the ccssm, math practice standards, and the shifts in classroom practices

THE CCSSM, MATH PRACTICE

STANDARDS, AND THE SHIFTS IN CLASSROOM PRACTICES

Presented by Taryn DiSorbo and Kim Vesper

POSITION Developing students’ capacity to engage in

the mathematical practices specified in the Common Core State Standards will ONLY be accomplished by engaging students in solving challenging mathematical tasks, providing students with tools to support their thinking and reasoning, and orchestrating opportunities for students to talk about mathematics and make their thinking public. It is the combination of these three dimensions of classrooms, working in unison, that develop students habits of mind and promote understanding of mathematics.

TASKS, TOOLS, AND TALK FRAMEWORK

The tasks or activities in which students engage should provide opportunities for them to “figure things out for themselves.” (NCTM, 2009, pg. 11), and to justify and communicate the outcome of their investigation;

TASKS, TOOLS, AND TALK FRAMEWORK

Tools (i.e. language, materials, and symbols) should be available to provide external support for learning. (Hiebert, et al, 1997); and

Productive classroom talk should make students’ thinking and reasoning public so that it can be refined and/or extended. (Chapin, O’Conner, & Anderson, 2009).

CHARACTERISTICS OF TASKS ALIGNED WITH SMP Significant Content (i.e. they have the potential to

leave behind important residue)(Hiebert et al, 1997). High Cognitive Demand (Stein et. al, 1996; Boaler &

Staples 2008) Multiple ways to enter the task and show competence

(Lotan, 2003) Require justification or explanation (Boaler & Staples

2008) Make connections between different representations

(Lesh, Post, & Behr, 1988) Provide a context for sense making (Van De Walle,

Karp, & Bay-Williams, 2013) Provide opportunities to look for patterns, make

conjectures, and form generalizations. (Stylianides, 2008; 2010)

RELATIONSHIP BETWEEN TASKS, TOOLS AND TALK AND THE STANDARDS FOR MATHEMATICAL PRACTICES

Practice Task Features Tools Talk

1 Make sense of problems and persevere in solving them

• High Cognitive Demand

• Multiple Entry Points

• Requires Explanation

2 Reason Abstractly and quantitatively

Contextual

3 Construct Viable arguments and critique the reasoning of others

Requires justification or proof

4 Model with mathematics Contextual

5 Use appropriate tools strategically

6 Attend to precision

7 Look for and make use of structure

Opportunity to look for patterns and make conjectures

8 Look for and express regularity in repeated reasoning

Opportunity to make generalizations

Teacher Actions







Make resources available that will support entry and engagement


Contextual Encourage use of different representational forms





Can be solved using different tools

Make tools available that will support entry and engagement






Teacher Actions

TALK Students must talk, with one another as

well as in response to the teacher. When the teacher talks most, the flow of ideas and knowledge is primarily from teacher to student. When students make public conjectures and reason with others about mathematics, ides, and knowledge are developed collaboratively, revealing mathematics as constructed by human beings within an intellectual community.

NCTM, 1991, p. 34





• Multiple Entry Points• Requires Explanation


• Ask students to explain their thinking

• Ask questions that assess and advance student understanding


Contextual











Teacher Actions








• Ask students to explain their thinking

• Ask questions that assess and advance student understanding


Contextual Encourage use of different representational forms

Prompt students to make connections between symbols and what they represent in context



Ask students to argue for their point of view and evaluate and make sense of the reasoning of their peers

4 Model with mathematics Contextual Ask students to justify their models



Make tools available that will support entry and engagement

6 Attend to precision Encourage students to be clear and use appropriate terminology



Encourage students to look for patterns



Prompt students to consider the connections between the current task and prior tasks

Teacher Actions

OVERALL No matter where you live-Pennsylvania,

California, Virginia, or Canada-students need opportunities to engage in the habits of practice embodied in the Standards for Mathematical Practices in CCSSM

The features of the tasks that you select for students to work on set the parameters for opportunities they have to engage in these practices-high-level tasks are necessary but not sufficient conditions.

The way in which you support students-the questions you ask and the tools you provide-help good tasks live up to their potential.

DEFINITION OF RIGOR According to Webster, RIGOR is strict

precision or exactness. According to mathematicians RIGOR is

having theorems that follow from axioms by means of systematic reasoning.

According to most people RIGOR means “too difficult” and “only limited access is possible.”

Also they believe RIGOR is an excuse to avoid high quality math teaching and learning

IN SCHOOLS… Rigor is

teaching and learning that is active, deep, and engaging.

ACTIVE

DEEPENGAGING

Active learning involves conversation and hands-on/minds-on activities. For example, questioning and discovery learning goes on.

ACTIVE

DEEPENGAGING

Deep learning is focused, attention given to details and explanations, via problem solving or projects. Students concentrate on the intricacies of a skill, concept, or activity.

ACTIVE

DEEPENGAGING

When learning is engaging, students make a real connection with the content. There is a feeling that, although learning may be challenging, it is satisfying.

ACTIVE

DEEPENGAGING

Rigor is a process—not a problem

ACTIVE

DEEPENGAGING

USING THE NCTM PRACTICE STANDARDS AND THE CCSSM

HOW DO WE GET STUDENTS ACTIVE

AND ENGAGED, THINKING DEEPLY

ABOUT MATH?

THIS IS WHAT RIGOR LOOKS LIKE

1. Name the polygon2. Describe the polygon

using the following terms: congruent, parallel, perpendicular, angle, measure, base, height, sides.

3. Label the vertices using the letters A-F

4. Describe the relationship between and

5. Identify congruent sides using appropriate notation.

6. For each angle, provide an estimate, with justification, of its measure.

A B

C

DE

F


7. Is this a regular or irregular polygon? Write a descriptive paragraph to support your answer. Include diagram.

8. Explain a method you would use to find the perimeter of the polygon.

9. Using a ruler, determine the perimeter to the nearest centimeter.

10. Describe a method to describe the area. Label your steps in sequential order. Use pictures to describe your steps if you want.

A B

C

DE

F


11. Formulate an expression that represents the area of the polygon.

12. Implement your method to solve for the area.

13. If the lengths of the sides were doubled predict how that would affect the perimeter of the figure.

14. If the lengths of the sides were doubled predict how that would affect the area of the figure.

15. If the measures of some angles increased, how would the lengths of the sides change? Justify your answer.

A B

C

DE

F


16. Measure each angle and find the sum of the angle measures. Compare the sum of the angle measures to the sum of the angle measures in a triangle, a quadrilateral, and a pentagon. What pattern do you notice?

17. If the polygon were the base of a 3 dimensional figure, what type of figure could it be? Explain your answer.

18. If the polygon is the base of a hexagonal prism, what would its sides look like?

A B

C

DE

F


19. How many faces, vertices, and edges would the hexagonal prism have?

20. Explain how you could determine the volume of the hexagonal prism. Compare your method to a classmate’s. How are the two methods alike? How are the two methods different?

21. How many lines of symmetry can you draw in the polygon?

22. Name a line segment that shows a line of symmetry.

23. Use mathematical notation to identify parallel sides.

A B

C

DE

F


24. Draw the polygon in Quadrant I of a coordinate plane.

25. Identify the coordinate pairs of each vertex of the polygon.

26. If you translated the polygon 2 units to the right and 3 units down, what would the new coordinate pairs be for each vertex?

27. If you rotate the polygon 90o, in which quadrant would it be located?

28. Draw a 90o rotation.

A B

C

DE

F


29. Reflect the original polygon in Quadrant I over the x-axis. Identify the coordinate pairs of the image polygon.

30. What type of transformation would have occurred if the image of the original polygon in Quadrant I were in Quadrant 3? Illustrate your answer.

31. If the original polygon in Quadrant I were dilated by a scale factor of ½, what would the coordinate pairs of the new polygon be?

32. Draw a similar figure and write a proportion that shows the scale factor.

A B

C

DE

F

Overwhelming evidence suggests that we have greatly

underestimated human ability by holding expectations that are too low for too many children, and by holding differential expectations where such differentiation is not

necessary.

TEACHING MATHEMATICAL PRACTICESPROBLEM…

Say “Hi” to everyone at your table and shake hands.How many handshakes were

exchanged at your table?If we exchange handshakes for the

entire room, how many handshakes would there be?

PROBLEM SOLVING 25 PEOPLE SHAKING HANDS

We could actually have the 25 people perform the activity of shaking each other’s hands

and keep track!

PROBLEM SOLVING 25 PEOPLE SHAKING HANDS

Number of Students, n

Number of handshakes

, Hn

2 13 34 65 106 157 218 289 3610 45. .. .. .

H4=H3+3=3+3=6

H6=H5+5=10+5=15

Hn=Hn-1+(n+1)

PROBLEM SOLVING 1st student shakes hands with 24 other

students. 2nd student shakes hands with 23 other

students. 3rd student shakes hands with 22 other

students. And so on…

Therefore, the total number of handshakes will be…

24+23+22+…+3+2+1=300

PROBLEM SOLVING A

Geometric/spatial representation of the problem

4+3+2+1=10

Triangular numbers can beRepresented by dots arrangedIn a triangle

Triangular number

1 2 3 4 …

Number of dots

1 3 6 10 …

LESSON PLANNINGCreate instructional strategies that will address:

1. common misconceptions2. errors3. differentiation of instruction4. student engagement5. reflection opportunities6. mathematical communication7. vocubalury8. multiple representations of

mathematical concepts.PRO EQUITY MODEL

SOME USEFUL ALGEBRAIC PROPERTIES1. Identity Property of Addition (0+n=n+0=n)2. Zero Property of Multiplication3. Identity Property of Multiplication (1xn=nx1=n)4. Golden Rule of Equations5. The Distributive Property of Multiplication over

Addition6. Commutative Property of Addition and

Multiplication7. Associative Property of Addition and

Multiplication.8. Isolate the Unknown

Solve: x+4=16

x+4=16 -4 -4

x+0=12 x=12

TOOLSSTATEMENTS1. Solve: x+4=162. x+4-4=16-43. x+0=124. x=12

REASONS1. Given2. Golden Rule of Eq3. Simplify (CLT)4. Identity

property of addition

TOOLSSTATEMENTS1. Solve: 3x=152. 3x÷3=15÷33. 1x=54. x=5

REASONS1. Given2. Golden Rule of Eq3. Simplify

(divide)4. Identity

property of multiplication

TOOLSSTATEMENTS1. Solve: 3x+4=162. 3x+4-4=16-43. 3x+0=124. 3x=12

5. 3x÷3=12÷36. 1x=47. x=4

REASONS1. Given2. Golden Rule of Eq3. Simplify (CLT)4. Identity property

of addition5. Golden Rule of Eq6. Simplify (divide)7. Identity property

of Multiplication

TOOLSSTATEMENTS1. Solve: 8(x-2)=3x+42. 8x-16=3x+43. 8x+16-16=3x+4+164. 8x+0=3x+205. 8x=3x+20

6. 8x-3x=3x+20-3x7. 5x=20+08. 5x=20

9. 5x÷5=20÷510. 1x=411. X=4

REASONS1. Given2. Simplify (distribute)3. Golden Rule of Eq4. Simplify (CLT)5. Identity Property of

addition6. Golden Rule of Eq7. Simplify (CLT)8. Identity Property of

addition9. Golden Rule of Eq10. Simplify (divide)11. Identity Property of

Multiplication

CREDIT CONTACT US

Lee V. Stiff, North Carolina State University and EDSTAR Analytics, Inc.

http://www.nctm.org/uploadedFiles/Professional_Development/Institutes/High_School_Institute/2013/Workshop_Materials/NCTM_HS_Institute_Presentation_DC_August2013_PDFVersion.pdf

Peg Smith (2013, August). Tasks, Tools, and Talk: A Framework for Enacting Mathematical Practices.

http://www.nctm.org/uploadedFiles/Professional_Development/Institutes/High_School_Institute/2013/Workshop_Materials/Smith-Tasks,%20Tools,%20Talk.pdf

Taryn DiSorbo

[email protected]

Kim Vesper

[email protected]

the ccssm, math practice standards, and the shifts in classroom practices

Documents

engaging students

challenging mathematical

frameworkthe tasks

characteristics of tasks

investigation tasks

students habits of mind

reasoning public

andproductive classroom