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Fractions My Algebra Students Can't Do.

Brian D. Rude

(Revised) 2008

       I have not taught college algebra all my life, but I've got a few years experience in it, starting when I entered graduate school in math in 1998. Since that time I have become aware that I could not take a fluent knowledge of fractions for granted in college freshmen. When explaining algebraic fractions I used to tell students that algebraic fractions are the same as the fractions they learned in elementary school, just with some complications caused by manipulating algebraic symbols. This didn’t seem to make much impression, and over time I began to wonder if the students really knew fractions. So finally this year, 2008, I decided I would start the semester with a fractions quiz in my two lower level algebra classes. (This level is not supposed to be remedial. Students are given regular college credit. However the next level class is the regular college algebra that is the required math class for most majors.) On the very first day of the semester last January I gave this quiz in the last ten minutes of the class period. I am writing this in September 08, so I have the results of the fall semester as well as the spring semester.

       I told the students that we could not spend class time on fractions, but if they are deficient they should try to catch up. They should try to learn how to do fractions, and I would help them if they needed help. This worked well with another decision I made this semester, to make all quizzes retakable. So a student who doesn’t know fractions could study and retake the fractions quiz (different version, same type problems) until they get them figured out and raise their score. The fractions quiz consisted of ten problems, and a loose time limit of ten minutes. The average score in one class this spring semester was a little over three out of the ten. In the other class the average score was a little under three out of ten. That is abysmal. I expected to scores would be low, but I hadn't really expected that low. The results in the fall semester with two more sections of the same class were the same, an average score of a little over three out of ten.

       Quizzes are a regular part of my teaching. I try to give at least one, and preferably two, quizzes each week to each class, the last ten minutes of the hour. I have gotten into the habit now of making version 2 of a quiz at the same time I make the original. All it takes is changing some numbers on the original quiz, and then shuffling the order the problems around a little. For this fractions quiz I have eight versions now. I did spend a considerable amount of time before the start of the spring semester deciding just what I wanted on the fractions quiz, and how to make sure what I wanted was on every version of it. When a student needs version 2 of this fractions quiz I just go to a folder and pull out a copy. For other retakes I usually have to bring up the quiz on my computer and print it out, which just takes a minute.

       All quizzes I give are worth ten points. Since there are ten problems then each problem is all or nothing in the grading. I don't give any partial credit. This is not an ideal system. There were some points missed because the student did everything right, except to reduce the result to lowest terms. So it is very possible that some of the scores are misleading. And a colleague pointed out that pulling this on students totally unexpectedly on the very first day of class may not be conducive to getting an accurate assessment. Granting all of this, I feel I am still left with the inescapable conclusion that these students don't have the mathematical knowledge/skills/abilities that we ought to be able to take for granted. These are high school graduates. This is the kind of results I would expect of high school drop outs, not college freshmen.

       "College freshmen" might be a little misleading here. I would estimate that no more than about half of the students come to my algebra classes right out of high school. There are many older students. Many have children and jobs. A fair number of them have grown children. I don't think I have any senior citizens among them, but apparently students in their fifties are not too rare. So the mathematical background these students are drawing on must go back for a couple of decades at least. I don't know if this is relevant to the issue or not.

       Here is the original version of the fractions quiz I gave on the first day of the semester in this course.


       I put this in as a graphic, not text, so it should come out just as it originally appeared. I normally put quizzes on a half page, 5 1/2 inches by 8 1/2 inches, just as it appears here. This format makes for a little crowding on the paper, but that does not seem to have been a problem.

       We might guess that more students would have made good scores if the quiz were not presented abruptly on the first day of class. However that hypothesis does not seem to be supported by any evidence. When watching students retake this quiz I often get the impression that they have never developed any fluency in doing fractions. Rather each problem involving fractions is something new to figure out. This fits with the NCTM perspective, of course, and I have strong disagreements with NCTM’s way of doing lots of things in math. The NCTM is the National Council Of Teachers Of Mathematics. This organization has come up with some definite ideas about the teaching and learning of math. I do not agree with much of what they advocate. I will discuss this more as I go along.

       I don't recall any student doing really badly on the original quiz and then making a high score on the first retake. The really conscientious students usually take more than one retake to get a perfect score, or even an imperfect score caused by a simple error that is not a result of a lack of understanding. This would seem to indicate that their knowledge of fractions really is deficient. It seems to take some serious effort to improve, not just a quick review. And often in checking retakes I do some careful explaining. I draw diagrams like when explaining fractions in elementary school. Their deficits in fractions seem very substantial. It doesn't seem to be a matter of being a little rusty.

       Actually this fall two people, out of approximately 45, made 10 out of 10 on the original quiz. As I recall no one did that in January.

       In both the spring and fall semester a number of students have come in to retake this quiz, though not nearly as many as should. As I normally grade these quizzes immediately when they are done, and in the student's presence, I have been getting some feedback on what’s going on in their minds.

       One type of problem in particular interests me, exemplified by 6 ¾ - 2 3/8 (corresponding to number 9 above, addition or subtraction of mixed numbers with or without borrowing and unlike denominators). I expect them to convert ¾ to 6/8 and then subtract the 3/8. Almost invariably that is not the way they do it. They change 6 ¾ to 27/4, change the 2 3/8 to 19/8, and then, (at least the ones who can actually do the problem) get a common denominator and proceed. I ask them if that is the way they were taught. Usually they say they can’t remember, it was a long time ago. Then I show them that they can leave the whole numbers alone, just change ¾ to 6/8, subtract the 3/8, and then deal with the whole numbers. They seem to make sense of this, but don’t necessarily think of it as a better way.

       This seems a hard way to do this type of problem, but I am glad when they can do it by any means. They do get a common denominator by this method. A fair number of students simply cannot add or subtract fractions with unlike denominators. Some will simply add numerators and then add denominators, apparently knowing nothing else to do.

       By the time students come in for retakes most of them have figured out multiplication of fractions, and perhaps division. Often they know that division is just like multiplication but you do something more, but they are not always sure just what that something is.

       But, we might ask, is all this really so bad? It's only calculation that is involved, is it not? And we have calculators now to do the routine calculations. So maybe we don't need to know fractions any more. Or do we?

       I would argue in the strongest terms that it is not just calculation that is involved. Understanding is involved. From thinking about fractions and algebra and my students over the past year I have concluded that a thorough and fluent knowledge of fractions is a necessary foundation for one to learn algebra. Being weak in fractions is a serious detriment to learning algebra. I will give a few examples. I'm not claiming that these examples are adequate to prove my case, but they are at least suggestive.

       People make mistakes. It is sensible to try to analyze mistakes, but it is not easy to know just what can be concluded from them. I make lots of mistakes myself. If you could look over all the mathematics I have put on paper in the last ten years, you would probably find that a time or two I added two and two and got five. But it would be a mistake to conclude from that that I can not add whole numbers on the level of what we would expect from second graders. But examples provide some food for thought, so I will give a few examples of student mistakes that I think support my case. More specifically I will try to show from these few examples that fractions computation is inextricably entwined with algebra, that it is not sensible to think we can separate routine computation from an understanding of algebraic ideas. The handwriting in these examples is my own. I do scan and save interesting examples from the work of students, but in the interests of protecting privacy I will not use scans of actual student work. But I will copy actual student work.

       What are we to make of this example?



       This is from the first hourly test in the course in which I gave the fractions quiz. This student set the equation up accurately, x + (x/2 + 14) = 104. The rest should be easy. Combine the x's, move the 14, divide, and there's your answer. But apparently there is confusion. Does this mean that this student cannot simply add x + x/2 and get 3/2 x? Isn't that adding of fractions? Shouldn't it be easy and automatic? I think it should. We teach combining of like terms rather early in algebra. We explain that 3x + 4x equals 7x just like three apples and four apples make seven apples. However, we explain, 3x and 4y are not like terms. If you add three apples and four oranges then you don't have seven apples and you don't have seven oranges. Or we might say, if you add three five dollar bills and four twenty dollar bills you don't have either seven five dollar bills or seven twenty dollar bills. This is all understandable to students, I think. Do we therefore assume that they understand that x and 1/2 x make 3/2 x? Should we assume this? I always assumed we should. I did not make up the above example. As I say, the handwriting here is my own, but I am copying from a scan of the student's original test paper.

       It is apparent in the above example that the student attempted to clear the equation of fractions, which is a good idea. But why didn't he succeed? We cannot read his mind, of course, but we might certainly guess that a deficiency in fractions contributed to his lack of success. Rather than simply doing the computation, it looks to me that the student was unsure of what to do, and simply abandoned the problem with a guess and a hope.

       But now let us analyze this situation from the perspective that knowing fractions is only important in calculation. Then how are students to combine like terms in this example? Should they use a calculator to add 1 and 1/2? That doesn't seem sensible. Should they just know from their general sense of number and 1 plus 1/2 = 3/2. It's not that uncommon to have situations in everyday life that involve such numbers. If a person thinks he can paint a room in an hour, but it takes him an hour and a half, does his number sense handle these quantities without trouble? I would assume so. But does the fraction 3/2 make sense to that person?

       Maybe we should say that the answer to x + x/2 should come from a general number sense, but that the answer to 3/8 x + 2/3 x should come by converting to decimals and then using a calculator. Then we might ask, should the students understand how fractions can be converted to decimals? Can we just give them a recipe for doing this? Actually some calculators handle fractions directly, so can we argue that they don't even have to know how to convert a fraction to a decimal? But wouldn't it be much better if we gave them an understanding of both fractions and decimals?

       I am not sure of the answers to all these questions. But I do always come back to the conclusion that the ideal solution is for everyone to know fractions (and all of arithmetic) well. Then these questions do not come up.

       Now consider this example. This was from another student on the same problem.


       This student obviously had no trouble combining like terms and getting 1 1/2 x. However she did not divide correctly. 90 divided by 1 1/2 is 60, not 135. How did the student get this? My guess is that it came from her general number sense. So we might ask if she used her general number sense to combine x and 1/2 x. Of course if one knows fractions well than it is a moot point to ask whether one got 1 1/2 x by getting a common denominator, or by just knowing that obviously x + 1/2 x = 1 1/2 x.

       Does this student know that 1 1/2 x is the same as 3/2 x? Before this year it would not occur to me to ask this question. I would just assume that any student in college algebra knows that. Now the question seems very sensible.

       Here is an example unrelated to the previous ones.


       Most of the actual problem involved here is not shown. It takes several paragraphs to state. But the immediate task is to find the equation of the line that goes through the two points (0, 0) and (.2, 12). So first we find the slope, using the slope formula m = (y2 - y1) / (x2 - x1), which this student has done. However he makes a mistake in simplifying 12/.2. He gets .6, but it should be 60. How did he do this, and how should he have done it?

       I would prefer that he do it in his head. He could think 12 divided by 2 is 6, but .2 is small so the answer is 60. Or he could think 12 divided by 1/5 is the same as 12 times 5, which is 60. But he could also punch it in the calculator. Did he do that? We can only guess, of course. Maybe he punched it in wrong. Maybe he did it in his head but made a simple mistake in some way. My best guess is that he did it in his head, and thought he was doing it right. We should also consider his lack of willingness to check it. My point is that a high degree of fluency in dealing with arithmetic would help him a lot in algebra, and my guess is that he does not have it. Of course it is not just fractions that are involved in this example. But if he is weak in fractions we might guess that he is also weak in decimals.

       Or possibly I am all wrong. Possibly he is a smart enough student, and knows arithmetic well, but just doesn't care, at least not at the moment. We could take the view that fractions and decimals are just computation, and he should have his calculator handy, and he should have used it, instead of being lazy and guessing. So maybe this is just one example among many that I come across about every day. It is suggestive, but proves nothing. In the past year I have become very sensitized to mistakes that indicate a possible lack of knowledge of arithmetic, fractions in particular. I find plenty of them, but they usually do not prove anything.

       Here is another example, unrelated to previous ones. In a problem of finding slope, the rise is computed to be 1/4, and the run is computed to be 1/2. Therefore the slope, rise over run, is 1/8, according to one student's paper. Of course that is wrong. 1/4 divided by 1/2 is 1/2. This particular student, I think, didn't really think about it. His number sense, perhaps, told him the answer was 1/8 so he put it down and turned his mind to something else. How should this problem be handled? Is it a calculator problem? Obviously it could be. Or it could be a visual problem. If you visualize a triangle with a rise of 1/4 and a run of 1/2, then the slope is obviously 1/2, if indeed you can easily visualize that, and if it is obvious that 1/4 is half of 1/2. But however one does it, rightly or wrongly, one should have a technical and fluent ability to do fractions as a back up to any other short cut one might have available.

       Remember that the big question that I am trying to address here is whether or not a deficit in fractions can be a serious detriment to learning algebra. The examples above obviously do not definitely answer this question. However these questions certainly suggest that a detailed technical knowledge of arithmetic would aid in the study of algebra

       Here is a situation that I think better makes the case that it's not just computation that we are concerned with, but understanding. In our upper level college algebra course an important topic is transformation of functions. If you know what the graph of y = x2 looks like, you know what the graph of y = x2 + 2 looks like. It's just the same shape lifted vertically two units. The graph of y = (x - 2)2 is the same shape moved horizontally to the right two units, and so on. This is an important idea. By completing the square we can take a quadratic function like y = x2 + 2x + 10 and change it into y = (x + 1)2 + 9, which is the basic parabola of y = x2 moved one unit to the left and nine units up. These calculations do not require the use of any fractions.

       But now consider this function . The graph of this function can be sketched very easily if we can change it into , which, if we are familiar with the basic functions is simply a vertical shift, an inversion, and a stretch factor applied to the basic function . But how are students to understand this? If they are fluent with fractions we can just show this sequence of steps . But how do you explain that if the student doesn't understand fractions?

       There are two general possibilities here. First, you can explain it in terms of fractions, and if they students don't know fractions they must learn them now. The other possibility is that you can come up with some explanation that doesn't directly use fractions. Perhaps you could simply say that it's just a rule that you can split a numerator, and put each part of the numerator over the denominator. But should that be a ad hoc rule, a rule made up on the spot, which students put in their notes and try to remember without understanding? Surely it would be much better if the students understood it in a much broader sense, as a part of the general topic of fractions which is thoroughly known and understood before they ever get to algebra.

       Could we somehow understand the above sequence of steps as simple computation, to be done with a calculator? I don't see how. Could we somehow say that a student understands algebra if they cannot follow this simple series of algebraic steps? If this is not algebra, what is?

       I keep coming back to the basic conclusion that an understanding of fractions is absolutely necessary for an understanding of algebra. Fractions are not just a matter of computation, that can be separated from an understanding of algebraic ideas. A knowledge of fractions is a knowledge of the real number system. This is not to say that the relatively concrete understanding of fractions that an eighth grader has is equivalent to a more sophisticated knowledge of the real numbers that a mathematician has, but would anyone argue that that more sophisticated knowledge could come by some means other than through the usual sequence of learning arithmetic thoroughly before studying algebra? Would anyone argue that we can learn calculus without going through algebra?

       I first presented several actual student examples that show that fraction problems do arise in doing algebra. Then I gave an example that I claim illustrates that an understanding of fractions is necessary for an understanding of algebra. This is not a very complete argument, but I think will be convincing to many math teachers because it reflects their own experience.

       But if fractions are so important to understanding algebra, and if many students do not understand fractions, how do they ever pass algebra? Or do they?

       They do. I think they do it all the time. However I think they usually struggle to get a grade of C. Surely they are not the A students. But how can they even scrape together a C?

       Lets go back to the first two examples of student work that I gave earlier. I did a survey on this particular test problem. I had 19 tests to grade. Out of those 19 tests there were three students who definitely showed evidence of being derailed by a lack of understanding of fractions. Those are the examples I gave. Two of them did not adequately add x and 1/2 x, and the third obviously did not divide correctly by 1 1/2. Seven students out of nineteen showed evidence of correctly both adding and dividing the fractions in this problem. They either got the problem entirely right, or at least did not make any serious mistakes that came from a lack of understanding. Four students avoided the problem by multiplying through by two and thus clearing the equation of fractions (which is a good thing to do). Five students failed to get far enough in the problem to do anything with fractions. So what does this show?

       I would argue that it shows that a substantial deficit can be camouflaged I think this is very important. Of the 19 students whose work I analyzed on that one problem, only three show somewhat clear evidence of a problem with fractions, and even that evidence is equivocal. But if we are believe the results of the fractions quiz, we would expect many more than three to have problems with fractions. Should we expect a deficit in fractions to totally prevent anyone from being successful on this problem? Should we expect that only a deficit with fractions would prevent success with this problem? And is this problem, and what it tells us about knowledge of fractions, typical of algebra problems in general?

       I'm not sure I can answer all these questions, but my hypothesis is that unless we know how to look very closely, a lot of deficits can go undetected. They are hidden in some way. A deficit, such as not understanding fractions, can do serious harm to a student's progress in a course, but does not necessarily doom the student to failure. Students can still learn how to do a lot of problems. They still can learn a fair amount of mathematical ideas. Perhaps most importantly they can learn a lot of recipes that enable them to do problems and pass tests.

       A knowledge of fractions, like so much in math, is not a link in a chain so much as a part of a net. A net can be broken in a place or two but still hold. In many situations a deficit in one area can be compensated for in some way. Students still manage to do problems. They don't do problems perfectly, but we do not expect them to. Thus a serious deficit can be camouflaged. We don't notice it because we don't suspect it. We are not looking for it, because we have no idea that we should.

       This camouflage effect could extend to the people in my sample who handled the fractions correctly. Suppose a person has a good general sense of numbers. If 1 1/2 of a number is 90, maybe common sense, or a number sense, tells you that the number is 60. Those seven people who got the whole problem right could still include some who don't really know how to do fractions. Those three who multiplied through by 2 could include some who realize they can't deal with fractions effectively. And of course those five who failed to get far enough in the problem to get to any need of fractions problem probably include a fair number who can't do fractions.

       So I think that it is indeed possible for people who have a poor grasp of fractions to somehow scrape through a typical college algebra course with a grade of C. Their lack of understanding is a detriment, indeed a serious detriment, but many detriments can be overcome, at least to some degree. There may be many ways to compensate. The students, understandably, will use whatever they can to get a problem right. They may not think at all about their ability with fractions. They may think no deeper than that math is hard, but they usually manage to get by some way.

       The parallel with illiteracy comes to mind. We are told by people who struggle with a serious inability to read that there are many ways they cover it up. We are told that by the people who try to help these people. How can a person be successful as a salesman, a businessman, even a teacher, and yet not be able to read? It sounds incredible, but we are told that it happens. I don't know how. But if it does happen it seems reasonable that probably a lot of people get through college algebra with a lot less understanding than we infer by grading their tests.

       However there may be an important difference. People who cannot read, we are told, are painfully aware of their disability. Is that the same with students who don't understand fractions? I would assume not. I would assume that most people who have a lot of trouble with math are not the type to analyze carefully just what their trouble is. Rather, I would think, they know that math is generally hard and frustrating, without knowing particularly why. Much of this remains a mystery to me. It's hard to imagine not being able to read, and it's hard to imagine not being able to do simple arithmetic.

       I think that a disability in fractions does not show up in college algebra very well for many reasons, not the least of which is because we are not looking for it. We are looking for algebra. On any algebra test there will be a lot of missed problems. We may have a very limited understanding of why these problems are missed. As a teacher I explain a lot. I explain in class, and students, at least many of them, nod at the appropriate times. I get the feeling they are following what I am saying. Then I grade homework. I put a lot of red ink on their papers, hoping to help students understand their mistakes. But that is limited. Writing is slow and laborious. Sometimes I will do a part of an assignment on paper, scan it, and post it on D2L, which is our online course management system. Students can download what I post and hopefully benefit from it. And I help students individually. They come to my office and ask for help. I explain as best I can, and usually the student seems to understand at least enough to make some progress. But then I give a test, and there are a lot of mistakes. I give partial credit when I can, as most teachers do. But after all is said and done, in spite of my best efforts, a lot of mistakes are made on a lot of problems and I don't understand why. I can say they don't study enough, and there's a lot of truth to that. I can say a lot of people just don't have much ability in math, and there's a lot of truth in that. But to the basic question, why after all I've done and the student has done, was the student unable to do such and such a problem, I don't know.

       But amidst a lot of mistakes on a test there appears to be a lot of knowledge also. At the end of the day and after all is said and done I am comfortable with the basic judgment that when students get a 70% test average in my course they deserve a grade of C and three hours of college credit. I hope that I have given them something more than just a grade. I hope every student takes away from my class something of intrinsic value, an appreciation of some sort or another. I know this happens for some students, but certainly not for all.

       So what can we do to improve things? I think my fractions quiz is a start. It forces a recognition of the problem, at least to some extent. And by offering retakes I give conscientious students a method of at least partially remediating that problem.

       Should every math course include some attempt to remediate the deficits that students bring to that course? With my repeatable quizzes, and my repeated invitations throughout the semester to students to take advantage of them, I am making an effort, limited though it may be, to remediate their deficits in fractions. But what about other deficits, including deficits that I may have never even thought about?

       I think the answer is clear, at least on the college level. We do not have a responsibility to provide remediation for every possible deficit a student might bring to a math course. That is the student's responsibility. It has to be the student's responsibility. I think it is also clear that the answer to this question at the lower levels of math is different. Elementary and high schools do have at least some responsibility to try to remediate all deficits as long as students are still in school. But no school can promise to do everything. I don't know just how responsibilities of remediation should be defined and allocated.

       But, at the college level, I think it is sensible to take some responsibility of providing avenues of remediation to those students who need help and are willing to work. My system of repeatable quizzes seems worthwhile to me. It takes a little time and effort, to be sure, but not too much.

       But there is a bigger issue. Surely we have to ask why there should be such a lack of understanding of fractions. Two possibilities come to mind. Possibility one is that we are doing something wrong in the teaching of fractions. Possibility two is that our teaching is okay, but the human species is just not that mathematically inclined. Perhaps under the best of circumstances only a minority of people can actually gain a useful understanding of fractions.

       Obviously I don't like the pessimism of this second possibility. I prefer to think that probably we are doing something wrong. Of course I have been saying that for a long time. I think the perspective of math offered by the NCTM in the 1989 and 2000 standards are not well thought out. I think this NCTM perspective has to be doing substantial damage.

       Here is a picture that comes to mind. In a fifth grade classroom the teacher starts an arithmetic lesson by sending about eight, possibly ten students up to the blackboard, depending on how much blackboard space there is in the room. For about ten minutes she calls out problems, and the students race to complete the problems, either at the blackboard or at their desks. Then the teachers rotates the group, sending the first group to their desks and bringing a new group to the blackboard. This goes on for some period of time, perhaps a half hour or so. This scenario, I think, was a regular part of my elementary education back in the 1950's. Actually my memory is not very good. I can't remember if this sort of thing would happen about every day, or maybe just a few times in a semester. I took it for granted when I was young.

       But I wonder if it happens now days. I mentioned that I have many disagreements with the ideals being advocated by the NCTM in the last few decades. Of course I must admit that I don't know too much about what they have been doing for the past 50 years. I have not been a teacher all of my life, and even when I was I didn't always keep up with educational trends. But I do know what the followers of NCTM ideas would think of the scenario I have described. The would call it "drill and kill". They would call it "soul deadening" or "mind numbing", or whatever terms of opprobrium they could come up with. But I don't remember it that way. In my memory "going to the board" was just one more routine part of everyday life, just as drilling on the addition flash cards was a routine part of everyday life in second or third grade.

       In about 2003 I took a long look at the "Standards" published by the NCTM in 2000. I decided I had some very serious disagreements, which I wrote up and put on my website. Here's a link.

       So what goes on in elementary and middle school math classes now days? Is it pretty much like what I had when I was young, including a certain amount of drill on fractions at the blackboard. I would certainly hope so. Some would disagree with my view. They would think it should not be the same now as then. But what about football? When I was young boys drilled hard in the football training season. (I didn't, to be sure. I had no interest in athletics.) Do they drill hard today? Should it be any different now? Did piano students drill when I was young? Of course they did. Should it be any different now? Did conscientious college students in the 1960's cram before tests? Of course they did, at least the ones who made a success of college. I certainly did. Should it be any different today? Practice and drill, of one sort or another, is a necessary part of many things that are well worthwhile doing.

       I cannot help but think that probably the unrealistic ideals of the NCTM (and progressive education in general) have made unfortunate inroads into educational practice. I hope I'm wrong.

       I will end this article with the idea of the "stagnant spiral syndrome", which I think describes what is happening in many cases. I realized in recent years that many of the topics that I teach in college algebra are supposed to be taught early in high school, or even junior high. This was dramatically brought home to me when browsing the internet one day. I had just covered slope my lower level algebra course, and felt it had gone pretty well. Slope usually does. I think I had chanced on the California math standards on the internet, or maybe it was the NCTM standards. Anyway I noticed that slope was included in the seventh grade standards. What happened? Why am I teaching in college a concept that presumably should come much earlier? Were my students exposed to the idea of slope in the seventh grade, or eighth grade? Did it do any good?

       I think one thing that is going on is a stagnant spiral syndrome. The "spiral" here refers to the "spiral method", by which students return to previous topics, perhaps on a yearly basis, but take a deeper cut into the subject matter on each spiral. This idea, the "spiral method", has been in and out of favor over the years. My view is that spiraling is inevitable in many subjects, just by the nature of the subject matter itself. I will use grammar in the elementary grades as an example of both how spiraling should work and how it shouldn't. It has a personal relevance to me, because I have given some thought to it over the years. I remember getting the same basic grammar year after year in elementary school. We would start with nouns, and I would remember that we learned that last year. Then we'd study verbs, and I would remember that we studied that last year. Then we'd go on to adjectives. I remember being a bit frustrated after a few years. My thinking on this crystallized when I finally hit eighth grade and had a really good grammar teacher. This, I thought, is the way grammar should have been taught every year. We learned a lot, and it was satisfying.

       I suppose grammar should be taught every year in elementary school, and it's probably sensible to start each year with a review of last year's work. But if we are not careful we might not go beyond last years work. We may not make a deeper cut on each successive spiral, and that can be frustrating. At least it frustrated me in elementary school.

       I did not use the term "stagnant spiral" until a year ago. I was teaching the graphing of rational functions in the upper level algebra course. On this particular day students seemed especially daunted by the subject. The subject was hard to understand for them, to be sure, but something about their reaction seemed to border on anxiety. Students often listen intently when I'm explaining a topic in algebra, and they were listening intently on this day. But this seemed more than the usual concentration. What was going on?

       After thinking about this particular class for a day or so I formed an opinion about what was going on. My analysis was that most everything in the course up to that point was not new to the students. That does not mean that the students knew everything up to that point. Obviously they did not. That was painfully evident when I would give a test. But there are lots of little clues that the students have been exposed to every topic we covered sometime before in their mathematical education. That gives them some sense of comfort. For every topic students can think, "This is not new. I've done it before. It'll be okay." When a student has some reason, good or bad, to skip a class, this sense of comfort can be drawn on. It is a false sense of comfort, of course, but understandable. But apparently graphing of rational functions was not something they had encountered before, and it caused them some actual anxiety.

       I conclude from this example, and many other little examples too numerous to mention, or even recall, that to one extent or another most of my students must be caught up in a stagnant spiral syndrome. Every topic we cover, through much of the first half of the course anyway, is not new to them. Therefore they should know it. But they don't. So therefore they should learn it now. Some do, of course, but for a substantial number of students, that is not how it works. They learned it a little bit the first time it was presented years ago. They learned it a little bit in several other math course before they got to mine. But some how they never learned it they way they should. And when I give a test I conclude that many of them have not learned it the way they should in my course either. They have spiraled through much the same topics year after year, apparently, but with little progress.

       What could account for a stagnant spiral syndrome? When I was a kid complaining about grammar being boring, my mother would say something to the effect that we have to learn the same topics every year because apparently the kids don't know it. That did not satisfy me then, and it doesn't satisfy me now.

       I suppose a stagnant spiral can happen for many reasons. I don't know why it happened in grammar in elementary school. I think it has happened in math at least partly because the "fuzzy math" of recent decades doesn't work very well. By “fuzzy math” I am referring to math instruction that tries to follow the recommendations of the NCTM. This type of instruction, I strongly suspect, tends to be inefficient and ineffective, so kids never do learn fractions, among other things, when they should, or as well as they should. My hypothesis is that fractions are often poorly taught, which makes it hard to teach decimals well, which makes it hard to teach per cents well, which makes it hard to teach a lot of applications of arithmetic, which means arithmetic is seldom understood very well by most students, which makes a poor foundation for algebra, which means a lot of algebra must be repeated year after year, which means that I end up teaching in college algebra simple things like equation solving and the meaning of slope which presumably were covered years back.

       At least that's the way it looks to me now.