Teachers' ways of listening and responding to students' emerging mathematical models

In this paper, I present the results of a case study of the practices of four experienced secondary teachers as they engaged their students in the initial development of mathematical models for exponential growth. The study focuses on two related aspects of their practices: (a) when, how and to what extent they saw and interpreted students' ways of thinking about exponential functions and (b) how they responded to the students' thinking in their classroom practice. Through an analysis of the teachers' actions in the classroom, I describe the teachers' developing knowledge when using modeling tasks with secondary students. The analysis suggests that there is considerable variation in the approaches that teachers take in listening to and responding to students' emerging mathematical models. Having a well-developed schema for how students might approach the task enabled one teacher to press students to express, evaluate, and revise their emerging models of exponential growth. Implications for the knowledge needed to teach mathematics through modeling are discussed.     

Preparing K‐8 Preservice Teachers in a Content Course for Standards‐Based Mathematics Pedagogy

Many K–8 preservice teachers have not experienced learning mathematics in a standards‐based classroom. This article describes a mathematics content course designed to provide preservice teachers experiences in learning mathematics that will help build a solid foundation for a standards‐based methods course. The content course focuses on developing preservice teachers' mathematical knowledge, as well as helping them realize what it means to learn mathematics that is taught using the pedagogy in the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Furthermore, findings are presented from a study on this course that describe students' pre‐ and postcourse beliefs, attitudes, and perceptions of what it means to learn and teach mathematics. These findings provide evidence that the students in the study are beginning to understand what is meant by a standards‐based classroom. Data were collected from surveys and interviews. Quotes from the students who aspire to be elementary teachers are used throughout the article to support the points.     

Teachers' Reflections on Their Subject Matter Knowledge Structures and Their Influence on Classroom Practice

Research has indicated that experts' subject matter knowledge structures (SMKSs) differ from those of novices in that they contain more cross‐linking, interconnections, and overarching thematic elements, characteristics that are in accordance with those espoused in current reform documents. Unfortunately, teachers' SMKSs are not necessarily translated into classroom practice, for either novice or more experienced classroom teachers. A means to facilitate the translation of teachers' SMKSs into practice would ensure that those desired characteristics of experts' subject matter knowledge manifest themselves in teachers' classroom practice. Four experienced physics teachers diagrammed their SMKSs, which were then compared to those inferred from their classroom practice. Prior to instruction, two teachers, as part of the explicit‐reflective treatment, were asked to reflect at multiple time points on congruence between their SMKSs and classroom practice focusing on the presence of essential concepts, interconnections, and overarching thematic elements. No discernible difference was apparent between control and treatment groups, as teachers from both groups showed a high‐degree of congruence between inferred and diagrammed SMKSs. Results further substantiate the challenges in identifying a means for both developing and facilitating the enactment of coherent, connected, and dynamic SMKSs or, in effect, accelerating teachers' pedagogical content knowledge.     

“Mathematicians Would Say It This Way”: An Investigation of Teachers' Framings of Mathematicians

Although popular media often provides negative images of mathematicians, we contend that mathematics classroom practices can also contribute to students' images of mathematicians. In this study, we examined eight mathematics teachers' framings of mathematicians in their classrooms. Here, we analyze classroom observations to explore some of the characteristics of the teachers' framings of mathematicians in their classrooms. The findings suggest that there may be a relationship between a teachers' mathematics background and his/her references to mathematicians. We also argue that teachers need to be reflective about how they represent mathematicians to their students, and that preservice teachers should explore their beliefs about what mathematicians actually do.     

From formal standards to everyday practice of mathematics learning

This paper uses the example of six Japanese teachers and their mathematics lessons to illustrate how clear, high standards for mathematics instruction are combined with teachers' holistic concern for students. We draw upon data from the Third International Math and Science Study Case Study Project in Japan that was designed to elucidate the context behind the high achievement of Japanese students. Using everyday examples of classroom practice, we illustrate both flexibility in teachers' approach to teaching and adherence to Monbusho's (Ministry of Education, Science, Sports, and Culture)Course of Study. Our purpose is to emphasize how flexibility and attention to individual needs by Japanese teachers combine with quality mathematics instruction based on the detailed Japanese curricula. Six teachers' characteristics and lessons (two teachers at each educational level—elementary, junior high, and high school) are described in order to show the variety of teachers who exist in Japan. These teachers use their understanding of theCourse of Study and are supported by their school environment to enhance their students' conceptual understanding of the fundamentals of mathematics. Characteristics of their teaching include: 1) involving the whole class in learning. 2) using extremely focused curriculum guidelines that expect mastery of concepts at each grade level, 3) thoroughly covering mathematics units in an organized and in-depth manner, 4) leading classes as facilitators or guides more often than as lecturers, and 5) focusing on problem solving with the primary goal of developing students' ability to reason, especially to reason inductively. The examples in this paper show how these methods develop in individal classrooms.     

Mathematical Connections and Their Relationship to Mathematics Knowledge for Teaching Geometry

Effective competition in a rapidly growing global economy places demands on a society to produce individuals capable of higher‐order critical thinking, creative problem solving, connection making, and innovation. We must look to our teacher education programs to help prospective middle grades teachers build the mathematical habits of mind that promote a conceptually indexed, broad‐based foundation of mathematics knowledge for teaching which encompasses the establishment and strengthening of mathematical connections. The purpose of this concurrent exploratory mixed methods study was to examine prospective middle grades teachers' mathematics knowledge for teaching geometry and the connections made while completing open and closed card sort tasks meant to probe mathematical connections. Although prospective middle grades teachers' mathematics knowledge for teaching geometry was below average, they were able to make over 280 mathematical connections during the card sort tasks. Curricular connections made had a statistically significant positive impact on mathematics knowledge for teaching geometry.     

The Road to Reform: A Grounded Theory Study of Parents' and Teachers' Influence on Elementary School Science and Mathematics

Though elementary teacher educators introduce new, reform‐based strategies in science and mathematics methods courses, researchers wondered how novices negotiate reform strategies once they enter the elementary school culture. Given that the extent of parents' and veteran teachers' influence on novice teachers is largely unknown, this grounded theory study explored parents' and teachers' expectations of children's optimal science and mathematics learning in the current era of reform. Data consisted of semi‐structured, open‐ended interviews with novice teachers (n = 20), veteran teachers (n = 9), and parents (n = 28). Researchers followed three stages of coding procedures to develop a logic model connecting participants' discrete designations of the landscape, regulating phenomena, contextual orientation, and desired outcomes. This logic model helped researchers develop propositions for future research on the interactive nature of parents' and teachers' influential role in elementary science and mathematics education. Implications encourage science and mathematics teacher educators—as well as school administrators—to explicitly develop and support novice teachers' ability to initiate and sustain parent/family engagement in order to create a school climate where teachers and parents are synergistically motivated to change.     

Studying sample lessons rather than one excellent lesson: A Japanese perspective on the TIMSS videotape classroom study

The findings of the TIMSS Videotape Classroom Study include aspects of mathematics lessons showing a strong resemblance between Germany and the US in difference to Japan. This paper discusses some of the features that appear to make Japanese lessons different from the other two countries. In particular, the paper examines the goals of lessons described by Japanese teachers, how lessons are structured and implemented, and the emphasis on alternative solutions to a problem in the teaching and learning processes. The characteristics of Japanese lessons identified by the TIMSS Videotape Classroom Study can naturally be interpreted as indications of teachers' efforts to foster students' mathematical thinking in the classroom.     

Math Autobiographies: A Window into Teachers' Identities as Mathematics Learners

Mathematics autobiographies have the potential to help teachers reflect on their identities as mathematics learners and to understand their role in the development of their students' mathematics identities. This paper reports on a professional development project for K‐2 teachers (n = 41), in which participants were asked to write mathematics autobiographies. Using an adaptation of an existing framework for characterizing teachers' mathematics stories, we describe the consistencies among the participants' experiences as mathematics learners and the events that are identified as being the impetus for a transition from a negative to a positive attitude toward mathematics. Implications for both teachers and teacher educators are presented.     

Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

U.S. and Chinese Teachers' Constructing,Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the "outline and worksheet" format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

Mathematics in Children's Thinking

The human mind inevitably comprehends the world in mathematical terms (among others). Children's informal and invented mathematics contains on an implicit level many of the mathematical ideas that teachers want to promote on a formal and explicit level. These ideas may be innate, constructed for the purpose of adaptation, or picked up from an environment that is rich in mathematical structure, regardless of culture. Teachers should attempt to uncover the mathematical ideas contained in their students' thinking because much, but not all, of the mathematics curriculum is immanent in children's informal and invented knowledge. This mathematical perspective requires a focus not only on the child's constructive process but also on the mathematical content underlying the child's thinking. Teachers then can use these crude ideas as a foundation on which to construct a significant portion of classroom pedagogy. In doing this, teachers should recognize that children's invented strategies are not an end in themselves. Instead, the ultimate goal is to facilitate children's progressive mathematization of their immanent ideas. Children need to understand mathematics in deep, formal, and conventional ways.     

Challenging Preservice Teachers' Mathematical Understanding: The Case of Division by Zero

Preservice elementary school teachers' fragmented understanding of mathematics is widely documented in the research literature. Their understanding of division by 0 is no exception. This article reports on two teacher education tasks and experiences designed to challenge and extend preservice teachers' understanding of division by 0. These tasks asked preservice teachers to investigate division by 0 in the context of responding to students' erroneous mathematical ideas and were respectively structured so that the question was investigated through discussion with peers and through independent investigation. Results revealed that preservice teachers gained new mathematical (what the answer is and why it is so) and pedagogical (how they might explain it to students) insights through both experiences. However, the quality of these insights were related to the participants' disposition to justify their thinking and (or) to investigate mathematics they did not understand. The study's results highlight the value of using teacher learning tasks that situate mathematical inquiry in teaching practice but also highlight the challenge for teacher educators to design experiences that help preservice teachers see the importance of, and develop the tools and inclination for, mathematical inquiry that is needed for teaching mathematics with understanding.     

Problem solving heuristics,affect, and discrete mathematics

It has been suggested that activities in discrete mathematics allow a kind of new beginning for students and teachers. Students who have been “turned off” by traditional school mathematics, and teachers who have long ago routinized their instruction, can find in the domain of discrete mathematics opportunities for mathematical discovery and interesting, nonroutine problem solving. Sometimes formerly low-achieving students demonstrate mathematical abilities their teachers did not know they had. To take maximum advantage of these possibilities, it is important to know what kinds of thinking during problem solving can be naturally evoked by discrete mathematical situations—so that in developing a curriculum, the objectives can include pathways to desired mathematical reasoning processes. This article discusses some of these ways of thinking, with special attention to the idea of “modeling the general on the particular.” Some comments are also offered about students' possible affective pathways and structures.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Rural teacher leadership in science and mathematics

This study adds to our understanding of science and mathematics teacher leadership in rural schools. Through In Vivo and Concept coding of teacher interviews, we investigated 20 secondary science and mathematics teachers' perceptions of rural teacher leadership during their participation in a three-year professional development program. As the teachers developed as teacher leaders, they broadened their focus from improving their own students' learning to sharing new knowledge learned through the program with other teachers both informally and formally. We compared our program components to the Teacher Leader Model Standards and added an emphasis on the importance of disciplinary content knowledge. We also identified patterns in science and mathematics teacher leadership that are contextually connected to teachers' instruction in rural high poverty schools. Rural teacher leadership included the importance of building strong teacher–student relationships, providing new academic opportunities for students, encouraging students' success, and building community connections.     

Teachers attending to student reasoning: Do beliefs matter?

We present the results of a quasi-experimental study of pre-service elementary teachers' learning to recognize students' mathematical reasoning from classroom videos. Researchers examined the nature of participants’ beliefs regarding mathematics education. We found that pre-service elementary teachers whose beliefs were consistent with NCTM Process Standards (NCTM, 2000), or that transitioned in the direction of consistency with the Standards, regarding the teaching and learning of mathematics, were more successful in recognizing students' reasoning than those whose beliefs were generally inconsistent. Predictive Analytics and Generalized Linear Regression modeling were used to quantify the magnitude of experimental pre-service teachers’ reasoning growth and combined pre/post study assessment reasoning success in contrast to that of the comparison groups. The resulting model explained nearly 90% of the variability in success on the reasoning assessment, showing that beliefs do indeed matter for recognition of reasoning.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Conceptual Representations of Flu and Microbial Illness Held by Students,Teachers, and Medical Professionals

This study describes 5 th, 8th, and 11th‐grade students', teachers', and medical professionals' conceptions of flu and microbial illness. Participants constructed a concept map on “flu” and participated in a semi‐structured interview. The results showed that these groups of students, teachers and medical professionals held and structured their conceptions about microbes differently. A progression toward more accurate and complete knowledge existed across the groups but this trajectory was not always a predictable, linear developmental path from novice to expert. Across the groups, participants were most knowledgeable about symptoms of microbial illness, treatments of symptoms, and routes of transmission for respiratory illnesses. This knowledge was tightly linked to participants' prior experiences with colds and flu. There were typically large gaps in participants' (children and teachers) understandings of vaccines, immune system responses, treatments (including the mechanisms of pain medications and the functions of antibiotics), and transmission of non‐respiratory microbial illness. A common misconception held by students was the belief that antibiotics can cure viral infections.