AND PSYCHOLOGICAL ASPECTS
(Received January 9, 1995)
What would happen if a sixth grader were asked to estimate the sum of 12/13 + 7/8? If the sixth grader were typical, chances are very good that he or she would provide an answer of about 20. On the 1980 NAEP exam, 28% of the sixth graders said that the answer would he about 19, and 27% said that the answer would be about 21 (cf., Post, 1981b). Looking at 12/13 and 7/8, one can see that these children either just added the numerators or the denominators to get 19 or 21. These are common problem of rational number learning: children's use of incorrect rote algorithms, their failure to think of a symbol like "12/13" as representing one number, and, more generally, their shaky understanding of rational number size and order concepts (Behr & Post, 1988).
Though it is too easy to blame teachers for children's misconceptions, unfortunately many elementary teachers are deficient in the mathematical knowledge that they teach (Lacampagne et al., 1988; Post et al., 1988). For example, from a sample of over 200 elementary teachers in a large, Midwestern city, a little over 50% could divide 1/3 by 3, and less than 50% could divide 3 by 4/3 (Post et al., 1988). These and other results provide incentive for examining teachers' knowledge of rational number concepts.
This paper synthesizes research dealing with content knowledge of rational numbers, the learning of rational number concepts, and teachers' pedagogical reasoning about these concepts. The goal is to illustrate a formative model of pedagogical reasoning that has the potential to maximize both knowledge growth about teaching and student learning of rational number concepts. The basic idea in the model is to use what is known about rational number concepts as a normative base for the assessment of teachers' pedagogical reasoning. Teachers' reasoning is evaluated by assessing its harmony with the research base. The evaluation will be illustrated by providing a positive and a negative example of harmony with what is known about rational number concepts.
The specific content area to be considered is rational number order and equivalence concepts. The next section will describe some of the key difficulties that children have with these concepts.
RATIONAL NUMBER ORDER AND EQUIVALENCE CONCEPTS
One of the problems in the learning rational numbers is that whole number counting strategies fail to work (Behr & Post, 1988). With whole numbers, the idea of counting the "next number" makes sense. The next number after four is five. This idea does not make sense in the rational numbers, which makes rational numbers difficult to count. For example, there is no next fraction after 1/2. Put formally, the rational numbers are "dense," meaning that between any two rational numbers there is always at least one (actually, an infinite number) of rational numbers.
A related problem is that children have difficulty developing a concept of the size or magnitude of a rational number. With whole numbers, children can more easily grasp the size of a number such as four. They can see that four is one more than three and one less than five. However, because the counting strategies fail to work in the rational numbers, children have difficulty determining the size of rational number. The acuteness of this problem is evident from the NAEP results, where over half of the children surveyed estimated the sum of 12/13 + 7/8 to be about 20 (Post, 1981b).
Notions of rational number order, equivalence, and size are deep concepts which require a substantial amount of pedagogical skill to teach effectively. This knowledge will be described as a synthesis of formal notions of rational numbers and research on how rational numbers are learned. The next section will lay the groundwork for this synthesis.
LOGIC, PSYCHOLOGY, AND REPRESENTATION
Any attempt to relate content knowledge with the way in which a subject matter is learned must proceed with caution. This caution was expressed by Dewey (1902, 1916, 1933) with his distinction between the "logic" of a subject matter and its "psychology". Intuitively, the logic of a subject matter facilitates the growth of knowledge about it, whereas the psychology enables those not yet initiated in the subject matter to learn it. For the particular example considered here, the logic of rational numbers defines their essential mathematical properties, and the psychology of rational numbers includes results of empirical investigations of children's mathematical learning.
The "royal road to leaming," however, is a stumbling block to the immature. The "perfected form" of a subject matter's logic can get in the, way of learning.
The next section will describe a particular result from the logic of rational number order and equivalence concepts that can impede learning, if presented formally. The subsequent section will then explore this same result from a "psychological" point of view. A tool that will be useful in describing this contrast between logic and psychology is the notion of a "representation". From a logical point of view, the rational numbers can be represented as a set of axioms and order concepts can be represented by a formal deduction from these axioms. From a psychological point of view, "representation" can be used to describe the learning of rational number order concepts. Mathematical understanding can be described as the ability to "translate" among different embodiments or representations of an idea (Lesh, 1979; Lesh et al., 1987). This view of "representation" has implications for the design of rational number instruction as well as for pedagogical reasoning about rational number concepts.
A LOGICAL VIEW OF RATIONAL NUMBER CONCEPTS
"Representation" is a useful notion for describing, not just the logic of rational number concepts, but the nature of mathematics more generally. Mathematically, a "representation" can be thought of as a "model". A large part of mathematics can be understood as the building of models, solving problems within models, and then translating these solutions back into the "real world". Kaput argues:
At an informal level, mathematicians represent ideas with the help of symbols, diagrams, and pictures. At a more formal level, they represent groups by other groups, functions by their approximations, and syntactical features of structures using logical models. Kaput (an algebraist) argues that "the idea of representation is continuous with mathematics itself' (Kaput, 1987, p. 25).
The rational numbers can also be described using the mathematical notion of "representation". Namely, the rational numbers can be represented as a structure satisfying the axioms for an ordered field. From these axioms, which number less than 10, it is possible to deduce all that is known (mathematically) about the rational numbers. The notion of "density," which causes so much difficulty for elementary children, is simply the provable statement:
Between any two elements of the field, there exists at least one other element, This definition would be hardly illuminating to a third grader!
Front the formal representation of rational numbers, it is possible to deduce the validity of a strategy for comparing any two rational numbers. One can compare two fractions by finding equivalent fractions with a common denominator. For example, to compare 2/5 and 7/18, one can change both 2/5 and 7/18 to equivalent fractions with denominator equal to 90. In this case, the common denominator is found by multiplying the two denominators, 5 and 19, together. Continuing the procedure:
Since two fractions with common denominators can be compared by comparing the numerators, the problem is solved. 7/18 must be less than 2/5.
Unfortunately, this procedure is often not very meaningful to fifth graders (nor to many high school seniors, cf., Robitaille & Travers, 1992)!
A PSYCHOLOGICAL VIEW OF RATIONAL NUMBER CONCEPTS
The above, formal procedure is often not the most intuitive way to compare rational numbers. The Rational Number Project (an NSF funded project, henceforth abbreviated RNP) has investigated the strategies that children invent for comparing rational numbers (cf., Behr & Post, 1988). These invented strategies might be viewed as "psychological counterparts" to the formal, common denominator procedure.
Several examples of children's strategies can be given. When comparing 5/8 and 2/5, some children would argue that 5/8 is greater than 1/2, and 2/5 is less than 1/2. Thus, 5/8 is greater than 2/5. This can be called a "transitive strategy". When comparing 7/8 and 12/13, some children would argue that both fractions are close to 1, but 7/8 is 1/8 away from 1 and 12/13 is 1/13 away from 1. Since 12/13 is closer to one, it is larger. This can be called a "reference point strategy" (Behr & Post, 1988. P. 208). When comparing 4/8 and 7/8, some children would resort directly to physical models. For example, they would cut a rectangle into 8 parts and show that 4 of these parts covers less area than 7 of these parts (Behr & Post, 1988, p. 207).
The last example, employing physical models, highlights the sense in which "representation" is often used to describe mathematics learning. When teaching and learning mathematics, it is often useful to represent mathematical ideas using physical or symbolic models (cf., Post, 1981a). For example, when learning to count, the child represents a one-to-one correspondence using physical models or number words (numerals). The number, which is an abstract feature of the world or of human consciousness, is represented with the help of concrete objects or symbols. Different media can be used to represent different number concepts. For example, multibase blocks can be used to represent place value ideas, and centimeter rods can be used to represent rational numbers.
Lesh has described a "translation model," wherein a child's understanding of a mathematical concept is viewed as that child's ability to translate within and among different representations of the concept (Lesh, 1979; Lesh et al., 1987). The RNP has studied the ways in which children are able to translate among different representations of rational number concepts (Post et al., 1985). The development of children's understanding of rational number concepts has been related to three characteristics of children's thinking.
The first characteristic is a child's ability to translate between different expressions of rational number concepts. For example, the rational number 2/3 can be expressed symbolically as the symbol set, 2/3, or as a circular region cut into three parts with two of these parts shaded. The child who "understands" rational number concepts can make translations between these different embodiments or representations of the fraction concept.
A second characteristic is that a child can translate among different ways to express a concept within the "same form" of representation. For example, if a child is using chips to represent fractions, she can recognize that 4/6 is equivalent to 2/3 by restructuring an array of four black chips and two white chips (representing 4/6) into an array of two equal sets of black chips and one equal set of white chips (representing 2/3).
The third characteristic is the child's ability to progressively move away from concrete representations of rational numbers to more symbolic modes. For example, when asked to compare 5/6 and 2/3, a child might devise a plan without actually using the physical chips. This planning without the manipulative anticipates a more formal understanding of the ordering concept.
These three characteristics, as well as the informal strategies invented by children for comparing rational numbers. will be used to assess teachers' pedagogical reasoning in the next section.
TEACHERS' REASONING ABOUT
The logic and psychology of rational numbers, and the role that representation plays in rational number learning can both be used to conceptualize teachers' pedagogical content knowledge. The basic idea is to use knowledge derived from the RNP as a "data base" for evaluating teacher knowledge. In other words, the approach is to view research knowledge, not as another body of content to be mastered by teachers, but as a standard for assessing or evaluating teachers' reasoning.
Both the logic of rational number concepts and the psychology of rational number learning are a part of this plan. The logic provides a view of the terminal point in any inquiry and thus provides direction as to where instruction should ultimately go. As Dewey argues, the logic of a subject matter "enable [s] the educator to determine the environment of the child, and thus by indirection to direct" (Dewey, 1902, p.31). The psychology of the subject matter provides guidance for the sequence and development of actual instructional materials. In the particular case considered here, results from the RNP can be used to assess the degree to which teachers' understand how rational number concepts are learned.
An example of a teacher embodying this synthesis of logical and psychological aspects of rational number learning and an example where this synthesis is not evident will be described below. The data come from a battery of tests given to 238 intermediate teachers from two Midwestern sites. The tests were given in connection with an NSF project to generate profiles of mathematical understanding of teachers (Post et al., 1988). The examples are selected from interview transcripts with teachers who were asked to reason like a hypothetical child or to explain a problem to a hypothetical child. In one task, for example, teachers were asked how children would solve a series of ordering problems that involved rational numbers. In the response described below, teachers were asked how children would answer the question: "What is larger, 3/4 or 3/5". Then, in a later question, teachers were confronted with erroneous responses that children might make to the comparison problem and then asked how they would respond. For example, teachers were asked to respond to a child who argued: "3/4 is less than 3/5 because there are less pieces".
An example of "harmony"
When asked how a child who understood fractions would compare 3/4 and 3/5, one teacher responded as follows:
This example shows a number of points that are in harmony with the research base of the RNP. The teacher's reasoning is consistent with the three characteristics of translation among representations, described in the previous section. The hypothetical child both draws a picture of the fraction situation and speaks about symbols (or writes down symbols), thereby showing two different embodiments of the fraction concept. The child also translates between different representations. When explaining that 3/4 is larger, she translates from a pictorial to a symbolic mode.
In an interview 4 months later, the same teacher also showed an understanding of the need for children to "anticipate" the solution to an ordering problem without actually using manipulatives. The teacher was asked, again, how a child who understood fractions would compare 3/4 and 3/5. S/he responded:
Though the hypothetical child does not actually draw a picture here, she anticipates how a picture of the situation would look if drawn.
An example of "lack of harmony"
Not all teachers gave responses that were in such agreement with the research base of the RNP. When asked how a child who understood fractions would compare 3/4 and 3/5, one teacher responded as follows:
This example shows lack of harmony between the teacher's reasoning and what is known about the learning of rational number concepts. The hypothetical child appeals to a formal notion of representation, tied into the logic of the subject matter. The child is thinking about the common denominator algorithm for comparing two rational numbers. Though the common denominator algorithm is likely to be used when comparing, say, 2/5 and 7/18, it is not likely to be used when comparing 3/4 and 3/5. A child who "understands fractions" would be more likely to use a transitive, reference point, or some other informal strategy shown by the RNP to be important in comparing rational numbers.
Summary of the two examples
The first example of teacher reasoning shows harmony with the research base of the RNP in two ways, each related to the difference between the logic and psychology of rational numbers and the role of "representation" in learning rational number concepts. The positive example shows sensitivity to the role that alternative representations play in the learning of rational numbers. This example also downplays, but anticipates, the logic of rational numbers by focusing on the psychology or development of the subject matter. The second example of teacher reasoning shows lack of harmony with the results of the RNP in two ways. The negative example emphasizes the symbolic understanding of rational number ordering, which is tied into the formal logic of the rational numbers. Furthermore, this example shows little understanding of the rich strategies that children have been seen to use in the research conducted by the RNP.
TOWARD A MODEL OF PEDAGOGICAL REASONING
The above positive and negative examples of harmony between the learning of rational number concepts and teachers' reasoning are heavily weighted in the direction of teachers' descriptions. It might be objected that what teachers do, not what they say, is important. Though a teacher might not be able to explain how a child who understands fractions would compare 3/4 and 3/5, he or she might be able to provide appropriate interventions, on the spot, which would help the child. Focusing on just what a teacher is able to say about his or her thinking seems to miss the essence of teaching.
Teachers' actions may speak louder than teachers' descriptions of what they would do. However, what a teacher is able to say about his or her thinking is also important, and "justifying" the use of teachers' descriptions of their own reasoning will help to illustrate the normative conception of pedagogical reasoning that is being proposed.
Formative inquiry and the growth of knowledge
A theory about a domain such as teacher reasoning is always constructed for some purpose. Dewey (1929) argues that it is important to keep this purpose in mind so that inquiry does not become cut off from its roots (in experience) and forget how to find its way back to reality. Accordingly, when thinking about a model of pedagogical content reasoning, the purpose for building a model must be kept in mind.
At least two purposes for studying pedagogical reasoning might be proposed. These purposes are not necessarily distinct, but they differ somewhat in their emphases. One purpose would be primarily summative. That is, one might design a model of pedagogical reasoning to assess what teachers know for some agency, such as a state department. This purpose would lead in the direction of teacher testing, certification, and perhaps content standards. Another purpose would be primarily formative. That is, one might study teachers' pedagogical reasoning to help design a teacher education program. What would be important, in this formative case, is that the model of pedagogical reasoning have instructional value. It might be used, for example, in helping prospective teachers learn how to teach rational number concepts.
From a summative point of view, emphasis on what a teacher is able to describe about his or her thinking seems out of place. The ability to discuss the learning of students with a third party might be connected with the ability to use language effectively in the classroom. However, the bottom line, from a summative point of view, is the learning of the students. An agency such as a state department is probably interested in student achievement, not in teachers' ability to verbalize children's understanding. The latter does not appear to be related to standard notions of "accountability".
However, from a formative point of view, examining what a teacher is able to say about his or her thinking makes more sense. For teacher educators to be sure that their students are learning, it is necessary to be able to ask their prospective teachers what they are thinking and why. Even in the most observation-based teacher education programs, discussion of the pros and cons of various teaching behaviors would facilitate the candidates' learning how to teach. The formative aspect of a model of pedagogical reasoning would require verbalization, as well as observation, so that the knowledge and experience of master teachers and good teaching situations could be passed on to students.
The importance of discourse in a formative conception of teacher reasoning can be amplified by considering more closely the way in which knowledge "grows". Popper's model for the evolution of knowledge is apropos here (Popper, 1963, 1972). Popper views the growth of knowledge as occurring through the process of formulating bold conjectures followed by attempts to refute these conjectures. More specifically, he describes the process of knowledge growth as starting with some problem, which leads to a tentative theory about the problem, which is followed by error elimination, which leads to new problems. The descriptive and critical features of language are central to this process.
The most important of human creations, with the most important feed-back effects upon ourselves and especially upon our brains, are the [descriptive and critical] functions of human language... It is to this development of [these] functions of language that we owe our humanity, our reason. For our powers of reasoning are nothing but powers of critical arguments (Popper, 1972, p.119; pp. 120-121). In other words, the evolution of new knowledge will be enhanced if the relevant beliefs are described and critically discussed. In the particular case of teachers' knowledge, placing teacher beliefs to the test in the classroom and in discussions with other interested parties would increase the chance that knowledge would grow.
This leads a conception of pedagogical reasoning that is both formative and normative. Teachers attempt to state their knowledge and beliefs as clearly as possible. They also provide reasons for their beliefs, thereby encouraging criticism and the growth of knowledge. Beliefs are exposed to the evolutionary push and pull that enables the best and fittest to survive.
Teachers' descriptions of how they would teach rational number concepts fit into a model of the evolution of knowledge. Exploring teachers' descriptions of how a hypothetical student would understand rational numbers can facilitate the growth of knowledge. In the particular case considered here, studying the harmony between teachers' reasoning and the research base provided by the RNP can provide a starting point for critical discussion.
In this paper, the "harmony" between teachers' explanations and the research base has been explored using two theoretical tools. The first is the distinction between the logic of a discipline and its psychology. What is of interest is the way in which a teacher successfully uses the end product of inquiry, here the formal procedure for comparing two rational numbers, in describing how a hypothetical student understands rational numbers. This might be described as the extent to which a teacher either connects or short circuits the growth process associated with rational number understanding. The second tool is the notion of a "representation". Representations grounded in the logic of rational number learning are different, at least in degree, from those rooted in the psychology of this discipline. The learning of rational number concepts can be facilitated when "psychological" representations are used in mathematics teaching.
An example of teachers' reasoning about rational number order and equivalence concepts that is in harmony and another that is not in harmony with the research base have been presented. An example has been assessed as "not in harmony" if it stresses the logic of learning the formal procedure for comparing two rational numbers at the expense of its psychology. An example has been assessed as "in harmony" if the teacher's explanation is sensitive to multiple representations that the RNP has shown to be important in student learning.
A teacher's synthesis of logical and psychological notions of rational number learning is useful for building a model of teacher rationality. The positive and negative examples of teachers who make or do not make use of results from the RNP can be used as springboards to begin discussions about the teaching of rational numbers. In this way, the public criteria for assessing reasoning can be used to assess teachers in a formative way. This formative influence may make it less likely that students will estimate the sum of 7/8 and 12/13 to be about 20!
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