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The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general scientific thinking skills that are often taken for granted (and rarely taught) in physics classes. In this paper, I give an overview of my analysis of these additional skills. I propose specific tools for helping students develop these skills in subsequent papers. [ABSTRACT FROM AUTHOR]

Transfer of structural math knowledge to physics is difficult for students. While research suggests various improvement techniques, enhancing parallelism of algebraic structures used in physics to those studied in mathematics courses seems underrepresented. This paper proposes an alternative way of introducing wave functions as a set of parametric equations. Research shows that the mathematical underpinnings of the mechanical wave function are problematic for students. As seen from the math perspective, there can be several drawbacks that likely impede scientific analysis and interpretation of the wave function: y (x , t) = A cos ( 2 π λ x − 2 π T t). [ABSTRACT FROM AUTHOR]

An important step in learning to use math in science is learning to see symbolic equations not just as 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 propose "anchor equations" as a construct to support teaching and learning in introductory physics. I define anchor equation, provide examples, and suggest ways anchor equations can be used in instruction to support the development of students' mathematical sense-making. [ABSTRACT FROM AUTHOR]

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 the dimensionality of measured quantities and dimensional analysis (DA) 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. [ABSTRACT FROM AUTHOR]

Learning to create, use, and evaluate models is a central element of becoming a scientist. In physics, we often begin an analysis of a complicated 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. [ABSTRACT FROM AUTHOR]

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 (with a unit). 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. [ABSTRACT FROM AUTHOR]

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 (co-vary). 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 emphasized 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 learn to use math productively in science. [ABSTRACT FROM AUTHOR]