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Brian D. Rude    2004

       This article relates to two other articles I have written. “Thoughts On My Teaching at NDSU”, written about six months ago, is meant to be only a short description of the teaching situation of freshman math courses at North Dakota State University, not an in depth analysis. In this article I will repeat some of that article, but I will try to analyze the rationale of the curriculum and practices of these freshman math courses much more deeply. The other article, “Chicago Math”1, was written in 1996, and was an attempt to understand a math book that seemed very strange to me, but which entered my life only very peripherally. In this article I will attempt to do for NDSU math what I tried to do for Chicago Math. NDSU math also seems very strange to me. However my perspective is quite different in the two situations.

       When I wrote “Chicago Math” I had only the book to go by, two books actually, the seventh and eighth grade books of that series. I was not a math teacher at that time, though I had been for a few years in the nineteen sixties. I knew nothing about what was going on in the field of math education. My daughter was in the seventh grade when I first took a look at her math book. She was in the eighth grade by the time I had done much writing about it. From those two books I got enough information to form some opinions. For this article the situation is quite different. I have a book, titled “College Algebra”, but unfortunately I have a whole lot more. I have experience with this book. For the academic year 2003-04 I taught from this book at NDSU. It was a very frustrating experience. I was not rehired after that one year. I can’t say I was fired, for it was a one year job. I completed that year, so there was nothing to be fired from. But it felt like I was fired, for I had evidence that the decision makers in the Math Department at NDSU were not satisfied with my work. I was certainly not satisfied with them.

       I am not the only one frustrated with current ideas in math education. The “math wars” are ample evidence of that.2 The reasons for this frustration are varied, perhaps, but I think there is a common core. That common core is the lack of reality grounding of many of the ideas of what might be called “reform math”, or “NCTM math” (for the National Council of Teachers of Mathematics, which strongly promotes these ideas.) These ideas, I would argue, do not fit real people and real situations. Some of my frustrations at NDSU, I believe, derive from the Math Department’s attempts to implement unrealistic ideas and ideals, though not necessarily NCTM ideas and ideals. So hopefully what I say in this article will be of benefit to others frustrated with other math programs.

       When I wrote the Chicago Math article I could only guess how the program was implemented. When my daughter would bring the book home, which was not everyday, I would read it a bit and try to figure it out, but I didn’t know what the teacher was doing with it in class. In writing this article I have a lot more information. I know how it was implemented to quite an extent. I know how I tried to implement it, and I gave it a strenuous good faith effort. And I have some idea how it was implemented by others. I sat in on other’s classes a few times, though there is a limited amount of information that can be gained that way, and I had at least a little bit of discussion with others who taught it. But many mysteries remain. It was a frustrating year for me, a very painful transition from what I had been used to the previous five years. I also picked up a lot of clues that math at NDSU is very frustrating to many students, which I will discuss at more length later.

       Actually I taught two courses at NDSU, College algebra, which is superficially like the college algebra I had taught at two other institutions, and Finite Math, which is a collection of mathematical topics for students who will not take any more math courses. The subject matter was different in these two courses, but the administration of the two courses was the same. The textbooks were written (or compiled) by two members of the NDSU Math Department staff. Both books share the same perspective. I can’t claim I ever quite understood that perspective, and the purpose of this article is to try to unravel that perspective as best I can. There is also a trigonometry book which I presume shares the same perspective, but I have no experience with that. It is mainly the college algebra book that I will try to analyze.

       And actually there are two college algebra books. We got a new edition of the college algebra book the second semester. It is very similar to the first semester book, but with some changes and a lot of corrections. The first semester book was really bad with typographical errors. This was frustrating to all concerned, and when students were discontent it served as a focus of their discontent. A large number of typographical errors is something that is easy to identify, understand, and complain about. But I think there are bigger problems that can be identified. One bigger problem is coherence. And I think perhaps even a bigger problem than that is the effort to use some ideals of modern math education in ways and in contexts that just don’t work.

       “NDSU Math” is not a program or a project. So far as I know, it is not being marketed outside of NDSU. Therefore it is obviously not important that it be critiqued like some of the programs of “reform math” (sometimes called “new new math” to distinguish it from the “new math” of the 1960’s) which are commercially produced and are aspiring to shape the way math is taught in America. However it has at least some pretensions of being aligned to the NCTM Standards of 1989 and 2000. I will try to show that it adopts some of the ideas in the Standards, but, at least as implemented, very much violates some of the ideas in the Standards. In fact I wonder if it is not a bad combination of the worst of both worlds.

       As I did in my Chicago Math article, I will try to identify the premises that, either consciously or unconsciously, are used. Some of these premises, I believe, come directly from the NCTM ideals. But other premises do not, or even seem directly opposed to some of the NCTM ideals.

       One might also wonder about the motivation of this book. There are lots of perfectly adequate college algebra books available. Why write a new one? I would assume that part of the reason is to implement some ideas and ideals. This again makes it important to identify these ideas and ideals, and if they are faulty, to analyze why.

       It is commonly charged that “reform math” waters down the math, resulting in a “dumbing down” of standards. I think this does apply to the NDSU math in some ways, but it is not the most important thing. Some topics are slighted that ought not to be, but something more is added on. This something more makes it a tough program. It is demanding. It is frustrating.

       My central conclusion is that overall it includes a component of intelligence that does not fit our student population. Smart students can compensate for all the shortcomings and come out with a decent grade. But to me that doesn’t solve the problem. Less smart students do well to pass it with a D. Of course it can be argued that this is always the case. There will always be students who make D’s and F’s. But I think the NDSU situation is substantially different than at similar schools. It seems to me that there is a large number of students of reasonable intelligence who are blindsided by NDSU math. That would be the type of student who could go to another state university, work hard on a math course such as college algebra, and come out with a B or a C. The experience would be routine for them, if a bit challenging. When such a student is distracted by personal troubles, work troubles, a party habit, or any of a number of similar problems that beset college students, they may not pass. But absent those troubles, and with reasonable diligence, such a course is routine. But when that same type of student comes to NDSU, college algebra is not routine. It is more than a bit challenging. It is daunting, frustrating, and threatening, even traumatizing. All this would be acceptable, perhaps, if they really got a superior product, a solid course of study that forms a foundation for more math, or a feeling of accomplishment, or a feeling of having learned something of intrinsic worth. Unfortunately that seems not to be the case. It appears to me that the students get an inferior product, at a very inflated price. We claim to teach for understanding, of course, but I don’t think that is what we deliver. Students do not get an understanding of algebra so much as bits and pieces of algebraic concepts coupled with some recipes for calculator use.

       I will digress for just a moment to describe what I consider two major aims of education, utility and enlightenment. First there is the utilitarian aim. Education prepares one for everyday life. Reading and arithmetic have an important place in everyday life for just about anyone. But many subjects in college, even in high school, are not really taught for their utilitarian value. They will not be used in daily life. College algebra, for most people is not very utilitarian. Most people who take it will not only never use it in a practical sense, and they will also forget most of it. However the enlightenment aim of education is also important. Many subjects, including algebra, are seen as well worth learning, to at least some extent, because they contribute to the general perspective of an educated person.

       The “opening doors” idea is an important part of this enlightenment aim of education. The idea is that by learning a little bit of a subject we open doors to the student. By learning some algebra we open doors to more mathematics. We open doors to a world that could not otherwise be known at all. When a door is opened to students, it does not follow that they will ever choose to go through that door. Often they simply look through the open door, and go somewhere else. But the benefit of having that door open can be tremendous.

       A hybrid of both the utility and enlightenment aims of education, which can be called the “stepping stone” rationale, is also important. Many things that are worthwhile learning are worthwhile primarily because they are necessary prerequisites for learning something else. A particular topic may not be very important in itself, it may not be either useful or enlightening, but it may be a “stepping stone” to another topic that is worthwhile.

       In my opinion NDSU math does not do a very good job for any of these three aims of education. On the utility side, students do acquire some calculator skills and knowledge, but this is not too important. College algebra is not taught primarily for utility. For enlightenment, for opening doors to the world of mathematics, which I consider much more important than utility, I don’t think we do very well. We don’t do very well because we don’t teach for understanding. We claim to, of course, but I don’t think we succeed, and I will talk more later about why we don’t succeed. The “stepping stone” aim of education, acquiring skills and knowledge in order to learn other topics, I think is similarly perverted because we don’t teach for understanding. College algebra is supposed to be a preparation for taking calculus. It seems to me it is more of a weeding out process than a preparation.

       Unfortunately I feel part of the problem is what we might call the “culture of the math department”. There are some attitudes that were expressed in one way or another during the year that I feel need discussion. There is no parallel to this in my Chicago Math article, as I derived my information there entirely from the textbooks. And these attitudes have little parallel in two other math departments that I have some experience with. I will discuss these attitudes later in this article.

       Coherence of the College Algebra Text

       It’s a little hard to know where to begin, so I will begin somewhat as I experienced the book myself. When starting a new teaching job I ask for copies of the book I will use as early as I can, at least a month before school starts, so I can leisurely make my plans for my classes. The finite math book seemed to be just a regular math book. There was some new material that I had not seen before, so I was glad I had gotten a head start. However that was misleading. I think it was just a day or two before classes started that I learned we would have a new book for finite math, a book like the college algebra book, written by members of the NDSU math department, and sharing the perspective and methods that I will try to describe. This book was somewhat frustrating, but somehow less so than the college algebra book.

       The college algebra book, as I began to persue it about a month before classes started, seemed not to be a regular book. Is it a supplementary book, I wondered, a book of problems and examples to accompany the regular text book? It did not seem to have a straightforward presentation of algebra. If it is only a supplementary book, then what is the regular textbook?

       No, there was not another book. The book I had was the textbook for the course. I did not attach the term “lack of coherence”to the book at that time, but I think I should have. I will describe much of the first chapter of the book. In particular I will focus on the matter of coherence in this chapter. Coherence can mean a number of different things, but one thing it would have to include would be an orderly succession of the topics. If a book presents material in chapter 3 that is logically necessary to develop the ideas in chapter 2, then the book may fairly be accused of having a degree of incoherence. However I do not think it is accurate to say that coherence is totally a matter of logic, at least not formal logic, and is not just a matter of the sequence of topics. There are other considerations that also have to do with coherence, or its lack. I will try to show how the choice, treatment, and arrangement of topics is apparently guided by some current ideas in math education, but which result in a lack of coherence that makes it very difficult to teach or learn from.

       Section 1.1 is titled “Concept of a line” The term “linearity” is not stressed, but the term “linear relationship” is used. Linearity is an important topic in math. We start analytic geometry with linear equations simply because they are the simplest. Analytic geometry is an important part of algebra, though apparently the term has gone out of style. Also we don’t necessarily call it analytic geometry at the beginning. We just call it graphing equations. A definition of “linear” may be given when graphing on the xy plane is first introduced, but I think it is fair to say that for the vast majority of students the term “linear” makes sense only when they later encounter non-linear equations. After learning something about quadratic equations and functions, “linear” is more meaningful.

       Is linearity a good topic in itself, or would it be better to be considered just a facet of other topics? Should it be demoted to simply an adjective that is useful at times, and not at other times? For two semesters now I have tried to do it justice, devoting at least a full class period to developing the idea of linearity and discussing some implications of the idea. Within the first week I would try to arrive at the conclusion that the following four statements are equivalent:

       1. The graph of the equation is a straight line

       2. The graph has constant slope.

       3. There are no squares, square roots, cubes, or other functions in the equation.

       4. Equal increments of y values correspond to equal increments of x values.

       These statements seemed a good way to prepare to do the problems. The very first problem at the end of section 1.1 gives four short tables of data and asks which tables represent linear relationships. But maybe spending time explaining linearity is a mistake. I am inclined to think it would have been better to throw it out, and devote the time to something else. But how are students to do problem 1 in the exercises?

       Apparently this first section is based on the idea that a linear relationship may be presented as an initial value and a rate of change. Thus Example 1 in Section 1.1 presents an initial value of 15,000 people in a city and an increase of 500 people per year. From this verbal statement we are asked to derive, apparently by means of arithmetic and common sense, a table of population of the city over time, . There will be 15,500 people in the city one year later, then 16,000 the year after that, and so on. Then from the table we can draw a graph, and then, by some means, we get an equation. Example 2, concerning fixed costs and variable costs of producing a product follows the same pattern. We use the initial description of the problem, arithmetic, and common sense to produce a table of values, and only then get an equation. I will call this the “initial-value-rate-of-change” approach, and contrast it to what I would call an algebraic approach. I will try to show that the initial-value-rate-of-change approach is an arithmetic approach, and doesn’t work out very well.

       I am going to reproduce example 2 in full here, in order to be able to discuss it more thoroughly.

       Example 2 Creating a linear model

       A factory that makes rubber bands has fixed costs (for building, machinery, recycling, etc.) of $3,100. The variable costs (for material and labor) for making 1 dozen rubber bands is $1.27.

       (a) Calculate the factory’s total cost, C, to make 1, 5, 10, and 50 dozen rubber bands.

       (b) Find a formula for C in terms of d.

       (c) What is the average rate of change in cost per dozen rubber bands manufactured?

       (d) What does the average rate of change found in part (c) above tell us about the factory’s expenses?

       Solution (a) Start-up costs of the project are $3,100. When the factory manufactures rubber bands, the additional costs incurred depend upon the number of dozen made. Thus, the total cost . . .

       for one dozen        C = 3.1 + 1.27(1) = $3,101.27

       for five dozen        C = 3.1 + 1.27(5) = $3,106.35

       for ten dozen        C = 3.1 + 1.27(10) = $3,112.70

       for fifty dozen        C = 3.1 + 1.27(50) = $3,163.50

       (b) From part 1) a formula for C in terms of d is:

       C = 3,100 + 1.27(number of dozen d)

       = 3,100 + 1.27d, or

       = 1.27d + 3,100

       (c) The average rate of change is $1.27 per dozen rubber bands. In other words, it costs an average of $1.27 to manufacture one dozen rubber bands.

       (d) It is the additional cost for one more dozen rubber bands. (For each increase inn dozens of rubber bands manufactured, there is a cost increase of $1.27.

       In both of these first two examples a certain order is explicitly asked for. First use the verbal description to compute some sample values. Then find a formula. This seems strange to me. Just how are students to get the formula? Notice in example 2 that solution part a uses the formula, and then part b gives the formula. It can be argued that it is common mathematical sense, and to quite an extent I would agree. It can be argued that we have explained that a linear relationship can be defined by a starting point and a slope, so students should simply apply that idea. But it is not clear from the book just what line of reasoning is expected.

       But why not start with algebra, and then show how it can be made to apply to a problem like city growth? Which is the more basic concept, linearity or graphing of equations with two variables? The first semester that I taught this course I basically presented the material from the problem. The second semester I preceded the problem with a 10 minute reminder of what we learned in algebra. In beginning algebra we first get used to the idea of letting x be an unknown number, and develop methods for solving simple equations for x. Then we get equations with two unknowns, which have many solutions, each solution being a pair of numbers, an x value and a y value, and then we graph these solutions by letting one variable, x, be the horizontal distance from the origin and the other variable, y, be the vertical distance from the origin. Once we have the idea of graphing equations by plotting points, we develop the idea of slope. Then we develop a way to get an equation from a given slope and one point, or from two given points. Then we present the distance formula and the midpoint formula, and the relationship of slopes of lines that are either perpendicular or parallel to each other. All this follows from the arbitrary conventions of graphing (x is horizontal, y is vertical) and common sense. All this is not learned immediately. It takes time, careful thought, and practice to get the ideas straight in one’s mind. But once this much algebra is reasonably well understood it becomes a powerful tool that can be applied to a wide variety of problems, like city growth of example 1, or manufacturing costs like in example 2.

       Or we can just use arithmetic. It would be possible to take a group of students who are ignorant of algebra and expose them to the idea that growth can be thought of as starting with an initial value, then defining a constant rate of change, and then, on the basis of arithmetic, being able to do problems that are posed. I think of this as being the level of math attained at approximately the eighth grade, at least for students of normal ability when they are well taught throughout elementary school. Eighth grade math, in my view, should include all the number relationships covered by the operations of addition, subtraction, multiplication, and division. It should include a thorough understanding of fractions, because fractions are necessary for division to be closed. It should include decimals because that is the most useful and customary form of fractions, and the form of fractions that fits into our place value number system. It would include per cents because that is again a very useful and customary form of fractions. And, perhaps most importantly, it would include a lot of practice in applying these ideas to practical problems. When I was young the eighth grade level of math, I believe, did not usually include signed numbers. But perhaps now signed numbers is a regular part of arithmetic.

       I would argue that the idea of an initial value combined with a fixed rate of change is arithmetic. It could certainly be given to eighth graders. Eighth graders could do problems based on this idea. However I think it is important that we would not expect eighth graders to do such problems after reading section 1.1 and listening to 40 minutes of explanation in class. I would think of it as more like a week’s effort, perhaps even two weeks. Eighth graders could do the problems, if taught right, because the number relations can be understood on the basis of the arithmetic they have learned. But they could do these problems only if taught to do these problems. A few, but probably only a very few, eighth graders could figure it out just from the examples as presented.

       Now, how does all this fit in with college students? College students, presumably, are a little smarter, on average, than eighth graders, and have a few more years of math, including algebra, in high school. Shouldn’t they be able to handle the number relations of the initial-value-and-rate-of-change idea, and whip out the answers with ease? I would like to think they could. My experience is that not too many can. What’s wrong here?

       One hypothesis is that in the last few decades, perhaps since the “modern math” of the 1960’s, we have deemphasized arithmetic. In fact I’m not sure the word is even used any more. I don’t recall seeing it used in the NCTM Standards. But arithmetic is number relationships, and everyone needs to be fluent in number relationships. Presumably we have advanced. Hopefully we teach more than arithmetic. But I think we must consider the possibility that currently we teach less than arithmetic. The NCTM Standards very definitely deemphasizes practice.3 But it takes a lot of practice to attain fluency in just about anything we do. Perhaps students of the 1950’s, or even the 1910’s, would be better equipped to make sense out of the problems of Section 1.1 than today’s students. I don’t know.

       I think college students would be able to do these problems very well if they were presented to them as if they were eighth graders. Spend a week on the topic instead of a day. Present it as a certain type of problem that they are to learn to do. But this is still teaching arithmetic.

       But whether the idea should or should not be easily handled by the students, we still may ask if this is a good topic to begin a book on college algebra. And why is this topic number one in our algebra book?

       I think my problem is that I am looking for logic for the answer, and I am looking to the nature of the subject matter for the answer. Probably I should instead look to current trends in math education for the answer.

       One of the important trends in math education of at least the last 20 years is the idea of problem based learning, or PBL. I doubt that this idea has been carefully thought out, but

       it is the only reason I can think of for starting out with the initial-value-rate-of-change approach. So it makes sense to consider the possibility that what seems so incoherent at first may turn out to be coherent from a different perspective. Perhaps the college algebra book is reasonably coherent from the PBL perspective.

       I considered that possibility most of the year. But most of the year I didn’t have time to sit down and think these things out. Only in the last few months have I had time to do that. In my mind the possibility remains open that the book is coherent from the perspective of PBL, but that seems mostly a theoretical possibility. My impression remains that it is not a well written book and is difficult to teach from. And if it is coherent from a PBL perspective then my conclusion is that PBL is a seriously flawed premise. PBL seems to make some sense, but it has not worked out well.

       Even if one accepts what I think is the basic idea of PBL, that problems are the best way to approach a subject, the goal is still to get to the subject. What is the subject to which these problems are leading? The subject, it would seem, are linear equations and their graphs. But is this algebraic knowledge assumed at this point, or is it being developed? This is unclear.

       Following example 2, the rubber band problem, is an explanation that may be claimed to tie everything together, and thereby justify the problem based approach. It defines m as delta y over delta x, but does not discuss the idea of slope or define the delta notation. It arrives at the slope intercept form of the linear equation, y = mx + b and concludes:

       In other words,

       Value of y = (Average rate of change) x (Value of x ) + Initial Value

       So I think it can be argued that all the information needed to do this type of problem is there. If we assume that the basic rationale being applied is that problems should be used to lead to math, then the presentation makes some sense. But my experience was that the text was hard to teach from. Students get bits and pieces of understanding, along with bits and pieces of procedures to apply without understanding. It still remains unclear whether this is supposed to be a review of algebra or a way of developing the algebra.

       There are a some details that one can criticize in this section and the concluding explanation. There is the use of x to mean multiplication in the very last statement. Normally that notation is immediately dropped when students start algebra, so the cross of multiplication will not be confused with the x as the variable. There is the use of the delta notation, which seems out of place with the “arithmetic problem” approach, and depends on that concept being previously explained. There is the use of “dozens of rubber bands in the example, instead of units of some product. And $1.27 (from solution part c in example 2) does not seem like a realistic figure. It seems like it ought to be more like 5 cents to manufacture a dozen rubber bands. And the statement of solution part c seems to say something different than the statement in solution part d.

       Details are important. Or, more accurately, details can be important. Sometimes details are not important. It can certainly be argued that no text has all the details perfect, and that students can use their brains to interpret and overcome such minor defects. But it can also be argued that the effect of troublesome details can mount up quickly. It seems to me that the general approach is more important here. Apparently the authors intend to start Section 1.1 with problems, and the problems will lead to algebra. I think this general approach works very poorly. Certainly mathematics is intimately linked to problems, but it does not follow that any particular problem can be thrown in at any particular time, or that starting with problems is the best way to approach and develop a mathematical idea.

       Section 1.2 is titled “Geometric and Algebraic Properties of a Line”, which seems to imply that the algebra of linear functions is not taken for granted at this point. The section begins with a discussion of slope, and continues, mostly through examples, with slope-intercept form, the relationship of the slopes of parallel and perpendicular lines, the idea of graphing, and point-slope form, in that order. Section 1.2 I originally took to be a review of algebra, though I wondered why we got the review of algebra after Section 1, because obviously the algebra is used in Section 1.1.

       Or were we supposed to stick to arithmetic in Section 1.1 and now develop the algebra in section 1.2?

       Section 1.3 presents linear regression. This seems quite a leap from the algebra of Sections 1.1 and 1.2. It seems primarily a matter of learning a series of calculator steps by rote. In fact four and a half pages of the text are devoted to recipes for the different graphing calculators. Hopefully the algebra is secure enough at this point for the idea to make sense to the students, at least in general terms. Linear regression, apparently is considered to be a logical successor to linearity. But one might argue that it fits more with statistics, and that statistics ought to come after algebra, and maybe after a few ideas in calculus also, and that it comes after a discussion of mean, standard deviation, etc. The idea of the line of best fit is developed to some extent. It is explained that the line of best fit of graphed data may be approximated by finding the equation of the line going through the first and last points of data, or by visually estimating the line of best fit on the graph. Then linear regression on the graphing calculator is presented as a way to get the best fit.

       An understanding of linear regression can mean learning how to punch it in the calculator. Or an understanding of linear regression can mean using calculus. Given a set of xy data points we can assume an equation exists of the form y = ax + b. We can write an expression for the deviations squared of each y data value from the y value computed by the ax + b equation. Then we can write an expression for the sum of all these deviations squared. This expresion will be a function of a. Differentiate with respect to a, set to zero, and solve for a. Then do similarly for b. Then the formula for finding a and b makes sense, and of course we want to use a calculator to do the arithmetic. Obviously all this calculus cannot be done in a college algebra course. Therefore the whole process is reduced to rote learning of a calculator procedure. I think it has some meaning to the students - they understand the basic idea that there is a line of best fit and the calculator can find it, but it is quite a stretch to claim on the basis of what we have students do in this section that we are teaching for understanding.

       But why do we have this section at all? It’s not really algebra. It certainly is mathematics, but mathematics by recipe. It includes algebra, but is not really on the main line of development of algebraic concepts.

       My hypothesis is that again the writers of the text are trying to follow modern trends.

       “Data analysis and probability” is one of the “content standards” of the 2000 Standards of the NCTM. And an emphasis on technology is part of the Standards. But even if linear regression is determined to be a topic to be included in in the course, I would question that this is the place for it. It might be a good topic when students have a good understanding of graphing, and have some experience with the graphing calculator, but not before.

       Section 1.4, inexplicably, starts out with the general idea of graphing by point plotting. Why now? Had the students not already understood this basic idea, then sections 1.1, 1.2, and 1.3 would have to be meaningless. A discussion of intercepts follows the point plotting, and then a discussion of symmetry. Finally, in section 1.4, there is a short discussion of graphing equations on a calculator. Why is calculator graphing presented here when it is obviously needed in section 1.3?

       Chapter One is titled “Lines, Linear Models, and Graphing”. Why is it not simply titled “Basic Algebra”, or “A Review of Algebra”? The question remains open as to whether chapter one is meant to be a review of algebra, or whether it is meant to be a development of the algebra. If it is meant to be a review of basic algebra it seems very chaotic. Topics are not presented in the order that makes sense. If it is meant to be a development of algebra then it doesn’t fit into one chapter or a week or so of class time. A third possibility is that it is meant to be an abbreviated review of how the algebra was presented and developed in high school. My observation of the reaction of the students does not support this possibility. A fourth possibility is that the writers of the book were attuned to the ideals of the NCTM, but had insufficient time to do it right, to work out a set of topics and the sequence to present them in an order and in a way that would amount to a coherent presentation of the material.

       Problem based learning is one of the current ideas in the teaching of math. If we assume that this is a guiding principle in this book, then the general approach, at least, is explained. This premise is simply the idea that every topic must be introduced and developed in the context of a problem, and that the algebra will come out of the problem. In one sense this is sort of like starting in the middle of a story and then using flashbacks to bring in relevant information. I think there is something to be said for this general idea. In any subject we must by necessity start in the “middle of the story” in various ways. However that does not obviate the need for a careful arrangement of topics to be presented. It does not explain why calculator graphing is presented in section 1.4 after it is already used in section 1.3. Perhaps it does explain the arrangement of topics of sections 1.1 and 1.2. Perhaps the arithmetic approach of section 1.1 is supposed to lead to the ideas of Section 1.2.

       There are several powerful ideas that ought to be in a basic algebra course, probably at the eighth or ninth grade level. One of those ideas is the idea of a variable. A letter can stand for a number, and since it is a number it can be operated on by addition, subtraction, etc. A second very basic idea is that of the equation, and that there are specific techniques that can be used to solve those equations, and that those specific techniques are not magic - they have rationales, they can be understood - but they also need to be learned to a high level of fluency. A third basic and powerful idea of algebra is putting two variables in the same equation, and graphing the solutions on an xy plane. A fourth basic idea is that a real life situation can be translated to an equation, and then the techniques of equation solving can lead to an answer. Now days we would call that modeling. Years ago we would just call it “word problems”.

       Starting the study of “the linear relationship” with the initial-value-rate-of-change idea does not use all of these powerful basic ideas of algebra. It uses the idea of a variable, and indeed it makes some use of the idea of solving equations and graphing equations, but it is primarily an arithmetic approach. Is problem based learning supposed to throw out much that was previously learned when each new topic is taken up? Certainly one can learn the basic ideas of algebra that I described, and then revert to arithmetic to handle “the linear relationship”, but why would one want to? Arithmetic, of course, contains powerful ideas. Arithmetic is a wonderful invention of the human mind, well worth the time we spend on it in elementary school. But algebra transcends it. Those basic ideas I described are not that hard to learn. Most students have learned them to one extent or another in high school. They form a much better basis to learn about “the linear relationship” than arithmetic. But if these ideas fall into disuse they will be forgotten. And to be used they might have to be refreshed a bit in the students’ minds. That means students must be reminded about concepts such as slope, intercept, graphing by plotting points, how to get the slope from two points, how to get an equation from a point and a slope or from two points, and so on. Thus most college algebra books spend a chapter or two reviewing it. This is all to the good. These topics are basic. They will be used for much more than just “the linear relationship”. But such a review ought to be coherent.

       For several years I have had reason to suspect that in many math course (many courses of any subject probably), many students scrape together enough facts, recipes, and snippets of understanding to pass tests, but really don’t get the basics. Their understanding is superficial. I think many teachers are aware, to one degree or another, of this possibility. There is no easy, quick, and guaranteed way to avoid this superficiality on the part of the students. But using arithmetic to approach topics that really work much better than algebra might contribute to this superficiality. And using a problem based approach indiscriminately might also contribute to this. Problems, I presume, are supposed to lead to mathematical ideas. But those mathematical ideas need explanation, emphasis, and practice.

       Algebra is generalized arithmetic. The best basis to learn algebra, I will hypothesize, is a high level of fluency in arithmetic. If this is the case then it makes sense to concentrate on arithmetic first, and then concentrate on algebra second, not mix them up. This would be consistent with the “principle of concentration”,4 which I think is an important general principle of learning most subjects. You have to concentrate on one topic, one detail, at a time. Of course there is also a counterbalancing idea, which I call the “spread and relate” principle. When details are estabilished by concentration then they must be related to the rest of the subject matter. But in general the “spread and relate” principle cannot be applied to a topic until that topic has been established, and that requires concentration. A problem based approach can detract from this concentration if used too indiscriminately.

       I do not claim to fully understand the approach used in Chapter One of this book, but my best guess is that an attempt to use problem based learning leads the authors to use arithmetic in Chapter One when they should use algebra, and the result is a very poor presentation of what should be a straightforword review of basic algebra.

       I will not attempt to analyze the remaining chapters of the book to the extent that I have Chapter One. Much of these remaining chapters consists of an adequate presentation of the material, quite useable as a text for the topics presented. And the topics presented are what we expect for a college algebra course. There are plenty of details that one might choose to complain about, but that can be said of any text. I will turn to other aspects of the course that I think very much need to be analyzed.

       The Graphing Calculator

       We are expected to make much use of the graphing calculator in this course. It is asserted often in the NCTM Standards that technology can enhance learning, but it is not at all clear how this best might be done. I was not at all satisfied with how we handled the calculator in College Algebra. I think what we did is not a good model of calculator use in a course such as this. This is partly due to a problem of coherence in presenting calculator topics. But even more importantly, I think calculator topics have displaced some more important algebraic topics.

       It would seem that putting an equation in the calculator and finding a suitable window for a good picture would come before linear regression. But in the book this comes informally in the “Interlude” in the middle of Section 1.4, after we had struggled to do a lot more with the calculator in section 1.3. Linear regression could have been a lot easier if it came at a point when graphing a function was mastered.

       The problem of coherence in this book is again illustrated by the treatment of “zoom” and “trace” on the calculator. In the interlude of section 1.4, the emphasis is on setting the window, which apparently is to be done mostly by trial and error with the clues given about how many “peaks” and “valleys to expect. Also included are the admonitions “(no “zooming” allowed)” and “(DO NOT “zoom and trace”)”. No explanation is given for these admonitions. I explained to the students that the goal here is to learn to set the window directly, that trace and zoom are features that many students learn to use but are sort of a lazy way of doing things that really take up too much time in the long run. Similar admonitions are given when the minimum and maximum finding routines of the graphing calculator are presented. I explained that zoom and trace will not provide the accuracy needed quickly and easily. I assumed that a lot of students had learned to use zoom and trace previously, and are inclined to use them in places where they are not the best approach to the problem, so now we want to avoid them entirely, at least for the moment. Then on page 162, in section 3.1, which is about quadratic functions, suddenly the text presents zoom and trace, as if they were brand new topics. And they are presented as a way to find maxima or minima

       Is there some rationale for the treatment given to trace and zoom? Or is it just a result of lack of care in the writing of the book? My best guess is that it is a lack of care. Probably under pressure of time they had to just slap it together and get it off to the publisher. If that is the case there is not much purpose in analyzing further.

       Getting a suitable window has caused a lot of problem for students. Generally it is because they have no way of knowing about the function they entered. Time and again a student would put the function in the calculator, push the graph button, and see something meaningless, perhaps a totally blank screen, or perhaps a set of axes, and nothing more. With no idea how to get a suitable window, the calculator is useless.

       In addition to the problem with finding a good window, there seemed to be a number of other ways that students can get stuck when trying to graph an equation. Perhaps the most common problem is having a plot left on that requires a different window that the equation the student is trying to graph. But there are other problems too. They can inadvertently set xmin the same as xmax, or they can be in the wrong mode because they have no idea the mode key exists or what it might be for. It occurred to me that we need to have some list of such problems printed out for the students to refer to when they hit a snag in trying to graph on their calculator. This would be useful, but more importantly, I think, is having the calculator topics in the right order. Having linear regression as the first calculator topic in the book did not work well.

       If we insist that calculator usage be made an important part of the course, it still is not necessary that it be emphasized right at the beginning. I wonder if teaching algebra through quadratic functions, or perhaps through polynomial functions, or even through rational functions, and then bringing in the calculator would be a better way to do it. Min and max might have more meaning after working with parabolas. After working through the chapter 2 on functions, graphing in general should be more meaningful. Then when going into exponential and logarithmic equations the calculator skills could be applied, and might actually aid in the learning of these topics. After progressing through the course and covering linear, power, rational, exponential, and then logarithmic equations one is then in a good position to find a good window. And all those glitches that plague students could be dealt with in much simpler situations.

       The central premise, it seems to me (and if indeed there is a premise), of using the graphing calculator is that it allows students to quickly and easily “see” what a function looks like. In one sense this is obviously true. A graphing calculator can plot points, and it can do it at a very high rate of speed. It may certainly be asserted that a graphing calculator may do in seconds what would take hours, even hundreds of hours, for a student to do without it. But this begs the question of whether those points really need to be plotted.

       To get the idea of a graph of a function, and the idea of graphing in general, we typically have students in elementary algebra plot some points. Typically we set up a “t-table”, x values in a column on the left and y values on a column on the right, and then plot each xy pair on graph paper. At the very beginning of this process we point out that we can choose a value of x and compute y, or we can choose a value of y and complete x, or we can do several points each way. And we usually start with a linear equation, to keep things simple and manageable. To plot six or seven points by hand probably takes five to ten minutes. Then we explain the rationale for connecting the points by a smooth line.

       I have always told my classes that this process is a very primitive beginning to graphing, but a very important beginning. A student who fails to get this basic idea of graphing cannot advance in algebra. This important beginning, in my opinion, can easily be subverted by using the graphing calculator. The idea that the calculator can plot hundreds of points in seconds, whereas we take five minutes to plot perhaps five points, is utterly irrelevant to the idea of understanding graphing. There is no advantage to plotting hundreds of points in seconds. There is an advantage of plotting five points by hand. The advantage is understanding.

       But then where do we go from here. Students often want to plot points forever. Many times when given an equation on tests, students who don’t understand the problem will set up a t-table and plot a few points, which is often irrelevant to whatever is being asked at the moment. We need to get beyond this primitive beginning of plotting a few points by hand. We need to learn about the different types of functions. The graph of a linear equation will always be a straight line, so we learn about slope, intercepts, and so on. A quadratic function, however, is different, so we must learn what we can about quadratic functions. It’s true that a calculator can plot hundreds of points in seconds and show the resulting parabola. But why not instead learn what we can about parabolas in general?

       The college algebra books I have used in the last six years have surprised me in one way. They all have taken the idea that we graph all equations by pretty much the same strategy. We find intercepts and symmetry, and plot several points by hand if needed. If we are dealing with rational functions we also look for asymptotes. This strategy certainly makes some sense, but I think it is limited. It is very inferior to the strategy of learning the nature of the different functions. If we want to graph the equation x + y = x2 + 5, we should apply our knowledge of quadratic functions, not look for intercepts and symmetry. If we need to graph the equation y = 5e^(x-2) then we need to know something about exponential equations. And of course we have to understand transformation of functions. But all of this is standard fare for college algebra.

       It may be asserted that the graphing calculator will help students learn the characteristics of the different functions that we teach. I think there is only a bit of truth to this, but that bit of truth is outweighed by a larger truth. The larger truth is that the graphing calculator can subvert the learning of the characteristics of the different functions. I think this idea of subversion of learning has parallels in lower levels of math. It can equally be asserted that a calculator can help students learn the multiplication tables, or it can subvert that learning. A calculator can help or subvert the learning of fractions. It can help or subvert the learning of percents.

       It is true that “technology can enhance learning”, but that is not a reason to make the course “calculator based”. A pencil is technology, and a pencil enhances learning - pencil and paper is definitely better than a hornbook of the 1700’s or chalk on a slate from whatever time that was. But it is not good to say that we are going to make a course “pencil based.” A copier is technology, and it very much enhances learning. We need it. But it would be foolish to make a course “copier based”. The internet is technology, and it also enhanceds learning. At NDSU we had “Blackboard” which is a course management system, a computer program that allows us to to post assignments and other documents on a website to which the students have access. I used it extensively and appreciated it. The year previous to my year at NDSU I had a web site, made available by the institutuion to each factulty member. I used it much as I used it much as I used Blackboard at NDSU. The only difference is that my website was available to anyone worldwide, whereas Blackboard is available only to students in the course. Any kind of a website like that can enhance learning, but it would be a big mistake to say my course is “website based”.

       So what is the optimum role of the calculator in teaching math? What is the optimum role of the pencil in teaching math? It is a subservient role. Teaching math consists of many things. The careful explanation of each topic in turn, in my opinion, is the most important thing we do. If a pencil, or a calculator, or an opaque projecter, or a website contributes to this, then we should make use of the pencil, or the calculator, or the opaque projecter, or the website. I leave it to the students to know when to use the pencil. I’ve seldom felt the need for an opaque projector. I used the graphing calculator very minimally when teaching college algebra in places other than NDSU. And the website is a wonderful convenience for both students and the teacher.

       Problems can be made up for which the graphing calculator is the only practical way to solve them. We had a lot of such problems in College Algebra at NDSU. In particular we had a lot of max/min problems. When one gets into the habit of thinking a lot about these types of problems one may think of them as the typical type of problem, or the “target problem” for college algebra. Therefore a large role for the graphing calculator in a college algebra course may seem quite reasonable. I think that type of thinking is a trap that we have fallen in to, and I think we need to dig ourselves out. Max/min problems are a natural for some topics in elementary calculus, but not for algebra. Problems that are natural for algebra would be problems that use the ideas we are trying to teach, which showcase these ideas, which aid in the understanding of these ideas. Problems in which the ideas we are trying to teach must be stretched to fit, or problems which are only peripherally related to the ideas, or are minor consequences of these ideas, do not serve the student well. Of course this leaves a lot of room for opinion, but I think we have drifted into an unhealthy emphasis on the graphing calculator. We have drifted away from the basic concepts of algebra.

       From a PBL perspective problems come first. The opposite of a PBL perspective would be, in my opinion, a “structure of knowledge” perspective. By this perspective we first think about the knowledge we wish to impart, and then carefully choose the problems to promote this end. In both perspectives, of course, problems have an important place. But there is a difference of emphasis. PBL requires big problems, problems which lead to a whole complex of learning. The structure-of-knowledge perspective leads to smaller problems, problems that apply one simple idea at a time. And the idea that a problem is meant to apply comes from the emerging structure of knowledge. Problems must be chosen carefully to fit the immediate context. There is no such thing as a “good problem” or a “poor problem” A problem is good if it fits the immediate context, the immediate topic that the students are trying to learn, and if it is not burdened by characteristics, complications, or difficulties that do not promote the immediate learning goal. A problem that is a poor fit one week may be the problem that “hits the nail on the head” the next week, and then may be the problem that is “beating a dead horse” the week after. Problems must be carefully chosen to meet the immediate learning needs of any given topic. Problems should follow, not lead.

       I mentioned “PBL”, problem based learning, earlier. I conclude, after a year of trying to figure out what motivates the textbook, that PBL is one of the “guiding lights” of NDSU math. Problems must be used to approach a topic in math. And problems must be used to test for the knowledge and skills we are trying to impart. I will talk more about the testing situation momentarily. For the moment I want to put PBL together with the idea of “using technology”, meaning the calculator, and we come up with something like

       the “box problem”, more or less as follows:

       A box is to be contructed with a square base, and it must hold 9 cubic feet. The cost of the material for the base is $.20 per square foot. The cost of the material for the sides is $.15 per squqare foot. The cost of the material for the top is $.10 per square foot. Find the dimensions of the box of minimum cost.

       A problem like this occurs in Chapter One. A probem such as this is likely to occur on several of the tests in the course. One can argue that this makes a lot of sense. It’s the sort of thing that ought to define success in college algebra. It requires students to put together several steps, to translate a problem into an equation (or a function), and then use the calculator to find the minimum of that function, and then to interpret the results. I recognized early in the year that few students seem to be able to do this type of problem successfully. But it was not until the last week of the second semester that I zeroed in in this problem. I was helping a student prepare for the final exam. We had gone over several topics that he asked questions about, and he seemed to understand pretty much what he should understand. The he said, “Oh, and what about the box problem. I have trouble with that.”

       This was a reasonably intelligent and diligent student. He was well motivated. Why couldn’t he get it all together for this problem. In helping him with this “box problem” for perhaps fifteen minutes it became clear that the geometry was not well established in his mind. I explained that if the side of the box has length x and width h, then the area must be xh. His response seemed to be that this was some idea to dredge from distant memory, not something that has been obvious since eighth grade. He had to think a minute to remind himself about the difference between area and perimeter. This surprised me. I thought the basic ideas of area and perimeter of rectangles ought to be obvious to everyone since about the eighth grade. If it is not obvious for this student, does that say something about him, or about the way he was taught math in elementary and junior high school? I don’t know, of course. But it is obvious that if this basic geometry is weak in a person’s mind, then the “box problem” is not going to be easy.

       Another major part of this problem is translating a written problem to algebraic language, getting an equation in other words. I will have more to say about this in the next section . A third major part of this problem is using a graphing calculator to find the minimum or maximum of a function. This is the only major part of the “box problem” that is directly taught in this course. Assuming it is new, or relatively new, to most students then it is understandable that it might be a little weak in their minds.

       And then a fourth major part of this problem is interpreting graphs. Finding a minimum makes sense only if one is reasonably fluent in interpreting graphs. I have become aware that this is a weak spot for many students. I would not have expected this. I would have thought that interpreting graphs is normally accomplished by the end of eighth grade by working with bar and line graphs. Then after graphing linear equations in elementary algebra, it should be pretty easy. But it was not for many of my students.

       So out of four major competencies required in this “box problem”, only one of them, finding the minimum on a calculator, is directly addressed in this course. The other three are considered prerequisite competancies. I will not quarrel with this expectation. It makes sense. Yet very few students could seem to do the box problem, even the third or fourth time they encounter it. I am not sure what to make of this.

       Written Problems

       This book emphasizes written problems (called modeling, of course). How are we to teach students how to do written problems? This is always a problem in teaching math. I have had many students confide to me in my office that “word problems” have been difficult and frustrating for them all their lives. I think math teachers in general tend to short circuit the problem. It seems obvious to us that if you know the math you can apply it. If you know how to add two and five then you can solve a problem such as “If John has two apples and Jane has five apples, how many apples do they have altogether?” Or if you can solve a simple equation you can solve the problem “Two times a certain number minus eight is equal to three times the number plus two.” If you really understand the math, it would seem, then its application is obvious. But to very many students it is not obvious. It is a big problem, and a big frustration. So how do we teach written problems?

       On a superficial level one answer comes to mind. We teach by example. We present written problems and work through a few of them. Then the students can work through others on their own. And we remind them that practice is necessary. They should work through a lot of them on their own. I will call this the “few general examples” method. But when a test is given they may be largely unsuccessful. Often this method involves a fair amount of pretence. We pretend to teach them and they pretend to learn. Then when they can’t do written problems on a test we all pretend it was just a bad day.

       In the previous several years I did develop my ideas on teaching written problems a little bit, at least in a course such as college algebra. After the chapter on solving simple equations I would present “distance problems”, carefully working through and explaining each part of the problem. The next day I would explain “mixture problems”. Then the third day, if we could spend another day on written problems, I would present “work problems” which were always harder. But that is all the types of written problems we would do. Limitations of time would dictate that. And on tests I would give only one or more of these three types. I will call this the “limited types” method. The only way it is different from “few general examples” idea is that we strictly limit written problems to those three types. We make no pretense that students can do a lot of other types of problems. But with careful explanation most students can do at least these three types, and they show evidence that they understand what they are doing. It is not rote memorization by any means.

       The NDSU college algebra book generally uses the “few general examples” approach.

       I don’t think it works for most students. I think the “limited types” method does work, but of course it succeeds only in a very limited domain - the specific types of problems that we teach. It is understandable that math teachers are not too happy with this. If we test only the specific types of written problems that we have taught, it feels we are painfully close to teaching by rote.

       My preferred solution ( if I may call it a solution) is to use the “limited types” approach, and accept its limitations.

       We would like our students to learn math thoroughly. We would like them to think like mathematicians. They are hardly thinking like mathematicians, it can be argued, if we limit ourselves to only three types of written problems. But let us turn the question around a bit. History teachers would probably like their students to think like historians. Art teachers would probably like their students to think like artists. Is this a realistic expectation? Can we, as mathematicians or math teachers say that when we took history or art we thought like historians or artists. I can’t. Based on personal experience I can say definitely that that would be expecting too much from me.

       Mathematicians are not smart in every subject (thought admittedly many of them are). I was not particularly adept at learning history or art. Those are the subjects in which I got C’s in high school. I was more of a B student generally, and an A student in math and science. But surely my history teacher wanted all of us to think like historians. And surely my art teacher wanted all of us to think like artists. But I didn’t do a very good job thinking either like a historian or an artist.

       My feeling toward the two different subjects, art and history, are quite different. I had a college history teacher who admonished us that we must “learn to think”. He aggravated me enough to cause me to reflect back on my 11th grade history teacher. I didn’t care too much for 11th grade American history. At the time I didn’t analyze why, but the college history teacher prompted me to think about it. Their methods of “teaching us to think” were to give us essay tests and to make us feel inadequate when we didn’t do very well on them. My ninth grade art teacher was more practical. She had no rhetoric about us thinking like artists. She just taught us what she could. I didn’t think much about the art course when I took it. I knew I was not much of an artist, and I wasn’t disappointed in getting C’s. But looking back on it, I realize it was a pretty good course. It opened some artistic doors to me. After the year was over I can’t say that I ever chose to go through those doors, or even to look through those doors again. But I do consider that one art course in the ninth grade part of my liberal education.

       I consider it an important goal of most courses to “open doors”, to give students at least a glimpse of a world they could not otherwise know. But I do not consider it realistic to expect students to think like masters of the subject think. We do a disservice to students to tell them they should. It’s not possible. Thinking like a practitioner of a subject is a realistic expectation only of those who advance in a subject, who actually become, or are capable of becoming, practitioners of the subject. But we can teach students some things about a subject, and we can teach them some thought patterns of that subject. In math we can teach the average student to do a limited number of types of written problems. I think this is well worthwhile. When done well students see some intrinsic worth in the experience, though they seldom verbalize it, just as I can look back at my ninth grade art course and feel that I got something of intrinsic worth from the experience.

       NDSU math obviously does not agree with my perspective on such things.

       The Students’ Perspective

       Students come to college eager to learn, do they not? That is the whole point of a liberal education, to have one’s mind and perspective expanded, to have whole new worlds opened up, and to partake fully of the free sharing of ideas with their colleagues. And everyone knows you learn more from dormitory bull sessions than you do in class, don’t they? Therefore they should be eager to take on an intellectual challenge like college algebra.

       I have described a very idealistic perspective in the above paragraph. But I do not subscribe to that perspective. I set it up as a “straw man”, to be knocked down. There is some truth and validity to this perspective, perhaps. Some students are indeed eager to take intellectual challenges and have their worldview expanded. And students do indeed learn some important things in dormitory bull sessions. But I consider the idea that they learn more in dormitory bull sessions than in the classroom pure nonsense. And my experience leads me to think that the idealistic perspective I described is not very important or prominent in the thinking of the average student. The perspectve that I believe is very prominent in most students’ minds is what I call the “contract perspective”. This is not an idealistic perspective, but neither is it a perspective to be dismissed. It is a very useful perspective. It motivates a lot of accomplishment and all levels of education beyond perhaps the primary grades. The contract perspective is simply the idea of an exchange. Students are quite willing to do a number of things, in return for which they receive something valuable to them, a grade on their transcript. Just why this grade on their transcript is valuable varies widely with different individuals. But for the vast majority of students that is the payoff. They don’t think too much in idealistic terms. They just want a decent grade at the end of the semester. In return for this grade they are willing to do a lot, most of them anyway. They expect to come to class, though perhaps not with the regularity we would like. They expect to listen carefully and take notes, though perhaps many of them are not too good at that. They expect to put in hours outside of class, though not as many hours as we admonish them that they should. They expect to learn something, and they expect to demonstrate that learning on tests. They are willing to put up with a fair amount of what they might consider as extraneous demands, such as group projects or term papers, and they do it mostly without complaint. But in return for all this they expect the course to be manageable. The expect the things required of them to be possible, to lead to the acquiring of knowledge that has some worth, and to be reasonably efficient toward that goal. A fair number of students know that success is very problematic. They may fail, and if they fail they make their plans accordingly, sometimes to take the course again, sometimes to change majors, sometimes to drop the dream of getting a college education.

       Students do have some idea of what they consider fair for us to ask of them. We should not ask the impossible. Whatever we ask of them should be possible for the normally tallented student. Of course that is a very loose standard. If a student complains, “I flunked because what you ask is not possible for a normally tallented student.” we can easily reply, “Well, you must not be normally tallented, because a lot of people did pass the course, some with high grades.” But a loose standard does not mean there is no standard. Students will feel cheated when the feel the standard is obviously not fair.

       The contract perspective may not be very idealistic. In fact it is almost reduces learning to a commercial transaction. But I think it is very important. It motivates a lot of education, and we are all the beneficiaries of that. I think we should respect it. And especially I think we should recognize it as the dominent thinking for most students. It should be taken into account whenever we set up a course. It is not the only thing we should take into account, of course, but it should not be ignored.

       The Tests

       I will describe the tests we gave. There are about 20 sections of college algebra, with about 30 to 35 students in each section, making a total of somewhere around 600 students taking college algebra each semester. After the attrition of a few weeks or a month that number falls substantially, though I have no statistics. There are three evening tests, which I will call “mid-course tests”, and then the final. The mid-course tests are from 6:30 to 8:00 on three evenings during the semester. Each mid-course tests consists of two parts, the regular part and the take-home part. The regular part of the test consists of nine problems each worth ten points. One of these is labeled a “bonus problem”, and may be a bit more difficult than the others. Thus there are 90 points possible, but the score is reported on an 80 point basis. The top score then would be 90/80, which is over a hundred percent. The other twenty points come from the take-home problems which are to be done cooperatively in groups of two to four students and turned in the night of the test. The “in class” part of each mid course test is printed on a single sheet of paper. The students bring blue books to write their answers in.

       All problems on the test are written problems. This certainly makes some sense. What is math for if not for applying it? But, of course that is not all there is to it. It is certainly possible to speak of “pure math”, or “pure algebra” as a body of abstract ideas. In fact it is a point of pride among some research mathematicians that there need be no practical use whatsoever for the mathematical ideas they develop. A perspective such as this has little applicability in the lower levels of math, but I think it is very important that algebra is a body of knowledge. It exists outside of its applications Of course we want to be able to solve practical problems with the math that we learn, but we also want to understand math purely as math. I don’t think there is a conflict between pure math and applied math. The practical aids the understanding, and the understanding aids the practical. But there certainly is the question of find the right balance, or the optimal combination, of the two. And there is the further problem of finding a coherent combination of problems and ideas.

       “Problem Based Learning” (PBL) is one of the newer ideas among educators. This idea asserts that problems are the way to get to mathematical ideas. Obviously there is some truth in this, but it does not follow that every mathematical idea is best approached through a problem. My experience leads me to believe that most algebraic ideas are best approached as common sense extensions of previous algebraic ideas, not through problems.

       And I think it is even less true that mathematical ideas must be assessed only through problems. Of course problems have always been the regular way that mathematical knowledge is assessed. Problems apply the learning, and how can you test for knowledge other than to apply it? But there are problems, and then there are problems. More specifically we seem to have the idea that tests must consist only of high level problems.

       Our tests, I would argue, are the mathematical equivalent of “essay tests”. I am aware that some teachers think essay tests are the the only valid tests, or the only type of test worthy of thinking people. This was the attitude in the only college history class I ever took, and I am aware that some teachers in other subjects feel the same way. I have never agreed with this idea. In fact I have argued5 that there are very good reasons to ask for lower level responses on a test. A high level problem is “top loaded”, It becomes an all-or-nothing situation, it requires that everything be in place for success. That is certainly a goal of our teaching, to have “everything in place”, to have a thorough understanding of the mathematical ideas involved. However there are many students who have much accomplishment, but that accomplishment is not brought out by a high level problem. I have argued elsewhere that a test should be approximately “linear”. That is the output, the test score, outght to be approximately proportional to input in effort and knowledge.6 This is primarily accomplished by having a mix of both high level and lower level test questions.

       Our tests had all the questions on one sheet and the students write their answers in bluebooks. This fits the essay question format. I cannot argue that every question was a high level question. We even had a few multiple choiced questions. But even something seemingly as low level as multiple choice can be changed to a higher level problem in two ways. Students may be asked to explain their answer (which sounds sensible enough, but there is a lot to analyze and consider) and the answer may be made indefinite. In other words a multiple choice question may be made into a high level question, another “essay question”.

       The grading of the tests is a team effort. Each teacher is assigned one or two problems to grade. All the tests for each section are kept in an large envelope. These envelopes are kep in a file drawer in the computer room. Each teacher takes one envelope at a time to grade his or her assigned problems, then returns the envelope of tests to the computer room so it is available to other teachers. This is a sensible system. It’s essential purpose is to insure that everyone’s test is graded the same, and therefore fairly. It does give rise to some irritating complications, but for the most part those complications are quite manageable.

       There is another part of how we grade the tests that I do have some problem with. We meet and discuss our “rubric” before the tests are graded. If I am assigned problems four and five to grade, then I will decide to give so many points for this and so many points for that. In practice my “rubric” is subject to change by the committee, which in practice means subject to change by the course supervisor. I don’t object to meeting and discussing how we plan to grade each problem. There ought to be some system to our grading, and discussing the matter can definitely be helpful. However I concluded we carried things too far. Having agreed how we are going to grade each problem in the meeting, we do not thereafter feel free to modify the plan, but often we should modify the plan. I have several examples.

       On the very first test a problem asked the students to look at a scatterplot and choose among five equations for the line of best fit, and explain why. This is a good example of a multiple choice question not at all being a low level question. It is high level, an essay question, because students must explain their answer. They, presumably, must put together a line of reasoning and put it into words. They could say that one equation is the best choice because it is obtained from the first and last points on the graph. Or they could argue that another equation is the best fit because one or two of the graphed points are “outliers” that ought to be thrown out.

       But this problem was not to be graded subjectively. It was to be graded according to the rubric previously decided on. As a result the unfortunate teacher who graded this problem according to the rubric was beseiged by students who felt they were not given the credit they deserved. I personally went to this teacher on behalf of one of my students, who I thought had a good case. However in talking with the teacher who graded the problem I quickly concluded that she was very uncomfortable with the situation. She felt bound to apply the rubric, though I got a distinct impression that she did not think it was a good rubric. She also felt obligated to defend the rubric. But there was to be no reconsidering of the rebric. That would require regrading that problem on all the tests, which would prompt another spate of complaints from students. But also the culture of the math department, which I will discuss shortly, was not conducive of any sort of reconsideration.

       I will describe another problem with the grading of a test question which I graded. The problem is a chart showing the depth of a river at points from 0 to 100 feet from the left bank The right bank is at 100 feet. Part a of the question says to choose the type of equation to fit the data - linear, quadratic, cubic, or exponential. Part b says to get the regression equation from the calculator. Part c says to find the depth from the formula at 70 feet from the left bank. The problem says to round to three decimal places.

       The three decimal places, it seems to me, is a problem. The coefficient of the x squared term was something like -.008283. When rounding you get only one significant figure. If you keep all decimal places the curve goes very accurately through the points. If you round to three decimal places, according to instructions, then the curve is not a good fit. And then which do you use for part c, the rounded equation or the unrounded equation. My solution is basically to give full credit either way. But unfortunately that is not the end of it. The poor fit with the rounded coefficients prompted a number of students to change their choice of types of equations. A quadratic equation would fit with a common sense idea of the shape of a river bottom. But the bad fit from rounding to three decimal places prompted some students to decide that a cubic or exponential equation was a better fit. I applied the rubric, or course, but not everyone was happy with the result, least of all myself.

       What is most discouraging is that we had this same problem both semesters. I don’t recall the details, and for some reason it affected only a small number of students the first semester when it happened. What I do remember well is talking with a student in my office, a very polite, but also resourceful and persistent student. I thought he had a reasonably good case, but I didn’t have much to offer him. I had applied the rubric as required. It was an uncomfortable situation for me.

       Two points come out of this. We need to have tests set up ahead of time, tests in about 10 versions that do not change from year to year. Tests need a lot of debugging. But we don’t debug them, at least not very well. We just start from scratch each year or each semester. So of course we have a lot of trouble. The other point is the rubric idea doesn’t work that well. I get the impression that others think that we can decide on a rubric in five or ten minutes that will cover all cases and that can be applied equally by all teachers. I think it is not at all that simple, especially with the problems we give. An essay type question does not lend itself to mechanical type grading. An essay type question requires subjective grading.

       I start with the rubric that I feel is mandated by the meeting, but of course I run into any number of situations not covered by the formal rubric. In order to be consistent I make notes of how many points I gave in each situation. Sometimes I will end up with a whole page of such notes. And most importantly, I will spend a lot of time on the first set of tests. I will look at my problem on every test (about 30 of them), leaving each bluebook open to that problem. Only after I have some feeling of what to expect do I begin applying red ink.

       I like the advantages of dividing up the grading. Indeed it only makes sense if we are going to have group testing. But I have concluded that we would do a lot better not to mandate how the grading be done. A meeting to discuss each problem is good. But settling on a concrete rubric, not subject to change, is not good. We would do much better to let each teacher develop his or her own rubric. I think most teachers would do what I do, apply no red ink until after looking at a number of responses, and then decide on a rubric. One must keep a record of points given in each variation of student response in order to be fair. We need to be able to say that we graded everyones test the same. But that is not hard to do

       Group Projects

       I never warmed up to the idea of group projects in a course such as college algebra. I tried to. It was made plain at my initial interview that they were required. So I tried to make the best of them.

       People do like working together in many ways. Spontaneously formed study groups are common in any class, at least after a month or so and students get to know each other. I would estimate that probably in about 50% of the cases our group projects were a positive experience for the students. They either enjoyed working together, or felt the experience was worthwhile, or at least didn’t think of it as a bad experience. But there is a downside to group projects also. Negative experiences range anywhere from mild feelings of irritation to real trauma. For many students, perhaps half, the projects are seen as “jumping through hoops”, meaning something extraneous, something they have to do to get a grade, but not something that is intrinsicly of any value. By the contract perspective students expect to learn things and to demonstrate that learning on tests. Most anything else is seen as extraneous.

       The rationale for requiring group projects was never made too clear. One of my colleages would tell students, “When you get out into the real world in a job you’ll be working together in teams, so you should learn how to do it. So that’s why we require group projects.” This sounded ridiculous to me. I was tempted to say, “But isn’t that what we went to kindergarten for, to learn how to work together?” I don’t know if this colleague really believed this rationalization, or whether it was just rhetoric. Nor do I know to what extent other teachers of college algebra supported the idea. I know there were several who, like me, would not give a moment’s thought to having group projects if it were not required. Perhaps several of my colleagues really believed in them. I don’t know.

       Are group projects supposed to be a way of learning algebra, or are they just supposed to be applications of what we learned? In the perspective of James Kilpatrick, as I understand it, who wrote “The Project Method” in 1918, the project would be open ended. It would be a way of learning material, not just an application of what was learned in class. And more importantly, the project always would be of the learner’s choosing. It might be a project suggested by the teacher, but definitely not an assignment imposed by the teacher. In this way the project would have the student’s “whole hearted” support.7 This idealistic perspective of projects is quite the opposite of what we had in college algebra.

       Actually there was only one project required in College Algebra in the fall semester, and none in the spring semester. The fall project was data processing. It was about five or six pages long, giving information and instructions. The subject of the project was the number of websites on the internet over a period of years. The instructions told how to take this information and with the aid of the graphing calculator project website growth. It was not a bad project, I felt. The information and instructions were clear. There were plenty of details that were difficult, but diligent students could work it out, at least with a little help. It was suitable for a group of three or four students to do together.

       “Doing it together”, in my view, primarily means the smart one in the group figures it out and explains it to the others. I don’t know if this is what is envisioned by advocates of group projects. From other situations and contexts I know there is an ideal of equality in doing group projects. In a previous institution teachers who assigned group projects had each member of the group sign a statement that they each contributed equally to the result. That hardly seems realistic to me. It would be a minority of groups in which it would be true. More commonly in a group of three or four people there will be one or two who understand much more quickly than the others. They then help the others. A student not good in math could be lucky and be in a group with someone much more talented in math, and get a free ride.

       My best insight as to how group projects actually worked would be when a group of three or four students would come to my office during office hours and tell mey that are stuck solid. As I recall the rule was that students were to receive help only from members of their group, not from the teacher or anyone not in the group. This rule was bent a little, sometimes more than a little. My method, when a group of three or four students come to my office for help, would be to explain that I am not supposed to help them, but I could listen to their ideas to make sure they understood the problem. But then I would nudge their thinking this way or that to get them going. And when that failed I would just cheat and help them directly. In these situations I would be acutely aware that the situation would be predominently a conversation btween the group’s informal leader and me. The others would watch and listen intently, and might even have a question or two, but it was obvious that when the leader understood, we were done. He or she would then explain it later to the other members. “Equal contribution” by each member of the group was a a fiction. It was not a realistic expectation. Indeed in most cases it was simply not possible.

       Logistical problems were always a concern. Students have busy lives. Many of them have jobs. Many of them have children. Finding time to meet as a group therefore was often difficult. I think surely several meetings would be required to really make it a group effort. Diligent students would figure out the project themselves in advance of the first group meeting. Less diligent or less capable students would procrastinate.

       I was surprised the number of times students would come to me with a problem - they couldn’t contact their other group members. “What are their names?” I would ask. “Do you have their phone numbers?” They would reply, “Well, one was Joe, and I didn’t get the girl’s name, and no, we didn’t exchange phone numbers.” For teachers who really believe in group projects this doesn’t mean much, I suppose. For me it seemed to somehow reinforce my distaste for the whole idea.

       In Finite Math there were about three group projects each semester, and they were not always very successful. I will describe one in particular that I felt was a bad one. Probability is an important topic in finite math, and this was a probability problem. It was only about one paragraph long. I will paraphrase, but the problem is easily stated. In a game show a contestant is shown three doors and told that behind one of them is a new car. Behind each of the other two is a goat. The contestant gets to choose one door, and will receive as a prize whatever is behind that door. However there is a complication. After the contestant makes his choice, one of the other doors is opened revealing a goat. At this point the contestant is given a second choice. He can stick with his original choice of door, or he can switch. He can choose the other unopened door if he wishes, and receive what is behind that door, instead of his original choice. The question is this. Does he have a better probability of winning the new car if he stays with his original choice, or does he have a better probability of winning the new car if he switches his choice to the other unopened door.

       This problem, unlike the internet growth problem I described for college algebra, is definitely not a “follow the recipe” problem. It definitely requires thinking. In fact, I wonder if it is not the type of project Kilpatrick would envision back in 1918. Ideally it would send the students to look for resources to learn the basic ideas of probability, and then they would come up with a way to put these ideas together and come up with an answer. Of course in the original idealistic idea of group projects the teacher would only suggest this problem. Students would be free to pursue it or not. But our situation is different When this project was assigned we had finished probability. Therefore the students should be way ahead of the students in the idealistic scenario who would start out knowing nothing about probability. Therefore it would seem that it should be relatively easy.

       But it was not easy. I struggled with it myself for a while before figuring it out to my satisfaction. A reminder had come out from the finite math course supervisor that we were not to give help to students on projects. It’s like a test, not homework. As a result very few students really figured out the math. I had three sections of finite math, totaling about 75 students. One student figured it out right. I could say one group figured it out, so therefore three students figured it out. I don’t think I am being overly cynical to say that one group of three figuring it out means that one person figured it out. A few other groups had some success in getting on the right track.

       From the original idealistic perspective of group projects this would not necessarily be grounds to consider the project a failure. More important would be whether or not the project was successful in prompting learning. It would be quite possible for a project to be successful in prompting some important learning even though the actually problem was not totally solved.

       To my mind the experience with this group project points up a lot of failures. I would say that it did not prompt learning for my students. Presumably the learning was done. At best this project could be an application of that learning, and perhaps worthwhile in that way. But when so few figured it out, surely it must be more of a frustration than anything else. Remember that the “contract perspective” is very prominent in students minds. Their goal is to be successful and get a good grade. A project in which success is difficult or impossible must be viewed by most students as breaking of the contract on our part. Students are quite willing to work hard to fulfill their part of the contract, but they take it as implicit that what they are asked to do in the contract must be possible. It must yield to effort.

       I think this experience also suggests a failure of the original idea of group projects. I can well imagine this problem being used in a textbook of how to teach middle school math. I sounds like it would be about right for fifth graders. But after seeing what college students do with the problem, the original idea that an interesting problem will send students looking for resources, from which they will gain knowledge, and then apply that knowledge to the problem, seems highly unlikely.

       It might be argued that only one of my students could do this because I did a poor job in teaching probability. However I had a chance see the projects of another teacher. I was in her office to ask about her experience with the project and she showed me the projects from her classes. I looked through a number of them, and again and again they had it wrong. It didn’t take long to look at a paper and find the conclusion. Paper after paper concluded that it made no difference whether the contestant stayed with the first choice or switched. That was wrong. The probably of winning by staying with the first door is one-third (three doors, only one of which conceals the new car). The probability of winning by switching to the other door is two-thirds. (Think of two groups of doors. Group A consists of one door, the one chosen. Group B consists of the other two doors, with a two-thirds probability that one of those doors conceals the new car. The contestant can now chose between group A and group B. If he chooses group B he will obviously not choose the open door revealing the goat. Therefore the full two thirds probability is on the closed door of group B.)

       I do not care for group projects, either as a learner or a teacher. I determined to do my best with them, since they were required. But also throughout the year I also determined to look for the “smoking gun”, evidence or a line of reasoning to reject projects. I never found what I wanted, but I was able to analyze why I, and many students, dislike them. Group effort dilutes individual effort. This is not readily apparent to “groupers”8, those to whom it comes natural to want to work together. But it i painfully apparent to, though usually not verbalized by, non-groupers. Effort is diluted, and feedback is diluted. And I would also argue that focus is made more difficult. All this requires elaboration, but I am not prepared to do that at the moment.

       The Culture Of The Math Department


       I now have at least some experience with three university math departments. NDSU seems different than the other two, and different from what I would expect.

       I always felt there was a bit of a militaristic bent on the part of some members. There are several things I can mention, none of which proves anything, of course, and all of which it can be argued are irrelevant to the teaching of math. But I think there is some relevance.

       There was a hierarchy. We were divided into “faculty” and “lecturers”. I first noticed that on the NDSU Math Department website before I was hired. On the department website there is one link to “faculty” and another link to “lecturers”. Somewhat later I discovered that there is the claim that all the math faculty at NDSU have their doctorates. But how can this be? They’re advertising a position that only requires a masters degree. Well, we are not faculty. We are just lecturers. Even though we teach the freshman courses, which probably includes about half of all enrollment in math courses, we are not “faculty”. It seemed duplicitous to me. I felt that a parent reading that “all members of the math faculty have doctorate” could reasonably expect that the person who teaches math to his freshman daughter would therefore have a doctorate. For the vast majority of freshman that is not the case, for “faculty” in the math department obviously does not mean the people who teach the students. I decided to forget about it. The terminology seemed strange, but that happens all the time, and unfortunately I could easily think of similarly duplicitous statements by other state universities (When I was a freshman at U of MO in 1961 we didn’t pay “tuition”. But we had an “incidental fee” that was obviously the same thing), and similarly duplicitous statements by governments, so I can’t single out NDSU.

       Of course every institution has ranks. Every institution has a hierarchy. But the clear demarcation into an upper and a lower status, faculty and lecturers, was new and unexpected to me. Lecturers did not attend departmental meetings. Lecturers were not expected to do research (which actually was welcome, I make no pretense of being able to do genuine mathematical research). There was even a hierarchy among the lecturers. We had a senior lecturer, who apparently had considerable authority, and each course had a “supervisor”, who had some authority over the teachers of that course. I learned “through the grapevine” that at NDSU you don’t circumvent the chain of command. It just isn’t done. My contact with the chairman of the department was next to non-existent.

       At the beginning of the year there were two meetings. One was of the entire math department. It was, as I recall, really nothing more than a formality, a chance to introduce everyone. It was not a planning meeting in any way. Then there was a meeting of the lecturers. Included on the agenda were the usual items, but also at 10:00, according to the schedule, our chairman would come in with a few words, “and a warning”. I don’t remember just what the few words were, or that they included much of a warning about anything. It seemed more to be a matter of a formality. But something in the department mentality required that there be a “warning”, that the lecturers, apparently, are to be kept in their place.

       I suspect there was a rhetoric of team effort, though little was said along this line during the year. At my initial interview there was some talk about teamwork, though I do not recall just what was said.. Of course there are different ideas of what constitutes a “team”. One idea of a team is that decisions are discussed and made jointly. Another idea of a team is that the leaders lead and the followers are expected not to question. This latter idea fits the military. A squad of soldiers giving instant obedience to the orders of their leader may certainly be said to be a team. But usually, I think, when people talk “teamwork” they are talking about a much more cooperative situation, a situation in which input from all members is not only accepted, but expected. Any teamwork in the math department seemed much more of the military form.

       I made up my mind early in the year that I would keep my mouth shut and, in effect, follow orders. Even when there is a genuine culture of cooperative teamwork it is sensible for new members to defer to the old members for a period of time. During the second semester, when I was still expecting to be back the next year, I did give some thought to being less passive the next year. I didn’t know if I could make much of a difference in how things are done, but I figured after a year of experience I at least had the standing to question how things are done, and have a say now and then. Unfortunately I never got the chance.

       The attitude toward the students did not seem to me to be what it ought to have been. We were told very early in the course, weeks before the first scheduled evening test, that we did not allow make up tests. In particular we were to make it plain to students that their work schedule was not a reason to schedule a makeup test for them. We were told to explain to them, “You have to be a student first. You have to schedule your work around your schooling, not the other way around.” As a general principle maybe there is something to be said for this, but when carried too far I think this is nothing short of pig headed. These are real people we are dealing with. A lot of them are struggling hard. Many simply can’t go to school unless they work, so for them work has to come first. I think we should try to accommodate them. “What happens” I asked my course supervisor before the first test, “if a student says flat out, ‘I’ll lose my job if I come to the test instead of going to work. Then I can’t go to school.’ What should we tell them? Tough luck? . . . .?” My supervisor did not want to hear this. The attitude in the math department seemed to be that we have made the arrangements. We have made the rules. There shouldn’t be any problem.

       Test averages often came out low, sometimes even failing. A cutoff line of 60% for failing actually means 60 out of 110 possible points, which would be 54.5%. And for many students the 20 points from the take home group problems represented another unearned bonus. Common sense and simple decency would indicate to me that when the average is that low something is wrong. But the attitude of the department was that it is up to us to set standards and up to the students to meet them. Once we had set standards that was it. There was to be no talk of lowering the curve. One time in the spring semester a student confronted one of our lecturers bluntly, saying “If the average comes out that low, doesn’t it tell you something?”. The lecturer, quite understandably, turned it back on the student. “It should tell you that students need to study more, not that there’s anything wrong with the test.” However this same attitude was taken when discussing the situation with colleagues, which surprised me. I might feel obliged to use about the same rhetoric with a student. But when the student has gone and I am discussing the situation with colleagues, my attitude would be quite different, and quite clear - of course it says something about the test, or how we teach, or what we teach, or something. It is a situation of great concern to me.

       In my previous teaching experience occasionally a test average would come out particularly low, somewhere in the D range. I would tell the class that I was not going to do anything about curving the grades at the moment, but I always reserve the right to curve a few points down at the end of the semester if it seems needed to be fair. But I would also say that in my past experience I have very seldom departed from the 90-80-70-60 curve, and when I did it was only by a very few points.

       This attitude, that it is reasonable and legitimate to adjust the curve at times, just seems like common sense to me. Teacher made tests are not standardized. They cannot be standardized. The are made with a lot of subjective judgments, and they are graded with a lot of subjective judgments. To stubbornly insist that no adjustments are to be considered, ever, does not seem realistic or educationally sound. Students also expect that sometimes a curve will be adjusted. A number of times in both semesters students would come to me and ask, “But what happens if the class average is way down at the end of the semester?” I would be perfectly honest about it. I would tell them, “At NDSU teachers do not have the authority to lower the curve on their own. I am told that there will be no departures from the 90-80-70-60 curve. That’s all I can tell you. That’s all I know.” And that was the simple truth. Of course it leaves open the possibility that at the end of the semester we would then be allowed to adjust the curve. I did wonder during the first semester if there wouldn’t be some adjustment allowed at the end of the semester, but there wasn’t.

       The math department might argue that my sections had lower scores than most other sections. That is true, and was heavy on my mind the whole year. I cannot claim to have been very successful in the situation I was in. But it was also mentioned to me by a sympathetic colleague that the more experienced teachers know how to move a substantial number of the less able students out of the course before the first test, thereby raising the class average. Obviously this might be the case, and obviously it would be impossible to prove one way or the other.

       I do not know what the statistics are about the per cent of students who successfully complete the course. I have heard some numbers, something like 70% of students are successful, perhaps. But the numbers mean little without some definitions. Should “successful completion of the course” mean a grade of anything but F, or perhaps anything but D and F? There are some students who enroll, but frivolously drop after a class period or so. Should they be included in the calculations? There are other students who persist longer before quitting in frustration, perhaps waiting until after the first, or even second, test. Should they be included? I would certainly argue that to say something like“Seventy per cent of final grades are A’s, B’s, or C’s” doesn’t mean much. Many drop outs, in my opinion, should not have been drop outs. They are not all comparable to the drop outs from college algebra in the other two state universities in which I taught.

       I don’t have the numbers or the definitions. What I do have are memories and feelings about many students who I felt got a raw deal. My judgment, admittedly subjective, is what I stated early in this article. Things are different here, and things are not good. Many students who would find college algebra a routine experience at a similar university are blindsided by college algebra at NDSU. We give an inferior product at a high price.

       I have dwelled on the attitudes of the math department to make the point that though we adopt some of the ideas and ideals of math teaching, we put them in a context that is not at all like the context envisioned when these ideas and ideals developed. The project rationale and the “activity method” goes back at least to the early twentieth century. But the idea then was that projects are to be of the student’s choosing, and are to be open ended. Assessment, when it was mentioned at all, was also to be open ended. None of this fits the “contract perspective”. In other words, we may have stumbled on the worst possible combination of the new and the old.

       Premises, With Some Criticism

       Premise # 1 Problem based learning (PBL) will be used.

       This doesn’t sound like a bad premise, but I think the results were not good. And in comparison to the premise “The nature of the subject will be the organizing theme” it is not at all good.

       Premise #2: We will not teach procedures.

       I first realized this when discussing completing the square with a colleague, but it seems to be a general rule. “We don’t teach completing the square” said my colleage, but didn’t say why. After thinking about it for a day or so I concluded that the premise is that we don’t teach procedures. Anything that can be interpreted as a procedure is beneath our dignity. I agree that completing the square can be thought of as a procedure, but it can also be thought of as a mathematical idea of some importance, relevance, and even aethestic appeal. If this premise were translated to a refusal to give rote recipes, it would be good, perhaps. But if translated to denigrating anything that can be interpreted as a procedure, then it is definitely not good. Many things that can be labeled a procedure can also be labeled as a train of thought, a rationale, or even a strategy. Which label best characterizes completing the square? I don’t know. It doesn’t matter. There are many procedures that merit teaching.

       Premise #3: Technology will be used and emphasized. Anything that can be done with a calculator need not be done algebraically.

       This has proven to be a very bad premise, in my opinion, at least in our case. I conclude from my year at NDSU that we should not emphasize technology. I think we should emphasize ideas. Our emphasis on the graphing calculator does not add to the teaching of ideas. We would have done a lot better had we forgotten the calculator and just taught algebra. In order to teach calculator use, we neglected algebra.

       Premise #4: Collaborative learning will be stressed.

       Group projects are not the only way to collaborate. I think this was a net detriment to the course.

       Premise #5 :Our standards are not flexible.

       One might consider “flexible standards” to be an oxymoron. But it seems to me that we are too inflexible in many ways. It is one thing to be inflexible on standards that haven proven their value. It is quite another to decide that the task of the moment is a such a standard, to decide that the test we just wrote out is such a standard, perfect in every way, and not subject to discussion, or to decide that the grading rubric we discussed for ten minutes is such a standard.

       Premise #6. Details are not important. We don’t have to worry about them.

       This may be the most important one of all. And of course it is just my opinion. Others might have quite different opinions. I have tried to analyze the book in terms of ideas, premises, and rationales. But I think a lot of the book, its lack of coherence and the large number of errors, is best explained as simply a rush to get it in print. I understand most of the book is a compilation from another source. I don’t know why. A denigration of details is not new in educational ideas. There seems to always be a search for some magic key that will unlock the learner’s natural development and we can just stand back and let learning happen. But in general education demands a lot of attention to details. We must concentrate on one very small topic at a time, get it right, and then build on it by adding the next small topic. This is not to say that the big picture can be neglected, of course. But the small picture can not be neglected either.

       Each of these premises have some merit, and some problems. As in most any area of life there can be bad combinations of good things. Premise 4, about collaborative learning, and premise 5, about inflexible standards, are like pickles and ice cream. Students know about inflexible standards. That’s built into their contract perspective. So why do we waste time with group projects? Premise 3, about technology, and premise 6, about details, are another bad combination. Using the graphing calculator requires mastering a lot of details. Presumably the book gives these details, but not too coherently. But we don’t have a troubleshooting guide for the graphing calculators, and we don’t really teach for understanding. Students are left trying to scramble for details that are not held together by a coherent big picture.

       NDSU math, it seems to me, cannot be fairly accused of being NCTM math. But the fingerprints are there. Most obvious are the group projects, the graphing calculator emphasis, and the PBL orientation.


       1. Click here for Chicago Math.

       2. Click here for a short description of the math wars and a few links of interest.

       3. Click here, for ="Some Disagreements With The Standards" which develop this argument that the Standards don’t recognize the importance of practice. See my main page,, for a bit of background on this article.

       4. The “concentration mode” and the “spread and relate” mode will be more fully developed in Chapter 17 of my book, but I do not yet have it posted on my website.

       5. For a discussion of the need for lower level prompts as well as higher level prompts click here for Chapter 15.

       6. Linearity of tests will be in Chapter 16 of my book, which I do not yet have posted on my website.

       7. My information comes from the book “Left Back” by Diane Ravitch, pages 178-180. I have not read the original 1918 article by Kilpatrick. (This as a wonderful book, by the way, but I never quite understood the title. I think “Bad Advice From Teachers’ College” would be a better title for it.)

       8. Click here for's-do.htm “Let’s Do It Together”, for a lot more thoughts along this line.