Math Modelsprogression in the early grades

Becky Paslay2015 IEA Summer Institute

Session 1 Focus

● Why Use Models?--The Research ● Models for:

o counting and cardinalityo adding / subtracting

● K-2 model progression● Enactive, Iconic and Symbolic Trajectory

Session 2 Focus

● Sample Student Work

● Practice Categorizing

● Rubric Rough Draft

Session 1Becky Paslay2015 IEA Summer Institute

4

DMT Framework

http://dmt.boisestate.edu/

Encouraging Multiple Models and Strategies

A Review of the Literature

Bruner (1964)

Amplifiers of Sensory Capacities

(Iconic)

Amplifiers of Motor Capacities(Enactive)

Amplifiers of Ratiocinative Capacities(Symbolic)

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)

Mapping Instruction

Generalized Modeling

Gravemeijer & van

Galen (2003)

Encouraging Multiple Models and Strategies● Sociomathematical

norms for explanationso differento sophisticatedo efficiento acceptable

Cobb, 2000

http://youtu.be/fe2kolrcKSo

Encouraging Multiple Models and Strategies● Realistic Mathematics Education (RME)

o Theory by Cobb, 2000o student’s models can evolve into the abstract

● DMT Frameworko Enactive - Iconic - Symbolico Brenerfur et al, 2015

● Model to “concretize expert knowledgeo Gravemeijer & van Galen, 2003

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)

Romberg & Kaput (1999)

mathematics is more like a banyan rather than a palm tree

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

Modeling StagesFosnot1 - Realistic Situation

2 -Computational strategies as students explain

3 - Tools to THINK with…...

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

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.

Addition & Subtraction Problem Types

JRUJoin Result Unknown

JCUJoin Change Unknown

JSUJoin Start Unknown

SRUSeparate Result Unknown

SCUSeparate Change Unknown

SSUSeparate Start Unknown

PPW:WUPart-Part-Whole: Whole Unknown

PPW:PUPart-Part-Whole: Part Unknown

CDUCompare Difference Unknown

CSUCompare Set Unknown

CQUCompare Quantity Unknown

CRUCompare Referent Unknown

CountingForward and Backwards by 10s & 100

22

*an exercise presented by Brenderfur

Counting Forward

23

1 2 3 4 5 6 7 8 9 10

Counting Forward

24

1 ten

10

ten ones

Counting Forward

25

111 ten What is staying the same? What is changing?

11

ten ones

Counting Forward

26

11 121 ten What is staying the same? What is changing?

12

ten ones

Counting Forward

27

11 12 131 ten What is staying the same? What is changing?

13

ten ones

Counting Forward

28

11 12 13 14

14

ten ones

1 ten What is staying the same? What is changing?

Counting Forward

29

11 12 13 14 15

15

ten ones

1 ten What is staying the same? What is changing?

Counting Forward

30

11 12 13 14 15 16

16

ten ones

1 ten What is staying the same? What is changing?

Counting Forward

31

11 12 13 14 15 16 17

17

ten ones

1 ten What is staying the same? What is changing?

Counting Forward

32

11 12 13 14 15 16 17 18

18

ten ones

1 ten What is staying the same? What is changing?

Counting Forward

33

11 12 13 14 15 16 17 18 19 What is staying the same? What is changing?

19

ten ones

1 ten

Counting Forward

34

11 12 13 14 15 16 17 18 19 20 What is staying the same? What is changing?

20

tens ones

1 ten

Counting Forward

35

1 ten 2 tens

20

tens ones

10

ten ones

Counting Forward

36

21

tens ones

211 ten 2 tens

Counting Forward

37

22

tens ones

21 221 ten 2 tens

Counting Forward

38

23

tens ones

21 22 231 ten 2 tens

Counting Forward

39

24

tens ones

21 22 23 241 ten 2 tens

Counting Forward

40

25

tens ones

21 22 23 24 251 ten 2 tens

Counting Forward

41

26

tens ones

21 22 23 24 25 261 ten 2 tens

Counting Forward

42

27

tens ones

21 22 23 24 25 26 271 ten 2 tens

Counting Forward

43

28

tens ones

21 22 23 24 25 26 27 281 ten 2 tens

Counting Forward

44

29

tens ones

21 22 23 24 25 26 27 28 291 ten 2 tens

Counting Forward

45

21 22 23 24 25

2+10

tens ones

26 27 28 29 301 ten 2 tens

10

ten ones

20

tens ones

Counting Forward

46

30

tens ones

3 tens1 ten 2 tens

20

tens ones

10

ten ones

Counting Backward

47

3 tens1 ten 2 tens

20

tens ones

10

ten ones

30

tens ones

Counting Backward

48

29

tens ones

21 22 23 24 25 26 27 28 291 ten 2 tens

Counting Backward

49

28

tens ones

21 22 23 24 25 26 27 281 ten 2 tens

Counting Backward

50

27

tens ones

21 22 23 24 25 26 271 ten 2 tens

Counting Backward

51

26

tens ones

21 22 23 24 25 261 ten 2 tens

Counting Backward

52

25

tens ones

21 22 23 24 251 ten 2 tens

Counting Backward

53

24

tens ones

21 22 23 241 ten 2 tens

Counting Backward

54

23

tens ones

21 22 231 ten 2 tens

Counting Backward

55

22

tens ones

21 221 ten 2 tens

Counting Backward

56

21

tens ones

211 ten 2 tens

Counting Backward

57

1 ten 2 tens

20

tens ones

10

ten ones

Counting Backward

58

11 12 13 14 15 16 17 18 19

19

ten ones

1 ten

Counting Backward

59

11 12 13 14 15 16 17 18

18

ten ones

1 ten

Counting Backward

60

11 12 13 14 15 16 17

17

ten ones

1 ten

Counting Backward

61

11 12 13 14 15 16

16

ten ones

1 ten

Counting Backward

62

11 12 13 14 15

15

ten ones

1 ten

Counting Backward

63

11 12 13 14

14

ten ones

1 ten

Counting Backward

64

11 12 13

13

ten ones

1 ten

Counting Backward

65

11 12

12

ten ones

1 ten

Counting Backward

66

11

11

ten ones

1 ten

Counting Backward

67

10

ten ones

1 ten

Counting Backward

68

1 2 3 4 5 6 7 8 9

9

ones

Counting Backward

69

1 2 3 4 5 6 7 8

8

ones

Counting Backward

70

1 2 3 4 5 6 7

7

ones

Counting Backward

71

1 2 3 4 5 6

6

ones

Counting Backward

72

1 2 3 4 5

5

ones

Counting Backward

73

1 2 3 4

4

ones

Counting Backward

74

1 2 3

3

ones

Counting Backward

75

1 2

2

ones

Counting Backward

76

1

1

ones

Counting Backward

77

0

zero

Go Noodle!

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

Addition & Subtraction Problem Types

JRUJoin Result Unknown

JCUJoin Change Unknown

JSUJoin Start Unknown

SRUSeparate Result Unknown

SCUSeparate Change Unknown

SSUSeparate Start Unknown

PPW:WUPart-Part-Whole: Whole Unknown

PPW:PUPart-Part-Whole: Part Unknown

CDUCompare Difference Unknown

CSUCompare Set Unknown

CQUCompare Quantity Unknown

CRUCompare Referent Unknown

Sample ProblemEllie 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.

Models *Bar/Tape Model*Number LinePicturesTen FrameVenn DiagramTree DiagramGraphs

Tools*Unifix CubesRekenrekDice, Cards, DominoesBase Ten BlocksGeoboard*Graph PaperMisc. Manipulatives

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

Modes of RepresentationEnactive

Physical or action-based representations

IconicVisual image(s) of a situation that is relatively proportionally accurate

SymbolicAbstract representations where the meaning of the symbols must be learned

Bruner, J. (1964)

EnactiveIconic

Symbolic

Enactiveconcrete, physical, manipulatives, cubes, fingers (objects)

Iconicvisual, picture, drawing, diagram, bar model, number line, graph

Symbolicnumbers, 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?

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?

DMT Framework

http://dmt.boisestate.edu/

Greg Tang http://gregtangmath.com/index

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

ENACTIVE-ICONIC-SYMBOLIC Model TRAJECTORY Discussion

To Analyze Student Thinking

Stategies vs. Models

strategy = the mental process we use to solve

model = the method of notation used to explain our strategy

Solve Multiple Ways

a. 3 + 5

b.38 + 7

c. 492 + 263

Discuss

a. 3 + 5

b.38 + 7

c. 492 + 263

Compare within your group. We will

return to discuss whole group later.

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?

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?

THE ENACTIVE-ICONIC-SYMBOLIC TRAJECTORY

As Instructional Scaffolding

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?

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

E-I-S as Instructional Scaffolding

One instructional progression from an informal iconic drawing to a more formal iconic drawing

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?

E-I-S as Instructional Scaffolding

One potential instructional progression from an informal iconic drawing to a more formal iconic drawing

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?

More Practice Sorting

Samples from Idaho State Department Web

http://www.sde.idaho.gov/site/math/mtiWebinarsArchived.htm

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 thinkinghttp://youtu.be/xZk5Zo2L5oU

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.

Creating a Math Rubric

Copyright ©2001, revised 2015 by Exemplars, Inc. All rights reserved.

Four Point Rubric from Exemplars Inc.

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) Adapted from Exemplars Standards-Based Math Rubric. Retrieved from: http://www.exemplars.com/assets/files/Standard_Rubric.pdf (Exemplar S)

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

Go Noodle

https://www.gonoodle.com

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

Rate the Model and Tools

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.”

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.”

DMT Framework

http://dmt.boisestate.edu/

ReferencesBrendefur, 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.

ReferencesRomberg, 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.com Go Noodle

http://gregtangmath.com/index

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