This research was supported by the United States-Israel Binational Science Grant (#88-00213/1). 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.
Mathematics educators involved in elementary teacher education programs often voice their dissatisfaction with students' content knowledge of and attitudes toward mathematics. Prospective elementary teachers' mathematical knowledge is frequently described as insufficient (Ball, 1990; Even 1990; Graeber, Tirosh, and Glover, 1989; Reys, 1974; Simon, 1993; Wheeler, 1983). It is also often reported that prospective teachers view mathematics as a static body of knowledge composed of facts, methods and rules understood by only a very small portion of the population, while the rest dislike mathematics and believe they are doomed to failure in it (Becker, 1986; Baxter, 1983; Bulmahn and Young, 1982, Rees and Barr, 1984; Steffe, 1990; Tirosh, 1990).
This gloomy picture sets a real challenge to those teacher educators who believe that students' ways of thinking and attitudes toward a subject are a main component in any form and level of instruction. Clearly, comprehensive knowledge of prospective teachers' conceptions of the specific mathematical topics included in their curricula is essential for instruction in teacher education in much the same way as knowledge about children's conceptions is crucial in school instruction. In fact, it makes no sense to base teacher education programs on the assumption that teachers are tabula rasa than to assume that students enter their classrooms void of a wide range of conceptions if mathematics (Cooney, 1994). Thus some major, related goals for mathematics educators are: (1) to analyze prospective elementary teachers' ways of thinking about particular mathematical topics, (2) to develop didactic approaches that take account of these conceptions while aiming to help prospective teachers extend their own mathematical conceptions of these topics, and encouraging them to expand their notion of mathematics as a human activity, and (3) raising prospective teachers' confidence in their ability to do mathematics. Fulfilling these goals is of special importance as today's prospective teachers are tomorrow's teachers, and teachers are key figures in bringing about educational improvement (NCTM, 1989; NCTM, 1991).
A major part of the curricula of elementary schools, in most countries, is devoted to non-negative rational numbers. There is a considerable body of research on students' conceptions of rational numbers (e.g., Behr, Harel, Post, and Lesh, 1992; Carpenter, Fennema, and Romberg, 1993; Greer, 1992). Many researchers indicate that children experience difficulties with rational numbers and several reasons for these difficulties has been suggested: (1) children do not have the same everyday experience in using rational numbers as they do with natural numbers (Greer, 1994), (2) many children find it difficult to accept a given fraction as a number and tend to view it as two whole numbers (Hart, 1981; Kerslake, 1986), and (3) children often incorrectly attribute observed properties of operations with natural numbers to those with rational numbers (Bell, Fischbein and Greer, 1984; Hiebert and Wearne, 1986; Nesher, 1988; Owens, 1987; Sowder, 1988). Another source of children's confusion is the need to recognize and work with various interpretations of, and notations for, rational numbers (e.g., part-to-whole comparison, decimal, ratio, operator, indicated division, and measure of continuous and discrete quantities; Behr, Lesh, Post, and Silver, 1983; Freudenthal, 1983; Greer, 1992). In the light of these findings, it is notable that national and international studies consistently show that non-negative rational numbers are a stumbling block for many students (Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981; Carpenter, Lindquist, Brown, Kouba, Silver, and Swafford, 1988; Hart, 1981; Hiebert, 1988; Kieren, 1988; Travers and Westbury, 1990).
Given the central role of rational numbers in school mathematics, students' widely acknowledged difficulties in learning this topic, and the crucial role that teachers' subject matter knowledge of particular school topics plays in the instructional process (Ball, 1988; Leinhardt, Putnam, Stein and Baxter, 1991), it is somewhat surprising that only a few researchers have begun to study the knowledge base prospective teachers bring to the instruction of rational numbers. These researchers investigated the correctness and justifications of prospective elementary teachers' knowledge of division by fractions (Ball, 1990; Simon, 1993; Wheeler, 1983), their performance in solving multiplication and division word problems involving fractions (Greer and Mangan, 1986; Graeber and Tirosh, 1988) and their beliefs about multiplication and division (Tirosh and Graeber, 1990a). It has been shown that prospective elementary teachers' knowledge of rational numbers is procedural and sparsely connected. Although many prospective teachers could do calculation with rational numbers, they had significant difficulties in solving multiplication and division word problems, and their beliefs about the operations of multiplication and division were based solely on operations with natural numbers, and not adjusted top operations with rational numbers (Ball, 1990; Simon, 1993).
Each of these studies on prospective teachers' mathematical knowledge of rational numbers related to one or two specific aspects of this knowledge. In this chapter we shall describe and discuss a project one of whose main goals was to develop a conceptual framework for analyzing prospective teachers' mathematical knowledge of rational numbers. The Conceptual Adjustments in Progressing from Whole to Non-negative Rational Numbers (CAPWN) project is funded by the United States-Israel Binational Science Foundation, and four principal investigators are involved in it: James Wilson, from the University of Georgia, Anna Graeber, from the University of Maryland, and Efraim Fischbein and Dina Tirosh, from Tel-Aviv University. The project takes place in Israel and is currently in its third year.
In what follows, we shall present the major goals of our project, the conceptual framework that was used to analyze prospective teachers' mathematical knowledge of rational numbers, and some of the main findings. Then we shall provide an example of one situation taken from sessions with a class of prospective teachers that was particularly illuminating in terms of gaining a better grasp of the processes that take place when prospective teachers amend their mathematical content knowledge and revise their conceptions of the nature of mathematical activities, and of themselves as learners of mathematics.
The four main aims of this project are:
1. To understand children's and prospective elementary teachers' conceptions of rational numbers.
2. To develop didactic approaches aimed at helping prospective teachers extend their own mathematical conceptions of rational numbers and their knowledge about children's ways of thinking about them.
3. To involve prospective teachers in activities that encourage them to expand their notion of mathematics as a human activity.
4. To help prospective teachers decrease their mathematics anxiety and raise their confidence in their mathematics ability.
We shall deal, in this chapter, only with the part of the research that refers to prospective teachers. Let us start with describing the conceptual perspective we used to analyze prospective teachers' mathematical knowledge.
The conceptual framework for analyzing children' and adults' content knowledge of specific mathematical topics used in our project is based on the assumption that learners' mathematical knowledge is embedded in a set of connections among algorithmic, intuitive and formal dimensions of knowledge (Fischbein, 1983).
The algorithmic dimension is basically procedural in nature -- it consists of the rules and prescriptions of mathematics. This dimension also involves students' capability to explain the successive steps included in various, standard procedural operations.
The intuitive dimension comprises our ideas and beliefs about mathematical entities and the mental models we use for representing number concepts and operations with them. Intuitive knowledge is characterized as the type of knowledge that we tend to accept directly and confidently as being obvious, without feeling that it needs proof, and as having an imperative power, that is, it tends to eliminate other alternative representations, interpretations or solutions (Fischbein, 1987). Fischbein, Deri, Nello and Marino (1985) suggested, for example, that students' performance in solving mathematical word problems may be guided by primitive, behavioral models that do not apply to new concepts they encounter (e.g., repeated addition is a model for multiplication that works well for whole numbers but is too constraining for rational numbers).
The formal dimension includes the definitions of the concepts and of the operations, structures, and theorems relevant to a specific content domain. This type of knowledge is represented by axioms, definitions, theorems and their proofs. Both the formal and the algorithmic dimensions of knowledge can, for the student, become highly procedural and rote. Their vitality depends upon the student's constructing consistent connections among algorithms, intuitions and concepts.
Ideally, these dimensions of knowledge should cooperate in the processes of concept acquisition, understanding and problem solving. In reality, though, this is not always the case -- often there are serious inconsistencies between students' algorithmic, intuitive and formal knowledge. Such inconsistencies could be the source of common difficulties learners encounter in their mathematical activities, such as misconceptions, cognitive obstacles and inadequate usage of algorithms.
The three dimensions of knowledge are not discrete; they overlap considerably. In fact, any mathematical activity requires the use of all three of them. Nonetheless, it is useful to focus on them separately when the aim is to assess a student's mathematical knowledge with regard to a certain domain.
The following section, which describes prospective elementary teachers' knowledge of rational numbers, illustrates how the suggested framework is used to analyze learners' knowledge of rational numbers.
Prospective elementary teachers' mathematical knowledge of rational numbers was assessed during the first and second years of the project. We shall briefly describe the methodology and some of the findings related to this part of the project.
Subjects
One-hundred-forty-seven elementary prospective teachers in their first year of teachers' college participated in this part of the study. Twenty-six of them were mathematics majors while the other 121 students had chosen a different main subject (biology, music, English, etc.).
The diagnostic questionnaire
All subjects were asked to complete a diagnostic questionnaire which examined their formal, algorithmic and intuitive understanding of rational numbers. The diagnostic questionnaire examined the following aspects of prospective teachers' knowledge of rational numbers:
a) The algorithm dimension: Ability to compute with rational numbers and to explain the successive steps of the standard algorithms used for operations with fractions and decimals. Other components examined, but not reported here, are related to transformation from fractions to decimals and vice versa, and undefined mathematical expressions involving fractions.
b) The formal dimension: Ability to define rational and irrational numbers, knowledge related to the hierarchy of several sets of numbers (natural numbers, integers, rational numbers, irrational numbers, real numbers), to the set membership of various numbers and to the density of the rational numbers, and familiarity with the commutative, associative and distributive laws.
c) The intuitive dimension: Capability to produce adequate intuitive models for representing number concepts and operations with them, and competency in evaluating the adequacy of statements related to arithmetic operations (e.g., "division always makes smaller", "multiplication always makes bigger").
Some examples of the various types of items included in the diagnostic questionnaire are provided in Appendix 1 and Appendix 2.
The interviews
Three phases of interviews were conducted at the first stage of the study. These interviews were conducted to further our understanding of the prospective teachers algorithmic, formal and intuitive dimensions of knowledge and their interconnections.
Phase 1: Algorithmic and intuitive understanding of rational numbers.
Twenty five prospective teachers who exhibited difficulties in solving multiplication and division problems had at least three 20-40 minute interviews.
Each interview consisted of two parts: The main part consisted of questions outlined specifically for each subject, based on her responses to the multiplication and division algorithmically-based problems. During the interview, each subject was asked to (1) perform drills which were similar to the ones she missed on the diagnostic questionnaire, and to explain her performance, and (2) to explain the rationale of the various algorithmic procedures. The second part included tasks presented to all interviewees. They were asked to determine the correctness of several statements related to the operations of multiplication and division.
A sample outline of an interview for one of the subjects is provided in Appendix 1.
Phase 2: Prospective teachers' formal and intuitive understanding of rational numbers.
Eighteen subjects who exhibited serious deficiencies in their formal understanding of rational numbers were interviewed. The interviews were semi-structured, that is, an "interview program" was outlined for all subjects, yet additional probes were made during interviews to better understand the subjects' conceptions. Each prospective teacher had at least three 20-45 minute interviews.
A full description of the interview program is provided in Appendix 2.
Phase 3: Prospective teachers' facility in representing rational numbers.
Thirty-two non-mathematics majors, constituting all the students in two of the classes involved in this study, were selected as participants in this phase of the interview. The participants were asked to:
- construct as many representations as possible of
- find the unit of a given representation (area, number line
and set) of the stated fractional part 
- identify critical features of representations of 
, and
of 
- indicate how they would respond to students who gave particular
representations of 
(some of the representations
were appropriate while others were not).
Each prospective teacher had at least 20-45 minute interviews. A sample of some of the tasks included in these interviews are provided in Appendix 3.
A. Prospective teachers' algorithmic knowledge of rational numbers
Most prospective teachers were successful in performing addition,
subtraction and multiplication with fractional numbers and with
decimals. Prospective teachers who incorrectly solved these problems
showed the same, incorrect response patterns as did school students
on these tasks (e.g., adding numerators to obtain the numerator
of the sum, and similarly, adding the denominators). The situation
with respect to solving division problems with fractions and decimals
was less satisfactory (see Table 1). The mathematics majors performed
better than the non-mathematics majors in these computational
tasks, but neither of the two groups performed satisfactorily.
The most problematic items were those involving division of decimal
numbers -- most of the mathematics-major prospective teachers
who solved these items correctly first wrote the decimals in their
fractional form and then solved these items. The most frequent
incorrect responses to the decimal problems involved incorrect
replacement of the decimal points, or a statement to the effect
that this problem could not be performed, the dividend being smaller
than the divisor. Incorrect responses to division in its fractional
form included taking the inverse of the dividend instead of that
of the divisor, and, in cases such as 
: 3,
multiplying both the numerator and the denominator by 3.
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The prospective teachers were generally unable to justify the successive steps of the standard algorithms of arithmetic operations with decimals and fractions. We shall refer, for instance, to prospective teachers' responses to the item which relate to the placement of the decimal point in multiplication of decimal numbers (see Appendix 1, problem 2). Although all participants knew that the student's answer was correct, only one was able to justify the place of the decimal point in the product. The others said that they simply "do it" this way. They were unable to call upon their knowledge of fractions or their knowledge of multiplication with whole numbers as possible sources for answers. In fact, they were quite surprised there should be a need to explain why certain algorithmic steps are performed the way they are. During the interviews, when they were encouraged to attempt to inquire why, when multiplying decimal numbers, the decimal point is put, at the end of the process, at a certain place, the vast majority of them expressed beliefs to the effect that "If you don't know how to explain it you won't be able to find it by yourself, no matter how hard you think about it. Such knowledge could not be reached by us. Someone who knows the reasons has to tell us; then we shall remember them and tell them to our students".
With regard to division of decimals, the prospective teachers were presented with the following item:
A student solves the expression 5 : 0.8 in the following manner:
1st step
2nd step
3rd step
1. Explain steps 1 and 2.
2. What is the answer to the original division problem (5 : 0.8)?
Most prospective teachers knew that the student's answer is correct, yet only eight provided adequate explanations for steps 1 and 2. The most frequent, inappropriate answer was 0.625, as "we need to divide the answer by 10 in order to compensate for the previous multiplication of the dividend and the divisor by 10".
B. Prospective teachers' formal knowledge of rational numbers
Substantial differences were identified between the performance of the mathematics majors and that of the non-mathematics majors on each of the items included in this category. While the vast majority of the mathematics majors (92%) correctly defined rational and irrational numbers, only 23% of the other were able to do so; 81% of the mathematics majors and 25% of the non-mathematics majors drew an adequate Venn diagram to describe the relations between the natural numbers, the integers, the rational numbers, the irrational numbers and the real numbers; 92% of the mathematics majors correctly described the commutative law, while only 54% of the non-majors did so; 85% of the mathematics majors gave a correct description of the distributive law while only 12% of the non-majors did so; and finally, 77% of the mathematics majors were able to state the associative law while only 3% of the non-mathematics majors did so.
By and large, the mathematics-major prospective teachers showed adequate formal knowledge of rational numbers. The non-major mathematics prospective teachers' formal knowledge, however, was insufficient, incomplete, and dull.
In the second year of this study, prospective teachers' knowledge
of two additional components related to the formal dimension were
explored among 37 non-mathematics major prospective teachers:
identification of the set membership of various numbers, and the
density of the rational numbers. Prospective teachers were asked
to determine the set membership of 10 numerical expressions. For
instance, they had to determine if -3 is a natural number, an
integer, a rational number, an irrational number, and/or a real
number. Their performance on this item was very poor. Only 8%
and 22% of the prospective teachers, respectively, knew that 0
and 0.251 are rational numbers. Twenty-four percent argued that
2/0 is an integer. Confusion reigned concerning the notion of
real numbers -- the majority of the prospective teachers mistakenly
identified "real numbers" with "positive numbers",
and consequently argued that all the given numerical expressions,
except 
and -3, were real numbers. Others argued that real
numbers were "nice numbers", namely, that 0.42, 
, 0.251, 
, and -3 were real numbers,
while 0.121221222, ¸, and 0 were
not.
Only few prospective teachers were aware of the density of
rational numbers: 24% knew that between 
and
there are infinite amounts of numbers. In fact, 43%
claimed that there are no numbers between and
, and 30% claimed that is the
successor of . With regard to decimals, the results were somewhat
better. For instance, 40% knew that between 0.23 and 0.24 there
is an infinity of decimals and could even present some of them.
C. Prospective teachers' intuitive knowledge of rational numbers
Many prospective teachers held primitive beliefs concerning the results of multiplication and division. These primitive beliefs were widespread both among those who did and those who did not major in mathematics. In fact, 60% of the prospective teachers incorrectly argued that "multiplication always makes bigger", and 51% incorrectly argued that "division always makes smaller".
As expected, the vast majority of the prospective teachers used common ways of illustrating fractions, including graphic representations (either disks or rectangular regions), concrete representations (e.g., apples, chocolate bars) or verbal representations (i.e., word problems) to construct representations of fractions and operations with fractions. Table 2 shows that most prospective teachers drew appropriate representations of one-third, in which the unit was evident, the three parts appeared to be reasonably equal and one of the three parts was designated. The representations of operations involving fractions were much more difficult tasks for the prospective teachers. Only few constructed appropriate representations of multiplication and division expressions involving fractions. No substantial differences were found between mathematics major and non-mathematics major prospective teachers' intuitive knowledge of rational numbers.
Most prospective teachers lacked the ability to construct appropriate representations of fractions and operations with rational numbers.
| Appropriate Representations | Inappropriate Representations | |||||
| Verbal | Graphic | Concrete | Verbal | Graphic | Concrete | |
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(*) some of the prospective teachers provided more than one representation for the given expression
In order to better understand the prospective teachers' difficulties
with these items, we conducted intensive individual interviews
with the subjects. They were asked, among other things, to construct
as many different representations as possible for 
and
and to comment on the appropriateness of given
representations (area, set, number line, and ratio) of 
and . Some of the appropriate and inappropriate representations
that were used in these interviews were taken from the prospective
teachers' responses to the diagnostic questionnaires. Other representations
used during this interview were adapted from the research literature
on children's conceptions of mathematical operations (see Appendix
3, item 2 for a description of some of the representations used
during this phase of the interview).
It was found that prospective teachers rely primarily on the
part-whole interpretation of fractions. Most of them drew appropriate
area models for 
and 
. Despite
the instruction to give as many representations as possible, most
prospective teachers gave only one, the area model representation
of these fractions. When asked to give alternative models, most
prospective teachers simply changed the geometrical shape used
in the area models. Some referred to concrete objects such as
watermelons or chocolate bars. Very few constructed set models,
and no number line or ratio models were produced.
The predominance of the part-whole interpretation and of the
area model of fractions seriously limited the subjects' ability
to represent the fraction 
. Only two prospective teachers
drew successful models of ; both were area models. The
approach and comments of those who were unsuccessful revealed
some of the difficulties with representing improper fractions.
A typical approach was to draw a rectangle, portion it into three
parts, and then append two more same-sized parts. This approach
resulted in a rectangle partitioned into five equal sections.
Many then looked at their drawing and evaluated it as incorrect,
making comments such as "You can't illustrate out
of the whole".
When the prospective teachers were asked to comment on given
number line and set model representations of 
and 
, the predominance of the area model was much in
evidence. The prospective teachers tended to reject a set model
of 
in which one of three triangles with unequal areas
was marked. They justified this response with the argument that
"the areas of the three figures included in the set are not
equal, and thus one triangle can't represent on-third." In
commenting on the appropriateness of number line models, many
prospective teachers readily accepted a number line on which points
0, 1, 2, and 3 were labeled and the segment between zero and one
was shaded, as a model of one-third. They justified this by saying
that the segment from 0 to 1 was one of the three parts (0 to
1, 1 to 2, and 2 to 3). None of them mentioned that on the number
line the unit is generally defined by the distance from zero to
one and that one-third is associated with a point situated one-third
of the way from 0 to 1.
The representation of was a more difficult task for
the prospective teachers than that of , or
even . Representations were considered appropriate if
1) the unit for the two addends were the same and evident, 2)
one-half was distinguished; one-third was distinguished, and 3)
the union of the two parts was identified by some means other
than use of a plus sign between two disjoint parts. Only about
a third of the prospective teachers gave representations that
met these criteria, and all of these were area models. Most subjects
whose responses were classified as inappropriate did not recognize
the need to use the same unit for the two addends.
The predominance of the part-whole interpretation and of the area model interferes with many other representations of rational numbers. This restricted perception directed prospective teachers towards a narrow view of rational numbers. Clearly, it is important to promote prospective teachers' familiarity with various representations of rational numbers, as different representations emphasize different aspects of these numbers.
D. Interconnections between algorithmic, formal and intuitive knowledge
In the course of analyzing the data from the diagnostic test and from the various interview sessions, we became more and more aware of apparent contradictions between prospective teachers' algorithmic performance, their formal knowledge and their intuitive beliefs about the arithmetic operations. The most prevalent inconsistencies became apparent when comparing computational skills with their intuitive knowledge about the mathematical operations. For instance, all the subjects who argued that division always makes smaller(about a half of the prospective teachers), correctly computed at least some of the division problems that resulted in a quotient which was greater than the dividend.
The various interview sessions gave insight into possible sources of these inconsistencies. we shall not be able to thoroughly discuss issues related to these inconsistencies within the framework of this paper. The reader could refer to some other papers that deal with such discrepancies (Ball, 1990; Simon, 1993; Tirosh and Graeber, 1990a; Tirosh and Graeber, 1990b).
Generally speaking, the prospective teachers' mathematical knowledge was found to be rigid and segmented. Most of them operated almost totally with rigid algorithms and procedural knowledge about rational numbers concepts, were generally unable to justify the successive steps of the algorithms, held primitive models of the operations, could not produce adequate representations of rational number concepts or operations with rational numbers, and lacked a representation of mathematics as an organized and structured body of knowledge. Mathematics for most of them was a mere collection of computational techniques not well mastered, unjustified formally, indeed often even intuitively.
The mathematics-major prospective teachers' performance on most algorithmic and formal items somewhat exceeded that of the non-majors. These differences in performance may be explained by two main factors: (1) the mathematical background of mathematics majors is stronger than that of non-mathematics majors (the former took more mathematics in high school), and (2) in general, mathematics majors exhibited more positive attitudes toward mathematics and towards their ability to study it than did non-mathematics majors. However, as noted previously, the mathematics knowledge of both these groups of prospective teachers was insufficient, rigid and segmented.
We have tried to explain the reasons for this lack of knowledge on the part of the prospective teachers. The most probable explanation we can come up with is that the curricula in the teachers' colleges do not, in general, sufficiently emphasize the mathematical topics taught in elementary school class. Prospective teachers learned these notions in their childhood, when the accent may have been (excessively at times) on concrete representations and particular solving techniques. Consequently, they will only have accumulated fragmentary knowledge without an integrating perspective, and obscured behavior with rational numbers, therefore, is often limited to recalling and applying memorized symbol manipulation rules.
This explanation is supported by the fact that the difficulties our subjects experienced with rational numbers are often similar to the ones children are known to confront. This may also indicate that the same mechanisms of reasoning are at the source of both children and adults' limited understanding of rational numbers. It seemed that students' and prospective teachers' conceptions of numbers were almost entirely based on natural numbers. This allowed them to partially operate on the algorithmic level (i.e., using memorized algorithms to perform given, computational expressions), but blocked their ability to cope with tasks that demanded adequate formal and/or intuitive knowledge of rational numbers.
The main conclusion we draw from this part of the research is that elementary arithmetic topics should be included in the curriculum of the teachers' colleges, but considered from a different perspective. Such a curriculum should attempt to expand prospective teachers' formal, algorithmic and intuitive understanding of the concepts and processes included in the elementary school curricula and the interactions among them. Prospective teachers should be exposed to historical perspectives on the concepts at issue and to the course of their evolution. They should become acquainted with the research findings that are relevant to the understanding of the learning processes of these concepts. They should be encouraged to adopt the view that the mistakes they and their future students make may stem from a systematic line of thinking and that some of them are to be regarded as conceptual obstacles whose identification and treatment is essential to the process of learning these tasks. Prospective teachers should be encouraged to reflect on their own thinking and to analyze that of their students.
In respect to rational numbers, it is essential that the curricula for elementary school teachers attempt to increase their familiarity with common representations of and with them; boost their understanding of the similarities and differences of these representations, and their ability to use them flexibly. Such curricula should also encourage prospective teachers to engage in activities aimed at improving their computation skills and their understanding of the rationale behind the algorithms. It is also essential that prospective teachers become acquainted with the definitions of the set of real numbers and its subsets, and develop a clear idea of the interrelations between the various sets of numbers. Such knowledge, on the part of the teacher, seems necessary for allowing him/her to introduce rational number concepts in a manner that will help students develop strong conceptual underpinnings of rational numbers and of the processes underlying their operations.
In the course of the study we observed that the prospective teachers experienced significant difficulty when reflecting on their own thinking and analyzing the possible sources of their own mistakes. The prospective teachers were generally unable to list students' common ways of thinking about rational numbers. It was decided that the curricula for prospective teachers should reflect the accumulating knowledge related to children's understanding of rational numbers and to their difficulties in grasping the pertinent concepts and processes. More specifically, the various stumbling blocks related to the transition from natural to rational numbers -- difficulties that have been discussed quite extensively in the research literature during the past two decades -- have to be addressed by the curricula for elementary school teachers (e.g., the natural tendency to transfer the constraints of operations with natural numbers [such as 'the dividend is never smaller than the divisor'] to operations with rational numbers).
In light of these observations, we concluded that a main goal of teacher education should be to develop prospective teachers' mathematical and pedagogical content knowledge of rational numbers. Such a demanding task necessitates (1) more comprehensive knowledge about prospective teachers' mathematical and pedagogical content knowledge of rational numbers, and (2) the development of a special course for prospective elementary teachers, which attempts to improve prospective teachers' algorithmic, formal and intuitive knowledge of rational numbers, as well as increasing their knowledge about students' conceptions and ways of thinking about rational numbers and their possible sources. The second, (current) stage of our study was devoted to the elaboration and the experimental teaching of a prototype module on rational numbers.
We have argued that the conceptual framework that was used in our study is applicable not only to the domain of the rational numbers, but to other mathematical domains as well. This, of course, deserves further research.
Not less important is the need to learn more about the three dimensions of knowledge described in this model. Questions related to the development of each of these dimensions of knowledge, to the nature of the interconnections, and to implications of this model on the development of didactic materials to be used in elementary teacher education were not given the attention they deserve in this paper. We hope to elaborate more on this issue during the symposium.
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