MULTIPLICATION OF FRACTIONS 1. Preservice Elementary Teachers Understanding of Fraction Multiplication

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1 MULTIPLICATION OF FRACTIONS 1 Running head: MULTIPLICATION OF FRACTIONS Preservice Elementary Teachers Understanding of Fraction Multiplication Presented at the 2014 NCTM Research Conference, New Orleans, LA April 9, 2014 Jihwa Noh Karen Sabey University of Northern Iowa, Cedar Falls, IA, USA Authors Note Jihwa Noh (Jihwa.noh@uni.edu) and Karen Sabey (co-author, karen.sabey@uni.edu) are at Department of Mathematics, University of Northern Iowa, Cedar Falls, IA Correspondence concerning this article should be addressed to Jihwa Noh. Mailing address: Wright Hall 220, Department of Mathematics, University of Northern Iowa Cedar Falls, IA USA TEL: (319) FAX: (319)

2 MULTIPLICATION OF FRACTIONS 2 Preservice Elementary Teachers Understanding of Fraction Multiplication Abstract In this study 164 preservice elementary teachers understanding of fraction multiplication was examined in a context where they were asked to write a story problem and interpret drawn models for symbolic expressions illustrating multiplication with fractions. Participants were selected from an entry level course and an exit level course of their teacher preparation program to observe any growth in their assessed understanding. Preservice teachers interpretations of fraction multiplication used in writing story problems and their impacts on the ability to interpret drawn models were analyzed. In addition, different types of math textbooks were reviewed to compare instructional approaches to fraction multiplication. Findings and implications are discussed and further research is suggested. Keywords: drawn model, fraction multiplication, mathematical knowledge for teaching, teacher preparation

3 MULTIPLICATION OF FRACTIONS 3 In recent years, increased attention has been given to the adequacy of teachers content knowledge (Common Core State Standards Initiative [CCSSI], 2010; Conference Board of the Mathematical Sciences, 2001, 2012; National Council of Teachers of Mathematics [NCTM], 2000; National Research Council, 2001) and teachers mathematical knowledge has been identified as one of the key components of teacher effectiveness and for quality learning (Charalambous, Hill, & Mitchell, 2012; Hill, Rowan, & Ball, 2005; Newton, 2008). Within the related literature, many words, such as deep, conceptual, connected, flexible, and profound are used to clarify what is meant by the kind of mathematical knowledge teachers need to have. Ball, Thames, and Phelps (2008) used the term specialized content knowledge. In comparison to common content knowledge that is expected to be held by any educated adult (for example being able to get a correct answer when multiplying 1/3 times 1/2), specialized content knowledge is unique mathematical understanding and reasoning on which teachers draw to effectively perform mathematical tasks of teaching such as selecting appropriate tasks, choosing and developing useable definitions, recognizing what s involved in a particular representation, and linking representations to underlying ideas and to other representations (p. 400). Teachers who lack specialized content knowledge may be unable to help students understand underlying meanings such as why it makes sense to multiply the numerators and the denominators when multiplying fractions or develop flexibility in moving among multiple representations such as diagrams, graphical displays, and symbolic expressions. Thus, for students to be able to learn and use sophisticated mathematical ideas and procedures, teachers must hold well-developed specialized content knowledge and their learned specialized content knowledge needs to be assessed to identify the support they need.

4 MULTIPLICATION OF FRACTIONS 4 In the present study reported here (which is part of a larger study of teacher knowledge), preservice teachers specialized content knowledge of fraction multiplication was examined in a context where they were asked to write a story problem and interpret drawn models for symbolic expressions illustrating fraction multiplication. Another aspect of the present study was to explore how the ability to interpret drawn models is related to the approaches to interpreting multiplication of fractions. The domain of fraction multiplication was chosen because previous research suggests that preservice and inservice teachers understanding of fraction operations is limited (Ball, 1990; Ball et al., 2008; Thompson & Saldanha, 2003; Tirosh, 2000; Tirosh & Graeber, 1990) and studies of preservice teachers knowledge of fraction operations primarily focuses on fraction division (Ball, 1990; Ma, 1999; Son & Crespo, 2009; Tirosh, 2000; Young & Zientek, 2011). Asking preservice teachers to pose problems such as writing relevant story problems and reason about drawn models to given symbolic situations provides information about what meaning they attach to those situations because problems they pose and interpretations they use reflect the mathematical understanding, skills, and beliefs they have (Toluk-Uçar, 2009). Such mathematical understandings and meanings they use to make sense of the given situations are the essential elements in performing those mathematical tasks of teaching described above. Also, uncovering preservice teachers existing understandings creates an opportunity to question and revise these understandings and meanings. Therefore, tasks in which preservice teachers are asked to create appropriate story problems and interpret drawn models can be useful tools for the learning of specialized content knowledge and assessing their learned specialized content knowledge. Problem Posing

5 MULTIPLICATION OF FRACTIONS 5 Teachers ability to generate story problems has been identified as important as it helps students connect the mathematics they are learning to real-life situations and understand how the theoretical language of mathematics and the everyday language of story problems are related (Ball, Hill, & Bass, 2005; McAllister & Beaver, 2012; National Mathematics Advisory Panel, 2008; Rittle-Johnson & Koedinger; 2005). Previous studies examined the mathematics problems posed by preservice and inservice teachers (Barlow & Cates, 2006; Crespo, 2003; Toluk-Uçar, 2009; Whitin, 2004). While some of these studies were concerned with the role of problem posing in shaping teachers beliefs, content knowledge, and teaching, others focused on teachers ability to pose problems during instruction in order to gain insights into their students mathematical understanding. Whereas these studies used problem posing as a teaching technique to help teachers reflect their own understandings and practices, this present study used problem posing primarily as an assessment tool to reveal features of preservice teachers specialized content knowledge regarding the concept of fraction multiplication by classifying the types of interpretations used in the story problems they created. There are a few studies that used problem posing to examine teachers understanding of operations with fractions (McAllister & Beaver, 2012; Toluk-Uçar, 2009). These studies highlighted errors made by teachers. The present study considered teachers approaches valid or invalid to provide insight into the knowledge preservice teachers rely on for interpreting fraction multiplication. It is beneficial to examine not only erroneous approaches but also valid approaches because expertise involves having greater flexibility in approaches to solving problems as well as having more elegant ways of using a single approach (Lee, Brown, & Orrill, 2011; Smith, disessa, & Roschelle, 1993). This helps teacher educators provide appropriate feedback and develop lesson plans to bolster the strengths and compensate for the weaknesses.

6 MULTIPLICATION OF FRACTIONS 6 Drawn Models Knowing underlying meanings of and having flexibility with representations are characteristics of a competent problem solver (CCSSI, 2010; Goldin, 2002; NCTM, 2000). Teachers understanding of representations boost or limit their ability to support the development of student understanding of rational numbers (Izsák, 2008). While literature has suggested the positive contribution of drawn representations, as an example, moving from concrete to abstract understandings of mathematical concepts (e.g., Post, Wachsmuth, Lesh, & Behr, 1985), many studies done on representations have focused on graphs (e.g., Gagatsis & Shiakalli, 2004); but few investigated the ability to interpret drawn diagrams such as area and number line models. Also, although the transitions from visual representations to symbolic representations are often addressed, the inverse (addressing the transitions from symbolic representations to visual representations) is seldom touched upon (Luo, Lo, & Leu, 2011). The present study attempted to address these gaps. In addition to the problem posing task, preservice teachers in the present study were given a task that required them to analyze different drawn models for a given fraction expression modeling fraction multiplication. To examine preservice teachers knowledge of fraction multiplication, this study attempted to address the following questions: 1. What patterns in preservice teachers interpretations of multiplication of fractions are evidenced in a problem-posing task? 2. How do patterns in the interpretations of multiplication of fractions evidenced in a problemposing task compare between preservice teachers at the entry level and the exit level of their teacher preparation program?

7 MULTIPLICATION OF FRACTIONS 7 3. How do preservice teachers interpret drawn models for a given fraction multiplication problem? 4. How does the ability to interpret drawn models for fraction multiplication compare between preservice teachers at the entry level and the exit level of their teacher preparation program? 5. What relationship, if any, exists between the observed patterns in the problem posing task and the ability to interpret drawn models for fraction multiplication? Method Participants Data for this study were collected from 164 pre-service elementary teachers in different phases of their teacher preparation program offered at a public university in the Midwestern United States. Sixty-five of the participants were selected from an entry level course of their teacher preparation; the remaining 99 were enrolled in an elementary mathematics methods course which was at the exit of their program. This selection was made to observe any growth in the assessed knowledge of preservice teachers at the entry level and exit level of their teacher preparation program. The entry level course used in this study was the first of three mathematics content courses required of elementary education majors at the university from which data were collected. Depending on when participants declared their major, the exit level participants had taken two or three mathematics content courses before taking the methods course. In either case, the first content course focused on number sense in whole numbers and rational numbers. Instruction on fractions and fraction operations typically lasted 4-5 semester weeks with some variation among instructors teaching the course. Instrument

8 MULTIPLICATION OF FRACTIONS 8 The data reported in this study were gathered as part of a larger study of teacher knowledge in which an assessment of preservice elementary teachers mathematical knowledge for teaching over a variety of topics including numbers and operations and proportions was developed and administered. Items on the assessment were a combination of multiple choice and constructed response items and included items taken from existing literature (Hill, Schilling, & Ball, 2004) as well as several items created by the researchers of this study. In this study, the results of two of the assessment items that required preservice teachers to write a story problem and make sense of drawn models for multiplication of fractions were analyzed. Figure 1 displays the two assessment items (Item A and Item B) used in this study. Item A Write a story problem that can be answered by: Item B Which model below cannot be used to show that 1 =1? Figure 1. Assessment items Data Coding and Scoring Each response on the problem posing task (Item A) was analyzed based on both correctness and type of interpretation used to create the story problems. Story problems were first coded according to the degree to which they illustrate a multiplication situation and give the

9 MULTIPLICATION OF FRACTIONS 9 correct numerical answer to the given multiplication expression. Criteria for coding on correctness were as follows: (i) A story problem models a multiplication situation and if successfully solved produces the correct answer to the given fraction expression (ii) A story problem models a multiplication situation but does not represent the given multiplication problem, has an error and/or is incomplete (iii) A story problem models a different operation, solvable (iii-1) or unsolvable (iii-2) unsolvable (iv) A story problem translates the multiplication sign into words in a non-sense making way (v) no response or an attempt was made without substantial information Independent from the coding results on its correctness, each response was then paired with a type of interpretation for fraction multiplication that appeared to be used in the story problem. A list of possible types of interpretations was created based on related literature and the researchers prior experience. The initial list was refined and extended as the coding progressed. The final list of criteria used to analyze interpretations of multiplication of fractions follows: (a) area of a rectangular shape; (b) part-of-part; (c) multiplication rule of probability which computes the probability of two independent events occurring simultaneously; (d) scaling as shrinking or enlarging; and (e) responses that are irrelevant or unable to be analyzed on interpretations of fraction multiplication (e.g., responses modeling a different operation that is not multiplication, translating the multiplication sign into words, or written without substantial information). Types a, b, c, and d were considered valid interpretations. The drawn model task (Item B) was scored as correct or incorrect. The suggested correct answer was option C. Options A, B, and D exhibited models that could be used to represent the

10 MULTIPLICATION OF FRACTIONS 10 given fraction expression. The diagram in option C was not a valid model because the two shapes (one rectangle and one circle) used to represent the unit whole would not guarantee the same area. Coding and scoring of data were carried out by all three of the researchers. The researchers worked independently on coding each response. Upon completion, the researchers collaborated as a group until a consensus was reached. From the Problem Posing Task Results On correctness. Table 1 presents a classification of the story problems written in response to the problem posing task (Item A) based on correctness within each course as well as overall. Of all the story problems that fell under iii, the most common operation used was addition. Story problems that were classified as partially correct (labeled ii) illustrate a multiplication situation but have an error or are incomplete. Consider the following example. If 2/3 of the students ate 5/7 of an apple, how many apples would be eaten total? Although this problem models a situation requiring multiplication to calculate the answer, to solve this problem, the number of students would have to be specified. The calculation would be 5/7 2/3 N where N is the number of students. Table 1 Correctness of Story Problems Written Correctness Entry (out of 65) Exit (out of 99) Total (out of 164) i fully correct (11%) ii partially correct (18%) iii incorrect operation (29%) iii-1 solvable (25%) iii-2 unsolvable (4%) iv multiplication sign into words (7%)

11 MULTIPLICATION OF FRACTIONS 11 v unanalyzable (35%) Types of interpretations. Regardless of whether the story problems written contain errors, mathematical or non-mathematical, the approaches to interpreting the fraction multiplication expression evidenced in story problems were analyzed and presented in Table 2. Sample story problems from responses include the following: If she uses 5/7 of a stick of butter per pie, how many sticks of butter are necessary for 2/3 of a pie? (part-of-part); Bobby is planting a small garden in his backyard. The length of the garden is 2/3 meters, while the width is 5/7 meters. Bobby wants to know what area he has available to fill with plants (area of a rectangular shape); I have 2/3 cup of [flour]. My grandma says she ll give me 5/7 times more. How much do I have now? (scaling), I have wrecked 2 out of the 3 cars I have owned. I have also bought 7 new tires for these cars, and 5 out of 7 were junk. What are the chances of me wrecking a car and blowing a tire? (multiplication rule of probability). These examples contain errors: Some are subtle and others are fundamental. Correctness of story problems was not considered, however, when the approaches used in the story problems were examined. Table 2 Types of Interpretations Used in Story Problems Written Entry Types of Interpretations (out of 65) Exit (out of 99) Total (out of 164) a area of a rectangular shape (1%) b part-of-part (17%) c multiplication rule in probability (1%) d scaling (11%) e unanalyzable (e.g., problems that modeled a different operation, provided insufficient information or were left blank) (70%)

12 MULTIPLICATION OF FRACTIONS 12 To reveal any possible effect of a particular type of interpretation on the ability to write a story problem for fraction multiplication, the problems that modeled a multiplication situation involving the two fractions given in the task, whether they were solved by the calculation that was asked to be represented (i.e., fully correct problems) or not (i.e., partially correct problems), were examined to classify the types of interpretations used in them. The results are presented in Table 3. Part-of-part and scaling were predominantly utilized in those problems, with part-of-part being the prevalent type of interpretation used in the problems that were fully correct. However, very few students from the entry course used either type of interpretation. Table 3 Types of Interpretations Used in Story Problems Modeling Fraction Multiplication Types of Interpretations i fully correct ii partially correct (out of 18) (out of 29) Entry Exit Total Entry Exit Total a area of a rectangular shape b part-of-part c multiplication rule in probability d scaling From the Drawn Model Task On the task involving drawn models (Item B), the diagram in option C cannot be used because the relative sizes of the rectangle and the circle are uncertain. A majority of the preservice teachers chose either option B (involving two identical rectangles with each being divided into 6 parts) or option D (showing a number line model instead of an area model). Option D was the most frequently chosen answer overall as well as in the entry course, while the suggested answer, option C, was the most chosen answer in the exit course. A distribution of the responses on this item is presented in Table 4. Table 4 Distribution of Responses on the Task Involving Drawn Models

13 MULTIPLICATION OF FRACTIONS 13 Item B Options Entry (out of 65) Exit (out of 99) Total (out of 164) A (8%) B (28%) C (suggested answer) (31%) D (32%) Note: There was one no attempt response in each course. Thus, the total sum is two less than the total number of participants. Comparison of Results from Two Items After Item A and Item B were analyzed individually, their results were compared to reveal any relationship that might exist between the observed patterns in the problem posing task and the ability to interpret drawn models for fraction multiplication. To do so, first, the ways preservice teachers who wrote fully correct story problems (Item A) answered Item B were examined and presented in Table 5. Of 18 that wrote problems that modeled a multiplication situation involving the original fractions and would give the correct numerical answer if solved successfully, 9 preservice teachers chose the suggested answer for Item B. All nine of these preservice teachers were from the exit course. There was only one student in the entry course who wrote a fully correct story problem. This preservice teacher chose option D for Item B. Table 5 Distribution of Item B Responses When Story Problems Written Were Fully Correct Item A Responses containing a fully correct story problem Item B Options Entry Exit Total (out of 18) A (0%) B (11%) C (suggested answer) (50%) D (39%)

14 MULTIPLICATION OF FRACTIONS 14 Second, to reveal how useful a particular type of interpretation of fraction multiplication may have been to interpreting drawn models, types of interpretations used in writing story problems were examined among the 51 preservice teachers who correctly answered the drawn model task (Item B). The results are presented in Table 6. Sixteen of the preservice teachers used an interpretation that is appropriate to reason about multiplication of fractions. Although the partof-part interpretation can be seen as one of the most useful ways to reason about drawn models for fraction multiplication, only 9, all of whom were from the exit course, used the part-of-part interpretation. Table 6 Distribution of Types of Interpretations From Item A When Item B Responses Were Correct Item B responses that were correct Entry Exit Total Item A: Types of Interpretations (out of 51) a area of a rectangular shape (4%) b part-of-part (18%) c multiplication rule in probability (2%) d scaling (8%) e unanalyzable (68%) Discussion The analysis shows only 49 out of 164 preservice teachers used a valid interpretation when writing their story problems. Eight were from the entry course while the remaining 41 students were from the exit course. While part-of-part was the most frequently used valid interpretation, several tendencies were observed in written story problems. One common way in which the preservice teachers from both courses interpreted the problem-posing task was through the use of choosing an arbitrary whole number. It seemed to

15 MULTIPLICATION OF FRACTIONS 15 give them direction in writing the story problem. Consider the following example: There are 9 apples in each group. It takes 2/3 of a group to make a pie. Adam eats 5/7 of a pie. How many apples would Adam eat if they were spread evenly throughout the pie? Solving this problem would give the answer of 4 2/7 apples, which is not the answer to the given multiplication problem. While this story problem seemed to use the part-of-part interpretation for multiplication of fractions, the fact that the preservice teacher inserted the whole number, 9, changed the original problem. There could be many reasons why preservice teachers would opt to use a whole number in a fraction multiplication problem: to help find a context, they are more comfortable with whole numbers than with rational numbers, or they have better number sense with whole numbers than with rational numbers. Even though the preservice teacher may have thought this would lead to the correct answer, it can easily be noted that he/she did not understand how 9 would affect the written story problem. The preference of using whole numbers instead of fractional numbers implies that preservice teachers are not comfortable using fractions which may be due to their lack of understanding of fractions. Another tendency was to interpret the given multiplication problem by using the scaling interpretation as resizing. One such example is I have 2/3 cup of [flour]. My grandma says she ll give me 5/7 times more. How much do I have now? This problem, however, would be solved by a calculation involving multiplication and addition (i.e., 2/3 + (2/3 5/7)), which is different from the one that was asked to be represented. It was also a prevalent tendency that many preservice teachers merely translated the multiplication sign ( ) into words such as times by or multiply by in their story problems. An example of this tendency is Ben has 2/3 of his candy bar left and Sally has 5/7 of her candy bar left. If you multiply their candy bars together,

16 MULTIPLICATION OF FRACTIONS 16 how much would they have? This story problem shows no operational sense of fraction multiplication. Several preservice teachers wrote a story problem that involved fraction multiplication but erroneously assumed that it would result in a larger amount. Consider the following example: Tom has 2/3 of an apple and needs to make a big [batch] for a recipe by 5/7. How much apples will he have now? In this case, the preservice teacher may have treated the given fraction multiplication problem like it was a whole number multiplication problem. As noted by Isiksal and Cakiroglu (2011), the student may have followed the intuitive rule that multiplication always makes numbers bigger while division makes them smaller (p. 216). One of the most frequently observed errors among the written story problems was in relation to the unit whole. Either the whole was not specified or the whole was referenced incorrectly. Many preservice teachers wrote a story problem that did not specify what the whole was, which makes it impossible to determine what the students intended. An example of this error is You have 2/3 of a pie and want 5/7 more. How much pie would you have? To correctly solve this story problem, it must be specified whether the individual wants 5/7 more of a whole pie or if he/she wants 5/7 of 2/3 of a pie. Specifying the whole is crucial when writing a correct story problem. Not having a clear whole can lead one to assume something other than what the author intended. That it is why it is essential to understand the role of the whole in any given fraction multiplication story problem. As with the misunderstanding of the whole, a few responses contained an incorrect whole. For example, the answer to 2/3 of my day is spent at school. If 5/7 of my time at school is spent problem-solving, how much of my school day is spent problem-solving? is 5/7 of a day. It would have correctly represented the given multiplication problem if the question asked was

17 MULTIPLICATION OF FRACTIONS 17 how much of my day (rather than my school day) is spent problem-solving? In this problem, 2/3 of the day was at school so the whole is a day; 5/7 of my time at school so the whole is time at school which is 2/3 of the entire day. Although the student was careful to specify units, the question asked in the problem changed the solution. Being able to use an applicable context in a written story problem is an essential component of generating a reasonable story problem, given any fraction operation. Contexts used in some of the story problems written were unrealistic. For example, Jamie made a pan of brownies. The pan s length was 2/3 cm and the pan s width was 5/7 cm. How big is the pan? is mathematically correct; however, the measurements used in the story problem are unrealistic. Also, the word big is ambiguous because it might mean the area of the pan (2/3 x 5/7) or the perimeter of the pan (2(2/3 + 5/7)). In addition to the mathematical errors revealed in this study, many story problems exhibited language issues such as spelling or grammar errors in the written story problems. On the task involving models for fraction multiplication, 31% of the preservice teachers correctly answered Item B. Recall that the correct answer to the models for fraction multiplication task was the figure that does not represent the given multiplication problem. Some reasons for why preservice teachers may have chosen one option over the others can be speculated. Option D, which was the most popular answer chosen in both groups of preservice teachers, may have been chosen due to the fact that option D was a number line model rather than an area model as seen in each of the other choices. Preservice teachers may have thought that option D was not similar in this aspect, leading them to choose this option over the others. By selecting option B, preservice teachers may have noticed that this area model contained nine parts that were shaded with six double shaded. Options A and C, however, each contained three

18 MULTIPLICATION OF FRACTIONS 18 parts with two of those parts in each area model being double shaded. The preservice teachers that selected option B may have deduced that options A and C must have been correct because they were similar in format. Although students in the exit course outperformed those in the entry course in the task verifying drawn models for fraction multiplication, approximately 65% of the preservice teachers in the exit course could not answer the task correctly. This finding further confirms that operations with fractions are challenging concepts to comprehend. Before sorting through the data, it was hypothesized that part-of-part would be the most useful interpretation to reason about drawn models for fraction multiplication. Of a total of 28 preservice teachers that used the part-of-part interpretation in their story problems, only 9 were able to answer which model could not be used to show the given fraction multiplication problem. Even though part-of-part seemed to be an appropriate and logical interpretation to use when interpreting drawn models for fraction multiplication, apparently it did not influence all 28 preservice teachers reasoning about them. This suggests that while preservice teachers may recognize the part-of-part interpretation and know how to write a story problem using the interpretation, they may not be fluent in applying that knowledge to models for fraction multiplication. Being able to translate this interpretation from a story problem to a diagram such as drawn model would indicate a more complete understanding of fraction multiplication (Luo, Lo, & Leu, 2011). A majority of preservice teachers did not have the conceptual understanding of what fraction multiplication entails: meaning of the numerator, denominator, and the unit whole; knowing that when a fraction multiplication problem involves a fraction that is less than one, then the product will be smaller rather than larger; and knowing the different types of interpretations of fraction multiplication. They may know the procedural aspect of fraction

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