1. Math Models progression in the early grades Becky Paslay 2015 IEA Summer Institute

2. Session 1 Focus ●Why Use Models?--The Research ●Models for: o counting and cardinality o adding / subtracting ● K-2 model progression ●Enactive, Iconic and Symbolic Trajectory

3. Session 2 Focus ●Sample Student Work ●Practice Categorizing ●Rubric Rough Draft

4. Session 1 Becky Paslay 2015 IEA Summer Institute 4

6. DMT Framework http://dmt.boisestate.edu/

7. Encouraging Multiple Models and Strategies A Review of the Literature

8. Bruner (1964) Amplifiers of Sensory Capacities (Iconic) Amplifiers of Motor Capacities (Enactive) Amplifiers of Ratiocinative Capacities (Symbolic)

9. Encouraging Multiple Models and Strategies “The most important thing about memory is not storage of past experience, but rather the retrieval of what is relevant in some usable form. This depends upon how past experience is coded and processed so that it may indeed be relevant and usable in the present when needed.” (Bruner, 1964)

10. Mapping Instruction Generalized Modeling Gravemeijer & van Galen (2003)

11. Encouraging Multiple Models and Strategies ● Sociomathematical norms for explanations o different o sophisticated o efficient o acceptable Cobb, 2000 http://youtu.be/fe2kolrcKSo

12. Encouraging Multiple Models and Strategies ● Realistic Mathematics Education (RME) o Theory by Cobb, 2000 o student’s models can evolve into the abstract ● DMT Framework o Enactive - Iconic - Symbolic o Brenerfur et al, 2015 ● Model to “concretize expert knowledge o Gravemeijer & van Galen, 2003

13. Encouraging Multiple Models and Strategies ● longer-term memory ● better understanding of concepts ● “mapping instruction” versus generalized modeling o step by step process with ready made manipulatives o elaborate from own ideas, self-developed and reflect number sense understandings (Brenderfur, Thiede, Strother and Carney, 2015; Gravemeijer & van Galen, 2003; Resnick & Omanson, 1987)

14. Romberg & Kaput (1999) mathematics is more like a banyan rather than a palm tree

15. Shift from traditional math towards human mathematical activity Topics = Same Approach Changing Math Worth Teaching ➔ model building ➔ explore patterns ➔ powerful analytical problem solving ➔ relevant ➔ invite exploration ➔ inquire ➔ justification ➔ flexible technology use ➔ creative attitudes, habits & imagination ➔ enjoyment and confidence

16. Modeling Stages Fosnot 1 - Realistic Situation 2 -Computational strategies as students explain

17. 3 - Tools to THINK with…...

18. Go Noodle! https://www.gonoodle.com/channels/gonoodle/mega-math-marathon

19. Encouraging Multiple Models and Strategies Different contexts generate different models which allow teachers to take student ideas seriously, press students conceptually, focus on the structure of mathematics and address misconceptions.

20. Addition & Subtraction Problem Types JRU Join Result Unknown JCU Join Change Unknown JSU Join Start Unknown SRU Separate Result Unknown SCU Separate Change Unknown SSU Separate Start Unknown PPW:WU Part-Part-Whole: Whole Unknown PPW:PU Part-Part-Whole: Part Unknown CDU Compare Difference Unknown CSU Compare Set Unknown CQU Compare Quantity Unknown CRU Compare Referent Unknown

22. Counting Forward and Backwards by 10s & 100 22 *an exercise presented by Brenderfur

23. Counting Forward 23 1234 56 7 8 910

24. Counting Forward 24 1 ten 10 ten ones

25. Counting Forward 25 11 1 ten What is staying the same? What is changing? 11 ten ones

26. Counting Forward 26 1112 1 ten What is staying the same? What is changing? 12 ten ones

27. Counting Forward 27 1112 13 1 ten What is staying the same? What is changing? 13 ten ones

28. Counting Forward 28 1112 13 14 ten ones 1 ten What is staying the same? What is changing?

29. Counting Forward 29 1112 13 14 15 ten ones 1 ten What is staying the same? What is changing?

30. Counting Forward 30 1112 13 14 15 16 ten ones 1 ten What is staying the same? What is changing?

31. Counting Forward 31 1112 13 14 15 16 17 ten ones 1 ten What is staying the same? What is changing?

32. Counting Forward 32 1112 13 14 15 16 17 18 ten ones 1 ten What is staying the same? What is changing?

33. Counting Forward 33 1112 13 14 15 16 17 18 19 What is staying the same? What is changing? 19 ten ones 1 ten

34. Counting Forward 34 1112 13 14 15 16 17 18 19 20 What is staying the same? What is changing? 20 tens ones 1 ten

35. Counting Forward 35 1 ten 2 tens 20 tens ones 10 ten ones

36. Counting Forward 36 21 tens ones 21 1 ten 2 tens

37. Counting Forward 37 22 tens ones 2122 1 ten 2 tens

38. Counting Forward 38 23 tens ones 212223 1 ten 2 tens

39. Counting Forward 39 24 tens ones 21222324 1 ten 2 tens

40. Counting Forward 40 25 tens ones 21222324 25 1 ten 2 tens

41. Counting Forward 41 26 tens ones 21222324 2526 1 ten 2 tens

42. Counting Forward 42 27 tens ones 21222324 2526 27 1 ten 2 tens

43. Counting Forward 43 28 tens ones 21222324 2526 2728 1 ten 2 tens

44. Counting Forward 44 29 tens ones 21222324 2526 2728 29 1 ten 2 tens

45. Counting Forward 45 21222324 25 2+10 tens ones 26 2728 29 30 1 ten 2 tens 10 ten ones 20 tens ones

46. Counting Forward 46 30 tens ones 3 tens 1 ten 2 tens 20 tens ones 10 ten ones

47. Counting Backward 47 3 tens 1 ten 2 tens 20 tens ones 10 ten ones 30 tens ones

48. Counting Backward 48 29 tens ones 21222324 2526 2728 29 1 ten 2 tens

49. Counting Backward 49 28 tens ones 21222324 2526 2728 1 ten 2 tens

50. Counting Backward 50 27 tens ones 21222324 2526 27 1 ten 2 tens

51. Counting Backward 51 26 tens ones 21222324 2526 1 ten 2 tens

52. Counting Backward 52 25 tens ones 21222324 25 1 ten 2 tens

53. Counting Backward 53 24 tens ones 21222324 1 ten 2 tens

54. Counting Backward 54 23 tens ones 212223 1 ten 2 tens

55. Counting Backward 55 22 tens ones 2122 1 ten 2 tens

56. Counting Backward 56 21 tens ones 21 1 ten 2 tens

57. Counting Backward 57 1 ten 2 tens 20 tens ones 10 ten ones

58. Counting Backward 58 1112 13 14 15 16 17 18 19 ten ones 1 ten

59. Counting Backward 59 1112 13 14 15 16 17 18 ten ones 1 ten

60. Counting Backward 60 1112 13 14 15 16 17 ten ones 1 ten

61. Counting Backward 61 1112 13 14 15 16 ten ones 1 ten

62. Counting Backward 62 1112 13 14 15 ten ones 1 ten

63. Counting Backward 63 1112 13 14 ten ones 1 ten

64. Counting Backward 64 1112 13 ten ones 1 ten

65. Counting Backward 65 1112 ten ones 1 ten

66. Counting Backward 66 11 ten ones 1 ten

67. Counting Backward 67 10 ten ones 1 ten

68. Counting Backward 68 1234 56 7 8 9 9 ones

69. Counting Backward 69 1234 56 7 8 8 ones

70. Counting Backward 70 1234 56 7 7 ones

71. Counting Backward 71 1234 56 6 ones

72. Counting Backward 72 1234 5 5 ones

73. Counting Backward 73 1234 4 ones

74. Counting Backward 74 123 3 ones

75. Counting Backward 75 12 2 ones

76. Counting Backward 76 1 1 ones

77. Counting Backward 77 0 zero

78. Go Noodle! https://www.gonoodle.com/channels/youtube/count-by-2s-5s-and-10s

79. Addition & Subtraction Problem Types JRU Join Result Unknown JCU Join Change Unknown JSU Join Start Unknown SRU Separate Result Unknown SCU Separate Change Unknown SSU Separate Start Unknown PPW:WU Part-Part-Whole: Whole Unknown PPW:PU Part-Part-Whole: Part Unknown CDU Compare Difference Unknown CSU Compare Set Unknown CQU Compare Quantity Unknown CRU Compare Referent Unknown

81. Sample Problem Ellie has 22 apples. She gives 13 to Mark. How many apples does she have left? -How should your students model this problem? -Write them on index cards.

82. Models *Bar/Tape Model *Number Line Pictures Ten Frame Venn Diagram Tree Diagram Graphs Tools *Unifix Cubes Rekenrek Dice, Cards, Dominoes Base Ten Blocks Geoboard *Graph Paper Misc. Manipulatives

83. Identify whether the model is enactive, iconic, or symbolic and how you know. - Include the models you created on the index cards.

84. Modes of Representation Enactive Physical or action-based representations Iconic Visual image(s) of a situation that is relatively proportionally accurate Symbolic Abstract representations where the meaning of the symbols must be learned Bruner, J. (1964)

85. Enactive Iconic Symbolic

86. Enactive concrete, physical, manipulatives, cubes, fingers (objects) Iconic visual, picture, drawing, diagram, bar model, number line, graph Symbolic numbers, symbols, table, equation, algorithm, notations, abstract, words What words are used to connect to the enactive, iconic and symbolic representations? What words do the CCSS use?

87. Using the E-I-S Trajectory to Diagnose Student Understanding Enactive Iconic Symbolic One potential trajectory for how students may come to represent their understanding of subtraction. - How is this similar or different to how you sequenced the models?

88. DMT Framework http://dmt.boisestate.edu/

89. Greg Tang http://gregtangmath.com/index

90. Session 2 Focus ●Discuss Strategies vs. Models ●Practice Categorizing sample student work ●Work to develop a very rough draft rubric ●Rate various models and tools

91. ENACTIVE-ICONIC-SYMBOLIC Model TRAJECTORY Discussion To Analyze Student Thinking

92. Stategies vs. Models strategy = the mental process we use to solve model = the method of notation used to explain our strategy

93. Solve Multiple Ways a.3 + 5 b.38 + 7 c.492 + 263

95. Discuss a.3 + 5 b.38 + 7 c.492 + 263 Compare within your group. We will return to discuss whole group later.

96. 1. How would you sequence these student solutions from informal to formal (include your index card examples also)? 2. If time allows, identify how the student thinking is similar or different among models?

97. Using the E-I-S Trajectory to Diagnose Student Understanding Enactive Iconic Symbolic One potential trajectory for how students may come to represent their understanding of subtraction. - How is this similar or different to how you sequenced the models?

98. THE ENACTIVE-ICONIC-SYMBOLIC TRAJECTORY As Instructional Scaffolding

99. E-I-S as Instructional Scaffolding How do you take a student who is here....................... to here? Ellie has 22 apples, she gives 13 to Mark, how many apples does she have left?

100. Line up the ‘cubes’ horizontally so the ‘drawing’ looks like the following. Set up as a bar model. Draw the number line off the bar model. Represent jumps on bar model/number line combination

101. E-I-S as Instructional Scaffolding One instructional progression from an informal iconic drawing to a more formal iconic drawing

102. E-I-S as Instructional Scaffolding How do you take a student who is here....................... to here? Ellie has 22 apples, she gives 13 to Mark, how many apples does she have left?.......... or here?

103. E-I-S as Instructional Scaffolding One potential instructional progression from an informal iconic drawing to a more formal iconic drawing

104. E-I-S as Instructional Scaffolding What is the mismatch between taking a student who is here........... to here? Ellie has 22 apples, she gives 13 to Mark, how many apples does she have left?

105. More Practice Sorting Samples from Idaho State Department Web http://www.sde.idaho.gov/site/m ath/mtiWebinarsArchived.htm

106. Discuss ● Bar model with and without individual numbers and number line ● Base Ten Blocks - number line (enactive) ● Base Ten Blocks - number tree (iconic) *Listen to student thinking http://youtu.be/xZk5Zo2L5oU

107. Big Ideas for Take Away ● There isn’t a perfect addition progression. ● We can have general ideas but models and strategies may fit in different places based on the students, the task or the number set.

108. Creating a Math Rubric

109. Copyright ©2001, revised 2015 by Exemplars, Inc. All rights reserved. Four Point Rubric from Exemplars Inc.

110. 1 Point: Little Accomplishment 2 Points: Marginal Accomplishment 3 Points: Substantial Accomplishment 4 Points: Full Accomplishment ●No attempt is made to construct representations (Exemplar S) ●No evidence of a strategy, or uses a strategy that does not help solve the problem(Exemplar C) ●Applies procedures incorrectly (Exemplar C) ●No evidence of mathematical reasoning (Exemplar C) ●An attempt is made to construct representations (Exemplar S) ●A partially correct strategy is chosen, leading some way toward a solution but not to a full solution of the problem (Exemplar C, S) ●Could not completely carry out procedures (Exemplar C) ●Some evidence of mathematical reasoning (Exemplar C) ●Appropriate and mostly accurate mathematical representations (Exemplar S) ●A correct strategy is chosen based on the mathematical situation in the task (Exemplar S) ●Applies procedures with minor error(s) (Exemplar C, Van de Walle, 2006) ●Uses effective mathematical reasoning (Exemplar C) ●Appropriate and accurate mathematical representations (Exemplar S) ●Uses an efficient strategy leading directly to a solution (Exemplar C) ●Applies procedures accurately to correctly solve the problem (Exemplar C) ●Employs refined and complex reasoning (Exemplar C) Adapted from Van de Walle, J. (2004) Elementary and Middle School Mathematics: Teaching Developmentally. Boston: Pearson Education. pages 78-82 Adapted from Exemplars Classic Exemplars Rubric. Retrieved from: http://www.exemplars.com/assets/files/math_rubric.pdf (Exemplar C) http://www.exemplars.com/assets/files/math_rubric.pdf Adapted from Exemplars Standards-Based Math Rubric. Retrieved from: http://www.exemplars.com/assets/files/Standard_Rubric.pdf (Exemplar S) http://www.exemplars.com/assets/files/Standard_Rubric.pdf

111. Three Point Rubric that evolved from the previous attempts and adapted from Van de Walle and Exemplars.

112. Go Noodle https://www.gonoodle.com https://www.gonoodle.com/channels/think-about-it/make-a-wish

113. Rate the Model and Tools

127. Which statement are you leaving with? 1.“I need to teach the models that are appropriate for my grade level.” 1.“I need to find contextual problems that will encourage students to use the models that are appropriate for my grade level.”

128. Which statement are you leaving with? 1.“I need to teach the models that are appropriate for my grade level.” 1.“I need to find contextual problems that will encourage students to use the models that are appropriate for my grade level.”

129. DMT Framework http://dmt.boisestate.edu/

130. References Brendefur, J., Thiede, K., Strother, S., and Carney, M. (2015). DMT Framework and Classroom Structure. Department of Education, Boise State University, Boise, Idaho. Bruner, J. S. (1964). The course of cognitive growth. American psychologist,19(1), 1. Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In R. Lesh & A. Kelly (Eds.), Handbook of research design in mathematics and science education (pp. 307-334). Mahwah, NJ: Lawrence Erlbaum. Imm, K. L., Fosnot, C. T., & Uittenbogaard, W. (2007). Minilessons for operations with fractions, decimals, and percents: A yearlong resource. firsthand/Heinemann. Dolk, M., & Fosnot, C,T, (2002). Young Mathematicians at Work: Constructing Fractions, Decimals and Percents: Heinemann, 1- 19. Gravemeijer, K., & van Galen, F. (2003). Facts and algorithms as products of students’ own mathematical activity. A research companion to principles and standards for school mathematics, 114-122.

131. References Romberg, T. A., & Kaput, J. J. (1999). Mathematics worth teaching, mathematics worth understanding. Mathematics classrooms that promote understanding, 3-17. Smith, M.S., & Stein, M.K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: NCTM. Thurston, W.P. (1990, January). Letters from the editors. Quantum, 6-7. https://www.gonoodle.comhttps://www.gonoodle.com Go Noodle http://gregtangmath.com/index

132. FOR COPIES

140. GRAPHICS

141. 0214 3 658790101211131514 18 16 17 2221 20 19

142. -3-2-5-6-4-8-7-11-10-9-12-13

143. -3-2-5-6-4-8-7-11-10-9-12-13 0214 3 658790101211131514 18 16 17 2221 20 19 -10 -3