Designing Learning Strategy to Improve Undergraduate Students ’ Problem Solving in Derivatives and Integrals : A Conceptual Framework

Derivatives and integrals are two important concepts of calculus which are precondition topics for most of mathematics courses and other courses in different fields of studies. A majority of students at the undergraduate level have to master derivatives and integrals if they want to be successful in their studies However, students encounter difficulties in the learning of derivatives and integrals. Most of these difficulties arise from the students’ weakness in problem solving. This paper presents a learning strategy which has been designed to overcome these difficulties based on mathematical thinking and generalization strategies with prompts and questions.


INTRODUCTION
Calculus is an important subject since it exists in most of university courses such as; economy, engineering, statistics, science, and all mathematical courses like; analysis, numerical analysis, statistic, differential equation and operation research (Tall, 1992(Tall, , 1993(Tall, , 1997(Tall, , 2010a;;Tall and Yudariah, 1995;Karamzadeh, 2000;Tarmizi, 2010).According to Tall (2012), imagining university courses without calculus is unfeasible.Therefore, derivatives and integrals are two important topics in university mathematics which are prerequisites in order to learn other concepts in the different fields of studies (Tall, 1993(Tall, , 1997(Tall, , 2004a(Tall, , 2011)).
Many university courses depend on the knowledge on derivatives and integrals and also their applications (Tall, 2004a(Tall, , 2010b(Tall, , 2011;;Metaxas, 2007;Pepper et al., 2012).In fact, traces of derivatives and integrals are visible in the advanced mathematics and even other subjects.Therefore, learning derivatives and the integrals can be helpful and useful for students in order for them to learn other mathematical courses at university level (Tall, 2010a(Tall, , 2011;;Tarmizi, 2010).However, there are some obstacles in the learning of calculus and its concepts especially derivatives and integrals (Tarmizi, 2010;Tall, 2010aTall, , 2012;;Pepper et al., 2012).Derivatives and integrals are two difficult concepts of mathematics for many undergraduate students.The difficulties in learning derivatives and integrals among undergraduate students are due to their weakness in solving problems involving these concepts (Tall, 1993(Tall, , 1997(Tall, , 2011;;Willcox and Bounova, 2004;Yazdanfar, 2006;Metaxas, 2007;Roknabadi, 2007;Tarmizi, 2010;Rubio and Chacon, 2011;Pepper et al, 2012;Azarang, 2012).
Many researchers (Tall, 1992(Tall, , 1997(Tall, , 2012;;Stacey, 2006;Metaxas, 2007) have noted that students possess difficulties in the learning of derivatives and integrals concepts, because the teachers and students focus on symbolic aspect rather than graphical.Moreover, the inability to make connection and relation between graphical aspect and symbolical aspect is another reason for the weakness (Tall, 1997(Tall, , 2012;;Metaxes, 2007;Shahshahani, 2012).The need for a heuristic and appropriate framework or plan to solve problems and weakness in using previous knowledge and information in new areas are other reasons for the difficulties.(Polya, 1988;Tall and Yudariah, 1995;Tall, 2001Tall, , 2004aTall, , 2007;;Kirkley, 2003;Villers and Garner, 2008;Mason, 2010;Tarmizi, 2010).
Mathematical thinking is related to improving generalization in the learning of mathematics (Watson and Mason, 2006;Mason et al, 2010;Tall, 2012).Tall (2002a) asserts that generalization strategies in mathematical thinking worlds are expansive, reconstructive and disjunctive generalization.In fact, generalization is an important element of the mathematical thinking process and problem solving methods.It can be used to support students to overcome their difficulties in the learning of calculus especially derivatives and integrals (Polya, 1988;Cruz and Martinon1998;Larsen, 1999;Karamzadeh, 2000;Tall, 2002bTall, , 2004b;;Sriraman, 2004;Mason et al, 2010;Kabael, 2011).

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Considering the mathematical thinking activities, Watson and Mason (1998) have developed various general questions and prompts which can be used to motivate the development of Mathematical sense among students.The prompts and questions offered examples of some generated questions which are used as a way to know how suitable questions and structure should be chosen (Roselainy, 2008).The main delivery methods for applying prompts and questions through mathematical thinking are expounding and explaining.In addition, effective learning can be achieved through other interaction styles and six important interaction modes (Mason, 1999(Mason, , 2002;;Roselainy, 2008) and thus they should be used for effective teaching based on prompts and questions.
This study is designed based on specialized and modified forms of generalization strategies and mathematical thinking to support undergraduates in improving their problem solving in the learning of derivative and integral.Prompts and question are suitable methods to improve the learning strategy.
The lack of methods to make strong connections between graphical aspect and symbolic aspect is seen among undergraduates yet.Therefore, there is a necessity to use the properties of graphical aspect in teaching and learning of derivatives and integrals among undergraduate students.In addition to these reasons for problem solving, the absence of problem solving plan is another important difficulty among students in the learning of derivatives and integrals (Polya, 1988;Yudariah and Tall, 1995;Tall, 2001Tall, , 2004aTall, , 2007;;Kirkley, 2003;Villers and Garner, 2008;Parhizgar, 2008;Mason, 2010;Tarmizi, 2010;Azarang, 2012;Ghanbari, 2012).

Figure 1. Difficulties in solving problems of derivatives and integrals
Lack of problem solving framework or pattern and the inability to recall previous knowledge in new situations are other causes of the difficulties faced by students.Hence, students need to be aware of problem solving framework in order to follow the process of solving problems of derivatives and integrals.Most undergraduate students fail to get the answers because they are clueless on what should be done.Therefore, there is a strong requirement to introduce and use a problem solving plan in the learning and teaching of derivatives and integrals.The above weaknesses are highlighted in Figure 1.Recalling previous knowledge is difficult for students due to some factors such as such as they are imagining concepts in separate areas, they are unable to make connection between earlier and later information, the inability to use suitable method and strategy to recall information and so on.Figure 1 shows more information about the difficulties in problem solving.

Appropriate Methods to Overcome Students' Difficulties
Based on the points in the previous heading, there is a strong necessity to find suitable strategy to improve students' problem solving abilities within the learning of derivatives and integrals.Mathematical thinking (Tall, 2004a;Mason et al, 2010) and generalization (Tall, 2002a) have been proposed to rectify these difficulties in learning of derivatives and integrals.Generalization strategies such as expansive and reconstructive allow connection to be made between graphical aspect and symbolic aspect.Generalization has the potential to increase focus on graphical aspect, because the expansive as a strategy of generalization can be used in graphical aspect to work more on that aspect (Tall, 2008;Villiers and Garner, 2008).
Mathematical thinking is a suitable method to help people to use generalization strategies (Tall, 2004a;Mason et al, 2010), because using generalization can conduct students to avoid using disjunctive strategy to use expansive and reconstructive strategies in graphical and symbolical worlds of mathematical thinking (Tall, 2002a).Generalization can be used to recall previous knowledge and making connections between concepts (Polya, 1982;Karamzadeh, 2000;Stacey, 2006), because generalization involves making connection between previous concepts and news.
Mathematical thinking is a suitable method to help people to use generalization (Tall, 2004a;Mason et al, 2010).Three worlds of mathematics (Tall, 2008) and mathematical thinking process (Mason et al, 2010) can be helpful and beneficial to overcome students' difficulties in problem solving.Mathematical thinking worlds cover both graphical and symbolic aspects of derivatives and integrals through embodied and symbolic worlds (Tall, 2012).Students can use the properties of these worlds to make connection between them to rectify the mentioned difficulties.Besides, mathematical thinking process (Watson 2002;Mason et al, 2010) such as specializing and generalization can be used as a problem solving framework.More details can be found in Table below.

Designing Learning Strategy
Mathematical thinking can be a useful method to reduce students' difficulties in the learning of derivatives and integrals, because it can cover both graphical and symbolic aspects.Moreover, it can support generalization to help students in recalling previous knowledge in the problem solving process.Therefore, mathematical thinking worlds and generalization strategies need to be introduced more when designing a learning strategy.

Mathematical Thinking Worlds
Tall and Watson compare the embodied and symbol at university level with formal view or formal approach and introduce it as an axiom (Watson, 2000;Tall, 2004aTall, , 2004bTall, , 2012;;Stewart, 2008).They emphasize that those three points of view (geometric, symbolic and axiomatic) cannot be considered as mathematical concepts.However, they are three different cognitive developments which happen in mathematical thinking in three separate worlds.
In the development of mathematical thinking theory, Tall has studied and carefully used a majority of mathematical topics (Tall, 2004b).Tall (2004b) establishes the theory of mathematical thinking worlds namely embodied world, symbolic world and formal world; he also invents the word of procept which is a combination of the word process and concept in symbolic world and this word is important for conceptual understanding.The word of world has been chosen to emphasize distinct ways of thinking about thinkable concepts in mathematics (Tall, 2005;Mejia-Romas, 2006).
Table 2 shows the postures of derivative and integral through three worlds of mathematical thinking.Derivative and integral and their properties are shown with graphs and histograms in the embodied world of mathematical thinking (Stewart, 2008;Tall, 2008Tall, , 2012)).In the symbolic worlds, the postures of derivatives and integrals are the symbols of them.These symbols can be their general forms and properties which are shown with numbers and algebraic letters.Although, mathematical thinking involves the formal world, the main emphasis in the learning calculus is on embodied world and symbolic world and its concepts such as derivatives and integrals (Tall, 2004a(Tall, , 2008(Tall, , 2012)).
Using generalization strategies is another useful method to improve students' problem solving.Generalization can make a connection between graphical and symbolic aspects (Tall, 2008;Kashefi et al, 2013).It can be a useful method for recalling previous knowledge in problem solving activities (Polya, 1988;Karamzadeh, 2000).Tall (2002aTall ( , 2004bTall ( , 2012) ) asserts that mathematical thinking can support generalization in mathematics.Generalization strategies are used in mathematics to show processes at broader contexts.It can help problem solvers to know about the products of those processes (Tall, 2002a).According to Tall the components of generalization can appear in three types; expansive, reconstruction and disjunctive.

Generalization and Its Strategies
First of all, expansive generalization involves extending the existing information of learners without any change of their previous ideas.On the other hand, in expansive generalization; the new information should be similar to the current information in the same area.For example, in learning vector space firstly students are taught on R and R 2 then educators will add another component to plane (R 2 ) and introduce apace (R 3 ).
Furthermore, when a person extends a concept by changing his or her previous ideas or knowledge, (s)he  (Harel and Tall, 1991;Tall, 2008).According to Tall (2002aTall ( , 2008)), this generalization can be used as an ideal generalization to make a connection between embodied and symbolic aspects to overcome students' difficulties in recalling previous knowledge in new situations (Polya, 1982;Karamzadeh, 2000).
Next, disjunctive generating problems can be solved and analyzed in advanced knowledge, but this generalization has less effects on students learning in comparison to expansive and reconstructive (Tall, 2002a).In order to use this kind of generalization, students solve new problems by adding numbers of disconnected pieces of information illogically.Mason and colleagues (2010) have tried to correct this generalization in the third stage (review) of their framework.Using disjunctive generalization leads students to solve problems routinely instead of applying suitable problem solving framework (Tarmizi, 2010).
To sum up, generalization from Tall's viewpoint has three components such as; expansive, reconstruction and disjunctive.The relationship between the components of generalization according to Tall's perspective on generalization is shown in Figure 2 (Harel and Tall, 1991;Tall, 2002).
Consideration on generalization (Tall, 2002a) is important in the teaching of mathematical concepts such as derivative and integral.Besides, generalization in the problem solving process such as specialization, conjecturing and generalization (Mason, Burton and Stacey, 2010) are useful in overcoming students' difficulties in the learning of derivative and integral (Roselainy, 2008;Kashefi et al, 2013).Hence, specialization and conjecturing are foundations to achieve generalization.
Specialization in the teaching of mathematics can be used in various cases or examples (Roselainy, 2008;Mason et al, 2010;Mason, 2012;Kashefi et al, 2013).In other words, specialization means referring to more examples in order to learn the concepts or solve the problems.These examples are specific and particular instances of more general situations in a concept (Mason et al, 2010).According to Mason, Burton and Stacey (2010) if specialization happens successfully via a useful conjecturing, it can be helpful in making Generalization is the main process after specialization.When students face new concepts or new problems after specializing and conjecturing, formulation is formed in their minds.It is the beginning point in utilizing generalization (Mason and colleagues, 2010).Generalization activity can be extended to previous knowledge or to new related concepts (see Figure 3).

Embedding Mathematical Thinking Worlds and Generalization Strategies
Based on the results of Table 1, mathematical thinking and generalization strategies can be useful methods to overcome students' difficulties in the learning of derivatives and integrals.It is important to possess the quality of combining and blending these methods together to design a suitable learning strategy for derivatives and integrals.
The generalization strategies (Tall, 2002a) have been integrated with generalization process (Mason et al, 2010) to establish modified generalization strategies.The modified generalization is a combination of generalizations.The proposed framework of learning strategy in derivatives and integrals contains the theory of mathematical thinking (Tall, 2004b), the perspective of Tall about generalization (Tall, 2002a), and the framework of Mason, Burton and Stacey (2010)  Mason generalization process with two preliminary processes (specialization and conjectured) has been modified by embedding Tall's three generalization strategies.Generalization can be achieved in Mason's process by using at least one strategy of Tall's.In other words, the problem solving framework of Mason is being improved when the generalization process is modified.Although specialization and conjecturing belong to the design strategy, the main focus is on generalization process that has been modified.Thus, specialization and conjecturing are pre-processes of generalization.The modified generalization strategies was then mapped onto the Tall's three worlds of mathematical thinking (embodied, symbolic and formal) to form MGSDI.
Figure 5 provides more details of the learning strategy.MGSDI is designed by considering problem solving, mathematical thinking and generalization.The quality of linkage between generalization strategies and mathematical thinking is illustrated in Figure 6.In addition, the relations between problem solving with mathematical thinking worlds are also shown.
MGSDI is proposed to rectify difficulties in the learning of these concepts.After designing MGSDI, it can be improved through dynamic activities.Based on

Designing and Developing MGSDI through Prompts And Questions
Prompts and questions are suitable methods to design strategies in mathematical thinking approach (Roselainy, 2008;Mason et al, 2010;Kashefi et al, 2013).Watson and Mason (1998) and Watson (2002) assert that prompts and questions can be used by teachers as guidance for developing mathematical thinking in the classroom within problem solving process.
Questions help students to focus on particular strategies and to see patterns and relationships (Mason et al, 2010).It will build the foundation of a strong perceptual network.In addition, questions can be used to prompt students when they become "stuck".Teachers are often tempted to turn these questions into prompts to encourage thinking and incorporate students' problem solving activities (Watson andMason, 1998, 2006;Mason et al, 2010).Therefore, this study proposes that the contents of design strategies which depend on prompts and questions should cover generalization strategies, mathematical thinking process as problem solving framework and three worlds of mathematics.
Therefore, prompts and questions have been applied to improve the designed learning strategy (MGSDI).Table 3 presents the posture of appropriate prompts and questions based on mathematical thinking worlds.
The postures of using prompts and question should be shown in different mathematical worlds and different activities of mathematical thinking.It means that the prompts and questions activities are designed to develop

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MGSDI in embodied and symbolic worlds of mathematical thinking.In addition, for each concept such as derivatives and integrals the activities of prompts and questions are designed through different worlds.
Table 4 presents prompts and questions which can be used for derivatives in embodied world.Students are guided to transit from embodied world to symbolic world.It should be noticed that Tall (2004aTall ( , 2008) ) believes that formal world of mathematical thinking is not applied for calculus.However, it can be used in mathematic analysis which is at a higher level than calculus.
Another example of derivatives using prompts and questions in the symbolic world is offered in Table 5.Most researchers (Tall, 1997;2011;Roselainy, 2008, Kashefi et al, 2013) mention that derivative and integral are related with each other in calculus.According to Roselainy ( 2008), prompts and question can be used to teach integrals.Teaching integral in the embodied world via prompts and question is shown in Table 6.
The activities in Table 6 introduce and highlight the meaning and properties of integrals.The table shows another example of Integrals which is offered in the embodied world with specialization and generalization.
In addition, Table 7 illustrates the infinite integrals in the symbolic world based on specialization and generalization by using prompts and questions.This table also shows some activities to recall information from derivatives in integrals.
Table 8 shows the finite integrals in the symbolic world and some activities to make the connection between embodied and symbolic worlds.

CONCLUSION
Mathematical thinking by using both graphical and symbolic aspects of derivatives and integrals is an important method that can support students.Moreover, mathematical thinking process such as specialization, conjecturing and generalization can be a suitable framework in problem solving of derivatives and integrals.Using generalization strategies based on mathematical thinking can make the connection between these aspects.Thus, it seems that students' difficulties can be rectified by using mathematical thinking and generalization.
Different learning strategies can be designed to help students in the learning of derivatives and integrals based on generalization strategies.The MGSDI can help students to rectify their difficulties by modifying generalization strategies and combining them with three worlds of mathematics in the learning of these topics.
This study has presented the quality of using prompts and questions for MGSDI.Appropriate prompts and questions have been introduced for derivatives and integrals in embodied and symbolic worlds of mathematical thinking.Also, the activities of prompts and questions highlight specialization and generalization and their relationships through mathematical thinking worlds.Many researchers such as Yudariah (1995), Watson andMason (2006), Roselainy (2008) have attempted to use mathematical thinking process for improving students' difficulties, but they did not use mathematical thinking worlds.They only used the framework of Mason and his colleagues (2010).However, one of the students' difficulties is the inability to connect embodied world and symbolic world.In addition, most researchers have used generalization in the learning of derivatives and integrals, but they used generalization strategies in symbolic world for problem solving (Karamzadeh, 2000;Tall, 2002a;Yudariah and Roselainy, 2004;Stacey, 2006;Roselainy 2008;Kashefi et al, 2013).Moreover, this study has proposed a modified generalization strategies for derivatives and integrals (MGSDI) to overcome the difficulties of problem solving through learning these concepts.It should be emphasized that the MGSDI supported the previous studies (Tall, 1992(Tall, , 1997(Tall, , 2011(Tall, , 2012;;Willcox andBounova, 2004Stacey, 2006;Yazdanfar, 2006;Metaxas, 2007;Roknabadi, 2007;Metaxas, 2007;Javadi, 2008;Tarmizi, 2010;Rubio and Chacon, 2011;Pepper et al, 2012;Ghanbari, 2012).
Therefore, MGSDI is recommended to be used in real classrooms.This study also suggests that researchers have to design and develop suitable activities based on MGSDI to be implemented in the teaching process of derivative and integral.In addition, the impact of the MGSDI can be evaluated on students' problem solving.

Figure 3 .Figure 4 .
Figure 3. Generalization's process in using generalization.Figure 4 illustrates the design strategy.This study has designed learning strategies called Modified Generalization Strategies in derivatives and integrals (MGSDI) specifically to support undergraduates in improving their problem solving skills in calculus topics.Modified generalization strategies which consist of a series of complex learning activities are first formulated based on Tall's three components of generalization (expansive, reconstructive and disjunctive) and Mason's generalization process namely specialization, conjecturing and generalization.

Figure 5 .
Figure 5.The detail strategy of MGSDI

Table 5 .
Presenting derivatives with prompts and questions based on mathematical thinking Topicsimilarity and difference in this example?b-Give exact rate for t. c-When the rate of t can be the best and what is the exact solution for velocity?d-Write the general form for this example.(what do you mean.e-What information do you need to write general forms from other topics such as function and limit?f-Give the general form of deprivation and describe it.g-Can you describe this example and the general form with the curve or graph of this example?Table 6.Presenting integral with prompts and question based on mathematical thinking Topicthe area between the graph and x-axes from x=-2 to x=2.ii) Find the answer in several ways.iii) Describe how can find the exact solution of area in this example.iv) Please give another example.v) Please give a more general example.a-Compare examples 1 and 2. b-What are the similarities and differences?c-What is the main property for both of them?d-What information in example 2 do you need to solve it?e-Sketch examples 1 and 2 in one coordinate axes and find the area between them and x-axes from x= -2 to x= 2. f-How can you relate the solution of these examples (1and 2) to algebraic aspect?g-Describe how you can connect this example to algebraic form.Table 7. Prompts and question for infinite integral in symbolic world Topicto find the original function from derivative function.d-Give the general form of founded original function symbolically.

Table 1 .
Students' difficulties in learning derivatives and integrals, the reasons and proposed solutions Students

Table 2 .
Posture of derivatives and integrals in mathematical thinking worlds reconstructs his or her cognitive area of knowledge.This change is named reconstruction generalization.To illustrate, in transitions from figure to symbol (or vise versa), this kind of generalization is happened

Table 3 .
Suitable questions for the strategy based on mathematical thinking

Table 4 .
Derivatives in embodied world with prompts and questions

Table 8 .
Presenting finite integrals with prompts and question based on mathematical thinking Topic