Showing papers in "Mathematics Education Research Journal in 2020"

An exploration of teacher and student perceptions of blended learning in four secondary mathematics classrooms

Abstract: The COVID-19 pandemic forced many teachers around the world to make a sudden switch from face-to-face to online teaching. This shift in practice has provided an opportunity to reconsider how technology use in mathematics education can be utilised to improve student engagement. In this study, we explore four case studies of Australian secondary mathematics classrooms conducted prior to the COVID-19 pandemic to examine how teachers are using blended learning approaches and how their students perceive these pedagogical practices. Findings across all four sites indicate that technology use expands student opportunities to engage with mathematics learning through the provision of multiple pathways and methods of access. Specifically, we find evidence supporting the use of blended classroom teaching strategies to provide differentiation and personalised learning approaches; visualisation and dynamic manipulation of mathematics concepts; and alternative methods for teacher-student feedback and communication. We argue that the student learning experience in mathematics can be enhanced through a variety of blended learning approaches by allowing for diverse points of access to learning opportunities which are more closely aligned to individual learning needs and free from the temporal constraints of the classroom.

Exploring an innovative approach to teaching mathematics through the use of challenging tasks: a New Zealand perspective

Abstract: This paper reports on a New Zealand iteration of the Encouraging Persistence, Maintaining Challenge (EPMC) project, which proposes that students learn mathematics best when they build connections between mathematical ideas for themselves. This iteration explores the actions, perceptions and learning of 12 primary teachers and their 281 students during the implementation of a set of challenging tasks related to geometric reasoning. The teachers launched the suggested tasks, ensuring that the challenge was maintained. The students explored these tasks with minimal input from the teacher, and learning was summarised and extended. The teachers were positive about the intervention. The challenging task approach enabled students’ thinking became visible and, at times, the teachers’ prior perceptions of their students’ ability were challenged. A highly significant difference between the students’ pre- and post-assessment scores was found. The students were supported to have autonomy in their learning and make mathematical connections themselves. The students became less reliant on their teachers’ help and were positive about their involvement in the project.

The mathematical progress of students with an intellectual disability in inclusive classrooms: results of a longitudinal study

Abstract: Little is known about the mathematical development of students with intellectual disabilities (ID) in inclusive classrooms. It is important to have a research-based understanding of the subject since inclusive education is becoming the norm in many countries, and an increasing number of students with an ID now attend mainstream schools. We investigated the learning gains of 38 students with ID from 31 grades 2 and 3 inclusive classrooms. Data on mathematics achievement were collected at the beginning and at the end of one school year. A cluster analysis revealed four homogeneous groups that differed significantly in their mathematical progress. Students in the same cluster improved in the same subskills. Prior knowledge is a significant predictor for progress and explains more variance than IQ. In addition, the acquisition of the quantity-number concept, especially the linkage of quantities and numbers, seems to be an important factor for mathematical development. These results show that mathematics instruction needs to be tailored to the specific knowledge profiles of students.

Is the key to better PISA math scores improving spatial skills

Abstract: The Program for International Student Assessment (PISA) testing is conducted every 3 years by the Organization for Economic Co-operation and Development (OECD; oecd.org ) with 15-year-olds across the globe. Students complete a battery of tests in science, reading, and mathematics and complete an attitudinal survey. The goal of the PISA testing is to determine, broadly, if children are prepared for successful adulthoods. This means that items on the test are application- and not theoretically based. The mathematics portion of the PISA test includes problems in four content subtest areas: (1) change and relationships, (2) quantity, (3) uncertainty and data, and (4) space and shape. The subscale most closely related to spatial cognition is space and shape. Developers of the mathematics portion of the test indicate that student scores on subset areas 1 through 3 also depend on a child’s understanding of space and shape; thus, the space and shape construct is a key to overall performance on the test. The goal of this paper is twofold. First, it examines PISA data by subtest for several countries by analyzing trends between high-performing countries and other countries of interest with a particular focus on the subtest scores for space and shape. Second, the paper examines the relationship between scores on individual released items from PISA and scores on standard instruments designed to assess spatial skills for students from the USA. Findings indicate that success on PISA items is significantly correlated with scores on tests of spatial cognition. This finding means that improving spatial skills could be an overlooked strategy to improve student performance on large-scale tests of student achievement and has the potential to ultimately better prepare children for successful adulthoods.

The role of spatial, verbal, numerical, and general reasoning abilities in complex word problem solving for young female and male adults

Abstract: This study analyzed the relative importance of different cognitive abilities for solving complex mathematical word problems (CWPs)—a demanding task of high relevance for diverse fields and contexts. We investigated the effects of spatial, verbal, numerical, and general reasoning abilities as well as gender on CWP performance among N = 1282 first-year university engineering students. Generalized linear mixed models unveiled significant unique effects of spatial ability, β = 0.284, verbal ability, β = 0.342, numerical ability, β = 0.164, general reasoning, β = 0.248, and an overall gender effect in favor of male students, β = 0.285. Analyses revealed negligible to small gender effects in verbal and general reasoning ability. Despite a gender effect in spatial ability, d = 0.48, and numerical ability, d = 0.30—both in favor of male students—further analyses showed that effects of all measured cognitive abilities on CWP solving were comparable for both women and men. Our results underpin that CWP solving requires a broad facet of cognitive abilities besides mere mathematical competencies. Since gender differences in CWP solving were not fully explained by differences in the four measured cognitive abilities, gender-specific attitudes, beliefs, and emotions could be considered possible affective moderators of CWP performance.

Language as an including or excluding factor in mathematics teaching and learning

Abstract: This article explores the power of language to either include or exclude certain groups of students from genuine opportunities for mathematical sense-making. The substantial increase worldwide in the number of students learning mathematics through a language other than their primary language makes this a particularly urgent issue. This paper focuses on the South African situation, where, because English is widely perceived as the language of opportunity, it is, by grade 4, overwhelmingly the chosen language of learning and teaching. The epistemological and pedagogical consequences of this choice are evidenced in the poor performance of the country’s students on national and international assessments of mathematical proficiency. Drawing on research literature around language immersion education models and the extent to which these align with certain key principles of second language acquisition, this position paper motivates for a stronger and more sustained commitment to providing students, particularly those from marginalized and vulnerable communities, with opportunities for becoming both bilingual and biliterate. Empirical data from two South African grade 4 mathematics classrooms are used to illuminate aspects of the mathematical sense-making challenges students and their teachers face without such commitment.

Spatial reasoning skills about 2D representations of 3D geometrical shapes in grades 4 to 9

Abstract: Given the important role played by students’ spatial reasoning skills, in this paper we analyse how students use these skills to solve problems involving 2D representations of 3D geometrical shapes. Using data from in total 1357 grades 4 to 9 students, we examine how they visualise shapes in the given diagrams and make use of properties of shapes to reason. We found that using either spatial visualisation or property-based spatial analytic reasoning is not enough for the problems that required more than one step of reasoning, but also that these two skills have to be harmonised by domain-specific knowledge in order to overcome the perceptual appearance (or “look”) of the given diagram. We argue that more opportunities might be given to both primary and secondary school students in which they can exercise not only their spatial reasoning skills but also consolidate and use their existing domain-specific knowledge of geometry for productive reasoning in geometry.

Does disability matter in mathematics educational research? A critical comparison of research on students with and without disabilities

Abstract: Using a Disability Studies in Education framework, this systematic review analyzed international research published in English (2013–2017) on the teaching and learning of mathematics from the prekindergarten to 12th-grade level, comparing research on students identified as having disabilities to research on students without disabilities. Coding articles (N = 2477) for methodology, participants, mathematical domain, and theoretical orientation, we found that research on students with disabilities was overwhelmingly quantitative (81%) and tended to use behavioral and medical theoretical orientations. Research on students without disabilities was both qualitative (42%) and quantitative (42%) and tended to use constructivist and sociocultural theoretical orientations. In addition, research on mathematical learning that included students with disabilities lacked sustained qualitative inquiry documenting learning processes of students with disabilities and rarely included the teacher as an explicit focus. Following Gervasoni and Lindenskov (2011), we contend that these pronounced differences in research contribute to the segregation of students with disabilities and low-achieving students in lower quality mathematics instruction and may lead to low expectations of the mathematical competence of students with disabilities. We call for increased attention to research that considers how disability is produced and enacted in the complex context of mathematics classrooms.

In search of the mechanisms that enable transfer from spatial reasoning to mathematics understanding

Abstract: Spatial reasoning, the ability to mentally represent and transform objects and their relations, is considered so closely connected to mathematics that there is no longer a question of whether the two are related Instead, there remains debate about how to answer the question of why This paper explores the way the two fields of mathematics education and psychology define, assess and examine this important relationship We find that both fields rely extensively on psychometric tests to define spatial reasoning skills and do not characterise spatial reasoning in context We look at the points of connection and propose that the best way forward is to build on the complementary approaches currently undertaken We suggest better alignment between the spatial and mathematics reasoning skills that have been identified (theoretically and for assessment development), and a focus on how spatial reasoning interventions influence how students think and learn With the growing interest in spatial intervention for mathematics achievement, we analyse the different ways this has been undertaken and suggest directions for future research For example, how might laboratory-based studies be scaled up within a range of classroom contexts, and how might classroom-based studies offer more control settings and systematic variation?

Threats and opportunities in remote learning of mathematics: implication for the return to the classroom

Abstract: Australian schools, like schools elsewhere, have been through a period of closure The closure creates both threats and opportunities for teachers and students In the context of a project exploring approaches to teaching in early years, we outline some considerations and offer advice to teachers and educators on strategies for welcoming students back to school

Finding community and overcoming barriers: experiences of queer and transgender postsecondary students in mathematics and other STEM fields

Abstract: Although there is little research on the experiences of queer and/or transgender postsecondary students in science, technology, engineering, and mathematics (STEM) fields, we can infer from current literature that these subjects may be less welcoming than the humanities and social sciences. We conducted two studies to investigate this possibility: (1) a narrative inquiry study with postsecondary transgender students and (2) a grounded theory narrative study with undergraduate queer students. Transgender students who had transitioned indicated that they were subjected to lower expectations when presenting as female, but transgender women experienced this change as positive, since their treatment by others was no longer accompanied by gender dysphoria. Queer students experienced mathematics and other STEM fields as objective and independent of identity, yet simultaneously exclusionary of their queer identities. Many of the queer students in these studies found strength and resilience in queer communities, but there were some transgender women who did not view being queer as a central facet of their identity and did not feel the same sense of community. In general, those who were more gender-nonconforming felt a greater need for community with other queer people. We draw connections between gender category oppression and gender transgression oppression, and suggest strategies to make academic STEM fields more inclusive of queer and transgender students.

Evaluating the Impact of a Spatial Reasoning Mathematics Program (SRMP) Intervention in the Primary School.

Abstract: As part of the Connecting Mathematics Learning through Spatial Reasoning project, a Spatial Reasoning Mathematics Program (SRMP) intervention was implemented with one cohort of 30 students in grades 3 through 4. The SRMP embedded transformation skills in learning sequences comprising repeating and growing patterns, 2D and 3D relationships, structuring area and perimeter, directionality and perspective-taking. Analysis indicated a significantly better gain by the experimental group on the PASA-2 measure of awareness of pattern and structure and on the PASA-Sp assessment of spatial ability at the post-SRMP period. However, there were no significant differences found between groups on the PATMaths4 test of mathematics achievement. Qualitative analyses indicated that students demonstrated the development of complex spatial concepts well beyond curriculum expectations. The SRMP highlighted the important role of patterning and spatial structuring in the formation and representation of spatial concepts.

Supporting primary school teachers’ classroom assessment in mathematics education: effects on student achievement

Abstract: In a three-phase study, with a total of 40 third-grade teachers and their 830 students, teachers were supported to use classroom assessment techniques (CATs) to reveal their students’ knowledge of number operations. In phase I, four teachers and 66 third-grade students participated in five monthly workshops in which CATs were co-designed and their use was discussed. In phase II, the first phase was replicated with four workshops with six different teachers and 148 third-grade students. In these two exploratory phases, we evaluated student achievement on a standardized national mathematics test in a pre-/posttest design and compared changes herein to changes in the national norm sample. In phase III, a control condition was added to the design to experimentally investigate the effect on student achievement with 30 teachers and 616 third-grade students. Teachers were randomly assigned to participate in 0, 1, 2, or 3 1-hour workshops. In all three phases, we found a significant increase in students’ mathematics achievement scores on the standardized mathematics test. In phase III, the increase was significantly larger in the classes of teachers participating in three workshops than in classes with less workshops. Additionally, results from the analysis of classroom observations, feedback forms, and interviews indicate that teachers could easily integrate the CATs into their practice and could gather valuable information on their students. The results from the different phases of this study combined indicate that supporting teachers in their development and use of classroom assessment in mathematics may contribute to the improvement of students’ mathematics achievement.

Shifting towards Equity: Challenging Teacher Views about Student Capability in Mathematics.

Abstract: Persistent inequities in mathematics teaching and learning for specific groups of learners are a key challenge for researchers and educators. The use of ability grouping has been common practice in New Zealand mathematics classrooms. However, many researchers (e.g. Boaler and Wiliam 2001; Zevenbergen in 2003) highlight the negative effects of grouping and lack of opportunities for students placed in low-ability groups. The Developing Mathematical Inquiry Communities (DMIC) project is a whole-school professional development intervention designed to support a shift towards more inclusive and equitable pedagogy in the mathematics classroom. This article examines the changes in teacher beliefs about ability grouping in mathematics classrooms over time as teachers reconstruct their pedagogical practices as part of DMIC. Analysis of teacher and student interview data illustrates the ways in which the shifts in practice at the beginning of the re-invention contrasted with those at the end of the first year. We use an adaptation of Valsiner’s zone theory (Valsiner 1997) to explain how the changed classroom practices supported teachers to enact more equitable practices while also constructing a more expansive view of student capability.

Factors considered by Western Australian Year 10 students in choosing Year 11 mathematics courses

Abstract: In Australia, many Year 10 students choose not to enrol in Year 11 and 12 intermediate and advanced mathematics courses, despite having the ability to complete these courses successfully. Girls, in particular, remain significantly underrepresented in such courses. In this study, Year 10 students identified by their teachers as capable of succeeding in an advanced Year 11 mathematics course attended one of eight focus group sessions. In these sessions, they discussed the factors they considered in choosing their Year 11 mathematics courses. The data were analysed through a combination of directed and conventional content analysis, to capture both established importance factors in mathematics course choice, and potentially new or context-specific factors that arose within this group. Intrinsic value, utility value, generalized mathematics self-efficacy and sociocultural influences were cited most frequently as important factors in students’ decision-making. Findings not only affirmed the relevance of expectancy-value theory for describing the motivational factors that influence students’ mathematics course choices, but also underscored the need to consider the sociocultural contexts within which these choice decisions are made.

The construction and validation of a geometric reasoning test item to support the development of learning progression

Abstract: This article presents preliminary analysis of a test item in a large-scale study design to promote the development of geometric reasoning progression. Two sets of data were analysed to validate the item designed to assess secondary school students’ knowledge of a rectangle. The first data set involved 155 Year 4–10 students from seven trial schools across social strata. The second data set involved 585 Year 7–10 students from eleven project schools situated in lower socio-economic areas. The aim was to audit Australian students’ knowledge of hierarchy of shapes and document the process of validating a test item. The findings indicated that an iterative process of design, test and redesign, incorporating Sfard’s mathematical discourse framework and a multi-stage Rasch analysis, is vital in validating the results. A distinct change in students’ reasoning about rectangle is observed and this is not due to age. Moreover, Rasch analysis identified eight distinct thinking zones to assist in mapping out a learning progression for developing geometric reasoning.

The influence of spatial reasoning on analysing about measurement situations

Abstract: Measurement concepts such as volume and surface area provide rich contexts for real-world applications of number processes. Despite their importance, many students and prospective teachers show superficial understanding of measurement concepts. A lack of spatial reasoning and integration of geometric knowledge in problem solving situations may be the cause. This study seeks to determine the connection between spatial reasoning and discourse in influencing students’ mathematical reasoning in measurement situations. We analysed 118 year 8 to 10 students’ responses on two scenarios: (1) manipulation of 3D relations in 2D format and (2) explanation of changes in the volume and surface area of a shoe box after enlargement. The results showed that successful reasoners demonstrated a connected, integrated abstraction between numerical and geometric schemes, leading to success in reasoning about volume and surface area situations.

“Well, they understand the concept of area”: pre-service teachers’ responses to student area misconceptions

Abstract: The purpose of this study was to explore the ways elementary pre-service teachers responded to hypothetical student misconceptions about area measurement topics, framed in the context of their existing understanding and the Mathematical Knowledge for Teaching framework. Data collection consisted of written pre-assessments, followed by semi-structured interviews with 24 pre-service teachers enrolled in a geometry and measurement course. Findings included a frequent misattribution of area understanding to students, tendencies to provide alternate procedural strategies or re-explain concepts, and key differences in pedagogical strategies depending on initial content knowledge or apparent correctness of the student response. While there existed a tendency among some pre-service teachers to encourage procedural approaches, several others were able to leverage their own understanding towards conceptual, student-centered instructional responses. Such responses placed the student at the center of the instructional interaction, paving the way for exploration and mathematical discovery. Recommendations for supporting pre-service teachers in navigating the intersection between content and pedagogical knowledge in area measurement are discussed.

Assessing spatial reasoning during play: educator observations, assessment and curriculum planning

Abstract: Children are innately mathematical and explore mathematical concepts through play. However, educator beliefs about mathematics can impact the inclusion of mathematics in early childhood education (ECE). Recent research has suggested that spatial reasoning is a key concept which forms the foundations of mathematics learning. The theoretical argument underpinning this research is that young children benefit from intentional teaching specifically focused on supporting the development of children’s spatial reasoning skills during play. This mixed-methods research project investigated the effects of the implementation of a suite of play-based, spatial reasoning activities on educators’ teaching practices—including observations, assessment and evidence-based planning—and educator beliefs about mathematics in ECE. Twenty-seven participants were educators from 15 early childhood centres for children age 3–5 years, based within culturally and socio-economically diverse populations. Participant qualifications included diplomas, graduate and postgraduate degrees in early childhood and primary education, with teaching experience ranging from 6 months to 35 years. The investigations found reciprocal influences between the three key areas of the research project which included: the implementation of the activities by educators, educator beliefs about mathematics, spatial reasoning and mathematics teaching practice. The findings have implications for further research and curriculum design and practice. These include the need for research methodologies which contribute to sustained professional learning outcomes and the uptake of research findings in practice, play-based spatial reasoning assessment strategies and the contribution of a focus on spatial reasoning to early childhood curricula.

Towards a framework for spatial reasoning and primary mathematics learning: an analytical synthesis of intervention studies

Abstract: The connection between spatial reasoning and mathematics learning and pedagogy in primary school children has been the subject of an increasing number of studies in recent years. There has been no comprehensive analysis, however, of how studies based on spatial reasoning interventions may lead to improvements in students’ mathematics learning in school classroom environments. This article considers 18 studies selected from a combined systematic literature review of 133 studies, from Scopus and Education Research Complete (ERC) using PRISMA, and 23 studies recommended by the research team from bibliographies of major international research centres with a spatial reasoning dedication. This combination approach has allowed a synthesis of research and practice in an analytical way, assisting construction of a framework for spatial reasoning interventions for consideration in developing core knowledge and skills within the primary school mathematics curriculum. The findings highlight the importance of designing and evaluating spatial reasoning programs for primary school children in order to improve students’ mathematics classroom learning, including evidence from standardized tests, as they progress through the school system. The article supports the need for further research on interventions that provide sustainable school-based spatial reasoning programs.

Supporting indigenous primary students’ success in problem-solving: learning from Newman interviews

Abstract: This paper reports on part of a large study investigating how Indigenous students’ mathematical proficiency can be effectively supported in the primary years. Within this study, the problem-solving processes of thirty-six primary aged students in an Indigenous community school were evaluated. This was examined by ascertaining students’ problem-solving proficiency on a written test, conducting error analysis from written test scripts, followed by Newman interviews. The findings indicated that supporting students’ strategic competence and productive dispositions were critical in fostering students’ problem-solving success. When students’ strategic competence was supported through a scaffolded Newman interview, students’ problem-solving proficiency increased. Furthermore, the oral and personal nature of Newman interviews acted to significantly increase students’ productive dispositions towards the tasks. These findings have implications for teaching practice and provide tangible points of action that classroom teachers can implement to support Indigenous students’ success in problem-solving.

Word problems associated with the use of functional strategies among grade 4 students

Abstract: This article discusses the characteristics of word problems that are associated with students’ use of functional strategies and their ability to represent the generalization of functions. In the context of a broader research project designed to explore and foster functional thinking among elementary school students, twenty-five grade 4 (9- to 10-year-old) students were asked to identify functional relationships in five problems involving specific or indeterminate quantities. Their responses to a number of questions involving the generalization of the relationships in the problems were analyzed and associated to the characteristics of the problems. The type of representation of generalization used (verbal, generic, or symbolic) was also identified. Our findings indicate that grade 4 students showed potential for functional thinking prior to receiving instruction on variables and their notation. Such thinking was most effectively prompted when they worked with word problems that explicitly involved an additive function. When students generalized functional relationships, they represented them verbally or with generic examples. None of the students used symbolic representation. The originality of this study lies in the description of the specific characteristics of word problems that are associated with functional thinking; this information will prove useful to both teachers and curriculum designers. Identifying these characteristics could help build and propose tasks that encourage students to use more than one and more sophisticated strategies.

Inclusive practices in the teaching of mathematics: some findings from research including children with Down syndrome

Abstract: Children with Down syndrome can and do learn important mathematics and increasingly, this is occurring in regular school classrooms. The research literature offers little about the practices of effective primary school teachers who are including children with Down syndrome in the teaching of mathematics. In this paper we present findings from a project that followed teachers’ journeys through a school year as they taught mathematics in primary classrooms including a child with Down syndrome. In particular, we focus on one aspect of the project: the nature of the teachers’ practice and the specific challenges that emerged during teaching and planning. Data were collected through classroom observations, interviews with teaching team members and examining learning artefacts such as work samples and reflection journals. Classroom experiences in such contexts often require adjustments both in the planning and in the implementation. Vignettes illustrate such adjustments and provide insights into the teacher decision-making involved. Themes that emerged from analysis of the data are discussed. These themes were based on researcher reflections and meetings following each round of observations and were reported back to the teachers for verification. We explore the themes, reflecting on current policy influencing inclusive education in Australia and internationally, highlighting challenges that emerged in seemingly straightforward concepts. Our aim in this paper is to explore examples of an innovative approach to mathematics education for students with Down syndrome—the technique of teaching all students the mathematics from their year level, with adjustments. The careful study of the practices of teachers offers implications for effective inclusive mathematics education for the diverse classrooms that are increasingly the norm across Australia and around the world.

Prospective primary school teachers’ competence for analysing the difficulties in solving proportionality problem

Abstract: In this paper, we describe the design, implementation and results of a formative intervention, meant to develop prospective primary school teachers’ competence in the analysis of the difficulties emerging in the resolution of proportionality tasks. Solving problems by different methods, identifying the knowledge put at stake in each problem and taking this information into account to recognize the difficulties that pupils may encounter in solving the problems using each strategy are essential aspects of the epistemic and cognitive facets of didactic-mathematical knowledge. The experience has been carried out with a sample of 88 prospective primary school teachers during the third year of their studies, by applying a didactic model that includes work in teams, institutionalization, and assessment of the individual learning achieved. To analyse the participants’ responses, we used some theoretical and methodological tools of the onto-semiotic approach in mathematics education. The identification of pupils’ potential difficulties in addressing problem-solving, based on the objects involved in the mathematical activity, was a challenging task for prospective teachers. The difficulties most frequently identified by the prospective teachers were those concerned with understanding the statement requirements, or the problem-situations context, and the mathematical procedures involved in the resolution of the task. They did not discriminate the difficulties according to the resolution strategies. Time constraints conditioned the degree of learning achieved. We conclude that it is necessary that teacher education programs consider this type of didactical analysis and should be articulated with situations focused on developing other complementary knowledge and competencies.

Teachers’ responses to instances of student mathematical thinking with varied potential to support student learning

Abstract: Teacher responses to student mathematical thinking (SMT) matter because the way in which teachers respond affects student learning. Although studies have provided important insights into the nature of teacher responses, little is known about the extent to which these responses take into account the potential of the instance of SMT to support learning. This study investigated teachers’ responses to a common set of instances of SMT with varied potential to support students’ mathematical learning, as well as the productivity of such responses. To examine variations in responses in relation to the mathematical potential of the SMT to which they are responding, we coded teacher responses to instances of SMT in a scenario-based interview. We did so using a scheme that analyzes who interacts with the thinking (Actor), what they are given the opportunity to do in those interactions (Action), and how the teacher response relates to the actions and ideas in the contributed SMT (Recognition). The study found that teachers tended to direct responses to the student who had shared the thinking, use a small subset of actions, and explicitly incorporate students’ actions and ideas. To assess the productivity of teacher responses, we first theorized the alignment of different aspects of teacher responses with our vision of responsive teaching. We then used the data to analyze the extent to which specific aspects of teacher responses were more or less productive in particular circumstances. We discuss these circumstances and the implications of the findings for teachers, professional developers, and researchers.

Theorems or procedures? Exploring undergraduates' methods to solve routine problems in linear algebra

Abstract: While the lion’s share of mathematics education research in linear algebra has been concerned with the conceptual aspects of students’ learning, this study focuses on procedural knowledge that undergraduates can develop in a typical first-year course. The study analyzes the methods that a large cohort of second-semester students applied in a final exam to solve three routine problems: finding a basis for a matrix column space, solving a linear system of equations, and devising a normal to a subspace. The results indicate an inverse relation between the efficiency of the methods that students used and the number of students who used them. This was especially evident in the case of theorems, the application of which was avoided by all but a few students in favor of compound algebraic procedures. Furthermore, the study shows that at the end of an instructional sequence, many students can struggle with choosing an appropriate problem-solving method and carrying it out without mistakes. The analysis of students’ mistakes is leveraged to draw university teachers’ attention to common difficulties in linear algebra.

Motivators or conceptual foundation? Investigating the development of teachers’ conceptions of contextual problems

Abstract: Teachers are encouraged to connect mathematic instruction to the real-world by posing tasks that are situated in rich, relevant contexts, but research has found that many teachers integrate contextual problems (CPs) as motivators rather than as supports for conceptual development. To provide insight into how teachers’ conceptions about CPs shift as they teach through contextual problem solving, we interviewed six teachers before and after they taught from a unit designed from principles of realistic mathematics education, an instructional design theory which positions realistic contexts as learning supports. Our findings indicate that teachers initially viewed CPs primarily as affective or motivating enhancements, but after teaching the RME unit with university-based support, the teachers articulated integrated understandings of how CPs can function as supports for conceptual development. The teachers articulated how CPs provide initial access, sites for progressive representational formalization, and references to which students can fall back in order to interpret subsequent tasks. The authors identify connections between these ideas and the support provided by the university team and the teachers’ guides.

Exploring approaches for inclusive mathematics teaching in Danish public schools

Abstract: The Danish MINK (Mathematics and Inclusion) project explored inclusive mathematics teaching in regular classes in ordinary public schools, with a focus on teacher professional development and classroom experiments as the main elements. The project started as a reaction to the challenges for practice arising from a political reform. The MINK project was a design study with genuine collaboration between teacher educators and upper primary school teachers. Participating teachers had opportunities to influence all phases of the project, including which themes to explore in depth. The article is a result of a follow-up elaboration, with specific research questions, which goes beyond the goals and questions in MINK. The article presents which themes emerged during the project as most relevant to explore when focusing on inclusive mathematics teaching and insights about the emergent themes. In “Discussion,” we relate insights from the project to the general need for professionals in mathematics education to engage in discussions on inclusion, and we sketch a number of anticipated conclusions about strategies and further needs for teacher education and teacher-in-service education.

What does teaching of spatial visualisation skills incur: an exploration through the visualise-predict-check heuristic

Abstract: A major challenge in the field of spatial reasoning is the translation of decades of accumulated research findings from the domain of psychology and mathematics education into the school mathematics curriculum. This study aimed to operationalise a heuristic termed visualise-predict-check (VPC) for the teaching of spatial visualisation and explore the affordances for enabling teachers to foster spatial reasoning. We conducted moment-by-moment and fine-grained analysis of teaching-learning transactions of two teachers who implemented a lesson devised on the basis of the VPC heuristic. The data sources included video-taped recordings and classroom observations. The VPC heuristic provided teaching trajectories to explicitly prompt students to engage in spatial visualisation while learning mathematics and in so doing the class routine was remarkably spatially elevated. However, subtle disparities in the implementation of the lesson lead to differences in students’ engagement. The more deliberate VPC fostered students’ spatial thinking whereas the limited application of VPC tended to inhibit opportunities for reasoning spatially. VPC offers much potential to enhance spatial skills both in classroom routine teaching and in specialised interventions.