Showing papers in "Mathematics Education Research Journal in 2019"
TL;DR: In this article, a conceptual framework for teacher expertise in language-responsive mathematics teaching is proposed, starting from typical situational demands that teachers face in language responsive mathematics teaching and the orientations, categories, and pedagogical tools they need to cope with these situational demands, especially the demand to identify mathematically relevant language demands.
Abstract: In spite of the widely accepted need for language-responsive subject-matter teaching, few teachers are prepared for this challenge due to the lack of empirically founded subject-specific professional development (PD) programs for language-responsive classrooms. The design research study presented in this article pursues the dual aim of (a) promoting teachers’ expertise in language-responsive mathematics teaching using PD courses and (b) investigating teachers’ developing expertise in qualitative case studies. Both aims are pursued based on a conceptual framework for teacher expertise in language-responsive mathematics teaching, starting from typical situational demands that teachers face in language-responsive mathematics teaching and the orientations, categories, and pedagogical tools they need to cope with these situational demands, especially the demand to identify mathematically relevant language demands. For language-responsive teaching, the interplay of categories for mathematical goals and language goals turns out to be of high relevance.
42 citations
TL;DR: The Teacher Moves for Supporting Student Reasoning (TMSSR) framework as mentioned in this paper ) is a framework of teacher moves specific to inquiry-oriented instruction, which is based on the analysis of four instructors' implementations of a middle grades (ages 12-14) research-based unit on ratio and linear functions.
Abstract: Teachers play a critical role in supporting students’ mathematical engagement. There is evidence that meaningful student engagement occurs more often in student-centered classrooms, in which the teacher and the students mutually share mathematical authority. However, teacher-centered instruction continues to dominate classroom discourse, and teachers struggle to effectively support student inquiry. This paper presents a framework of teacher moves specific to inquiry-oriented instruction, the Teacher Moves for Supporting Student Reasoning (TMSSR) framework. Based on the analysis of four instructors’ implementations of a middle grades (ages 12–14) research-based unit on ratio and linear functions, the TMSSR framework organizes pedagogical moves into four categories, eliciting, responding, facilitating, and extending, and then places individual moves within each category on a continuum according to their potential for supporting student reasoning. In this manner, the TMSSR framework characterizes how multiple teacher moves can work together to foster an inquiry-oriented environment. We detail the framework with data examples and then present a classroom episode exemplifying the framework’s operation.
34 citations
TL;DR: In this article, the authors examine patterns of participation and achievement in senior secondary school mathematics in Victoria, Australia and reveal that correlations between mathematics performance and socioeconomic status are far weaker in the non-metropolitan school population than the metropolitan school population, suggesting that a nonmetropolitan location has a moderating effect on the impact of SES.
Abstract: In many countries, there is pressure for schools to increase student engagement and skills in mathematics, in particular for disadvantaged students. This is certainly true in Australia. This study repurposes school level data to examine patterns of participation and achievement in senior secondary school mathematics in Victoria, Australia. It confirms that school socioeconomic status (SES) is strongly tied to participation and achievement in these subjects, and that nonmetropolitan schools tend to perform more poorly than metropolitan schools in these areas. It shows that nonmetropolitan schools are less likely to offer advanced mathematics subjects than metropolitan schools, and where they do, their students are less likely to choose those options. This study also reveals that correlations between mathematics performance and SES are far weaker in the nonmetropolitan school population than the metropolitan school population. This suggests that a nonmetropolitan location has a moderating effect on the impact of SES, pointing the way for potentially fruitful lines of future inquiry.
20 citations
TL;DR: In this article, a case study of a teacher who stated "making things explicit" as an ostensible goal of his instructional practice, how the explicitation is realised in teaching mathematics was examined.
Abstract: The phrase ‘make it explicit’ is a common advice given to teachers. It is, however, not clear to us what this actually means when translated into classroom practice. Our review found that we are not alone: “explicit” is used in different ways in the education literature. This paper explores, through a case study of a teacher who stated “making things explicit” as an ostensible goal of his instructional practice, how the explicitation is realised in teaching mathematics. In particular, we examine how he used the instructional materials that he crafted to fulfil his goal of explicitation. We were able to uncover three strategies he used: explicit-from, explicit-within, and explicit-to.
13 citations
TL;DR: This paper investigated preservice teachers' own personal approaches to solving fraction division problems and compared their ability to interpret student work on the same task with the ability of students to interpret their own work.
Abstract: The mathematical domain of fraction division continues to be an area of great difficulty for many teachers and preservice teachers (Lo and Luo, Journal for Mathematics Teacher Education 15:481–500, 2012; Newton, American Educational Research Journal 45:1080–1110, 2008; Rizvi and Lawson, International Education Journal 8(2):377–392, 2007; Young and Zientek, Investigations in Mathematics Learning 4(1):1-23, 2011). In this study, preservice teachers were given a fraction division task that sought to investigate their own personal approaches to solving a fraction division problem. Their results were contrasted with their ability to interpret student work on the same task. The purpose was to uncover information and gain greater insight into conceptualizations of fraction division they used and links or connections they made among verbal, diagrammatic, and algebraic representations as they solved the task themselves and then analyzed sample student solutions to the same task. Our findings have implications for future research and instruction on fraction division.
13 citations
TL;DR: In this paper, the critical factors of enacting the structured problem-solving lessons used in Japanese Lesson Study to elicit and develop primary students' capacity to generalise are explored.
Abstract: Structured problem-solving lessons are used to explore mathematical concepts such as pattern and relationships in early algebra, and regularly used in Japanese Lesson Study research lessons. However, enactment of structured problem-solving lessons which involves detailed planning, anticipation of student solutions and orchestration of whole-class discussion of solutions is an ongoing challenge for many teachers. Moreover, primary teachers have limited experience in teaching early algebra or mathematical reasoning actions such as generalising. In this study, the critical factors of enacting the structured problem-solving lessons used in Japanese Lesson Study to elicit and develop primary students’ capacity to generalise are explored. Teachers from three primary schools participated in two Japanese Lesson Study teams for this study. The lesson plans and video recordings of teaching and post-lesson discussion of the two research lessons along with students’ responses and learning are compared to identify critical factors. The anticipation of students’ reasoning together with preparation of supporting and challenging prompts was critical for scaffolding students’ capacity to grasp and communicate generality.
12 citations
TL;DR: For example, this paper found that for a given level of performance in Mathematics in Year 10, girls see greater improvement than boys in Year 12 for all levels of Mathematics except the most advanced course.
Abstract: Students in many jurisdictions can study Mathematics at different levels in their final 2 years of secondary education. The levels of Mathematics range from standard (not involving calculus), through basic calculus, to more advanced treatments of calculus and algebra. In this context, some students can elect to study Mathematics at a level below their ability. We consider the situation in New South Wales (NSW), Australia, where most Year 12 students who apply to university are awarded a percentile ranking, namely the Australian Tertiary Admission Rank (ATAR). The ATAR reflects students’ results in the final 2 years of secondary education and frequently determines what they can study at university. As the study of Mathematics is often segregated by gender, it is of interest to explore how boys’ and girls’ choices about level of Mathematics study affect their ATAR. We analyze administrative data for 46,000 senior secondary students in NSW who completed their Year 12 in 2011 and the Longitudinal Survey of Australian Youth (LSAY) for the same cohort. Using two-level regressions that control for relevant student and school characteristics, we find that, for a given level of performance in Mathematics in Year 10, girls see greater improvement than boys in Year 12 for all levels of Mathematics except the most advanced course. Girls who study basic Mathematics achieve ATAR increments as high as girls in some advanced courses. We discuss how awareness of these results may influence students’ decisions on what level of Mathematics to study in Years 11 and 12.
10 citations
TL;DR: In this paper, an epistemological theory in mathematics education is presented that offers the opportunity to trace students' conceptual development in both its individual and social facets through analyzing patterns of reasoning.
Abstract: The purpose of this article is to show how the philosophical theory of inferentialism can be used to understand students’ conceptual development in the field of mathematics. Based on the works of philosophers such as Robert Brandom, an epistemological theory in mathematics education is presented that offers the opportunity to trace students’ conceptual development in both its individual and social facets through analyzing patterns of reasoning. A design experiment on decimal numbers serves as a paradigmatic example. The overall goal is to illustrate the relationship between mathematical standard and individual ways of reasoning in conceptual development processes.
10 citations
TL;DR: In this article, the authors investigated the gender, technology, engineering, and mathematics (STEM) underrepresentation of females in STEM and found that one in four girls in NSW currently undertakes no mathematics in Year 12.
Abstract: With science, technology, engineering, and mathematics (STEM) heralded as pivotal to Australia’s future prosperity, declining participation in Year 12 mathematics and science has attracted nationwide concern. While the national statistics certainly provide clear evidence of declining enrolments and the underrepresentation of females in STEM, we wondered if possible jurisdictional differences had been overlooked. To investigate this issue, we compiled Year 12 enrolment data from 1991 to 2017 in New South Wales, Australia’s most populous state. Three complementary analyses were conducted: (1) changes in the Year 12 student cohort, (2) male and female rates of participation in Year 12 STEM courses, and (3) differences between STEM and non-STEM course enrolments. These analyses confirm declining enrolments in digital technologies and mathematics, especially for girls. In contrast, enrolments in almost all NSW science courses have been increasing since 2001, at a rate faster than many non-STEM courses. Declining enrolments in advanced mathematics were less substantial than nationally, and participation in intermediate level mathematics increased in 2017 for the first time since 1991. Despite these promising signs, our analysis also shows that students are selecting less challenging courses, while one in four girls in NSW currently undertakes no mathematics in Year 12. These results indicate the need for continued policy work on gender, mathematics, and digital technologies if key STEM targets are to be met. We argue that understanding key differences between state jurisdictions may be critical to developing interventions with greater impact.
10 citations
TL;DR: In this article, the authors analyse interview and observation data of four grade 3 teachers, with the view to identifying the mechanisms conditioning the expression of teachers' identities, and point out the need for teacher development that creates enablers that lead to changes in classroom practices that align with policy-designated, progressive roles in teaching mathematics.
Abstract: This article addresses the question: Why teachers of mathematics have yet to ‘take up’ progressive roles? Drawing on the philosophy of critical realism and its methodological equivalent, social realism, we analyse interview and observation data of four grade 3 teachers, with the view to identifying the mechanisms conditioning the expression of teachers’ identities. In so doing, we show how post-apartheid changes in systemic roles of teachers create contradictory tensions for teachers as these bring their own mathematical learning and teaching experiences into contradiction with the new post-apartheid roles they are mandated to enact. We examine how this contradiction, together with beliefs about mathematics, pedagogy and learners, is expressed in the teaching of grade 3 mathematics. We maintain that the complementarity between teachers’ beliefs and old systemic roles provides an explanation for why teachers of grade 3 mathematics have yet to ‘take-up’ progressive roles. The implications point to the need for teacher development that creates enablers that lead to changes in classroom practices that align with policy-designated, progressive roles in teaching mathematics.
10 citations
TL;DR: In this article, a survey of secondary mathematics teachers in Australia, the USA and Korea was conducted to investigate how they view proof and the teaching of proof in the context of secondary education.
Abstract: Despite the recognised importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia, the USA and Korea perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including purposes, characteristics, reasons for teaching and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students.
TL;DR: In this paper, a replication study of prospective elementary teachers' conceptions of multidigit number and examined the main claim that most elementary pre-service teachers think about digits incorrectly at least some of the time.
Abstract: Replication studies play a critical role in scientific accumulation of knowledge, yet replication studies in mathematics education are rare. In this study, the authors replicated Thanheiser’s (Educational Studies in Mathematics 75:241–251, 2010) study of prospective elementary teachers’ conceptions of multidigit number and examined the main claim that most elementary pre-service teachers think about digits incorrectly at least some of the time. Results indicated no statistically significant difference in the distribution of conceptions between the original and replication samples and, moreover, no statistically significant differences in the distribution of sub-conceptions among prospective teachers with the most common conception. These results suggest confidence is warranted both in the generality of the main claim and in the utility of the conceptions framework for describing prospective elementary teachers’ conceptions of multidigit number. The report further contributes a framework for replication of mathematics education research adapted from the field of psychology.
TL;DR: In this article, the authors compared the math contents of Indonesian and Singaporean curricula based on math contents tested in PISA 2012 items and enrich the comparison with the examination of the role of problem solving in the intended learning processes of the two curricula.
Abstract: The results of international comparative studies like PISA and TIMSS attract researchers to conduct studies comparing math contents of curricula directly to each other. Unlike those studies, we compared math contents of Indonesian and Singaporean curricula based on math contents tested in PISA 2012 items. We also enrich the comparison with the examination of the role of problem solving in the intended learning processes of the two curricula. The results of the comparison of math contents show that there are differences in the breadth of the math contents of the two curricula. The results also indicate that the Singaporean curriculum covers math contents tested in the PISA 2012 items which are missing in the Indonesian curriculum. However, it is still difficult to identify possible reasons for the distinct performances between Indonesia and Singapore in the PISA 2012 study from the comparison. Enriching the comparison with the examination of the role of problem solving in the intended learning processes of the two curricula gives us additional insight into a possible reason. The results show that the intended learning process of the Singaporean curriculum highlights problem solving much more than does that of the Indonesian curriculum. These results are suggestive of the need for further research.
TL;DR: This paper explored the extent to which this challenging yet accessible financial context might stimulate students' mathematical exploration of multiplicative thinking and proportional reasoning and found that the majority of the students had some intuitive understanding of how to solve a financial problem that involved rate.
Abstract: What could two people stand to gain from sharing a taxi ride? We aimed to explore the extent to which this challenging yet accessible financial context might stimulate students’ mathematical exploration of multiplicative thinking and proportional reasoning. Through teaching experiment methodology, data were collected from 37 Year 5 and 6 students (10–12 years of age) in suburban Melbourne. The findings reveal that the majority of the students had some intuitive understanding of how to solve a financial problem that involved rate, and at least half of them used either multiplicative thinking or proportional reasoning. While the study reported is small and cannot claim to be representative, the findings confirm that well-designed financial problems have the potential to unveil sophisticated mathematical understandings among primary school students. This research demonstrates what young adolescents can do prior to formal exposure to ratio and proportion as part of the curriculum.
TL;DR: In this article, the authors report the findings of a study using tasks where a family orders and considers sharing the cost of a Fish n Chips meal, where literacy and numeracy must be applied to make sense of tabulated price information.
Abstract: This article reports the findings of a study using tasks where a family orders and considers sharing the cost of a Fish n Chips meal. Purchasing take-away food is an example of an everyday situation where literacy and numeracy must be applied to make sense of tabulated price information. Originally developed for use in Australia, the tasks were modified so that they might be challenging yet accessible to 10–12-year-old Pāsifika students in New Zealand. Working collaboratively with two teachers in two different schools, we aimed to explore the ways and means by which Pāsifika students drew on social and cultural norms and practices as they applied mathematics to make financial decisions and interacted with each other in justifying solutions that offered what they deemed to be value for money and a fair approach to paying the bill. The students’ and teachers’ reactions to the lessons revealed that as students worked to reconcile social, cultural and mathematical funds of knowledge, their teachers gained meaningful insights into the values about money being learned within families and communities.
TL;DR: A teaching unit is designed and implemented using modal clumps, medians, and hatplots to guide primary school students (grade 4, aged 10–11) to compare groups with TinkerPlots to investigate howPrimary school students compare groups in large authentic data sets using TinkerPlot.
Abstract: Comparing groups is a fundamental activity in statistics since it includes many basic concepts like center, variability, and representation and can pave the way to inferential statistics. Preliminary work involving group comparisons can be undertaken at an early age, e.g., at primary school using proto-concepts like modal clumps, precursor representations like hatplots, or formal concepts like medians. We have designed and implemented a teaching unit using modal clumps, medians, and hatplots to guide primary school students (grade 4, aged 10–11) to compare groups with TinkerPlots. Consequently, a case study with six of these primary school students was conducted to investigate how our primary school students compare groups in large authentic data sets using TinkerPlots. One finding is that all six primary school students make sophisticated use of modal clumps, medians, and hatplots to compare groups in large and authentic data with TinkerPlots.
TL;DR: In this article, the authors report on qualitative data collected over four years from two typical pre-service teachers whose developing mathematical content knowledge was investigated during their primary and secondary programme during their education, using the four dimensions of the knowledge quartet: foundation knowledge, transformation, connection and contingency.
Abstract: There is a consensus that we need to improve the quality of pre-service teacher education, and teachers’ mathematical content knowledge is critical for teaching. Identifying opportunities and influences that assist pre-service teachers to extend their mathematical content knowledge throughout their teacher education programme is important. This paper reports on qualitative data, collected over 4 years from two typical pre-service teachers whose developing mathematical content knowledge was investigated during their primary and secondary programme. These data were analysed and reported using the four dimensions of the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. The results highlight the consequences of programme structure in order to help pre-service teachers to establish and sustain a positive mathematics learner identity, build teacher identity and develop mathematical content knowledge.
TL;DR: In this article, the authors conducted one-to-one interviews with 100 primary grade children as they interacted with the Montessori Number Base-10 Blocks iPad app and found that children's actions when interacting with app features can affect their engagement with designed mathematics constraints and affordances.
Abstract: The purpose of the inquiry was to understand how children in primary grades (i.e., preschool, kindergarten, and second grade) engaged with affordances and constraints of features in a Base-10 Blocks virtual manipulative mathematics app designed to promote learning opportunities. Researchers conducted one-to-one interviews with 100 primary grade children as they interacted with the Montessori Number Base-10 Blocks iPad app. The video data were qualitatively analyzed using open descriptive, thematic, and structured coding. Results show that children’s actions when interacting with app features can affect their engagement with designed mathematics constraints and affordances. Results also identified three emergent themes around children’s engagement with the app’s simultaneous linking features: verification, self-correction, and making connections. Findings suggest the importance of helping children identify and reflect on affordances within virtual manipulative mathematics apps. These findings also indicate that as designers, educators and researchers design or select virtual manipulative mathematics apps for classroom use; they should to consider children’s prior achievement as well as in-app perceptions and engagement with design features in the apps.
TL;DR: In the context of reported declines in the proportion of students opting to study higher-level mathematics, a qualitative study sought insights into seven Australian mathematics teacher leaders' decision-making processes and actions in their particular school contexts as discussed by the authors.
Abstract: School leaders employ various school-based actions to influence students’ subject enrolments at senior secondary levels (Years 11 and 12), which in turn affect students’ entrance into tertiary courses and career choices. In the context of reported declines in the proportion of students opting to study higher-level mathematics, this qualitative study sought insights into seven Australian mathematics teacher leaders’ decision-making processes and actions in their particular school contexts. It aimed to relate their actions to particular attributions for enrolment declines and their goals for students’ learning and achievement. The leaders’ attributions included students’ lack of ability, changes in university courses’ pre-requisites, students’ lack of effort or persistence, and negative attitudes towards mathematics. The leaders described a variety of school-based actions; some school leaders had actually chosen opposing actions but expressed similar reasons for implementing them, and vice versa. Tensions among external pragmatic constraints, the actions of other school staff, and the teacher leaders’ own goals for student learning in mathematics framed the findings of this study.
TL;DR: In this article, the authors explored how problem solving is approached in two 6th grade mathematics textbooks and national elementary mathematics curriculum in Turkey and compared three approaches of "teaching for, about and through problem solving".
Abstract: This paper explores how problem solving is approached in two 6th grade mathematics textbooks and national elementary mathematics curriculum in Turkey. The curriculum and textbooks’ problem solving approaches are analysed and compared in the light of three approaches of ‘teaching for, about and through problem solving’. Four learning objectives from different content areas (e.g. algebra, data processing) from the curriculum are determined and their treatments in the textbooks are specifically analysed. The analysis showed that the curriculum did not have a prescriptive stance on any one of problem solving approaches. Similarly, the textbooks also did not follow any approach faithfully. ‘Teaching for problem solving’ approach was not employed while ‘teaching through problem solving’ approach was used for only one learning objective in the textbooks. ‘Teaching about problem solving’ approach was encountered in terms of only one problem solving heuristic (table drawing). We discuss the implications of these findings and note some further research areas.
TL;DR: In this paper, the authors investigated the psychometric characteristics of the Math Anxiety Scale (MANX) with data collected from 952 middle school students in Turkey and found that although the MANX was sensitive to detect students with moderate levels of math anxiety and it was not targeted to identify those with very high and low math anxiety levels, it had high reliability and validity.
Abstract: The purpose of this study was to investigate the psychometric characteristics of the Math Anxiety Scale (MANX; Erol 1989, Unpublished master thesis, Bogazici University) with data collected from 952 middle school students in Turkey. The Rasch Rating Scale model was used to examine the MANX at the item level. The results revealed that although the MANX was sensitive to detect students with moderate levels of math anxiety and it was not targeted to identify those with very high and low math anxiety levels, it had high reliability and validity. Moreover, the majority of the MANX items were of good quality. The results of this study provide strong evidence for the validation of the MANX despite the need for deletion of eight misfit items and three items with the same item difficulties. Future research should consider possible revision or development of new items to capture gradations of challenges at the very high and low ends of the continuum.
TL;DR: It is argued that pre-service teachers need to be aware of the connections between specific geometrical facts when they construct a proof by linking visualisation to formal reasoning.
Abstract: The study aimed at characterising the shift from configural reasoning to proof construction in geometry. One hundred eighty-two preservice primary teachers solved two geometry problems in which they had to generate a proof from the information provided by a geometrical configuration. Results indicated that proof construction was linked to the way pre-service teachers coordinated the different apprehensions, as identified by Duval (1995), mediated by the existence of strategic knowledge. Strategic knowledge is undertood as the ability to see some specific geometrical statements as premises of a geometrical proposition that can be used to deduce intermediate statements or the conclusion. We argue that pre-service teachers need to be aware of the connections between specific geometrical facts when they construct a proof by linking visualisation to formal reasoning. We conclude with implications for teacher education programmes.
TL;DR: The authors investigated preservice teachers' curricular noticing as they draw upon curriculum materials to design instruction, with a focus on how to launch that lesson, and found that the curriculum materials not mentioned in the curriculum to draw children into the lesson.
Abstract: Translating a mathematics lesson from curriculum materials into instruction is not straightforward, and launching, or beginning, a lesson such that students will become productively engaged in the target mathematics is nontrivial. The purpose of the study was to investigate preservice teachers’ curricular noticing as they draw upon curriculum materials to design instruction, with a focus on how to launch that lesson. We engaged preservice elementary licensure students at two universities in a four-part process of analyzing mathematics curriculum, planning a lesson, demonstrating their visualization of enactment through animating their lesson launch, and reflecting on the process. Findings indicate that the focal case study pair modified the curricular materials to model mathematical aspects of fractions, adapted the introduction of key academic vocabulary, and introduced materials not mentioned in the curriculum to draw children into the lesson. We discuss implications for preservice teachers’ planning of the lesson launch and their curricular noticing.
TL;DR: The authors argue that critique and skepticism may be valuable scholarly tools, but such practices should be balanced with openness and belief towards ideas from worldviews different than our own, which can provide new and creative interpretations and extend our collective research power.
Abstract: Differing research worldviews have typically resulted in interpretations at odds with one another. Yet, leveraging distinct perspectives can lead to novel interpretations and theoretical construction. Via an empirically grounded research commentary, we describe the value of such activity through the lens of previously reported findings. This synthesis of research from dissimilar scholarly traditions is one example of how paradigms in related but sometimes disconnected fields were used to provide a more comprehensive model of foundational numeracy development. While critique and skepticism may be valuable scholarly tools, we argue that such practices should be balanced with openness and belief towards ideas from worldviews different than our own. This balance can provide new and creative interpretations and extend our collective research power.