ethanporcaro/eric-fulltext-meta huggingface dataset:
Examining K-12 Prospective Teachers&apos; Curricular Noticing
Author: Males, Lorraine M.
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Preservice Teachers,Elementary Secondary Education,Mathematics Curriculum,Instructional Materials,Comprehension,Difficulty Level,Educational Research,Attention,Decision Making

This paper explores the construct of curricular noticing, defined as the act of teachers making sense of the complexity of content and pedagogical opportunities in written or digital curricular materials (Dietiker, Amador, Earnest, Males, &amp; Stohlmann, 2014), and reports the results of four exploratory studies aimed to examine the Curricular Noticing framework. Taken together, these studies capture work done with 62 PSTs in elementary and secondary mathematics methods courses at four universities. Findings illuminate what PSTs attend to in curriculum materials and how they interpret and respond to these materials. Irrespective of level (i.e., elementary, secondary) and materials, PSTs can learn to notice aspects of curriculum materials in order to make decisions about what to do and how to do it, and activities within methods courses can facilitate this development. [For the complete proceedings, see ED583989.]
A Mathematical Modeling Lens on a Conventional Word Problem
Author: Czocher, Jennifer A.
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Word Problems (Mathematics),Mathematics Instruction,Mathematical Models,Language Proficiency,Teaching Methods,English (Second Language),Second Language Learning,Middle School Students,High School Students,Interviews,Thinking Skills,Spanish,Language Usage,English Language Learners,Native Speakers,Schemata (Cognition),Qualitative Research

Given the Common Core's dual emphases on mathematical modeling, there is a need to understand modeling as a practice and content standard to develop students' mathematical modeling skills. This study of 12 students from differing levels of mathematics instruction and English Language proficiency includes analysis of their modeling with mathematics and a focus on their transitions through a mathematical modeling cycle. Findings suggest that students were engaging in critical processes that support mathematical modeling. We posit that conventional word problems can augment the benefits of using mathematical modeling tasks and can help educators explore a process-oriented approach to mathematical modeling. [For the complete proceedings, see ED583989.]
Categorizing Statements of the Multiplication Principle
Author: Lockwood, Elise
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Multiplication,College Mathematics,Textbooks,Textbook Content,Computation,Mathematical Concepts,Mathematical Logic,Textbook Evaluation,Coding

The multiplication principle is a fundamental principle in enumerative combinatorics. It underpins many of the counting formulas students learn, and it provides much-needed justification for why counting works as it does. However, given its importance, the way in which it is presented in textbooks is surprisingly varied. In this paper, we document this variation by presenting a categorization of statement types we found in a textbook analysis. We also highlight mathematical and pedagogical implications of the categorization. [For the complete proceedings, see ED583989.]
Students&apos; Strategies for Assessing Mathematical Disjunctions
Author: Dawkins, Paul Christian
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Learning Strategies,College Mathematics,College Students,Calculus,Mathematical Concepts,Mathematics Skills,Mathematical Logic,Comparative Analysis,Educational Experiments,Problem Solving,Mathematics Activities

This paper presents results from three teaching experiments intended to guide students to reinvent truth-functional interpretations for mathematical disjunctions. The initial teaching experiments revealed that students' emergent strategies for assessing disjunctions did not entail or facilitate the development of a relevant partitioning of example space (comparable to Venn diagrams). Students were unable to form generalizable strategies for finding relevant exemplars to evaluate quantified disjunctions. The latter teaching experiment, in contrast, successfully prompted students' to attend to reference and partitioning of the referent space through an alternative instructional sequence. I set forth the methodology and findings of this study to demonstrate how conventions of mathematical logic can emerge within students' mathematical activity toward the end of their apprenticeship into proof-oriented mathematics. [For the complete proceedings, see ED583989.]
Theorization of Instrument Design in Research on Mathematical Problem Solving
Author: Zhang, Pingping
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Problem Solving,Secondary School Mathematics,Middle School Students,Mathematical Logic,Interviews,Geometry,Grade 6,Beliefs,Concept Formation,Schemata (Cognition)

In this paper we report the interview questions used in a study of middle school students' mathematical problem solving behaviors which were chosen based on Vygotsky's concept formation theory and Berger's appropriation theory. We discuss the task design/selection process along with the findings associated with the use of these tasks so to provide new direction for gauging research on mathematical problem solving. [For the complete proceedings, see ED583989.]
Teachers&apos; Expectation about Geometric Calculations in High School Geometry
Author: Boileau, Nicolas
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Teacher Expectations of Students,Geometry,Secondary School Mathematics,High School Students,Algebra,Mathematics Teachers,Regression (Statistics),Problem Solving,Equations (Mathematics),Mathematics Instruction,Questionnaires

This paper reports on a study of the instructional situation in high school Geometry that Hsu (2010) called Geometric Calculation in Algebra (GCA). In particular, we conducted a virtual breaching experiment in order to examine the extent to which high school teachers recognized breaches of two norms that we conjectured to describe geometry teachers' expectations of this work context. The results of our analysis of the data (using z-tests and mixed effect regression models) provide evidence that, in the situation of GCA, (1) teachers appear not to take issue with giving students tasks that require them to set-up and solve equations whose solutions have no geometric meaning (e.g., the length of a side of the figure is zero), and (2) teachers do not appear to expect students to document the geometric theorem or property that justify the setup of those equations (highlighting the contrast between the situation GCA and that of doing proofs). [For the complete proceedings, see ED583989.]
Changing Cones: Themes in Students&apos; Representations of a Dynamic Situation
Author: Stevens, Irma E.
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Semi Structured Interviews,Preservice Teachers,Secondary School Teachers,Mathematics Teachers,Mathematical Concepts,Concept Formation,Graphs,Problem Solving

Researchers have identified challenges students face when modeling dynamic situations. This report discusses the results of semi-structured clinical interviews with ten prospective secondary mathematics teachers who were provided with a dynamic image of a growing and shrinking cone. We asked the students to graph the relationship between the surface area and the height of the cone. We identify four themes in the students' solution approaches and discuss the implications of these approaches. Specifically, we discuss the themes with respect to relationships between the students' solutions and their images of the growing and shrinking cone, including the extent that they leveraged this image to determine their solutions. [For the complete proceedings, see ED583989.]
Indigenous Knowledge Provides an Elegant Way to Teach the Foundations of Mathematics
Author: Lipka, Jerry
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Teaching Methods,Indigenous Knowledge,Alaska Natives,Eskimos,Concept Formation,Numbers,Number Concepts,Mathematical Logic,Culturally Relevant Education

This unlikely cast of characters, by working collaboratively in a trusting learning community, was able to identify an approach to teaching rational numbers through measuring from the everyday practices of Yup'ik Eskimo and other elders. &quot;The beginning of everything,&quot; as named by a Yup'ik elder, provided deep insights into how practical activities were conceptualized and accomplished by means of body proportional measuring and nonnumeric comparisons. These concepts and practices shed light on the importance of measuring as comparing and the importance of relative units of measure, and helped us imagine a way to establish an alternative learning trajectory and school-based curriculum that begins with the insights gained from Yup'ik and other elders. This approach may well provide teachers a way to teach aspects of elementary school mathematics in an integrative and elegant way. [For the complete proceedings, see ED583989.]
Student Understanding of Directional Derivatives of Functions of Two Variables
Author: Martínez-Planell, Rafael
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematical Concepts,Concept Formation,Mathematical Logic,Schemata (Cognition),Semi Structured Interviews,Calculus,Conventional Instruction,Learning Activities,Graphs,Theories

Action-Process-Object-Schema (APOS) Theory is applied to study student understanding of directional derivatives of functions of two variables. A conjecture of the main mental constructions that students may do in order to come to understand directional derivatives is proposed and is tested by conducting semi-structured interviews with 26 students who had just taken multivariable calculus. The interviews explored the specific constructions of the genetic decomposition that student are able to do and also the ones they have difficulty doing. The conjecture, called a genetic decomposition, is largely based on the elementary notion of slope and on a development of the concept of tangent plane. The results of the empirical study suggest the importance of constructing coordinations of plane, tangent plane, and vertical change processes in order for students to conceptually understand directional derivatives. [For the complete proceedings, see ED583989.]
Productive Use of Examples for Proving: What Might This Look Like?
Author: Zaslavsky, Orit
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Validity,Mathematical Logic,Middle School Students,High School Students,Undergraduate Students,Interviews,Concept Formation,Problem Solving,Productivity

The study reported in this paper is part of a larger study on the roles of examples in learning to prove. We focus here on manifestations of students' productive use of examples for proving in the course of exploring conjectures and proving or disproving them. In this context, we define productive use of examples for proving as students' utterances that indicate that working with examples led them to realize and gain insights into some aspects of the key ideas for proving (or disproving) the conjecture. There were a total of 39 participants (12 middle school, 17 high school, and 10 undergraduate students). Each took part in an individual one-hour task-based interview. We identified 77 cases of productive use of examples, 41 based on an interviewer's provision of example(s) and 36 based on students' spontaneous generation of examples. These cases serve to characterize students' strengths that are not directly fostered in school. [For the complete proceedings, see ED583989.]
Jake&apos;s Conceptual Operations in Multiplicative Tasks: Focus on Number Choice
Author: Risley, Rachael
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Case Studies,Mathematics Instruction,Mathematics Teachers,Concept Formation,Teaching Methods,Multiplication,Computation,Mathematical Logic,Elementary School Mathematics,Number Concepts,Grade 4,Educational Games,Video Technology,Observation

This case study examined how a teacher's choice of numbers used in tasks designed to foster students' construction of a scheme for reasoning in multiplicative situations may afford or constrain their progression. This scheme, multiplicative double counting (mDC) is considered a significant conceptual leap from reasoning additively with units of one (1s) and composite units. A researcher-teacher's work with Jake allowed us to center on his gradual cognitive advance as different numbers chosen for the unit rate in problems (e.g., 5 cubes-per-tower) were used in the context of the Please Go and Bring for Me platform task. Our findings show that a child's use of an evolving scheme may initially depend on the numbers used in the task. We discuss the key recognitions that (a) a new way of operating does not evolve in a &quot;once-and-for-all&quot; way for all numbers and (b) the support our study provides for Pirie and Kieren's core notion of folding-back. [For the complete proceedings, see ED583989.]
Students&apos; Conceptions of Derivative Given Different Representations
Author: Cetner, Michelle
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Calculus,Mathematical Concepts,Problem Solving,Interviews,Mathematical Logic,High School Students,Secondary School Mathematics,Theories,Concept Formation,Schemata (Cognition)

In Calculus, students are often both presented problems with and taught to use three interconnected types of representations: symbolic, graphic, and numeric. However, students often fail to notice the relationship between mathematical objects (and even the same object) that are presented using different types of representations. Using the APOS framework and student interviews, this study explores ways in which students conceive of tasks involving derivatives that are posed using the different representational types. Patterns are drawn among: problem conception and representation type; problem conception and task type; schema level and representation type; and schema level and problem conception. Though there were patterns between representation and student conception, stronger patterns emerged between task type and student conception. [For the complete proceedings, see ED583989.]
Mental Mathematics and Enactment of Specific Strategies: The Case of Systems of Linear Equations
Author: Proulx, Jérôme
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mental Computation,Mathematics Activities,Learning Strategies,Equations (Mathematics),Mathematics Instruction,Problem Solving,Algebra,Case Studies,Secondary School Mathematics,Secondary School Teachers,Mathematics Teachers,Visualization

This study on the mental solving of systems of linear equations is part of a larger research program, aimed to gain a better understanding of the potential of mental mathematics activities with topics/objects other than numbers. Through outlining the study details and the activities engaged with, the paper reports on the variety of strategies developed for mentally solving systems of linear equations. The analysis pays attention to specificities and particularities of these strategies, focusing on their economical, tailored, and spontaneous nature to solve the tasks. In reflecting on the context of mental mathematics, the paper closes with discussions of the potential and richness of these strategies for exploring systems of linear equations. [For the complete proceedings, see ED583989.]
Curricular Treatment of Fractions in Japan, Korea, Taiwan, and the United States
Author: Son, Ji-Won
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Foreign Countries,Fractions,Addition,Subtraction,Mathematics Instruction,Common Core State Standards,Cultural Differences,Cross Cultural Studies,Teaching Methods,Elementary School Mathematics,Number Concepts

This paper investigates how the selected three East Asian countries--Japan, Korea, and Taiwan-- introduce and develop ideas related to fractions and fraction addition and subtraction compared to the Common Core State Standards of Mathematics and EngageNY. Looking at curricular approaches used across countries can provide a better picture of what is of importance in instruction aimed at developing students' mathematical proficiency. Understanding how the aforementioned three Eastern Asian materials treat fractions will offer both mathematics teachers and teacher educators some concrete images of the visions of the Common Core State Standards of Mathematics and specific ideas on teaching and learning of fractions. [For the complete proceedings, see ED583989.]
Student Conceptions of What It Means to Base a Proof on an Informal Argument
Author: Zazkis, Dov
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics Instruction,Validity,Mathematical Logic,Persuasive Discourse,Undergraduate Students,College Mathematics,Mathematical Concepts,Concept Formation,Interviews

In this paper we explore how students construe what it means for an informal argument to be the basis of a formal proof and what students pay attention to when assessing whether a proof is based on an informal argument. The data point to some undergraduate mathematics students having underdeveloped conceptions of what it means for a proof to be based on an argument. These underdeveloped conceptions limit what students pay attention to during informal-to-formal comparison tasks and may have adverse effects on students' ability to use their own informal arguments to construct proofs. [For the complete proceedings, see ED583989.]
A Refinement of Michener&apos;s Example Classification
Author: Cavey, Laurie O.
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics,Professional Personnel,Mathematics Instruction,Mathematical Logic,Classification,Interviews,Teaching Methods,Models

In this paper we propose a refinement of Michener's (1978) well-known example classification based on data from university mathematicians. The refinement takes into account the mathematician's perspective on the role of examples in doing mathematics. More specifically, our work provides insight into the ways in which mathematicians talk about using examples in their scholarly work and their work with students. The proposed classification has the potential to inform our work as teachers as we strive to create opportunities to engage students in authentic mathematical work. [For the complete proceedings, see ED583989.]
Generalizing Average Rate of Change from Single- to Multivariable Functions
Author: Dorko, Allison
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematical Concepts,Mathematics Instruction,Generalization,College Mathematics,Semi Structured Interviews,Calculus,Mathematical Logic,Problem Solving

This paper explores students' ways of thinking about the average rate of change of a multivariable function and how they generalize those ways of thinking from rate of change of single-variable functions. I found that while students thought about the average rate of change of a multivariable function as the change in the independent quantity with respect to the changes in the dependent quantities, they had difficulty determining a process to assign a value to that rate of change. Most tried to represent the average rate of change as a singular expression, generalizing the ?y/?x expression to create expressions of the form ?z/[?x and ?y], yet did not appear to have a sense of what they believed they were measuring. This suggests that quantitative reasoning, or lack thereof, was at the heart of the students' generalizations. A pedagogical implication of this research is that students' natural tendency to try to determine a singular expression for the average rate of change of a multivariable function could serve as useful as motivating the need to hold a variable fixed. [For the complete proceedings, see ED583989.]
When &quot;Half an Hour&quot; Is Not &quot;Thirty Minutes&quot;: Elementary Students Solving Elapsed Time Problems
Author: Earnest, Darrell
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Elementary School Mathematics,Elementary School Students,Problem Solving,Time,Student Evaluation,Concept Formation,Mathematical Concepts,Grade 2,Grade 4,Mathematics Tests,Responses

This paper presents assessment study results addressing the question: Do students treat elapsed time problems differently if phrased as &quot;half an hour&quot; versus &quot;thirty minutes&quot;? A paper-and-pencil assessment was given to second (n = 292) and fourth (n = 205) grade students in six New England elementary schools. I compare responses on tasks presented in hour units and minute units. Results indicate that children respond differently to elapsed time questions as a function of the units provided in the question (half hour or thirty minutes) depending on the provided starting time (e.g., on the half hour versus on the second half of the clock). [For the complete proceedings, see ED583989.]
Designing Formative Assessment Lessons for Concept Development and Problem Solving
Author: Swan, Malcolm
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Formative Evaluation,Teaching Methods,Concept Formation,Evaluation Methods,Secondary School Mathematics,Common Core State Standards,Foreign Countries,Cooperative Learning,Problem Solving,College Mathematics

Formative assessment is the process by which teachers and students gather evidence of learning and then use this to adapt the way they teach and learn. I describe a design research project in which we integrated formative assessment strategies into lesson materials that focus on developing students' conceptual understanding and their capacity to tackle non-routine problems. A theoretical framework for assessment task design is presented, together with an analysis of research-based principles for formative assessment lesson design. Particular aspects are highlighted: the roles of pre-assessment, formative feedback questions and sample work for students to critique. While there are some early signs that these lessons provide an effective model for teachers to introduce formative assessment into everyday classroom practice, the materials require a radical shift in the predominant culture within most classrooms. [For the complete proceedings, see ED583989.]
Linguistic Norms of Undergraduate Mathematics Proof Writing as Discussed by Mathematicians and Understood by Students
Author: Lew, Kristen
Date: 2015-11-00
Type: Speeches/Meeting Papers
Subjects: Mathematics,Professional Personnel,Validity,Mathematical Logic,Undergraduate Students,College Mathematics,Language Usage,Writing (Composition),Qualitative Research,Interviews

We studied the linguistic norms of mathematical proof writing at the undergraduate level by asking two mathematicians and five mathematics undergraduate students to read seven partial proofs based on student-generated work and to identify and discuss uses of mathematical language that were out of the ordinary with respect to standard mathematical proof writing. By asking participants to discuss the seriousness of each breach, we not only identify and discuss some of these linguistic norms, but also describe important differences between the ways in which mathematicians and students understand them. [For the complete proceedings, see ED583989.]