Prospective Secondary Mathematics Teachers Research Articles

Prospective secondary teachers’ noticing of students’ thinking about the limit concept: pathways of development

AbstractResearch has shown that there is a need to examine prospective teachers’ development trajectories related to noticing expertise. An important content in the Spanish high school curriculum (16–18 years old) is the limit concept. Given the importance of this concept in the curriculum and the difficulties some prospective teachers have, developing their noticing of students’ mathematical thinking of this concept in teacher education programs is crucial to achieve high school student mathematics achievement. This study examines how prospective secondary school mathematics teachers (PTs) notice students’ mathematical thinking about the limit concept as they participated in a teaching module. PTs had to anticipate and interpret students’ mathematical understanding and make instructional decisions to support students’ conceptual progression using information about high school students’ understanding of the limit concept. We examined PTs changes related to how they anticipated, interpreted and made instructional decisions during the teaching module. We identified a change in how PTs conceived the understanding of the dynamic limit concept: from all-or-nothing dichotomy to progression; and a change in the instructional decisions they made: from decisions focused on changing the type of discontinuity to conceptual decisions. These changes allow us to characterise development noticing pathways. Our findings also help to identify the teaching module characteristics that support the development of PTs noticing.

Successfully preparing future mathematics teachers for digital technology integration: a literature review

Over the last several decades, digital technology (DT) integration in teaching and learning mathematics has received considerable attention, as such integration has not led to the expected results. Teacher education is a critical factor – even more important than access to tools and devices – in technology-enhanced teaching and learning. The purpose of this literature review is to examine and systematise core findings of studies that inform mathematics teacher education that supports the integration of digital technologies. We explore key considerations in mathematics teacher education for successfully preparing Prospective Secondary Mathematics Teachers (PSMT) for digital technology integration. This review includes 25 theoretical and empirical articles found using the keywords ‘Digital Technology’ and ‘Mathematics Education’. The review explores different topics that impact teachers’ technology integration classified into six categories according to their varying nature: (1) Mathematical digital competence, (2) Empirical and theoretical justifications, (3) Stages in DT integration, (4) Pedagogies for DT integration, (5) Mathematical skills, and (6) Systemic factors. It articulates and connects several findings reported in the extant literature previously seen as isolated and disconnected.

Comparing teacher noticing on paper and pencil and technology tasks

AbstractWork in teacher noticing has captured to what extent teachers notice students' thinking on paper and pencil tasks and technology tasks, yet no one study has considered how the same group of teachers notices across task types. This study used Jacobs et al.'s (Journal for Research in Mathematics Education, 2010, 41(2), pp: 169–202.) professional noticing of students' thinking framework to guide the design and analysis of prospective secondary mathematics teachers' (PSMTs) noticing of high school students' thinking on two geometry tasks: a paper and pencil task and a technology task. High school students' written work and a video clip were shared as the artifacts from which to notice for each task. PSMTs responded to a set of noticing prompts to capture how they were attending, interpreting, and deciding to respond to students' thinking for each task. Their written responses to these questions were then open coded for each noticing component and codes from each task were compared to reveal any differences in the content of PSMTs' noticing across task types. Results revealed differences between tasks for each component skill of noticing. Implications of these findings for mathematics teacher educators and future research are discussed.

Examination of Projects Prepared by Prospective Secondary School Mathematics Teachers Using Scratch

The aim of this study is to examine the projects prepared by prospective secondary school mathematics teachers using Scratch for the learning outcomes in the Mathematics Course Curriculum (Secondary School 5th, 6th, 7th, and 8th grades) in terms of the proficiency level of programming concepts and computational thinking concepts. 73 prospective secondary school mathematics teachers participated in this research in which the case study design was used. Prospective teachers prepared Scratch projects related to the 5th, 6th, 7th, and 8th-grade level learning outcomes of the Mathematics Course Curriculum. A total of 292 Scratch projects were examined within the scope of the research. Scratch projects were evaluated through the "Scratch Projects Assessment Rubric" and the "Dr. Scratch Assessment Tool". The result of the research showed that most of the criteria in the "Scratch Projects Assessment Rubric" were provided in more projects than at the beginning as new projects were developed by the prospective teachers at different grade levels every week. Nevertheless, it was evident from the Dr. Scratch evaluation that as the prospective instructors utilized the Scratch program, the level of proficiency of the projects they created is improved. It is recognized that this beneficial development has occurred because of the rise in the number of potential instructors who have experience with the Scratch program.

Prospective teachers’ unified understandings of the structure of identities

United States curriculum standards advise mathematics teachers to teach students to attend to structure and understand how mathematical concepts are related. This requires teachers to have a structural perspective and a coherent, unified understanding of mathematical structures that span curricula. This study explores Prospective Secondary Mathematics Teachers’ (PSMTs) unified understandings of identities and characterizes the structural features of identities that PSMTs attend to. I contribute a theoretical framework of three ways in which PSMTs reason about identities: a do-nothing element, a result of undoing something, and a coordination with inverse, binary operation, and set. I classify the level of coherence of their identity schemas demonstrated as they reasoned about the structural connections among additive, multiplicative, and compositional identities. I illustrate how having unified, coherent understandings of identities can lead PSMTs to reason productively about inverse and identity functions, while having incoherent understandings of identities can lead to inaccurate reasoning about inverse and identity functions. I conclude with teaching implications for fostering PSMTs’ unified understandings of algebraic concepts.

Supporting prospective mathematics teachers’ planning and teaching technology-based tasks in the context of a practicum course

The introduction of digital technologies into mathematics classrooms requires the creation of suitable technology-based mathematics resources, as well as educating and encouraging teachers to use these resources. This study examines four prospective secondary mathematics teachers' planning and teaching of technology-based mathematical tasks in the scope of a 14-week practicum course within a four-year mathematics education programme in Türkiye. Dynamic Geometry Task Analysis and Instrumental Orchestration frameworks have been chosen as the conceptual basis of the study. Action research towards a cyclical process is used to analyse how the incorporation of two frameworks in the process guide or support the development of the prospective teachers' classroom practices with the use of technology. The findings of the study indicate that using the technology specific frameworks in a practicum course was helpful to improve prospective teachers’ designing and teaching technology-based tasks. They showed different development, change and evaluation in their tasks based on their teaching experiences through the process.

Collaboration in Mathematics Teacher Education: the What, How, and Why of Mathematical Modeling

In this paper, we share our collaboration across the disciplines of mathematics and mathematics education to develop and implement a mathematical modeling task for prospective secondary mathematics teachers. Through this collaboration, we identified three key components of mathematical modeling: the what, how, and why. In this paper, we outline these components from the literature and how each framed our development and implementation of the Sprinkler Task in our mathematics content and mathematics methods courses for secondary teachers. These three components show that mathematical modeling is a particularly fruitful space for collaboration between the disciplines of mathematics and mathematics education in teacher education.

Unveiling belief and pedagogical content knowledge of prospective secondary mathematics teachers

Prospective teachers' beliefs have a beneficial effect on their knowledge and actions. Furthermore, teachers' pedagogical content knowledge is an essential contributor to teachers' impact on students' academic outcomes. However, examining how prospective teachers' beliefs influence their pedagogical content knowledge is still unclear. The current study investigates prospective mathematics teachers' beliefs and pedagogical content knowledge. We recruited three undergraduate students in the mathematics department consisting of one male and two female participants in this study. By assigning two interviews, a test, and observation, we examined three prospective mathematics teachers' pedagogical content knowledge and their beliefs about the nature of mathematics, mathematics learning, and mathematics teaching. Prospective mathematics teachers' test, interview, and observation results were analyzed in three stages: data condensation, presentation, and conclusion drawing and verification. The findings point out that traditional beliefs (instrumentalist and platonist views) are more dominant in prospective teachers' beliefs about the nature of mathematics. Moreover, the more sophisticated prospective teachers' beliefs about the nature of mathematics, mathematics learning, and mathematics teaching are, the higher one's content knowledge and pedagogical content knowledge levels are. Implications of this study, mathematics teacher educators should assist and foster prospective teachers in developing constructivist or problem-solving beliefs.

Promoting Prospective Mathematics Teachers' Understanding of Derivative across Different Real-life Contexts

This study investigates prospective secondary school mathematics teachers' developing conceptions about the meaning of derivative (as rate of change) while working on a series of tasks involving different non-motion real-life contexts. The study was conducted by employing a case study method. The participants of the study were 25 prospective secondary school mathematics teachers enrolled in the mathematics teaching methods course. A series of contextual tasks involving different real-life situations about derivative was designed and implemented in the form of problem-based learning as a regular part of the course. Pre and post questionnaires, written group solutions to the tasks, individual reflection papers, and video-recorded online classroom discussions were the data sources. A constant comparative analysis method was employed in analysing the data. The results showed that prospective teachers have difficulties in interpreting the meaning of derivative in different real-life contexts. While they were frequently using “amount of change” and “slope” interpretations, they started to use “rate of change” interpretation. However, the classroom discussions showed that most of the prospective teachers were using the “rate of change” expression as a memorized term without engaging in its contextual meaning. We made some inferences about the teaching and learning of derivative and also the professional development needs for teachers.

Improving Prospective Teachers’ Beliefs About a Double Discontinuity Between School Mathematics and University Mathematics

Prospective teachers often perceive a “double discontinuity” between school mathematics and university mathematics. The first discontinuity can be described as the belief that there is no coherence between school mathematics and university mathematics, which forms part of the notoriously problematic transition from school to university. The second discontinuity can be described as a belief about the lack of relevance of university mathematics for the later professional practice of prospective teachers. Beliefs about coherence and relevance have been known to impact prospective mathematics teachers’ interests and academic success. In this paper, we discuss an intervention involving 72 prospective secondary school mathematics teachers, aimed at influencing their beliefs about coherence and relevance. For this, we refer to the construct of beliefs as the main part of our theoretical framework, as well as the sub-constructs of beliefs regarding coherence and relevance. We then describe an intervention implemented within the first two years of mathematics courses, involving so-called “teacher-oriented” tasks that aim to trigger reflection on the benefit of university mathematics for teaching mathematics in school. The effect of the intervention was measured with a pretest-posttest experimental design using a questionnaire concerning teachers’ beliefs about coherence and relevance. Our results show that the prospective teachers’ beliefs about coherence and relevance generally decrease during the semester. However, statistically significant differences between the treatment group and a control group were found, especially regarding their beliefs about relevance.

Geometric Refutations of Prospective Secondary Mathematics Teachers

Increasing students’ exposure to mathematical reasoning (Lannin, Ellis, Elliot, 2011) has been a constant call from mathematics education reform efforts. However, studies have raised concerns about whether teachers are prepared to teach reasoning. This paper explores one aspect of reasoning, refutations, by examining the geometric refutations of thirteen pre-service secondary teachers (PSTs). Findings suggest that the PSTs did struggle to create culturally accepted refutation, but their results were less concerning than previous studies suggest. Primary challenges include that the PSTs separate the work they do for the different subprocesses of reasoning, reason differently depending on the subprocess, and favor non-culturally accepted means of refutations. Additionally, when the PSTs created culturally accepted refutations, they often used specific counterexamples, which are less pedagogically powerful than general counterexamples (Zaslavsky Peled, 1996). However, the PSTs in this study did provide general counterexamples more often than the participants of other studies (Potari, Zachariades, Zaslavsky, 2009). Suggestions for ways to improve PSTs understanding of reasoning are provided.

Development of Prospective Secondary Mathematics Teachers' Noticing Skills in School Experience Course

This study investigates the development of prospective secondary mathematics teachers' (PSMTs') noticing skills in a course conducted at the last year of their teacher education program. PSMTs watched and analysed different instructional videos, including an unknown teacher's, their mentor teacher's, and their own teaching videos. The data were collected from five PSMTs through reflection papers and group discussions about what and how they noticed in the videos. The results showed that the PSMTs had different levels of noticing at the beginning of the course and developed their noticing skills over time through the research. It was found that PSMTs' noticing skills differed across different video types. This study revealed the importance of experience in real classroom environment for the development of PSMTs' noticing skills. Based on the results reached in the research, some suggestions were included.

Including School Mathematics Teaching Applications in an Undergraduate Abstract Algebra Course

We describe the design and implementation of lessons in undergraduate abstract algebra that integrate applications to teaching high school mathematics. Each lesson consists of a pre-activity, class activity, homework set, assessment questions and detailed instructor notes. The included applications intend to bridge advanced mathematics concepts to secondary school mathematics content for prospective secondary mathematics teachers, as well as provide instructors with a useful resource. Each lesson includes four guiding features: addressing meaningful content; making connections to mathematics teaching; emphasizing active learning; and providing comprehensive notes for instructors. Interviews with instructors and their students document how the instructors adapted the lessons to their classrooms and how the lessons influenced undergraduates' mathematical understanding.

The Effect of Proof Format on Reading Comprehension of Geometry Proof: The Case of Indonesian Prospective Mathematics Teachers

This study aims to investigate the effects of the use of multiple geometry proof formats on Indonesian students’ reading comprehension of geometry proof (RCGP). Four classes of prospective secondary mathematics teachers (N=125), aged 18 to 19 years, participated in this quasi-experimental study. While the experimental group was instructed in three proof formats (paragraph, two-column and flow-chart proof), the control group was instructed in only the two-column proof format. Similar pre- and post-tests, based on Yang and Lin’s (2008) RCGP test, were administered to both groups. N-Gain scores were used to determine the improvement of both groups. The N-Gain scores showed significantly more improvement of students’ RCGP in the experimental group. More detailed analysis indicated that the use of multiple proof formats supports the students’ understanding of the facets of logical status of statements and the critical ideas in the proof. This study shows the benefits of offering multiple proof formats to support prospective mathematics teachers’ RCGP.

From Micro Teaching to Real Classrooms: Examination of Prospective Mathematics Teachers' Technology-Based Teaching

This study examines the changes and development of prospective secondary mathematics teachers‟ technology-based tasks through teaching practices. The Dynamic Geometry Task Analysis model consisting of the components of mathematical depth and technological action has been chosen as the conceptual framework of the study. In this study, a qualitative research paradigm has been adopted and action research methodology involving a cyclical process has been used. Participants of the study were four prospective secondary mathematics teachers, who were enrolled in a secondary mathematics education program at a state university in Turkey. This research was carried out within the scope of a 14-week Practicum course and focused on prospective mathematics teachers‟ implementations of their technology-based tasks through micro-teaching and classroom teaching. Data mainly consisted of each prospective teacher‟s one technology-based task, video recordings of their teaching practices (micro-teaching and classroom teaching) and interviews. Data was analyzed by using the video analysis method to examine how and why mathematical depth and technological action of the planned tasks changed or developed during micro-teaching and classroom practices. Findings of the study indicated that prospective mathematics teachers improved the levels of mathematical depth of their tasks by making use of their planned technological actions in a more effective way from micro-teaching to classroom teaching. In particular, it became apparent that micro-teaching supported the development of prospective mathematics teachers‟ technology-based tasks in terms of content and also the implementation processes of their tasks in classrooms.

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

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

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain restrictions, and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.

An Evaluation of ULTRA; an Experimental Real Analysis Course Built on a Transformative Theoretical Model

Most prospective secondary mathematics teachers in the United States complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically-motivated instructional model to design modules for a real analysis course that could inform secondary teachers’ actionable content knowledge and pedagogy. The theoretical model and designed curriculum launches the study of advanced mathematics content, in this case, real analysis, via authentic 7–12 classroom situations, abstracts the secondary mathematics, and uses that content to motivate the presentation of the advanced mathematics content. Subsequently, the curriculum then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. This study evaluated the success of the curriculum in terms of the students’ proficiency with real analysis, challenging secondary mathematics content, and the use of that content in teaching situations via both written post-tests and interviews and showed the viability of the model and experimental curriculum.

과제 대화록에 나타난 중등수학 예비교사들의 수학적 주목하기

The present study attempts to explore mathematical noticing of prospective secondary mathematics teachers in the area of graphing functions through analyzing the Task Dialogue which is a written script of hypothetical student-teacher conversation for a given task. Using the constant comparative method, 117 task dialogues written by 39 prospective secondary mathematics teachers were analyzed. As a result, differences were apparent in what the prospective teachers focused on and how they reasoned with the focused mathematics. Specifically, most of the prospective secondary mathematics teachers chose the time as an independent variable and focused on the formal causal relationship between the two variables without considering the problem situation. Moreover, most teachers thought of the relationship between the two variables as a change in one quantity or as a rate of change, and a few noticed the amount of change in two quantities as a way to rationalize the relationship between the two. Given that mathematics teachers would face with various levels of students in class, we argue it is important for prospective teachers to be aware of ways students might conceive of the relationship between two quantities in terms of both the amount of change and rate of change in functional situations. We finalize our discussion by providing implications for prospective secondary mathematics teacher education and for studying noticing of mathematics in a finer grain size.

Prospective mathematics teachers’ expectations for middle grades students’ arguments

AbstractIn this study, 15 prospective secondary (grades 6–12) mathematics teachers were asked to situate themselves as middle school (grades 6–8) mathematics teachers and validate arguments purported to prove that the sum of the first n odd natural numbers is. We examined their stated expectations for middle school students’ mathematical arguments and the rationales given for these expectations. Even though the participants were situated as middle grades teachers when evaluating hypothetical student arguments, not all of the participants embraced this perspective when discussing their expectations for middle school students’ arguments. In fact, only six participants embraced the teacher perspective, whereas nine participants relied, at least in some part, on their own experiences as students to determine their expectations for student work. In this article, we report five representative participants’ expectations and rationales to illustrate three emerging perspectives on prospective teachers’ expectations for middle school students’ arguments.