Exploring Discipline-Based Responsive Teaching Strategies in High School Mathematics

Algebraic thinking and reasoning are important skills for students to develop as they transition from arithmetic to algebra. The Common Core State Standards for Mathematics links these ideas through their mathematical practices where students make sense of mathematical problems, reason abstractly, model with mathematics, and use appropriate tools that engage them in algebraic thinking and reasoning. This chapter focuses on how a specific tool—the spreadsheet—supports students in building their understandings of spreadsheet features and capabilities while also engaging them in thinking more logically about numbers as needed in preparation for formal algebra. The challenge for teachers is to scaffold students' learning about and with spreadsheets as mathematics learning tools for exploring patterns, variables, and linear problems. This chapter describes this transition through multiple problems where students learn to design dynamic and dependable spreadsheets in ways that concurrently develop their algebraic reasoning and thinking through middle grades mathematics problems.

Central in the frameworks that describe algebra from K-12 is the idea that algebraic thinking is not a single construct, but consists of several algebraic thinking strands. Validation studies exploring this idea are relatively scarce. This study used structural equation modeling techniques to analyze data of middle school students’ performance on tasks that correspond to four algebraic thinking strands: (i) Generalized Arithmetic, (ii) Functional Thinking, (iii) Modeling Languages, and (iv) Algebraic Proof. The study also examined the role that cognitive abilities play in students’ algebraic thinking. Results emerging from confirmatory factor analysis showed that the proposed model adequately explains students’ algebraic thinking. Additionally, results emerging form latent path analysis showed that students are first able to solve Functional Thinking tasks and only when this is achieved, they proceed to solve Generalized Arithmetic tasks, then Modeling Languages tasks, and finally Algebraic Proof tasks. Lastly, the quantitative analyses indicated that students’ cognitive abilities (analogical, serial, and spatial reasoning) predict students’ algebraic thinking abilities.

<p class="Afiliasi">Algebra is a fundamental topic in the mathematics curriculum at Madrasah Tsanawiyah (MTs), yet many students continue to face challenges in solving algebraic problems. This study aims to examine the algebraic thinking skills of eighth-grade students at MTs Unggulan PP Amanatul Ummah Surabaya, particularly through number-oriented and structure-oriented approaches. A descriptive qualitative method was employed, involving 61 students as participants. The instruments included an algebraic thinking test incorporating visual illustrations and in-depth interviews with six selected students representing high, medium, and low ability levels. The findings reveal that only high-ability students were able to effectively apply structure-oriented algebraic thinking. The majority of students demonstrated moderate-level skills, with primary difficulties in constructing mathematical models based on problem structures. These results underscore the importance of instructional strategies that cultivate students’ structural reasoning in algebra. The study provides valuable insights for educators to design learning activities that support the development of students' algebraic thinking, and it serves as a reference for future research to explore targeted interventions aimed at enhancing algebraic problem-solving abilities in MTs students.</p>

The aim of this study is to investigate the relationship between ninth grade students’ symbol sense behaviors, algebraic thinking skills and academic achievement. To achieve this aim, a qualitative research approach known as case study design was employed. A total of three students studying in a high school in Gümüşhane province constituted the study group of the research. In the study, considering the opinions of the mathematics teacher conducting the course and the academic achievement levels of the students in the mathematics course, one student from each achievement level was selected as low, medium and high academic achievement level. The data were acquired from five research inquiries in the literature and adapted in line accordance with expert perspectives. The data were analyzed using thematic coding with an analysis table prepared in line with expert opinions. Students with high achievement level showed symbol sense behaviors at the desired level by using symbols in a flexible and fluent way, while students with low and medium achievement level could not exhibit symbol sense behaviors at the desired level. Students with high academic achievement have advanced algebraic thinking skills. The algebraic thinking skill behaviors of students with low academic achievement are more in the form of rote stereotyped signs. As a result, students' algebraic thinking skills and symbol sense behaviors were found to be compatible with their academic achievement levels. In addition, it was concluded that there was a positive relationship between algebraic thinking skills and symbol sense behaviors.

Contents: Preface. Skeptic's Guide to Algebra in the Early Grades. Part I: The Nature of Early Algebra. J.J. Kaput, What Is Algebra? What Is Algebraic Reasoning? J.J. Kaput, M.L. Blanton, L.M. Armella, Algebra From a Symbolization Point of View. J. Mason, Making Use of Children's Powers to Produce Algebraic Thinking. J.P. Smith III, P.W. Thompson, Quantitative Reasoning and the Development of Algebraic Reasoning. E. Smith, Representational Thinking as a Framework for Introducing Functions in the Elementary Curriculum. Part II: Students' Capacity for Algebraic Thinking. V. Bastable, D, Schifter, Classroom Stories: Examples of Elementary Students Engaged in Early Algebra. C. Tierney, S. Monk, Children's Reasoning About Change Over Time. N. Mark-Zigdon, D. Tirosh, What Is a Legitimate Arithmetic Number Sentence? The Case of Kindergarten and First Grade Children. T. Boester, R. Lehrer, Visualizing Algebraic Reasoning. D.W. Carraher, A.D. Schliemann, J.L. Schwartz, Early Algebra Is Not the Same as Algebra Early. B.M. Brizuela, D. Earnest, Multiple Notational Systems and Algebraic Understandings: The Case of the Best Deal Problem. I. Peled, D.W. Carraher, Signed Numbers and Algebraic Thinking. Part III: Issues of Implementation: Taking Early Algebra to the Classrooms. M.L. Franke, T.P. Carpenter, D. Battey, Content Matters: The Case of Algebra Reasoning in Teacher Professional Development. M.L. Blanton, J.J. Kaput, Building District Capacity for Teacher Development in Algebraic Reasoning. B. Dougherty, Measure Up: A Quantitative View of Early Algebra. D. Schifter, S. Monk, S.J. Russell, V. Bastable, Early Algebra: What Does Understanding the Laws of Arithmetic Mean in the Elementary Grades? P. Goldenberg, N. Shteingold, Early Algebra: The MW Perspective. Afterword: A. Schoenfeld, Early Algebra as Mathematical Sense-Making.

This paper builds on our previous research and investigates how students’ fractional competence and reasoning can provide clear evidence of non-symbolic algebraic thinking and its progressive transition towards fully generalised algebraic thinking. In a large-scale study, 470 primary students completed a written paper and pencil test. This included three reverse fraction tasks which required students to find an unknown whole when presented with a quantity representing a fraction of that whole. Seventeen students from one participating primary school undertook a semi-structured interview which included reverse fraction tasks, similar to those on the written test, but with progressive levels of abstraction, starting with particular instances and becoming more generalised. Two important products of the study are the Classification Framework for Reverse Fraction Tasks and the Emerging Algebraic Reasoning Framework. The interview results highlight two critical transition points for the emergence of students’ algebraic reasoning. The first is the ability to transition from additive strategies to multiplicative strategies to solve reverse fraction problems. Students reliant on diagrams and additive strategies struggled to solve more generalised tasks that required multiplicative rather than additive strategies. The second transition is the shift from multiplicative thinking to algebraic reasoning where students could generalise their multiplicative knowledge to deal with any quantity represented in a reverse fraction task.

This book chapter illustrates how the use of languages (Spanish and English) mediate middle school Latinx bilingual students’ and (co)facilitators’ explorations of algebraic thinking through computer programming. The participants include two groups of 3 students (one predominantly using English and one both languages), a facilitator—typically an undergraduate engineering student—and two co-facilitators who had participated as students learning the first level of the curriculum through an afterschool program. The setting for the afterschool program was in a bilingual middle school that enrolled primarily Latinx students. Data sources include twelve video recordings that lasted 1.5 hours of students’ interactions in each group while coding and artifacts such as students’ work and curriculum tasks. We draw from one of the twelve sessions in each group that focused on algebraic thinking. Drawing from sociocultural theories of learning, we show how bilingual students manipulated variables through computer programming to make meaning of algebraic concepts that involved exploring variables, evaluating expressions, and generating conjectures. The chapter ends with a discussion and implications for the classroom. The quote above illustrates how Juanita leveraged her bilingualism in Spanish and English as she co-facilitated the teaching of a curriculum that integrated mathematics and computer programming with an undergraduate engineering student in an after-school program. She refers to the students struggling a lot but, at the same time, “they got it (the content) right away.” Her views of bilingual students as “smart learners” and that they can “possibly understand the code better” reflect the central aims of this after-school program. As the State of Computer Education reports (Code.org Advocacy Coalition & CSTA, 2018), rarely do students enrolled in Title I schools have the opportunity to learn computer programming. Zong (2022) reports that the Latinx population is projected to grow to 111.2 million by 2060, making it 28% of the U.S. population. At the same time, there are approximately 4.5 million multilingual student learners in the United States, with three fourths of this student population who speak Spanish as their first language. This steady growth of the Latinx population and the limited opportunities that Latinx students have in engaging with computer programming has consequences for students like Juanita who attend rural bilingual middle schools with large numbers of English Learners. The majority of students in rural schools tend to be on free and reduced-price lunch and also remain underrepresented in science, technology, engineering, and mathematics (STEM) fields. Thus, this study sought to address the following research question: “How does the use of language (Spanish/English/translanguaging) mediate middle school Latinx bilingual students’ and (co)facilitators’ explorations of algebraic thinking through computer programming?”

Students’ algebraic reasoning, at the beginning of their schooling years, includes the development and promotion of functional thinking and the understanding of mathematical properties, which can be stimulated by solving problems. In the latest Portuguese Program for Mathematics Elementary Education, we do not see the topic Algebra in the first year of school although some other topics include objectives of algebraic nature. This fact showed the importance of research about the introduction of concepts and development of algebraic skills by elementary school teachers. We investigated the concept of algebraic thinking and how it has been addressed, by interviewing 50 teachers from Portugal. The results showed that the respondents agree with algebraic experiences in the early years of school; however, the data showed a considerable distance when compared to the concept of “algebraic reasoning” adopted by the current scientific community have of algebraic reasoning. With regard to activities that contribute to its promotion, problem solving was considered to play an important role in the development of algebraic skills as well as its representation and generalization.

Trigonometry is an area of the high school mathematics curriculum strictly related to algebraic, geometric, and graphical reasoning. In spite of its importance to both high school and advanced mathematics, research has shown that trigonometry remains a difficult topic for both students and teachers. A new approach the teaching of trigonometry – calibrated for the 21 st century - opens the question of the place and nature of trigonometry in contemporary high school mathematics. In this prospective, the authors have carried out a study aimed at explaining connections between research and teaching practice of trigonometric functions emphasizing conceptual understanding, multiple representation and connections, mathematical modelling, and mathematical problem-solving. This paper shows a teaching approach for meaning construction of trigonometric functions with the aid of technological artefacts.

Students still have limited algebraic reasoning skills, so a learning model called Problem-Based Learning (PBL) is utilized as a teaching strategy to help student develop their algebraic thinking abilities. This study uses the learning model Problem-Based Learning (PBL) to the algebraic thinking skills of class VIII students. The method of research is descriptive qualitative research. The students in class VIII students of SMPN 1 Lahat participated in this study. Problem-Based Learning (PBL) worksheet in accordance with three algebraic thinking abilities: generational activities, transformational activities, and global, meta-level, and mathematical activities. Data for the study came from observations, interviews and written exams. Data from the results of observation sheets are used to see the emergence of indicators of students' algebraic thinking during learning with a total of 2 sessions; data from the results of written tests are used to categorize students' algebraic thinking abilities after learning, taking 2 students from each category, and interviews to clarify the results of students' answers. The findings of this study show that students possess a range of low, medium, and high levels of algebraic thinking ability. The result of the implementation learning model showed that 14 students were in the medium category and 17 students were in the low category.

This systematic literature review (SLR) investigates transformative teaching strategies for enhancing students’ algebraic thinking, a foundational competency in mathematics education. Recognizing the long-term significance of algebraic reasoning, this review synthesizes findings from 25 peer-reviewed articles published between 2022 and 2024, identified through comprehensive searches in Scopus and Web of Science, and selected using the preferred reporting items for systematic reviews and meta-analyses framework. The sample size aligns with accepted standards for SLRs in emerging educational domains. Drawing from recent scholarship, this study constructs a structured framework across four interrelated themes: (1) cognitive transitions in algebra learning, illuminating how students shift from arithmetic to abstract reasoning; (2) innovative pedagogical strategies that promote active, student-centered learning; (3) representational fluency, highlighting the role of visual, symbolic, and contextual tools in bridging conceptual gaps; and (4) developmental alignment in curriculum and assessment design, advocating for instructional sequencing tailored to learners’ cognitive growth. The synthesis reveals that integrating cognitive, pedagogical, and curricular dimensions significantly strengthens algebraic reasoning. Despite its methodological rigor, the study is limited to English-language journal articles and excludes grey literature, which may constrain the comprehensiveness of findings. Moreover, the literature reflects developments only up to 2024, and more recent innovations may not be captured. Nonetheless, this review contributes a timely, evidence-based model for guiding instructional reform in algebra education and underscores the need for targeted, flexible strategies that support students’ conceptual progression toward higher-level mathematics.

In this chapter, we analyze the process of developing algebraic thinking in children, as it relates to the necessary conceptualization giving meaning to the ideas underpinning the basic rules of algebra. In recent years, the National Council of Teachers of Mathematics and the Ontario Ministry of Education continue to provide resources for the development of algebraic reasoning, starting in early childhood. In early childhood, the relationship between equivalence and equality is a key element that integrates different facets of the development of numerical. We analyze algebraic thinking by examining mathematics-related tasks completed by twenty-one 5-year-old children. Our purpose is to highlight the use of landmark strategies, big ideas, and models, in regard to equivalence and equality and their role in the development of early algebraic reasoning.

This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities provided for PSMTs to engage in proving- related activities, including making conjectures, investigating conjectures, developing arguments, evaluating arguments, and disproving by using counterexamples. We also analyzed instructional strategies reported by the instructors. We found the inconsistency between instructors’ perceptions of the importance of reasoning and proof in algebra and instructor-reported opportunities to learn. Findings also indicated that developing arguments was reported the most frequently. In addition, instructors reported more pedagogy- focused general teaching strategies than proof-specific teaching strategies.

This article applies anthropological theory in exploring how Mathematics teachers imbed indigenous epistemologies in the teaching of pure Mathematics in high schools in South Africa. I spent two months staying in one of the Venda communities sharing at times food and other amenities with the three boys, at the same time collecting data and understanding cultural practices. I observed the participants on a daily basis, playing a mathematical game called “mutoga” in their local language. I also joined them in class where they were taught Mathematics at school. Findings seem to indicate that ethnomathematics epistemologies can successfully be embedded in the teaching of pure Mathematics in South African high schools. I therefore argue that the use of indigenous mediation experiences must assume culturally responsive pedagogy to open up the curriculum and assessment practices to allow for different ways of knowing and being. Initially, a generic argument for the inclusion of indigenous content within the Mathematics curriculum is suggested. Secondly, several exemplar scenarios of teaching praxis including indigenous content are discussed. Finally, evidence on the utility of such exemplar scenarios for students in learning about indigenous peoples and key processes and skills for working with indigenous communities from student feedback are discussed. DOI: 10.5901/mjss.2014.v5n15p327

With the deepening development of mathematics education, the cultivation of algebraic thinking has gradually become one of the core tasks of junior high school mathematics teaching. As a fundamental ability in mathematics, it not only has a positive promoting effect on students logical thinking and abstract ability, but also provides important tools and methods for students to further learn advanced mathematical knowledge and solve practical problems. Based on the understanding of mathematical algebraic thinking, this article mainly discusses the importance of students algebraic thinking and how to cultivate students algebraic thinking ability in general operations, logical reasoning, mathematical modeling, etc. and explores new methods and approaches to cultivate students algebraic thinking in junior high school teaching.