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Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence

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Classroom Assessment Techniques
Mathematical Thinking

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Instruction in mathematics should help students:

If students are to achieve these goals, then an appropriate intellectual environment in which they learn mathematics must be created. The MathCATs provide materials which support the creation of such learning environments.

Theory and Research

On the nature of mathematics
Mathematics... today is a diverse discipline that deals with data, measurements, and observations from science; with inference, deduction, and proof; and with mathematical models of natural phenomena, of human behavior, and of social systems...

In addition to theorems and theories, mathematics offers distinctive modes of thought which are both versatile and powerful, including modeling, abstraction, optimization, logical analysis, inference from data, and use of symbols. Experience with mathematical modes of thought builds mathematical power -- a capacity of mind of increasing value in this technological age that enables one to read critically, to identify fallacies, to detect bias, to assess risk, and to suggest alternatives. Mathematics empowers us to understand better the information-laden world in which we live. (National Research Council, 1989, pp. 31-32).

This quotation describes a number of distinct features of mathematics; a body of knowledge; a set of tools to be used in everyday life and by the scientific community; and a set of powerful thinking tools. Mathematics is seen as discipline which empowers its students. It follows that education in mathematics should set out to address each of these features.

Challenges facing teaching
Teachers of undergraduate mathematics and other disciplines that rely on mathematics face a number of challenges. Serious conceptual problems have been documented in students who seem appropriately qualified. Armstrong and Croft (1999) set diagnostic tests of mathematics on entry to engineering programs, and showed several areas of weakness. For example, around 20% of students had problems dealing with significant figures, and around 15% had problems with decimals. Student ratings of their confidence that they understood and could apply mathematics ranged from about 60% for the graph of a linear function, to about 10% for polar co-ordinates. Faculty are often dissatisfied with student knowledge on entry to freshman mathematics classes (the Royal Society, 1998). Particular problems are associated with: student fluency and accuracy; failure to understand the connectedness of mathematics; a lack of understanding of proof and the need for rigorous argument; and an inability to solve non-standard problems (e.g., Tall, 1992).

A number of commentators have identified problems with conventional approaches to the teaching of mathematics in high schools and colleges. Weaknesses are related to pedagogy and assessments which emphasize the mastery of mathematical techniques, but emphasize neither the conceptual side of mathematics nor the development of the habits of mind that characterize mathematical thinking. Traditional testing methods in mathematics have often provided limited measures of student learning, and equally importantly, have proved to be of limited value for guiding student learning. The methods are often inconsistent with the increasing emphasis being placed on the ability of students to think analytically, to understand and communicate, or to connect different aspects of knowledge in mathematics (e.g., Ridgway, 1988; Brown, Bull and Pendlebury, 1997).

One consequence of this type of curriculum and assessment system is that students learn in school that problems mostly have neat, unique solutions, and that methods to solve problems will be provided to them. For example, in the 1983 National Assessment of Educational Progress, nine students out of ten agreed with the statement "There is always a rule to follow in solving mathematics problems" (NAEP, 1983, pp. 27-28). Over time students come to adopt a passive role, and think of mathematics as a dead body of knowledge which they have to memorize, rather than as a set of higher-order thinking tools which will increase their abilities to deal with a complex world. (e.g., Carpenter, Lindquist, Matthews, and Silver, 1983; Schoenfeld 1992).

Developing Mathematical Thinking
...the reconceptualization of thinking and learning that is emerging from the body of recent work on the nature of cognition suggests that becoming a good mathematical problem solver - becoming a good thinker in any domain - may be as much a matter of acquiring the habits and dispositions of interpretation and sense-making as of acquiring any particular set of skills, strategies, or knowledge. (Resnick, 1989, p. 58).

Thinking mathematically depends on a number of different components (Schoenfeld, 1992), notably core knowledge, problem solving strategies, effective use of one's resources, having a mathematical perspective, and active engagement in the practice of mathematical thinking. Mathematics instruction must present experiences which develop student knowledge in each of these areas.

Mathematics in the classroom should model these elements if students are to come to understand and use mathematics and to learn to think mathematically. Learning mathematics is about learning to work in the ways that mathematicians work, and is about acquiring the thinking skills that mathematicians use. These skills are important for scientists as well as mathematicians. Pólya (e.g., 1954, 1957) argued that mathematics resembles the physical sciences in its dependence on conjecture, insight, and discovery. He argued that for students to understand mathematics, their experience with mathematics must be consistent with the way mathematics is done by mathematicians.

There is an extensive body of knowledge comparing the knowledge of experts and novices (e.g., Ericsson and Charness, 1994 for a review across disciplines; Schoenfeld, 1985 for studies in undergraduate mathematics) which can be mined for ideas on appropriate teaching strategies; and a great many studies which show the effectiveness of particular teaching methods (e.g., Palinscar and Brown, 1984). Accessible accounts of the literature are provided by Schoenfeld (1983, 1985) and Bransford, Brown, and Cocking (Eds.) (1999).

Developing Assessment
Improved assessment systems may help with these problems. There is evidence that educational attainment can be raised by better assessment systems (Black and William, 1998; Dassa, Vazquez-Abad, and Ajar; 1993). Such assessment systems are characterized by: a shared understanding of assessment criteria; high expectations of performance; rich feedback; and effective use of self-assessment peer assessment, and classroom questioning.

The intention of the MathCATs is to improve the quality of both formative and summative assessment systems, and thereby to improve the quality of undergraduate mathematics teaching and learning.

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Mathematical Thinking CATs || Fault Finding and Fixing || Plausible Estimation
Creating Measures || Convincing and Proving || Reasoning from Evidence

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