Below you will find my paper enlarged edition from ATCM98.
If you have any questions or comments or want further details, please do not hesitate to write to me or send me an E-mail.

Yours sincerely
Sten Asgaard Andersen
Sollerod Park 7, 2
DK-2840 Holte
Denmark.
 
 

ATCM98, Tsukuba.

Danish students and the TI-92 graphic calculator.

Sten Asgaard Andersen,  teacher  at Virum Gymnasium, an upper secondary high school near Copenhagen, Denmark,  where I teach Mathematics and Physics.

Abstract:
Since 1996 experiments with the TI-92 graphic calculator have been made in a few classes in Denmark. The main purpose is to investigate the influence of teaching Mathematics by using an advanced calculator with symbolic manipulation. The following questions naturally arise:
Is the students understanding improved ? What about the basic concepts of Mathematics, are they still understood by the students ? How do students respond to a calculator like the TI-92 ?  Is it compatible with the curriculum ?
In my paper I will try to answer these and other questions. Some conclusions are made about teaching Mathematics with the  TI-92 in Upper Secondary High School in Denmark.
 
Introduction
 Over a period of 30 years the electronic calculator has evolved from a machine that could only perform simple four-form operations into one that can execute algebraic symbolic manipulations accurately and instantly. At the same time, the cost of a calculator has dropped so low that every student in Denmark can afford one.
Calculators allows students access to mathematical concepts and experiences from which they were previously limited with only paper and pencil. Because calculators make possible mathematical exploration, experimentation and enhancement of learning mathematical concepts, it is recommended that appropriate calculators be made available for use by students at every grade from high school to university.
As a matter of facts there are still many sceptics who worry that calculators use will impair students mathematical ability and may result in increased mathematical illiteracy.
The reality is that calculators are valuable educational tools that allow students to reach a higher level of mathematical power and understanding. By reducing the time that previously was spent on learning and performing tedious paper and pencil arithmetic and algebraic algorithms, calculators use today allows students and teachers to spend more time developing mathematical understanding, reasoning and applications. Graphing calculators as well as calculators with symbolic algebra manipulation capability provide new pedagogical enhancement opportunities. They afford students learning tools that complement - not replace - mental and paper-and-pencil skills and they give scope to the teaching (ex. multiple solution techniques).
Calculator technology allows students who would ordinarily be frustrated or bored by tedious manipulations to have access to the real mathematics itself, thus gaining a higher level of mathematical understanding rather than giving up. The fact is, calculators are better tools to do some of the computations and manipulations that were once done with paper and pencil. In the past, paper and pencil were the only tools available. Appropriate use of technology and associated pedagogy will get more students thinking and reasoning mathematical. Thus more people will develop useful mathematical understanding and mathematical power.
Calculators do not only allow students who would ordinarily be turned off by traditional mathematics tedious computations and algorithms to experience true mathematics, but they also help students to more quickly and readily develop number sense, gain mathematical insight, and reasoning skills, value mathematics and cultivate mathematical understanding, while they enjoy what they are leaning.

Why teach mathematics
Mathematics is as old as the human being. It is found everywhere, in our surroundings, in our homes, in nature, and we use it every day in different contexts. Its application varies from one group to another based upon their needs and struggle for survival.
One may say that mathematics is the most important factor within the technical and social development. In a modern IT society like ours nobody can function without reading and writing. And that applies to mathematics, too.

Danish school system
Before I go into detail about my project, let me very briefly give you some information about the Danish school system in general, and about mathematics in particular.
Children start school when they are about 6 years old and spend 9 years in primary school.
Afterwards 25 % of the pupils go to upper secondary high school for another 3 years, that means 10th  to12th  grade.

Curriculum in mathematics for 10th to 12th grade
In Upper Secondary High School, students can do mathematics at two levels: the B-level is obligatory, the A-level (high level) is optional. The first year, that is 10th grade, is common to both levels. After that the students have to choose between one more year leading to the B-level or two more years leading to the A-level.

The main subjects are:
1.  Number theory
2.  Geometry
3.  Functions
4.  Differential calculus
5.  Statistics and probability theory
6.  Integral calculus and differential equations
7.  Vectors and geometry in 2 and 3 dimensions
8.  An optional subject
 

Aspects:
1.  The historical aspect
2.  The model aspect
3.  The inner structure of mathematics
 

 
In mathematics at level A the characteristic features of mathematics should be further emphasized/investigated. The A-level should be a challenge for all students, including the brightest ones.

It is important to underline that both the theory and the inner structure of mathematics are very essential teaching material for 12th grade students of mathematics. We spend a lot of time working with the theory and go into many details about assumptions and proofs.

My experiment.
My project - Danish students and the TI-92 graphic calculator -  took place in a class at 12th grade, level A.

Graphic calculators like the TI-92 with the possibility of symbolic manipulation are normally not allowed at high schools in Denmark. However, these tools are available, so it was a great challenge for me to use the TI-92 in the teaching of my class.
Since August 1996, experiments with the TI-92 graphic calculator have been made in only 4 classes. 3 of them started in 10th grade and continued in 11th grade.
The 4th class was mine. My students joined this project only in 12th grade. They had never seen a TI-92 before. In 10th and 11th grades they used the TI-82 and were used to doing differential calculus manually. From the very beginning all my students were very positive and enthusiastic, most of them had a feeling of being something out of the ordinary.

An experiment like this requires permission from the Danish Ministry of Education. Furthermore -as you cant expect the students to buy another calculator - you have to find a sponsor. Texas Instruments in Denmark was very helpful and lent me 27 TI-92 including manuals and a view-screen for the classroom. I want to my gratitude to Texas Instruments.

For an A-level-class (the highest level), the subjects mainly consisted of
1)  integral calculus including calculating areas and rotation volumes
2)  differential equations, 1st order,  simple 2nd order, linear homogeneous and inhomogeneous equations
3)  vectors and geometry in 2 and 3 dimensions
4)  modelling as an optional subject

Now lets have a look at similarities and differences between my students and other students in 12th grade:
1.  The curriculum is the same for all students at 12th regardless of the kind of graphic calculator used.
2.  The final written examinations: My students  spent 1+3 hours, other students 4 hours. I will go into detail about this point later.
3.  The daily teaching: My students used TI-92, other students TI-82/83.

During 12th grade the students have 5 lessons of mathematics per week. Besides, they have homework including weekly written assignments, which means solving problems using their calculator.

Concerning the two first subjects of the curriculum, the TI-92 is an excellent advanced calculator. As a built-in software package you will find the Derive package. This was really a great help to all my students. The brightest ones become much better and - this is important - the weak students also become better not only at calculating but also  at the underlying theory. In classes without this tool, you very often see that the weak students get lost, because at first they cant solve integrals. etc. Now, with this tool, my students very soon got the right answers to their assignments, so they felt better and got more self-confidence. Seeing the right solutions and trying to find a way out on the calculator, they see it as a challenge to solve the problems without calculators and to understand the underlying theory. Consequently they acquired a better understanding of the subjects and also of the formulas.

Besides The Derive package, the TI-92 also has the Capri-geometry package. Using the geometry programs it is possible to illustrate or perhaps prove geometry theorems by means of drawings. However, this software-package is not as useful as the Derive programs in respect of 12th grade mathematics in Denmark.
In details the 3rd part of the curriculum-  vectors and geometry in 2 and 3 dimensions -  consists in finding equations for lines , planes, ball surfaces, vector functions (parameter curves), distances between point and line, point and plane, dot-products, cross-products, and calculating angles, points of intersection between two lines, a line and a plane or a line and a ball surface, etc. Furthermore, examining conic sections involving geometry and other characteristics. Working with these subjects it is very important that the students can visualize figures and diagrams, in order to to solve a given problem in 3 dimensions.  Here the TI-92 is not very useful, because you cant have genuine 3 dimensions on a 2 dimensional screen. But as soon as the students understand what to do, they can calculate dot-products, cross-products, angles etc. when solving  a specific problem.
As optional subject. my students chose traffic-modeling. Here too, the TI-92 turned out to be very helpful.
From a pedagogical point of view, the TI-92 motivates the students, it has many advantages, hardly any disadvantages, gives scope to the teaching and opens new approaches

Educational material:
Because a project like this has never been performed before, no educational material or teaching aids exist. I wrote a book on the subject myself. Unfortunately it is written in Danish, so it makes no sense to you.
 

Final written examination.
The final written examination took place in the middle of May. On standard terms the students have 4 hours for their assignments and they are allowed to use a TI-82/83 or a similar tool.
My students did the written examination in a different way. The period of 4 hours was split up into two parts. During the first part - of 1 hours duration -  my students were not allowed to use any tool at all except a pencil. The main purpose was to evaluate simple problems by using their  mathematical commen sense.

Because all students have got a better understanding, you dont have to hesitate to make demands on them.

The second part of the written examination - immediately after the first, without any break - was of 3 hours duration. All tools, the TI-92, manuals, books, everything except mobile phones were allowed.

The positive effect of using the TI-92 became apparent at the final written examination. Most students did very well. The assignments were constructured in such a way that understanding of the problems was both essential and necessary. Without understanding of the problems, the TI-92 was not useful to the students. If one compares the score at the final written examination for my students with that of other students, one will see a score about 10-15 % higher for my students. This does not prove anything , but it indicates that a tool like the TI-92 or something like it might improve the learning process and give the students a better understanding.
 

Conclusion:

As you may have understood from what I have said, I am very enthusiastic about this tool, the TI-92. In my opinion that is where the future is. I have been asked to give a written report to the Danish ministry of education. My conclusion in that report will be both positive for the teaching process and optimistic for the future with regard to the use of advanced graphic calculators with symbolic manipulation.
The students feel comfortable with this calculator.
As a matter of fact, you cant beat tools like the TI-92, so you have to join them. What is important, is the way of joining them. An optimal way with such tools may be to adjust the curriculum and change the focus -  the teaching of mathematics  - from just teaching skills to communicating a basic understanding . All assignments should depend on understanding.
 
After this experiment, I am convinced that mathematics is taught better this way and the students came to regard mathematics as an exciting science.