Interpreting Middle School Students' Proportional Reasoning Strategies: Observations From Preservice Teachers

This paper reports the findings of an investigation of 11 preservice secondary school teachers' interpretations of the development of proportional reasoning strategies used by middle school students. The preservice teachers examined samples of solution strategies generated by middle school students in proportional reasoning situations and prepared written responses of their views concerning the developmental levels indicated in the students' work. Each preservice teacher also participated in an hour‐long interview, in which the researchers asked for elaboration and clarification of the written responses and, in some cases, challenged the preservice teachers to consider alternative interpretations for the middle school students' work. The interviews were audiotaped for later analysis by the investigators, and key aspects of both the written and audiotaped responses were entered into a spreadsheet and later tabulated into categories indicating trends in the preservice teachers' interpretations. Some implications for the preparation of preservice middle school science and mathematics teachers are included.     

What Matters Most: A Comparison of Expert and Novice Teachers' Noticing of Mathematics Classroom Events

In this study, we examined 10 expert and 10 novice teachers' noticing of classroom events in China. It was found that both expert and novice teachers, who were selected from two cities in China, highly attended to developing students' mathematics knowledge coherently and developing students' mathematical thinking and ability; they also paid attention to students' self‐exploratory learning, students' participation, and teachers' instructional skills. Furthermore, compared with novice teachers, expert teachers paid greater attention to developing mathematical and high‐order thinking, and developing mathematics knowledge coherently, but paid less attention to teachers' guidance. Moreover, we further illustrated the qualitative differences and similarities in their noticing of classroom events. Finally, we discussed the findings and relevant implications.     

Preservice Elementary Teachers' Conceptions of What Causes Night and Day

Two different procedures were utilized to study 50 preservice elementary teachers' conceptions of what causes night and day on earth. The numbers of responses representing alternative conceptions for the models with written and verbal explanation procedures were 27 and 14, respectively. The alternative conception most frequently expressed for both procedures was that night and day are caused by the earth revolving around the sun. The frequency with which alternative conceptions were expressed by future teachers is a problem which should be addressed with instruction using models.     

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.     

Investigating the Strategies Used by Pre‐Service Teachers in Taiwan When Responding to Number Sense Questions

This study examined the strategies used by pre‐service teachers when responding to number sense related questions. 15 pre‐service teachers from one University in Southern Taiwan were interviewed. Results indicated that about one‐third of these pre‐service teachers were able to use number sense strategies (such as recognizing the number size, using benchmarks, etc.) and the other two‐thirds relied heavily on written algorithms to solve problems. This is consistent with the findings of the earlier studies ( Reys & Yang, 1998 ; Yang & Reys, 2002 ; Yang, 2003 ), which state that fifth, sixth and eighth graders in Taiwan rely heavily on the written method when responding to number sense related questions. This implies that the performance of pre‐service elementary teachers on number sense is low. If we want to improve elementary students' knowledge and use of number sense, then we should try to improve the ability of their future teachers' number sense. This supports the statement of Ma (1999) which stated that “to empower students with mathematical thinking, teachers should first be empowered (p. 105).”     

Mathematics teachers’ conceptions about modelling activities and its reflection on their beliefs about mathematics

The current study examines whether the engagement of mathematics teachers in modelling activities and subsequent changes in their conceptions about these activities affect their beliefs about mathematics. The sample comprised 52 mathematics teachers working in small groups in four modelling activities. The data were collected from teachers' Reports about features of each activity, interviews and questionnaires on teachers' beliefs about mathematics. The findings indicated changes in teachers' conceptions about the modelling activities. Most teachers referred to the first activity as a mathematical problem but emphasized only the mathematical notions or the mathematical operations in the modelling process; changes in their conceptions were gradual. Most of the teachers referred to the fourth activity as a mathematical problem and emphasized features of the whole modelling process. The results of the interviews indicated that changes in the teachers' conceptions can be attributed to structure of the activities, group discussions, solution paths and elicited models. These changes about modelling activities were reflected in teachers' beliefs about mathematics. The quantitative findings indicated that the teachers developed more constructive beliefs about mathematics after engagement in the modelling activities and that the difference was significant, however there was no significant difference regarding changes in their traditional beliefs.     

“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.     

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.     

Learning to Teach: Prospective Teachers' Evaluation of Students' Written Responses on a 1992 NAEP Graphing Task

This article describes how prospective elementary teachers examined, analyzed, and evaluated four students' written responses on a graphing task for an end‐of‐course performance assessment in a mathematics methods course. Also, they described teaching strategies that built on what students know and do not know, as shown in the fourth‐grade students' work. This course assessment provided evidence of the prospective teachers' pedagogical content knowledge. Two themes emerged in the context of this final course project: the importance of process and correct answers and the usefulness of creating rubrics.     

Making Connections: Elementary Teachers' Construction of Division Word Problems and Representations

If teachers make few connections among multiple representations of division, supporting students in using representations to develop operation sense demanded by national standards will not occur. Studies have investigated how prospective and practicing teachers use representations to develop knowledge of fraction division. However, few studies examined primary (K‐3) teachers' learning of contextual division problems, making connections among representations of division, and resolving the ambiguity of representing quotients with remainders. A written post‐course assessment provided evidence that most teachers created partitive division word problems, used a set model without splitting the remainder, and wrote equations with limited success. Post‐course written reflections demonstrated that many teachers developed pedagogical knowledge for helping students make connections among multiple representations, and mathematical knowledge of unit fractions. These findings suggest two areas that have implications for mathematics teacher educators who design professional development courses to facilitate teachers' learning of mathematical content and pedagogical knowledge of division and fraction relationships.     

Comparing Science Teaching Styles to Students' Perceptions of Scientists

Many educational researchers seem to concur with the idea that, among other factors, the teacher's teaching style has some impact on student learning and the perceptions students develop about science learning and the work of scientists. In this study, nine middle grades teachers' teaching styles were assessed using the Draw‐a‐Science‐Teacher‐Teaching Test Checklist (DASTT‐C) and categorized along a continuum from didactic to inquiry/constructivist in orientation. Students' (n = 339) perceptions of scientists were determined using the Draw‐a‐Scientist‐Test Checklist (DAST‐C). Teachers' teaching styles and their students' perceptions of scientists were then compared using nonparametric correlational methods. Results showed that no significant correlation existed between the two measures for the population studied. Although the study provides no understanding about when or how relationships developed between teachers' teaching styles and students' perceptions of scientists, trends in the results give rise to some concerns regarding the preparation of future science teachers and the in‐service development of practicing teachers.     

Turkish Prospective Chemistry Teachers' Alternative Conceptions about Acids and Bases

The purpose of this study was to obtain prospective chemistry teachers' conceptions about acids and bases concepts. Thirty‐eight prospective chemistry teachers were the participants. Data were collected by means of an open‐ended questionnaire and semi‐structured interviews. Analysis of data indicated that most prospective teachers did not have difficulties about macroscopic properties of acids and bases. However, despite chemistry instruction, most of the prospective teachers were found to have problems in understanding the neutralization concept, the distinction between strength and concentration of acids and linking the acids and bases topic to daily life. These findings have some implications for teacher education programs.     

The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

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.     

Are You Convinced? Middle‐Grade Students' Evaluations of Mathematical Arguments

Students learn norms of proving by observing teachers generating proofs, engaging in proving, and generalizing features of proofs deemed convincing by an authority, such as a textbook. Students at all grade levels have difficulties generating valid proof; however, little research exists on students' understandings about what makes a mathematical argument convincing prior to more formal instruction in methods of proof. This study investigated middle‐school students' (ages 12–14) evaluations of arguments for a statement in number theory. Students evaluated both an empirical and a general argument in an interview setting. The results show that students tend to prefer empirical arguments because examples enhance an argument's power to show that the statement is true. However, interview responses also reveal that a significant number of students find arguments to be most convincing when examples are supported with an explanation that “tells why” the statement is true. The analysis also examined the alignment of students' reasons for choosing arguments as more convincing along with the strategies they employ to make arguments more convincing. Overall, the findings show middle‐school students' conceptions about what makes arguments convincing are more sophisticated than their performance in generating arguments suggests.     

Using Video‐Based Pedagogy in an Elementary Mathematics Methods Course

The study explored the impact of using video‐based pedagogy on preservice teachers' cognitions about teaching mathematics. The use of video‐based pedagogy was integrated into the course, Methods for Teaching Elementary School Mathematics. A variety of written and interview data were collected during the course and, in the following semester, during student teaching. Evidence from case studies of three preservice teachers indicates that they engaged in reflection and reconstruction of their beliefs about how children learn mathematics and moved from a more didactic perspective of teaching mathematics toward a student‐centered perspective. Such movement appears to have been influenced by the use of video‐based pedagogy.     

Beginning Secondary Science Teachers' Conceptualization and Enactment of Inquiry‐Based Instruction

This study investigates the conceptions and use of inquiry during classroom instruction among beginning secondary science teachers. The 44 participants were beginning secondary science teachers in their first year of teaching. In order to capture the participants' conceptions of inquiry, the teachers were interviewed and observed during the school year. The interviews consisted of questions about inquiry instruction, while the observations documented the teachers' use of inquiry. All of the interviews were transcribed or coded in order to understand the conceptions of inquiry held by the teachers, and all of the observations were analyzed in order to determine the presence of inquiry during the lesson. The standard for assessing inquiry came from the National Science Education Standards. A quantitative analysis of the data indicated that the teachers frequently talked about implementing “scientific questions” and giving “priority to evidence.” This study found a consistency between the way new teachers talked about inquiry and the way they practiced it in their classrooms. Overall, our observations and interviews revealed that the beginning secondary science teachers tended to enact teacher‐centered forms of inquiry, and could benefit from induction programs focused on inquiry instruction.     

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.     

Self‐Directed Teacher Learning in Collaborative Contexts

Two related case studies of secondary mathematics teachers examine the roles and conditions helpful in initiating, directing, and/or supporting teachers' own professional development. Using multiple data sources from school‐based and professional settings, we applied analytic induction to identify patterns of similarities and differences in teachers' experiences and developmental trajectories. Findings revealed how the teachers served as brokers in their own development process and how attitude, attention, and awareness played important roles in teacher learning. As the teachers initiated learning processes, they demonstrated awareness of broader educational contexts that support learning by identifying external policies and priorities that aligned with teachers' attitudes. They then used these external priorities as boundary objects to broker for teachers' learning, including other teachers' learning. They attended to learning needs in directing a plan for learning activities, rather than using externally mandated goals for changing practice (e.g., an external focus on improving students' tests scores). These cases suggest that in structuring professional learning, opportunities to self‐identify learning needs should be part of the process, and external supports can become more powerful when teachers play a brokering role in relating these supports to their immediate contexts. Further implications for professional development processes are explored.