This research was supported by the United States-Israel Binational Science Grant (#93-xxxxxxx). Any opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of this foundation.
This teaching module on rational numbers for prospective elementary teachers discusses the main concepts related to rational numbers and operations with rational numbers from various perspectives including mathematical , relevant historical, typical reported difficulties and their causes, and didactics. The module emphasizes the coherence of mathematics, its internal structure and consistency, and discuss these issues in the context of the rational number system. The main topics included in this teaching module are: mathematics and reality, mental models and their impact on mathematical reasoning, the concept of schema, the role of consistency in mathematics, the numerical system, and the rational numbers.
The teaching module has been used in our project for teaching experiments with individuals, small groups and whole classes. Special attention was given to the specific needs of prospective teachers in terms of their position between "being merely students" and not yet "real teachers", and their feelings about it. Typical sessions began with an opening activity in which prospective teachers solved mathematical and/or pedagogical content problems individually or in groups. These problems, aimed to expose the subjects' own ways of thinking, constitute the basis for meaningful discussions about the mathematical, pedagogical and didactic aspects of the domain of rational numbers. Here we present and discuss one situation, which exemplifies what took place, taken from a session with an entire class of prospective elementary teachers.
Many studies have reported that a substantial number of students tend to add fractions by "adding the tops and the bottoms". Various possible causes of this behavior have been offered:
- The students do not view fractions as representing quantities but see them as four separate whole numbers to be combined in one way or another. Each fraction is viewed as two numbers separated by a line, and it seems reasonable to add the numerators to obtain the numerator of the sum and to add the denominators similarly (Carpenter, Coburn, Reys, & Wilson, 1976).
- The students confuse the rule of adding fractions with that of multiplying fractions (Herskowitz, Vinner, & Bruckheimer, 1978).
- The students view the four numbers involved in addition of fractions as two numerators and two denominators. They believe that the adequate way to perform addition is to add "alike items", that is, numerator to numerator, and denominator to denominator. They view this as similar to adding the way it is done with whole numbers, in the sense that in both cases only alike terms are added -- ones to ones, tens to tens, numerators to numerators, and so on (Herskowitz, Vinner, & Bruckheimer, 1978).
- There are some life situations in which such a way of operating is appropriate (Borasi, 1987; Kline, 1980; Meyerson, 1976; Mochon, 1993).
Most of the prospective teachers who participated in the first stage of our study correctly solved the problems that dealt with addition of fractions. Yet, all those who incorrectly responded to these problems (about 15% of the participant prospective teachers) added the numerators and the denominators.
The activity on addition of fractions, which was developed for this study, was aimed at enhancing prospective teachers' algorithmic, formal, and intuitive content knowledge and the interconnections between these types of knowledge. Another main aim was to increase prospective teachers' related pedagogical knowledge. We shall first describe this activity.
Task 1.
a. Solve the following problems:
b. List the arithmetic operations you executed while solving these problems. What should one know in order to be able to solve these problems?
c. Solve the following problems:
Task 2. Multiplication of fractions is simple:
Addition of fractions is much more complicated.
Why wasn't it decided to add fractions in the following way:
(Try to give more than one reason)
Task 3. Ran, Dan, Dorit and Alon solve the problem 
in the following way:
Ran explains that "It is reasonable to add the numerators and the denominators, as in this way we do the same thing in both addition and multiplication".
Dan explains that "When we add, we always add things of the same kind, for instance, ones to ones, tens to tens. In this case, we shall do the same; add numerators to numerators, and denominators to denominators".
Dorit explains that: "It is easy and simple to add this way".
Alon gives an example: "In a basket-ball game, Miki Berkovitz {a well-known Israeli basketball player) hit, during the first part of the game, 5 out of 8 attempts, in the second part, he succeeded in 7 out of 12. So, during the entire game he hit 12 out of 20 trials, that is
Assume that you are the teacher in this class. How would you react to each of these responses?
Task 4. Here are three different ways of writing a solution
to the problem 
Is the first way correct?
Is the second way correct?
Is the third way correct?
Discuss the pros and cons of each of these ways.
The 14 prospective teachers who participated in this activity were asked to think about each of these four tasks. Then, in respect to Task 3, they were asked to work in pairs, to simulate the four situations, each participant acting as a student in two of the cases and as the teacher in the other two cases. Afterwards, they presented their simulation in class.
This activity opened the stage to a thorough discussion on algorithmic, formal and intuitive aspects of addition of fractions and their interconnections. A large part of the discussion was devoted to various issues concerning the nature of mathematical operations and definitions, children's conceptions and ways of thinking and the learning and teaching of mathematics. We shall briefly describe only some of the issues (and emotions) evolving in class.
Of the 14 prospective teachers who participated in this session, 12 solved the addition problems presented in the first task correctly, and were surprised to find out how much knowledge was needed for getting there (e.g., addition, multiplication, division, finding the common denominators, etc.). The other two prospective teachers first added the tops and the bottoms, but after discussing their solutions with other members of the class, they gave the appropriate solution. These subjects explained that for a moment they thought about each fraction as two different numbers. Later on, when the class watched the video that documented their activities during the lessons, a discussion developed as to the frequency of such reactions among children. It was then suggested to the participating prospective teachers to read several articles on this issue (e.g., Herskowitz, Vinner, and Bruckheimer, 1978; Carpenter, Coburn, Reys, and Wilson, 1976).
All prospective teachers argued, in response to the second task, that it was impossible to add fractions in the suggested way (i.e., adding the tops and the bottoms). They were surprised to encounter such a question, and commented that they never thought about the sources of the definitions of operations, or about the possibility of an alternative definition that would be totally different from the one currently used. Aviv, for instance, explained that "Such questions should not be asked. Addition is defined in a certain way. There are some rules in mathematics, and we operate accordingly. These rules often do not seem reasonable, but still we have to operate according to them ... We have to accept these rules as they are, and not to wonder about them."
The other prospective teachers used two types of arguments
to justify their conclusion that addition could not be defined
in the suggested way: mathematically-based arguments and practical
arguments. Prospective teachers whose reactions were classified
as mathematically-based explained that the suggested definition
does not satisfy certain, mathematical principles. Anat, for instance,
explained that "
should be greater than 
, since you added a positive number to a given
number. But, if we use the suggested way to add these numbers,
we get 
, and is not greater than ".
Other reactions included in this category related to the consistency
between the two definitions: the conventional definition of fractions,
and the suggested one. Yael, for instance, explained that "adding
the tops and the bottoms leads to an answer that is inconsistent
with the answer you get when you use the correct way of adding
fractions, and thus, it is impossible to add fractions in the
suggested way". Yael was not able to free herself from the
definition she was familiar with and to consider the suggested
definition on its own merit.
Practical arguments showed that the suggested definition does
not match real-life situations in which addition of fractions
is the appropriate operation. Miriam, for instance, suggested
to consider the recipe for a cake, in which 
kilogram
wheat is added to 
kilogram sugar. "Altogether
we'll have 
kilogram, and not 
kilogram".
The third task was very demanding for all prospective teachers. The request to react to students' conceptions and ways of thinking encouraged them to reflect on their own conceptions of addition of fractions and to appreciate the complexities of understanding and explaining this operation. Their attempt to respond to each of the students, and especially to Alon's suggestion, challenged their own conceptions of numbers, mathematics definitions, mathematics and mathematics instruction. While working on preparing the simulation games, they raised various issues and discussed them among themselves. Some of these issues were:
1. Issues related to addition of fractions
- Why do we look for the common denominator and not the common numerator?
- Why do we teach addition of fractions in the usual, complicated way, when the other, easier way of adding tops and bottoms works too?
- How do we determine whether, in a given context, we need to add fractions in the usual way or differently?
- Sometimes the correct solution to 
, at
other times it is 
. Thus, there are two solutions
to this problem. However, mathematical operations should result
in only one number. What should be done in such situations?
2. Issues related to definitions of operations
- How do mathematicians make decisions about definitions of operations? What are the reasons behind choosing a certain definition?
- What characterizes definitions of operations?
- Should definitions be proved?
- What is the relationship between mathematics and real-life situations?
3. Issues related to children's ways of thinking and to mathematics instruction
- What are the sources of students' intuitive solutions to addition of fractions?
- Should students be allowed to provide their own suggested definitions? When?
- How should a teacher respond when he does not know the answer?
- Should we allow students to listen to incorrect, apparently reasonable suggestions made by other students in the class?
- Should we present, in class, anticipated common incorrect solutions even before they have been made by the students in the specific class?
- What type of justifications should be used in elementary classes (e.g., should students be exposed only to practical arguments or also to mathematically-based ones)?
- Which of the issues discussed in our class could (or should) be discussed in elementary classes?
In class, as the various pairs presented their simulation games,
audience became greatly involved and offered comments that could
have been made by students, to further explore the nature of each
argument. (The teacher of the prospective teachers' class also
acted as a student, and presented some challenging questions).
The discussions around Alon's suggestions were particularly revealing.
The prospective teachers who acted as Alon's teachers struggled
with his argument and felt frustrated by the fact that the answer
provided by adding the tops and the bottoms was correct while
the usual way of adding fractions led to an inappropriate answer.
At a certain point, the entire class started working on this issue,
and various suggestions were made for dealing with it. In this
attempt, the class decided, at a certain stage, to form some problems
for which the usual way of adding fractions does work, and problems
for which it does not, and then to attempt to identify the similarities
and differences between these problems. While working on creating
this, they realized that in the problems for which the usual way
of adding fractions was effective, the unit of reference needs
to be the same for the two addends, and that this does not hold
for the other type of problem. The participants got deeply involved
in this inquiry, and continued this investigation in various directions
(e.g., Can 
be greater than 
? Do other mathematical
operations also require the same unit of reference?). The entire
class cooperated on these issues, and learned a great deal of
mathematics in doing so.
The fourth task, much like the third one, urges the prospective teachers' to consider their usual manner of adding fractions in an attempt to explain its rationale. Some of them realized that they were unable to go beyond describing the successive steps they apply to this type of problem (find a common denominator, divide it by the denominator of the first fraction, multiply the number you got by the numerator, etc.). Some of them used different manipulatives in order to construct a meaningful way of adding fractions.
The discussion on the pros and cons of each of the suggested ways of adding fractions raised various pedagogical issues. One debate centered on the use of "general method" (i.e., multiplying the two denominators to get the common denominator) versus "specific methods" (i.e., finding the lowest common denominator). Both these terms were coined by the prospective teachers to designate the differences between the methods, to imply their characteristics, and to allow for fluent communication. Some prospective teachers argued that the general method was more powerful, and always preferable in mathematics. Their justification was that it is a main aim of mathematics to form generalizations that can be used regardless of the nature of numbers involved in the specific task. Other prospective teachers argued in favor of the specific method, that finding the lowest common denominator requires the application of previously acquired knowledge related to multiplication, division, and prime numbers. They explained that the task of finding the common denominator offers the learners the opportunity to apply their knowledge in a new situation.
In an attempt to decide on this issue, the prospective teachers discussed the essential differences between the two types of arguments and several other issues regarding the criteria to be used to determine the usefulness of algorithms. Some of the questions raised during this conversation were: Should students be encouraged to discover, by themselves, methods of adding fractions? Can students be expected to do this? is it necessary to agree on one strategy for adding fractions, or can different members of a group use their own, preferred strategy? Is it recommended to expose elementary school students to various strategies for adding fractions?
This fourth task, much like the ones preceding it in the initiating activity, urge the prospective teachers to rethink the rules and assumptions which they generally accepted as self-evident and unquestionable. Often, the discussion went beyond the specific case of addition of fractions to more general mathematical as well as pedagogical issues.
In the last decade, a number of studies have reported on educational activities which aimed to enhance prospective and/or inservice teachers' mathematical and pedagogical content knowledge (Arcavi, Tirosh and Nachmias, 1989; Ball, 1988; Confery, 1990; D'Ambrosio and Campos, 1992; Even and Markovits, 1991; Fenemma, Carpenter, & Peterson, 1989; Maher and Alsoton, 1990; Simon, 1993, Steffe, 1991; Tirosh, 1993; Wood, Cobb, and Yackel,19XX). Some of these used mathematical topics or representations that the prospective teachers had never before encountered so as to engage them in studying mathematical topics they had not studied before and in reflecting on the cognitive processes involved in these activities. Others attempted the same goals by presenting prospective teachers with real-world problems which involved them in generating a mathematical model for the problem context. Still other studies engaged the prospective teachers in research on children's knowledge of mathematical concepts.
The activity presented in this chapter applied research findings on children's ways of thinking about addition of fractions as a means of enhancing prospective and/or inservice teachers' mathematical and pedagogical content knowledge about rational numbers. Other aims of this activity were to encourage prospective teachers to perceive mathematics as creative, dynamic and vital, and to improve their self-esteem regarding their mathematics potential. Our impression, after having tried out the activity, is that it has several potential uses in teacher education.
First, it can serve as a means of exploring prospective teachers' conceptions of addition of fractions, their understanding of related mathematical concepts, and their beliefs about mathematics and mathematics instruction. In line with previous results (Ball, 1990; Even and Tirosh, 1995), our findings indicate that when asked to respond to specific suggestions made by students, teachers are pushed to articulate their own understanding. Thus, in turn, they provide teacher educators with an opportunity to study adult learners' cognitive processes and conceptions.
Second, the activity can serve as a means to raise prospective teachers' awareness of their own inadequate conjectures, and to encourage modification. Our findings indicate that teachers' exposure to students' suggested definitions encouraged many of the former to unclog their knowledge, to reflect on their responses and to reconsider and re-evaluate their judgments.
Third, this activity could be used to elicit discussions on issues related to mathematical operations, definitions, and related concepts. The request to respond to students' suggested definitions led the majority of the teachers to reflect on their own understanding of this topic and to realization that they could only repeat certain conjectures but not explain them, either to themselves or to others. This encouraged them to raise questions related to mathematical operations and to discuss them in class.
Fourth, the activity serves as a springboard for discussing
a number of central issues related to the nature of mathematics
and mathematics instruction. One of these concerns the difference
between internal, mathematically-based arguments and external,
practically-based justifications. With reference to addition of
fractions, an internal, mathematically-based argument for not
defining addition of fraction as "add tops, add bottoms"
is that this definition is not single-valued (e.g., according
to this definition 
does not equal to 
). External,
practically-based arguments, which are commonly used during instruction,
depend on the described situations, i.e., on the particular contexts.
This led prospective teachers to a discussion on the nature of
the arguments they would like to use in their future classes.
Issues related to the relationship between stages of development
(in Piagetian terms), maturity, and types of argumentation's were
raised.
Finally, this activity, which is based on students' known difficulties, serves as a means of increasing teachers' awareness of students' understanding and misunderstanding, and thus improves their pedagogical content knowledge. The prospective teachers were curious to learn about other cognitive obstacles related to this specific topic as well as to other topics they intended to teach.
While working on the different assignments included in this initiating activity or watching the video that documented their work on it, the prospective teachers realized the vast diversity among them in terms of the methods they used for solving the addition problems, their conceptions of fractions and of operations with fractions, and their beliefs about mathematics and mathematics instruction. Attention was also paid to the social interaction and the social climate in the class, and, in particular, to the teacher's role in attempting to encourage students to describe their arguments in a way that will be comprehensible to the entire class. The prospective teachers acknowledged that the open, accepting atmosphere created a feeling that it was legitimate to make mistakes. They moreover felt that these mistakes were often used as a springboard for enhancing the entire class's mathematical understanding as well as their acquaintance with children's ways of thinking. It was our impression that the prospective teachers came to realize that knowing mathematics, much like other types of human knowledge, is not a matter of all or nothing; that there are different, legitimate ways of exploring mathematical situations; and perhaps most important: that they themselves can learn mathematics and enjoy it.
Many comments of the participants prospective teachers at the beginning of our sessions reflected their perception that their main role as elementary teachers of mathematics is that of transmitting to their students the information that is printed in mathematical textbooks; that there is one adequate, best way to introduce a given mathematical content to the students, that the ultimate goal of teaching a certain topic is that students will use the (one) correct way to reach the expected answer, and that the teaching of mathematics is a "step by step" process (see also Cooney, 1994). Gradually, the prospective teachers started considering the applicability of the type of instruction they experienced in the presently described class, to their own class to their future classes. In doing so, they related to the (various) conceptions children bring to the learning situation, to the importance of attending to students' ways of thinking, and to the viability of letting students struggle to resolve confusion while the teacher acts as a facilitator in this process. The participating prospective teachers learnt a great deal of (and about) mathematics during these sessions. Yet, so far we have not followed them into their own classes, and thus at this stage of our study we lack the information needed to evaluate the effects of the approach on participants' own teaching. We agree with the claim that teachers educators "Takes too great a leap of faith in assuming that a study of advanced mathematics, of children's misconceptions, of philosophies of mathematics, and of strategies for teaching mathematics can provide, by itself, a foundation from which changes in the teaching of mathematics can be realized (Cooney, 1994). Thus, the next stage of our research deals with this central issue.
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