Investigating peer-assessment strategies for mathematics pre-service teacher learning on formative assessment

Formative assessment practices for secondary mathematics have been advocated as valuable for students, but difficult for teachers to learn. There have been calls in the literature to increase the emphasis on formative assessment in mathematics teacher preparation courses. This study explored the use of peer-assessment strategies for helping pre-service secondary mathematics teachers (PSTs) cultivate formative assessment principles and practices for assessing school students. Twenty-seven PSTs participated in a peer-assessment cycle comprised of: sourcing a rich mathematics task; constructing an assessment rubric for it; and collecting and analysing a selection of secondary student responses to the task. Each PST then provided written and verbal feedback to a peer on his/her rubric and student solution assessments. We draw on theoretical conceptions of Teacher Assessment Literacy in Practice to characterize the PSTs’ perceptions of their experience of formative assessment processes for learning to assess school students, in terms of cognitive and affective dimensions of their conceptions of assessment. The cohort evidenced a wide range of levels of confidence with the various aspects of formative assessment practices but on average less confidence in assessing school student task responses themselves than in assessing peer work. In addition to highlighting specific changes to different types of assessment knowledge, the PSTs also evidenced an awareness of shifts in their attitudes, in coming to view student task responses with more appreciation and humility.

     

How preservice teachers interpret and respond to student errors: ratio and proportion in similar rectangles

Interpreting and responding to student thinking are central tasks of reform-minded mathematics teaching. This study examined preservice teachers’ (PSTs) interpretations of and responses to a student’s error(s) involving finding a missing length in similar rectangles through a teaching scenario task. Fifty-seven PSTs’ responses were analyzed quantitatively and qualitatively. Analysis results revealed that although the student’s errors came from conceptual aspects of similarity, a majority of PSTs identified the errors as stemming from procedural aspects of similarity, subsequently guiding them by invoking procedural knowledge. This study also revealed two different forms of address and teaching actions in PST interventions along with three categories of acts of communication barriers. The broader implications of the study for international communities are discussed in accordance with the findings.     

The Relationship between Pre-service Mathematics Teachers’ Focus on Student Thinking in Lesson Analysis and Lesson Planning Tasks

This study explored whether pre-service teachers’ (PSTs’) lesson analysis skills during a teacher education course in the country of Turkey were related to their skills of lesson planning. PSTs’ lesson analysis skills during fieldwork were assessed by their attention to and interpretation of student thinking and learning, and how it is influenced by the teachers’ instructional decisions. The PSTs’ lesson analysis scores were significantly and positively correlated with scores in lesson planning task focusing on student thinking. The findings contribute to the literature on whether PSTs’ lesson analysis skills may be transferred to one of the core activities of teaching.     

Prospective elementary teachers’ responses to unanticipated incorrect solutions to problem-solving tasks

Eliciting and responding to student thinking are high-leverage practices that have powerful implications for student learning. However, they are difficult to enact effectively, particularly for novices, and more research is needed to understand how teacher education can support teachers in developing these skills. This study examined ways prospective elementary teachers (PSTs) responded to unanticipated incorrect student solutions to high-demand problem-solving tasks and how their responses changed over a 6-week field experience embedded in a practiced-based mathematics methods course. Data were collected in 6 weekly cycles of planning (written plans), enactment (video of problem-solving sessions with students), and reflection (written reflections on video). Problem-solving task implementations were analyzed using cognitive demand and math-talk frameworks. Of the three collaborating PST groups, two groups improved at responding to unanticipated incorrect solutions, but these two groups also developed a tendency to shut down anticipated solutions. The third group showed no patterns in their responses to unanticipated incorrect solutions, but did maintain the cognitive demand when responding to anticipated solutions. I present a case of one group of collaborating PSTs who made improvements in responding to unanticipated incorrect solutions in terms of the pedagogical moves they employed, cognitive demand associated with their responses, and how their responses changed over time. Implications for teacher education are discussed.     

Prospective teacher learning: recognizing evidence of conceptual understanding

This study examined prospective teachers’ (PSTs) ability to recognize evidence of children’s conceptual understanding of mathematics in three content areas before and after an instructional intervention designed to support this ability. It also investigates the role PSTs’ content knowledge plays in their ability to recognize children’s mathematical understanding. Results of content knowledge assessments administered at the beginning of the study revealed that content knowledge did seem to support PSTs’ analyses of children’s understanding when the child’s response demonstrated understanding or demonstrated a misconception. Content knowledge did not seem to support PSTs’ analyses of children’s procedural responses, as many PSTs with good content knowledge initially characterized procedural solutions as evidence of conceptual understanding. Similarly, content knowledge did not seem to support PSTs’ analyses of children’s responses with features commonly associated with understanding but not evidence of understanding. After the instructional intervention consisting of three multifaceted lessons in which PSTs examined many examples of student thinking, they showed improved ability to analyze responses with conceptual features and no evidence of conceptual understanding and responses demonstrating procedural knowledge. Results suggest that content knowledge is not sufficient for supporting PSTs’ analysis of children’s thinking, and that building activities such as the intervention into content courses may help develop this ability. Implications for teacher education programs and future research are considered.     

Pre-service teacher planning for differentiation of instruction in mathematics classrooms

This study investigated how 46 pre-service teachers (PSTs) planned for differentiation of instruction in mathematics. Content analysis was utilized to explore the differentiation strategies included and student characteristics considered in PST plans and how PSTs used differentiation strategies and student characteristics to differentiate lesson content, process, product, and environment. In addition, a rubric was designed and utilized to analyze the level of detail PSTs provided in their plans. Results indicated that overall, PSTs were developing in their planning for differentiation of instruction, using general terms to describe the modifications they would make to meet student needs. Moreover, results revealed that PSTs included strategies to differentiate lesson content and environment most often in their plans and frequently considered student readiness levels when planning for differentiation. Findings also revealed that PSTs need support in learning how to plan for differentiation based on student cultural backgrounds. Implications for mathematics teacher preparation are discussed.

     

Assessment training effects on student assessment skills and task performance in a technology-facilitated peer assessment

This study examines the impact of an assessment training module on student assessment skills and task performance in a technology-facilitated peer assessment. Seventy-eight undergraduate students participated in the study. The participants completed an assessment training exercise, prior to engaging in peer-assessment activities. During the training, students reviewed learning concepts, discussed marking criteria, graded example projects and compared their evaluations with the instructor’s evaluation. Data were collected in the form of initial and final versions of students’ projects, students’ scoring of example projects before and after the assessment training, and written feedback that students provided on peer projects. Results of data analysis indicate that the assessment training led to a significant decrease in the discrepancy between student ratings and instructor rating of example projects. In addition, the degree of student vs. instructor discrepancy was highly predictive of the quality of feedback that students provided to their peers and the effectiveness of revisions that they made to their own projects upon receiving peer feedback. Smaller discrepancies in ratings were associated with provision of higher quality peer feedback during peer assessment, as well as better revision of initial projects after peer assessment.     

Opportunities to notice: Chinese prospective teachers noticing students’ ideas in a distance formula lesson

This paper investigates the noticing of six Chinese mathematics prospective teachers (PSTs) when looking at a procedural error and responding to three specific tasks related to that error. Using video clips of one student’s procedural error consisting of exchanging the order of coordinates when applying the distance formula, some variation was found in how PSTs attended to, interpreted, and responded to this error. A more important finding is represented by the inconsistent responses that individual PSTs provided to the three related tasks. This finding suggests that, to some extent, prior learning experience, beliefs, and orientations inform what PSTs notice. But the finding also suggests the centrality of selecting tasks that provide accurate representations of PSTs’ emerging professional noticing. Implications for teacher educators are discussed.     

AHA Moments of Science and Mathematics Pre-service Teachers

This research study explored pre-service teachers' (PST) reflections of their student teaching experiences through AHA moments. Participants included 37 pre-service teachers enrolled in mathematics and science student teaching seminars. Qualitative methods were used to analyze PSTs' written and verbal responses to questions regarding AHA experiences. Four themes emerged related to PSTs' AHA moments: a greater awareness of PSTs' identity as teachers, the importance of knowing their students, the realization of inconsistencies in their own and their students' beliefs, and the importance of anticipating students' misconceptions. The potential for using an AHA moment assignment, as presented here, appears to support reflection among PSTs; however, it is not clear these PSTs used judgments and analysis, both integral elements of the reflective process.     

A Cohort-Driven Assessment Task for Scientific Report Writing

A formative assessment task was developed to improve the scientific report writing skills of university students. Students undertaking this task typically possessed varying levels of scientific literacy and were drawn from a cohort of mixed abilities. The assessment task involved the construction of a scientific report that included feedback from instructor to students before final submission of the assessment piece. After initial submission of a scientific report, the instructor developed a cohort-specific marking scheme based on the deficiencies that were evident within the class group. Using a mixture of peer and self-review against specific criteria, the students were required to resubmit an amended report. This resulted in elevated marks compared with those that would have been obtained after first submission, thus rewarding the student for the application of feedback. This technique proved to be efficient for both parties and also resulted in improvement of skills of the entire student population, regardless of the ability of the student prior to the assessment task. Using this methodology, students of varying aptitudes were able to measure their own skill improvement against tangible criteria, and enjoy a degree of learning success independent of the ranking within their group.     

A Student-centred Approach: Assessing the Changes in Prospective Science Teachers’ Conceptual Understanding by Concept Mapping in a General Chemistry Laboratory

Although researchers in higher education propose alternatives to traditional approaches to assessment, traditional methods are commonly used in college or university science courses. The purpose of this study was to explore the feasibility and validity of Prospective Science Teachers’ (PSTs) concept maps as authentic assessment tools in a student-centred approach to describe the changes in the conceptual understanding of the PSTs in general chemistry laboratory investigations. After the PSTs (n = 47) decided on important issues, such as who would assess their concept maps and what scoring strategy and criteria would be used, they practiced assessing their own and peers’ concept maps during the first five laboratory investigations. They subsequently constructed and assessed pre- and post-laboratory concept maps in a student-centred approach consisting of self, peer, and instructor assessments for the five remaining laboratory investigations. The results of the study showed using pre- and post-laboratory concept maps as authentic assessment tools in a student-centred approach was valid and reliable for describing the conceptual understanding of the PSTs in a university general chemistry laboratory course. The results of individual interviews indicated most PSTs had positive views of their assessment practices in the laboratory course. This study also provides pedagogical implications for the training of science teachers.     

Do students value feedback? Student perceptions of tutors’ written responses

The topic of feedback to students is an under‐researched area, and there has been little empirical research published which focuses on student perceptions. This study explores student perceptions of written feedback and examines whether feedback received demonstrated a student‐centred approach to learning. A multi‐method approach of qualitative and quantitative data collection and analysis was used to survey 44 students in the faculties of Business and Art & Design. Student responses show feedback is valued, but believed tutor comments could be more helpful. Survey results indicate that students may need advice on understanding and using feedback before they can engage with it. Content analysis of feedback samples and student responses uncovered four main themes of feedback considered unhelpful to improve learning: comments which were too general or vague, lacked guidance, focused on the negative, or were unrelated to assessment criteria. It is suggested that by focusing on messages conveyed by their writing, providing feedback set in the context of assessment criteria and learning outcomes, and by ensuring that it is timely, tutors could greatly improve the value of feedback.     

Preservice Secondary Science Teachers’ Teaching and Reflections During a Teacher Education Program

The 39 preservice teachers (PSTs) who participated in this study were enrolled in a masters program for secondary science teacher certification. Initially they held broad ideas about teaching and learning gleaned from their own experiences. Guided by the program course work, some PSTs embraced the pedagogical approaches introduced in the program, applied them in their teaching, and reflected on the outcomes. Their reflections showed that they were focused on keeping all of their students interested in science and on student participation in the process of meaning-making. Some PSTs embraced the program goals but struggled to achieve them in teaching. Others focused on transmission of content and did not attempt to develop an environment of student agency. There were nine career-changer PSTs and most of them remained teacher-centered throughout the program. The implications of student- and teacher-centered approaches adopted by the PSTs and the rationales provided by them are discussed in the paper.     

Examining and elaborating upon the nature of elementary prospective teachers’ conceptions of partitive division with fractions

The purpose of this study was to examine and elaborate upon elementary prospective teachers’ (PSTs) conceptions of partitive division with fractions. We examined the degree to which PSTs’ conceptions were connected (i.e., capable of translating between representations correctly; aware that partitive division generates a unit rate for its quotient) and flexible (i.e., capable of differentiating between opportunities to partition or iterate (or both) when solving a partitive division task; aware that partitioning or iterating (or both) could be associated with the operation of division, as appropriate). Seventeen PSTs participated in task-based interviews prior to instruction in a mathematics content course for teachers. These PSTs demonstrated disconnected conceptions of partitive division with fractions when they incorrectly translated between representations and either inconsistently or did not express awareness that the purpose of the task was to generate a unit rate. These PSTs demonstrated rigid conceptions of partitive division with fractions such that they did not express awareness that the process of iterating could be associated with the operation of division, even when they obtained a correct answer by iterating. Results extend prior research by looking beyond PSTs’ performance on tasks to elaborate upon PSTs’ conceptions of the operation of partitive division. This study contributes new insights into PSTs’ conceptions that can be used by mathematics teacher educators to inform the design of future instructional interventions.     

Students’ geometrical perception on a task-based dynamic geometry platform

This paper describes a task-based dynamic geometry platform that is able to record student responses in a collective fashion to pre-designed dragging tasks. The platform provides a new type of data and opens up a quantitative dimension to interpret students’ geometrical perception in dynamic geometry environments. The platform is capable of generating a collective image map of student geometrical perceptions for a pre-designed dragging task. This map is interpreted as students’ qualitatively different ways of perceiving a geometrical phenomenon under the drag mode, ways which are quantified and categorized in a collective way. The idea of task perceptual landscape is proposed to facilitate discussion on the pedagogical significance of this platform. Specifically, a task case is presented and analysed in which a methodology is developed that provides a way to classify students’ geometrical perceptions with respect to the task. The task perceptual landscape is interpreted as a collective example space of student perception of a task. Furthermore, an idea of personal example space is developed through the findings from a qualitative study for the same task. This brings about discussion on possible pedagogical correlation between the quantitative and qualitative aspects of the platform.     

Evolving through tensions: preservice teachers’ conceptions of social justice teaching

ABSTRACT

This study utilized cultural historical activity theory to explore the evolution of nine preservice teachers’ (PSTs’) conceptions of social justice teaching while enrolled in a social justice-oriented teacher education program. From three interviews conducted over one year, findings show that tensions PSTs encountered while student teaching in high-poverty schools predominantly shaped their thinking. PSTs’ conceptions of social justice teaching evolved to include navigating inequitable systems, loving students critically, and viewing social justice teaching as uniquely personal. Implications include the importance of teacher educators leveraging inevitable student-teaching tensions as learning opportunities to further PSTs’ commitment to social justice teaching.     

‘I can’t believe I didn’t learn this in school’: ‘refusing secondly’ as an anti-racist English education framework

ABSTRACT

This study examines the responses of pre-service teachers (PSTs) to the young adult novel All American Boys in light of their viewing the 2016 documentary 13^th. In this paper, I use anti-racist English education scholarship to discuss how these two texts helped PSTs ‘refuse to start with secondly.’ I examine how Adichie’s concept of ‘refusing secondly’ within readings of literature both affords and constrains the anti-racist possibilities of literature teacher preparation courses. Using qualitative methodologies, I analysed student reflections, recorded class discussions, and co-constructed class documents. Students connected the historical and the contemporary in considerations of race and racism. They also implicated societal institutions before situating themselves within the continuing legacies of race and racism. These findings demonstrate the ways that ‘refusing secondly’ may offer space for PSTs and teacher educators to use literature to navigate the continuous and indeterminate process of becoming more anti-racist.