Sixth graders' oral retellings of compare word problems.

Abstract

In this study, we explored the use of oral retellings as a problem solving strategy for compare word problems in mathematics. Twenty-nine students in two sixth-grade mathematics classes, one class labeled as average achieving and the other labeled as low achieving, received instruction in the use of oral retellings as a strategy for solving word problems. Following four 30-minute instructional sessions, eight students from each class were selected to participate in one-on-one interviews. During the interview, we asked the student to retell and solve six compare word problems, three containing consistent language and three containing inconsistent language. To investigate the nature of oral retellings when used as a problem solving strategy, we analyzed responses for problem representation, strategy development, and language. Results showed that inconsistently-worded items were more difficult for students to solve than items with consistent language. Students who translated inconsistent items into consistent language more frequently selected an appropriate arithmetic operation and were more successful than those students who appeared to rely on key words rather than a meaningful representation of the problem, That is, their retelling.

Sixth Graders' Retellings of Compare Word Problems

A train leaves Chicago at 1:27 p.m. Another train leaves St. Louis at 4:30 p.m. At what time will the students reading this word problem (a) consider the exercise pointless, (b) begin doubting their ability to solve it, (c) lose interest altogether, or (d) all of the above?

Teachers struggle to motivate, empower, and enhance students' confidence as problem solvers. While the ability to solve problems is imperative to all areas of life, in mathematics, problem solving is uniquely essential; it is "a hallmark of mathematical activity and a major means of developing mathematical knowledge" (National Council of Teachers of Mathematics, 2000, p. 116). Indeed, problem solving is the substance of mathematics, with word problems "front and center" as problem solving contexts for developing mathematical power as well as analytical thinking and cognitive abilities (Latterell & Copes, 2003; Parmer, Cawley, & Frazita, 1996). Reform-minded teachers seek to create a culture of problem solving wherein students communicate mathematically about problems and their strategies for approaching them. Word problems offer a natural ingress into problem solving and a culture of mathematical power and communication.

Word problems and the strategies associated with solving them have been the subject of abundant research for many years (Lester, 1994; Trafton & Midgett, 2001). Compare word problems have been the focus of a large portion of that research. Lewis and Mayer (1987) defined compare problems as those containing a "static numerical relation between two variables" (p. 363). The static nature of the compare problem makes it the most difficult type of addition/subtraction word problem to solve (see Carpenter, Fennema, Franke, Levi, & Empson, 1999; Fuson, Carroll, & Landis, 1996; LeBlanc & Weber-Russell, 1996; Mwangi & Sweller, 1998; Okamoto, 1996; Pape, 2003; Riley & Greeno, 1988). Limited research has been conducted to assess the potential of oral retellings in measuring students' comprehension of compare word problems (Verschaffel, 1994). The purpose of this study is to tell a story of retelling that is, to examine the nature of oral retellings of compare word problems as evidenced in interviews with 16 sixth grade students.

Key Terms

Word Problems

Also called "story problems," word problems tell a mathematical story. An example of a word problem follows:

Jose has 11 butterflies in his collection. On a family vacation, he catches 4 more butterflies. How many butterflies does Jose have in his collection now?

Compare Word Problems

Compare word problems involve the static comparison of two disjoint sets and usually contain a relational statement or term (more, less or fewer). Compare problems have been labeled using a variety of classification systems (see Briars & Larkin, 1984; Carpenter et al., 1999; Fuson et al., 1996; Riley, Greeno, & Heller, 1983). This study focused on the nature of retellings of compare problems without grouping them according to the categories proposed by these researchers. Instead, problems were classified according to the manner in which the relational term was used in the context of the problem. The research of Lewis and Mayer (1987) regarding consistent and inconsistent language guided this classification.

Subject and Object of Inconsistently-worded Compare Problems

Lewis and Mayer (1987) used the terms subject and object to identify the two elements of the complex sentence usually occurring as the second sentence in an inconsistently-worded compare problem. The tradition established by Lewis and Mayer and continued by subsequent researchers allows for the use of these terms; although problematic to grammarians, the current study maintained this precedent. This specialized use is illustrated in the following word problem:

  Amanda has 11 cupcakes. She has 5 fewer cupcakes [subject] than
  cookies [object]. How many cookies does she have?

Consistent Versus Inconsistent Compare Problems
  Lewis and Mayer (1987) illustrated consistent and inconsistent
  language using the following word problems:
  Consistent: Joe has 3 marbles. Tom has 5 more marbles than Joe. How
  many marbles does Tom have?
  Inconsistent: Joe has 8 marbles. He has 5 more marbles than Tom. How
  many marbles does Tom have? (p. 354)
These researchers described the language issues associated with these
problems with the following explanation:
  In consistent language problems the unknown variable (e.g., Tom's
  marbles) is the subject of the second sentence, and the relational
  term in the second sentence (e.g., more than) is consistent with the
  necessary arithmetic operation (e.g., addition). On the other hand, in
  inconsistent language problems the unknown variable is the object of
  the second sentence, and the relational term (e.g., more than)
  conflicts with the necessary arithmetic operation (e.g., subtraction).
  (p. 363)

In this study, word problems in which the relational term more, less or fewer produces a conflict between the action cued by the term and an appropriate arithmetic operation for solving the problem were deemed to be inconsistently-worded. The following problem serves as an example:

Amanda has 11 cupcakes. She has 5 fewer cupcakes than cookies. How many cookies does she have?

To the problem solver, the word fewer may suggest subtraction of the two numbers given; however, the inverse operation (addition) leads to the correct solution.

Translation Procedurefor Inconsistently-worded Compare Problems

Lewis and Mayer (1987) described a translation procedure for inconsistently-worded compare problems that successful problem solvers frequently use, i.e., changing the relational term to its opposite and switching the subject or object of the second sentence. This procedure changes inconsistent language into consistent language and maintains the meaning of the original problem. For example:

Problem:

Patrick rode his bike for 27 minutes. He biked for 13 minutes less than he jogged. How many minutes did Patrick jog? A possible retelling using this translation procedure:

A boy rode his bike for 27 minutes. Then he jogged for 13 minutes more than he biked. How many minutes did he jog?

Oral Retellings

In a general context, an oral retelling may involve recalling a story or a section of expository text. For a reading passage, an oral retelling may emphasize important elements of the story structure such as plot, inferences, and details (Reutzel & Cooter, 2004). Used in this way, oral retellings of stories assess overall comprehension.

In this study, an oral retelling was a verbal reconstruction of a word problem. Students were asked to read a word problem, retell it in their own words, and solve the problem. Instead of using the oral retelling as an assessment, the retellings in this study served as windows into how students represented the problem, selected computation strategies, and responded to linguistic issues such as consistent and inconsistent wording.

Methodology

An elementary school in a suburban city in the western United States was selected for the site of this study. The sixth grade mathematics teacher sought a culture of problem solving. She had instructed students in nine specific strategies for approaching word problems. These strategies were: making a list, drawing a diagram, working backward, using trial and error, using manipulatives, using deductive logic, breaking the problem into subsets, looking for a pattern, and constructing a graph or table. In addition to providing explicit instruction in these strategies, the teacher used a problem of the week to review the strategies, give students opportunities to apply them, and cultivate the culture of problem solving.

Two classes of sixth graders served as the participant pool for this study. Students were assigned to these classes at the beginning of the academic year according to similar mathematical achievement. Mathematics proficiency was determined using a combination of fifth grade teacher recommendations; Standford Achievement Test scores; and scores on a mathematics computation test comprising basic addition, subtraction, multiplication, and division of whole numbers, operations on fractions, and decimal operations. Based upon proficiency in mathematics as determined by these measures, students were grouped as either above average, average achieving, or low achieving. Because of school scheduling conflicts, no students from the class identified as above average in mathematics achievement participated in the study.

Participants and Instruction by the Researcher

Twenty-nine students participated in instruction in oral retellings as a problem solving strategy. The instruction took place during four 30-minute periods. Students first practiced retelling narrative and expository text selections; then they were introduced to retelling mathematical word problems. The students also wrote original word problems and retold them with partners. Some of the techniques introduced and practiced during the instructional period included: organizing information from the story into segments such as beginning, middle, and end; using elaboration to create a mental image of the physical process taking place in the story (such as giving something away); highlighting the main ideas; and emphasizing accuracy. This instructional approach loosely followed the practice sessions developed by Gambrell, Kapinus, and Koskinen (1991) in which students were instructed in using retellings of prose as an instructional strategy in language arts.

Interview/Observation

We randomly selected eight students from each class to participate in an individual interview. The first author conducted each of the interviews, modeling the retelling protocol on a warm-up problem as shown in the interview protocol found in Appendix A. The word problems used in the interview/observation are found in Table 1 along with the consistent or inconsistent language designation, relational term, and the percentage of participants who arrived at a correct numerical solution for each item. Each word problem was written on a separate index card and was administered in the same order to each of the participants. The students were told that their responses would be audio recorded.

The interview format was modeled after the immediate free-recall protocol employed by Gambrell et al. (1991) and Verschaffel (1994) and was closed-ended (Fetterman, 1998), meaning that no follow-up questions were asked after the students' initial response. Paper and pencil as well as manipulatives (i.e., counters) were provided, and the student was asked to solve the problems using any method he or she chose. The student was instructed to read the word problem as many times as needed to understand it and, when ready, turn the card over, restate the word problem and solve it.

In the instructions given prior to the interview/observation, the researcher told students they would not receive feedback as to the correctness of their retelling or numerical answer but would simply be thanked for their response. There was no time limit invoked; however, prompts such as "Can you retell it in your own words?" were used when the child paused for longer than 10 seconds. If a student could not complete a retelling, he/she was allowed to turn the card over and reread the word problem one additional time. Each interview lasted about 20 minutes and was tape recorded. The researcher took notes about the students' physical responses, including use of manipulatives, and collected scratch paper used by students. Transcriptions were made and students' paper-and-pencil work was triangulated with the verbal data from the interviews and observation notes.

Data Presentation

An assessment using the retelling provides the teacher with qualitative data about language development. Used with English as a second language students, Bernhardt's (1991) recall protocol illustrated that the difficulties students have understanding the second language may be a result of a number of influences: metacognition, syntactic features, intratextual perceptions, phonemic or graphemic elements, and/or word recognition. Bernhardt's qualitative analysis employed a spreadsheet format to organize the retelling data. In other research, Barnett-Foster and Nagy (1996) used similar procedures to analyze response strategies employed for various test formats and looked for differences in types and frequencies of problem-solving behaviors. We developed a matrix similar to Bernhardt's, with heavy influence from Barnett-Foster and Nagy, to organize and analyze retelling data. The matrix (found in Appendix B) served as a guide in coding the retellings according to the amount and kinds of information that were restated. Retellings were analyzed as to problem representation, strategy development, and language usage. We examined types of strategies and behaviors in relation to correct and incorrect solutions of the compare problems. Consider, for example, Adrienne's (a pseudonym, as are all other names of students) retelling of Item 3:

Item 3: Patrick rode his bike for 27 minutes. He biked for 13 minutes less than he jogged. How many minutes did Patrick jog for?

Adrienne's Retelling:

Patrick rode his bike for 27 minutes. And then he went jogging after that, and he jogged for 13 minutes more than he biked. (Writes 27 + 13 = 40 vertically on scratch paper.) 40.

The scoring guide coding Adrienne's retelling follows:

Problem Representation                                Item 3

Repreated original problem word-for-word (or nearly)
Restated the problem in own words                     [square root of]
Retelling changed meaning of problem
Correctly defined the goal of the problem

Strategy Development
Referred to an algorithm                              [square root of]
Referred to a memorized math fact
Appeared to use a counting strategy
Appeared to use a key word strategy (uses + for
more,--for less or fewer) for inconsistent language
items
Used an appropriate arithmetic operation              [square root of]
Appeared to rely on stereotypical expectations or
patterns (uses subtractions for all items)

Language
Conserved language
Recalled incorrect numbers
Elaborated beyond original problem                    [square root of]
Used pronoun referents
Translated inconsistent langauge to consistent        [square root of]
Seemed confused by structure (consistent,
inconsistent)
Correctly solved problem                              [square root of]

From the matrix, we see that Adrienne represented the problem using her own words rather than repeating the problem verbatim. Her computation strategy involved referring to a written algorithm for which she selected an appropriate arithmetic operation, addition. She elaborated beyond the original problem by adding elements of time and order, saying that Patrick rode his bike and then went jogging. She translated inconsistent language to consistent language by changing the relational term to its opposite (i.e., less to more) and switching the subject and object of the second sentence (biked tojogged). Finally, Adrienne's numerical solution to the problem was correct.

In the overall analysis of the data, the use of the matrix format allowed for a visual method of recording information. In addition, the matrix facilitated observing patterns in behavior among the 16 participants.

Data Analysis

The purpose of this study was to tell a story of retellings that is, to examine the nature and usefulness of oral retellings as a potential problem-solving strategy for compare word problems. Table I shows each word problem with its consistent or inconsistent language designation, relational term, and the percentage of participants who arrived at a correct numerical solution. Inconsistently-worded problems containing the terms less and fewer provided the greatest challenge. This information provides a context for interpreting the remaining data on sixth graders' oral retellings.

Overall Problem Representation

In the instructions for the interview, the researcher encouraged students to retell the problem in their own words rather than attempt to memorize it. Analysis of the retellings showed that rather than repeating a problem exactly as written (or almost exactly), most students did restate the problem in their own words. Students appeared to memorize consistently-worded items more easily and frequently than they memorized inconsistently-worded items. For example, Item 6, a consistently-worded item, was the problem most frequently repeated verbatim, followed by consistently-worded Item 1. Inconsistently-worded Items 3 and 5 were most frequently restated in the students' own words. Adrienne's retelling of Item 3 (above) was typical of a retelling in which the student restated the problem in his/her own words.

Students whose retelling changed the meaning of the problem seldom arrived at a correct solution. In particular, students struggled to maintain the meaning of inconsistently-worded items. Items 3 and 5 were most commonly misinterpreted, and these items were also the most often solved incorrectly (see Table 1). However, Bill's retelling of Item 3 shows that, despite maintaining the meaning of the problem, an incorrect operation may, nevertheless, be selected.

Bill: Okay, umm ... Patrick rode his bike for 27 minutes, and then he jogged. He rode his bike for 27 minutes then he jogged for 13 minutes more. And it says how long did Patrick jog.

Cutler: Okay.

Bill: He jogged for 14 minutes.

In this retelling, Bill's retelling, although accurate, does not appear to aid him in solving the problem.

Cherrie's retelling of Item 6 leaves out the relational term entirely. Nonetheless, Cherrie selects subtraction as the appropriate operation and correctly solves the problem despite her faulty retelling.

Item 6: Anew suit jacket costs 51 dollars. The pants that match cost 23 dollars less than the jacket. How much do the pants cost?

Cherrie: A new suit jacket costs 51 dollars. The pants that match it cost 23 dollars. (Figuring on scratch paper) 28?

Cutler: Thank you.

Closely related to maintaining the meaning of the problem was correctly stating the goal of the problem. The goal of many word problems is stated in the final sentence of the problem. For instance, the goal of Item 3 could be stated, "How long did Patrick jog?" or "How many minutes did Patrick jog for?" Consider Adrienne's retelling of Item 1.

Item 1: The happy-faced clown carries 16 balloons. The sad-faced clown has 7 balloons. How many more balloons does the happy-faced clown have than the sad-faced clown?

Adrienne: There's a sad-faced clown, and he has 16 balloons. No, that's the happy-faced clown that has 16 balloons. And the sad-faced clown has 7 balloons. And they want to know how many more balloons the happy-faced clown has. (Figures on scratch paper). 9.

Adrienne directly states the goal of the problem, saying "And they want to know how many more balloons the happy-faced clown has." Bill's retelling of Item 1 also directly states the goal of the problem.

Bill: The happy face clown has 19, no, 16 balloons and the sad one has 7. And it is asking how many the happy clown has than the sad clown.

Overall, retellings of consistently-worded items more frequently included a direct statement of the goal of the problem than retellings of inconsistently-worded items. Of the retellings that included a correct goal statement, 91% also resulted in a correct numerical solution. Students who organized their retelling to include a correct goal statement appeared to create a mental representation of the problem that was meaningful and aided their comprehension and success in solving the problem.

Overall Strategy Development

The majority of the students in this study recorded a standard algorithm for the problem on their scratch paper; therefore, the operation they chose was apparent. Students were more likely to select an appropriate operation for consistently-worded problems, with inconsistently-worded items, particularly items 3 and 5, giving students the most difficulty. This result supports the research of Verschaffel (1994), Lewis and Mayer (1987), and others that inconsistent language may miscue students' operation selection.

For four interview items, subtraction was a standard algorithmic procedure; the remaining two items called for addition. About one third of the students appeared to rely on stereotypical expectation or patterns, selecting subtraction for all interview items. There are several possible explanations for this phenomenon. They may have wondered about which operation to use, looked at their scratch paper and seen subtraction (as was actually required by items 1, 2, 4, and 6), and continued with the same operation. Some students may have been misled by the example item the researcher modeled which required subtraction, thinking subtraction was the focus of the assessment. Other students may have considered each item individually and selected subtraction based upon their beliefs about which operation the problem required.

Students who appeared confused by inconsistent language may have relied inappropriately upon a key-word strategy in which the terms fewer or less invoke subtraction and more implies addition (e.g., Verschaffel, 1994). In this study, if a student did not translate inconsistent language into consistent language and selected an incorrect operation, we assumed that the student was relying on a key word to select an arithmetic operation. Consider Item 3 and Cherrie's retelling of it.

Item 3: Patrick rode his bike for 27 minutes. He biked 13 minutes less than he jogged. How many minutes did Patrick jog for?

Cherrie: Patrick rode his bike for 27 minutes, and hejogged for 14 minutes less than he jogged ... (writes 37 - 13 = 24).

Cherrie may have focused on the word less and made subtraction her operation selection according to a key-word strategy. Cherrie also incorrectly recalled the numerosities used in the problem.

Language of Retellings of Word Problems

Had Cherrie performed the translation procedure described by Lewis and Mayer (1987), i.e., changing the relational term to more and switching the subject and object of the second sentence, her retelling may have maintained the meaning of the problem. Adrienne's retelling of Item 4 illustrates this translation process.

Item 4: At the car show, Jorge saw 55 black cars. He counted 29 more black cars than red. How many red cars did Jorge see?

Adrienne: 'Kay. There's this boy, and he went to a car show, and he saw 55 black cars. And then he looked at the red cars, and he saw 29 fewer red cars than ... (computes on scratch paper) 26.

In her retelling, Adrienne switched the subject and object of the second sentence and translated the relational term from more to fewer, resulting in a consistently-worded retelling of the problem. She subsequently selected an appropriate operation and correctly solved the problem. Students who performed this translation procedure more frequently selected the appropriate operation and correctly solved the problem, appearing to rely less on key words and more on a meaningful representation of the problem, that is, their retelling.

Further analysis of the students' retellings of inconsistently-worded items indicated that the students most often switched the subject or object of the second sentence but retained the original relational term, perhaps because it appeared to be a predictable key word, signaling a specific arithmetic operation. Termed a reversal error by Verschaffel (1994), this type of keyword strategy results in an erroneous retelling and an incorrect numerical solution.

Correct solutions were influenced by more than operation selection. Other variables, such as computational errors and recalling incorrect numbers, came into play as well. Similar to Cherrie's retelling of Item 5, Chris' retelling of Item 5 illustrates students' frustration at not correctly or quickly recalling the numbers or characters' names in a word problem.

Item 5: Mandy has 47 marbles. She has 29 fewer marbles than Kent. How many marbles does Kent have?

Chris: Mandy has 47 marbles. Kent has, I mean Mandy has 17 fewer than Kent. How many marbles does Kent have? Forgot the numbers ...' kay ... I think ... um ... George, or whatever his name is, has ... he has about. Wait. I need to do it again. I think that um, Fred or George, or whatever his name is, has 76 marbles. But I'm probably wrong. I don't know though. Forgot the numbers.

Issues of memory and cognitive overload may have affected some students' ability to recall these details. Analysis of the interview transcripts showed that most students had difficulty recalling at least 2 of 12 numbers used in the problems, often substituting numbers from previous problems. Because of this difficulty, while many students' retellings may have maintained the meaning of the original problem and students may have selected the appropriate operation, they may still have computed an incorrect numerical solution.

Though elaboration was actively taught in the instructional period as a tool for enhancing recall of narrative and mathematical texts, few children elaborated upon the interview items beyond slight alteration of the context of the problem (e.g., "at the toy store" rather than "at the store"). Students who elaborated may have created a more substantive mental image of the problem; they appeared to be picturing the events of the problem and were able to retell them in a logical manner. For example, Zach made a personal connection to Item 5 in which one of the characters' names is Kent (see Table 1).

Zach: My dad's name is Kent.

Cutler: Well, that will be easy for you to remember?

Zach: Mandy has 47 marbles. She has 29 fewer than Kent. How many marbles does Kent have? (Figures). Kent has 75.

Although Zach made a computational error that resulted in an incorrect numerical solution, his elaboration of the problem may have resulted in a more personally meaningful representation. This phenomena supports the findings of Swanson, Cooney, and Brock (1993), whose study examined the verbatim information students recalled. Swanson et al. asked children to write all they could remember from the word problem and scored the recall of numbers, relational terms, the question within the problem, and extraneous information. Their findings suggest that the process of encoding a word problem allows for elaboration, which may be more important in predicting solution accuracy than verbatim recall of the information encoded.

Highlights Contrasting the Retellings of AA and LA Students

The initial intent of the study did not include a comparison of the retellings of the students in the LA (low achieving) group with those of students in the AA (average achieving) group. However, examining the retellings according to mathematics achievement level offers additional insights as well as further questions regarding the nature and usefulness of oral retellings.

As discussed in the section describing overall problem representation, students from both groups more frequently retold inconsistently-worded problems. Analysis of the problem representations according to group show that AA students who retold the problem in their own words appeared more successful in maintaining the meaning of the problem than their LA peers who retold the problem in their own words (see Table 2). Students whose retelling maintained the meaning of the problem also had a higher level of success in problem solving.

As shown in Table 3, little substantive difference appeared between the computation and problem-solving strategies exhibited by LA students and AA students. Both groups relied heavily upon algorithms rather than memorized facts or counting strategies. For the LA group, 81% of the items were solved using an algorithm. Among the AA students, 96% of the items were solved using an algorithm. In this study, achievement level in mathematics or type of word problem did not appear to be a factor in students' preferred method of computation.

Both groups experienced a high level of success on consistently-worded items in which a key word or relational term was consistent with the appropriate arithmetic operation. As indicated in Table 4, an apparent reliance on key words (e.g., choosing subtraction for less and fewer and addition for more) occurred in 29% of AA students' retellings and 41% of LA students' retellings of inconsistently-worded items. Further, students in either achievement group who appeared to use a key-word strategy for inconsistently-worded items less frequently selected an appropriate operation than students who did not rely on key words. This interpretation supports Verschaffel's (1994) finding that students who used a key-word strategy had no intention of understanding or representing the problem; rather they proposed to memorize or look for words that would prompt them to select a specific arithmetic operation.

An important language issue associated with retellings of compare problems involves the translation of inconsistent language into consistent language (Lewis & Mayer, 1987). Most students had difficulty with the translation process, with only 37% correct translations for the AA group and 29% correct translations for the LA group. Further, those who correctly performed the translation process were not guaranteed to select an appropriate operation. These students may not have created a meaningful representation with their retellings.

Implications and Limitations

This study has implications for mathematics educators and researchers who use oral retellings to assess students' comprehension of word problems and could have implications for teachers who want to use retellings as an instructional tool. In this study, retelling was approached as a problem solving strategy rather than as an assessment technique. Our findings suggest that children who regularly use retellings as a strategy may become more interested in the goal and meaning of a problem than in its key words. A teacher who instructs students in using retellings to organize and make meaning of the information within a word problem may find that children do, with experience, learn to translate inconsistent language into consistent language, thus basing their operation selection on an accurate depiction of the meaning of the problem rather than relying on key words.

Given the parameters of working in the context of actual school conditions under limited instructional time, the results of this study cannot be generalized to other populations or beyond compare word problems. Three consistently-worded items and three inconsistently-worded items were used in each interview/observation. Therefore, only those types of problems were considered in telling the tale of oral retellings.

The interview scoring guide we developed for this study may become a map for teachers, helping them determine what to look and listen for in a retelling of a word problem. A retelling may provide a teacher with an informal classroom assessment that informs instruction while aiding children's creation of a meaningful representation of a word problem.

Recommendations for Future Research

Our findings suggest that instruction in oral retellings may enhance students' abilities to approach and solve compare word problems successfully Further research into oral retellings of compare word problems should be conducted to clarify the utility and role of retellings in the problem solving process. Because this study was limited to students labeled as having average or below-average achievement in mathematics, suggestions for future research include using oral retellings with high achieving students, with younger populations, with mixed ability groupings, and among English as a second language populations.

Additional research efforts may consider investigating the usefulness and effects of oral retellings with other categories of word problems, original problems created by students, and other types of mathematics texts. Future research should also examine which instructional techniques are effective for helping students develop meaningful retellings of word problems.

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Vygotsky, L.S. (1978). Mind in society: The development of higher psychological processes. (M. Cole, V. John-Steiner, S. Scribner, & E. Souberman, Eds. and Trans.). Cambridge, MA: Harvard University.

Appendix A Interview Protocol

Today we are going to do some work with math word problems. I am going to give you a word problem similar to those we talked about when I came to your class a few weeks ago. You can read the problem to yourself or out loud as many times as you would like. When you are sure that you're completely ready and have a picture of what's happening in the problem, turn the card over and retell the problem in your own words. This is the most important part: I don't want you to memorize the problem. I simply want you to tell me in your own words what the problem is asking. Then you will try your best to solve the problem. You aren't being graded; this is just for me to learn about how kids solve word problems. I won't be able to tell you if you got the correct answer or not. I will just simply thank you for what you do. Let me show you how this is done:

Maria has 37 apples. She shares 18 apples with her classmates. How many apples does she have left?

Now that I have read the problem, I am going to turn the card over and tell you in my own words what the problem is asking.

Maria starts out with 37 apples, but she gives 18 apples away. They want to know how many apples she has left.

Notice that I retold the problem, but I used my own words. I didn't try to memorize it. I am going to give you 6 word problems. I want you to do just as we have done. Read the problem and retell it in your own words. Do you understand what I am asking you to do?

Appendix B Interview Scoring Guide

Problem Representation

Item Number             1       2        3       4       5        6
(relational term)       (more)  (fewer)  (less)  (more)  (fewer)  (less)

Repeated original
problem word-for-word
(or nearly)
Restated the problem
in own words
Retelling changed
meaning of problem
Correctly defined the
goal of the problem

Strategy Development

Item Number             1       2        3       4       5        6
(relational term)       (more)  (fewer)  (less)  (more)  (fewer)  (less)

Referred to an
algorithm
Referred to a
memorized math fact
Appeared to use a
counting strategy
Appeared to use a key   NA      NA                                NA
word strategy
(uses + for more, -
for less or fewer)
for inconsistent
language items
Used an appropriate
arithmetic operation
Appeared to rely on
stereotypical
expectations or
patterns (used
substraction for all
items)

Language

Item Number             1       2        3       4       5        6
(relational term)       (more)  (fewer)  (less)  (more)  (fewer)  (less)

Conserved language
Recalled incorrect
numbers
Elaborated beyond
original problem
Used pronoun referents
Translated              NA      NA                                NA
inconsistent language
to consistent
Seemed confused by
structure (consistent,
inconsistent)
Correctly solved
problem

Carrie S. Cutler, Doctoral Candidate, Department of Curriculum and Instruction, University of Houston

Eula Ewing Monroe, Professor, Department of Teacher Education, Brigham Young University 

Table 1. Interview/observation word problems, types, and correct
numerical responses.

                                                               Correct
                                                               Numerical
   Word Problem                          Type                  Answer

1. The happy-faced clown carries 16      Consistently-worded   76%
   balloons. The sad-faced clown has 7   with relational term
   balloons. How many more balloons      more
   does the happy-faced clown have than
   the sad-faced clown?
2. At the store, Lin counted 34 beanie   Consistently-worded   69%
   babies that were cats. She counted    with relational term
   18 fewer dog-type beanie babies than  fewer
   cats. How many dog beanie babies did
   Lin count?
3. Patrick rode his bike for 27          Inconsistently-       19%
   minutes. He biked for 13 minutes      worded with
   less than he jogged. How many         relational term less
   minutes did Patrick jog for?
4. At the car show, Jorge saw 55 black   Inconsistently-       51%
   cars. He counted 29 more black cars   worded with
   than red. How many red cars did       relational term more
   Jorge count?
5. Mandy has 47 marbles. She has 39      Inconsistently-       38%
   fewer marbles than Kent. How many     worded with
   marbles does Kent have?               relational term
                                         fewer
6. A new suit jacket costs 51 dollars.   Consistently-worded   82%
   The pants that match cost 23 dollars  with relational term
   less than the jacket. How much do     less
   the pants cost?

Table 2. Retelling and maintaining the meaning of problem.

                             Retold Problem in  Retelling Maintained
Item Number                  Own Words          Meaning of Problem
(relational term, language)  AA    LA           AA    LA

1 (more, consistent)          62%   75%         100%  62%
2 (fewer, consistent)         87%  100%          87%  62%
3 (less, inconsistent)       100%  100%          62%  50%
4 (more, inconsistent)        75%  100%         100%  62%
5 (fewer, inconsistent)      100%   87%          75%  50%
6 (less, consistent)          75%   62%          87%  87%
Group Mean                    83%   87%          85%  62%

Table 3. Strategy and operation selection.

                                     Referred to a  Used a
Item Number              Used an     Memorized      Counting
(relational term,        Algorithm   Math Fact      Strategy
language)                AA    LA    AA   LA        AA   LA

1 (more, consistent)      50%   75%  50%  0%         0%  25%
2 (fewer, consistent)    100%  100%   0%  0%        12%   0%
3 (less, inconsistent)    62%  100%   0%  0%        37%   0%
4 (more, inconsistent)   100%  100%   0%  0%         0%   0%
5 (fewer, inconsistent)   87%  100%   0%  0%         0%   0%
6 (less, consistent)      87%  100%   0%  0%         0%   0%
Group Mean                81%   96%  10%  0%         8%   4%

Table 4. Key word strategy and arithmetic operation selection for
inconsistently-worded items.

                         Appeared to use Key
                         Word Strategy for    Used an Inappropriate
Item Number              Inconsistent Items   Arithmetic Operation
(relational term)        AA   LA              AA   LA

3 (less, inconsistent)   50%  62%             50%  87%
4 (more, inconsistent)   12%   0%              0%  12%
5 (fewer, inconsistent)  25%  62%             25%  50%
Group Mean               29%  41%             25%  50%