Using math in physics -- 1. Dimensional analysis: https://arxiv.org/abs/2011.12760

Edward F. Redish

Making meaning with math in physics requires blending physical conceptual knowledge with mathematical symbology. Students in introductory physics classes often struggle with this, but it is an essential component of learning how to think with math. Teaching dimensional analysis (DA). figuring out what measurements were combined to create a symbolic quantity, is a valuable first step in helping them learn to appreciate this difference. In this paper I discuss some of the issues associated with learning dimensional analysis and show some ways we can modify our instruction to help. This paper is one of a series on how to help students develop the scientific thinking skills required for learning to use math in science.

Comments: 5 pages, 3 figures

Using math in physics -- 2. Estimation: https://arxiv.org/abs/2011.12699

Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification: that physical quantities can be assigned specific numbers. A second is to develop an intuition for scale. One way to help students develop these skills is to teach estimation: the ability to consider a physical situation and put reasonable approximate numbers to it

Comments: 5 pages, 2 figures

Using math in physics -- 3. Anchor equations: https://arxiv.org/abs/2011.12761

An important step in learning to use math in science is learning to see physics equations as not just calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper I discuss the role of basic anchor equations in introductory physics and show some examples of how to help students learn to use them.

Comments: 6 pages, 4 figures

Using math in physics -- 4. Toy models: https://arxiv.org/abs/2011.12700

Learning to create, use, and evaluate models is a central element of becoming a scientist. In physics, we often begin an analysis of a complex system with highly simplified or toy models. In introductory physics classes, we tend to use them without comment or motivation. Some students infer that physics is irrelevant to their understanding of the real world and are discouraged from making the cognitive blend of physics concepts with math symbology essential for making sense of physics. In this paper, I discuss the often hidden barriers that make it difficult for our students to accept and understand the value of toy models, and suggest instructional approaches that can help.

Comments: 6 pages, 5 figures

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Using math in physics: 5. Functional dependence: https://arxiv.org/abs/2012.00794

Edward F. Redish

When students are learning to use math in physics, one of the most important ideas they need to learn is that equations are not just calculational tools; they represent relationships between physical variables that change together (covary). How much a change in one variable or parameter is associated with a change in another depends on how they appear in the equation: their functional dependence. Understanding this sort of relationship is rarely taught in introductory mathematics classes, and students who have not yet learned to blend conceptual ideas with mathematical symbols may not see the relevance and power of this idea. We need to explicitly teach functional dependence as part of our effort to help students to learn to use math productively in science.

Comments: 5 pages, 4 figures