Pre-service teachers in mathematics: Struggling with the Standards

The purpose of this paper is to describe a two-year investigation of pre-service teachers in mathematics. Of particular interest is the degree to which the NCTM (1989) Curriculum and evaluation standards for school mathematics, the NCTM (1991) Professional standards for teaching mathematics, and other reform agendas impact pre-service teachers' thinking and practices. The beliefs, perceptions, philosophies, and classroom actions of four cohorts of student teachers (n = 44) were examined throughout this 2year study. Of particular interest were the pressure student teachers felt to implement the Standards in their teaching, their perceptions of their own teaching with respect to the Standards, the content of their lessons, and the impact of cooperating teachers on the development of the student teachers. Data was collected through a variety of sources, organized and analyzed in the spirit of ethnographic work. Data was generated through classroom observations, pre- and post-lesson conferences, lesson plans, seminar sessions, survey questionnaires, and informal conversations. A minimum of four classroom observations were made for each participant. Analysis of the data revealed general agreement among the student teachers in their regard for the Standards. Although they appeared to affirm the goals, content and recommendations contained therein, they almost uniformly suggest that they lack the tools to implement the Standards appropriately in the classroom. The consistency in the types and content of lessons observed appeared to validate this concern. An overwhelming majority of the lessons observed bore little or no resemblance to the values so highly espoused by the student teachers. Student teachers perceived notable pressure from the education program to include the Standards in their teaching practices, while feeling little pressure from cooperating teachers. They also point to cooperating teachers as the most significant influence on their teaching.     

Integrative assessment: reframing assessment practice for current and future learning

The requirement to provide timely formative tasks that are designed to facilitate student learning and autonomy has provoked a wider examination of the role of assessment in higher education and encouraged further investigation of the alignment of learning, teaching and assessment in curriculum design frameworks. Many current authors have proposed that the primary purpose of assessment is to enhance current and future learning and that current practice tends to overemphasise the importance of assessment for progression and certification purposes. This paper proposes that a clearer distinction be made between assessment tasks designed to facilitate and test current learning through the use of formative and summative assessments, and those tasks primarily designed to enhance future learning, which could be better termed integrative assessments. This distinction would allow students and teachers to have greater clarity around the proposed outcomes and reward mechanisms associated with assessment tasks and feedback. This paper proposes that teachers should strive to incorporate four different types of assessment tasks throughout a programme of study, namely diagnostic, formative, integrative and summative tasks, and that the outcomes and reward mechanisms for different assessment types be explained more clearly to students.     

Data-based individualization in mathematics to support middleschool teachers and their students with mathematics learning difficulty

Project STAIR (Supporting Teaching of Algebra: Individual Readiness) aims to develop and iteratively refine a framework for using data-based individualization (DBI) to support the readiness for algebra of middle-school students with mathematics learning difficulty. Project STAIR was tested in four middle schools in the United States to assess the impact of intensive professional development, ongoing coaching, and frequent student progress monitoring on teachers’ DBI practices, school-level culture and climate, and students’ algebra readiness. Findings from this exploratory research point to the promise of Project STAIR for improving teacher perceptions of school culture and climate related to support of students experiencing mathematics learning difficulty as well as increasing specific DBI practices, including teachers’ ratings of importance, understanding, and confidence with assessment practices. Students’ mathematics outcomes improved on several algebra-readiness progress monitoring measures, along with a diagnostic measure, but not on a more distal measure of mathematics performance.     

Preparing future teachers for diverse schools in an online learning community: Perceptions and practice

This study provides an example of one institution's efforts to design coursework that meets the simultaneous challenges of supporting the aims of increasing access to online courses and simultaneously better preparing teachers to work in diverse classrooms. Based on online pre- and post-surveys and monthly open-ended writing prompts administered to students in an introductory teacher preparation course, the study sought to discover students' motivation to select online or blended courses, student perspectives on the benefits and challenges to taking this course online, characteristics of the learning environment that promoted or interfered with students' learning, instructor's perspective of learners' reaction to topics addressing K-12 classroom diversity, and the impact of an online format on students' discussions of issues related to learner diversity. Findings suggest that online courses should include a classroom placement component in which students experience a diverse classroom in order to best prepare students for diverse teaching assignments. Thus, the best online teacher preparation courses maybe those that blend virtual and face-to-face interaction rather than being strictly online.     

Pre-service and in-service teachers’ perceptions on the integration of children’s literature in mathematics teaching and learning in Ireland

The beneficial role that children’s literature plays in facilitating the meaningful integration and advancement of literacy and numeracy in the primary mathematics classroom has been well validated by research findings internationally. In Ireland, supporting the development of literacy and numeracy is a key educational priority. Consequently, a myriad of policy initiatives such as the Literacy and Numeracy for Learning and Life strategy have been introduced. All aim to address concerns about young people’s lack of basic literacy and numeracy skills and to consider new teaching and learning modalities to enhance same. Despite this, no official emphasis is given to incorporating literature in the Irish primary school mathematics curriculum. Therefore, it is pertinent and timely that this study seeks to ascertain pre-service and in-service teachers’ views on the use of literature to support mathematics teaching and learning and to investigate perceived barriers to and enablers for the integration of children’s literature in the mathematics classroom in Ireland. The analysis of the findings will be framed using Ajzen’s (Ajzen, Icek. 1991. “The Theory of Planned Behavior.” Organizational Behavior and Human Decision Processes 50 (2): 179–211) Theory of Planned Behavior (TPB) model. This research is part of a large international research collaboration (see www.mathsthroughstories.org), in which the beliefs of teachers with respect to children’s literature are investigated.     

Opportunities for learning given to prospective mathematics teachers: between ritual and explorative instruction

The mathematics education field, including prospective teacher education program, has seen a continuous effort to change teaching practices to be more cognitively demanding, conceptually oriented and student centred. Our goal in this study was to examine how certain underlying assumptions about mathematical learning, as reflected in a skilled instructor’s discourse, align with opportunities to learn. The data included a set of fully transcribed 11 lessons from an introductory algebra course. The method of analysis was built upon the communicational (commognitive) framework and included discerning between the instructor’s mathematizing and identifying talk. This framework was extended to quantify the instructor’s identifying talk over the whole set of lessons. Our findings showed that at the surface level, the instruction in the class seemed to align with “explorative” goals. On a deeper level, however, it was more aligned with “ritual” goals that are concerned with producing narratives about people, not about mathematics.     

Structural exclusion through school mathematics: using Bourdieu to understand mathematics as a social practice

Research on learner errors in mathematics education is beginning to focus on how teachers can learn to identify and engage with the reasoning behind these errors. Research on professional learning communities is beginning to show that they present powerful opportunities for on-going teacher collaboration and learning. In this paper, I bring the two areas of research together. Drawing on data from one professional learning community in the Data Informed Practice Improvement Project, I show how teachers in this community came to understand key concepts about learner errors and shifted their ways of talking about learner errors. I identify three important shifts that the teachers made in their learning about learner errors: from identifying to interpreting errors; from interpreting to engaging errors; and from focusing on learner errors to focusing on their own knowledge. I argue that these three shifts suggest a deepening of teachers’ thinking in relation to learner errors.     

Mathematics for the future: developing a Head Start curriculum to support mathematics learning

While mathematics instruction for very young children needs to be age-appropriate in format and content, it also needs to prepare children conceptually for the kinds of mathematics learning that will be expected of them in future years. This perspective, informed by the work of Russian psychologists and educators on a measurement-based approach to early mathematics instruction [e.g., V.V. Davydov, Children’s Capacity for Learning Mathematics, National Council of Teachers of Mathematics, Reston, VA, pp. 109–205], was the basis for an experimental mathematics curriculum which focused on the concept of unit as it applies to enumeration, measurement, and the identification of relations among geometric shapes. The curriculum particularly emphasized two ideas about units that derive from a measurement perspective: first, that the numerical result we obtain from counting or other measurement operations will depend on our choice of a unit; and second, that units of one kind can be combined to form higher-order units or taken apart to form lower-order ones. The curriculum included a weekly project activity conducted by the Head Start teachers, suggestions for supplementary activities, and a weekly home activity for a parent or other family member to carry out with the child. It was implemented with children in three Head Start centers (N=46; age range 2 years, 9 months–4 years, 7 months at the beginning of the program). To evaluate the curriculum, two assessment instruments (the mathematics subscale of the DSC and a supplementary instrument constructed especially for this study) were administered, at the beginning and again at the end of the school year, to these children and to two comparison groups. One comparison group (N=48; age range 2 years, 6 months–4 years, 7 months) received a literacy intervention rather than a mathematics one; the other (N=29; age range 2 years, 8 months–4 years, 7 months) did not receive any experimental intervention. Results showed significant, albeit modest, positive effects of the intervention. The importance of reexamining current beliefs about what is possible—and desirable—within a preschool mathematics curriculum is emphasized.     

What they notice in video: a study of prospective secondary mathematics teachers learning to teach

Most teacher preparation programs have embraced the use of video as an effective methodology for developing teachers’ noticing skills. This study focused on learning about what secondary mathematics prospective teachers (PSTs) were able to notice when viewing video of their own co-teaching, particularly in a microteaching setting that consisted of peers. PSTs documented their observations on an observation tool while re-watching their video and then identified and ranked their top three observations. The ranked noticing statements were analyzed based on a grounded theory approach. Overall, PSTs’ ranked observations were more likely to attend to students and had a strong focus on mathematics and student learning. Ranked observations equally demonstrated both broad and specific understanding of video moments and often made suggestions that something they noticed could be improved in the implementation stage, versus improvements in planning or changes in themselves. Results support PSTs’ use of video for developing noticing skills in teacher education programs.     

Becoming critical: the failure to connect

This article is a retrospective critique of Becoming Critical by Carr & Kemmis. Into it are woven responses to other retrospective reviews of the book by Susan Groundwater–Smith, Susan Noffke, and Carr & Kemmis themselves. The basic argument is that Becoming Critical was overdependent on the work of Jurgen Habermas and was therefore vulnerable to the same critique. The author argues that just as Habermas did not resolve the dualism between theory and practice, nor did Becoming Critical. Like the work of Habermas, its theory of rationality is very weak in creating a link between ideology critique and the organisation of strategic action. This explains why, in spite of its attractiveness to many education academics, it was unable to support critical action research at an operational level. The article refutes suggestions that the problem with the book was that times have changed. It argues that Becoming Critical anticipated these changes, but was conceptually unable to address them. This was largely because it differentiated a hierarchy of distinct types of action research grounded in quasi–transcendental categories of knowledge constitutive interests. In the course of the critique of Becoming Critical the author proposes a more philosophically pragmatic account of action research that integrates the ‘critical’ as an intrinsic dimension of practical discourse.     

Equipping novice teachers with a learning map to enhance teaching practice

Using tools to support learning design has been proven feasible in improving the integration of technology into the curriculum. However, novice teachers are faced with two major issues, including their limited experience in learning design and limited ability in using new technologies. Learning map is explored and developed in e-Textbooks to serves as a scaffolding for novice teachers during their development into expert teachers. This paper presents a case study in which learning map is integrated into learning design with associated resources in an e-Textbook authoring environment. The participants were instructed to complete a learning design task for an e-Textbook during a one-hour session and then complete a questionnaire afterward. The results indicate that novice teachers value the learning map for learning design in e-Textbooks regardless of their background in ICT teaching and learning design. Novice teachers adopt positive attitudes toward the use of templates because of their numerous benefits. The teachers also suggested that better quality templates should be integrated into the authoring tool to distinguish the advantage of e-Textbook learning design from that of a traditional book and to provide them with additional options.     

Autonomy,beliefs and the learning of elementary mathematics teachers

Seven elementary teachers participated in a project designed to help them learn to teach mathematics according to reform recommendations. Teachers were provided opportunities to learn through both private reflection and public inquiry about their teaching and children's learning. The teachers’ instruction, reflection, and beliefs were studied. All of the teachers adopted some reform-based procedures including having children report problem-solving strategies. However, only three of them developed more complex practice in which children were involved in inquiry into one another's strategies. The groups had different beliefs about the autonomy of children to construct mathematics and their own autonomy to make instructional decisions.     

Only connect: Troubling oppositions in gender and mathematics

This paper focuses on the ways in which many researchers working in the area of gender and mathematics make sense of their data. In particular, it is argued that their use of the oppositional framing, separation versus connection (and others, such as cognition versus affect and objective versus subjective), operates to fix difference, and so to fix gender and mathematics within a structure of binary thinking that ultimately serves to re/produce gender inequalities. The aim is to suggest a more productive approach to understanding the continued gendering of participation in mathematics. This approach is based in deconstructing these oppositional patterns. This theoretical approach is illustrated using readings of interviews with two young mathematics students, Analia and Phil, talking about their relationships with the subject. The paper ends by looking at what this unfixing of difference means for mathematics pedagogy.     

Preservice teachers and parents: using a reading course to change perceptions and practice

The authors sought to understand preservice teachers’ views about parents of students who struggle with reading and about their own preparedness to deal with such parents. Research, including surveys, student evaluation and tutoring intervention, indicates that before their work with parents and students, preservice teachers held strong beliefs about parents’ role and responsibilities with respect to their children who find reading daunting. After a semester‐long reading course which provided opportunities to work with struggling readers and their parents, the preservice teachers in this study expressed beliefs that were contrary to the ones they offered at the beginning of the course.     

Coach learning to help teachers learn to enact conceptually rich,student-focused mathematics lessons

The past decade has witnessed a strong, standards-based call for improving what mathematics is taught and how it is taught. In the USA, districts have hired instructional coaches to help teachers shift their teaching from algorithm-based instruction to instruction that is more student-centered and conceptually focused. The purpose of this study was to contribute to the field’s understanding of (a) the specific coaching practices that help teachers enact more conceptual-based forms of instruction; and (b) how coaches learn to enact those practices. Using a design-based implementation research approach, we trained coaches using a particular model for one-on-one coaching (Content-Focused Coaching); the coaches then worked with teachers to plan lessons aligned with the coaching model. Data consisted of videotapes of pre-lesson conferences that were transcribed and coded according to the model. Analyses of 32 coaches’ practice over a 2-year period suggest that each of the three components of our coaching model (attention to student thinking, pedagogy, and mathematics) demonstrated statistically significant improvement over time. An illustrative analysis of five coaching sessions of one coach revealed a progression over five sessions from planning discussions that stayed at the level of general strategies to more specific conversations about teaching a particular task and then to deeper discussions that integrate attention to mathematical concepts, student thinking, and pedagogical moves. We view this delineation of coach learning as an important first step in laying the groundwork for the design of future coach training.

     

Learning environment and anxiety for learning and teaching mathematics among preservice teachers

Learning Environments Research - The learning environment has been found to be related to mathematics anxiety at a variety of educational levels, including higher education, but to date has not...     

An unexpected relationship between failure and subsequent mathematics learning

135 high schoolSs were presented with one of three treatment programs: rule (R); rule, then discovery (RD); and discovery, then rule (DR). AllSs were tested on the training rules and on three transfer problems. No significant differences were found on the transfer problems, but the R and RDSs learned the training rules better than the DRSs (p<0.01). It was proposed that failure on the discovery task may have had an inhibiting effect on subsequent learning on the R program.The authors wish to thank the students who participated in the experiment, the members of a teachers' research seminar supported by a government grant to the Montgomery County Instructional Center, and particularly to Mr. Philip Beck and Mr. Philip O'Neil without whose help and cooperation the study could not have been conducted.     

Pre-service teachers teaching about and across cultures using digital environments: the case of eTutor

Increasingly, school teachers are expected to be able to teach about and across cultures, as well as teach in digitally enriched classrooms. Teacher educators are expected to prepare their teaching students for such challenges. One university designed the eTutor online environment to enhance and support the development of pre-service teachers’ (PSTs) intercultural capability, as well as enhance their capability to use digital technologies in their own teaching practices. Over 450 PSTs and school students from eight schools across four countries interacted with each other online for a 14-week period. Participants posted blogs, comments, wall posts and shared images and videos centred on cultural exchange. Despite the challenges of access to technology and conflicting calendars, eTutor was seen by participating schools as a potentially useful way to engage their students in intercultural learning. The experience of interacting with children from multiple cultures in an online environment contributed positively to a shift, for the majority of PSTs, from an ethnocentric to a more ethnorelative worldview. However, PSTs found establishing relationships with students in the online environment challenging.