Promoting Productive Struggle With Juicy Problems

Bring Common Core Math to life in your classroom through rich problems that help students productively

struggle to connect expressions, equations and representations to explain real world situations.

CMC 2014Vicki Vierra, Ph.D. vvierra@vcoe.org

Agenda

Welcome & IntroductionsTraffic JamMath ModelingProductive StruggleCounterfeit BillStar NumbersDealing Down

Additional Resources

Take Off, Touch Down• If the description pertains to you, “Take Off”.– See who else shares your characteristic

and “Touch Down”• K - 5th grade• 6th grade• 7th grade• 8th grade• HS• Advanced Learners• English Learners• Struggling problem solvers

Table Introductions

Give each person a 20-second “spotlight” to share:NameGrade(s) / Course(s) / Role(s)LocationSomething you struggled to learn

Never Tell An Answer

Please remember the enormous responsibility we all have as learners not to spoil anybody else’s fun.

The quickest way to spoil someone else’s fun is to tell them an answer before they have a chance to discover it themselves.

Susan Pirie

“Telling You the Answer Isn’t the Answer”

(Rhett Allain, Wired Magazine)

“What if a person was having trouble doing a pull up for exercise? Instead of giving them some other exercise, I could help them by doing the pull up for that person. Right? No, that wouldn’t actually be useful. However, if I push on the person’s feet a little bit, they can still struggle and still exercise.”

Cautions & Bonuses

Productive struggles will consume class time and cause student resistance

Frictionless learning, however does not promote “stickiness” and retention.

In the long run, struggle saves re-teaching time and promotes the growth of new dendrites.

Struggle allows time and space for new learning to take place and solidify

Mathematical Modeling

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Standards for

Mathematical Practice – Habits of Mind

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1.Make sense of problems and persevere in solving them

2.Reason abstractly and quantitatively3.Construct viable arguments and critique the

reasoning of others4.Model with mathematics5.Use appropriate tools strategically6.Attend to precision7.Look for and make use of structure8.Look for and express regularity in repeated

reasoning

Model with mathematics: Apply the mathematics they know to solve problems arising in everyday life, society, and the

workplace. In early grades, … as simple as writing an addition equation to describe a situation. In middle grades, … apply proportional reasoning to plan a school event or analyze a problem

in the community. By high school, … use geometry to solve a design problem or use a function to describe how

one quantity of interest depends on another. Comfortable making assumptions and approximations to simplify a complicated situation,

realizing that these may need revision later. Able to identify important quantities in a practical situation and map their relationships using

such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation Reflect on whether the results make sense, possibly improving the model if it has not served

its purpose.

Math Practice #4

Mathematical Modeling

•“Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.”

•Process:▫ Identify variables and select those that are essential▫Formulate a model to describe the relationships▫Analyze and perform operations to draw conclusions▫ Interpret results in the light of the context▫Validate the conclusions

What Is New and Different About Modeling Problems?

• Not well formulated▫ Real world situations are not organized and labeled for analysis;

formulating tractable models, representing such models, and analyzing them is appropriately a creative process.

• More open-ended (involving creativity & assumptions)• Require decisions to be made about what to keep and what to

ignore▫ Choices, assumptions, and approximations are present

throughout this cycle• No one “correct” answer• Requires time

▫ “Substantial investment in time” (NCTM, 2000)

Modeling in a Typical Textbook

The function f(t) = -16t2 + 400 represents the height of a freely falling ballast bag that starts from rest on a balloon 400 feet above the ground. Find f(t) when t = 5. What does this solution mean with regard to the height of

the object?

What’s missing here for actually doing mathematical modeling?

How is this different from the “Traffic Jam” problem?(Dan Meyer, 2013)

Counterfeit Bill

A customer enters a store and purchases a pair of slippers for $5, paying for the purchase with a $20 bill. The merchant, unable to make change, asks the grocer next door to change the bill. The merchant then gives the customer the slippers and $15 change.

After the customer leaves, the grocer discovers that the $20 bill is counterfeit and demands that the shoe-store owner make good for it.

The shoe-store owner does so, and by law is obligated to turn the counterfeit bill over to the FBI. How much does the shoe-store owner lose in this transaction?

(Sobel & Maletsky, 1999)

Star Numbers These dots are arranged in patterns that represent the first three star numbers.

A. What are the first three star numbers?B. Use the dot patterns and your answers for Part A to find the next

three star numbers.C. Write an equation you could use to calculate the nth star number.

(CMP 8th Frogs, Fleas and Painted Cubes)

What Do You Notice? What Do You Wonder About?

5 Powerful Questions Teachers Can Ask Students:1. What do you think?

• Keeps us from telling too much. Provides opportunities for students to make sense of an apply new information

2. Why do you think that?• This pushes students to provide their reasoning

3. How do you know this?• Helps students make connections between their thoughts and prior

experiences.

4. Can you tell me more?• Extends students thinking so they can share further evidence

5. What Questions do you still have? • Allows students to wonder and ask deeper questions(Edutopia, Oct. 31, 2013)

Problem to Solve

• If you could hop like a frog, how far could you hop?

• What do you need to know in order to answer this question?

• What assumptions are you making?• What mathematics are you using?• Be ready to share your answer – Must justify your thinking!

“Dealing Down” Test your skill at writing expressions.

• Groups of 3-4; Shuffle the 25 cards • Deal four cards in the center of the table• Each player uses the 4 cards to write an expression

with the least possible quantity. The numbers can be used in any order and with any operation.

• Compare answers. Each player explains why their answer is accurate and the least possible quantity.

• Each player with the least quantity gets 1 point• Complete the recording sheet for each round.

Write a report about your strategies• Discuss the properties you used, e.g., Commutative,

Associative, and Distributive• Tell how you used the Order of Operations

(CMP3 7th Accentuate the Negative)

Additional Resources

• Sum Patterns• Square Numbers• Triangular Numbers• Skeleton Tower• The Gliding Ghost• Reflecting on Counterfeit Bill (www.nctm.org )

Epiphany!

“Allowing productive struggle to occur, using artistic and scientific instruction, modeling symphonic thinking, and encouraging students to lean into constructivist problem solving, can lead to the holy grail of transformational teaching: epiphany.”

(Rhett Allain, Wired Magazine)

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“Allowing productive struggle to occur, using artistic and scientific instruction, modeling symphonic thinking, and encouraging students to lean into constructivist problem solving, can lead to the holy grail of transformational teaching: epiphany.”

(Rhett Allain, Wired Magazine”

How Do You See This Growing?

Why Students in the US Need Common Core Math

https://www.youtube.com/watch?v=pOOW0hQgVPQ

Jo Boaler, Ph.D., Stanford University

How to Encourage Questioning1. Make it Safe

a. Make questioning welcomed and desired

Visual Representation