Mathematical Culture Shock

Trent KullWinthrop University

MAA SE 2010 ConferenceMARCH 27, 2010

Motivation

• Teaching teachers to work with English language learners

• Modern language job search• Advising students

OverviewSudden immersion in the world of mathematics can lead to a

"culture shock" of sorts for the undergraduate student. The symbols, vocabulary, language, and logic associated with a formal study of subjects such as calculus, analysis, and linear algebra may overwhelm and demotivate aspiring mathematicians. We will look at several instances where this can occur, and discuss ways to enhance student and instructor success in these situations.

Culture• An integrated pattern of human knowledge,

belief, and behavior that depends upon the capacity for symbolic thought and social learning

• The set of shared attitudes, values, goals, and practices that characterizes an institution, organization or group

Culture shock• The difficulties and frustrations of sudden exposure

to a foreign culture. Negative, passive reactions to a set of noxious circumstances. [Oberg, 1960]

• Responses to unfamiliar cultural environments; an active process of dealing with this change. [Ward, Bochner, Furnham; 2001]

Exposure to second-culture influences

• Affective model• Behavioral model• Cognitive model

Affective model

• How people think• Confusion, anxiety, disorientation• Suspicion, bewilderment, perplexity• Intense desire to be elsewhere

Behavioral model

• How people act• Culturally inappropriate behavior• Leads to poor performance• Study skills, motivation, aversion to technology, need

for immediate results

Cognitive component

• How people perceive• Perception of members of the culture• Stereotypes• Male dominated, natural talent, special

insight, computational expertise

Cultural readjustment scale[Spradley, Phillips; 1972]

1. The language spoken.2. The general pace of life.3. How punctual most people are.4. Ideas about what offends

people.5. How ambitious people are. 6. The degree to which your good

intentions are misunderstood by others.

7. The amount of privacy.

8. Ideas about what is funny.9. Types of recreation and leisure

time activities.10. How formal or informal people

are.11. Ideas about friendship.12. The amount of reserve people

show in relationship to others.13. Discussion topics.14. Ideas about what is sad.

Teaching and learning:The language

• Mathematics is no one’s natural language• MSL: “Mathematics second language”• Language register: words, structures, and meanings

associated with a language [Cuevas, 1984]• Mathematics language register (MLR)– “Almost totally nonredundant and relatively

unambiguous” [Brunner, 1976]• Handbook of Mathematical Discourse [Wells, 2009]

The language: discourse• “…provide a basis for discovering the ways

in which students and non-mathematicians misunderstand what mathematicians write and say. Those misunderstandings are a major…reason why so many educated and intelligent people find mathematics difficult and even perverse.”

• http://www.cwru.edu/artsci/math/wells/pub/abouthbk.html

The language: high density

The limit of the function f as the input x approaches c is equal to L means: For every epsilon greater than zero there exists a delta greater than zero such that if the absolute value of the difference between x and c is between zero and delta, then the absolute value of the difference of f evaluated at x and L is less than epsilon.

.0

0,0

lim

Lxfcx

thatsuch

thatmeans

Lxfcx

The language: high density

In mathematics you don’t understand things. You just get used to them.

.0

0,0

lim

Lxfcx

thatsuch

iffLxfcx

[Cool t-shirts, yesterday]

High density

'' yxyydx

d

dx

dy

fLy

tbdt

dta

dt

dL

2

2

High density

jj dxxKx 1

0

dtaBaBtgdtaBaBtf

dxdttxwtxvx

T

j

T

j

xx

T

0 011*,10 000

*,0

0

1

0,,

Teaching and learning:Logic

• If a set contains more vectors than entries in each vector, then the set is linearly dependent.

• If a set is linearly dependent, then the set contains more vectors than entries in each vector.

• If a set is linearly independent, then the set does not contain more vectors than entries in each vector.

• has a nontrivial solution if and only if the equation has at least one free variable.

0Ax

Logic

• Negative: Some dislike mathematics because of the necessary emphasis on precision, meaning, structure, lack of assumptions

• Positive: Some like the subject for exactly the same reasons• In any case, necessary and requires time for many to use

willingly and properly• Language, symbols and rules of logic may lead to aspiring

mathematicians floundering in abstractness

• Liberal arts students– Concrete applications (science, statistics, debate, music, organization,

optimization)– Logic, axiomatics: Help students become conscious of their actual

assumptions when they make a decision or adhere to a belief in any sphere.

– “Proof is a check on our intuition. It also sharpens the intuition, much as argumentation with an adversary on, say, a political issue often reveals defects in our thinking.” [Klein, 1976]

• Undergraduate mathematics majors: The language, symbols, concise (high density) presentations can hide the context

Teaching and learning:Context

Providing context• Not just physical applications• History of mathematics

– Calculus concepts exceptionally difficult and unnatural (infinity); mathematicians understood intuitively and built a logical foundation after hundreds of years of study

– Differential equations solution techniques often begin with an assumption that hides tremendous research involved

• Highlight the sophisticated nature of mathematics– Construction of real number line– Definition of a tangent line

• Complete understanding comes with great effort, often multiple attempts

References

Slides and references will be available at

http://faculty.winthrop.edu/kullt/ .

Thank you!