BOX 3–5 Typical Student Beliefs About the Nature of Mathematics

  • Mathematical talent is innate— “either you have it or you don’t,” —and effort doesn’t make much of a difference.

  • Mathematics problems have one and only one right answer.

  • There is only one correct way to solve any mathematics problem—usually the rule the teacher has most recently demonstrated to the class.

  • Ordinary students cannot expect to understand mathematics; they expect simply to memorize it and to apply what they have learned mechanically and without understanding.

  • Mathematics is a solitary activity, done by individuals in isolation.

  • Students who have understood the mathematics they have studied will be able to solve any assigned problem in 5 minutes or less.

  • The mathematics learned in school has little or nothing to do with the real world.

  • Formal proof is irrelevant to processes of discovery or invention.

SOURCE: Greeno, Pearson, and Schoenfeld. (1996b, p. 20).

standing so that instructional strategies can be selected to support an appropriate course for future learning. In particular, assessment practices should focus on identifying the preconceptions children bring to learning settings, as well as the specific strategies they are using for problem solving. Particular consideration needs to be given to where children’s knowledge and strategies fall on a developmental continuum of sophistication, appropriateness, and efficiency for a particular domain of knowledge and skill.

Practice and feedback are critical aspects of the development of skill and expertise. One of the most important roles for assessment is the provision of timely and informative feedback to students during instruction and learning so that their practice of a skill and its subsequent acquisition will be effective and efficient.

As a function of context, knowledge frequently develops in a highly contextualized and inflexible form, and often does not transfer very effectively. Transfer depends on the development of an explicit understanding of when to apply what has been learned. Assessments of academic achievement need to consider carefully the knowledge and skills required to understand and answer a question or solve a problem, including the context in

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