Student Errors in Dynamic Mathematical Environments

This study investigates interactions between calculus learning and problem solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students’ problem solving abilities in a common US calculus course design that included traditional lecture and assessment with problem solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem solving as well as multiple mathematical registers and quantitative modeling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students’ ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students’ performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviors that characterized top-performance or attrition in the course, respectively. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less-sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of much US calculus reform. We provide explanations for these patterns in students’ problem solving performance in view of both their preparation for university calculus and the courses’ assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students’ problem solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction.

Journal of Mathematical Modelling and Application, 2014

This paper presents some concepts, principles, and techniques for automated testing of real-time reactive software systems based on attributed event grammar (AEG) modeling of the environment in which a system will operate. AEG provides a uniform approach for automatic test generation, execution, and analysis. Quantitative and qualitative assessment of the system comprised of the software under test and its interaction with the environment, can be performed based on statistics gathered during automatic test execution within an environment model.

The Journal of Mathematical Behavior, 2005

ABSTRACT Introduces this special issue of the Journal of Mathematical Behavior. This special issue originated from the 10th International Congress of Mathematics Education's Topic Study Group 18: Problem Solving in Mathematics Education. The general aims of the Topic Study Group were to provide a forum for those who are interested in any aspect of problem-solving research at any educational level, to present recent findings, and to exchange ideas. We set up three specific goals for the Problem Solving Topic Study Group: (1) to examine the understanding of the complex cognitive processes involved in problem solving; (2) to explore the actual mechanisms by which students learn and make sense of mathematics through problem solving, and how this can be supported by teachers; and (3) to identify future directions of problem-solving research, including the use of information technology. The Topic Study Group received a good response. Most of the papers in this special issue are from those who presented at the Topic Study Group. In addition, we invited a few other researchers to submit papers in order to cover various aspects of problem-solving research that we wished to be represented in this issue. This special issue includes 12 papers, each addressing at least one of the three goals listed above. The first six papers that appear are empirically based; in these papers, the authors present the results of the fieldwork that they have conducted and also raise research questions for future studies. The remaining six papers are essays discussing issues about problem solving, and how these issues have been, or should be, the subjects of research. In this article, we briefly highlight the contributions of each of the 12 papers. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

2010

Research Questions: One way students may develop conceptual understanding is through working on strands of related mathematical tasks and thus developing and refining their understanding of the underlying mathematical concepts contained in the tasks. The purpose of this study is to illuminate this process by detailing the inherent mathematical structures in such a strand and discuss what aspects of it facilitated student learning. The research questions addressed are: (1) What mathematical structures can be uncovered by exploring/engaging with the combinatorics tasks used in the Rutgers longitudinal study? (2) In what ways are these mathematical structures revealed during students' problem-solving processes? Methodology: Ten tasks from the combinatorics/counting strand are selected from the Rutgers longitudinal project for this qualitative study. The data available for analysis are in the form of digitized video tapes, verified transcripts, and students' written work. The analysis focuses on decoding students' solutions into formal mathematical definitions and theorems. Concept maps are used to illustrate the overall hierarchy of the presented mathematical structures. Findings: There are a total of sixty-three inherent mathematical structures extracted from the formal solutions of ten selected combinatorics tasks. These structures are categorized as definitions, notations, axioms, properties, formulas, and theorems. When classified with respect to the seven relevant sub-domains of mathematics, these structures pertain to: set theory, enumerative combinatorics, graph theory, sequences & sets, general algebraic system, probability theory, and geometry. The analysis suggests that the participating students uncovered many of these mathematical structures primarily in the following ways: (1) Manipulating a concrete model, (2) Listing all possible combinations, (3) Inventing different representations, (4) Seeking patterns, and (5) Making connections. These findings support the following suggestions for practice: (1) Teachers may benefit from studying the underlying structures of a task thoroughly before assigning the task to students, (2) In determining the order of related tasks within a strand, teachers need to consider the sophistication level and the coherence of the underlying structures across tasks, (3) Using concrete models can help students to both develop and verify solutions to complex problems, and (4) Tasks whose inherent structures belong to a variety of mathematical sub-domains can help students build an increasingly interconnected view of mathematics. Significance: This study outlined a method of extracting inherent mathematical structures from mathematical tasks. The results suggest that students have natural abilities to uncover these structures by themselves. It is hoped that this will motivate mathematics teachers to improve the way they think about using problem solving in their teaching. I want to express my gratitude to several people for their support over the years. Marjory F. Palius and Robert Sigley were of great help by facilitating my access to data CDs, tapes, transcripts, and other useful information. My fellow graduate students provided encouragement and patience when I most needed them. My family were accommodating of my efforts towards this degree. I am grateful to my committee members. The chair and my advisor Carolyn A. Maher was the one who succeeded time and time again in convincing me to stay in the program when I was ready to walk away. Throughout the dissertation process she provided tons of ideas, advice, and support. Alice S. Alston's inquiries always made me think about my writing over and over again. Elizabeth B. Uptegrove kindly gave suggestions and corrected my grammatical errors. Special thanks must go to Iuliana Radu, my dearest friend, peer, and mentor. Her continuous support was instrumental in my persevering in this endeavor. She has proofread everything I wrote since I entered the program. She corrected my English errors, gave me timely and valuable feedback, found more articles for me to read, and at times took care of my personal needs so that I could devote more time to study. I can not imagine the present accomplishment without these admirable and unforgettable people. My gratitude goes to you all.

Abstract This paper shares the findings of an exploratory, qualitative investigation of elementary school students' problem solving strategies. Twelve fourth-grade students were given three mathematical tasks about fractions and interviewed during task completion about their problem solving strategies, and their understanding of how to solve the problems. Students were selected to provide variance across their mathematical achievement on the state-wide test and the curricula used in their classroom.

International Journal of Mathematical Education in Science and Technology, 1993

Transformation, 2016

Traditional large-scale and high-stakes assessments have focused largely on whether test takers give the correct or incorrect answers to questions. Early instructional software followed this paradigm. The introduction of intelligent tutoring systems (ITSs) led to an emphasis on discovering where students were making mistakes and explaining the mistakes through matching them to pre-defined error catalogs. The deficiencies with this approach were an emphasis on identifying only procedural mistakes and not validating whether the matched errors were, in fact, the true causes of mistakes. The present paper describes an ITS-style assessment software that diagnoses causes of errors by assessing underlying and prerequisite concepts a student needs to solve a problem. The assessments focus on a variety of knowledge types: abstract concepts, procedures, and ability to apply concepts to problems. The software even assesses whether a mistake was caused by carelessness or mistyping information from the problem. The software was evaluated by comparing its agreement in diagnosing causes of students' mistakes with that of experienced teachers. Results showed that the software's agreement percentage was in the 90s and statistically equal to that of experienced teachers' inter-rater agreement.

Instructional Science, 2010

This paper reports on a quasi-experimental study comparing a "productive failure" (Kapur, 2006, in press) instructional design with a traditional "lecture and practice" instructional design for a two-week curricular unit on rate and speed. Participants comprised 75, 7 th -grade mathematics students from a mainstream secondary school in Singapore. Students experienced either a traditional lecture and practice teaching cycle or a productive failure cycle, where they solved complex, ill-structured problems in small groups without the provision of any support or scaffolds up until a consolidation lecture by their teacher during the last lesson for the unit. Findings suggest that students from the productive failure condition produced a diversity of linked problem representations but were unable to produce good quality solutions, be it in groups or individually. Expectedly, they reported low confidence in their solutions. Despite seemingly failing in their collective and individual problem-solving efforts, students from the productive failure condition significantly outperformed their counterparts from the lecture and practice condition on both well-and ill-structured problems on the post-tests. After the post-test, they also demonstrated significantly better performance in using structured-response scaffolds to solve problems on relative speed-a higher-level concept not even covered during instruction. Findings and implications are discussed.

2016