanation. But I can't imagine it would make much sense to eighth graders. It would make sense to eighth graders, I believe, if it were preceded by a number of problems that carefully, and intuitively, develop the idea. But it is not. The section on compound interest is a lead in to the more general idea of exponential growth and decay. Again, I think this can be made understandable to eighth graders ( but I wonder if there are not more basic topics that are more important to develop at this level), but only if carefully developed by a series of more concrete problems. Explanation only by mathematical language and logic at this level can not work for the vast majority of student at this point.

      In learning mathematics one goes from the concrete to the abstract. Abstractness is a long term goal, and a very important goal, but jumping toward that goal too quickly can undercut the foundation of mathematics.

Premise 2 - Drill

      The second premise I identified, that drill will never be used, seems perhaps to be the heart of the problem with this program. A general pattern of any math text, it seems to me, is that an idea is presented and then a page of problems is given in which that idea can be applied. The Chicago program follows that to a degree, but in some very important ways it fails to follow that pattern. It fails in that it seems to purposely avoid simply presenting a page of identical problems. Very often, I would argue, a page of identical problems is what is needed. We need drill.

      We do not learn a new idea with one application, by doing one problem. We learn by doing many problems. That, of course, is drill. Perhaps we don't like to use the word. It seems somehow old fashioned. Of course it is old fashioned. Drill has been around for centuries, probably millennia. It has been around simply because it works. Drill is simply practice. You don't learn to swing a golf club without practice. You don't learn to drive a car without practice. You don't learn to play a clarinet without practice. You don't learn the history of Cuba by reading one book through one time. You don't learn to understand and appreciate a sonnet by reading it through one time. The human brain is not like a computer than can accept information from a disk drive and a high rate of speed with 100% accuracy. Rather the brain builds up knowledge from the information it receives. When the brain of a first grader is presented with a page of ducks and stars and circles to count and group together it builds up knowledge. When a college student pores over a calculus text the brain is building up knowledge. Drill is not the only way we acquire knowledge, of course. When we read an explanation and struggle to understand, that is not drill. But in many diverse fields of knowledge drill is an essential part of the learning process. Drill will not take the place of explanation and understanding, but it is just as true that understanding will not take the place of drill. In most academic learning understanding is quickly lost if it is not followed up with practice. When managed well, drill is simply food for the brain.

      It is sometimes said that the student must "master" one topic before going on to the next. I think this is an oversimplification. There is some truth to it, but it should not displace a more important truth. The more important truth is that an appropriate level of fluency must be attained in one topic before going on to the next topic. An "appropriate level of fluency", of course is a vague term. I would simply argue that "mastery" is also ill defined. Mastery could be defined as nothing less than 100% on all homework, quizzes, and tests. Few would defend such a definition. If mastery is defined as some lesser attainment, then that, it seems to me, is equivalent to my point that an "appropriate level of fluency" must be attained. In order to know what an appropriate level of fluency is in a given situation the teacher must be aware of different levels of fluency.

      Fluency comes by degrees. To illustrate various degrees of fluency consider the following example. A seventh grade class in general math is to learn to reduce fractions. Though they have been introduced to fractions as early as the fourth grade, the concepts are still new to most students. Increasing levels of fluency in reducing fractions might be represented by:

      1. As the teacher explains it, it "clicks" in the students' mind.

      2. The students have begun a twenty problem assignment when the bell rings to end the class period.

      3. The class has completed and discussed two daily homework assignments on reducing fractions and is ready to go on to the next topic which is the addition of fractions.

      4. The class learns to do the opposite of reducing. They learn to take fractions to higher terms so that fractions with unlike denominators can be added and subtracted.

      5. The class takes the chapter test on fractions.

      6. The class continues on through the chapters on decimals, percents, business problems, formulas, areas and volumes, and so on. In each of these topics fractions frequently occur that must be reduced.

      7. The students go through the eighth grade and repeat a considerable amount of what they learned in the seventh grade.

      This is not the only way levels of fluency could be illustrated of course. But the point is that the degree of fluency at the low levels are very different from the high levels. This can be a bit invisible if not carefully considered. At level one in this example Johnny may see 4/8 and say 1/2. At level seven in this example Johnny may see 4/8 and say 1/2. But if the teacher thinks, based on this evidence, that Johnny has learned to reduce fractions at level one in this illustration then she is very wrong. If reducing fractions is left at level one the knowledge will be gone without a trace in a matter of days or weeks.

      I expect level one in this listing is routinely achieved by students using the Chicago program. Levels two and three are crucial. I feel the Chicago program does not give the teacher good materials to achieve these levels. These levels are achieved by practice of one sort or another. Often this practice is best described as drill. The Chicago program, trying hard to foster understanding by using mathematical language and logic, scrambles too much into each group of exercises. Drill is slighted.

      Drill may be put together well or poorly. Some drill, for one reason or another, may be mostly a waste of time. Other drill, for other reasons, may be exactly what the brain needs. Educators certainly ought to be very much concerned with the form of drill. But to think we can dispense with drill seems foolish. Understanding, as important as it may be, cannot substitute for drill.

      It could certainly be argued that the Chicago Math simply tries to make drill in a optimally beneficial form. I would argue that it has failed, because in many cases, a simple page of identical problems are needed, but are not given. Instead a page of obscure thinking problems are given, some of them focusing on process. The primary idea of the section is lost in a collection of other related little ideas that are brought up in the problems. I will give examples of this when I discuss Chapter 6 in more detail.

      The higher levels of fluency in my listing above come not from drill, but from extending and applying the mathematics, and from review. The Chicago programs seems to have lots of review. I think this is less than optimally beneficial, however, because the levels of fluency that depend on drill are not attained.

      The "assumption of fluency", I maintain, is the number one systematic error that teachers make, especially in math. The "assumption of fluency" is just the idea that it is easy to assume fluency on the part of students when it is not really there. "I explained that to them," the teacher thinks, "so they know it." Of course it is not that simple. Explanation is very important, but fluency comes only with a sufficient amount of drill. The Chicago program, by de-emphasizing drill and using an inappropriate level of mathematical language and logic makes an appropriate level of fluency difficult to attain.

Premise 3 - Pattern of Organization

Mathematicians like patterns. Scientists like patterns. Indeed to find a pattern in what seems to be chaos is basic to any field of inquiry. But sometimes we force patterns. We insist on sticking to a pattern when the subject itself doesn't fit that pattern very well. I think this is what the Chicago program does. The result is a text that lacks coherence. This is ironic, in that a great deal of time and effort was obviously spent in organizing the text. My view is that that time and effort were directed toward organizing the text to a pattern that does not fit the subject well.

      The seventh grade book starts out with a chapter on addition. When my daughter first started this book I thought a chapter on addition was too much. We expect each school year at this level to start with some review. To have a whole chapter just reviewing addition seemed excessive. It also suggested that math had been considerably "dumbed down" in the last few decades. But the writers of the Chicago program would disagree that they have "dumbed down" math. They thought they were improving math. The difference, of course, is that Chapter One on addition is not meant to be just a review of adding whole numbers, rather it is to be a mathematical inquiry into the idea of addition. Thus they combine several perspectives on addition into the chapter. They use the "putting together model" and the "slide model", and perhaps one or two more models. They also include the associative and commutative principles. And of course they also give review in addition. Fractions, decimals, and percents are included in the chapter on addition, since fractions, decimals, and percents can be added.

      I expected to find a chapter on fractions that would include reducing fractions, expanding fractions, adding fractions, subtracting fractions, multiplying fractions, and dividing fractions, and so on. Instead we have a chapter on addition that includes adding fractions. We have a chapter on subtraction that includes subtracting fractions. Is this a better pattern of organization than what I expected? If so, why?

      I am very much in agreement with the idea of presenting models of addition, subtraction, and so on. I never had trouble with written problems when I was a child, but I was quite aware that others did. After I was grown up I tried to analyze why written problems can be hard for some people. I observed that some teachers would try to give recipes to work written problems. "If you see the words 'in all' or 'total' then you add." I thought this was a big mistake. It can be misleading. A written problem requiring subtraction or division could easily have the words "in all" in it. This approach does not lead to understanding. But how do you know to add in one problem and subtract or multiply in another? The answer is that you add if the situation presented in the problem fits the meaning of addition. This is primarily a matter of language, not math, but you have to know the meaning of addition. Addition means putting together and determining the quantity of whatever is put together. At least it does on an intuitive level to young children. Thus if the problem requires putting together then you use addition. Of course this is an over simplification. There is a little more too it. Addition only works for two quantities, usually unequal, at a time. If you have a number of equal quantities then you multiply, because that's what multiplication means. And of course the statements I just made can be further challenged in various ways. However I think I have summarized what addition and multiplication means to a child just learning them. The point is that you know how to do written problems by knowing which arithmetic operation fits the situation presented by the problem. Written problems often cause problems because the students forget just what the operations are for. Therefore presenting "models" for these operations makes a lot of sense.

      However I would argue that these models are best presented when the various techniques are presented. When learning to add 2 + 3 in the first grade the model is obvious. We use the "putting together model". That is how we learn what 2 + 3 makes. But when learning to add 9 + 5 and other combinations that result in two digit answers, that meaning can begin to get lost. It needs to be reinforced. Students must be reminded that 9 + 5 = 14 only because you can take 9 apples and 5 apples and count them up and you will always get 14 apples. It is essential to memorize the answer of course. Nine and five make fourteen, and we need to have that fact on an instant recall level. We can't draw apples or circles every time we need to add nine and five. But memorization does not preclude understanding. As we progress we must always make sure that basic understandings are not lost.

      When students learn to carry and add multidigit numbers it is again easy to forget the meaning of addition. Learning the technique, the process, the algorithm, is a big job. When we learn to add fractions and decimals it is again easy to lose the forest for the trees. Observant teachers will be quick to notice when this happens and give explanation when needed. Whether we talk about the "putting together model" or the "meaning of addition", or "what addition is used for" is not too important. What is important is that the students understand addition, and continue to understand it even when caught up with a mass of details that can obscure it.

      Multiplication presents an even greater bewildering array of details that can obscure the basic meaning. An intuitive understanding of multiplication starts out with the idea of adding equal quantities. If you have three baskets, and four eggs in each basket then you can count them up and find there are twelve eggs. Or you can just learn that three times four is twelve. Multiplication is a short cut to adding when you have equal quantities. This is a simple definition of multiplication that makes intuitive sense to children. It is not the only way that multiplication can be defined by advanced mathematicians, but it is an appropriate way to define it for third graders. This intuitive understanding is not adequate for more advanced mathematics, or even more advanced arithmetic. When multiplying fractions the "short way of adding" idea doesn't seem to explain anything. However a slight modification of the meaning of multiplication is all that is needed. At this level an intuitive understanding of multiplication could be stated something like this: In multiplication one number tells you how many of the other number you have. In the problem 1/2 x 3/4, the first number, 1/2, tells you how many of the other number you have. It says you only have a part of the other number. You don't have 3/4, you have only a part of it. So of course your answer will be less than 3/4. This meaning of multiplication, of course, can be equally well applied to the simpler problem of 3 x 4. One number tells you how many of the other number you have. Three tells you how many fours you have.

      Meaning and understanding are very important. Students should not forget what addition means just because they are caught up in the details of how to do it. But does it follow, then, that a chapter on addition makes sense in a seventh grade math text? Does it follow that there is no chapter on fractions in the same text?

      The eighth grade text, titled "Algebra", also starts out with a chapter on addition. It also presents the same "models". Is this sensible? There is no chapter on signed numbers in the algebra book. Is this sensible? Presumably the students learned to handle signed numbers the previous year. Of course there was no chapter on signed numbers. They were introduced to adding signed numbers in the chapter on addition, and so on. Does this work? Or does it work better to have a chapter on signed numbers in one of the texts?

      I think it works better to have a chapter on signed numbers, and to present all the operations, addition, subtraction, multiplication, and division, in that chapter. I confess, however, that this may appeal to me simply because that is what I remember having in ninth grade algebra. I also remember it being difficult for some of my classmates. Perhaps there is a better way, but I do not believe the Chicago program has found it.

      There are many ways in which any subject can be organized. Some of these ways will be good and others will be not so good. I would argue that the organization of the Chicago program is not so good, and I would argue that the main reason for this is that the writers have tried to make the subject fit into their choice of pattern, rather than organizing the text to fit the subject. I don't believe a chapter on addition has any place in a text on algebra.

      There are two educational ideas that are worth mentioning at this point, the "point of immediate need", and the "spiral approach".

      There is a case to be made that signed numbers, or anything else, should be taught at the point of immediate need. The "point of immediate need" principle has wide applicability, and a certain amount of validity and benefit. The principle rests on the idea that if a concept, fact, skill, or other element of learning is introduced when it is needed by the student then it will be better understood and retained by the student. The student is motivated to learn it, according to this principle, because he has an immediate need for it. Consider how this perspective can be applied to signed numbers and equations. You can do simple equations without understanding signed numbers. The equations, must be carefully selected however. You can teach how to solve X - 10 = 25, without signed numbers, but you cannot give the equation X + 10 = 5. The two equations are very similar, however, which gives a good point of entry for getting into signed numbers. After the students can do some simple equations then the equations that require signed numbers can be introduced, and thus demonstrate the need for signed numbers. Since the students need signed numbers at this point, presumably they will be motivated to learn them. Signed numbers, then, are introduced at the point of immediate need.

      I don't think the Chicago series tries very hard to introduce topics at the point of immediate need. Rather they usually introduce topics in order to fit their preset pattern of organization.

      The "spiral approach" is the idea that a subject can be taught by going through a series of topics at one level, then returning to the same series of topics at a deeper level, and so on. There is much to be said for this approach. It is done naturally in many subjects. When teaching arithmetic in the lower grades we do it quite a bit. Each year we review much of the previous year's work and try to extend it a bit. It would be very hard to try to teach arithmetic to children without spiraling. However it can be carried too far. A common fault of the spiral approach is that again and again students return to the same topic without taking a substantially deeper "cut" in the topic. It becomes inefficient because the students spend considerable time and effort reorienting to each topic without substantially extending the topic.

      The Chicago program, so far as I am aware, does not attempt to be a spiral approach. But it does seem to exhibit the faults of the spiral approach. The chapter on addition in the eighth grade algebra text and the seventh grade text seems very much a spiral approach. My argument is that a text should follow the natural organization of the subject. When spiraling comes naturally it should be used. When spiraling is forced it should not be used. The addition chapters seem to me to be forced spiraling.

Premise 4 - Mixing topics

      The fourth premise, that all topics should be mixed up together because everything is related to everything else, violates a more important principle. That more important principle is what I will call the "concentration principle". This is simply the idea that the learning of most topics requires a certain level of isolation and concentration. Yes, everything is indeed related to everything else, but we can't learn everything at one time. We have to concentrate on one very small part at a time. This concentration may mean different things for different topics. Often it means repetition. Consider this example: A class of second graders is learning about the discovery of America. The teacher may present a variety of information through a variety of activities. She knows that most of it will be forgotten, but she wants a few basics to stick in mind. Among these basics are the time, the person, the setting, and so on. The name "Columbus" and the date "1492" should not be forgotten. Somehow these and a few other concepts must be given greater concentration. Much of this, I believe, is simply by repetition. The essential body of knowledge that she wants the children to retain could probably be stated in about thirty seconds. But if it is stated only once and then followed by other information and activities, it would surely be lost.

      In learning to play piano one concentrates on one piece at a time. Sometimes one concentrates on one or two measures, repeating it again and again to get the hand and finger motions right. In learning a sport one similarly isolates and concentrates on one thing at a time.

      In the learning of math we concentrate in various ways. In the third grade the teacher may drill the class on the subtraction combinations by using flash cards. In the fifth grade the students concentrate on reducing fractions by doing a page of problems, or by doing problems competitively at the board. In algebra the teacher concentrates on the learning of "completing the square" by lecture. In each case we concentrate on one topic by isolating it and practicing it in some way.

      But isolation and concentration is only half the story. By isolation and concentration it is possible to produce a bit of learning cut off from all other learning. If one learns how to "complete the square" but has no idea what that has to do with anything else, then not much is gained. "Completing the square" connects forward to the quadratic formula, which connects forward with more advanced algebra. It connects backwards to previous knowledge of algebra and arithmetic. These connections to other knowledge must not be left to chance.

      Much of learning of most any subject, therefore, follows the pattern of an alternation between the "isolate and concentrate" mode and the "spread and relate" mode. We must isolate in order to concentrate. We must concentrate both for understanding and for retention. Drill and levels of fluency are very important in this "isolate and concentrate" mode.

      But that is still not enough. The topic must be related to the rest of the emerging body of knowledge. Any given learning is useless if it is totally in isolation. It must be related to other knowledge. Sometimes this relating is obvious and easy. Other times it is not so obvious and easy. Both concentration and relating are essential. Without concentration the topic is not formed in the learner's mind. But without relating, the topic is isolated in the learner's mind and is useless.

      For example when one first learns to reduce fractions one is concerned only with that task. Forget long division. Forget the war of 1812. Forget nouns and verbs, and everything else. One must concentrate. One cannot learn to reduce fractions and to find a common denominator at the same time. One must isolate a small segment of a topic and concentrate on it sufficiently to learn it. Then when the isolated element is learned to a sufficient degree of fluency it must be related to other knowledge. One must spread one's focus so that the isolated element can be related to other topics. Learning to add fractions cannot be done the same class period in which the idea of fractions are introduced. One must first isolate and concentrate on the meaning of fractions. This requires some drill to reach an appropriate level of fluency. Then one must understand the idea of reducing fractions. This requires time and effort to produce understanding, and once that understanding has been acquired, more time and effort to attain an appropriate level of fluency. Next one must learn to expand fractions to higher terms. Then, finally, one may address the idea of adding or subtracting fractions. All of these topics, the meaning of fractions, reducing fractions, expanding fractions, comparing fractions, and finally adding fractions are related. They must be related in the learners' minds. But before they can be related they must be brought into existence. Failing to isolate and concentrate will prevent these topics from being built up in the learner's mind. Failing to spread and relate will prevent these separate topics from joining together into a coherent whole. Thus the learner must do both. The learner must alternate between the isolate-and-concentrate mode and the spread-and-relate mode.

      The Chicago program seems heavily weighted on the spread-and-relate mode. It lacks concentration. As I have mentioned before, often an appropriate assignment is simply a page of problems all of the same type. This provides the needed concentration. Then, after this concentration, one must spread and relate. Then a page of identical problems is not at all appropriate. Then an assignment must include different types of problems so that one can learn to distinguish them. But concentration must come first. Without concentration learning is superficial. This leads to frustration and lack of achievement. Without sufficient isolation and concentration, the "spread and relate" part cannot be done. You can't fit the pieces of a puzzle together when many of the pieces are either faulty or entirely missing.

      I might point out that this concentration-relation principle does not fit well at all with the "point of immediate need" idea. My view is that the former is essential and basic to serious learning, while the latter is superficial. The "point of immediate need" principle is desirable to use when it can be used, but that is all. It provides momentary relevance and motivation, but it does not provide concentration. Without concentration there is nothing to relate. When we alternate between concentration and relating, topics often do not come at a point of immediate need to the learner, but they do come in a form that can be used to build up a coherent and reasonably durable structure of knowledge.

      Another example of organizing to a preset pattern is found in the "SPUR" motif used in the review at the end of each chapter. "SPUR" stands for skills, properties, uses, and representations. I expect there is some benefit to teachers to analyze the teaching and learning of math in these terms, but I question whether they make a good organizing principle for a chapter review. "Representations", for example, does indeed relate to the teaching and learning of math. We use all kinds of representations to teach and learn math. Every time we draw a diagram to illustrate a written problem we are using representations. But can we directly teach children to use representations? Should we try to do so? I think this sort of thing can be analyzed by teachers with considerable benefit, but putting it in the review section in the student's book simply complicates the chapter review. It would be better left out.

Premise 5 - Broad Focus

      The fifth premise, that the focus must be broad from the start, is rather close to the fourth premise, but I think it is still distinct. I will give some examples. Chapter 6 is titled, "Linear Sentences" This is very broad. The actual learning of the chapter, for the successful student, is limited to simple equations. It is perfectly true that they are "sentences" and that they are linear. But what meaning can "linear" have at this point. Of course the students can learn to parrot back the answer that "linear means there are no powers of a variable above the first". But I would argue that until the learner has done a few quadratic equations that answer can not really be meaningful. And why are they called linear "sentences" instead of linear equations? Again this follows the premise of being as broad as possible. If we start out with sentences then we will never have to broaden out from equations to inequalities, and from there to statements. This may be appealing to the mathematician who can appreciate that equations are only one type of sentence, but I would argue that it is poison to the normal eighth grader. It can only cause confusion. It cannot, at this point, be intuitively meaningful. It may be understood to some extent, but it cannot possibly have the meaning that it will have eventually when the student has gone beyond calculus. And what value might it have to the eighth grader if it cannot be understood in any but a very superficial level? I would argue that it is much more of a detriment than an benefit. I think chapter 6 should simply be titled "Simple equations".

      I will not argue that a narrow focus is always better than a broad focus. Sometimes a broad focus is more beneficial and sometimes a narrow focus is more beneficial. It depends on the individual situation. I just argue that always trying to have a broad focus is not good.

Applying the premises

      Now I will turn to a page by page analysis of Chapter 6 and try to give more concrete examples to the general arguments I have put forth above. Chapter 6, it seems to me is the heart of a course in algebra. It is about solving simple equations.

      Page 261 introduces the idea of equations by presenting four different situations that all lead to the equation, 3x + 1000 = 1800. I have no strong complaints about this, but I would argue that only problem 2 is a good one. In problem one the idea of saving $3 a week doesn't really seem like a way to get to $1800. It would take 267 weeks, or about five years. In problem three, the school almost doubles its enrollment in one year, which doesn't make it very realistic.

      In problem four a perceptive student might point out that no quadrilateral, much less a rectangle, can have one side that exceeds the sum of the other three sides. This is the kind of error that is fun to find, but I hesitate to bring it up, for it is not the kind of thing I am criticizing the book for. I'll probably have a few such errors in my own writing.

      Page 262 uses a graphic representation of the equation 4W + 3 = 11. I think this is a good idea. In the bottom half of the page they do the same problem with mathematical logic. I would argue that the "Addition Property of Equality" is a clumsy way to express how it is done. It has been introduced to the students, and presumably they understand what it means. It means you can add the same number to each side of an equation and the equation is still true. This is very important, of course. The question then, is, does the term, "Addition Property of Equality" add to understanding? Or does it simply confuse things? I feel the confusion outweighs the benefit for the vast majority of students. I think it is more important to emphasize the more general idea that whatever you do to one side of the equation you must do the same to the other side, in order for it to remain an equation. This is intuitively meaningful. Furthermore it is more broad. The "addition property of equality" is a special case. There are also the subtraction, multiplication, and division properties of equality.

      Giving a special name to an idea may or may not be beneficial. Many times it is. In history the term "manifest destiny" seems a useful term for the idea that in the eighteen hundreds Americans felt their country was destined to grow very big, perhaps even from coast to coast. It is an easy term to learn, and helps nail down a concept that is a little abstract. In grammar such terms as "declarative", "interrogative", and so on seem worthwhile. The concepts could be learned without the terms, but the terms seem helpful. In music such terms as "enharmonic" or "polyphonic" seem worthwhile. Again the concepts could be learned without the terms, but the terms seem worthwhile. However in this case, the term "addition property of equality" seems a little clumsy for an idea that need not be very abstract. It is indeed mathematical language and logic. But I don't think it promotes intuitive understanding. Intuitive understanding is better promoted by pointing out the more general rule that whatever you do to one side of the equation you must do to the other side also.

      On page 263 "worksteps" are introduced. I have no argument with this, but I wonder if it wouldn't have been better to lead off with this page and dispense with page 262.

      Of course what the book presents is not necessarily the students' primary guide. The teacher is the primary guide. The teacher may insist that both pages be carefully studied, or she may emphasize page 262 and ignore page 263, or she may do the opposite, or she may ignore both pages and explain things in her own words.

      Pages 264, 265, and 266 are problems in which the students may practice solving equations. But they are presented as "questions". This fits the premise of avoiding drill. It also fits the premises of mixing all topics together and following a preset pattern. But what is needed here, it seems to me, is simply a page of problems, a set of about ten or fifteen equations to solve. If I were the teacher I would assign problems 1a, 4 - 7, 9 - 12, 14 - 17, 19, and 20. I would not assign problem 18, but would mention it as something to look at for a bit of fun. These problems give an appropriate amount of practice in solving simple equations of this form. The problems I left out, of course, are what is most relevant to this discussion. Problems 1b, 2, and 3 seem an excessive focus on the process. It requires a degree of verbalization that seems more detrimental than beneficial. Such focus on the process may certainly be used by the teacher in explanation, but once the idea is understood it doesn't seem worth while to put this sort of thing into problems.

      When I first looked at problem 8 I thought it was just a cumbersome way to giving more practice in solving equations, a way to avoid the appearance of drill. The criticism applies to problem 13. However on closer examination I see that problem 8 is asking the students to observe that the same equation can be rearranged in some ways without changing it. This idea certainly ought to be discussed, but I'm not sure this is a very good way to do it. Some students will laboriously solve each equation and miss the point. Other students will superficially glance at the equations and answer with choice d. A few will accept a friend's answer without considering any of the equations. One hopes a few students will actually get the point of the problem.

      Perhaps problem 18 is put in to give the students another chance to notice that the problem is unrealistic.

      Problem 21 is a bit of a mystery. A whole section in this chapter, Section 6.3, is devoted to combining like terms. Problem 21 requires combining of like terms. Is there any good reason for this problem to be in this section?

      The rest of the problems are review. I think an excess of review is provided in the book. This is not bad in itself. If the teacher will simply use her own judgment it does no harm. But if she feels obligated to assign all the review provided, then I think the review is excessive. Review can be important as part of the "spread and relate" mode, but if it prevents adequate concentration then it can be detrimental.

      Section 6-2, at first seems to be a repeat of Section 6.1. What is the difference between these two sections? Section 6.1 gives introductory examples, a graphic illustration, "worksteps", and then problems. Section 6.2 starts out with another example, then gives general rule: "To solve ax + b = c for x, add -b to both sides. Then multiply both sides by 1/a." It seems that some of the explanation of section 6.1 is repeated, and then a recipe is presented - "To solve ax . . . . . . . . ." If understanding is to be emphasized then why is the explanation reduced to a recipe? Why is some of the explanation repeated in this section?

      There is a difference, however, between these two sections. In Section 6.1 only whole numbers were used. In Section 6.2 fractions and decimals are found in the equations. This seems sensible enough. Fractions and decimals do present some problems in computation that are not found when only whole numbers are used. However this distinction is not made very plain to the student. The section could have been titled "Fractions and Decimals In Simple Equations". The only nod in this direction is the phrase "even when a, b, and c are fractions or decimals" near the top of page 267. This does not have nearly the emphasis needed. I missed it when I first looked over the section. I would expect this section to make several points about dealing with fractions and decimals. It is pointed out, in example 2 on page 267, that if both fractions and decimals are found in one equation it usually makes sense to convert to all fractions or all decimals. It should also be pointed out that an equation with fractions or decimals can often be simplified by "multiplying through" to convert it to whole numbers only. Thirdly, it could be pointed out that rounding off must sometimes be done. These ideas are not presented.

      Section 6.2 should be the logical successor of Section 6.1. It is, to be sure, but I don't think this is at all apparent to the student of less then ideal diligence. There seems to be a lack of coherence.

      I think the main problem with both sections, however, remains what I already discussed. The important principle that should be emphasized is lost in a less than coherent blizzard of details. That important principle is:

      Whatever you do to one side of the equation you have to do it to the other side.

This principle makes intuitive sense. It is easily understood. It is basic, much more basic than the recipe given on page 267. Why is it not clearly stated and emphasized? I believe the premises I mentioned are the reason.

      Premise One is that mathematical logic and language will be used. In theoretical algebra this "do the same thing to each side" principle may seem too basic, too simple, too intuitive. It is on too low a level of abstraction, apparently, to be used in the Chicago program. Instead we have the "additive principle of equality", etc.

      I also think Premise One also detracts from a more important idea. That is the hierarchical nature of mathematical logic. Some principles of math are much more basic than others. That is the whole idea of having postulates, theorems, and corollaries. The Chicago program, I presume, does not try to give the complete logical derivation of algebra, because it is recognized that is beyond the scope of what eighth graders can do. But they try to bring in the language and some of the principles of a logical derivation of algebra. The result is a mish mash of logic and language that, in my opinion, is a poor way to present algebra. Abstractions are used that are more detrimental than beneficial, and simple ideas that could be very beneficial, such as the principle of doing the same thing to each side of the equation, are left out. Mathematical logic and language is indeed used. But the mathematical logic is not well developed. Some details are there, but the grand design is not. The principle that you do the same thing to each side of the equation is high in the hierarchy of intuitive principles of beginning algebra. It is the basis of the whole chapter, or should be.

      The authors of the text would probably disagree with me on this. They would say they gave a lot of thought to the grand design. But I think their grand design is summed up in the premises, and I think the premises are faulty.

      Premise Five says the focus will be broad from the beginning. This premise is violated a bit by having only whole numbers in section 6.1 and fractions and decimals in Section 6.2. I won't complain about this violation of a premise. I think it is a good idea. It is just not carried out very well.

      The exercises starting on page 269 show the usual mish mash. At this point I think a simple page of equations to solve would be most appropriate.

      Section 6.3 is about the distributive property. This is obviously important in math, but again I do not think it is handled very well. In accordance with Premise One, mathematical logic is used, but I think intuitive understanding would be more appropriate at this stage. Mathematically I would have to agree that the section is about the distributive law. But intuitively this section is about combining like terms in equations. Therefore I think the section should be titled "Combining Like Terms". The distributive law can be brought in. Indeed I think it should be, but it should follow, not lead.

      The first two sentences do mention adding like terms, but emphasis is given to the distributive property. It is presented and then used in the examples. The distributive property leads. Abstractness leads. Concreteness is supposed to follow. The expression ac + bc = (a + b)c leads. Then the expression -3x + 5x = 2x follows.

      I think there is one more step to be taken in concreteness. 3x + 4x can be compared to three apples and four apples. This is intuitively meaningful. Instead of starting with the abstract distributive law, why not start with three apples and four apples, then go to 3x + 4x, and then to the distributive law?

      I would argue that the distributive law should be regarded as simply as a consequence of what we mean by addition and multiplication. In higher mathematics it might be true that the distributive law is taken as a postulate, but that does not mean it is best presented to eighth graders as a postulate. To eighth graders it is intuitively meaningful as an algebraic expression of the simple idea that three of something and four of something make seven of that something.

      The "Multiplicative Identity Property of 1" is logically necessary, perhaps, to add k + 8k. But do eighth graders benefit from the terminology and logic? I think it is much more intuitively meaningful to simply explain it as a matter of mathematical language. It is just a short cut to write k when we mean 1k. We are lazy. Short cuts are fine if we just agree on what short cuts we are going to use and how we are going to use them. It doesn't have to be any more complicated than that.

      At about this point another question arises in my mind. Isn't it time to treat equation solving as a skill, rather than just as application of principles? In the middle of page 271 the equation of the example is reduced to 2x + 4 = -6. The next step is to add -4 to each side:

           2x + 4 + -4 = -6 + -4

Principles are applied here, and certainly students should be aware of mathematical principles and how they are used. This is in accordance with Premise One, and also in accordance with the idea of intuitive understanding. But I would also argue that it is very desirable to learn to solve equations in an efficient routine manner. To treat equation solving as a skill means to switch from an understanding approach to speed and efficiency approach. We don't need to restate every principle we use every time we use it. We don't have to go through the derivation of the quadratic equation every time we use it. We don't have to read the entire Internal Revenue Service code every time we pay our income taxes. Understanding is always desirable, but we shouldn't have to demonstrate that understanding forever.

      I expect the term "transposition" has gone out of style in mathematics. When I was in the ninth grade "transpose" meant simply to take a number to the other side of the equation and change its sign. I understand children now days don't learn to "carry" and "borrow" when they learn to add and subtract whole numbers in the lower grades. Instead they "regroup". I understand the rationale for this. We want children at all ages to understand and apply principles, not just learn and apply recipes. But I would argue that recipes should not be totally thrown out. If the terms "carry", "borrow", and "transpose" do supplant understanding and have no counterbalancing benefit, then certainly they ought to be thrown out. I would argue, however, that any tendency to supplant understanding is outweighed by the benefit of efficiency. I think all three of those terms are intuitively meaningful at the age and in the context in which they are presented, and they confer a very substantial benefit of efficiency.

      Terms can aid understanding in some cases, and they can supplant understanding in other cases. My view is that the term "manifest destiny" aids understanding in history. I also think the terms "carry", "borrow", and "transpose" aid understanding in math. I am not at all sure that the term "Multiplicative Identity Property Of One" aids understanding of algebra in the eighth grade.

      I think the idea of treating equation solving as a skill is avoided by the Chicago program because it violates both Premise One and Premise Two. By these two premises the teacher would not send kids to the board and have a speed drill in solving simple equations. I think this is a mistake. In the third grade we expect children to become efficient in adding and subtracting. In the eighth or ninth grade I think we should expect them to become efficient at solving simple equations.

      Example 2, on page 272, does use the distributive law, of course. And it does require adding like terms. To me, however, what is more important is that it combines two parts of math. There is the matter of solving equations, and there is the matter of translating a situation into an equation. It can certainly be argued that the matter of translating a situation into an equation is an artificial construct. If you understand equations and if you understand the problem then the translation from the words of the problem to the equation is obvious. In one sense this is true, of course. But being true in a logical sense is not necessarily the same as being optimum for teaching.

      I may be showing my past history in this. I remember in the ninth grade learning to solve equations in one chapter, and then having the whole next chapter devoted to translating problems into equations. This seemed sensible at the time, and seems sensible now. Having a separate chapter to the problem of translating problems into equations gives emphasis to it. Perhaps integrating the translating of problems with the solving of problems into one chapter is a better arrangement. But I would still argue that they don't have to be integrated on every page.

      I have mentioned the principle of concentration, and the idea that the learner needs to alternate between the "isolate and concentrate" mode and the "spread and relate" mode. Of course this can be overdone. The more capable the student is the more he or she can integrate topics. My argument would be that the Chicago series integrates far too much. Learning to solve equations and learning to translate written problems into equations at the same time may work for a few very bright students. But for most students it is too much integration. I think it works much better to first isolate and concentrate on learning to solve equations, and then work on translating written problems into equations. Premises four and five would argue against this. I argue against the premises.

      There is another interesting point about example 2 on page 272. One of the steps is "Divide each side by 3.5." From my perspective this makes perfect sense. It is in accordance with everything I have been saying. However it is not in accordance to what the book has been doing. Always before we neither subtracted nor divided on both sides of the equation. Instead of subtracting a number, we added the negative of the number. And instead of dividing by a number we multiplied by its reciprocal. I always felt this was clumsy, and intuitively not good because it can needlessly add confusion. Now, suddenly, this pattern is unceremoniously dropped.

      Example 3 on page 272 is preceded by a paragraph of explanation. The explanation sounds like this is the first time the topic has been introduced. Last year's book, so far as I know, did not isolate and concentrate on this topic, as I would think would be needed. But last year's book probably had a similar explanation. A bit of this topic in one book and a bit of it in the next book does not add up to an adequate treatment of the topic. It needs isolation and concentration at some point (preferably last year, in the context of a chapter on per cent applications following a chapter on principles of per cents).

      Example 4 on page 273 again mixes the solving of equations with translation of written problems into equations.

      The problems on page 274 again seem a bit of a mish mash, but perhaps are generally appropriate as an opportunity to apply the math they are supposed to be learning. However the section is titled "The Distributive Law". Somehow this doesn't seem to follow well.

      Section 6.4 is titled "Repeated Addition and Subtraction". It is primarily translating written problems into equations. I'm not sure the topic is worthwhile. It seems to me that it is more of a puzzle to be solved rather than building blocks of a structure of knowledge.

      Section 6.5, a mid-chapter review, seems quite sensible. I have no criticisms of it.

      Section 6.6 "Solving ax + b = cx + d" seems a good topic. It not only follows logically from the previous sections, but it follows intuitively. The pictorial explanation on page 284 seems sensible to me. I have no great quarrel with the examples on page 285 and 286, though again I would not use the language of "addition property of equality", etc. The problems on page 287 are not bad, but I would leave out the questions of process, and just assign the equations and written problems.

      At this point it seems to me that the process of reducing equation problem solving to a skill, at least for simple equations, should be complete. It certainly can be argued that the text does that at the end of example one on page 285. However it appears to go too far. It first gives principles (addition property of equality, etc.), and then jumps to a recipe (In the first step, . . . .) I think we need an intermediate step, which would consist of a generalized strategy and a few tactics. The generalized strategy is to set up an equation to reflect the situation given in the problem and then to solve it. The general strategy for solving an equation is to move all the x's to one side of the equation and all the numbers to the other side. The tactics by which this is accomplished is transposition and then multiplying or dividing to get rid of the coefficient of the x. The principle underlying all this is that whatever you do to one side of the equation you must also do to the other side of the equation.

      I find the puzzle on page 289 interesting, but I would not assign it, and I wonder why it is given here. Basically it consists of three equations with three unknowns. It is not, in any way that I can see, a situation in which the current topic can be applied. I think it should be saved for the book in the series in which systems of equations are studied. Then it would be very relevant. At this point it is something that might be solved by blind trial and error, but is not something that can be expected to yield to diligence. To me it seems a classic example of what I call a "puzzle" as opposed to an "exercise". An exercise will yield to effort. A puzzle may yield to pure dumb luck, or it may not.

      I am undecided about section 6.7 on inequalities. It is in keeping with the premise that the focus should be kept broad right from the beginning. I maintain that trying to keep the focus too broad militates against the concentration that is needed. I wonder if it wouldn't be better to avoid inequalities until equations are mastered. Then it is a very simple matter to learn one extra thing about inequalities - when you multiply by a negative you reverse the inequality. At this point, it may be more confusing than helpful.

      In Section 6.8 the distributive property is revisited. I think the illustration given on page 296 is a good one. But why is there another section on the distributive law. The reason is that now it will be used for a slightly different reason. Now it is used to get rid of parentheses. Certainly the distributive law applies to both section 6.3 and Section 6.8. However this is thinking mathematically. Certainly we want to think mathematically, but I have argued that the degree of mathematical thinking must fit the learner. We want to go from the concrete to the abstract, but if we go to fast we lose the learner. We want to go from intuitive meaning to mathematical meaning, but if we go too fast we lose the learner. I would also argue that at any level of mathematics (at least until we get into non-Euclidean math) intuitive meaning should never be slighted. At this level I think it would be much better to title Section 6.3 "Combining Like Terms" and to title Section 6.8 "Parentheses in Equations". In both sections the distributive law can be brought in, but it should follow intuitive understanding, not lead it.

      I think example 4, page 297, is understandable to most students, but I think they would understand it by mental manipulation of numbers, not as a consequence of the distributive law. Again the distributive law should follow, not lead. Also I would question whether it fits in the section. If the distributive law leads, then it fits. But if the section is concerned with handling parenthesis in equations, then example four does not fit. I think the section should be about handling parenthesis in equations.

      Example 5, page 297, seems even more questionable. Yes, it follows from the distributive law. But surely most students will wonder why the book moves from equations with parentheses to multiplication of a fraction times a whole number when they learned in the sixth or seventh grade. The connection to the distributive law is important mathematically, but not, at this level, intuitively. It may make some sense to the student, but I would think it would mainly be a diversion, not an element to add to a structure of knowledge.

      In Example 6, page 298, we are again using the basic principles, that I maintain should give way to equation solving as a skill. Premise One is followed. Mathematical language and logic is used. But I think Premise One is faulty. Intuitive understanding and skill development must also be considered, and at this point should take precedence

      I would not use questions 1 and 2 on page 298. The real learning is in "questions" 3 through 8. On page 299 I would dispense with questions 9 through 12 for the same reasons. Problems 13 through 16 are the meat of the section. Problem 18 may fit the section in a mathematical sense, but I find it hard to believe that most students would be aware of this, or care about it. I think nearly all students would simply multiply 8 1/2 by 6 and be done with it. The more perceptive students might wonder why this problem is given.

      I think problems 19 and 20 are worthwhile. But I do not recall that the idea of clearing an equation of fractions or decimals was presented before. I think it should have been in section 6-2. Problem 21 seems just right. It applies the ideas learned in the section. It could not be done without algebra. It is a good level of concreteness versus abstractness. Problems 22 and 23 likewise seem quite appropriate.

      Section 6-9 is about dealing with a minus sign in front of parentheses. I would title it "Parentheses Preceded By a Minus Sign". One could dispense with understanding and simply give the students the rule, "If you have a minus in front of parentheses, drop the parentheses and change the sign of each term." This would certainly violate the idea that students should understand what they are doing, an idea accepted by both the premises I have identified and by the principle of intuitive understanding. I would not advocate this approach, but I think it should be recognized that it is the way most students would verbalize the lesson after they have learned it. I also think it is quite appropriate for the teacher to verbalize it in these words. The only question is how much understanding we can attach to this end result. I think it is fine to use the distributive law as part of the explanation, but again I think the distributive law should follow intuitive understanding, not lead it. An intuitive explanation would go something like this: "A minus in front of parentheses means that everything inside the parentheses is to be subtracted. So if you want to get rid of the parenthesis then you have to subtract everything inside, and that simply means changing the sign of each term inside. Remember that a number outside the parenthesis means to multiply each term inside the parentheses by that number. That's what the distributive law is. A minus outside means minus one times everything inside. And minus one times anything changes its sign. You learned that when we learned how to multiply signed numbers."

      I would not bring in the "Multiplication Property of -1". Nor would I use the phrase "the opposite of the sum is the sum of the opposites of its terms". Nor would I put "quantity" in bold print. "Quantity" is used in its everyday meaning. To put it into bold print implies that there is some special mathematical meaning that is relevant at this point. That is misleading.

      The summary on page 306 and the review that begins on page 307 seem to quite reasonably follow what was presented in the chapter. The review seems to have less stress on process and more stress on simply applying what has been learned. An analysis of each problem would repeat much of what I have already said.

Some Heretical Premises

      It is not too often that my daughter asks for help on math, but now and then she does. One particular incident made an impression on me. I don't remember the exact problem, but it was something like this:


           Simplify: 6a + b + 2a -2b

I asked her what answer she had gotten. "6", she replied. I asked her how she got it. She said she had let a = 1 and b = 2 and worked it out.

      It only took a minute or so to straighten out the problem for her, but I spent much longer reflecting on what I had observed. I felt something was being violated here, some basic principles of math, or of learning math. I could see where she was coming from. We often assign arbitrary values to variables for one reason or another. In this particular case, however, that stratagem was particularly inappropriate. There was something about my daughter's lack of recognition of this inappropriateness that impressed me. I felt that perhaps something deeply significant was missing in her approach to math.

      Looking back on my own experience in learning math in elementary and junior high school I identified three premises that I operated on. They are:

      1. Each problem will have one and only one right answer.

      2. There is one correct way to get that answer.

      3. The process will make sense.

      I think the authors of the Chicago Math program would want to burn me at the stake for seriously suggesting these premises. Premises one and two seem to suggest that I am advocating a shallow "recipe" approach to arithmetic. Without premise three, I would certainly agree. Premises one and two certainly do lead to recipe thinking, and I don't like recipe thinking. But taken as an entire set I think these premises are beneficial.

      Mathematicians would certainly consider them simplistic. They do not fit all of mathematics. But I would argue that they don't have to. Arithmetic in elementary school consists primarily of learning about number relations. These three premises are effective in promoting this learning. Certainly not all of mathematics is number relationships, but arithmetic is. The more advanced mathematics that do not fit these premises, I would argue, arise out of these basic number relations. Without a fluency in basic number relations the concepts of advanced mathematics would have little meaning and even less use. For the goal of simply learning basic number relations these premises are very valuable.

      One might argue that there comes a point in which these premises ought to be discarded, at least the first two. Again I agree. They are not constructive in many topics of advanced mathematics. One might also argue that if they are to be discarded at some point then they ought to be discarded immediately. With this I would very much disagree. In fact I think this is what the Chicago program tries to do, and I think the result is a hard way to learn arithmetic, and a poor foundation for learning algebra and geometry. I think my daughter developed an unconscious expectation that math is sort of a meaningless game. You do what the teacher wants, and you make good grades. Everyone is happy. But this expectation, the conceptual habit represented by this expectation, is not conducive to really learning math. It led my daughter to decide to plug in numbers, crank out an answer, and hope that it was right.

      One must always be on guard not to lose premise three. I think I probably had pretty good arithmetic teachers when I was a child. Math usually made sense to me. But let me make it very clear that it made intuitive sense to me. It did not make sense in the way that the authors of the Chicago math want it to make sense. I don't believe it can. I don't remember much about my elementary school days, but I can conjure up a memory of a diagram in my second or third grade arithmetic book of three stacks of two cups each as an explanation of multiplication. I remember learning to add fractions in the fourth or fifth grade. I remember being rather impressed in the eighth grade with a diagram that divided a circle up into slender wedges, unrolling these wedges into a row of slender triangles, and applying the formula for the area of a triangle to come up with the formula for the area of a circle. Of course not everything made sense to me. I never understood and rationale for the "invert and divide" rule. But in general I did well in math and usually enjoyed it. The "one correct answer by one correct method" perspective made arithmetic into a long series of rather enjoyable puzzles to be unraveled. I sometimes speak of puzzles as having no place in serious learning because they can be frustrating and wasteful of time and effort. In fact I define "puzzle", in a teaching and learning context, as a task that cannot reasonably be expected to yield to effort. But that is not the "puzzle" I am taking about here. Arithmetic was a puzzle that would always yield to diligent effort. It would always yield to diligent effort because it was carefully laid out for me, both by the teacher and the text. Every puzzle, every new topic, would arise as a logical extension of what came before. But this was common sense concrete logic, not the mathematical language and logic that the Chicago program tries to use. To me the phrase, "It will make sense" means I feel confident that I know what I am doing, that I can get the right answer and recognize and appreciate the rightness of that answer. The "understanding" that seems to come from the Chicago program, at least judging by what I see in my daughter, is more a matter of a tentative confidence that the answer will be judged by the teacher as correct. That seems a pale imitation of the understanding that I enjoyed.

      Educators have many ideas about children and learning. Some of these ideas are fallacious. "Children want to understand why." is a sentiment that I have come across many times in educational literature. I think it would be fair to say that the Chicago Math writers would express something like this as one of the premises behind their writing. But I believe the sentiment is fallacious. I wouldn't argue that it is all wrong, but I would vigorously argue that it is mostly wrong. Sometimes a small truth can serve only to mask a larger and much more important truth. The larger and more important truth, in my opinion, is that children first want to learn "how"; then they want to learn "what", and finally as a distant third they are willing to learn "why". The evidence of this is the simple everyday observation of what children do, how they respond to instruction. I would argue that when explanation quickly leads to "how" their eyes light up. When explanation dwells on "why" their eyes glaze over. I don't have the opportunity to see how kids respond to the Chicago math program, other than through my daughter, so I will leave it to others to test this idea.

      My premises, heretical though they may be, very well fits the "how-what-why" idea. Intuitive understanding supports this perspective. It allows the "how" to dominate, but without degenerating into mindless recipe thinking. The mathematical understanding attempted by the Chicago program not only fails to support this "how-what-why" perspective, I believe it is simply impossible at this level. Intuitive understanding may begin to give way to more mathematical understanding at this level, but the process takes many years to complete.

Some Final Thoughts

      Toward the end of writing this I began reviewing a calculus text. This was partly because I was thinking hard about the Chicago series, but also because I enjoy math. This was not the text with which I originally learned calculus as a freshman in college. That book, which I'll call the "old text", disappeared over the years. Rather it is a calculus text, which I'll call the "new text", that I picked up a few years ago for reference, but never studied to any great extent. After a few days of informal study I began to realize that many of the faults I described in the Chicago series are also present in this new text. Set notation is occasionally used, which I do not find helpful. There is a heavy chapter on limits, developing the idea with a lot more detail and rigor than seems needed to understand derivatives. This reminds me of the Chicago programs emphasis on mathematical language and logic. I don't remember being mired down on minutiae of limits when I first learned calculus. What I remember again very much fits the "intuitive understanding" that I have discussed. At the time most of it made sense. But I also realize that there were some serious problems with my education in calculus. The first and second semesters of calculus seemed fine, but the third semester seemed a bit less satisfying. When I took differential equations I really felt I understood very little. I began to think that I had some gaps in my knowledge of calculus. Would my new calculus text, with the "faults" that seem to parallel the Chicago series' faults, have been better? Did I have good teachers in calculus, or not so good?

      After thinking on these questions for a few weeks I have a definite opinion. No, I did not have particularly good calculus teachers. But I think this is because most college teachers are not particularly good, at least compared to high school and elementary teachers. I believe my problems in calculus cannot be attributed to my old calculus text. I think my problems would have been only magnified if I had had to learn from the new text. I do not keep up with educational trends, but I expect both my new calculus text and the Chicago program reflect a general trend in mathematics education toward what is believed to be greater emphasis on understanding. But again I would argue that mathematical language and logic should fit the level of the math being learned. If it is either too abstract or too concrete it is less than optimum. Greater logical rigor does not always translate into deeper understanding.

      Another thought seems very relevant as I finish this article. When I was in high school and college I didn't care much for history. Freshman American history in college was especially irksome. The professor and the small group teaching assistants liked to spout the rhetoric of teaching us to think. "Don't worry about memorizing facts." they admonished us. "Try to understand the broad patterns of American history." We have to "learn to think" they told us. I scraped through the course, with a gentlemanly B as I recall, and gave little thought to history for a number of years. Later in life I discovered two things. One thing I discovered was that a well written history book can be very interesting. A book on the Spanish Armada was an eye opener for me. I have no idea why I chose to read this book, but it made quite an impression. History could make sense. History could be interesting. Why did I discover this only in middle age?

      The other thing I discovered as time went on was that my wife had a vastly superior knowledge of history than I had. Now and again she would surprise me with a bit of knowledge that I would expect only history professors would know. Yet our educational experiences in history were, on the surface at least, identical. We had both had freshmen American history in college, and no other history courses in college. We both had American history in the eleventh grade and world history in the tenth grade. We both had American government (labeled "Civics") in the ninth grade. Why did she know so much of history and I knew so little?

      Part of the answer is that her father was a history professor. But another part of the reason is relevant here. I concluded, after some years of reflection, that I had not had a good history teacher, or at least a good history course, since elementary school. My wife had had several good history teachers along the way. I'm not sure whether I should blame my teachers or the educational ideology of teaching history under which they labored at that time. One of my teachers, to be sure, was a jerk and eventually got himself fired, but I expect the others were as conscientious and well intentioned as most teachers. I think there was an educational ideal at that time in history education that perverted teaching and learning. The ideal, in brief, was that if we could learn to think we wouldn't have to worry about learning a lot of facts. The history teachers I had apparently tried to follow this ideal. The history teachers my wife describes apparently did not.

      My college history professor was going to "teach us to think". The Chicago program tries hard to promote understanding by using high level mathematical language and logic. In each case, I believe, we have let the goal of understanding lead us into the mistake of bending the subject to fit our preconceived notions, rather than bending our teaching to the nature of the subject.

      I feel I was cheated in my education in history. I fear my daughter may be cheated in math in somewhat the same way. That is perhaps the driving force behind this article.