Canadian Journal of Science, Mathematics and Technology Education
ISSN: 1492-6156 (Print) 1942-4051 (Online) Journal homepage: http://www.tandfonline.com/loi/ucjs20
Using Modelling Tasks to Facilitate the Development of Percentages Juhaina Awawdeh Shahbari & Irit Peled To cite this article: Juhaina Awawdeh Shahbari & Irit Peled (2015): Using Modelling Tasks to Facilitate the Development of Percentages, Canadian Journal of Science, Mathematics and Technology Education, DOI: 10.1080/14926156.2015.1093201 To link to this article: http://dx.doi.org/10.1080/14926156.2015.1093201
Accepted online: 23 Sep 2015.
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ACCEPTED MANUSCRIPT Using Modelling Tasks to Facilitate the Development of Percentages
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Juhaina Awawdeh Shahbari,1,2,3 and Irit Peled1
1
University of Haifa
2
The College of Sakhnin
3
Al-Qasemi Academy- Academic College of Education
Address correspondence to Juhaina Awawdeh Shahbari, Department of Mathematics Education, University of Haifa, Haifa 3498838, Israel. E-mail:
[email protected] Abstract This study analyses the development of percentages knowledge by seventh graders given a sequence of activities starting with a realistic modelling task, in which students were expected to create a model that would facilitate the reinvention of percentages. In the first two activities, students constructed their own pricing model using fractions, and then extended the model while experiencing reinvention and extension of their knowledge of percentages. In the last two activities, they coped with a realistic changing reference situation. A control group used a traditional instructional unit. A pre-test and two post-tests showed between-group differences.
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ACCEPTED MANUSCRIPT Résumé Cette étude analyse l’acquisition de la connaissance des pourcentages par des élèves de 7e année qui doivent effectuer une suite d’activités débutant par une tache de modélisation réaliste dans laquelle ils doivent créer un modèle qui faciliterait la réinvention des pourcentages. Au cours des
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deux premières activités, les élèves ont construit leur propre modèle de tarification à l’aide de fractions, puis ils l’ont étendu au fur et à mesure qu’ils faisaient l’expérience de la réinvention et de la progression de leurs connaissances en matière de pourcentages. Dans les deux dernières activités, ils ont dû gérer un changement de situation de référence réaliste. Un groupe de contrôle a utilisé, pour sa part, une unité d’instructions traditionnelles. Un test préliminaire et deux tests postérieurs ont mis en évidence les différences entre les deux groupes.
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ACCEPTED MANUSCRIPT INTRODUCTION Modelling tasks are defined in this study as tasks that involve an authentic situation, and whose solution requires a process of organizing and simplifying the situation and making decisions with regard to its mathematical structure and representation. This process encourages children's
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spontaneous use of their existing mathematical knowledge, and thereby strengthens and adds meaning to this knowledge. A good modelling task thus serves as an ―effective prototype‖ (Lesh et al., 2000, p. 626), meaning that it guides students toward understanding other structurally similar situations. In our earlier work (Peled & Shahbari, 2009), we showed that strengthening mathematical knowledge around the concept of fractions, learned previously, can improve decimal number knowledge even without any new learning of decimals. This is possible because the same general coordination schema (based on the size and number of parts) that is involved in ordering fractions is also helpful in understanding order relation in decimals. In this study we show that strengthening students’ knowledge of fractions with a focus on a mediating schema can also facilitate construction of the percent concept. We present a sequence of four tasks, beginning with a modelling activity where a fraction model is expected to emerge. The sequence then requires students to make decisions using fractions (and later also percentages) in new situations where students are not told when and how to apply this mathematical knowledge. The tasks thus guide students towards understanding the use of percentages.
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ACCEPTED MANUSCRIPT THEORETICAL BACKGROUND The modelling process is defined by researchers in a variety of ways, but there is general agreement that modelling can be described as the "the entire process leading from the original real problem situation to a mathematical model" (Blum & Niss, 1991, p. 39). This process
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includes simplifying and organizing the situation, translating the organized structure to mathematical terms, working mathematically, interpreting mathematical output and validating (Kaiser, 2007). Educational researchers offer two main rationales for including modelling tasks that evoke such processes in the school curriculum. The first is the idea that learners can be motivated by making mathematical concepts meaningful. Under this reasoning, modelling is used to strengthen existing curricular goals by facilitating students’ construction of mathematical concepts. The second involves the claim that it is not enough for students to learn mathematical concepts and apply them when told to do so; they must also learn to make their own decisions on fitting mathematical concepts and tools to given situations. This goal calls for using modelling tasks to develop modelling competencies, making modelling a goal in itself rather than a means for developing mathematical concepts (Blum & Niss, 1991; Niss, Blum, & Galbraith, 2007; Stillman et al., 2007). Engagement in modelling tasks can lead to knowledge change in already acquired concepts, and also to the emergence of new concepts. The latter is more complex and demanding for the task designer. In the Dutch Realistic Mathematic Education (RME) approach, new concepts are introduced using specially designed realistic tasks.
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ACCEPTED MANUSCRIPT RME is rooted in Freudenthal's (1968) approach to teaching mathematics as a human activity, a means of organizing information and solving problems, rather than as a closed system of rules and procedures (Gravemeijer & Terwel, 2000; Rasmussen & King, 2000). Under RME, mathematics is taught via appropriately sequenced situations, such that organizing these
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problems then allows the emergence of new models (Gravemeijer& Stephan, 2002; Rasmussen & King, 2000). In this way, models of [a given situation] are initially situated in the context and become abstracted over the learning trajectory, turning into models for [new situations] and enabling the rise of formal mathematics (Gravemeijer & Stephan, 2002). Naturally, a modelling process that is directed by the second rationale (modelling as a goal) is more open, while the first (modelling as a means) calls for a more structured design aimed at the reinvention of specific knowledge. In this study we combine both open and structured activities in an effort to promote the construction of percentages as a part of an integrated mathematical knowledge system. We focus on the concept of percent because it is an important topic (Gay & Aichele, 1997) that nonetheless seems to confuse many, including both children and adults (Berry, 2002; Dole, 2000). Learning-teaching of percentages: Different methods were proposed in the literature to the teaching of percentages; some of these methods suggest connecting or converting percentages to other concepts. Dole (2000), for example, suggested representing a situation with percentages as a statement of proportion with the use of a dual-scale number line. Others recommended connecting percentages and proportion
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ACCEPTED MANUSCRIPT through building doll activity (Moss & Caswell, 2004); or connections between percent, fractions and decimals via the halving strategies and benchmark (Moss & Case, 1999). Other strategies adapted static models as area model (Scaptura, Suh & Mahaffey, 2007). Our instructional trajectory offers a dynamic model that is constructed by the students
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themselves. This model leads to the emergence of the percent concept and to connecting percentages with components in existing knowledge. It starts with a modelling task where children use their existing mathematical knowledge of fractions and ratios. The models that emerge create a basis for coping with a more structured task that is expected to guide pupils towards using percentages. While their initial knowledge of percentages is very basic, the tasks encourage reinvention of further knowledge of this concept. Our research investigates how this combined sequence facilitates growth in children's knowledge of the percent concept, how it promotes the integration of percentages with fractions, decimals, ratio and proportion, and whether it has a better effect on children's knowledge than a conventional textbook unit. METHOD Research setting and participants The study was carried out with three seventh grade classes totaling 96 students (12-13 years old). All the classes were in one school located in the north of Israel. The three classes were chosen from among seven seventh-grade classes in the same school because their mathematics teachers reported that their achievements in mathematics were similar, and they had similar student
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ACCEPTED MANUSCRIPT populations. Two classes were assigned to be the experimental groups and the third class was the control group. All participants had been introduced to the percent concept very briefly (over two or three lessons) in the sixth grade. At the time of the study, all were familiar with the percent sign and
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could transform simple familiar fractions (
1 1 , ) to percentages. A pretest (Appendix A) given 2 4
at the beginning of the study showed that the students had no knowledge of the concept beyond this basic understanding, and could not perform any tasks beyond transformations. For example, they could not solve simple change problems involving the increase or decrease of a given price by a certain percentage. Data collection and analysis A mixed design was used, comprising both qualitative and quantitative data collection and analysis; the qualitative data related only to the two experimental classes and the quantitative data related to all three research classes. The qualitative data sources included videotapes of the experimental classes as well as students' notes and worksheets. Students worked in small groups of 4-5 students. Five of these groups were selected for more detailed analysis and were also videotaped during the study. The transcripts of student engagement in the activities were reviewed several times to identify and characterize salient patterns, elements, operations and processes. Students' work sheets and notes were reviewed simultaneously with the transcripts in each group to support the characterization of the observed processes.
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ACCEPTED MANUSCRIPT The quantitative data comprised the results of three similar tests: a pretest, post-test 1 and posttest 2. All three tests included items measuring procedural and conceptual knowledge of common fractions, decimal numbers, and proportion (22 items, Cronbach's = .84, .91 and .90 for the pre-test, post-test1 and post-test 2 scores respectively), and items on percentages that include
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knowledge the students were expected to acquire during the intervention (23 items, Cronbach's = .88, .91 and .94 for the pre-test, post-test1 and post-test 2 scores respectively). Test item examples are presented in Appendix A. The quantitative analysis was conducted by repeated measures ANOVA tests comparing the effect of the task sequence on knowledge of fractions and percent in the experimental and control groups, and identifying the source of differences between measures. Procedure The pre-test was administered prior to the learning process. The experimental groups then engaged with the designed instruction sequence over nine lessons of 45-50 minutes each. The control group learned with the conventional textbook over the same period. Post-test1 was conducted immediately after the learning process, and post-test 2 was conducted three months after the end of the instructional process. The design process of the instructional sequence for the experimental groups The activities were designed using the principles suggested by Lesh et al., (2000) for constructing model eliciting activities (MEA). According to their principles, the problem situation should be relevant to the students promoting making sense of it based on their personal
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ACCEPTED MANUSCRIPT knowledge. It should elicit the construction of a mathematical model that can be generalized and used flexibly in other situations. The study was conducted using Design-Based Research methodology that bridges between theoretical research and educational practice (Design-Based Research Collective, 2003). The
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activities were first implemented in a pilot study with students that did not take part in the later study. They were revised through iterative cycles where both problem context and problem quantities were modified on the basis of the observations.
The instructional sequence for the experimental groups The experimental groups were given four activities in a sequence designed to facilitate knowledge growth and further reinvention of the percent concept. Table 1 describes briefly each activity in the sequence and details the specific goals of its design. The first two activities were designed to promote the construction and elaboration of a splitting structure. This structure was considered as pivotal in the instructional sequence, i.e., as a crucial factor in guiding students towards reinvention. The remaining two activities were designed to strengthen students’ knowledge, and specifically, their awareness of a changing reference. This knowledge was expected to affect their work with percentages, expanding their concept of percent. The instructional unit for the control group The control group studied percentages through a conventional mathematics textbook unit. The unit included transforming fractions and decimals to percentages and vice versa, tasks that
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ACCEPTED MANUSCRIPT required presenting shaded areas in percentages and vice versa, and increase–decrease problems. These tasks were similar to tasks that were classified by Parker and Leinhardt (1995) as conventional. FINDINGS
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Overall, students’ work over the four-step sequence followed our expected trajectory. During "The Bedding Set" activity (Appendix B), students constructed mathematical structures based on their knowledge of fractions. They expanded and implemented their knowledge in "The Tableware Set" activity (Appendix C), in which the mathematical structure was transformed from fractions to decimals and then to percentages. During "The Bag of Fries" activity (Appendix D), the students came to grips with the notion of a changing reference using the familiar concept of fractions. Finally, this was expanded to percent terms in the last activity, "The Pet Weight loss" activity (Appendix E). Below, we elaborate on the modelling process for each step. Step A: Building a general pricing model based on fraction knowledge The mathematical structure was built initially through coping with "The Bedding Set" activity. At this stage, three main phases of the modelling process can be distinguished. In the first phase, students organized and analyzed the context, in this case identifying and listing the items in the bedding set. In the second phase they made mathematical calculations appropriate for the context. They began this phase by dividing the price of the full set equally between the items, and then recalculated and adjusted the prices, creating an unequal split based on the size of each item. In the third phase the students engaged in mathematical discussion of mathematical objects,
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ACCEPTED MANUSCRIPT defining each item in the set as a certain fraction of the whole. The phases are described in Table 2, with examples from students' discourse.
Step B: Using the general pricing model in reinvention of the percent concept
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The second activity, "The Tableware Set", gave the students an opportunity to extend the mathematical structure they had built in the previous activity. This task led them to change the fractional representation used in "The Bedding Set" activity to decimal representation. The students were thus guided towards the concept of hundredths and, from there, towards new units – percentages – with a sign and a name. Expanding the mathematical structure with fractional representation In "The Tableware Set" situation, the students adapted the mathematical structure developed in "The Bedding Set" activity– i.e., the general model by which they priced individual items when given the price of the whole set. The students discussed how to adapt the general pricing model to a case with more components (30 pieces) by using a larger denominator. Episode 1 shows that they first adapted the pricing model by assigning particular fractions to certain items, as suggested by Ayat (line 1.1). In addition, episode 1demonstrates that students were aware of the need to expand the denominator so that the items could be appropriately ranked. This is exhibited in Hisham's (line 1.2) and Malak's (lines 1.3 and 1.4) suggestions. Episode1:
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ACCEPTED MANUSCRIPT 2 3 and .... (Pointing to the words ―big plates‖ and ―small plates‖) 10 10
1.1
Ayat:
1.2
Hisham: It is not enough.
1.3
Malak: We can divide the price to 15 parts, it will be easier.
1.4
Malak: Perhaps 20 parts, 20 pieces must fit.
1.5
Ayat: For the small plates the price will be
1.6
plates it should be larger…
1.7
Malak: Yes, it is possible that here
3 and for the larger 20
5 . 20 1 2 and here .... 20 20
The students continued the pricing process by assigning a fraction to each single item and each category of items while checking their results. As expressed in Ayat's words (line 2.1), they knew that the sum of all the fractions must be equal to one. Episode 2: 2.1
Ayat: The sum of all parts should be
2.2
Malak: Good!
20 . 20
Transformation of the mathematical structure from fractional to decimal representation
The need for decimal representation appeared when the students tried to price each item in each category and to rank the items by their prices. Some groups continued using a fractional
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ACCEPTED MANUSCRIPT representation when pricing one item in each of the different sub-categories of the complete set. Then, in ordering items by their prices they switched to using a decimal representation. Other groups used a decimal representation from the beginning. The former pattern is exemplified in episode 3. While the students found the price of each item by pricing the category and then
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dividing by the number of items in each category, they initially tried to do this using fraction, as is evident from Muhammad's comments (lines 3.1 and 3.2). The decimal representation appeared when students needed to order the prices of the items, as seen in Muhammad's suggestion (lines 3.8 and 3.9). Episode 3 demonstrates the difficulty of converting fractions to decimals, Students confused the fraction line with the decimal point, as exhibited by Hussein (line 3.10). As can be seen in the episode, he overcame this difficulty with the help of his classmate. Episode 3: 2 so we must divide by 4, so 24
3.1
Muhammad: If the price of 4 bowls is
3.2
the price of 1 bowl is
3.3
Hussein: How can we know the price of 1 plate if the price of 8 plates 3.4 is
3.5
Dima: We can use a calculator.
3.6
Hussein: (using a calculator with the option of a fractional display)
3.7
5 divided by 24 divided by 8, it's
1 . 48 5 ? 24
5 192
….
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ACCEPTED MANUSCRIPT 3.8
Muhammad: We can use decimals to rank the prices of the items, it's
3.9
easier.
3.10
Hussein:
3.11
Suleiman: No, no, for one half [1/2] we write 0.5 and not 0.2...
3.12
Hussein: (uses the calculator for converting 1/48) It's 0.02…
1 is 0.48 48
Episode 4 shows that some of the students used the decimal representation at an earlier stage, as can be seen in Zohra's contribution (line 4.3).
Episode 4: 3 [of the total price of the set]. How 10
4.1
Aram: The price of 8 plates is
4.2
much is one plate?… We can divide it by 8.
4.3
Zohra (using a calculator):
4.4
Hasnaa: Ah, I understand, so 6 trays are
3 divided by 8 is 0.0375. 10 4 [of the total set price], so 4.5 [to 10
figure the part for one tray] - divided by 6. 4.6
Beidaa:
4 divided by 6 is 0.0666. This number is called cyclic. 10
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ACCEPTED MANUSCRIPT Emergence of the percent sign The percent sign emerged in two steps. First, the effort to assign prices guided the students to use two digits after the decimal point. This led the students to the word "hundredth", as expressed in lines 5.1, 5.2, 5.3 and 5.4.
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Episode 5: 5.1
Doaa: The big plates are 16 hundredths. (Writes 16 hundredths)
5.2
Miriam: The bowl is 6 hundredths. (Writes 6 hundredths)
5.3
Doaa: The small plate is 2 hundredths and the small bowl is 1
5.4
hundredth. (Writes 3 hundredths)
The students then tried to substitute a sign for the word hundredths. They explored a number of possibilities, such as a dollar sign (lines 6.1 and 6.2) or some version of the number hundred (line 6.5) before coming up with the percent sign. Episode 6: 6.1
Dima: There is a sign "$".
6.2
Mohammad: This sign is for money, it’s dollars.
6.3
Dima: We can draw a flower...
6.4
Muhammad: Maybe we can write a colon ":".
6.5
Hussein: We can write 100 from the top.
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ACCEPTED MANUSCRIPT 6.6
Dima: There's this sign - %.
6.7
Riemann: This is percentage.
6.8
Suleiman: It's like in 50% discount.
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Acquisition of the percent concept According to the pre-test findings, although the students were aware of the percent sign prior to engaging in "The Tableware Set" activity, they had only very basic knowledge of percentages. Thus, their use of percentages can be considered as construction of new knowledge. The students internalized the concept as they engaged with the remaining parts of the activity. They seemed to be constructing its meaning, using percent as both part-whole and operator, as expressed by Dima (line 7.1) and Hussein (line 7.6) respectively. In addition, episode 7 shows the students' calculation skills, as exhibited by Hussein (lines 7.4and 7.6). Episode 7: 7.1
Dima: All the set is 100% of the price.
7.2
Riemann: If you buy the whole set except the bowls, then subtract the 7.3 bowls…. 500 minus….
7.4
Hussein: We can do100% minus 2.08%.
7.5
Dima: Yes, and then calculate…
7.6
Hussein: 97.92% of the 500.
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ACCEPTED MANUSCRIPT The students’ discourse later in the instructional sequence, during "The Pet Weight Loss" activity, supported the conclusion that they had acquired a deep understanding of the concept of percentage as a ratio, as exemplified in episode 8. Students, like Saleh (lines 8.1 and 8.2), chose percentages as a mathematical tool to express the decrease in ratio. In addition, the episode
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shows that some students could use percentages flexibly as exhibited in Saleh's (lines 8.4, 8.5 and 8.6). Episode 8: 8.1
Saleh: We can calculate the percentage decrease of each animal and 8.2
not by
kilos. 8.3
Duah: We want to calculate how much the weight of each animal fell.
8.4
Saleh: Do it not by kilos, but in percentages, for instance, how heavy 8.5 was
the rabbit? It was 20 kg and now it’s 18 kg, so in percentage terms 8.6
20 kg is 100%
and 10 kg is 50%, 5 kg is 25%. 8.7
Duah: So 2 kg is 10%.
Step C: Revealing and overcoming common misconceptions in markup-markdown problems The findings from the pretest showed that nearly all the students (97%) had difficulty with changing reference problems, whether using fractions or percentages. These findings were verified in "The Bag of Fries" activity, where the students were tasked with reconstructing an original paper model after the original model was enlarged by a third or a fourth. All the students first reduced the enlarged model by the same fraction used to enlarge it, ignoring the changed
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ACCEPTED MANUSCRIPT reference. For example, when a group was told that the model they were holding had been enlarged by a third, they tried making it smaller by a third, as presented in Episode 9 (lines 9.5, 9.6, 9.8 and 9.9).
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Episode 9: 9.1
Hussein: We need to remove a third of it.
9.2
Dima: First, we should measure it.
9.3
Hussein: Ok, do that.
9.4
Dima (after measuring the enlarged model): It is 27 cm.
9.5
Hussein: 27 divided by 3 is 9. So we should remove 9 cm from the
9.6
enlarged model.
9.7
Dima: Why would you divide it by 3?
9.8
Hussein: Because it (the enlarged model) is enlarged by a third; now, I 9.9 want to
know how much the third is.
Cognitive conflict emerged when the students compared the original model that was constructed earlier and the reconstructed model; the models were not the same. The subsequent discussions leading to conflict resolution are detailed in Shahbari and Peled (2014). Step D: Coping with a changing reference using percentage In "The Bag of Fries" activity, the students learned to cope with a changing reference using fractions. The final activity of the sequence, "The Pet Weight Loss" activity, allowed them to transfer their understanding to percentages. Episode 10 show that students became aware of the
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ACCEPTED MANUSCRIPT changing of the reference in percent terms in the context of the activity, as explained by Ayat (lines 10.1, 10.2 and 10.3), who referred to the specific case and then generalized (lines 10.5, 10.6 and 10.7). Episode 10:
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10.1
Ayat: If an animal loses 5% of its weight and then gets fatter by 5%, it 10.2 will not go back to what it was before, it will go back to something
10.3
smaller; the 5% in each case are not equal.
10.5
Ayat: If you increase something by a certain size, and then subtract the 10.6 same percentage, it doesn't come out the same because the size has
10.7
changed.
Constructing the percent concept In order to examine if the knowledge of percentages is general and not related to (or situated in) the specific activities, quantitative analyses was conducted, using three tests: a pretest and two post-tests. The results point to significantly improved percent knowledge in the experimental groups compared with the control group. The changes in average scores by group and across the three tests are presented in Figure 1.
A repeated measurements analysis indicates a significant difference between the pretest and the first post-test regardless of group )F(2, 186) = 345.30, p>0.001), a difference that can be attributed chiefly to the rise in scores following the learning process in the experimental groups. The
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ACCEPTED MANUSCRIPT difference between the experimental and control groups regardless of the test was also statistically significant )F(2, 93) = 6.30, p>0.01), with the two post-tests providing the source of the difference. The findings obtained from post- test 2 indicate that this change was maintained with time.
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Connection between the percent concept and existing knowledge The students’ discussions show that the instructional sequence provided them with the opportunity to organize different components of their existing knowledge of fractions, decimals, ratio and proportion, and to build on this existing knowledge to acquire new knowledge. The quantitative data point to considerable improvement in the experimental groups compared with the control group. Average scores by group and across the three tests are presented in Figure2. A repeated measurements analysis indicates a significant difference between the pre-test and the first post-test regardless of the group )F(2,
186)
=100.962,p>0.001), with the difference again
attributable to the rise in scores in the experimental groups. The groups also differed significantly regardless of the test (F(2,93)=5.20, p
