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

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.

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.