Revisiting the Relationship of Arithmetical Thinking and Letter-Symbolic Algebra
Demetra Pitta‐Pantazi · Maria Chimoni· Constantinos Christou
The article investigates how arithmetical thinking and algebraic thinking are linked to
each other. It attempts to determine whether algebra helps to understand arithmetic, whether
arithmetic is the basis of algebra, or whether there is a clear distinction between the two.
Previous studies conducted regarding the issue have been inconclusive. Hence, the study aims
to provide empirical evidence by analyzing students’ performance using different arithmetical
tasks (use of properties of numbers, operations, and equivalence) and algebraic tasks (same
concepts but with the representation of letter-symbolic to unknown quantities). The study was
focused on three questions: 1) How does students’ performance in arithmetical tasks differ
from their algebraic tasks? 2) What strategies do students use when they solve arithmetical
tasks related to properties of operations, equivalence properties, and properties of numbers?
3) What is the effect of students’ strategies in arithmetical tasks on their performance in
algebraic tasks? The study was conducted on 203 Grade 6 students from 4 urban schools
using convenience sampling to choose the schools. A researcher-made paper-and-pencil test
was used to collect data, including two forms of tasks (arithmetic and algebraic). The test
consisted of ten algebraic and arithmetic questions that covered three different topics: 1)
properties of operations, 2) properties of equivalency, and 3) properties of numbers. The
findings showed that sixth-grade pupils outperformed their algebraic equivalents on
arithmetic tasks, i.e., if the expressions contained only known quantities. Also, most students
used calculations to solve the arithmetical tasks, and only a few used structure-sense.
Students who used structure-sense as a strategy successfully solved the arithmetical and
algebraic tasks. Hence, a significant effect was found on this matter. The findings suggest that
structure sense is a common characteristic of arithmetical and algebraic thinking and is
essential for success in arithmetic and algebra (letter-symbolic) contexts.
Different statistical tools were used to answer the research questions. A paired sample
t-test was used to determine the significance between the mean scores of students in
arithmetical and algebraic tasks. A paired-sample t-test was used since the data (scores) are
intervals from the same individual who answered the two tasks. Since the study was
conducted to determine the difference between the two tasks, a paired-sample t-test was
appropriate. An Analysis of Variance (ANOVA) was also used in the study to determine the
effect of the different strategies used in solving arithmetical tasks on their algebraic tasks.
Before using the ANOVA, normality and homogeneity tests were done, and since all
assumptions were met, the ANOVA was carried out. ANOVA was used in this study because
it aims to determine the effect of the independent variables (frequency of strategies based on
calculations and the frequency of strategy based on structure sense) on the dependent variable
(students’ performance mean on the algebraic task). The independent variables in the case are
categorical (high and low frequency), and the independent variable is continuous (mean
scores). Hence, ANOVA, specifically two-way ANOVA, was used.
The study has a solid theoretical foundation and provides the gap between the existing
literature. It also has an excellent methodological approach, ensuring that each research
question is thoroughly addressed. The sample size and the tasks are also clearly defined,
making it easier for readers to follow. The study’s findings are very relevant as they highlight
how structure-sense strategies can be used in bridging the gap between arithmetic and
algebra. These results allow teachers to investigate how structure-sense in arithmetic can help
students prepare themselves for algebra. ANOVA and the paired-sample t-test were suitable
methods for analyzing the data, and the results were well-explained and highlighted.
