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Joe Garofalo

Recent research in mathematics education has shown that success or failure in solving mathematics problems often depends on much more than the knowledge of requisite mathematical content. Knowing appropriate facts, algorithms, and procedures is not sufficient to guarantee success. Other factors, such as the decisions one makes and the strategies one uses in connect ion with the control and regulation of one's actions (e.g., deciding to analyze the conditions of a problem, planning a course of action, assessing progress), the emotions one fee ls while working on a mathematical task (e.g., anxiety, frustration, enjoyment), and the beliefs one holds relevant to performance on mathematical tasks, influence the direction and outcome of one's performance (Garofalo and Lester 1985; Schoenfe ld 1985; McLeod 1988). These other factors, although not explicitly addressed in typical mathematics instruction, are nonetheless important aspects of mathematical behavior.

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Christine P. Trinter and Joe Garofalo

Nonroutine function tasks are more challenging than most typical high school mathematics tasks. In our classes of precalculus students and preservice mathematics teachers, we have found that nonroutine tasks encourage our students to expand their thinking about functions and their approaches to problem solving. As a result, they gain greater appreciation for the power of multiple representations and a richer understanding of functions.

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Joe Garofalo and Christine P. Trinter

Students think resiliently about using the quadratic formula, analyzing factors graphically, finding the shortest distance between two points, and finding margin of error.

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Christine P. Trinter and Joe Garofalo

If students suspect that a problem is solvable, they will persevere in their efforts to analyze and solve it.

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Joe Garofalo and Christine Trinter

In this article, we present 2 technology-involved tasks that we use in our mathematics pedagogy courses to ostensibly give preservice secondary mathematics teachers (PSMTs) sample activities they can use in their teaching or use to assess their own future students' ability to apply trigonometric functions in contextual situations using technology. However, we have two other purposes for posing these tasks. One purpose is to provide occasions for PSMTs to self-assess their mathematical and technology knowledge, and subsequently take action to learn mathematics and technology features. The other purpose is to use such tasks as springboards for substantive discussions about teaching, learning, technology, and assessment. Such simulation tasks have engaged PSMTs and helped them develop their knowledge base for teaching mathematics.

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Beth L. Cory and Joe Garofalo

This study investigates 3 preservice secondary mathematics teachers' understandings of limits of sequences and their changing conceptions of limit during and after instruction involving interactive, dynamic sketches embodying the formal definition of the limit of a sequence. Manipulating a coherent visual representation of the formal definition in the contexts of various sequences, coupled with answering carefully chosen questions and completing interview tasks before, during, and after technology-enhanced instruction, gave the participants opportunities to investigate and reflect on their own concept image as they compared their understandings to the results of the actions they performed on the sketch.

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Kimberly Corum and Joe Garofalo

Incorporating modeling activities into classroom instruction requires flexibility with pedagogical content knowledge and the ability to understand and interpret students' thinking, skills that teachers often develop through experience. One way to support preservice mathematics teachers' (PSMTs) proficiency with mathematical modeling is by incorporating modeling tasks into mathematics pedagogy courses, allowing PSMTs to engage with mathematical modeling as students and as future teachers. Eight PSMTs participated in a model-eliciting activity (MEA) in which they were asked to develop a model that describes the strength of the magnetic field generated by a solenoid. By engaging in mathematical modeling as students, these PSMTs became aware of their own proficiency with and understanding of mathematical modeling. By engaging in mathematical modeling as future teachers, these PSMTs were able to articulate the importance of incorporating MEAs into their own instruction.

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Leroy G. Callahan and Joe Garofalo

Metacognition refers to the knowledge and control one has of one's cognitive functioning, that is, what one knows about one's cognitive performance and how one regulates one's cognitive actions during performance. In recent years, a growing number of psychologists and educators have been discussing and investigating the role and significance of metacognition on various aspects of academic performance (Flavell 1979). Most of the theorizing and research concerning metacognition has been in reference to performance on reading and memory tasks (Baker and Brown 1984; Schneider 1985), but lately mathematics educators have begun studying the role of metacognition in the performance of mathematical tasks. These mathematics educators are convinced that what one knows or believes about oneself as a learner and doer of mathematics and how one controls and regulates one's behaviors while working through mathematical tasks can have powerful effects on one's performance (Garofalo and Lester 1985; Kilpatrick 1984; Schoenfeld, in press; Silver 1985).

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Joe Garofalo and Jerry Bryant

Our friend Pat received a telephone bill of slightly more than $60000 and called the telephone company to report the error. When the company's representative suggested that she may have made more calls than she remembered during the month, Pat pointed out that it would take a very large number of calls to run up a bill of $60 000. The representative agreed and told Pat that her bill would be investigated. The representative then told her that in the meantime she should pay the full amount of the bill and that when the telephone company determined the correct amount due, any excess payment would be credited toward her next bill!

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Joe Garofalo, Christine P. Trinter and Barbara A. Swartz

Logical arguments use examples and existence to prove or disprove four statements.