Click here for home page, brianrude.com

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 = (y