1. NEKSDC Thursday, October 11, 2012 Presenter: Elaine Watson, Ed.D.
Principal's Workshop: Looking for Evidence of the Standards for Mathematical Practice
NEKSDC
Thursday, October 11, 2012
Presenter: Elaine Watson, Ed.D.

2. Introductions Share What feeds your soul personally?
What is your professional role?
What feeds your soul professionally?

3. Volunteers for Breaks
I need volunteers to remind me when we need breaks!
Every 20 minutes, we need a 2-minute “movement break” to help our blood circulate to our brains.
Every hour we need a 5-minute bathroom break.

4. Formative Assessment
How familiar are you with the CCSS Standards for Mathematical Practice?

5. Goals for this Workshop
You will leave with a deeper understanding of:
The 8 Common Core Practice Standards
Recognizing the Practice Standards in action by the STUDENTS in a math class.
Recognizing TEACHER MOVES that elicit the Practice Standards being used by students.
The types of tasks that build students’
ability to “practice” the Practice Standards

6. CCSSM Equally Focuses on…
Standards for Mathematical Practice
Standards for Mathematical Content
Same for All Grade Levels
Specific to Grade Level

7. Hunt Video:

8. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Explain to self the meaning of a problem and look for entry points to a solution
Analyze givens, constraints, relationships and goals
Make conjectures about the form and meaning of the solution

9. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Plan a solution pathway rather than simply jump into a solution attempt
Consider analogous problems
Try special cases and simpler forms of original problem

10. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Monitor and evaluate their progress and change course if necessary…
“Does this approach make sense?”

11. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Persevere in Solving by:
Transforming algebraic expressions
Changing the viewing window on a graphing calculator
Moving between the multiple representations of:
Equations, verbal descriptions, tables, graphs, diagrams

12. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Check their answers
“Does this answer make sense?”
Does it include correct labels?
Are the magnitudes of the numbers in the solution in the general ballpark to make sense in the real world?

13. 1. Make Sense of Problems and Persevere in Solving
Mathematically proficient students:
Check their answers
Verify solution using a different method
Compare approach with others:
How does their approach compare with mine?
Similarities
Differences

14. 2. Reason Abstractly and Quantitatively
Mathematically proficient students:
Make sense of quantities and their relationships in a problem situation
Bring two complementary abilities to bear on problems involving quantitative relationships: The ability to…
decontextualize
to abstract a given situation, represent it symbolically, manipulate the symbols as if they have a life of their own
contextualize
to pause as needed during the symbolic manipulation in order to look back at the referent values in the problem

15. 2. Reason Abstractly and Quantitatively
Mathematically proficient students:
Reason Quantitatively, which entails habits of:
Creating a coherent representation of the problem at hand
considering the units involved
Attending to the meaning of quantities, not just how to compute them
Knowing and flexibly using different properties of operations and objects

16. 3.Construct viable arguments and critique the reasoning of others
Mathematically proficient students:
Understand and use…
stated assumptions,
definitions,
and previously established results…
when constructing arguments

17. 3.Construct viable arguments and critique the reasoning of others
Mathematically proficient students:
Understand and use…
stated assumptions,
definitions,
and previously established results…
when constructing arguments

18. 3.Construct viable arguments and critique the reasoning of others
10 minute video

19. 4. Model with Mathematics
Modeling is both a
K - 12 Practice Standard
and a
9 – 12 Content Standard.

20. 4. Model with Mathematics
Mathematically proficient students:
Use powerful tools for modeling:
Diagrams or graphs
Spreadsheets
Algebraic Equations

21. 4. Model with Mathematics
Mathematically proficient students:
Models we devise depend upon a number of factors:
How precise do we need to be?
What aspects do we most need to undertand, control, or optimize?
What resources of time and tools do we have?

22. 4. Model with Mathematics
Mathematically proficient students:
Models we devise are also constrained by:
Limitations of our mathematical, statistical, and technical skills
Limitations of our ability to recognize significant variables and relationships among them

23. Modeling Cycle
The word “modeling” in this context
is used as a verb that describes the process of transforming a real situation into an abstract mathematical model.

24. Modeling Cycle
Problem
Formulate
Validate
Report
Compute
Interpret

25. Modeling Cycle Problem Identify variables in the situation
Select those that represent essential features

26. Modeling Cycle Formulate
Select or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables

27. Modeling Cycle Compute
Analyze and perform operations on these relationships to draw conclusions

28. Modeling Cycle Interpret
Interpret the result of the mathematics in terms of the original situation

29. Validate the conclusions by comparing them with the situation…
Modeling Cycle
Validate
Validate the conclusions by comparing them with the situation…

30. Report on conclusions and reasoning behind them
Modeling Cycle
Validate
Re - Formulate
Report on conclusions and reasoning behind them

31. Modeling Cycle
Problem
Formulate
Validate
Report
Compute
Interpret

32. 6. Attend to precision Use clear definitions
Mathematically proficient students:
Try to communicate precisely to others:
Use clear definitions
State the meaning of symbols they use
Use the equal sign consistently and appropriately
Specify units of measure
Label axes

33. 6. Attend to precision Calculate accurately and efficiently
Mathematically proficient students:
Try to communicate precisely to others
Calculate accurately and efficiently
Express numerical answers with a degree of precision appropriate for the problem context
Give carefully formulated explanations to each other
Can examine claims and make explicit use of definitions

34. What SMP’s do you see?
Even or Odd Video

35. 7. Look for and make use of structure
Mathematically proficient students:
Look closely to discern a pattern or structure
In x2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as 2 + 7
Can see complicated algebraic expressions as being composed of several objects:
5 – 3 (x – y)2 is seen as 5 minus a positive number
times a square, so its value can’t be more than 5 for
any real numbers x and y

36. 8. Look for and express regularity in repeated reasoning.
Mathematically proficient students:
Notice if calculations are repeated
Look for both general methods and for shortcuts
Maintain oversight of the process while attending to the details.

37. What SMPs Do You Observe Maya Practicing?
See Maya Video

38. Let’s Practice Some Modeling
Students can:
start with a model and interpret what it means in real world terms OR
start with a real world problem and create a mathematical model in order to solve it.

39. Possible or Not?
Here is an example of a task where students look at mathematical models (graphs of functions) and determine whether they make sense in a real world situation.

40. Possible or Not?

41. Questions:
Mr. Hedman is going to show you several graphs. For each graph, please answer the following:
A. Is this graph possible or not possible?
B. If it is impossible, is there a way to modify it to make it possible?
C. All graphs can tell a story, create a story for each graph.

42. One
A. Possible or not?
B. How would you
modify it?
C. Create a story.

43. Two
A. Possible or not?
B. How would you
modify it?
C. Create a story.

44. Three A. Possible or not? B. How would you modify it?
C. Create a story.

45. Four A. Possible or not? B. How would you modify it?
C. Create a story.

46. Five A. Possible or not? B. How would you modify it?
C. Create a story.

47. Six
A. Possible or not?
B. How would you
modify it?
C. Create a story.

48. Seven A. Possible or not? B. How would you modify it?
C. Create a story.

49. Eight A. Possible or not? B. How would you modify it?
C. Create a story.

50. Nine A. Possible or not? B. How would you modify it?
C. Create a story.

51. Ten
A. Possible or not?
B. How would you
modify it?
C. Create a story.

52. All 10 Graphs
What do all of the possible graphs have in common?

53. And now... For some brief notes on functions!!!!
Lesson borrowed and modified from Shodor.
Musical Notes borrowed from Abstract Art Pictures Collection.

54. Pyramid of Pennies
Here is an example of a task where students look at a real world problem, create a question, and create a mathematical model that will solve the problem.

55. Dan Meyer’s 3-Act Process
Act I
Show an image or short video of a real world situation in which a question can be generated that can be solved by creating a mathematical model.

57. Dan Meyer’s 3-Act Process
Act I (continued)
1. How many pennies are there?
2. Guess as close as you can.
3. Give an answer you know is too high.
4. Give an answer you know is too low.

58. Dan Meyer’s 3-Act Process
Students determine the information they need to solve the problem.
The teacher gives only the information students ask for.

59. Dan Meyer’s 3-Act Process
What information do you need to solve this problem?

62. Dan Meyer’s 3-Act Process
Act 2 continued
Students collaborate with each other to create a mathematical model and solve the problem.
Students may need find text or online resources such as formulas.

63. Dan Meyer’s 3-Act Process
Go to it!

64. Dan Meyer’s 3-Act Process
The answer is revealed.

65. Dan Meyer’s 3-Act Process

66. Standards for Mathematical Practice
Describe ways in which
student practitioners
of the discipline of mathematics
increasingly ought to engage
with the subject matter
as they grow in mathematical maturity

67. Standards for Mathematical Practice
Provide a balanced combination of
Procedure and Understanding
They shift the focus to ensure
mathematical understanding
over
computation skills

68. Inside Mathematics Videos

69. Standards for Mathematical Practice
Students will be able to:
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.

70. Think back to the Pyramid of Pennies.
At what point during the problem did you do the following?
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.

71. Resources for Rich Mathematical Tasks
The “Go-To” site for looking at the Content Standards and finding rich tasks, called “Illustrations” that can be used to build student understanding of a particular Content Standard.

72. Resources for Rich Mathematical Tasks
is a website with a plethora of  resources  to help teachers transition to teaching in a way that reflects the Standards for Mathematical Practice.
It’s worth taking the 6:19 minutes to watch the Video Overview  of the Video Tours to familiarize yourself to all of the resources.  There are more video tours that can be accessed by clicking on a link below the overview video.

73. Resources for Rich Mathematical Tasks
There are several names that are associated with the website: MARS, MAPS, The Shell Center…however the tasks are usually referred to as The MARS Tasks. The link above will show tasks aligned with the Practice Standards
They have been developed through a partnership with
UC Berkeley and the University of Nottingham

74. Resources for Rich Mathematical Tasks
Tools for the Common Core is the website of Bill McCallum, one of the three principle writers of the CCSSM.
Highlights of this site are the links (under Tools) to the Illustrative Mathematics Project, the Progressions Documents, and the Clickable Map of the CCSSM.

75. Resources www.watsonmath.com
Northeast Kingdom School Development Center Standards of Mathematical Practice Workshop and Dinner