Making sense of qualitative geometry: The case of Amanda

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning.     

Preservice elementary teachers’ knowledge for teaching the associative property of multiplication: A preliminary analysis

This study examines preservice elementary teachers’ (PTs) knowledge for teaching the associative property (AP) of multiplication. Results reveal that PTs hold a common misconception between the AP and commutative property (CP). Most PTs in our sample were unable to use concrete contexts (e.g., pictorial representations and word problems) to illustrate AP of multiplication conceptually, particularly due to a fragile understanding of the meaning of multiplication. The study also revealed that the textbooks used by PTs at both the university and elementary levels do not provide conceptual support for teaching AP of multiplication. Implications of findings are discussed.     

The parametric nature of two students’ covariational reasoning

Researchers have argued that covariational reasoning is foundational for learning a variety of mathematics topics. We extend prior research by examining two students’ covariational reasoning with attention to the extent they became consciously aware of the parametric nature of their reasoning. We first describe our theoretical background including different conceptions of covariation researchers have found useful when characterizing student reasoning. We then present two students’ activities during a teaching experiment in which they constructed and reasoned about covarying quantities. We highlight aspects of the students’ reasoning that we conjectured created an intellectual need that resulted in their constructing a parameter quantity or attribute, a need we explored in closing teaching episodes. We discuss implications of these results for perspectives on covariational reasoning, students’ understandings of graphs and parametric functions, and areas of future research.     

Elementary and middle grade students’ constructions of typicality

This study addresses the measures chosen by students when selecting or constructing indices to properties of distributions of data. A series of individual teaching experiments were conducted to provide insight into the development of five 4th to 8th grade students’ conceptualizations of distribution over the course of 8 weeks of instruction. During the course of the teaching experiment (emergent) statistical tasks and analogous teacher activities were created and refined in an effort to support the development of understanding. In the process of development, attempts were made by students to coordinate center and variability when constructing measures to index properties of distributions. The results indicate that consideration of representativeness was a major factor that motivated modification of approaches to constructing indices of distributions, and subsequent coordination of indices of variation and center. In particular, the defining features of student's self-constructed “typical” values and notions of spread were examined, resulting in a model of development constituting eight “categories” ranging from the construction of values that did not reflect properties of the data (Category 1) to measures employing conceptual use of the mean in combination with other indices of center and spread (Category 8).     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Supporting Preservice Teachers' Understanding of Place Value and Multidigit Arithmetic

This article provides an analysis of a teaching experiment conducted in the context of teacher education designed to support preservice teachers' understandings of place value and multidigit addition and subtraction. The experiment addresses the following research question: Can the results from research conducted in elementary mathematics classrooms guide preservice elementary teachers' development of conceptual understanding of the same concepts? In both cases, the students (e.g., elementary students and preservice teachers) participated in activities from an instructional sequence designed to support conceptual understanding of both place value and multidigit addition and subtraction. Analyses of the episodes from the teaching experiment document the learning of the preservice teachers and how that learning was supported by initial conjectures grounded in the research on elementary students' ways of reasoning.     

Dylan’s units coordinating across contexts

Units coordination has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. To explore students’ units coordination development, we conducted an eleven-session constructivist teaching experiment with a pair of sixth-grade students, investigating how they coordinated whole number and fractional units in discrete and continuous settings. In this paper we focus on one student, Dylan, who reasoned with whole number units but not fractional units at the beginning of the teaching experiment. We describe Dylan’s development of units coordination as he continued to reason with whole number units in fractional situations, and we discuss implications for instruction.     

Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program

We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiple-groups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-groups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.     

Spatial structuring and the development of number sense: A case study of young children working with blocks

This case study discusses an activity that makes up one of five lessons in an ongoing classroom teaching experiment. The goal of the teaching experiment is (a) to gain insight into kindergartners’ spatial structuring abilities, and (b) to design an educational setting that can support kindergartners in becoming aware of spatial structures and in learning to apply spatial structuring as a means to abbreviate and ultimately elucidate numerical procedures. This paper documents children's spatial structuring of three-dimensional block constructions and the teacher's role in guiding the children's learning processes. The episodes have contributed to developing the activity into a lesson that could foster children's use of spatial structure for determining the number of blocks. The observations complement existing research that relates spatial structuring to mathematical performance, with additional insight into the development of number sense of particularly young children in a regular classroom setting.     

From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis

Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their whole-number knowledge with their fractional schemes whereby whole-number division was regarded as the same as multiplication by the reciprocal fraction.     

Service Learning Within a Secondary Math and Science Teacher Education Program: Preservice MAT Teachers’ Perspectives

Previous literature suggests that service learning may offer new opportunities to support the development of preservice science and math teachers, but few studies examine service learning beyond isolated teaching events. In this qualitative study, we attempt to improve upon this literature by following Master of Arts in Teaching (MAT) students’ views of their service learning experiences throughout their MAT program and first two years teaching. Data sources included audiotaped individual interviews, focus group field notes, and surveys with seven preservice teachers over a three‐year period. Three major findings emerged from the data analysis. First, participants identified characteristics of service learning teaching events that made them particularly useful, and these included the timing of events, targeted grade level, exposure to high‐needs contexts, and opportunities to practice pedagogical skills. Second, participation in the service learning events improved preservice teachers’ confidence and comfort teaching in high‐needs contexts, but several concerns and deficit perspectives about teaching in high‐needs contexts remained. Third, participants indicated specific ways that the service learning teaching events impacted their readiness to teach in high‐needs contexts. These findings may inform other science and math teacher educators seeking to embed service learning opportunities into their programs.     

Attention to meaning by algebra teachers

Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential.     

From fractions to proportional reasoning: a cognitive schemes of operation approach

Four seventh grade students participated in a constructivist teaching experiment in which manipulatives within a computer microworld were used to solve fractional reasoning tasks followed by tasks that involve concepts of rate, ratio and proportion. Through a retrospective analysis of video tapes, their thinking processes were analyzed from the perspective of the types of cognitive schemes of operation used as they engaged in the given problem situations. One result of the study indicates that the modifications of the students’ available schemes of operation when solving the fractional reasoning tasks formed a basis for the cognitive schemes of operation used in their solutions of tasks involving proportionality.     

SCAFFOLDING FOR SUCCESS: REFLECTIVE DISCOURSE AND THE EFFECTIVE TEACHING OF MATHEMATICAL THINKING SKILLS

This paper describes a research study into the teaching of mathematical thinking skills. Nine classes of students (in total) who had followed a course emphasising metacognitive skills outperformed their control groups on assessments of those skills and were also more successful on measures of their mathematical development. However, participant observation data revealed that there were important variations in teaching style between teachers and the success of their classes varied considerably. Observational data was used to classify the teaching styles into four groups. The teaching styles of the two most successful groups, the ‘dynamic scaffolders’ and the ‘reflective scaffolders’, are analysed here.     

Learning to teach science in urban schools: Context as content

The purpose of this phenomenological study was to explore how science teachers who persisted in urban schools interpreted and responded to the unique features of urban educational contexts. With 17 alumni who taught in metropolitan areas across seven states, the Science Educators for Urban Schools (SEUS) program provided a research setting that offered a unique view of science teachers’ development of knowledge of urban education contexts. Data sources included narratives of teaching experiences from interviews and open‐ended survey items. Findings were interpreted in light of context knowledge for urban educational settings. Findings indicated that science teaching in urban contexts was impacted by the education policy context, notably through accountability policies that narrowed and marginalized science instruction; community context, evident in teacher efforts to make science more relevant to students; and school contexts, notability their ability to creatively adjust for resource deficiencies and continue their own professional growth. Participants utilized this context knowledge to transform student opportunities to learn science. The study suggests that future science education research and teacher preparation efforts would benefit from further attention to the unique elements of urban contexts, specifically the out of classroom contexts that shape science teaching and learning.     

From Fractions to Rational Numbers of Arithmetic: A Reorganization Hypothesis

Nathan and Arthur, 2 children in a 3-year teaching experiment on children's construction of the rational numbers of arithmetic (RNA), developed their operations for multiplying, dividing, and simplifying fractions over the last 2 years (Grades 4 and 5) of the experiment. The 2 children worked in the context of specially developed computer microworlds with a teacher/researcher for approximately 45 min a week for 50 weeks over the 2-year period. The children's construction of multiplicative fractional schemes was investigated in a retrospective analysis of each of the 50 videotaped teaching episodes. Four distinct modifications of the children's fractional schemes were discerned that contributed to their construction of the RNA. The investigation suggested that the operations and unit types associated with the children's whole-number sequences did not interfere with the reorganization of their fractional schemes but rather contributed to those schemes. The reorganization involved an integration of their whole-number knowledge with their fractional schemes whereby whole-number division was regarded as the same as multiplication by the reciprocal fraction.     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

Middle school students’ proportional reasoning in real life contexts in the domain of geometry and measurement

The purpose of this study was to examine middle school students’ proportional reasoning, solution strategies and difficulties in real life contexts in the domain of geometry and measurement. The underlying reasons of the difficulties were investigated as well. Mixed research design was adopted for the aims of the study by collecting data through an achievement test from 935 sixth, seventh and eighth grade students. The achievement test included real life problems that required proportional reasoning, and were related to the measurement of length, perimeter, area and volume concepts. In addition, task-based interviews were conducted on 12 of these students to collect more comprehensive data and to support the findings of the achievement test. Findings revealed that although students were mostly successful in giving correct answers, their reasoning lacked a clear argument of the direct and indirect proportional relationships between the variables and that they approached the problems by superficial characteristics of the problems.     

Increasing engineering students' awareness to environment through innovative teaching of mathematical modelling

This article presents the results of two studies on using aninnovative pedagogical strategy in teaching mathematical modellingand applications to engineering students. Both studies are dealingwith introducing non-traditional contexts for engineering studentsin teaching/learning of mathematical modelling and applications:environment and ecology. The aims of using these contexts were:to introduce students to some of the techniques, methodologiesand principles of mathematical modelling for ecological andenvironmental systems; to involve the students in solving real-lifeproblems adjusted to their region emphasizing the aspects ofboth survival (short term) and sustainability (long term); toencourage students to pay attention to environmental issues.On one hand, the contexts are not directly related to engineering.On the other hand, the chances are that many graduates of engineeringwill deal with mathematical modelling of environmental systemsin one way or another in their future work because nearly everyengineering activity has an impact on the environment. The firststudy is a parallel study conducted in New Zealand and Germanysimultaneously with first-year students studying engineeringmathematics. The second study is a case study of the experimentalcourse Mathematical Modelling of Survival and Sustainabilitytaught to a mixture of year 2–5 engineering students inGermany by a visiting lecturer from New Zealand. The modelsused with the students from both studies had several specialfeatures. Analysis of students’ responses to questionnaires,their comments and attitudes towards the innovative approachin teaching are presented in the article.