The effect of two intervention courses on students’ early algebraic
thinking
Maria Chimoni and Demetra Pitta-Pantazi
University of Cyprus, Department of Education; chimoni.maria@ucy,ac,cy, [email protected]
This aim of this study is to describe the nature and content of instruction that facilitates the
development of students’ early algebraic thinking. In total, 96 fifth-graders attended two different
intervention courses. Both courses approached three basic content strands of algebra: generalized
arithmetic, functional thinking, and modeling languages. The courses differed in respect to the
characteristics of the tasks that were used. The first intervention included real life scenarios, and
semi-structured tasks, with questions which were more exploratory in nature. The second
intervention course involved mathematical investigations, and more structured tasks which were
guided through supportive questions and scaffolding steps. The findings, yielded from the analysis
of pre-test and post-test data, showed that the first course had better learning outcomes compared
to the second, while controlling for preliminary differences regarding students’ early algebraic
thinking.
Keywords: early algebraic thinking; teaching intervention; tasks.
Introduction
In response to calls for spreading the teaching and learning of algebra throughout K-12 grades (e.g.
NCTM, 2000), a wealth of studies focused on the design and implementation of instructional
interventions that facilitate the development of early algebraic thinking (e.g. Blanton & Kaput,
2005; Irwin & Britt, 2005; Warren & Cooper, 2008). These studies offered strong evidences that
students are able to develop algebraic thinking as early as the primary grades. Moreover, these
studies highlighted the key role of teachers in providing their students with rich opportunities to
investigate and understand algebraic ideas from elementary school.
As Kieran, Pang, Schifter and Ng (2016) highlighted, a large number of past studies identified
pattern generalization and functional thinking as important routes that foster the development of
early algebraic thinking; little research has involved other aspects of algebra, such as properties of
numbers and operations. This bring us to the question of the effectiveness of intervention courses
that might involve a range of algebra content strands, such as functional thinking and generalized
arithmetic. There is, therefore, still a need for extending our understanding of supportive instruction
that aims to improve students’ early algebraic thinking and further clarifying the content of a corpus
of lessons that capture the core content strands of algebra.
The current study addresses this issue. Furthermore, considering the suggestions of recent literature
regarding the impact of different types of tasks on students’ learning (e.g. Swan, 2011), this study
raises the question of whether the nature and content of the tasks used in instructional interventions,
regarding their structured or semi-structured nature, and the association of their context to real life
scenarios or not, might formulate the effect on students’ early algebraic thinking.
Theoretical Framework
The notion of early algebraic thinking
Several research studies addressed the multidimensional nature of early algebraic thinking. Kaput
(2008) claimed that there are two core aspects of algebraic thinking: (i) making generalizations and
expressing those generalizations in increasingly, conventional symbol systems, and (ii) reasoning
with symbolic forms, including the syntactically guided manipulations of those symbolic forms.
According to Kaput (2008), these two aspects denote reasoning processes that are considered to
flow through varying degrees throughout three content strands of algebra: (i) generalized arithmetic,
(ii) functional thinking, and (iii) the application of modeling languages. Generalized arithmetic
involves generalizing rules about relationships between numbers, manipulating operations and
exploring their properties, transforming and solving equations, and understanding the equal sign in
number relations. Functional thinking refers to the identification and description of functional
relationships between independent and dependent variables. Modeling refers to the generalization of
regularities from mathematized situations or phenomena inside or outside mathematics.
The notion of early algebraic thinking has also been associated through literature with several
mathematical processes. For example, Kieran (2004) suggested that early algebraic thinking is
linked to problem solving, modeling, working with generalizable patterns, justifying and proving,
making predictions and conjectures, analyzing relationships, and identifying structure.
Early algebraic thinking, therefore, is expected to emerge through intervention courses that capture
a variety of areas and contexts related to algebra and assist students to use a range of mathematical
processes. Existing literature pertaining the positive impact of instructional interventions on
students’ early algebraic thinking involves diverse approaches, such as functional approaches,
multi-representational approaches, equation approaches, and generalization approaches (Watson,
2009).
Sources of meaning in algebraic problems and the importance of the nature of tasks
Radford (2004) specified that there are three main sources of meaning within algebraic problems
that trigger the development of early algebraic thinking: (a) the algebraic structure itself (e.g. the
letter-symbolic representations, graphical representations), (b) the problem context (e.g. word
problems, modeling activities) and (c) the exterior of the problem context (e.g. social and cultural
features, such as language, body movements, and experience). Hence, the specific characteristics of
these sources might facilitate or not the construction of meaning when students participate in
algebra lessons.
Additionally, existing literature on the importance of the tasks that students are engaged with, has
shown that the nature and features of mathematical tasks influence learning, since they direct
students’ attention to specific content and specific ways of processing information (Jones & Pepin,
2016). For example, Sullivan, Clarke and Clarke (2012) suggested that problem-like tasks have a
positive effect on students’ mathematical thinking rather than step-by-step procedures. In this
perspective, the extend to which a task involves problems that are more or less structured, is
associated with an open question or a series of scaffolding questions, and represents situations
related to students’ experiences within real life contexts or not, may influence students’
development of early algebraic thinking.
Aim of the study
The aim of this study is the investigation of the effect of two different intervention courses in
improving students’ early algebraic thinking. Both courses involved the three content strands of
algebra suggested by Kaput (2008), had the same duration, were based on the inquire-based
learning approach and were cognitively demanding. Nevertheless, they were elaborated through
different types of tasks. The first course, which was named “Semi-structured problem situations”,
used semi-structured tasks connected to real life scenarios and required students to identify the
mathematics involved in order to answer to their main question. The second course, which was
named “Structured mathematical investigations”, used more mathematical tasks that were assisted
with scaffolding questions. Hence, the two teaching experiments were compared in relation to the
types of the tasks through which algebraic thinking was expected to emerge.
Methodology
Participants
The participants were 96 fifth-graders from 4 classes in 2 urban schools. The classes were selected
by convenience. Two of the classes (one class from each school) formed the group that participated
in the first course and the other two classes formed the group that participated in the second.
Test on early algebraic thinking
The same test was administered to the students before and after the conduction of the courses in
order to measure their early algebraic thinking. The test consisted of 22 tasks that were accordingly
categorized into three groups which reflected Kaput’s (2008) three content strands of algebra. The
first group (generalized arithmetic) involved tasks, such as determining whether the sum of two
numbers will be odd or even, using the distributive property of multiplication for examining errors
in performing the multiplication algorithm, describing movements in the hundredths’ table, and
solving equations and inequalities. The second group of tasks involved finding the nth term in
geometrical and numerical patterns, interpreting graphs, and describing correspondence
relationships among quantities. The third group of tasks (modeling languages) required the
generalization of regularities by observing the relationships involved in realistic situations and
analyzing information presented symbolically, graphically or diagrammatically. The internal
consistency of scores measured by Cronbach’s alpha was satisfactory for the test (a=0.87).
Teaching experiments
Both intervention courses addressed the same algebraic concepts, and were developed through ten
lessons of 80-minutes duration. The researcher had the role of the teacher in all lessons. Table 1
presents the objectives of the lessons in each strand. The problems used in both interventions were
adapted from previous studies or online resources.
The “Semi-structured problem situations” course used semi-structured problems arising from real
life situations. Students were confronted with a general question and were given time to explore the
problem situation, analyze and combine information and apply their own strategies for solving the
task. These tasks employed some features of modeling-like tasks. Specifically, modeling-like tasks
were considered as appropriate for enhancing the development of algebraic thinking because they
involve the description and interpretation of complex systems of information through the
application of processes such as, constructing, explaining, justifying, predicting, generalizing,
conjecturing, and representing (English, 2011).
The “Structured mathematical investigations” course reflected mathematical contexts that aimed to
direct students to develop relational thinking, through identifying and understanding structure in
mathematical concepts. Specifically, the tasks were more mathematical in nature, involved
scaffolding steps and pathways which guided students to the extraction of an explicit conclusion.
This kind of activities were considered as relevant and important for enhancing algebraic thinking
since they apply fundamental processes, such as formulation and expression of relationships and
generalizations, and progressive symbolization.
Lessons Content strand
3,4
Objectives
Generalized
arithmetic
Apply properties and relationships of whole numbers,
apply properties of operations on whole numbers, treat
numbers by attending structure rather than computations
1,2,6,7 Functional thinking
Encode information graphically for analyzing a functional
relationship, identify correspondence among quantities or
co-variation relationships, identify and describe
numerical and geometrical patterns
5,8,9,10 Modeling languages
Generalize regularities from mathematized situations
inside or outside mathematics
Table 1: Structure of Instructional Interventions and Objectives for each Lesson
Figure 1: Semi-structured problem situation (left) and Structured mathematical investigation (right)
The task on the left was adapted from a lesson presented in the website
https://illuminations.nctm.org. Using a context of arranging chairs around tables, students were
exposed to two different linear patterns. As specified in the website, this activity leads to an
intuitive understanding of how to extend and describe a pattern using words or symbols. The task
on the right was adapted from a lesson presented in the website www.explorelearning.com .
Students studied three patterns of squares in a grid. Each pattern was more complex compared to
the previous pattern (The pattern presented in Figure 1 was the first pattern). As stated in the
website, this activity aims to the extension of figural patterns and the extraction of a general rule. In
this sense, both tasks targeted on the description and generalization of figural and numeric patterns.
However, the first task introduced from the beginning a complex pattern; the second started from a
simple pattern and moved to more complex patterns.
Analysis
The SPSS statistical package was used to analyze the results. Since the tasks in the pre-test and
post-test were the same, gain scores were used (the difference between post-test and pre-test scores)
as the dependent variable. The Kolmogorov-Smirnov and Shapiro-Wilk tests showed that the gain
scores were normally distributed (p>.01). The P-P and Q-Q plots did not show crucial variations. In
order to compare the early algebraic thinking abilities of the two groups prior to the intervention, a
multivariate analysis of variance (MANOVA) was conducted. MANCOVA was used to examine
the impact of the intervention courses on participants’ early algebraic thinking. The type of
intervention was the independent variable, students’ performance in early algebraic thinking pre-test
was considered as the covariate, and the performance differences between the pre- and post- tests as
the dependent variables. Moreover, paired-sample t-test was performed in order to measure the
differences in the performance of students of the same group in the pre- and post-tests.
Results
The results of the MANOVA analysis suggested that the two groups did not have any statistically
significant differences in their early algebraic thinking abilities prior to the intervention (F=.576,
p>.05). Table 2 presents the results of the MANCOVA analysis, regarding the comparison of the
impact of the two teaching experiments on the groups’ performance in the early algebraic thinking
post-test, controlling for their pre-test scores.
Ability
Overall Early
Algebraic
Thinking
Generalized
Arithmetic
Functional
Thinking
Modeling
Structured Mathematical
Investigation
Mean1
SE
Semi-structured
Problems
Mean1
SE
df F
p
np2
.452
.206
.570
.179
1
6.452
.013*
.088
.663
.213
.647
.246
1
.081
.777
.001
.369
.225
.547
.270
1
26.845 .000** .286
.291
.291
.509
.319
1
9.804
.003*
.128
1
Estimated Marginal Means
*p<.05, **p<.01
Table 2: Results of the Multiple Covariance Analysis Between the Two Intervention Groups Post-test
Performance in Early Algebraic Thinking
The analysis indicated significant overall intervention effects, controlling for pre-test scores in the
early algebraic thinking test (Pillai’s F=9.586, p<.05). The students in the “Semi-structured problem
situations” group had a significantly higher overall performance in early algebraic thinking to
students in the “Structured mathematical investigations” group. The effect size indices for the
overall algebraic thinking ability (partial n2=.088) suggested that the effect of the “Semi-structured
problem situations” course over the “Structured mathematical investigations” course was moderate.
The performance of the “Semi-structured problem situations” group in the generalized arithmetic
tasks did not have any significant difference in relation to the performance of the “Structured
mathematical investigations” group (Pillai’s F=.081, p>.05). The “Semi-structured problem
situations” group had significantly higher performance in the functional thinking tasks (Pillai’s
F=26.845, p<.01) and the modeling tasks (Pillai’s F=9.804, p<.05) in comparison to the “Structured
mathematical investigations” group. The effect size indices for the functional thinking tasks (partial
n2=.286) and the modeling tasks (partial n2=.128) suggested that the effect of the “Semi-structured
problem situations” course over the “Structured mathematical investigations” course was moderate.
Table 3 presents the results of the paired-samples t-test regarding the differences in the pre- and
post-test scores within the same group. The results showed statistically significant differences
between the pre- and post-tests performance means of the “Structured mathematical investigations”
group. Students in this group had a significant increase in their overall algebraic thinking ability and
in the generalized arithmetic tasks. The results also showed that no statistically significant
differences existed between pre- and post-tests performance means in the functional thinking and
modeling tasks.
Regarding the “Semi-structured problem situations” group, the results showed statistically
significant differences in the mean difference between the pre- and post-tests means of
performance. These students had a significant increase in their overall ability and in all types of
tasks.
Ability
Overall
Early
Algebraic
Thinking
Generalized
Arithmetic
Pre-test
Post-test
M
M
SD
SD
T(df)
p
Structured Mathematical
Investigations
Semi-structured Problems
.337 .195
.452 .206 -5.519(33)
.000**
.368 .151
.570 .179 -10.147(34)
.000**
Structured Mathematical
Investigations
Semi-structured Problems
.467 .326
.663 .213 -4.112(33)
.000**
.473 .235
.647 .246 -4.818(34
.000**
Functional
Thinking
Modeling
Structured Mathematical
Investigations
Semi-structured Problems
Structured Mathematical
Investigations
Semi-structured Problems
.302 .263
.369 .225 -2.774(33)
.09
.404 .228
.223 .241
.547 .270 -5.663(34)
.291 .291 -1.231(33)
.000**
.227
.183 .202
.509 .319 -9.926(34)
.000**
**p<.01
Table 3: T-test Comparisons between Pre-test and Post-test Performance of the two groups
Discussion and Conclusion
This study compared the effect of two intervention courses on students’ early algebraic thinking.
The results showed that instruction with “Semi-structured problem situations” had better learning
outcomes compared to instruction with “Structured mathematical investigations”, while controlling
for preliminary differences regarding students’ early algebraic thinking. Specifically, students who
received instruction that was developed through “Semi-structured problem situations” outperformed
students who received instruction that was developed through “Structured mathematical
investigations” in the algebraic thinking post-test. Nevertheless, more detailed results regarding the
effect of the two types of courses have shown that both of them had positive impact in the
generalized arithmetic strand. What seems to have influenced the overall outcome of the
comparison between the two courses is the fact that students involved in the “Semi-structured
problem situations” course had significantly higher performance in the functional thinking and
modeling strands to students that were involved in the “Structured mathematical investigations”
course.
A possible explanation for this result seems to be the fact that the two intervention courses involved
different types of tasks in respect to the way algebraic thinking was expected to emerge. While both
interventions had high cognitive demands and were developed through activities that entailed
cooperative learning, use of manipulatives, and technological tools, it appears that the nature and
type of the tasks used had a significant role regarding the learning outcomes. As suggested by Stein
and Lane (1996), the tasks determine not only the concepts and knowledge that students acquire but
also the way students will come to process, use and make sense of those concepts and knowledge.
On the one hand, the tasks that were included in the “Semi-structured problem situations” course
shared common features with modeling approaches to mathematical problem solving. As English
(2011) described, modeling-like tasks offer enriched learning experiences that require students to
extract meaning from open situations by mathematizing the situations in ways that are meaningful
to them. This kind of processes are linked to early algebraic thinking. As Kieran (2004) supported,
early algebraic thinking is related to several processes, including problem solving, modeling,
justifying, proving, and predicting. Hence, modeling-like tasks seem to involve the majority of the
processes that are related to early algebraic thinking. On the other, “Structured mathematical
investigations” tasks appeared to be effective in helping students to notice the structure in
arithmetical contexts and engage students to learning experiences that are mostly focused on the
generalized arithmetic strand.
As Radford (2004) argued, the algebraic structure of a problem (e.g. the letter-symbolic
representations), the problem context (e.g. word problems, modeling activities) and the exterior of
the problem context (e.g. social and cultural features, such as language, body movements, and
experience) constitute basic sources that students utilize in order to extract meaning. The results of
the current study indicated that the “Semi-structured problem situations” tasks encompassed all of
these sources in an effective way and enabled students to construct their own meaning and develop
understanding of various algebra aspects. Thus we may say that the positive effect of an
intervention course is in a great extend related to the design and implementation features of the
tasks involved.
Future research might further investigate whether the effect of “semi-structured” or “structured”
tasks is different with younger or older students. The effect of an intervention course that makes use
of both “semi-structure” and “structured” tasks might also be addressed. Moreover, the qualitative
characteristics of students’ behavior while participating in this kind of intervention courses needs to
be investigated in detail, in order to better understand the nature of thinking that they develop and
the strategies they apply.
References
Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes algebraic
reasoning. Journal for Research in Mathematics Education, 36(5), 412–446.
English, L.D. (2011). Complex Learning through cognitively demanding Tasks. The Mathematics
Enthusiast, 8(3), 483–506.
Jones, K., & Pepin, B. (2016). Research on mathematics teachers as partners in task design. Journal
of Mathematics Teacher Education, 19(2–3), 105-121. doi:10.1007/s10857-016-9345-z
Irwin, K.C., & Britt, M.C. (2005).The Algebraic nature of students’ numerical manipulation in the
New Zealand Numeracy Project. Educational Studies in Mathematics Education, 58(2),
169–188. doi: 10.1007/s10649-005-2755-y
Kaput, J. (2008). What is algebra? What is algebraic reasoning? In J. Kaput, D.W. Carraher, & M.L.
Blanton (Eds), Algebra in the early grades (pp. 5–17). New York: Routledge.
Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick, &
M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study
(pp. 21–34). Dordrecht, The Netherlands: Kluwer.
Kieran, C., Pang, J., Schifter, D., & Ng, S.F. (2016). Early Algebra: Research into its nature, its
learning, its teaching. In Kaiser, G., ICME-13 Topical Surveys. Springer Open.
doi:10.1007/978-3-319-32258-2
Radford, L. (2004). Syntax and meaning. In M.J. Høines & A.B. Fuglestad (Eds.), Proceedings of
the 28th Conference of the International Group of Psychology of Mathematics Education, 1
(pp. 161–165). Bergen, Norway.
Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to
think and reason: An analysis of the relationship between teaching and learning in a reform
mathematics project. Educational Research and Evaluation, 2(1), 50-80.
Sullivan, P., Clarke, D., & Clarke, B. (2012). Teaching with tasks for effective mathematics
learning. Berlin: Springer.
Warren, E. & Cooper, T. (2008). Generalizing the pattern rule for visual growth patterns: Actions
that support 8 year olds’ thinking. Educational Studies in Mathematics, 67(2), 171–185. doi:
10.1007/s10649-007-9092-2.
Watson, A. (2009). Key understandings in mathematics learning, Paper 6: Algebraic Reasoning.
London: Nuffield Foundation. Available online at: http://www.nuffieldfoundation.org/keyunderstandings-mathematics-learning