Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach

Algebra is considered key to success in secondary mathematics, yet instruction remains mostly teacher-centered and procedurally oriented, with limited opportunities for students to develop algebraic understanding. This study draws on a large sample of video-recorded ninth grade U.S. algebra lessons to explore the nature of learning opportunities that may help deepen students’ algebraic understanding. I highlight aspects of opportunities to learn algebraic procedures and describe instruction that can enrich these opportunities. I posit that this holds promise as an incremental improvement approach—relatively small adjustments in teachers’ current practices that can serve as a bridge to more ambitious teaching.     

STUDENTS’ WAYS OF REASONING ABOUT NONLINEAR FUNCTIONS IN GUESS-MY-RULE GAMES

There is a growing movement toward the introduction of algebra in early grades. This is supported by an increasing number of research studies that have reported success in getting young students to “do” mathematics considered beyond their reach. Yet, the consensus is that more research is needed to provide insights into the processes by which students make sense of algebraic ideas. In particular, more studies are needed on how nonlinear functions can be introduced in early grades. This article reports an algebra research strand that introduced seventh grade students to quadratic functions using Guess-My-Rule games. The article describes several instances of the students engaging successfully with ideas and forms of reasoning involving quadratic functions. The purpose is to contribute to the debate on the introduction of algebra in early grades by providing further evidence of young students’ ability to engage with algebraic ideas usually considered to be beyond their reach.     

Parsing the notion of algebraic thinking within a cognitive perspective

There is a growing consensus that algebra is an important aspect of mathematics teaching and learning and several abilities are required in order students to have successful performance in algebra. The present study uses insights from the domain of psychology to enrich what is currently known in the domain of mathematics education about the relationship of algebraic thinking with abilities involved in fundamental cognitive processes. In total, 190 students between the ages of 13–17 years old were tested through two tests. The first test addressed four types of cognitive systems which are responsible for the representation and processing of different types of relations in the environment: the spatial-imaginal, the causal-experimental, the qualitative-analytic and the verbal-propositional. The second test addressed algebraic thinking. The results support the key role of the four types of cognitive processes in students’ algebraic thinking. The results also suggest that abilities involved in the four types of cognitive processes predict algebraic thinking abilities, irrespective of the age of the students.     

Teaching Algebra to Students with Learning Difficulties: An Investigation of an Explicit Instruction Model

Abstract. Thirty‐four matched pairs of sixth‐ and seventh‐grade students were selected from 358 participants in a comparison of an explicit concrete‐to‐representational‐to‐abstract (CRA) sequence of instruction with traditional instruction for teaching algebraic transformation equations. Each pair of students had been previously labeled with a specific learning disability or as at risk for difficulties in algebra. Students were matched according to achievement score, age, pretest score, and class performance. The same math teacher taught both members of each matched pair, but in different classes. All students were taught in inclusive settings under the instruction of a middle school mathematics teacher. Results indicated that students who learned how to solve algebra transformation equations through CRA outperformed peers receiving traditional instruction on both postinstruction and follow‐up tests. Additionally, error pattern analysis indicated that students who used the CRA sequence of instruction performed fewer procedural errors when solving for variables.     

Stuck on convention: a story of derivative relationships

In this article, we explore the responses of a group of undergraduate mathematics students to tasks that deal with areas, perimeters, volumes, and derivatives. The tasks challenge the conventional representations of formulas that students are used to from their schooling. Our analysis attends to the specific mathematical ideas and ways of reasoning raised by students, which supported or hindered their appreciation of an unconventional representation. We identify themes that emerged in these responses and analyze those via different theoretical lenses—the lens of transfer and the lens of aesthetics. We conclude with pedagogical recommendations to help learners appreciate the structure of mathematics and challenge the resilience of certain conventions.     

高中教师代数教学知识的调查研究

代数是中学数学课程的重要部分.在高等代数知识方面,教师能理解一些基础而重要的概念,并掌握公式算法.但理解涉及逻辑知识的一些符号表示的复合命题时有困难.在学校代数知识方面,大多数教师显示出扎实的代数基础知识与技能,特别是常规问题的解法明确,运算准确.高中数学教师具体较为扎实的学校代数与高等数学中有关代数的知识,但在理解学生的学习困难,错误概念等方面有待提高。     

Meta-Representation in an Algebra I Classroom

We describe how 1 Algebra I teacher and her 8th-grade students used meta-representational knowledge when generating and evaluating equations to solve word problems. Analyzing data from a sequence of 4 lessons, we found that the teacher and her students used criteria for evaluating equations, in addition to other types of knowledge (e.g., different interpretations of the equal sign) previously reported in the literature. Moreover, the teacher and her students had trouble understanding one another's proposed algebraic models of problem situations due to differences in the criteria that each applied, and this impeded learning. These findings (a) extend an accumulating body of evidence for the role meta-representation plays in mathematics and science learning and (b) add a new dimension to researchers' growing understanding of what teachers must know in order to teach algebra and other complex mathematics and science topics effectively.     

Effects of a reform high school mathematics curriculum on student achievement: whom does it benefit?

This study compared the effects of an integrated reform-based curriculum to a subject-specific curriculum on student learning of 19,526 high school algebra students. Using hierarchical linear modelling to account for variation in student achievement, the impact of the reform-based Core-Plus Mathematics curricular materials on student test scores is compared to the subject-specific curriculum. Findings from this study indicate that students enrolled in integrated mathematics outperformed subject-specific students on an Algebra I exam (highly aligned with content), and performed equally on an Algebra II exam (poorly aligned). High minority students in high-need schools demonstrated higher performance when they were enrolled in integrated mathematics.     

The influence of race–ethnicity and physical activity levels on elementary school achievement

The authors used structural equation modeling to map the relationships between student race–ethnicity via the mediating variable physical activity on English language arts (ELA) and mathematics achievement among 964 fourth- and fifth-grade students. The students attended a New York City Metropolitan area school district and completed the Physical Activity Questionnaire for Older Children, which measured weekly average activity levels. Confirmatory factor analysis validated the use of this instrument. Physical activity had a significant, substantive effect on both ELA and mathematics achievement for students, but was most pronounced among White students on ELA and among Black students on mathematics. Hispanic ethnicity had significant direct and indirect negative associations with ELA and mathematics achievement via their decreased physical activity levels relative to White and Black students. These findings help confirm the important link between physical activity and academic achievement and the need to foster more healthy physical activities for students of all races and ethnicities.     

On the role of representations and artefacts in knowing and learning

This article provides a critical commentary on the concepts of representation and digital artefacts in Morgan and Kynigos’s article of this Special Issue. To set the context, in the first part, I examine some of the tensions that arose during discussions through the 1980s and 1990s about representation in mathematics education research. Then, I comment on the conceptual differences between Morgan’s and Kynigos’s approaches. These differences point to different epistemological assumptions that lead to different conceptualizations of artefacts in learning processes. In the last part, I argue that Morgan’s and Kynigos’s approaches have the merit of moving the discussion about representations to new theoretical horizons. I suggest, however, that a discussion about representations and digital artefacts requires a thematized account of the manner in which the phenomenological artefact- and representation-mediated knowledge produced by students in the classroom relates to the target cultural mathematical knowledge. Such an account, I contend, requires an explicit ontological conception of knowing and knowledge. I conclude the article with an example in which knowledge is considered as codified movement and knowing as the event of its enactment in concrete practice. Within this Hegelian materialist viewpoint, representations are neither predicated in terms of an adequacy between ideas and their representations nor as heuristic devices in meaning making processes. Representations are rather an integral part of the activity of knowledge presentation.     

Mathematical Modelling: the Interaction of Culture and Practice

Using a sociocultural approach we analyse the results of a Mexican/British project which investigated the ways in which mathematics is used in the practice of school science and the role of spreadsheets as a mathematical modelling tool. After discussing the different school cultures experienced by two groups of pre-university 16–18 year old students we analyse how these different cultures influenced their practice of mathematics, as well as their work with mathematical spreadsheet modelling activities. There were clear differences between the two groups of students in their preference for external representations, in their understanding of the kind of answers they were expected to produce and in the way they conceived the role of mathematics in the practice of science. Although students preferences for a particular representation were not significantly modified by the use of a spreadsheet as a modelling resource, at the end of the study the students recognised the value of using a more diverse set of representations. The results obtained suggest the possibility of enhancing students' capability to shift between a wider range of representations, including graphical, algebraic and numeric ones, using a modelling approach embedded in a computer environment such as a spreadsheet. This revised version was published online in July 2006 with corrections to the Cover Date.     

Perspectives on the Origins of Life in Science Textbooks from a Christian Publisher: Implications for Teaching Science

This study analyzes six seventh grade Israeli mathematics textbooks, examining (1) the extent to which students are required to justify and explain their mathematical work, and (2) whether students are asked to justify a mathematical claim that is stated by the textbook or a mathematical claim that they themselves generated when solving a problem. Two different units of analysis were used to analyze two central topics in the seventh grade curriculum as follows: (1) equation solving in algebra and (2) triangle properties in geometry. The findings indicate that all six textbooks had considerably larger percentages of geometric tasks than algebraic tasks, which required students to justify or explain their mathematical work. Moreover, considerable differences were found among the six textbooks regarding the percentages of tasks that required students to justify and explain in both topics, but more so with the algebraic topic. Analysis of whether the textbook tasks required students to justify a mathematical claim that is stated by the textbook or a mathematical claim that the students themselves generated also revealed substantial differences among the textbooks. These findings are discussed, as well as the research methods used, in light of relevant literature.     

Discourses of time and maturity structuring participation in mathematics and further mathematics

Abstract

This paper examines how young people account for choosing mathematical subjects, and how these processes sustain, or not, their continued participation. It draws on a 2-year qualitative study of 24 young people’s accounts of following advanced mathematical pathways within a widening participation programme. Working within a post-structural framework, I combine two arguments: firstly, that local discourses of time, age and maturity position contemporary adolescence as a time of ‘becoming’ that aligns personal aspirations with mathematical progress, and secondly that students’ accounts of choice and aspiration require multiple imaginings of present and future selves. I identify distinct discourses –moving/improving and getting ahead - that structure the intelligibility of participation in mathematics and further mathematics respectively. I argue that tracing the alignments between students’ accounts of themselves and/ in mathematics offers potential to understand emergent practices in mathematics participation but also how exclusions are re-inscribed along classed and gendered lines.     

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network.     

Keep DRAGging ON: Is solving more problems in DragonBox 12+ associated with higher mathematical performance during the COVID-19 pandemic?

Prior research has shown that game-based learning tools, such as DragonBox 12+, support algebraic understanding and that students' in-game progress positively predicts their later performance. Using data from 253 seventh-graders (12–13 years old) who played DragonBox as a part of technology intervention, we examined (a) the relations between students' progress within DragonBox and their algebraic knowledge and general mathematics achievement, (b) the moderating effects of students' prior performance on these relations and (c) the potential factors associated with students' in-game progress. Among students with higher prior algebraic knowledge, higher in-game progress was related to higher algebraic knowledge after the intervention. Higher in-game progress was also associated with higher end-of-year mathematics achievement, and this association was stronger among students with lower prior mathematics achievement. Students' demographic characteristics, prior knowledge and prior achievement did not significantly predict in-game progress beyond the number of intervention sessions students completed. These findings advance research on how, for whom and in what contexts game-based interventions, such as DragonBox, support mathematical learning and have implications for practice using game-based technologies to supplement instruction.

Practitioner notes

What is already known about this topic

• DragonBox 12+ may support students' understanding of algebra but the findings are mixed.
• Students who solve more problems within math games tend to show higher performance after gameplay.
• Students' engagement with mathematics is often related to their prior math performance.

What this paper adds

• For students with higher prior algebraic knowledge, solving more problems in DragonBox 12+ is related to higher algebraic performance after gameplay.
• Students who make more in-game progress also have higher mathematics achievement, especially for students with lower prior achievement.
• Students who spend more time playing DragonBox 12+ make more in-game progress; their demographic, prior knowledge and prior achievement are not related to in-game progress.

Implications for practice and/or policy

• DragonBox 12+ can be beneficial as a supplement to algebra instruction for students with some understanding of algebra.
• DragonBox 12+ can engage students with mathematics across achievement levels.
• Dedicating time and encouraging students to play DragonBox 12+ may help them make more in-game progress, and in turn, support math learning.

     

Learning from comparison in algebra

Mastery of algebra is an important yet difficult milestone for students, suggesting the need for more effective teaching strategies in the algebra classroom. Learning by comparing worked-out examples of algebra problems may be one such strategy. Comparison is a powerful learning tool from cognitive science that has shown promising results in prior small-scale studies in mathematics classrooms. This study reports on a yearlong randomized controlled trial testing the effect of an Algebra I supplemental comparison curriculum on students’ mathematical knowledge. 141 Algebra I teachers were randomly assigned to either implement the comparison curriculum as a supplement to their regular curriculum or to be a ‘business as usual’ control. Use of the supplemental curriculum was much less frequent than requested for many teachers, and there was no main effect of condition on student achievement. However, greater use of the supplemental curriculum was associated with greater procedural student knowledge. These findings suggest a role for comparison in the algebra classroom but also the challenges of supporting teacher integration of new materials into the curriculum.     

JUSTIFICATIONS AND EXPLANATIONS IN ISRAELI 7TH GRADE MATH TEXTBOOKS

This study analyzes six seventh grade Israeli mathematics textbooks, examining (1) the extent to which students are required to justify and explain their mathematical work, and (2) whether students are asked to justify a mathematical claim that is stated by the textbook or a mathematical claim that they themselves generated when solving a problem. Two different units of analysis were used to analyze two central topics in the seventh grade curriculum as follows: (1) equation solving in algebra and (2) triangle properties in geometry. The findings indicate that all six textbooks had considerably larger percentages of geometric tasks than algebraic tasks, which required students to justify or explain their mathematical work. Moreover, considerable differences were found among the six textbooks regarding the percentages of tasks that required students to justify and explain in both topics, but more so with the algebraic topic. Analysis of whether the textbook tasks required students to justify a mathematical claim that is stated by the textbook or a mathematical claim that the students themselves generated also revealed substantial differences among the textbooks. These findings are discussed, as well as the research methods used, in light of relevant literature.

     

Curriculum reform in Irish secondary schools – a focus on algebra

Algebra has long been identified as an area of difficulty in the teaching and learning of mathematics. Evidence of this difficulty can be found in Irish secondary-level classrooms. Chief Examiner Reports have consistently identified algebra as an area of student weakness in State examinations. In light of poor student performance, and as part of a nationwide reform of secondary mathematics curricula, a functions-based approach to teaching algebra has been adopted in Irish schools. It was introduced in September 2011 in place of the transformational (rule and procedure)-based approach which was previously used. Through comparing the diagnostic test scores of incoming students in an Irish university in the years before and after the reform, this study finds that the reformed approach has coincided with a decline in students’ technical algebraic skills. However, interviews with practising mathematics teachers reveal that this decline is not a direct result of the functions-based approach, but rather of a mixture of approaches being implemented in classrooms. Such divergence of approaches can be linked to the common mismatch between the intended curriculum prescribed by policy-makers and the implemented curriculum that is actually carried out by teachers in their classrooms.     

PROCESSES AND REASONING IN REPRESENTATIONS OF LINEAR FUNCTIONS

This study examined student actions, interpretations, and language in respect to questions raised regarding tabular, graphical, and algebraic representations in the context of functions. The purpose was to investigate students’ interpretations and specific ways of working within table, graph, and the algebraic on notions fundamental to a conceptualization of linear functions. Through a case study method which investigated individual representations and student articulations within them, the study revealed that students can make a transition from a given representation of linear function to another and yet demonstrate limited understanding of linear functions.     

从算术到代数

从算术向代数过渡,是学生数学学习过程中极为重要的转变阶段.符号是代数不同于算术的典型特征,学生从算术向代数的过渡,是从对数的思考向对符号的思考的转变,是从算术思维向代数思维的转变,是思维层次从个别到一般、具体到抽象的飞跃.