Patrick W Thompson

Notations, Conventions, and Constraints: Contributions to Effective Uses of Concrete Materials in Elementary Mathematics.

Twenty 4th-grade children were matched according to performance on a whole-number calculation and concepts pretest and assigned at random to one of two groups: wooden base-ten blocks and computerized microworld. Instruction in each group was designed to orient students toward relationships between notation and meaning. Instruction given the two groups was based upon a single script that extended whole number numeration to decimal numeration, and emphasized solving problems in concrete settings while inventing notational schemes to represent steps in solutions. Neither group changed in regard to whole-number notational methods. Blocks children understood decimal numerals as if they were funny whole numbers; Microworld children attempted to give meaning to decimal notational methods, but were largely in a state of disequilibrium at the end of the study. Notations, Conventions, and Constraints 1 The use of concrete materials in elementary mathematics instruction has been widely advocated. Textbooks on methods for teaching elementary school mathematics and college textbooks on mathematics for elementary school teachers promote liberal use of concrete materials. Yet, the research literature on effectiveness of instruction involving uses of concrete materials is equivocal at best (Fennema, 1972; Labinowicz, 1985; Resnick, 1982; Resnick & Omanson, 1987; Sowell, 1989; Suydam & Higgins, 1977; Wearne & Hiebert, 1988). The equivocacy of past research on the beneficial effects of concrete materials is illustrated by comparing studies by Resnick and Omanson (1987), Fuson and Briars (1990), Labinowicz (1985), and Wearne and Hiebert (1988). In Resnick and Omanson’s (1987) study, the use of base-ten blocks had little impact on children’s understanding of and skill with multi-digit place-value subtraction. Fuson and Briars (1990) reported that children achieved a high level of skill with multi-digit addition and subtraction through the use of base-ten blocks. Labinowicz (1985) reported that third-graders using base-ten blocks developed relatively little computation skill. Wearne and Hiebert (1988) reported that the use of base-ten blocks had a perceptible impact on fourth, fifth, and sixth graders’ development of meaning for decimal numeration and for symbolic addition and subtraction with decimal fractions. Some of the equivocacy can be attributed to the goals by which the studies were conducted and evaluated (computational facility versus problem solving) or to their orientation toward algorithms (prescription versus invention). However, not all the differences in results can be attributed to these sources. The nature of students’ engagement with concrete materials and their orientation toward materials in relation to notation and numerical value is another possibility (Fuson & Briars, 1990). We can place students’ use of concrete materials in a larger context—their development of an orientation toward using notation to express their reasoning as it occurs in concrete settings. In this regard we must be sensitive to students’ image of their activities. Resnick and Omanson (1987) observed that students’ active participation in a Notations, Conventions, and Constraints 2 prescribed activity may have little effect if students think that they are following a prescription. Students’ reenactment of a prescribed procedure does not give them opportunities to construct constraints in their meanings and reasoning—they meet constraints only because they are obliged to adhere to prescription, and it matters little that the prescriptions entail use of concrete materials. In reenacting prescribed procedures, students do not experience constraints as arising from tensions between their attempts to say what they have in mind and their attempts to be systematic in their expressions of it. As students come to be systematic in their expressions of reasoning and make a commitment to express their reasoning within their system, that same systematicity places constraints on the reasoning they wish to express. When students are aware of the constraining influence exerted by their arbitrary use of notation, they may feel freer to modify their standard uses of notation to express better what they have in mind. Also, when students are aware of reciprocal relationships between notation and reasoning they may be more inclined to concentrate on their reasoning when experiencing difficulty and concentrate less on performing correct notational actions. A person’s meaningful use of notation can be highly idiosyncratic, it can be creative expression constrained by convention, or it can be an automated use of convention. In the first case the individual is engaging in personal expression. In the second case the individual is conforming to convention with the awareness of conforming. In the third case the individual is using convention unthinkingly—perhaps unknowingly. Conventions, Curriculum, and Research One way to approach notational and methodological convention in school mathematics is to allow students to create their own problem-solving methods in concrete settings and to treat notational expressions of method as a mathematical problem. One aim of this approach is for students to understand that any notational method is but one of many valid ways to express one’s reasoning. Another aim is that reasoning and expressing be joined dialectically. As students attempt to refine expressions of their reasoning, they have Notations, Conventions, and Constraints 3 occasion to clarify their reasoning. As they clarify their reasoning and attempt to express it, they inform the systematicity of their notational schemes. It is hoped that under these circumstances students will find occasions to experience the negotiations that establish notational conventions, and only then would they be in a position to appreciate the natural, productive tension between creativity and conventionality. To understand a convention qua convention, one must understand that approaches other than the one adopted could be taken with equal validity. It is this understanding that separates convention from ritual. To have students recreate conventionality wherever it might occur is neither practical nor desirable (Cobb, 1991). This would be like having them create their own languages to appreciate the conventionality of language. On the other hand we cannot ignore convention; to ignore convention in our teaching can lead students to think of mathematics ritualistically. We must choose judiciously those curricular sites where we address matters of convention honestly and directly. One principle to guide our choice is: address matters of convention in areas that are (a) conceptually central to the mathematics curriculum, (b) amenable to students’ appreciation of convention within the area, and (c) propitious for students’ recognition of convention generally. The areas of representing numerical value and representing methods of numerical evaluation are natural sites in the elementary curriculum fitting these criteria. We cannot investigate the relative benefit of all possible treatments of convention in mathematics even within areas for which we decide convention might be treated productively. On the other hand, mathematics education research can attempt to make explicit those conventions that are assumed and treated as given, those conventions that are assumed and presented as conventions, and those conventions that are meant for students to recreate or to create in some idiosyncratic form. THE STUDY The study was designed to investigate what features of students’ engagement in tasks involving base-ten blocks contribute to students’ construction of meaning for decimal Notations, Conventions, and Constraints 4 numeration and their construction of notational methods for determining the results of operations involving decimal numbers. The study was conducted over nine days—one day for pretest, seven days of instruction, and one day for posttest. Conventions of Notation and Method Base-ten numeration as a convention was not addressed directly. Rather, it was addressed thematically by encouraging students to refer to types of blocks by the numbername of their numerical value (e.g., hundred instead of flat). Base-ten numeration as a system for denoting numerical value was addressed as principle. It was not developed as one of many alternative numeration systems. Nothing was taken as conventional about methods for solving arithmetic problems using base-ten blocks (e.g., start with blocks of smallest value). Students were free to approach problems of addition and subtraction without constraint, except that they were to solve the problems posed and they were to remain within the base-ten numeration system. Placement of initial terms in a notational statement of an addition or subtraction problem were explained as convention. The discussion is summarized in Figure 1. The purpose of having this discussion with students was to orient them toward a conceptual organization of the notational schemes they already possessed. Teachers used this convention initially, but they did not demand that students use it. Notations, Conventions, and Constraints 5 2317 456 The place where you write the amount you start with. The place where you write the amount you intend to remove. The place where you write the number of blocks of a place value after you add/subtract blocks having that value. 2317 + 456 The places where you write the amounts you intend to combine. The place where you write the number of blocks of a place value after you add blocks having that value. Addition

Quantitative reasoning, complexity, and additive structures

Six fifth-grade children participated in a four-day teaching experiment on complex additively-structured problems, which was followed by in-depth interviews of individual children. The teaching experiment was meant to investigate childrens difficulties in holding in mind at once situations in which one or more items played multiple roles. Two important difficulties were identified: (1) distinguishing between “difference” as the result of subtracting and “difference” as the amount by which one quantity exceeded another; and (2) indirect evaluation of an additive comparison. Sources of these difficulties, along with pedagogical and curricular recommendations for addressing them, are discussed.

Foundational Reasoning Abilities that Promote Coherence in Students' Function Understanding

The concept of function is central to undergraduate mathematics, foundational to modern mathematics, and essential in related areas of the sciences. A strong understanding of the function concept is also essential for any student hoping to understand calculus – a critical course for the development of future scientists, engineers, and mathematicians. Since 1888, there have been repeated calls for school curricula to place greater emphasis on functions (College Entrance Examination Board, 1959; Hamley, 1934; Hedrick, 1922; Klein, 1883; National Council of Teachers of Mathematics, 1934, 1989, 2000). Despite these and other calls, students continue to emerge from high school and freshman college courses with a weak understanding of this important concept (Carlson, 1998; Carlson, Jacobs, Coe, Larsen & Hsu, 2002; Cooney & Wilson, 1996; Monk, 1992; Monk & Nemirovsky, 1994; Thompson, 1994a). This impoverished understanding of a central concept of secondary and undergraduate mathematics likely results in many students discontinuing their study of mathematics. The primarily procedural orientation to using functions to solve specific problems is absent of meaning and coherence for students and has been observed to cause frustration in students (Carlson, 1998). We advocate that instructional shifts that promote rich conceptions and powerful reasoning abilities may generate students’ curiosity and interest in mathematics, and subsequently lead to increases in the number of students who continue their study of mathematics. This article provides an overview of essential processes involved in knowing and learning the function concept. We have included discussions of the reasoning abilities involved in understanding and using functions, including the dynamic conceptualizations needed for understanding major concepts of calculus, parametric functions, functions of several variables, and differential equations. Our discussion also provides information about common conceptual obstacles to knowing and learning the function concept that students have been observed encountering. We make frequent use of examples to illustrate the ‘ways of thinking’ and major understandings that research suggests are essential for students’ effective use of functions during problem solving, and that are needed for students’ continued mathematics learning. We also provide some suggestions for promising approaches for developing a deep and coherent view of the concept of function.

THE CONCEPT OF ACCUMULATION IN CALCULUS

The concept of accumulation is central to the idea of integration, and therefore is at the core of understanding many ideas and applications in calculus. On one hand, the idea of accumulation is trivial. You accumulate a quantity by getting more of it. We accumulate injuries as we exercise. We accumulate junk as we grow older. We accumulate wealth by gaining more of it. There are some details to consider, such as whether it makes sense to think of accumulating a negative amount of a quantity, but the main idea is straightforward. On the other hand, the idea of accumulation is anything but straightforward. First, students find it is hard to think of something accumulating when they cannot conceptualize the “bits” that accumulate. To understand the idea of accomplished work, for example, as accruing incrementally means that one must think of each momentary total amount of work as the sum of past increments, and of every additional incremental bit of work as being composed of a force applied through a distance. Second, the mathematical idea of an accumulation function,

Didactic Objects and Didactic Models in Radical Constructivism

This chapter discusses ways in which conceptual analyses of mathematical ideas from a radical constructivist perspective complement Realistic Mathematics Education’s attention to emergent models, symbolization, and participation in classroom practices. The discussion draws on examples from research in quantitative reasoning, in which radical constructivism serves as a background theory. The function of a background theory is to constrain ways in which issues are conceived and types of explanations one gives, and to frame one’s descriptions of what needs explaining.

Integers as Transformations

To investigate whether elementary school students can construct operations of thought for integers and integer addition that are crucial for understanding elementary algebra, 2 sixth graders were taught for 6 weeks in eleven 40-minute sessions using a computerized microworld that proposed integers as transformations of position, integer addition as composition of transformations, negation as an operator upon integers or integer expressions, and representations of expressions as defined words. By the final session, both students had constructed mental operations for negating arbitrary integers and determining the sign and magnitude of a sum and had constructed a rule of substitution that allowed them to negate integer expressions. One student could negate represented expressions.

In the Absence of Meaning

There are many diagnoses of the bad state of U.S. mathematics education, ranging from incoherent curricula to low-quality teaching. In this chapter I will address a foundational reason for the many manifestations of failure—a systemic, cultural inattention to mathematical meaning and coherence. The result is teachers’ inability to teach for understanding and students’ inability to develop personal mathematical meanings that support interest, curiosity, and future learning. In developing this argument I discuss the subtle ways in which actual meanings with which teachers currently teach and actual meanings students currently develop in interaction with instruction contribute to dysfunctional mathematics education. I end by proposing a long-term strategy to address this situation.

A Conceptual Approach to Calculus Made Possible by Technology

Calculus reform and using technology to teach calculus are two longtime endeavors that appear to have failed to make the differences in student understanding predicted by proponents. We argue that one reason for the lack of effect is that the fundamental structure of the underlying curriculum remains unchanged. It does not seriously consider students’ development of connected meanings for rate-of-change functions and accumulation functions. We report an approach to introductory calculus that takes coherence of meanings as the central criterion by which it is developed, and we demonstrate that this radical reconstruction of the ideas of calculus is made possible by its uses of computing technology.

Researching mathematical meanings for teaching

In The Age of Discontinuity: Guidelines to Our Changing Society (1992), Professor of Management Peter Drucker lays out ways in which technologies are transforming, and will continue to transform, industries throughout the world economy; for many workers, what characterizes work life now is the continual need to adapt to technological change. Such changes are not limited to the world of work: technology is transforming interactions with media, and this also relates to books. This chapter focuses on one way in which technology may transform educational processes and bring about new educational dynamics. Specifically we examine ways in which e-book technology might influence one genre of book, the (mathematics) textbook

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