1. Standards of Mathematical Practice

3. Today’s Goals
To explore the mathematical standards for Content and Practice
To discuss and learn strategies for getting students engaged in the Standards of Mathematical Practice
To explore how we can use existing resources to focus on the Practices

4. Common Core State Standards
Mathematics
Standards for Content
Standards for Practice

5. Standards for 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 use of structure
Look for and express regularity in repeated reasoning
This is there the shifts really come in. Yes, students need to know and understand certain skill and concepts. But our classrooms must be structured to enable students to gradually develop those standards of mathematical practice which will allow them to apply those skill and concepts they have learned in college and their careers.
© Institute for Mathematics & Education 2011

6. SMP 1: Make sense of problems and persevere in solving them.
Gather Information
Make a plan
Anticipate possible solutions
Continuously evaluate progress
Check results
Question sense of solutions
Mathematically Proficient Students:
Explain the meaning of the problem to themselves
Look for entry points
Analyze givens, constraints, relationships, goals
Make conjectures about the solution
Plan a solution pathway
Consider analogous problems
Try special cases and similar forms
Monitor and evaluate progress, and change course if necessary
Check their answer to problems using a different method
Continually ask themselves “Does this make sense?”
© Institute for Mathematics & Education 2011

8. SMP 2: Reason abstractly and quantitatively
Decontextualize
Represent as symbols, abstract the situation
Contextualize
Pause as needed to refer back to situation 5
Mathematical Problem P
x x x x
© Institute for Mathematics & Education 2011

10. SMP 3: Construct viable arguments and critique the reasoning of others
Use assumptions, definitions, and previous results
Make a conjecture
Build a logical progression of statements to explore the conjecture
Analyze situations by breaking them into cases
Recognize and use counter examples
Distinguish correct logic
Communicate conclusions
Explain flaws
Justify conclusions
Ask clarifying questions
Respond to arguments
© Institute for Mathematics & Education 2011

12. SMP 4: Model with mathematics
Problems in everyday life…
…reasoned using mathematical methods
Mathematically proficient students
make assumptions and approximations to simplify a situation, realizing these may need revision later
interpret mathematical results in the context of the situation and reflect on whether they make sense
Images: asiabcs.com, ehow.com, judsonmagnet.org, life123.com, teamuptutors.com, enwikipedia.org, glennsasscer.com
© Institute for Mathematics & Education 2011

14. SMP 5: Use appropriate tools strategically
Proficient students
are sufficiently familiar with appropriate tools to decide when each tool is helpful, knowing both the benefit and limitations
detect possible errors
identify relevant external mathematical resources, and use them to pose or solve problems
© Institute for Mathematics & Education 2011

16. SMP 6: Attend to precision
Mathematically proficient students
communicate precisely to others
use clear definitions
state the meaning of the symbols they use
specify units of measurement
label the axes to clarify correspondence with problem
calculate accurately and efficiently
express numerical answers with an appropriate degree of precision
Comic:
© Institute for Mathematics & Education 2011

18. SMP 7: Look for and make use of structure
Mathematically proficient students
look closely to discern a pattern or structure
step back for an overview and shift perspective
see complicated things as single objects, or as composed of several objects
© Institute for Mathematics & Education 2011

20. SMP 8: Look for and express regularity in repeated reasoning
Mathematically proficient students
notice if calculations are repeated and look both for general methods and for shortcuts
maintain oversight of the process while attending to the details, as they work to solve a problem
continually evaluate the reasonableness of their intermediate results
© Institute for Mathematics & Education 2011

22. Over emphasis on computation obscures understanding of the operations.
Focus on behavior of the operations
supports computational fluency.
Deborah Schifter

23. Content Goals
These standards of practice will be examined in the context of the following content standard:
Understand properties of the operations.
Grade 1: p. 14 Grade 2: p. 18
Grade 3: p. 22 Grade 4: p. 28
Grade 5: p. 33 (paragraph #2)
“The practice standards, then, must be addressed in the context of work on deep mathematical content. Today we will focus on two of the practice standards in the context of students’ explorations of the properties of the operations.”
If participants are unsure of what the word “operations” means, clarify that it refers to addition, subtraction, multiplication, and division.
“The phrase ‘properties of the operations’ is strewn throughout the K-5 content standards. The document notes that students need not learn the formal words for these properties. Rather, the emphasis should be on students having opportunities to investigate how the operations behave and apply that understanding in their computation.”
Ask teachers of grades 1-5 to turn to the page for their grade to see references to the properties of the operations.

24. What generalization is suggested by these problems?
“In this session, we will examine a case from a third-grade classroom and a case from a second-grade classroom. Each case illustrates two of the mathematical practices—Practice 8: Look for and express regularity in repeated reasoning and Practice 3: Create viable arguments and critique the reasoning of others. In both classrooms, students explore properties of the operations.
“To start, consider these problems Ms. Kaye presented to her third graders. What do you notice about these pairs of problems? What generalization is suggested by them? State the generalization in common language.”
Participants talk in small groups for 2-5 minutes to write out a generalization. This will be followed by a brief discussion in which they share their statements.
If they choose, they may write the statement in algebraic notation after they write it in English.
When the group comes together, have some participants share their articulations. After some articulations are offered in common language, some participants might choose to state the generalization using algebraic notation. Algebraic notation should not become the focus of the session. Rather, point out that, for those who are ready to think about it, the notation is being used to express an idea. The emphasis should be on the idea.
“This is an example of what is meant by MP8: Look for and express regularity in repeated reasoning. You noticed regularity in these addition computations and expressed a generalization about what you noticed.”

25. Video 1 – Will it always work?
Ms. Kaye’s 3rd Grade
Norms for Watching Video
Video clips are examples to allow for discussion of teaching and learning, not for criticism or evaluation of the teacher. What is in the video is a very limited piece of the lesson, so be wary of assumptions we may make about what came before or after.
All comments should be made respectfully.
Always assume another’s comments are not intended to offend.
Show this video (about 50 seconds) of Ms. Kaye’s class’s statement of the generalization. Ask for comments about the students’ claim. “What is important about students making a claim that something will always work, that it will work for all numbers?”
“This is an illustration of Mathematical Practice 8: Look for and express regularity in repeated reasoning. Students noticed what was in common across all the problems—they noticed the same thing was happening in all of the problems—and they expressed the regularity.”
“Once the generalization has been articulated, there is still more work to do, as is demonstrated in the next clip.”
[The video on this slide is from the project, Using Routines as an Instructional Tool for Developing Students’ Conceptions of Proof, Susan Jo Russell, Deborah Schifter, and Virginia Bastable, Copyright © 2012, TERC. Used by permission. All Rights Reserved.]
CTB/McGraw-Hill; Mathematics Assessment Resource Services, 2003

26. Video 2 – Will it always work?
Show this clip (4.5 minutes), in which Ms. Kaye explains the next task to the class: to use a story context or cubes or a number line to explain why it always has to work.
After showing the clip, have participants paraphrase what the task is. Have them articulate what Vera’s question is and explain why it is an important question. “Notice that the class is being asked to construct an argument for their claim. This is Mathematical Practice 3.”
Then give participants 6-8 minutes to work in pairs to do the task. If participants have difficulty, ask them to come up with a representation for Then ask them to change the representation just enough to show What needs to change? What if you start with 9 + 4? What if you start with any two addends?
Spend 6-8 minutes sharing their representations. Show one cube representation, one number line representation, and one story context (if the group has come up with each of these). Ask, “How can we look at this representation to see that your claim works for any two addends?”
[The video on this slide is from the project, Using Routines as an Instructional Tool for Developing Students’ Conceptions of Proof, Susan Jo Russell, Deborah Schifter, and Virginia Bastable, Copyright © 2012, TERC. Used by permission. All Rights Reserved.]

27. Video 3 – Will it always work?
Show this video (3 minutes) which shows two students explaining their representation and making a generalization.
After the video, ask participants to demonstrate the students’ representations. “How do these students use their representations to create a general argument for their rule, an argument that doesn‘t depend on particular examples?”
“These kinds of representations—representations that show the action of the operations—are accessible to elementary students and allow them to make arguments for generalizations that apply to an infinite class of numbers.”
[The video on this slide is from the project, Using Routines as an Instructional Tool for Developing Students’ Conceptions of Proof, Susan Jo Russell, Deborah Schifter, and Virginia Bastable, Copyright © 2012, TERC. Used by permission. All Rights Reserved.]

28. What Standards for Mathematical Practice were focused on in this task?

29. Adding 1 to a Factor Writing prompt:
“Ms. Kaye gave her students the task of investigating adding 1 to an addend in order to prepare them for the next task, which takes them into the heart of third-grade content—understanding multiplication. She wants them to compare what happens with multiplication to what they found out about what happens with addition.
The next task is: What if 1 is added to a factor? Ms. Kaye posed this task to her students.” Read the prompt.
Have participants work for 5 minutes in small groups to respond to the prompt themselves. Then have them share how they articulated the generalization.
Writing prompt:
In a multiplication problem, if you add 1 to a factor, I think this will happen to the product…

30. Students’ Articulation 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.”
Here are some of the ways individual students in Ms. Kaye’s class responded to the prompt.

31. Sample task
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 x 5 = 35
New equations x 6 = 42
8 x 5 = 40
Have participants read the task. Give them minutes to work on the problem in pairs or in their small groups.
Have them share representations and arguments.

32. Frannie’s Story Context
There are 7 jewelry boxes and each box has 5 pieces of jewelry. There are 35 pieces of jewelry altogether.
“Now we turn to what the students did in Ms. Kaye’s class. After the students worked on the task, they came together to share their work. Ms. Kaye started the discussion with Frannie’s story context.”
READ STORY FROM SLIDE.
“The next slides lay out the ideas explicated in the discussion of Frannie’s story context. As we go through these slides, consider why Ms. Kaye might have chosen this story to begin the discussion with the whole group.”

33. Jewelry Boxes Seven boxes with five pieces of jewelry in each box
“Visualize what will happen with the boxes of jewelry if we change the expression to 8 x 5.”

34. Jewelry Boxes 7 x 5  8 x 5 Eight
Seven boxes with five pieces of jewelry in each box
35 pieces of jewelry
+ 5 pieces of jewelry
40 pieces of jewelry
“The students in Ms. Kaye’s class say that when the first factor is increased (to 8 boxes)—the product (number of pieces of jewelry) increases by the number in one box.”

35. Jewelry Boxes Seven boxes with five pieces of jewelry in each box
“Now visualize how the picture will change if we change the expression to 7 x “

36. Jewelry Boxes
7 x 5  7 x 6
six
Seven boxes with five pieces of jewelry in each box
35 pieces of jewelry
+ 7 pieces of jewelry
42 pieces of jewelry
“The students explained that when the second factor is increased (to 6 pieces in a box), the product increases by the number of boxes because there is one new piece of jewelry in each box.”
“Why do you think Ms. Kaye may have chosen to start the whole class discussion with this example?” Give 3-5 minutes to discuss this question with a partner.

37. Other Stories Baskets of bouncy balls Tanks with salmon eggs
Baskets of mozzarella sticks
Rows of chairs
“The lesson continued as they worked through all of the posters produced by the class. Students presented different story contexts.”

38. 9 x 4 = 36
9 x 5 = 45
10 x 4 = 40
“They looked at diagrams like this and discussed how the story contexts map onto it. In this diagram, they were working with different numbers.”
Have participants point out where in the diagram they see the change from 9 x 4 to 9 x 5 and from 9 x 4 to 10 x 4.

39. Making Sense of Multiplication
“And they looked at arrays.”
Have participants explain how the array changes from 7 x 5 to 8 x 5
and from 7 x 5 to 7 x 6.
“How could you use any of these representations—a story context, a diagram, or an array—to support the general claim, about any whole-number factors, not just specific numbers?”
Give participants 8-10 minutes to work on this question in small groups. Then discuss in whole group.
Explain how the array changes from 7 x 5 to 8 x 5 and from 7 x 5 to 7 x 6.

40. What Standards for Mathematical Practice were focused on in this task?

41. Strategies for Implementing the Practices

42. Use Rich Tasks

43. Where to find rich tasks.

44. Rich Problems Worksheet
Read through the tasks on the Rich Problems worksheet
With a partner discuss what similarities you see in these problems

45. Creating Open Ended Questions

47. Creating open ended questions
With a partner choose an upcoming topic you will be teaching.
Create 3 open ended questions to engage your students?
For each questions what Math Practices would you expect your students to be engaged in?

48. Additional Strategies

50. Where do I find these opportunities in my resources?

52. With a partner, pick an upcoming lesson and discuss how you are going to integrate the standards of Mathematical Practice.
Focus on 1 or 2 Practices
What will you see the students doing if they are engaging in the Practices?
How will you facilitate the Practices?