# Rational Numbers: An Integration of Research

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

5
PROTOQUANTITATIVE ORIGINS OF RATIO REASONING

Lauren B. Resnick Janice A. Singer University of Pittsburgh

This chapter lays the groundwork for a theory of the intuitive origins of proportion and ratio reasoning. We argue that children have a set of protoquantitative schemas that allow them to reason about ratio- and proportionlike relations without using numbers. Among others, (a) a fittingness schema--or the idea that two things go together based on an external dimension--and (b) a covariation schema -- or the idea that two size-ordered series covary, either directly or inversely -- form the basis of the protoquantitative knowledge. In the course of elementary schooling, children also learn, separately, about properties of numbers, including their factorial structure. At the heart of our theory is the proposal that these two types of knowledge -- protoquantitative schemas about physical material in the world, and factorial number sense -- eventually must merge to give children a means to model quantitatively situations that require the use of ratios and proportions.

We know that ratio and proportion are difficult concepts for children to learn. They constitute one of the stumbling blocks of the middle school curriculum, and there is a good possibility that many people never come to understand them. What makes ratios so hard to learn? What resources exist for teaching them more effectively and learning them more easily?

The hypotheses developed in this chapter represent an extension of work by Resnick and Greeno ( Resnick, 1989; Resnick & Greeno, 1990) on the intuitive origins of mathematical concepts of number. By intuitive we mean knowledge that does not depend on formal instruction, knowledge that children construct on the basis of their everyday experience in the world. The broad psychological theory within which our nations about mathematical development have been formulated holds that learning is both situation-

-107-

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
Notes

#### Cited page

Style
Citations are available only to our active members.
Buy instant access to cite pages or passages in MLA 8, MLA 7, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 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

Rational Numbers: An Integration of Research

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.

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

Help
Full screen
Items saved from this book
• Bookmarks
• Highlights & Notes
• Citations
/ 378

### 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 8, MLA 7, 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." (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.