Computing in Differential Equations with Mathematical Thinking Approach among Engineering

Students Zaleha binti Ismail Faculty of Education Universiti Teknologi Malaysia Johor, Malaysia p-zaleha@utm.my

Fereshteh Zeynivandnezhad Faculty of Education Universiti Teknologi Malaysia Johor, Malaysia f_zeynivand@yahoo.com

Yudariah binti Mohammad Yusof Faculty of Education Universiti Teknologi Malaysia Johor, Malaysia Yudariah@utm.my

Eric David Weber College of Education Oregon State University weberer@onid.orst.edu

Abstract— Differential equations are indispensible for

engineers as they help to produce models which to study various phenomena. Research has shown that students experience difficulty in interpreting graphical solutions of differential equations and symbolic solutions. Therefore, mathematical thinking powers should be developed in differential equations in classrooms as the current situation in differential equations is concerned with procedural knowledge such as finding symbolic solutions with an emphasis on conceptual knowledge rather than procedural knowledge. However, very few studies have investigated how engineers use differential equations in real situations. This study reports the computation of real life problems, applying differential equations. Data was collected through in-depth interview with seven engineering students from chemical engineering faculty at a public university in Malaysia. Findings showed that to solve real life problems, students needed both procedural and conceptual knowledge of differential equations. Moreover, they used software packages, specifically a computer algebra system, to compute the problems and to interpret the solutions using their mathematical thinking powers such as specializing, generalizing, conjecturing, and convincing. The findings may lead to an examination of the current situation in differential equation classrooms to ensure a balance between procedural, conceptual, and technological knowledge with mathematical thinking approach.

Keywords—Mathematical thinking, Differential equations, Procedural knowledge, and Conceptual knowledge

I. INTRODUCTION The majority of mathematical phenomena in engineering

courses are modeled by using differential equations[1]. Equation f(x)= dy/dx is a simple kind of differential equation and function I(x) is a solution that I'(x)=f(x). I(x) is an anti-derivative which is calculated by reversing the differentiation process. The general solution of equation 1

2sec 2 1dyy x ydx

= − (1)

can be obtained by separating the variables and then integrating them in the form of

2ln 1 sin 2y x c− − = + (2)

But what does this solution mean? If a student changes c, every solution of the differential equation can be found. According to Tall [2], one has lost the two solutions of differential equations y=1, y=-1. Visualization can help here. Fig. 1 shows the graph of the general solution of the equation (1).

The current situation of differential equations involves the blind manipulation of formulae to obtain the solution. In most cases, the teaching of differential equations is done in a very procedural manner [3]. The arrival of the computer has given us the opportunity to take a fresh look at the theory and to get a clearer insight into some fundamental ideas. Using simple numerical methods allows us to show visually how the theory works as well as their difficulties under various circumstances[2]. Using simple specific cases, educators try to make the theory easier for students to understand rather than providing them with an in-depth, technical analysis. The development of mathematical thinking is more than just adding new experiences to the structure of fixed knowledge. It is a continuous reconstruction of mental connections that have evolved to enhance the structure of sophisticated knowledge over time[4].

Fig. 1. The graph of the general solution of equation (1)

2014 International Conference on Teaching and Learning in Computing and Engineering

978-1-4799-3592-5/14 $31.00 © 2014 IEEE

DOI 10.1109/LaTiCE.2014.39

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II. CONCEPTUAL AND PROCEDURAL KNOWLEDGE IN MATHEMATICS

Most research tries to focus on conceptual knowledge, although currently the emphasis is on procedural knowledge. But what is the appropriate focus, conceptual or procedural knowledge? Only a few studies have been conducted to investigate how engineering students use conceptual and procedural knowledge in real situations. As mentioned above, the use of technology in differential equations is inevitable. However, few studies investigate the ways that technology affects the relation between procedural and conceptual knowledge [5]. This study reports how chemical engineering students use procedural, conceptual, and technical knowledge in differential equations.

Separating conceptual and procedural knowledge is not always possible. Conceptual knowledge is referred to as the knowledge of concepts, either implicitly or explicitly [2]. This kind of knowledge is sometimes called conceptual understanding. Conceptual knowledge demonstrates understanding such as interpreting and applying mathematics concepts to a situation, translating between mathematical, verbal, visual, and formal expressions and linking relationships. Procedural knowledge means manipulating mathematical skills including calculations, formulae, algorithms and symbols [6]. A procedure is defined as a series of steps which must be taken to achieve a goal. Procedural knowledge is referred to knowledge of procedures. Procedures, according to Paraskakis, have been characterized through constructs such as skills, strategies, productions, and internalized actions [3]. In this study, the definition was provided by Engelbrecht et al [7] is considered as conceptual and procedural knowledge.

Measuring conceptual and procedural knowledge is critical evidence used to interpret the relationship between conceptual and procedural knowledge. The first one has been assessed in several different ways; however, procedural knowledge has been measured in fewer ways. Measuring conceptual knowledge depends on whether tasks require knowledge of implicit or explicit concepts. Evaluating implicit knowledge on tasks in which students make a categorical choice, for instance the correctness of an answer or procedure or making a quality rating such as rating a procedure as very smart or not-so-smart. Moreover, implicit measures involve translating between representational formats such as symbolic fractions into pie charts and comparing quantities. Explicit measures of conceptual knowledge include providing definitions and explanations. For example, explanations as to why procedures work or drawing a concept map [6].Variation in the measurement of procedural knowledge is much less than that of conceptual knowledge. The final measures are accuracy of the answers or procedures. Procedural tasks includes those problems that have been solved before, therefore, people should know solution procedures. However, sometimes the tasks consist of transfer problems which are unfamiliar problems that need either the recognition of a relevant known procedure or small adoptions of a known procedure to accommodate the unfamiliar problem [8].

The development of mathematics concepts and procedural knowledge has been made, although a model of their relationship is still primitive due to the consideration of components such as instructional methods, learning environments, and differences in mathematics topics.

III. MATHEMATICAL THINKING According to Tall [9], mathematical thinking develops in

three different ways including conceptual embodiment, operational symbolism, and axiomatic formalism. Table I shows the relationships between these three worlds.

TABLE I. THREE WORLDS OF MATHEMATICS [10:6]

Worlds of mathematics description

Conceptual embodiment

Builds on human perceptions and actions by developing mental images that are verbalized in increasingly sophisticated ways and become perfect mental entities in our imagination

Operational symbolism

Grows out of physical actions into mathematical procedures. While some learners may remain at a procedural level, others may conceive the symbols flexibly as operations to perform and also to be operated on through calculation and manipulation.

Axiomatic formalism

Builds formal knowledge in axiomatic systems specified by set-theoretic definition, whose properties are deduced by mathematical proof

Students do not cope with moving into the three worlds of mathematics in the same manner. For example, some operate procedurally within the world of operational symbolism. However, they may be less proficient in dealing with symbols as manipulatable concepts. While some students may build naturally from embodied and symbolic experience, others may build naturally based on written definitions. Other students may try to pass examinations by learning proofs procedurally in axiomatic formalism world [5].

However, courses at the undergraduate level, such as differential equation courses, only emphasize procedural fluency [11, 12] .The findings of the study [12]showed that candidates' learning development in procedural learning did not lead them to develop conceptual knowledge.

Many researchers believe that procedural knowledge enables conceptual development. Therefore, they use procedural knowledge and reflect on the outcome, which is known as the developmental approach [13] .However, some researchers believe that conceptual knowledge enables the development of procedural knowledge. Educational approach is an instructional implication which focuses on building meaning for procedural knowledge before mastering it. How can these two be linked together in mathematics?

Gray and Tall believe that proceptual thinking provides a link between procedural and conceptual knowledge. They believe that a procept, which involves combining mental objects, consists of a process and a concept produced by a process, and a symbol used to denote both, which are related

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through using procedures and manipulation objects appropriately [14]. Procept expresses that different procedures can yield the same concept. For example, 4+3,2+5,14/2 can be exchanged because they all show number 7[15]. Therefore cognitive difficulties of understanding of mathematical object can be dealt with through the use of Procept theory [14].

Students can expand the complexity of their ideas using a mathematical thinking process. This process includes specializing, generalizing, conjecturing, and convincing [16] . Several mental activities can characterize mathematical thinking including exemplifying, specializing, completing, deleting, correcting, comparing, sorting, organizing, changing, varying, reversing, altering, generalizing, conjecturing, explaining, justifying, verifying, convincing, and refuting [17]. Mathematical thinking is considered as four processes including specializing, generalizing, conjecturing, and convincing. They are powers possessed by every human being, though whether they are employed is another matter. They are also processes because they take place in time. Therefore, they can be seen as processes to be initiated or undertaken and as powers to be activated, more or less skilfully. Mathematical thinking [16] was chosen in this research due to its converting in teaching and learning mathematics instruction [18]. However some prompts and questions were adapted to use in differential equation classroom [17].

Mathematical thinking can be improved in a specific environment such as using a computer algebra system (CAS) due to their potential to manipulate mathematics expressions symbolically and graphically. But the main function of CAS is manipulation of mathematical expression in symbolic form. How can CAS become an effective instrument of mathematical thinking for students? The relationship between conceptual insight and technical skills as the process of instrumentation has attracted researcher's attention [19, 20] .

IV. INSTRUMENTAL GENESIS Apparently, the use of tools does not happen in a vacuum.

Tools are applied in an act, practice or a context. However, how individuals look at activities and practices is very important. An artifact becomes an instrument through a process called instrumental genesis [21] . Instrumental genesis combines two processes including instrumentalization and instrumentation. In the process of instrumentation, tools shapes the user's action. This means that users find their own schemas and schemes to use the tool. The use of tools and the objects to be investigated are shaped by users [22]. The former is directed toward the artifact and the latter is directed toward the subject’s behavior (See Fig. 2).

In simple terms, instrumentation looks at how tools shape the action of the tool-using subject, but instrumentalization denotes the way the users shape the tool. However, the constraints induced by the instrument must be identified[19]. More precisely, an instrument can be considered an extension of two components: first, the body, which is a functional organ, consists of an artifact component such as an artifact or part of an artifact organized in the activity and second, a

psychological component. The construction of this organ is called instrumental genesis, which is a complex process related to the characteristics of the artifact such as potentialities and its constraints to the activity, and knowledge of the subjects former work method. The psychological component is referred to as a scheme [22] which was introduced as the invariant organization of behaviour in a situation. A scheme has some main functions: a pragmatic function which allows the agent to do something; a heuristic function that allows the agent to anticipate and plan actions; and an epistemic function, which allows the agent to understand what he is doing [22].

Fig. 2. Instrumentation and instrumentalization [22:144]

All schemes have two levels: usage or utilization scheme of an artifact, which is referred to as the management of the artifact such as turning on a calculator, adjusting the screen contrast, choosing a particular key; and the instrumented action scheme which is the ability to carry out a specific task or to be oriented by an activity [22, 23] such as computing a function’s limit. Some parts of the schemes formation are visible; for example, gesture is as an elementary student behaviour which may be observed in the use of schemes; moreover, instrumented technique is the observation of instrumented action schemes [24] , For example, what can be described for limit computation by teachers in a CAS environment [25] . Within didactic institutions, mathematical objects through four practice components are brought into play such as task, technique, technology, and theory [26]. Usually tasks are expressed in terms of verbs; for example, "solve the second order differential equation". Techniques are defined as a way of carrying out tasks, which is separate from the discourse that produces, explains, and justifies, which is considered as technology. However, Chevallard [27] believes that this type of discourse is integrated into technique and this type can be characterized as theoretical progress. Moreover, theory is a form of abstract distance from the empirical. Therefore, discourse can be considered as a bridge connecting technique to theory [26]. The relationship between 4's T has been expressed by Lagrange, who believes that technique plays a role as epistemic value through understanding the mathematical objects that they handle. It also serves as an object for conceptual reflection when compared with other techniques and when discussed with

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regard to consistency. Artique [21:248] adapted the anthropological theory of Chevallard by collapsing technology and theory into one theory, and she emphasized that technique has to be given a wider meaning than usual in educational discourse: "a technique is a manner of solving a task, and as soon as one goes beyond the body of routine tasks for a given institution, each technique is a complex assembly of reasoning and routine work".

Therefore, in this research, based on instrumental genesis

[24], the mathematical activity of student organized as task (T), carried out using a technique ( )τ , a theory is the

understanding of mathematical objects ( )Θ . A

praxeology ( ), ,T τ Θ is considered as a mathematical activity [25]. The main point of using CAS is to provide an inventory of instrumented techniques, which provide meaningful or correct answers to a restrained type of task. Modification of praxeology is supposed to be developed by students and teachers. These modifications have to be understood relative to the affordance of the involved techniques as their pragmatic value - the efficiency for solving tasks-and their epistemic value - provide insight into the mathematical objects and theories to be studied.

V. METHODOLOGY

Teaching experiment was chosen as a methodology to explore the students’ construction of mathematics knowledge followed by allowing students to gain experience through interactions with researchers. In addition, constructivist teaching experiment is appropriate to attribute the context within which students construct mathematical knowledge [28]. Considering differential equations is a requirement of undergraduates in science and engineering, One differential equation classroom which included 37 engineering students at one of the public universities in Malaysia was purposely chosen. One typical intervention session included 30 minutes integrating differential equations using Maxima as a computer algebra system. The intervention session started after finishing a 70 minute lecture on differential equations using pen and paper by one of the research member. The first was conducted by a lecturer and the latter by a first author.

The instruments for intervention sessions were worksheets that were made based on concepts of mathematical thinking by Mason [18] and instrumental genesis [29]. The definition was chosen for this research, due to be converted into the activities in the classroom. The instruments included both written and computer lab activity to solve a specific differential equation to use in the intervention sessions. Maxima, a free and open source software, was chosen as a CAS to use in this research. After finishing the intervention sessions, seven in-depth interviews were conducted to determine how students compute real life problems in chemical engineering. In doing so, participants were asked to give an example from chemical engineering. The questions were used during the interview to probe or for clarification were based on Mason's definition of

mathematical thinking [17]. Seven students, Ckin, Philip, Wendy, Penny, John, Ade, and Una, were purposely chosen for an in-depth interview. Data were collected using student interviews to look for the development of mathematical thinking processes in computing real life problems. Data were analyzed according to qualitative data analysis [30]. Thus, interview transcriptions were coded based on the mathematical processes in Mason’s definition of mathematical thinking [31]( see TABLE II). Typical interview coded is shown in TABLE III. Finding of the research will be presented in the next session.

TABLE II. INITIAL CODEBOOK FOR CODING DATA

Constructs Code Sub-construct Specializing

SP1 Reading the question SP2 Identify the features of the facts that

makes it an example of a specific differential equation

SP3 Identify the facts, theorems and properties, the techniques to solve

SP4 Classify and sort information be alert to ambiguities - Organizing data or information

SP5 Introducing images, diagrams to understand the problem

SP6 Introducing symbols, representation, notation to understand the problems

SP7 Trying some specific cases to get an idea of what the answer might be

SP8 Identifying any similarity or analogous questions–identify possible underlying pattern - Develop a sense of why the answers may be correct

Generalizing GN1 Check the calculations to make sure the generalization is true

GN2 Check the argument to ensure that the computations are appropriate

GN3 Look for pattern and relationships GN4 Extend the result to a wider context by

generalizing GN5 Extend by seeking a new path to the

resolution GN6 Extend by altering some of the

constraints Conjecturing CJ1 Predicting relationships and results

CJ2 Check the consequences of conclusion to see if they are reasonable

CJ3 Reflect on implications of conjectures and arguments

Convincing CV1 Convince yourself- verification CV2 Convince a friend – explanation

(Finding and communicating the reasons why something is true).

CV3 Convince an sceptic -justification

VI. FINDINGS This study is a part of the research which investigated how

engineering students use procedural, conceptual, and technical

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knowledge in real life situations that include differential equations. Findings showed students used specializing powers to identify the factual features of examples of specific differential equations. For example, Ade wrote the real life problem in chemical engineering as shown in Fig. 3.

Fig. 3. Ade's example of a real life problem

Ckin, one of the participants in this research used the specializing powers to give an example using Internet (see Fig. 4).

Fig. 4. Ckin's example of a real life problem

This exemplifying is illustrated through both high and low achievers. For example, John, a low achiever, presented the example (see Fig. 5).

Fig. 5. John's answer to a real life problem

Almost all participants used an example from a chemical engineering course. However, the complex and detailed problems of higher achievers were greater than those of others. For example, Philip, a high achiever, presented a comprehensive example from the transport process course. Fig. 6 shows the problem that he mentioned from a chemical engineering course.

Fig. 6. Philip's example of a real life problem

TABLE III. INITIAL CODEBOOK FOR CODING DATA

To solve the examples that students gave, they first needed to be fluent in procedural knowledge (see Fig. 7). However, this fluency needs to be integrated with conceptual knowledge. Wendy found the solution not in the manner that was applied in the differential equation classroom. First, she wrote the differential equation that could model the problem and then she substituted the values in the equation to find the unknown variable.

Fig. 7. Wendy's answer using pen and paper

Ade SP2

Researcher What does it mean? Ade

(Ade verbally explain all symbols above)

SP3

SP6

Ade cp is so much. is constant. (writing the differential equation in her area)

SP2

Researcher you need specific no to Cp. how much is it?

Ade 5/3R SP2 Researcher ok, now solve one example using Maxima

and draw the graph?

Ade Let me check the help sheet Researcher what is your differential equation? Ade Ade Copied and modified it : eq'diff(y,x)=-y

Then ask about Cp, after understanding how to assign first assign then write the equation.

SP2

SP3

SP6

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Students showed a way to solve the problems using Maxima, which is a computer algebra system environment. For example, when the researcher asked Wendy to solve the problem using Maxima, she exchanged the pen and paper method in order to solve the problem in the Maxima environment. Fig.8 shows her solution through Maxima.

Fig. 8. Wendy's solution of a real life problem using Maxima

Students tried to find the general equation using a differential equation, Penny answered the researcher's question by saying that they did not need the relation between T and x (line 2, 4).

1 Penny: T dependent and x independent, we look forward to q

2 Penny: We do not need the relation between T and x in this problem

3 Researcher: If you want to know the relationship between x and T what do you do?

4 Penny: No need to do that

However, software helped the students to visualize the result, which made the interpretation easier (see Fig. 9).

Fig. 9. Philip's graph of a solution to a real life problem

When the researcher asked Philip how he interpreted the solution, he used convincing powers using graphs and he introduced symbols as well ( line 1,3).

1 Philip: The heart exponentially is reduced. 2 Researcher: How do you understand this?

3 Philip: First through the graph and more assurance using the symbolic solution

In solving differential equations, students showed

difficulties in the procedures that they wanted to use to get the solution. Some of them forgot the procedure that they wanted to use to solve the given differential equation. However, they could use the help sheet, provided by researcher about using Maxima to get a solution. In a computer algebra system such as Maxima, students used several procedures to obtain the solution of a given differential equation. Una described the procedure to solve a given differential equation as first defining it, followed by a general solution, a particular solution, and by drawing the graph of the solution. A frequent problem that happens in a Maxima environment was the assignment of values to variables. After the researcher's guided the assignment of values, students used Maxima to get the solution to real life problems, using Maxima as a powerful calculator. Fig. 10 shows how Maxima was used to generate general solutions.

Fig. 10. Philip 's solution of real life differential equation using Maxima

VII. DISCUSSION AND CONCLUSION This research was conducted to study how engineering

students apply procedural, conceptual, and technical knowledge to real life problems involving differential equations. To do so, seven students of the faculty of chemical engineering were chosen for an in-depth interview based on the information derived from teaching experiment sessions. Findings showed that students started solving problems with procedural knowledge. In the process, they used mathematical thinking powers, particularly specializing and generalizing. The type of knowledge required to solve a problem is not predetermined by formulation of the problem. This findings confirm previous research which have been classified as conceptual in character, students prefer to follow procedures in doing tasks [7]. The reasons for doing procedurally varies from cognitive preference to the attempt to do mathematics properly. However, students showed that they use of conceptual knowledge of graphs to justify why the solutions are correct. According to Engelbrecht, if a procedure is defined as a pre-given sequence of steps to apply, the solution was mentioned, (e.g. drawing the graph by Philip),through the

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finding of the graphical solutions are truly procedural. However, this solution indicates a clear conceptual understanding of the geometrical interpretation of derivative. The findings confirm Engelbrecht 's study, that Philips's solution proceeds after translation of the graph into an algebraic solution. The observation shows that the students ' interpretations are strongly procedural. However, the strategies to solve the problem lie in the approach used to solve the problem, which is based on a conceptually model of the situation rather than the use of a pre-determined procedure.

Student used procedural knowledge in pen and paper due to the need to find general solutions. This does not mean they have poor mastery of mathematics. As it can be seen in the examples shown by students, procedural reasoning requires quite sophisticated mathematical work. This confirms the finding by Engelbrecht, the move from conceptual to procedural makes the solution process more complicated than a cleaner conceptual approach.

To solve differential equations using Maxima, students have to know the specific instrumented technique related to the knowledge that they got in a pen and paper environment, this has been noted by Atigue [32] . Therefore, students must know how Maxima operates, which affects their decision making in terms of the usefulness of the outputs and the use of the utility provided by the instrument. However, using a help guide can control the organizational constraints. Furthermore, instrumentation of the tools has a strong impact on mathematical reasoning [33]. However, the findings confirm the literature for the potential of computer algebra systems in the process of teaching and learning mathematics such as high calculation speed, visualization, environment for exploration, and pattern identification [33-35]. Moreover, the computer algebra environment can promote an understanding of the solution process [36]. There is a rich body of articles suggest that the use of a computer algebra system brings conceptual gains in mathematics classes. But, new instrumented assisted tasks and techniques must be created to new theoretical discourse.

To sum up students' behavior in differential equation using a computer algebra such as Maxima, findings showed that students solved the problem procedurally. they used their mathematical thinking powers such as indentifying the facts and technique to solve, which were proceeded through conceptual knowledge. Moreover, Maxima capabilities such as visualization helped students to understand and interpret the solutions of a differential equation. However, the focus of assessment in differential equations is procedural, which caused a limitation in this research and students tried to make effort on procedural knowledge of the differential equation. The researchers and educators can take advantage of the research findings. The findings can be used to make a framework to develop mathematical thinking in differential equations at undergraduate level using a computer algebra system, which will play a role to have a balance in procedural, conceptual, and technical of knowledge to develop mathematical thinking.

ACKNOWLEDGMENT We are very grateful to the Ministry of Higher Education,

Malaysia and Universiti Teknologi Malaysia for providing the research funding (Grant Vote No. 4L068) and support that enabled this study to be carried out. The authors acknowledge Pr. John Mason for having reviewed all the material in this research.

REFERENCES

[1] Camacho-Machín, M., J. Perdomo-Díaz, and M. Santos-Trigo, An exploration of students’ conceptual knowledge built in a first ordinary differential equations course (Part I). The Teaching of Mathematics, XV, 2012. 1: p. 1-20.

[2] Tall, D., Lies, damn lies... and differential equations. Mathematics Teaching, 1986. 11: p. 54-57.

[3] Paraskakis, I. Rethinking the teaching of differential equations through the constructivism paradigm. in Advanced Learning Technologies, 2003. Proceedings. The 3rd IEEE International Conference on. 2003. IEEE.

[4] Tall, D., How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. 2013: Cambridge University Press.

[5] Kaput, J.J., Technology and mathematics education. 1992: Macmillan.

[6] Rittle-Johnson, B., et al., Assessing Knowledge of Mathematical Equivalence: A Construct-Modeling Approach. Journal of Educational Psychology, 2011. 103(1): p. 85-104.

[7] Engelbrecht, J., C. Bergsten, and O. Kågesten, Undergraduate students' preference for procedural to conceptual solutions to mathematical problems. International Journal of Mathematical Education in Science and Technology, 2009. 40(7): p. 927-940.

[8] Rittle Johnson, B., Promoting transfer: Effects of self explanation and direct instruction. Child development, 2006. 77(1): p. 1-15.

[9] Tall, D., The transition to advanced mathematical thinking: Functions, limits, infinity and proof. Handbook of research on mathematics teaching and learning, 1992: p. 495-511.

[10] Tall, D., The transition to formal thinking in mathematics. Mathematics Education Research Journal, 2008. 20(2): p. 5-24.

[11] Breen, S. and A. O’Shea, The use of mathematical tasks to develop mathematical thinking skills in undergraduate calculus courses–a pilot study. 2011.

[12] Arslan, S., Traditional instruction of differential equations and conceptual learning. Teaching Mathematics and its Applications, 2010. 29(2): p. 94-107.

[13] Kadijevich, D. and L. Haapasalo, Linking procedural and conceptual mathematical knowledge through

169

CAL. Journal of Computer Assisted Learning, 2001. 17(2): p. 156-165.

[14] Tall, D., MATHEMATICAL AND EMOTIONAL FOUNDATIONS FOR LESSON STUDY IN MATHEMATICS.

[15] Gray, E. and D. Tall, Abstraction as a natural process of mental compression. Mathematics Education Research Journal, 2007. 19(2): p. 23-40.

[16] Mason, J., L. Burton, and K. Stacey, Thinking mathematically. 2010: Addison-Wesley London.

[17] Zeynivandnezhad, F., Z. Ismail, and Y. Mohammad Yosuf, Mathematical Thinking in Differential Equations Among Pre-Service Teachers. Jurnal Teknologi, 2013. 63(2).

[18] Watson, A. and J. Mason, Questions and prompts for mathematical thinking. 1998: Association of Teachers of Mathematics.

[19] Kidron, I., Abstraction and consolidation of the limit procept by means of instrumented schemes: the complementary role of three different frameworks. Educational Studies in Mathematics, 2008. 69(3): p. 197-216.

[20] van Herwaarden, O.A. and J.L. Gielen, Linking computer algebra systems and paper-and-pencil techniques to support the teaching of mathematics. International Journal of Computer Algebra in Mathematics Education, 2002. 9(2): p. 139-154.

[21] Artigue, M., Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 2002. 7(3): p. 245-274.

[22] Trouche, L., Managing the complexity of human/machine interactions in computerized learning environments: Guiding students’ command process through instrumental orchestrations. International Journal of Computers for Mathematical Learning, 2004. 9(3): p. 281-307.

[23] Verillon, P. and P. Rabardel, Cognition and artifacts: A contribution to the study of though in relation to instrumented activity. European Journal of Psychology of Education, 1995. 10(1): p. 77-101.

[24] Trouche, L., From artifact to instrument: mathematics teaching mediated by symbolic calculators. Interacting with Computers, 2003. 15(6): p. 783-800.

[25] Trouche, L., An instrumental approach to mathematics learning in symbolic calculator

environments. The didactical challenge of symbolic calculators, 2005: p. 137-162.

[26] Kieran, C., Conceptualizing the learning of algebraic technique: role of tasks and technology. Lecture presented at ICME-11, Monterrey, Mexico, 2008.

[27] Chevallard, Y., L'analyse des pratiques enseignantes en théorie anthropologique du didactique. Recherches en didactique des mathématiques, 1999. 19(2): p. 221-265.

[28] Steffe, L.P., The constructivist teaching experiment: Illustrations and implications. Radical constructivism in mathematics education, 2002: p. 177-194.

[29] Zeynivandnezhad, F., et al., Mathematical Thinking in Differential Equations through a Computer Algebra System : a Theoretical Framework, in 4th IGCESH 20132013, Universiti Teknologi Malaysia: Johor p. 1001-1005.

[30] Creswell, J.W., Educational research: Planning, conducting, and evaluating quantitative and Qualitative Research. Vol. 4th. 2012: Pearson.

[31] Zeynivandnezhad, F., et al., Can Syntax in Computer Algebra System Promote Mathematical Thinking in Differential Equations?, in 7th International Computer & Instructional Technologies Symposium2013, pegem Akademi: Erzurum-Turkey. p. 275.

[32] Artigue, M., The integration of symbolic calculators into secondary education: some lessons from didactical engineering. The didactical challenge of symbolic calculators, 2005: p. 231-294.

[33] Ruthven, K., Instrumenting Mathematical Activity: Reflections on Key Studies of the Educational Use of Computer Algebra Systems. International Journal of Computers for Mathematical Learning, 2002. 7(3): p. 275-291.

[34] Lagrange, J.-B., Transposing computer tools from the mathematical sciences into teaching, in The didactical challenge of symbolic calculators. 2005, Springer. p. 67-82.

[35] Nabb, K.A. CAS as a restructing tool in mathematics education. in Proceedings of the 22nd International Conference on Technology in Collegiate Mathematics. 2010.

[36] Drijvers, P. and K. Gravemeijer, Computer Algebra as an Instrument: Examples of Algebraic SchemesThe Didactical Challenge of Symbolic Calculators, D. Guin, K. Ruthven, and L. Trouche, Editors. 2005, Springer Netherlands. p. 163-196.

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