Cognitive Demands and Second-Language Learners: A Framework for Analyzing Mathematics Instructional Contexts

The issues involved in teaching English language learners mathematics while they are learning English pose many challenges for mathematics teachers and highlight the need to focus on language-processing issues related to teaching mathematical content. Two realistic-type problems from high-stakes tests are used to illustrate the complex interactions between culture, language, and mathematical learning. The analyses focus on aspects of the problems that potentially increase cognitive demands for second-language learners. An analytical framework is presented that is designed to enable mathematics teachers to identify critical elements in problems and the learning environment that contribute to increased cognitive demands for students of English as a second language. The framework is proposed as a cycle of teacher reflection that would extend a constructivist model of teaching to include broader linguistic, cultural, and cognitive processing issues of mathematics teaching, as well as enable teachers to develop more accurate mental models of student learning.     

Preservice teachers’ mathematics task modification for emergent bilinguals

Implementing mathematically challenging tasks is difficult for teachers when working with emergent bilinguals because cognitively demanding tasks in mathematics commonly have high language demand. Currently, inadequate teacher preparation for teaching emergent bilinguals is becoming a significant concern in the United States as this population of students is rapidly growing. This study investigated how two mathematics preservice teachers (PSTs) support middle school emergent bilinguals to understand cognitively demanding mathematical problems through task modification. Fieldwork with a concurrent intervention was designed for the PSTs to work with emergent bilinguals in a one‐on‐one setting. The PSTs modified cognitively demanding mathematics tasks and designed a lesson for the emergent bilinguals based on the modified tasks. The results revealed that the task modification made by the PSTs tended to shift from reducing cognitive demands in mathematics and language to maintaining the demands through learning strategies of contextual support.     

A study of early career teachers’ practices related to language and language diversity during mathematics instruction

The role of language in mathematics teaching and learning is increasingly highlighted by standards and reform movements in the US. However, little is known about teachers’, and especially early career teachers’ (ECTs) practices and understandings related to language in mathematics instruction. This multiple case study explored the language-related understandings and practices of six ECTs in diverse elementary classrooms. Using iterative cycles of analysis, we found that all ECTs regularly attended to students’ mathematical vocabulary use and development. Yet, there was variability in ECTs’ focus on how to teach mathematical vocabulary, expectations for students’ precise use of mathematical terminology, and the use of multiple languages during instruction. These findings indicate that ECTs need more targeted support during teacher preparation and early career teaching in order to better support all students’ language development in the mathematics classroom.     

FROM ONE LANGUAGE TO ANOTHER: A SEMIOTIC INTERPRETATION OF THE TRANSLATION OF MATHEMATICAL WORDS

The teaching and learning of Primary school mathematics in Malta involves the use of code-switching between the local language Maltese, and English Mathematical terms themselves are usually retained in English and teachers may use various strategies to share the meaning of these words with their pupils. One strategy that may be used in a bilingual situation is translation from one language to another. In this paper I explore how a teacher used this strategy to teach her 7 to 8-year-old pupils mathematical vocabulary related to the topic'Money and Shopping'. While Maltese equivalents for these words exist, it is the English versions that form part of the school mathematics register. I develop a semiotic model where a mathematical word is considered to be a sign, and the process of translation is viewed as a chain of signification from one language to another.     

Word problems in mathematics education: a survey

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

     

Exploring the quality of the mathematical tasks in the new Turkish elementary school mathematics curriculum guidebook: the case of algebra

To take its due place in the world of education, Turkey has been through serious reform initiatives in the curriculums of various school subjects since 2003. The new Turkish elementary school curriculum was prepared considering the research studies conducted in Turkey and in other countries, as well as the educational systems of developed countries and previous experiences with mathematics education in Turkey. This study attempts to provide a perspective on the nature of the instructional tasks in the new elementary school mathematics curriculum. In particular, our focus is to explore the level of cognitive demands (LCD) in the algebra tasks provided in the national elementary mathematics curriculum guidebook. This curriculum document is a major resource for administrators, stakeholders, textbook publishers and ultimately for teachers. For every learning objective, it provides sample tasks to be used in mathematics instructions. In this study, our purpose is to explore the LCD of each of these tasks by utilizing a framework developed by Smith and Stein (Math Teach Middle School 3:344–350, 1998). The framework classifies mathematical tasks according to the level of demands: lower-level and higher-level demands. While the lower-level demands are related to memorization and procedures without connections, the higher-level demands are related to procedures with connections and doing mathematics. The findings revealed that 60% of algebra tasks for each grade level required higher LCD and a great majority of the remaining tasks were at the level of procedures without connections. The findings of the study particularly inform curriculum developers about issues regarding the quality of the tasks given in the curriculum guide and provide possible suggestions to improve the implementation of the curriculum change process.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic "minority" parents.     

Language and communication in mathematics education: an overview of research in the field

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognised, but there is not agreement about what this role (or these roles) might be or even about what the term ‘language’ itself encompasses. In this issue of ZDM, we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this survey paper, we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We identify four broad areas of concern in mathematics education that are addressed by language-oriented research: analysis of the development of students’ mathematical knowledge; understanding the shaping of mathematical activity; understanding processes of teaching and learning in relation to other social interactions; and multilingual contexts. A further area of concern that has not yet received substantial attention within mathematics education research is the development of the linguistic competencies and knowledge required for participation in mathematical practices. We also discuss methodological issues raised by the dominance of English within the international research community and suggest some implications for researchers, editors and publishers.     

Teachers learning the registers of mathematics and mathematics education in another language: an exploratory study

Many mathematics teachers around the world teach in a language different from the one in which they studied or completed their teacher education. Often these teachers must learn both the registers of mathematics and of mathematics education to teach in the additional language. This paper examines the factors that help teachers to learn these registers in Māori, the Indigenous language of New Zealand. Many of these teachers are second-language learners of the Māori language and attended English-medium schools and teacher-education programmes. After a brief discussion about the key role of language in teaching mathematics, this paper examines data from teachers at two Māori-immersion schools and a professional development facilitator. The analysis provides initial understanding of the factors that support or hinder their learning of the mathematics registers. Finally, a research agenda is suggested for further investigation of this issue.     

The stability of learners’ choices for real-life situations to be used in mathematics

One of the efforts to improve and enhance the performance and achievement in mathematics of learners is the incorporation of life-related contexts in mathematics teaching and assessments. These contexts are normally, with good reasons, decided upon by curriculum makers, textbook authors, teachers and constructors of examinations and tests. However, little or no consideration is given to whether students prefer and find these real-life situations interesting. There is also a dearth of studies dealing explicitly with the real-life situations learners prefer to deal with in mathematics. This issue was investigated and data on students’ choices for contextual issues to be used in mathematics were collected at two time periods. The results indicate that learners’ preferences for contextual situations to be used in mathematics remained fairly stable. It is concluded that real-life issues that learners highly prefer are not normally included in the school mathematics curriculum and that there is a need for a multidisciplinary approach to develop mathematical activities which take into account the expressed preferences of learners.     

Exploring Images of Parental Participation in Mathematics Education: Challenges and Possibilities

In this article, we draw on research within a large project on parental involvement in mathematics education in working-class Latino communities. Our research is situated within a sociocultural framework and, in particular, the concept of funds of knowledge. We also draw on research on parental involvement in education, particularly that which critically examines issues of power and perceptions of parents. We build on the concept of dialogic learning and on the characterization of parents as intellectual resources and present a model for parental involvement in mathematics in which parents engage as (a) parents, (b) learners, (c) facilitators, and (d) leaders. In particular, in this article, we focus on the third component—parents as facilitators of mathematics workshops for the community at large—centering on some of the challenges as parents and teachers engage in this type of collaboration. We also look at the possibilities afforded by a model for parental involvement that views parents as intellectual resources. By looking at examples of interactions among parents and teachers, and among parents and children, in mathematics workshops, we challenge conventional notions about parental involvement—in particular, as they apply to working-class, language/ethnic “minority” parents.     

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.     

Mathematics university teachers' perception of pedagogical content knowledge (PCK)

Teaching mathematics in university levels is one of the most important fields of research in the area of mathematics education. Nevertheless, there is little information about teaching knowledge of mathematics university teachers. Pedagogical content knowledge (PCK) provides a suitable framework to study knowledge of teachers. The purpose of this paper is to make explicit the perception of mathematics university teachers about PCK. For this purpose, a phenomenological study was done. Data resources included semi-structured interviews with 10 mathematics university teachers who were in different places of the mathematics university teaching experience spectrum. Data analysis indicated a model consisting of four cognitive themes which are mathematics syntactic knowledge, knowledge about mathematics curriculum planning, knowledge about students' mathematics learning and knowledge about creating an influential mathematics teaching–learning environment. Besides, it was found out that three contextual themes influenced on PCK for teaching mathematics in university levels which were the nature of mathematics subjects, university teachers' features and terms of learning environment.     

United States teachers’ views of effective mathematics teaching and learning

This study investigates US teachers’ cultural beliefs concerning effective mathematics teaching using semi-structured interviews with 11 experienced teachers. For US teachers, effective teaching is student-centered. Cognitively appropriate mathematical content should be understood through many hands-on activities that allow students to explore by themselves the relationship between mathematical knowledge and their life experiences. Correspondingly, the US teachers view an effective teacher as a facilitator who is sensitive to student social and cognitive needs and is skillful at organizing collaborative learning. The result of this study helps researchers and educators understand the student-centered learning model in US classrooms.     

Digital technologies and the challenge of constructing an inclusive school mathematics

This article addresses research related to the use of digital technologies in the teaching and learning of mathematics in Brazil. Its scope is limited to the context of school mathematics and, more specifically, to an ongoing research programme which involves the development of collaborative research partnerships with teachers of mathematics. The paper begins with a brief presentation of the introduction of computers into the Brazilian educational scenario in the 1980s, highlighting how computer technology was heralded as a key to permitting new pedagogical approaches appropriate to the constructivist philosophy of that time. It goes on to consider recent developments in the theoretical frameworks used to interpret mathematics learning in the presence of digital technologies and the importance of focusing on the learning system as a whole, considering epistemological, cognitive and pedagogic dimensions concomitantly. In this vein, it is argued that for any real integration to take place, the mathematical practices afforded by digital tools must be considered legitimate by all the actors in this process and, perhaps most notably, by teachers. The rest of the paper focuses on our approaches to involve teachers in making decisions about technology use in their own classrooms. The strategy used was based on the realisation of research activities underpinned by the idea of the collaborative design of learning situations and the goal of including the wide diversity of learners that characterises Brazilian mathematics classrooms.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Solution representations of percentage change problems: the pre-service primary teachers’ mathematical thinking and reasoning

Solution representations can reveal how problem solvers communicate mathematical thinking and reasoning in problem-solving process. The present study examined the solution representations used by 20 pre-service teachers for the percentage change problems. The pre-service teachers were invited to solve a combination of simple and complex percentage change problems. The score for the majority of simple problems was 75% or above, but the score for the complex problems was below 75%. The highest percentage error occurred when the pre-service teachers encountered a percentage greater than 100% in the percentage change problems. Irrespective of their level of mathematics qualifications, the equation approach demonstrating two-step problem-solving process was the predominant strategy adopted by the pre-service teachers. The equation approach imposes low cognitive load and, therefore, is more accessible and efficient than the unitary approach. A few pre-service teachers used the unitary approach. The findings indicate that the pre-service teachers possessed relevant mathematical knowledge for percentage change problems. Furthermore, the inclusion of the equation approach in mathematics textbooks would provide an alternative perspective regarding the teaching and learning of percentage change problems.     

Integrating technology into mathematics teaching at the university level

The emergence of new computing technologies in the second half of the twentieth century brought about new potentials and promised the rapid transformation of the teaching and learning of mathematics. However, despite the vast investments in technology resources for schools and universities, the realities of schooling and the complexities of technology-equipped environments resulted in a much slower integration process than was predicted in the 1980s. Hence researchers, together with teachers and mathematicians, began examining and reflecting on various aspects of technology-assisted teaching and learning and on the causes of slow technology integration. Studies highlighted that as technology becomes increasingly available in schools, teachers’ beliefs and conceptions about technology use in teaching are key factors for understanding the slowness of technology integration. In this paper, I outline the shift of research focus from learning and technology environment-related issues to teachers’ beliefs and conceptions. In addition, I highlight that over the past two decades a considerable imbalance has developed in favour of school-level research against university-level research. However, several changes in universities, such as students declining mathematical preparedness and demands from other sciences and employers, necessitate closer attention to university-level research. Thus, I outline some results of my study that aimed to reflect on the paucity of research and examined the current extend of technology use, particularly Computer Algebra Systems (CAS) at universities, mathematicians’ views about the role of CAS in tertiary mathematics teaching, and the factors influencing technology integration. I argue that due to mathematicians’ extensive use of CAS in their research and teaching, documenting their teaching practices and carrying out research at this level would not only be beneficial at the university level but also contribute to our understanding of technology integration at all levels.     

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.