Problem Solving:
How Can We Do
It Better?
September 20, 2012
Presented by
Lois Williams, Ed.D.
Tidewater Team
College of William and Mary
What do we know about
problem solving?
Research
Students’ “ability to solve word problems falls
far below their ability to compute.”(Burns,
2000)
Research
• This discrepancy is not because children have
poor computation or reading skills, but
because children “do not know how to
choose the correct operation to apply to the
problem.” (Burns, 2000).
Research
• The more problems you solve, the better the
problem solver you become.
• Students construct their own rules for solving
problems.
Example
Megan had $45 on Monday. On Tuesday she earned
$16 babysitting and on Thursday she earned $25
cutting grass. If she did not earn any more money
that week, how much money did Megan have by
Friday?
Megej khj $45 on mosnfn. Onjbf kishfbj knnb
$16 njdfh skf ,mkldfhjkn jnjkjf m sjhe jkdhh $25
jhdfif ufhhjf . Jgdffjhdfb kfhdfkjhdfn ufhkjdhj
fjkdhf ldjfhjhgnjfcnfi;zpfj iur eiy ?
Example
• Mhe kjh ckikh kjh kj khk 96, 79 and 87 oj
injhk kihe kjnj kjhej. Iohh lj ;pwe rfh kkkke
h?
• Megan received a 96, 79 and 87 on her most
recent math quizzes. What is her average on
these quizzes?
Research
Students more accurately solve
problems with diagrams than
without.
Research
• About one-fifth of students disagreed with
the statement that a mathematical problem
can be solved in different ways (Lindquist,
1989).
Research
• Good problem solvers monitor their thinking
regularly and automatically. This is called
metacognition. (Schoenfeld, 1992)
• There is evidence that metacognitive
behavior can be learned. (Garofalo, 1987;
Thomas, 2006)
Strategies
Key Word Strategy
What is it?
Teaching students to scan a problem for
specific words that suggest an operation such
as “in all” means “add”.
Disadvantages of Key Word
Strategy
1. Key words are often misleading. (Drake and
Barlow, 2007)
There are three boxes of chicken nuggets on the
table. Each box contains six chicken
nuggets. How many chicken nuggets are
there in all?
Disadvantages of Key Word
Strategy
2. Many problems have no key words.
The rope is 25-feet long. How many 7-foot
jump ropes can be made?
Disadvantages of Key Word
Strategy
3. The key word strategy sends the wrong
message about doing mathematics. (van de
Walle, 2010)
What message are we
trying to send?
Working with Diagrams
Working with Diagrams
THURS.
FRIDAY
SATURDAY
SUNDAY
SUPERMAN
234
250
275
200
SPIDERMAN
250
255
260
225
BATMAN
240
275
250
200
Two-Step Problems
Tony bought three dozen eggs for 89 cents a
dozen. How much was the bill?
How much change did Tony get back from $5?
Tony bought three dozen eggs for 89 cents a
dozen. How much change did Tony get back
from $5?
Two-Step Problems
1. Give students a one step problem and have
them solve it.
2. Before discussing the answer, have each
child or group use the answer to the first
problem to create a second problem.
3. Collect the second problems.
4. Discuss answer to first.
5. As a class, start answering the second
problems.
Two-Step Problems
7. Select one first and second question. Put
them on the board and show the students how
to make a “hidden” question.
8. Have students do it.
9. Share the “hidden question” problems with
the class. Ask students to solve AND identify
the “hidden questions”.
Singapore Bars
• 1995 TIMSS placed the United States and
Singapore near the bottom of the list of
countries in mathematics and science.
• The next TIMSS places Singapore in first
place in mathematics.
What Happened?
Singapore: Join/Separate
(part/whole)
At the Virginia Museum of Art there are 135
European artists represented and 119
American artists represented. How many
European and American artists are
represented?
135
119
Singapore: Difference
(part/whole)
There are 253 artists represented in the Virginia
Art Museum. 134 are European artists. The
rest are Virginia artists. How many are
Virginia artists?
?
134
253
Singapore: Add/Subtract
Comparison
154 women took part in an art competition. 25
fewer men than women took part. How many
men took part in the competition?
154 women
? men
25
Singapore: Add/Subtract
(comparison model)
154 women and 127 men took part in an art
competition. How many more women than men
took part in the competition?
154 women
127 men
?
Singapore: x/÷
(part and whole model)
Lois saved $80 a week for 6 weeks. How much
did she save altogether?
$80
Singapore: x/÷
(part/whole model)
Lois saved $120 in 6 weeks. How much did she
save each week?
$120
Singapore: x/÷
(part /whole model)
Lois saves $20 each week, How many weeks
will it take her to save $120?
$120
$20
Singapore: x/÷
(comparison model)
• There are 9 green apples. There are 3 times
as many red apples as green apples. How
many red apples are there?
9 green
9
9
9
Singapore: x/÷
(comparison model)
There are 27 red apples. There are 3 times as
many red apples as green apples. How many
green apples are there?
27 red apples
?
Singapore: x/÷
(comparison model)
There are 27 red apples and 9 green apples.
How many times as many red apples as green
apples are there?
9
27
Singapore: Fractions
Lauren bought 24 flowers. 3/4 of them were
white. How many white flowers did Lauren
buy?
24
?
Try One!
Alex spent 4/5 of his money on a storybook.
The storybook cost $20. How much money
did he have to begin?
Lois Williams
bichon3@comcast.net