Crete, ICTM2, July 1-6, 2002

Interactive Animated Problem Sets in Learning
Functions and Binary Operations

Martti E. Pesonen and Hanna Lehtola
Department of Mathematics
University of Joensuu
Finland
http://www.joensuu.fi/mathematics/

Abstract

We describe the computer based learning activities meant to extend and reinforce our new students’ understanding of the definitions of function and binary operation, and discuss some aspects of the outcomes, e.g. how the students managed in doing the activities and what they thought about them.

A more complete description is found as:
WWW document
MS-Word document
RTF Document (2,8 MB, zip-version 0,1 MB)

1. Background

emphasis on maths teacher education
enriching teaching and learning methods
towards enabling distance learning
pedagogical elements:

concept formation through phases:

orientation, definition, identification, production, reinforcement

using verbal-symbolical-gaphical representations (VSG)

mixture of ideas: Dreyfus, Dubinsky, Haapasalo, Tall, Vinner, ...

2. WWW-Javasketchpad worksheets

on a html-page can be anything else and any number of "sketches":
<applet>constructions as parameter</applet>

based on Java applet Javasketchpad (Geometer's Sketchpad)

CODE="GSP.class" the applet code

ARCHIVE="jsp4.jar" file in local folder or on a server

CODEBASE="path to the folder"

the page can contain form structures; enables answer input and send
Example

3. Structure of the whole demo

74 first semester students on an Analysis course just started
1/2 hour paper&pencil Pre-test

+ 2 hour exercise session (Function Demo) in computer lab

+ online Post-Test a week later

The Pre-test
- simple VSG definition identification problems
- three statements:
a) For each value of the variable the value of the function is always different.
    correct 61, wrong 13
b) No element of the domain set can be mapped to an element of the same set.
    correct 36, wrong 38
c) No element can have more than one image.
    correct 46, wrong 27, no answer 1
- what is a function: give it's definition:
    correct 18, half correct 19, wrong 30, no answer 7

The Function Demo
- online www-form (see above)
- dynamical graphic representations (Javasketchpad)
- identification + production tasks
- traditional problems (some familiar from school)
- student feedback

The Post-test
- some problems similar to Pre-test
- evaluation (student opinions) of a single sketch
- more feedback

4. The Function Demo
    Problem sets 1-5
    - most elementary Venn-diagram-like
    Problem sets 6-10
    - axis-to-axis representations
    Opinions about axis-to-axis-representation
    Problems 11-12
    - traditional coordinate system representations, with curve and without curve
    Problems 13-15
    - power&root, exponential, logarithm
    Problems 16-17
    - parametric plane curves, planar motion
    Student Feedback
    - "I think that ... "

5. Comparison of Pre-test and Post-test
In most comparable problems there was considerable increase in ability and knowledge, especially in those containing graphical representations.

6. Binary Operations and Scaling
Experiments in Linear Algebra

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