Int J of Sci and Math Educ DOI 10.1007/s10763-015-9702-x
Modelling in Primary School: Constructing Conceptual Models and Making Sense of Fractions Juhaina Awawdeh Shahbari 1,2,3 & Irit Peled 1
Received: 3 September 2014 / Accepted: 24 October 2015 # Ministry of Science and Technology, Taiwan 2015
Abstract This article describes sixth-grade students’ engagement in two modeleliciting activities offering students the opportunity to construct mathematical models. The findings show that students utilized their knowledge of fractions including conceptual and procedural knowledge in constructing mathematical models for the given situations. Some students were also able to generalize the fraction model and transfer it to a new situation. Analysis of the students’ work demonstrates that they made use of four fraction constructs—part-whole, operator, quotients, and ratio. The activities also revealed difficulties in the students’ knowledge of fractions, some of which were overcome in the process of organizing and mathematizing the problem. Keywords Fractions . Modelling . Model-eliciting activity . Operator . Part-whole
Introduction Mathematical knowledge and the competencies required for using it in different situations are regarded as essential learning goals in today’s world (Kaiser & Schwarz, 2010). This recognition is reflected in the Organization for Economic Cooperation and Development (OECD) Program for International Student Assessment (PISA), a set of periodic tests designed to measure students’ Bcapacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage in mathematics in ways that meet the needs of that individual’s life as a constructive, concerned, and reflective citizen^ (OECD, 2004: 15).
* Juhaina Awawdeh Shahbari
[email protected] 1
University of Haifa, Haifa, Israel
2
Al-Qasemi Academy - Academic College of Education, Baqa al-Gharbiyye, Israel
3
The College of Sakhnin, Sakhnin, Israel
J. A. Shahbari, I. Peled
While typical school word problems require the application of previously learned procedures (English, 2006), modelling problems demand new decisions and constructions. These include processes of interpretation, identification, quantification, conjecture, and justification, along with the creation of conceptual tools (models) (Lesh & Doerr, 2003). These models are intended to be reusable in other situations and adaptable to other purposes (Lesh & Harel, 2003). Modelling processes are thus regarded as developing the competencies called for in today’s technological age of dynamic information systems (Lesh & Doerr, 2003). From a pedagogic perspective, modelling activities are also characterized by a strong thought-revealing aspect, meaning that teachers’ observations of student modelling activities provide them with a rich source of information about students’ knowledge (Doerr & English, 2006; Lesh, Hoover, Hole, Kelly, & Post, 2000). Although these issues are more prevalent in secondary than primary school mathematics curricula, it is important and possible to introduce modelling at an early age (English, 2010). Realistic mathematical modelling in elementary and middle school is an essential means of guiding children to recognize the role of mathematics as a tool for interpreting reality, while at the same time enhancing their understanding of mathematical concepts (Bonotto, 2010). This study deals with integrating modelling into primary school mathematics via activities that are designed to promote the building of mathematical models and at the same time to strengthen learners’ mathematical knowledge. The study’s aim was to observe and report on the extent to which fractions were used by students who engaged in two modelling activities. Put differently, the study represents an effort to better understand how modelling tasks create opportunities for developing and strengthening fraction knowledge. The two activities used here were drawn from an earlier study (Shahbari & Peled, 2015), in which we used a sequence of modelling tasks to facilitate the extension and reinvention of percentages. The findings of that study indicated that the sequence of modelling activities also led to a better understanding of fractions. Thus, following principles of design-based research, we redesigned the first two tasks of the original sequence by changing the context of the activities while retaining their structure. We then set out to observe children’s spontaneous use of fractions in these two redesigned activities.
Theoretical Background Modelling Approach Mathematical modelling is a cyclic process that starts from analysis and structuring of the real situation, proceeds to the mathematical model and results, and then returns to the real situation, where the results are validated (Blum & Leiss, 2005). Model-eliciting activities (MEAs) are a relatively complex type of modelling task involving real situations with incomplete, ambiguous, or undefined information (English & Fox, 2005), designed to enable the transfer of learning to other situations (Kaiser & Schwarz, 2010). They thus tend to focus on different understandings and abilities than those found in traditional textbooks (Lesh & Lehrer, 2003). In MEAs, students must produce a meaningful symbolic description of real-world situations, and then test and iteratively revise the model’s givens as they proceed (Lesh & Harel, 2003). MEAs require active thought and organizational processes (constructing, explaining, justifying, predicting, conjecturing, representing, quantifying, coordinating, and organizing)
Constructing Conceptual Models and Making Sense of Fractions
rather than the mere application of previously learned procedures (English, 2010; English & Walters, 2004). In addition, MEAs are normally designed for small-group work in which learners are required to share responsibility for constructing models (English & Walters, 2004; Zawojewski & Lesh, 2003). They therefore promote the creation of common mental models and conceptual tools (Lesh & Doerr, 2003). While the construction of mathematical concepts and the development of modelling competencies are two different goals, they are nonetheless closely related. In contrast to the Dutch Realistic Mathematics Education approach, which employs modelling activities in order to develop new concepts, the construction of good models for the situations described in modelling tasks is mostly based on using the existing repertoire of mathematical tools. However, it is also expected that the use of mathematical tools will in turn promote and strengthen students’ understanding in these areas. Given the importance of modelling competencies in today’s world, mathematical modelling can therefore serve as a vehicle both for developing students’ capacity to cope with real-world problems, and for the transmission of content needed to deal with mathematical concepts (Julie, 2002; Julie & Mudaly, 2007). In the current study, we focus on the use of modelling activities that facilitate students’ own decision to use fractions, and thereby strengthen their knowledge of this concept. Fractions Fractions are regarded as difficult to both learn and teach (Behr , Lesh, Post, & Silver, 1983; Smith, Solomon, & Carey, 2005) due to their multifaceted nature (Kieren 1993; Kilpatrick, Swafford, & Findell, 2001) and because learners assume algorithms, procedures, and properties of whole numbers (Siegler, Fazio, Bailey, & Zhou, 2013). Kieren (1976, 1980) identified five mathematical sub-constructs of fractions—measure, quotient, ratio, operator, and part-whole. The part-whole construct is the idea that a continuous amount or set of discrete items can be divided into parts of equal sizes (Charalambos & Pitta-Pantazi, 2007). Ratios convey the notion of relative magnitude (Behr et al., 1983)— 3/4, for example, being derived from the difference in magnitude between three and four (Bill, 2002). Operators shrink or stretch quantity (Marshall, 1993). Quotients relate to a situation of division (Charalambos & Pitta-Pantazi, 2007): three pizzas shared between five people generate equal shares of three fifths (3÷5= 3/5) (Behr et al., 1993). Finally, measures pertain to the process of counting the number of units that cover a region, derived by subdividing a unit into smaller equal parts (Kieren, 1980)—a necessary tool for extending the whole-number system (Hart, 1981). Of the five sub-constructs, the partwhole sub-construct is most directly relevant to the understanding of fractions, and is incorporated into the other sub-constructs, all of which are linked to operations relating to fractions, fraction-equivalence, and problem solving (Behr et al., 1983). Understanding how pieces of knowledge are interrelated constitutes a form of conceptual knowledge (Hallett, Nunes, Bryant, & Thorpe, 2012) that generates Bimplicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain^ (Rittle-Johnson, Siegler, & Wagner Alibali, 2001: 346). Conceptual knowledge is Bknowing that^—or more specifically, Brelational representations^ (Byrnes, 1992,p. 236)—as compared with procedural knowledge, which pertains to Bknowing how^ (Byrnes, 1992). With respect to fractions, conceptual knowledge constitutes an understanding of the magnitudes of the fractions used in the computation (Siegler, Thompson & Schneider, 2011). A
J. A. Shahbari, I. Peled
supportive relationship between conceptual and procedural knowledge in fractions is commonly assumed to exist, increasing competence in each augmenting that in the other (Hallett et al., 2012; Hecht & Vagi 2012; Rittle-Johnson et al 2001). Fraction Difficulties The difficulties students encounter in learning fractions relate to both procedural and conceptual knowledge. As reported by Brown and Quinn (2006), 48 % of 148 ninth- and tenth-grade students in Australia were unable to find the correct sum of two fractions with unlike denominators, either adding or subtracting the numerators and denominators, and 46 % of the student respondents were unable to compute the operator algorithm. Also in Australia, Clarke and Roche (2009) found that only 10.8 % of 323 sixth-grade students could compare ¾ and 7/9 and explain which was larger or smaller. In the USA, 65 % of the students studied by Silver and Kenney (2000) failed to correctly order five sets of fractions of less than one, and Barlow and Drake (2008) found that 67 % of 45 sixth-graders failed to correctly divide 6 by ½ (dividing fractions). Stafylidou and Vosniadou (2004) found that only a third of 80 fifth- and sixth-grade Greek students understood that a fraction consists of two independent numbers. Utilization of problem-solving skills can help students discover ways of employing and applying fractions (Naiser, Wright, & Capraro, 2004), and realistic situations encourage them to use their informal knowledge to construct meaning for themselves (Tzur, 1999). While real-world problems in traditional textbooks generally serve merely as a Bcover-story^ whose solution requires the application of a particular rule or procedure (English, 2003), the present study employs model-eliciting activities that require realistic situation analysis. The Present Study’s Goals and Questions Integrating modelling activities into the curriculum demands a great deal of investment in the design of appropriate activities. The present study examines one way in which modelling can be integrated into primary school curriculum content via activities focusing on the fraction concept. The study’s primary goal was to observe the learning opportunities created by the modelling activities for sixth-grade students. Specifically, we addressed two research questions: 1) Would the two activities trigger the spontaneous use of fractions in building mathematical models for the given situations? 2) What is the nature of fraction knowledge exhibited during engagement with the tasks, and how do students handle fraction difficulties encountered during their work?
Method Research Setting and Participants The study was conducted in two sixth-grade classes (ages 11–12) in a primary school in the north of Israel. The classes had 31 and 34 students, respectively, and included
Constructing Conceptual Models and Making Sense of Fractions
children from various socio-demographic backgrounds and a range of ability levels. This grade level was chosen because by the end of the sixth grade, students in Israel have encountered all five sub-constructs of fractions. Israeli children are introduced to unit fractions in the third grade. They learn the part-whole sub-construct with discrete and continuous wholes in the fourth grade; the measure, ratio, and quotient subconstructs in the fifth grade; and the operator sub-construct in the sixth grade. The students in both classes worked in groups of 4–5 students each, as recommended by English and Walters (2004) for modelling activities (there were eight groups in the class of 34 and seven in the class of 31). The group members were chosen by the mathematics teacher so as to ensure that the groups were heterogeneous in terms of the children’s mathematical achievement. Each group engaged in two activities—the Bgarage set^ activity followed by the BSnow White and the Seven Dwarves set^ activity—over three lessons. Data Collection The data sources included the students’ worksheets, notes, and final models. In addition, four groups (two in each class) were selected for closer observation. These focal groups were similar in their student ability mix to the rest of the groups, and had no special features. The focal groups were videotaped, and their discussions were transcribed and analyzed. The Model-Eliciting Activities As mentioned earlier, the two activities were designed on the basis of our earlier research on using modelling tasks to promote the construction of percentages (Shahbari & Peled, 2015). Following the principles of design experiment, the new problems kept the structure of the original problems while using a new context that was expected to be more relevant for the current purpose (fractions). Activity 1 (the garage set; Appendix A) requires students to price individual items from a five-item set in order to sell them separately while keeping the sum of the prices equal to the price of the whole set. This activity prompts students to elicit a mathematical model that ensures fair pricing. Activity 2 (the Snow White and the Seven Dwarves set; Appendix B) is similar to the garage set activity but contains more components. In this case, students are expected to utilize and generalize the mathematical model (i.e. the pricing model) elicited in the first activity. The two MEAs were both designed to emphasize different meanings and operations of fractions. Each activity requires consideration of two types of whole: the discrete whole, represented by the set with its individual parts, and the continuous whole, expressed by the price of the set, which the prices of the individual items must equal. The activities were expected to elicit mathematical models expressed in terms of fractions, and to promote the use of fraction operations such as addition, multiplication, and fraction-equivalence. The activities thus presented an opportunity to expose the children’s difficulties in these areas. The two MEAs were designed on the basis of the six principles of model-eliciting activities outlined by Lesh et al. (2000), as detailed in Table 1.
J. A. Shahbari, I. Peled Table 1 Manifestation of MEA design principles Principle
Manifestation of the principle in the problems
Reality: The situation should be relevant to students, Students in sixth grade are familiar with similar toy allowing them to make sense of it based on their sets, and with the specific contents of the sets (a personal knowledge. garage, Snow White and the Seven Dwarves). While students may not have encountered the particular task (pricing individual items), the parameters of pricing are within the realm of their experience and personal knowledge. Self- assessment: The situation should include clear criteria to assess efficacy of the solution without having to ask an external source.
After pricing the items (using the model they constructed) students can check if the relative values are realistic, and whether the sum of the item prices is equal to the price of the whole set.
Model construction: The situation should be interesting and motivate students to experience a mathematical process that requires construction, expansion, and improvement of a mathematical model.
The processes of organizing and ranking data promote development of a mathematical model. The items in the garage set are different in their values, so they need to be ranked. In addition, the need for the repeated pricing process promotes the development of a general pricing model.
Documentation: The situation should provide whatever is needed for students to document their working.
The students need to document the different prices through the pricing of the garage items in different sets.
Model generalization: The situation should encourage The students can generalize the mathematical model students to elicit models that are reusable, sharable, that elicited in the garage set activity to the Snow and modifiable. White set activity. The models elicited are usable models to other activities which require a process of ranking and quantifying. Effective prototype: The situation should provide a useful and efficient prototype.
The garage set activity provides a prototype situation for other activities.
Data Analysis The data were analyzed in three steps. First, we examined the worksheets, notes, and final models produced by all the groups to determine whether the activities led to a useful generalizable model which would indicate an understanding of fractions (their meaning and operations). Then, we performed a fine-grained analysis of the transcripts from the four focal groups with an eye toward answering our research questions focusing on the use of fractions in the constructed model and on the different difficulties and different uses of fractions along the modelling process. To answer the first question, we analyzed the transcripts for modelling processes employed by the children, using as our guide the stages described by Blum and Leiss (2005), in which interpretation of the situation and non-mathematical discussion of the situation lead to construction, verification, and, finally, generalization of a mathematical model. Then, to answer the second question, we classified conversational statements from the transcripts into two categories: (1) statements involving the sub-constructs of fractions as identified by Kieren (1976, 1980), and (2) statements dealing with difficulties which arose during the activity and that students were able to resolve by applying fractions.
Constructing Conceptual Models and Making Sense of Fractions
Findings The findings obtained from analysis of the worksheets and notes showed that all the groups in the two classes succeeded in building mathematical models to deal with the given situations. As it turned out, the models they produced and generalized utilized their knowledge of fractions. In the following sections, we first present the results obtained from the four focal groups relating to the modelling process, and then continue with findings pertaining to fraction constructs and fraction difficulties experienced during student work.
Results Relating to the Modelling Process In general, the analyses of the transcriptions indicated that all four groups employed the same modelling processes in the garage set activity: interpreting the situation, identifying factors for pricing, working mathematically, and constructing and verifying the mathematical model. When the groups were faced with the Snow White set activity, only one group replicated the same modelling processes in full. The other three groups were able to move from interpreting the situation and identifying factors for pricing to generalizing the mathematical model elicited in the garage set activity. The following sections present our findings from the two activities in depth with examples from students’ discussions. The four groups are labeled A (Momen, Adi, Alaa, and Hassan), B (Tala, Rema, Sameh, and Ramah), C (Amir, Naden, Saeed, and Eman), and D (Mona, Manal, Omer, and Rana). The letters and numbers in the examples indicate the group and the numbered line in the transcript, respectively. The Modelling Process Exhibited in the Garage Set Activity Interpreting the Situation. Having read the problem, the students sought to interpret the short description and turn it into a more elaborate real-world situation by adding missing data as needed so that the situation will make sense. Students discussed different components that might be included in a garage set, naming individual items. Each group suggested different items, the garage being common to all, as can be seen in the following excerpt: [A.1] Momen: That means that there is a set with five items. [A.2] Adi: The main item is the garage …yes? [A.3] Momen: Yeah … we need to record four other items …like tools or a car… [A.4] …a crane …
Identifying Factors for Pricing. Once the situation is understood, the nonmathematical discussion continues as students decide what factors should determine the prices of individual items. At this point, the students had to identify relevant factors for pricing the items when the garage set sold for $50. Initially, the students in all four
J. A. Shahbari, I. Peled
groups intuitively priced the items equally. After assessing their solution by thinking of real examples, they adopted an unequal division, ranking the prices of the different items based on relevant factors. The students in group A offered two principle factors— the size and importance of each item. They then set an appropriate price for each item: [A.53] Alla: It must be related to the importance of the item …the garage is the most [A.54] important …so it will have the highest price … [A.55] Momen: Okay, but the price is related to the size … [A.56] Alaa: The garage is the largest item… [A.57] Adi: So the screwdriver will be less because it is smaller. Here, Alla [A.53 and 54] and Momen [A.55] presented two factors that will affect the price of the items—the size and importance of the item in the set. Working Mathematically. At this point, students initiated processes that include mathematical objects and operations. The students priced the garage set items in accordance with the principles they had adopted (for group A, the principle that the prices should reflect the items’ size and importance). The groups had to price two sets, one selling at $50 and the other at $60 (see Appendix A). Pricing the two sets demanded the use of arithmetic operations (division, subtraction, and addition): [A.73] Alla: We can divide the $50 into the garage and the other items—the [A.74] garage at $25 and the other items at $25… [A.75] Hassan: No, that’s too much. The garage should be $20, the crane $10… the [A.76] rest $20. At first, the students priced the items in the $50 and $60 sets in isolation. The implications of this became clear as they continued working mathematically through the process of pricing and validation. The students employed two methods to validate their work: numerical value and relative value. When they compared the prices for the same item in the two sets, they promptly adopted a numerical basis: [C.193] Naden: The screwdriver in this set [priced at $50] is $8 and in [C.194] this set [priced at $60] it is $6. [C.195] Amir: It isn’t right…there is something … not logical! Naden [C.193 and 194] compared the prices on a numerical basis, noting that the screwdriver would cost $8 in the cheaper set and $6 in the more expensive set. Amir [C.195] immediately noted the inequity.
Constructing Conceptual Models and Making Sense of Fractions
Comparing via the relative value of the items takes into account not only the price of the individual item, but also that of the set as a whole. The students thus compared the two ratios: [A.181] Adi: The garage in this set [priced at $50] is $20 and in this set [A.182] [priced at $60] it is $25… so we can see that the two are less than half. [A.183] Hassan: But how can we know which is more? [A.184] Adi: In both sets, the garage is $5 less than half the price … so we [A.185] can say that they are the same… [A.186] Momen: No, they’re not the same, the $5 in the two sets are different … the [A.187] $5 here [the set priced at $50] is 1/5 of the price but here [the set [A.188] priced at $60] it is 1/6 of the price. [A189] Adi: Yeah…it is not the same. Here, Adi [A.181 and 182] conducted a comparison using a ratio, as indicated in her solution. Momen, on the other hand [A.186, 188, and 189], used ratios to compare $5 against the prices of the two complete sets, though none of the students noticed that he did not do so correctly (Momen may have intuitively thought of the triplet 5, 10, 50, and then divided both 50 and 60 by 10 instead of by 5.) Nonetheless, the two methods of comparison both revealed the inefficiency of the pricing process used thus far and the need for a more complicated method that would maintain not only the ranking of items within each set, but also comparability between the sets. Constructing a Mathematical Model. Students attempted to construct a general pricing model. At this point, a mathematical model emerged as an answer to the inequities produced by the original pricing process. Comparison between the items by way of their relative pricing led the students to price the items in terms of parts—an explicit use of fraction knowledge: [A.207] Momen: We must keep the ratio equal in all of the items … for example, the [A.208] garage is always ½ … okay? [A.209] Hassan: The garage is always ½ so the other four items together are ½. [A.210] Alaa: How do we divide ½ by 4? Momen [A.207 and 208] demonstrated his understanding that the ratio should remain constant across the different sets, using fraction terms in order to express this.
J. A. Shahbari, I. Peled
The groups continued discussing the problem until they arrived at a mathematical model involving fractional parts. Here, it should be noted that while most groups produced five fractions with the same denominator, others ended up with different denominators, often as a result of reducing fractions. This was true not only for the focal groups, but for the sample as a whole, as seen in their worksheets and notes. The following example shows the more common model using the same denominator to express all the prices: 7 2 4 1 2 , wheel= 16 , crane= 16 , screwdriver= 16 , and cables= 16 . Garage= 16 The transcripts and notes suggest that in all of the groups, the initial denominator was chosen intuitively. Then, following deliberations, the students realized that a bigger number would allow a more flexible ranking of prices. The groups all began with the same denominator, as this was the easiest way to approach the problem. Some groups then reduced fractions where possible, as in this example, where the initial denominator was 50 (e.g. 10/50 became 1/5): 4 1 1 , crane= 15, mechanic= 10 , and screw= 25 . Garage= 12, car= 25 Verifying the Mathematical Model. All the groups initially validated the mathematical model by considering whether the model produced a mathematically reasonable outcome. That is, the students checked to confirm that the sum of the fractions was one, as exemplified in the following discourse: [D. 292] Manal: It is 38/40… it is wrong, it must be 40\40. [D.293] Mona: Why is it wrong? [D. 294] Manal: The sum of the parts must be one. [D.295] Omer: We can add one here and one here. At this point, the students were given the task of computing item prices for a garage set with an overall price of $140. This task allowed them to verify their model by applying it in a new case: [A.354] Hassan: The garage is ½ so in this set [priced at $140] it will be $70. [A.355] Alaa: The mechanic…is easy to know. 1/10 of 140 is… [A.356] Momen: It is 14. The students computed the price of all the items when the set was priced at $140: [A.368] Hassan: Now we can total all the prices … 70 and 14 and 28 and 5.6 and 22.4 [A.369] Momen: It is okay… it is 140.
Constructing Conceptual Models and Making Sense of Fractions
The students verified their model for the set priced at $140, Hassan [A.354] and Alaa [A.355] used their mathematical model to compute the prices of the items, and Hassan [368] then proceeded to ensure that the sum of the individual prices equaled the overall price. The modelling processes that emerged via engagement in the garage set activity are presented in Fig. 1. The students interpreted the situation to produce a real model. They then constructed a mathematical model and worked through it in order to verify it in a new situation. Finally, they generalized the mathematical model using the Snow White and the Seven Dwarves set activity, as described next.
The Modelling Process Exhibited in the Snow White and the Seven Dwarves Set Activity The modelling phases exhibited by group C in the Snow White and the Seven Dwarves set activity resembled those observed in the garage set activity. These began with a realistic model (interpretation and identifying factors for pricing), followed by a mathematical analysis and a mathematical model, and then verifying the mathematical model within the given reality. While the students in group C did not take the model elicited by the garage set activity into consideration, groups A, B, and D adapted the pricing criteria they had established in the previous activity and generalized the mathematical model they had created (Fig. 2). In order to avoid repetition, we shall focus here on the generalization of the realistic and mathematical models, and then on verification of the model for the new context. Generalization of the Real Model and Mathematical Model. As noted above, after reading the new task and interpreting the context, the students in groups A, B, and D realized they could adapt the pricing criteria and mathematical model they had employed in the garage set activity. In groups A and B, the students immediately Identifying ffactors for pricing
Interpreting the situation
ki Working mathematically
The garage set activity Constructing a mathematical model Verifying the mathematical model Generalization of the mathematical model in the Snow White and the Seven Dwarves set
Fig. 1 The modelling process in the garage set activity
J. A. Shahbari, I. Peled Interpreting the situation Generalization the real model
The snow white and the seven dwarves set Generalization the mathematical model Verifying the mathematical model
Fig. 2 The modelling process of groups A, B, and D in the Snow White and the Seven Dwarves set activity
recognized that there was no need to set specific prices for different items in the set, and that they could simply implement an expanded version of the mathematical model they had previously created: [A.452] Hassan: To price each item again … [A.453] Adi: We can follow what we did in the garage activity … using fractions … [A.454] Momen: Yes, we don’t need to do all of that all over again … we can use [A.455] another denominator. Adi [A.453] and Momen [A.454] both suggested adapting the previous model in order to price each item rather than adopting an arbitrary pricing. In group D, the students began by pricing the individual items in the $90 set, but one student quickly realized that this was unnecessary: [D.473] Omer: I will start with the house, it will be $20… the most expensive. [D.474] Manal: The princess will be the second item; it will be $10 maybe? [D.475] Omer: Each dwarf… [D.475] Rana: But this isn’t good… we can use a fraction for each item like in [D.476] the garage … [D.477] Omer: Yeah… we can use that for all three prices… In the discussion between Omer [D.473] and Manal [D.474], the students started pricing the BSnow white and seven dwarves^ items, but then their classmate Rana [D.475] suggested constructing a mathematical model like the BGarage^ activity.
Constructing Conceptual Models and Making Sense of Fractions
Verifying the Mathematical Model. The students verified their models for the Snow White set activity much in the same way as in the garage set activity. First, they checked that the sum of the fractional parts was one. They then used the mathematical model to price items in new cases (Snow White sets costing $120 and $240). Analysis of all the groups’ worksheets and notes shows that most of the groups succeeded in coping with both modelling activities. In the garage set activity, they supplemented the initial story with missing data, priced the items in each set with awareness of internal and external rankings, and constructed a mathematical model. In the Snow White set activity, all but four of the 15 groups generalized the models that had been elicited in the garage set activity (the other four repeated the same processes as in the first activity). These findings are summarized in Table 2.
Results Relating to Knowledge of Fractions Engagement in the MEAs exposed and enhanced the students’ knowledge and understanding of fractions. Here, we first describe the students’ understanding of fraction sub-constructs as this emerged from the transcripts. We then describe the difficulties they encountered and their efforts to resolve these problems. Fraction Sub-Constructs Both the activities accentuated the part-whole, operator, and ratio sub-constructs of fractions. It should be noted that the categorization into the different constructs is not always clear-cut. We, therefore, explain the nature of the different examples. The importance of the analysis is in the showing the richness and variety of these different uses. Part-Whole. The activities forced students to consider the part-whole sub-construct in two ways—in terms of the set as a collection of discrete parts, and in terms of the Table 2 The modelling processes across all groups in the two MEAs Modelling process
Garage set activity
Groups
Interpretation of situation and completion of missing data
Snow White and the Seven Dwarves set activity
Pricing based on item rankings
Construction of a mathematical model
Generalization Pricing of the previous using the mathematical model model
Replication of the modelling processes used in the first activity
1, 2, 4, 5, 6, ✓ 9, 14 , 15
✓
✓
✓
✓
–
3, 10, 12
✓
✓
✓
✓
–
✓
7
✓
✓
✓
–
–
✓
8
✓
✓
–a
–
✓
–
11
✓
✓
✓
–
✓
–
13
✓
✓
–
–
✓
–
a
Not clear from analysis of the worksheets
J. A. Shahbari, I. Peled
overall price of the set as a continuous whole, with the prices of individual items as parts. The latter meaning was brought to the fore via the building of the mathematical 7 2 model. Recall the mathematical models presented earlier: Garage= 16 , wheel= 16 , 4 1 2 crane= 16, screwdriver= 16, and cables= 16. 4 1 1 Garage= 12, car= 25 , crane= 15, mechanic= 10 , and screw= 25 . In the first example, the same denominator was applied to all the fractional parts, while in the second, the fractional parts had different denominators. However, both models represent unequal partitioning of the whole. Operators. The students specified the price for each item as a fraction of the whole price: [C.232] Naden: The garage is always ½ of the price. [C.233] Amir: No, it is too much; it should be 1/3 of the price. These statements reflect the fraction as operator, specifying fractional parts (½ or 1/3) of the original price as indicators of the individual item price. Quotient. The students had to divide two quantities—the price of an item and the price of the whole set: [D.203] Mona: The price of the garage is $27 in this set [priced at $60], so … we can [D.204] divide 27 by 60 to get 27 60. Mona [D.203] recognized that the fraction quantities—27 and 60.
27 60
is a product of the division of two
Ratio. The students compared the prices for each item in the two garage sets, expressing the ratio between the price of each individual item and the price of the set as a whole: [D.209] Manal: The garage price in the two sets is not the same ratio, in this set 27 [D.210][priced at $50] the price is 25 50 and in this set [priced at $60] it’s 60.
Manal [D.209 and 210] referred to the fractions her understanding of this concept.
25 50
and
27 60
as ratios, thus indicating
Fraction Difficulties Revealed and Overcome The principal difficulties the students encountered involved the sum of the parts not being equal to the whole, procedural knowledge relating to fraction comparison, operator algorithms, and reduction and expansion.
Constructing Conceptual Models and Making Sense of Fractions
The Sum of the Parts not Equal to the Whole. In constructing the mathematical model, the students specified a fraction for each item. Two of the focal groups recognized that the sum of the parts was not equal to one: [B.294] Rema: We can say the price of the screw is 18 and the screwdriver is 28 of the [B.295] price. [B.296] Tala: Yeah, the price of the car will be price.
3 8
and the crane is
4 8
of the
[B.297] Sameh: No, it’s wrong … all the parts together must be one… it’s more than [B.298] one … you see 3 and 4 and 1 and 2 [pointing to the fractions for each item] [B.299] are … 10 8 . We can divide the price into smaller parts in order to keep the rankings. Sameh’s comment [B.297-299] that the sum of the fractional parts being 10/8, they were not aware to take note of the fact that the sum of the parts must be one. He thus understood that the pricing process was incorrect. Those groups then proceeded to construct a mathematical model in which the sum of the fractions was one. They adopted the same process for the second activity: [B.512] Tala: We need a big number for the denominator … I think 40 is okay. [B.523] Rema: I will write it here [pointing to the list of items] [B.524] beside the items’ names … the largest part is the house. … 3 39 1 [B.582] Tala:… 10 40 and 40… it’s 40. We have to add 40 to the price of one of the items.
Tala [B.582] checked that the sum of all the fractional parts was equal to one (40/40). 1 Discovering that the sum of the parts was less than one (39 40), she suggested adding 40 to one of the items. Operator Algorithm. Some of the students found it difficult to understand the use of fractions as a functional operation expanding or reducing the original amount in the garage set activity: [B.218] Sameh: We can always say that the garage is half of the price, so it’s always [B.219] fair whatever the price of the set is. [B.220] Ramah: I don’t understand what you mean by Bhalf of the price^!
J. A. Shahbari, I. Peled
[B.221] Sameh: What is half of 10? 5. What is half of 20? 10 … You do it like that [B.222] whatever the price is; you calculate half of it … Ramah [B.220] had difficulty understanding the operator algorithm, not recognizing the notion of half the price. Sameh [B.221] explained the idea to her. She was then able to apply the operator algorithm in order to compute the price of the set items according to the model when the overall price is given, evidencing her ability to employ procedural knowledge related to this aspect of fractions: [B.382] Rema: If the whole price is 140, how much is the price of each item? 4 [B.383] Ramah: The price will be … [writing] 140* 24 …It’s … [computing] it’s $23.33.
Ramah [B.383] here demonstrated her ability to use fractions as operators, computing the price of each item when the overall price is given. This is also evinced in her statements during the Snow White set activity: [B.531] Tala: The house is more expensive … [B.532] Ramah: So the house can be the largest fraction part … 4 [B.533] Ramah: Try the house as 10 24 of the price and the princess as 24…
[B.534] Rema: I think we must specify a smaller fraction for the house. Here, Ramah [B.533] used the fraction as an operator, indicating that she has now understood the operator algorithm. Reduction and Expansion Fractions. Some students found it difficult to compare the fractional parts for items in the different sets. This was especially true in the groups whose final mathematical models employed more than one denominator: [C.708] Eman: Which is more expensive, the mirror, the witch, or the prince? [C.709] Naden: How do we compare them? [C.710] Saeed: First, we have to compute the fractional parts of these [the mirror 1 1 3 and [C.711] the witch]. It will be 24 þ 48 ¼ 48 . [C.712] Naden: Why? 1 2 1*2 2 [C.713] Amir: Because 24 is 48 [he writes 24*2 = 48 ]. If you multiply the
[C.714] numerator and the denominator by the same number the magnitude [C.715] doesn’t change.
Constructing Conceptual Models and Making Sense of Fractions
Naden [C.712] exhibited difficulty with the reduction and expansion of the fraction. She overcame this difficulty with Amir’s help, as observed in other tasks requiring the reduction and expansion of fractions.
Discussion Modelling activities offer students an opportunity to develop modelling competencies, while at the same time enriching their knowledge and making the mathematical concepts used in the modelling process more meaningful (Bonotto, 2010). Following observations of improved fraction knowledge in our earlier study on the reinvention of percentages (Shahbari & Peled, 2015), we designed the present study to investigate more closely how this improvement through modelling takes place. The findings show that all the studied groups succeeded in eliciting a mathematical model for the first activity (the garage set). They met the demands of the situation by interpreting the situation, identifying the factors relevant for the task (pricing), working mathematically, and constructing and verifying a mathematical model. Their success in constructing an appropriate model confirms previous studies that demonstrate primary students’ capacity to engage in model-eliciting activities (English, 2006, 2010; Eric, 2008), and the processes displayed were consistent with Blum and Leiss’s (2005) cyclic modelling stages. Our findings also confirm the benefits of modelling activities highlighted by other studies (English, 2006; Zawojewski & Lesh, 2003). First, working in small groups allows responsibility for constructing the mathematical model to be shared, through communication and discussion of various mathematical and modelling components (Mousoulides, Sriraman, & Lesh, 2008). Second, the opportunity for self-assessment allows students to validate their mathematical model without reference to an external authority (a teacher). Third, these activities foster the development of several different modelling competencies, such as ranking and quantifying. Finally, the activities show students how a model can be generalized and transferred to a new activity. More generally, modelling activities teach students how to find mathematical solutions for assigned problems and refine them in order to meet real-world conditions (Doerr & English, 2003). With respect to fractions, the present findings indicate that the part-whole, operator, quotient, and ratio sub-constructs were emphasized in both activities, thereby providing an opportunity for constructing connections between the subconstructs of fractions and deepening understanding of this concept (Hiebert & Carpenter, 1992). The garage set activity provided a platform for identifying problems in the students’ procedural knowledge—the sum of the parts must equal the whole, comparison, addition, the operator algorithm, and reduction and expansion. The second activity provided an opportunity for verifying that the students had overcome earlier difficulties. Overall, analysis of the students’ discourse demonstrated that, in line with findings from other studies (Hallett et al., 2012; Hecht & Vagi 2012, Rittle-Johnson et al. 2001), the modelling activities provided an opportunity for developing components in students’ conceptual as well as procedural knowledge, and that the two continually stimulated each other without either necessarily taking precedence. The activities thus
J. A. Shahbari, I. Peled
served as a vehicle for developing students’ capacity to engage mathematically in realworld situations, while promoting and strengthening their understanding of fractions. Our findings support Julie and Mudaly’s (2007) argument that students can best learn to think and reason about mathematical situations by Bmaking sense of situations^ rather than being given an algorithm.
Concluding Remarks There is some inner contradiction in claiming that we are offering modelling activities that elicit fraction models and are supposed to enrich fraction knowledge. Modelling activities are expected to involve students’ own decisions. They differ from traditional tasks especially with regard to the choice of mathematical models. Thus, the tagging of activities as fraction tasks might seem odd. As mentioned in the introduction, the two main and different reasons for using modelling activities include the development of modelling competencies and the construction or strengthening of mathematical knowledge. In the initial stage of constructing concepts, modelling tasks might be introduced in a guided and planned nature leading to expected emerging models. However, when a more open nature is desired, the mathematics used by the students is less Bcontrolled,^ making it difficult to integrate modelling activities into the curriculum for strengthening specific mathematical concepts. In this article, we show that it is possible to allow students to build their own models, and at the same time expect them to use fractions, as though Ball is foreseen but freedom of choice is given.^ Thus, our findings show that modelling activities can be designed in a way that would keep the open nature of the problem, and at the same time aim at strengthening a specific mathematical concept.
Appendix A: The Garage Set Activity In a toy shop, customers wish to purchase individual items of a garage set which only comes in sets of five items. While the sets all contain the same items, they differ in some respects and so are priced differently. The shop owner decides to break up the sets in order to sell each item separately. The owner decides that the sum of the individual items in each set must equal the price of the whole set. He asks his employees to price the individual items in two sets selling at $50 and $60. The prices of each item must be comparable between the two sets. Students, you are employees of the toy shop, so the decomposition and pricing task is your responsibility. [After they finish coping with the activity and constructing the mathematical model, students are presented with a final task.] Final task: The shop owner found another garage set in the store that costs $140. You are responsible for pricing the individual items in the new set.
Constructing Conceptual Models and Making Sense of Fractions
Appendix B: The Snow White and the Seven Dwarves Set Activity A toy shop has a lot of unsold BSnow White and the Seven Dwarves^ sets consisting of 20 items. All the sets have similar composition, and are priced based on the material they are made of—canvas, plastic, wood, cardboard, etc. The owner decides they will sell better if each item is sold separately, but decides that the sum of all the items must equal the price of the whole. Students, as employees of the toy shop, you are responsible for the decomposition and pricing tasks. The sets are priced at $90, $120, and $240. The price of each item must be comparable between the three sets.
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