Rational Numbers: An Integration of Research

By Thomas A. Romberg; Elizabeth Fennema et al. | Go to book overview

knower, there are the complementary dimensions of the subconstructs -- at least quotient, measure, operator, and ratio numbers.

The first of the complementarities described earlier is related logically to the existence of the 1/n fractions for any nonzero integer n. Thus, the unit fractions along with the quotient and ratio nature form a mathematical base for rational numbers. This base also is seen in the mathematical actions of young persons. Considering rational numbers as a humanly knowable activity means taking into account the properties indicated and the many others that distinguish rational from natural numbers.

Seeing humans as capable of knowing rational numbers brings into play ideas from two models. The first argues for the efficacy of seeing rational number knowing as one of four types: ethnomathematical, intuitive, technical symbolic, and axiomatic deductive. The challenge is to find the interrelationships among these knowing types and their appropriate places in the curriculum. The second model, a more general model of the growth of mathematical understanding, portrays an interweaving, nonlinear leveled structure. Each level is characterized by a complementarity of process and form. Because of this, outer-level knowing can derive from either the processes or the forms of inner levels. This model also suggests four bases for mathematical knowing: action, image, form, and structure. These are not to be confused with traditional concrete, pictoral, and symbolic modes. Under this model, understanding rational numbers is characterized as a dynamic whole, of knowing rational numbers at many levels at once.

What are the rational number knowledge structures and the methods of organization that characterize an understanding of rationals toward which one might guide young learners? To know and understand rational numbers is to know numbers that are at once quotients and ratios and to know them, simultaneously, in their many forms. This knowing is organized in many embedded levels that are transcendent but recursive, and in which folding back to go ahead characterizes the interweaving of intuitive and formal understanding.

REFERENCES

Behr, M. J., Lesh, R., Post, T., & Silver, E. A. ( 1983). "A mathematical and curricular analysis of rational number concepts". In R. Lesh & M. Landau (Eds.), The acquisition of mathematics concepts and processes (pp. 92-98). New York: Academic.

Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. ( 1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15( 5), 323-341.

Bergeron, J., & Herscovics, N. ( 1987). "Unit fractions of a continuous whole". In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Psychology of mathematics education, PME-XI (Vol. 1, pp. 357-365). Montreal: PME.

-81-

Notes for this page

If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.
Citations (0)
Some of your citations are legacy items.
Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
• Saved book/article
• Highlights
• Quotes/citations
• Notes
• Bookmarks
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Bookmark this page
Rational Numbers: An Integration of Research
Table of contents

Settings

Settings

Typeface
Text size Reset View mode
Search within

Look up

Look up a word

• Dictionary
• Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen
/ 378

Questia reader help

How to highlight and cite specific passages

1. Click or tap the first word you want to select.
2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

Cited passage

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

Buy instant access to save your work.

Already a member? Log in now.

Advanced search

Oops!

An unknown error has occurred. Please click the button below to reload the page. If the problem persists, please try again in a little while.