What does it mean to critique? An exploration of the types of critiques promoted in seventh-grade mathematics curriculum and assessment materials

While research in mathematics education has shown that mathematics assessments are highly consequential, traditional assessments often lag behind advancements in instructional methods. One such advancement is the promotion of mathematical habits of mind such as students' abilities to critique others' reasoning. This study explored the use of student work embedded in seventh-grade curriculum-based mathematics assessment tasks as a mechanism for critiquing others' thinking. The researcher investigated the prevalence and nature of student work in assessment tasks as compared to textbook tasks from five seventh-grade, Common Core State Standards for Mathematics-aligned curriculum series. The text analyses findings revealed that while there were multiple critique types in student work across both assessment and textbook tasks, there were substantial differences between students' opportunities to make sense of someone else's mathematical thinking in curriculum-based assessments as compared to the student textbooks. These findings reinforced prior curriculum and assessment research that found assessment often lags behind instructional methods.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that "works," the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.     

Developing and refining a framework for mathematical and linguistic complexity in tasks related to rates of change

We present and develop a preliminary framework for describing the relationship between the mathematical and linguistic complexity of instructional tasks used in secondary mathematics. The initial framework was developed through a review of relevant literature. It was refined by examining how 4 ninth grade mathematics teachers of linguistically diverse groups of students described the linguistic and mathematical complexity of a set of tasks from their curriculum unit on linear functions. We close by presenting our refined framework for describing the interaction of linguistic complexity and mathematical complexity in curriculum materials, and discuss potential uses of this framework in the design of more accessible classroom learning environments for linguistically diverse students.     

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.     

Planning District‐Wide Professional Development: Insights Gained From Teachers and Students Regarding Mathematics Teaching in a Large Urban District

This paper describes the mechanism used to gain insights into the state of the art of mathematics instruction in a large urban district in order to design meaningful professional development for the teachers in the district. Surveys of close to 2,000 elementary, middle school, and high school students were collected in order to assess the instructional practices used in mathematics classes across the district. Students were questioned about the frequency of use of various instructional practices that support the meaningful learning of mathematics. These included practices such as problem solving, use of calculators and computers, group work, homework, discussions, and projects, among others. Responses were analyzed and comparisons were drawn between elementary and middle school students' responses and between middle school and high school responses. Finally, fifth‐grade student responses were compared to those of their teachers. Student responses indicated that they had fewer inquiry‐based experiences, fewer student‐to‐student interactions, and fewer opportunities to defend their answers and justify their thinking as they moved from elementary to middle school to high school. In the elementary grades students reported an overemphasis on the use of memorization of facts and procedures and sparse use of calculators. Results were interpreted and specific directions for professional development, as reported in this paper, were drawn from these data. The paper illustrates how student surveys can inform the design of professional development experiences for the teachers in a district.     

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.     

A ��task-centric approach�� to professional development: enhancing and sustaining mathematics teachers�� ability to implement cognitively challenging mathematical tasks

In this article, we describe a task-centric approach to professional development for mathematics teachers in which teachers?? learning experiences are focused on the selection and implementation of cognitively challenging mathematical tasks. We examined teachers?? selection and implementation of cognitively challenging tasks at three points in time: before and after their participation in the professional development initiative and during a follow-up data collection 2?years later. Data included instructional tasks, samples of student work, and classroom observations, and were compared between the time points to identify changes in teachers?? task selection and implementation and to determine whether these changes were sustained over time. Results indicate that teachers increased and sustained their ability to select high-level instructional tasks and to maintain the level of cognitive demand during instruction. All teachers, however, did not exhibit this pattern. Portraits of teachers who continued to select and enact tasks at a high level are contrasted with those who did not, and factors are identified to account for teachers?? current practices.     

Exploring the Use of New Representations as a Resource for Teacher Learning

An important goal of mathematics education reform is to support teacher learning. Toward this end, researchers and teacher educators have investigated ways in which teachers learn about mathematical content, pedagogical strategies, and student thinking as they implement reform. This study extends such work by examining how one elementary school and one high school teacher learned from students' interpretations of new conceptually based representations contained in instructional materials aligned with the Principles and Standards for School Mathematics ( National Council of Teachers of Mathematics, 2000 ). Results indicated that teaching with new representations provided a rich context for teacher learning at both the elementary and high school level, and three dimensions were identified along which such learning occurred. The results suggest that pedagogical content knowledge with respect to representations is an important facet of teacher cognition that should be studied in greater depth.     

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.     

Sequence limits in calculus: using design research and building on intuition to support instruction

This article is based on research completed within an ongoing project to develop a calculus course which serves as the foundation for the mathematical education of undergraduate students who are training to become elementary teachers. Several research-based activities have been developed, tested, and refined. In this paper we discuss how the design research approach was used to create and implement an instructional task that introduces the concept of limit of a sequence using popular characters from a children’s television show. We present the intuition that students brought to the instructional sequence, the development of the tasks based on the instructional design theory of Realistic Mathematics Education, and the evolution of the intuition that students displayed after instruction. Results include the instructional task developed and student work which reveals that students use context, informal notions of limit, and the notion of “arbitrarily close” to write about their limit understandings.     

The influence of curriculum material design on opportunities for student learning

This paper explores how the design features of curriculum materials might influence potential opportunities to learn and student outcomes through a comparative, documentary analysis of four curriculum materials commonly used in the US. The four curriculum programs were developed to align with different theories of learning and mathematical goals and have strongly differing design features. Using findings of a large-scale, experimental study of the effects of the four curricula on student achievement as a backdrop, we analyzed the designs of the curriculum materials with respect to three analytical categories: (a) mathematical emphasis, (b) instructional approach, and (c) support for teachers. Results of the analysis reveal substantially different types of opportunities to learn across the four sets of materials. We hypothesize how these differences could explain the achievement results of the experimental study and point to additional influential factors to consider.     

The Impact of Enacted Mathematics Curriculum Models on Prospective Elementary Teachers' Course Perceptions and Beliefs

This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.     

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.     

Impact of the Science and Technology for Children Curriculum in the Oshkosh Area School District

This study examines the instructional impact of National Science Resources Center's Science and Technology for Children curriculum in the Oshkosh Area School District. Specifically, the instructional effectiveness of four physical‐science units for grades 1‐4 were investigated. Students were pre‐ and posttested using a multiple‐choice exam containing items adapted from the Third International Mathematics and Science Study, National Association of Educational Progress, TerraNova, and other widely recognized sources. Results were compared with existing instructional materials. Extensive resources for data interpretation include audiotaped lessons, classroom observations, interviews with students and teachers, student‐attitude surveys, and observations of teacher training. Results suggest that the adoption of this curriculum among experienced teachers in the district will provide little or no immediate gains on student achievement and potentially a slight decrease in student attitudes toward science.     

New directions in differential equations: A framework for interpreting students' understandings and difficulties

The purpose of this paper is to offer a framework for interpreting students' understandings of and difficulties with mathematical ideas central to new directions in differential equations. These new directions seek to guide students into a more interpretive mode of thinking and to enhance their ability to graphically and numerically analyze differential equations. The framework reported here is the result of investigating in depth six students' understandings through a series of task-based individual interviews and classroom observations. The two major themes of the framework, the function-as-solution dilemma theme and students' intuitions and images theme, extend previous research on student cognition at the secondary and collegiate level to the domain of differential equations and reflect the increased recognition of situating analyses of student learning within students' learning environment. For new areas of interest such as differential equations, mapping out students' understandings of important mathematical ideas can be an important part of curricular and instructional design that seeks to refine and build on students' ways of thinking.     

Investigating Elementary Teachers' Conceptions of Animal Classification

Numerous studies have been conducted regarding alternative conceptions about animal diversity and classification, many of which have used a cross‐age approach to investigate how students' conceptions change over time. None of these studies, however, have investigated teachers' conceptions of animal classification. This study was intended to augment the findings of past research by exploring the conceptions that elementary teachers possess about animal classification. Using interviews and written items, we documented teachers' conceptions about animal classification and compared them with student conceptions identified in previous research studies. Many of the teachers' conceptions observed in this study were similar to students' conceptions in that they were often too limited or too general compared with scientifically accepted conceptions. Also, the teachers in this study frequently used “non‐defining” characteristics, such as locomotion and habitat, to classify animals. As a result, several misclassifications were observed in the teachers' responses to the written items. Notably, the results of our study demonstrate that teachers often have the same alternative conceptions about animal classification as students. We explore some possible explanations for these alternative conceptions and discuss the instructional implications of the findings.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Local Instruction Theories as Means of Support for Teachers in Reform Mathematics Education

This article focuses on a form of instructional design that is deemed fitting for reform mathematics education. Reform mathematics education requires instruction that helps students in developing their current ways of reasoning into more sophisticated ways of mathematical reasoning. This implies that there has to be ample room for teachers to adjust their instruction to the students' thinking. But, the point of departure is that if justice is to be done to the input of the students and their ideas built on, a well-founded plan is needed. Design research on an instructional sequence on addition and subtraction up to 100 is taken as an instance to elucidate how the theory for realistic mathematics education (RME) can be used to develop a local instruction theory that can function as such a plan. Instead of offering an instructional sequence that “works,” the objective of design research is to offer teachers an empirically grounded theory on how a certain set of instructional activities can work. The example of addition and subtraction up to 100 is used to clarify how a local instruction theory informs teachers about learning goals, instructional activities, student thinking and learning, and the role of tools and imagery.