TEACHING MATHEMATICAL PROBLEM SOLVING
R A J S H A H , P H . D . F O U N D E R M AT H P L U S A C A D E M Y
nine
eighteentwenty-seven
thirty-six
forty-five
fifty-four
sixty-three
seventy-twoeighty-one
ninety
9 18 27 36 45 54 63 72 81 90
➙ 0 + 9 = 9➙ 1 + 8 = 9 ➙ 2 + 7 = 9 ➙ 3 + 6 = 9 ➙ 4 + 5 = 9 ➙ 5 + 4 = 9 ➙ 6 + 3 = 9 ➙ 7 + 2 = 9 ➙ 8 + 1 = 9 ➙ 9 + 0 = 9
Math is an ADVENTURE– PA U L L O C K H A RT, M AT H E M AT I C I A N ’ S L A M E N T
P R O B L E M S V S . E X E R C I S E S
• Exercise - You know a way to solve the problem
• Problem - You do NOT know a direct way to solve the problem
• Example: Sally has 24 cookies. If she shares equally among 3 people, how many will each person get?
J U S T I F Y I N G P R O B L E M S O LV I N G
1. Inquiry & problem solving are at the heart of mathematics and science
2. Provides opportunities to develop perseverance
3. Students who experience problem based learning perform BETTER on standardized tests.†
† Boaler, J (1998) Open and Closed Mathematics: Student Experiences and Understandings
Y O U C A N ’ T A F F O R D N O T T O D O T H I S !
W I T H O U T P R O B L E M S O LV I N G …
• Collection of algorithms, formulas, and definitions to be memorized
• Reduces to just “answer-getting”
• Intrigue, discovery, and joy are removed
F R O M P R O C E D U R A L F L U E N C Y T O P R O B L E M S O LV I N G
S H I F T I N G P R A C T I C E
• Start with “rich” tasks
• Launch with curiosity
• Encourage productive struggle
• Orchestrate a productive classroom discussion
W H AT I S A R I C H TA S K ?
• Low barrier to entry
• Fosters productive struggle, creativity
• Open to multiple strategies and representations
• Allows for making connections and conceptual understanding
• Generally done collaboratively
A farmer has a total of 13 chickens and cows. His wife notices that there are 42 legs. How many of each animal are on the farm?
P R O B L E M - C E N T E R E D C L A S S R O O M S
• Students often lack problem solving strategies beyond guess and check
• Teachers find it difficult to anticipate more than one way to solve a problem
T E A C H I N G A P R O B L E M S O LV I N G F R A M E W O R K
P R O B L E M S O LV I N G F R A M E W O R K
JAMES TANTON GEORGE POLYA JOHN MASON
E N T RY
• What do I KNOW?
• What do I WANT?
• What can I introduce?
• Get started with easy wins
• Start with an easy case
• Add / remove a constraint
AT TA C K
• Draw a Picture or Diagram
• Make a table or organized list
• Go to Extremes
• Experiment & Be Creative
• Seek Pattern & Structure
• Solve a Simpler Problem
• Wishful Thinking
• Guess, Check and Revise
R E V I E W
• Is your solution answer reasonable?
• Compare to conjectures and estimates
• Share strategies and solutions
• Check for efficiency and generality
E X T E N D
• Generalize the solution
• Use “What-If-Not” to explore similar problems
W H AT - I F - N O T
• List all attributes of the problem including “givens”
• Ask “what if not” for each attribute?
• Choose a new question to investigate
A farmer has a total of 13 chickens and cows. His wife notices that there are 42 legs. How many of each animal are on the farm?
PROBLEM SOLVING PROCESS
What do I KNOW?
What do I WANT?
What can I INTRODUCE?
Write theessentials of
the problem in YOUR OWN
WORDS
TRY A STRATEGY
Ask yourself:
DRAW A PICTURE OR DIAGRAM
MAKE ANORGANIZED LIST
EXPERIMENT& BE CREATIVE
GO TOEXTREMES
B O X O F C H O C O L AT E S
• A box of chocolates adorned the kitchen counter.
• When Jake saw it, he ate 1/6 of the box.
• Along came Joe and he ate 1/5 of what Jake left.
• Then came Jill who ate 1/4 of what remained.
• Later, Jeff ate 1/3 of the remaining chocolates.
• I got there and I ate 1/2 of what Jeff left.
• After I left, only 4 chocolates were uneaten.
M O R E T H A N A L G E B R A …
Tina had 3 times as much money as Timmy. After Tina spent $60 and Timmy spent $10,
they each had an equal amount of money left. How much money did Tina have at first?
Tina
Timmy
$ 1 0
$ 6 0
1
2
3
4 . . .
20 How many stars in this row?
H O W M A N Y S TA R S ?
W O U L D Y O U R AT H E R … ?
1/3 off the regular price
1/3 more at regular price
regular size regular price
A B
P R O B L E M S O LV I N G I S A M E A N S O F L E A R N I N G M AT H — N O T J U S T A WAY O F A P P LY I N G I T
G E T T I N G T H E M O S T O U T O F P R O B L E M S O LV I N G
D I S C U S S I O N S ( S M I T H & S T E I N 2 0 1 1 )
1. Anticipate student strategies
2. Monitor student work
3. Select students who should contribute
4. Sequence contributions (e.g. concrete to abstract)
5. Connect - help students see connections
C H A L L E N G E S I N FA C I L I TAT I N G D I S C U S S I O N S
• Lack of teacher’s control
• Demands quick reactions
• Requires deep, interconnected knowledge of content, pedagogy, and students
P U R P O S E O F T H E 5 P R A C T I C E S
• To make student-centered instruction more manageable by moderating the degree of improvisation required by the teachers and during a discussion.
TA S K # 1
• A fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would the students need each day for 12 caterpillars?
• Use drawings, words, or numbers to show how you got your answer.
• Do this problem in as many ways as you can, both correct and incorrect
( S M I T H & S T E I N 2 0 1 1 )
M O N I T O R I N G
S T R AT E G Y W H O / W H AT O R D E R
Q U E S T I O N S
• How might carefully selecting and sequencing students’ responses affect the quality of the discussion?
• How would these practices give you more control over the discussion?
• Why is connecting important?
• What is the teacher’s role in helping students make connections?
N O A H ’ S A R K• Can you figure out how many SEALS are needed to
balance the 3rd deck?
• On each deck the animals on the left exactly balance the animals on the right.
FAW N N G U Y E N
?
P R O B L E M S O LV I N G T E A C H I N G T I P S
• “NEVER” tell an answer
• Teach by ASKING rather than teach by TELLING
• Make students justify EVERYTHING
• “I don’t know, why do you think?”
• “Convince me.”
• “How do you know?”
• “Please explain.”
• Be their PEER in problem solving
S T U D E N T S L E A R N B E S T B Y M A K I N G S E N S E O F T H I N G S T H E M S E LV E S
D R AW A P I C T U R E
• You are buying supplies to make s’mores. You spend 1/5 of your money on chocolate, half of what’s left on marshmallows, and the remaining $12 on graham crackers. How much money did you start with?
M A K E A TA B L E
• The first four rows of a pattern of stars are given as shown. How many stars will there be in 20th row?
G O T O E X T R E M E S
• When purchasing a bottle of water, which is a better deal: getting 1/3 more water at regular price OR getting the regular bottle at 1/3 off the price?
E X P E R I M E N T & B E C R E AT I V E
• Can you connect a 3×3 square grid of dots using four straight lines without picking up your pen?
S E E K PAT T E R N S & S T R U C T U R E
• What is the ones-digit of 250?
S O LV E A S I M P L E R P R O B L E M
• How many squares are on a chess board?
W I S H F U L T H I N K I N G
• When purchasing a bottle of water, which is a better deal: getting 1/3 more water at regular price OR getting the regular bottle at 1/3 off the price?
G U E S S , C H E C K A N D R E V I S E
• A farm has cows and chickens. You see 18 heads and 50 feet. Can you figure out how many chickens and cows there are?
