Computer-aided mathematics instruction with Mathematica 3.0

The author discusses research to test the motivation of students using Mathematica

by Andreas Holzinger

This article consists of three units. The first is a short overview of new technologies being used for mathematical instruction. The second is a presentation of a study in which the motivation of Austrian mathematics students using computer-based teaching was analyzed. Within the third, a unique school experiment conducted in Graz, Austria is introduced: mathematics instruction conducted using Mathematica 3.0 and Maths & Fun notebooks.

TECHNOLOGY AND EDUCATION

TECHNOLOGY AND TOOLS

A sophisticated interactive learning system like Mathematica 3.0 brings a dimension of activity and motivation to the classroom unparalleled by any other teaching medium. By putting Mathematica 3.0 to use in the classroom, opportunities for learning - which are not at all possible via traditional school lessons - can be realized. 
However, Mathematica 3.0 should not be compared to historical "teaching machines" of pedagogic cybernetics in the early sixties. MANDL & HRON (1985) found that students with "weak" computer skills make the most significant gains when using computer-based learning (cf. WEIDENMANN (1994), p.548). However, BOSCO (1984) states emphatically that there are no guarantees when completely new technologies are implemented in new fields. Furthermore, it is possible - even probable - that newly introduced technology becomes used in unsuitable and trivial manners (ibid., p. 549).

EFFECTS OF COMPUTERS

The effects that interactive software like Mathematica 3.0 have on learning have yet to be completely investigated due in large part to the relative age of the software industry and Mathematica 3.0 in particular. However, KEITEL & RUTHVEN (1993) cit. from RETSCHITZKI (1997), p.142, confirm that the computer favors the acquisition of automatism in the field of arithmetic. That is, computational practice "become less of a pain" for students and they may work longer and with more enthusiasm at the learning objective. YERUSHALMY (1991) also confirms an improvement in mathematical competence in the field of algebra when instruction has been aided by the computer (cf. ibid.).

Since computers have penetrated into the school landscape, a pedagogical and psychological topic must be discussed: how does the effort of the computer modify the student´s perception? According to WEIDENMANN (1994), p.549, there are two opposing opinions: a) extensive computer usage promotes linear thinkers who practice less intuitive, less divergent thinking, and b) the computer promotes problem solving ability and flexible thinking.
CLEMENTS & MERRIMAN (1988) even made the determination that operation with the computer increases the creativity of the students. Their investigation was conducted using LOGO, a programming language for children developed by SEYMOUR PAPERT (1980) at the Massachusetts institute of Technology (cf., RETSCHITZKI (1997), p.146). A similar result regarding the motivation of students for real world problem solving by means of Mathematica has been confirmed by the study presented momentarily.

NEW ROLE OF THE TEACHERS

When discussing computer usage in the classroom, a natural question arises: "does the computer threaten the function of the mathematics teachers?" Investigations of SCHOFIELD ET.AL. (1994) cit. from RETSCHITZKI (1997), p.141 show that students - content to work with computers - prefer interaction with a teacher during more complex question formulation. In addition, teachers who use computer-aided instruction have more individual time for their students than those using conventional instructional techniques (cf. reference analysis of teacher behavior of SCHOFIELD ET.AL (1990), ibid., p.141). These statements reaffirm the results found during this study.

Furthermore, the results of this study show that teachers are not at all superfluous during the interactive work of the students with the computer.
On the contrary, because teachers must explain "the basics", they determine the emphases, answer questions and help in the case of occurring problems. Therefore, computers do not replace teachers but create a new innovative instructional medium, like e.g. an overhead projector at former times. 
Thus, during instructional lessons, the mathematics teacher slips into a completely new and pedagogically very interesting role. The teachers are no longer in the role of "university lecturers" that report the subject in frontal instruction, but are active helpers, attendants and consultants.

ADVANTAGES AND DISADVANTAGES OF COMPUTER-AIDED INSTRUCTION

The advantages of computer-aided mathematics instruction lay certainly in its practice relatedness, the possibility of obtaining a solution to complex, "real life" problems, the improvement of mathematical problem solving ability, and the possibility of simulation and visualization.  As a result, mathematics does not become easier for the students - as this investigation showed - but student motivation increases and, as a result, supports increased mathematical learning.

The weaknesses of computer-aided mathematics instruction can be found in hardware dependence and the problems associated with system malfunctions. However, this weakness tends to eliminate itself as hardware becomes increasingly more reliable and less expensive. Also, immediate deterioration of pure arithmetic knowledge occurs on student side, a topic which is out of the scope of this paper.

STUDY: MATHEMATICA IN TEACHING

INTRODUCTION

This study was conducted at Handelsakademie (director Dr. KUSS), a high school in Graz-Austria which specializes in preparing students for management and banking jobs. Mathematics instruction was carried out in 14 classes (350 students) with the aid of Mathematica 3.0.

The curricular goals of the mathematics instruction implemented during this study are more or less set by the Austrian Government and are similar in nature to that which can be found in many U.S. text books. Teaching goals explicitly mentioned in the curriculum, such as simulation, model formation and solution of problems which arise in nature, technology and economy, are handled especially well by Mathematica (cf. austrian curriculum BGBl. 895/1994).

HOW DOES A COMPUTER HOUR PROGRESS?

At Handelsakademie, the second age-groups have three hours per week of mathematics instruction. One hour is dedicated to more conventional "chalk
instruction" where mathematical instruction occurs on a more theoretical basis. During this hour, two things typically occur: mathematical basics are elaborated in "dry training" and conventional "arithmetic" is practiced (i.e. examples are solved without computers). The pure basics for subsequent work are elaborated at the computer simultaneously. The last two hours occurs as a "double hour" and always takes place in the computer lab. When in the lab, two students typically share a computer.

STUDENT QUESTIONAIRE AND STATISTICS

INTERVIEWS

Here are some excerpts from the student interviews:

Female, 17 years old.
Interview question: "How do you, from a student's viewpoint, find this instructional experiment? How do you like mathematics at the computer?"
A:"I think that this experiment is totally super. One can learn a lot and one gladly learns. Yes, I like it a lot."

Female, 16 years old.
I: "You are an acting participant in computer-aided math instruction. How do you find this school experiment from your point of view? How satisfied
are you with it?"
A: "I do not find the experiment so bad. I don't need to do a lot of computations as the computer does the calculation work for me. This makes mathematics a lot more of fun. Yes I am very satisfied with it."

Male, 17 years old.
I: "How do you find computer-aided math instruction?"
A:"I like this experiment very well. We are not only in the classroom and I like to work on the computer, especially together with other partners."

Female, 16 years old.
I: "How do you find computer-aided math instruction?"
A:"Well, much better than conventional chalk-instruction!"

Male, no specification.
I: "What about this teaching style?"
A:"I find this teaching style very good, I am more enthusiastic - most of all because it is fun, too."

Female, 16 years old.
I: " What´s your opinion about  computer-aided math instruction?"
A:"It is surely better than the normal manual lessons."

Female, 17 years old
I:"How do you feel about these lessons ?"
A: "I am more or less satisfied, only the program is a little complicated for me to handle."

Overall, the bulk of the students that were asked these questions (basic set N = 95) were very satisfied with the instruction procedure of computer-supported mathematics instruction and with using Mathematica in the classroom. This acceptance is very high.

EVALUATION METHOD

The following is presented as an example excerpt to support the following results of the questionaire

Question (asked before the study began, before the students have had any computer based teaching experience):
Do students believe that their problem solving capability will improve after computer-supported math instruction and Mathematica? The hypothesis and result are shown below:

Hypothesis H1:

The students will believe that their problem solving capability will improve after computer-supported math instruction and Mathematica.

ITEM 16: 
I think, my problem solving capability will improve while using Mathematica (BEFORE)

ITEM 16 „I think, my problem solving capability will improve while using Mathematica (BEFORE)“

Freq            Valid    Cum

Value Label   Value  uency Percent  Percent Percent


strongly agree   1    11     11,6     11,6    11,6

agree            2    29     30,5     30,5    42,1

do not know      3    24     25,3     25,3    67,4

disagree         4    16     16,8     16,8    84,2

strongly disagr. 5    15     15,8     15,8   100,0
                   -------  -------  -------
         
          Total        95    100,0    100,0


Mean   2,947   Std err  ,129    Std dev   1,258


Valid cases    95   Missing cases      0

The next question was asked six months later, after the students have had first experiences with computer-supported math instruction and especially with Mathematica.

ITEM 162 I think, my problem solving capability improved by using Mathematica (AFTER)

ITEM 162 „I think, my problem solving capability 
	improves by Mathematica (AFTER)“

                                Freq           Valid     Cum
Value Label  Value  uency Percent Percent  Percent

strongly agree   1    17    17,9     17,9     17,9
agree                 2    29    30,5     30,5     48,4
do not know      3    34    35,8     35,8     84,2
disagree             4    15    15,8     15,8    100,0
strongly disagr.  5      0      0         0          0
                    -------  -------  -------
         Total               95    100,0    100,0

Mean   2,495   Std err   ,099    Std dev      ,966
Valid cases      95      Missing cases      0

The next step was the test of significance by using the Wilcoxon matched-pairs signed-ranks test:

Wilcoxon Matched-Pairs Signed-Ranks Test



ITEM16 My problem solving capability 
improves by Mathematica (BEFORE) with 

ITEM162 My problem solving capability 
improves by Mathematica (AFTER)



Mean Rank/Sum of Ranks/Cases


33,45     1338     40  - Ranks(ITEM162 LT ITEM16)

27,95      615,0   22  + Ranks(ITEM162 GT ITEM16)

                   33  0 Ties (ITEM162 EQ ITEM16)

                   --
                   
		   95    Total



Z =   -2,5737      2-Tailed P =  ,0101


04 Feb 97 SPSS for MS WINDOWS Release 6.1

Verification of the hypothesis:

Since it is a question of a very significant difference (error probability p = 0,0101) the hypothesis H1 can be held:

"The students believe that their problem solving capability improved after computer-supported math instruction and Mathematica."

RESULTS

The study brought the following results (in brief):

1. Higher motivation.
The interest of the students for solving mathematics problems is greater after their work in the computer-assisted math instruction then before.

Mean (Before): 3,147; Std. err. 0,105; 
Std. dev.  1,021; Valid cases 95
Mean (After): 2,663; Std. err. 0,101; 
Std. dev. 0,985;Valid cases 95
2-Tailed P =  ,0011

2. Mathematical problems do not become easier when using the computer.
The solution of mathematical problems does not become easier for the student after exposure to computer-aided math instruction.

Mean (Before):3,158; Std.err. 0,111;
Std.dev. 1,085;Valid cases 95

Mean (AFTER) 3,400;Std.err. 0,109; 
Std.dev.1,066; Valid cases 95;

2-Tailed P =  ,1653

3. Perceived improvement of problem solving ability.
The belief that problem solving ability of the students is much better after computer-aided math instruction. 
(data see example in unit before)

4. Pure arithmetical problems are not felt more merrily.
The fun of pure arithmetical problems after computer-aided maths instruction on processing does not increase.

Mean (BEFORE)3,063; Std.err. 0,106;
Std.dev.1,029;Valid cases 95;

Mean (AFTER) 2,979; Std. err. 0,103; 
Std.dev. 1,000; Valid cases 95;
2-Tailed P =  ,6311

5. Pure arithmetical problems are not felt more easily
The students feel solution of pure arithmetical problems as not easier after computer-aided maths instruction.

Mean (BEFORE) 3,337; Std.err. 0,113;
Std.dev. 1,097; Valid cases 95;

Mean   (AFTER) 3,611; Std.err.0,095; 
Std.dev. 0,926; Valid cases 95;
2-Tailed P =  ,1090

6. Pure arithmetical abilities are not felt better.
The students believe that their abilities in arithmetic after computer-aided maths instruction did not correct themselves.

Mean (BEFORE) 3,305; Std.err.0,130;
Std.dev. 1,264; Valid cases 95;

Mean (AFTER)3,463;Std.err. 0,112; 
Std.dev. 1,090; Valid cases 95;

2-Tailed P =  ,4152

7. Promotion of the team ability.
The students find after maths instruction that the computer supports teamwork.

Mean (BEFORE) 1,642; Std.err. 0,095; 
Std.dev. 0,922

Mean (AFTER) 1,316; Std.err. 0,052; 
Std.dev. 0,511; Valid cases 95;
2-Tailed P =  ,0056

8. Instruction becomes more practical
The students consider this instruction more practical after computer-aided maths instruction.

Mean (BEFORE) 2,558; Std.err.0,119; 
Std.dev.1,155; Valid cases 95

Mean (AFTER) 2,137; Std.err. 0,100;
Std.dev. 0,974;Valid cases 95;
2-Tailed P =  ,0145

ABOUT MATHS & FUN

35 different Mathematica 3.0 notebooks were developed by WERNER CYRMON, GÜNTHER GSALLER, DR. ROLF MERTIG, DR. HANS WILDING AND DR. REINHARD SIMONOVITS for use in this study. They are now currently available on CD-ROM for sale.

HISTORY OF  "MATHS & FUN"

Since 1992, Professor DR. SIMONOVITS has been using Mathematica in Austria for mathematics instruction. During this time, he has  developed interactive learning units for use with Mathematica. Recognizing the effectiveness of student teamwork, he allows students to take exams as a team and considers this "a great pedagogic step". In 1993, Professor DR. SIMONOVITS and Professor DR. WILDING started the instructional experiment officially entitled: "Mathematics on the personal computer in the commercial college".

MATHEMATICS PROJECTS DONE WITH Mathematica

Mathematica is also very well-suited to conducting and presenting classroom mathematics projects. For example, students of the 3bk class presented the Mathematica project "Magic tower", to more than 200 parents, students and teachers (for detailed information see: "http://www. mathsnfun.ac.at"). The following year, students of the 4bk class presented an astronomy project "Mission Startrek", also to approximately 200 parents, students and teachers. (for detailed information see: "http://www.mathsnfun.ac.at/ Astronomie/Astrohome.htm").
Students of the 4ak class worked in 1997 under DR.WILDING on the software project "Maths Help". "Maths Help" is a multimedia post-auxiliary teacher which uses Mathematica as an engine (for detailed information check "http:// www.mathsnfun.ac.at/Mathshelp"). For this software, the class and DR. WILDING received a prize from the Austrian Innovation Agency in May
1997.

REFERENCES


WEIDENMANN, B.; KRAPP, A., HRSG. (1994): Pädagogische Psychologie; 3.Auflage; Weinheim: Beltz, Psychologie Verlags Union.


RETSCHITZKI, J.; GURTNER, J.L. (1997): Das Kind und der Computer; 1.Auflage; Göttingen; Bern; Toronto; Seattle: Huber.


GATES, W.H. (1995): Der Weg nach vorn: die Zukunft der
Informationsgesellschaft; 1.Auflage; Hamburg: Hoffmann und Campe.


CLEMENTS, D.H.; MERRIMAN, S.(1988): Componential developments in Logo programming environments. In: R.E.Mayer (Ed.), Teaching and learning computer programming. Multiple research perspectives; Hillsdale (NJ):Erlbaum.
MATHEWS, SUSANN; MCCALLISTER, LATITIA (1995): MATHEMATICA in the Middle School and High School: An Aid to Solving Word Problems in Algebra I :Research to test the effectiveness of MATHEMATICA in helping Algebra I students solve systems of equations resulting from word problems in two variables; In: MATHEMATICA in Education and Research, Vol.4 No.4 1995.

ABOUT THE AUTHOR

ANDREAS HOLZINGER has taught physics and psychology in high school (while earning the MPh in Pedagogics at the University of Graz); electronics, telecommunications and informatics in further education (while earning his MSc in Physics and Psychology at the University of Graz). He is completing his PhD in Education with research concerning problem solving in physics-education.

Andreas Holzinger

University of Graz-Austria

Heinrichstraße 26
A-8010 GRAZ

email: andreas.holzinger@kfunigraz.ac.at
http:\\www.kfunigraz.ac.at\gepwww\holzinge