Conceptions of mathematics and student identity: implications for engineering education

Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students’ conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students’ conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.     

Electronic vs. paper textbook presentations of the various aspects of mathematics

Based in part on our work in adapting existing paper textbooks for secondary schools for a digital format, this paper discusses paper form and the various electronic platforms with regard to the presentation of five aspects of mathematics that have roles in mathematics learning in all the grades kindergarten-12: symbolization, deduction, modeling, algorithms, and representations. In moving to digital platforms, each of these aspects of mathematics presents its own challenges and opportunities for both curriculum and instruction, that is, for the content goals and how they connect with students for learning. A combination of paper and electronic presentations may be an optimal solution but some difficulties with such a complex solution are presented.     

Students’ sense-making frames in mathematics lectures

The goal of this study is to describe the various ways students make sense of mathematics lectures. Here, sense-making refers to a process by which people construct personal meanings for phenomena they experience. This study introduces the idea of a sense-making frame and describes three different types of frames: content-, communication-, and situating-oriented. We found that students in an abstract algebra class regularly engaged in sense-making during lectures on equivalence relations, and this sense-making influenced their note-taking practices. We discuss the relationship between the choice of frame, the students’ sense-making practices, and the potential missed opportunities for learning from the lecture. These results show the importance of understanding the ways students make sense of aspects of mathematics lectures and how their sense-making practices influence what they might learn from the lecture.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

The potential of recreational mathematics to support the development of mathematical learning

ABSTRACT

A literature review establishes a working definition of recreational mathematics: a type of play which is enjoyable and requires mathematical thinking or skills to engage with. Typically, it is accessible to a wide range of people and can be effectively used to motivate engagement with and develop understanding of mathematical ideas or concepts. Recreational mathematics can be used in education for engagement and to develop mathematical skills, to maintain interest during procedural practice and to challenge and stretch students. It can also make cross-curricular links, including to history of mathematics. In undergraduate study, it can be used for engagement within standard curricula and for extra-curricular interest. Beyond this, there are opportunities to develop important graduate-level skills in problem-solving and communication. The development of a module ‘Game Theory and Recreational Mathematics’ is discussed. This provides an opportunity for fun and play, while developing graduate skills. It teaches some combinatorics, graph theory, game theory and algorithms/complexity, as well as scaffolding a Pólya-style problem-solving process. Assessment of problem-solving as a process via examination is outlined. Student feedback gives some indication that students appreciate the aims of the module, benefit from the explicit focus on problem-solving and understand the active nature of the learning.     

Opportunity to learn problem solving in Dutch primary school mathematics textbooks

In the Netherlands, mathematics textbooks are a decisive influence on the enacted curriculum. About a decade ago, Dutch primary school mathematics textbooks provided hardly any opportunities to learn problem solving. In this study we investigated whether this provision has changed. In order to do so, we carried out a textbook analysis in which we established to what degree current textbooks provide non-routine problem-solving tasks for which students do not immediately have a particular solution strategy at their disposal. We also analyzed to what degree textbooks provide ‘gray-area’ tasks, which are not really non-routine problems, but are also not straightforwardly solvable. In addition, we inventoried other ways in which present textbooks facilitate the opportunity to learn problem solving. Finally, we researched how inclusive these textbooks are with respect to offering opportunities to learn problem solving for students with varying mathematical abilities. The results of our study show that the opportunities that the currently most widely used Dutch textbooks offer to learn problem solving are very limited, and these opportunities are mainly offered in materials meant for more able students. In this regard, Dutch mainstream textbooks have not changed compared to the situation a decade ago. A textbook that is the Dutch edition of a Singapore mathematics textbook stands out in offering the highest number of problem-solving tasks, and in offering these in the materials meant for all students. However, in the ways this textbook facilitates the opportunity to learn problem solving, sometimes a tension occurs concerning the creative character of genuine problem solving.     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

Investigating Elementary Mathematics Curricula: Focus on Students With Learning Disabilities

The purpose of this study was to investigate three elementary mathematics curricula to examine the accessibility for students with learning disabilities (LD) with regards to challenges associated with working memory. We chose to focus on students' experiences when finding the area of composite shapes due to the multiple steps involved for solving these problems and the potential for these problems to tax working memory. We conducted a qualitative analysis of how each curriculum provided opportunities for students with LD to engage with these problems. During our analysis, we focused on instruction that emphasized visual representations (e.g., manipulatives, drawings, and diagrams), facilitated mathematical conversations, and developed cognitive and metacognitive skills. Our findings indicated a need for practitioners to consider how each curriculum provides instruction for storage and organization of information as well as how each curriculum develops students' thinking processes and conceptual understanding of mathematics. We concluded that all three curricula provide potentially effective strategies for teaching students with LD to solve multi‐step problems, such as area of composite shapes problems, but teachers using any of these curricula will likely need to supplement the curriculum to meet the needs of students with LD.     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

Reasoning-and-Proving in School Mathematics Textbooks

Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.     

Distinguishing Features of Mechanical and Human Problem-Solving

Over the years, research in mathematical problem-solving has examined expert/novice problem-solving performance on various types of problems and subjects. In particular, DeFranco examined two groups of Ph.D. mathematicians as they solved four mathematics problems and found that although all were content experts, only one group were problem-solving experts. Based on this study, this article posits the notion that one distinguishing feature between experts and novices is that experts tend to look for special features of a problem and use algorithms only as a “fail-safe” system while novices act like “machines” relying on algorithms to solve the problems. Why? The article explores the idea that novice problem solvers learned to solve problems the way they learned proof, that is, in a formal, abstract and mechanizable way. Beliefs about proof and the culture in which it is practiced help frame a mathematician's view of the discipline and ultimately impacts classroom practice. The authors believe that current classroom instruction tends to create a culture that fosters algorithmic proficiency and a “machine-like” approach to the learning of mathematics and problem-solving. Further, they argue that mathematicians need to be aware of the distinction between knowing a proof is true and explaining why it is true. When these distinctions are appreciated and practiced during classroom instruction, then and only then will students begin to acquire the mathematical knowledge to become better problem solvers.     

When an argument is the content: Rational number comprehension through conversions across registers

This article reframes previously identified misconceptions about repeating decimals by describing these misconceptions as limited understandings of how mathematics concepts are referenced. In particular, misconceptions about repeating decimals and their quotient of integer representations are recast as limited understandings of mathematics as a discipline that derives its content from representational systems and the denotations they provided. Under this framework, arguments (e.g., proofs) that convert repeating decimals to their quotient of integer representations provide content for “rational number,” which is represented in multiple ways, each offering distinct opportunities for mathematical activity. The notion of an argument as content is illustrated as arguments providing access to a concept. One Grade 8 student’s struggle with understanding rational number is used to illustrate this framework and its implications for teaching and learning.     

Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

Beliefs and Attitudes Toward Mathematics Among Third- and Fifth-Grade Students: A Descriptive Study

A survey on attitudes toward mathematics of third- and fifth-grade students enrolled in a large urban school district was conducted in the spring of 1994 as part of a review of the mathematics program. Student responses to survey items were analyzed with respect to gender and grade level. Although girls and boys were equally likely to indicate that they like mathematics, in both Grades 3 and 5, boys were more likely than girls to report being good at mathematics. This result has often been reported for older students but has rarely been explored in younger children. Some gender and grade differences were observed in students' attitudes towards the specific strands of mathematics. Both girls and boys in Grade 5 were more likely than students in Grade 3 to believe that mathematics was relevant to their lives. Virtually all children thought that both girls and boys needed to study mathematics. No gender or grade differences in students' beliefs regarding the process of learning mathematics were observed. The results support the need for further research to identify variables that influence the development of student perceptions and attitudes toward mathematics during elementary school.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

Learning to See: Making Sense of the Mathematics of Change in Middle School

In this article, we will describe the results of a study of 6th grade students learning about the mathematics of change. The students in this study worked with software environments for the computer and the graphing calculator that included a simulation of a moving elevator, linked to a graph of its velocity vs. time. We will describe how the students and their teacher negotiated the mathematical meanings of these representations, in interaction with the software and other representational tools available in the classroom. The class developed ways of selectively attending to specific features of stacks of centimeter cubes, hand-drawn graphs, and graphs (labeled velocity vs. time) on the computer screen. In addition, the class became adept at imagining the motions that corresponded to various velocity vs. time graphs. In this article, we describe this development as a process of learning to see mathematical representations of motion. The main question this article addresses is: How do students learn to see mathematical representations in ways that are consistent with the discipline of mathematics? This revised version was published online in July 2006 with corrections to the Cover Date.     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Teaching and learning of mathematics: what really matters to teachers and students?

The learner’s perspective study, motivated by a strong belief that the characterization of the practices of mathematics classrooms must attend to learner practice with at least the same priority as that accorded to teacher practice, is a comprehensive study that adopts a complementary accounts methodology to negotiate meanings in classrooms. In Singapore, three mathematics teachers recognized for their locally defined ‘teaching competence’ participated in the study. The comprehensive sets of data from the three classrooms have been used to explore several premises related to the teaching and learning of mathematics. In this paper the student interview data and the teacher interview data were examined to ascertain what do students attach importance to and what do teachers attach importance to in a mathematics lesson? The findings of the student interview data showed that they attached importance to several sub-aspects of the three main aspects, i.e., exposition, seatwork and review and feedback of their teachers’ pedagogical practices. The findings of the teacher interview data showed that they attached importance to student’s self assessment, teacher’s demonstration of procedures, review of prior knowledge and close monitoring of their student’s progress in learning and detailed feedback of their work. It was also found that teachers and students did attach importance to some common lesson events.