Search

In this note we describe the mathematics that emerges from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of algebra, geometry and calculus. [ABSTRACT FROM AUTHOR]

This study reports the academic benefits of peer tutoring in algebra for middle school students. A total of 380 students enrolled in grades 7th and 8th participated in the study. Two peer tutoring sessions took place during each week (10 weeks). Interactions between peers lasted 20 to 25 minutes for each session. The typology of tutoring was fixed and same-age. A pretest posttest with control group design was used. Statistical significant improvements were reported in the academic achievement variable after the implementation of the peer tutoring program for 7th and 8th grade courses separately and altogether. Over 87% of the students in the experimental group improved their marks. The overall effect size for the experience was reported to be medium (Hedge's g = 0.48). The main conclusion of this study is that fixed and same-age peer tutoring in algebra may be very beneficial for middle school students. [ABSTRACT FROM AUTHOR]

There is a compelling need to develop an algebra teacher self‐efficacy instrument (ATSEI) as algebra continues to be considered a gatekeeper course for postsecondary educational and career opportunities, which is seen as a crucial piece in closing the achievement gap. This paper reports on the development and validation of the ATSEI, an instrument that measures two domains, Efficacy To Do School Algebra (Knowledge Efficacy, KE‐A) and Efficacy to Teach Algebra (Personal Teaching Efficacy, PTE‐A) along six categories. Four of the categories represented content standards (variables, functions, patterns, and modeling) and two of the categories represented process standards (technology and concrete models, and multiple representations). Through conducting an exploratory factor analysis across two phases, the instrument was reduced and refined from an initial 118 items developed from a curriculum analysis to 36 items that reflected two significant categories, Functions and Technology. The ATSEI measure is validated for in‐service mathematics teachers and thus provides an instrument to examine need and impact in professional development venues. The specificity of the ATSEI allows those working with teachers to be better able to support them in the field and in return positively influence the learning outcomes of the students they teach. [ABSTRACT FROM AUTHOR]

We present two empirical studies with 241 and 211 pre-service teachers that evaluate the explanatory power of word order matching and static comparison as models for the reversal error. We used tasks consisting of generating an algebraic equation representing a comparison given in a verbal statement. We introduce the types of magnitude involved in the statement as variables of analysis, something that was not previously tackled in previous works. Our results show that there are no statistical differences in the production of reversal errors depending on the information included in the name used to designate the variable, and that there are statistical differences depending on the syntactic configuration as well as the type of magnitude involved in the statement. The interpretation of these results indicates that both word order matching and static comparison have some potential as explanatory models for the reversal error, and that neither one of them, alone, is enough to completely explain the phenomenon. [ABSTRACT FROM AUTHOR]

Abstract: Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras. [ABSTRACT FROM AUTHOR]

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change leaves some relationships or properties invariant, these properties prove to be inherently interesting to teachers and students. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM PUBLISHER]

Rational number understanding is a critical building block for success in more advanced mathematics; however, how rational number magnitudes are conceptualized is not fully understood. In the current study, we used a dual-task working memory (WM) interference paradigm to investigate the dominant type of strategy (i.e., requiring verbal WM resources vs. requiring primarily visuospatial WM resources) used by adults when processing rational number magnitudes presented in both decimal and fraction notation. Analyses revealed no significant differences in involvement of verbal and visuospatial WM, regardless of notation (fractions vs. decimals), indicating that adults rely upon a mix of strategies and WM resources when processing rational number magnitudes. However, this pattern interacted with algebra ability such that those performing better on the algebra assessment relied upon both verbal and visuospatial WM when engaging in rational number comparisons, whereas rational number performance by adults with low algebra fluency was affected only by a simultaneous verbal WM task. Together, results support previous work implicating the involvement of WM resources in rational number processing and is the first study to indicate that the involvement of both verbal and visuospatial WM, as opposed to relying primarily on verbal WM, when processing rational number magnitudes may be indicative of higher mathematical proficiency in the domain of algebra. [ABSTRACT FROM AUTHOR]

Over one million K-12 students pursue virtual education every year, but researchers know very little about the effectiveness of such programs. This paper exploits a district policy change that suddenly shifted advanced eighth graders into a virtual classroom for Algebra I. After the policy, higher-ability eighth graders in the treatment district began taking Algebra I in the virtual classroom at rates similar to the statewide average of their peers in traditional classrooms. The change in course delivery provides a unique opportunity to study effects of a virtual course on academic outcomes. The analysis uses variation in program uptake across performance quintile, district, and year in a difference-in-difference-in-difference approach to estimate the causal effect of the virtual course, finding that eighth grade virtual students tend to underperform relative to eighth graders who took Algebra I in a traditional classroom and relative to pre-policy, same-district students who had to take the course in ninth grade. [ABSTRACT FROM AUTHOR]

This study explores how students learn to create, discuss, and reason with representations to solve problems. A summer school algebra class for seventh and eighth graders provided opportunities for students to create and use representations as problem-solving tools. This case study follows the learning trajectories of three boys. Two of the three boys had been low-achievers in their previous math classes, and one was a high achiever. Analysis of all three boys’ written work reveals how their representations became more sophisticated over time. Their small group interactions while problem-solving also show changes in how they communicated and reasoned with representations. For these boys, representation functioned as a learning practice. Through constructing and reasoning with representations, the boys were able to engage in generalizing and justifying claims, discuss quadratic growth, and collaborate and persist in problem-solving. Negotiating different student-constructed representations of a problem also gave them opportunities to act with agency, as they made choices and judgments about the validity of the different perspectives. These findings have implications for the importance of giving all students access to mathematics through representations, with representational thinking serving as a central disciplinary practice and as a learning practice that supports further mathematics learning. [ABSTRACT FROM AUTHOR]

The number of students who leave U.S. schools mathematically underprepared has prompted widespread concern. Low achieving students, many of whom have been turned off mathematics, are often placed in low tracks and given remedial, skills-oriented work. This study examines a different approach wherein heterogeneous groups of students were given responsibility and agency and asked to engage in a range of mathematical practices collaboratively. The teaching intervention, which was introduced in the first paper, took place as part of a summer class on algebra, and it gave students the opportunity to participate with mathematics in changed ways. This paper will report evidence that the vast majority responded with increased engagement, achievement, and enjoyment. The students chose collaboration and agency as critical to their improved relationships with mathematics. [ABSTRACT FROM AUTHOR]

Recent reform movements have emphasized students making meaning of algebraic relationships; however, research on student thinking and learning often remains disconnected from the design of widely used curricular materials. Although a previous examination of algebra textbooks (Nathan, Long, & Alibali, 2002) demonstrated a preference for a symbols-first approach, research has demonstrated that Algebra I students' performance on verbally presented problems is better than on symbolic equations, consistent with cognitive theories suggesting the value of concreteness fading. The present study investigates whether current textbooks used in Algebra I courses demonstrate a formalisms-first approach using five different analyses. Results show that despite nearly 2 decades of research on student learning, the conventional textbooks used in most classrooms have been resistant to change and emphasize manipulation with symbols prior to making sense of verbal scenarios. [ABSTRACT FROM AUTHOR]

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x , y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial–numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. [ABSTRACT FROM AUTHOR]

Two different interpretations of algebra that differ in the ontological status assigned to variables are distinguished. Variables may either be viewed as meta-mathematical tools to express generality or as objects similar to numbers and other members of the mathematical ontology. Both interpretations are detailed and linked with the literature and the use of variables in computer programming. Furthermore, it is analyzed how these two conceptualizations lead to two different understandings of the process of change of values. Some evidence from algebra assessment on the understanding of change by students is given that that illustrate that the theory is useful in analyzing students work. [ABSTRACT FROM AUTHOR]

An empirical study has been carried out to evaluate the potential of word order matching and static comparison as explanatory models of reversal error. Data was collected from 214 undergraduate students who translated a set of additive and multiplicative comparisons expressed in Spanish into algebraic language. In these multiplicative comparisons we used a format that can be translated from Spanish word-for-word as ' n times more than' (increasing comparison) and ' n times less than' (decreasing comparison) instead of ' n times as many', which is usual in other studies. Data analysis shows a significantly lower incidence of reversal error in the decreasing comparisons compared to the increasing ones. Additionally, no significant differences were found between additive and multiplicative comparisons. These results cannot be explained by the static comparison model. [ABSTRACT FROM AUTHOR]

Flexibility in problem solving has been widely recognized as an important skill for students' mastery of mathematics. Here we utilize the Opportunity-Propensity framework to investigate student characteristics, teacher characteristics, and teacher instructional practices that may be associated with students' gains in flexibility in algebra. Teacher and student data were collected from 8th and 9th grade Algebra I teachers in Massachusetts as part of a larger study on the impact of a researcher-developed year-long supplementary curriculum that focused on improving students' flexibility. We explore student demographics, teacher background characteristics and teacher instructional practices as predictors of student gains in flexibility. We further investigate instructional practices associated with flexibility gains through an analysis of teacher questioning in the classroom for teachers whose students achieved the greatest gains in flexibility and those whose students achieved the least gains. Our results indicate that prior knowledge is a reliable predictor of flexibility gains and that gender is an important student background characteristic associated with the development of flexibility. In addition, although high and low gain teachers did not differ in their implementation fidelity, high flexibility gain teachers asked more open-ended questions that prompted students to verbalize the main ideas of the lesson. [ABSTRACT FROM AUTHOR]

There exists an increased focus on school mathematics, especially first-year algebra, due to recent efforts for all students to be college and career ready. In addition, there are calls, policies, and legislation advocating for all students to study algebra epitomized by four rationales of the Algebra for All movement. In light of this movement, there must be a clear consensus about what is taught in the name of algebra. Yet, researchers documented this is not the case. The present research proposes to unify the leading algebra standards and assessment framework documents to identify the key ideas of algebra. The analysis resulted in six key ideas: (a) Variables, (b) Functions, (c) Patterns, (d) Modeling, (e) Technology, and (f) Multiple Representations. Outlined is the research process and resulting unification of existing algebra framework documents, and consideration is given for its uses in educational policy regarding algebra and potential directions for future research. [ABSTRACT FROM PUBLISHER]

This paper uses a multilevel analysis of mathematical reasoning rooted in Cultural Historical Activity Theory to examine how mathematical discourse and student reasoning about linear functions developed across 3 weeks in a ninth grade bilingual algebra class. Despite the teacher's expertise teaching with a conceptual focus, and her stated intention to focus on slope as a rate of change, the case study students in her class appeared to appropriate a procedural understanding of slope. By examining the nested activity systems of the students' group discussions, the classroom, school, and district, this analysis shows how external assessment pressures shaped the teacher's selection of tasks, her and her students' use of mathematical discourse, and ultimately, her students' opportunity to learn a critical algebraic concept. [ABSTRACT FROM AUTHOR]

Students hold many misconceptions as they transition from arithmetic to algebraic thinking, and these misconceptions can hinder their performance and learning in the subject. To identify the errors in Algebra I which are most persistent and pernicious in terms of predicting student difficulty on standardized test items, the present study assessed algebraic misconceptions using an in-depth error analysis on algebra students' problem solving efforts at different points in the school year. Results indicate that different types of errors become more prominent with different content at different points in the year, and that there are certain types of errors that, when made during different levels of content, are indicative of math achievement difficulties. Recommendations for the necessity and timing of intervention on particular errors are discussed. [ABSTRACT FROM AUTHOR]

This paper contains a discussion of how the concept of critical aspects and the principle of compositionality can provide a powerful tool to analyse and understand the communications that occur in the classroom. It is grounded in data collected in a longitudinal study. The content chosen is algebra. It is argued that the critical aspects and the principle of compositionality should be considered as a methodological principle that describes how communication in the classroom should be designed. Here, I present the power of using variation theory whose main purpose is to generate an understanding of critical aspects and compositionality in practice. [ABSTRACT FROM AUTHOR]

In this paper, we present a cognitive analysis of the relationship between the argumentation process leading to the construction of a conjecture and its algebraic proof in solving Calendar Algebra problems. To solve this kind of problem, students encounter two sources of potential difficulties: the shift from using arithmetic in the argumentation to using algebra in the proof and the shift from an inductive argument towards a deductive proof. Thus, the aims of this article are to describe these cognitive difficulties and to show how students overcome them. Methodologically, we compare students' problem solving process corresponding to three problems presented in the first four lessons of a teaching experiment. The analysis and comparison between these three resolution processes is performed using Toulmin's model. [ABSTRACT FROM AUTHOR]

This study is a longitudinal look at the different mathematics placement profiles of Black students and White students from late elementary school through 8th grade. In particular, this study utilizes the Early Childhood Longitudinal Study--Kindergarten Class of 1998-1999 (ECLS-K) data set to analyze the impact of teacher evaluation of student performance versus student demonstrated performance on the odds of being placed into algebra in the 8th grade. Results revealed that Black students had reduced odds of being placed in algebra by the time they entered 8th grade even after controlling for performance in mathematics. In addition, teacher evaluations of student performance were shown to play a greater role, albeit adversely, for Black students than for their peers. These results are discussed in terms of both implicit theory research and critical race theory. An important implication of this study is that placement recommendations must be monitored to ensure high-achieving students are placed appropriately, regardless of racial background. [ABSTRACT FROM AUTHOR]

Highlights: [•] Reasoning-and-proving in exposition and tasks in two reform algebra texts examined. [•] Statistically significant differences between texts in tasks and exposition sentences. [•] Statistically different treatment across algebra subtopics in each text. [•] Few patterns or conjectures were tied to the development of arguments. [•] Technology not well represented across categories in either text. [Copyright &y& Elsevier]

This paper presents the partial results of a research on the professional knowledge of mathematics olympiad evaluators teachers put into action when analyzing errors relating to algebra. By means of two tests on errors interpretation and by using analysis inspired by the grounded theory, we obtained dimensions for the characterization of knowledge in understanding about errors. In the results section, we describe a teacher's knowledge related to practice and knowledge about students' learning. [ABSTRACT FROM AUTHOR]

The article shows the theoretical elements related with the conception of the curricular strategies and its instrumentation in the process of the student's of the Mathematical career formation - Physics. Examples are presented that demonstrate how to deal with the computer science's curricular strategy from the teaching process - learning of the subject Algebra I in the third year of the career. They give the possibility that the formation process be more effective, they facilitate the systematizing knowledge and abilities as well as the development of the integral general culture in the future professors of Mathematics - Physics. [ABSTRACT FROM AUTHOR]

Adaptive learning technologies are emerging in educational settings as a means to customize instruction to learners' background, experiences, and prior knowledge. Here, a technology-based personalization intervention within an intelligent tutoring system (ITS) for secondary mathematics was used to adapt instruction to students' personal interests. We conducted a learning experiment where 145 ninth-grade Algebra I students were randomly assigned to 2 conditions in the Cognitive Tutor Algebra ITS. For 1 instructional unit, half of the students received normal algebra story problems, and half received matched problems personalized to their out-of-school interests in areas such as sports, music, and movies. Results showed that students in the personalization condition solved problems faster and more accurately within the modified unit. The impact of personalization was most pronounced for 1 skill in particular--writing symbolic equations from story scenarios--and for 1 group of students in particular--students who were struggling to learn within the tutoring environment. Once the treatment had been removed, students who had received personalization continued to write symbolic equations for normal story problems with increasingly complex structures more accurately and with greater efficiency. Thus, we provide evidence that interest-based interventions can promote robust learning outcomes--such as transfer and accelerated future learning--in secondary mathematics. These interest-based connections may allow for abstract ideas to become perceptually grounded in students' experiences such that they become easier to grasp. Adaptive learning technologies that utilize interest may be a powerful way to support learners in gaining fluency with abstract representational systems. [ABSTRACT FROM AUTHOR]

Highlights: [•] We present a review of literature on prerequisite algebra skills and associated misconceptions of middle grade students. [•] We use the Common Core State Standards for Mathematics content domains and standards for mathematical practice as an organizing framework. [•] This review sheds light on the need for pre-service and in-service teachers of mathematics at the middle grades to be fully aware of student misunderstandings. [•] Implications for practice and future research are discussed. [Copyright &y& Elsevier]

Learning communities can strengthen early undergraduates' learning, but planning them can be daunting for instructors. Learning communities usually rely on integrative assignments that encourage interdisciplinary analysis. This article reports on our experiences using microloans as an interdisciplinary assignment in a learning community that united algebra with world literature. Students used the microfinance website to make small loans of real money to entrepreneurs in low-income countries. Four themes emerged from student reaction to the assignment: awareness of Africa, community impact, mathematical consciousness, and productive collaboration among students. Our reflections suggest that microfinance assignments may support integrative learning across many other disciplines. [ABSTRACT FROM AUTHOR]

We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment. [ABSTRACT FROM AUTHOR]

This note presents demonstrations of quadratic expressions that come about when particular problems are posed with respect to matchsticks that form regular triangles, squares, pentagons and so on. Usually when such ‘matchstick’ problems are used as ways to foster algebraic thinking, the expressions for the number of matchstick quantities are linear and not quadratic. It will be shown that a pedagogy of ‘summing all the matchsticks’ is central to the emergence of quadratic expressions. This pedagogy involves generational and transformational activities which are considered as some of the main activities of algebra. Key elements to these activities are processes such as recognizing and extending patterns, and specializing and generalizing particular functional relationships. Implications of these processes in terms of algebraic thinking are considered. [ABSTRACT FROM PUBLISHER]

The article reports aspects of the evolution of a teaching approach over repeated trials for beginning symbolic algebra. The teaching approach emphasized the structural similarity between arithmetic and algebraic expressions and aimed at supporting students in making a transition from arithmetic to beginning algebra. The study was conducted with grade 6 students over 2 years. Thirty-one students were followed for a year, and data were analysed as they participated in the three trials conducted that year. Analysis of students' written and interview responses as the approach evolved revealed the potential of the approach in creating meaning for symbolic transformations in the context of both arithmetic and algebra as well as making connections between arithmetic and symbolic algebra. Students by the end of the trials learnt to use their understanding of both procedures and a sense of structure of expressions to evaluate/simplify expressions and reason about equality/equivalence of expressions both in the arithmetic and the algebraic contexts. [ABSTRACT FROM AUTHOR]

In this article we report on the effects of a digital intervention on the development of algebraic expertise of 17–18 year old students in the Netherlands. The question to be answered was whether the intervention would be effective and what factors influenced the outcome. With notions of formative assessment and symbol sense as guiding theoretical concepts, the intervention’s design principles included the concepts of crises, formative scenarios and feedback. The intervention aimed to improve algebraic expertise and was deployed in fifteen grade 12 mathematics classes in nine schools. Data included results from pre- and posttests, scores, questionnaires and log files of the students’ digital work, and responses to a student survey. Results from the effect study, analyzed with multilevel models, showed that the intervention was effective in improving algebraic expertise. Factors that significantly contributed to the post-test score were pre-test results, the amount of time invested in digital self-tests and attitude towards mathematics. The intervention’s success was not significantly influenced by other variables. We conclude that these types of intervention have a potential for the acquisition of versatile algebraic expertise. [Copyright &y& Elsevier]

Although research has shown that self-efficacy beliefs predict academic achievement across all academic subjects and levels, little is known about the effect of using spreadsheets on self-efficacy beliefs in mathematics. This study is an investigation of the effect of instruction that includes spreadsheet-based purposeful activities on secondary school students' self-efficacy beliefs for algebra. The context of the study is a 10th grade mathematics course in a public secondary school. Analysis of the data indicated that students who received spread-sheet-based instruction had significantly higher self-efficacy for algebra than those who received conventional instruction. [ABSTRACT FROM AUTHOR]

Algebra progress monitoring is a research-based practice that extends a long history of research in curriculum-based measurement (CBM). This article describes the theoretical foundations and research evidence for algebra progress monitoring, along with critical features of the practice. A detailed description of one practitioner’s implementation of algebra progress monitoring in her classes and her school district offers many insights regarding how this assessment strategy is useful and was translated into typical school practice. [ABSTRACT FROM PUBLISHER]

This study examined how literal symbols affect students' understanding of algebraic expressions. Middle school students (N = 322) were randomly assigned to 1 of 3 conditions in which they were asked to interpret an expression (e.g., 4c + 3b) in a story problem. Each literal symbol represented the price of an item. In the c-and-b condition, the symbols used were the 1st letters of the items (e.g., price of a cake in dollars = c; price of a brownie in dollars = b). In the other 2 conditions, c and b were replaced with nonmnemonic English letters (x and y) or Greek letters (Φ and ψ). Incorrect interpretations of the expression were most common among students in the c-and-b condition. Moreover, students in this condition were more likely than students in the other conditions to misinterpret the symbols as labels for objects (e.g., c stands for cake). An analysis of participating students' textbooks revealed that mnemonic symbols were used correctly and were not uncommon. Results suggest that the use of mnemonic symbols may hinder students' interpretation of algebraic expressions. [ABSTRACT FROM AUTHOR]

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12 × 12 multiplication table showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are compressed. Insights obtained from the various methods can be used to deepen teachers' and students' understandings of possible ways to bridge arithmetic and algebra. [ABSTRACT FROM AUTHOR]

The amount of overt structure in the presentation of a task affects students' engagement, creativity, and willingness to tolerate frustration. In a professional development project, with algebra teachers from nine American schools, we tried to help teachers make judicious decisions in their use of structure by having them facilitate low-structure tasks, remove structure from overly structured tasks, and observe "at-risk" students engaged in learning through low-structure tasks. Project schools that worked on structuring generally improved their algebra passing rates, both overall and for African-American students. [ABSTRACT FROM AUTHOR]

Solving arithmetic and algebraic word problems is a key component of the Singapore elementary mathematics curriculum. One heuristic taught, the model method, involves drawing a diagram to represent key information in the problem. We describe the model method and a three-phase theoretical framework supporting its use. We conducted 2 studies to examine teachers' perceptions and children's application of the model method. The subjects were 14 primary teachers from 4 schools and 151 Primary 5 children. The model method affords higher ability children without access to letter- symbolic algebra a means to represent and solve algebraic word problems. Partly correct solutions suggest that representation is not an all-or-nothing process in which model drawing is either completely correct or completely incorrect. Instead, an incorrect solution could be the consequence of misrepresentation of a single piece of information. Our findings offer avenues of support in word problem solving to children of average ability. [ABSTRACT FROM AUTHOR]

This article examines findings on student perceptions of individual interventions based on the principles of universal design for learning (UDL). The examination includes a comparison of the reported perceptions of mainstreamed students with high incidence disabilities (i.e., learning disabilities, behavioral disorders, or other health impairments under Section 504 of the Rehabilitation Act) to that of their general education peers. Findings showed that relative to their other academic classes, both groups of students had high levels of satisfaction and expressed similar themes as to what they perceived to be the best and worst parts of the interventions and ideas for improvement. Both groups also reported near unanimous agreement as to wanting their teachers to use more UDL interventions. The reported perceptions and subsequent comparison forms the basis for discussing the implications of UDL in high school settings. [ABSTRACT FROM AUTHOR]

This study analyses the relationship between solving exploratory and investigation tasks involving functional relationships and the development of algebraic thinking in grade 8 students, giving special attention to the way they interpret and use the algebraic language. The methodology, qualitative and interpretative, is based in two case studies of students involved in a teaching unit of 16 classes which included the study of numerical sequences, functions and first degree equations. The data collection involved two interviews (one carried before and the other after the teaching unit), the participant observation of the classes by the teacher, registered in her diary, and gathering of students' written records. The results show that the emphasis in the study of functional relationships based on exploratory and investigation tasks promotes the development of meaning for the algebraic language and the construction of a wider vision regarding the use of symbols. [ABSTRACT FROM AUTHOR]

In this article, the author explores the "algebra for all" issue to raise awareness about the many facets of this dilemma facing educators at the middle and high school levels. She discusses both sides of this controversial issue, especially regarding its impact on students' futures relative to higher education and employment. The author concludes that unless dramatic reform occurs in the delivery of algebra so that all students benefit from such instruction, students must make the decision to take algebra based on individual needs, interests, and desires. [ABSTRACT FROM AUTHOR]

In this article, we deal with students' algebraic generalizations set in the context of elementary geometric-numeric patterns. Drawing from Vygotsky's psychology, Leont'ev's Activity Theory, and Husserl's phenomenology, we focus on the various semiotic resources mobilized by students in their passage from the particular to the general. Two small groups of Grade 9 students are investigated through a fourdimensional analysis: video, audio, transcripts, and written material. The resulting qualitative analysis shows how discourse, gestures, actions, and rhythms orchestrate one another and how, through a complex and subtle coordination of them, the students objectify different aspects of their spatial-temporal mathematical experience. The analysis also suggests connections between the syntax of the students' algebraic formulas and the semiotic means of objectification through which the formulas were forged, thereby shedding some light on the meaning of students' algebraic expression. Some implications for the teaching and learning of mathematics are discussed. [ABSTRACT FROM AUTHOR]

Abstract: We examine the rhetorical methods of Leonardo of Pisa in his exposition of single false position in Liber Abbaci. For example, Leonardo makes extensive use of formulaic phrases in his solutions. Some of these formulas also seem to indicate whether a particular solution needs further justification. Although he prefers proofs in terms of the pseudo-Euclidean canon of al-Khwārizmī, sometimes such proof eludes Leonardo and he resorts instead to justification by experiment. We also look at the extent to which using symbolic representations might distort our view of Leonardo''s thinking. [Copyright &y& Elsevier]

Research investigating algebra students' abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students' errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students' generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students' engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c) generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing. [ABSTRACT FROM AUTHOR]

Abstract: This paper is inscribed within the research effort to produce evidence regarding primary school students’ learning of algebra. Given the results obtained so far in the research community, we are convinced that young elementary school students can successfully learn algebra. Moreover, children this young can make use of different representational systems, including function tables, algebraic notation, and graphs in the Cartesian coordinate grid. In our research, we introduce algebra from a functional perspective. A functional perspective moves away from the mere symbolic manipulation of equations and focuses on relationships between variables. In investigating the processes of teaching and learning algebra at this age, we are interested in identifying meaningful teaching situations. Within each type of teaching situation, we focus on what kind of knowledge students produce, what are the main obstacles they find in their learning, as well as the intermediate states of knowledge between what they know and the target knowledge for the teaching situation. In this paper, we present a case study focusing on the approach adopted by a third grade student, Marisa, when she was producing the formula for a linear function while she was working with the information of a problem displayed in a function table containing pairs of inputs–outputs. We will frame the analysis and discussion on Marisa''s approach in terms of the concept of theorem-in action (Vergnaud, 1982) and we will contrast it with the scalar and functional approaches introduced by Vergnaud (1988) in his Theory of Multiplicative Fields. The approach adopted by Marisa turns out to have both scalar and functional aspects to it, providing us with new ways of thinking of children''s potential responses to functions. [Copyright &y& Elsevier]

Information processing researchers have assumed that algebra symbol skills depend on mastery of the abstract rules presented in the curriculum (Matz, 1980; Sleeman, 1986). Thus, students' ubiquitous algebra errors have been taken as indicating the need to embed algebra in rich contextual settings (Kaput, 1995; National Council of Teachers of Mathematics [NCTM] Algebra Working Group, 1998). This study explored a nonrepresentational account of symbolic algebra skills as feature correlation within the visual field. We present evidence that algebra students respond spontaneously to the visual patterns of the notational display apart from engagement with the declarative content of the rules. Thus, persistent algebra errors may reflect disengagement from declarative content rather than an inability to deal with it. We sketch a Lexical Support System designed to sustain students' engagement with the declarative content of algebraic rules and processes, thus complementing the exciting curricular possibilities being developed for referentially rich algebra. [ABSTRACT FROM AUTHOR]

ABSTRACT: This article presents essential principles for mathematics instruction for students with disabilities and those who are at risk for school failure. These students have great difficulty in applying algorithmic knowledge to solve problems and even greater difficulty with abstract problem solving associated with higher level mathematics. A procedural and cognitive model of problem solving at the algebraic level is presented. [ABSTRACT FROM AUTHOR]

Focuses on the use of worked examples as an instructional supplement in algebra classes. Translation of English expressions into algebraic equations; Worked examples as support for homework; Individual instruction for remedial mathematics class.

Much of the neuroimaging research has focused on how mathematical operations are performed. Although this body of research has provided insight for the refinement of pedagogy, there are very few neuroimaging studies on how mathematical operations should be taught. In this article, we describe the teaching of algebra in Singapore schools and the imperatives that led us to develop two neuroimaging studies that examined questions of curricular concerns. One of the challenges was to condense issues from classrooms into tasks suitable for neuroimaging studies. Another challenge, not particular to the neuroimaging method, was to draw suitable inferences from the findings and translate them into pedagogical practices. We describe our efforts and outline some continuing challenges. [ABSTRACT FROM AUTHOR]