Chapter 6, page 90
Similarly, the idea that you can no longer add numbers by counting them (1, 2, 3, 4, etc.) makes it
hard to understand the entirely different rules for adding fractions. The idea that multiplying two fractions
can yield a number smaller than the two numbers multiplied makes no sense to Desmond, who has
mastered the natural-numbers idea that multiplying two numbers always yields a product greater than or
equal to the two numbers multiplied. In sum, because Desmond is trying to understand fractions using the
rules of the natural number system (shown on the left side of Table 6.5), he has difficulty understanding
fractions, which are governed by a different set of rules (shown on the right side of Table 6.5).
To teach students like Desmond, a teacher cannot simply start teaching the rules for fractions,
because the new conceptual basis for these rules will make no sense to these students. The thoroughness
with which students have successfully learned the natural number system now interferes with learning
about fractions. Instead, the teacher will need to devote considerable time helping students understand how
the fractional number system differs from the natural number system, so that they learn that they cannot
apply their ideas about natural numbers to fractions. We will address in later chapters how to help
students learn new conceptual systems of this sort. For now, the critical point is that when teachers
understand students’ alternative conceptual systems, they will realize that they need to help students
understand the various differences between their prior conceptions and the target conceptions that they are
learning.
Table 6.6
Examples of alternative conceptions
Topic Target conception Common alternative conceptions
Light and
vision
We see objects because light
(such as light from the sun)
bounces off of that object and
then strikes our eyes.- We see objects because a visual emanation of some kind
travels from our eyes to the object. - We see objects because a visual emanation of some kind
travels from our eyes to the object, then bounces off the
object and travels back to our eyes.
Matter Matter is composed of
molecules which are elastic
and do not individually have
the same properties as the
substance (e.g., individual
water molecules are not wet). - Matter is simply matter through and through. For
example, water is water, and it is not composed of any
smaller particles. - Matter is composed of small particles that have the same
properties as the whole substance. For example, water
molecules are tiny, wet drops of water, just like water.
Photosyn-
thesis
Plants get their energy from
photosynthesis. In
photosynthesis, light triggers
a reaction in which carbon
dioxide and water combine to
produce glucose and oxygen.
The mass of a plant comes
mainly from the carbon
dioxide and the water.- Plants get their energy (or food) from the soil. The plant
absorbs its mass from the soil. - Plants get their energy from plant food given to the plants
by humans.
Prices Prices are determined by an
interaction of supply and
demand. Supply, in turn, can
depend on the cost of
producing an item.
- Children younger than 7 think that the price of things
depends on size. Expensive things are expensive because
they are large (e.g., large furniture and automobiles). A
diamond is inexpensive because it is small. - At around 10, many students think that the price of a
product depend on the amount of work that goes into
producing the product.