Ed
_________________________________________
A TEST WITH THE SYMBOLIC CALCULATOR, TI-92 MATHEMATICS, 3MX" from Norway
Responsible of the project: Tor Jan Aarstad email: <aarstad@robin.no>
Strand Upper Secondary School - 1996/1997
A summary of the report:
CONTENTS:
1. The background for the test, page 3
2. A limit to the test, page 4
3. The progress in the test, page 6
4. TI-92 used as data logger in CBL-experiments, page 7
5. Differential equations, page 7
6. Relevant WEB-addresses on the Internet, page 8
7. Arguments for and against the use in the school, page 9
8. Conclusion, page 10
I) From the examination papers, page 11
II) The average of the marks at the public written examination, page 12
III) The average marks based on class work, page 12
Comments, page 12
Experiences, page 13
Professional, page 13
Educational / methodical, page 14
Technical, page 14
9. Disadvantages, page 15
10. General, page 16
11. Hypothesis, page 16
12. Questions, page 17
13. Approach of current interest in 3MX, page 17
14. The students own evaluations, page 18
1. THE BACKGROUND FOR THE TEST
In the circular SUOA/V, June 15. 1993, it was given the opportunity to use
graphical calculators as help at the final exams in mathematics. The
circular contains some demands to the calculators, where one of the demands
sounds like this: The calculator must not be able to perform symbolical
operations.
In the connection with Reform'94, the graphical calculators became a
compulsory aid in mathematics for general subject and
economics/administration subject.
At this time the graphical calculators were only equipped to perform
numerical calculations, but Texas Instruments had announced that they were
working on a new graphical calculator that would be equipped with the
opportunity to perform symbolical calculations and therefore the
opportunities to get exact calculations.
The computer programs DERIVE, MathCad, Math Plus, Mathematica and Cabri
Geometry have been on the market for several years. Many of these programs
have the technology for symbolical calculations, and the programs have
gained great fitness among mathematicians.
At Strand upper secondary school, the programs MathCad and DERIVE have been
used for several years in the optional subject differential calculus two
hours a week in the upper class. We received clear response from the pupils,
that the use of the programs led to enthusiasm for the subject and that it
were relatively understandable.
When Texas Instruments presented the TI-92, where DERIVE was the basis of
the calculators symbolical operations, the interest for this technical
wonder became very high. The experiences we have received when using DERIVE
in math all these years gave us high expectations for using the TI-92
calculator in the education. A small and easy removable calculator would
have great advantages rather that a stationary PC.
It was known that the CBL system (Calculator-Based Laboratory System) also
was adjusted for use of the TI-92. Therefore exercises in physics and
chemistry, which are adjusted to the TI-82 and the TI-83 calculators, can be
performed by the TI-92. We knew that there were a lot of help for the TI-92
calculator on the Internet, for example a program to manage the CBL system,
and therefore it was important to put a limit to the experiment.
The conditions for having an experiment with the TI-92 calculator at Strand
upper secondary school were good. When a great deal of the students in math
(3MX) had chosen the subject 3FY in physics and the optional subject
differential calculus as well, and when they had the same teacher in all the
subjects, there were great enthusiasm at school when KUF/SUE approved the
experiment with the TI-92 calculator. \Good conditions for testing the TI-92
in several subjects gave us frankness and expectations.
We are grateful to Texas Instruments and Ess Data - importer in Norway, for
letting us have sufficient numbers of TI-92 calculators to the teacher and
students free for disposal!
2. A LIMIT TO THE EXPERIMENT
The Primary part of the calculator test was to test the use of a graphical
calculator with technology for symbolical calculations, TI-92, and compare
it with graphical calculators that performs only numerical calculations.
KUF/SUE gave the opportunity to use the TI-92 calculator as an aid for the
exam in math, 3MX, 1997 under the circumstances that the exercises are
adjusted to the calculators possibilities.
In the connection with the pupils in math (3MX) at Sandnes upper secondary
school also were allowed to use the TI-92 calculators for the exam in 1997,
the teachers from the two schools could cooperate in the test period.
The TI-92 calculator opened for great opportunities in mathematics which are
very impressive. We were aware of the temptation of expanding the schedule's
limits on several areas to use the calculator's possibilities to show
mathematical connections. In this experiment it wasn't relevant to introduce
more subjects or other subjects that the ordinary pupils also have to learn.
Therefore it was only relevant to consider the following problems:
Will the pupils motivations, calculation skills and technical understanding
increase? To what extent must the exercise express be changed?
The TI-92 will also be a central tool in the optional subject differential
calculus. To use the calculator's technical possibilities, we will expand
the calculator with relevant software (adv.92g) from the Internet. Therefore
there must be established a procedure to transfer and to implement the
programs on the calculator.
An essential part of the work in this subject will be studies of the way the
programs from the Internet works.
In physics (3FY) the TI-92 will be used with the CBL system. The CBL systems
interface was downloaded from the Internet, but some adjustments had to be done.
3. THE PROGRESS IN THE TEST
WHICH TOPICS HAVE TURNED OUT WELL ON THE TI-92?
Vector analysis: The TI-92 disclosed several opportunities for performing
vector analysis.
Function analysis: Treatment of different types of functions: Derivation and
integration: Definite and indefinite integrals: Space and volume, rotating
objects: In these fields the TI-92 turned out very well. When the
mathematical operations were performed symbolic, we got a great profit from
using the calculator, because the mathematical expressions could be
recognized and identified in each calculation.
Calculation of probabilities: Simulations: Statistics and testing of
hypothesis. In these fields there were no special functions or programs
installed. We had to acknowledge that the TI-83 was superior on this sort
of calculations. We could however rationalize the calculations in relation
to the numerical calculators by being able to define separate functions with
many variables.
Differential Equations: It became apparent that a special program-pack from
the internet had to be implemented to get benefit from working with
differential equations on the TI-92. When this program-pack was installed on
the TI-92, the calculator disclosed opportunities for calculations which
satisfy mathematicians on University standard.
Geometry and Conic sections: TI-92 uses the program Geometry Capri as a
regular program, but we didn't get an opportunity to explore the possibilities.
4. TI-92 USED AS DATA LOGGER IN CBL-EXPERIMENTS
The following internet-addresses were of great help: The home-address to the
company that is the main supplier of CBL-probes: Vernier Software
5. DIFFERENTIAL EQUATIONS
All the students in the experiment group have also had two hours of optional
subject a week - Differential equations.
The program-pack ADV.92G proved to be a suitable aid! The program-pack
ADV.92G can be found on the internet. The program-pack can be loaded down
from the following address: ftp://ftp.derive.com/pub/adv.92g
We make studies of the general swinging equation. Where m is the body's mass
D is the spring-stiffness q is the damping-factor.
6. RELEVANT WEB-ADDRESSES ON THE INTERNET
"National center for Teaching aid" in Norway uses this address:
<http://skolenettet.nls.no>
In Texas Instrument's web-sites you can find several possibilities for
program- and tool-menus which can be transferred by using TI-92?s
Graph-Link: http://www.ti.com
Some companies try to make general home-pages which can be the starting
point for contact with different parts of the same trade abroad and inland.
For mathematics- and science-teachers we recommend:
http://www.sol.no/teknodidakt
For searching on the internet we recommend: http://kvasir.sol.no/no
7. ARGUMENTS FOR AND AGAINST THE USE IN THE SCHOOL
Advantages: Disadvantages:
The user can see the needs of the basic knowledge of mathematics - among
other things because the user ought to assess the reliability of the answers
given by the calculator.
After a short introduction to the basic mathematical methods, the principles
can be applied to relatively complicated problems.
The principles of mathematical formulas can be studied in the applications
where we make simple variations in the inputs.
The work can be concentrated on understanding of the mathematical principles
before time-consuming calculations.
The machine can in a rapid and safe way, do "heavy" mathematical calculations.
The user will discover the need of routines for control or alternative
calculations.
The applications of the theories will be more interesting.
The machine can generate general mathematical functions in subjects like
physics by use of collections of data (by CBL) and regression analysis.
The machine has a great capacity for work because the permanent menu can be
exchanged by a packet with programmes - specially from Internet - and a
following menu for the application.
The machine activity can be changed by replacing the permanent machine menu
with the programme packet ADV.92G, which can be transmitted from Internet,
and relatively complicated mathematical problems, like e.g. differential
equations of first and second order, can be solved and we get exact answers.
The geometry programme (Cabri) has a lot of possibilities for descriptive
proofs of theorems in geometry.
The users have only a little motivation for drill and repetition of
analytical calculations, which the calculator can do faster and more safely.
The users can simply get dependent on the calculator - even when doing
relatively simple calculations.
Insufficient input can have dramatic consequence for the answer.
The settings of the calculator may get great consequence for the success of
the treatment on the calculator.
The calculator can give the answer in an unfamiliar manner.
The use of the calculator in upper secondary school can lead to insufficient
basic knowledge of mathematics, and the users can get some difficulties in
higher education without the calculator.
Some users may have insufficient knowledge to use the calculator effectively.
8. CONCLUSION
In the evaluation of the experimental use of TI-92 in the mathematics
lessons has been the plan for the education and some sets of exercises given
by the National Council of Secondary Education in Norway.
The use of TI-92 in math has been a great inspiration and motivation for the
pupils. TI-92 has been of great value for the pupils on every level of
knowledge.
This can be verified with some reservations, however, by referring to the
results from the public written examination and assessment grade in the
following manner:
I) FROM THE EXAMINATION PAPERS
Text:
Solution on test schools:
Solution on ordinary schools:
1a)1) Find the derived function to the function f without use of the
calculator:
2p: 95,1%
1p: 0%
0p: 4,9%
2p: 71,0%
1p: 9,0%
0p: 20,0%
1a)2) Find the derived function to the function g without use of the
calculator:
2p: 92,7%
1p: 0%
0p: 7,3%
2p: 52,3%
1p: 12,2%
0p: 35,5%
1b)Solve the integration problem without use of the calculator:
2p: 95,1%
1p: 4,9%
0p: 0%
2p: 89,0%
1p: 2,6%
0p: 8,4%
Note: p means point. 2p means quite right answer, 1p means "something" right
and 0p means a worthless/ missing answer.
II) THE AVERAGE OF THE MARKS AT THE PUBLIC WRITTEN EXAMINATION:
For the pupils from the test groups: 41 examination papers -
The average mark: 3,90 Ordinary pupils: 155 examination papers - The average
mark: 3,48
III) THE AVERAGE MARKS BASED ON CLASSWORK:
The test schools:
REMARK: The marks are calculated for the same group of pupils - 2MX and 3MX
are throughout two years of school.In a ordinarily course the marks would
have shown a declining tendency the second year.
COMMENTS:
There are several interesting aspects that gets relevant by studying the
number material above.
The basis for the representation are pupil-papers from a 3MX-external
examiner in 1997. All pupils in both test groups were sit for an exam in
writing in 3MX, and results from these papers are also included in the
evaluated material.
Even though the number of the participants in this experiment is too low to
draw any certain conclusions, does the survey above give us relatively
strong signals of what to expect if we start using the symbolic calculator
in the mathematics training.
There seemed to be documentation for following claims:
The calculation skill increases.
Motivation for working with the subject increases.
The ability of problem solving is stimulated.
The professional understanding increases, and then it becomes easier to find
solutions in professional problems.
The claims above can possibly be discussed, but in this report there is no
opportunity of giving any thorough reasons. A couple of essentially traits
can still be pointed out:
There are demonstrable more correct derivation- and integration solutions in
the test groups than among ordinary pupils. The average mark achieved at the
exam is considerably higher in the test group than among ordinary pupils.
The average mark for general proficiency is higher in 3MX than achieved
average mark for general proficiency in 2MX in the test groups.
There can of course be several other reasons for this, and the speculations
are until later left to every single reader.
EXPERIENCES:
Professional:
It has been a pleasure working with TI-92 in math (3MX). The
mathematics-subject will obviously become more understandable when using a
symbolic calculator. The greatest use of the calculator was obtained by
studying how different changes in "input" was important for the answer in
"output".
We could concentrate our attention completely on studies of professional
phenomena instead of using a lot of time on analytical calculations.
We had to continual repeat the procedure for the analytical calculations -
and they still belong to the foundation of mathematics - but sometimes long
and elaborate analytical calculations will seem like a derailment from the
process of learning.
For the physics lessons, working on the physical formulas were specially
relevant, but essentially the calculator wasn't much of help in physics.
The optional subject differential-calculations became considerably more
understandable when the program package ADV was downloaded from the
Internet. The pupils had especially high profit from the programs that
solved general first- and second- degree equations, found the
direction-graphs and then found one particular solution.
The pupils experienced that it wasn't quite easy to solve
differential-equations in the analytical, manual way, and therefore the
calculator released the subject from a big burden.
Educationel / Medthodical
There were a lot of new challenges that had to be defeated, but in an open
dialogue with the pupils we got the procedure into a constructive trail
after a while. Even though it very often was tempting for the teacher to
demonstrate the "latest technical finding" on the calculator, the taken
effect became best when the pupils found the technical possibilities
themselves and demonstrated this for the rest of the group. Since most of
the calculators were equipped with a switch for transferring the graphical
calculator-image to an overhead, there were pupils that wanted to show
possibilities that they had found themselves all the time. Some pupils
reached very high competence in using the calculator after a while.
It proved to be relatively easy to motivate the pupils to study at home when
the problem was to be solved on the calculator. The taken effect of learning
was very high.
Technical
It has been interesting to see how some pupils, especially some of the
girls, have changed their attitude to the calculator as a technical
instrument during the year. The first weeks were influenced by an almost
negative relationship to the machine's possibilities, but after a while we
saw a noticeable change of attitude that had a positive effect on the whole
group. It actually seemed like "something" had to mature before the
realities were taken seriously, and then it got clear that frequently use of
the calculator was a condition to be able of using it as a technical support
in given situations. Most pupils became competent users of the calculator
during the year.
All necessary downloading from the Internet, and transferring between the
machines proved to be more troubled than assumed. Since the machine's memory
is organized in folders, it was very exciting to see where the saved files
ended every time. The program packages on the Internet have however got
detailed descriptions for both downloading and using, so all the problems
were solved.
DISADVENTAGES
When we are going to evaluate advantages and drawbacks in allowing to use
symbolical calculator in mathematical education in the Norwegian high
school, it's important to evaluate the situation from the pupils stand. The
fact is that the symbolical calculator is an inalienable aid for the pupils
who control the basic of analytical mathematics manipulations, and then also
can evaluate calculated answers and progresses. It's however not so clear
that pupils, who often doesn't have necessary insight and practice in
manipulating with help of the math basic axiom, can simplify and rationalize
the way to a mathematical understanding with help from a technical aid. This
is an important point in evaluation which must be given a huge importance to
the result.
In individual cases it appeared like it was almost an insoluble problem for
the pupils to prove that manual calculations were in agreement with the
answer that the calculator came up with. Specific derivation and integration
exercises came up with an answer on the calculator which was a prepared
answer of the manual calculation, and it could be problematic to prove the
correspondence. In this way it raised a constant doubt, and the absence of
analytical knowledge was clearly demonstrated as a drawback. On the other
side have the mathematical education over the latest decades been to much
concerned about doing exercises over and over again and the understanding
have in many cases suffered. Now we have access to a technical aid, which in
combination with a basic analytical education, will certainly make an
importance for the mathematical education in the school.
GENERAL
The discussion about the symbolical calculator should be allowed in the
mathematical education is some times an emotionally charged discussion. The
argumentation covers the whole spectrum from keeping the sides of the clean
math (skeptical) to strengthen the useful mathematics (the enthusiastic). It
seems to be important to spend a relative great amount of resources on an
information activity among the one entitled to give a opinion. In the next
round it will be naturally to discuss the situation before a decision is
made. It have been launched many ideas on how to test the symbolical
calculator in the school, and it's apparently many who believe in this
technical aid. In the experiment however we chose a variant were the
originally exam tasks in 3MX were adjusted to the use of TI-92. It appeared
only to be necessary with some minor adjustments, and then we had shown that
it didn't get any decisive meaning for the central given exam tasks if any
of the pupils had access to a symbolical calculator. From the experience of
this year can considerable aspects attached to the use of symbolical
calculator in the teaching, sum up in one hypothesis part and one question part.
HYPOTHESIS:
The value of algebra and analytical basic knowledge must be evaluated
independent from the technical options. Analytical basic knowledge is
superior to the opportunity for technical manipulations. The value of the
machine as a technical aid in some situations will be based on the pupils
knowledge about the opportunities. The motivation and the technical
understanding increase. The tasks will be more descriptive. Parameter will
play a bigger part. The tasks will generally be more complex.
QUESTIONS:
Will it be of no use to use TI-92 as a tool in the education, and then
forbid it to the exam?
Should it be tasks which should be solved without use of the calculator on
the exam?
Should technical manual be allowed to use on the exam?
Should we in math give contents to the concepts numerical, graphic and
analytical?
What is central math now and in the future?
To what extent should the (symbolical) calculator influence the mathematical
education?
Will the development be fair compared to the pupils/students who don't by
the symbolical calculator?
How should the teachers and other who is entitled to give a opinion be
informed and trained to create a constructive debate over this topic before
it's decided if the symbolical calculator should be allowed as an aid on the
exam?
13. APPROACH OF CURRENT INTEREST IN 3MX
How much insight and knowledge about TI-92 is reasonable to expect for an
exam in 3MX? - Most students accumulate large competence to use of TI-92,
but what can be expected...?
Are there fields which are of very great present interest? ...fields of no
present interest?
How should the problems be formulated so the solution will show current
technical insight and competence? - Will it be of current interest to ask
for several solution methods on a problem as for instance graphic,
numerical, analytic, ..., without use of graphic calculator, ...?
How should the problems be formulated?
How actively is it reasonable to expect that TI-92 is used in the solution
of the problem?
- To what extent can we expect and accept that TI-92 is used in the solution
process of the problems?
How should it be directed in the answer about current use of TI-92 in the
solution process?
Which knowledge will be important and central in mathematics when TI-92 is
available as technical help? (numerical calculation of approximation values
to square roots, algebraically calculations, sundry techniques for
calculation of integrals and differentials, ...
How should the mathematics be introduced to get as large profit as possible
with TI-92 as technical help?
How can we frame intellectual challenges in science and mathematics based on
the use of graphic calculator/TI-92?
14. THE STUDENTS OWN EVALUATIONS
1. Technical use
TI-92 is practical and perspicacious with functional menus. That menus are
placed strategic, and that same function exist several places makes it
easier to use TI-92. The entry to ordinary use is reasonable low. The
calculator also has functions calculated for "higher" mathematics, and the
use of these are not for beginners. The programs can be difficult to make,
to find them again on the calculator and to use them.
2. How do you experience TI-92 as help in 3MX?
Working with understanding mathematics can be given maximal attention
because the calculation work becomes easier. The mathematical theories and
formulas can be explored in an efficacious way. By doing small changes in
the parameter values, we are able to see, by pushing a key, how the changes
effect the result. With large manual calculations it helps to have access to
the answer.
3. Has mathematics become easier to understand when using TI-92?
When using TI-92 the possibilities tied up to theories and formulas can be
explored by doing small changes in the "input"-values in order to study the
effect this has on the "output"-values. The analytical calculation process
is important, but when using TI-92 we get to see the material from different
angles. The mathematical understanding have, when both traditional and
technical solutions can be presented and discussed parallel, obtained
maximal foundation.
4. How should TI-92 be used in mathematical-training?
The work with TI-92 can in some fields experiences as "extra-work" when both
technical and traditional methods must be learned. The competence depend on
regular and active use of the calculator
5. How does the manuals function?
It is a great advantage that the manuals are in Norwegian and that they are
available in two versions, an easy version for beginners and a fuller
version for those who have come further in use of the calculator.
6. General comments
It is easy to become dependent of TI-92.
It is hard to become motivated for manual and more cumbersome solution
methods when TI-92 is available.
The calculator is 'great" in mathematics. It is brilliant for mathematical
calculations and accounts.
There are only positive effects of the calculator:
The opportunity to use TI-92 in 3MX experiences as a great inspiration in
the work.
How will more advanced science studies (science and mathematics) turn out
without use of TI-92?
Abolishing of the calculator should be of no present interest:
Norwegian version.
Date: 20.11.97
Contact Tor Jan Aarstad for more information.
Edward D. Laughbaum
The Ohio State University
Department of Mathematics
231 West 18th Avenue
Columbus, OH 43210
Associate Director, The Ohio State University Technology College Short
Course Program
Associate Director, Ohio Early Mathematics Placement Testing Program
Emeritus Professor of Mathematics
(O) (614) 292-7223 or (614) 292-9504
(FAX) (614) 292-0694
http://www.math.ohio-state.edu/shortcourse
http://www.empt.org
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