Looking at the Structure
of Arithmetic Story
Problems Through the
Eyes of Children’s
Solution Strategies
Consider the following
problems—How would you
solve each?
•
•
•
Eliz had 8 cookies. She ate 3 of them.
How many cookies does Eliz have left?
Eliz has 3 dollars to buy cookies. How
many more dollars does she need to earn
to have 8 dollars?
Eliz has 3 dollars. Tom has 8 dollars.
How many more dollars does Tom have
than Eliz?
• Most adults would solve all three of
these problems by subtracting 3
from 8.
• To young children, however, these
are 3 different problems, which they
solve using different strategies.
Research
• Children can solve many
mathematical problems that
involve +, -, x or / without
being shown how to solve them.
• Initially, most children use a
strategy researchers refer to
as “direct modeling.”
• That is, the children model the
action and relationships they
see in the story.
Additive Structures
• Four Basic Problem Contexts:
–
–
–
–
Join,
Separate,
Part-Part-Whole, and
Compare.
• Although the number size, themes, and
contexts of the problems may vary, the
basic structure involving actions and
relationships within each class remains the
same.
Join and Separate
problems involve action.
– In join problems, elements are added to
a given initial set.
– In separate problems, elements are
removed from a given initial set.
Part-Part-Whole and
Compare-No action
• Part-Part-Whole problems involve static
relationships among a set and its two
distinct subsets.
• Compare problems involve the comparison
of two distinct, disjoint sets rather than
the relationship between a set and its
subsets.
Join, Result Unknown
• Four birds were singing on a branch.
Three more flew in to join them.
How many birds were singing now?
• Join (an action), Result Unknown
• 4 + 3 = __
Join, Change Unknown
• Four birds were singing on a branch.
Some more flew in to sing with them.
Now there are 7. How many birds
are on the branch?
• Join (an action), Change Unknown
• 4 + __ = 7
Join, Start Unknown
• Some birds were singing on a branch.
Three more flew in to sing with them.
Now there are 7. How many birds
were already there?
• Join (an action), Start Unknown
• __ + 3 = 7
Separate, Result
Unknown
• Seven birds were singing on a branch.
Three flew away. How many birds
are left?
• Separate (an action), Result Unknown
• 7 - 3 = __
Separate, Change
Unknown
• Seven birds were singing on a branch.
Some flew home. Now there are 3.
How many birds went home?
• Separate (an action), Change
Unknown
• 7 - __ = 3
Separate, Start Unknown
• Some birds were singing on a branch.
Three flew home. Now there are 4.
How many birds were there to begin
with?
• Separate (an action), Start Unknown
• __ - 3 = 4
Part-Part-Whole,
Whole Unknown
• Some birds were singing on a branch. Three were
blue and 4 were red. How many birds were on the
branch?
• Part-Part-Whole, Whole Unknown (Note—no
action! This is a relationship between 2 parts
of one set.)
• There is no specific number sentence for partpart-whole problems. Can you see why?
Part-Part-Whole,
Part Unknown
• Seven birds were singing on a branch.
Three were blue and the rest were red.
How many birds were red?
• Part-Part-Whole, Part Unknown (Note—
no action! This is a relationship between
2 parts of one set.)
Compare, Difference
Unknown
• The oak tree has 7 birds. The pine tree
has 4 birds. The oak tree has how many
more birds than the pine tree?
• Compare, Difference Unknown (Note—no
action! This is a relationship between 2
distinct, disjoint sets.)
Compare, Larger
Quantity Unknown
• The oak tree has 3 more birds than the
pine tree. The pine tree has 4 birds. How
many birds does the oak tree have?
• Compare, Larger Quantity Unknown
(Note—no action! This is a relationship
between 2 different sets.)
Compare, Smaller
Quantity Unknown
• The oak tree has 3 more birds than the
pine tree. The oak tree has 7 birds. How
many birds are in the pine tree?
• Compare, Smaller Quantity Unknown
(Note—no action! This is a relationship
between 2 different sets.)
As children mature, their
strategies become more
abstract and efficient.
– Direct Modeling strategies are replaced by-– Counting strategies, which in turn are replaced
with-– Derived Facts strategies (invented strategies
with single-digit numbers), which are replaced
by-– Recall of Number Facts strategies.
Other Notes
• Direct Modeling provides a basis for
children’s learning of other, more
efficient, strategies.
Other Notes
• Strategies at different levels of
abstraction are related----which
leads to flexible use of strategies.
• Children in any classroom will be at
different levels of understanding and
will use different strategies to solve
the same problems.