Learning Fractions with Understanding: Building on Informal Knowledge

Learning Fractions with Understanding: Building on Informal KnowledgeAuthor(s): Nancy K. MackSource: Journal for Research in Mathematics Education, Vol. 21, No. 1 (Jan., 1990), pp. 16-32Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749454 .

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Journal for Research in Mathematics Education 1990, Vol. 21, No. 1, 16-32

LEARNING FRACTIONS WITH UNDERSTANDING: BUILDING ON INFORMAL KNOWLEDGE

NANCY K. MACK, Northern Illinois University

Eight sixth-grade students received individualized instruction on addition and subtraction of fractions in a one-to-one setting for six weeks. Instruction was designed to build on the student's informal knowledge of fractions. All students possessed a rich store of informal knowledge of fractions that was based on partitioning units and treating the parts as whole numbers. Students' informal knowledge was initially disconnected from their knowledge of fraction symbols and procedures. Students related fraction symbols and procedures to their informal knowledge in ways that were meaningful to them; however, knowledge of rote procedures frequently interfered with students' attempts to build on their informal knowledge.

"Two of the most compelling issues currently in the cognitive science of instruction are (a) settings for instruction that engage students as active learners and (b) relations between intuitive understanding and knowledge of symbolic procedures" (Greeno, 1986, p. 343). Research has begun to address these issues by focusing on the knowledge related to real-life situations that students construct and bring to instruction and its role in students' learning and teachers' instruction (Brown, Collins, & Duguid, 1989; Carpenter & Fennema, 1988). This type of knowledge has been discussed under the guise of several names: "children's informal mathematics" (Ginsburg, 1982), "intuitive" knowledge (Leinhardt, 1988), "situated" knowledge (Brown et al., 1989), and "prior" or "informal" knowledge (Saxe, 1988). Whatever name is used, this type of knowledge can be characterized gener- ally as applied, real-life circumstantial knowledge constructed by the individual student that may be either correct or incorrect and can be drawn upon by the student in response to problems posed in the context of real-life situations familiar to him or her (Leinhardt, 1988). This type of knowledge is referred to in this article as informal knowledge.

A substantial body of literature has documented that both children and adults possess a rich store of informal knowledge related to a variety of mathematical content domains that they are able to successfully draw upon to solve everyday problems outside of school settings (Carraher, Carraher, & Schliemann, 1987; Lave, Murtaugh, & de la Rocha, 1984; Saxe, 1988; Scribner, 1984). This literature further suggests that informal knowledge is largely unrelated to knowledge of mathematical symbols and procedures. Although a gap may exist between types of knowledge, Hiebert (1988) proposes that in school settings students' informal knowledge can serve as a basis for the development of understanding of mathe-

This article is based on the author's doctoral dissertation completed at the University of Wisconsin-Madison in 1987 under the direction of Thomas P. Carpenter. A previous ver- sion of this paper was presented at the annual meeting of the American Educational Research Association, New Orleans, LA, April 1988.



17

matical symbols and procedures, regardless of the content domain. He cautions however, that this development may be constrained to situations where formal symbols and procedures are similar to properties embedded in prior knowledge.

Insights into the potential role of students' informal knowledge in learning and instruction stem primarily from studies concerning students' understanding of whole number arithmetic (Carpenter, Fennema, Peterson, Chiang, & Loef, in press; Carpenter & Moser, 1983; Fennema, Carpenter, & Peterson, in press; Lam- pert, 1986; Riley, Greeno, & Heller, 1983). Therefore, it is not yet clear whether students' informal knowledge can provide a basis for developing understanding of formal symbols and procedures in other content domains.

Fraction Research

Studies concerning students' understanding of fractions have focused primarily on students' misconceptions rather than on their informal knowledge. Results from the Rational Number Project (RNP) (Behr, Lesh, Post, & Silver, 1983; Behr, Wachsmuth, Post, & Lesh, 1984; Behr, Wachsmuth, & Post, 1985), the National Assessment of Educational Progress (NAEP) (Carpenter, Corbitt, Kepner, Lind- quist, & Reys, 1981; Kouba, Brown, Carpenter, Lindquist, Silver, & Swafford, 1988), and Strategies and Errors in Secondary Mathematics (SESM) (Kerslake, 1986) documented numerous common errors students make when operating on fractions represented symbolically and suggested that many students' understanding of fractions is characterized by a knowledge of rote procedures, which are often incorrect, rather than by the concepts underlying the procedures. These studies examined students' understanding after they had received formal instruction and left unexamined the issue of students' informal knowledge of fractions.

In contrast to the research documenting students' misconceptions regarding fraction symbols and procedures, evidence is beginning to accumulate that suggests students bring to instruction a rich store of informal knowledge of fractions. Pothier and Sawada (1983) and Kieren (1976) have documented that students come to instruction with informal knowledge about partitioning and equivalence, and Behr et al. (1983; 1985) have found that students come to instruction with informal knowledge about joining and separating sets and estimating quantities involving fractions. Additionally, Gunderson and Gunderson (1957) and Leinhardt (1988) have shown that students could successfully perform operations on fractions by drawing upon informal knowledge when problems were presented in the context of real-life situations. Although these studies provide valuable insights into the nature of students' informal knowledge of fractions, they did not address questions related to the ways that students can build on informal knowledge to give meaning to fraction symbols and procedures. Therefore, it is not clear if or how students' informal knowledge can be used to develop understanding of fraction symbols and procedures during instruction.

Furthermore, there is a question of how knowledge of symbolic algorithms af- fects children's ability to draw on informal knowledge during instruction to give meaning to fraction symbols and procedures. Hiebert and Wearne (1988) and



18 Building on Informal Knowledge

Resnick, Nesher, Leonard, Magone, Omanson, and Peled (1989) have suggested that knowledge of rote procedures may hinder students from successfully building on prior knowledge; therefore, they assert that investigations concerning students' learning during instruction should focus in part on the influence that knowledge of rote procedure has on students' ability to build on prior knowledge.

The purpose of this study was to examine the development of students' understanding about fractions during instruction from two perspectives: (a) the ways that students are and are not able to build on informal knowledge to give meaning to fraction symbols and procedures and (b) the influence of knowledge of rote procedures on students' ability to build on informal knowledge.

METHOD

Sample

The sample consisted of eight sixth-grade students of average mathematical ability who were identified by their teachers as having limited understanding about fractions. On an initial screening test all eight students demonstrated little understanding of fraction symbols and algorithmic procedures. All subjects came from a middle school that draws students predominately from middle to upper-middle income families in Madison, Wisconsin. Prior to, and during this study, none of the students received instruction on fractions in their sixth-grade mathematics class. All students received fraction instruction in their fifth-grade mathematics class that followed the traditional textbook series sequence and focused on procedures for finding equivalent fractions, converting mixed numerals and improper fractions, and adding and subtracting fractions with both like and unlike denominators.

General Characteristics of the Instruction

Each student was regarded as an independent case study and received instruction in a one-to-one instructional session. All instructional sessions lasted 30 min- utes and occurred during regular school hours. I met with each student from 11 to 13 times over a period of six weeks, with one exception. One student, Aaron, covered the instructional content by the middle of the fifth week, and his explanations reflected a strong understanding of fraction symbols and procedures; therefore, I concluded his instructional sessions at that time.

All instructional sessions combined clinical interviews with instruction based on guiding principles described by Carpenter, Fennema, and Peterson-viewing students' learning and teachers' instruction as problem solving and the student and teacher as cooperative problem solvers (Carpenter et al., in press; Fennema et al., in press). The majority of the problems were presented to the students verbally. Students were encouraged to think aloud as they solved problems. If they failed to think aloud, they were asked to explain what they had been thinking as they solved the problems.

The instructional content deviated from the topics covered in chapters on fractions in traditional textbook series in two important ways: (a) Students' informal



Nancy K. Mack 19

knowledge about fractions provided the basis for instruction (Carpenter, 1988), and (b) estimation with fractions was emphasized (Hiebert & Wearne, 1986; Na- tional Council of Teachers of Mathematics, 1980, 1989; Reys, 1984). The specific situations where instruction emphasized estimation consisted of three types: (a) approximating the quantity represented by concrete materials, real-world situations, and symbolic representations, (b) estimating sums and differences involving fractions and assessing the reasonableness of answers through estimation, and (c) using fractions to construct sums or differences close to, but not equal to, one.

Concrete materials in the form of fraction circles and fraction strips were available for students to use, and their use was encouraged as long as the students thought they were needed. However, in situations where their solutions remained dependent on the concrete materials at the beginning of the fifth week, the students were gradually encouraged to make the transition to working with symbolic representations. Pencil and paper were available; however, their use was not encouraged until the students successfully solved problems posed in the context of real- world situations by drawing on informal knowledge or using concrete materials.

After each instructional session, a lesson was planned for the student's next session based on the student's informal knowledge, misconceptions related to fraction symbols and algorithmic procedures, and responses to problems presented in previous sessions. Because the purpose of the instructional sessions was to aid the student in drawing on informal knowledge, the lessons were designed to be flex- ible in the topics covered, the amount of time spent on a topic, whether the student was required to master a topic before moving on to another, and the sequence in which the student covered specific fraction topics.

All instructional sessions were audiotaped. Each day I wrote out detailed notes from the student's audiotaped session and transcribed critical protocol segments. The notes and protocols were used in the development of appropriate lessons for following sessions and in the data analysis. I reviewed the student's protocols several times during the study and after its conclusion to identify situations where the student successfully and unsuccessfully gave meaning to fraction symbols and procedures by relating them to his or her informal knowledge. In situations where the student successfully related fraction symbols and procedures to informal knowledge, the protocol review focused on determining if this was accomplished through the student's own initiative or with the assistance of instruction.

Assessment Tasks

The first instructional session focused on assessing the student's knowledge of a variety of fraction topics to determine what knowledge the student initially possessed. Assessment was not limited to the first session but continued throughout all sessions. Each question the student was given during any instructional session was regarded as an assessment task. In general, the tasks were based on four cen- tral ideas: (a) The more parts a unit is divided into the smaller the parts become, (b) a fraction represented symbolically is a single number with a specific value rather than two independent whole numbers, (c) selected ideas of equivalence re-




late to concrete and symbolic representations, and (d) addition and subtraction of fractions represented symbolically requires common denominators.

The specific tasks the student received were based on the student's responses to previous questions and on his or her choice of context for the problems (symbolic representations or real-world situations familiar to the student). The purpose of each task was to encourage the student to draw on informal knowledge related to specific fraction concepts and to relate fraction symbols and procedures to informal knowledge.

The tasks were used not only to assess the student's ability to relate informal knowledge and knowledge of fraction symbols and procedures, but also to provide directions for instruction. In general, in situations where the student was unable to successfully solve a problem, the student was given a simpler problem. In situations where the student successfully solved a problem presented in the context of a real-world situation, the student was given a corresponding problem represented symbolically. If the student successfully solved a problem represented symbolically by relating fraction symbols and procedures to informal knowledge but the relationship appeared to be tenuous, he or she was given a similar task. If the student appeared to understand the relationship between fraction symbols and procedures and informal knowledge, the student was given a problem that was closely related but more complex. For example, if the student successfully solved the following word problem: "Suppose you had four cookies and you ate seven-eighths of one cookie, how many cookies would you have left?" the problem 4 - 7/8 was presented. If the student successfully solved this problem, he or she was given a problem such as 4 1/8 - 7/8 or a problem posed in the context of a real-world situation involving 4 1/8 - 7/8.

Individualizing Instruction

The specific manner in which I used instruction to assist students in relating fraction symbols and procedures to their informal knowledge varied. In general, I continually assessed the student's thinking and adjusted my instruction to make the problems that drew on the students' informal knowledge and knowledge of fraction symbols and procedures more and more similar. This often involved moving back and forth between concrete materials, problems posed in the context of real- world situations, and problems represented symbolically as well as moving back and forth between specific fraction topics.

Results

All eight students came to instruction with misconceptions related to their knowledge of fraction symbols and procedures; however, they also came with a substantial store of informal knowledge about fractions that enabled them to solve numerous problems presented in the context of real-world situations. Five themes characterized the nature of the students' informal knowledge of fractions and the ways they were able to build on informal knowledge during instruction: (a) Stu- dents' informal solutions involved partitioning units; (b) students' informal knowl-



Nancy K. Mack 21

edge of fractions was initially disconnected from their knowledge of fraction symbols, procedures, and concrete representations; (c) students could build on informal knowledge when they could match problems represented symbolically to problems presented in the context of real-world situations they understood; (d) students often encountered interference from knowledge of rote procedures when attempting to solve problems represented symbolically and in real-world situations; and (e) limited transfer of knowledge was observed even in students who were able to build on informal knowledge in other contexts. The results are organ- ized into five sections based on these themes.

Informal Solutions Involving Partitioning In general, students' informal solutions of fraction problems involved separat-

ing units into parts and dealing with each part as though it represented a whole number rather than a fraction. Because all eight students approached fraction problems in this manner throughout the study, they continually referred to fractions in terms of the "number of pieces" rather than the size of fractions and solved numerous problems in a rational and relatively error-free way. The following protocol, which was taken from Ned's fourth instructional session, illustrates this approach.

I: Suppose you have two lemon pies and you eat 1/5 of one pie, how much lemon pie do you have left?

Ned: You'd have 1 4/5. First of all you had 5/5 to start with. Then if you ate one you'd have four pieces left out of the five, and you still have one whole pie left.

Ned's responses "you ate one" and "four pieces left out of five" suggested that he was thinking of 5/5 as five parts and was treating each part as an independent unit or whole number. Ned's response also illustrated that by dealing with the parts in this manner, he solved the problem in a way that was meaningful to him.

Initial Disconnection Between Knowledge of Symbols, Procedures, and Concrete Representations

At the beginning of the study all students successfully solved numerous problems presented in the context of real-world situations and consistently explained their solutions in terms of informal knowledge about fractions. They could not, however, solve many problems represented symbolically that were similar to the real-world problems. Furthermore, they often explained their solutions using faulty knowledge related to formal symbols and algorithmic procedures.

One situation where students' responses suggested this initial disconnection between their informal and symbolic knowledge involved comparing fractional quantities. During the first or second instructional session, each student was asked a question like "Suppose you have two pizzas of the same size, and you cut one of them into six equal-sized pieces and you cut the other one into eight equal-sized pieces. If you get one piece from each pizza, which one do you get more from?" All students responded that they would get more from the pizza cut into six pieces. Bob explained his answer in a manner representative of all the students: "The one




cut into six has fewer pieces so there's more in each piece." Each student also was asked a question like "Tell me which fraction is bigger,

1/6 or 1/8." Five students were presented this problem immediately after solving a related problem presented in the context of a real-world situation, and three students were presented this problem prior to solving a similar real-world problem. Four of the students who first solved the real-world problem and the three students who first worked with the symbolic representation responded, "One eighth is bigger." Julie explained her answer in a manner representative of these students: "[Eight] is a bigger number I think. [Eight] is bigger than [six]."

The differences between the students' explanations for the problem presented in different contexts, and the fact that four of the five students unsuccessfully com- pared fractions represented symbolically immediately after successfully comparing fractions presented in the context of a real-world problem, suggested that the students' informal knowledge related to comparing fractional quantities was initially disconnected from their knowledge of fraction symbols. Further, the students' explanation of why 1/8 is bigger than 1/6 suggested that their knowledge of fraction symbols was characterized in part by misconceptions that stemmed from attempting to apply rules and operations for whole numbers to fractions.

There were several situations early in the study where students' responses suggested that their knowledge of fraction symbols, procedures, and concrete representations was initially unrelated to their informal knowledge about fractions. In each situation, students explained their answers in a manner similar to that described above. Informal knowledge about fractions was initially used only for problems presented in the context of real-world situations, and knowledge of symbols and procedures was used for symbolic and concrete representations.

All students' informal knowledge allowed them to determine the appropriate unit in a real-world problem. However, they had difficulty identifying the unit in situations represented symbolically and concretely. Thus, students treated symbolic and concrete representations involving collections of units as though they were a single unit. Aaron's response illustrates this misconception: "Fractions are a part of a whole.... They're always less than one whole." Even when situations were presented concretely, students tended to consider all the parts represented as the unit.

The following protocol, which was taken from Julie's second instructional session, illustrates this error.

1: (showing Julie the picture in Figure 1) How much is shaded?

Figure 1: Julie's picture to identify



Nancy K. Mack 23

Julie: Five-eighths. I: Suppose I said those were pizzas. Julie: (interrupting) Oh, 1 1/4!

Julie's response "5/8" suggested that she initially interpreted the unit as the eight pictured parts. However, her response "1/4" suggested that she was able to determine the appropriate unit when the problem was presented in the context of a real- world situation that made the context clear. All the other students initially responded in a manner similar to Julie when encountering problems involving mul- tiple units.

Building on Informal Knowledge

Instruction began building on the student's informal knowledge during the first instructional session. Each student was presented with problems in the context of real-world situations to determine what prior knowledge the student possessed for various topics and then the student was presented with problems represented symbolically that matched his or her informal knowledge. The student's thinking was continually assessed, and problems were adjusted to match the changes that occurred in the student's knowledge of fraction symbols and procedures.

When appropriate problems were used, all eight students attempted to use their informal knowledge of fractions to construct meaningful algorithms for problems represented symbolically. All students encountered some situations where they successfully used their informal knowledge on their own to give meaning to fraction symbols and procedures. In general, this occurred when students first were presented problems in the context of real-world situations and then were presented symbolic problems that were closely related. In instruction I frequently had to move back and forth between problems represented symbolically and problems in the context of real-world situations before students successfully related fraction symbols and procedures to their informal knowledge. In general, this occurred when the problems that drew on students' informal knowledge and the problems involving fraction symbols and procedures were less similar. Thus, students' ability to build on informal knowledge of fractions appeared to be dependent on a close match between problems represented symbolically and those that drew on their informal knowledge.

One of the earliest situations where students were able to build on their informal knowledge of fractions involved subtracting a fraction from a whole number, i.e., 4 - 7/8. The following protocol, which was taken from Aaron's first and second instructional sessions, illustrates the manner in which students extended their informal knowledge to symbolic procedures.

First Instructional Session

I: When you add fractions, how do you add fractions? Aaron: Well, you go across.You add the top numbers together and the bottom numbers

together.




I: Now I want you to solve this problem (shows Aaron a piece of paper with 4 - 7/8 printed on it).

Aaron: (Writes 4 - 7/8 on his paper) Well, you change this (the 4) to 4/4. I: Why 4/4? Aaron: 'Cause you need a whole, so you have to have a fraction and that's that fraction, and

then you have to reduce, or whatever that's called, that (the 4) times two, so you'll have 8/8. Eight eighths minus seven, so it's 1/8.

I: Now suppose I told you you have four cookies and you eat 7/8 of one cookie, how many cookies do you have left?

Aaron: You don't have any cookies left. You have an eighth of a cookie left.

I: If you have four cookies... Aaron: (interrupting) Oh! Four cookies!

I: ...and you eat 7/8 of one cookie, how many cookies do you have left? Aaron: Seven-eighths of one cookie? Three and one eighth. I: Now how come you got 3 1/8 here (referring to what Aaron had just said) and you got 1/

8 there (referring to paper)? Aaron: (Pauses, looking over problem) I don't know. (Contemplates problem; repeats prob-

lem.) Well, because on this you're talking about four cookies, and on this you're talking about one.

Second Instructional Session

I: Last time we were working on the problem 4 - 7/8. Aaron: (Immediately writes 4/4 - 7/8) That's impossible! This (4/4) is smaller than that (7/

8), in fraction form it is. This (4/4) actually equals one.

I: Suppose you have a board four feet long and you cut off a piece 7/8 of a foot long to make a shelf. How much of the board do you have left?

Aaron: (Looks at the problem he had written earlier, 4/4 - 7/8.) I: Don't look at your problem [on paper]. (Repeats board problem). Aaron: (Draws a line for the board, first thinking 7/8 of the whole board. Instructor repeats

the problem; Aaron marks off the board to show four feet.) Oh, I know now, 3 1/8 feet.

I: Very good. Now you said the problem couldn't be worked.

Aaron: You have to multiply to find the same denominator, which is eight, so four times two is eight and this four times two is eight, so it's 8/8. (Writes 3 8/8 - 7/8 = 3 1/8 on his paper.)

I: Now where'd you get this 3 8/8? Aaron: This used to be 3 4/4, and 4/4 is one, and I need that so I can take a piece away. I: You couldn't figure that problem out last time.

Aaron: I thought four was the same as 4/4, but it's really the same as 3 4/4, 3 8/8, 3 2/2, 3 1/1....

I: I want you to solve this problem: 4 1/8 minus 5/8. Aaron: (Immediately writes 3 8/8 - 5/8 = 3 3/8.) I: Let's use the [fraction] pieces to see if that's right.



Nancy K. Mack 25

Aaron: (Puts out 3 1/8 circles.) Wait, 3 1/8 (looks at his paper)...I can't change it to 8/8 for some reason.

I: Why not? Aaron: It's gotta be changed to 9/8. (Changes 3 8/8 to 3 9/8 on his paper.) I: Why 9/8? Aaron: Because there's one piece over there and you have to add it. I: How'd you figure that out? Aaron: It just seems like that's the right thing, 'cause if there's 8/8 and there's one over here,

you have to add it. You can't just forget about it, and that'd make 4/8, 3 4/8, well, 3 1/2. (Writes answer on his paper.)

Although Aaron's prior knowledge of fractions included the effects of formal instruction, his comment "add top numbers together and bottom numbers together" suggested how fragile this knowledge was and how disconnected it was from his informal knowledge about adding and subtracting fractions. Subsequently he used his informal knowledge to solve the problem in a way that was meaningful to him. The movement back and forth between problems in different contexts appeared to provide a match between the symbolic representations and Aaron's informal knowledge that enabled him to construct a meaningful algorithm for the real-world problem and then to extend the algorithm to solve 4 1/8 - 5/8. Furthermore, Aaron's comments "so I can take a piece away" and "there's one piece over there" suggested that he thought of the fraction as parts and treated each part as a sepa- rate unit.

The other seven students responded to problems involving subtracting a fraction from a whole number in a manner similar to Aaron's. All eight students progressed through this sequence of subtraction problems (4 - 7/8, 4 1/8 - 5/8, 4 1/8 - 1 5/8) before discussing equivalent fractions and solving problems involving adding and subtracting fractions with unlike denominators. All students also constructed meaningful algorithms for converting mixed numerals and improper fractions on their own initiative as they solved subtraction problems with regrouping.

During the early instructional sessions I encouraged students to match symbolic representations to problems that drew on their informal knowledge. All students soon began sharing this responsibility as they related fraction symbols and procedures to informal knowledge in meaningful ways. On their own initiative, all eight students explained their solutions in terms of problem contexts that were meaningful to them.

A major consequence of building on students' informal knowledge of fractions was that students often invented alternative algorithms. Some of these alternative algorithms involved several steps and were rather inefficient, whereas others proved to be very efficient. Whether the alternative algorithms were efficient or not, they differed from algorithms that are traditionally taught. Students suggested that these algorithms were meaningful to them and favored them over traditional algorithms.

Students commonly invented alternative algorithms for subtraction problems




with regrouping. Six students invented the following procedure for solving problems such as 4 1/8 - 1 5/8: First subtract one from four to get three, next subtract 5/8 from three ("because you can't subtract 5/8 from 1/8") to get 2 3/8, then add 1/8 to 2 3/8 ("because that's still left from what you started with") to get 2 4/8 or 2 1/2. The sequence of steps suggests that to construct their algorithm the students first matched the problem to their knowledge of whole numbers and then to their knowledge of problems involving the subtraction of a fraction from a whole number.

Another common alternative algorithm was for converting mixed numerals and improper fractions. Students are traditionally taught to convert mixed numerals to improper fractions by multiplying the denominator and whole number together, then adding the numerator to obtain the new numerator for the improper fraction. They are taught an inverse process from that described above for converting improper fractions to mixed numerals. All students in this study invented a similar alternative algorithm for converting mixed numerals and improper fractions based on their knowledge of fractions equivalent to one. The following protocol, which was taken from Teresa's fifth instructional session, illustrates this alternative algorithm.

I: I want you to write 3 5/8 as an improper fraction. Teresa: Twenty nine-eighths, eight goes into three, I mean 8/8 goes into one, so it's 8, then

16, then another one is 24, plus 5 is 29. I: Now write 14/3 as a mixed numeral. Teresa: (Writes 3/3 3/3 3/3 3/3 2/3.) Four and two-thirds. I had to write it down or else

I'd get it mixed up in my head.

Four students in this study also invented an alternative algorithm for comparing fractions by building on the knowledge that the more parts a unit is divided into the smaller the parts become. Students are traditionally taught to compare fractions with unlike denominators by first finding common denominators, changing the fractions to equivalent fractions, and then comparing the numerators. The students in this study however, used knowledge of unit fractions to compare certain fractions. The following protocol, which was taken from Teresa's tenth instruction session, illustrates this alternative algorithm.

I: Tell me which of these fractions is the largest, 2 2/3 or 2 5/6? Teresa: Two and five sixths...Well...because, umm...2 2/3,..., 2/3 is close to one, and 5/6 is

close to one, but sixths are smaller than thirds, and so you have littler, less way, or less...umm (pause), littler pieces to get to one.

These four students utilized their alternative algorithm throughout the study even after they used the traditional algorithm for other problems. The students' responses suggested that they viewed their algorithm as being more efficient than finding common denominators.

The students invented a variety of alternative algorithms throughout the study. One common characteristic of all student-invented algorithms, with the exception



Nancy K. Mack 27

of the alternative algorithm for comparing fractions, was that in general, they were not utilized for an extended period of time. Students soon discovered quicker ways of solving the problems, frequently through their own efforts or after asking me how I would solve the problem. As soon as they discovered these quicker algorithms, they abandoned their alternative algorithms in favor of the more efficient ones, which often reflected those that are traditionally taught in schools.

Interference of Rote Procedures

All the students came to instruction with isolated knowledge of procedures for operating with fractions which were often faulty. This knowledge frequently interfered with their attempts to give meaning to fraction symbols and procedures in two ways: (1) Knowledge of procedures often kept students from drawing on their informal knowledge of fractions even for problems presented in the context of real- world situations, and (2) initially, students often trusted answers obtained by applying faulty procedures more than those obtained by drawing on informal knowledge. All eight students exhibited these difficulties at some time during the study.

Even after students were encouraged to draw on informal knowledge, some of them continued to solve the problems and explain their answers in terms of previ- ously learned symbolic procedures. The following protocol, which was taken from Laura's first instructional session, illustrates the influence of these procedures.

I: (Shows Laura a paper with 7/8 + 5/6 printed on it.) Now I want you to estimate the answer to this problem, 7/8 plus 5/6.

Laura: (Immediately writes on her paper: 35/48 + 35/48 = 100/96) I: Where did you get the 35/48 plus 35/48? Laura: Well, you try to figure out what number 5 and 7 can go into, which is 35. That's

where I got the 35, and then you find out what 8 and 6 go into, which is 48, and then they were like, if they were under each other then you'd put where it equals, and then you put the number here and then you add them.

Both Laura's and one other student's thinking continued to be dominated by their knowledge of rote procedures in this manner throughout much of the study.

At the beginning of the study all eight students often obtained different answers for problems presented in different contexts. When this happened, all the students often resolved the inconsistencies in favor of the faulty procedures. The following protocol, which was taken from Tony's first instructional session, illustrates the way that knowledge of faulty procedures initially interfered.

I: When you add fractions, how do you add them? Tony: Across. Add the top numbers across and the bottom numbers across. I: I want you to think of the answer to this problem in your head. If you had 3/8 of a pizza

and I gave you 2/8 more of a pizza, how much pizza would you have? Tony: Five-eighths. (Goes to his paper on his own initiative and writes 3/8 + 2/8 =, gasps,

stops, then wites 5/8.) I don't think that's right. I don't know. I think this (the 8 in 5/8) just might be 16. I think this'd be 5/16.




I. Let's use our pieces to figure this out. (Tony gets out 3/8 and then 2/8 of the fraction circles and puts the pieces together.) Now how much do you have?

Tony: Five eighths. It seems like it would be sixteenths.... This is hard.

The other seven students also attempted to resolve inconsistencies by applying knowledge of faulty procedures. The interference of faulty procedures could be overcome, but this was not easily accomplished. All the students required some assistance in overcoming the interference of faulty procedures. Students first were presented with problems in the context of real-world situations and encouraged to draw on their informal knowledge and use concrete materials. After they explained their solutions to the real-world problem in terms of their informal knowledge, they were encouraged to record the problem and answer on paper. When students continued to apply faulty procedures when recording problems symbolically, they were given real-world problems and asked to explain their answers in terms of their informal knowledge and to model the problem with concrete materials. Three students overcame the interference of knowledge of faulty procedures rather quickly when instruction proceeded in this manner. The other five students took longer to overcome this interference. For these five students it was necessary to spend a great deal of time solving real-world problems, then modeling the problems with concrete materials, and finally, recording the problems symbolically.

Limited Transfer of Knowledge

Limited transfer of knowledge was observed even in students who were able to build on their informal knowledge of fractions in other contexts. As students de- veloped meaning for fraction symbols and procedures by relating them to informal knowledge, they continued to focus on breaking fractions into parts and treating the parts as whole numbers rather than as fractions. They continued to refer to fractions in terms of "the numbers 6f pieces" rather than the size of the fraction. The following protocol, which was taken from Julie's ninth instructional session, illustrates how thinking of fractions as the number of parts rather than as fractions limited students' ability to successfully relate fraction symbols to informal knowledge.

I: Tell me which of these two fractions is the smallest, 4/5 or 5/6? Julie: Four fifths. I: Why 4/5? Julie: 'Cause the denominator...wait, would you say them again? I: Four fifths and 5/6. Julie: They're the same. I: Why are they the same? Julie: Because there's one piece missing from each-1/6 missing from 5/6 and there's 1/5

missing from 4/5.

Three other students responded in a manner similar to Julie's when comparing fractions such as 4/5 and 5/6. They too focused on the number of missing pieces



Nancy K. Mack 29

rather than on the size of the fractions as they solved the problem. The responses of six students suggested that their informal knowledge of frac-

tions did not readily transfer to discrete sets. During the first instructional session, these six students were asked to partition a discrete set of objects, such as showing 2/3 of six cookies. Only one student successfully partitioned the set. Five students responded that "two cookies" were 2/3 of all the cookies. Jason explained this response saying, "You want two out of three." This type of situation was not included during most of the instructional sessions, but each student received a task of this type near the end of the study. Only two students successfully partitioned the set on their own. Four students solved the problems after it was suggested that they think of the cookies as one big cookie rather than as several small cookies.

DISCUSSION

This study provides a different picture of students' understanding about fractions than has been presented by many previous studies. Although a growing number of studies have suggested that students possess informal knowledge of fractions, these studies and others have continued to suggest that many students have little understanding of fraction symbols and algorithmic procedures. This study shows students coming to instruction with a rich store of informal knowledge about fractions that they are able to build on to give meaning to formal symbols and procedures. However, there appear to be limits to students' informal knowledge.

The results suggest that initially students' informal knowledge of fractions is disconnected from their knowledge of fraction symbols and procedures and is limited to a partitioning conception of rational number. Some researchers may argue that students' informal partitioning approach may severely limit students' understanding of rational number and cause them difficulties in the long run (Behr et al., 1983; 1984; 1985; Kerslake, 1986). The students in this study, however, demonstrated that they were able to utilize their partitioning approach to solve a variety of problems in meaningful ways. These results suggest that students' understanding is not severely limited by their partitioning approach to fraction problems, and despite the limitations that do exist, it is possible for students to relate fraction symbols to informal knowledge in meaningful ways, provided that the connection between the informal knowledge and the fraction symbols is rea- sonably clear. The results add evidence in support of Hiebert's (1988) proposal that informal knowledge can provide a basis for developing understanding of mathematical symbols and procedures in complex content domains.

As students used their partitioning approach to solve fraction problems, they demonstrated that they were able to solve numerous problems early on that have traditionally been considered to be some of the most difficult problems students encounter, such as subtraction problems with regrouping and converting mixed numerals and improper fractions. From a mathematical perspective, these types of problems depend on more prerequisite skills than most other types of problems involving fractions. Therefore, traditional textbook series present these problems




last in the sequence of topics for addition and subtraction. The results of this study, however, suggest that as students build upon informal knowledge in meaningful ways, the development of their understanding may differ from the sequence in which fraction topics are traditionally taught.

The results suggest that students can build on informal knowledge to develop an understanding of fraction symbols and procedures, but can they expand on this core of knowledge to develop a broad conception of rational number? Several researchers propose that students' understanding of rational number is dependent on first developing a broad conception of rational number (Behr et al., 1983; 1984; 1985; Kieren, 1976) and then progressing through a sequence of topics within each strand of rational number that are based on mathematical prerequisites (Behr et al., 1983; 1984; 1985; Kerslake, 1986). The results of this study concerning the nature of students' informal knowledge and the different sequencing of topics suggest that there may be alternative ways for developing a broad conception of rational number. A viable alternative may be to develop a strand of rational number based on partitioning, and then to expand that conception to other strands once students can relate mathematical symbols and procedures to their informal knowledge and can reflect on the relations. Investigations of this issue are needed to gain deeper insights into the potential role of informal knowledge in the development of students' understanding of rational number.

Although students demonstrated that they were able to build on their informal knowledge by relating symbolic representations to problems that drew on their informal knowledge of fractions, the results suggest that knowledge of rote procedures interferes with students' attempts to construct meaningful algorithms. Stu- dents focused on symbolic manipulations, whether correct or incorrect, in situations where they possessed knowledge of rote procedures. The results, however, do not suggest that the influence of rote procedures cannot be overcome, but a great deal of time and directed effort is needed to encourage students to draw on informal knowledge rather than use rote procedures. The results add more evidence to the argument in favor of teaching concepts prior to procedures (Hiebert & Wearne, 1988; Resnick et al., 1989), and suggest that students can construct meaningful algorithms by building upon informal knowledge.

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AUTHOR NANCY K. MACK, Assistant Professor, Department of Mathematical Sciences, Northern Illinois Uni-

versity, DeKalb, IL 60115



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Learning Fractions with Understanding: Building on Informal Knowledge