# Rational Numbers

**Overview**

• A brief introduction

• A small problem

• A few examples

• How it fits together

• Why it matters so much

• For more information

**A Brief Introduction**

• Taught middle school mathematics to children

considered at risk to not graduate from high

school

• Taught high school algebra to children considered

at risk to not graduate from high school

• Taught mathematics to undergraduate and

graduate students preparing to become

elementary and secondary teachers

• Taught mathematics education to graduate

students preparing to become elementary and

secondary teachers

**My perspective**

• Rational numbers are difficult for

students because of the way in which

the mathematics intersects with the

way people think about rational

number situations

**A Small Problem ( or two )
**

• Think about each of the situations on

the handout. What is the meaning of ¾

in each situation?

Draw a picture to represent the

meaning of ¾ in each situation

**A few examples**

• New Units and Notations

1. 45 minutes of 60 minutes

in an hour, ¾ of one hour

2. 3(¼ cookies)

3. ¼(3 cookies)

**Another examples**

Same notation means different things

¾ as a ratio (not part whole)

**Return**

**And yet another example**

¾ is an operator that

maps one quantity

to another

**Examples I won’t
elaborate on**

• Rational number as decimal

• Rational number as percent

• Rational number as rate

**Fitting it all together**

• Why are rational numbers hard for

children?

– In teaching, we don’t take enough time to help

students understand the meaning of rational

numbers

**• The same number can mean different things,
depending on the unit
• Rational numbers often represent a relationship
between quantities, not absolute numbers
• Rational numbers require multiplicative ( not additive )
reasoning**

**Other kinds of
difficulties**

• Notational difficulties – not everything we

write as a “fraction” is a rational number

(e.g., π/2)

• Over-reliance on part-whole metaphors for

fractions

• Interference of generalizations made

about whole numbers

• Use of “fraction” in everyday language

**Why it continues to
matter**

• General quantitative literacy

– “Do the math: An 11-(percentage) point

Democratic lead on the generic ballot test,

minus 5 points for the gauge's Democratic

skew, translated into a 6-point Democratic

victory. When the 6-point Democratic popular

vote win is measured against the GOP's 5-point

win in 2002 and its 3-point win in 2004, it

clearly constituted a wave.” (Cook, 2006)

• Another Interpetation

– “Wow. So in 2002, a humdrum, non-wave

election, the GOP won by 5 points. But

this year, in a "wave election that

rivaled the 1994 tsunami," the Dems won

by 6 points. See? No wave: 5. Wave: 6!”

(Kaus, 2006)

**Why it matters
academically**

• My previous research in

undergraduate student learning of

differential equations

– Students had difficulty with the

concept of a

**rate-of- change equation**

because they had underlying difficulties

with the concept of rate

**Conclusion**

• Rational numbers are difficult for students

because of the ways in which the mathematics

intersects with the ways people think and learn

• Instruction often compounds these difficulties by

not acknowledging and making sure students have

experiences with the multiple ways to think about

rational numbers

• Weak understanding of rational numbers have long

term repercussions for students as they continue

in life and in academics

**For More Information**

• Lamon, S. (2006). Teaching Fractions and

Ratios for Understanding: Essential

content knowledge and instructional

strategies for teachers (2nd Edition).

Mahwah, NJ: Lawrence Erlbaum Publishers.

• The Rational Number Project

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