This article was downloaded by: [University of Connecticut]On: 12 October 2014, At: 12:41Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Journal of Research onTechnology in EducationPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/ujrt20

Prospective High SchoolMathematics Teachers’Attitudes toward IntegratingComputers in Their FutureTeachingOrit Hazzana

a Israel Institute of TechnologyPublished online: 25 Feb 2014.

To cite this article: Orit Hazzan (2002) Prospective High School MathematicsTeachers’ Attitudes toward Integrating Computers in Their Future Teaching,Journal of Research on Technology in Education, 35:2, 213-225, DOI:10.1080/15391523.2002.10782381

To link to this article: http://dx.doi.org/10.1080/15391523.2002.10782381

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of allthe information (the “Content”) contained in the publications on ourplatform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy,completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views ofthe authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoever

or howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

Prospective High School Mathematics Teachers' Attitudes toward Integrating Computers in Their Future Teaching

Orit Hazzan Israel Institute ofTechnology

Abstract This article presents the findings of a study on the attitudes of prospective high school mathemat­ics teachers toward integrating computers into their future classroom teaching. Ninety-four pro­spective teachers in four classes that focused on didactic and cognitive aspects of learning math­ematics with computers were asked to present pro and con arguments that would influence their use of computers in their future mathematics teaching. A two-dimensional ftamework is used to present an analysis of the prospective teachers' arguments, which were collected through written questionnaires and class discussions. One dimension relates to the class components of a lesson: learner, teacher, mathematical content, learning environment, and class atmosphere. The second dimension is made up of the psychological aspects: cognitive, affective, and social. (Keywords: computers, learning and teaching mathematics, prospective teachers' attitudes.)

Hundreds of papers arguing that computers should become an integral part of our life, as well as our educational systems, have been published (Edelson, Pea, & Gomez, 1996; Eden, Eisenberg, Fischer, & Repenning, 1996; Flake, 1996). Because such integration requires a change in teaching methods, teach­ers should have a central role in such a transition. Despite the benefits gained from using computers to teach mathematics, the number of high school math­ematics teachers who integrate computers into their math classes remains rela­tively low (Sfard & Leron, 1996).

This article explains this situation by analyzing prospective high school math­ematics teachers' attitudes toward the integration of computers in their future mathematics classes. This topic was one of the issues discussed during a course that focused on both didactical and cognitive aspects of learning mathematics using computers. More specifically, 94 prospective teachers (from four classes) were asked to present pro and con arguments that would influence their use of computers in their future mathematics teaching. The prospective teachers' argu­ments, which were collected through written questionnaires and class discus­sions, are presented in a two-dimensional scheme. One dimension relates to the class components of a lesson: learner, teacher, mathematical content, learning environment, and class atmosphere. The second dimension is made up of the psychological aspects: cognitive, affective, and social.

In this article, I present the research background, explain what each category means, present some of the prospective teachers' arguments, and end with gen­eral reflections on the research findings.

BACKGROUND The data presented in the article were collected during a course I taught to

prospective high school mathematics teachers on the use of computers in teach-

journal of Research on Technology in Education 213

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

ing mathematics. The course focused on general computational environments (such as the Web), including possible implementations in mathematics classes, and in mathematical software tools (such as dynamic geometry environments). Emphasis was placed on general principles and ideas of integrating computers in mathematics classes.

The course was carried out in a computer lab. Most of the time, the prospec­tive teachers worked in pairs in one of the computational environments, guided by activities I prepared in advance. Each class session ended with a class discus­sion and a reflection on what was learned during the lesson. These discussions addressed both mathematical and pedagogical issues (mainly cognitive and so­cial). Sometimes these discussions took up the entire lesson. In such cases, pro­spective teachers worked first in small groups, focusing on some dilemma, and then presented the groups' opinions in front of the class. Homework assign­ments included designing activities for high school pupils in various computa­tional environments, reading papers, and discussing theoretical questions.

The data presented in this article were collected from written responses and class discussions in four classes attended by a total of 94 prospective high school mathematics teachers. During the course, several class discussions focused on the benefits as well as the pitfalls of integrating computers into mathematics classes. At the end of the courses, the prospective teachers were asked to list ar­guments supporting and opposing the integration of computational environ­ments in the teaching and learning of mathematics.

RESULTS

The two-dimensional scheme shown in Figure 1 was based on the analysis of the prospective teachers' pro and con arguments with respect to the integration of computer activities in their future teaching.

The first dimension is that of class components: learner, teacher, mathemati­cal content, learning environment, and class atmosphere. The second dimension is that of psychological aspects: cognitive, affective, and social. Thus, for ex­ample, in the cell where the learner and the affective factor intersect, arguments

Components of the Lesson

Mathematical Learning Class

-l:2 Learner Teacher Content Environment Atmosphere '-' ~

~ Cognitive ~ .......,. Factors ~ -~ Affective -...;;; ~ Factors --s ~ Social

Factors

Figure 1. Two-dimensional scheme for describing prospective mathematics teachers' attitudes toward the integration of computers into their future teaching.

214 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

describing the learner from an affective point of view are found; in the cell where the teacher and the social aspect intersect, there are arguments that ad­dress, from a social point of view, the teachers' role in a class where pupils learn with computers. In the rest of this section, the prospective teachers' arguments toward integrating computers in their future teaching are presented according to this scheme.

The cells in Figure 1 do not represent disjoint categories. Some of the argu­ments may belong to more than one cell. For example, one of the arguments was that working in a computational environment increases pupil interest in mathematics. This argument belongs to both the cognitive aspect and the affec­tive aspect. Moreover, it is possible to classifY it both in the learner category and the ::natiematical content category. In my efforts to find a coherent classifica­tion, I f.Jllowed the written formulation and classified each argument by its sub­ject as presented by the prospective teachers.

Tnere are cases in which a specific argument (e.g., "Each pupil learns at his or her own speed") appeared in the prospective teachers' responses both as a pro argument and as a con argument. Thus, with respect to the argument about the pace of learning, some of the prospective teachers appreciated the fact that this way of learning takes pupil differences into account, while others presented it as a con argument, saying that it would not bring all pupils to the same level. This is an interesting point, because it reminds us that there is no single best way for teaching and learning mathematics in general or for teaching and learning mathematics with computers in particular.

In reading the prospective teachers' arguments, the reader may be inclined to "confront" some of the arguments. A common response to a pro argument would be: "this can also be done without computers"; a cDmmon response to a con argument would be: "this also happens in the traditional teaching method." \'\That I would like to suggest is for the reader to accept the prospective teachers' arguments as a tool that may help us understand their perspective regarding the integration of computational environment in the teaching and learning of mathematics.

It is interesting to note that, though the arguments specifically address the in­tegration of computational tools into mathematics classes, many of the prospec­tive teachers' arguments are general and can be put forward with respect to the integration of computers into other subject matters as well.

I now present the prospective teachers' arguments along the first dimension. Within each category a sample of the arguments is presented, classified accord­ing to the three psychological aspects: cognitive, affective, and social. In each category, I present only the predominant arguments that appeared in many of the prospective teachers' reports and were main issues in class discussions. In ad­dition, I explain briefly what each category means and emphasize main observa­tions with respect to each category.

~ll.fote: Quotes from the prospective teachers' written responses are presented in italics. Other descriptions, which mainly summarize topics that were elaborated on in dass discussions, are presented in regular font. All the quotes were trans­lated from the original Hebrew for the purposes of this report.

journal of Research on Technology in Education 215

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

Arguments Focusing on Learners In this category a::-e the arguments revolving around the pupils in the class,

their ways of thinking, their attitudes, their system of learning values, and their behavior.

Pros Cognitive Aspect

\Vtlys of learning: 1earners can conjecture, check their conjectures, improve their solution without being embarrassed by a mistake, work in teams conducting a "mathematical conversation, "and explore mathematical ideas as far as their curios­ity guides them. Allrhis is done while the learner knows that the teacher is in the class should e:zdditional help be required.

Learners' difference!: Learning with computers enables each pupil (or pair of pupils) to progress indiz1idually, without being dependent on other classmates. The reason is the set­ting up of computer c!t::sses: each pair is invited to continue to the next activity just after the previous one is completed, and there is no need to wait for the whole class as in the traditional teaching method [with the teacher standing in front of the room}. Moreover, learning with computers encourages difforent learning styles to be expressed.

Cons Cognitive Aspect

Learners and mathematical content: There are cases in which learners may progress without understanding previous stages.

Pupils who miss classes in which computers were used may encounter difficulties in catching up with the dass.

Each pupil learns ar his or her own speed, so not all of them learn the same mate­rial.

Learners and computers: Learners become dependent on the computers (for ex­ample, their work mil}' be lost if the computer shuts down). They may lose the ability to write, calculate, or perform simple algebraic operations. Moreover, the computer shows the product of a mathematical operation (a graph, a solution of a system of equations, etr:.), but d?es not present the process used to arrive at the solution.

Social Aspect Interactior.: Arguments in this category address the lack of interaction between

classmates (in contrast to working in small groups), the lack of interaction be­tween the teacher and the pupils (because the teacher cannot attend to each and every pupil during a 45-minute lesson), and the lack of human response (versus the machine response) that some pupils need.

Notes The cognitive aspect of the pupils was predominant when the prospective

teachers presented pro arguments for integrating computers in their future teaching. Tr_is fact i::-_dicates that the prospective teachers appreciate the plau­sible contribution o: learning with computers to the understanding of math­ematics by their future pupils. Moreover, the prospective teachers specifically in­dicate ways of thinking and learning that bring about this improvement, such as the contribution of pupil interaction to the learning of mathematics.

216 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

Learner differences was presented as both a pro and con argument for inte­grating computers in mathematics classes. When presented as a pro argument, it addressed the cognitive aspect of learning mathematics; whe::1 presented as a con argument, it addressed the social aspect of learning mathematics. More spe­cifically, as a con argument, it referred to an increasing gap that such a way of learning might create between the advanced pupils and the weaker pupils (for further elaboration, see the Discussion section).

Several prospective teachers raise the con argument, which addre5ses the diffi­culties that pupils who miss a lesson with computer activities may encounter in their attempts to catch up with the class. They claim that these difriculties result because it is difficult to accomplish a learning process if one does not attend a lesson. The reason is that taking notes from a friend is not the same as exploring by oneself. In my opinion, such arguments reflect the prospec::ive teachers' rec­ognition of the nature of mathematical content, learned through exploration on a computer, as being deeper and more sophisticated in some sense. Hence, one should be active in order to learn these ideas.

Arguments Focusing on Teachers In this category there are arguments focusing on the teacher of tie class in

which pupils learn mathematics with computers.

Pros Cognitive Aspect

Teacher's role: The teacher becomes a guide and is not the focus of the lesson any­more. Moreover, being released .from his or her traditionally centered mle, the teacher can follow the pupil's work by looking at the computer screens.

Cons Cognitive Aspect

Teacher's role, status, and position: The teacher's role, status, and position change in several aspects. Here are some quotes that refer to this change: "The computer becomes the class's 'brain. ' The teacher's role is just to navigate, guide, and connect pupils' knowledge with the official mathematical knowledge. v "The com­puter is conceived as smarter than the teacher because of its numerous abilities and the fact that it does not make mistakes. "

Affective Aspect Teachers and mathematical content: It takes much more time to prepare a les­

son in a computer lab (since it requires creativity and finding appropriate tasks) than in the traditional class setting. Moreover, teachers have to be very "cre­ative" if they want to integrate computers in the current school reality, when they have to deal with factors such as unavailable computer labs and the ma­triculation exams. Some prospective teachers express their .frustration with the fact that the knowledge they gain at the university is useless in the real world (schools).

Moreover, many of the prospective teachers expressed their concerns that les­sons in computer labs are not managed traditionally and that traditional obedi­ence is replaced by some noise. More specifically, they present th~s kind of argu-

Journal of Research on Technology in Education 217

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

ment: The computer takes on the central and dominant role over the teacher in the classroom.

Social Aspect Teachers and pupils: During a computer activity, the teacher cannot devote

time to all pupils, and weaker pupils may suffer from this.

Notes It is easy to observe that the prospective teachers refer mainly to the teachers'

role in a lesson that is conducted in a computer lab. Moreover, the arguments that deal with this topic are mainly con arguments. I would like to suggest two sources for this tendency. First, the prospective teachers do not see how they them­selves may benefit from integrating computers into their future teaching; second, the prospective teachers do not feel safe in this mode of teaching. Sorr.e obser­vations on activities that address these topics-the prospective teachers' profes­sional development and their concerns-are presented at the end of this article.

As we shall see later, both in the teacher and in the learning environ:::nent cat­egories, the number of pro arguments is significantly lower than the number of con arguments. Moreover, the situation is reversed in the learner and the math­ematical content categories (i.e., the number of pro arguments is significantly higher than the number of con arguments). This can be explained by -::he pro­spective teachers' lack of teaching experience. Thus, it is reasonable to assume that, in addition to their anxieties as new teachers, the prospective teachers ex­press concerns that stem from the integration of computers. These concerns are especially expressed in the teachers and the learning environment categories. The question to be asked now is: Why do they see the benefits when they think about the learners and the mathematical content? I would like to suggest the following plausible explanation, which is based on the class discussions in which the prospective teachers' elaborated on their arguments: The prospective teach­ers are faced with a conflict. On the one hand, they believe that teaching and learning with computers may improve the learning of mathematics. This belief is reflected in the number of pro arguments in the learner and the learning ma­terial categories. On the other hand, they feel that such integration may change the traditional teachers' role, a role that they are familiar with from their experi­ence as school pupils. These concerns are reflected in the number of con argu­ments in the teacher and the learning environment categories.

To convey the spirit of this conflict, let me describe it through the following statement, which takes the prospective teachers' perspective: We know that learn­ing with computers may have a positive influence on learners and their understand­ing of the mathematical content. However, the integration of computers ir:.to math­ematics classes may change the teacher's role and the learning environment in unfomiliar ways. Thus, we should not allow change in the teacher's status tmd the class atmosphere, even in cases when the main actors (the pupils and the mathemati­cal content) may benefit from the integration of computers. Of course, su .. :h state­ments are unconsciously declared. However, being aware of such flow of thought, we can help the prospective teachers in their attempt to cope wi-rh this conflict (see the Discussion section).

218 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

Arguments Focusing on the Mathematical Content In this category are arguments that put the mathematics at the focal point

and address the effect of the computers on the way in which mathematical ideas are presented, learned, and examined.

Pros Cognitive Aspect

Mathematical thinking: There are activities that are solved better by computers. The way mathematical ideas are represented on the computer stimulates the pupils' thinking, leads them to connect among mathematical topics, and enables them to think broadly without being distracted by technical problems. Moreover, the com­puter may visualize mathematical ideas (such as the concept of derivative) and lay out a construction of complicated mathematical objects (such as 3D objects).

Improved presentation medium: Computers save time in drawing graphs, solv­ing equations, and calculating. The computer performs all these functions pre­cisely and quickly.

Social Aspect Kind of activities: Computers enable us to solve problems that would other­

wise be difficult to solve (e.g., checking a geometrical conjecture for many cases by dynamic geometry software). Moreover, the fact that the computer may serve as a cognitive partner influences the kind of activities suggested to pupils: problems with more than one solution that can be solved in more than one way. Such problems encourage pupils' creativity, cooperation, motivation, and inven­tiveness and improve their mathematical thinking.

Cons Cognitive Aspect

Evaluation: Since learning with computers emphasizes learning processes rather than final results, it is much more difficult to assess pupils' mathematical knowledge.

Improved presentation medium: This, again, is one of the arguments that stands both as a pro and con argument. Presenting it as a con argument, the prospective teachers claimed that the computer does too much work for the pupils. Even in cases when pupils do work on sophisticated mathematical activities with comput­ers, they do not know basic mathematical operations, such as drawing graphs.

Notes In this category the prospective teachers mainly analyze ways in which the math­

ematical content is viewed and conceived by pupils. I believe that the fact that the prospective teachers present the mathematics as the subject of some of their ar­guments is important, because such arguments deal with abstract topics and ideas.

From the con column we can learn about the prospective teachers' anxieties with respect to the fact that their future pupils will lose some of the ability to do basic calculations and drawings. This is similar to the case of elementary school pupils no longer learning long division. This procedural calculation was replaced with more advanced mathematical activities, such as the evaluation of the calculators' calculations as a reasonable answer. I believe that the same rec­ognition should influence the teaching and learning of high school mathemat-

Journal of Research on Technology in Education 219

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

ics. We should suggest advanced mathematical activities to our pupils, and we should use the computer as a cognitive partner that is sometimes used as a ma­chine for performing technical calculations and drawings.

Arguments Focusing on the Learning Environment The arguments in this category deal with the nature of working with comput­

ers, the opportunities they provide, and the barriers pupils encounter while learning with computers.

Pros Cogni-.:ive Aspect

~y_.. of learning: Computers provide a world of mathematical experience, includ­ing sinmlations for raising conjectures and precise tools for investigating them, per­sonal experience for all pupils and an ability to see, feel, move, construct, and ma­nipulate ''things. "

Problem solving: Some mathematical software tools offer a great source for both new problems and ways ofproblem solving. For example, the computer is a useful tool for solving problems that require data collection, data processing, and data analysi!. Computers enable one to solve problems gradually, by way of ongoing feed­back tc the pupils, both as to their success and foilure.

Social Aspect Pupil discussion following an activity that was worked out with computers

improves the pupils' mathematical language.

Cons Affective Aspect

Technical problems: There can be problems with the operation of computers. When we use computers in mathematics classes, we may waste time learning the software (even i:t cases in which we integrate the learning of the mathematics with the learn­ing of the computational environment). Moreover, we all know that we waste time whenez;er new versions appear.

Less(ms based on a computer activity may be interrupted by many technical prob­lems, such as electric power foilure; technical problems with the computers; and problems with the network, viruses, or bugs.

Logistic problems: Such problems deal mainly with the school system. Some­times u>e do not have computer labs at all· in other cases, we do not have enough compurers and jive pupils have to share one computer. Such a setting is not effi­cient for most of the pupils. In a broader perspective, the prospective teachers say that, because lessons with computers take more time than traditional lessons, we should add more learning hours or reduce the amount of learning material.

Social Aspect Computer-learner relations: There are some disadvantages to computers that de­

rive fr(Jm their being a machine. Computers do not follow learning processes, do not judge learners' performance, do not answer learners' questions, and are unable to en­courage pupils.

220 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

Notes As mentioned previously, the number of con arguments in this category is sig­

nificantly higher than the number of pro arguments. A closer look may reveal the source of this gap in the prospective teachers' concer::1s over technical prob­lems. This is an important observation because it has nothing to do with the learning, the teaching, or the mathematics itsel£ Making the prospective teach­ers aware of these concerns may help them cope with them. As soon as they dis­cover that they can either deal with such problems by themselves or have tech­nical support, they are able to turn their energy to the teaching of mathematics.

The idea of learning by successive refinements in interactive computational environments is one of the important factors that the prospective teachers men­tion as a pro argument. Indeed, it may be a result of the special emphasis that the lecturer (me) puts on that idea and the prospective teachers' attempts to meet my expectations. The idea of learning by successive refinements refers to a gradual mental construction of complex (mathematical) concepts that stems from an ongoing interaction with the computer, in which learners improve their solution to a given problem via the examination of the computer's reaction to their current solution. A general discussion about this idea can be found in Leron and Hazzan (1997); a specific discussion about the expression of succes­sive refinements in learning with dynamic geometry environments is presented in Hazzan and Goldenberg ( 1997).

Arguments Focusing on the Class Atmosphere The prospective teachers' arguments in this category refer to what is going on

during a lesson based on computer activities.

Pros Social Aspect

W'tlys of learning: Computers provide an opportunity to communicate with pup­ils around the world and an opportunity for teamwork and joint effort between classmates.

The lesson time: Time flies, and this is an opportunity to change the routine.

Cons Social Aspect

Social gap: Since a computer is an expensive device, not all pupils may be able to afford it. Thus, pupils who have a computer at home might progress faster than pu­pils who do not own a PC and gain an advantage over the latter.

Technical gap: Pupils have different levels of expertise in operating computers. This fact may cause some pupils to fall behind, making them feel uncomfortable.

The lesson time: Due to the friendly interface of the computer, lessons in a com­puter lab may be conceived as social events and pupils may not take them seriously.

Notes The prospective teachers' arguments grouped into this category belong mainly

in the social aspect. Indeed, class atmosphere is a social topic. Within this aspect the prospective teachers focus mainly on two issues:

journal of Research on Technology in Education 221

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

• the possible interaction among pupils during a lesson that is conducted in a computer lab (a pro argument)

• the social gap that may be a result of economic circumstances (a con argument)

Indeed, many articles that support the use of computers in learning processes base their arguments on Vygotsky's theory (1986), which discusses the role of social interaction during a learning session. The second central point the pro­spective teachers address here-the social economic gap-is an important factor not traditionally discussed in relation to mathematics education. However, the prospective teachers' awareness of this factor is important, and some attention should be given to this topic.

DISCUSSION The preceding analysis describes prospective teachers' attitudes toward the in­

tegration of computers in their future teaching of mathematics. From the pro­spective teachers' answers we can learn that they refer to the main components of the lesson: learners, teachers, the learning material, and the interaction among them, as well as the learning environment and the class atmosphere. Moreover, we can see that the presented arguments deal with cognitive, social, and affec­tive aspects. It is interesting to note that, in many of the written responses, after specifying the pro arguments and the con arguments, many of the prospective teachers have added a remark in the following spirit: It is worth integrating learning with computers together with learning and teaching without computers. Such a statement indicates that the prospective teachers do not take it for grant­ed that computers would solve all problems embodied in the complexity of learning and teaching processes, but rather that the integration should be given serious thought in each individual case. In regard to this, one of the prospective teachers wrote: The computer can supply information, but we may lose its potential if we do not educate our pupils to use that information. Many databases are accessi­ble now but our target should be the analysis of these data and not just getting it.

In management theories, it is well known that motivated employees support positive changes in their organizations. Unmotivated employees are satisfied with any existing situation within their organizations (lnamori, 1985, as cited in Senge, 1997). Thinking about schools as organizations, we should ask our­selves such questions as: How do we motivate prospective teachers to integrate computers in their future classes? Where and when it is appropriate to do this? How does it correspond to our educational values? How do we make the inte­gration of computers into teaching more attractive to prospective teachers?

We should not, however, ignore a particularly significant problem. Some of the prospective teachers who had a teaching experience in school systems said that veteran teachers discouraged them when they tried to initiate the idea of teaching mathematics with computers. In such cases, the class discussion was directed toward two main issues:

1. how to integrate computers in a gradual way so that veteran teachers who do not feel safe with computers will not feel threatened

222 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

2. how to prepare themselves for a situation when they themselves become the mathematics coordinators in their schools and young novice teachers who join the school staff will want to introduce innovations into tie school sys­tem in general and into the mathematics classroom in particular.

One of the outcomes of the research is a set of activities currently under de­velopment designed to help prospective teachers cope with their concerns and to encourage them to be guided by their beliefs when they feel that the integra­ti::m of computers in mathematics classes may improve the learning of math­ematics. Here are some examples of such activities.

Simulation Games We have developed simulations of situations that may occur in classes where

comp1..1ters labs are used to teach mathematics. The idea is to let the prospective teachers engage in unfamiliar situations and thus let them get a sense of such situations in a supportive environment. For example, one such simulation is a "~mpCs" requirement to get from the "teacher" final answers of an investigation that is then carried out in a computational environment. In such a situation, tie prospective teachers are asked to suggest ways to guide that "pupil" without r:roviding the final answers. In such cases, it is quite clear that giving the pupil tne final answer is not the appropriate way to guide a pupil. Thus, the prospec­tive teachers are required to look for ways to guide the pupil who is stuck, with­C•ut missing the mathematical essence of the original investigation the pupil is working on.

Prospective teachers continue to face a conflict. On the one hand, they believe that using computers may improve the learning of mathematics. On the other hand, they worry about the anticipated change in the teacher's status in a computer-based learning environment. One class discussion that occurred after ] had mentioned this conflict focused on the class of the prospective teachers as university students in our course, who were learning ways to integrate comput­ers into mathematics classes. As noted in the Background section, the course ·was organized around computer activities followed by class discussions andre­flections. Thus, I asked the prospective teachers in the class to analyze our class ~n general and to describe their view about my role in the class. When they re­=erred to their own experience as students in the course and described their view :;bout my role in the course, they realized that being a teacher of a class in -..vhich pupils learn with computers is not as frightening as they had originally ;::xpected. Still, we should not ignore their concerns and should keep referring to these concerns and discuss them when they are expressed.

Locus-Based Activities To let the prospective teachers discover how they might also gain in-depth

learning with computers, I designed a set of activities based on the concept of locus. The prospective teachers came across this concept mainly in high school when they learned the concept of circle. However, they had not seen any appli­cation of this notion, nor had they had any opportunity to work with this con­~ept meaningfully. Mter the prospective teachers had worked on the activities

journal of Research on Technology in Education 223

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

in one of the dynamic geometry environments, a class discussion took place that addressed mathemat1cd ideas (such as the concept of locus and its con:J.ec­tion to other mathematiol concepts), cognitive issues (such as what they have learned from the activities), and pedagogical considerations (such as in what sense this learning experien:e was different from other learning situations they had experienced). The idea was to let them learn a concept that was neither too simple nor too difficult so -:hat they would be able to reflect on their own learning processes by thinking about what they get out of such learning.

Professional Development Activities Another activity was de1eloped to help the prospective teachers learn how the

use of computers could promote their professional development as teachers. Ar the stage where prospectiYe teachers are still university students, this discussion is theoretical and hypothetical. However, there are ways that using computers can promote prospective ar:::l inservice teachers' professional development. Here are some of the suggestions offered during that discussion:

• Combining school work 'Nith a job in a company that develops educational software tools.

• Coordinating a discussion group on teaching mathematics. (The member~ of this discussion group may be the teachers at their own school, the teachers in their country, or another group of teachers.)

• Joining an academic teaxn that develops learning material.

Another important issue discussed in class and mentioned in the prospective teachers' written reports was the influence of social gap on learning mathemlt­ics with computers. In the class discussions, we tried to identifY ways to help re­duce the social gap betweer_ classmates, as far as owning computers goes. The idea is to work out this problem with the prospective teachers so that, if they face such a gap in the future:, they will have some solutions. For example, the prospective teachers suggest~d dividing the pupils into groups with a mix of pu­pils having home computers and pupils not having home computers. They c_lso suggested opening the scho::ll computer labs in the afternoons.

I propose the following drrections for future research:

• Research on inservice teachers' attitudes toward the integration of comput~rs in mathematics classes. Such a study may provide us with additional insights since the research repon:ed in this article focuses on how prospective teach·~rs who are not yet part of a school system see this issue before entering the world of teaching and learning.

• Research on the distinctions among different kinds of computational environ­ments. My study describes the prospective teachers' attitudes toward the ir:te­gration of computers in n:athematics lessons in general. I suggest that this topic be extended by investigating particular computational environments. such as the Web and dynamic geometry environments, in use in the teaching of mathematics. Such infcrmation may help us understand to what extent we can generalize the prospective teachers' arguments.

224 Winter 2002-2003: Volume 35 Number 2

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4

• Research on the distinctions among different kinds of teaching methods. Such studies would help us understand what methods of teaching with com­puters the prospective teachers prefer (investigations, calculations), why they prefer one teaching method over another, and which part of the lesson they like the most. Ill

Contributor

Dr. Orit Hazzan is a senior lecturer at the Department of Education in Technology and Science at the Technion-Israel Institute ofTechnology. Her PhD dissertation analyzes undergraduate students' understanding of abstract algebra concepts. In the last five years, she has worked on and researched the integration of information technologies in institutions of higher education. (Address: Dr. Orit Hazzan, Department of Education in Technology and Sci­ence, Technion-Israel Institute ofTechnology, Haifa 32000, Israel; oritha@ techunix.technion.ac.il.)

References

Edelson, D. C., Pea, R. D., & Gomez, L. M. (1996). The collaboratory note­book. Communications of the ACM, 39(4), 32-33.

Eden, H., Eisenberg, M., Fischer, G, & Repenning, A. (1996). Making learn­ing a part oflife. Communications of the ACM, 39(4), 40-42.

Flake, J. L. (1996). The World Wide Web and education. Computers in the Schools, 12(112), 89-100.

Hazzan, 0., & Goldenberg, E. P. (1997, July). An expression of the idea of suc­cessive refinement in dynamic geometry environments. Proceedings of the 21st In­ternational Conference for the Psychology of Mathematics Education, 3, 49-56.

Leron, U., & Hazzan, 0. (1997, October). Computers and applied construc­tivism. In D. Tinsley & D. C. Johnson (Eds.), IFIP WG 3.1. Working Confer­ence-Secondary School Mathematics in the World of Communication Technologies: Learning, Teaching and the Curriculum (pp. 195-203). Grenoble, France.

Senge, P.M. (1990). The fifth discipline. New York: Doubleday/Currency. Sfard, A., & Leron, U. (1996). Just give me a computer and I will move the

earth: Programming as a catalyst of a cultural revolution in the mathematics classroom. International journal of Computers for Mathematical Learning, 1(2), 189-195.

Vygotsky, L. (1986). Thought and language. Cambridge, MA: The MIT Press.

journal of Research on Technology in Education 225

Dow

nloa

ded

by [

Uni

vers

ity o

f C

onne

ctic

ut]

at 1

2:41

12

Oct

ober

201

4