© Joan A. Cotter, Ph.D., 2015
How to Teach Your Child to Solve Problems!
Sioux Empire Christian Home Educators Homeschool Conference
Sioux Falls, SD Saturday, May 2, 2015 3:00 p.m.– 4:00 p.m.
Joan A. Cotter, [email protected]!
© Joan A. Cotter, Ph.D., 2015
Math Education is Changing • Mathematics is doubling every 7 years.
• Math is used in many new ways. We need analytical thinkers and problem solvers.
• Brain research is providing clues on how to better facilitate learning, including math.
• Increased emphasis on mathematical reasoning, less on rules and procedures.
• Educators and parents are learning that rote memorizing is not effective.
© Joan A. Cotter, Ph.D., 2015
What is a Problem? Dictionary says:
“a matter or situation regarded as unwelcome or harmful and needing to be dealt with and overcome”
Oops! That’s not what we’re looking for!
© Joan A. Cotter, Ph.D., 2015
Math Problems • Computation drills are actually exercises. • A true math problem: solution is not obvious. • Two types of math problems:
1. Story problems and 2. Other problems involving math.
© Joan A. Cotter, Ph.D., 2015
Problem Solving
• is NOT rote memorizing.
• is NOT following rules blindly.
• is NOT passive learning.
• is the major reason we study math.
© Joan A. Cotter, Ph.D., 2015
Problem Solving Problem solving is not about:
• trying to remember a similar problem • or looking for key words, • but thinking carefully about the situation, • discovering what is given, • figuring out what is needed, • and deciding on methods to get there.
© Joan A. Cotter, Ph.D., 2015
Chinese Checkerboard
How many dots are there?
© Joan A. Cotter, Ph.D., 2015
Chinese Checkerboard
Twelve 10s plus 1 in the middle gives 121.
© Joan A. Cotter, Ph.D., 2015
Chinese Checkerboard
Six 6s plus 21 fours plus lone 1 gives 121.
© Joan A. Cotter, Ph.D., 2015
Chinese Checkerboard
Nine 9s plus four 10s gives 121.
© Joan A. Cotter, Ph.D., 2015
Chinese Checkerboard
(13 × 14)/2 plus three 10s gives 121.
© Joan A. Cotter, Ph.D., 2015
Some Barriers to Solving Problems
• Trying to remember what rule to apply.
• A superficial understanding of math in general.
• Difficulty in reading.
• Thinking you’re not good in math.
• A fear of failure (math anxiety).
• Expecting instant success without hard work.
© Joan A. Cotter, Ph.D., 2015
Helping Young Children Solve Problems
• Ask young child to decide between two options. Do you want to wear blue socks or red ones? Which book shall we read for bedtime tonight?
• Jigsaw puzzles are good for learning persistence. • Encourage by praising effort. • Speak positively about problems you have solved. • Ask child to explain how they solved the problem. • Encourage children to make up their own problems.
© Joan A. Cotter, Ph.D., 2015
Problem Solving in Japan
• Japanese teachers discuss one problem in depth rather than four problems superficially.
• They encourage multiple solutions. It is a check on correctness.
• Wrong solutions are discussed.
• The students do not look for “key” words.
© Joan A. Cotter, Ph.D., 2015
Braid Model of Problem Solving
• Understanding the problem/Reading the story
• Planning how to solve the problem
• Carrying out the plan/Solving the problem
• Looking back/Checking
© Joan A. Cotter, Ph.D., 2015
Some Solving Problems Strategies • Read the problem at least twice. • Draw a sketch. • List what is known and unknown. • Use smaller numbers. • Estimate before calculating. • In algebra try numbers in place of variables. • Expect some frustration. • Take a break and come back.
© Joan A. Cotter, Ph.D., 2015
After Solving a Problem • Did you answer the question?
• Does the answer make sense?
• Is the answer properly rounded and labeled?
• Explain your work. (Not show computations.)
• How else could the problem be solved?
• Which method is most efficient?
• Think how this problem leads to other problems.
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles
Whole
Part Part
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles
What is the whole?
3 2
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Missing addend problem
Lee received 3 goldfish as a gift. Now Lee has 5. How many goldfish did Lee have to start with?
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Missing addend problem
Lee received 3 goldfish as a gift. Now Lee has 5. How many goldfish did Lee have to start with?
Is 3 the whole or a part?
3
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Missing addend problem
Lee received 3 goldfish as a gift. Now Lee has 5. How many goldfish did Lee have to start with?
5
3
Is 5 the whole or a part?
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Missing addend problem
Lee received 3 goldfish as a gift. Now Lee has 5. How many goldfish did Lee have to start with?
2
5
3
What is the missing part?
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Missing addend problem
Lee received 3 goldfish as a gift. Now Lee has 5. How many goldfish did Lee have to start with?
2
5
3
Write the equation. 2 + 3 = 5 3 + 2 = 5 5 – 3 = 2
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles
• Research shows part-whole circles help young children solve problems. Writing equations do not.
• Do not teach “key” words.
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Compare problems
larger set
small- er set
differ- ence
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Compare problems
Alex has 4 apples. Morgan has 7 apples. How many more apples does Morgan have?
Is 4 the larger, smaller, or the difference?
4
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Compare problems
Alex has 4 apples. Morgan has 7 apples. How many more apples does Morgan have?
7
4
Is 7 the larger, smaller, or the difference?
© Joan A. Cotter, Ph.D., 2015
Part-Whole Circles Compare problems
Alex has 4 apples. Morgan has 7 apples. How many more apples does Morgan have?
What is the difference?
3
7
4
© Joan A. Cotter, Ph.D., 2015
Rows and Columns
Good games can provide a chance for children to think of alternatives and make decisions to improve their chance of winning.
© Joan A. Cotter, Ph.D., 2015
Rows and Columns Objective: To give the players practice in adding up to 15. Object of the Game: To collect the most cards in
rows or columns that add up to 15.
© Joan A. Cotter, Ph.D., 2015
Rows and Columns 8 7 1 9
6 4 3 3
2 2 5 6
6 3 8 8
© Joan A. Cotter, Ph.D., 2015
Rows and Columns 1 9
6 4 3 3
6 3 8 8
© Joan A. Cotter, Ph.D., 2015
Rows and Columns 7 6 1 9
6 4 3 3
2 1 5 1
6 3 8 8
© Joan A. Cotter, Ph.D., 2015
Rows and Columns 1
6 4 3 3
1 5 1
3 8 8
© Joan A. Cotter, Ph.D., 2015
Math Balance
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
© Joan A. Cotter, Ph.D., 2015
Math Balance
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
7 = 7"
© Joan A. Cotter, Ph.D., 2015
Math Balance
10 = 2 + 8"
10 9 8 7 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
© Joan A. Cotter, Ph.D., 2015
Math Balance
10 9 8 6 5 4 3 2 1 1 2 3 4 5 6 7 8 9 10
8 + 6 = 14"
7
© Joan A. Cotter, Ph.D., 2015
Equation Puzzles
1 2 3 4
Write equations using mathematical symbols, such as +, –, =, ×, ÷. (The order of the digits cannot change.)
© Joan A. Cotter, Ph.D., 2015
Equation Puzzles
1 2 3 4 =" ×"
Write equations using mathematical symbols, such as +, –, =, ×, ÷. (The order of the digits cannot change.)
© Joan A. Cotter, Ph.D., 2015
1 2 3 4
Equation Puzzles
–"+"=
Write equations using mathematical symbols, such as +, –, =, ×, ÷. (The order of the digits cannot change.)
© Joan A. Cotter, Ph.D., 2015
1 2 3 4
Equation Puzzles
–"= –"
Write equations using mathematical symbols, such as +, –, =, ×, ÷. (The order of the digits cannot change.)
© Joan A. Cotter, Ph.D., 2015
1 2 3 4
Equation Puzzles
= ÷"
Write equations using mathematical symbols, such as +, –, =, ×, ÷. (The order of the digits cannot change.)
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Problem 1. Thirteen children are going on a
field trip. If 4 children can ride in a car, how many cars are needed?
[4 cars]
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Problem 2. Pauline has 13 petunias to plant. She
wants exactly 4 in a row. How many rows can she plant?
[3 rows]
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Problem 3. Four children have $13 to split
evenly. How much does each one receive?
[$3.25]
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Problem 4. Four children divide 13 candy bars.
How much does each one receive?
[3 bars] 1 4
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Problem 5. Jack packages 13 cookies with 4 per
bag and eats the leftovers. How many does he eat?
[1 cookie]
© Joan A. Cotter, Ph.D., 2015
Remainder Problems
Five division problems with the same numbers, but different answers.
[4 cars] [3 rows] [$3.25]
[3 bars] [1 cookie]
1 4
© Joan A. Cotter, Ph.D., 2015
Some Problems 1. Chris and Pat have 100 cubes together. Chris
has 6 more than Pat. How many does Chris have?
2. Use four 4s to equal 1, 2, . . . , or 10.
44 44 = 1 4 4
4 4 + = 2
© Joan A. Cotter, Ph.D., 2015
A Geometry Problem
What is the area?
© Joan A. Cotter, Ph.D., 2015
A Geometry Problem
What is the area?
w1
w2
h
© Joan A. Cotter, Ph.D., 2015
A Geometry Problem
What is the area?
w1
w2
h
w1
© Joan A. Cotter, Ph.D., 2015
A Geometry Problem
What is the area?
w1
w2
h
© Joan A. Cotter, Ph.D., 2015
A Geometry Problem
What is the area?
w1
w2 – w1
h
© Joan A. Cotter, Ph.D., 2015
Tens Fractal
How many little black triangles do you see?
© Joan A. Cotter, Ph.D., 2015
Good Mathematical Practices 1. Make sense of problems and persevere in
solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the
reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure.
© Joan A. Cotter, Ph.D., 2015
How to Teach Your Child to Solve Problems!
Sioux Empire Christian Home Educators Homeschool Conference
Sioux Falls, SD Saturday, May 2, 2015 3:00 p.m.– 4:00 p.m.
Joan A. Cotter, [email protected]!