Making Meaning with Math in Physics: A Semantic Analysis Edward F. Redish and Ayush Gupta

Journal Club – March 1st 2010, Claire Hotan

Why did I choose this paper? 

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Wanted something it didn't matter much if you didn't have time to read. Physics education is of importance to all of us! I love mathematics, but I'm aware I sometimes see things differently to “physicists”, and am interested in why? Educational psychology can be cool too :) About the paper: presented at a physics education conference in August 2009.

Abstract

We can all do it, but we don't always know how.

Physics makes powerful use of mathematics, yet the way this use is made is often poorly understood. Professionals Been closely integrate their mathematical symbology with physical doing meaning, resulting in a powerful and productive structure. But it so because of the way the cognitive system builds expertise long through binding, experts may have difficulty in unpacking their you well-established knowledge in order to understand the forget difficulties novice students have in learning their subject. This new is particularly evident in subjects in which the students are learning to use mathematics to which they have previously people been exposed in maths classes in complex new ways. In this find it paper, we propose that some of this unpacking can be hard! facilitated by adopting ideas and methods developed in the field of cognitive semantics, a sub-branch of linguistics devoted to understanding how meaning is associated with language.

Theory paper: no ACTUAL hints on how to teach :)

Have the tools, just don't know to use them.

Introduction 

Physics relies on understanding and use of maths to “integrate our understanding of the physical world with symbolic relations of maths”. 

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Physics instructors often feel students do not have a satisfactory grasp of the maths required to solve problems. 

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Compare to maths class - learn algorithmic processes for various types of problem solving (no physical meaning).

In fact they may have a good grasp of the maths (see their marks), but not the physical intuition to use it appropriately in physics – this needs to be learned!

Use cognitive semantics to learn how we “make meaning” with maths – how the equations reflect and explain the physical world similar to language.

Physics models the physical world using maths 

“Translations from physical-causal relations between entities in the real world to mathematical relations between symbols is what we call mathematical modelling.”

Math. modelling ↔ Causal physics 

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Start from lower left: Identify variables and parameters. Determine mathematics that will describe the features we wish to model. Model the system by mapping measurables to symbols and expressing physicalcausal relations in terms of mathematical operations. Mathematics gives rules and methods to transform relationships and solve equations. 

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Can now process to solve problems that physically didn't make immediate sense by working in the mathematical regime.

We are in the mathematical regime, so we now need to interpret our results to understand what it means in the physical world. To complete the loop, we evaluate our results to consider whether our choice of model was valid – do our results match observations? Do we need to modify the model? Scientists use these 4 skills to mathematically describe physical behaviours. Thus the teacher should focus on all 4 skills, not just the mathematical processing as often happens in physics courses. 

Combining maths and physics isn't necessarily this simple or sequential, they get

Re-interpret the maths depending on the physics 

Use maths to do physics but use physics to do maths.

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eg. Corinne's Shibboleth

(how to tell a physicist from a mathematician) One of your colleagues is measuring the temperature of a plate of metal placed above an outlet pipe that emits cool air. The result can be well described in Cartesian coordinates by the function T(x,y) = k(x2 + y2) where k is a constant. If you were asked to give the following function, what would you write? T(r,θ) = ?

Novices often pay attention only to mathematical grammar

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Mathematicians often focus on the mathematical grammar of a statement, ignoring the physical meaning – but so do novice physics students. Here is an example problem students frequently get wrong for this reason. Remembering that the strength of an electric field is independent of a test charge, we are asked the following question:

Mathematical grammar A very small charge q is placed at a point somewhere in space. Hidden in the region are a number of electrical charges. The placing of the charge q does not result in any change in the position of the hidden charges. The charge q feels a force, F. We conclude that there is an electric field at the point that has the value E = F/q. If the charge q were replaced by a charge –3q, then the electric field at the point a) Equal to –E

d) Equal to E/3

b) Equal to E

e) Equal to some other value not given here.

c) Equal to –E/3

f) Cannot be determined from the information given.

would be

More than 1/2 of 200 students pick (c)!  F is the force on the test charge due to interaction with source charge, so increases ∝ to test charge increase, giving same field as before.  Novices tend not to link algebraic symbols with physical meaning, and are vulnerable to such errors. Writing F(q) might help them realise that the symbol F is a function of q, but not that these symbols stand for real physical quantities. 

The link to linguistics 

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These examples, and the paper contains another on the photoelectric effect, demonstrate how experts (physicists, other scientists and engineers) use additional (physical) knowledge when solving problems. In linguistics, study is done in how people understand language and the meanings of words (semantics) - and the importance of extra knowledge of the language. In understanding how we make sense of the language of maths in the context of physics, we consider what is known about how people make sense of language in the context of daily life.

Cognitive Semantics 

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Fundamental principals from Evans & Green (2006): Encyclopaedic knowledge: Ancillary knowledge is critical to determining the meaning of words. Conceptualisation: Meaning is constructed dynamically. Embodied cognition: The meaning of words is grounded in physical experience. Conceptual grounding: Semantic structure expresses conceptual structure

Encyclopaedic knowledge 

Our definitions of words rely on knowledge of a wider web of concepts, ideas and objects. 

Eg “hypotenuse” 

We need “triangle”, “side”, “right angle”, “longest” 

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For these, need ideas of “plane”, “shape”, “line”, “angle”, “length”, and so on.

May rely on all our other senses too. 

Eg “banana” 

Colour, taste, smell, knowledge of edibility, source etc.

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No axiomatic base to language (unlike maths?!).

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In maths, meaning in equations, y=mx+c.  

Equality, variables, addition, multiplication... Further meaning comes from “knowing” that this is the equation of a straight line, only x and y are variables... 

Thus get meanings like line, slope, intercept...

Conceptualisation 

Meaning is constructed dynamically depending on the context in which a word is used. 

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Encyclopaedic knowledge is used to package concepts together to select the most appropriate meaning.

Eg “safe” 

The child is safe / The beach is safe 

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We construct meaning based on the structure of the sentence and our understanding of all the other words in it (and may require winder context).

In physics/maths, we can associate different concepts with x, y etc. and so T(r,θ) to find our answer to the previous question.

Conceptual grounding 

Changing semantic structure can change conceptual structure. 

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Which form of a word is used, where it is placed in a sentence.

Meaning of a word may not change, but our interpretation of it does. In maths (physics), how we write something can predispose us to think of that thing in a certain way.  

eg. F=ma vs a = F/m The first is a mathematical structure which students tend to use to define “force”. The second tends to make one think of a causal relationship between F and a.

Embodied cognition 

Suggests that ultimately all our concepts, thus meanings, form from interactions with the physical world. 

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This is our axiomatic approach to language – axioms are our experiences (touch, sight...). 

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Abstract meaning is probably connected to and derived from physical experience.

Although many of the concepts we deal with in day to day life seem abstract, eventually they can (probably) be traced back to perceptual experiences.

Understanding many mathematical concepts relies on everyday ideas: orientation, grouping, bodily motion, physical containment, instantaneous change, etc. 

Set theory ↔ experience with containers, collections of objects. 

What about ring theory? How is a mathematical “ring” like that on my finger? How do we get physical intuition from abstract maths? how does algebra of supersymmetric groups tell us about subatomic particles?

Interpreting partial or corrupted text 

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Evidence supporting this complex view of making meaning may be the ease with which we (native English speakers at least (??)) make meaning of corrupted text and speech, and may not even be aware that we have mentally corrected what the speaker says. Another example is of course Jabberwocky (Lewis Carroll, 1872). 

About half the non-filler words (conjunctions, articles, prepositions, pronouns) are nonsense words, and yet we have a mental image of the poem!

Jabberwocky 'Twas brillig, and the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.

He took his vorpal sword in hand: Long time the manxome foe he sought— So rested he by the Tumtum tree, And stood awhile in thought.

One, two! One, two! and through and through The vorpal blade went snicker-snack! He left it dead, and with its head He went galumphing back.

'Twas brillig, and the slithy toves Did gyre and gimble in the wabe; All mimsy were the borogoves, And the mome raths outgrabe.

"Beware the Jabberwock, my son! The jaws that bite, the claws that catch! Beware the Jubjub bird, and shun The frumious Bandersnatch!"

And as in uffish thought he stood, The Jabberwock, with eyes of flame, Came whiffling through the tulgey wood, And burbled as it came!

"And hast thou slain the Jabberwock? Come to my arms, my beamish boy! O frabjous day! Callooh! Callay!" He chortled in his joy.

Lewis Carroll, Through the Looking-Glass and What Alice Found There, 1872.

Meaning Making with Equations: contrasting two students 

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Two students are asked how they would explain what v=v0+at means. Both are competent at solving physics problems using maths. Student 1 focuses on the mathematical meanings (in the derivative/integral sense) of the terms. Student 2 talks about defining variables, dimensional analysis and the effect of term 2 on term 1. While both are competent students, Student 2 demonstrates a deeper conceptual understanding of the physical meaning of the mathematics, allowing them to more easily adapt to different solution methods, and make predictions without doing calculations.

Conclusion Our point here is that understanding equations in physics is not just about learning the mathematical operations. It is about making many connections to stores of knowledge about the physical system; not just at the level of variable-definitions but at level of the bases of the mathematical operations and how those operations connect to physical meaning. Expectations that the mathematics describing a physical system should connect to our intuitive sense about how the physical system behaves can also provide a valuable safety net against possible errors in the calculations and help us make sense of new or unfamiliar equations. This perspective, enhanced by the tools developed recently by cognitive semanticists, should help give us new and deeper insights into the use of mathematics in physics and the other sciences.