Primary teachers’ subject matter knowledge: decimals

The main objective of this study was to investigate primary teachers’ subject matter knowledge in the domain of decimals and more elaborately to investigate their performance and difficulties in reading scale, ordering numbers, finding the nearest decimal and doing operations, such as addition and subtraction. The difficulties in these particular areas are analysed and suggestions are made regarding their causes. Further, factors that influence this knowledge were explored. The sample of the study was 63 primary teachers. A decimal concepts test including 18 tasks was administered and the total scores for the 63 primary teachers ranged from 3 to 18 with a mean and median of 12. Fifty per cent of the teachers were above the mean score. The detailed investigation of the responses revealed that the primary teachers faced similar difficulties that students and pre-service teachers faced. Discrepancy on teachers’ knowledge revealed important differences based on educational level attained, but not the number of years of teaching experience and experience in teaching decimals. Some suggestions have been made regarding the implications for pre- and in-service teacher training.     

Why it is important for in-service elementary mathematics teachers to understand the equality .999… = 1

Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers’ reactions to arguments that .999… = 1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to .999…. This paper reports on one instance of a teacher whose responses indicate that the teacher's sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. .333… and .666…) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality .999… = 1.     

Concept Development of Decimals in Chinese Elementary Students: A Conceptual Change Approach

The aim of this study was to examine the concept development of decimal numbers in 244 Chinese elementary students in grades 4–6. Three grades of students differed in their intuitive sense of decimals and conceptual understanding of decimals, with more strategic approaches used by older students. Misconceptions regarding the density nature of decimals indicated the progress in an ascending spiral trend (i.e., fourth graders performed the worst; fifth graders performed the best; and sixth graders regressed slightly), not in a linear trend. Misconceptions regarding decimal computation (i.e., multiplication makes bigger) generally decreased across grades. However, children's misconceptions regarding the density and infinity features of decimals appeared to be more persistent than misconceptions regarding decimal computation. Some students in higher grades continued to use the discreteness feature of whole numbers to explain the distance between two decimal numbers, indicating an intermediate level of understanding decimals. The findings revealed the effect of symbolic representation of interval end points and students' responses were contingent on the actual representations of interval end points. Students in all three grades demonstrated narrowed application of decimal values (e.g., merchandise), and their application of decimals was largely limited by their learning experiences.     

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.     

Revisiting decimal misconceptions from a new perspective: The significance of whole number bias in the Chinese culture

The aim of this study was to investigate Hong Kong Grade 4 students’ understanding of the decimal notation system including their knowledge of decimal quantities. This is a unique study because most previous studies were conducted in Western cultural settings; therefore we were interested to see whether Chinese students have the same kinds of misconceptions as Western students given the Chinese number naming system is relatively transparent and explicit. Three hundred and forty-one students participated in a written test on decimal numbers. Thirty-two students were interviewed to further explore their mathematical reasoning. In summary, the results indicated that many students had mastered reasonable knowledge of decimal notation and quantities, which may be attributed to the Chinese linguistic clarity of decimal numbers. More importantly, the results showed that some students’ construction of decimal concepts have been adversely affected by persistent misconceptions arising from whole number bias. Two kinds of whole number misconceptions, namely “-ths suffix error” and “reversed place value progression error”, were revealed in this study. This paper suggests that a framework theory approach to conceptual change may be an alternative approach to addressing students’ learning difficulties in decimals.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

Pre-service mathematics teachers’ concept images of radian

This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.     

An exploration of the role natural language and idiosyncratic representations in teaching how to convert among fractions, decimals, and percents

Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N = 581).In addition to using geometric figures and manipulatives, teachers used natural language such as the words nanny and house to characterize mathematical procedures or algorithms. Some teachers used the words or phrases bigger, smaller, doubling, and building-up in the context of equivalent fractions. There was widespread use of idiosyncratic representations by teachers and students, specifically equations with missing equals signs and not multiply/dividing by one to find equivalent fractions. No evidence though of a relationship between representational forms and degree of correctness of solutions was found on student work. However, when students exhibited misconceptions, those misconceptions were linked to teachers’ use of idiosyncratic representations.     

Characteristics of Pre-Service Primary School Teachers’ Configural Reasoning

The goal of this study is to identify the characteristics of pre-service primary teachers’ configural reasoning, understood as the relationships between concepts and figures set to solve geometrical proof problems. Ninety-seven primary teachers were asked to solve two geometrical proof problems in which a geometrical figure was provided. The results suggest the existence of two levels of the pre-service primary teachers’ discursive organization. The first, when pre-service primary teachers link the geometrical facts to the figure by discursive apprehensions, and the second, when several geometrical facts are related by logical chains to infer new information. The identification of a relevant configuration and the way in which geometrical facts are logically organized from discursive apprehensions are key factors in the shift between these moments. We contend that these factors help to explain how the image component stimulates thought and how conceptual constraints control the formal rigor of the process.     

Using dynamic software in mathematics: the case of reflection symmetry

This study was carried out to examine the effects of computer-assisted instruction (CAI) using dynamic software on the achievement of students in mathematics in the topic of reflection symmetry. The study also aimed to ascertain the pre-service mathematics teachers’ opinions on the use of CAI in mathematics lessons. In the study, a mixed research method was used. The study group of this research consists of 30 pre-service mathematics teachers. The data collection tools used include a reflection knowledge test, a survey and observations. Based on the analysis of the data obtained from the study, the use of CAI had a positive effect on achievement in the topic of reflection symmetry of the pre-service mathematics teachers. The pre-service mathematics teachers were found to largely consider that a mathematics education which is carried out utilizing CAI will be more beneficial in terms of ‘visualization’, ‘saving of time’ and ‘increasing interest/attention in the lesson’. In addition, it was found that the vast majority of them considered using computers in their teaching on the condition that the learning environment in which they would be operating has the appropriate technological equipment.     

Secondary teachers’ meanings for function notation in the United States and South Korea

This study investigates what teachers in U.S. reveal about their meanings for function notation in their written responses to the Mathematical Meanings for Teaching secondary mathematics (MMTsm) items, with particular attention to how productive those meanings would be if conveyed to students in a classroom setting. We then report South Korean teachers’ responses to see whether the meanings U.S. teachers demonstrated are shared with South Korean teachers. The results show that many U.S. teachers use function notation to name rules instead of to represent relationships. The data from South Korean teachers indicates that the problematic meanings in U.S. teachers’ responses are shared with a minority of South Korean teachers. The results suggest a need for attention to ideas regarding function notation in teacher education for pre-service teachers and professional development programs for in-service teachers.     

A coloured window on pre-service teachers’ conceptions of rational numbers

In undergraduate mathematics courses, pre-service elementary school teachers are often faced with the task of re-learning some of the concepts they themselves struggled with in their own schooling. This often involves different cognitive processes and psychological issues than initial learning: pre-service teachers have had many more opportunities to construct understandings and representations than initial learners, some of which may be more complex and engrained; pre-service teachers are likely to have created deeply-held–and often negative–beliefs and attitudes toward certain mathematical ideas and processes. In our recent research, we found that pre-service teachers who used a particular computer-based microworld, one emphasising visual representations of and experimental interactions with elementary number theory concepts, overcame many cognitive and psychological difficulties reported in the literature. In this study, we investigate the possibilities of using a similarly-designed microworld that involves a set of rational number concepts. We describe the affordances of this microworld, both in terms of pre-service teacher learning and research on pre-service teacher learning, namely, the helpful “window” it gave us on the mathematical meaning-making of pre-service teachers. We also show how their interactions with this microworld provided many with a new and aesthetically-rich set of visualisations and experiences.     

How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

Mathematical paradoxes as pathways into beliefs and polymathy: an experimental inquiry

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.     

The use of cross multiplication and other mal–rules in fraction operations by pre-service teachers

Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.     

Revisiting the hybrid real number system

The hybrid real number system consisting of terminating and nonterminating decimals, dark numbers, dual dark numbers, involving the notions of personal infinities and the impersonal infinity has been discussed. Some algebraic properties of dark numbers and dual dark numbers are discussed.     

Using narrative inquiry for investigating the becoming of a mathematics teacher

This article presents narrative inquiry as a method for research in mathematics education, in particular the study of how pre-service teachers’ views of mathematics develop during elementary teacher education. I describe two different, complementary approaches to applying narrative analysis, one focusing on the content of a narrative, the other focusing on the form. The examples discussed are taken from interviews with and teaching portfolios compiled by four pre-service teachers. In analysing the content of the students’ narratives, I use emplotment to construct a retrospective explanation of how one pre-service teacher’s own experiences at school were reflected in the development of her mathematical identity. In analysing the form of the narratives, I also look at how the students told their stories, using linguistic features, for example, to identify core events in the accounts. This particular focus seems to be promising in locating turning points in the trainees’ views of mathematics.     

Measurement model for division as a tool in computing applications

The paper describes the use of a spreadsheet in a mathematics teacher education course. It shows how the tool can serve as a link between seemingly disconnected mathematical concepts. The didactical triad of using a spreadsheet as an agent, consumer, and amplifier of mathematical activities allows for an extended investigation of simple yet intriguing properties of whole numbers. The authors argue that revisiting elementary content in a technological context enables pre-service teachers to appreciate the role that conceptual knowledge can play in the development of a spreadsheet-enabled pedagogy.