Download abstract - ACT-R

Progress Towards an ACT-R/PM Model of Algebra Symbolization
Kevin A. Gluck, John R. Anderson, Scott A. Douglass, and Michael D. Byrne
Department of Psychology
Carnegie Mellon University
Pittsburgh, PA 15213
([email protected])
The Inductive Support Effect
The process of translating the relationships specified in a
problem statement into an algebraic expression is called
symbolizing. Koedinger and Anderson (1998) reported what
they described as the inductive support effect in algebra
symbolization. Their results show that students who
symbolized after solving two result-unknown problems were
able to symbolize faster during tutoring and also showed
better overall learning gain from pretest to posttest than
students who symbolized before solving the result-unknown
problems. Their interpretation of this result was that students
are able to induce the appropriate algebraic symbolization of
the problem statement out of the arithmetic operators used in
solving the result-unknown problems. Designing problems
such that they allow this induction to take place scaffolds, or
supports, the symbolization process. Thus, it is termed the
inductive support effect.
Ohlsson (1998) correctly notes that K&A do not offer a
cognitive analysis of the symbolization process itself, and
this provides a motivation for the research and modeling
work described here. The goal is to arrive at a better
understanding of the symbolization process, and especially
the benefit gained from an inductive support design, through
empirical study and cognitive modeling.
Materials, Procedure, and Results
The worksheet tool from the Algebra 1 tutor was
re-implemented in a computationally efficient manner so as
to interface well with an eye tracker. The appearance of the
tutor interface was modified in order to provide more
reliable point-of-regard estimates, but the basic functionality
remained intact. Special attention was paid to reproducing
the help and feedback messages accurately. All of the
problems were two-operator problems of the form a ± bx.
Participants were 18 middle and high school students
from Pittsburgh schools. Half of the subjects were in an
inductive
support
condition
(symbolization
after
result-unknowns) and half were not (symbolization was done
first). All subjects solved four problems on each of four
consecutive days, for a total of 16 problems.
The first question that needs to be addressed is whether
the results show a replication of the inductive support effect.
There is no effect on learning gain, as there was in the K&A
(1998) data, but there is an effect on performance during
learning. Table 1 shows the mean time and accuracy data
across inductive support (IS) and non-inductive support
(Non-IS) participants on expression symbolization (Exp),
result-unknown (RU) questions, and start-unknown (SU)
questions during tutoring.
Only the differences between IS and Non-IS participants
on expression symbolization (shown in bold) are statistically
significant. The effect on time replicates the K&A result.
However, they did not get an effect on accuracy, whereas the
results from this study show that IS students are more
accurate symbolizers.
Time
Acc.
Cond
Non-IS
IS
Non-IS
IS
Exp
M
(SD)
25.1
10.9
14.8
4.2
58.8
14.4
90.3
19.8
RU
M
(SD)
18.1
6.9
18.8
4.5
78.4
18.0
72.6
12.8
SU
M
42.2
33.6
53.6
55.7
(SD)
20.1
9.2
20.5
19.9
Table 1. Time and accuracy data. All N's = 9.
A second question that needs to be addressed is what the
eye movement data suggest about differences in cognitive
process that might shed some light on these performance
differences. Such analyses are now underway, and the latest
data will be presented at the workshop.
ACT-R/PM Model
Because it is anticipated that the eye movement data will
be informative with respect to cognitive process in this task,
and we will want to account for important patterns of visual
attention, the architecture chosen for the modeling effort is
ACT-R/PM (Byrne & Anderson, 1998). As of the
preparation of this abstract, the device interface, which
provides a communication pathway between the model and
the tutor, is complete. The cognitive modeling is now
underway and will be informed by upcoming analyses
regarding errors and eye movement patterns. The main goal
of the workshop presentation will be to describe the
then-current state of the ACT-R/PM model of
symbolization.
References
Byrne, M. D., & Anderson, J. R. (1998). Perception and action.
In J. R. Anderson & C. Lebiere, The atomic components of
thought (pp. 167-200). Mahwah, NJ: Erlbaum.
Koedinger, K. R., & Anderson, J. R. (1998). Illustrating
principled design: The early evolution of a cognitive tutor for
algebra symbolization. Interactive Learning Environments, 5,
161-179.
Ohlsson, S. (1998). Representation and process in learning
environments for mathematics: A commentary on three systems.
Interactive Learning Environments, 5, 205-215.