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- 1. + Teaching Physics Standing on Your Head: Mathematics and Epistemology in Physics Edward Redish Department of Physics, U of MD July 27, 2015 AAPT-College Park
- 2. + Scientific reasoning and the role of math in physics n Science is all about epistemology – deciding what we know and how we know. n In physics, math has been closely tied with our epistemology since Galileo and Newton. n As a result, mathematics plays a significant role in physics instruction, even in introductory classes. (Not always in an optimal way, however.) n We don’t just calculate with math, we converse with it, “make meaning” with it, think with it, and use it to create new physics. July 27, 2015AAPT-College Park 2
- 3. +... Using Physics Education Research n My research group has been studying learning physics reasoning at the university level for ~20 years in many contexts. n Engineering students in introductory physics n Physics majors in advanced classes n Biology students in introductory classes, both with mixed populations and in a specially designed class for bio and pre-med students. n Data (mostly qualitative) n Videos of problem-solving interviews n Ethnographic data of students solving real HW problems in real classes, either alone or in groups. n Some multiple-choice questions on exams or with clickers. n Theory n Resources Framework* – built on ideas from education, psychology, neuroscience, sociology, and linguistics research. July 27, 2015AAPT-College Park * Redish, “How should we think about how our students think?”, Am. J. Phys. 82(2014) 537-551. 3
- 4. + A big problem students often have with physics is math. n We often say that “math is the language of physics”, but what physicists do with math is dramatically different from what mathematicians do with it. n For many students, this transition is a serious hurdle to learning physics. n Let’s unpack this in a few examples. July 27, 2015AAPT-College Park * Redish & Kuo, “Language of physics, language of math”, Sci. & Ed 25:5-6 (2015) 561-590. 4
- 5. + Example: Functional dependence A very small charge q0 is placed at a point somewhere in space. Hidden in the region are a number of electrical charges. The placing of the charge q0 does not result in any change in the position of the hidden charges. The charge q0 feels a force, F. We conclude that there is an electric field at the point where q0 is placed that has the value E0 = F/q0. If the charge q0 were replaced by a charge –3q0, then the electric field at the point would be a) Equal to –E0 b) Equal to E0 c) Equal to –E0/3 d) Equal to E0/3 e) Equal to some other value not given here. f) Cannot be determined from the information given.July 27, 2015AAPT-College Park Nearly half of 200 students chose this answer. Given in lecture in algebra-based physics .5
- 6. + Huh? n The topic had been discussed in lecture and students had read text materials showing a mathematical derivation. n When asked, most students could cite the result, “The electric field is independent of the test charge that measures it.” July 27, 2015AAPT-College Park ! E ! r( )= ! Fq0 E net q0 6
- 7. + What’s going on? n Many students treated the physics as a pure math problem: If A = B/C what happens to A if C is replaced by -3C? n They ignored the fact that F here is not a fixed constant, but is the force felt by charge q0 and therefore depends on the value of q0. July 27, 2015AAPT-College Park 7
- 8. +The processing of equations in physics is more complex than in a math class. n We link our equations with physical systems — which adds information on how to interpret the equation n We use symbols that carry extra information not otherwise present in the mathematical structure of the equation. n Our math is not just math: It’s a model of a physical system. July 27, 2015AAPT-College Park 8
- 9. + The structure of mathematical modeling: July 27, 2015AAPT-College Park • Often these all happen at once – intertwingled. (The diagram is not meant to imply an algorithm.) • In physics classes, processing is often stressed and the remaining elements shortchanged or ignored. 9
- 10. + Integrating math with physical knowledge does work for us n It lets us carry out chains of reasoning that are longer than we can do in our head, by symbolic reasoning n Calculations n Predictions n Summary and description of data n Development of mathematical theorems and laws n But math also codes for conceptual knowledge. n Math becomes a critical piece of how we decide we know something (our epistemology). July 27, 2015 AAPT-College Park 10
- 11. + Packing Concepts into Equations: Equations as a conceptual organizer July 27, 2015AAPT-College Park aA = FA net mA Force is what you have to pay attention to when considering motion What matters is the sum of the forces on the object being considered The total force is “shared” to all parts of the object These stand for 3 equations that are independently true for each direction. You have to pick an object to pay attention to Forces change an object’s velocity Total force (shared over the parts of the mass) causes an object’s velocity to change When we just write “F=ma” our students often miss the rich set of conceptual associations hidden in the equations and mis-use them. 11
- 12. + Example: A vector line integral* n A square loop of wire is centered on the origin and oriented as in the figure. There is a space-dependent magnetic field n If the wire carries a current, I, what is the net force on the wire? July 27, 2015AAPT-College Park ! B = B0yˆk * Griffiths, Introduction to Electrodynamics (Addison-Wesley, 1999). From a video of two physics majors working together to solve a problem in a third-year E&M course. 12
- 13. + Two paths to a solution July 27, 2015AAPT-College Park n Student B n I’m pretty sure they want us to do the vector line integral around the loop. n It’s pretty straightforward. n The sides do cancel, but I get the top and bottom do too, so the answer is zero. ! F = I d ! L × ! B " #∫ n Student A n Huh! Looks pretty simple – like a physics 1 problem. n The sides cancel so I can just do on the top and bottom where B is constant. n Gonna get ! F = I ! L × ! B ! F = IL2 B0 ˆj What do you think happened next? 13
- 14. + No contest! n Student A immediately folded his cards in response to student B’s more mathematically sophisticated reason and agreed she must be right. n Both students valued (complex) mathematical reasoning (where they could easily make a mistake) over a simple (and compelling) argument that blended math and physics reasoning. n The students expectations that the knowledge in the class was about learning to do complex math was supported by many class activities. July 27, 2015AAPT-College Park 14
- 15. +A language for analyzing epistemology July 27, 2015AAPT-College Park n Epistemological resources* n Generalized categories of “How do we know?” warrants. n Epistemological framing** n The process of deciding what e-resources are relevant to the current task. (NOT necessarily a conscious process.) n Epistemological stances n A coherent set of e-resources often activated together *Bing & Redish, Phys. Rev. ST-PER 5 (2009) 020108; 8 (2012) 010105. ** Hammer, Elby, Scherr & Redish, in Transfer of Learning (IAP, 2004) 15
- 16. + Careful! July 27, 2015AAPT-College Park n These are NOT intended to describe distinct mental structures. They emphasize different aspects a unitary process: activating a subset of the knowledge you have for a particular situation. n Warrant – focuses on a specific argument using particular elements of the current context. n “Since the path integral of a conservative force is path independent, these two integrals will have the same value.” n Resource – focuses on the general class of warrant being used. n “You can trust the results in a reliable source such as a textbook.” n Framing – focuses attention on the interaction between cue and response. n You decide you need to carry out a calculation. 16
- 17. +Some physics e-resources July 27, 2015AAPT-College Park Knowledge constructed from experience and perception (p-prims) is trustworthy Algorithmic computational steps lead to a trustable result Information from an authoritative source can be trusted A mathematical symbolic representation faithfully characterizes some feature of the physical or geometric system it is intended to represent. Highly simplified examples can yield insight into complex mathematical representations Physical intuition (experience & perception) Calculation can be trusted By trusted authority Physical mapping to math (Thinking with math) Value of toy models There are powerful principles that can be trusted in all situations Fundamental laws Except for the first, each of these often involve math. 17Intro Physics context Often a theorem or equation Often a theorem or equation
- 18. + An a meta-epistemological result: Coherence July 27, 2015AAPT-College Park Coherence Multiple ways of knowing applied to the same situation should yield the same result 18
- 19. + How did previous examples mix math and epistemology? E = F/q q ! -3q July 27, 2015AAPT-College Park Calculation can be trusted Physical mapping to math (Thinking with math) 19 Calculation can be trusted
- 20. +Epistemological framing n Depending on how students interpret their situation and their expectations, they may not think to call on resources they have and are competent with. n Their valuation of different e-resources can lead them to ignore or suppress valid modes of reasoning. n This becomes particularly important when students and faculty prefer different ways of knowing. July 27, 2015AAPT-College Park 20
- 21. + The language of epistemology n This language provides nice classifications of reasoning – both what we are trying to teach and what students actually do. n But can it provide any guidance for instructional design? July 27, 2015AAPT-College Park 21
- 22. + NEXUS/Physics: An introductory course for life science majors n Create prototype materials n An inventory of open-source instructional modules that can be shared nationally . n Interdisciplinary n Coordinate instruction in biology, chemistry, physics, and maths. n Competency based n Teach generalized scientific skills so that it supports instruction in the other disciplines. July 27, 2015AAPT-College Park * Redish et al., NEXUS Physics: An interdisciplinary repurposing of physics for biologists, Am. J. Phys. 82:5 (2014) 368-377. http://www.nexusphysics.umd.edu 22
- 23. +Epistemological resources July 27, 2015AAPT-College Park Knowledge constructed from experience and perception (p-prims) is trustworthy Physical intuition (experience & perception) Information from an authoritative source can be trusted By trusted authority The historical fact of natural selection leads to strong structure- function relationships in living organisms Many distinct components of organisms need to be identiﬁed Comparison of related organisms yields insight Learning a large vocabulary is useful Categorization and classiﬁcation (phylogeny) There are broad principles that govern multiple situations Heuristics Living organisms are complex and require multiple related processes to maintain life Life is complex (system thinking) Function implies structure In intro bio, typically none of these often involve math. 23Intro Biology context
- 24. + Missing! n These are critical components woven deeply into every physics class! n These are not only weak or missing in many bio students, they see them as contradicting resources they value. n And math is not well developed as a way of knowledge building . July 27, 2015AAPT-College Park Value of toy models Fundamental laws Life is complex (system thinking) Function implies structure 24
- 25. + This requires dramatic changes! n We cannot take for granted that these students will value toy models to build mathematical insight as a basis for more elaborate examples. We have to justify their use. n We cannot take for granted that these students understand or appreciate the power of principles like conservation laws (energy, momentum, charge). We have to teach it explicitly. n We cannot take for granted that these students can begin with math and translate it into physics. We have to help them learn how to do this. July 27, 2015AAPT-College Park 25
- 26. + Epistemological stances – “Go-to” e-framings July 27, 2015AAPT-College Park n Both students and faculty may have developed a pattern of choosing particular combinations of e-resources. n The epistemological stances first chosen by physics instructors and physics students may be dramatically different – even in the common context of a physics class. 26
- 27. + The figure shows the PE of two interacting atoms as a function of their relative separation. If they have the total energy shown by the red line, is the force between the atoms when they are at the separation marked C attractive or repulsive? C BA Total energy r Potential Energy July 27, 2015AAPT-College Park Example: Epistemological stances Given as a discussion question in a class for introductory physics for bio students. (A year of calculus was a pre- requisite for the class.) 27
- 28. + How two different professors explained it when students got stuck. July 27, 2015 AAPT-College Park n Remember! (or here) n At C, the slope of the U graph is positive. n Therefore the force is negative – towards smaller r. n So the potential represents an attractive force when the atoms are at separation C. F = − ∇U F = − dU dr This figure was not actually drawn on the board by either instructor. 28
- 29. +Wandering around the class while students were considering the problem, I got a good response using a different approach. July 27, 2015AAPT-College Park n Think about it as if it were a ball on a hill. Which way would it roll? Why? n What’s the slope at that point? n What’s the force? n How does this relate to the equation F = − dU dr 29
- 30. + A conflict between the epistemological stances of instructor and student can make teaching more difficult. July 27, 2015AAPT-College Park Calculation can be trusted By trusted authority Physical mapping to math (Thinking with math) Physical intuition (experience & perception) Physical mapping to math (Thinking with math) Mathematical consistency (If the math is the same, the analogy is good.) Physics instructors seem most comfortable beginning with familiar equations – which we use not only to calculate with, but to code and remind us of conceptual knowledge. Most biology students lack the experience blending math and conceptual knowledge, so they are more comfortable beginning with physical intuitions and only later mapping to math. 30
- 31. + Teaching physics standing on your head July 27, 2015AAPT-College Park n For physicists, math is the “go to” epistemological resource – the one activated first and the one brought in to support intuitions and results developed in other ways. n For biology students, the math is decidedly secondary. Structure/function relationships tend to be the “go to” resource. n Part of our goal in teaching physics to second year biologists is to improve their understanding of the potential value of mathematical modeling. This means teaching it rather than assuming it. 31
- 32. + Conclusion n An analysis of how math is used in physics, including both unpacking what professionals do and analyzing how students respond, can give insight into student difficulties with learning to reason scientifically. n This analysis has implications for how we understand what our students are doing, what we are actually trying to get them to learn, and (potentially) how to better design our instruction to achieve our goals. July 27, 2015AAPT-College Park 32

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