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Some Ideas On The Teaching Of Fractions

Brian Rude, 2009

      This article is a follow up on my previous article, "Fractions My Students Can't Do". In that article I described how I have begun using a fractions quiz on the first day of class in my lower level algebra courses. This was prompted by my observation that apparently many college students don't know fractions. They should. They should have fractions well under control before they leave eighth grade. But they don't, at least many of the students in the community college where I teach. Before giving this quiz I explain that we will not spend class time on fractions. That should come before algebra. But if the quiz shows they don't know fractions then they should work on them, and take advantage of the unlimited retakes on this quiz, and learn them.

      Only a minority of students take advantage of this opportunity. Most of the students in the class should, but, for one reason or another, they don't. However the minority that does makes for a steady stream of students through my office throughout the semester, and that gives me a lot of opportunity to explain simple ideas about fractions, again and again. This article is an attempt to relate what I have learned, or conjectured, about teaching fractions.

      I will describe one recent example of this process. Cal (not his real name of course) is not a good student in algebra. He might pull a C out of the course by the end of the semester, or he may not. It was just past midterm when he came in and said he'd like to try the fractions quiz again. I printed off version two of the quiz from the computer for him. I think he said something about not understanding them, and I suggested that he try the quiz anyway and then we'd have something to work with. He agreed and set to work. However it was just a very few minutes when he said that he really didn't know how to do any of them. So I looked at his paper. It was mostly blank. As I recall he did have one problem correct, and had written down a thing or two on a couple of others.

      My approach in situations like these is to pick out some point to begin. If at all possible I will point out something that the student did right. As I recall in this case one of the problems was 2 3/8 + 1/2. He had the right answer, 2 7/8. So I started with that. I told him it was correct, and asked if it made sense to him. His reply, as expected, was noncommittal. So I spent probably two minutes or so explaining it to him, explaining in a way that he could interpret as simply confirming what he had done. As usual I drew circles, showing the half a circle and the five eighths of a circle were different, how the half circle can be redrawn as four eighths of a circle, how eighths added to eighths are like apples added to apples, not apples added to oranges (two apples plus three oranges do not make five apples and they don't make five oranges, so one half and three eighths do not make four halves and they don't make four eights.) Then I looked for the next easiest problem of addition or subtraction and continued.

      I probably did not try to explain every problem at this time. Can I condense two or three months of sixth grade instruction about fractions into a fifteen minute session with a student? I often do, and in many cases I can because there is a lot of residue in the student's mind from previous instruction. Sometimes a student will catch on very quickly, come back in a day or two for a retake of the fractions quiz, and do well. Other times it is a longer process. Quite commonly, unfortunately, the process never goes very far. I can explain carefully, sometimes at length, and invite the student to come in another day and try another version of the quiz. (I have a rule that you have to wait a day before trying it again.) Many students leave it at that. A few, a very few, have the persistence to come back again and again, and thereby make considerable progress. This small minority can be very gratifying to me. They know they are weak in fractions, or in arithmetic in general, and are appreciative of filling in that gap in their education. But this minority is pretty small. A larger group of students are aware that they don't really understand fractions, and probably many other math concepts that should have been learned in elementary school, but they don't care that much. They would like to pass the algebra course, and will work to do so, but they are not greatly motivated beyond that limited goal.

      And of course there are students who do know their fractions. They may miss problems on the fractions quiz for many reasons, but it is apparent when I am grading their quizzes that they pretty well know what they should. And when I work with them individually, which happens occasionally, it is apparent that they do have a good understanding of fractions, and probably all of elementary school math. However this group of students seems to be a minority. The majority of students in my lower level algebra classes seem to have moderate to serious deficits in their understanding of elementary school math.

      What I have described is the immediate background of my perspective on teaching fractions. I do not have any experience teaching, or even explaining, fractions to a normal class of fifth or sixth grade students. I do have two years experience teaching seventh and eighth grade math in 1964-66. That is not fresh in my memory.

      I also have two years experience teaching in a prison school in 1970-72. That, likewise, is not fresh in my memory. However it was in that situation that I began seriously thinking about teaching and learning and formulated some ideas that I believe have stood the test of time. In that situation I worked at length with individual students, sometimes for as long as a year before the student had served his time and was released, or finished the general math course. This situation allowed me to see things up close that I think are not at all visible in a normal classroom setting. Thus that prison school experience is very important to my ideas of teaching and learning math.

      And I also have some experience as a parent teaching fractions to my children when they were elementary school age.

      But in general I am not writing this article based on experience teaching fractions to elementary school students. Elementary school is where fractions should be taught, taught well and thoroughly. Hopefully fractions will be understood pretty well by the end of the sixth grade, leaving time for a lot of applications of arithmetic in the seventh and eighth grade. And I would think that if we want decimals, per cents, and applications of arithmetic to be understood well by the end of the eighth grade we should aim for a good understanding of fractions by the end of the sixth grade.

      I know it has been the goal for some decades now to push algebra down into eighth or even seventh grade. That has always sounded good to me, but I had always assumed that arithmetic would be well understood before algebra is started. But my experience teaching college students suggests that assumption might be unfounded. This raises the possibility that in trying to push algebra down a few grades we might be doing a poor job with both arithmetic and algebra.

      There also seems to be the idea in recent decades, and I think this is made explicit in the NCTM's "Standards", that algebraic ideas should be slowly developed in the lower grades, rather than being developed suddenly at the start of algebra. I wonder if this is a mistake. Perhaps by introducing algebraic ideas early we just make them seem abstract, disconnected from common sense, and thus actually harder to learn when we get serious about them. And if we neglect arithmetic in favor of trying to teach a little algebra early, we may be making a bad choice.

      At any rate I am writing based on my experience teaching fractions remedially, a very limited experience. And I am writing based on my conjectures about how it ought to be done, conjectures untested in the real world.

      It would seem sensible to divide this discussion into three parts. First I will discuss teaching the meaning of fractions. Second I will discuss teaching addition and subtraction of fractions. And thirdly I will discuss teaching multiplication and division of fractions. Another very important topic, which I will try to weave into these discussions, is the matter of understanding the operations of arithmetic, and especially what I will call the "necessary redefinition of multiplication".

      How should fractions be introduced to students at the very beginning? And when should this be? It would seem sensible that we start out by teaching the meaning of fractions. I would assume that in elementary school this would not be done by mathematical definition. In keeping with the general idea that we go from the concrete to the abstract, I would assume that we should start with a meaning of fractions that makes sense in the world of the third, fourth, or fifth grade child.

      I believe it was the end of fourth grade when I first encountered fractions. My memory is not good on things of so many years ago, but I do have a vague memory of pictures in a book meant to show fractions - apples cut into halves, pies cut into sixths, and so on. I presume this type of thing is an important beginning point. I assume fractions need to be explained in terms of parts of a whole. Is there any sensible alternative? Can we teach fractions right from the start as a ratio? I don't know. But I would think that the idea of a ratio would derive from the idea of parts of a whole.

      A general pattern in teaching is to first present a new idea in some form, and then to ask the student to deal with this new idea in some way. Teaching usually involves a lot of prompts and responses. After a new idea or concept is explained, either concretely or abstractly, either definitionally or operationally, the student is given prompts of some sort, prompts that requires responses, and the responses must be judged by very concrete criteria. The response is either right or wrong. One form this might take would be a worksheet in which various pictures or diagrams are presented and the student writes the fraction represented by that picture or diagram. In this process students learn what responses are desired and what responses are not. Hopefully this process leads to knowledge, and hopefully that knowledge can be verbalized. But if it cannot be explicitly verbalized, at least it needs to be used to dependably give the responses we want.

      Perhaps the most basic understanding of fractions is accomplished when it is obvious to the student that three thirds make a whole, because that's what we mean by a third, and five fifths make a whole, because that's what we mean by fifths. How do we teach this? I'm not sure, beyond saying that we explain and we give problems for students to do.

      Every once in a while when working with my algebra students I will discover that they do not know how many thirds there are in a whole, or how many fourths there in one. This doesn't happen very often. I presume the average student, even those who flunk my fractions quiz, understand this very basic idea of a fraction. What does one third mean? It means that three of them make a whole. Yet apparently that understanding is not universal among my algebra students. Here is a recent example that shows this. The context is composition of functions and inverse functions. The problem was this:

f(x) = 3x - 2 and g(x) = 1/3 x + 2. Find (fog)(x) and from the result tell whether or not f and g are inverses of each other.

      The way to do this is to replace x in f(x) = 3x - 2 by 1/3 x + 2, and simplify. If the result simplifies down to x, then the functions are inverses of each other. But if the result does not simplify down to x, then the functions are not inverses of each other. So the essential step in this problem is to simplify 3(1/3 x + 2) - 2. The student I was helping is very conscientious, but math comes hard for her. She seemed to understand the concepts of inverse functions and composition of functions, but was stumped, at least for a few seconds, in simplifying 3(1/3 x + 2) - 2. "Clear the parenthesis" I told her. "Multiply 3 by each number inside the parentheses." That seemed to ring a bell of recognition in her mind, but then she was immediately stopped by multiplying 3 times 1/3. She seemed to realize that this should be easy, but she hesitated. I don't remember just what happened next. I was a bit dumbfounded that she didn't seem to know how to multiply 3 by 1/3, but I also recognized that this was not the first time I had seen this problem. Perhaps she recovered from her hesitation by herself and did it right, or perhaps I helped her. All I remember was the behavior clues that she was genuinely struggling with the idea, and my sense of astonishment at her struggle.

      How is it possible that this algebra student didn't have that basic meaning of fractions? Does it mean she was poorly taught? Does it mean that her brain is particularly bad at math? I don't know. I suspect that it is an idea so obvious that it is easy to overlook. Teachers should learn not to overlook the obvious. I would guess that an important part of the answer to "How do we teach this basic idea?" is that we return to it now and then. We should be careful not to assume that it is obvious to the student

      After students get an idea of what we mean by fractions, I presume the next stage would be a consideration of the idea of equivalent fractions, reducing fractions, and taking fractions to higher terms. This lays the groundwork to make sense out of adding fractions.

      How do you teach that three fifths and six tenths are the same fraction? I presume you do it with examples, including diagrams, and explanation of what to look for in those diagrams. The idea of equivalent fractions is a logical consequence of the idea of fractions. I would think this idea would come more operationally than definitionally. Again we start, I presume, with the concrete.

      I would expect that considering parts of a whole would lead to the idea of ratio. Or would it? I'm not sure. Is it the same idea, or a related idea? I would presume that the word, "ratio" should be introduced rather early, along with the idea of fractions as parts of a whole, but I would expect that time would be required for the word to begin to have meaning.

v An important related idea, either at this stage or sometime very soon, is that a fraction is a number. A fraction represents a quantity. Can we assume that this is automatically evident when we first explain the meaning of fractions? I would think not. Can we assume that when students begin to understand the idea of a ratio, then they understand that a fraction is a number? Again I would think not. Here is an illustration of the meaning of fractions that possibly might be used.

      Mrs. Jones has five apples in her refrigerator. She uses three of them to make a salad. What fraction of her apples does she use?

      I would expect that students could think about this, conclude that three-fifths is the answer the teacher is looking for, and thereby gain in understanding what fractions are. But from this example it does not seem to follow at all that students would think of three-fifths as either a ratio or as a number. Here is another example.

      The telephone company has five miles of cable in their storehouse. They use three miles of it to string a new line to a new factory being built outside of town. What fraction of their cable do they use?

      Here again this example might be useful and understandable to students. Is it reasonable to expect students, in response to examples such as these, to think of the three-fifths in one example as a ratio, or as a number? The mental picture the students get from these two examples are quite different. Both examples are understandable in considering parts of a whole, and I presume would lead to the consideration of three-fifths as a ratio, but I would think these two examples would do little to promote consideration of three fifths as a number.

      In other words I would expect that students would first need to make sense of fractions as parts of a whole, then as a ratio, which makes a consideration of equivalent fractions possible, and finally as a number. If students understand that three out of five apples and three out of five miles of cable have something in common, then the teacher can explain the idea of ratio. When the students have some understanding of fractions as ratios, it is time to talk about equivalent fractions. Then when equivalent fractions make some sense, the teacher can start talking about fractions as numbers.

      I would think, then, that the meaning of a fraction is not something that can be immediately explained to students. I would think understanding of what a fraction is would be a long process. I would expect the teacher would need to do a lot of explaining with a lot of different examples and problems.

      At some point in this process the teacher could do something like the following. The teacher makes a few large cards with fractions on them. She holds up the card with 1/2, and the card with 3/4 and asks which is bigger. She then puts them in order from small to large. She would ask why 3/4 must go to the right of 1/2. And she would explain the answer. She would use circle diagrams to visually show that 3/4 is bigger than 1/2. (Probably this has been done, and will be done, many times to explain how to add or subtract 1/2 and 3/4.) Perhaps she would talk about one half of a football field is not smaller than three-fourths of an apple, but one half of an apple is smaller than three-fourths of an apple, and one-half of a football field is smaller than three-fourths of a football field. Then this type of explanation can lead into homework assignments in which a problem consists of five or six fractions to put in order from left to right, or fractions are to be put on a number line.

      To add fractions, they must be numbers. But I would expect it to take a while before fractions really become numbers. Is it necessary for fractions to be understood as numbers before they can be added? I'm not sure. Perhaps learning to add fractions is an important step on the way to understanding fractions as numbers. Is it possible that some teachers can't teach fractions as numbers because they don't understand it themselves? Probably. Is it possible that some teachers are in the opposite situation, they cannot understand what the issue is because it is so obvious to them that fractions are numbers? Again, probably.

      If we don't stress early that every fraction is a number, we may find that we lose track of the need to explain it at some point. But it would seem if we try to stress the idea of a fraction as a number to early we risk being too abstract. One resolution of this dilemma is to keep coming back to the idea. We come back to the idea to explain and develop the idea, but we also come back to the idea just to check that the students' knowledge and understanding have not deteriorated. Do students in the eighth grade, after they have been dealing with fractions for several years, know that one third is bigger than one fifth, and one fifth is bigger than one tenth? I would think that is something that should not be taken for granted. It should be checked a number of times as the students progress through the several years in which they are learning fractions.

      My college students who come to retake the fractions quiz have the most trouble with addition and subtraction. They may or may not know they need to get a common denominator. And when they do know they need to get a common denominator, they often seem to know it only as a rule to be followed. They often don't seem to understand the reason for the rule.

      It is not too unusual for one of my algebra students to try to add fractions by adding the numerators and then adding the denominators. This used to seem very strange to me, and totally incomprehensible. But over time I realized it makes some sense. If a team wins two out of three ball games one week, and the next week wins one out of four games, how should we combine the results of the two weeks? It is true that over the two weeks they won three out of seven games. So in that way at least we can in a sense add parts of a whole, and we do it by adding the numerators and then adding the denominators. But that is not what we mean by adding fractions. In mathematics we do not add 2/3 and 1/4 to get an answer of 3/7, even if it does make sense in figuring a win-loss record. Should we talk about this with fifth graders? I don't know. Maybe it could help give them perspective on what they are doing. Maybe it would cause a lot more confusion than help.

      So what is the reason needing a common denominator for adding and subtracting fractions? I explain it in terms of adding apples and oranges. If you have two apples and three oranges you don't have five apples and you don't have five oranges. This is the basic idea of combining like terms in algebra, and it also fits in with the distributive law. Of course some students are quick to point out that you do indeed have five pieces of fruit. It can be pointed out that "pieces of fruit" is a common denominator. This approach has the risk that students may think that a common denominator is just a matter of semantics. But other examples can show it more clearly, such as money. If you have two dimes and three nickels you don't have five dimes and you don't have five nickels. And this is not just semantics. It is value. Again the student may reply that you have five coins, but I think it is pretty hard to ignore the value. You may have five coins, but you do not have to value of five dimes and you do not have the value of five nickels.

      I draw diagrams to illustrate a common denominator. As an example consider the problem of adding 2 1/2 and 1 3/8. There are some students at the college level who will know nothing to do with the fractions except add the numerators and then add the denominators, and come up with 3 4/10. The diagrams I draw are almost always circles. They don't have to be very big circles, dime size circles usually are adequate, and they don't have to be particularly well drawn. It's easy to draw a half a circle, and it's easy to draw in lines to turn that half of a circle into four eighths. I will sometimes tell the student to imagine each circle as a pumpkin pie.

      This approach generally works to explain the problem. But we have to ask about the history of the student's knowledge of fractions. If this explanation works, why was it not learned in the sixth grade? If it was learned, complete with all the diagrams, lots of practice, problems, applications, quizzes and tests, then why is it not understood and remembered now? If it was learned and forgotten then why should we expect it will stick with the student any better now than it did in the past? Or was the student's past instruction faulty?

      And are they thinking of something like adding the games won and played as I just described? Is there more sense to the student's faulty reasoning that is apparent on the surface? And why would they think this way, if they previously knew and understood the right way to add fractions.

      My guess, and it is only a guess, is that students get plenty of faulty instruction in many ways, and one of those ways is the teacher allows students to lose sight of the idea that a fraction is a number, and as such, a quantity. Adding fractions may make perfect sense to students when diagrams are put on the board and explained. But it is very possible that they lose sight of this as time goes on. They may do a lot of homework. They may gain proficiency in adding fractions, but as a procedure to be learned mechanically, while at the same time they lose sight of what they are really doing. Thus I argue that the idea of a fraction as a number (and lots of other ideas) need to be revisited now and then.

      Can we revisit a topic too many times? We probably can. Can we revisit a topic too few times. Again, we probably can. How do we find the right balance? I presume we must make judgments from experience and the feedback we get from students.

      There are two related ideas that I think are relevant here, and which I have developed elsewhere. These are the ideas of “fragile structure“ and “levels of fluency“. We work hard for understanding, but understanding is not an all or nothing thing. Students can have a level of understanding, but still have a “fragile structure”. Ideas may be connected in one’s mind to form concepts, but not in a dependable way. Understanding that seems to be present one minute may fall apart and not be present even minutes later. And understanding may be attained for a short span of time to enable students to learn to do a certain type of problems. The understanding may slip out of mind, while a simple memory of how do to problems remains. Thus we continue, thinking the original understanding remains when in fact it doesn’t.

      The idea of "levels of fluency" is that we may ostensibly know something, yet have a low level of fluency in that knowledge. If the teacher asks a student what six and eight are, and the student answers "fourteen", there is still room for a lot of variability. The student may think hard to remember, perhaps counting mentally or physically on his fingers, before tentatively replying. Or the student might answer as easily as if the teacher had asked him his name. Teachers need to develop some sensitivity to these levels of fluency. (And I am sure they do. I don't mean to be suggesting something new here. I just think it is an idea that deserves more recognition.)

      Awareness of these two concepts, I believe, help us to realize and understand how learning can be less than it often seems.

      When students are in college they have some ability to judge their own understanding and to take their own remedial action. I would expect this ability in fifth graders, though not entirely absent, would be comparatively undeveloped. It would make sense that we, their teachers, must check for understanding as we progress. So we must revisit various important ideas to make sure students really do have the understand that we think they do.

      Can it happen that a fifth grade teacher teaches reducing of fractions and addition and subtraction of fractions without herself having fluent understanding of the topic? Obviously it is possible. Indeed I am cynical enough to think it probably happens often. But that is another story.

      You can't go far with fractions without dealing with mixed numbers and improper fractions. After students have learned the basic idea of adding fractions some problems will sometimes produce improper fractions which need to be changed to mixed numbers. I would expect that this would not be the first time they have encountered mixed numbers, but it is certainly a time when they should be understood. And when students progress into multiplying and dividing fractions mixed numbers should be thoroughly understood.

      How should mixed numbers be explained? I would expect that again circle diagrams would be very useful. I would expect that a consideration of five fifths, six fifths, and so on would be an easy continuation of learning about two fifths, three-fifths and four-fifths. Circle diagrams I would think would make the transition to mixed numbers easy. And I would assume that the reverse, starting with a mixed number and changing it to an improper fraction would again be learned with diagrams. Students can learn the rule "multiply the whole number by the denominator and add the numerator, then put that over the denominator", but that should be understood, not just memorized. How many diagram examples does it take? Probably a lot, spread over a period of time.

      I think of the fourth grade as when the serious study of fractions begins. But I think of fifth grade as a time when a lot of serious study of fractions takes place. Does it take a year for students to gain an understanding of the basics firm enough to really understand addition and subtraction of fractions. I am guessing that it probably does.

      The terms "carrying" and "barrowing" have been out of favor for many years I understand. I think we would do best by ignoring what's in and out of favor. The terms arose, I presume, because they are intuitively meaningful. I would expect to use them freely in explaining addition and subtraction of fractions.

      It can be argued that a procedure, such as adding fractions should always be understood. It should never be just a mechanical process to be carried out without understanding. It would seem that that's what I have been arguing in the previous paragraphs. However that is not the totality of my perspective. Indeed I would argue there is a place for the opposite to at least some extent. Understanding is important. Understanding should lead the students learning. It is a good goal that everything be understood. But it can also work the other way, for two reasons. Mechanically learning a procedure so that it is firmly fixed in memory can be an aid to understanding. And remembering is sometimes a lot faster than understanding. Memory can bridge over inadequate understanding, and memory can be can be quick at times when understanding is not so quick. Understanding is putting the parts together. Memory can keep the parts available so that they can be assembled in the right way. And memory will eliminate the need for reassembling the parts again every time the knowledge is needed.

      So I think there is a place for learning procedures sometimes even when understanding is not attained. Adding fractions should be practiced until it can be carried out in a mechanical fashion. Hopefully it would not be done mechanically until it is well understood, but sometimes it must be. When it is well understood, and then practiced until it can be done easily and mechanically, then we have a good basis for saying the procedure is "mastered", and not until then.

      I have argued before that implied elements in a structure of knowledge are held in place by memory, as well as by understanding That is very important. Memory is immediate and efficient. But implied elements in a structure of knowledge are also cemented in place by understanding. This provides support and permanence to memory. An understood process can always be thought out again to make sure it is right. Understanding is a back up that always immediately available to check one's work. Thus the combination of memory and understanding work together.

      Levels of understanding can vary widely. What we mean by understanding can vary widely. I would like to think that everything about fractions would be understood by every learner, but that doesn't seem like a realistic goal. Some things can be hard to understand. In those cases we often make trade offs. Some things are easy to understand, and so of course they should be understood. But other things are hard to understand, and sometimes learning a procedure without full understanding is a sensible trade off.

      Followers of educational fads have always charged that teachers always used to teach by "rote memorization". That seems totally rhetorical to me. How would a fifth grade teacher teach the adding of fractions in 1910? Would she draw diagrams to show that 1/2 can be changed to 4/8 in order to add 1/2 + 3/8? Common sense would indicate so, I would think. The idea that she would teach a recipe for doing that doesn't make any sense. The idea that students would not ask for some rationale doesn't make much sense. How would teacher explain carrying and borrowing in 1910? Wouldn't they explain? How could you teach carrying and borrowing without explaining? How would you teach algebra, or calculus, without explaining?

      Practices vary with times and circumstances, of course. A third grade teacher in 1910 with a class of 40 students would probably make trade offs that we don't have to make today. But the idea that teachers in past times made no attempt to explain anything just doesn't fit with my experience of human nature.

      I have often said that the claim that children want to first know "Why?" is not very realistic. Learners tend to first want to know how to do the problems at hand. But that is not to say that they are content to receive recipes without understanding. "How come?", and "I don't get it" and many similar phrases are heard every day by teachers. What do teachers expect to do when a student asks for help? I think it is pretty much a universal urge to explain, to give understanding. And it is pretty much a universal urge on the part of learners to expect things to make sense on at least some level. The level of understanding that a student is satisfied with may not be the level of understanding that we would like, but it is not a zero level either.

      Teachers want to explain. So we do. But in our eagerness to explain an idea or a problem, we may lose sight of other limitations to learning. Understanding is often only a first step in a learning process. Initial understanding must usually be followed by application and practice. This takes time. An important part of my present job is to maintain office hours, time for students can come by to get help. And they do. Not a day goes by when I don't have students come by for help, usually on homework problems. So what do I do? I explain, just as any teacher would do. Over the years I have come to realize that that is sometimes the easy part. In the community college where I teach the student who comes to me for help is likely to be a parent and a worker. The fact that she comes today for help on a problem means that at least for today she can steal a few minutes, maybe a half hour, to do so. In such situations the common pattern is that I explain the math of immediate concern. The student, after a few minutes, gets an "aha moment", and I think I have done wonderful stuff. But afterwards I reflect a bit and realize that probably the student can't do this everyday. She struggles to find time to study and do homework. She struggles with her other responsibilities just to make it to class with enough regularity to make success possible. When she got that "aha" expression on her face I thought my job was done, and in a sense it was. But her job was not done. She must practice on problems if that understanding is to remain with her. Until she gets in enough practice her understanding is a very fragile structure of knowledge. Her level of fluency is low. My explanation, either in class or in my office, is aimed toward understanding. Understanding is a very important first step in learning a topic of math, but we should never forget that it is only that.

      Now let us turn to the multiplication and division of fractions. I think understanding the multiplication of fractions requires a major shift in thinking about multiplication. And, of course, it requires an understanding of the relation between multiplication and division.

      What does multiplication mean to a third or fourth grader, before they start to learn about fractions? What can multiplication mean? What are the possibilities? Is it obvious? I can't really answer these questions. I don't know what goes on in third grade classrooms. So I will have to conjecture a bit.

      I would presume we first teach the idea of multiplication rather concretely, with objects. If you have three coins in each pile, and you have four piles, then you can simply count all the coins to discover there are twelve in all. Then we can say that four times three is 12, and that's what multiplication is. We can then, over time, show that this is a precise and reproducible idea. If you have four bags with three apples in each bag you can count them all up and there will be twelve. If you have three books on each of four shelves there will be twelve books in all. If you have four beans in each of three cups you have twelve beans. Through any number of examples we can conclude that three times four is twelve. As in all math at least through high school, we abstract from reality. Our only proof is conformity to reality, but that is a very important proof, and an understandable proof.

      What I have described I would call an "operational" definition of multiplication. Is there any alternative to that definition? Could we argue that right from the start we should explain multiplication as a closed binary operation on integers? My opinion is no, that would be a waste of the students' time.

      The idea of so many objects in a group and so many groups can be called the "array definition" of multiplication. Instead of real objects we can just draw x's or circles in an array, so many in each row and so many rows. Along with the array definition I think we should explain multiplication by a second definition, as repeated addition. This fits intuitively with the array definition. It is easily understandable with a few examples and a bit of guided thought.

      Are these two ideas distinct in the students' minds when they are learning the multiplication table? I'm not sure, and I'm not sure that it matters. I would think we want an operational definition of multiplication, not a formal verbal definition. Both of these perspectives of multiplication are understandable to students providing they have a good understanding of counting and addition. Using either, or both, or a generalized mixture of these two definitions, students can understand the times table, and can have some appreciation of the necessity of knowing the times table. I would expect at this point that a student who forgets the product of five times six could immediately pick up a pencil and draw 6 x's in a row, and then another row, and pretty quickly get the correct answer. I would expect the student should also be able to add five and five and five and five and five and five, or perhaps remember that five fives are twenty-five so six fives must be thirty, or count by fives on his fingers, or in perhaps yet other ways apply the meaning of multiplication to figure out the product of five times six. To "apply the meaning of multiplication" would have to mean to apply one's operational definition of multiplication.

      I would expect that written problems would be appropriate at this point, problems that simply use the meaning of multiplication. If Mary has five paper bags with eight pieces of candy in each bag, how many pieces of candy does she have altogether. This very simple type of problem lays the basis for more complicated problems.

      At some point as we proceed the distinction between numbers and quantity becomes more important. Just how distinct these two concepts are in the minds of elementary students I cannot say. I'm not sure it's important in the minds of young students. But I think it is important in the "necessary redefinition" of multiplication, which I will now explain.

      What does a student think of the problem 1 3/8 x 2/3? I think it would probably have little meaning when multiplication of fractions is first introduced. I assume that this would be after students have been adding and subtracting fractions for a while. So I would expect that the corresponding addition problem, 1 3/8 + 2/3, would seem pretty concrete to them. They could visualize that many apples, or conceptualize that many dollars. Their imagination would make it quite reasonable that the answer must be somewhere around 2, apples, or dollars or anything else. But what about the multiplication problem, 1 3/8 x 2/3? Can that be visualized with apples or dollars or any other objects?

      Can we take this problem back to the array definition of multiplication, or the repeated addition definition? This seems not so easy. Can we say that it means 1 3/8 rows with 2/3 object in each row? Maybe, but that seems to really be stretching imagination to the limit.

      All of these thoughts can come together with a slightly refined definition of multiplication. And that refined definition is this:

      When you multiply, one number tells you how many of the other number you have.

      This, I believe, can be made quite understandable to students at this age. In the problem 1 3/8 x 2/3 we can consider that the first number, 1 3/8, tells us how many of the 2/3 that we have. We have a bit more than one 2/3, so the answer must be a little more than 2/3. Or we could say that the 2/3 tells us how many of the 1 3/8 we have. We don't even have one of those, so the answer must be less than 1 3/8. With this definition, or redefinition, we don't have to try to visualize objects in rows with so many in each row. Instead we have to think about quantity.

      It might be argued that logically this new definition is equivalent to the previous definitions. But I would expect that this logic would be irrelevant to children. The previous definitions were not in any way restricted to objects, and to whole numbers, but I would think that students would think in terms of objects, much more than thinking in terms of quantity. The early examples they were given would be of objects, not quantities. The counting numbers are for objects more than for quantities. The concept of quantity may not be dependent on the concept of discrete objects, but I would think that mentally dealing with quantities would have to arise out of mentally dealing with objects.

      This redefinition of multiplication, one number tells you how many of the other number you have, is not restricted to either objects or quantity, but is very adaptable to thinking in terms of quantity. To understand multiplication in a broad sense, children must often think in terms of quantity, rather than just objects. I would expect that this ability, and this propensity, would develop over time. It would take many examples, and it would take careful explanation of those examples, for it to make sense. That means it takes time. But what is the alternative?

      One alternative, and I expect this is very common, is to teach the procedure for multiplying fractions and just accept that it doesn't really make sense. I think in my elementary school days that was more or less how it was done. In the example above, 1 3/8 x 2/3, students can certainly learn the procedure and follow that procedure. But surely we would want it to be more than a rote process eventually. I would argue that understanding would have to be in terms of this redefinition of multiplication. One number tells you how many of the other number you have

      At some point the students can be given problems such as "It takes 1 3/8 gallons of gas to fill the tank of John's lawn tractor. If John fills the tank only 2/3 full, how many gallons will that be?" I think it was in problems such as this that multiplication of fractions made sense to me, but I don't think I ever heard the simple explanation, "multiplications means one number tells you how many of the other number you have". That would have helped. In this example 1 3/8 is pretty concrete, a quantity of gasoline, one and three-eighths gallons. That can be easily visualized, at least by students who have some experience with the gallon as a measure of volume. How many of those quantities do you have? Five tankfulls? Two tankfulls? Even less than that? Less than one tank full of gas? Of course, you don't even have one tank full of gas, so obviously the answer is less than 1 3/8. "Obviously" that is, when the students get it.

      I think this would be understandable to children when they are learning to multiply fractions, but I have no experience to confirm it. Perhaps I am too pessimistic. Perhaps most teachers routinely explain multiplication in concrete, understandable terms with concrete problems.

      I do remember as a child pondering the mystery of how you can multiply two numbers and come up with an answer smaller than either number, or smaller than just one of the numbers. This is understandable with the new definition of multiplication. It also shows that the old definition of multiplication was probably not nearly as precise or abstract as we might assume. Logically the array definition would not preclude the idea that you could have less than one row of objects, or that you could have less than one object in each row. But I would expect that practically every child would have these as hidden premises. Hidden premises can cause a lot of trouble if not exposed. Just how deeply a teacher might want to take the class into thinking about less than one row, and so on, or how explicitly a teacher might ask students to compare these definitions, I wouldn't have much of an idea. Understanding is always a desirable goal. But trade offs must often be made. It may certainly be sensible to accept imperfect understanding in trade for simply being able to do problems. This is sometimes true, in my opinion, in all levels of math that I am aware of. (And other times understanding is relatively easy. We should not lose sight of that.)

      Fractions can bring together the ideas of multiplication and division. Indeed it should. Thus we must ask what division means to students at this age, or what it should mean, or could mean.

      Probably the most basic meaning of division might be called the “cutting into parts“ If you have fifteen pieces of candy and three children who each deserve an equal share, then those fifteen pieces of candy may be physically separated into three piles. This is very concrete. It involves objects. I presume it is understandable to children. Fifteen divided by three is 5. Lay out the candy, draw the circles, think about it, or whatever. That's what division means.

      I wonder how many children at this stage might ask what would happen if there are sixteen pieces of candy. At some point the teacher would explain that each pile would contain five pieces and a third of the last piece. Is this something that ought to be done early, long before the children even start fractions? Or is it something to bring up only when the children have some ability to work with fractions? I really don't know.

      A very similar but not identical meaning of division would be the "contained in" meaning. Again we are thinking in terms of objects and whole numbers. How many sixes are contained in 18? Well, you can drawn 18 x's, draw a circle around six of them, draw a circle around another six of them, and draw yet another circle around the last six of them. Then you have a visual confirmation that there are three sixes contained in eighteen.

      Or have we demonstrated repeated subtraction? When we drew a circle around the first six x's, it can be argued we mentally subtracted six. Then with the second six we mentally subtracted another six. And with the third circle we have subtracted six again, and now observe there are no x's left. So obviously there must be three sixes in eighteen. And so, we might, argue, division is a matter of repeated subtractions.

      All of these ideas of division are closely related and can be demonstrated by the first example I gave, fifteen pieces of candy and three children. But I would think these three ideas are different enough to be explained and discussed separately.

      A fourth definition, or meaning, of division is the opposite of multiplication. Hopefully by this time students can begin to think rather abstractly about operations. Addition is an operation. Two numbers go together to produce a third. Multiplication is also an operation. Two numbers go together to produce a third. These ideas derive from counting objects, but hopefully at this stage are not confined to counting objects. Division can be defined as the opposite of multiplication, just as subtraction can be defined as the opposite of addition. In a more concrete manner we may say "If you have an addition problem, and the answer, but are missing one of the addends, then subtraction gives the missing addend". That is not a formal explicit definition of subtraction. It is more of an operational definition. But with a few examples and a bit of practice it can make sense to children. Similarly we may say that division is the opposite of multiplication. We can say, "if you have a multiplication problem, and the answer, but are missing one of the factors, division will give you the missing factor". Again understanding this depends on doing examples and problems.

      I have some personal experience with teaching this concept. When I was teaching in the prison school most of my students were working their way through a general math workbook, about eighth grade level. Circumstances dictated individualized instruction. Each week I would lose a student or two out of each class, and gain a new student or two. This situation allowed me to see all phases of this curriculum simultaneous. In a single class of perhaps twelve students I would have a student or two in the early lessons, lessons 2, 3, or, 4, and probably a several students in the middle, maybe lesson 18 or 25, and a student or two about to finish up, lesson 37 or whatever the last lesson was. Over a period of months I could see what worked, and what didn't work in this sequence of lessons. One thing I observed was that in the later lessons students would not understand why you divide in a certain context. The most important context for this was probably what I call type 3 percent problems. "Joe spent $3.30. This was 15% of the money he received for his birthday. How much money did Joe get for his birthday?" To do this problem you divide $3.30 by .15, but why? In algebra you can simply tell students to set up an equation. But in general math, without any algebra, that doesn't apply. Then what does apply? I decided the best way to think of it was that you divide because you have a multiplication problem. To find a per cent of a number you multiply. To find 30% of 15 you multiply .3 times 15. This is what I have always called a type I problem. In a type III problem this is turned around. You have the answer, but you are missing one of the factors. So you divide, but students didn’t seem to understand just why. To develop that understanding I wrote out a lesson, and duplicated it (spirit duplicator in those days), and inserted it in an appropriate place early in the course. It seemed to work. After reading an explanation and doing a few problems students seemed quite capable of understanding how division was related to multiplication. And when students would need that understanding later in the course, it was a familiar idea.

      This is an important idea. If you have a multiplication problem with the answer, but one of the factors missing, you can find that missing factor by division - no algebra required. I don't remember this idea being taught when I was in sixth, or seventh, or eighth grade. Should it be? I think it should be.

      Thus we have four definitions, or four perspectives, of division. They are all equivalent, of course, but that does not mean they should not be explained separately. At some point the student needs to recognize them as all the same, or as slightly different ways of looking at the same thing. But, again, I would expect that to come over a period of time and with a lot of experience and a lot of problems, not immediately.

      Both the “cutting into parts” and the "contained in" meaning of division are closely connected to the "array" meaning of multiplication. An array of eighteen x's arranged in rows of six clearly shows this. The “repeated subtractions” perspective is very similar to these first two definitions. All of these meanings of division are very much object oriented. They are less quantity oriented. The fourth meaning of division, as the opposite of multiplication is not so object oriented. It is as much quantity oriented as object oriented, and more abstract

      With the redefinition of multiplication more or less in place, and with some appreciations of the various ways of looking at multiplication and division, then I would expect the student is ready to proceed into the multiplication and division of fractions.

      I would think that there is one key idea that is not too hard for students to understand and that brings together all that I have talked about above. I will call it the "reciprocal principle". That is the simple idea that multiplying by 1/n is the same as dividing by n. Multiplying by one fourth is the same as dividing by four. Dividing by 20 is the same as multiplying by 1/20.

      Is this obvious to students? Should it be? It is a logical consequence of what we mean by fractions, but is that enough? I would expect that it is not at all obvious at the beginning of the study of fractions. I would further expect that it at some point becomes obvious to many students, perhaps all students who are normally capable and diligent. But I would also expect that it needs to be made explicit by the teacher. I am not sure this was done when I was in elementary school learning fractions. Perhaps it was, and I just took it for granted. Or perhaps it was not. I think it needs to be made explicit, and explicitly verbalized, because it serves as an anchor and a starting point. It allows us to explain why we multiply and divide fractions as we do.

      We started out explaining fractions in terms of parts of a whole. This leads to the idea of a fraction as multiplication and division rolled together into one symbol. Thus 2/5 means to multiply by two and divide by five. From here we can go to the idea that a fractions means division. The fraction 2/5 means two divided by five.

      The basic rule for multiplying fractions is you multiply across the top and then multiply across the bottom. Why is this? The answer, once again, is that is a logical consequence of what has come before. If we can show students how it is a logical consequence then we have accomplished understanding. However at this level the "logical" can not be too abstract. Rather it is an intuitive logic that we are after. And, as always, the proof is conformity to reality. We want students to get to the point where it seems reasonable and sensible. Formal and abstract logic are not much concern at this point.

      Now we can use the meaning of fractions to multiply fractions. Consider 2/5 times 7. By the redefinition of multiplication 2/5 tells you how many sevens you have. You have less than one seven, so your answer must be less than seven. Since 2/5 means 2 x 1/5 you can multiply by 2 first and then multiply by 1/5, or you can multiply by 1/5 first and then multiply by 2. Doing it the first way you get 14 which must then be taken times 1/5, which means 14 divided by 5, which can be interpreted as 14 cut into five parts, which is 2 4/5. Doing it the second way you get 1/5 x 7 which means 7/5 which can then be multiplied by 2 to get 14/5 which is 2 4/5.

      The procedure for multiplying fractions is easy enough to learn. You just multiply across the top, multiply across the bottom, and simplify. But do students understand why? Do teachers understand why? I vaguely remember learning to multiply fractions as a child, and I mostly learned the method with minimal understanding. I think it did make sense over time, but perhaps not as explicitly as it should have been, had it been better taught.

      Understanding why is an important part of learning math. Indeed a great deal of mathematics teaching consists of explaining why. But that is not the totality of teaching math. The old idea that students want to know why has some truth to it, but not as much as we would like. In general students want to know how, what, and why, in that order. However it does not follow that we should give them only what they want at the moment. If we tell them only how to do the problems, and nothing more, we are settling for a pale imitation of what math should be. But if we insist that we will teach only for understanding and never tell students simply how to do the problem, we may be courting needless frustration. Often the information of how to do a problem provides a framework in which the learner can work for understanding. And there is no clear cut separation in the students' minds about the "how", "what", and "why". The student who at one moment is quick to demand, "just tell me how to do the problem", may seconds later demand with equal urgency, "How come . . . . . ? ! !". Learners, in most situations, have many motivations. They want to be able to do problems, but they also want math to make sense. If we can explain why, to at least some extent, for everything we teach them, we foster the expectation of understanding in the students mind, and that is a good foundation for future

      So why do you multiply across the top and multiply across the bottom? Is it really understandable? I think it can be made understandable at the elementary school level. And if we can explain multiplying fractions then perhaps we can later tackle the even greater mystery of why we invert when we multiply.

      These two questions (why do we multiply as we do, and why do we invert when we divide) both have basically the same answer, the meaning of a fraction as division (which includes the reciprocal principle), and both depend on the redefinition of multiplication to be understood.

      Take the example of 5/8 x 3/5. The fraction 5/8 tells you how many of the other number you have. You don’t even have one of the other number. You have only a part of it. The part you do have can be arrived at by multiplying by 5 and dividing by 8, or by first dividing by 8 and then multiplying by five. But how do you take 5 of 3/5? You multiply. You can switch it around, and take 3/5 of 5, which means you multiply the 5 by 3 and then divide by 5. So you have multiplied across the top and multiplied across the bottom. And you did it as a logical consequence of the meaning of fractions.

      At some point these ideas can be formalized, or summed up, in this way.

      5/8 x 3/6 means

     (5 x 1/8) x (3 x 1/6),

which can be rearranged into any order:

5 x 3 x 1/8 x 1/6 or

1/8 x 1/6 x 5 x 3 etc.

      When you multiply across the top and then multiply across the bottom you are simply doing this.

      Do students at this stage know about the associative and commutative laws? I don’t think they have to know them by name, but they should certainly be aware of what they mean. There should be many times in the past in which the teacher has pointed out that order doesn’t matter in addition and multiplication. It should be common sense to students that order doesn’t matter in multiplication and division. At some point it is surely sensible to give this idea a name, the commutative law, but I am not at all sure what the best time to do this would be.

      It is in these terms, I think, that we explain why we do what we do in multiplying fractions. I would think this would be done mostly by guided thinking. It would be good if we could express succinctly and explicitly why we multiply across the top and multiply across the bottom I don’t think that’s easy, and I don’t think I have done it here. But I think we can explain it so that, over time at least, it makes sense to students.

      Why do we invert when we divide? Because, as I have mentioned, to multiply by a fraction such as 1/3 is the same as to divide by 3 - the reciprocal principle. Therefore we can suggest to students that to multiply by a number and to divide by its reciprocal produce the same result, or that to divide by a number and to multiply by its reciprocal produce the same result. This is rather abstract. It will not be understandable just by hearing it stated. But with examples and problems I think it can become quite understandable to students.

      Here is an example that may or may not be useful in explaining these things to students. Three young friends find a pot of gold at the end of the rainbow. They may divide it among themselves immediately, or after they get it home. However on the way home they know they will encounter a troll who lives under a bridge and demands one tenth of the wealth of all who pass over the bridge, and, of course, the three friends must pass over the bridge in order to get home. Also on the way home they know they will encounter a good fairy who always multiplies by seven fold the wealth of all whom she meets. Let us say that there are 120 nuggets of gold in the pot at the end of the rainbow. The question is how many nuggets of gold will each friend have after they have arrived home and divide everything? And then the next question, a very important one, is this. Does it matter in which order they 1) divide the gold, 2) meet the troll, and 3) meet the good fairy? If it does make a difference the friends will want to divide their gold in such a way that they will maximize their profit. Perhaps they will do best if they divide the gold as soon as they find it. Or perhaps they will do best to wait till they get home, or after they have crossed the bridge with the troll and before they meet the good fairy. Or perhaps it makes no difference when they divide the gold.

     Possibility number one: First they divide the gold, second they meet the troll, and third they meet the good fairy. So each friend gets 40 nuggets, of gold, gives 4 to the troll, resulting in 36, and ends up with 252 nuggets in the end.

Possibility number two: They first divide the gold, second they meet the good fairy, and third they pass over the bridge and pay the troll's tax of ten per cent. Each friend has 40, then 280, then 252 nuggets.

     Possibility number three: First the troll, then divide, then the good fairy. The 120 becomes 108, then 36, then 252.

     Possibility number four: First the troll, then the good fairy, then divide. The 120 becomes 108, then 756, then 252.

     Possibility number five: First the good fairy, then divide, then the troll. The 120 becomes 840, then 280, then 252.

     Possibility number six: First the good fairy, then the troll, then divide. The 120 becomes 840, then 756, then 252.

     Or you can set up the problem as 120 [ (1/10) (7) (1/3) ]. Hopefully by this time students are aware that the order of operations doesn’t matter. These ideas combined with the idea that multiplying by 1/n is the same as dividing by n, can be applied to problems such as the pot of gold example to explain, both concretely and abstractly, both intuitively and mathematically, why we multiply fractions by multiplying across the top, multiplying across the bottom, and then simplify as needed.

      So why do we multiply across the top and then multiply across the bottom to multiply fractions? Because that is just a short cut way to do what is meant by a fraction, a logical consequence of the meaning of fractions. And why do we invert and multiply when we divide fractions? For the very same reason, because that is just a short cut way to do what is meant by a fraction, a logical consequence of the meaning of fractions.

      What I have written above is obviously not a recipe for teaching fractions. I'm not sure there should be a recipe for teaching fractions, or anything else. Hopefully I have given some ideas that are useful in teaching fractions.