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The main purpose of this study is to investigate mathematics teachers' mathematical thinking process while they are constructing the length of [cube root of 2] by paper folding. To carry out this aim, two teachers--who are PhD. students--were interviewed one by one. During the construction, it was possible to observe the consolidation process of Pythagorean and Thales Theorem. All interviews were videotaped and analyzed through descriptive methods, according to abstraction in context, characteristics of consolidation and mathematical habits of mind. It was found that both two teachers constructed the knowledge of Pythagorean and Thales Theorems before the study and also these knowledges were consolidated. In addition, it was determined that these two approaches (abstraction in context and mathematical habits of mind) were close and corroborated each other. Moreover, consolidation process corroborated mathematical habits of mind.

It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to elementary and middle school teachers and students in the learning and understanding of fraction multiplication and division.

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA, 2014). [This paper was first published in the "Australian Senior Mathematics Journal," in 2014 (EJ1093386).]

In this article, the author presents a paper folding activity that can be used for teaching fractions. This activity can be used to describe areas of folded polygons in terms of a standard unit of measure. A paper folding fractions worksheet and its corresponding solutions are also presented in this article. (Contains 2 figures.)

In this study I examine the issue of how to support students' coordination of computer algebra systems (CAS) and paper-and-pencil in solving mathematics problems. Together with a classroom teacher I designed and conducted a collaborative teaching experiment in a ninth-grade algebra classroom to understand how to support students' coordination of tools. I introduce a predict-act-reflect-reconcile activity structure for coordinating CAS and paper-and-pencil in solving problems involving linear expressions and equations and adopt this as an analytic lens to examine the role of curriculum materials on impacting students' learning. Findings suggest the activity structure of predict-act-reflect-reconcile is a potentially fruitful starting point for coordinating CAS and paper-and-pencil in classroom activity, but not sufficient on its own as a task design principle without focused classroom discussion and support during the enacted curriculum. This research points to the importance of teaching students how to communicate about their process of using multiple tools while solving problems, especially the mathematical thinking that students engage in while reconciling differences across tool-based representations.

This study examined comparability of student scores obtained from computerized and paper-and-pencil formats of the large-scale statewide end-of-course (EOC) examinations in the two subject areas of Algebra and Biology. Evidence in support of comparability of computerized and paper-based tests was sought by examining scale scores, item parameter estimates, test characteristic curves, test information functions, Rasch ability estimates at the content domain level, and the equivalence of the construct. Overall, the results support the comparability of computerized and paper-based tests at the item-level, subtest-level and whole test-level in both subject areas. No evidence was found to suggest that the administration mode changed the construct being measured. (Contains 5 tables and 8 figures.)

This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is targeted as a rich and pressing context to study this issue. We report results of a collaborative teaching experiment in which we designed for and tested a functions approach to solving equations with ninth-grade algebra students, and link to results of semi-structured interviews with students before and after the experiment. Results of analyzing the five-week experiment include instructional supports for students' representational fluency in solving linear equations: (a) sequencing the use of graphs, tables, and CAS feedback prior to formal symbolic transpositions, (b) connecting solutions to equations across representations, and (c) encouraging understanding of equations as equivalence relations that are sometimes, always, or never true. While some students' change in sophistication of representational fluency helps substantiate the productive nature of these supports, other students' persistent struggles raise questions of how to address the diverse needs of learners in complex learning environments involving multiple tool-based representations.

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard's (2008) communication approach and on Châtelet's (2000) notion of the "virtual" in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of "material entanglement," which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables.

College algebra fulfills general education requirements at many colleges in the United States. The study reported here investigated differences in mathematics achievement between undergraduates in college algebra classes using one of two homework methods: "WeBWorK," an open-source system for web-based homework, or traditional paper-and-pencil homework. We assessed learning for 439 students in 19 college algebra classes at a large public university in the United States. Twelve classes used "WeBWorK" and seven had traditional paper-and-pencil homework. Analysis of covariance revealed no significant differences in algebra performance or achievement gain by homework group, ethnicity or gender when statistically controlling for previous mathematics achievement. Results support the conjecture that "WeBWorK" is at least as effective as traditionally graded paper-and-pencil homework for students learning college algebra in moderately sized lecture-based classes.

This study aims to investigate plane geometry problem-solving strategies of prospective mathematics teachers using dynamic geometry software (DGS) and paper-and-pencil (PPB) environments after receiving an instruction with GeoGebra (GGB). Four plane geometry problems were used in a multiple case study design to understand the solution strategies developed by 2 prospective teachers. The results revealed that although the participants mostly used algebraic solutions in the PPB environment, they preferred geometric solutions in the GGB environment even though algebraic solutions were still possible (the software did not preclude them). Furthermore, different proofing strategies were developed in each environment. This suggests that changing the environment may prompt students to seek for additional solutions, which, in turn, results in a deeper understanding of the problem. As such, using both environments simultaneously in solving the same problems appears to bring about important benefits.

How can the key concept of equivalent expressions be addressed so that students strengthen their representational fluency with symbols, graphs, and numbers? How can research inform the synergistic use of both paper-and-pencil analysis and computer algebra systems (CAS) in a classroom learning environment? These and other related questions have important implications for any beginning algebra course. This article focuses on two approaches to equivalence of expressions: (1) developing fluency within and among numeric, graphical, verbal, and symbolic representations; and (2) coordinating the symbolic features of CAS and paper-and-pencil techniques.

The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem-solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA 2014).

We comment on the paper "The co-emergence of machine techniques, paper-and-pencil techniques, and theoretical reflection: A study of CAS use in secondary school algebra" by Carolyn Kieran and Paul Drijvers. We look at aspects of Kieran and Drijvers' analysis with regard to "task-technique-theory" in the light of a model of abstraction in context and suggest that this re-viewing brings mutual analytic benefits.

This note demonstrates multiple representations (numerical and graphical) of spreadsheet-based quadratic sequences together with prospective teachers' pencil-paper transformations of these numerical sequences into a corresponding symbolization as algebraic expressions. With the majority of prospective teachers, the experience of school mathematics is one of disaffection. They are in a teacher education programme that offers them certification to teach up to grade nine in the public schools. Their work illustrates processes, such as recognizing and extending patterns, by specializing and generalizing particular functional relationships. Insights gained from various methods used by them, together with the instructional affordances, i.e. the instructional benefits of spreadsheets, represent a curricular continuity between quadratic, numerical sequences and related algebraic expressions and suggest a possible change to the order in which the two are introduced. (Contains 6 figures.)

This paper addresses the dialectical relation between theoretical thinking and technique, as they co-emerge in a combined computer algebra (CAS) and paper-and-pencil environment. The theoretical framework in this ongoing study consists of the instrumental approach to tool use and an adaptation of Chevallard's anthropological theory. The main aim is to unravel the subtle intertwining of students' theoretical thinking and the techniques they use in both media, within the process of instrumental genesis. Two grade 10 teaching experiments are described, the first one on equivalence, equality and equation, and the second one on generalizing and proving within factoring. Even though the two topics are quite different, findings indicate the importance of the co-emergence of theory and technique in both cases. Some further extensions of the theoretical framework are suggested, focusing on the relation between paper-and-pencil techniques and computer algebra techniques, and on the issue of language and discourse in the learning process.

Homework gives students an opportunity to practice important college-level physics skills. A switch to Web-based homework alters the nature of feedback received, potentially changing the pedagogical benefit. Calculus- and algebra-based introductory physics students enrolled in large paired lecture sections at a public university completed homework of standard end-of-the-chapter exercises using either the Web or paper. Comparison of their performances on regular exams, conceptual exams, quizzes, laboratory, and homework showed no significant differences between groups; other measures were found to be strong predictors of performance. This indicates that the change in medium itself has limited effect on student learning. Ways in which Web-based homework could enable exercises with greater pedagogical value are discussed. (Contains 2 notes, 7 tables, and 4 figures.)

Several activities involving area and volume using empty paper rolls are presented. The relationships of parallelograms to cylinders are illustrated. Teaching suggestions are provided. (CW)

An activity in which students use manipulative materials to help solve simple algebraic equations using the operations of adding inverses, removing opposites, and sharing equally is presented. Directions, examples, the rationale, and cautions are included. (KR)

The paper describes the results of a study whose main aim was to find the interrelationship between pupils' school grades in Czech language (native), mathematics, and physics and pupils' cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils' performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables.

Gaming the system is a persistent problem in Computer-Based Learning Platforms. While substantial progress has been made in identifying and understanding such behaviors, effective interventions remain scarce. This study uses a method of causal moderation known as Fully Latent Principal Stratification to explore the impact of two types of interventions--gamification and manipulation of assistance access on the learning outcomes of students who tend to game the system. The results indicate that gamification does not consistently mitigate these negative behaviors. One gamified condition had a consistently positive effect on learning regardless of students' propensity to game the system, whereas the other had a negative effect on gamers. However, delaying access to hints and feedback may have a positive effect on the learning outcomes of those gaming the system. This paper also illustrates the potential for integrating detection and causal methodologies within educational data mining to evaluate effective responses to detected behaviors.

In commemoration of the 100th volume of the "Mathematics Teacher," the Editorial Panel of "Teaching Children Mathematics" invited the author to read and reflect on an article published in "MT" in 1912. Her article compares the current approach to number and algebra content to that presented by Shipley. In addition, the article discusses the change (or lack thereof) of broader issues in mathematics education over the past hundred years.