1. *Author and researcher
Email address: mutie@usc.edu
1
Comparative Cognitive Task Analysis (CTA) Study
Mathematics - Solving Quadratic Equations by Factorization
Acquillahs M. Mutie, Ed.D.a,b*
, Kenneth A. Yates, Ed.D.b
, Camille Ramos-Beal, Ed.D.a
a
Pomona Unified School District
b
Rossier School of Education, University of Southern California
Introduction
Cognitive task analysis (CTA) seeks to elicit the highly automated and often-unconscious
knowledge experts use to solve difficult problems and perform complex tasks. Students taking algebra find
solving and understanding quadratic equations very challenging yet quadratic equations are a major
component of building mastery in algebra. The purpose of this pilot study was to perform a comparative
experimental study for comparing the CTA guided instruction using a gold standard protocol with the
regular instruction. As part of the comparative analysis, the results will also provide an initial assessment of
the extent to which CTA guided instruction will translate into a K-12 classroom teaching learning
environment.
Pilot Study Question and Methodology
The overriding study question for this pilot study is: Do participants in the experimental group
perform the procedural action and decision steps of solving quadratic equations through the factorization
procedure more accurately and completely than participants in the control group?
Methodology
Eighty-nine high school students participated in the study. A pre-assessment of prior knowledge
was administered to the students in both the control and experimental group with no significant difference
in the results. The experimental group of 29 high school 9th grade algebra students were taught using CTA
techniques that strictly followed a CTA developed protocol for teaching solving quadratic equations while
the control group of 60 high school students were taught using the traditional methods by a highly qualified
teacher with 4 years’ experience in teaching high school mathematics. Only one procedure for solving
quadratic equations was taught, the factorization procedure. This researcher and the other teacher planned
this lesson together and together came up with the lesson notes and examples to be taught to students.
Guided practice for both groups of students used the same questions. This researcher and the other teacher
met twice a week to compare notes and discuss how the students were fairing on.
Post-assessment. Once the control and experimental groups were taught how to solve quadratic
equations using the factorization procedure, both groups were given an assessment to find out if there was
any significant difference in the performance between the students that were taught using the traditional
teaching methods and those taught following CTA guided instruction.
Results and Data Analysis
The results demonstrated a statistically significant difference between the experimental group and
the control group. The mean score of the experimental group was 9.2 points out of a possible score of 10
points with a standard deviation of 1.265 points while the mean of the control group was 6.6 points with a
standard deviation of 2.808 points.
We performed a statistical test at the 𝛼 = 0.01 significance level of the null hypothesis 𝐻!: 𝜇! =
𝜇! where the mean score of the control group (𝜇!) and the mean of the experimental group (𝜇!) were equal
against the alternative hypothesis 𝐻!: 𝜇! < 𝜇!. The sample test statistic was 𝑡 = −5.862, and the p-value =
4.19x10-8
.
Since the experimental group had a mean score of 9.2 on the post assessment, and the control
group had a men score of 6.6. The standard deviations for the two groups, experimental and control were
1.265 and 2.808 respectively. The effect size (Cohen’s d) for this study was calculated to be 1.19. This
means that the average score in the experimental is 1.19 of a standard deviation larger than the mean score
of the control group.
Conclusion
Since the p-value is approximately zero and is less than the significance level, these results give
sufficient evidence to reject the null hypothesis and conclude that the mean score of the experimental group
is significantly greater than the mean score of the control group. In fact an average effect size of 1.19 shows
that students in the experimental classroom improved by 38 percentile points. That is, students scoring at
the 50th
percentile on standardized tests would be predicted to score at the 88th
percentile after being taught
using the CTA methodology and following a gold standard protocol.