PME-NA "J Contreras"

PROSPECTIVE ELEMENTARY TEACHERS' DOMINANT SITUATIONS AND KNOWLEDGE ABOUT REPRESENTATIONS OF RATIONAL NUMBERS

José N. Contreras	Armando Moises Martínez-Cruz
The University of Southern Mississippi	California State University, Fullerton
Jose.Contreras@usm.edu	Armando.Martinez@nau.edu

Abstract: In this paper we examine 92 prospective elementary teachers' dominant situations and knowledge about representations of rational numbers. We found teachers' knowledge of story-problem representations for 3/5 very limited since only 142 of the 275 generated responses were categorized as well-posed story-problems with solution 3/5. The data also revealed that the dominant situation for 3/5 was the part-whole subconstruct followed by the measure subconstruct. Although most students could represent 3/5 pictorially with units of different sizes using a continuous model, most students were not able to conceptualize that a given shaded region could represent different fractions from the ones most naturally suggested by the given diagram.

Students' understanding of rational numbers is critical to the development of their mathematical competence. Mathematical competence includes both procedural and conceptual knowledge (Hiebert & Lefevre, 1986). However, as reported by national assessments (e.g., Kouba, Brown, Carpenter, Lindquist, Silver & Swafford, 1988) and more in-depth research studies (e.g., Markovits & Sowder, 1991; Vinner, Hershkowitz & Bruckheimer, 1981; Wearne & Hiebert, 1983) students' understanding of rational numbers seems to be quite limited. To improve students' performance and understanding, teachers need to put more emphasis on helping them develop their conceptual knowledge of mathematical ideas in general and rational numbers in particular. To teach rational numbers for understanding, teachers need to have a strong conceptual knowledge of the underlying mathematical principles. Therefore, it is imperative to examine prospective teachers' conceptual knowledge of rational numbers. Since conceptual knowledge is knowledge that is rich in relationships (Hiebert & Lefevre, 1986), it is necessary to examine teachers' understanding of connections between multiple representations. The purpose of this study is to investigate prospective elementary teachers' dominant situations and explicit knowledge about representations of rational numbers. To this end, the following research questions were formulated:

1) Do prospective elementary teachers know story problems whose solutions are 3/5?

2) What are prospective elementary teachers' dominant situations for 3/5?

3) What types of pictorial representations do prospective elementary teachers draw to represent 3/5?

4) What is prospective elementary teachers' understanding of the arbitrary nature of a unit?

Theoretical and Empirical Background

The concept of rational number is a complex and multifaceted construct. The theoretical background of this study is based on cognitive and conceptual analyses of rational numbers performed by other researchers (Behr, Lesh, Post, & Silver, 1983; Kieren, 1976; Marshall, 1993; Ohlsson, 1988) as well as on the concept of unitizing. Among the subconstructs (Behr et al. 1983), situations (Marshall, 1993), or interpretations of rational number identified by these researchers are: part-whole, fractional measure, quotient, rate, ratio, and operator. The part-whole situation involves the physical or mental partitioning of some continuous whole or a set of discrete objects into equal-size parts. The fractional measure situation addresses the question of "how much there is of a quantity relative to a specified unit of that quantity" (Behr et al., 1983, p. 99). This interpretation of a rational number focuses on the notion of a rational number as a number rather than on part-whole relationships. The quotient situation represents an indicated quotient or division. The ratio situation represents a relationship between two quantities of the same dimension or with the same units. The rate situation defines a new quantity as a relationship between two quantities with different units. The operator situation of rational number treats a rational number as a transformation: it operates on a value to produce another value. Its main function is that of a multiplier or divider. These and other researchers suggest that students need to have a complete understanding of the rational number subconstructs as well as an understanding of the connections among the subconstructs for students to have a complete understanding of rational number. The learning of whole numbers, fractions, proportions, and other mathematical constructs involves the construction and coordination of abstract units (Reynolds & Wheatley, 1996). Lamon (1996) defines unitizing as the "cognitive assignment of a unit of measurement to a given [or a certain] quantity" (p. 170). She cites research that supports the idea that "the nature of the unit largely accounts for the cognitive complexity entailed in linking meaning, symbols, and operations" (p. 170) for rational numbers. Several researchers (e.g., Simon, 1993; Post, Cramer, Behr, Lesh, and Harel, 1993) have studied some aspects of prospective elementary teachers' knowledge about the representations of the construct of rational number. Their findings suggest that prospective elementary teachers' knowledge of both pictorial and story-problem representations of operations with fractions tends to be very limited. This study extends their research by examining prospective elementary teachers' understanding of the arbitrary nature of a mathematical unit and of basic representations of fractions.

Methodology and Data Sources

A total of 92 prospective elementary teachers enrolled in three sections of a mathematical content course for elementary majors were the participants of the study. The students were asked to complete the four tasks described in the appendix. The tasks were selected to assess prospective elementary teachers' explicit knowledge or ability to create or describe representations for 3/5 and asses their understanding of the arbitrary nature of a unit. Within each of the three groups, the tasks were given one at a time. The students were given each subsequent task after all the students had completed the previous task. The purpose of the first task was to stimulate students to write story problems involving a variety of interpretations (e.g., part-whole, measure, quotient, and ratio). The purpose of the second task was to assess students' knowledge of pictorial representations as well as to investigate the extent to which prospective teachers use units of different sizes. The purpose of tasks three and four was to examine prospective elementary teachers' understanding of the arbitrary nature of the unit.

Results

A content analysis of each student's response for each task was performed using the aspects of the conceptual framework or the purposes of the tasks.

Prospective elementary teachers' knowledge of story-problem representations for 3/5

The participants generated a total of 275 responses of which 234 were problems and 41 were not problems (e.g., verbal statements, symbolic statements, pictures, etc.). Only 142 (51.6%) of the responses were categorized as well-posed story problems with solution 3/5. The remaining responses were not story problems, or were not well-posed story problems, or were well-posed story problems whose solution is not 3/5.

Prospective elementary teachers' dominant situations about rational numbers

Students generated 142 well-posed story problems whose solution is 3/5. Out of these 142 story problems, 90 (63.4%) involved part-whole relationships, 44 (31%) involved the measure situation of rational number, 4 (2.8%) interpreted 3/5 as a ratio, and 3 (2.1%) used a quotient interpretation for 3/5.

Prospective elementary teachers' knowledge of pictorial representations for 3/5

Thirty-five (38%) students generated three different correct pictorial representations for 3/5 involving units of different size. Thirty-nine (42.5%) students generated three correct pictorial representations but only two representations involved units of different sizes. Seven (7.5%) students created 3 correct pictorial representations for 3/5 but they all involved a unit of the same size. Ten (11%) students created at least one incorrect or unclear pictorial representation for 3/5.

Prospective elementary teachers' understanding of the arbitrary nature of a unit

Students were asked what fraction or fractions could be represented by the shaded portion of a given diagram (See Appendix, Task 3). Twenty (21.5%) students believed that the given diagram could only represent 3/4, 12 (13%) students stated that the diagram could only represent 3/5 and 40 (43.5%) students thought that the given diagram could represent 3/4 or 3/5. A total of 14 (15%) students thought that the given diagram could represent fractions equivalent to 3/4 or to 3/5, or to both 3/4 and 3/5. To some degree, these findings are not surprising because the diagram suggests these two representations. However, it is worthwhile to mention that only three students indicated and provided a correct justification that the given diagram could represent fractions different from 3/4 or 3/5. To gain further insight into students' conceptualization of a unit, students were asked to complete Task 4. Seventy-five (81.5%) students indicated that the given diagram could represent 3/5. It is interesting to note that most of these students said or implied that the given diagram could represent 3/5 if the circle were completed or they assumed that the figure was a circle. That is, it seems that students could not conceptualize that the diagram could represent 3/5 by itself, without the need to have 5 parts. For example, the diagram could represent 3/5 of a pound of beef. It is interesting to notice that almost all the students (88 or 95.5%) indicated and justified correctly that the given diagram could represent 3/4. Fourteen (15%) students said that the diagram could represent 3/10 but none of them provided a clear and correct justification for that fact. Interestingly, only 17 (18.5%) students indicated and justified correctly that the diagram could represent 1 1/2.

Discussion and Conclusion

Rational numbers are an important component of the school curriculum. The research reported in this paper provides insight into teachers' dominant situations and explicit knowledge of representations of rational numbers. It was discouraging to find that only 142 out of the 275 teachers' responses could be categorized as well-posed story problems with solution 3/5. Many of the responses were categorized as statements, as no story problems, as no well-posed story problems, and as well-posed story problems but with solution different from 3/5. These findings seem to indicate that some teachers still have an underdeveloped understanding of the meaning of 3/5 and when it is appropriate to talk about 3/5. Not surprisingly, the dominant situations for 3/5 were the subconstructs of part-whole and measure. This is so because preservice teachers' experiences are probably dominated by these two subconstructs of rational number and the fact that the explicit notation of 3/5 suggested 3/5 (as opposed to 3:5 or 3÷5). A second important finding was that most students (74 or 80.5%) could represent 3/5 pictorially with at least two units of different size using a continuous model. However, the third and probably the most interesting and significant finding was that most teachers have not developed a complete understanding of the concept of fraction and its relationship to the arbitrary and abstract nature of the unit. To reiterate, none of the prospective teachers stated and provided a correct justification that the given diagram could represent 3/10 and only 17 students indicated and correctly justified that the given diagram could represent 1 1/2. The findings of this study indicate that prospective elementary majors need instructional interventions to develop a better conceptualization of a fraction and its relationship to the arbitrary and abstract nature of the unit. Examination of some mathematics textbooks for elementary teachers revealed that representations of fractions using composite units are lacking (Contreras, in preparation). It seems that textbook authors assume, at least implicitly, that prospective elementary teachers understand simple and composite representations of fractions and the arbitrary and abstract nature of the unit. In this paper we have examined prospective elementary teachers' knowledge of representations of rational numbers. Studies are also needed to examine both elementary and secondary teachers' mathematical knowledge and its relationship to instruction, especially cases of teachers who have strong mathematical knowledge. Contreras (1997) and Contreras and Martínez (1996) have done some work in this area but further studies are needed to gain a more profound understanding of teachers' knowledge.

APPENDIX

1. Create three different story problems whose solution can be represented by . Use a different meaning or interpretation of for each problem.

2. Think of the following teaching situation

In a mathematics course students are asked to draw a diagram to represent . Students are provided with graph paper.

Draw three different correct representations that students could draw. Your representations should be as different as possible. [Three 5 by 15 rectangular grids are provided]

3. What fraction or fractions could the shaded portion of the figure represent? Explain your responses and draw diagrams if appropriate.

4. a) Could the following diagram be used to represent ? Justify your answer [The same diagram as in question 3 was shown]. Provide diagrams if appropriate. [Similar questions were asked for , , and 1]

Notes

For space limitations, the instrument is presented in a simplified and compressed form.

A group of students was asked to provide 4 story problems for 3/5.

References

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Contreras, J. (In preparation). A content analysis of representations of fractions in mathematics textbooks for elementary teachers.

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