Algebraic Reasoning Research Articles

In Pursuit of a Connected Way of Knowing: The Case of One Mathematics Teacher

In this paper, we offer illustrations of a mathematics teacher’s difficulties with content knowledge when trying to find connections between school mathematics and science; we do so by describing the development of this teacher’s thinking and learning in her pursuit of connections between the concepts of slope of a line and density of matter. The paper is based on a sub-study that is part of a larger Colombian project, PROMESA (Creating Science and Mathematics Connected Learning Experiences that Open Opportunities for the Promotion of Algebraic Reasoning), which incorporated a Professional Learning Programme (PLP) seeking to integrate school science and mathematics teachers into working teams, in order to create science and mathematics connected learning experiences that considered the promotion of algebraic reasoning. The ‘challenging questions’ that emerged for this teacher, during the workshops of the induction stage of the PLP, became the driving force for her continued engagement in learning mathematics content in a connected way, as opposed to the compartmentalised content-item thinking that she had experienced as a school student. We provide illustrations of first steps in the development of a teacher’s mathematical understanding, which can support growth of mathematical knowledge for teaching.

Preservice Elementary Teacher's Thinking about Algebraic Reasoning

Preservice Elementary Teacher's Thinking about Algebraic Reasoning

Exploring the use of a dynamic online software programme in the teaching and learning of trigonometric graphs

Trigonometry is an important section in mathematics that links algebraic, geometric and graphical reasoning. Many secondary school learners are not familiar with these types of reasoning; thus, trigonometry presents a challenge for these learners. The traditional ‘chalk and talk’ method of teaching trigonometry is used by many teachers; however, this article is based on a case study that explores the innovative use of a dynamic online software program in the teaching and learning of Grade 10 trigonometric graphs. The participants in this study were 25 Grade 10 mathematics learners from one school in the south of Durban, KwaZulu-Natal, South Africa. Data were collected by using two trigonometry worksheets, semi-structured interviews with selected learners and lesson observations. The worksheets were constructed based on the following focus areas: behaviour of graphs, interpretation of graphs, modelling and overall performance of the learner. In all focus areas, the participants’ scores improved significantly when working within the dynamic online environment. Further analysis revealed the following five themes: easy access, learners in control, technology is the future, motivation and meaningful interaction. These findings would be of interest to mathematics teachers, learners, curriculum developers and mathematics teacher educators.

Reasoning on the complex plane via inscriptions and gesture

Using a diagrammatic reasoning framework about inscriptions, we explored undergraduates’ reasoning about complex-valued equations. Our findings suggest that reasoning geometrically requires first reasoning algebraically about algebraic inscriptions. We found students tended to create algebraic and geometric inscriptions when their verbiage could no longer support geometric reasoning. Furthermore, they incorporated similar iconic gestures for reasoning about their geometric inscriptions, which reduced to deictic gestures as they applied their previously developed reasoning to subsequent tasks. Contrary to other research, our participants’ gestures did not taper off with future tasks. Rather, their gestures transformed as concepts were automatized. Moreover, our research suggests that gestures serve as a link between verbiage and inscriptions rather than inscriptions serving as a link between verbiage and gesture as other researchers claim. In promoting synchronicity of algebraic and geometric reasoning, teachers may want to capitalize on the fact that students tend to implement similar gestures as they reason.

Studying technology-based strategies for enhancing motivation in mathematics

BackgroundDuring the middle school years, students frequently show significant declines in motivation toward school in general and mathematics in particular. One way in which researchers have sought to spark students’ interests and build their sense of competence in mathematics and in STEM more generally is through the use of technology. Yet evidence regarding the motivational effectiveness of this approach is mixed. Here we evaluate the impact of three brief technology-based activities on students’ short-term motivation in math. 16,789 5th to 8th grade students and their teachers in one large school district were randomly assigned to three different technology-based activities, each representing a different framework for motivation and engagement and all designed around an exemplary lesson related to algebraic reasoning. We investigated the relationship between specific technology-based activities that embody various motivational constructs and students’ engagement in mathematics and perceived competence in pursuing STEM careers.ResultsResults indicate that the effect of each technology activity on students’ motivation was quite modest. No gains were found in self-efficacy; for implicit theory of ability, a lower incremental view of ability was found; we found modest declines in value beliefs. With respect to math learning, students in all three inductions had modest improvements in their scores on the math learning measure. However, these effects were modified by students’ grade level and not by their demographic variables. In addition, teacher-level variables did not have an effect on student outcomes.ConclusionsThe present findings highlight the importance of tailoring motivational experiences to students’ developmental level. Our results are also encouraging about developers’ ability to create instructional interventions and professional development that can be effective when experienced by a wide range of students and teachers. Further research is needed to determine the degree, duration of, and type of instructional intervention necessary to substantially impact multi-dimensional, deep-rooted motivational constructs, such as self-efficacy.

Development of abstract mathematical reasoning: the case of algebra.

Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.

Does Early Algebraic Reasoning Differ as a Function of Students' Difficulty with Calculations versus Word Problems?

According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2nd- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.

Algebraic Thinking: Conceptions of Elementary School Teachers

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.

Middle school pre-service mathematics teachers' guidance for sixth grade students in problem solving process

The mathematics of the middle school has an important place in developing of algebraic thinking. Middle grades correspond to transition from arithmetic to algebra. According to Thompson and Smith (2007), quantitative reasoning provides a smooth transition from arithmetic reasoning to algebraic reasoning as a glue. Sixth grade and verbal problems are important in developing quantitative reasoning. On the other hand, As Harel (2008) emphasized, the important thing in problem solving is the ways of thinking rather than answering the problems correctly in school mathematics. Therefore, developing thinking ways of students is one of the responsibilities of mathematics teachers. This responsibility can get more difficult for pre-algebra students like sixth grade students. In this study, it was aimed to investigate pre-service middle school mathematics teachers’ guidance for sixth grade students in problem solving process. This is a qualitatively designed study in which the data was collected through 10 pre-services teachers’ clinical interviews with sixth grade middle school students. Data was analysed qualitatively by using content analysis technique. It was found that pre-service middle school mathematics teachers focused on students’ getting the result rather than guiding their thinking ways. The abstract should be typed here (maximum 250 words).

The development and nature of problem-solving among first-semester calculus students

This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students’ problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students’ ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students’ performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students’ problem-solving performance in view of both their preparation for university calculus and the courses’ assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students’ problem-solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction.

Developing learning environments which support early algebraic reasoning: a case from a New Zealand primary classroom

Current reforms in mathematics education advocate the development of mathematical learning communities in which students have opportunities to engage in mathematical discourse and classroom practices which underlie algebraic reasoning. This article specifically addresses the pedagogical actions teachers take which structure student engagement in dialogical discourse and activity which facilitates early algebraic reasoning. Using videotaped recordings of classroom observations, the teacher and researcher collaboratively examined the classroom practices and modified the participatory practices to develop a learning environment which supported early algebraic reasoning. Facilitating change in the classroom environment was a lengthy process which required consistent and ongoing attention initially to the social norms and then to the socio-mathematical norms. Specific pedagogical actions such as the use of specifically designed tasks, materials and representations and a constant press for justification and generalisation were required to support students to link their numerical understandings to algebraic reasoning.

Research of Mutual Relationships between Addition and Subtraction: From a Viewpoint of Algebraic Reasoning

Research of Mutual Relationships between Addition and Subtraction: From a Viewpoint of Algebraic Reasoning

Teaching Quantitative Reasoning: A Better Context for Algebra

This editorial questions the preeminence of algebra in our mathematics curriculum. The GATC (Geometry, Algebra, Trigonometry, Calculus) sequence abandons the fundamental middle school math topics necessary for quantitative literacy, while the standard super-abundance of algebra taught in the abstract fosters math phobia and supports a culturally acceptable stance that math is not relevant to everyday life. Although GATC is seen as a pipeline to STEM (Science, Technology, Engineering, Mathematics), it is a mistake to think that the objective of producing quantitatively literate citizens is at odds with creating more scientists and engineers. The goal must be to create a curriculum that addresses the quantitative reasoning needs of all students, providing meaningful engagement in mathematics that will simultaneously develop quantitative literacy and spark an interest in STEM fields. In my view, such a curriculum could be based on a foundation of proportional reasoning leading to higher-order quantitative reasoning via modeling (including algebraic reasoning and problem solving) and statistical literacy (through the exploration and study of data).

RELATIONSHIPS BETWEEN FRACTIONAL KNOWLEDGE AND ALGEBRAIC REASONING: THE CASE OF WILLA

To investigate relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. The students were interviewed twice, once to explore their quantitative reasoning with fractions and once to explore their solutions of problems that required explicit use of unknowns to write equations. As a part of the larger study, the first author conducted a case study of a seventh grade student, Willa. Willa’s fractional knowledge—specifically her reversible iterative fraction scheme and use of fractions as multipliers—influenced how she wrote equations to represent multiplicative relationships between two unknown quantities. The finding indicates that implicit use of powerful fractional knowledge can lead to more explicit use of structures and relationships in algebraic situations. Curricular and instructional implications are explored.

A Mathematics Intervention: The Case of 4 Year-old Rylan and Hilda!

This paper provides an overview of a three-year study with 60 early childhood educators and approximately 240 children aged 4 to 5 years currently being conducted with 15 Australian Indigenous communities. International research provides strong evidence for investing early for improved health and wellbeing of young children (Barnett, 2011). The numeracy Intervention, Patterns and Early Algebra Preschool (PEAP) Professional Development (PD) Program, reported in this paper focuses on developing children's early algebraic and mathematical reasoning skills. A case study of two children in the study exemplifies how PEAP has the potential to advance young children's mathematical problem solving and thinking skills. The broader aim of the project is to work towards closing the gap in numeracy achievement for children from low socio-economic and disadvantaged backgrounds.

Elementary and middle school students’ analyses of pictorial growth patterns

Research has suggested the importance of incorporating algebraic thinking early and throughout the K-12 mathematics curriculum. One approach to help children develop algebraic reasoning is through the examination of pictorial growth patterns, which serve as a context for exploring generalization. The purpose of this study was to compare how elementary and middle school students analyze pictorial patterns, with a focus on whether students used figural or numerical reasoning. Task-based interviews were conducted with a second grader, fifth grader, and eighth grader in which they were asked to describe, extend, and generalize two pictorial growth patterns. Using a phenomenographic approach, analyses showed younger students used figural reasoning more than older students, but all students did not exclusively use figural or numerical reasoning. The students’ generalizations included informal notation, descriptive words, and formal notation. The findings suggest that pictorial growth patterns are a promising tool for young students’ development of algebraic thinking.

An Analysis of Algebraic Reasoning Test of the Life Situation of Multiple-Representation by Using Grey Relational to Assess

This study uses Grey relational analysis (GRA) to evaluate a portion of the algebraic reasoning that uses multiple-representation of the life situation of multiple-representations. Current results of this study not only identify the easiest and hardest questions for students, but also show how proposed GRA provides a novel approach to educational testing.

A Bargain Price for Teaching about Percentage

Develop mathematical knowledge by analyzing common student errors and discussing arithmetic, algebraic, and pictorial reasoning.

Deriving real-time action systems with multiple time bands using algebraic reasoning

The verify-while-develop paradigm allows one to incrementally develop programs from their specifications using a series of calculations against the remaining proof obligations. This paper presents a derivation method for real-time systems with realistic constraints on their behaviour. We develop a high-level interval-based logic that provides flexibility in an implementation, yet allows algebraic reasoning over multiple granularities and sampling multiple sensors with delay. The semantics of an action system is given in terms of interval predicates and algebraic operators to unify the logics for an action system and its properties, which in turn simplifies the calculations and derivations.

How Does Physical Space Influence the Novices' and Experts' Algebraic Reasoning?

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