Pre-service mathematics teachers become full participants in inquiry investigations

Research is described concerning the effectiveness of inquiry-based laboratory environments created in US mathematics/science education programme courses. Laboratory projects were conducted using a framework that allowed pre-service teachers to explore, analyse, and communicate ‘investigable’ realms of physical phenomena. Goals were for pre-service teachers to experience the value of learning in an inquiry-enhanced environment and to engage in contextualized mathematics so they would utilize this instruction in their future classrooms. It is proposed that inquiry-based laboratories are needed within the mathematics classroom in order to allow students the opportunity to contextualize, to connect to other disciplines, and to experience mathematical concepts. Pre-service teachers were expected to pursue conjectures, collect data, think critically, and communicate findings. This qualitative research shows how the use of inquiry can complement the learning of mathematical content and educational strategies for pre-service teachers. Results provide detailed information for teacher educators regarding instructional design of contextualized mathematical inquiry.     

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.     

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.     

MobileMath: exploring mathematics outside the classroom

Computer games seem to have a potential for engaging students in meaningful learning, inside as well as outside of school. With the growing availability of mobile handheld technology (HHT), a number of location-based games for handheld mobile phones with GPS have been designed for educational use. The exploitation of this potential for engaging students into meaningful learning, however, so far remains unexplored. In an explorative design research, we investigated whether a location-based game with HHT provides opportunities for engaging in mathematical activities through the design of a geometry game called MobileMath. Its usability and opportunities for learning were tested in a pilot on three different secondary schools with 60 12–14-year-old students. Data were gathered by means of participatory observation, online storage of game data, an online survey and interviews with students and teachers. The results suggest that students were highly motivated, and enjoyed playing the game. Students indicated they learned to use the GPS, to read a map and to construct quadrilaterals. The study suggests learning opportunities that MobileMath provides and that need further investigation.     

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.     

Calculators in Elementary School Mathematics Instruction

Due to the increased availability of hand-held calculators, teachers at all grade levels must now face the prospect of having to change both how they teach mathematics as well as what mathematics they teach. Since most teachers did not learn mathematics with the help of technology, they need time to adjust to both a new learning environment and a new teaching one. Through federal funds, the Texas Education Agency has created mathematics staff development modules which help teachers learn about calculators, mathematics, and the integration of calculators in mathematics instruction. This article presents games based upon those included in the staff development modules. Each game was designed to promote exploration of mathematical relationships via a calculator, specifically, Texas Instrument's Math Explorer.     

Teaching-Learning of Evaluative Criteria for Mathematical Arguments Through Classroom Discourse: A Cross-National Study

Videotaped lessons of 5th graders on equivalent fractions from 7 American and 6 Japanese classrooms were analyzed in terms of a recurrent pattern in public discourse among a teacher and students. This pattern—called inquiry, response, feedback—occurs when a teacher initiates discourse (mostly with an inquiry), a student or students respond (often with an answer to the teacher inquiry), and the teacher provides feedback to the student's response. We found2 approaches to the teaching-learning of the criteria for evaluating mathematical arguments. In the Japanese classroom, students were encouraged to offer their own argument to the whole class and evaluate arguments proposed by other students. They seldom were given direct evaluation by their teacher. In contrast, American teachers often gave individual elaboration as well as direct evaluation to the student's responses, and some of the teachers offered their own opinions about mathematics, about valid ways of argumentation, or about both. The Japanese approach would help students acquire evaluative criteria indirectly through participating in mathematical discourse, whereas the American approach would help students learn modes of arguments through direct instruction.     

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.     

The Relationship of Teacher‐Facilitated,Inquiry‐Based Instruction to Student Higher‐Order Thinking

Commissions, studies, and reports continue to call for inquiry‐based learning approaches in science and math that challenge students to think critically and deeply. While working with a group of middle school science and math teachers, we conducted more than 100 classroom observations, assessing several attributes of inquiry‐based instruction. We sorted the observations into two groups based on whether students both explored underlying concepts before receiving explanations and contributed to the explanations. We found that in both math and science classrooms, when teachers had students both explore concepts before explanations and contribute to the explanations, a higher percent of time was spent on exploration and students were more frequently involved at a higher cognitive level. Further, we found a high positive correlation between the percent of time spent exploring concepts and the cognitive level of the students, and a negative correlation between the percent of time spent explaining concepts and the cognitive level. When we better understand how teachers who are successful in challenging students in higher‐order thinking spend their time relative to various components of inquiry‐based instruction, then we are better able to develop professional development experiences that help teachers transition to more desired instructional patterns.     

Conceptual learning in a principled design problem solving environment

To what extent can instructional design be based on principles for instilling a culture of problem solving and conceptual learning? This is the main focus of the study described in this paper, in which third grade students participated in a one-year course designed to foster problem solving and mathematical reasoning. The design relied on five principles: (a) encouragement to produce multiple solutions; (b) creating collaborative situations; (c) socio-cognitive conflicts; (d) providing tools for checking hypotheses; and (e) inviting students to reflect on solutions. We describe how a problem solving task designed according to the above principles, promoted students' understanding of the area concept. We show that the design afforded the surfacing of multiple solutions and justifications in various modalities (including gestures) and initiated peer argumentation, leading to deep learning of the area concept.     

Enhancing language-responsive meaning-making processes as an epistemic catalyst for developing multiplicative reasoning in young children

Multiplicative reasoning involves the ability to coordinate bundled units on a more abstract level (“unitizing”; Lamon, 1994). As it is considered a “cutoff point” for students’ future mathematical learning, teachers must provide equitable access to mathematical conceptual understanding for all students on all mathematical achievement levels. The study presented in this paper investigates to what extent a preventive and a language-responsive instructional approach can have an effect on the outcome of students on different mathematical achievement levels. Three German second grade teachers introduced multiplication to students (n = 66, aged 7–8 years) in their classes using meaning-related phrases (e.g., “6 times 4 means 6 fours”), while teachers in the control group (n = 58) did not focus on using these phrases. Analyses of both a multiplication posttest and a follow-up test showed significant differences between the intervention and control groups on all achievement levels for both conceptual and procedural items.     

A Comparison of Textbooks' Presentation of Fractions

In the United States, fractions are an important part of the middle school curriculum, yet many middle school students struggle with fraction concepts. Teachers also have difficulty with the conceptual understanding needed to teach fractions and rely on textbooks when making instructional decisions. This reliance on textbooks, the idea that teaching and learning of fractions is a complex process, and that fraction understanding is the foundation for later topics such as proportionality, algebra, and probability, makes it important to examine the variation in presentation of fraction concepts in U.S. textbooks, especially the difference between traditional and standards‐based curricula. The purpose of this study is to determine if differences exist in the presentation of fractions in conventional and standards‐based textbooks and how these differences align with the recommendations of National Council of Teachers of Mathematics, Common Core State Standards, and the research on the teaching and learning of fractions.     

The Impact of Professional Noticing on Teachers' Adaptations of Challenging Tasks

This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.     

Towards a framework for developing students' fraction proficiency

The importance of the knowledge of fractions in mathematical learning, coupled with the difficulties students have with them, has prompted researchers to focus on this particular area of mathematics. The term ‘fraction proficiency' used in this article refers to a person's conceptual comprehension, procedural skills and the ability to approach daily situations involving fractions. In the area of fractions, there has been a call for more research to study how, and where, efforts should be focused in order to integrate the various aspects of fraction knowledge for students, and even for teachers, to help them develop proficiency in fractions. Thus, the article presents a theoretical synthesis of the specialized literature in the learning and teaching of fractions, with the aim of proposing a framework for developing students' fraction proficiency. The frameworks presented in the article may shed light upon the implications for the design of fraction instruction, which should focus on developing a multi-faceted knowledge of fractions, rather than simply isolating one facet from the others.     

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

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.     

Emergent modeling: discrete graphs to support the understanding of change and velocity

In this paper we focus on an instructional sequence that aims at supporting students in their learning of the basic principles of rate of change and velocity. The conjectured process of teaching and learning is supposed to ensure that the mathematical and physical concepts will be rooted in students’ understanding of everyday-life situations. Students’ inventions are supported by carefully planned activities and tools that fit their reasoning. The central design heuristic of the instructional sequence is emergent modeling. We created an educational setting in three tenth grade classrooms to investigate students’ learning with this sequence. The design research is carried out in order to contribute to a local instruction theory on calculus. Classroom events and computer activities are video-taped, group work is audio-taped and student materials are collected. Qualitative analyses show that with the emergent modeling approach, the basic principles of calculus can be developed from students’ reasoning on motion, when they are supported by discrete graphs.     

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.     

Ways in which engaging with someone else’s reasoning is productive

Classrooms which involve students in mathematical discourse are becoming ever more prominent for the simple reason that they have been shown to support student learning and affinity for content. While support for outcomes has been shown, less is known about how or why such strategies benefit students. In this paper, we report on one such finding: namely that when students engage with another’s reasoning, as necessitated by interactive conversation, it supports their own conceptual growth and change. This qualitative analysis of 10 university students provides insight into what engaging with another’s reasoning entails and suggests that higher levels of engagement support higher levels of conceptual growth. We conclude with implications for instructional practice and future research.