Math Instruction and the Common Core--How to help your students Master the Content By Clare Wurm

2. Math Instruction and the
Common Core:
How to Help Your Students
Master the Content
Clare Wurm, Consultant
State Education Resource Center

3. Agenda:
o Review Common Core State
Standards for Mathematics
o Learn effective instructional strategies
o Practice strategies
o Questions

4. Three Minute Video
Overview of CCSS

5. CCSS Math: 3 Instructional Shifts
• Focus strongly where the standards
focus
• Coherence: Think across grades and
link to major topics within grades
• Rigor: In major topics, pursue
conceptual understanding, procedural
skill and fluency, and application

6. Focusing attention within Number and Operations
Operations and Algebraic
Thinking
Expressions
and
Equations
Algebra
 
Number and Operations—
Base Ten

The Number
System

Number and
Operations—
Fractions

K 1 2 3 4 5 6 7 8 High School
6

7. Grade
Focus Areas in Support of Rich Instruction and
Expectations of Fluency and Conceptual Understanding
K–2
Addition and subtraction, measurement using
whole number quantities
3–5
Multiplication and division of whole numbers
and fractions
6
Ratios and proportional reasoning; early
expressions and equations
7
Ratios and proportional reasoning; arithmetic
of rational numbers
8 Linear algebra and linear functions
Priorities in Mathematics
7

8. CCSS Math: 3 Instructional Shifts
• Focus strongly where the standards
focus
• Coherence: Think across grades and
link to major topics within grades
• Rigor: In major topics, pursue
conceptual understanding, procedural
skill and fluency, and application

9. CCSS Math: 3 Instructional Shifts
• Focus strongly where the standards
focus
• Coherence: Think across grades and
link to major topics within grades
• Rigor: In major topics, pursue
conceptual understanding, procedural
skill and fluency, and application

10. Required Fluencies in K-6
Grade Standard Required Fluency
K K.OA.5 Add/subtract within 5
1 1.OA.6 Add/subtract within 10
2
2.OA.2
2.NBT.5
Add/subtract within 20 (know single-digit sums from memory)
Add/subtract within 100
3
3.OA.7
3.NBT.2
Multiply/divide within 100 (know single-digit products from
memory)
Add/subtract within 1000
4 4.NBT.4 Add/subtract within 1,000,000
5 5.NBT.5 Multi-digit multiplication
6 6.NS.2,3
Multi-digit division
Multi-digit decimal operations
10

11. CCSS Math Organization
Two distinct sets of standards:
 Content Standards (math
concepts)
GRADE or COURSE SPECIFIC
 Practice Standards (habits of mind)
APPLY to K-12

12. CCSS Math 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.

13. How do the CCSS change instruction?
• Three Aspects of Rigor
• Mathematical Practices
• Content Standards

14. How do the CCSS change instruction?
• Three Aspects of Rigor
• Mathematical Practices
• Content Standards

15. CCSS Math Rigor
ConceptualUnderstanding

16. How do the CCSS change instruction?
• Three Aspects of Rigor
• Mathematical Practices
• Content Standards

17. CCSS Math 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.

18. How do the CCSS change instruction?
• Three Aspects of Rigor
• Mathematical Practices
• Content Standards

19. How do the CCSS change instruction?
5.G.4 Classify two-dimensional figures in a hierarchy based
on properties.
Rather than: Define the following:
SQUARE
RECTANGLE
POLYGON
RHOMBUS
QUADRILATERAL

20. How do the CCSS change instruction?
POLYGON
QUADRILATERAL
RECTANGLE RHOMBUS

21. CCSS Considerations for Students
Receiving Special Education Services
 Standards are for ALL students.
 Standards are intended to make all
students college and career-ready.
 IEP goals should align to grade-
appropriate standards

22. Level of Support
• General Education “as designed”
• Supplemental Instruction
•Direct instruction
•Address gaps
• With Accommodations (How)
•Changes in instruction and materials
•Create access (remove barriers)
• With Modifications (What)
•Change in curriculum (amount or performance)

23. Common Core
QUESTIONS?
QUESTIONS?

24. CCSS Math Instruction
1. Ask student to explain thinking.
2. Break problem down into discrete tasks.
3. Use a variety of models and manipulatives.
4. Use scaffolded instruction.

25. Brainteaser!
How many different rectangles are in this figure?

26. 1. Ask student to explain thinking.
 clarifies vocabulary
 allows speaker to hear and clarify his/her
thoughts
 allows others to hear different solution and
compare to their own
 informs teacher about student’s understanding

27. 1. Ask student to explain thinking.
o Discourse facilitates process that
deepens learning. (Barnes, 1993;
Cazden; 2001)
o Constructivists believe higher-level
thinking is enhanced through
discourse with peers and teacher.
Vygotsky (1934, 1978); (Almasi, 2002)

28. 1. Ask student to explain thinking.
1. Ask student to explain thinking.
Usually not
enough

29. 1. Ask student to explain thinking.
Keep asking! Students need practice articulating ideas.
How did you start?
What do you need to find out?
Does your method always work?
How is your strategy different than _______?
(book’s, classmate’s, teacher’s)
Convince me that you’re right.

30. 2. Break problem down into
discrete tasks.
AKA: Task Analysis
Each discrete task can be taught separately.

31. 2. Break problem down into
discrete tasks.
“Task analysis can be used effectively with all
children, regardless of cognitive level and/or
expressive communicative abilities.”
“This evidence-based practice can be used for any
skill that can be broken down into smaller steps,
including academics, behaviors, communication, and
social skills.”
Franzone, E. (2009b). Overview of task analysis.

32. 2. Break problem down into
discrete tasks.
Example: Adding fractions

33. Example: Adding fractions

36. 2. Break problem down into
discrete tasks.
Example:
7.RP.2 Recognize and represent proportional relationships between
quantities.
b. Identify the constant of proportionality (unit rate) in tables, graphs,
equations, diagrams and verbal descriptions of proportional
relationships. Represent proportional relationships by equations.

37. A student is making trail mix. Create a graph to determine if the
quantities of nuts and fruit are proportional for each serving
size listed in the table. If the quantities are proportional, what is
the constant of proportionality or unit rate that defines the
relationship? Explain how the constant of proportionality was
determined and how it relates to both the table and graph.

38. Tasks involved:
Read and interpret table.
Create a graph from data in table.
Determine if quantities are proportional.
Find unit rate or constant of proportionality.
Explain how you found it.
A student is making trail mix. Create a graph to determine if the
quantities of nuts and fruit are proportional for each serving size
listed in the table. If the quantities are proportional, what is the
constant of proportionality or unit rate that defines the
relationship? Explain how the constant of proportionality was
determined and how it relates to both the table and graph.

39. Read and interpret table.
Create a graph from data in
table.
Determine if quantities are
proportional.
Find unit rate or constant of
proportionality.
Explain how you found it.

40. 3. Use a variety of models and
manipulatives.
Students think and learn differently so
diverse models will “click” for different
students.
Seeing a concept in different forms helps
students make connections among math
concepts.

41. 3. Use a variety of models and
manipulatives.
4. Use a variety of models and manipulatives.
Concrete Representational Abstract

42. 3. Use a variety of models and
manipulatives.
Concrete = manipulatives
Representational = drawing, diagram
Abstract = mathematical symbols

43. 3. Use a variety of models and
manipulatives.
3.MD.7.a:
Relate area to the operations of
multiplication and division.
5 x 3

44. 3. Use a variety of models and
manipulatives.
5 x 3
Concrete:

45. 3. Use a variety of models and
manipulatives.
5 x 3
Concrete:

46. 5 x 3
Representational:

47. 3. Use a variety of models and
manipulatives.
5 x 3
:
http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?from=category_g_1_t_1.html

48. 5 x 3
Representational/
Abstract:

49. 3. Use a variety of models and
manipulatives.

50. 3. Use a variety of models and
manipulatives.

51. 3. Use a variety of models and
manipulatives.

52. 3. Use a variety of models and
manipulatives.
A.CED.2
Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.

53. 3. Use a variety of models and
manipulatives.
A.CED.2
Create equations in two or more variables to represent
relationships between quantities; graph equations on
coordinate axes with labels and scales.
Seniors are selling candygrams for $3 each. The
principal donates $5 to start off their fundraising.
Write an equation to show how much money they
will have(y) after selling x candygrams.

54. 3. Use a variety of models and
manipulatives.
X =
candygrams
Y=
Money raised
0 5
1 8
5 20
10 35
25 80
50
155 0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60
Moneyraised
Candygrams sold
Seniors' Fundraiser

55. 3. Use a variety of models and
manipulatives.
Seniors are selling candygrams for $3 each. The principal
donates $5 to start off their fundraising. Write an equation to
show how much money they will have(y) after selling x
candygrams.

56. 3. Use a variety of models and
manipulatives.
Seniors are selling candygrams for $3 each. The principal
donates $5 to start off their fundraising. Write an equation to
show how much money they will have(y) after selling x
candygrams.

57. 4. Scaffold Instruction
Instructional Scaffolding…
..is support that teachers provide to
facilitate students’ development of
math proficiency.
…is essential for teaching students with
or at risk for mathematics disabilities.
Doabler, et al. Enhancing Core Mathematics Instruction for Students at Risk for Mathematics
Disabilities

58. 4. Scaffold Instruction
As students become more
independent in their learning,
the scaffolding is gradually
withdrawn.
Doabler, et al. Enhancing Core Mathematics Instruction for Students at Risk for Mathematics
Disabilities

59. 4. Scaffold Instruction
Gradual Release
I do.
We do.
You do.

60. Gradual Release
Teacher or
Para
Role I do it. We do it.
You do it
together. You do it alone.
Examples • Think-
Alouds
•Modeling
•Demon-
stration
•Coaching
•Guided practice
•Small group
• Partner work
• Group work
• Prompting
• Independent
work
Watch it Practice it Try it
Student Does,
Teacher Watches
Teacher Does,
Student Watches
Lots of Guided
Practice
60

61. 4. Scaffold Instruction
Scaffolded instruction will
incorporate some of the other
instructional techniques that we
have discussed.

62. Brainteaser!
How many different rectangles are in this figure?

63. How many different rectangles
are in this figure?
• Think aloud
• Create a table
• List some sizes; ask student(s)
to find other sizes
• Make multiple copies to
color/highlight possibilities

64. 4. Scaffold Instruction
A student is making trail mix. Create a graph to determine if the
quantities of nuts and fruit are proportional for each serving
size listed in the table. If the quantities are proportional, what is
the constant of proportionality or unit rate that defines the
relationship? Explain how the constant of proportionality was
determined and how it relates to both the table and graph.

65. 4. Scaffold Instruction
1.OA.a.1Use addition and subtraction within 20 to solve word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing,
with unknowns in all positions, e.g., by using objects, drawings, and equations with
a symbol for the unknown number to represent the problem.
TASK:
a. There are 8 children and 6 chairs. A child sits in each
chair. How many children won’t have a chair?
b. There are 8 children and some chairs. A child sits in
each chair. 2 children don’t have a chair. How many
chairs are there?

66. CCSS Math Instruction
1. Ask student to explain thinking.
2. Break problem down into discrete tasks.
3. Use a variety of models and
manipulatives.
4. Use scaffolded instruction.

67. Common Core
QUESTIONS?
QUESTIONS?

68. Practice
There are math problems posted around the room.
Choose one at your students’ level and try one (or more)
of the instructional strategies to plan instruction. Be ready
to share.
Share questions that you have on QUESTIONS board.
Peruse the Common Core State Standards for
Mathematics.

70. Bibliographical Notes
Doabler, et.al. Enhancing Core Mathematics Instruction for Students at Risk
for Mathematical Disabilities, Teaching Exceptional Children, Mar/Apr 2012.
Franzone, E. (2009b). Overview of task analysis. Madison, WI: National
Professional Development Center on Autism Spectrum Disorders, Waisman
Center, University of Wisconsin.
Wakeman, Karvonen & Ahumada, Changing Instruction to Increase
Achievement for Students With Moderate to Severe Intellectual Disabilities,
Teaching Exceptional Children, Nov/Dec 2013.

71. Math Resources
Achieve the Core -- CCSS basics, background, examples http://www.achievethecore.org/
Illustrative Mathematics -- Standards, progressions, problems and tasks
http://www.illustrativemathematics.org/
National Library of Virtual Manipulatives -- Java interactive math activities http://nlvm.usu.edu/
CCSS basics High School Mathemtics Instructional Toolkit (p. 5= math discussion tips)
http://celenza.wikispaces.com/file/view/Math_Instructional_Toolkit.pdf
IXL Learning -- Practice problems by standard http://www.ixl.com/math/
NCTM Illuminations http://illuminations.nctm.org/
Reproducible Resources (Problem-Solving Organizer) http://www.solution-tree.com/free-
resources/specialneeds

72. Clare Wurm, Consultant
tel: 860-632-1485, x 383
email: wurm@ctserc.org
SERC LIBRARY, Offers more than 10,000 resources www.ctserc.org/library
– Books
– Instructional materials
– Tests
– Journals
– Online databases
– DVDs, videos, CD-ROMs
– Professional development materials for staff
Contact Us or Visit the SERC Library