What makes a problem mathematically interesting? Inviting prospective teachers to pose better problems

School students of all ages, including those who subsequently become teachers, have limited experience posing their own mathematical problems. Yet problem posing, both as an act of mathematical inquiry and of mathematics teaching, is part of the mathematics education reform vision that seeks to promote mathematics as an worthy intellectual activity. In this study, the authors explored the problem-posing behavior of elementary prospective teachers, which entailed analyzing the kinds of problems they posed as a result of two interventions. The interventions were designed to probe the effects of (a) exploration of a mathematical situation as a precursor to mathematical problem posing, and (b) development of aesthetic criteria to judge the mathematical quality of the problems posed. Results show that both interventions led to improved problem posing and mathematically richer understandings of what makes a problem ‘good.’     

Designing a Pedagogy of Inquiry in Teacher Education: Moving from resistance to listening

This paper reports a study of preservice teachers who investigated their own teaching during a field-based component of a mathematics education methods course. The course was designed to engage the preservice teachers in both mathematical and pedagogical inquiry. Analysis of video recordings of course discussions, audiotaped interviews with preservice teachers, audiotaped discussions of instructor's planning meetings, and copies of the instructor's and preservice teachers' journals identified two critical incidents that depict students' resistance to the course directions. Analysis of these critical incidents suggests that prospective teachers' interactions with their students can become the mirror through which we can investigate their interactions with us, as teacher educators, and with our course activities. In this way we might reframe the problem of resistance to one of listening—listening to the students, to each other, and to ourselves.     

Balancing competing demands: Enhancing the mathematical problem posing skills of prospective teachers through a mathematical letter writing initiative

Responding to mathematical problems is a core activity in classrooms. The problems that teachers select determine the mathematical content, processes and nature of mathematical inquiry occurring in classrooms and thereby contribute to the development of mathematical skills and dispositions. Selecting, designing or reformulating mathematical problems is a critical skill, then, for prospective and practising teachers. This study explores the influence of a mathematical letter writing initiative in developing the problem posing skills of 28 prospective primary teachers. We examine the characteristics of mathematical problems designed by prospective teachers, and their understandings of what constitutes a good mathematical problem, prior to and following completion of a 12-week letter writing initiative with 10–11-year-old children. Analysis of the data reveals the benefits of engaging in the initiative as evidenced in improvements in several problem characteristics. There was an increase in the number of multiple approach and multiple solution problems and in the level of cognitive demand of problems posed. The challenge of posing non-traditional problems, alongside the competing demands of building in opportunities for success, may have diminished participants’ ability to evaluate and attend to the cognitive demand of problems.

     

Teachers implementing mathematical problem posing in the classroom: challenges and strategies

This paper reports a study about how a teacher educator shared knowledge with teachers when they worked together to implement mathematical problem posing (MPP) in the classroom. It includes feasible methods for getting practitioners to use research-based tasks aligned to the curriculum in order to encourage children to pose mathematical problems. Techniques, challenges, and strategies of implementing an MPP focus were also reported.     

The Role of Content Knowledge and Problem Features on Preservice Teachers’ Appraisal of Elementary Mathematics Tasks

The objective of this study was to examine the nature of preservice teachers’ evaluations of elementary mathematics problems using the Mathematical Tasks Framework (MTF), a model designed to discriminate among tasks according to their cognitive complexity. We also tested the relationship between mathematics content knowledge and problem length on the preservice teachers’ evaluations. Twenty-six undergraduate students enrolled in an elementary mathematics methods course at a large urban university were introduced to the MTF and cognitive complexity during a class lecture and were subsequently required to sort 32 mathematics problems according to the framework. Results demonstrated that overall, the preservice teachers had more difficulty accurately classifying problems considered to represent high levels of cognitive complexity compared to problems that were less complex. Those with strong mathematics content knowledge, as measured by a standardized test, were able to sort the problems more accurately than those with weaker content knowledge. Two open-ended items assessing content knowledge were not related to sorting performance. Finally, the preservice teachers were influenced by the surface characteristic of task length; the data indicated that the teachers tended to label short problems as less cognitively demanding and long problems as more so. Implications for preservice professional development include an increased emphasis on mathematics content knowledge as well as expert modeling of the identification of deep conceptual principles at the heart of the mathematics curriculum.     

AN INVESTIGATION ON CHINESE TEACHERS�� REALISTIC PROBLEM POSING AND PROBLEM SOLVING ABILITY AND BELIEFS

In the present study, which is a part of a research project about realistic word problem solving and problem posing in Chinese elementary schools, a problem solving and a problem posing test were administered to 128 pre-service and in-service elementary school teachers from Tianjin City in China, wherein the teachers were asked to solve 3 contextually challenging division-with-remainder (DWR) word problems and pose word problems according to 3 symbolic expressions. Afterwards, they were also given 2 questionnaires wherein they had to evaluate 3 different pupil reactions to, respectively, 1 problem solving item and 1 problem posing item about DWR. First, our results revealed that teachers behaved quite ‘realistically’ not only when solving and posing DWR problems themselves but also when evaluating elementary school pupils’ DWR problem solving and problem posing performance. Second, we found a correspondence between teachers’ own performance on the tests and their evaluations of pupils’ reactions. Third, the present study provides some further insight into the role of one of the instructional factors that is generally considered responsible for the strong and worldwide tendency among elementary school children to neglect real-world knowledge and realistic considerations in their endeavours to solve and pose mathematical word problems, namely the teachers’ conceptions and beliefs about this topic.     

Curriculum-Dependent and Curriculum-Independent Factors in Preservice Elementary Teachers’ Adaptation of Science Curriculum Materials for Inquiry-Based Science

In this nested mixed methods study I investigate factors influencing preservice elementary teachers’ adaptation of science curriculum materials to better support students’ engagement in science as inquiry. Analyses focus on two ‘reflective teaching assignments’ completed by 46 preservice elementary teachers in an undergraduate elementary science methods course in which they were asked to adapt existing science curriculum materials to plan and enact inquiry-based science lessons in elementary classrooms. Data analysis involved regression modeling of artifacts associated with these lessons, as well as in-depth, semester-long case studies of six of these preservice teachers. Results suggest that features of the existing science curriculum materials, including measures of how inquiry-based they were, have a relatively small influence on the preservice teachers’ curricular adaptations, while teacher-specific variables account for a much greater percentage of the variance. Evidence from the case studies illustrates the critical impact of the preservice teachers’ field placement contexts as an explanatory, teacher-specific factor in their curricular adaptations. These findings have important implications for science teacher educators and science curriculum developers, in terms of not only better understanding how preservice teachers engage with curriculum materials, but also how programmatic features of teacher education programs influence their ability to do so.     

Teaching in inclusive classrooms: efficacy and beliefs of Canadian preservice teachers

This study provides a Canada-wide snapshot of the self-efficacy and beliefs of 1490 preservice teachers in relation to inclusive teaching using the Teacher Efficacy for Inclusive Practice and the Beliefs about Learning and Teaching Questionnaire. At the time of data collection, these preservice teachers were in the final stages of their teacher education programmes. Based on the results of surveys in 11 Faculties of Education, men have higher self-efficacy than women for managing behaviour in the classroom, elementary preservice teachers have higher self-efficacy in collaboration than those teaching in secondary schools, those enrolled in the 1-year post-degree programme show lower self-efficacy than those enrolled in programmes of other durations, and those who have experience with people with special education needs show higher self-efficacy than those who do not. Differences illustrated that women were more inclusive than men, 1-year post-degree preservice teachers were less inclusive than others, and those who have experience with people with special education needs are more inclusive than those who do not. Recommendations for teacher education and limitations surrounding the practical significance of the findings are discussed.     

What Preservice Teachers and Knowledgeable Others Professionally Notice During Lesson Study

The purpose of this study was to understand what preservice teachers and knowledgeable others professionally notice as they engaged in repeated cycles of a modified version of lesson study, as a component of a field experience in a teacher education program. The study also centered on comparing the professional noticing practices of preservice teachers with other lesson study participants, including classroom teachers and university facilitators. Data analyzed included videos of weekly lesson study analysis meetings for seven weeks for each of four teams. Each team included six preservice teachers, a classroom teacher, and a university facilitator. Findings indicate that preservice teachers primarily noticed elements about the classroom environment and teacher pedagogy, but included instances of noticing centered on students' mathematical thinking. In contrast, classroom teachers and university facilitators, as knowledgeable others, typically noticed general events and were less focused on students' mathematical thinking. Analysis of noticing trends over the seven weeks indicated that noticing levels remained steady initially, dropped in the fourth and fifth week, and resumed original status in the final weeks. Results that the preservice teachers' noticing comments were at higher levels than the knowledgeable others are contrary to other research studies and indicate that incorporating lesson study with appropriate scaffolds into a field experience for preservice teachers may be a viable option for encouraging noticing of students' mathematical thinking.     

Preparing Preservice Teachers to Become Teacher Leaders

Since PreK–12 student achievement is the primary focus of schools, all teachers are called to serve as teachers leaders and improve learning on their campuses. Rather than waiting until they have gained experience, teachers can begin acquiring the knowledge, skills, and dispositions of teacher leaders during their preservice programs. Drawing upon published literature, this synthesis paper recommends reflection, service learning, and involvement in university-level student professional organizations as ways to prepare preservice teachers to become teacher leaders.     

Facilitating preservice teachers’ development of mathematics knowledge for teaching arithmetic operations

Mathematical tasks, centered on arithmetic word problems, are discussed as the basis of an approach to facilitate preservice elementary teachers’ development of mathematical knowledge for teaching arithmetic operations. The approach consists of three groups of tasks that allow students to reflect on their initial knowledge, explore arithmetic word problem situations from multiple perspectives, and apply their resulting knowledge, respectively. Findings from the investigation of the approach indicate that the tasks can provide an effective and meaningful basis to help preservice teachers to develop deeper understandings of the arithmetic operations and their relationships and pedagogical-mathematical knowledge for teaching the arithmetic operations conceptually.     

A comparison of preservice and in-service earth and space science teachers' general mental abilities,content knowledge,and problem-solving skills

The purpose of this study was to compare in-service and preservice earth and space science teachers on their general mental abilities, their content knowledge or declarative knowledge of earth and space sciences, the Gagnean levels of their content knowledge or declarative knowledge, and the procedural knowledge used in solving earth and space science problems. This study used a contrast-group design to compare in-service (n = 30) and preservice (n = 30) earth and space science teachers. The in-service earth science teachers (a) bring more declarative knowledge to the problem-solving situation, (b) use fewer steps while problem solving, (c) generate more subroutines and alternate hypotheses, and (d) possess different structural knowledge than do preservice earth science teachers. Findings from this study support Norman's theory of learning that experts (in-service teachers) function at the tuning mode of learning, whereas novices (preservice teachers) function in an accreting or structuring mode. In-service earth science teachers exhibited smoothness, automaticity, and decreased mental effort not exhibited by preservice earth and space science teachers.     

Preservice elementary school teachers’ knowledge of fractions: a mirror of students’ knowledge?

This research analyses preservice teachers’ knowledge of fractions. Fractions are notoriously difficult for students to learn and for teachers to teach. Previous studies suggest that student learning of fractions may be limited by teacher understanding of fractions. If so, teacher education has a key role in solving the problem. We first reviewed literature regarding students’ knowledge of fractions. We did so because assessments of required content knowledge for teaching require review of the students’ understanding to determine the mathematics difficulties encountered by students. The preservice teachers were tested on their conceptual and procedural knowledge of fractions, and on their ability in explaining the rationale for a procedure or the conceptual meaning. The results revealed that preservice teachers’ knowledge of fractions indeed is limited and that last-year preservice teachers did not perform better than first-year preservice teachers. This research is situated within the broader domain of mathematical knowledge for teaching and suggests ways to improve instruction and student learning.     

Problem posing based on investigation activities by university students

This paper reports a classroom-based study involving investigation activities in a university numerical analysis course. The study aims to analyse students' mathematical processes and to understand how these activities provide opportunities for problem posing. The investigations were intended to stimulate students in asking questions, to trigger their thinking processes, to promote their ability to investigate and to support them in learning numerical analysis' concepts and procedures. The results show that the investigations provided opportunities for students to experience mathematical processes, including posing questions, formulating and testing conjectures and, to some extent, proving results. They also provide some understanding about the role of problem posing in these processes. Posing questions occurred mainly in an implicit way, in the interpretation of tasks and in identifying regularities, analysing graphs and testing cases. The conjectures were often based on pattern identification or data manipulation, and the students tended to accept them without testing or proving. The students also proposed alternative formulations for the initial questions and posed new problems from their explorations and attempts to refine previous conjectures.     

Using mathematical analyses activities to improve manipulative discrimination in early childhood teacher education

This article reports on the findings of manipulative analyses performed by preservice and in-service teachers in an early childhood teacher education program mathematics methods course. The activities are intended to model and promote mathematical analyses for better discrimination between mathematics manipulatives by early childhood teachers. Findings suggest that these activities elicit varied analyses by the teachers related to practical and pedagogical issues, but are initially lacking in mathematical considerations. Through modeling and practice, they do include mathematical considerations, possibly shifting their beliefs regarding manipulatives away from the common assumption that manipulatives will result in automatic learning. The depth of these analyses, though, suggests that greater focus is required in building a stronger content knowledge base.     

ETHNICITY,GENDER, AGE,SES, AND ACTIVITY TYPE AS FACTORS CONTRIBUTING TO EARLY CHILDHOOD PRESERVICE TEACHERS’ INTERACTIONS WITH INTERNATIONAL AND U.S. PRESCHOOL CHILDREN

The interactions of 67 early childhood preservice teachers with International and U.S. preschool children in structured story time and unstructured table activities were videotaped and analyzed. Overall, preservice teachers displayed significantly more facilitative (e.g., ego boosting, teaching, questioning, and attending) than nonfacilitative (e.g., commanding, negatively responding, ignoring, and physical restraint) behaviors toward children. However, analyses of data relative to different ethnicity, gender, and activity type groups also revealed preservice teachers were relating to children in differential ways. Preservice teachers displayed significantly less facilitative and more nonfacilitative behaviors toward International children than U.S. children. In addition, preservice teacher‐child interactions with International children were rated as less positive and more negative than their interactions with U.S. children. Still also, preservice teachers displayed significantly less facilitative and less nonfacilitative behaviors toward girls than toward boys. Furthermore, preservice teacher‐child interactions with girls were rated as significantly less positive and less negative than with boys. Finally, preservice teachers displayed significantly more nonfacilitative behaviors toward children in structured story time activities than in unstructured table activities.     

I felt so guilty: emotions and subjectivity in school‐based teacher education

Research in the field of emotions in relation to teaching is relatively new, but expanding. However, studies addressing the emotional dimension of preservice teacher education, particularly with respect to the role of school‐based teacher educators are currently under‐represented in the literature. This paper reports findings from a study focussed on the emotional dimension of the practicum for school‐based teacher educators as they support preservice teacher colleagues. It adopts a qualitative method informed by feminist post‐structural theory in an attempt to give meaning to teachers’ narratives of their personal responses to supporting a less than successful preservice teacher. The study investigates teachers’ shifting sense of agency throughout the experience as they work within apparently intersecting discursive frames. The case study reveals the depth of emotions experienced by teachers and examines the impact of the emotions on teacher identity. It appears that the tertiary sector has failed to recognise the emotional costs of such experiences and the associated needs of school‐based teacher educators. Finally, the paper asks in what ways can staff in universities work collaboratively with teachers to address the concerns being raised by a study such as this, as there appears to be a genuine need to assist teachers copes with the emotional outcomes of working with problematic preservice teachers.     

Dissecting success stories on mathematical problem posing: a case of the Billiard Task

“Success stories,” i.e., cases in which mathematical problems posed in a controlled setting are perceived by the problem posers or other individuals as interesting, cognitively demanding, or surprising, are essential for understanding the nature of problem posing. This paper analyzes two success stories that occurred with individuals of different mathematical backgrounds and experience in the context of a problem-posing task known from past research as the Billiard Task. The analysis focuses on understanding the ways the participants develop their initial ideas into problems they evaluate as interesting ones. Three common traits were inferred from the participants' problem-posing actions, despite individual differences. First, the participants relied on particular sets of prototypical problems, but strived to make new problems not too similar to the prototypes. Second, exploration and problem solving were involved in posing the most interesting problems. Third, the participants' problem posing involved similar stages: warming-up, searching for an interesting mathematical phenomenon, hiding the problem-posing process in the problem's formulation, and reviewing. The paper concludes with remarks about possible implications of the findings for research and practice.     

Assessment for equity: learning how to use evidence to scaffold learning and improve teaching

Abstract

This article examines evidence regarding the assessment learning of preservice teachers (PTs) in a new Master of Teaching designed to prepare teachers to address the less than equitable outcomes of certain groups of students in New Zealand. The assessment curriculum was integrated across all of the courses and the in-school experiences as one of six interconnected facets of practice for equity. Evidence about the assessment learning of 27 preservice teachers was collected using a survey, interpretive analysis of three assignments and a focus group interview. The findings demonstrated that preservice teachers combined theory and practice encountered in many contexts to build the assessment understanding and competence needed to address equity issues. We argue that this was facilitated by incorporating the assessment curriculum within each course, intertwining university and school experiences, and the specific focus on addressing equity throughout the programme.     

Teaching Science Inquiry in Urban Contexts: The Role of Elementary Preservice Teachers’ Beliefs

This paper reports on a study of elementary preservice teachers’ inquiry-based practices, their efficacy beliefs, and the role beliefs had on two preservice teachers’ practices in urban classrooms. Results show inquiry-based practices can be cultivated through field-based experiences and preservice teachers’ efficacy beliefs, as it relates to practice in urban settings, are malleable. Specifically, personal efficacy beliefs about teaching science improved or were sustained for one cohort of preservice teachers. However, beliefs about students’ ability to learn science, that is outcome beliefs, were less stable. The results of two case studies show that science content knowledge was a factor in preservice teachers’ inquiry-based practices. However, why preservice teachers’ beliefs about student learning declined is less clear. More research is needed, along with follow-up data on teacher induction, to learn how preservice teachers’ beliefs impact urban students’ science education.