1. science
2. mathematics

A summary about the experiences
how to integrate CAS hand-held computers (TI-89/92)
in Mathematical Education in Austria.
Otto Wurnig
Institute of Mathematics
University of Graz, Graz, Austria
[email protected]
Key words: Austrian CAS Projects, module principle, observation window, spiral of
creativity, tests, minimum knowledge, examination practice.
Abstract. Up to 1990 the minimum equipment in high schools in Austria was two computer
labs, one of which always met the requirement of even large classes where very often two
students were working per computer. But they suffered from a lack of really useful software.
In spring 1991 DERIVE 2.x was bought with a general licence for all schools in Austria
preparing students for university level. In the spring of 1992 ACDCA (Austrian Centre for
Didactics of Computer Algebra) was founded in order to create a forum for ongoing
discussions and research concerning the use of CAS in teaching mathematics, especially in
grammar schools. Conferences, meetings and publications are intended to offer a framework
for university teachers, teacher trainers and school teachers to exchange their experiences
and to do research projects (Homepage: http://www.acdca.ac.at). Three Research Projects
have already been carried out: the Austrian CAS I Project (PC-DERIVE) in 1993/94
involving 17 schools, the Austrian CAS II Project (TI92-DERIVE) involving 46 schools and
the Austrian CAS III Project (Electronic Media in teaching Mathematics) involving 60
schools. A fourth project named Technology in Mathematical Education was started in
September 2001. In my lecture I will speak as one of the project leaders and refer to the
results of the last projects with TI92-DERIVE as seen by the participating students and
teachers.The teachers appreciated the White Box/Black Box Principle, the module method,
the graphical possibilities and in combination with it the Window-Shuttle-Techniques. During
the CAS II Project the teachers realised that they had to change their examination practice. In
the CAS-III-Project some new models of assessment were used and controlled.
1 Introduction
In 1993/94 the Austrian CAS I Project (DERIVE-Project) involving 700 students,
17 schools and 28 mathematics teachers was carried out. The leading principle for the
research teachers was the white-box/black-box principle of B. Buchberger (Research
Institute for Symbolic Computation of the university of Linz) [1].
One topic where the white-box/black-box principle can be used in a wonderful way is
Algebra. First the students learn to transform terms, afterwards to solve a linear equation
stepwise with or without DERIVE. When they have done well they are allowed to take the
command SOLVE to get the solution at once. When they have to solve a system of linear
equations for the first time they first have to reduce the system to one linear equation with one
variable. They are already allowed to solve this equation with SOLVE. Once they can do such
reductions in a reasonable time, they are allowed to use the command SOLVE for a system of
equations too [2].
B. Buchberger also admits that there are exceptions to the white-box/black-box-principle. "It
may sometimes be reasonable to play, first, with an algorithm as a black box in order to
obtain a better understanding of the problem before one goes into the details of teaching the
underlying theory. This may well be reasonable in certain situations." (Black-box/white-boxprinciple) This modification was first suggested at a Spring School on Didactics of
Computer Algebra in Krems (Austria) 1992. It was exemplified in the lecture “Introducing
Calculus with DERIVE” [3].
2 TI-92 Project
In the German report of the Austrian DERIVE-Project H. HEUGL writes: „It would be ideal if
every student had a portable CAS-calculator, which could be linked up with the CAS in a
computer lab, in his school bag.“ [4]
In Autumn 1996 the TI-92 came on the market and the planning of an Austrian CAS II
Project (TI-92-Project) involving 1500 students, 46 schools and 70 mathematics teachers
began. It was carried out in 1997/98. Half of the research teachers came from the CAS I
Project, half of them were new. All teachers had to be trained handling the TI-92. It was great
luck that just at the right moment the world wide T³-Project “Teachers Training with
technology” started in Austria. The didactical ideas with the TI-92 in the start phase were
written down in the book “Der TI-92 im Mathematikunterricht.” [5]
The mathematics teachers in the CAS-Projects were supported through the research of
K. Aspetsberger, who reported about his own experiments with CAS for the first time in 1989
[6]. He created the module principle. He discovered that the intensive use of functions in
mathematics lessons using computer algebra systems led the students to a new understanding
of functions. In contrast to regular mathematics lessons where students see functions like
objects having certain characteristics, functions in computer algebra systems are seen as tools
for computing certain tasks. In the project several sorts of modules could be distinguished.
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DERIVE-modules used as black boxes or sometimes made “white” when teachers are using
the black-box/white-box-principle e.g. CROSS(a, b) to get the cross product of two vectors
or modules like ODE1 to solve differential equations.
Modules created by the students in the white box phase. The main goal is the cognitive
learning process during the production of the module.
Modules created by the teacher and used by the students as black boxes (utility files).
K. Aspetsberger reports that good students can sometimes create their own system by defining
new functions with the experience of programming.. But he also admits that less able students
often make use of modules without a detailed understanding of the problem and use functions
without reflecting upon whether the functions are appropriate for solving the particular
problem. Students and teachers have more difficulties to detect errors if students do not know
the way the functions work [7].
Beside the classical use of computer algebra systems for symbolic differentiation and
integration TI92-DERIVE can also be used to work simultaneous in two windows. By solving
the following problem many students find that it is preferable to split the screen. This multiple
window technique is named window-shuttle principle in Austria [4]. It is a technique which
many students like to use to get new ideas or to control their calculations.
Problem:
The curve, which a flying object makes in the air is defined through the graph
of the following function:
f(t) = 45 + 20t - 5t² . h means the height and t the time.
Draw the curve of the flying object in the (t, h) - co-ordinates system by using of the TI-92 in
your test book! (a) At what time does the object reach the highest point of the trajectory and
when does it hit the ground (h=0)? (b) In which height does the trajectory start and when
does the object reach the same height again?
Figure 1: The students split the window to work simultaneously.
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3
Observation Window
The TI-92-PROJECT was designed with the experience gained with the design of the CAS I
Project. It is useful for investigations concerning the use of computer algebra systems in
mathematics to concentrate on a few topics only. Those topics in the CAS I Projects are called
observation windows [8]. In every form two observation windows were taken. In the 9th
form, where thirty teachers were engaged, the first observation window e.g. was to find out
the various methods the students were taking with the TI-92, when they had to solve a
quadratic equation for the first time. They had to do this in at least three different ways
within a time limit of twenty minutes [9]. It was surprising which variants of right ways the
students found.
Different ways
percent
number of ways
percent
SOLVE
Graph
Table
by hand
FACTOR
93%
50%
39%
4%
3%
0 way
1 way
2 ways
3 ways
more than 3 ways
7%
35%
18%
30%
10%
Table 1: Quadratic equation - ways of solution for the first time
Studying the table you can see that nearly all students (93%) used the command SOLVE. The
students first entered the equation (x+6)² = (x+3)²+x² and then most of them took the TIcommand SOLVE to get the result. As hoped for, 58% of the students found at least a second
way (table or graph).
It was surprising though, which variants of right ways were found by the students. The
following two variants of solution strongly influenced the further mathematical education.
Some students saved the equation (x+6)²=(x+3)²+x² without changing it as function y1(x). In
this case the table showed true for x=9. This way infrequently leads to the right solution with
equations, but it is a very good way for inequalities!
Table 2:The saved quadratic function is true for x=9.
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Figure 2: equation and inequality as a function.
Table 3: Truth table of equation and inequality.
Some students saved the expression (x+6)² under y1(x) and the expression (x+3)²+x²
under y2(x) and they found that the value of y is the same (=225) in both tables when
x=9. The students found this way in analogy to the way of solution for a system of linear
equations with two variables!
Figure 3: one equation – two functions
Table 4: for x=9 follows y1(x)=y2(x)
The way with two tables produces the basis for the right understanding of the graphic way for
the intersection of two curves (TI-command INTERSECTION).
Figure 4: One equation – two functions
Figure 5: Two graphs – one intersection point
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This graphic way for the intersection of two curves (TI-command INTERSECTION) is very
important for the solution of algebraic equation such as x = 2 . sin(x) in the 10th form!
The statistics show that the method with the graphic window (Intersection and Zero) and with
the table window was very often taken. Both variants of solution strongly influenced the
further mathematical education in the research classes.
Beside the observation windows the other goals of the project were to study: the basic
principles of teaching mathematics with the TI-92, the basic skills which students have to
learn with it in mathematics and the influence of the TI-92 on oral & written exams.
4. The Influence of CAS in Learning Mathematics
One possible model to describe “how we do mathematics” is the spiral of creativity by
B. Buchberger [10]:
experimental facts
problem solving
computing
observing
algorithm
conjecture
programming
proving
theorems
Figure 6: The spiral of creativity
The spiral begins with the observation of data materials or with a problem, the solution of
which can be found in the development of algorithms or in the creation of new concepts.
Through analysing, experimenting or generally through heuristic strategies conjectures are
found, theorems are formulated and ideas of proof sought.
If you are content with the result after entering the circle once or repeatedly, you can go up
to a higher level of the spiral. You are in the next level because you have found a better
knowledge or a better algorithm.
B. Buchberger notes that his answer is “recursive” and can govern maths education at all
levels and areas of mathematics including the elementary first steps in mathematics at high
school as well as the most advanced areas in university mathematics [11].
H. Heugl, the chairman of ACDCA and organizer of the DERIVE-Project, recognizes three
stages in learning mathematics if students are using symbolic computation systems in the
classroom [12]:
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Stage I – the heuristic stage
Typical activities: finding conjectures, trying to devise problem solving strategies, student
oriented experimental ways of learning, methods of trial and error, testing models, testing and
interpreting results.
Stage II – the exact stage
Typical activities: corroborating assumptions, deducing algorithms, proving theories.
Stage III – the application stage
Typical activities: applying algorithms, solving problems, modelling, looking for more
suitable models for real life problems, using the CAS as a black box for operations which
students developed themselves in stages I and II and then testing and interpreting.
H. Heugl notices that in traditional mathematics education the experimental or heuristic phase
often does not exist. In the Austrian DERIVE-Project, however, the growing importance of
the student oriented experimental phase in learning mathematics is an important result.
5. Comparison of the Use of DERIVE and the Use of TI-92
In addition to the observations made by the research teachers, an independent centre of
school development (Zentrum für Schul-Entwicklung), which is an institute of the ministry of
education, questioned the teachers and students. The results were published in three reports
[13],[14],[15]. The ZSE-Reports contain some interesting and sometimes surprising results:
A u s m a ß d e r F r e u d e a m M a t h e m a tik u n t e r r ic h t
5
v o r d e r E i n f ü h r u n g v o n D E R IV E 1 9 9 3 /9 4
n a c h d e m E in s a tz v o n D E R IV E 1 9 9 3 /9 4
wenig Freude - viel Freude
v o r d e r E i n f ü h r u n g d e s T I 9 2 1 9 9 7 /9 8
4
n a c h d e m E in s a tz d e s T I9 2 1 9 9 7 /9 8
3
2
1
7. S tufe
9. S tufe
1 0 . S tu f e
1 1 . S tu f e
Figure 7: Maths and fun with PC-DERIVE and TI92-DERIVE
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With the use of the TI-92 fun in mathematics lessons increases at every level, with
DERIVE this effect only turns up from grade 9 (15 years old students) onwards.
The largest increase of fun with DERIVE was noticed in grade 9, with the TI-92 in grade 7
and grade 11 respectively.
In both, the PC-DERIVE and the TI-92 project, boys derived more fun from it than girls.
Whereas in the TI-92 project the TI-92 was a great help for more than two thirds of the
students in at least three fields (tests, home assignment and problem solving in school), only
about half of the students in the DERIVE project thought DERIVE helped them and that only
for the problem solving at school. So the advantage of the TI-92 is its availability at all
times.
6. Tests in Mathematics with the Use of the TI-92
In the Austrian CAS II Project (TI92-Project) 1997/98 the students of 70 research teachers
wrote their tests in mathematics with the TI-92. At their final meeting in 1998 the teachers
collected their most important results [16]:
• the problems in tests have to be more goal oriented → text longer instead of shorter.
• TI-92 has no floppy → much documentation in test book, therefore fewer examples.
The point P = [-8;-3] is to be reflected on the line g: 3x+2y+4=0 (Sketch a diagram on the
work sheet!) What are the co-ordinates of the reflected point?
(a) First solve the problem geometrically in your test book!
(b) Solve the problem in an algebraic way with the methods of the analytic geometry on the
work sheet! Write down your plan in the left column and document the important interim
results in the right column! Do the calculations without the TI-92 on the flip side!
left: steps of the plan!
right: TI-92 documentation!
[-8;-3] → p
P
g1: 3x + 2y + 4 = 0
[3;2] → n
g2 = g( ∋ P, ⊥ g1 )
X=P+t.n
[x = 3t - 8, y = 2t - 3]
3.(3t - 8) + 2.(2t - 3)+4=0
SOLVE(ans(1), t) t = 2
S = [-2;1]
P* = [4; 5]
g1 ∩ g2 = {S}
S=(P + P*)/2 → P* = 2S - P
Table 5: Text longer – exact planning – good documentation
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difficult decision: What is to be the minimum knowledge in mathematics without the TI-92?
Solve the two formulas per hand according to the indicated variables!
a)
v = ? (no double fraction in the result)
b)
h=?
2
2
g·. t
m·. v
y = v·. t - ——————
e = m·g·h - ——————
2
2
Table 6: Minimum knowledge: to solve a simple equation by hand
difficult decision: What minimum knowledge of TI92-commands is an absolute must?
(1) Substitute the number 7 for x in the term 12x-20:
IL:______ 12x-20x=7___________________OL:______________64_________________
(2) Multiply the following terms:
(3x²+7) (2x-3)
IL:___ EXPAND((3x²+7).(2x-3)____________OL:____ 6x³-9x²+14x-21________________
(3) Convert the result of (2)into the input of (2):
IL:____ FACTOR(6x³-9x²+14x-21)__________OL:_____ (3x²+7).(2x-3)_______________
(4) Solve the equation
3x-7 = 4(4x+2)
using the TI-command SOLVE:
IL:_____ SOLVE(3x-7=4.(x+2),x)__________OL:_______x=-15/13___________________
Table 7: Minimum knowledge with the TI-92
• for solving problems it is very important not always to insist on the use of the TI-92.
• students find new ways with the TI-92 → more work for the teacher
modules/programmes are a good chance for good students→a new problem for bad students.
Calculate the perpendicular distance of the point P from the line AB with the formula taught:
[0;-1] →
a
:
[-3;4] → n
:
[-5;4] → p
IL:___dotp(p-a,unitv(n))_________________OL:______7___________________________
Table 8: To work with modules – a good chance for good students
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7. The influence of CAS on the Examination Practice.
After a meeting of the project management five topics were chosen for the CAS III Project.
2000 students in 70 experimental classes from grade 7 to 11 took part in this project. It was
carried out in 1999/2000. The general report (in German) has been finished in Autumn 2001
(http://www.acdca.ac.at/). One topic was the influence of CAS on the Examination Practice.
In spring 1999 a team of teachers under the leadership of H. Heugl developed some variants
of a new model of assessment [17]. In accordance with the Ministry of Education
experimental studies to test the new model were carried out in 1999/2000. I chose to use the
following variant in form 11: The fundamental idea of this variant is to use the pre-set time
for written tests in a school year - 350 minutes in form 11 - in different ways [18]:
For short tests - up to a maximum of 25 minutes - to check reproductive skills or
reproductive knowledge with or without CAS.
1) Circle:
x² + y² - 8x + 10y = 0
Calculate M = (m, n) and r
I: x²-8x+16+y²+10y+25=16+25
I: factor(x²-8x+16,x)
I: factor(y²+10y+25,y)
O:
O:
(x-4)²
(y+5)²
2) What is the distance from line g: 3x - 4y = 12 to k: M=[-5, -3], r = 2
I: dotp([-5,-3]-[4,0],unitv([-3,4])
O:
3
3) Calculate one of the two intersection points of the line g with the circle k!
g: X = [7, -4] + t . [3, -2]
k: (x-2)² + (y+5)² = 65
I: (x-2)² + (y+5)² = 65 | x=7+3t and y=-4-2t
I: solve( 13t² + 26t + 26 = 65, t)
I: [x, y] = [7, -4] + t . [3, -2] | t = 1
X = [x, y]
O: 13t²+26t+26=65
O: t = 1 or t = -3
O: [x = 10 y = -6]
4) Find the equation of the circle tangent in point B of the circle k:
k: ([x, y] - [7, -2])² = 20
B = [3, -4]
I: dotp([x,y]-[7,-2], [3,-4]-[7,-2])=20
O: -4x -2y + 24 = 20
Table 9: A short test of 25 minutes (I…Input, O…Output)
The four problems were prepared in the mathematics lessons. The aim of the 25-minute-test
was to check the minimum knowledge in analytic circle geometry in a reproductive way.
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For one longer test per half-term, e. g. 100 minutes, - to check problem solving skills. There
should be sufficient time to experiment, and to use materials which have been worked out at
school or at home.
After dealing with the conic sections in the mathematics lessons the next test was a problemoriented test of 100 minutes. The three problems were intentionally taken from the three
chapters: analytic geometry in R³, conic section geometry and calculus.
The following problem [19] could either be solved with the new concept of calculus or with
the possibilities of the TI-92.
Supposing a point object is moving on the curve of the function
f: x → (1/4) . (x³ + 2x² - 3x), [-2;2] → R
and is striking a wall (x=2). Find out the size of the angle of impact?
Sketch a diagram oriented by the graphic window of the TI-92!
3 students calculated the slope k of the tangent for x=2 in the home window (first derivative)
and afterwards the angle of impact with tan-1(k).
3 students tried to make a drawing with the help of the co-ordinates given in the TRACEMode of the graphic window and then tried to measure the angle.
9 students produced the tangent inclusive the equation in the graphic window with the
command tangent.
Figure 8: The solution in the algebraic window
Figure 9: The solution in the graphic window
2 students of the 9 plotted the tangent with the help of this equation and then measured the
angle.
3 students of the 9 saw k in the equation of the tangent and used the command tan-1(k).
The last 4 of the 9 took k (= 4.25) of the equation of the tangent and with the direction vectors
[1, k] and [0, 1] they calculated cos(α) with the command dotp.
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For working out a short chapter of mathematics, which has not been dealt with at school.
Each student should prepare his short chapter in written form at home and present it to his
classmates at school.
The preparation of a short chapter of mathematics at home and the ensuing presentation at
school proved to be the most difficult part on behalf of the students. Most of the students had
never before prepared a disposition for a theme at school. It took me two lessons and many
discussions to make even good mathematicians understand how to prepare an acceptable
written and oral presentation.
The problems of the Examination Practice in CAS supported classes are of such great
importance that they led to an international ACDCA-Conference which took place in
Portoroz, Slovenia, in July 2000. Its theme was “Exam Questions & Basic Skills in
Technology-Supported Mathematics Teaching”. In the centre of the conference was the
question “What are the indispensable Manual Calculation Skills in a CAS-Environment?”
[20]. The lecture was published and initiated an intensive discussion in Europe and every
mathematics teacher using CAS in his mathematical education is continuously faced with it.
References
[1] B. Buchberger, Why Should Students Learn Integration Rules?, RISC Linz, Technical
Report No. 89-7.0, University of Linz, Austria, (1989).
[2] B. Kutzler, DERIVE – The Future of Teaching Mathematics, The International DERIVE
Journal, Vol. 1, No. 1, (1994), 37-48.
[3] A. J. Watkins, Introducing Calculus with DERIVE, In J. Böhm (ed.), Teaching
Mathematics with DERIVE, Chartwell Bratt, Bromley, England, (1992), 1-19.
[4] H. Heugl, W. Klinger, J. Lechner, Mathematikunterricht mit Computeralgebra-Systemen,
Addison-Wesley, Bonn, (1996).
[5] K. Aspetsberger, F. Schlögelhofer, Der TI-92 im Mathematikunterricht, Texas
Instruments, Freising, (1996).
[6] K. Aspetsberger, B. Kutzler, Using a Computer Algebra System at an Austrian High
School, in J. Collins, N. Estes, W. Gattis, D. Walker (eds.), Proc. The Sixth International
Conference on Technology and Education, Orlando, U.S.A., (1989).
[7] K. Aspetsberger, Investigations on the use of DERIVE for students at the age of 17
and 18, The International DERIVE Journal, Vol. 3, No. 1, (1996), 58-72.
[8] K. Fuchs, The planning of observation windows when using CAS in mathematics
teaching, The International DERIVE Journal, Vol. 3, No. 1, (1996), 39-55.
[9] O. Wurnig, Using TI-92 in the 9th-Grade of Austrian Grammar Schools - Hypotheses,
Experiences, Results, Problems. In ACDCA 5th Summer Academy, Gösing , Proceedings,
ACDCA, (1999).
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[10] B. Buchberger, The Creativity Spiral in Mathematics, RISC Linz, Technical Report No.
92, University of Linz, Austria, (1992).
[11] B. Buchberger, Why Should Students Learn Integration Rules?, RISC Linz, Technical
Report No. 89-7.0, University of Linz, Austria, (1989).
[12] H. Heugl, Symbolic computation systems in the classroom, The International DERIVE
Journal, Vol. 3, No. 1, (1996), 1-10.
[13] G. Grogger, Der Einsatz von DERIVE im Mathematikunterricht an AHS. Ergebnisse
einer bundesweiten Schülerbefragung (1993/94). Zentrum für Schulentwicklung, ZSEReport 6, Graz, (1995).
[14] E. Svecnik, Der Einsatz von DERIVE im Mathematikunterricht an AHS. Ergebnisse einer
bundesweiten Lehrerbefragung (1993/94) sowie vergleichende Darstellung mit
Ergebnissen einer Schülerbefragung, ZSE-Report 12, Graz, (1995).
[15] G. Grogger, Evaluation zur Erprobung des TI-92 im Mathematikunterricht an AHS.
Ergebnisse einer bundesweiten Schüler- u. Lehrerbefragung (1997/98). ZSE-Report 40,
Graz, (1999).
[16] J. Lechner, O. Wurnig, Schularbeiten, 5. Klasse. In W. Klinger (ed.), Der Mathematikunterricht im Zeitalter der Informationstechnologie, ACDCA, (1998).
[17] H. Heugl, I. Schirmer-Saneff, Leistungsmessung/Leistungsbeurteilung. In Endbericht für
das ACDCA Projekt III, ACDCA, (2001).
[18] O. Wurnig, New Ways of Assessment in CAS-oriented mathematical Education - New
Experiences, First Results. In T. C. Etchells, L.C. Leinbach, D. C. Pountney (eds), Proc.
of the 4th int. DERIVE & TI-92 Conference (Liverpool 2000), bk teachware, Hagenberg,
Austria, (2001).
[19] H. BÜRGER, R. FISCHER, G. MALLE, Mathematik Oberstufe 3, Verlag HölderPichler-Tempsky, Wien, (1991)
[20] W. Herget, H. Heugl, B. Kutzler, E. Lehmann, Indispensable Manual Calculation Skills
in a CAS-Environment, In V. Kokol-Voljc etal. (eds.), Exam Questions & Basic Skills in
Technology-Supported Mathematics Teaching, bk teachware, Hagenberg, Austria, (2000),
3-6.
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