Exploring students' basic knowledge and skills

by using VSG problems -

experiments in WebCT environment

Yliopistopedagogiikan seminaari TKT- & matematiikan opettajille 17.-18.5. 2004

University pedagogy camp, May 17-18, 2004

Mekrijärvi, Finland

Revised March 10, 2005

Contents

Introduction
WebCT-tests

WebCT

Interactive dynamical www-worksheet (Geometria applet)

In English

In Finnish

Discussion

1 Introduction

In tertiary level mathematics studies the role of concepts and abstraction abilities become most essential. In Abstract Algebra and Linear Algebra the axiomatic structures like group, ring, field and linear space extend the basic ideas of arithmetics as well as vector and matrix algebra.

It is known that generalization is a far more elementary process than abstraction. By generalization we mean for example proceeding from two dimensional vector calculus to n-dimensional, and by abstraction the jump from concretely defined vector system like the euclidean Rⁿ to a set theory based axiomatic system in which we just can do what is allowed through the axioms.

1.1. The mathematical concept Binary Operation

Sets of elements and relations between sets are the fundamental ingredients of algebraic structures. The relation is a special kind of two-variable function called binary operation, which takes two variable values and gives one precise result, all from the same set. Therefore we can call it an internal binary operation. An external binary operation is also a two-variable function, but here we allow two different sets, one for the main elements of the structure and another for “scaling” these elements.

Definition of an internal binary operation in a set
Let A be a non-empty set. We call an internal binary operation every function A×A ®A.

Definition of an external binary operation in a set
Let A and K be non-empty sets. We call an external binary operation (or scaling function) every function K×A®A.

To be able to fully appreciate these definitions we must have a solid and imaginative understanding of the function concept. A great deal of research on this has been done during the last 25 years even in tertiary level context ([Vinner & al.]). Also learning binary operations is well studied and reported (e.g. [Dubinsky & al.]). In general, these approaches have been quite algebraic in nature.

1.2. The purpose of testing

The purpose of this development work study and has been to try to find ways to enrich the selection of representing these objects by using new technology, and how to implement a systematic learning material for distance and self learning use. The emphasis is in abstract Linear Space theory, since dynamical graphics very suitable for representing vectors and other linear elements and one of the authors is experienced in teaching elementary Linear Algebra. The examined concepts are the internal and external binary operations, and the learning environment is now the course management program WebCT.
However, WebCT does not support the idea of ''learning while tested'', since all problems in a WebCT-test must be answered before seeing results. There should be possibility to have intermediate and immediate feedback and advice (adaptivity?) while going through the learning module, especially when the concept is ''familiar'' to the student (like the function concept). Also the teacher should have records about how the students manage.

The test material was divided into two WebCT-tests.
1. BINARY OPERATIONS consisted of

an Introduction (orientation module) with text and examples,
a “Pre-test” with 11 APOS-VSG problems, and
a collection of Definition Identification VSG-problems (definition module).

This document contained only multiple-choice (MC) questions, so that it could be evaluated by the system immediately, and it was allowed to do as many times as the student needed to achieve the predefined limit 25/50. This limit should have been higher, since it was too easy to achieve even by pure guessing.
2. LASKUTOIMITUS 2 contained MC, matching and short answer questions, including one for feedback/evalution. The content was

VSG Identification problems,
VSG Production problems,
a glimpse to external binary operations (this last part is to be separated into another learning module).

The theoretical background for developing test items are the VSG-Scheme and the APOS-Theory, which are described more thoroughly in e.g. [Haapasalo] and [Breidenbach et al.].

APOS-theory based problems might be useful in pre-testing and post-testing the depth of the students' understanding of the concept.
(APOS = Action - Process - Object -Schema model of the maturity of the knowledge)

This first attempt to use APOS problems in pre-test was not very successful, since these new problems turned out to be too easy.

Word document (rtf)