Algebraic Thinking Research Articles

Interview study on preservice teachers’ knowledge for teaching early algebra

This study aims to identify features of knowledge about teaching early algebra that preservice teachers have, and
 to suggest what preservice teacher education needs to improve teachers’ knowledge for teaching early algebra. We
 measured the knowledge of 109 preservice teachers using the assessment of mathematical knowledge for teaching
 elementary students as part of the measures of effective teaching project, and conducted in-depth interviews with
 18 of them. We report the findings from the interviews in terms of big ideas of algebra. Based on the results, we
 discuss teacher education programs with regard to the connections between the mathematical content, operation
 laws as the strand of algebraic thinking, and mathematical and didactical decisions from the teacher’s perspective.

Promoting Prospective Mathematics Teachers’ Professional Vision on a Whole-class Reflective Discussion: Contributions of Digital Resources

The present study aims to understand how the professional vision of prospective teachers (PTs) regarding whole-class reflective discussion develops as they explore a multimedia case (MC) in a mathematics teacher education course. The data come from the written productions of seven groups of PTs (N=15) related to whole-class discussion of one 6th grade lesson in two sections of the MC: first, they have focused on the teacher’s lesson plan and on her intentions for the whole-class discussion aiming at the students’ algebraic thinking; and in the second, they have analysed two video episodes of that lesson phase. The results show an integrated development of two processes of PTs’ professional vision – noticed events and the nature of the reasoning about them – with significant evolution from one section to another as they explore the MC. The study shows PTs’ alternative ways of accessing and making sense of the complexity of both ambitious and authentic teaching practices that can also be employed in at distance teacher education and mitigate the consequences of the impossibility to access the classrooms, such as that resulting from the recent COVID-19 pandemic.

Relational Thinking And Problem Solution Strategies Beginning Algebra High School Students

Many students' difficulties in learning early algebra are associated with deficiencies in learning during the student's transition from arithmetic to algebra. relational thinking; a way of thinking that can be a gateway to algebraic thinking, is touted as one of the solutions to these difficulties. This study aims to determine the profile of relational thinking and early algebra problem solving strategies for junior high school students. Through the phenomenological design, the researcher selected six participants who fit the research criteria. The data analyzed refers to several theoretical frameworks on relational thinking from several experts. This study focuses on the indicators of interpreting the equals sign , using the relationship between numbers and applying the basic properties of number operations, as well as the methods students use in solving initial algebraic problems (algebraic forms and linear equations of one variable). As a result and its implication, students' relational thinking is seen in several items, namely the first 3 indicator items: each item is seen in S1, S4, and S6; The second 2 indicator items are visible on the S4; and the third 3 indicator items are seen in S1. The majority of subjects do computing dominantly and or apply the nature of operations based on procedural knowledge. Similarly, in solving early algebra problems, it was found that the dominant tendency of students to consider algebraic expressions from a procedural perspective rather than a structural perspective (based on mathematical concepts or structures). From these results, it is deemed necessary to develop relational thinking since students study arithmetic in elementary school, including by familiarizing students with number sentences in various forms and encouraging understanding of mathematical structures (relationships between elements and understanding of the nature of number operations).

Relational Thinking And Problem Solution Strategies Beginning Algebra High School Students

Many students' difficulties in learning early algebra are associated with deficiencies in learning during the student's transition from arithmetic to algebra. relational thinking; a way of thinking that can be a gateway to algebraic thinking, is touted as one of the solutions to these difficulties. This study aims to determine the profile of relational thinking and early algebra problem solving strategies for junior high school students. Through the phenomenological design, the researcher selected six participants who fit the research criteria. The data analyzed refers to several theoretical frameworks on relational thinking from several experts. This study focuses on the indicators of interpreting the equals sign , using the relationship between numbers and applying the basic properties of number operations, as well as the methods students use in solving initial algebraic problems (algebraic forms and linear equations of one variable). As a result and its implication, students' relational thinking is seen in several items, namely the first 3 indicator items: each item is seen in S1, S4, and S6; The second 2 indicator items are visible on the S4; and the third 3 indicator items are seen in S1. The majority of subjects do computing dominantly and or apply the nature of operations based on procedural knowledge. Similarly, in solving early algebra problems, it was found that the dominant tendency of students to consider algebraic expressions from a procedural perspective rather than a structural perspective (based on mathematical concepts or structures). From these results, it is deemed necessary to develop relational thinking since students study arithmetic in elementary school, including by familiarizing students with number sentences in various forms and encouraging understanding of mathematical structures (relationships between elements and understanding of the nature of number operations).

The role of variables in relational thinking: an interview study with kindergarten and primary school children

Relational thinking and dealing with variables are two essential aspects of algebraic thinking. Relational thinking means viewing mathematical expressions and equations as a whole rather than as individual computing processes. It is characterized by using relationships between mathematical objects, and refers to the relations of equality and inequality. In this study, to examine the relational thinking of kindergarten and primary school children, this perspective was applied using non-symbolic representations in the form of boxes and marbles. Using multiple variables is a very powerful but also difficult tool of algebra. The study had the aim of examining how kindergarten children and primary school children establish relationships between several variables which are represented with real materials. The interview study was conducted with children aged 5–10 years. Marbles and different colored boxes represented equations with unknowns and quantities depending on each other. Initially, two approaches could be differentiated, namely, number-oriented and structure-oriented approaches. It could be shown that certain conceptualizations of variables were related to children’s ability to show relational thinking. Kindergarteners are stimulated to think relationally by unknown quantities which can be determined. This process was observed in primary school children dealing with quantities that depended on each other. In addition, the conceptualization of the variables represented as boxes was examined. The concepts of general number and variable as changing quantity were categorized. Further conceptualizations resulted from the interview data, namely, categories of the undeterminable, the specific number, and the quasi-general.

What Was and Is Algebraic Thinking Skills at Different Education Levels?

The aim of this study is to determine the algebraic thinking levels, misconceptions and understanding levels of algebraic expressions of 6th, 7th and 8th grade students. In this research, developmental research methods, which is one of the descriptive studies, were used. This study was carried out with secondary school students (6th, 7th and 8th grades) studying in a public school located in a province of the Eastern Anatolia Region in Turkey. The sample of the study consists of 82 students studying at secondary school in the second semester of the academic year 2020-2021. Algebraic thinking levels determination form (ATLDF), consisting of 8 questions and 4 levels, and interview were used as data collection tools. The test was administered to 33 students from 6th grades, 21 students from 7th grades and 28 students from 8th grades, and then interviews were conducted. From the tests and interviews, it was seen that the students had 30 misconceptions about algebraic expressions, that their algebraic thinking levels were quite low, that there were only 3 students who could reach Level-4 and all of these students were female students. From the results, it was concluded that the algebraic thinking level of the students who have problems in the transition from arithmetic to algebra is low; and therefore, it was suggested that the connection between arithmetic and algebra should be established firmly, and studies should be done in this direction.

Decoding fact fluency and strategy flexibility in solving one-step algebra problems: An individual differences analysis

Algebraic thinking and strategy flexibility are essential to advanced mathematical thinking. Early algebra instruction uses ‘missing-operand’ problems (e.g., x – 7 = 2) solvable via two typical strategies: 1) direct retrieval of arithmetic facts (e.g., 9 – 7 = 2) and 2) performance of the inverse operation (e.g., 2 + 7 = 9). The current study investigated the strategies people choose when solving these problems, and whether some people are more flexible in their choices than others. U.S. undergraduates (n = 59) solved missing-operand problems and made speeded verifications of arithmetic sentences corresponding to the direct- and inverse-matched facts. To ‘decode’ their strategy as direct or inverse, each participant’s response times (RTs) for missing-operand problems were regressed on their RTs for the corresponding direct and inverse facts. Our findings replicated the problem size effect for the arithmetic verification task and extended this effect to missing-operand (i.e., one-step) algebra problems, suggesting that the two tasks draw on common representations and processes in the addition (but not subtraction) context. We found individual differences in strategy choice and flexibility such that participants varied both in terms of fluency for retrieving the direct fact and sensitivity to the potential benefit of switching to the inverse fact, which was validated by self-report. We did not find a predicted relation between strategy flexibility and standardized mathematical achievement. These findings inform our understanding of the cognitive processes involved in strategy flexibility in algebra and establish an RT-decoding paradigm for future examination of individual differences in students’ learning of early algebra concepts.

Identifying and promoting young students’ early algebraic thinking

Algebraic thinking is an important part of mathematical thinking, and researchers agree that it is beneficial to develop algebraic thinking from an early age. However, there are few examples of what can be taken as indicators of young students’ algebraic thinking. The results contribute to filling that gap by analyzing and exemplifying young students’ early algebraic thinking when reasoning about structural aspects of algebraic expressions during a collective and tool-mediated teaching situation. The article is based on data from a research project exploring how teaching aiming to promote young students’ algebraic thinking can be designed. Along with teachers in grades 2, 3, and 4, the researchers planned and conducted research lessons in mathematics with a focus on argumentation and reasoning about algebraic expressions. The design of teaching situations and problems was inspired by Davydov’s learning activity, and Toulmin’s argumentation model was used when analyzing the students’ algebraic thinking. Three indicators of early algebraic thinking were identified, all non-numerical. What can be taken as indicators of early algebraic thinking appear in very short, communicative micro-moments during the lessons. The results further show that the use of learning models as mediating tools and collective reflections on a collective workspace support young students’ early algebraic thinking when reasoning about algebraic expressions.

“Learning models”

The overarching aim of this article is to exemplify and analyse how some algebraic aspects of equations can be theoretically explored and reflected upon by young students in collaboration with their teacher. The article is based upon an empirical example from a case study in a grade 1 in a primary school. The chosen lesson is framed by the El’konin-Davydov curriculum (ED Curriculum) and learning activity theory in which the concept of a learning model is crucial. Of the 23 participating students, 12 were girls and 11 boys, approximately seven to eight years old. The analysis of data focuses on the use of learning models and reflective elaboration and discussions exploring algebraic structures of whole and parts. The findings indicate that it is possible to promote the youngest students’ algebraic understanding of equations through the collective and reflective use of learning models, and we conclude that the students had opportunity to develop algebraic thinking about equations as a result of their participation in the learning activity.

Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School

In this study, we analyze how 9–10-year-old pupils work with equations, a central aspect of algebraic thinking in early grades and a cornerstone for more formal learning of algebra. Specifically, we seek: (a) to describe the main characteristics of the tasks that support algebraic thinking through a translation process from arithmetic word problems to algebraic language and vice versa, and (b) to identify how pupils refer to indeterminate quantities in these contexts and what meaning they give to them. The analysis focuses on the semantic congruence of the expressions proposed by them and on the dialogue they held during the translation process. We analyzed the oral discussion in the pools and the written responses to the problem that pupils posed. The results show that arithmetic word problems allow the indeterminate to become an object of thought for pupils, who represent it in multiple ways and refer to it when proposing equations that represent the structure of each problem. Another finding highlights that reflection on the interpretation of the equations supports the identification of two meanings associated with indeterminate quantities, namely, unknown and variable.

Coordinating visual and algebraic reasoning with quadratic functions

Quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students’ difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old) students investigated the potential of figural pattern generalisation—a context not traditionally used for teaching quadratics—to stimulate students’ coordination of visual and algebraic reasoning and attention to quadratic function concepts. Theorisations of embodied visualisation, algebraic thinking, and student noticing were drawn on to analyse the pairs responding to 19 quadratic figural pattern generalisation tasks interspersed throughout their class topic on quadratic equations. It was found that students became adept at connecting the generality of different types of structural aspects of figures (square, rectangular, linear, constant/invariant) to their symbolic expression in quadratic equations. Students’ construction of numeric instantiations of figural aspects was found to support pairs in moving towards symbolic generalisation. Task prompts to find different (but equivalent) algebraic equations for the same pattern evidenced pairs beginning to distinguish among general, factorised and standard forms of quadratic equations. One pair’s attention to first and second differences (between total quantities of figures in a sequence) highlighted both the difficulty of and potential for connecting quadratic rate-of-change concepts and parameters visually. Implications for including figural pattern generalisation when teaching quadratics and suggestions for further research are shared.

TRANSFORMATION OF PRE-SERVICE MATHEMATICS STUDENT'S ALGEBRA AND CALCULUS THINKING IN SOLVING DIFFERENTIAL EQUATION PROBLEMS

This study aims to analyze the transformation of algebraic thinking and calculus of preservice the mathematics students based on SOLO taxonomy in solving differential equations problems. The research subjects were 86 students in the mathematics education study program. Subject selection uses purposive sampling (students who take courses in differential equations). Data were collected using problem-solving tests and interviews which were then analyzed using the descriptive qualitative method with the following stages: (1) transcribing test and interview data, (2) coding segmentation, (3) analyzing student thinking transformations, and (4) concluding. The results showed that the transformation of algebraic and calculus thinking was used by students at each level of thinking to solve problems. The higher the level of thinking achieved, the better and the maximum transformation of algebraic and calculus thinking used by students. These results indicate that students need to be well supported and facilitated in problem-solving to achieve higher levels of thinking, such as the relational and extended abstract levels.

Variables in early algebra: exploring didactic potentials in programming activities

In this paper we consider implications of the current world-wide inclusion of computational thinking in relation to children’s development of algebraic thinking. Little is known about how newly developed visual programming environments such as Scratch could enhance early algebra learning. The study is based on examples of programming activities used by mathematics teachers in Sweden, teaching students aged 10–12 years during the first two years of implementing programming in the mathematics curriculum. Informed by Chevallard’s praxeology in terms of praxis and logos, we describe, unpack, discuss and expand these activities. Core issues related to algebra found in the three activities are as follows: making implicit variables explicit; using a counter variable; and identifying parameters as a specific type of variable. Our findings show that, in addition to already identified uses of variables in early algebra, programming activities in the early years bring in new aspects and new ways of treating variables that could, potentially, enhance students’ understanding of variables and generalization, provided that programming praxis is embedded in an appropriate algebra logos.

Seeds of algebraic thinking: a Knowledge in Pieces perspective on the development of algebraic thinking

In this paper, we elaborate the seeds of algebraic thinking perspective, drawing upon Knowledge in Pieces as a heuristic epistemological framework. We argue that students’ pre-instructional experiences in early childhood lay the foundation for algebraic thinking and are a largely untapped resource in developing students’ algebraic thinking in the classroom. We theorize that seeds of algebraic thinking are cognitive resources abstracted over many interactions with the world in children’s pre-instructional experience. Further, we provide examples to demonstrate how the same seeds of algebraic thinking present in early childhood can be invoked in reasoning across contexts, grade levels, and different levels of formality of algebraic instruction. The examples demonstrate how the seeds perspective differs from other accounts of the relationship between children’s early activity and their engagement in algebraic reasoning processes. We anticipate this new theoretical direction for characterizing the nature and development of algebraic thinking will lay the foundation for a robust agenda that sheds light on the development of algebraic thinking and informs algebra instruction, particularly how teachers notice and respond to children’s developing algebraic thinking.

From process to object in teachers’ introductory algebra discourse

Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field’s understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers’ use of words, symbols, routines and narratives when they introduce various algebraic concepts. Our results show that there exist instances of objectification in the teachers’ discourses, but these are all implicit, which can make them difficult for students to notice. Thus, teachers need to be made more aware of the importance of making the objectification process explicit. Furthermore, our results show that instances of objectification are tightly connected to the use of formal algebraic symbols. This highlights the importance of the transition from concrete/iconic to formal/arbitrary representations.

Walking through Algebraic Thinking with Theme-Based (Mobile) Math Trails

Tasks are a key resource in the process of teaching and learning mathematics, which is why task design continues to be one of the main research issues in mathematics education. Different settings can influence the principles underlying the formulation of tasks, and so does the outdoor context. Specifically, a math trail can be a privileged context, known to promote positive attitudes and additional engagement for the learning of mathematics, confronting students with a sequence of real-life tasks, related to a particular mathematical theme. Recently, mobile devices and apps, i.e., MathCityMap, have been recognized as an important resource to facilitate the extension of the classroom to the outdoors. The study reported in this paper intends to identify the principles of design for mobile theme-based math trails (TBT) that result in rich learning experiences in early algebraic thinking. A designed-based research is used, through a qualitative approach, to develop and refine design principles for TBT about Sequences and Patterns. The iterative approach is described by cycles with the intervention of the researchers, pre-service and in-service teachers and students of the targeted school levels. The results are discussed taking into account previous research and data collected along the cycles, conducing to the development of general design principles for TBT tasks.

Strategies Employing Prospective Primary Education Teachers in a Functional Relationship Task

Background : The early algebra curricular proposal seeks to promote algebraic thinking from the first educational levels. Functional thinking is an approach intended to promote algebraic thinking in students. At present, various curricular programs include notions related to this thinking. However, this type of thinking has had little influence on prospective teachers, thus arising the need to investigate how they approach tasks to prepare them adequately for their teaching. Objective : To analyse the strategies that prospective Chilean primary education teachers use when they solve a task that implies the generalisation of a functional relationship. Design : We follow a qualitative methodology at an exploratory-descriptive level. Setting and participants : The sample included 18 prospective primary education teachers from a Chilean university. Data collection and analysis : We collected the data through a written test in a group of 18 prospective teachers who were in the first year of training and analysed the data. Results : We highlight the diversity of arithmetic and functional strategies used by prospective teachers. In addition, we highlight their trajectories to solve the task, finding that most of them begin using arithmetic strategies and end using functional strategies. Conclusions : This research will identify the strategies used by prospective primary education teachers and whose data could be useful for the creation of teacher education programs.

An Activity Based on Figurative Patterns: Moroccan Students’ Reasoning

The aim of this paper is to analyse the reasoning and symbolisations used by sixth grade Moroccan students in solving a task based on figurative patterns. Our analysis aims at identifying the systems of actions elaborated by the students to give the general expression of the sequence, according to their perceptions of the sequence of its patterns. The results of this study show that the students can produce a wide variety of generalisations, without using the letter, by resorting to verbal expressions, the majority of which remain in the class of a factual or contextual generalisation. However, several students were not able to determine the general term of the sequence in the question, through an expression that is detached from the context of the situation. This study also contributes to raising the consciousness of Moroccan teachers and researchers of mathematics didactics about the important role played by algebraic thinking, in particular the generalisation coming from an activity based on the figurative patterns, which generates several equivalent and meaningful numerical or algebraic expressions. This article also sheds light on ways for programme designers to enrich primary school curricula, by proposing arithmetic activities with algebraic potential, like the one proposed in our study.

Generalization strategies and representations used by final-year elementary school students

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.

Profile of Student Algebraic Thinking with Polya's Problem-Solving Strategy: Study on Male Students with Field Independent Cognitive Style

This study aims to describe the profile of students' algebraic thinking with Polya's problem solving strategy in completing a linear program conducted on male students with field independent cognitive style. This research is exploratory research with qualitative approaches. In accordance with the purpose of the study, the subject (single subject) were male student with field independent cognitive style. Single subject were screened from 119 participants (male students in grade X) at a high school in the city of Mataram, Indonesia. The criteria for determining a single subject are male students with field independent cognitive style, and having the highest math scores on linear algebra material. The research instrument consisted of the main instrument (researcher/human instrument) which interacted directly with the subject to explore the subject's algebraic thinking profile, and research aid instruments consisting of the GEFT test, mathematical ability test, linear programming tasks, and interview guidelines. Through Polya's problem solving, the students' algebraic thinking profile has been described, where the subject has represented mathematical ideas using algebraic expressions, the subject interprets algebraic expressions; the subject uses symbolic representations, formulations, and expressions of equations using algebraic conventions; and the subject can interpret the solution.