Mathematical cognition

数学，尤其是算术，是每个人每天都会用到的一项技能。购物、银行业务、家庭装修，甚至追随喜爱的运动队，都需要激活和操纵数量和数字。知情公民每天通过电视、杂志和报纸体验大量的定量数据。因此，理解人们用来完成数学任务的表征和过程具有明显的实际重要性。从理论上讲，数学是一个易于处理的领域，可以在其中测试有关思想家如何能够处理多重心理任务的理论。我正在进行的研究项目涉及人们如何存储和访问他们的数学知识的问题。目前的建议集中在我们如何使用有限的精神资源来管理复杂的数学活动。考虑以下问题：你需要估算一个13 × 8.5米的房间的平方英尺。你有几个选择。你可以用学校教的解算法。你可以用计算器。你可以想象这些数字，然后心算。或者，你可以把8.5四舍五入到10，然后给出一个近似的答案，“大约130平方米”。你选择哪一种解决方法取决于具体情况、你的数学能力以及对答案精度的要求。在我的研究中，我们将算术任务与其他已知需要特定心理资源的任务结合起来，以确定人们如何解决数学问题。例如，参与者可能会被要求在解决上述问题时记住视觉或口头信息，然后我们测试他们的心理过程是如何受到特定记忆需求的影响的。我对数学中的个体差异特别感兴趣。哪些认知能力或技能有助于数学技能？总而言之，在过去的20年里，我对数学认知的研究提供了对成年人用于解决数学问题的心理过程的灵活性和多样性的见解。这些见解和本提案中描述的进一步研究将使我们对人们如何解决数学问题的理解取得进展。

Mathematics, especially arithmetic, is a skill that everyone uses on a daily basis. Shopping, banking, home renovations, and even following a favorite sports team necessitate the activation and manipulation of quantities and numbers. Informed citizens experience a daily barrage of quantitative data through TV, magazines, and newspapers. Thus, understanding the representations and processes that people use to accomplish mathematical tasks has obvious practical importance. Theoretically, mathematics is a tractable domain in which to test theories about how, for example, thinkers are able to manage multiple mental tasks. My ongoing research program addresses questions about how people store and access their mathematical knowledge. The present proposal is focused on how we use limited mental resources to manage complex mathematical activities. Consider the following: You need to estimate the square footage of a room that measures 13 x 8.5 m. You have several options. You could use the solution algorithm taught in school. You could use a calculator. You might visualize the numbers and mentally calculate. Or, you could estimate by rounding 8.5 to 10 and responding with an approximate answer, "about 130 square m". Which solution approach you choose will depend on the situation, on your mathematical competencies, and on the precision required for the answer. In my research, we combine arithmetical tasks with other tasks known to require specific mental resources to determine how people solve mathematical problems. For example, participants might be asked to remember visual or verbal information while solving the above problem and then we test how their mental processes are affected by the specific memory demands. I am particularly interested in individual differences in mathematics. What cognitive abilities or skills contribute to mathematical skill? In summary, over the last 20 years, my research on mathematical cognition has provided insights into the flexibility and diversity of mental processes that adults use to solve math problems. These insights and the further research described in this proposal will allow advances in our understanding of how people solve mathematical problems.