1. K-2 Mathematics: Ensuring Mathematical Success for All
Denise Schulz, NCDPI
NC Department of Public Instruction

2. Welcome “Who’s in the Room?”
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3. maccss.ncdpi.wikispaces.net
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4. NC Department of Public Instruction

5. Guiding Principles for School Mathematics Teaching and Learning
Access and Equity
Curriculum
Tools and Technology
Assessment
Professionalism
Guiding
Principles
for
School
Mathematics
Although such teaching and learning form the nonnegotiable core of successful mathematics programs, they are part of a system of essential elements of excellent mathematics programs. Consistent implementation of effective teaching and learning of mathematics, as previously described in the eight Mathematics Teaching Practices, are possible only when school mathematics programs have in place—
a commitment to access and equity;
a powerful curriculum;
appropriate tools and technology;
meaningful and aligned assessment; and
a culture of professionalism.
NC Department of Public Instruction

6. Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
NC Department of Public Instruction

7. Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
NC Department of Public Instruction

8. Tasks & Representations Fluency from Understanding
What might be the math learning goals?
Math Goals
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
How might we question students and structure class discourse to advance student learning?
Discourse & Questions
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
How might we check in on student thinking and struggles and use it to inform instruction?
Struggle & Evidence
NC Department of Public Instruction

9. Learning Goals should:
What might be the math learning goals?
Math Goals
Learning Goals should:
Clearly state what it is students are to learn and understand about mathematics as the result of instruction.
Be situated within learning progressions.
Frame the decisions that teachers make during a lesson.
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10. Principles to Action – pg. 16
Now, let’s think about the standards for mathematical practice.
What do you notice?
Principles to Action – pg. 16
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11. How Many Marbles?
20 marbles are in the container. 9 are red and the rest are green. How many green marbles are in the container?
Thinking like a 1st grader, how would you solve this problem?
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12. How Did You Solve the Problem?
Share your representation with the other people at your table.
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13. A Possible Solution:
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14. Another Possible Solution:
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15. Another Possible Solution:
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16. Use and connect mathematical representations.
Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.
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(National Research Council, 2001, p. 94)

17. Use and connect mathematical representations
Illustrate, show, or work with mathematical ideas using diagrams, pictures, number lines, graphs, and other math drawings.
Use concrete objects to show, study, act upon, or manipulate mathematical ideas (e.g., cubes, counters, tiles, paper strips).
Record or work with mathematical ideas using numerals, variables, tables, and other symbols.
Solve the problem
How did you solve the problem? Find the type of representation and gather
Compare your work with your group.
Find someone from another group and compare how you each solved it
What do you notice about how most people solved it?
Situate mathematical ideas in everyday, real-world, imaginary, or mathematical situations and contexts.
Use language to interpret, discuss, define, or describe mathematical ideas, bridging informal and formal mathematical language.
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18. Important Mathematical Connections between and within different types of representations
Contextual
Physical
Visual
Symbolic
Verbal
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Principles to Actions (p. 25) (Adapted from Lesh, Post, & Behr, 1987)

19. How are these solutions connected?
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20. Proficient problem solvers frequently use representations to solve problems and communicate results.
Boaler, What’s Math Got To Do With It?, 2015
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21. Recommendations for Implementation
Select visual representations that are appropriate for students and the problems they are solving (Woodward et al., 2012).
Make explicit connections between representations.
Strengthen students’ representational competence by:
Encouraging purposeful selection of representations.
Engaging in dialogue about explicit connections among representations.
Alternating the direction of the connections made among representations.
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(NCTM, 2014)

22. Principles to Actions pg. 29
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Principles to Actions pg. 29

23. Fluency from Understanding
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
Procedural Fluency should:
Build on a foundation of conceptual understanding.
Over time (months, years), result in known facts and generalized methods for solving problems.
Enable students to flexibly choose among methods to solve contextual and mathematical problems.
Baroody, 2006; Fuson & Beckmann, 2012/2013; Fuson, Kalchman, & Bransford, 2005; Russell, 2006
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24. How does this task develop a foundation of conceptual understanding for building toward procedural fluency?
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25. Research Shows:
Students who use memorization as their primary strategy are the lowest-achieving students in the world.
Students who were achieving at high levels were those who had worked out that numbers can be flexibly broken apart and put together again.
NC Department of Public Instruction
Boaler, What’s Math Got To Do With It?, 2015

26. Build procedural fluency from conceptual understanding.
A rush to fluency undermines students’ confidence and interest in mathematics and is considered a cause of mathematics anxiety.
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(Ashcraft 2002; Ramirez Gunderson, Levine, & Beilock, 2013)

27. Compression of Concepts vs. Procedural Ladder
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Boaler, What’s Math Got To Do With It?, 2015

28. Fluency builds from initial exploration and discussion of number concepts to using informal reasoning strategies based on meanings and properties of the operations to the eventual use of general methods as tools in solving problems.
Principles to Actions (NCTM, 2014, p. 42)
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29. Principles to Action - page 47
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Principles to Action - page 47

30. Butterflies in the Garden
Mrs. Hope’s class saw 76 butterflies in the garden. Some of the butterflies flew away. Now there are 49 butterflies in the garden. How many butterflies flew away?
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31. Tasks & Representations Fluency from Understanding
What might be the math learning goals?
Math Goals
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
NC Department of Public Instruction

32. Mrs. Hope’s class saw 76 butterflies in the garden
Mrs. Hope’s class saw 76 butterflies in the garden. Some of the butterflies flew away. Now there are 49 butterflies in the garden. How many butterflies flew away?
76 − ∎=49
76 −49= ∎ 76
- 49
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33. You cannot take a bigger number from a smaller number.
Expires: Grade 7 (7.NS.1)
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34. What misunderstandings might students have if they make the assertion, “You can’t take a bigger number from a smaller one?”
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35. Although students in K-2 are not learning how to subtract a bigger number from a smaller number, it doesn’t mean that in mathematical operations, it can’t be done.
A student’s future learning should not be confused by emphasizing a misconception.
Ma, 1999
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36. How do you talk about addition and subtraction?
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37. Janet and her dad baked cookies for the class bake sale
Janet and her dad baked cookies for the class bake sale. On Saturday morning they baked 2 dozen cookies. In the afternoon, they baked 3 dozen cookies. How many cookies will Janet bring to the bake sale?
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Bamberger, Oberdorf, & Schultz-Ferrell, 2010

38. What possible answers do you think students had?
Turn and Talk
What possible answers do you think students had?
Student answers: 5 dozen, 50, 60, and 510
How did students arrive at these answers?
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39. 5 dozen
These students added the 2 and 3 to get 5 dozen, but did not know how many individual cookies that totaled. 50
“There are 24 cookies in 2 dozen and 36 cookies in 3 dozen. We wrote 24 over the top of We added and that equals 10. Since you can’t have more than 1 digit in a place we just put the zero in the answer. Then we added 2 tens plus 3 tens and that equals 5 tens. So, the answer is 50.”
510
“We used connecting cubes and paper and pencil. First, we made 2 stacks of 10 and 4 ones for the 2 dozen cookies. Then we made 3 stacks of 10 and 6 ones for the 3 dozen cookies. We put the ones together and that equaled 10 ones. We put the tens together and that equaled 5 tens.” 60
“We put together stacks of cubes, too, but after we put the ones together we knew that we could make a stack of 10 with them. Then we had 6 tens altogether. The answer is 60.”
NC Department of Public Instruction
Bamberger, Oberdorf, & Schultz-Ferrell, 2010

40. What error are students
making in this problem?
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41. Students did not see 54 as a total amount with 48 as one part of the total.
They thought about each digit in the numerals, subtracting the smaller digit from the larger digit regardless of its place in the expression.
Bamberger, Oberdorf, & Schultz-Ferrell, 2010
NC Department of Public Instruction

42. “When children focus on following the steps taught traditionally, they usually pay no attention to the quantities and don’t even consider whether or not their answers make sense.”
NC Department of Public Instruction
Richardson, 2009

43. Expired Mathematical Language: Borrowing or Carrying
Vocabulary such as borrowing and carrying focuses on procedures rather than understanding
These terms have the potential to create problems for students, including:
Thinking they can arbitrarily change a number in a subtraction problem if it is too small by “borrowing” from another number
Focusing on digits rather than number value
Not reinforcing understanding of the base-ten number system
Use words like regrouping, trade, or equal exchange to indicate the action of trading or exchanging one place value unit for another unit.
This language prepares students for situations beyond 10’s and 1’s, and for fractions and beyond
NCTM, 2011
Faulkner, 2013
NC Department of Public Instruction

44. Expired Mathematical Language: Borrowing or Carrying
NC Department of Public Instruction

45. Research:
The other answers listed are ones that are considered unreasonable.
The class that was taught algorithms came up with the most unreasonable answers, while the class that was taught with no algorithm gave the most reasonable answers
The algorithm is convenient for adults, who already know that the 5 of 52 stands for 50. However, for primary-school children, who have a tendency to think that the 5 means five, and so on, the algorithm serves to reinforce the error.
The children in the algorithms class did not notice that their answers of 29, in the 900s, and so on were unreasonable for this problem.
Kamii, 2000
NC Department of Public Instruction

46. This becomes more than just a set of procedures…
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47. Butterflies in the Garden
Mrs. Hope’s class saw 76 butterflies in the garden. Some of the butterflies flew away. Now there are 49 butterflies in the garden. How many butterflies flew away?
NC Department of Public Instruction

48. Use keywords to solve problems.
Expires Grade 3 (3.OA.8)
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49. Keywords encourage students to strip numbers from the problem and use them to perform a computation outside of the problem context.
Many keywords are common English words that can be used in many different ways.
Keywords become particularly troublesome when students begin to explore multistep word problems, because they must decide which keywords work with which component of the problem.
NC Department of Public Instruction
Karp, Bush, & Dougherty, 2014

50. John had 14 marbles in his left pocket
John had 14 marbles in his left pocket. He had 37 marbles in his right pocket. How many marbles did John have?
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51. John had 14 marbles in his left pocket
John had 14 marbles in his left pocket. He had 37 marbles in his right pocket. How many marbles did John have?
NC Department of Public Instruction

52. Start-Unknown Problem Types
Susie has 16 marbles. This morning she gave away 2 marbles. How many did she start with?
Susie has 16 marbles. Her friend just gave her 2 more marbles. How many marbles did she start with?
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53. How do you think students respond to these questions if they’ve been taught a key word strategy?
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54. Comparison Problem Types
Susie has six marbles. This is two more than Deante. How many marbles does Deante have?
Deante has four marbles. This is two fewer than Susie. How many marbles does Susie have?
Susie has six marbles. Deante has four marbles. How many marbles does Deante need to have the same number of marbles as Susie?
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55. Student’s math reasoning…
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56. Apples in the Box
There are 36 fewer apples in the box than apples on the ground. There are 50 apples in the box. How many apples are on the ground?
NC Department of Public Instruction

57. Tasks & Representations Fluency from Understanding
What might be the math learning goals?
Math Goals
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
NC Department of Public Instruction

58. A sense making strategy will always work.
Avoid Key Words
Key words are misleading.
Many problems have no key words.
The key word strategy sends a terribly wrong message about doing mathematics.
A sense making strategy will always work.
Often the key word or phrase in a problem suggests an operation that is incorrect. For example: Maxine took the 28 stickers she no longer wanted and gave them to Zandra. Now, Maxine has 37 stickers left. How many stickers did Maxine have to begin with?
Especially when you get away from overly simple problems found in primary textbooks and a child that has been taught to rely on key words is left with no strategy.
The most important approach to solving any contextual problem is to analyze the structure and make sense of it. The key word approach encourages students to ignore the meaning of the problem and look for an easy way out.
NC Department of Public Instruction
Van de Walle & Lovin (2006)

59. When students are taught the underlying structure of a word problem, they not only have greater success in problem solving but can also gain insight into the deeper mathematical ideas in word problems.
Peterson, Fennema, & Carpenter, 1998)
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60. Problem Types
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61. Problem Type Sort Add To-Result Unknown Take From-Result Unknown
Put Together/Take Apart-Total Unknown
Compare-Difference Unknown
Add To-Change Unknown
Take From-Change Unknown
Put Together/Take Apart-Addend Unknown
Compare-Bigger Unknown
Add To-Start Unknown
Take From-Start Unknown
Put Together/Take Apart-Both Addends Unknown
Compare-Smaller Unknown
NC Department of Public Instruction

62. Mathematics Teaching Practices
Establish mathematics goals to focus learning.
Implement tasks that promote reasoning and problem solving.
Use and connect mathematical representations.
Facilitate meaningful mathematical discourse.
Pose purposeful questions.
Build procedural fluency from conceptual understanding.
Support productive struggle in learning mathematics.
Elicit and use evidence of student thinking.
NC Department of Public Instruction

63. Tasks & Representations Fluency from Understanding
What might be the math learning goals?
Math Goals
What representations might students use in reasoning through and solving the problem?
Tasks & Representations
How might we question students and structure class discourse to advance student learning?
Discourse & Questions
How might we develop student understanding to build toward aspects of procedural fluency?
Fluency from Understanding
How might we check in on student thinking and struggles and use it to inform instruction?
Struggle & Evidence
NC Department of Public Instruction

64. What questions do you have?

65. Follow Us! NC Mathematics www.facebook.com/NorthCarolinaMathematics
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66. DPI Mathematics Section
Kitty Rutherford
Elementary Mathematics Consultant
Denise Schulz
Lisa Ashe
Secondary Mathematics Consultant
Joseph Reaper
Dr. Jennifer Curtis
K – 12 Mathematics Section Chief
Susan Hart
Mathematics Program Assistant
NC Department of Public Instruction 67

67. For all you do for our students!
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68. References
Bamberger, H.J., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math Misconceptions: From Misunderstanding to Deep Understanding
Faulkner, V. (2013). Why the Common Core Changes Math Instruction, Kappan Magazine
Kamii, C (2000). Young Children Reinvent Arithmetic: Implications of Piaget’s Theory. Second Edition
Karp, K.S., Bush, S.B., & Dougherty, B.J. (2014). 13 Rules That Expire, Teaching Children Mathematics
Ma, L (1999). Knowing and Teaching Elementary Mathematics
NCTM (2011). Developing Essential Understanding of Addition and Subtraction PreK-Grade 2
Richardson, K (1999). Developing Number Concepts: Place Value, Multiplication, and Division
Van de Walle, J. & Lovin, LH (2006). Teaching Student Centered Mathematics Grades K-3
NC Department of Public Instruction