Teaching Mathematics Online

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Chapter 2

A Blended Learning Approach in Mathematics B. Abramovitz ORT Braude College, Israel M. Berezina ORT Braude College, Israel A. Berman Technion Israel Institute of Technology, Israel L. Shvartsman ORT Braude College, Israel

ABSTRACT In this chapter we present our work aimed at interweaving e-learning and face-to-face learning in Calculus courses for undergraduate engineering students. This type of blended learning (BL) contains the best properties of e-learning and face-to-face learning and helps overcome many obstacles in traditional teaching. We use our approach in order to improve students’ conceptual understanding of theorems. We describe online assignments specifically designed to help students better understand the meaning of a theorem. These assignments are given to students in addition to traditional lectures and tutorials with the objective that they can learn to learn on their own. Students “discover” the theorem and study it independently, by using a “bank” of examples and a lot of theoretical exercises we supply. The assignments are built in such a way that students receive feedback and instructions in response to their Web-based activity. DOI: 10.4018/978-1-60960-875-0.ch002

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A Blended Learning Approach in Mathematics

INTRODUCTION Using today’s advanced computers and Internet networks, a wide spectrum of different ways of learning and teaching are being developed: from supplying students with learning materials in traditional universities through to studying in new purely online universities. We are Mathematics lecturers in a traditional engineering university where face-to-face teaching is prevalent; however, some elements of online learning are widely used. Almost all courses have websites through which the lecturer stays in touch with students and uploads different materials for them in addition to face-to-face lectures, tutorials and office hours. In attempting to resolve problems related to our face-to-face teaching (for details, see the “Problems and Trends” section), we exploited the opportunities offered by the worldwide web. We knew that our students needed additional special material to supplement our lectures. They were simply not getting enough “face time” with us in class, owing to the shortage of lecture hours. Moreover, we wanted our students to get used to learning theory independently. In order to overcome these problems we developed a web-based learner-centered approach as a component of BL, which has thus far been implemented in Calculus courses. We constructed online theoretical assignments to help students better understand the meaning of a theorem. These self learning assignments are of an unusual type and given to the students in a special order to turn them from passive receivers of knowledge into active partners in the learning process. During our face-to-face teaching we used these assignments to complete the learning process. We called our approach the Self Learning Method (SLM). SLM has three main parts that are presented in detail in the “Description of the Method” section. In the first part of SLM students learn how to formulate a conjecture. We use the Integral

Mean Value Theorem as an illustration. Students “discover” the theorem independently, by using a “bank” of examples we supply. In the second part students study assumptions and conclusions of theorems. We illustrate this part by means of Lagrange’s Mean Value Theorem. Students conduct their own research by using a set of especially composed assignments. The third part of SLM focuses on proving a theorem. Three Calculus theorems were used to illustrate different types of assignments: “scattered puzzles”, “fill in puzzles” and “puzzles”. The main aim of these assignments was to teach students that a theorem’s proof has a logical order, and each its step is based on information given in a theorem’s assumptions. In the “SLM Implementation and Some Results” section we describe our positive experience in operating SLM as a part of BL. As an Internet self learning environment, we used the Webassign system (see www.webassign. net). Webassign was developed at North Carolina State University, and it is used by more than 300,000 students at over 1,500 institutions. In our teaching we apply the version that was adapted to Hebrew. Here we present examples translated from Hebrew to English.

PROBLEMS AND TRENDS We teach different Mathematics courses designed for engineering students. One of these courses is Calculus, which is an important part of the curriculum of most of the students in our university. Usually, this course consists of a four-hour weekly lecture (standard frontal teaching) and a two-hour weekly tutorial, plus homework assignments. Students take midterm and final pencil-and-paper exams. In our teaching we encounter problems that are difficult to solve in a face-to-face teaching framework. Many of our students are not interested in Mathematics: they are not intending to

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specialize in it, and they see it as a necessary evil. Some students came from schools that aim to make Mathematics teaching as simple as possible, focusing on standard methods and paying little attention to students’ understanding of basic mathematical concepts. This problem is particularly acute when students begin to study Mathematics at university. Similar problems were mentioned in the paper of Naidoo & Naidoo (2007). Our students generally find it difficult learning theory, and as a result, are frightened of it. Possible reasons are the abstractness of mathematical concepts, the formal way subjects are presented and the special language of a theorem or a definition. In addition, many students do not appreciate the importance of theory. They see the theoretical part of Mathematics as separate from the computing part. This distorted understanding of Mathematics is the result of students being taught Mathematics as a set of algorithms used in problem solving. Obviously, when Mathematics is learned this way, students have difficulties in solving problems that they have not yet seen. Another important problem we encounter is freshmen’s overloaded compulsory syllabi of mathematical courses. Engineering students are required to cover a lot of Mathematics material, which is essential for their professional studies. As a result, there are not enough teaching hours to ensure that students properly understand the material they have to cover. The modern world needs highly educated, qualified engineers who have a full grasp of mathematical concepts and advanced skills. Ensuring that the students who graduate from our college are indeed fully grounded engineers is our aim, but under the circumstances it is difficult to achieve this goal. Of course, some required material could be given to students to study on their own; however, experience shows that freshmen have a hard time accomplishing this because they do not have the necessary skills for independent study. Moreover, the books we recommend are usually written in a formal Mathematics style, which students also

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find daunting (see, for instance, Spivak (2006); Kon & Zafrany (2000)). In an attempt to solve these problems we developed a different approach in our teaching. We understood that students have to study additional advanced Mathematics material on their own in order to supplement face-to-face lectures and tutorials. Looking to the Internet as a medium that could help us achieve our target was only natural. We had already used some elements of e-learning in our teaching, for example, uploading teaching material onto course websites: syllabus, lecturer notes and lists of homework problems. We could have simply put additional material on the site, but that would just be duplicating existing textbooks. We wanted to develop an easy, understandable way for students to independently study theoretical material. We wanted our students to be active in their learning and to “discover” Mathematics, plus be given the feedback they needed as they work on their own. Thus our goal was to make their learning interactive; we did not want our students to perform, as in the words of Tall et al. (2008), “surface learning” (p. 13). Our aim was to reach “deep learning approaches… and conceptual learning” (p. 13). We understood that we could use Webassign to build a system for students’ online self learning that would complement the traditional learning of one of the most important parts of Mathematics, the learning of theorems. Theorems contain all the essential properties of concepts and relations between concepts. They justify the algorithms and formulate the conditions under which these algorithms are applicable. As we noted above, students, particularly freshmen, have difficulties in studying theorems. Students believe …that the theorem can be memorized as a “slogan”, and then it can easily be retrieved from memory under the hypnotic effect of a magic incantation. However, using a theorem as a magic incantation may increase the tendency to use it


carelessly with no regard to the situation or to the details of its applicability. (Hazzan & Leron, 1996, p. 25) To this end we developed SLM. We have been using it for about five years as a student online self learning system, in addition to traditional face-toface teaching experiences. We originally described the method in a 2007 paper, and in two subsequent papers (Abramovitz et al., 2009a, 2009b).

BACKGROUND A new culture, known variously as the digital, Internet or cyber culture, which transcends ethnic, national and regional boundaries, and allows millions to be part of a global virtual community has arisen (Fang, 2007). It has had an indelible effect on the academic world and initiated a transformation of the learning and teaching processes in universities. Moreover, today’s students, have grown up on cyber culture, and accept the Internet as a natural teaching medium. As noted by Cross (2006) “e-learning covers over the multiple possibilities born of the marriage of the learner and the Internet” (p. xxi). There are indisputable merits in e-learning, such as flexibility: learners can be independent of space and time (Garrison & Kanuka, 2004). Students’ activities can be carried out when the learners so desire, anytime and anywhere (Alonso et al., 2005). In this paper the authors also pointed out that e-learning is personalized: each student selects the activities most suitable for him/her (most relevant to his/her background). Online study provides more than one opportunity for satisfying each student’s learning style (Moore, 2006). Also e-learning helps students develop learning independence and fosters their self-reliance: much more time is spent on the assignment before consulting the teacher; students learn to trust their own judgment (Harding et al., 2005b; Garrison & Kanuka, 2004). E-learning

encourages shy learners to be confident (Naidoo & Naidoo, 2007). On the other hand, Engelbrecht and Harding (2005b) noticed that one e-learning problem is the lack of face-to-face-contact. Students are dissatisfied with the lack of personal attention when they have to confront a problem. They generally want weekly contact sessions so that they can get an opportunity to ask questions face-to-face (Harding et al., 2005b). Even teachers may feel isolated when they undergo the experience of being an online student (Czerniewicz, 2001). Traditional teaching will always be an effective means of learning, because of the face-to-face interaction with both the instructor and classmates (Alonso et al., 2005). Moreover, the teacher is the ultimate key to educational change and a significant part of the successful implementation of online learning, through his or her readiness to exploit the medium (Condie & Livingston, 2007). The problem is how to ensure that the benefits of e-learning are realized, e.g., keeping the student-teacher interaction. Blended learning (BL) could be a solution to this problem. Usually BL is described as learning with the convenience of online environments, without losing face-to-face contact (Harding et al., 2005b). At its simplest, BL is the integration of classroom face-to-face learning experiences with online learning experiences, against the backdrop of limitless design possibilities. Using BL does not mean that students are deprived of the traditional study format. BL only seeks how best to utilize face-to-face and online learning for purposes of higher education (Garrison & Kanuka, 2004). In the words of Graham (2006): “BL combines the best of both worlds” (p. 8). According to a report by the U.S. Department of Education (2009), BL has a larger advantage relative to purely online learning or purely face-to-face instruction. The report was based on reviewing a great number of research papers. In recent years the BL approach has been implemented in the teaching of Mathematics at

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the undergraduate level in many universities in different countries. Getting this to happen was not a simple enterprise; it can perhaps be blamed on the strong role of tradition in Mathematics teaching (Harding et al., 2005a). According to Bookbinder (2000), enhancing the mathematical curriculum with web-based technology takes time and effort, but the effort is well worth it. A wide overview of works on implementation of BL ideas at different levels in Calculus courses is given in the paper by Tall et al. (2008). They summarized their findings and said: “There is little evidence that the “brand” or type of technology makes any significant difference, beyond the obvious fact that some tasks require more powerful tools than others… The important thing is not which tools are used but how they are used” (p. 18). The authors argued that many approaches to students’ conceptual understanding of the subject can be developed based on the proper use of technology. Engelbrecht & Harding (2005a, 2005b) published comprehensive papers regarding a study aimed at analyzing Internet teaching of undergraduate Mathematics. In their first paper they described a variety of technologies used to teach undergraduate Mathematics online. They also mentioned some not-so-successful attempts at using online teaching, such as reading texts on screen and visiting a few web sites. Many of the Internet courses they reviewed could be characterized as attempts to digitally replicate traditional teaching. The authors also listed different types of cases that combined Internet teaching with face-to-face teaching and described the experiences of various universities. In their second paper they listed research issues connected to Internet teaching, among them – “Students as independent learners”. Outcomes of BL implementation in teaching of Mathematics at a collegiate level were studied by several researchers. Naidoo & Naidoo (2007) in their work on BL in an elementary Calculus course for engineering students, found that the students in an experimental BL group had deeper conceptual

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understanding and made fewer essential mistakes than the students in the control group. Iozzi & Osimo (2004) studied BL implementation in undergraduate Mathematics courses at Università Bocconi in Milan, which started with a notice board and ended with a virtual classroom for advanced studies. They also raised the issue of whether the students’ online participation is positively correlated with their overall performance in Mathematics. In their research they did not come to a definitive conclusion regarding gains in final grades, but did find that BL positively influenced the development of some cognitive abilities: cognitive empowerment, capacity for life-long learning, attitudes towards Information Technology and attitudes towards collaborative work. Groen & Carmody (2005) conducted a research experiment designed to answer the question of “How do learners respond to the blend?” Most of the students who participated in the BL experience learned how to approach the issue of knowledge construction, found that their cognitive skills had increased and developed a positive attitude to learning Mathematics. Ahmad et al. (2008) showed similar results. As a solution to the problems described in the “Problems and Trends” section, we constructed our approach, SLM, which is a model of BL implementation, and used it in a Calculus course. We related to research questions that are, to some extent, similar to the issues highlighted in the above mentioned papers. Particularly, we studied the issue of “students as independent learners,” which is actually a subject for BL research. SLM aims to improve students’ conceptual understanding of theorems. Based on our research (see, for instance, the “SLM Implementation and Some Results” section), we can argue that our BL approach results in students developing a positive attitude to the theoretical part of Mathematics.


DESCRIPTION OF THE METHOD SLM was constructed using the WebAssign system (http://www.webassign.net/how_it_works/), which is a provider of online instructional tools for faculty and students. Instructors create problems and compose assignments online within WebAssign and electronically transmit them to their class. Students enter their answers online, and WebAssign automatically grades the assignment and gives students instant feedback on their performance and hints about how to correct their errors. Webassign is highly suited to implement SLM. SLM gets students involved in the step by step process of learning the meaning of a theorem, which enables them to better understand the subject. The way in which the material is presented encourages them to take part in formulating, discussing and proving a theorem. Our approach is governed by the fundamental principle of classical teaching – the sequence of in which the material is learned is critical. We agree with Ramasamy’s (2009) assertion that in face-to-face learning/teaching the learners/teachers are there to elaborate and explain the steps. SLM has three main parts: 1. Formulation of a conjecture; 2. Study of the assumptions and conclusions of a theorem; and 3. Proving a theorem.

Mathematics education was reported in several studies (for example, Hazzan & Zazkis (1999); Bills et al. (2006)). Some Mathematics education researchers have proposed asking students to construct the examples (for instance, Watson & Mason (2005)). In our opinion, this is too difficult a task for freshmen. We see our “bank” of examples as the first step in students learning to construct an example. We believe that following our method they will be able to start constructing their own examples, including counterexamples. We would like to note that as they work on the assignments, students can get help if they want it. Help is built in. In some cases they may be given a simple hint, and in other cases – a full explanation; it depends on the complexity of the question. In an assignment, to access help, students move their cursor to the “click here” hyperlink. Thus, a student not only gets feedback to his or her answer, but also the needed help if the task becomes too difficult. This ensures that the student does not give up, and moves forward.

Formulation of a Conjecture First, students are asked to review the concepts and theorems that are needed to learn a new theorem. We use the Integral Mean Value Theorem as an example: If the function f (x ) is continuous on [a, b ] , then there exists a point c ,c ∈ (a, b), such that b

We constructed different kinds of web-based assignments for each SLM part. The assignments are written in a way that does not appear unusual to students, yet is intended to teach them theoretical knowledge. The assignments are supplemented by a “bank” of specially constructed simple examples. Students are asked to solve given theoretical problems regarding theorems by using these examples. Examples have been a part of Mathematics teaching throughout history. The significance of examples and their use in

∫ f (x )dx = f (c)(b − a) a

This theorem concerns the continuity and integrability of a function. In Assignment 1 students are asked to review these concepts. They are provided with examples, in our case, functions and their graphs. For each given function students have to determine if it is continuous and if it is integrable (see Figure 1). As the reader can see, students can check and correct their answers. In this

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Figure 1. Assignment 1 (Formulation of a Conjecture)

way they both verify their theoretical knowledge of the concepts and improve it. Particularly, the assignment re-emphasizes that every continuous function on an interval is integrable on it and that there exist functions that are integrable, but not continuous. By using these simple examples, students are reviewing the theoretical aspect of the relation between continuous functions and integrable functions once more. In Assignment 2, still using the same examples, students are asked to make calculations that will help them verify the theorem’s conclusion. Stu2

dents calculate the integral

∫ f (x )dx , as the area 0

under the graph of f (x ) for every one of the given functions. In the next step they find a rectangle with the base [0, 2] and height h , a rect2

angle whose area is equal to the integral ∫ f (x )dx 0

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. Toward the end of the assignment students need to determine if a point c , c ∈ [0, 2] , exists, such 2

that

∫ f (x )dx / 2 = h = f (c) . For some given 0

functions the conclusion is true, and for others, it is not. In this way students learn the sufficient conditions of the theorem on their own (see Figure 2). Assignment 3 contains a set of statements (see Figure 3). Students have to mark the statements as true or false. For every false statement a student can find a counterexample by using the “bank” of given functions. Only one given statement is true – the theorem. So, without their noticing, students analyze the obtained results and formulate the theorem. In such an active (creative) way, students, as part of their research, figure the theorem out on their own. The functions considered in these assignments are very simple, thus even students with a poor




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mathematical background will not find it difficult to solve them. Also, the required calculations are simple; they are based on the area of a triangle and a rectangle. Consequently, students are able to use these functions easily and check their answers at every step. Nevertheless, the functions provide students with a variety of possible cases: 1) there exists only one point c , such that b

∫ f (x ) d x = f (c)(b − a) ; 2) there exists more a

than one such point; or 3) no such points exist. The assignments are specially constructed so that students can work without the help of an instructor and “discover” for themselves the theorem.

Assumptions and Conclusions of a Theorem We start by presenting the theorem in a schematic way, emphasizing what is assumed and what is concluded. Then we provide students with different assignments that focus on the following questions: What are the assumptions of a theorem and what are its conclusions? What is the geometrical meaning of a theorem? What happens when one or more of the theorem’s assumptions are not fulfilled? What assumptions are necessary and which are sufficient? Generally speaking, what does the theorem mean? We use Lagrange’s Mean Value Theorem: If the function f (x ) is continuous on [a, b ] and differentiable on (a, b) , then there exists a point c , c ∈ (a, b), such that f ′(c) =

f (b) − f (a ) , b −a

as one example to show students how to understand a theorem as a set of conditions that are needed so as to reach a conclusion. In Assignment 1 of this section students are asked to check if the assumptions of the theorem hold true and to determine the number of points c satisfying the conclusion. This assignment is also given by Webassign. Here we present the

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non-electronic version, as it is a more convenient one for the reader. In Assignment 2 we focus students’ attention on the geometrical meaning of the theorem. Assignment 3 asks students to review certain geometrical concepts and their relations. Thus we invite our students to explore another aspect of the theorem – its geometrical interpretation, which helps them better understand the meaning of the theorem. Assignment 3 differs from the other ones because there are no examples (given functions). Here students are asked simple theoretical questions that were carefully chosen to ensure that students, even those with a poor mathematical background, succeed. Assignment 3 aims to teach students to “respect” a theorem’s assumptions and to build counterexamples. Students should understand that a function cannot be a counterexample if a conclusion is true. Such functions are f1(x ) , f2 (x ) and f4 (x ) . Also a function cannot be a counterexample of a statement if it does not satisfy the assumptions: for the first statement – functions f3 (x ) , f6 (x ) ; for the second statement – functions f3 (x ) , f5 (x ) . It has been our experience that students often replace the assumptions with the conclusion and make the inverse statement. Some of the given functions allow students to disprove the third statement, so we ask students to find appropriate counterexamples. We believe that over the course of time our students will be able to construct their own counterexamples. We realize that this assignment is not a simple one for freshmen, so we added a “click here” for help.

Proving a Theorem We purposely put the assignments of proving a theorem at the end of the process because the proof distracts students from understanding the theorem.


Box 1. Assignment 1 (Assumptions and Conclusions of a Theorm) Description Dear Student, you are already aware of Lagrange’s Mean Value Theorem. We invite you to study this theorem and to understand it more deeply. Enjoy your work! Instructions We recommend you download the assignments, solve them and then submit your answers. Remember that you can correct wrong answers, and try to figure out the reasons for your mistakes. Consider the following functions

= x 2 − 2x + 1 , x ∈ [0, 3] 3 2. f2 (x ) = x − x , x ∈ [−2, 2] 3. f3 (x ) = x + sin x , x ∈ [0, 2πn ] , where n 4. f4 (x ) = 2x + 5 , x ∈ [0, 4] . 1. f1 (x )

is a given natural number

For the graphs of the functions, click here. Mark each function V if it satisfies the assumptions of the theorem on the given interval, otherwise mark X:

1.

V

2.

V

If the function satisfies the assumptions, calculate the number of points

3.

V

c , such that f ′(c) =

4.

V

f (b) − f (a ) . Write down the number of points c b −a

that exist for each function. If the number is infinite, write 101. If the function does not satisfy the assumptions, write 0.

1.

1

2.

2

3.

2n

4.

101

Consider the following function: 2

x

5. f5 (x ) = x 2 , x ∈ [0, 1] . Prove that the function satisfies the theorem’s assumptions. Check if it is possible to find the exact values of the points

c.

Yes No For a detailed explanation of the solution, click here.

The assumptions were satisfied by all five examples. The meaning of the concept – there exists a point c – is not always clear to students. So the examples were chosen in order to show that there are several possibilities for the number of points c . Example 5 was chosen to illustrate that the exact value of c cannot always be determined even when its existence is known. This example also demonstrates the rationale of an existence theorem, which states that you do not have to compute a point in order to prove its existence.

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Box 2. Assignment 2 (Assumptions and Conclusions of a Theorm) Description Dear Student, we invite you to answer the following theoretical questions and deduce the geometrical meaning of the theorem. 1. Consider the straight line defined by the equation: y number k ? Mark the correct answer.

= kx + b , k, b

– given real numbers. What is the geometrical meaning of

k = 0 , then the straight line goes through the origin of coordinates (0, 0) . k = tan β , where β is the angle between the straight line and the positive direction of the y axis. k is the slope of the straight line: k = tan α , where α is the angle between the straight line and the positive direction of

If

the x axis.

(0, k ) . 2. Consider the straight lines defined by the equations: y = k1x + b1 , y = k2x + b2 The straight line intersects the y axis at the point

(complete the sentence):

k1k2 = −1

k1 = k2

3. Consider the straight line sentence):

x 2 − x1 y 2 − y1

y = kx + b

y2 y1

x2 x1

b1 = b2

. The lines are parallel if and only if

b1b2 = −1

passing through points (x 1 , y1 ) , (x 2 , y 2 ) ,

x 1 ≠ x 2 . The slope k

is (complete the

y 2 − y1 x 2 − x1

f (x ) defined on [a, b ] and the secant line y = kx + b connecting the ends of the function graph – ( a , f ( a )) ( b , f (b)) . What is k ? Mark the correct answer. points , f (b) b −a b f (b) − f (a ) k= k= k= k= f (a ) f (b) − f (a ) a b −a 4. Consider the function

[a, b ] and differentiable at a point c , c ∈ (a, b) . The straight line y = kx + b is the tangent line of the function graph at point (c, f (c )) . What is k (the slope of the tangent line)? Mark the correct answer. f (c) − f (a ) f (c) f (b) − f (c) k= k= k= k = f ′(c) c −a c b −c [ a , b ] 6. Consider Lagrange’s Mean Value Theorem: If the function f (x ) is continuous on and differentiable on (a, b) , then there f (b) − f (a ) exists a point c , c ∈ (a, b) such that f ′(c ) = . b −a 5. Consider the function

f (x )

defined on

Which of the following statements is equivalent to the conclusion of the theorem?

c is a point where the tangent line is orthogonal to the secant line connecting (a, f (a )) and (b, f (b)) . c is a point where the tangent line is parallel to the secant line connecting (a, f (a )) and (b, f (b)) . c is a point where the tangent line is parallel to axis x . c is a point where the tangent line is parallel to the line connecting (a, b) and ( f (a ), f (b)) . Click here for another formulation of the theorem and an illustration.

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Box 3. Assignment 3 (Assumptions and Conclusions of a Theorm) Description Dear Student, we are continuing to study Lagrange’s Mean Value Theorem. This theorem has two assumptions. If the assumptions are fulfilled, then the conclusion of the theorem is always true. Under other circumstances we are not able to know beforehand whether there exists a point

c , c ∈ (a, b) , such that f ′(c) =

f (b) − f (a ) b −a

for a given function. It will be interesting to find out what

happens if at least one assumption does not hold. Below you will explore this problem and along the way find out many interesting facts. Instructions Below there are two incorrect statements. Why are they false? Because for each statement there exists a function that satisfies the assumption, but the conclusion is not fulfilled. Such a function is called a counterexample for a statement: it disproves the statement. As you work on the assignment, you’ll find a counterexample for each statement. Good luck! Statements

f (b) − f (a ) f (x ) be a function continuous on [a, b ] . Then there exists c , c ∈ (a, b) , such that f ′(c) = b −a . 2. Let f (x ) be a function defined on [a, b ] and differentiable on (a, b) . Then there exists c , c ∈ (a, b) , such that f (b) − f (a ) f ′(c) = b −a .

1. Let

Consider the following functions 1.

 −π π  f1 (x ) = 3 x sin x , x ∈  ,   2 2 

2.

3.

f3 (x ) = [x ] , x ∈ [0, 2]

4.

f (x ) = 3 x 2 , x ∈ [−1, 1]

f2 (x ) =| sin x | , x ∈ [0, 2π ]  x 2 , x ≤ 0 f4 (x ) =  , x ∈ [−1, 1] 1 − x 2 , x > 0  x 2 , 0 ≤ x < 1 f6 (x ) =  , x ∈ [0, 1] 4 , x = 1

5. 5 6. For the graphs of the functions, click here. For each function mark V, if it is true, otherwise mark X:

Based on the results, find a counterexample for each statement among the given functions. Write down your answer. Function number 5 is a counterexample of the first statement. Function number 6 is a counterexample of the second statement. Look at the table with your answers and answer the following question: can the conclusion of the theorem be true when all the assumptions of the theorem are not satisfied? Yes No Which of the given functions can be a counterexample of the following incorrect statement? Statement If there exists

c , c ∈ (a, b) , such that f ′(c) =

f (b) − f (a ) , then the function f (x ) b −a

is continuous on

[a, b ]

and differen-

tiable on (a, b) . Write down your answer: Function number 4 is a counterexample of this statement. For help, click here.

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Box 4. Assignment 1 (Proving a Theorm) Description Dear Student, you have already seen Fermat’s Theorem:

If the function f (x ) is differentiable at x 0 and x 0 is a local extreme point of f (x ) , then f ′(x 0 ) = 0 . You have also seen the theorem’s proof, but do you really understand it? Let’s check. Instructions The steps of the proof are given below, but they are mixed-up and their order is illogical. We ask you to put the steps in the right order and get the correct proof. We consider the case when

x0

is a maximum point.

Steps Step 1

lim−

x →x 0

f (x ) − f (x 0 ) ≥0 x − x0

and

lim+

x →x 0

f (x ) − f (x 0 ) ≤0 x − x0

by the properties of the limit.

Step 2

f (x ) is differentiable at point x 0 , so there exists a finite limit f (x ) − f (x 0 ) lim = f ′(x 0 ) . x →x 0 x − x0

It is given that

Thus

f (x ) − f (x 0 ) f (x ) − f (x 0 ) = lim+ = f ′(x 0 ) . x →x 0 x →x 0 x − x0 x − x0 It follows that f ′(x 0 ) = 0 . lim−

Step 3

f (x ) − f (x 0 ) ≤0 x − x0 f (x ) − f (x 0 ) ≥0 x − x0

when

x − x0 > 0 .

when

x − x0 < 0 .

Step 4 It is given that

x0

is a maximum point, then

f (x ) − f (x 0 ) ≤ 0

for every

x

from a neighborhood of

x0

.

Write the proper order of the steps from left to right: 4

3

Normally, we present a proof during the lecture and ask students to solve the given web-based assignments independently. In this way we can find out whether students really understand the proof or have only learned it off by heart as they would a poem. In our teaching we use some Calculus theorems. We usually give students a proof of a theorem written as a chain of logical steps. The different

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1

2

assignments are designed to help students better understand and remember the theorem’s proof. Once they understand how a theorem is proved we hope that students will be able to prove simple statements independently. The first type of assignment is a “scattered puzzle” where the steps of the proof are written in the wrong order. Students have to rearrange the steps of the proof so that they are in the correct


Figure 4. Interweaving SLM and Face-to-Face Lectures (FTFL)

order. We consider here the assignment concerning Fermat’s Theorem. Assignment 1 has only four steps, so it is quite an easy one, even for freshmen. We use such assignments at the beginning of the Calculus course. During the course we give proofs with more steps or whose correct order is not unique. We also use a more advanced type of assignment such as a “fill in puzzle”. In the “fill in puzzle” assignment students have to fill in the missing parts of the proof. In our opinion, this assignment is comparatively more complicated and demands a deeper understanding of a proof than the rearrange the order assignment. Below we present the assignment focusing on the proof of the Main Theorem of Calculus: If f (x ) is a continuous function on [a, b ] , and g (x ) =

x

∫ f (t )dt , then g (x ) is differentiable on a

[a, b ] and g ′ (x ) = f (x ) at every point x

in [a, b ] .

In another type of assignment students are asked to explain why every step in the proof is correct. This requires students to figure out on which theorem or definition the step is based. Students receive a list of numbered theorems and definitions and they have to write the appropriate number from the list alongside each step of the proof. Obviously, the theorems and the definitions in the list are not given in the correct order; moreover, there are other, superfluous theorems or definitions in the list. We write the entire statements (not their names) in order to compel students to read these statements once more. This helps to improve students’ understanding of the statements.

SLM IMPLEMENTATION AND SOME RESULTS We have been implementing SLM in Calculus courses since 2006, intertwining its use with face-to-face teaching. We should note that the

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Box 5. Assignment 2 (Proving a Theorm) Description Dear Student, you have already studied the Main Theorem of Calculus. It’s now time to check whether you have completely understood its proof. Instructions Below you will find the incomplete proof of the theorem; some parts are missing. The list of the missing parts also appears below, but note that it also contains some additional unnecessary items. Write down the number of the missing part in the right place. You can use a number more than once. By the definition of the 7

g ′ (x ) = lim

g (x + ∆x ) − g (x ) ∆x

∆x → 0

.

4

g (x + ∆x ) − g (x ) = Suppose

∆x > 0

x < c (∆x )