Discovering advanced mathematics with calculator activities.
One of the benefits of "modern teaching methods" is the improved investigational skills of students who are encouraged to discover mathematical rules. It is this skill that we should be able to exploit with technology. We would like to demonstrate some of the investigational approaches to advanced mathematics that we are using in groups of students in a local college. There have been many studies on the use of graphics calculators and computer algebra systems in mathematics teaching and learning (see for example Mayes, 1994; Jaworski, 1993). These studies often use worksheets or laboratory based experiments as additional tasks to traditional teaching. For example, Watkins, in his research with foundation year engineering students, used DERIVE to introduce several topics in calculus while using a traditional (i.e. non technology approach) text book and DERIVE tasks. He found improved understanding and basic skills through the DERIVE based activities.
The role of calculator technology is changing. A few years ago we would use a scientific calculator to manipulate numbers and calculate values for expressions. The programmable calculator allowed simple programs to be written by students to carry out algorithms and recently the graphics calculator has changed our visual approach of mathematics teaching. The new range of calculators containing computer algebra systems, such as the Texas TI-92, has caused a shock wave among examiners in the UK. Now we have an "almost" pocket sized "calculator" with which students can do most of an examination paper without demonstrating their own understanding. Such technology may revolutionize the way that students "see" and "feel" mathematics.
The use of a calculator in the classroom can take on two roles
given values of m0, g and t (a scientific calculator)
(a graphics calculator)
(a symbolic algebra calculator)
, 
and 
are differentiated
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By leading students through a carefully designed selection of such exercises soon leads them to realize that
The UK curriculum has undergone a major upheaval during the last few years. In 1988 a new National Curriculum for children aged 5-16 was introduced into schools. For mathematics this has led to more emphasis on problem solving, investigations and number work and the demise of algebraic manipulative skills. To close the gap opened up at 16, new mathematics courses in upper secondary schools were introduced in 1994. These courses lead to the award of the "General Certificate of Education, Advanced Level".
The post 16 curriculum in schools and colleges is not prescribed by law as is the National Curriculum. Students choose two or three subjects and hence specialization begins at aged 16. For mathematics there is a common core curriculum on which all the Examination Boards base their syllabuses. This core makes up fifty percentage of an Advanced Level award. There is considerable choice in the remainder.
There are two ingredients of the core that are causing most concern among teachers and examiners. They are the following:
There are concerns for students progressing into higher education. In the UK many universities are introducing calculator policies which ban graphics calculators and calculators with a QWERTY keyboard. This will lead to an anomalous situation. Students will have been introduced to advanced calculator technology in schools, will be familiar with their use during a university course but will not be able to use this technology in the assessment for the course. It does appear that for some years to come we must ensure that our students are still capable at performing manipulations by hand in "the traditional way" so that they are not disadvantaged when they move on to use mathematics at the next level.
We will now demonstrate four examples of the use of calculator activities in our "Discovering Advanced Mathematics" project.
There is a PDF file available for this paper.
Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.
Mail comments to Pedro Gómez: pgomez@uniandes.edu.co