Components of Place Value Understanding: Targeting Mathematical Difficulties When Providing Interventions

Place value understanding requires the same activity that students use when developing fractional and algebraic reasoning, making this understanding foundational to mathematics learning. However, many students engage successfully in mathematics classrooms without having a conceptual understanding of place value, preventing them from accessing mathematics that is more sophisticated later. The purpose of this exploratory study is to investigate how upper elementary students' unit coordination related to difficulties they experience when engaging in place value tasks. Understanding place value requires that students coordinate units recursively to construct multi‐digit numbers from their single‐digit number understandings through forms of unit development and strategic counting. Findings suggest that students identified as low‐achieving were capable of only one or two levels of unit coordination. Furthermore, these students relied on inaccurate procedures to solve problems with millennial numbers. These findings indicate that upper elementary students identified as low‐achieving are not to yet able to (de)compose numbers effectively, regroup tens and hundreds when operating on numbers, and transition between millennial numbers. Implications of this study suggest that curricula designers and statewide standards should adopt nuances in unit coordination when developing tasks that promote or assess students' place value understanding.     

Evolution of Unit Fraction Conceptions in Two Fifth-Graders with a Learning Disability: An Exploratory Study

The literature seems limited in what is known about conceptual processes that underlie evolution of students with learning disabilities (SLD) conceptions of fractions. This exploratory study examines how a foundational scheme of unit fractions (1/n) may evolve through the mathematical activity of two fifth grade girls. We analyze data segments from episodes conducted during a teaching experiment grounded in the activity of iterating estimates of one person's equal share. Our findings include four distinct conceptual stages: (1) No Conception of the Nature of Adjustment to the Magnitude of a Unit Fraction, (2) Evolving Anticipation of the Nature of Adjustment but not of its Relative Amount, (3) Anticipation of the Nature of Adjustment with an Evolving Partial Amount, and (4) a Dual Anticipation of the Nature and Amount of Adjustment. Findings demonstrate each girl was able to use her constructed scheme to successfully solve and reason about novel problems. We discuss the need for more research to confirm the findings from this study, while offering a conjecture of the possibilities for more SLDs to advance their conceptions of fractions in future interventions.     

Graphical Representations of Tukey's Multiple Comparison Method

Abstract

Tukey's multiple comparison method is widely implemented in statistical packages. We list the inferential and perceptual tasks a graphical representation of Tukey's method may be required to perform, and show that the most commonly used graphical representations (underlining, line-by-line plotting, notched boxplots, and comparison circles) are all lacking in some respect. A new graphical representation, based on the mean-mean scatterplot, is proposed instead. This representation performs all the inferential and perceptual tasks not only in an unbalanced one-way model, but also in a general linear model. An event-driven, color-graphics implementation is presented. Finally, issues for future research, such as color perception and comparative experimentation, are briefly discussed.     

Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a Task–Technique–Theory Perspective

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further Lagrange’s (2000) conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the task-based activity of a pair of 10th grade students—activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.     

An Integrated Research on Children's Construction of Meaningful,Symbolic, Partitioning-related Conceptions and the Teacher's Role in Fostering That Learning

A teaching experiment was conducted with two fourth graders to study the co-emergence of teaching and children's construction of fraction knowledge. The children's learning, i.e., modifications in their fraction schemes, was fostered through working on tasks in a computer microworld. The children advanced from thinking about a unit fraction as one of several equal parts in a whole (the equipartitioning scheme) to operating with a unit fraction as a symbolized, iterable part the magnitude of which is based on the numerosity of the partitioned whole (the partitive fraction scheme). The paper interweaves an analysis of children's construction of partitioning-related symbolic conceptions of fractions with an analysis of the teaching—planning and using tasks—that fosters such an advancement by introducing fraction words and numerals in the context of the children's partitioning activities.     

Middle school students' understanding of the role sample size plays in experimental probability

In this study, we examined the impact of an instructional program on sixth-grade students' understanding of experimental probability as it relates to sample size. As the number of trials in an experiment increases, the experimental probability is more likely to reflect the parent distribution; thus, smaller samples are more likely to yield unusual results. Results of this study indicate that, while typical middle school students are seemingly unaware of the relationship between experimental probability and sample size, appropriate cognitive activity focused on results of simulations of random phenomena can foster conceptual development. We witnessed growth that occurred as a result of key instructional tasks and concomitant mental activity.     

The role of prediction in the teaching and learning of mathematics

The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students’ prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.     

How Students Use Physics to Reason About Calculus Tasks

The present research study investigates how undergraduate students in an integrated calculus and physics class use physics to help them solve calculus problems. Using Zandieh's (2000) framework for analyzing student understanding of derivative as a starting point, this study adds detail to her “paradigmatic physical” context and begins to address the need for a theoretical basis for investigating learning and teaching in integrated mathematics and science classrooms. A case study design was used to investigate the different ways students use physics ideas as they worked through calculus tasks. Data were gathered through four individual interviews with each of 8 ICP students, classroom participant‐observation, and triangulation of the data through student homework and exams. The main result of this study is the Physics Use Classification Scheme, a tool consisting of four categories used to characterize students' uses of physics on tasks involving average rate of change, derivative, and integral concepts. Two of the categories from the Physics Use Classification Scheme are elucidated with contrasting student cases in this paper.     

Understanding a Teacher's Reflections: A Case Study of a Middle School Mathematics Teacher

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students' thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions.     

Examining and developing fourth grade children’s area estimation performance

This study explored children’s area estimation performance. Two groups of fourth grade children completed area estimation tasks with rectangles ranging from 5 to 200 square units. A randomly assigned treatment group completed instructional sessions that involved a conceptual area measurement strategy along with numerical feedback. Children tended to underestimate areas of rectangles. Furthermore, rectangle size was related to performance such that estimation error and variability increased as rectangle size increased. The treatment group exhibited significantly improved area estimation performance in terms of accuracy, as well as reduced variability and instances of extreme responses. Area measurement estimation findings are related to a Hypothetical Learning Trajectory for area measurement.     

Applying Cognition-Based Assessment to Elementary School Students' Development of Understanding of Area and Volume Measurement

As part of a discussion of cognition-based assessment (CBA) for elementary school mathematics, I describe assessment tasks for area and volume measurement and a research-based conceptual framework for interpreting students' reasoning on these tasks. At the core of this conceptual framework is the notion of levels of sophistication. I provide details on an integrated set of levels for area and volume measurement that (a) starts with the informal, preinstructional reasoning typically possessed by students, (b) ends with the formal mathematical concepts targeted by instruction, and (c) indicates cognitive plateaus reached by students in moving from (a) to (b).     

Shortcomings of Mathematics Education Reform in The Netherlands: A Paradigm Case?

This article offers a reflection on the findings of three PhD studies, in the domains of, respectively, subtraction under 100, fractions, and algebra, which independently of each other showed that Dutch students' proficiency fell short of what might be expected of reform in mathematics education aiming at conceptual understanding. In all three cases, the disappointing results appeared to be caused by a deviation from the original intentions of the reform, resulting from the textbooks' focus on individual tasks. It is suggested that this “task propensity”, together with a lack of attention for more advanced conceptual mathematical goals, constitutes a general barrier for mathematics education reform. This observation transcends the realm of textbooks, since more advanced conceptual mathematical understandings are underexposed as curriculum goals. It is argued that to foster successful reform, a conscious effort is needed to counteract task propensity and promote more advanced conceptual mathematical understandings as curriculum goals.     

Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim

This paper critically examines the discrepancies among the pre-requisite fractional concepts assumed by a curricular unit on operations with fractions, the teacher's assumptions about those concepts and a particular student's understanding of fractions. The paper focuses on the case of one student (Tim) in the teacher's 6th grade class who was interviewed by one of the authors once a week during the teaching of the unit. The teaching materials and the teacher's instruction were based on the assumption that students understood the concept of a unit fraction as being one of several equal parts of a given whole. The teacher neither emphasized the need for equal parts nor the part-to-whole relation. The teacher's reasonable assumptions about her students’ understanding of fractions were severely challenged by the cognitive constructs that Tim exhibited during his first two interviews. When she viewed tapes of the class instruction and the interviews with Tim she realized Tim lacked essential constructs to make sense of her instruction. She subsequently made adjustments in her instruction, making effective use of more appropriate representations based on tasks from the unit that we modified and used with Tim in our interviews. These adjustments helped Tim to construct partitioning operations and an appropriate unit fractional scheme. This study illustrates the importance of coming to understand a student's mathematical activity in terms of possible conceptual schemes and modifying instructional strategies to build on those schemes. The coordinated design of the research study facilitated these instructional modifications.     

Karen in motion: The role of physical enactment in developing an understanding of distance,time, and speed

The role of direct kinesthetic experience in mathematics education remains relatively unexamined. What role can physical enactment play in mathematics learning? What, if any, implications does it carry for classroom teaching? In this article I explore the role that a third grader's kinesthetic experience plays in supporting her learning of the mathematics of motion, a content area typically for older students. Based on analyses of two individual interviews and classroom participation, I argue that Karen's ability to use physical enactment to inhabit motion trips, along with a thoughtfully emergent curriculum design, created a learning environment that enabled Karen to develop a deep, conceptual understanding of distance, time, and speed.     

Conflicting Beliefs: A Story of a Chemistry Teacher's Struggle with Change

One teacher's struggle to develop and implement a curriculum focused on student understanding of chemistry is explored in this case study of a high school chemistry teacher. Conflicting beliefs about her roles as a teacher in the classroom and her professional responsibilities are addressed. Three primary conflicts that emerged from data collected over a two year period include, (a) conflicts between state curriculum mandates and individual student understanding; (b) conflicts between theoretical and applicable chemistry content knowledge, and (c) conflicts between the students' goals and the teachers' goals for the course. The impact of the research process on the teacher's change process included reconceptualization of constraints and development of confidence in her professional judgment. The case study provides insights into contextual problems teachers face as they attempt to change practices.     

Evolving Communities of Mind-In Which Development Involves Several Interacting and Simultaneously Developing Strands

If a curriculum developer's goal is to create a single linear sequence of tasks that lead to the development of some important mathematical concept, then some researchers have suggested that these sequences should follow progressions similar to stages of development that have been identified in Piaget-like research on the relevant concept(s). These research-based sequences are referred to as learning trajectories. Other researchers emphasize that conceptual development can involve interactions among ideas expressed using a variety of representational media and can occur along a variety of "dimensions" such as concrete-abstract, simple-complex, or situated-decontextualized. Therefore, different paths can be appropriate for different students, and trying to funnel development along any single developmental path can be inappropriate for some students. These researchers often envision trajectories to be specific paths within a branching tree diagram that portrays the space of possibilities. This article emphasizes a third type of situation called model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, 2000). They are problem-solving situations in which goals include developing more powerful constructs or conceptual systems. Therefore, significant conceptual developments occur because students are challenged to repeatedly express, test, and revise their own current ways thinking-not because they were guided along a narrow conceptual path toward (idealized versions of) their teacher's ways of thinking. That is, development looks less like progress along a path, and it looks more like an inverted genetic inheritance tree in which great grandchildren trace their evolution from multiple lineages that develop simultaneously and interactively.     

Challenges of maintaining cognitive demand during the limit lessons: understanding one mathematician’s class practices

In this study, we examined five limit lessons using Mathematical Tasks Framework to understand students’ opportunities to learn cognitively challenging tasks and maintain cognitive demand during limit lessons. Our analysis of Dr A’s five lessons shows that students rarely had opportunities to maintain or increase cognitive demand. There are two main factors that shaped her instructional practices, students and time. These two factors greatly influenced how she selects and implements limit tasks in her classes. To serve her students’ needs of knowing more rules, formulas and procedures, she selected and discussed those simple tasks a lot. Although Dr A thinks challenging tasks and asking demanding questions can be potentially good instructional practices, she thinks these instructional practices would not serve her students well. With these factors, we made possible recommendations to have more student-centred teaching.     

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.     

Mathematics Teachers' Reasoning About Fractions and Decimals Using Drawn Representations

This qualitative study considers middle grades mathematics teachers' reasoning about drawn representations of fractions and decimals. We analyzed teachers' strategies based on their response to multiple-choice tasks that required analysis of drawn representations. We found that teachers' flexibility with referent units played a significant role in understanding drawn representations with fractions and decimals. Teachers who could correctly identify or flexibly use the referent unit could better adapt their mathematical knowledge of fractions validating their choice, whereas teachers who did not attend to the referent unit demonstrated four problem-solving strategies for making sense of the tasks. These four approaches all proved to be limited in their generalizability, leading teachers to make incorrect assumptions about and choices on the tasks.