Photovoice: understanding high school females’ conceptions of mathematics and learning mathematics

Photovoice is a participatory action research tool that is grounded in the literature for critical consciousness (Wang & Burris, 1997). Four creative high school girls who reported struggles with mathematics were given cameras and asked to take photographs to answer the following questions: (1) What is mathematics? (2) What is your ideal learning environment? (3) What things impede your learning of mathematics? Within-case and cross-case analyses of the photographs and interview responses were conducted. Each individual case was analyzed using the work of Belenky, Clinchy, Goldberger, and Tarule (Women’s Ways of Knowing, 1986) who investigated women’s epistemological perspectives. The girls were either silenced (disconnected from the mathematical knowledge of the teacher), received (lacked confidence to do mathematics independently), or “fragily” subjective (viewed mathematical knowledge as personal rather than imparted by the teacher) knowers of mathematics. The use of photovoice has the power to facilitate the nurturing of silence as it moves toward the “roar which is on the other side of silence” (Belenky et al., 1986, p. 4).     

Prospective middle school mathematics teachers’ preconceptions of geometric translations

This article reports an analysis of 44 prospective middle school mathematics teachers’ pre-existing knowledge of rigid geometric transformations, specifically the geometric translations. The main data source for this study was the participants’ responses to the tasks that were presented during semi-structured clinical interviews. The findings of the study revealed that prospective teachers had difficulties recognizing, describing, executing, and representing geometric translations. The results indicated that teacher candidates held various views about the geometric translations: (1) translation as rotational motion, (2) translation as translational motion, and (3) translation as mapping. The results further revealed various interpretations of the vector that defines translations: (1) vector as a force, (2) vector as a line of symmetry, (3) vector as a direction indicator, and (4) vector as a displacement. Although many of the teacher candidates interviewed knew that a vector has a magnitude and a direction, this knowledge did not generally lead them to conclude that vectors define translations.     

The nature and development of middle school mathematics teachers’ knowledge

In this article, we report on the use of a teacher profiling instrument with 62 middle school teachers at the start of a 3-year professional learning programme. The instrument was designed to assess the aspects of teachers’ knowledge identified by Shulman (1987) refined by Ball et al. (2008) and extended to include teachers’ confidence to use and teach various topics in the middle school mathematics curriculum and their beliefs about mathematics teaching and learning. Based on a hierarchical coding of items, the application of the partial credit Rasch model revealed that the profile items were measuring a single underlying construct and suggested that the various facets of teacher knowledge develop together. We describe the characteristics of four levels of the hierarchical construct measuring teacher knowledge and understanding for teaching mathematics in the middle years of schooling, and discuss the unique affordances of a holistic view of teacher knowledge in contrast to considerations of multiple knowledge categories.     

Pre-service primary teachers’ conceptions of creativity in mathematics

Teachers in the UK and elsewhere are now expected to foster creativity in young children (NACCCE, 1999; Ofsted, 2003; DfES, 2003; DfES/DCMS, 2006). Creativity, however, is more often associated with the arts than with mathematics. The aim of the study was to explore and document pre-service (in the UK, pre-service teachers are referred to as ‘trainee’ teachers) primary teachers’ conceptions of creativity in mathematics teaching in the UK. A questionnaire probed their conceptions early in their course, and these were supplemented with data from semi-structured interviews. Analysis of the responses indicated that pre-service teachers’ conceptions were narrow, predominantly associated with the use of resources and technology and bound up with the idea of ‘teaching creatively’ rather than ‘teaching for creativity’. Conceptions became less narrow as pre-service teachers were preparing to enter schools as newly qualified, but they still had difficulty in identifying ways of encouraging and assessing creativity in the classroom. This difficulty suggests that conceptions of creativity need to be addressed and developed directly during pre-service education if teachers are to meet the expectations of government as set out in the above documents.     

Sound Stuff? Naïve materialism in middle‐school students’ conceptions of sound

Few studies have dealt with students’ preconceptions of sounds. The current research employs Reiner et al. (2000 Reiner, M., Slotta, J. D., Chi, M. T. H. and Resnick, L. B. 2000. Naïve physics reasoning: A commitment to substance‐based conceptions. Cognition and Instruction, 18(1): 1–34. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) substance schema to reveal new insights about students’ difficulties in understanding this fundamental topic. It aims not only to detect whether the substance schema is present in middle school students’ thinking, but also examines how students use the schema’s properties. It asks, moreover, whether the substance schema properties are used as islands of local consistency or whether one can identify more global coherent consistencies among the properties that the students use to explain the sound phenomena. In‐depth standardized open‐ended interviews were conducted with ten middle school students. Consistent with the substance schema, sound was perceived by our participants as being pushable, frictional, containable, or transitional. However, sound was also viewed as a substance different from the ordinary with respect to its stability, corpuscular nature, additive properties, and inertial characteristics. In other words, students’ conceptions of sound do not seem to fit Reiner et al.’s schema in all respects. Our results also indicate that students’ conceptualization of sound lack internal consistency. Analyzing our results with respect to local and global coherence, we found students’ conception of sound is close to diSessa’s “loosely connected, fragmented collection of ideas.” The notion that sound is perceived only as a “sort of a material,” we believe, requires some revision of the substance schema as it applies to sound. The article closes with a discussion concerning the implications of the results for instruction.     

Teachers’ assessments of students’ learning of mathematics

This paper reports a study of the classroom assessment practices of 12 sixth form college mathematics teachers in Malta. It explores the extent to which these teachers are knowledgeable about their students’ learning of mathematics and the implications that this has for their classroom practices. It reveals that these teachers’ knowledge of their students’ understanding of certain mathematical concepts is fairly limited. It then goes on to discuss this phenomenon in terms of a process that can hinder rather than promote learning. The research illuminates the position of teachers who appear to lack certain detailed information about their students which, it can be argued, could inform more effective teaching strategies. The paper concludes by exploring possible implications for similar situations where learning could be enhanced by more effective classroom assessment strategies and their use to inform future teaching and learning activities.     

First‐year secondary school mathematics students’ conceptions of mathematical proofs and proving

The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.     

Pre-service teachers’ conceptions of creativity in elementary school English

Widely thought to be something worth encouraging in young learners, creativity has popularly been associated more with music and art than with other areas of the curriculum. There have been many studies of creativity but few that focus explicitly on what counts as creative thinking in specific subject areas. The aim of the research reported here was to determine pre-service teachers’ conceptions of creativity within the curriculum for English. The study involved analyses of primary school trainee teachers’ responses to questionnaires and follow-up focus group discussion to identify their conceptions. A group of 48 trainees in the final year of an undergraduate degree in primary education leading to qualified teacher status in England completed the questionnaire. Of these, eight volunteered to participate in a follow-up focus group discussion to further explore ideas. Responses were analysed quantitatively and qualitatively. Conceptions of creativity in English were found to be limited, focused mainly on naïve views of story writing and dramatic activity. Responses indicated that they were often unable to distinguish clearly between the concept of creativity, an example of its occurrence in the classroom, and what feature of that example made it creative. Consequently, their limited constructions of creativity were confused. It is important that teachers in schools as well as those responsible for training teachers in universities are advised that trainees’ conceptions of creativity in English may be inadequate in several respects and that they may not recognise opportunities for creativity. Pre-service training programmes could well benefit from structured courses on the forms and applications of creativity.     

Teachers’ perceptions of school connectedness and risk-taking in adolescence

School connectedness has been shown to be an important protective factor in adolescent development, which is associated with reduced risk-taking behavior. Interventions to increase students’ connectedness to school commonly incorporate aspects of teacher training. To date, however, research on connectedness has largely been based on student survey data, with no reported research addressing teachers’ perceptions of students’ connectedness and its association with student behavior. This research attempted to address this gap in the literature through in-depth interviews with 14 school teachers and staff from two Australian high schools. Findings showed that teachers perceived students’ connectedness to be important in regard to reducing problem behavior, and discussed aspects of connectedness, including fairness and discipline, feeling valued, belonging and having teacher support, and being successfully engaged in school, as being particularly important. This research enables the development of school-based intervention programs that are based on both student- and teacher-focused research.     

Growth orientation predicts gains in middle and high school students’ mathematics outcomes over time

Academic growth constructs, such as growth mindset and various forms of growth goals, have been of substantial focus in psycho-educational research. Recent research has sought to identify how such growth constructs are inter-related, finding that an underlying growth orientation (comprised of growth mindset, self-based growth goals, and task-based growth goals) was cross-sectionally associated with more positive outcomes for students. However, for such a construct to have meaningful relevance to education and educational research, it must be associated with actual growth in academic outcomes. Accordingly, using two-wave longitudinal structural equation modeling in a large sample of Australian middle and high school students (N = 2949), we examined the extent to which students’ growth orientation predicted growth in academic outcomes. We hypothesized that students’ growth orientation would be positively associated with gains in students’ mathematics engagement and achievement, while controlling for student demographic covariates (e.g., gender, age) and prior variance in each substantive factor measured one year prior. Results demonstrated that students' growth orientation in mathematics was a significant positive predictor of students' gains in mathematics engagement and achievement, above other sources of influence. Findings are discussed in terms of improving researchers’ understanding of how growth constructs are inter-related and how to promote students’ academic success in mathematics.     

Teachers’work: Beginning teachers’ conceptions of competence

Worldwide, teachers’ work continues to be the subject of discussion and debate, especially in terms of teacher competence and its appropriate appraisal. Available literature offers a number of perspectives of beginning teacher competence, yet the voices of beginning teachers themselves are not represented. This paper outlines the findings of a phenomenographic study that sought to uncover the conceptions of competence held by beginning teachers. The results of this study suggest that beginning teachers believe that teaching competence requires demonstration of thorough preparation, a sound knowledge base, effective classroom management, professional communication with a range of stakeholders, and an accurate sense of self-awareness in the role of teacher. These conceptions will be compared to a selection of the available literature about teacher competence and associated teaching standards.     

More than meets the eye: patterns and shifts in middle school mathematics teachers’ descriptions of models

Modeling is a major topic of interest in mathematics education. However, the field’s definition of models is diverse. Less is known about what teachers identify as mathematical models, even though it is teachers who ultimately enact modeling activities in the classroom. In this study, we asked nine middle school teachers with a variety of academic backgrounds and teaching experience to collect data related to one familiar physical phenomenon, cooling liquid. We then asked each participant to construct a model of that phenomenon, describe why it was a model, and identify whether a variety of artifacts representing the phenomenon also counted as models during a semi-structured interview. We sought to identify: what do mathematics teachers attend to when describing what constitutes a model? And, how do their attentions shift as they engage in different activities related to models? Using content analysis, we documented what features and purposes teachers attended to when describing a mathematical model. When constructing their own model, they focused on the visual form of the model and what quantitative information it should include. When deciding whether particular representational artifacts constituted models, they focused on how the representations reflected the system under study, and what purposes those representations could serve in further understanding that system. These findings suggest teachers may have multiple understandings of models, which are active at different times and reflect different perspectives. This has implications for research, teacher education, and professional development.     

Teachers’ assessment practices in mathematics: classrooms in the context of reform

This article responds to calls that have been made for research into how teachers incorporate new assessment ideas into their practice. We draw on a large‐scale study that examined the implementation of reform in mathematics in grades 7–10 in Ontario, Canada. We present teacher questionnaire data, and focus on data gathered from case studies for details of what new assessment practices look like in classrooms. We show teachers using a variety of forms of assessment and using assessment to improve student learning. Observed assessment practices went beyond tests to include the use of journals, observation, questioning, self‐assessment and unique forms of ‘quizzes’. These practices allowed teachers to examine students’ mathematical thinking and provided feedback to students and teachers to improve student learning. We also observed the important role of collaboration, coherence, and the teachers’ beliefs in supporting new assessment practices.     

Teachers use of writing to support students’ learning in middle school: A national survey in the United States

A random sample of middle school teachers (grades 6–9) from across the United States was surveyed about their use of writing to support students’ learning. The selection process was stratified so there were an equal number of English language arts, social studies, and science teachers. More than one-half of the teachers reported applying 15 or more writing to learn strategies at least once a month or more often. The most commonly used writing to learn strategies were writing short answers to questions, note taking for reading, note taking while listening, and completing worksheets. While teachers reported using a variety of writing to learn strategies, most of them indicated they received minimal or no formal preparation in college on how to use writing to learn strategies to support student learning, less than one-half of teachers directly taught students how to use the writing to learn strategies commonly assigned, and the most commonly used writing to learn strategies did not require students to think deeply about the material they were learning. We further found that teachers’ reported use of writing to learn strategies was related to their preparedness and the composition of their classroom in terms of above and below average writers, English Language Learners, and students with disabilities.     

An assessment of a social–cognitive model of academic performance in mathematics in Argentinean middle school students

This study tested a set of hypotheses derived from the model of academic achievement in mathematics of the Social Cognitive Career Theory in a sample of Argentinean middle school students. To this aim, 277 students (male and female; age: 13–15 years) were assessed using the following instruments: logical–mathematical self-efficacy scale, mathematics outcome expectations, mathematics performance goals, and mathematics ability test. All of these instruments had been adapted for use in Argentinean students. Academic achievement in mathematics (i.e., grades obtained on regular school exams) was the variable to be modeled through the path analysis technique. The analysis allowed identification of interrelations among the variables and identification of direct and indirect effects. Academic achievement in mathematics was partially explained by the model. Overall, the results support the theoretical postulates of Social Cognitive Career Theory.     

History of mathematics in secondary school teachers’ training: towards a nonviolent mathematics education

In the context of mathematics teachers’ training, the concept of dépaysement épistémologique (epistemological disorientation) emphasizes that the contact with the history of mathematics, particularly with the use of original sources, pushes aside commonplace students’ perspectives about the discipline and offers them a critical look towards mathematics’s historical, social and cultural aspects. Conceptually supported by the theory of objectivation, an emergent sociocultural theory in mathematics education, this study describes the dépaysement épistémologique lived by future mathematics teachers engaged in the reading of historical texts. A phenomenological approach allowed us to clarify various meanings associated with students’ lived experiences and a dialogical perspective provides a way to get these meanings in tension through a polyphonic narration. Our reading of this polyphonic narration suggests that dépaysement épistémologique associated with the reading of historical texts encouraged empathy from students towards the authors and their future learners, opening up the possibility for a nonviolent mathematics education.     

Pre-service teachers’ perceptions of middle school students

This report shares a study that explored the perceptions pre-service teachers have of middle school students. Participants were asked to complete the Adjective Checklist (ACL) by endorsing the words they considered most characteristic of a typical middle school student. Items most frequently endorsed indicated a predominantly negative perception (e.g. “awkward”, “confused” and “emotional”). Implications for pre-service teacher training are discussed.     

Exploring intrinsic and extrinsic motivational aspects of middle school students’ aspirations for their mathematics learning

Middle school students have been pervasively described in the research literature as exhibiting disaffection, disengagement, and a lack of interest in mathematics classrooms. This study investigated this notion empirically using students’ own voice on their wishes for mathematics learning to see if they characterise themselves in this way in their self-reports. Over 3500 Australian middle school students responded to a free-format prompt and reported diverse but overwhelmingly positive aspirations related to intrinsic motivational features inherent in their disposition and to extrinsic factors in their environment, such as the learning context and activities provided. Evidence of all four goal orientations was found in the students’ responses but most noticeably for the productive mastery-approach orientation. When these students described what they wish for in their mathematics classrooms, they mentioned challenging work, interesting work, and opportunities to collaborate.     

Teachers’ awareness of the semio-cognitive dimension of learning mathematics

While many semiotic and cognitive studies on learning mathematics have focused primarily on students, this study focuses mainly on teachers, by seeking to bring to light their awareness of the semiotic and cognitive aspects of learning mathematics. The aim is to highlight the degree of awareness that teachers show about: (1) the distinction between what the institution (school, university, society, etc.) proposes as a mathematical object (not in itself but as the content to be learned) and one of its semiotic representations; (2) the different aspects of a semiotic representation that the student able to handle the representation and the student who handles the representation with difficulty may focus on; (3) the semiotic conflicts generated by the contents of semiotic representations that are similar to each other in some respect. For this purpose, in this study, the semio-cognitive approach introduced by Raymond Duval was complemented with the semiotic-interpretative approach of the Peircean tradition. By embracing the pragmatist research paradigm, the methodology was based on the research questions, which guided the selection of the research methods within a qualitatively driven mixed methods design. The research results clearly show the need for a review of professional teacher training programs, as regards the role the semiotic handling plays in the cognitive construction of the mathematical objects and the learning assessment.