Turning ‘I don’t know’ into ‘Yes, I can’: Supporting Productive Struggle
TLQP Workshop, Binghamton University
12/19/15
Dr. Rachel Bachman
8 Standards of Mathematical Practice
- Make sense of problems and persevere in solving them
- Reason abstractly and quantitatively
- Construct viable arguments and critique the reasoning of others
- Model with mathematics
- Use appropriate tools strategically
- Attend to precision
- Look for and make sure of structure
- Look for and express regularity in repeated reasoning
8Mathematical Teaching Practices
- Establish mathematics goals to focus learning
- Implement tasks that promote reasoning and problem solving
- Use and connect mathematical representations
- Facilitate meaningful mathematical discourse
- Pose purposeful questions
- Build procedural fluency from conceptual understanding
- Support productive struggle in learning mathematics
- Elicit and use evidence of student thinking
Shared Strategies
Today’s Strategies
- Set up classroom norms
- Normalize mistakes
- Write down what you know
- Hurdle interviews
- Think Aloud in Pairs for Problem Solving (TAPPS)
- Pose problems of interest
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Example 1
Ali and Samy had an equal amount of money each. After Samy gave $15 to Ali, the ratio of Ali’s money to Samy’s money was 7 : 5. How much money did each boy have at first?
Example 2 – (Setting up class norms and normalizing mistakes)
Jenny was mixing herself a glass of chocolate milk. She enjoys a strong “chocolatey” taste. Her friend Kevin remarked, “You certainly have enough chocolate syrup in the glass.” Kevin was looking for a glass of his own so he could make himself some chocolate milk, too.
Jenny responded to Kevin, “I only have a third of my glass filled with chocolate syrup.” By this time, Kevin had found a glass that was twice the size of Jenny’s, he said, “Well, I am only going to fill mine one-fourth of the way with syrup.”
“But Kevin, your glass holds twice as much!”
Kevin, knowing that Jenny really liked “chocolatey” tasting milk said, “Jenny, tell you what let’s combine our drinks into a larger pitcher and then split the whole amount.”
Your task:
Will the new mixture be more “chocolatey” or less “chocolatey” tasting than Jenny’s original glass of chocolate milk?
Be sure that you have proof to support your conclusion.
Example 3 (Write what you know)
If the lengths of every side of the rectangular prism are doubled, how many times larger will the volume of the new box be?
Example 4 (Write what you know)
Determine, by eliminating possibilities, the number multiplied by itself three times that gives 68,921.
Example 5 (Hurdle Interviews)
Decide which of the following is true and be prepared to defend your answer.
Figure 1 has the most area not black
Figure 2 has the most area not black
Figure 3 has the most area not black
All three figures have the same amount of area not black
Example 6 (TAPPS)
A dart board allows players to score either 5 or 7 points each time they throw a dart. What is the largest score that cannot be earned in this game?
Example 7 (TAPPS)
You have $1000 in one-dollar bills. You have ten envelopes. How can you distribute the bills so that for any purchase for any dollar amount of money, some combination of the envelopes will add up to the purchase price? (You may use an envelope only once.)
Example 8 (Pose Problems of Interest)
How long will it take to fill up the water tank?
Example 9 (Pose Problems of Interest)
My brother is building fence for our cows for the summer. He has a piece of pasture that basically looks like an isosceles right triangle. He wants to make six equal size lots that cut across the triangle from one leg to the hypotenuse. How should be build the lots?
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