Promoting Preservice Elementary Teachers' Awareness of Learning and Teaching Mathematics Conceptually Through KTEM

In this article, we describe preservice elementary teachers' reactions to Liping Ma's (1999) book, Knowing and Teaching Elementary Mathematics (KTEM), from five universities. Ma's discussion of solely teaching elementary mathematics procedurally and its consequences awakens the preservice teachers' memories of learning elementary mathematics. Ma's analysis of and solution to the problem ignites strong emotions in the preservice elementary teachers and promotes a desire to teach elementary mathematics conceptually. Through the analysis of writing assignments, we summarize how reading and reflecting on KTEM gives preservice teachers an opportunity to examine their beliefs about teaching and learning elementary mathematics conceptually.     

Writing as a Tool to Demonstrate Mathematical Understanding

In this study, I examine how using a writers' workshop model in mathematics creates a space for students to write about their mathematical thinking and problem solving and how their writing impacts instruction. This case study of one classroom with one teacher spanned 6 weeks and included 18 implementations of an adapted version of the Writers' Workshop (WW) in a fourth‐grade mathematics class. On a biweekly basis, the data were reviewed and changes made to the model. The analysis of the students' writing revealed (a) their understandings and misunderstandings of the mathematical content, (b) their readiness for more challenging tasks, and (c) their connections to prior knowledge. Students used writing to demonstrate their understanding of mathematics and show their mathematical processes. In some cases, examining only the numerical work failed to illuminate the students' understanding, their writing provided deeper insight. Students recognized writing as a tool for learning; this was evident in interview responses.     

Investigating students’ perceptions of graduate learning outcomes in mathematics

ABSTRACT

The purpose of this study is to explore the perceptions mathematics students have of the knowledge and skills they develop throughout their programme of study. It addresses current concerns about the employability of mathematics graduates by contributing much needed insight into how degree programmes are developing broader learning outcomes for students majoring in mathematics. Specifically, the study asked students who were close to completing a mathematics major (n = 144) to indicate the extent to which opportunities to develop mathematical knowledge along with more transferable skills (communication to experts and non-experts, writing, working in teams and thinking ethically) were included and assessed in their major. Their perceptions were compared to the importance they assign to each of these outcomes, their own assessment of improvement during the programme and their confidence in applying these outcomes. Overall, the findings reveal a pattern of high levels of students’ agreement that these outcomes are important, but evidence a startling gap when compared to students’ perceptions of the extent to which many of these – communication, writing, teamwork and ethical thinking – are actually included and assessed in the curriculum, and their confidence in using such learning.     

Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions

In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies have provided evidence that students have difficulty with mathematical proofs. Few of these research studies, however, have focused on undergraduates’ abilities to produce proofs and counterexamples in the domain of continuous functions. The goal of this study is to contribute to research on student productions of proofs and counterexamples and to identify their abilities and mathematical understandings. The findings suggest more attention should be paid to teaching and learning proofs and counterexamples, as participants showed difficulty in writing these statements. More importantly, the analysis provides insight into the design of curriculum and instruction that may improve undergraduates’ learning in advanced mathematics courses.     

Disciplinary literacy in mathematics: One mathematician’s reading practices

Recent scholarship on disciplinary literacy calls for an emphasis on teaching discipline-specific language/literacy practices. An understanding of these practices is, therefore, essential to literacy instruction in secondary content areas such as mathematics. This case study examined one mathematician’s reading practices, with a focus on the strategies he used in text comprehension. Data collected include the mathematician’s think-alouds during reading, discussion of his reading think-alouds, and semi-structured interviews. These data were analyzed qualitatively through an iterative process involving multiple readings and identification and refinement of codes. The analysis revealed that the mathematician engaged in extensive reading and employed an array of strategies—rereading, close reading, monitoring and questioning, summarizing and paraphrasing, storying, drawing on prior knowledge and experience, evaluating and verifying, and note-taking and visualizing—to help him make sense of what he read. These findings provide important insights that can inform mathematics teachers’ efforts to support students’ mathematics reading/learning.     

God, king, and geometry: revisiting the introduction to Cauchy’s Cours d’analyse

This article offers a systematic reading of the introduction to Augustin-Louis Cauchy’s landmark 1821 mathematical textbook, the Cours d’analyse. Despite its emblematic status in the history of mathematical analysis and, indeed, of modern mathematics as a whole, Cauchy’s introduction has been more a source for suggestive quotations than an object of study in its own right. Cauchy’s short mathematical metatext offers a rich snapshot of a scholarly paradigm in transition. A close reading of Cauchy’s writing reveals the complex modalities of the author’s epistemic positioning, particularly with respect to the geometric study of quantities in space, as he struggles to refound the discipline on which he has staked his young career.     

The Effect of Journal Writing on Achievement in and Attitudes Toward Mathematics

A teaching experiment was conducted to investigate the effect of journal writing on achievement in and attitudes toward mathematics. Achievement variables included conceptual understanding, procedural knowledge, problem solving, mathematics school achievement, and mathematical communication. Subjects were selected from first intermediate students (11–13 years) attending the International College, Beirut, Lebanon, where either English or French is the language of mathematics instruction. The journal-writing (JW) group received the same mathematics instruction as the no-journal-writing (NJW) group, except that the JW group engaged in prompted journal writing for 7 to 10 minutes at the end of each class period, three times a week, for 12 weeks. The NJW group engaged in exercises during the same period. The results of ANCOVA suggest that journal writing has a positive impact on conceptual understanding, procedural knowledge, and mathematical communication but not on problem solving, school mathematics achievement, and attitudes toward mathematics. Gender, language of instruction, mathematics achievement level, and writing achievement level failed to interact with journal writing. Student responses to a questionnaire indicated that students found journal writing to have both cognitive and affective benefits.     

Building a Network to Empower Teachers for School Reform

Reform efforts in college mathematics teaching are often hindered by the fact that many instructors have never experienced instructional delivery methods other than lecture. Building a network of college mathematics faculty interested in reform has provided the impetus for faculty members to incorporate problem solving, cooperative learning, technology, manipulatives, writing, and alternative means of assessment into courses for preservice elementary teachers. Suggestions for setting up a network are intended to provide guidance for other departments wishing to stimulate reform movements.     

Communication in Mathematics: Preparing Preservice Teachers to Include Writing in Mathematics Teaching and Learning

“Math was strictly math, from what I remember.” This is a comment about using writing in mathematics from a preservice elementary teacher enrolled in a methods course. Comments such as these concern teacher educators who wish to prepare elementary teachers to include writing in mathematics instruction. A teacher development experiment was completed to discover how to improve preservice teachers’ abilities and attitudes toward using writing in mathematics. The preservice teachers made use of a graphic organizer to facilitate writing in the college math methods class, then practiced teaching writing with the same graphic organizer and in the math classes in an elementary classroom. Reflections of the preservice teachers illustrated this was a positive practice. The preservice teachers also concluded that writing in mathematics is valuable to instruction and would include it in their teaching.     

Student performance and attitudes in a collaborative and flipped linear algebra course

Flipped learning is gaining traction in K-12 for enhancing students’ problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.     

The development of instructional guidelines for elementary mathematical writing

Mathematical writing recently has been defined as writing to reason and communicate mathematically. But mathematics instructional resources lack guidance for teachers as to how to implement such writing. The purpose of this paper is to describe how methods of design-based research were used to develop an instructional resource when one does not currently exist. Thirty-four participants—including teachers, mathematics coaches, mathematics curriculum developers, literacy coaches, a mathematician, and academics in elementary mathematics education, mathematics education, writing education, and science education—participated in a multi-step process to recommend, revise, and confirm instructional guidelines for elementary mathematical writing. The development process began with thirty-two recommendations from science writing and language arts writing. Through multiple cycles of feedback, five instructional guidelines and related considerations and techniques for implementation of elementary mathematical writing emerged.     

Reading mathematics for understanding—From novice to expert

As students progress through the college mathematics curriculum, enter graduate school and eventually become practicing mathematicians, reading mathematics textbooks and journal articles appears to become easier and leads to increased proficiency and understanding. This study was designed to begin to understand how mathematically more advanced readers read for understanding in mathematical exposition as it appears in textbooks compared to first-year undergraduate students. Three faculty members and three graduate students participated in this study and read from a first-year graduate textbook in an area of mathematics unfamiliar to each of them. The observed reading strategies of these more mathematically advanced readers are compared to observed reading strategies of first-year undergraduate students from an earlier study. The reading methods of the faculty level mathematicians were all quite similar and were markedly different from those that have been identified for undergraduate students, as well as from those used by the graduate students in this study. A Mathematics Reading Framework is proposed based on this study and previous research documenting the strategies that first-year undergraduate students use for reading exposition in their mathematics textbooks.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

Some issues surrounding the use of algebraic calculators in traditional examinations

Algebraic calculators, with symbolic manipulation capabilities, are tools in some classrooms for 14–19-year-olds and their use in learning and teaching mathematics is likely to increase over time. At the time of writing their use in traditional examinations is permitted in Austria, France and the USA. Several other countries have considered their possible use in examinations. In the UK several government initiated working groups considered the implications of their use in academic mathematics examinations for 18-year-old students. The reports of these working groups have not been published. An overview of the issues they addressed is presented here and implications for future examinations are considered.     

Mathematics learning support and engagement in first year engineering

This paper considers the effects of both free optional mathematics learning support and engagement on the mathematics performance in a foundation mathematics subject of a cohort of engineering students entering university with poor mathematical skills. New engineering students were directed to either a foundation or standard mathematics subject based on the results of a placement test. For students in the foundation subject, it was found that high levels of learning support were associated with greater improvement over the semester. Some form of learning support was used by 57.9% of the students, a reasonably high proportion of the cohort. Some factors for this high level of use of learning support are considered. One possible factor, the placement test, appears to have had a positive effect. Engagement in the subject activities as measured by tutorial attendance and learning management system use was found to have a positive association with final mark. Students who utilized a high level of learning support were more likely to be more engaged with the subject, making it impossible to draw conclusions about improvements being solely due to the use of learning support.     

Mathematics teaching before and after the Meiji Restoration

This paper outlines mathematical education before the Meiji Restoration, and how it changed as a result. The Meiji Restoration in 1868 completely changed the social structure of Japan. In the Edo period (1600?C1868) Japan was divided into domains (han) governed by local lords (daimyo). Tokugawa Shogunate supervised local lords and governed Japan indirectly. In the Edo period there were no wars for more than two centuries and many people participated in cultural activities. Japanese mathematics developed in its own way under the influence of old Chinese mathematics. Japan also had a good education system so that the literacy rate was quite high. Each domain had its own school for samurai but mainly education was provided privately. Private schools for elementary education were called terakoya, in which mainly reading and writing and often arithmetic by the soroban (Japanese abacus) were taught. In the Edo period the soroban (abacus) was the only tool for computation and Arabic numerals were not used. The Meiji government was eager to establish a modern centralized state in which education played a key role. In 1872 the Ministry of Education declared the Education Order, whereby in elementary schools only western mathematics should be taught and the soroban should not be used. But almost all teachers only knew Japanese traditional mathematics ??wasan?? so they insisted on using the soroban. This was the starting point of a long dispute on the soroban in elementary education in Japan. Two Japanese mathematicians, KIKUCHI Dairoku and FUJISAWA Rikitaro, played a central role in the modernization of mathematical education in Japan. KIKUCHI studied mathematics in England and brought back English synthetic geometry to Japan. FUJISAWA was a student of KIKUCHI at the Imperial University and studied mathematics in Germany. He was the first Japanese mathematician to make a contribution to original research in the modern sense. He published a book on mathematical education in elementary school, which built the foundation of mathematical education in Japan.