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Mathematical Culture Shock Trent Kull Winthrop University MAA SE 2010 Conference MARCH 27, 2010

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Motivation Teaching teachers to work with English language learners Modern language job search Advising students

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Overview Sudden 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.

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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

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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]

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Exposure to second-culture influences Affective model Behavioral model Cognitive model

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

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Behavioral model How people act Culturally inappropriate behavior Leads to poor performance Study skills, motivation, aversion to technology, need for immediate results

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Cognitive component How people perceive Perception of members of the culture Stereotypes Male dominated, natural talent, special insight, computational expertise

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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.

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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]

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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/p ub/abouthbk.html http://www.cwru.edu/artsci/math/wells/p ub/abouthbk.html

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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.

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The language: high density In mathematics you don’t understand things. You just get used to them. [Cool t-shirts, yesterday]

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High density

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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.

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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

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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

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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

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References Slides and references will be available at http://faculty.winthrop.edu/kullt/http://faculty.winthrop.edu/kullt/. Thank you!

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