Algebraic Reasoning Research Articles

Grey Relational Analysis to Assess Algebraic Reasoning Test of the Pictorial Pattern

This study uses Grey relational analysis (GRA) to evaluate a portion of the algebraic reasoning that uses multiple-representation of pictorial patterns. 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.

The fractional knowledge and algebraic reasoning of students with the first multiplicative concept

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of three different multiplicative concepts participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed.

From Parental Involvement to Children's Mathematical Performance: The Role of Mathematics Anxiety

This study examined whether children's mathematics anxiety serves as an underlying pathway between parental involvement and children's mathematics achievement. Participants included 78 low-income, ethnic minority parents and their children residing in a large urban center in the northeastern United States. Parents completed a short survey tapping several domains of parental involvement, and children were assessed on mathematics anxiety, whole number arithmetic, word problems, and algebraic reasoning. Research Findings: The results indicated that parents influence children's mathematics achievement by reducing mathematics anxiety, particularly for more difficult kinds of mathematics. Specifically, the mediation analyses demonstrated that parental home support and expectations influenced children's performance on word problems and algebraic reasoning by reducing children's mathematics anxiety. Mathematics anxiety did not mediate the relationship between home support and expectations and whole number arithmetic. Practice or Policy: Policies and programs targeting parental involvement in mathematics should focus on home-based practices that do not require technical mathematical skills. Parents should receive training, resources, and support on culturally appropriate ways to create home learning environments that foster high expectations for children's success in mathematics.

The language of mathematics: Investigating the ways language counts for children’s mathematical development

This longitudinal study examined how language ability relates to mathematical development in a linguistically and ethnically diverse sample of children from 6 to 9years of age. Study participants were 75 native English speakers and 92 language minority learners followed from first to fourth grades. Autoregression in a structural equation modeling (SEM) framework was used to evaluate the relation between children’s language ability and gains in different domains of mathematical cognition (i.e., arithmetic, data analysis/probability, algebra, and geometry). The results showed that language ability predicts gains in data analysis/probability and geometry, but not in arithmetic or algebra, after controlling for visual–spatial working memory, reading ability, and sex. The effect of language on gains in mathematical cognition did not differ between language minority learners and native English speakers. These findings suggest that language influences how children make meaning of mathematics but is not involved in complex arithmetical procedures whether presented with Arabic symbols as in arithmetic or with abstract symbols as in algebraic reasoning. The findings further indicate that early language experiences are important for later mathematical development regardless of language background, denoting the need for intensive and targeted language opportunities for language minority and native English learners to develop mathematical concepts and representations.

Supporting Algebraic Reasoning through Personalized Story Scenarios: How Situational Understanding Mediates Performance

Context personalization refers to matching instruction to students' out-of-school interests and experiences. Belief in the benefits of matching instruction to interests is widely held in the culture of schooling; however, little research has empirically examined how interest impacts performance and learning in secondary mathematics. Here we investigate these issues with a series of problem-solving sessions where 24 Algebra I students were presented with story problems on linear functions, some of which were personalized to their interests. Our analyses focus on performance, strategy use, and mistake patterns. Results suggest that personalization supported situational reasoning (Nathan, Kintsch, & Young, 1992) about the actions and relationships in the scenario, improving performance for weaker students and on harder problems. However, personalized scenarios seemed to act as a distraction when stronger students in the sample worked on easier problems. Thus context personalization may have the potential to provide assistance and support performance as students learn new concepts.

Supporting Students’ Constructions of the Splitting Operation

Previous research has demonstrated the effectiveness of particular instructional practices that support students’ constructions of the partitive unit fraction scheme and measurement concepts for fractions. Another body of research has demonstrated the power of a particular mental operation—the splitting operation—in supporting students’ development of advanced fractional knowledge and algebraic reasoning. Steffe (2010) has hypothesized that students construct splitting through the unification of partitioning and iterating operations contained within the partitive unit fraction scheme. We used written assessments of 49 students, across sixth and seventh grades, to test this hypothesis. Our results show that students who have constructed a partitive unit fraction scheme go on to construct splitting within a relatively short period of time. Conversely, students who have not constructed a partitive unit fraction scheme generally do not construct splitting. We discuss these results and their implications for designing instruction and curricula that support students’ development of algebraic reasoning.

Examining number sense and algebraic reasoning through cognitive styles

Recently, a new cognitive style approach was introduced, which refers to two types of visualizers. This approach is based on neuropsychological evidence and neuroimaging results, which suggest the existence of two distinct imagery subsystems, the object and the spatial imagery subsystems. The goal of the study was twofold: first to examine a possible relationship between this new cognitive style approach and achievement in number sense and algebraic reasoning tasks, and second to explore a possible relationship between the strategies used in solving the aforementioned tasks and cognitive styles. A mathematical test on number sense and algebraic reasoning and the self-report Object–Spatial Imagery and Verbal cognitive style questionnaire were administrated to 83 prospective school teachers (PSTs). The results indicated that spatial imagery, in contrast to the object imagery and verbal cognitive styles, is related to achievement in number sense and algebraic reasoning. In addition to this, the results revealed that the higher the PSTs’ tendency towards spatial imagery cognitive style, the more conceptual and flexible strategies they employ in algebraic reasoning and number sense tasks.

Doceamus: Does Algebraic Reasoning Enhance Reasoning in General? A Response to Dudley

Doceamus: Does Algebraic Reasoning Enhance Reasoning in General? A Response to Dudley

Signs and Tools of Algebraic Reasoning: A Study of Models among Fifth Grade Students

This study focuses on the types of models created by students during algebraic pattern finding tasks. Attention is also given to the change in models over time. This is an important area of study because a closer look is needed to better understand the models created during mathematical activity, especially in the elementary classroom. It is reported here how fifth grade students used given concrete models and created new representations of models to reason algebraically about pattern finding tasks. Twenty-five fifth grade students participated in the three-day teaching experiment. Results indicate that students’ recursive models were abandoned and then transformed to explicit models, and finally adopted from others during whole class discussions. These adopted models in most cases were enduring over a six-week period.

Response to Tzourio-Mazoyer and Zago: yes, there is a neural dissociation between language and reasoning

In their commentary, Tzourio-Mazoyer and Zago caution against the conclusion, presented in Monti <i>et al.</i> (2012), that there exists a dissociation between algebraic and linguistic reasoning. In what follows we briefly summarize our experimental design and then show that Tzourio-Mazoyer and Zago's concerns spring from an inaccurate reading of our methods.

Towards an Algebraic Semantics of Software Evolution Process Models

随着大量的软件演化过程模型被软件演化过程元模型建模产生,如何验证过程模型的正确性,是摆在人们面前的一个重要任务.针对软件演化过程元模型,引入进程代数ACP(algebra of communicating processes)对其扩展,提出软件演化过程元模型代数,使用进程项指定软件演化过程模型的代数语义,在进程代数的统一框架下,基于等式推理验证软件演化过程模型的行为,使行为验证方式从模型推导变为代数推导.这种方法充分结合了Petri 网和ACP 的长处,可以有效地支持软件演化过程的形式验证.;As a number of software evolution process models increased, as modeled by EPMM (software evolution process meta model), verifying the correctness of these models becomes the important. This paper extends EPMM with ACP (algebra of communicating processes) and proposes EPMM-A (software evolution meta modelalgebra). In order to discuss behavior verification in the unified framework of EPMM-A, a process term is used to define an algebraic semantics of a software evolution process model. Based on equational reasoning of EPMM-A, behavior verification of a software evolution process model emphasizes algebraic reasoning as opposed to modelbased reasoning. This method combines the advantages of both Petri nets and ACP, which can effectively support software evolution process formal verification.

Hungry for Early Spatial and Algebraic Reasoning

Kindergartners and first-grade students listen excitedly to a modified storybook to guide their geometry activities.

Prospective middle grade mathematics teachers’ knowledge of algebra for teaching

This study examined prospective middle grade mathematics teachers’ knowledge of algebra for teaching with a focus on knowledge for teaching the concept of function. 115 prospective teachers from an interdisciplinary program for mathematics and science middle teacher preparation at a large public university in the USA participated in a survey. It was found that the participants had relatively limited knowledge of algebra for teaching. They also revealed weakness in selecting appropriate perspectives of the concept of function and flexibly using representations of quadratic functions. They made numerous mistakes in solving quadratic or irrational equations and in algebraic manipulation and reasoning. The participants’ weakness in connecting algebraic and graphic representations resulted in their failure to solve quadratic inequalities and to judge the number of roots of quadratic functions. Follow-up interview further revealed the participants’ lack of knowledge in solving problems by integrating algebraic and graphic representations. The implications of these findings for mathematics teacher preparation are discussed.

GEOMETRICAL CORRELATION AND MATCHING OF 2D IMAGE SHAPES

Abstract. The problem of image correspondence measure selection for image comparison and matching is addressed. Many practical applications require image matching "just by shape" with no dependence on the concrete intensity or color values. Most popular technique for image shape comparison utilizes the mutual information measure based on probabilistic reasoning and information theory background. Another approach was proposed by Pytiev (so called "Pytiev morphology") based on geometrical and algebraic reasoning. In this framework images are considered as piecewise-constant 2D functions, tessellation of image frame by the set of non-intersected connected regions determines the "shape" of image and the projection of image onto the shape of other image is determined. Morphological image comparison is performed using the normalized morphological correlation coefficients. These coefficients estimate the closeness of one image to the shape of other image. Such image analysis technique can be characterized as an “"ntensity-to-geometry" matching. This paper generalizes the Pytiev morphological approach for obtaining the pure "geometry-to-geometry" matching techniques. The generalized intensity-geometrical correlation coefficient is proposed including the linear correlation coefficient and the square of Pytiev correlation coefficient as its partial cases. The morphological shape correlation coefficient is proposed based on the statistical averaging of images with the same shape. Centered morphological correlation coefficient is obtained under the condition of intensity centering of averaged images. Two types of symmetric geometrical normalized correlation coefficients are proposed for comparison of shape-tessellations. The technique for correlation and matching of shapes with ordered intensities is proposed with correlation measures invariant to monotonous intensity transformations. The quality of proposed geometrical correlation measures is experimentally estimated in the task of visual (TV) and infrared (IR) image matching. First experimental results demonstrate competitive quality and better computational performance relative to state-of-art mutual information measure.

A STUDY OF THE INFLUENCE OF THE GAUGE GROUP ON THE DYSON–SCHWINGER EQUATIONS FOR SCALAR-YANG–MILLS SYSTEMS

The particular choice of the gauge group for Yang–Mills theory plays an important role when it comes to the influence of matter fields. In particular, both the chosen gauge group and the representation of the matter fields yield structural differences in the quenched case. Especially, the qualitative behavior of the Wilson potential is strongly dependent on this selection. Though the algebraic reasons for this observation is clear, it is far from obvious how this behavior can be described besides using numerical simulations. Herein, it is investigated how the group structure appears in the Dyson–Schwinger equations, which as a hierarchy of equations for the correlation functions have to be satisfied. It is found that there are differences depending on both the gauge group and the representation of the matter fields. This provides insight into possible truncation schemes for practical calculations using these equations.

The mediating role of a teacher’s use of semiotic resources in pupils’ early algebraic reasoning

This paper focuses on the semiotic resources used by an experienced sixth-grade teacher when her pupils are working on a mathematical task involving written text and the two inscriptions of figure and diagram. Socio-cultural analytical constructs such as semiotic bundle, space of joint action and togethering are applied in order to enable and frame the collective activity of the teacher and pupils. Four extracts from different situations in the classroom illustrate the important role of both teacher gestures and pupil gestures, interacting with other modalities such as speech and inscription, in the process of making sense of pupils’ appropriation of coordinating two dimensions in a diagram. It is argued that the nature of the mathematical task is an important entry point into early algebraic reasoning. The study emphasises the mediating role of the dynamics of semiotic bundles produced in teacher–pupil dialogues as a promising way to address the fundamental relationships between mathematics, pupil and teacher in a classroom context in order to provoke pupil involvement and engagement when experiencing mathematics.

A second grader's algebraic reasoning

A talkative second grader helps her teacher describe, extend, and generalize about growing patterns.

Naturaleza del razonamiento algebraico elemental

La introducción del razonamiento algebraico en educación primaria es un tema de interés para la investigación e innovación curricular en didáctica de las matemáticas, y presupone una visión ampliada de la naturaleza del álgebra escolar. En este trabajo proponemos una manera de concebir el razonamiento algebraico, basada en los tipos de objetos y procesos matemáticos introducidos en el enfoque ontosemiótico del conocimiento matemático. En síntesis, la consideración de una práctica matemática como algebraica se basará en la intervención de procesos de generalización y simbolización, junto con otros objetos usualmente considerados como algebraicos, tales como relaciones binarias, operaciones, funciones y estructuras. Esta forma de concebir el álgebra elemental es contrastada con las caracterizaciones dadas por otros autores. Asimismo, proponemos una tipología de configuraciones algebraicas que permite definir grados de algebrización de la actividad matemática.

Deciding Kleene Algebras in Coq

We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations instantaneously and properly scales to larger expressions. The decision procedure is proved correct and complete: correctness is established w.r.t. any model by formalising Kozen's initiality theorem; a counter-example is returned when the given equation does not hold. The correctness proof is challenging: it involves both a precise analysis of the underlying automata algorithms and a lot of algebraic reasoning. In particular, we have to formalise the theory of matrices over a Kleene algebra. We build on the recent addition of firstorder typeclasses in Coq in order to work efficiently with the involved algebraic structures.

A Second Grader's Algebraic Reasoning

A Second Grader's Algebraic Reasoning