Chapter 6, page 88
Table 6.5:
Differences between natural numbers and fractions
Feature Natural Number System Fractions
Locating positions
on the number line
Each position on the number
line is indicated by a single
number (1, 2, 3, 4, 5, 50, 389,
etc.).Two numbers are needed to indicate position on a
continuous number line, as in 2/5, 33/35, or
15/2.Ordering Larger digits mean larger
numbers. Every number has
exactly one number that comes
before it and exactly one
number that comes after it.
Larger digits do not mean larger numbers (1/15 is
not larger than 1/8 even though 15 is larger than
8). There is no single unique number that comes
before or after a number.Numbers between
numbers
There is no number between
two consecutive numbers (e.g.,
there is no number between 16
and 17).There infinitely many numbers between any two
other numbers (e.g., between 16 and 17, or
between 16.000001 and 16.000002).The smallest
positive number
One (1) is the smallest positive
number.There is no unique smallest positive number.Operations
Addition-
Subtraction
You can add by counting
combining objects and
counting. E.g., you can add 3
buttons to 4 buttons by putting
the buttons together and
counting them, giving 7
buttons.You cannot add by counting. 4/7 + 3/5 is not
equal to 4 + 3, or 7 + 5, or any other
straightforward sum of numbers.Subtraction You can subtract by counting
the number removed and then
counting what is left. If you
have 9 buttons, and remove 3,
then you can find what 9 minus
3 is by counting the buttons
that remain.You cannot subtract 4/7 minus 3/5 by any
straightforward subtraction of 4 minus 3, 7 minus
5, etc.Multiplication Multiplication makes the
number bigger.Multiplication makes the number either bigger or
smaller (e.g., ½ * ½ = ¼ vs. 2 x 3 = 6).
Division Division makes the number
smaller.Division makes the number either smaller or
bigger (e.g., ½ ÷ 1/5 = 5/2 vs.
5/2 ÷ 7/2 = 5/7).
Adapted from (Vamvakoussi & Vosniadou, 2004)
However, when students apply these conceptions about the natural number system to fractions,
they run into serious difficulties. The rules that govern fractions are not the same as the rules that govern
natural numbers, and this leads students to misunderstand fractions. Here is a transcript of a teacher
working with a seventh grader that illustrates some of these difficulties (based on research by Stafylidou
& Vosniadou, 2004; Vamvakoussi & Vosniadou, 2004).