This dissertation investigates the interplay between second and fourth graders’ use of rulers and paper strips to solve number line measurement problems. Drawing upon and extending Vygotsky’s (1978, 1986) approach to mediation in problem solving, and Saxe’s (2012) treatment of microgenesis of representational activity, I analyzed 36 second and 39 fourth graders’ measurement activity as they participated in videotaped semi-structured interviews. At each grade, students were randomly assigned to ruler, short paper strip, or long paper strip conditions, and solved five number line problems first without the tool available and then five similar problems with the tool available. For each problem, two numbers were labeled (e.g., 4 and 6) and students were asked to place a third number appropriately (e.g., 9). Problems and tools were designed to elicit unitizing strategies (splitting, iterating, counting) as students coordinated tool with target to produce a linear measure. I report findings related to the precision of students’ estimates contrasting tool absent and present conditions, and findings related to students’ adaptation of tools to serve measurement functions. Analyses of students’ precision revealed that with or without the support of tools, fourth graders were more precise than second graders, though the utility of tool varied with tool and problem type. The long paper strip was particularly difficult for second graders, and at both grades, many students rejected using the long paper strip. Students generally became precise with tools when a solving a problem type that displayed a linear unit of 1 instead of a larger unit, though this varied with tool type. On other problem types, tools interfered with precision, and on still other problem types, tools interfered with the precision of second, but supported the precision of fourth graders. Analyses of strategies revealed three levels in students’ ability to coordinate a unitization of tool and target as they tried to adapt tools to serve measurement functions. I argue that these findings can productively inform our understanding of the interplay between tool use in students’ developing measurement activity as well as the design of instructional problem environments to support students’ tool using in linear measurement.

Elementary school teachers in the United States are typically provided curricular materials by their school districts. However, little is known about how teachers may replace or supplement these provided materials with outside materials—materials found in their own curricular libraries or from external sources. This dissertation proposes a research framework on teachers’ use of outside materials that leads to the following research questions: What motivates teachers to consider using outside curricular materials? What approaches do they use to discover outside materials? What criteria do they use to evaluate outside materials? What steps do they take to prepare or adapt outside materials? And what external factors relate to teachers’ decisions to use outside materials? These questions were addressed through an online survey of 98 elementary teachers across two districts in Western Washington state, interviews with nine survey respondents who frequently considered using outside materials, and in-depth case studies of the decisions of three interviewees. The survey revealed that a large majority of teachers considered using outside curricular materials at least once in the 2015-2016 school year, that many of them did so frequently, and that their reasons for considering outside materials were linked to areas of dissatisfaction with the provided materials. A multiple linear regression found that higher teacher-reported feelings of curricular autonomy and more years of experience with their provided curriculum predicted higher frequency of considering outside materials. The survey also revealed that teachers used a wide variety of approaches to discover outside materials, including both online and offline methods. Interviews provided detail to these findings, revealing that teachers’ evaluation criteria were aligned with their initial motivation to consider outside materials, and that teachers tended to engage in minimal preparation of outside materials. Case studies of three focal teachers were used to illustrate different motivations for considering outside materials and different approaches for discovery, as well as to elaborate and refine the research framework that guided this investigation. This dissertation adds to literature showing that teachers actively participate with curricular materials in order to reach their professional goals, and makes a strong case for including attention to outside materials in curriculum development, district-level decisions, and future research.

Students bring a wealth of resources to science learning contexts, including knowledge of their everyday worlds. Yet within traditional approaches to K-12 science instruction, students typically are not given opportunities to draw upon this knowledge. In this dissertation, I examine the potential of place-based science curriculum and a digital mapping tool to support students’ conceptually rich science learning by leveraging their local knowledge.

Through design-based research, I developed an 18-lesson sequence and then analyzed 27 fifth graders’ participation within it to examine the potential of place-based science inquiry. Key features of the lesson sequence included (a) students’ use of a participatory geographic information system (GIS) mapping tool, Local Ground, to support investigating an ecological system - life underfoot on their schoolyard, and (b) students’ engagement in disciplinary practices of soil ecologists (e.g., observation at field sites, isolating and measuring variables, aggregating and visualizing data, and generating evidence-based arguments) to further support their understanding of both science knowledge-building practices and ecological systems.

Analyses of student dyads’ GIS map use in whole class presentations revealed that the map often afforded students opportunities to integrate their everyday knowledge with data collection experiences in sensemaking. Specifically, the map supported students as they drew upon data generated at different sites and different moments in time as they conjectured and contested arguments about relations between variables like soil moisture, earthworm counts, and shade, digging deep into student-generated data and reasoning about complex ecological relationships and processes.

Longitudinal analyses of two focal dyads over the 18 lessons revealed that student’s desires, their emergent affect-laden goals, shaped how they engaged in science disciplinary practices. Students’ desires sometimes led them deeper into reasoning about ecological systems and closer into alignment with ecologists’ sampling and representational practices, and sometimes led them further away. In contrasting the two pairs’ experiences within the same curriculum, this analysis offers insights into how science practices may become needed in formal science contexts.

This dissertation examines the potential of new forms and contexts to support conceptually rich science learning opportunities. This research contributes to the fields’ understanding of how we might engage students in science disciplinary practices, in ways that productively build on children’s extensive experiences, affect-laden goals, and varied perspectives

This dissertation explores fifth and eighth grade students' interpretations of three kinds of mathematical representations: number lines, the Cartesian plane, and graphs of linear functions. Two studies were conducted.

In Study 1, I administered the paper-and-pencil Linear Representations Assessment (LRA) to examine students' understanding of the three representations. The LRA had an experimental component that compared performance on routine problems to non-routine problems (problems not amenable to routine solution procedures). I administered the assessment to Grade 5 students (n=126) who had no formal instruction involving function graphs, and I compared their performances with those of Grade 8 students (n=131) enrolled in Algebra 1. A repeated measures ANOVA revealed students in each grade performed better on routine problems compared to non-routine problems, suggesting that routine problems may falsely indicate greater competence. Paired samples t-tests indicated no differences in performance between Grades 5 and 8 students on number line items, though Grade 8 students outperformed fifth graders on Cartesian plane and function graph items. Videotaped interviews with a subset of Grades 5 and 8 students revealed that students in each grade approached tasks across representations in similar ways, suggesting persisting misconceptions. Interviews also revealed patterns unique to each grade.

In Study 2, I examined the efficacy of a tutorial intervention. The intervention introduced written definitions to support principled understandings of the number line, the Cartesian plane, and function graphs. A repeated measures ANOVA that compared pre/posttest scores of Grade 5 students (n=20) to a matched control group (n=20) revealed significant gains from pre- to posttest in the experimental group, with no detectable gains in a control. At posttest, Grade 5 tutorial students performed significantly better on non-routine LRA problems than Grade 8 students who did not receive the tutorial. Video analysis revealed a correlation between tutorial students' appropriate uptake of definitions and gains from pretest to posttest.

Analyses across the two studies indicate that instruction that supports students' coordination of linear and numerical units can support students' learning with understanding. Potential applications include the development of curricula to support students' learning with understanding related to these representations and teacher professional development interventions.

The purpose of this dissertation is to explore the economic ideas of indigenous Triqui children between the ages of 5-15 who sell artisanal goods in Oaxaca, Mexico. I report findings from two studies that investigated (1) sellers' strategies for successfully selling goods, and (2) children's economic ideas linked to their selling strategies.

In Study 1, I documented two recurrent problems that sellers found themselves in and the selling strategies they used to solve these problems through observations, home visits, shadowing of sellers, and interviews. The recurrent problems were (1) how to convince a customer to purchase a good, and (2) how to interact with a customer that attempted to bargain. Sellers used sales pitches as solutions to these predicaments, where they attempted to convince the customer to purchase the good at the stated price by emphasizing different aspects of the goods. Some sales pitches emphasized the currency price of the good, others emphasized the quality/esthetic of the good, and yet others emphasized the utility of the good.

To better understand links between seller's engagement in recurrent problems and their developing economic ideas, I conducted systematic interviews with sellers (n=29) in Study 2. I sought to characterize children's economic ideas of exchange value (the value of goods in an exchange) and price setting through interview tasks involving fictional bartering and currency trades. I contrasted sellers' responses on the interview tasks with those of an age-matched control group of non-sellers (n=15). Findings revealed that sellers used price to determine the exchange value of goods more frequently than non-sellers. In contrast, some non-sellers assessed the exchange value of goods by referring to their own individual wants and needs, where goods were things to "own", not things to "sell", and therefore may not necessarily have a currency value. Sellers also revealed ideas about price setting that were based on rudimentary notions of profit.

This study made visible the knowledge that a historically non-dominant group of children develop, and thus contributes to literature that seeks to provide an alternative to deficit models through investigation of development in non-traditional contexts.

This dissertation targets young students’ developing knowledge relevant to a fundamental practice in academic mathematics: the use of diagrams to represent idealized mathematical objects whose properties are established by definition (as in the use of drawn dots to represent zero-dimensional points and drawn lines to represent one-dimensional lines with infinite extent). Initiation into this ‘definitional practice’ is critical to students' mathematical development. However, the practice is understudied in educational research. It also presents a significant source of confusion for students. Instead of using definitions, students may rely on the appearances of the diagrams and their knowledge of the physical world—an ‘empirical’ rather than definitional approach.

I have designed three studies to investigate students’ developing understanding of the definitional practice, using points and lines in Euclidean geometry as a potentially fruitful mathematical context. Adopting a design research approach that draws on Vygotsky’s method of double stimulation, these studies employ a pedagogical strategy that is rarely observed in mathematics classrooms: providing definitions and making explicit the distinction between drawn diagrams and the idealized objects they symbolize, pointing out that the defining features of points and lines are not perfectly embodied in their conventional representations. These studies investigate students’ uptake of this support and capture students’ developing understanding of the definitional practice.

The first study uses an experimental design to determine whether there are age/grade-related changes in students’ uptake of the intervention. Participants include students from a San Francisco Bay Area charter school in fourth grade (n=46), sixth grade (n=53), and eighth grade (n=43). Students are assigned to one of two treatment conditions whose purpose was to manipulate exposure to the definitions of points and lines. In the experimental treatment, students are presented with a sheet that contained mathematical definitions of points and lines, which are unavailable to students in the control condition. Students are then administered a paper-and-pencil assessment consisting of eight multiple choice items, some of which included diagrams of points and lines. Analyses suggest that with age, children shift towards relying on definitions rather than the appearances of the diagrams and knowledge about material objects.

The second study addresses unanswered questions about the general learning trend identified by the first study. Specifically, it uses structured one-on-one interviews to determine whether (a) students are indeed drawing on provided definitions when selecting idealized (rather than empirical) answers, and (b) whether students are constructing a conceptual differentiation between material diagram and idealized object. Participants include fourth (n=40) and sixth (n=37) grade students.

The third study explores how students make sense of the definitions of points and lines presented in Study 1 and 2. In particular, this study considers how students may be constructing analogies that draw on ideas related to the material world. Results demonstrate that students’ materially-based analogies incorporate other ideas typically learned in school to make sense of the definitions of points and lines. For example, students frequently refer to scientific ideas related to the microscopic world of atoms and molecules to explain the zero-dimensionality of a mathematical point. Participants were identical to those of the second study.

The insights generated by these three studies contribute to our understanding of sociocultural processes in cognitive development as well as to mathematics education research. This dissertation also has practical implications for mathematics instruction, suggesting potentially promising pedagogical possibilities for giving students a better ‘feel for the game’—an understanding of what academic mathematics is all about.

How do teachers and students create classroom environments in which mathematical errors are regarded as important opportunities for learning? What norms support students in learning from their errors, and how to these norms develop in a classroom community? This dissertation addresses these questions through a longitudinal case study investigating the emergence of classroom norms related to the treatment of errors. Classroom norms are here understood to be taken-as-shared expectations for behavior in a classroom. The fifth-grade case study classroom was selected because in prior research studies the experienced and highly-regarded teacher had engaged students in rich discussions of their mathematical errors and the opportunities presented by these errors for learning.

To answer the questions of (a) what norms related to mathematical errors were taken up, and (b) how these norms developed over the course of the school year, Saxe’s (2012) framework describing the relation of micro-, onto-, and sociogenetic processes was used as a guide for determining methods. Sociogenetic processes were the focus of this investigation, and Saxe’s framework points to microgenetic constructions, when studied collectively over time, as likely to illuminate otherwise difficult to observe sociogenetic processes, such as norm development. To provide information about microgenetic constructions related to errors, several types of evidence were collected throughout the school year during three data collection periods lasting two weeks each: at the beginning (September), middle (January), and end (April) of the 2014 to 2015 school year. During all three time periods, the teacher was interviewed, five “focal” students were interviewed, classroom mathematics lessons were video-recorded daily, and all students in the classroom answered a paper-and-pencil multiple choice survey about their expectations related to errors. Interviews were analyzed using grounded analysis methods, and video-recordings were analyzed using a focused coding procedure and StudioCode software. Sources of evidence were used in a triangulating fashion to identify norms in the classroom.

Through this analysis, seven norms were identified as having been taken up by the majority of members of the class by the end-of-year data collection period. Two of these norms were selected for in-depth description. The norm everyone has some mathematical understandings to which you should pay attention provides a good example of a norm that was closely tied to a specific collective practice, the “coaching” practice that was used frequently in the case-study classroom. The norm there are different types of errors, only some of which are acceptable provides an example of a norm that emerged part-way through the school year in response to a problem with the way errors were being treated. Classroom interactions and teacher and student interview statements exemplifying these norms are described. The process through which these two norms emerged in the case study classroom over the course of the school year is detailed, using evidence collected throughout the school year. In general, the teacher strongly promoted these norms by frequently and persistently modeling, describing, and praising behaviors consistent with these norms and by correcting inconsistent behaviors.

Implications for how Saxe’s framework may be productively applied in future investigations of classrooms norms are discussed. In particular, attention to ontogenetic processes – that is, individuals’ shifting expectations over time – was found to be useful as an access point for identifying the norms of a classroom community, and the teacher’s actions and expectations were found to be especially important indicators of classroom norms. Examination of collective practices related to errors was also useful for the identification of norms because some norms were strongly associated with collective practices, such as the “coaching” practice. The results of this study also have implications for teaching practice. The findings indicate that children are capable of taking up challenging practices related to the study of errors, and teachers who promote these practices in their classrooms may be successful if they are persistent in modeling, explaining, and praising the desired practices.

The findings of this descriptive study on the cultural aspects of the pretend play of upper middle class children in North Tehran are based on an-hour long interview with mothers of 38 young (n = 17) and old (n = 21) preschool children (half were girls). Influences of children's daily activities, cultural and family values and Tehran city-life conditions on children's pretend play were investigated.

Children's daily life influenced time available for play; mothers believed that play time had decreased due to increased participation at enrichment classes, more time spent watching TV, frequently going out along with mother, and reduced number of siblings and children in general. Children mostly played pretend indoors due to pollution and hazardous traffic. Despite dislike of clutter, mothers allowed children to play anywhere in the house as long as safe and comfortable. Children had a wide range of toys, which lend to all themes of play; mostly toys were related to Everyday Activities; girls played as frequently with Family Care and boys with Fantasy toys. Mothers appreciated toys for increasing creativity and imagination, and child's happiness and learning.

Majority of children's play partners were relatives and grownups. Their non-relative play partners were children's of mother's friends, and a few played with neighbors. Most mothers wished the child had more young play partners, however, would not invite a play partner if unsure of homogeneity of family values. Children's themes of play were mostly related to everyday activities. Girls, in addition, frequently played Family Care and boys Fantasy Themes and Danger in Environment. Themes did not vary by age, however, younger children more frequently engaged in thematic play. Mothers disliked themes of violence.

Mothers' perceived social functions of play were mostly related to bonding with relatives, having a fulfilling and enjoyable time, and learning appropriate social values and behaviors. They also valued increasing creativity and imagination. They believed that pretend play is essential for children's development and its deprivation will have negative developmental consequences.

This study examines fourth graders' understandings of arithmetic generalizations, the general properties of arithmetic that hold true for all numbers. Its focus is on three types of generalizations: (a) direction of change (e.g., addition of positive numbers increases the numerical value, while subtraction of positive numbers decreases the numerical value); (b) identity (e.g., the addition or subtraction of 0 to any number leaves its value unchanged); and (c) relationship between operations (e.g., addition and subtraction are inverse operations). Using a between subjects design, two interview studies were conducted to investigate the character of children's understandings and to understand how students' production of generalizations varied across different tasks (e.g., I am thinking about a number. If I multiply that number by 5 and then divide by 5, what will happen to my number?). Study 1 (n=24) focused on students' additive thinking in the context of addition and subtraction tasks, while Study 2 (n=24) focused on multiplicative thinking in the context of multiplication and division tasks.

In both studies, qualitative analyses of the video data revealed four levels in student thinking, levels that show a spectrum of increasing generality with which students treat arithmetic operations. At Levels 1 and 2, students rely on specific instances and substitution of values. At Levels 3 and 4 (advanced generalizations), students do not rely on any examples and make generalizations about the arithmetic operations. Further quantitative analyses revealed that student thinking was not always consistent and that students' production of advanced generalizations was affected by generalization type and domain type. In both studies, the identity tasks prompted the most advanced generalizations, while the relationship between operations tasks elicited the least advanced generalizations from students. Similarly, children were more likely to produce an advanced generalization in the additive domain than in the multiplicative domain. Although the multiplicative tasks may have been more challenging for students, the character of students' thinking and the difficulty of the tasks in relation to one another were similar across both domains. These parallel findings across the studies indicate that additive and multiplicative generalizations may involve similar developmental progressions.

This dissertation provides insight into a developmental model about children's construction of arithmetic generalizations. First, the four levels of generality describe qualitative shifts in student thinking and, second, these findings indicate that the development of students' understandings of arithmetic generalizations may be heterogeneous across a range of generalizations. This work can contribute to teachers' knowledge for teaching and inform the design of a developmentally appropriate instructional sequence.