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On 10/3/08 I posted a revised and much extended version of this article. This includes the results of the fractions quiz for my two fall classes, but mostly I discuss the results at considerably greater length. I try to make the case that a knowledge of fractions is essential for understanding algebraic principles.Check my home page for other articles about math education.
Click here for revised version of this article
Fractions My Algebra Students Can't Do.
Brian D. Rude
This is a very quickly written article, written in response to a particular blog. Here's a link. http://joannejacobs.com/2008/03/24/community-college-drop-outs/, though most people who find this article probably just came from there.
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 didnít seem to make much impression, and over time I began to wonder if the students really knew fractions. So finally this spring I decided I would start the year with a fractions quiz in my two lower level 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.) I told the students that we could not spend class time on fractions, but if they are deficient they should try to catch up. This worked well with another decision I made this semester, to make all quizzes retakable. So a student who doesnít know fractions could study and retake the fractions quiz (different version, same type problems) until they get them figured out. The fractions quiz consisted of ten problems, and a loose time limit of ten minutes. The average score in one class 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.
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 think I have four versions now. I did spend a considerable amount of time before the semester started 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 one out. 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. 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 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.
Here is the original version of the fractions quiz I gave on the first day of the semester in these two classes.
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, just as it appears here.
So far in the semester a number of students have come to my office to try a retake on the fractions quiz. (As I write this we are just about at the middle of the semester.) As I normally grade their quiz immediately when they are done, and in their presence, I have been getting some feedback on whatís going on in their minds. One problem in particular interests me, exemplified by 6 ĺ - 2 3/8. I expect them to convert ĺ to 6/8 and then subtract the 3/8. Almost invariably that is not the way they do it. They change 6 ĺ to 27/4, change the 2 3/8 to 19/8, and then, (at least the ones who can actually do the problem, and if I remember right) get a common denominator and proceed. I ask them if that is the way they were taught. Usually they say they canít remember, it was a long time ago. Then I show them that they can leave the whole numbers alone, just change ĺ to 6/8, subtract the 3/8, and then deal with the whole numbers. They seem to make sense of this, but donít think of it as a better way.
I donít know what is going on in k-12 math education. I donít know if they were taught this way or not. I donít consider their way a better way, but I can well imagine that some might.
Another thing strikes me as I watch these students doing fractions. I get the impression that they have never developed any fluency in doing them. 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 NCTMís way of doing lots of things in math.
I have tried repeatable quizzes before, but this semester is the first time I made a blanket rule that quizzes could be retaken without limit, other than I'll announce a cutoff date shortly before the end of the semester. I can't have a mob of students come in at the last minute to try to raise their bad grade by retaking quizzes. My grading is 60% of tests, 20% on homework and 20% on quizzes. So a conscientious student can pretty well count on an A average for 20% of his or her total grade.
There are not that many conscientious students, of course. I have not been swamped by demand for retakes. I had a retakable option on a limited number of quizzes several years ago, and experience indicated that most students won't take advantage of it. They should. The main benefit is not to get a better grade on that 20%. The main advantage is learning the material. To retake a quiz a student has to come to my office. I usually look at their graded quiz to make sure I choose a different version of it. Quizzes are supposed to be limited to ten minutes, but sometimes (often actually) I forget to pay any attention to the time. But then, usually I grade it quiz with the student right there. That provides a setting for feedback that can be very valuable. Not only can I explain mistakes to the student, but also I can get an idea of the student's thinking.
What are we to make of this? Through much of my life I assumed everyone could do fractions. It was sort of a passive assumption. I didn't think about it much. I feel that when I left the eighth grade I could do fractions as well as I can do them now. A bachelor's and a master's degree in math obviously added a lot to my knowledge of math, but I never felt it had anything to do with fractions. If I could do fractions well by the end of eighth grade, couldn't everyone?
Of course I realize now that, no, not everyone of my classmates in elementary and high school could do fractions like I could. But if the above fractions test were to be given to my high school graduating class in 1961, would the average score by about three out of ten? I guess I'll never know. And, of course, what really counts is doing the right thing now. Are students now days getting good instruction in math before they get to college?
Maybe the normal human mind is such that a useful knowledge of fractions, of arithmetic in general, is never going to be common among people at large. Maybe my algebra students in 2008 got just as good instruction in their elementary and high school education as I did in mine. Maybe my classmates can't do fractions now, and maybe they also couldn't in 1961 when we graduated.
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, sends the first group to their desks and brings 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 (National Council Of Teachers Of Mathematics) 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 bother to keep up with educational trends. But I do know that followers of what the NCTM promotes would think of the scenario I have described. The would call it "drill and kill". They would call it "soul deadening" or "mind numbing". I don't remember it that way.
In about 2003 I took a very serious 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. http://www.brianrude.com/disagr.htm
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 be the same now as then. 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?
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.