An experimental dynamical www-worksheet:
Basic linear transforms in the plane
in Educational on-line materials and methods 2002
Helsinki, August 15, 2002
|
Martti E. Pesonen
Department of Mathematics
University of Joensuu
Finland
http://www.joensuu.fi/mathematics/ |
The aims of these Linear Algebra WWW-based learning activities
were
-
to extend and reinforce the students’ understanding of the
definition of linear function
-
to improve the ability to switch between verbal, symbolical
and graphical (VSG) representations of transformations, especially
those given by a rule of geometrical nature
-
to test how well these problem sets built around a dynamical
figure support students' learning and learning motivation, especially in
order to acquire experience for producing distance learning material
We first take a look at the background
and some general features of the worksheets
and the structure
of this particular demo. Then we show how the students successed
in
their work, their attitudes compared with other demos, and finally
discuss some of the features that may have caused the failures.
1. Background
-
emphasis on mathematics teacher education
-
enriching teaching and learning methods
-
towards enabling distance learning
-
first year second semester linear
algebra course
-
ca. 80 -100 participants yearly
-
lectures + homework rather traditional
computer activities:
-
3 x Maple exercises in computer lab
-
interactive problem sets (some
10 hours, mostly as homework)
-
pedagogical elements behind the activities:
-
concept formation through phases:
orientation, definition,
identification,
production,
reinforcement
using verbal-symbolical-gaphical
representations (VSG)
-
mixture of ideas: Tommy Dreyfus, Ed
Dubinsky,
Lenni Haapasalo, David Tall, Shlomo
Vinner, ...
-
about our projects
(local)
2. WWW-Javasketchpad worksheets
-
on an html-page there 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 a local folder or on a server
CODEBASE="path
to the folder"
-
Example
of a Javasketchpad sketch
-
the page contains form structures that
enable input and send
(local)
-
The
Linear Transform worksheet (local)
-
The
original jsp-page (local)
3.
Structure of the Linear Function Demo
-
67 student assigments
-
online www-form, sending answers to
the teacher by email
-
dynamical graphical representations
of plane functions
-
multiple choice, symbolical and verbal
input
Part O: Sketches A, B, C, D
-
identification and production tasks:
-
variable - image - preimage
-
image of a segment or a circle
-
injection, surjection, bijection
-
functions depending on a parameter
(angle of rotation, constant of dilation)
-
function compositions
Part I: Puzzles 1, 2, 3, 4, 5
-
definition of linearity, example
-
identification of linearity
-
clarify the construction that produces
the image L(u) of the variable u
-
give a symbolical formula for L
-
give the matrix, if the function is
linear
Student feedback
4. Student success
The students got 75% of the problems correctly solved.
However, there was great variations in the success on
particular problems, ranging from 30% to 97%.
The
Linear Transform worksheet solutions with comments (local)
Most unsuccessful problems:
O
A 7 surjection? (57%) (local)
O
B 5 (LoL)(u) = ? (58%) (local)
O
C 4 Two dilations composed (47%) (local)
O
D 5 Matrix of a composition (31%) (local)
In the Puzzles the linearity was found out very
well.
In Puzzles 1-4 the symbolical formula was expressed
better
than the verbal form (in 5 only the verbal form was asked).
In Puzzle 4 the two Hints were reported to have been
used only by 24%, which may have happened
-
simply because they were not used to have Hints, or
-
they did not understand the word correspondence vihje
= Hint !
5. Student attitudes
A. The students' (relative) rates of positive attitude (or
like/dislike) was interpreted by the author from their open ended feedback
and divided in the following categories:
|
negative
attitude |
empty
or unclear |
"OK",
slightly positive |
positive
attitude |
| RATE |
-2 |
0 |
1 |
2 |
| FREQ |
10 |
15 |
32 |
10 |
During the course there were five demos containing dynamical
material.
The comparison is presented in the following table:
| Demo |
Type |
min |
attitude rate |
max |
| 1. Gaussian elimination (short) |
WWW |
-134 |
47
|
134 |
| 2. Function demo (in fact 2) |
WebCT |
-134 |
61
|
134 |
| 3. Linear space and independence |
WWW |
-134 |
50
|
134 |
| 4. Linear Transform |
WWW |
-134 |
32
|
134 |
| 5. Linear Transform II |
WWW |
-134 |
28
|
134 |
In cumulative sums, in the range between -20 to 20
- 10 students' rate was below zero
- 23 students' rate was > 4, the rest 34 being between
0 and 4.
6. Final remarks
Here the WebCT demo was different from the others
in many ways:
- the problems were simpler in general
- the problem focus was the definition of function, a
concept more familiar to them
- only 1/5 on the problems based on dynamical pictures
- the students got some credit for the whole course
- the students received better explaining feedback and
their "result" immediately after they had answered all questions
- the interface was more "professional"
The use of these dynamical worksheets was reported to
be very refreshing and fruitful change,
but not all were convinced with the usefulness in their
own learning.
Some ten students (mainly not male) did not like this
kind of activities at all, and most of them expressed their dislike against
computers clearly.
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