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Psychological Models for the Development of Mathematical Understanding: Rational Numbers and Functions
The domains of rational number and functions are foundational to many topics in advanced mathematics, and underpin the understanding necessary for participation in the pure and applied sciences (Lamon, 1999). Both domains are also known to be extremely difficult to master. Although many students do eventually learn to perform the basic operations and algorithms that the domains require, their conceptual knowledge remains remarkably weak, as does their ability to tackle novel problems (Carpenter, Fennema, & Romberg, 1993; Harel & Dubinsky, 1992). In this chapter, we discuss a program of research in which we have been trying to model the conceptual understanding that underpins novel performance in these two domains on the one hand, and to design improved curricular approaches for developing this sort of conceptual competence on the other.
The particular form of conceptual competence that we have been interested in has been characterized as number sense. As a number of authors have pointed out, the characteristics of good number sense include (a) fluency in estimating and judging magnitude, (b) ability to recognize unreasonable results, (c) flexibility when mentally computing, (d) ability to move among different representations and to use the most appropriate representation for a given situation, and (e) ability to represent the same number or function in multiple ways, depending on the context and purpose of this representation (Bereiter & Scardamalia, 1996; Case, 1998; Greeno, 1991; Sowder, 1992).
Our primary psychological assumption about number sense is that it depends on the presence of powerful organizing schemata that we refer to as central conceptual structures. We believe that these structures, which we model as complex networks of semantic nodes, relations, and operators, represent the core content in a domain of knowledge, help children to think about the problems that the domain presents, and serve as a tool for the acquisition of higher order insights into the domain in question. In an earlier series of articles, we proposed that central conceptual structures are normally assembled by the integration of two intuitive or “primitive” schemata. The first of these is primarily digital, verbal, and sequential; the second is primarily spatial, analogic, and nonsequential (Case & Okamoto, 1996; Griffin & Case, 1997; Kalchman & Case 1998; Moss & Case, 1999). In the first phase of children's learning (which we shall refer to as Level 1), these two core schemata are consolidated in isolation. In the second phase (Level 2), both of these two early schemata become more complex, while at the same time they are mapped onto each other. The result is that the students' understanding of the domain is transformed and a new psychological unit is constructed. During the next phase (Level 3), students slowly begin to discriminate among the different contexts in which the new unit can be applied, and to create slightly different representations of it–each with its own distinctive properties. In the final phase (Level 4), students build explicit representations of how these different variants of the core structure are related to each other, and learn to move among them freely and fluently depending on their purpose. More than anything else, it is this flexible movement that demonstrates that children have acquired true number sense, and not just a set of isolated conceptual understandings and algorithms.
In order to understand the way in which this general progression takes place in mathematics, it is useful to consider a concrete example. Consider, therefore, the domain of the whole number and the central conceptual structure on which children's number sense depends. According to the model proposed by Case and his colleagues (Griffin & Case, 1997; Okamoto & Case, 1996), the two primitive schemata on which the development of whole number depends are the schema for verbal counting (digital, sequential; Gelman, 1978), and the schema for global quantity comparison (spatial, analogic; Starkey, 1992). Although young children have strong intuitions for both counting and global quantity comparisons, these two schemata initially develop separately. Evidence in support of this assertion comes from several sources. In a recent factor analysis, Okamoto and her colleagues (personal communication, February 14, 2000) found two factors at this age level that corresponded with counting and quantity evaluation. Other evidence includes the fact that children have difficulty answering a question posed in one mode (e.g., verbal/sequential) that depends on the other mode for its answer. For example, they have a hard time answering the question “Which is more, 5 or 4?” although they can count to 5 without error, and can pick out an array of 5 objects as the larger, when it is contrasted with an array of 4 objects (Griffin, Case, & Siegler, 1994; Okamoto & Case, 1996; Siegler & Robinson, 1982).
As children make the transition to a higher level of cognitive development (Level 2) at about the age of 6, and as their thinking is stimulated by the numerical problems that they encounter at home and in school, they gradually elaborate on these two schemata and map them onto each other. Factor analytic studies now reveal a single factor, and children can answer a variety of questions requiring the coordination of the two initial schemata. Their new structure also permits them to solve a wide variety of cross-modal questions, including symbolically or verbally posed addition and subtraction problems, which they solve by counting forward and backward along the verbal counting sequence (Fuson, 1992; Siegler, 1996). In Okamoto's study, a single factor also emerged at this age level among the more advanced students.
As children begin to understand how mental counting works, as they continue to encounter problems that require mental counting, and as they continue to develop more generally, they move on to the next phase (Level 3), typically at about the age of 7 years. During this phase of their learning, they gradually form representations of multiple number lines, such as those for counting by 2, 5, 10, and 100. The construction of these representations gives new meaning to problems such as double digit addition and subtraction, which can now be understood as involving separate number lines, one for 10s and one for 1s.
In the final phase of their learning about whole number (Level 4), which typically begins around 9 or 10 years of age, children gradually develop both a generalized and an explicit understanding of the entire whole number system and the way in which different forms of counting are related to each other. Addition or subtraction with regrouping, estimation problems using large numbers, and mental math problems involving compensation are all understood at a higher level as this understanding gradually takes shape. The progression through the various phases is summarized in the second column of Table 1.1. The central numerical structure (the “mental number line”) that emerges in the second phase is illustrated in greater detail in Fig. 1.1. The
| Level of Understanding | Mathematical Domain | ||
| Whole Numbers | Rational Numbers | Functions | |
| Level 1: Consolidation of primitive schemata A: digital B: analog |
A: Counting schema B: Qualitative quantity schema (more/less; addition & subtraction). |
A: Formal halving and doubling schema, for numbers from 1 to 100 B: Qualitative proportionality schema, including visual halving and doubling. |
A: Recursive computation schema B: Bar graph schema |
| Level 2: Construction of new element A-B |
Mental number line, with counting as an operation that is equivalent to addition and subtraction |
Rational number line, with each number half of double previous one (e.g., whole, 1/2, 1/4, 1/8, or we believe more appropriately; 100%, 50%, 25%, 12.5%) |
Function schema, with line on Cartesian graph understood to represent results of iterative computation for different values of x |
| Level 3: Differentiation of new elements A1 – B1; A2 – B2 |
1s, 10s, 100s, and their relationship, understood and generalized to full whole number system |
Decimals, fractions, percents, and their relationship understood |
Confusable functions differentiated from each other, (e.g., y = 2x; y = x2; y = 2x. Function as object differentiated from function as sequence of operations |
| Level 4: Understanding of full system A1 – B1 × A2 – B2 × A3 – B3 |
1s, 10s, 100s, and their relationship, understood and generalized to full whole number system |
Decimals, fractions, percents, and their relationship understood |
Elements of polynomial (x, x2, x3) and the way they can relate, understood. |
FIG. 1.1. Central conceptual structure for whole number. The four rows indicate different representations. The horizontal arrows indicate an understanding of the relation between adjacent items within each different type of representations. The vertical lines indicate that subjects understand the equivalence of these different representations, and the relations between them.
integrative representation in the middle of the figure is one that we have found useful in helping children to differentiate, elaborate, and integrate their earlier analogic and digital representations of numbers in our instructional studies (Griffin & Case, 1997).
How general is the psychological progression that is illustrated in Figure 1.1? How general is the sort of reciprocal mapping that the structure illustrates? How general is the finding that an integrative external representation, appropriately used, can facilitate the internal process of integration, thus deepening children's conceptual understanding and improving their number sense? (Case, 1985; Case & Griffin, 1990). In the present study, we investigated all three of these questions, and came to the conclusion that all three processes are quite general indeed.
UNDERSTANDING IN THE DOMAINS OF RATIONAL NUMBER AND FUNCTIONS
Rational Number
Our model of the developmental sequence on which a deep and flexible understanding of rational numbers depends is formally identical to that for whole number. In the first phase (Level 1), we proposed that children develop two separate schemata; (1) a global, qualitative structure for proportional evaluation that is spatial and analogic, and that encodes one quantity in relation to another (Moss & Case, 1999; Resnick & Singer, 1993; Spinillo & Bryant, 1991); and (2) a numerical structure for splitting or doubling that is digital and sequential, and that includes the results of a number of familiar splitting operations (e.g., 2 × 50 = 100; 100/2 = 50; Case, 1985; Confrey, 1994; Kieren, 1992). Both of these schemata appear to be in place by about 9 to 10 years, and the latter depends on the availability of the central conceptual structure for whole number described earlier.
The new unit that is formed as these schemata are elaborated and integrated is the rational number line, a structure that permits children to solve a number of problems that involve ratio, rates, and/or equal sharing, provided that they involve simple halves or doubles. Once children understand a system for representing rational numbers (e.g., fractions or percents), they can gradually expand their understanding to include other forms of representation. Finally, after another period that is quite extended, they can come to understand the relationship among the various forms of representation, and can learn to move fluidly among them. This
structural progression is summarized in the third column of Table 1.1. The core structure for rational number is illustrated in Fig. 1.2.
The top line of the figure represents the fractional and percent language that students use to describe a sequence of halved and doubled quantities. The (left to right) arrows connecting the expressions in this row indicate the operations by which we presume children move from one element to the next in the sequence. The second line illustrates the perceptually based sequence of ratios that children learn to recognize and to order. This operation, which might best be termed visual-motor halving, is most easily executed by putting one's forefinger beside an object–then moving it up and down until one finds the point at which the top and bottom halves of the object are symmetrical. In the third row, we depict the number rectangle–the integrative representation that we use in our instruction, and that helps the digital and spatial schemata to merge. This integrative representation also facilitates children's learning of benchmark percent values and the numerical operations that connect them. Thus, they are able to compose and decompose percents that were calculated in this fashion (e.g., to determine the size of 75% by finding the sizes of 50% and 25% and then combining them). Finally, the bottom row of the figure is meant to represent the corresponding set of measurement techniques and formal arithmetic procedures that children learn to use when the goal is to express a ratio in some standard set of units such as milliliters. For example, if one knows that the total volume a beaker can hold is 120 ml, one can determine what 75% of that volume must be by first computing half of 120 (60), then computing half of the resulting total (30), then adding these two values.
In our instructional work, we have shown that once children possess a ratio measurement structure such as that diagrammed in Fig. 1.2, they are able to use this structure as a starting point for learning about decimals and fractions (Moss & Case, 1999).
MATHEMATICAL FUNCTIONS
We hypothesize that a formally similar progression takes place in the domain of functions, as summarized in the third column of Table 1.1. For this domain, the primitive schemata are; (1) a digital, sequential schema in which a series of iterative numerical calculations is made, resulting in a string of numbers with a clear pattern (e.g., 0, 4, 8, 12, 16, etc. that occurs from adding 4 to each successive number); and (2) a spatial or analogic schema in which quantities are represented as bars on a graph in such a fashion that a pattern is perceived by a left to right visual scan (e.g., each bar is longer than the previous bar). The bars on the graph are read off the vertical axis (y-axis) as discrete quantities, and the categories along the horizontal axis (x-axis) are qualitative ones (e.g., a graph of individual people represented along the x-axis, where each person has more money than the previous one). These schemata are hypothesized to be in place by the time children are 9 or 10 years of age, with the latter one dependent on the development of children's central conceptual structure for representing space (Case, Marra, Bleiker, & Okamoto, 1996; Okamoto & Case, 1996).
In the second phase (Level 2), these two schemata are elaborated and mapped onto each other. The elaboration of the digital/sequential schema is one in which the numerical operation is applied iteratively to a string of positive, ascending whole numbers in order to generate pairs of quantitative values (e.g., multiplying each number by 4 to generate the following pairs of numbers: 0–0, 1–4, 2–8, 3–12, 4–16, etc.). A parallel elaboration is presumed to take place in the analogic schema during the same phase. In this elaboration, the categories along the horizontal axis become continuous rather than discrete, and thus can be used to represent quantitative rather than qualitative data. Any pair of numbers with two values is now understood to be representable in this (Cartesian) space, and the pattern that these pairs yield is representable by joining up the points, and looking at the sort of line that results.
These two initial schemata, as they are elaborated, are also mapped onto each other in such a fashion that the base set of whole numbers on which a computation is executed becomes points that can be represented from left to right along the x-axis. The results of this computation (e.g., multiply by 4) become points that can be represented along the y-axis. The overall pattern can be seen in the size of the step that is taken from one point to the next when the points are joined up (e.g., “up by 4” in a table representing pairs of numeric values, and “up by 4”on the graph). The result of this elaboration and coordination is a new element, which corresponds to children's first bimodal representation of a functional relationship. When this new element has been formed, an algebraic representation such y = 4x can be constructed and can begin to have real meaning for children.
Figure 1.3 illustrates the structure of the central conceptual structure for function using the function y = 4x. The top row of the figure represents a string of positive, ascending whole numbers on which children operate to generate a second series of numbers. For this particular function, a constant multiplier of 4 is shown in the second row of the figure. The “ + 4” indicates that the values generated by this iterative operation consistently
increase by 4. The bars found along the next row are also seen to increase successively by 4. The integrative representation, on which we place a heavy emphasis in our instructional work (Kalchman & Case, 1999), shows how the digital and analogic schemata are mapped onto one another and that the constant increase of 4 is a numeric measure of the steepness of the line, that is, the slope. Finally, the last two lines of this figure show an increasing ability for students to algebraically abstract the newly differentiated representations, and to categorize types of functions accordingly. Our hypothesis is that the general structure of the model is maintained for all other types of functions, including nonlinear. All that differs is the notation and operations that must be employed. For example, for the nonlinear function y = x2, each kilometer in the first row of Fig. 1.3 is multiplied by itself, and the result is a set of values that grows exponentially and that is represented as a curved line in the integrative representation.
As children progress to the third phase in their learning (Level 3), they can begin to differentiate among families of functions, for example, linear and nonlinear functions. For a full differentiation to occur, however, it is necessary for children to understand integers, and to elaborate their Cartesian schema to the point where they can differentiate the four quadrants of the Cartesian plane, and can understand the relationship of these quadrants to each other.
Finally, at Level 4, children learn how linear and nonlinear terms can themselves be related and how to understand the properties of the resulting entities (polynomials) by analyzing these relations.
Our hypothesis is that the development of understanding in each of these individual domains (rational number and functions) progresses in a similar fashion to the way it progresses in the domain of whole number. In each case, we expect a recursive series of differentiations, integrations, and elaborations of increasingly complex cognitive structures.
INSTRUCTIONAL PROGRAMS
The question to which we now turn is how to design instruction that will facilitate children's movement through such a developmental sequence. At a general level, what we try to accomplish in our instruction is to help children move through the structural sequence that our developmental analyses have suggested underpin deep conceptual understanding and the achievement of “number sense.”Working within the framework of central conceptual structures, we must first consider the prerequisite sequential and analogic structures that children already possess for a domain, and determine what differentiation and elaboration of these structures is necessary.
Our next task is to create a core context–often one involving an analogy or a metaphor–that serves as a conceptual bridge between the initial, separate schemata and the integrated conceptual structure that is the ultimate target. To enable children to move from their existing structures to the desired structure as expeditiously as possible, we often attempt to create external, pencil and paper representations that contain both analogic and digital information and that foster their interlinkage. The board game plays an integrative role of this sort in our programs for teaching whole number. Other similar devices, described later, appear in our programs for teaching rational numbers and functions.
Often, in order to maximize the gains from these multimodal representations, we create situations in which quantities are represented as objects having locations that are fixed in physical space, and through which children can literally move. We also create a semantic context that makes this movement familiar to children, and relates it to their everyday lives. Under these circumstances, we believe that children often come to appreciate properties of the numbers that their spatial cognition naturally predisposes them to notice (adjacency, closeness, etc.), and that they might otherwise miss.
As our figures indicate, central conceptual structures contain a great deal of verbal as well as symbolic and spatial information. Thus, another important component of our instructional design is the way in which we encourage children to create and integrate their purely verbal representations with their symbolic and analogic representations. We do so by having them talk about the integrative representations that we use, and the other representations that they themselves create, using natural language. We then try to form a bridge between the natural language that children use spontaneously and the more formal linguistic and symbolic mathematical terms of each domain, in a fashion that maps the two as directly and simply as possible.
Finally, we try to design situations in which children themselves can choose how to move back and forth among the different representations to which they are exposed, thus getting practice from the start in the sorts of problems that are often considered criterial for demonstrating number sense.
Consider now how these general procedures have been applied in each of the two domains in question.
Teaching Rational Numbers
The standard approach to teaching rational numbers is to introduce children to pie chart representations of fractions in the third or fourth grade, and then to proceed to exercises involving the addition or subtraction of fractions. By the fifth or sixth grade, students are expected to be able to add and subtract fractions with different denominators, using a standard algorithm for conversion to a common denominator. As fractions are mastered, decimals are introduced. Finally, between sixth and eighth grade, percents are introduced.
In our own curriculum, the thrust is on building the underlying psychological structure whose development was previously described. Accordingly, our goal is to begin with the two isolated schemata on which we believe rational number understanding depends and then to foster children's movement through the general sequence that is summarized in Table 1.1.
Overview of the Curricular Sequence
Level 1. The two basic schemata whose presence must be established, in our view, and that must be elaborated in order to move from the first to second phase of rational number learning, are the intuitive proportionality schema and the formal halving and doubling schema for the whole numbers from 1 to 100. The context in which we introduce the first of these two schemata is one in which children are filling large tubes with water, and estimating and comparing how much different tubes contain in proportional terms. Young children have difficulty perceiving narrow, upright containers in proportional terms. Although they can see which of two such containers has more liquid in absolute terms, they can also see which has more in proportional terms. That is to say, they can see which one is fuller. The comparison of beakers thus provides a context in which children can naturally apply and elaborate this qualitative schema.
Further, by the time children are 10 or 11, they have also had 4 or 5 years of instruction in arithmetic with whole numbers and are familiar with the numbers from 1 to 100 and the operations of addition and subtraction. As well, students have by now acquired basic knowledge of multiplication and division, including the knowledge of certain halving/doubling pairs by heart. Thus, for example, they know that 50 is half 100, and that 25 is half of 50. Using their well-automated computation skills, they can figure out that 121/2 is half of 25. They can also partition 100 a variety of ways, such as 25/75 or 90/10.
Level 2. As previously mentioned, most programs begin children's instruction in rational numbers by teaching fractions. Moreover, they do so by introducing pizza pies, which must be divided equally for the purpose of sharing. The disadvantage of this is that children tend to encode each piece in absolute rather than in relative terms. They see a number like one fourth as being a thing, and three fourths as being three of these things, taken away from a whole pizza (Kerslake, 1986; Silver, 1997).
In our program, we create a context where children's proportional understanding of liquids and containers can be mapped onto their understanding of the whole number system from 1 to 100. We do so by beginning with the special case of percents. We begin the lessons with activities in which students estimate fullness of beakers using percent terminology. We then go on to computational problems such as figuring out what 50% or 75% of 120 ml would be, and so forth. The two strategies that we stress are numerical halving (100, 50, 25, etc.), which corresponds to the sequence of visual motor splits that children use naturally with their fingers, and composition (e.g., 100 = 75+25), which corresponds to visual motor addition of the results. Once children understand how percent values can be computed numerically, in a fashion that corresponds directly to intuitively based visual motor operations, we consider that they have moved into the second phase of their learning (Level 2) and have constructed the core rational number line that was indicated in Fig.1.2.
Level 3. Our next step is to move them to the third phase–where they begin to understand other ways of representing rational numbers. The next form of rational number we introduce is the two-place decimal. We do so in a measurement context by explaining that a two-place decimal number indicates the percent of the way between two adjacent whole number distances that an intermediate point lies (e.g., 5.25 is a distance that is 25% of the way between 5 and 6). We then gradually expand this original idea to include multiplace decimals, using a transitional double decimal notation that the children spontaneously invent (e.g., 5.25.25 is a number that lies 25% of the way between 5.25 and 5.26), and using both spatial and temporal measurement situations.
Level 4. Finally, as children become comfortable in understanding decimals and percents, we move on to the fourth phase, where children develop explicit representations of all possible rational numbers, including fractions, and begin to move among them freely and in a fashion that suits their particular purpose in the problems that they are confronting. At this point, a number of exercises are presented in which fractions, decimals, and percents are used interchangeably.
DETAILS OF INSTRUCTIONAL SEQUENCE FOR FOURTH GRADE
Each time the curriculum is implemented, we follow the general procedures described earlier. The following is a sequence of activities that took place in a mixed-ability fourth-grade classroom in one of our trials. The lessons that are summarized were presented over a 3-month period at a rate of 1 or 2 classes per week. Each class lasted for approximately 1 hour.
Estimating Percents (Lessons 1–3)
The first lesson started with an introduction to percents. Students were challenged to think about instances where percents occur in their daily lives and to report these instances to the class as a whole. Not only were they able to volunteer a number of different contexts in which percents appeared (their siblings' school marks, price reductions in stores having sales, and tax on restaurant bills were the ones most frequently mentioned), the students were able to indicate a good qualitative understanding of what different numerical values “meant,” for example, that 100% meant “everything,” 99% meant “almost everything,” 50% meant “exactly half,” and 1% meant “almost nothing.” Following these discussions, we presented the students with large drainage pipes of varying lengths covered with specially fitted sleeves made of flexible venting tube that fit around the pipes and could be pulled up and down and set to various levels. The children spontaneously estimated the various percentages of the pipes that were covered using the perceptual halving strategy. The students also continued percent estimations using beakers and vials filled with sand or water. These estimation exercises were designed to allow the students to integrate their natural halving strategies with percent terminology. The students were then introduced to a standard numerical form of notation for labeling percents.
Computing Percents (Lessons 4–6)
The visual estimation exercises using vials and beakers were continued with a new focus on computation and measurement. Children were instructed to compare visual estimates with estimates based on measurement and computation. For example, if a vial were 20 mm tall, 50% of that should be 10 mm. The children then began to estimate and mentally compute percentage of volume; for example, this vial holds 60 ml of water, 50% full should be 30 ml, and 25% full should be 15 ml. Other challenges included measuring objects in the classroom and then estimating and calculating different benchmark points such as 50%, 25%, 121/2%, and 75%.
The children were not given any standard rules to perform these calculations and thus they employed a series of strategies of their own invention. For example, to calculate 75% of the length of an 80cm desktop, the students typically considered this task in a series of steps. Step 1–find half, and then build up as necessary (50% of 80 is 40). Step 2–find the difference between 75% and 50% (75% – 50% = 25%). Step 3–find 25% of 80, (25% × 80 = 20). Step 4–sum parts (40 + 20 = 60). Other exercises included comparing heights of children to teacher, for example, and then assigning an estimated numerical value using the language of percents. For example, “Peter's height is what percent of Joan's?” or “What percent of your father's height is your height?” A series of specially made laminated cutout dolls ranging in height from 5 cm to 25 cm provide additional practice at comparing heights. Percent lessons were concluded with the students planning and teaching a percent lesson to a child from a lower grade.
Introduction to Decimals Using Stopwatches (Lessons 7–8)
In these two lessons, children were introduced to decimals as an extension of their work on percents. The lessons started with discussions of decimals and how they permit more precise measurement than do whole numbers. Two-place decimals were introduced as a way of indicating what “percent” of the distance between two whole numbers a particular quantity occupies. LCD stopwatches with screens that display seconds and hundredths of seconds (hundredths of seconds are indicated by two small digits to the right of the numbers) were used as the introduction to decimals. After lengthy discussions of what these small numbers represent quantitatively, the students came to refer to these hundredths of seconds as “centiseconds.” The stopwatch activities served to build up children's intuitive sense of small time intervals, and to give students the experience of the magnitude of “centiseconds.”
More importantly, use of these stopwatches provided the students with the opportunity to represent these intervals as decimal numbers. In the stopwatch activities, centiseconds indicated the percentage of time that had passed between any two whole seconds; they came to represent the temporal analogs of distance. Many activities and games were devised for the purpose of helping the students to actively manipulate the decimal numbers in order to illuminate the conceptually difficult concepts of magnitude and order. The first challenge that was presented to the students was “The Stop/Start Challenge.” In this exercise, students attempted to start and stop the watch as quickly as possible, several times in succession. They then compared their personal quickest reaction time with that of their classmates. In this exercise, they had the opportunity to experience the ordering of decimal numbers as well as to have an informal look at computing differences in decimal numbers (scores).
Another difficult initial aspect of using decimal symbols is the ordering of decimals when the numbers move from 0.09 to 0.10, for example. Some students are able to respond quickly enough to the challenge to achieve a score of .09 seconds. Therefore, such traditionally difficult rational number tasks such as, what is bigger, .09 or .40? can be naturally introduced. Another stopwatch game that offered active participation in the understanding of magnitude was “Stop the Watch Between.” The object of this game was for the student to decide which decimal numbers come in between two given decimal numbers and then to stop the watch somewhere in that span of decimal numbers. In the game “Crack the Code,” the students moved between representations as they were challenged to stop the watch at the decimal equivalent of, for example, ½ (.50). As an extension to these exercises, the students were encouraged to invent variations of these games to use as challenges for their classmates.
Learning About Decimals on Number Lines (Lesson 9)
A second approach to decimals was through the use of meter-long, laminated number lines that are calibrated in centimeters. This approach was based on students' work with percents using number lines. The first activities served as a review. Each child was given a small number line and was asked to find designated percents of the whole line by placing a unit block on the appropriate spot. (“Please place a unit block on the line that indicates 44% of your number line.”) The students were then told that these percent quantities could also be expressed as a decimal number; thus, for example, 44% could also be shown as 0.44. Other activities included “Percent/Decimal Walks” where several number lines (which are referred to as “sidewalks” by the students) were lined up end-to-end on the classroom floor with small gaps between each. Students walked a given indicated distance on the number lines, for example, “Can you please walk 3.67 sidewalks.”
Playing and Inventing Decimal Board Games (Lessons 10–13)
A board game called the “The Dragon game” was devised with the intention of giving the students the opportunity to learn about the magnitude of decimal numbers, as well as to add and subtract decimal numbers. The game board was approximately 60 cm × 90 cm and was composed of 20 individual laminated 10-cm number lines that were arranged as a winding path. Each number line was marked as a ruler; 10 thick black lines indicated cm measures, 10 slightly shorter blue lines highlighted the .5 cm measures, and 100 red lines provided the mm measures. This game directly followed from the “sidewalk” exercises mentioned earlier. The object of the game was to get from the beginning (the first sidewalk) to the end (the 20th sidewalk) before the other players. At each turn, a child picked two cards, an “add” or “subtract” card and a “number” card. Each number card had two digits written on it. The rule was that before making a move on the board, the player had to expand the two digits on the card by adding both a zero and a decimal point strategically so as to optimize the distance that the player would travel. For example, if a child picked a card with the digits 1 and 2, she had the options of calling that card .120, 1.20, 12.0, or 120. Three lessons followed where the students invented and planned their own rational number board game and then played each other's games.
Fractions (Lessons 14–17)
In keeping with the curriculum focus of translating among representations, fraction lessons were taught in relationship to decimals and percents. In these lessons, the children were challenged, for example, to represent the fraction 1/4 in as many ways as they could, using a variety of shaded geometric shapes as well as using formal fraction, decimal, and percent representations. They also worked on problems and invented their own challenges for solving mixed-representation equations involving decimals, percents, and fractions. For example, a student might compose the following challenge: “Is this true or false; 1/8 + 10% + 0.75 = 1?”
Review (Lessons 18–20)
Games were played where students had to add and subtract decimals, fractions, and percents by creating their own hands-on concrete materials. For example, students invented card games with mixed-representations and challenged their classmates to solve a variety of problems that were posed. As a final culminating project, students were invited to either invent their own rational number teaching strategies and lessons that could be taught to another group, or design a game or video presentation that incorporated specific rational number teaching objectives.
RESULTS FROM THE RATIONAL NUMBER STUDIES
In order to assess the effectiveness of the curriculum, we designed a series of measures that we administered both as pre- and posttests to students in the experimental programs as well as to students who served as control and comparative groups. To analyze results, items were assigned to subcategories that are generally taken to be indicators of rational number sense, and that are specific to the indicators of general number sense that were mentioned at the beginning of the chapter. These subcategories included comparing and ordering rational numbers; translating among decimals, fractions, and percents; solving problems that include misleading visual features; and inventing procedures for calculating with rational numbers. We also included items that were of a standard nature and that reflected traditional tasks. In the following section, we present items from each of these subcategories along with representative responses that students gave to these items at posttest. These examples are taken from three different empirical studies that we conducted where the experimental rational number curriculum was implemented and assessed. In the first, we presented the curriculum to a group of high-achieving fourth-grade students (n = 16) and compared their pre- and posttest performance to a well-matched treatment-control group (Moss & Case, 1999). In the second and third studies, the students who participated in the experimental program were from intact mixed-ability classrooms, one, a class of fourth graders (n = 21), and the other, a group of sixth graders (n = 16). For these latter studies, we compared the posttest performance of the experimental students to the performance of several traditionally instructed normative groups of students from fourth grade (n = 30), sixth grade (n = 36), and eighth grade (n = 26). The comparison group also included 32 preservice teachers in a postgraduate teacher training program, (Moss, 2000).
Quantitative analyses of the results revealed that all of the students in the experimental groups made significant gains from pre- to posttest, achieving effect sizes in the range of 2.3 to 3.5 standard deviations. The results also revealed that the posttest scores of the fourth and sixth-grade experimental students were higher than those obtained on the same measure by the eighth-grade comparison students and were equal to the scores that the preservice teachers achieved. Moreover, as we show in the examples that follow, the experimental students showed less reliance on whole number strategies when solving novel problems, and made more frequent reference to proportional concepts in justifying their answers than did the students in the normative groups.
The first item that we present was from the subcategory Interchangeability, which was comprised of items that required students to translate among the representations of rational number system–an important factor in rational number sense (Sowder, 1992). Although none of the students in the experimental groups was able to answer the following question at pretest, more than 80% of these students achieved a correct answer at posttest.
| Interviewer: | Do you know what one eighth is as a decimal? |
| Student: | Well one eighth is half of one fourth. And one quarter is 25%, so half of that is 121/2%. So as a decimal, that would have to be point 12 and a half. So that is point 12 point 5, so that means that it is .125. |
The reasoning of this student is representative of the kind of strategies that most of the students used to arrive at a solution to this problem. Furthermore, this solution strategy illustrates several features that became central to students' reasoning: First is the use of percents as a guide even when the problem does not contain the percent representation. Second, the student used the familiar 25% benchmark to convert from a fraction to a decimal. Third, this same protocol reveals that when working on problems with decimals, students often employ a mixed decimal and fraction representation. And fourth, the usefulness of the halving and doubling operation is clear. By contrast, over 50% of all the students in the comparison groups asserted that 0.8 must be the answer as 1/8 was the fraction to be translated.
Closely associated with interchangeability is the ability to compare and order rational numbers. Not only must students assign a quantitative referent to the rational numbers that are represented, but as is illustrated in the following example, they must also have an understanding of the density property of the rational number system.
| Interviewer: | Can any fractions fit between one fourth and two fourths? And if so, can you name one? |
| Student A: | Well, I know that one quarter is 25% and so two quarters is 50%. So, 40% fits between them. So that would be 40 hundredths. |
| Student B: | One quarter is the same as two eighths and two quarters is the same as four eighths so the answer is three eighths. |
This item that was answered correctly by 80% of the high-achieving students as posttest, (0% at pretest), was very difficult for the students from the comparison group, most of whom either did not know the answer or asserted that no such fraction existed. These students achieved a passing rate of 46%.
A third item that we present was designed to assess students' ability to overcome misleading features. The item that we now present that appeared on the measures used in both of the fourth-grade studies, challenged the students to ignore an irrelevant partitioning of a geometric region in order to correctly answer the question. At posttest, 90% of the students in the experimental groups were able to provide a correct answer. The following three examples of student reasoning are representative of the strategies that were used by these young students.
| Interviewer: | Can you shade three quarters of this pizza. (The pizza was partitioned into 8 sections). |
| Student A: | Well let me see …. This is a half (student shaded 4 sections), … so you would need 2 more to make three quarters. (Shades 2 more sections). |
| Student B: | There are 2 slices in a quarter so you need 6 [slices] to make three quarters (shades them). |
| Student C: | (Shades 6 sections) I just keep the quarters and forget about the eighths. |
By contrast, the students in the control group only achieved a passing rate of 50% and made the kind of error that is reported in the literature and is considered to be indicative of whole number interference. They asserted that “since it says ¾, you need to shade in three parts.”
Finally, the ability to invent procedures to solve standard and nonstandard computation problems is generally seen as an important feature of number sense. The types of errors that are consistently shown in the rational number literature demonstrate that students are overly dependent on the use of procedures. Even when uncertain of the rules, they misuse a procedure, preferring to accept an improbable answer rather than to invent an alternate strategy (Mack, 1995). The results of the previously mentioned studies reveal that the students acquired the ability to invent procedures for calculating with rational numbers. The next item that we present appeared on all of the measures in the three studies. Overall, 68% of the students were able to correctly answer this question. By contrast, virtually none of the school-age children and only 60% of the preservice teachers were able to answer this question.
The following two examples of students' reasoning is typical of the reasoning that the students used on this item.
| Interviewer: | What is 65% of 160? |
| Student A: | Okay, 50% of 160 is 80. Half of 80 is 40 so that is 25%. So if you add 80 and 40 you get 120 but that is too much because that's 75%. So you need to minus 10% and that's 16. So, 120 take away 16 is 104. |
| Student B: | The answer is 104. First I did 50% which was 80. Then I did 10% of 160, which is 16. Then I did 5% of it, which was 8. I added them (16 + 8) to get 24, and added that to 80 to get 104. |
The reasoning in these examples clearly indicates that using benchmark quantities when working with percents and translating among representations is an effective strategy for solving unfamiliar problems.
DISCUSSION OF RATIONAL NUMBER STUDIES
The foregoing examples of students' reasoning, while limited in number, are representative of the kinds of understandings that have emerged with each iteration of the curriculum (Moss, 2000). The scope of students' acquired understandings includes an overall understanding of the number system, which is illustrated in their ability to use the representations of decimals, fractions, and percents interchangeably; an appreciation of the magnitude of the rational numbers as seen in their ability to compare and order numbers within this system; an understanding of the proportionaland ratio-based constructs of this domain, which underpins their facility with equivalencies; an understanding of percent as an operator, which is evident in their ability to invent a variety of solution strategies for calculating with these numbers; and a general confidence and fluency in their ability to think about the domain, using the benchmark values that they have learned, which is a hallmark of number sense.
Recently, several other investigators have reported success in producing a deeper, more proportionally based understanding of fractions or decimals in the middle school years (e.g., Confrey, 1994; Kieren, 1995; Mack, 1995; Streefland, 1993). Not coincidentally, we believe, our program shares several important features with the programs designed by these other investigators, including a greater emphasis on the meaning or semantics of the rational numbers, a greater emphasis on the proportional nature of rational numbers, a greater emphasis on children's natural way of viewing problems, and their spontaneous solution strategies, and the use of an alternative form of visual representation (i.e., an alternative to the standard “pie chart”). In addition, our program shares several particular features with Confrey's program that, like ours, attempted to move children beyond the understanding of any single form of rational number representation toward a deeper understanding of the rational number system as a whole. These common features include a strong emphasis on continuous quantity and measurement, as opposed to discrete quantity and counting, on splitting as a natural form of computation that can be used in a measurement context, and on the equivalence between different forms of rational number representation.
We do, however, see our program as unique in several ways and it is our conjecture that the following three features are particularly crucial. (1) Our program begins with percents, thus permitting children to take advantage of and to combine their qualitative understanding of proportions and their knowledge of the numbers from 1 to 100 while avoiding (or at least postponing) the problems that fractions present, either on their own or as a basis for understanding decimals. (2) Our program uses a unidimensional form of number representation (the rational number line) in a context that emphasizes the global, proportional nature of any quantity, rather than the multiple units or “shares” into which it may be divided. (3) Our program emphasizes benchmark values for moving among equivalencies among percents, decimals, and fractions, an emphasis that permits students to think about problems in a much more flexible fashion and to use procedures of their own invention for approaching them.
CURRICULUM FOR MATHEMATICAL FUNCTIONS
The topic of functions is generally introduced in the ninth grade, where the graphic, numeric (tabular), and algebraic representations are typically taught in isolation. In this standard approach, algebraic equations are presented as the primary representation of functions, whereas graphs and numeric patterns found in tables of values are secondary and are loosely connected to the meaning of the algebra. The primary goal is for students to achieve competence in identifying and manipulating a few standard types of functions, such as linear functions, quadratic functions, and so forth. By contrast, in our experimental curriculum, we introduce students to the domain as early as sixth grade, with an emphasis that is considerably different. As is the case with curricula designed to foster number sense with whole numbers (Griffin & Case, 1997) or rational numbers (Moss & Case, 1999) our primary goal is to bring children through the developmental sequence that is summarized in Table 1.1.
Overview of Curricular Sequence
Levels 1 and 2. We begin by helping students elaborate their digital and analogic schemata, and map them onto each other so that they can begin to create a new psychological (and mathematical) object. To do so, the “bridging context” used is one in which both the digital or computational aspect of a function, and the graphic aspect, can be represented and understood simultaneously. The particular context we use for this purpose is a walkathon, in which the sponsorship rule varies from one participant to the next, and the results can be represented as tables of numbers or as a series of bar graphs. This context was chosen over other possible contexts because children have experience with and understand the variables in question (distance and money) as mathematical variables, and understand the functional (dependent) relationship between them; children are interested in the impact of different rules on the rate at which money accumulates; and children naturally think of distance as a continuous variable and are interested in what happens if they complete only part of the last kilometer.
The following is an example of how this walkathon context is used in the classroom. Children understand that when earning $1.00 per kilometer walked, their total earnings depend on the ultimate distance traveled, and are calculated by multiplying one times the number of kilometers walked. A table of values and a Cartesian grid–both of which are mathematical constructs with which sixth-grade children are familiar from previous schooling–are basic representations that children can use to keep track of kilometer by kilometer earnings. A symbolic representation is also easily and intuitively constructed by introducing the symbol of $ as the money earned and km as the distance walked. So, the sponsorship agreement of earning $1.00 per kilometer can be represented as $1.00 × km = $. Thus, from the onset, the graphic, tabular, and algebraic/symbolic representations of a function are seen as equivalent forms of the same mathematical relationship.
Level 3. Students move from the walkathon context to a computer lab where they are introduced to a spreadsheet technology for graphing. Here, they consolidate and apply the ideas from the first part of the program, and begin differentiating different functions by empirically varying the parameters of different formulae (e.g., the slope, the y-intercept, an exponent, etc.). We chose spreadsheet technology because it simultaneously displays the tabular, graphic, and algebraic representations with any change in one representation being instantly reflected in the others. The general idea is for students to understand the properties and behaviors of individual functions; to generalize these features to entire families of functions (e.g., y = mx = b and y = ax2 + b); to differentiate these families of function from each other (e.g., y = mx + b from y = ax2= b); and to understand the relationship between changes in the basic function, and changes in its various representations (e.g., tabular, graphic, algebraic).
Level 4. In the final component of the program, groups of students investigate further one particular type of function (e.g., linear, quadratic, or cubic) using the computer. They explore that function, and then use computer generated output of exemplary graphs, equations, and tables to illustrate the function's general properties and behaviors. Students then develop a presentation and share their new expertise with classmates. In sixth grade, we do not expect students to be able to understand quadratic equations, or to break them into components. Thus, these final exercises primarily serve to consolidate the understanding that they have already achieved. In higher grades, students go on to more complex functions including quadratic, cubic, and reciprocal functions.
Typical Instructional Sequence for Grade 8
The following sequence of instruction is set out lesson by lesson. However, because this curriculum is implemented differently in different schools and with different classes, the amount of time available to cover a single lesson varies, and one lesson does not necessarily correspond to one day's instruction. Still, we consider one lesson to involve approximately a one-hour period of time. Generally, we try to implement the lessons in 10 to 15 successive school days.
Lesson 1: Introduction to Functions and Slope. The walkathon context is introduced and children are given the example of being sponsored $1.00 for every kilometer walked for a 10 km walkathon. Using that rule, the first task is to record in a table of values the money earned at each kilometer walked. As one student completes the table for walking 0, 1, 2, 3, and so forth km, a second student marks on a large graph the coordinates identified in the table. For example, at 0 km, 0 times 1 equals 0, so $0 has been earned and a marker is put at the point where 0 km along the horizontal axis and 0 dollars along the vertical axis meet; the next marker is put at 1–1, and so on.
Here, we emphasize the physical action of walking along the horizontal axis from kilometer to kilometer and then going up to the number of dollars earned. This motion serves to enforce the dynamic aspect of a function–that is, change in one quantity brings about change in the other, and the left to right “movement” of a function, which is important for determining the direction of the slope of a linear function. Once the table of values and the graph are constructed, we consider ways of representing the function symbolically using km to represent the distance walked and $ to represent the money earned. Before settling on a representation, however, we first look at what operation we performed at each kilometer to get the money earned. In this first case, we multiplied each kilometer by 1 to get the money earned. This translates into the expression km × 1 = $. The significance of the “1” in the above symbolic expression is highlighted and its meaning considered in the corresponding graph and the table of values. In the table, we notice that the difference between consecutive $ values is always 1; and the line on the graph always goes up by one value as we go across by one value. Thus, students develop a feel for the spatial and numeric properties intrinsic to the y = 1 × x (y = x) line, and determine that it has an up by amount of 1. This up by amount corresponds to the mathematical term of slope, although this term is not introduced at this time.
The same procedures are then carried out for sponsorship agreements such as $2.00 per kilometer, $5.00 per kilometer, $10.00 per kilometer, 50¢ per kilometer, 25¢ per kilometer, and for rules created by the students. Before each rule is graphed, students predict the steepness of the line relative to the y = x line, thus establishing a sort of “benchmark.” Then, common characteristics of these functions are discussed. Characteristics typically noted by the students include: “They're all straight lines because the $'s all go up by the same amount each time [in the graphs and in the tables]”; “they all go through zero because zero times anything is zero”; and “the bigger the number you multiply by, the steeper the line is.” The term slope is introduced at this point, and students are asked to determine a value for the slope of each of the functions we explore. At no time is an algorithm for determining slope introduced.
Lesson 2: The Y-Intercept. The idea of a starting bonus is introduced and is explained as an initial amount of money that may be contributed before the walkathon even begins. This starting amount is called the “starter offer”–a phrase coined by students in pilot studies. We begin these lessons also with a sponsorship arrangement of $1.00 for every kilometer walked. Students graph this function, make a table of values, and write a symbolic representation for it. Students are then told that they will be given a $5.00 starter offer just for participating in the walkathon. As one student constructs the table, a second student places markers on the graph at each coordinate, and the markers are joined. Each student is then asked to construct a symbolic representation of the function (km × 1 + 5 = $).
Other rules in which students earn $1.00 per kilometer but have different starter offer amounts such as $2.00, $10.00, and $3.50 are given. For each new rule, the table and graph are created, and an equation written. Then, all of the starter offer rules are compared. The following common features typically emerge; (a) all are still straight lines, (b) all lines with the same per kilometer sponsorship rule are parallel, (c) all of the functions go up by one, and (d) changing the starter offer only changes the starting position of the line on the graph and not the slope of it. The term y-intercept is introduced as the mathematical name for the starter offer.
Next, a group discussion is initiated in which we talk about the effects of changing the starter offer on the graph versus the effects of changing the amount we multiply the kilometers by (the slope, or up by amount). We then review all of the functions with which we have worked and determine how the slope and y-intercept may be found in each of the representations used (tables, graphs, and equations).
Students, either individually or in pairs, then work on an activity where they invent two functions that will allow them to earn $153.00 at the completion of a 10 km walkathon. Both strategies must produce straight line functions. Tables, graphs, and equations are constructed to show their work. Students are also asked to identify the slope and y-intercept of each function they use. Individual or pairs of students then show their functions to the whole group. Students are also challenged to work “backwards,” that is, to find what the starter offer would have to be if the slope were 10, or what the slope would have to be if the starter offer were 20.
Lesson 3: Curved Lines. It is explained to students that there are some functions that generate curved-line rather than straight-line graphs. Students are asked to recall the numeric properties of those that produce straight line graphs. The notion of a constant up by amount in both a table of values and a graph is discussed, as well as the corresponding coefficient in a y = mx + b equation. Students are then shown a table of values for the function y = x2 and are asked to try and find the function. Students first notice that the sequence of values generated do not “go up by the same amount each time,” and thus, they cannot use their strategy of multiplying the first set of numbers by a constant value. The rule of multiplying each number by itself invariably is discovered and represented as $ = km = km2. Because younger students are not familiar with exponents as a mode of mathematical operation, it is explained that km2 is the same as km times km; likewise, km3 is the same as km times km times km, and so on. Thus, the equation for this function may be written as $ = km2. It is then explained that the coordinates of these functions are connected on a graph using a smooth curve rather than the series of line segments that students tend to spontaneously draw.
Next, we explore the idea of including a starter offer with these types of functions and students come to recognize that the inclusion of a starter offer has the same effect on this type of function as on the straight line functions. That is, the steepness of the line (or curve) is not altered, merely the place at which the function meets the vertical axis. A rule where one is being sponsored the number of kilometers by itself plus a starter offer of $10.00 is easily symbolized by students as $ = km2 = 10. Students are then challenged to come up with a function that would produce a curved line, and that they could use for earning $153.00 over 10 km.
Lesson 4: Negative Slopes and Y-intercept. Negative values along the y-axis are introduced by asking students to think about how the negative values along the vertical axis could be used. Students generally recognize that lines would contain a down by amount if a certain amount of money was given away for each kilometer walked. For example, if $2.00 were given away for each kilometer walked, a table of values would show that the $ or y values would go down by 2 each time, and in the equation it would be necessary to multiply the km by −2. Graphing the function would give a decreasing straight line with a slope of −2, and a starter offer of 0.
A context is then introduced for negative y-intercepts. The idea of debt is suggested whereby students have to pay off a starter offer. For example, students are asked to make a table, graph, and equation for paying off a $10.00 debt at $1.00 per kilometer walked. The equation for this type of function is written as $ = 1 × km + −10. Students are then challenged to invent several of their own functions in which they have to pay off a chosen amount of money over a certain number of kilometers. They are asked to make tables and graphs for these functions, and to write equations that represent their ideas.
Lessons 5–8 Computer Activities. Students spend the next several lessons working in pairs on computers using spreadsheet charting tools (see Fig. 1.4 for a sample computer screen). The activities for the computer were designed for children to consolidate and move beyond the tasks they have been doing in the classroom. In these activities, students are asked to change single parameters of functions, that is, just the slope, intercept, or exponent, of the y = x or the y = x2 function, in order to manipulate the graphic representation through preplotted colored points. They are then asked to change more than one parameter at a time in order to manipulate the given line or curve through the points.
Throughout, students are required to record the equations of the new functions, the numeric sequence found in the generated tables, and the graphic implications. In addition, students are asked to create their own functions that have certain visual and numeric characteristics (e.g., slopes steeper than 4, or inverted curves), program the equations for those functions into the computer, and record the results.
Lessons 9–10: In-Class Presentations. The final component of the curriculum requires students to prepare and then give a presentation on a specified type of function, which was featured in the computer activities. Students are asked to capture the general characteristics and behaviors of “their” function with exemplary graphs, equations, and tables, and to share their expertise with their classmates. When participating as a member of the audience, children are required to provide
feedback, ask clarifying questions, and challenge the presenters on any information that is inconsistent with their own understandings.
RESULTS FROM FUNCTION STUDIES
The test designed to measure students' understanding of function presents items in an order that is intended to reflect the levels of understanding hypothesized in the developmental model. Because the curriculum for functions is implemented with classes of children in the sixth, eighth, and 11th grades, expectations for posttest success are relative to the grade level of the students. Thus, examples of items that increase in difficulty are presented, along with corresponding responses from students who have participated in our experimental curriculum and who were in the sixth, eighth, and 11th grades, respectively. Control conditions have been used at the eighth- and 11th-grade levels only. This is because there is no local, standard sixth-grade curriculum that introduces comparable concepts. In the present work, sixth-grade responses were drawn from studies carried out with two different sixth-grade classes (n = 34). The eighth-grade students' responses are from a study in which there were both experimental and control conditions (Kalchman & Case, 1998, 1999 n = 24 and n = 21, respectively). Full results for a study conducted at the 11th-grade level where both control and experimental conditions were employed (n = 16 for both group) are in preparation (Kalchman, 2000). Some earlier analyses of 11th graders' reasoning about functions were done with students who had experienced a text-based program (Kalchman & Katz, 1999).
In the first example, students are responding to the item seen in Fig. 1.5. For this question, students must provide an equation for a function that, when graphed, will cross the given line within the Cartesian territory seen on the page. There are an unlimited number of possible linear and curvilinear functions that could cross this line. Linear solutions include functions that have a slope of approximately 3.5 or greater and a y-intercept of 0 (e.g., y = 6x; y = 10x); those that have a y-intercept less than 7 and a slope greater than 1 (e.g., y = 2x = 5; y = 4x = 1); those that have a negative slope and have a y-intercept greater than 7 and less than or equal to 10 (e.g., y = −3x + 8, y = −2x + 9).
Increasing and decreasing curvilinear graphs are also possible. For example, an increasing curvilinear function must either rise fast enough to
cross the given function if it has a y-intercept of 0 (e.g., y = x3), or must have a y-intercept great enough to pass the curve through the line (e.g., y = x2 + 5). Responses from students who have participated in our programs vary considerably and include all of the above types of functions. The following are samples of students' reasoning. In one case (Student B), the supporting context of the walkathon is evident in his thinking.
The previous responses from sixth-grade students must be considered against a backdrop of only 9% of students in an advanced level 11th-grade mathematics class (n = 33) giving correct responses to this item following their unit on functions (Kalchman & Katz, 1999). Seventy percent of the sixth-grade students who experienced our program gave a correct solution to this item following instruction (0% prior to instruction). Most of the older students tried unsuccessfully to apply the general equation for a straight line (i.e., y = mx + b) to the problem by simply inserting negative signs into the equation (e.g., y = −mx - b). Few of these older students even attempted to use numeric values for m and b.
In the next question, students were asked to look at the following sequence of numbers; 2, 5, 8, 11, 14, 17, …; and to write an equation for a function that would generate this pattern of values.
For success with this item, students must have clearly distinguished the domain (x values) of a function from its range (y values), and must also understand that a generated set of numbers provides key information about certain parameters of a function (i.e., what x is multiplied by and where the function will be when x is equal to 0). The most common response to this item from both younger students and a control group was y = x + 3 “… since the pattern goes up by 3.” Eighty percent of the eighth-grade students who participated in the experimental group were successful on this item following instruction, versus 4% in the control group (with 0% correct for both groups on the pretest). Responses from our eighth-grade students included the following reasoning:
| Student E: | “Well, it can't be y = x + 3, because the first number in the sequence would be 3 when x equals 0…. This sequence is increasing by 3 each time, which means there is a slope of 3 and when x equals 0, then you have to add 2 so the equation is y = 3x + 2.” |
| Student F: | “If I make a table of values I can see that the y-intercept is at 2 and that the y values go up by 3 each time. So, that means it's 3 times x and then plus 2. The equation would be y = 3x + 2.”1 |
In the final example, students were asked to give an equation for the graph of a function seen in Fig.1.6. For competence with this sort of item, students must be able to differentiate functions within families of functions (i.e., a particular representation of the general quadratic function) as well as to understand the meaning of the algebraic, graphic, and numeric parameters that comprise the function. Students who had experienced a text-based curriculum and younger students generally provided incorrect equations that included a negative exponent with the reasoning that it would “… make the curve come back down.”
The following response indicates the sort of reasoning we have found older students to use following our program:
DISCUSSION OF FUNCTION STUDIES
Results of studies carried out with the functions curriculum have shown that, first, even young students are able to use relatively sophisticated strategies when reasoning about functions. Second, there are some problems specific to the domain that are beyond the ability of some younger students, possibly because of their earlier stage of intellectual development and their limited experience with more advanced ideas in mathematics. Third, most of our students, regardless of age, emerge from the program with flexibility with respect to moving among representations and operating in the domain, an understanding of the mathematical meaning inherent to the individual representations of a function, and an ability to interconnect these isolated representations. We believe that introducing a powerful context (i.e., the walkathon) fosters these aforementioned abilities. In this context, students have the opportunity to merge their core digital and analogic schemata. They also have the opportunity to interconnect the numeric, graphic, and algebraic representations of a function early in the curriculum. This foundation for understanding function is expanded when students engage in computer activities, which allow them to make the more elaborate connections necessary for moving toward a complete conceptual structure for the domain.
GENERAL DISCUSSION
The results from the rational number and functions programs now join those that we reported earlier for whole number programs (Griffin & Case, 1997). We believe that they show how students develop a deeper understanding for the domain as a whole when they are brought through a natural developmental sequence, and are provided with a context in which the digital and analog representations on which the sequence is founded can be mapped onto each other–in a meaningful and familiar context–and where they themselves can choose to explore the conjoint space so created in a variety of fashions. An additional feature that the curricula embody is the use of natural language and the creation of unique products that children can display and talk about to others. Not only is there an improved curricular sequence, then, but an improved sense of ownership, and the ability to assume a metaposition in regard to their own and others' learning. Given the known challenges that these domains have presented historically for students of all ages, the accomplishments of the curricula that we have described seem noteworthy, as does the diversity, the fluency, and the sense of individual ownership that the protocols reveal.
A point with which we would like to conclude is one that has not been stressed in the foregoing account of the psychological structures on which the curricula are based. The relationship between structural modeling and curricular design is actually a circular one. Given a rich description of the central conceptual structures whose development is thought to underpin a domain, one can begin the task of designing a curricular sequence, and piloting various bridging devices immediately. But it may also happen that, as the curriculum is actually implemented, students' errors sensitize one to the fact that information that one thought students had automatic access to from other sources, must actually be included in the central conceptual structure itself. All of the structural sequences that have been presented in this chapter had their original origins in the developmental tradition, where underlying processes, rules, and structures were inferred from tests and test protocols. However, in each case, as we attempted to implement the original (relatively sparse) model via the designated curriculum, we discovered ways in which the model had to be expanded or modified in light of children's performance in a learning context. The models we ended up with thus became richer and richer, and included more numerical, linguistic, and iconic detail.
It is our hope that as the developmental models are refined further and the techniques of instruction are refined and generalized in parallel with them, we will be on our way to a deeper understanding of the psychological processes that are involved in successful learning and teaching in these subject areas, across the full range of ages that the elementary school and high school comprise.
ACKNOWLEDGMENTS
Joan Moss and Mindy Kalchman contributed equally to this chapter; order of authorship reflects alphabetical convention. This work was supported in part by the James S. McDonnell Foundation and by the Social Sciences and Humanities Research Council of Canada. We thank the teachers, children, and administrators of the schools in which this research was conducted. We also thank Cheryl Zimmerman for her technical and administrative support.
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1The function y – 3x – 1 was also a correct response that was given by students.