*NEXUS/Physics > Threads*

Using math in science is quite different from the way math is taught in typical math classes. Students often consider math as a way to calculate, but it is used in science to organize conceptual knowledge, build models, study functional and parameter dependence, consider extreme cases and many other things. For many students he key missing factor is learning to blend their knowledge of the physical world with math and to make physical meaning with mathematics. This lesson set identifies specific tools and approaches that can help students develop math-in-science skills. It provides a thread that can be called on and reinforced throughout the course.

The first competency specified in the AAMC/HHMI report *Scientific Foundations of Future Physicians* is to be able to apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world. The second competency specified by the AAAS in *Vision and Change* is the ability to use quantitative reasoning and mathematical modeling. Physics is an excellent place to learn mathematical reasoning and NEXUS/Physics has a specific thread running throughout the course on building student strengths in mathematical modeling and quantitative reasoning.

This page includes an instructional "thread" on mathematical reasoning. These materials have the goal of providing active-engagement activities to help students build their math-in-science skills and a greater understanding of the role of mathematical modeling in physics, chemistry, and biology. They include:

*Readings* -- text describing energy concepts from the macroscopic to the atomic scale described in a coherent way;
*Homework problems* -- tasks for out-of-class work for students to complete on their own;
*Lecture materials* -- PowerPoint slides discussing the issues of this thread.* *
*Peer instruction questions* -- clicker questions for encouraging in-class discussion;
*Group problem solving activities* -- guided activities for use in recitation sections with students working together in groups;
*Quiz and exam problems* -- problems that can be used for either formative or summative assessment throughout the class.
*References* -- the materials in this section are based on extensive education research. Some of the references to this literature are available here.

The readings discuss the role of mathematics in scientific modeling and reviews basic mathematics in a view oriented toward scientific modeling.

The following readings explain how using math in science differs from pure math. They stress the "why" behind issues like dimensional analysis, units, and estimation, and they discuss the nature of mathematical modeling.

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The following pages review the basic mathematical tools needed for the class, stressing interpretation and meaning making.

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The following pages identify specific general purpose methods that help students learn to use math to make sense of real world situations. In addition to the readings there are lecture slides available with examples under "Lecture materials" below.

The following pages describe how qualitative knowledge is coded in the core equations.

Reading the content in ...

**Homework problems**:

The out-of-class activities are intended to be done by students out of class working together. These are intended to be challenging problems that require deep thought about the fundamental concepts and a bringing together of concepts learned in biology and chemistry with the physics that is being studied. Of course, almost all of the problems in a physics class involve mathematical modeling in some way, but in the collection the focus is strongly on the role of mathematics and mathematical modeling.

One way that we address the issue of seeing the value of equations more broadly are with *problem triples*. These are set of three problems using the same physical situation in which students are first asked to use an equation to reason qualitatively, second to build a symbolic relationship, and only third to carry out an explicit calculation. One way to use these is to give a triple over a three week period. Multiple triples can be run simultaneously so during each week students are doing at least one problem of each type. Although these problems are mostly in standard "toy model" physics situations, the issues being dealt with are especially relevant for life science students.

We also have a large number of estimation problems available that can help students build their skills of quantifying their everyday experiences. See

Of course, the Representation Translation Thread also involves the mapping of mathematics into physical meaning and problems there also relate to the issue discussed here.

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These topics are built into all parts of the course. But specifically, here are a set of lecture slides that can be drawn from when the various items in the Toolbelt are introduced.

**Peer Instruction Questions**:

These are presented as slides in PowerPoint booklets to allow faculty to easily include them in their lectures. The clicker slides in this booklet are meant to be used as stimuli to encourage class discussion. They are intended for use in a class that attempts to help students develop a coherent and sophisticated understanding of scientific thinking. They are NOT intended as items to test whether students are “right or wrong” or “know” the correct answer by one-step recall if enough cues are given. (The direct links yield the PowerPoint versions some of which contain notes giving suggestions for use. The "pdf" links are PDF files of the slides without the notes.)

**Group Problem-Solving Activities**

These tasks are intended for small group problem-solving and discussion, and are intended to help students build strong conceptual fundamental understanding of the physics and to make the connection with what they know from biology and chemistry. They are not intended to be collected and graded. They are most effective when students work in groups of 3 to 5 and are given encouragement to think about the questions and bring in their knowledge from biology and chemistry where relevant. (Guidance for facilitating these activities are similar to those given for facilitating University of Washington-style Tutorials. See for example, *Facilitating in Tutorials* from the UMd-PERG.)

Quizzes are intended as formative assessment, not summative. It is intended for these to be challenging and for there to be potentially extended discussions about the answers when they are handed back. Exam questions are intended as summative. A few sample questions and formatting are provided here as document files for easy cutting and pasting into a printed quiz or exam. (*Use of these files are limited to faculty. Please contact **Joe Redish for access*.)

Some of the underlying research and the motivation, goals, and structure of this particular thread are discussed in greater detail in the papers,

- Problem Solving and the Use of Math in Physics Courses, E. F. Redish,
* Conference, World View on Physics Education in 2005: Focusing on Change, Delhi, August 21-26, 2005, *preprint.
- The Cognitive Blending of Mathematics and Physics Knowledge, T.J. Bing and E. F. Redish, in
*Proceedings of the Physics Education Research Conference, Syracuse, NY, August 2006*, *AIP Conf. Proc*. **883**, 26-29 (2007).
- Understanding students poor performance on mathematical problem solving in physics, J. Tuminaro and E. F. Redish, in
*Proceedings of the Physics Education Research Conference, Madison, WI, August 2003*, *AIP Conf. Proc*. **720** , 11-14 (2004)
- Elements of a Cognitive Model of Physics Problem Solving: Epistemic Games, J. Tuminaro and E. F. Redish,
*Phys. Rev. STPER*, **3**, 020101 (2007).
- Analyzing Problem Solving Using Math in Physics: Epistemological framing via warrants, T. J. Bing and E. F. Redish,
*Phys. Rev. STPER*, **5**, 020108 (2009). 15 pages
- Language of physics, language of math: Disciplinary culture and dynamic epistemology, E. F. Redish and E. Kuo,
*Science & Education*, **24**:5-6 (2015-03-14) 561-590. doi:10.1007/s11191-015-9749-7.
- Analysing the Competency of Mathematical Modelling in Physics. In: Greczyło T., Dębowska E. (eds)
*Key Competences in Physics Teaching and Learning*. Springer Proceedings in Physics, vol **190**. (2017, Springer, Cham). doi: 10.1007/978-3-319-44887-9_3 (free access to preprint through link)
- Blending physical knowledge with mathematical form in physics problem solving, Mark Eichenlaub & Edward F. Redish, to be published in GIREP book on Mathematics in Physics Education, G. Pospiech, ed. (Springer 2017 pr 2018).

For more information, contact Prof. E. F. Redish (redish@umd.edu)

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