Modeling Thinking ◆ 127often taught to solve the problem without even representing it or really
understanding it. Charles states that
the challenge is to identify statements in the problem that express
relationships between quantities, to understand those relationships,
and to choose an appropriate operation or operations to show those
relationships.
(p. 5)In the problem about John’s marbles, the relationships are:
There are 25 green marbles.
There are 10 more multicolored ones than green ones.Look at how this bar diagram of the word problem about John’s marbles
represents the quantities and their relationships.
Green marbles
Multicolored marblesThe diagram clearly illustrates what we are talking about when we say
“10 more than the green ones.” It makes the math visual. It brings the
problem into 3-D. The relationships among the quantities is clearly, visu-
ally apparent because of the bar diagram.
- There are 10 more multicolored marbles than green ones.
- The 2 boxes that contain 25 marbles and then the one that shows
10 more shows this relationship.
So now that we have established the relationships, we just have to do the
calculations. We have to know how to translate these relationships into
numerical expressions. We now have to rely on our understanding of
operations. So we can add the 25 plus 10 more. We can easily see where
we got this from by looking at the bar diagrams. So, the answer is John
had a total of 35 multicolored marbles altogether.
Bar diagrams help students to see the problem and give a 3-D view
of the relationships. They also help students to lay out the situation. They
encourage students to think before they try to solve. Teachers can introduce
bar diagrams as early as kindergarten.
When students understand how to make different diagrams, they can
use this strategy for solving multi-digit problems. So we must start out
with easy problems and single-digit numbers and scaffold up to multi-digit
numbers.