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The Roadmap Of Arithmetic: Summing It Up

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2006 Annual Conference & Exposition


Chicago, Illinois

Publication Date

June 18, 2006

Start Date

June 18, 2006

End Date

June 21, 2006



Conference Session

Mathematics in Transition

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Page Count


Page Numbers

11.1323.1 - 11.1323.9



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Paper Authors


Andrew Grossfield Vaughn College of Aeronautics

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Throughout his career Dr. Grossfield, has combined an interest in engineering design 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 night while working full time during the day, designing circuitry for aerospace/avionics companies. He is licensed in New York as a Professional Engineer and is a member of ASEE, IEEE, SIAM and MAA. is his e-mail address.

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NOTE: The first page of text has been automatically extracted and included below in lieu of an abstract

The Roadmap of Arithmetic: Summing it up


Notice this problem of human consciousness. Ideas and concepts enter our mind one at a time. Usually teachers and authors have the entire course content in their minds before they begin a course or write a text. However, a student can only confront one idea at a time and so the course content is arranged sequentially, one topic after another, one word after another. It is then left to the student to construct in his mind the best arrangement of the material. We must accept that courses and texts are sequentially ordered. Pages are numbered. A student, when confronted by a particularly difficult part of the text, may set the text aside never to see the remainder of the course. The problem is: what is the best arrangement for course material so that a student can visualize the major components of a course at the beginning and subsequently fill in the details?

This paper presents a roadmap in the form of a tree structure (See the chart at the conclusion.) that will allow young students to treat their study of arithmetic as a research project. The roadmap will provide a navigational aid to assist in the exploration of the world of numbers. Historically, the discovery of the periodic table guided scientists in the study of chemistry, indicating what was known and where gaps existed and raising questions about what remained to be explored. The periodic table has been immensely valuable, even though its final form may differ from the table that was originally proposed. This roadmap will serve students similarly in their study of arithmetic.

In every well-planned course, only one thing is studied. In arithmetic, the subject is numbers and so the word 'numbers' is placed at the top of the roadmap. At the end of the course, a student should know the kinds, forms, operations, properties and uses of numbers. In our society, this study takes a few years and is often disorganized. It should not be surprising that a young student might become confused and disenchanted. At the end of the course, the strategy should be reviewed so that the student can see what has been learned about numbers and how numbers are used and what remains for future study.

This roadmap suggests that including in the course presentation, a tree structure, giving equal importance to the kinds, operations, forms and properties of numbers, has conceptual advantages. The tree structure separates these concepts but highlights their interaction in the solution of quantitative problems. The visualization of the tree structure together with an understanding of the important concept of forms may aid students in mastering the subject.

The study of numbers is unending and so this roadmap cannot be exhaustive. Still it can provide a framework for introducing the study of numbers.

Grossfield, A. (2006, June), The Roadmap Of Arithmetic: Summing It Up Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--387

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