A third grader's way of thinking about linear function tables

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.     

Learning beginning algebra with spreadsheets in a computer intensive environment

This study is part of a large research and development project aimed at observing, describing and analyzing the learning processes of two seventh grade classes during a yearlong beginning algebra course in a computer intensive environment (CIE). The environment includes carefully designed algebra learning materials with a functional approach, and provides students with unconstrained freedom to use (or not use) computerized tools during the learning process at all times. This paper focuses on the qualitative and quantitative analyses of students’ work on one problem, which serves as a window through which we learn about the ways students worked on problems throughout the year. The analyses reveal the nature of students’ mathematical activity, and how such activity is related to both the instrumental views of the computerized tools that students develop and their freedom to use them. We describe and analyze the variety of approaches to symbolic generalizations, syntactic rules and equation solving and the many solution strategies pursued successfully by the students. On that basis, we discuss the strengths of the learning environment and the open questions and dilemmas it poses.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

Using an economics model for teaching linear algebra

We present an approach for teaching linear algebra using models. In particular, we are interested in analyzing the modeling process under an APOS perspective. We will present a short illustration of the analysis of an economics problem related to production in a set of industries. This problem elicits the use of the concepts of linear combination, linear independence, among other linear algebra concepts related to vector space. We describe cycles of students’ work on the problem, present an analysis of the learning trajectory with emphasis on the constructions they develop, and discuss the advantages of this approach in terms of students’ learning.     

Students’ use of technological features while solving a mathematics problem

The design of technology tools has the potential to dramatically influence how students interact with tools, and these interactions, in turn, may influence students’ mathematical problem solving. To better understand these interactions, we analyzed eighth grade students’ problem solving as they used a java applet designed to specifically accompany a well-structured problem. Within a problem solving session, students’ goal-directed activity was used to achieve different types of goals: analysis, planning, implementation, assessment, verification, and organization. As we examined students’ goals, we coded instances where their use of a technology feature was supportive or not supportive in helping them meet their goal. We categorized features of this applet into four subcategories: (1) features over which a user does not have any control and remain static, (2) dynamic features that allow users to directly manipulate objects, (3) dynamic features that update to provide feedback to users during problem solving, and (4) features that activate parts of the applet. Overall, most features were found to be supportive of students’ problem solving, and patterns in the type of features used to support various problem solving goals were identified.     

An exploration of students’ errors in derivatives in a university of technology

This paper reports on an exploration of errors that were displayed by students who studied mathematics in chemical engineering in derivatives of various functions such as algebraic, exponential, logarithmic and trigonometric functions. The participants of this study were a group of twenty students who were at risk in an extended curriculum programme in a university of technology in Western Cape, South Africa. The researcher used a qualitative case study approach and collected data from students’ written work. This research uses action, process, object, and schema (APOS) theory to classify errors into categories and to analyse and interpret the data collected. The students displayed five different kinds of errors, namely, conceptual, interpretation, linear extrapolation, procedural and arbitrary. The use of APOS theory as a framework revealed that several students’ errors might be caused by over-generalisation of mathematical rules and properties such as the power rule of differentiation and distributive property in manipulation of algebraic expressions. This study suggests that teaching of the standard rules of differentiation should put emphasis on its restrictions to eliminate common errors that normally crop up due to over-generalisation of certain differentiation rules.     

Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs

This research explored students’ views of geometric objects through the implementation of a curriculum module that allowed them to explore the relationships between transformational geometry and linear algebra. The majority of the students were middle and secondary mathematics education majors enrolled in a one-semester geometry course that is aimed at prospective teachers. A preponderance of the evidence suggests that the participating students, for the most part, viewed isometries operationally and viewed geometric objects (triangle, etc.) as “perceived.” Results also suggest that these two views influenced the students’ abilities to understand and to construct geometric proofs in transformational geometry.     

Toward a student-centred process of teaching arithmetic

This article describes a way toward a student-centred process of teaching arithmetic, where the content is harmonized with the students’ conceptual levels. At school start, one classroom teacher is guided in recurrent teaching development meetings in order to develop teaching based on the students’ prerequisites and to successively learn the students’ arithmetic. The students are assessed in interviews. Two special teachers participate and their current models of each student's arithmetic are tested when assessing the students. The students’ conceptual diversity and the consequent different content in teaching are shown. Further, the special teachers’ assessments and the class teacher's opinion of the new way of teaching are reported. A wide range both of the students’ conceptual levels and of the kinds of relevant problems was found. The special teachers manage their duties well and the classroom teacher has so far been satisfied with the new teaching process.     

The role of operating premises and reasoning paths in upper elementary students’ problem solving

The validity of students’ reasoning is central to problem solving. However, equally important are the operating premises from which students’ reason about problems. These premises are based on students’ interpretations of the problem information. This paper describes various premises that 11- and 12-year-old students derived from the information in a particular problem, and the way in which these premises formed part of their reasoning during a lesson. The teacher’s identification of differences in students’ premises for reasoning in this problem shifted the emphasis in a class discussion from the reconciliation of the various problem solutions and a focus on a sole correct reasoning path, to the identification of the students’ premises and the appropriateness of their various reasoning paths. Problem information that can be interpreted ambiguously creates rich mathematical opportunities because students are required to articulate their assumptions, and, thereby identify the origin of their reasoning, and to evaluate the assumptions and reasoning of their peers.     

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.     

Algebra word problem solving approaches in a chemistry context: Equation worked examples versus text editing

Text editing directs students’ attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three equation steps each of which comprises essential units of relevant information. In an experiment, students were randomly assigned to either text editing or equation worked examples condition in a regular classroom setting to learn how to solve algebra word problems in a chemistry context. The equation worked examples group outperformed the text editing group for molarity problems, which were more difficult than dilution problems. Empirical evidence supports the theoretical rationale in using equation worked examples to facilitate students’ construction of a coherent problem structure so as to develop problem skills and expertise to solve molarity problems.     

Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.     

Covariational reasoning and invariance among coordinate systems

Researchers continue to emphasize the importance of covariational reasoning in the context of students’ function concept, particularly when graphing in the Cartesian coordinate system (CCS). In this article, we extend the body of literature on function by characterizing two pre-service teachers’ thinking during a teaching experiment focused on graphing in the polar coordinate system (PCS). We illustrate how the participants engaged in covariational reasoning to make sense of graphing in the PCS and make connections with graphing in the CCS. By foregrounding covariational relationships, the students came to understand graphs in different coordinate systems as representative of the same relationship despite differences in the perceptual shapes of these graphs. In synthesizing the students’ activity, we provide remarks on instructional approaches to graphing and how the PCS forms a potential context for promoting covariational reasoning.     

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

Using interactive algebra software to support a discourse community

In this work we studied the impact of using NuCalc, an interactive computer algebra software, on the development of a discourse community in a college level mathematics class. Qualitative and quantitative data were collected over the course of 3 weeks of instruction. We examined the influence of the software on: group interactions; the mathematical investigations of learners; and the teacher’s interactions with students. Data points to four distinct ways in which the presence of NuCalc positively impacted the learning community we studied: (1) it served as a tool for extending students’ mathematical thinking, (2) it motivated students’ engagement in group discourse, (3) it became a tool for mediating discourse, (4) it became a catalyst for refining the culture of classroom, shifting the patterns of interactions between the teacher and learners.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Pedagogical representations to teach linear relations in Chinese and U.S. classrooms: Parallel or hierarchical?

This study investigates Chinese and U.S. teachers’ construction and use of pedagogical representations surrounding implementation of mathematical tasks. It does this by analyzing video-taped lessons from the Learner's Perspective Study, involving 15 Chinese and 10 U.S. consecutive lessons on the topic of linear equations/linear relations. We examined patterns of pedagogical representations that Chinese and U.S. teachers construct over a set of consecutive lessons, but also investigated the strategies of using representations to solve mathematical problems by Chinese and U.S. teachers. It was found that multiple representations were constructed simultaneously to develop the connection of relevant concepts in the U.S. classrooms while selective representations were constructed to develop relevant concepts in the Chinese classrooms. This study is significant because it contributes to our understanding of the cultural differences involving Chinese and U.S. students’ mathematical thinking and has practical implications for constructing pedagogical representations to maximize students’ learning.     

Linear algebra students’ modes of reasoning: Geometric representations

Main goal of our research was to document differences on the types of modes linear algebra students displayed in their responses to the questions of linear independence from two different assignments. In this paper, modes from the second assignment are discussed in detail. Second assignment was administered with the support of graphical representations through an interactive web-module. Additionally, for comparison purposes, we briefly talk about the modes from the first assignment. First assignment was administered with the support of computational devices such as calculators providing the row reduced echelon form (rref) of matrices. Sierpinska’s framework on thinking modes (2000) was considered while qualitatively documenting the aspects of 45 matrix algebra students’ modes of reasoning. Our analysis revealed 17 categories of the modes of reasoning for the second assignment, and 15 categories for the first assignment. In conclusion, the findings of our analysis support the view of the geometric representations not replacing one’s arithmetic or algebraic modes but encouraging students to utilize multiple modes in their reasoning. Specifically, geometric representations in the presence of algebraic and arithmetic modes appear to help learners begin to consider the diverse representational aspects of a concept flexibly.     

Motivating the concept of eigenvectors via cryptography

New methods of teaching linear algebra in the undergraduatecurriculum have attracted much interest lately. Most of thiswork is focused on evaluating and discussing the integrationof special computer software into the Linear Algebra curriculum.In this article, I discuss my approach on introducing the conceptof eigenvectors and eigenvalues, which I have used for the last3 years in my Linear Algebra course. I offer some examples onhow I have attracted the interest of our students via Hill ciphering,a method of cryptography. After emphasizing the effect of alinear transformation in a vector space and the importance ofeigenvectors, I show how students’ motivation and understandingtowards one of the abstract concepts in Linear Algebra; eigenvaluesand eigenvectors have grown positively.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.