EDUCATIONAL PSYCHOLOGY

Chapter 6, page 89

Transcript:

Teacher: Desmond, what is the smallest fraction that

you can think of? Write it down for me.

Desmond: [He writes: 1 ]

2

Teacher: One half? Why do you think so?

Desmond: Because both of the numbers are pretty

small.

Commentary:

Desmond has picked this number because 1

is the smallest natural number he knows, and

2 is the next smallest natural number he

knows. Because 1 and 2 are small, he thinks

that ½ must be a very small fraction.

Because 2 is smaller than 9999, Desmond

Teacher: Which of these is smaller? [She writes ½

and 1/9999]

Desmond: One half.

thinks that ½ is smaller than 1/9999.

Desmond is incorrectly applying ideas about

natural numbers to fractions.

Teacher: And what is the biggest fraction you can

think of?

Desmond: [Pauses a few seconds before writing:

99,999,999,999

999,999,999,999 ]

Desmond thinks that this fraction with lots of

9’s is a large fraction because both the

numerator and denominator are very large

numbers.

Teacher: Now, look at this list of numbers. Put these

in order from the smallest to the biggest.

[The teacher shows these numbers:

5/6 1 1/7 4/3 ]

Desmond: [Puts them in this order:

1 1/7 4/3 5/6 ]

Teacher: Why did you pick that order?

Desmond: Well, the first two numbers both have a

“1,” but the second one [1/7] also has a

“7.” And the next one has “4” on the top,

and then the last one is bigger, because it

has a “5.”

Desmond is ordering the numbers according

to how large the numerator is. When there is

a tie between 1 and 1/7, he judges 1/7 to be

larger because of the 7.

Teacher: And which of these is bigger? [She shows

him 4/15 and 4/7.]

Desmond: 4/15, because 15 is bigger than 7, and the

4’s are the same.

When evaluating this pair of numbers, the

numerators are the same, so Desmond ranks

them by the size of the denominators.

Now let’s reflect on what Desmond’s answers reveal about his thinking. Desmond has
successfully learned the natural number system, and he is trying to apply this understanding to fractions.
The fraction 4/15 is larger than 4/7 because the 4’s are the same and 15 is larger than 7. One half is a
small fraction because both numbers are small. The fraction 1/7 is larger than 1 because 7 is larger than 1.
Desmond does conceive of these numbers as fractions. He is simply using his basic understanding of
natural numbers--the numbers that he can count as 1, 2, 3, 4, and so on.
The problem for Desmond (and many other students) is that the rules that apply to natural
numbers do not apply to fractions. There are many important conceptual differences between the natural
number system and the number system of fractions (Jones, Langrall, Thornton, & Nisbet, 2002;
Stafylidou & Vosniadou, 2004; Vamvakoussi & Vosniadou, 2004). Table 6.5 shows some of the
important differences. Each of these differences is a source of difficulty in learning about fractions. For
example, in the natural number system, there is exactly 1 number between 8 and 10, and that number is 9.
But when fractions are included, there are infinitely many numbers between 8 and 10. This will make no
sense to students like Desmond who are applying the natural number system because Desmond cannot
envision having any fractions between 8 and 9 or 9 and 10. This makes the very idea of fractions quite
difficult for Desmond to grasp.