Teaching Calculus Using Dynamic Geometry Tools

Teaching Calculus Using Dynamic Geometry Tools

EU project: 118982-CP-1-2004-1-GR-COMENIUS-C21

EUROPEAN COMMISSION

SOCRATES PROGRAM

COMENIUS 2.1

Deliverable

Report on the Teachers Course

University of Cyprus

Teachers’ Training Course

Secondary school mathematics teachers’ network

The teachers’ network that was developed for the CALGEO program in Cyprus was comprised by 17 teachers.

These 17 teachers were from four different towns in Cyprus: Nikosia, Limassol, Larnaka, Paphos, Paralimni. All had been teaching mathematics in their schools and were able to use the computer laboratory for the lessons if they chose to do so.

The teachers had volunteered to participate in this network and also to participate in the CALGEO program.

During the first meeting with the secondary school teachers, information was provided about the aims of the project and what their involvement would be. Login codes were given so that all of them could have access to the CALGEO products (downloads, activities). In addition to this they were informed that they could enter the forum and ask questions, communicate with the researchers involved in the program, or present their own views and suggestions about the program.

The course

The members of stuff of the University of Cyprus that participate in the CALGEO research program, Prof. Constantinos Christou, Dr Demetra Pitta-Pantazi, Dr Nicholas Mousoulides, Mr Marios Pittalis and Mr Alkeos Soyoul from the University of Athens,organised a training course entitled “Teaching Calculus Using Dynamic Geometry Tools”between October 2006 and December 2006.

The course objective was twofold, first to initiate teachers to dynamic geometry applications and secondly to guide them how these tools may be used in the classroom to enhance students’ understanding in calculus.

The course was divided into two parts: (i) the practical issues of learning the various software and (ii) the discussions about the organisation of lessons that would benefit students the most. The participants had the opportunity to learn the softwares and also discuss possible lessons.

The training course aims were shared both by the University of Cyprus and University of Athens staff as well as with the participants. It was understood that by the end of the training course the teachers would have got to know the project members, learn the softwares and share ideas about the teaching of calculus with the use of dynamic software.

We believe that these aims were achieved. Good communication was established between those who carried out the course and the trainees. Such courses are very important since they give the opportunity to attendees to get together and in a small period of time be offered a complete package of information about new software as well as its applicability in the classroom.

All the produced material of the project was delivered to the teachers.

Methodology

The duration of the course wasseven3-hour sessions. The seven sessions took place during the period October to November 2006. All sessions were held at the Mathematics Computer Laboratory of the University of Cyprus. The Mathematics Computer Laboratory of the University of Cyprus has 18 computers all installed with the most recent versions of Geogebra, Authograph and Euclidraw.

The course was scheduled as follows:

Session one -Geogebra Software

The first session started with a presentation about the CALGEO project, its aims and partners. A brief presentation of all the participants followed, where each person gave some information about himself/herself.

During the first session the teachers were introduced to the Geogebra menus, processes and help. All participants had a computer in front of them, installed with the latest version of Geogebra software. The instructors were first showing one or two of these function (depending on their complexity) and then the attendees were asked to repeat the action or in some cases work on a simple task. At any point of the presentation the course participants were able to express their comments or questions.

Session Two – Geogebra Activities

In Session 2 participants were presented with Geogebra activities. Each activity was read carefully by the teachers and executed as if they were students themselves. Once this was done teachers expressed their questions, views and ideas about possible modifications. The secondary school teachers were also given the opportunity to read and discuss the rest of the activities and decide which ones to apply in their classrooms. The teachers made also useful recommendations for further improving the content and the format of the activities:

The topics that were presented were:

• Functions
• Tangent of a function
• Polynomial
• Integration

Session Three – Autograph Software

During the third session the secondary school teachers were introduced to the Autograph menus related to functions, processes and help. All participants had a computer in front of them, installed with the latest version of Autograph software. The instructors were first showing one or two of these function (depending on their complexity) and then the attendees were asked to repeat the action or in some cases work on a simple task. At any point of the presentation the course participants were able to express their comments or questions.

Session Four – Autograph Activities

In Session 4 participants were presented with Autograph activities. Each activity was read carefully by the teachers and executed as if they were students themselves. Once this was done teachers expressed their questions, views and ideas about possible modifications. The secondary school teachers were also given the opportunity to read and discuss the rest of the activities and decide which ones to apply in their classrooms. The teachers made also useful recommendations for further improving the content and the format of the activities:

The topics that were presented were:

• Parabola
• Functions
• Tangent
• Equation systems
• Polars
• Derivative
• Integration

Session Five – Euclidraw Software

During the fifth session the secondary school teachers were introduced to the Euclidraw menus related to functions, processes and help. All participants had a computer in front of them, installed with the latest version of Euclidraw software. The instructors were first showing one or two of these function (depending on their complexity) and then the attendees were asked to repeat the action or in some cases work on a simple task. At any point of the presentation the course participants were able to express their comments or questions.

Session Six –Euclidraw Activities

In Session Six the mathematics teachers were presented with activities that can be used in the classroom for the teaching of the following subjects:

• Introduction to infinite processes
• Introduction to the limit of a sequence
• Introduction to continuity of a function at a point

The mathematics teachers worked on the activities as if they were students and then expressed their views, critique and suggested possible modifications and extensions.

Session Seven – Euclidraw Activities

In Session Seven the mathematics teachers were presented with activities on:

• Derivative
• Tangent
• Theorem of mean valueof derivative and an application in a monotony function
• Integration

Again they had the opportunity to work on these activities, decide which ones to use in their classroom and express their views in regard to their content.

Teaching Calculus Using Dynamic Geometry Tools

EU project: 118982-CP-1-2004-1-GR-COMENIUS-C21

EUROPEAN COMMISSION

SOCRATES PROGRAM

COMENIUS 2.1

Deliverable

Report on Classroom Application

University of Cyprus

Report on Classroom Application

The implementation phase of the project took place in the period February – April 2007.

Table 1, presents in detail the number of students, the school, the district of each class and the total number of students involved in the study.

Table 1:

Class / Grade / District / Number of Students / Topic
Class 1 / Year 12 / Nicosia / 20 / Definite Integral
Class 2 / Year 12 / Paralimni / 19 / Definite Integral
Class 3 / Year 11 / Limassol / 9 / Definition of Derivative
Class 4 / Year 11 / Evrihou / 20 / Derivative and Tangent

The duration of each lesson was 80 minutes. Table 3 presents the main activities and objectives of the two lessons.

Table 23

Lesson / Objectives
1 /

1. To understand the concept of definite integral
2. To calculate the area below the graph

2 /

1. To understand the concept of definite integral
2. To calculate the area below the graph
3. To understand how we may find a more accurate measurement of area below the graph and xx´axis

3 /

1. To understand the concept of derivative
2. To understand the concept of tangent of a curve at a point A
3. To give the definition of tangent
4. To find the equation of a tangent at a point
5. To calculate based on the definition the tangent of a graph of a function at a certain point

4 /

1. To understand the concept of tangent of a curve at a point A
2. To give the definition of tangent
3. To find the equation of a tangent at a point
4. To calculate based on the definition the tangent of a graph of a function at a certain point

Investigation of students’ impressions of the lessons

The feedback that the students gave in regard to these lessons is presented below.

Students were first asked about their impression of the lesson, both in regard to its degree of difficulty as well as to whether they found it interesting. As shown in Figure 1, 30% found these lessons very interesting and 50% interesting. In Figure 2 we can see that more than half of the students claimed that the sessions were easy whereas around 30% thought that they was difficult. However,.

Figure 1: Impressions of the lesson of the day

Figure 2: Impressions of the level o f difficulty of the lesson

These findings become even more encouraging when we contrast the findings of these two questions with the two that follow. In the following two questions, we asked the same group of students to tell us how difficult they perceived mathematics to be in general and how interesting they found this subject. Most of the students claimed that they only like mathematics a little and that they tended to find them a little bit difficult. Thus, it may be argued that the lessons that they for the teaching of algebra with technological tools, they found them more easy and more interesting than they usually do.

Figure 3: Beliefs about mathematics

Figure 4: Belief about the level of difficulty of mathematics in general

When asked about their impression of the specific technological tools used in the classroom half of the students thought that they were relatively difficult to use, whereas over 45% thought that they were easy to use. It has to be noted here, that technological tools are seldom used in the mathematics classroom in Cyprus and in schools in general. Irrespective, if the difficulty that some of the students found, over than 40% liked these technological tools a lot and around 35% quite liked them. The percentage of students that claimed that they did not like them was around 3 %.

Figure 5: Impressions of the technological tools used in the classroom

Below we present some of the comments and more detailed responses that the students provided about the lessons:

Question 1: In what way did this lesson differ to all the previous ones that you have done?

Questionnaires. / Answers
1, 8, 14, 16, 17, 20, 21, 22, 27 / Use of computer
2 / We have used the computers to find the definition of integral
3 / It was more interesting since we can see how things really are and not only the operations. In addition to this, I believe that this lesson was more interesting to the students because we used the computers instead of the board.
4 / It was interesting
5 / It was done with the use of the computer and we had more accurate representations and we could see more correct results.
6 / We have used the computer and we have therefore avoided many calculations. Wehadmorefreedom.
7 / It was different since we did mathematics through the computer.
10, 19 / It was more interesting because we were on the computers.
11 / It was something different and it is more interesting.
12 / It was the first time we worked on the computers in the computer lab and we were given instructions about the way that we should work.
13 / With the computer we learned the meaning of τουαόριστουκαιορισμένου integral and we managed to create the curve F(x)=x2/4+3x and learn how to process it.
15 / It was special and interesting and more modern.
18, 30 / It helped us understand the chapter better and it was more interesting.
23, 29 / The use of the computer from our teacher.
24 / He was using visual aids that were more interesting.
25 / We had to reach some conclusions through various stages.
26 / There was activity, interest from all the students. It was something different, new and impressive and that is why we were are all participating.
28 / We have used the portable computer and the projector
31 / A presentations was done with the laptop
32 / It was very interesting.
33 / It was more interesting and we were more vigilant in the lesson.
35 / We had the laptop and we have completed a questionnaire.
9, 34 / Noanswer.

Question 2: Describe what you liked about today’s lesson.

Questionnaire. / Answer
1, 4, 17, 27 / Use of computer
2 / I liked everything because the lesson was different and more practical.
5 / The measurements were accurate and we had the correct picture of the function graphically.
6 / It was interesting to see the area and how much we can approach it.
7 / We saw the lesson through a different was and I liked the way that we have worked.
8 / I liked that we have used the computers because it was different from other lessons and it was more interesting.
10 / I discovered new things about mathematics that I did not know before.
11 / I generally liked the program, the construction of functions and their easy manipulation.
12 / The fact that we escaped from the regular routine of the lesson an used computer, something that lead to more interest on behalf of the students.
13 / I was impressed by the way that the program function and how it helps to understand some mathematical concepts.
14 / It was different to other lessons.
15 / The quick presentations that are constructed by the computer.
16 / I had something to do and I liked it.
18 / The representation on the computer of the calculation of area and volume and the completion of the leaflet helped us understand better the lesson.
19 / The way that we were adding more rectangles and we could see the difference of the top and bottom becoming smaller but never reaching zero.
21, 28, 31 / The presentation
22 / I liked that we brought computers and we could practically see what we do in theory and in this way they were becoming more understandable.
23 / The use f the computer since we could see the differences in the area as we were increasing the rectangles.
24 / The use of the computer. More interesting and more easy.
25 / The lesson basically “came from our responses”, through some steps we have reached some conclusion.
26 / By using the software on the computer we had easily constructed the graphical representation and we easily changed the number of rectangles without difficulty we found the area of the graph.
29 / The fact that we could see through the computer the calculation of the area of a parabolic space.
30 / The computer in the classroom and that we could see clearly the shapes.
32 / It was interesting to see with accuracy the graphical representations and the changes that we were making.
35 / The way in which the lesson was conducted.
20 / Nothing
3, 9, 33, 34 / Noanswer

Question 3: Describe what you did not like and why.

Αρ. Ερωτ. / Answer
5, 14 / There were not enough computers
9 / It was not interesting
11 / I did not like that at some point I did not know how to proceed and therefore I could not proceed.
12 / The presence of the inspector stressed both the students and the teacher, the teacher was proceeding in a fast way and he was also giving the instructions in a fast way and that is why I personally did not manage to do very well with the computer and I was behind the rest.
19 / The questionnaires
20 / Generally I did not like anything
22, 26, 30, 31, 35 / We did not have a computer each and therefore we could not do the procedures on our own.
23 / I did not like the fact that I was not using the computer myself.
27, 28, 29 / The fact that we were not using ourselves the computer
1, 2, 3, 4, 6, 7, 8, 10, 13, 15, 16, 17, 18, 21, 24, 25, 32, 33, 34 / No answer

Question4: Whattypesofdifficultiesdidyouface?

Questionnaire / Answer
2, 8, 27 / No difficulty
4, 11, 13 / To understand the program (The way of manipulating it – 1st time)
6, 21 / We were 3 people at the computer – not each one had his own computer.
12 / I did not understand very well the steps that I had to follow and I needed a better explanation from the teacher.
16 / The keyboard did not have the right buttons.
19 / I did not understand very well the previous questionnaire.
20 / Everything, when it is not interesting it is difficult for somebody to observe.
22, 25, 30 / Ididnotfaceanydifficultiesbecausewedidnotusetheprogram. Thepresentationwasdonebytheteacher.
28 / The transfer of various tools.
31 / We could not all see from the laptop.
35 / To see from a distance on the board
1, 3, 5, 7, 9, 10, 14, 15, 17, 18, 23, 24, 26, 29, 32, 33, 34 / No answer

Question 5: What changes would you make and why?

Questionnaire. / Answer
4, 10 / Nothing specific
5, 6, 14, 30 / More computers
7, 15 / Have more often lessons with the use of computer
9 / Exemption from the mathematics lesson
12 / To use the computers more often so that the students will have more interest in mathematics and the teacher will explain more analytically how to work with the computer.
13 / More training on the computers with the specific program that is directly or indirectly related to mathematics.
18 / If students themselves were using the computer it would have been much better.
19 / Three dimensional shapes, more practice, less theory. Noquestionnaires.
20 / The lesson should never be taught like this again.
21, 22 / Eachstudentshouldbeusinghisowncomputer.
26 / To install the program on the school computers so that the students will have the chance to get to know it.
27, 28, 29 / Weshouldbeusingthecomputer.
31 / More technology in school so thaw we can use it.
35 / Better projector
1, 2, 3, 8, 11, 16, 17, 23, 24, 25, 32, 33, 34 / No answer

Question 6: Ifyouwantedtoexplaintoafellowstudentthebasicmathematicalideathatyoudealtwithinthelessonwhatwouldyousay? Describe in a few words