Luis Saldanha

A Developing Approach to Studying Students' Learning through Their Mathematical Activity.

We discuss an emerging program of research on a particular aspect of mathematics learning, students’ learning through their own mathematical activity as they engage in particular mathematical tasks. Prior research in mathematics education has characterized learning trajectories of students by specifying a series of conceptual steps through which students pass in the context of particular instructional approaches or learning environments. Generally missing from the literature is research that examines the process by which students progress from one of these conceptual steps to a subsequent one. We provide a conceptualization of a program of research designed to elucidate students’ learning processes and describe an emerging methodology for this work. We present data and analysis from an initial teaching experiment that illustrates the methodology and demonstrates the learning that can be fostered using the approach, the data that can be generated, and the analyses that can be done. The approach involves the use of a carefully designed sequence of mathematical tasks intended to promote particular activity that is expected to result in a new concept. Through analysis of students’ activity in the context of the task sequence, accounts of students’ learning processes are developed. Ultimately a large set of such accounts would allow for a cross-account analysis aimed at articulating mechanisms of learning.

Intricacies of statistical inference and teachers’ understandings of them

Recent science reform efforts and standards documents advocate that students develop scientific inquiry practices, such as the construction and communication of scientific explanations. This paper focuses on 7th grade students’ scientific explanations during the enactment of a project based chemistry unit where the construction of scientific explanations is a key learning goal. During the unit, we make the explanation framework explicit to students and include supports or scaffolds in both the student and teacher materials to facilitate students’ in their understanding and construction of scientific explanations. Results from the enactment show significant learning gains for students for all components of scientific explanation (i.e. claim, evidence, and reasoning). Although students’ explanations were stronger at the end of the instructional unit, we also found that students’ still had difficulty differentiating between appropriate and inappropriate evidence for some assessment tasks. We conjecture that students’ ability to use appropriate data as evidence depends on the wording of the assessment task, students’ content knowledge, and their understanding of what counts as evidence. Having students construct scientific explanations can be an important tool to help make students thinking visible for both researchers and teachers. Appropriate and Inappropriate Evidence Use 2 Middle School Students’ Use of Appropriate and Inappropriate Evidence in Writing Scientific Explanations The National Research Council (1996) and the American Association for the Advancement of Science (1993) call for scientific literacy for all. All students need knowledge of scientific concepts and inquiry practices required for personal decision making, participation in societal and cultural affairs, and economic productivity. Science education should support students’ development toward competent participation in a science infused world (McGinn & Roth, 1999). This type of participation should be obtainable for all students, not just those who are educated for scientific professions. Consequently, we are interested in supporting all students in learning scientific concepts and inquiry practices. By scientific inquiry practices, we mean the multiple ways of knowing which scientists use to study the natural world (National Research Council, 1996). Key scientific inquiry practices called for by national standards documents include asking questions, designing experiments, analyzing data, and constructing explanations (American Association for the Advancement of Science, 1993; National Research Council, 1996). In this study, we focus on analyzing data and constructing explanations. These practices are essential not only for scientists, but for all individuals. On a daily basis, individuals need to evaluate scientific data provided to them in written form such as newspapers and magazines as well spoken through television and radio. Citizens need to be able to evaluate that data to determine whether the claims being made based on the data and reasoning are valid. This type of data evaluation, like other scientific inquiry practices, is dependent both on a general understanding of how to evaluate data as well as an understanding of the science content. Appropriate and Inappropriate Evidence Use 3 In this study we explore when students use appropriate evidence and when they use inappropriate evidence to support their claims. Our work focuses on an 8-week project-based chemistry curriculum designed to support 7 grade students in using evidence and constructing scientific explanations. We examine the characteristics of these students’ explanations, their understanding of the content knowledge, and the assessment tasks to unpack what may be influencing students use of evidence. Our Instructional Model for Scientific Explanations In our work, we examine how students construct scientific explanations using evidence. We use a specific instructional model for evidence-based scientific explanations as a tool for both classroom practice and research. We provide both teachers and students with this model to make the typically implicit framework of explanation, explicit to both teachers and students. Our instructional model for scientific explanation uses an adapted version of Toulmin’s (1958) model of argumentation and builds off previous science educators’ research on students’ construction of scientific explanations and arguments (Bell & Linn, 2000; Jiménez-Aleixandre, Rodríguez, & Duschl, 2000; Lee & Songer, 2004; Sandoval, 2003; Zembal-Saul, et al., 2002). Our explanation framework includes three components: a claim (similar to Toulmin’s claim), evidence (similar to Toulmin’s data), and reasoning (a combination of Toulmin’s warrants and backing). The claim makes an assertion or conclusion that addresses the original question or problem. The evidence supports the student’s claim using scientific data. This data can come from an investigation that students complete or from another source, such as observations, reading material, or archived data. The data need to be both appropriate and sufficient to support the claim. Appropriate data is relevant to the question or problem and relates to the given claim. Data is sufficient when it includes the necessary quantity to convince someone of a claim. The Appropriate and Inappropriate Evidence Use 4 reasoning is a justification that links the claim and evidence and shows why the data counts as evidence to support the claim by using the appropriate scientific principles. Kuhn argues (1993) that argument, or in our case scientific explanation, is a form of thinking that transcends the particular content to which it refers. Students can construct scientific explanations across different content areas. Although an explanation model, such as Toulmin’s, can be used to assess the structure of an explanation, it cannot determine the scientific accuracy of the explanation (Driver, Newton & Osborne, 2000). Instead, both the domain general explanation framework and the domain specific context of the assessment task determine the correctness of the explanation. Consequently, in both teaching students about explanation and assessing students’ construction of explanations we embed the scientific inquiry practice in a specific context. Student Difficulties Constructing Explanations Prior research in science classrooms suggests that students have difficulty constructing high-quality scientific explanations where they articulate and defend their claims (Sadler, 2004). For example, students have difficulty understanding what counts as evidence (Sadler, 2004) and using appropriate evidence (Sandoval, 2003; Sandoval & Reiser, 1997). Instead, students will draw on data that do not support their claim. Consequently, we are interested in whether students use appropriate evidence to support their claim or if they draw on evidence that is not relevant. Students’ claims also do not necessarily relate to their evidence. Instead, students often rely on their personal views instead of evidence to draw conclusions (Hogan & Maglienti, 2001). Students have a particularly difficult time reasoning from primary data, especially when Appropriate and Inappropriate Evidence Use 5 measurement error plays an important role (Kanari & Millar, 2004). Students can recognize variation in data and use characteristics of data in their reasoning, but their ability to draw final conclusions from that data can depend on the context. Masnick, Klahr, and Morris (this volume) concluded that young students who poorly understood the context of the investigation had difficulty interpreting data, particularly when the interpretation of that data contradicted their prior beliefs. Students will likely discount data if the data contradicts their current theory (Chinn & Brewer, 2001) and they will only consider data if they can come up with a mechanism for the pattern of data (Koslowski, 1996). When students evaluate data, more general reasoning strategies interact with domain-specific knowledge (Chinn & Brewer, 2001). Whether students use appropriate and inappropriate evidence may depend on their prior understanding of a particular content area or task. Students also have difficulty providing the backing, or what we refer to as reasoning, for why they chose the evidence (Bell & Linn, 2000) in their written explanations. Other researchers have shown that during classroom discourse, discussions tend to be dominated by claims with little backing to support their claims (Jiménez-Aleixandre, Rodríguez & Duschl, 2000). Our previous work supports these ideas. We found that middle school students’ had the most difficulty with the reasoning component of scientific explanations (McNeill, Lizotte, Krajcik & Marx, in review; McNeill, et al., 2003). Although students’ reasoning improved over the course of the 6-8 week instructional unit, it was consistently of lower quality than their claims or evidence. Students’ reasoning often just linked their claim and evidence and less frequently articulated the scientific principles that allowed them to make that connection. Similar to students ability to evaluate and use data, providing accurate reasoning is related to students understanding of the content. Students with stronger content knowledge Appropriate and Inappropriate Evidence Use 6 provide stronger reasoning in their scientific explanations (McNeill et al., in review). Previous research with students has found that their success at completing scientific inquiry practices is highly dependent on their understanding of both the content and the scientific inquiry practices (Metz, 2000). Both domain specific and general reasoning are essential for students’ effective evaluation of data and construction of scientific explanations. Although previous work has shown that students have difficulty with components of scientific explanations, there has been little research unpacking exactly when

Challenges of Developing Coherent Probabilistic Reasoning: Rethinking Randomness and Probability from a Stochastic Perspective

The concept of probability plays a vital role in mathematics and scientific research, as well as in our everyday lives. It has also become one of the fastest growing segments of high school and college curricula, yet learning probability within school contexts has proved more difficult than many in education realize.