Sandefur James, Manaster Alfred B
Georgetown University, Washington, DC USA.
University of California San Diego, La Jolla, CA USA.
ZDM. 2022;54(4):895-907. doi: 10.1007/s11858-022-01354-6. Epub 2022 May 3.
Recursive reasoning is a powerful tool used extensively in problem solving. For us, recursive reasoning includes iteration, sequences, difference equations, discrete dynamical systems, pattern identification, and mathematical induction; all of these can represent how things change, but in discrete jumps. Given the school mathematics curriculum's later emphasis on calculus-the mathematics of change in continuous contexts-it is surprising that the curriculum seems to neglect recursive thinking after the early grades. Research shows that recursion supports the learning of algebra among younger students, but the lack of similar research with older students is concerning. In this paper we suggest possible affordances from teaching recursive modeling, including a basic model of the spread of contagious diseases. We also discuss different ways to present these models at various points in the curriculum that might develop connections between mathematics and the real world, and support students' learning of mathematics. This leads to what we, as mathematicians, think would be interesting research questions for mathematical educators.
递归推理是一种在解决问题时被广泛使用的强大工具。对我们来说，递归推理包括迭代、序列、差分方程、离散动力系统、模式识别和数学归纳法；所有这些都可以表示事物如何变化，但变化是离散的跳跃式。鉴于学校数学课程后来强调微积分——连续情境下变化的数学——令人惊讶的是，该课程在低年级之后似乎忽视了递归思维。研究表明，递归有助于低年级学生学习代数，但缺乏针对高年级学生的类似研究令人担忧。在本文中，我们提出了教授递归建模可能带来的益处，包括一个传染病传播的基本模型。我们还讨论了在课程的不同阶段呈现这些模型的不同方式，这些方式可能会建立数学与现实世界之间的联系，并支持学生学习数学。这引出了作为数学家的我们认为对数学教育工作者来说有趣的研究问题。
