Generalization Strategies in the Problem Solving of Derivative and Integral

This study proposes a learning strategy of derivatives and integrals (LSDI) based on specialized forms of generalization strategies to improve undergraduate students’ problem solving of derivative and integral. The main goal of this study is to evaluate the effects of LSDI on students’ problemsolving of derivative and integral. The samples of this study were 63 undergraduate students who took Calculus at the Islamic Azad University of Gachsaran, Iran. The students were divided into two groups based on their scores in the pre-test of derivative and integral. The results indicated that there was a significant difference between the achievements of students in experimental and control groups after treatment. Thus, the findings reveal that using generalization strategies improves students’ achievements in solving problems of derivative and integral.

Some methods are introduced to support the students to overcome their problemsolving difficulties in the learning of derivative and integral. Researchers endeavour to support students' problem-solving in the learning of calculus by promoting mathematical thinking with or without a computer (Tall, 2008;Rahman, 2009;Kashefi, Ismail, & Yusof, 2012). There are extensive studies in promoting mathematical thinking to help students' understanding of calculus, especially derivative and integral (Dubinsky, 1991;Schoenfeld, 1992;Tall, 1995;Watson & Mason, 1998;Yusof & Tall, 1999;Gray & Tall, 2001;Mason, 2002;Rahman, 2009;Mason, Stacey, & Burton, 2010).
Mathematical thinking is an active process which improves students' understanding of highly complex activities such as specializing, conjecturing, and generalizing (Tall, 2002b;Yusof & Rahman, 2004;Stacey, 2006;Mason et al., 2010;Kashefi et al., 2012). According to Tall (2004b) and Tall (2008), mathematical thinking process occurs in three worlds of mathematics, such as the embodied world, the symbolic world, and the formal world. The embodied world of sensory meaning and action involve reflection on perception and action. The symbolic world contains computing and manipulating symbols in arithmetic and algebraic forms. The formal world is the world of axioms, formal definitions, and formal proof of theorems (Tall, 2002b). Therefore, in the teaching and learning of calculus specifically derivative and integral, the focus is on embodied and symbolic worlds (Tall, 2011).
Generalization as an important element of mathematical thinking process in problem solving can be supported to overcome students' difficulties in the learning of calculus especially derivative and integral (Polya, 1988;Cruz & Martinon, 1998;Larsen, 1999;Tall, 2002b;Tall, 2004b;Tall, 2011;Sriraman, 2004;Mason et al., 2010;Kabael, 2011). Tall (2002b) asserts that generalization strategies in mathematical thinking worlds are an expansive, reconstructive, and disjunctive generalization. First, expansive generalization happens when the notion expands the applicability range of an existing schema without reconstructing it. Second, reconstructive generalization occurs when the subject reconstructs an existing schema to widen its applicability range. Last, disjunctive generalization occurs when the subject moves from a familiar context to a new one. The subject constructs a new disjoint schema to deal with the new context and adds it to the array of schemas available (Harel & Tall, 1991). Mason et al. (2010) propose the three steps of problem-solving framework namely entry, attack, and review using mathematical thinking activities such as specializing, conjecturing, and generalization. Mason et al. (2010) believe that specialization and generalization as the main steps of mathematical thinking processes involve three problem-solving phases. Specialization involves entry and attack, and generalization covers attack and review. The key idea of their framework is a mulling circle between the entry and attack phases. Also, the activity of conjecturing has an important role in connecting specialization to generalization .
Tall believes that a majority of using generalization happens in the symbolic world as an expansive generalization and this kind of generalization is not able to make the connection between the graphical and symbolical world of mathematics for derivative and integral (Tall, 2002b;Tall, 2004b). Tall (2002b) and Tall (2011) believe that reconstructive generalization enables the making of relationship and connection between concepts of derivative and integral. It is because this kind of generalization changes the contracture of presentation, protects the meaning of concepts and transits it to the new world. Harel & Tall (1991) and Tall (2002b) state that disjunctive generalization also can protect the relationship and connection between concepts, but it has less effect as compared to expansive and reconstructive.
This study adopts a learning strategy of derivatives and integrals (LSDI) which is designed based on generalization strategies and mathematical thinking process through three worlds of mathematics to improve undergraduate students' problem solving of derivative and integral. Furthermore, the effectiveness of LSDI in enhancing students' problem solving is also being evaluated.

LEARNING STRATEGIES FOR DERIVATIVE AND INTEGRAL (LSDI)
This study uses the theory of three worlds of mathematics (Tall, 2004a;Tall, 2004b), viewpoint of Tall about generalization (Tall, 2002a), and the mathematical thinking process framework of Mason et al. (2010) in order to design strategies for learning derivative and integral (LSDI) to improve problem solving. In the designing of LSDI, the generalization process in the framework of Mason (specialization, conjecturing and generalization) was being modified along with the three generalization strategies namely; expansive, reconstructive and disjunctive (Tall, 2002b). Then, the modified generalization strategies were merged with three worlds of mathematics to design LSDI. Figure 1 represents the theory and the framework which were established in designing LSDI. Mathematical thinking worlds were selected because it covers both graphical and symbolical aspects of derivative and integral to overcome students' difficulties in problem-solving. Through mathematical thinking worlds, both graphical and symbolic aspects can be connected by reconstructive generalization. Moreover, expansive generalization can be supported in the embodied world to foster students' problem solving by the graphical aspect. Therefore, based on this study, mathematical thinking and generalization strategies should be considered in designing learning strategies for problem-solving. Scruggs & Mastropieri (1993) believe that learning strategies are established based on the use of tasks, and they involve how students structure and apply a collection of skills to learn content or to carry out a particular task more effectively and efficiently. Besides, Watson & Mason (1998) assert that learning strategies contain what we think such as planning, realizing and memorizing previous knowledge through doing the problem-solving process. According to Watson & Mason (1998) and Watson (2002), prompts and questions are more appropriate tasks use by teachers as guidance for developing mathematical thinking in the classroom within the problem-solving process.
The questions assist students in focussing on particular strategies and helping them to see patterns and relationships . These questions provide the establishment of a strong conceptual network. Consequently, the questions can be used as a prompt when students become 'stuck'. Teachers are often tempted to turn these questions into prompts to stimulate thinking and incorporate students in the problem-solving process activities (Watson & Mason, 1998;Watson & Mason, 2006;Mason et al., 2010). Therefore, referring to this study, the contents of designed learning Generalization (Tall, 2002b) strategies based on prompts and questions should cover generalization strategies, mathematical thinking process, and three worlds of mathematics. The learning tasks of derivative and integral were prepared based on the LSDI strategies through prompts and questions that are shown in Tables 1 and Table 2. Table  1 presents an example of the designed prompts and questions based on LSDI which was prepared for the derivative task. -What is the general rule for the derivative figure based on the original graph? -Can you show its general form graphically and algebraically?
-What is the same and the difference in this example? -What information do you need to write general forms of other topics, e.g. function and limit?
-Write general ideas for this example. -Classify the ideas.
-Give the general form and describe it? -When the average velocity gets close to absolute velocity? -Describe this example and the general form with the graph of this example.

Formal World
By using and , show that the function below are derivable: In Table 1, prompts and questions were used to design the derivative and integral tasks based on LSDI. These concepts were presented through different worlds of mathematics based on specialization, conjecturing and generalization as components of the mathematical thinking process. Moreover, the three generalization types were considered in the generalization phase of the mathematical thinking process of the problem-solving framework. Table 2 shows that the prompts and questions were designed as learning strategies of integral through mathematical thinking worlds based on specialization, conjecturing and generalization (involved three strategies namely; expansive, reconstructive and disjunctive). -Find the area between the graph and x-axes from = −1 to = 1 …(i) -Find the area between the graph and x-axes from = −2 to = 2 … (ii) -Compare (i)  -Please give a more general example. -Describe how you can connect this example to algebraic form. -Find the area between the graph and x-axes from = to = . -Give another example and find the area between the graph and xaxes from = to = .
-Give the general form of founded original function symbolically. -Connect the general form to its figure logically.

Formal World
Show that if is integrable then;

RESEARCH METHOD
Method of the study involves discussions on research design, sample, instruments, data collection and data analysis which are described sequentially.

Research Design
This experimental study tries to highlight the impacts of using LSDI especially generalization strategies on students' problem solving of derivative and integral. After comparing the scores between two groups of students as control and experimental, the rates of using generalization strategies due to their responds to open-ended problems were compared through pre and post-test of both groups.

Sample
Two classes were randomly selected among 12 classes which offered Calculus I at the University of Gachsaran in Iran. One of the two classes was selected randomly as an experimental group, and it involved 33 students. Another class acted as a control group and consisted of 30 students. Based on the results of pre-test, all students in both groups were in having the same level of problem-solving of derivative and integral. In the control group, derivative and integral were taught by using traditional and common methods used in the teaching of these concepts.
In contrast, derivative and integral taught in experimental class was based on designed strategies of LSDI. The duration of the teaching of derivative and integral in both groups was 8 weeks. A post-test was administered to both classes based on problem-solving of derivative and integral at the end of week 8.

Instrument
This study used two instruments for pre-test and post-test which were designed to understand students' problem-solving abilities of derivative and integral. Three experts who were familiar in this case verified the questions of problem-solving in this research. For pre-test, 9 problems (6 for derivative and 3 for integral) were given to the students, and 11 problems (6 for derivative and 5 for integral) were given in the posttest.

Data Analysis
The goal of the analysis was to see if LSDI affected students' scores and rates of using LSDI specially generalization strategies in the experimental group as compared to the control group. Students' scores were obtained based on their responses to the problems in the pre and post-tests. The students' responses were scored from 20based on Iranian scoring schema. The differences in the mean of those scores were compared within and between groups. This comparison was done with independent samples ttest because the data had the assumption of the t-test. Moreover, the Kappa test was used to see the agreements of students' scores in problem-solving between groups through a pre-test. After implementing LSDI in the experimental group and taking posttest, the two-mixed ANOVA design test and independent samples t-test were applied to highlight the improvement of students' scores in experimental group from the pre to post-test.
After scoring students' answer sheets and comparing them within each group and between two groups, the rates of using LSDI especially generalization strategies were compared within and between groups. A rubric was designed based on LSDI to compare the rates of its application specifically in the application of generalization strategies within and between groups. The rubric contained 13 items and was designed based on the blending of generalization strategies of Tall (2002b) and mathematical thinking process of Mason et al. (2010). The rubric and its items are presented in Table 3. Based on the rubric, the rates of using LSDI particular generalization strategies were investigated based on the students' responses to the open-ended questions. Students' answers to each question were checked to see if they had considered the items of LSDI specifically on a generalization or not. For each item, the score is either 0 or 1. If a student used that item, he or she would be given 1. A Student was given 0 if there was no consideration of the item in responding to that question. To illustrate, if a student considered E1 in his (her) response to the first problem, the given score was 1 for E1. This process was repeated for all of the items given in Table 3. Therefore, students' responses were checked to obtain the scores for the items. Finally, the given scores to all of the items were tabulated to compare within and between groups. The score for each component was a summation of its related items. For example, to know the score of specialization, the given scores of S1, S2 and S3 should be calculated. Therefore, the score for each student in responding to each problem can be from 0 to 3 for specialization. The scores in all components for each student were calculated based on their items and analyzed quantitatively within and between groups.

Findings of Students' Scores
The results of comparing pre-test scores indicated there was no difference between the mean of student' scores in the control group and experimental group based on independent samples t-test results. For more confidence about balancing of two groups, in the beginning, the Kappa test was done based on students' scores.
According to the results such as Value=0.786 and Approx. Sig.= 0.001, there were good agreements between students' scores of pre-test in problem-solving of derivative and integral in both groups. Pallant (2010) asserts that if the value of Kappa is bigger than 0.70, there is a good agreement between the data of different groups. Therefore, LSDI could be implemented in an experimental group to see its effects on this group by taking post-test at the end of the teaching process for the concepts.
The data which were collected from students' scores in problem-solving of derivative and integral were normal in the post-test. Also, the Levene's test indicated that there are significant differences between the variances of two groups based on collected data in post-test. Therefore, the parametric tests such as two-mixed ANOVA design and independent samples t-test could be used for comparing the students' scores between and within the groups through pre and post-test. The results of comparing the scores are presented in Table 4.

Table 4. Comparison of students' scores in pre-test and post-tests for PS of derivative and integral
The output of two-mixed ANOVA test indicates that students' scores between two groups and changing the scores from pre-test to post-test within groups were statistically significant (p= 0.001). Also, the results of independent samples t-test showed that there was no significant difference for students' scores of pre-test between the control group and the experimental group. Although the mean of scores in the control group was different from the mean of scores in the experimental group in the pre-test, the difference was not significant. However, independent samples t-test showed that there was a significant difference (with p= 0.001) between the mean of students' scores of post-test in the control group and the mean of students' scores of post-test in the experimental group. After analyzing students' scores, the differences in the rates of using LSDI especially generalization strategies were analyzed within and between groups through pre and post-test based on investigating students' responses to the problems.

Rates of Using LSDI in Solving of Derivative and Integral Problems
The components of LSDI involving the framework of Mason and the generalization strategies of Tall based on mathematical thinking such as specialization, conjecturing and generalization strategies (expansive, reconstructive and disjunctive) were investigated in the pre-test and post-test within and between groups. The data for the rates of applying LSDI in solving problems based on students' responses to the problems of derivative and integral were not normal data. Thus, non-parametric tests should be used to see the differences in utilization rate for LSDI within and between control and experimental groups through the pre and post-tests.
The rates of using generalization as the main part of LSDI in pre-test (PrG) and rate of its utilization in post-test (PoG) were compared within groups. The PrG and PoG are the summations of utilization rates of three generalization strategies in the pre and post-tests. Moreover, the differences in using each component of LSDI between the pretest and post-test were demonstrated within the group. Also, the rates of applying each component of LSDI were also compared between groups for the pre-test and post-test. To indicate differences within the group, the Wilcoxon test was used for non-parametric data, and to see differences between two groups Mann-Whitney test was administered (Pallant, 2010;Kirkpatrick & Feeney, 2012). In Table 5, the results of the comparison of LSDI components specifically generalization are given within each group. It should be noted that Pr is an abbreviation of pre-test and Po is an abbreviation of post-test, the first letter of each component added to Pr and Po. For examples in Tables 5 and 6, PrS means specialization in the pre-test, PoS means specialization in post-test. The results of the Wilcoxon test demonstrate that there is no significant difference in the rate of using components of LSDI especially generalization in the control group between pre-test and post-test. The results of this test such as Wilcoxon N=30and p>0.05 show no difference in the rates of applying generalization strategies between the pre and post-tests. However, the results of the Wilcoxon test in the experimental group show different postures of using items of LSDI. In this group, Wilcoxon N=33and p<0.05 indicate that there significant differences between the pre-test and post-test in the rates of using LSDI in the problem solving of derivative and integral within the experimental group. Furthermore, the rate of using items of LSDI was measured between two groups in the pre-test and post-test by using the Mann-Whitney U test. Also, the rates of using generalization strategies were compared between control and experimental groups as presented in Table 6. According to the results which are presented in Table 6, the Mann-Whitney U-Test reveals that the rates of using LSDI especially generalization strategies from control group are not significantly different from the experimental group in the pre-test. The results such as U, Zand two-tailed p>0.05 approve this assertion through the pretest in problem-solving of derivative and integral. Meanwhile, the Mann-Whitney Utest reveals that the rates of all components of LSDI specifically in the experimental group are significantly higher than the rates from the control group in the post-test (U, Zand two-tailed p<0.05). Furthermore, the mean of ranks from each group in different tests verifies this result.

Discussion
The results of comparing scores and rates of using generalization strategies are discussed to see if LSDI affected students' scores and their utilization rates of generalization strategies based on the responses to the problems of derivative and integral.
Although there is no significant difference between the mean of students' scores in control group with the mean of scores in the experimental group for the problems solving of derivative and integral in the pre-test, the scenario in responding to the posttest problems is different. The results indicate that the students' scores in the experimental group are better than the students' scores in the control group through solving post-test problems of derivative and integral. It indicates that the scores of students who learned derivative and integral based on LSDI especially generalization strategies were in the high range of qualification according to Iranian scoring rate at the undergraduate level. Also, the two-mixed ANOVA design test indicates that the improvements of students' scores within each group from the pre-test to post-test are dissimilar in different groups.
Based on the results, the improvement of scores from the pre-test to post-test for experimental group students is higher than the improvement of scores in the control group students from the pre-test to post-test in solving problems of derivative and integral. It can be identified that the achievements of the experimental group students who used LSDI especially generalization strategies were better as compared to the other group. Thus, to know the role of LSDI particularly generalization strategies in improving students' scores in problem-solving, the components of LSDI were investigated based on the responses to the problems of derivative and integral. Generalization strategies as main components of LSDI in problem-solving of derivative and integral were used remarkably in solving the post-test problems among experimental group students. Although students did not display the utilizations of generalization strategies in solving the pre-test problems, they tried to use the strategies many times through solving the post-test problems. Hence, generalization strategies such as expansive, reconstructive and disjunctive were applied in answering questions of problem-solving of derivative and integral. It should be mentioned that specialization and conjecturing as two fundamental bases for generalization were better considered in the post-test than the pre-test among students in the experimental group. However, in the control group, there was no difference in the utilization of specialization, conjecturing and generalization strategies when responding to the pretest and post-test of derivative and integral.
The control group students did not show applications of generalization strategies in responding to the pre and post-tests. There is no difference in the utilization rates of generalization strategies such as expansive, reconstructive and disjunctive and their bases such as specialization and conjecturing in answering the questions of problemsolving of derivative and integral between the pre-test and post-test in this group. However, the posture of the utilization rate for generalization strategies is different between the two groups.
Although the rates of using specialization, conjecturing and generalization strategies are not meaningfully different between two groups in solving the pre-test problems of derivative and integral, there are significant differences of using these strategies between groups in solving the problem of derivative and integral in the posttest. Based on the rubric and students' responses in their answer sheets, the experimental group students used more examples which were related to the problems when solving them. They tried to find the similar properties and ideas of examples in the entry phase of the problem solving framework. Also, the experimental group students categorized the ideas according to the properties of examples through the attack phase. Moreover, the students attempted to give a general guess to solve the problems based on related examples. Besides, they formulated the ideas based on the examples within the attack phase of Mason's problem-solving framework. They also chose the best method to find the answer for problems of derivative and integral within the attack phase.
The expansive generalization was used by the experimental group students extensively when responding to the post-test. The students found answers to some problems by using more related examples of those problems which belonged to the attack phase of problem-solving. The students in the experimental group tried to check the written solutions for problems in the same cases using review phase. However, control group students did not demonstrate these kinds of solutions when solving problems.
Furthermore, the experimental group students solved problems by generating ideas in both symbolic and embodied worlds of mathematics using properties of reconstructive generalization in the attack phase. Subsequently, they created new IJEME, Vol. 3, No. 1, March 2019, 77-92. 88 schemas which were based on applying the properties of the review phase of the problem solving framework. Also, in a few cases, the students attempted to extend the solution idea to higher level problems. In contrast, usages of these factors were not apparent in solving problems of derivative and integral among the control group students.
The utilization of disjunctive generalization was rated less as compared to other generalization strategies when solving problems by the experimental group. Some students tried to find answers by using familiar contexts. However, the control group students found the answers by using disconnected pieces of information, and also they generated the ideas of the solution in the wrong way in the same cases.
Based on the framework of problem-solving, three phases namely entry, attack and review was well considered among students in the experimental group. It can be concluded that using components of LSDI specifically generalization strategies has remarkable effectiveness in improving students' scores in the experimental group as compared to the control group students.

CONCLUSION
LSDI was implemented based on strategies such as presenting concepts within three worlds, using mathematical thinking process, prompts and questions to teach derivative and integral in the experimental group. Using components of LSDI namely; specialization, conjecturing, expansive generalization, reconstructive generalization and disjunctive generalization based on mathematical thinking worlds play an important role to enhance problem-solving of derivative and integral.
The results indicate that there is an improvement in the problem-solving scores of students who experienced strategies of LSDI when learning derivative and integral. The improvement of scores in the experimental group is better than the control group students' scores when comparing the pre-test and post-test. The components of LSDI are considered remarkable in solving problems among the experimental group students.
Specialization and generalization as main activities of mathematical thinking process were used in solving the problems of derivative and integral in the experimental group. Also, specialization involves two phases of problem-solving framework; entry and attack, and generalization cover attack and review. The utilization of problemsolving framework (entry, attack and review) among the experimental group students were more remarkable. Thus, using LSDI especially generalization strategies based on three worlds of mathematics improves students' problem-solving achievements in the learning of derivative and integral. Moreover, adopting a problem solving framework which involves entry, attack and review help to improve students' performance.
This study recommends evaluating the effectiveness of LSDI on other concepts of calculus for further research. Besides, investigating the process of using LSDI based on students' thinking can be used for future studies. Teachers should put more emphasis on the use of LSDI to teach calculus concepts, especially derivative and integral to enhance their students' performance.