Introducing Calculus to the High School Curriculum: Curves, Branches and Functions

Andrew Grossfield

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Conference: 2013 ASEE Annual Conference & Exposition
Location: Atlanta, Georgia
Publication Date: June 23, 2013
Start Date: June 23, 2013
End Date: June 26, 2013
ISSN: 2153-5965
Conference Session: Mathematics Division Technical Session 2
Tagged Division: Mathematics
Page Count: 18
Page Numbers: 23.815.1 - 23.815.18
DOI: 10.18260/1-2--19829
Permanent URL: https://strategy.asee.org/19829
Download Count: 457

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Andrew Grossfield P. E. Vaughn College of Aeronautics & Technology

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Throughout his career Dr. Grossfield, has combined an interest in engineering and mathematics. He earned a BSEE at the City College of New York. During the early sixties, he obtained an M.S. degree in mathematics at the Courant Institute of NYU at night while working full time as an engineer for aerospace/avionics companies. He studied continuum mechanics in the doctoral program at the University of Arizona. He is a member of ASEE, IEEE and MAA.

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Abstract

Introducing Calculus to the High School Curriculum Part 1: Curves, Branches and FunctionsAbstractThe number of high school students taking calculus and AP calculus is increasing and this trendappears to be accelerating. Examining this phenomenon leads to some questions: Is thememorization required by most students to pass calculus healthy for the individual student? Istraining a citizenry to memorize calculus facts psychologically healthy for American society as awhole? There has been criticism of various aspects of conventional algebra and calculus textswhich present barriers to the understanding of the concept of continuous smooth functions. Suchbarriers include lack of focus, lack of structure, unmotivated definitions, the clutter ofapplications, the number of pages in the texts and the emphasis on proofs at the expense of trueinsight and the essential needs of our society for a citizenry capable of understandingmathematical concepts.In our society calculus has had a reputation for being difficult to master. If this subject isintroduced to a much wider high school audience in the same way it has been taught in colleges,it might turn out to deter more students from entering the Science, Technology, Engineering andMath (STEM) disciplines when it is desired that STEM enrollments increase and that more of thenation s citizens acquire more insight into mathematical and statistical thinking. Althoughcalculus has traditionally been taught with an emphasis on proofs it does not mean that the bestinterests of society are served by continuing and extending to a larger audience thisunderperforming process. Alternative approaches must be considered.Basically, the concepts of calculus are neither abstract nor difficult. However, unlike calculus,the concepts of algebra are less obvious and more students have to struggle harder in order tounderstand essential algebraic concepts. Moreover, it is crucial that students acquire somedexterity in algebraic form changing manipulations and solving procedures, preferably beforestarting their study of calculus.This, the first in a set of three papers, is planned to provide the concepts of pre-calculus, visuallyand intuitively in order to reveal the intrinsic ultimate simplicity of calculus and spare a studentfrom having to read the entire 500 pages of conceptually cluttered verbose, disorganizedconventional text in order to acquire an overview. My hopes are that by providing a focus ofstudy, specifically algebraic and transcendental curves, as well as intuitive and visual definitionswhile maintaining an organized topic structure, and by delaying the proofs, which when placedprematurely, present a barrier to understanding, to create a conceptual environment where morestudents and teachers will gain insight relatively quickly into the nature of calculus.Subsequently a student, enabled with the goals and structure of the course in calculus, can referto conventional texts to fill in and expand on subordinate details.

Citation
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Grossfield, A. (2013, June), Introducing Calculus to the High School Curriculum: Curves, Branches and Functions Paper presented at 2013 ASEE Annual Conference & Exposition, Atlanta, Georgia. 10.18260/1-2--19829

TY  - CPAPER
AB  -                 Introducing Calculus to the High School Curriculum                      Part 1: Curves, Branches and FunctionsAbstractThe number of high school students taking calculus and AP calculus is increasing and this trendappears to be accelerating. Examining this phenomenon leads to some questions: Is thememorization required by most students to pass calculus healthy for the individual student? Istraining a citizenry to memorize calculus facts psychologically healthy for American society as awhole? There has been criticism of various aspects of conventional algebra and calculus textswhich present barriers to the understanding of the concept of continuous smooth functions. Suchbarriers include lack of focus, lack of structure, unmotivated definitions, the clutter ofapplications, the number of pages in the texts and the emphasis on proofs at the expense of trueinsight and the essential needs of our society for a citizenry capable of understandingmathematical concepts.In our society calculus has had a reputation for being difficult to master. If this subject isintroduced to a much wider high school audience in the same way it has been taught in colleges,it might turn out to deter more students from entering the Science, Technology, Engineering andMath (STEM) disciplines when it is desired that STEM enrollments increase and that more of thenation s citizens acquire more insight into mathematical and statistical thinking. Althoughcalculus has traditionally been taught with an emphasis on proofs it does not mean that the bestinterests of society are served by continuing and extending to a larger audience thisunderperforming process. Alternative approaches must be considered.Basically, the concepts of calculus are neither abstract nor difficult. However, unlike calculus,the concepts of algebra are less obvious and more students have to struggle harder in order tounderstand essential algebraic concepts. Moreover, it is crucial that students acquire somedexterity in algebraic form changing manipulations and solving procedures, preferably beforestarting their study of calculus.This, the first in a set of three papers, is planned to provide the concepts of pre-calculus, visuallyand intuitively in order to reveal the intrinsic ultimate simplicity of calculus and spare a studentfrom having to read the entire 500 pages of conceptually cluttered verbose, disorganizedconventional text in order to acquire an overview. My hopes are that by providing a focus ofstudy, specifically algebraic and transcendental curves, as well as intuitive and visual definitionswhile maintaining an organized topic structure, and by delaying the proofs, which when placedprematurely, present a barrier to understanding, to create a conceptual environment where morestudents and teachers will gain insight relatively quickly into the nature of calculus.Subsequently a student, enabled with the goals and structure of the course in calculus, can referto conventional texts to fill in and expand on subordinate details.
AU  - Andrew Grossfield P. E.
CY  - Atlanta, Georgia
DA  - 2013/06/23
PB  - ASEE Conferences
TI  - Introducing Calculus to the High School Curriculum: Curves, Branches and Functions 
UR  - https://strategy.asee.org/19829
DO  - 10.18260/1-2--19829
ER  -