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2. Standards For Mathematical
Practice
Third Grade
November 10, 2011
Presenter: Turtle Gunn Toms

3. Welcome!
• Thank you for taking time out of your day to join this
discussion.
• You should end the session today with at least 3
takeaways- something you can do tomorrow, a list of
resources, something to think about.
• I need your feedback at the end of this session.
Feedback helps me become a better teacher, and
helps you to reflect on your learning. Please enter
feedback in the chat box once we are done.

5. Clearing up confusion:
• This webinar is not about CCGPS content, it is about using the
CCGPS Mathematical Practices this year with GPS content.
• For information about how and why CCSS were developed
and adopted, watch: common core- teaching channel and
this: common core- math- teaching channel
• GPS is taught and tested 2011-12. CCGPS is taught and tested
2012-13.
• I will provide a list of resources mentioned during the session
and one of the documents that has been downloaded to your
computer is also a list of resources and future GPB broadcast
dates.

6. Think for 30 seconds, then share-
What is learning?
What defines an effective classroom?
How do students become proficient in
mathematics?

7. Answers from classroom teachers
• Learning happens when a student can make
connections.
• Learning happens when a student can make
sense of mistakes.
• Learning happens when students can think
about their thinking.
• An effective classroom is a place where
students are doing the work.

8. Is my classroom effective?
• Learning happens when a student can make
connections.
• Learning happens when a student can make
sense of mistakes.
• Learning happens when students can think
about their thinking.
• An effective classroom is a place where
students are doing the work.

9. So what does the teacher do?
• Focus on more on learning, less on teaching.
• Ask questions related to the ideas the students are
constructing, questions that illuminate the learner’s
thinking.
• Provoke disequilibrium.
• Allow productive struggle.
• Think differently. Many of us have seen math as
something to be learned, practiced, and applied. Now it
is understood as interpreting, organizing, inquiring, and
constructing meaning using a mathematical lens.

10. Chew on this for a moment:
“Am I really interested in
getting to know what is in their
heads, or,
do I just want them to know
what is in my head?”
Ann Shannon, 2011

11. How do we create a classroom
environment which encourages students
to take responsibility for their learning
and allows them to become proficient in
mathematics?
What changes and what stays the same?

12. What needs to go away:
• Problem solving Friday
• Enrichment for the few
• Just giving the answer (teacher or student!)
• Isolation of content from process
• GPS-ing students (what does that mean?)

13. Starting now: we can begin using
Standards for Mathematical Practice
• “The Standards for Mathematical Practice describe varieties of
expertise that mathematics educators at all levels should seek to
develop in their students. These practices rest on important
‘processes and proficiencies’ with longstanding importance in
mathematics education.”
(CCSS, 2010)
• The mathematical practices require a "re-negotiation" of the
classroom contract.
• 3 Major Shifts:
– Teachers cannot create learning-only learners can do that.
– Increased student responsibility- from receptive to active learner
– Teacher/student relationship shift- from adversarial to collaborative
Black and Wiliam, 2006

14. What happens if we continue to
sacrifice understanding?

15. In every classroom, in every
mathematical situation:
• Students must mathematize their world.
• Students must take responsibility for learning.
• Mathematics must be made explicit.
Hmmm….What does it mean to make
mathematics explicit?

16. Making the mathematics explicit:
• Children create and use graphic depictions
receiving guidance and feedback from the
teacher.
• Learner’s reasoning is made as explicit as
possible to help students see what another is
thinking.
• Student sharing of ideas and strategies is
paramount.
• Teacher looks for significant ideas to highlight.
LouAnn Lovin, 2011

17. Mathematizing Third Grade
• To mathematize, one sees, organizes, and interprets the world
through and with mathematical models.
• The potential to model the problematic situation
must be built in.
• Problematic situations must allow students to realize
what they are doing. They must be able to “imagine
concretely”.
• Problematic situations must prompt learners to ask
questions, notice patterns, wonder, ask why, and ask
what if.
(Young Mathematicians at Work, Catherine Twomey Fosnot, 2001)

18. “Only if children come to believe that there are
always multiple ways to solve problems, and
that they, personally, are capable of
discovering some of these ways, will they be
likely to exercise- and thereby develop-
number sense.”
Laura Resnick, 1990

19. CCGPS Standards for Mathematical
Practice
These are the
backbone of
the practices.

20. What teachers do:
1. Make sense of problems and persevere in solving them.
• Provide time and facilitate discussion in problem solutions.
• Facilitate discourse in the classroom so that students UNDERSTAND the
approaches of others.
• Provide opportunities for students to explain themselves, the meaning of
a problem, etc.
• Provide opportunities for students to connect concepts to “their” world.
• Provide students TIME to think and become “patient” problem solvers.
• Facilitate and encourage students to check their answers using different
methods (not calculators).
• Provide problems that focus on relationships and are “generalizable”.
6. Attend to precision.
• Facilitate, encourage and expect precision in communication.
• Provide opportunities for students to explain and/or write their
reasoning to others.

21. 1. Make sense of problems and persevere in solving
them.
• In third grade, students know that doing mathematics involves solving
problems and discussing how they solved them. Students explain to
themselves the meaning of a problem and look for ways to solve it. Third
graders may use concrete objects or pictures to help them conceptualize
and solve problems. They may check their thinking by asking themselves,
“Does this make sense?” They listen to the strategies of others and will try
different approaches. They often will use another method to check their
answers.
6. Attend to precision.
• As third graders develop their mathematical communication skills, they
try to use clear and precise language in their discussions with others and
in their own reasoning. They are careful about specifying units of measure
and state the meaning of the symbols they choose. For instance, when
figuring out the area of a rectangle they record their answers in square
units.

22. 1. Make sense of problems and persevere in solving
them.
• In third grade, students know that doing mathematics involves solving
problems and discussing how they solved them. Students explain to
themselves the meaning of a problem and look for ways to solve it. Third
graders may use concrete objects or pictures to help them conceptualize
and solve problems. They may check their thinking by asking themselves,
“Does this make sense?” They listen to the strategies of others and will try
different approaches. They often will use another method to check their
answers.
6. Attend to precision.
• As third graders develop their mathematical communication skills, they
try to use clear and precise language in their discussions with others and
in their own reasoning. They are careful about specifying units of measure
and state the meaning of the symbols they choose. For instance, when
figuring out the area of a rectangle they record their answers in square
units.

23. • What can we do to make sure students make
sense of problems and persevere in solving
them?
• How can we ensure students attend to
precision?

24. Word Problems vs Problematic
Situations
• Word problems: Teachers assign them after
they have explained operations, algorithms, or
rules, and students are expected to apply
these procedures to the problems.
• Problematic situations: Used at the
beginning- for construction of understanding,
generation and exploration of mathematical
ideas and strategies, offering multiple entry
levels, and supportive of mathematization.
(Young Mathematicians at Work, Fosnot, 2002)

25. Problem #1
• $1.33
- .74 or
• Etc…

26. Problem #1
18x 4
4x18
Here are two multiplication problems which have the same answer.
Find some other multiplication problems which have the same answer.
Show how you know.
(Adapted from Young Mathematicians at Work, Fosnot, 2001)

27. Problem #1 (round 2)
About how many children Danger:
DO NOT EXCEED
of your age would be safe to 950 pounds.
take the elevator at one
time?
(Young Mathematicians at Work, Fosnot, 2001)

29. Math Anchor Charts!

30. Reasoning and explaining
1. Reason abstractly and quantitatively.
What teachers do:
• Provide a range of representations of math problem situations and encourage various
solutions.
• Provide opportunities for students to make sense of quantities and their relationships in
problem situations.
• Provide problems that require flexible use of properties of operations and objects.
• Emphasize quantitative reasoning 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 and/or rules; and knowing and flexibly using
different properties of operations and objects.
6. Construct viable arguments and critique the reasoning of others.
• Provide ALL students opportunities to understand and use stated assumptions, definitions,
and previously established results in constructing arguments.
• Provide ample time for students to make conjectures and build a logical progression of
statements to explore the truth of their conjectures.
• Provide opportunities for students to construct arguments and critique arguments of peers.
• Facilitate and guide students in recognizing and using counterexamples.
• Encourage and facilitate students justifying their conclusions, communicating, and
responding to the arguments of others.
• Ask useful questions to clarify and/or improve students’ arguments.

31. Reasoning and explaining
1. Reason abstractly and quantitatively.
• Third graders should recognize that a number represents a specific quantity.
They connect the quantity to written symbols and create a logical
representation of the problem at hand, considering both the appropriate units
involved and the meaning of quantities.
4. Construct viable arguments and critique the reasoning of others.
• In third grade, students may construct arguments using concrete referents,
such as objects, pictures, and drawings. They refine their mathematical
communication skills as they participate in mathematical discussions involving
questions like “How did you get that?” and “Why is that true?” They explain
their thinking to others and respond to others’ thinking.

32. Reasoning and explaining
1. Reason abstractly and quantitatively.
• Third graders should recognize that a number represents a specific quantity.
They connect the quantity to written symbols and create a logical
representation of the problem at hand, considering both the appropriate units
involved and the meaning of quantities.
4. Construct viable arguments and critique the reasoning of others.
• In third grade, students may construct arguments using concrete referents,
such as objects, pictures, and drawings. They refine their mathematical
communication skills as they participate in mathematical discussions involving
questions like “How did you get that?” and “Why is that true?” They explain
their thinking to others and respond to others’ thinking.

33. • How can we encourage students to reason
abstractly and quantitatively?
• How can we support students in explaining
their thinking and examining the thinking of
others ?

34. Problem #2
120+ 73 = ___
73+ 120 = ___
• Etc…
(and what about relational equality?)

35. Problem #2
Ms. Breedlove loves to read! She is always reading. Right now she
is reading a book about math and her friend Molly is reading
the same book. Can you help her solve these problems?
The book Ms. Breedlove and Molly are reading has 107 pages. Ms.
Breedlove is on page 64. How many more pages does she need
to read to finish reading the book?
Molly is reading the same book, but she is only on page 43. How
many more pages does Molly need to read to finish the book?
(Young Mathematicians at Work, Fosnot, 2001)

37. Modeling and using tools
(what teachers do)
1. Model with mathematics.
• Provide problem situations that apply to everyday life.
• Provide rich tasks that focus on conceptual understanding, relationships, etc.
5. Use appropriate tools strategically.
• Provide a variety of tools and technology for students to explore to deepen
their understanding of math concepts.
• Provide problem solving tasks that require students to consider a variety of
tools for solving. (Tools might include pencil/paper, concrete models, empty
number line, ruler, calculator, etc.)

38. Modeling and using tools
1. Model with mathematics.
• Students experiment with representing problem situations in multiple ways
including numbers, words (mathematical language), drawing pictures, using
objects, acting out, making a chart, list, or graph, creating equations, etc.
Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed. Third graders should evaluate their results in the
context of the situation and reflect on whether the results make sense.
3. Use appropriate tools strategically.
• Third graders consider the available tools (including estimation) when solving
a mathematical problem and decide when certain tools might be helpful. For
instance, they may use graph paper to find all the possible rectangles that
have a given perimeter. They compile the possibilities into an organized list or
a table, and determine whether they have all the possible rectangles.

39. Modeling and using tools
1. Model with mathematics.
• Students experiment with representing problem situations in multiple ways
including numbers, words (mathematical language), drawing pictures, using
objects, acting out, making a chart, list, or graph, creating equations, etc.
Students need opportunities to connect the different representations and
explain the connections. They should be able to use all of these
representations as needed. Third graders should evaluate their results in the
context of the situation and reflect on whether the results make sense.
3. Use appropriate tools strategically.
• Third graders consider the available tools (including estimation) when solving
a mathematical problem and decide when certain tools might be helpful. For
instance, they may use graph paper to find all the possible rectangles that
have a given perimeter. They compile the possibilities into an organized list or
a table, and determine whether they have all the possible rectangles.

40. Problem #3

41. Problem #3
Miss Guy has a very energetic puppy. The puppy loves to play
outdoors, so Miss Guy decided to build a pen to allow her pet
to be outside while she is at school. She just happens to have
50 feet of fencing in her basement that she can use for the pen.
What are some of the ways she can set up the pen that uses all
the fencing? Children must
figure out for
What are the dimensions of the rectangular pen with the most themselves, in
space available for the puppy to play? their own
ways.
Write a letter to Miss Guy explaining her choices and which pen
you would recommend she build. Be sure to show how you
made your decisions and include a mathematical
representation to support your solution.
(Exemplars, Miss Guy’s Puppy Problem)

42. (Exemplars, Miss Guy’s Puppy Problem)

43. (Exemplars, Miss Guy’s Puppy Problem)

44. What tools and situations can we
provide?
• http://www.youtube.com/watch?v=7AmOJGb8fiA&
• “To do 26 + 37, I will first add 4 to 26 to get 30. I still have 33 to add on. Next I will add 30 to get 60, and finally
add the remaining 3 to get the answer 63.”

45. Children must figure out for
themselves, in their own ways.
Mackenzie burns
220 calories a day
running home from
school. How many
calories will she
burn in 5 days?
Thank you, Krystal and Henry County!

46. Seeing structure and generalizing
(what teachers do)
1. Look for and make sense of structure.
• Provide opportunities and time for students to explore patterns and
relationships to solve problems.
• Provide rich tasks and facilitate pattern seeking and understanding of
relationships in numbers rather than following a set of steps and/or
procedures.
8. Look for and express regularity in repeated reasoning.
• Provide problem situations that allow students to explore regularity and
repeated reasoning.
• Provide rich tasks that encourage students to use repeated reasoning to
form generalizations and provide opportunities for students to
communicate these generalizations.

47. Seeing structure and generalizing
1. Look for and make sense of structure.
• In third grade, students look closely to discover a pattern or
structure. For instance, students use properties of operations as
strategies to multiply and divide (commutative and distributive
properties).
8. Look for and express regularity in repeated reasoning.
• Students in third grade should notice repetitive actions in computation and
look for more shortcut methods. For example, students may use the
distributive property as a strategy for using products they know to solve
products that they don’t know. For example, if students are asked to find the
product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x
8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually
evaluate their work by asking themselves, “Does this make sense?”

48. Seeing structure and generalizing
1. Look for and make sense of structure.
• In third grade, students look closely to discover a pattern or
structure. For instance, students use properties of operations as
strategies to multiply and divide (commutative and distributive
properties).
8. Look for and express regularity in repeated reasoning.
• Students in third grade should notice repetitive actions in computation and
look for more shortcut methods. For example, students may use the
distributive property as a strategy for using products they know to solve
products that they don’t know. For example, if students are asked to find the
product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x
8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually
evaluate their work by asking themselves, “Does this make sense?”

49. How is this different from what
we used to think?
What are patterns in mathematics?
How can make them explicit?
• Subitization
• Ten frames
• Number word patterns
• Doubling
• 0-99 chart
• Benchmark (friendly) numbers
• Spatial patterns
• Patterning is the search for regularity and structure

50. Problem #4
Continue the pattern:
1x12=12
2x6=12
3x4=__
Etc…
(and what about relational equality?)

53. Where can we start?
• GaDOE Teaching Guides
• http://public.doe.k12.ga.us/ci_services.aspx?PageReq=CIServMath
• Learning Village
• https://portal.doe.k12.ga.us/LearningVillageLogin.aspx
• List Serve
• join-mathematics-k-5@list.doe.k12.ga.us
• join-mathematics-6-8@list.doe.k12.ga.us
• join-mathematics-9-12@list.doe.k12.ga.us
• join-mathematics-districtsupport@list.doe.k12.ga.us
• join-mathematics-administrators@list.doe.k12.ga.us
• join-mathematics-resa@list.doe.k12.ga.us
• Inside Mathematics
• http://www.insidemathematics.org/
• Teaching Channel
• http://www.teachingchannel.org/videos?categories=topics_common-core
• Arizona
• http://www.ade.az.gov/standards/math/2010MathStandards/
• New York City
• http://schools.nyc.gov/Academics/CommonCoreLibrary/SeeStudentWork/default.htm
• North Carolina
• http://www.ncpublicschools.org/acre/standards/extended/
• Ohio
• http://www.ode.state.oh.us/GD/Templates/Pages/ODE/ODEPrimary.aspx?page=2&TopicRelationID=1704

54. Resource to find in your school:
• Teaching Student-Centered Mathematics,
Grades K-3
• Teaching Student-Centered Mathematics,
Grades 3-5
• Teaching Student-Centered Mathematics,
Grades 5-8
By John Van de Walle and LouAnn Lovin
Provided courtesy of GA Dept. of Education

55. Turtle’s Recommended Reading
• Number Talks, Sherry Parrish
• My Kids Can, Judy Storeygard
• Young Mathematicians at Work, Catherine
Twomey Fosnot
• Thinking Mathematically, Carpenter, Franke,
and Levi
These are Turtle’s recommendations, not DOE
recommendations.

56. Two great resources, free online
for the moment…
• http://www.stenhouse.com/shop/pc/viewprd.asp?idProduct=9336&r=sb10r077
http://www.stenhouse.com/shop/pc/viewPrd.asp?idproduct=9509&idcategory=78

57. Downloads
Yes, there are a few.
• NCTM articles
• Ten frame and dot card packet
• Empty numberline explanation
• GA Dept. of Ed resources
• This powerpoint (you can watch again in elluminate recordings, by the way)
Enjoy and discuss with your colleagues.

58. Recommended Viewing
• Teaching Channel- CCSS videos-
http://www.youtube.com/watch?v=1IPxt794-yU&
• K-5 Standards for Mathematical Practice prezi-
http://prezi.com/zkopzkys49kk/k-5-ccgps-standard
• Learner.org- math videos-
http://www.learner.org/resources/series32.html

59. 3 things?
• Something to do tomorrow?
• Resources?
• Something to think about? Homework!
• Watch this-
http://www.learner.org/resources/series32.html#program
(#20- Shapes From Squares) Talk about it…try it!
(see the Third Grade Homework document for
details)
• Still hungry? Prezi.com- search for CCGPS- more
resources, more food for thought. Enjoy!

61. Feedback
• Choose one of these, and enter it into the chat
window. Please put the symbol next to the thought.
For example: “I never thought of gathering feedback
from students!” or, “I am already using journals! +”
• An AHA! (!)
• A question (?)
• Something positive from today’s session (+)
• Something you will change as a result of today’s
session, or that you wish I would change. (c)

62. Thank you for all
you do.
Keep calm and
carry on.