The purpose of this study was to gain insight into 30, first year calculus students' understanding of the relationship between the concept of vertex of a quadratic function and the concept of the derivative. APOS (action-process-object-schema) theory was applied as a guiding framework of analysis on student written work, think-aloud and follow up…

Transfer of learning, the application of learning to different contexts over time, is important to all learning for development. As 21st century skills specifically aim to be "generic," there is an assumption that they can be transferred from context to context. We investigate the process of transfer in problem solving, with specific…

Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs.…

In this article, we examine how students in a general education quantitative literacy course reason with public issues when unprompted to use quantitative reasoning. Michigan State University, like many institutions, not only has a quantitative literacy requirement for all undergraduates but also offers two courses specifically for meeting the…

This article is the result of research aims to describe the patterns and characteristics of the process of metacognition student of mathematics in solving calculus problems. Description was done by looking at changes in "awareness," "evaluation," and "regulation" as components of metacognition. The changes in…

Although test scores from similar tests in multiple choice and constructed response formats are highly correlated, equivalence in rankings may mask differences in substantive strategy use. The author used an experimental design and participant think-alouds to explore cognitive processes in mathematical problem solving among undergraduate examinees…

Introduction: Representations play an essential role in mathematical thinking. They favor the understanding of mathematical concepts and stimulate the development of flexible and versatile thinking in problem solving. Here our focus is on their use in optimization problems, a type of problem considered important in mathematics teaching and…

The purpose of this article is to investigate the mathematical practice of proof validation--that is, the act of determining whether an argument constitutes a valid proof. The results of a study with 8 mathematicians are reported. The mathematicians were observed as they read purported mathematical proofs and made judgments about their validity;…

This paper describes the implications of constructivism both for what to teach and for how to teach. The first part discusses three approaches on what to teach: (1) relating mathematical knowledge to other knowledge; (2) checking for self-consistency; and (3) putting thinking before facts. The second part is on how to teach based on…

Describes basic problem-solving skills and presents a protocol of a student solving a problem. Discusses some of the difficulties in obtaining a correct solution. Lists basic prompts for problem solving. (YP)

The ability to think critically in the presence of arguments with essential quantitative elements, most often graphical elements, will become an essential skill for educated citizens in the future. This paper takes one specific graphical display, a narrow series of observational and interpretational tasks related to a graph, and using a small set…

Teaching students to think mathematically or to apply mathematics to the solution of real-world problems has become a national priority. This paper explores a method of developing the basic problem-solving skills in a remedial mathematics course at the college freshman level. The first part of this paper describes some basic problem-solving skills…