Mathematical practices: What do they look like in instruction? Deborah Schifter Mathematical Sciences Research Institute March 22, 2012

Deborah Schifter, EDC Susan Jo Russell, TERC Virginia Bastable, MHC Funded in part by the National Science Foundation

Grade 1: Understand and apply properties of operations and the relationship between addition and subtraction. Grade 2: Use place value understanding and properties of operations to add and subtract. Grade 3: Understand properties of multiplication and the relationship between multiplication and division. Common Core State Standards

“Students need not use formal terms for these properties.”

= “I moved 1 from the 37 over to the 19, and that made Now it’s easy to add.”

Associative property of addition = “I moved 1 from the 37 over to the 19, and that made Now it’s easy to add.” (36 + 1) + 19 = 36 + (1 + 19)

What do you notice? = What is ?

Video 1 The child reads, “When we have an expression, we can change the numbers but still have the same answer. The numbers can go up and down. We change the numbers by making one less and the other one bigger. We can take away one and then add one.”

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. 8. LOOK FOR AND EXPRESS REGULARITY IN REPEATED REASONING. Mathematical Practice Standards

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. 8. Look for and express regularity in repeated reasoning. Mathematical Practice Standards

Associative property of addition On Saturday, 5 boys and 5 girls were in the swimming pool. How many children were in the pool? On Sunday, 5 boys and 6 girls were in the swimming pool. How many children were in the pool?

Associative property of addition On Saturday, 5 boys and 5 girls were in the swimming pool. How many children were in the pool? On Sunday, 5 boys and 6 girls were in the swimming pool. How many children were in the pool? How did your answer to the first problem help you solve this one?

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. 8. Look for and express regularity in repeated reasoning. Mathematical Practice Standards

Video 2 A child states, “If you have any addend and you change that by any number, the sum will change as much as it changed.” Mike goes over the claim with the class, asks students if they think this will work for all numbers, and then challenges them to come up with a diagram, a model, or a story context to demonstrate why it has to work, no matter what the numbers.

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. 8. Look for and express regularity in repeated reasoning. Mathematical Practice Standards

Video 3 The class is sitting in a circle with models they have created. Alice asks the class, as they look at all the models, what do they have in common? One child talks about how she can see 1 in all the models. Two pairs of children explain how their model demonstrates why the generalization works, no matter what addends they start with.

5 + 7 = = = 13

Video 4 Alice asks the class if this kind of thing will work with other operations.

Does this kind of thing work for other operations?

Adding 1 to a factor Writing prompt. In a multiplication problem, if you add 1 to a factor, I think this will happen to the product:

Students’ articulations of the claim The number that is not increased is the number that the answer goes up by. The number that is staying and not going up, increases by however many it is. I think that the factor you increase, it goes up by the other factor.

Distributive property of multiplication over addition 7 × 5 = 35 7 × 6 = 7 × (5 + 1) = (7 × 5) + (7 × 1) = × 5 = (7 + 1) × 5 = (7 × 5) + (1 × 5) =

Choose which of the original equations you want to work with. Then do one of these… Draw a picture for the original equation; then change it just enough to match the new equations. Make an array for the original equation; then change it just enough to match the new equations. Write a story for the original equation; then change it just enough to match the new equations. Example: Original equation 7 × 5 = 35 New equations 7 × 6 = 42 8 × 5 = 40

Video 5 A pair of boys present their story problem for 7 x 5 and show what happens when the story changes to 8 x 5 and to 6 x 7.

There are 7 groups of 5 fish living in the store. 7 × 5 = 35

One more group of fish came. 8 × 5 = 40

There are 7 groups of 5 fish living in the store. 7 × 5 = 35

All of the groups of fish got one more fish. 7 × 6 = 42

Video 6 Alice asks how the situation, where they add 1 to a factor, is different from when they were adding 1 to an addend. A girl explains that when they were adding 1 to an addend, they just added 1 to a stick of cubes. When they add 1 to an factor, they have to add a group or they add 1 to each group.

How different do these situations seem? We were talking about the addends changing by 1 and what happens to the sum. Now we’re talking about the factors changing by 1 and what happens to the product.

36 × 17 How would you perform this calculation without the standard algorithm?

Video 7 Duane explains to two classmates his strategy for multiplying 36 x 17.

36 × × × × 17

36 × × × × 17 (6 × 10) + (6 × 7)

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. 8. Look for and express regularity in repeated reasoning. Mathematical Practice Standards

Thomas’s strategy 36 × 17 (36 + 4) × (17 + 3)

Thomas’s strategy 36 × 17 (36 + 4) × (17 + 3) 40 × 20 = 800

Thomas’s strategy 36 × 17 (36 + 4) × (17 + 3) 40 × 20 = – 4 = – 3 = 793

If you have any addend and you change that by any number, the sum will change as much as it changed.

If you have any addend and you change that by any number, the sum will change as much as it changed. (36 + 4) + (17 + 3) = 60

If you have any addend and you change that by any number, the sum will change as much as it changed. (36 + 4) + (17 + 3) = = 60 – 4 – = 53

Video 8 Thomas presents his strategy to the class and shows how he came up with 793. Liz tells the class that when Thomas presented his strategy to the small group, she liked his strategy, but someone else in the group was saying it was wrong and he got another answer. Then she tells students she wants them to copy down Thomas’s strategy and for homework they should show how they could use Thomas’s strategy to come up with a different answer.

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. 8. Look for and express regularity in repeated reasoning. Mathematical Practice Standards

17 groups of 36 36

17 groups of

20 groups of

36 x

40 x

1. Content standards are not taught as discrete entities. 2. Enacting the practices isn’t business a usual. 3. Practices are enacted in the context of the content standards.

Professional Development Resources Developing Mathematical Ideas (each module has a casebook, facilitator’s guide, video, published by Pearson) ▫Building a System of Tens ▫Making Meaning for Operations ▫Reasoning Algebraically about Operations Connecting Arithmetic to Algebra (book and facilitator’s guide, published by Heinemann)