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

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

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

Session 1 Becky Paslay 2015 IEA Summer Institute 4

DMT Framework

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 explanations o different o sophisticated o efficient o acceptable Cobb,

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

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

3 - Tools to THINK with…...

Go Noodle!

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 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

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

Counting Forward

Counting Forward 24 1 ten 10 ten ones

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

Counting Forward ten What is staying the same? What is changing? 12 ten ones

Counting Forward ten What is staying the same? What is changing? 13 ten ones

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

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

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

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

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

Counting Forward What is staying the same? What is changing? 19 ten ones 1 ten

Counting Forward 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 tens ones 21 1 ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens

Counting Forward tens ones ten 2 tens 10 ten ones 20 tens ones

Counting Forward tens ones 3 tens 1 ten 2 tens 20 tens ones 10 ten ones

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

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones ten 2 tens

Counting Backward tens ones 21 1 ten 2 tens

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

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ten ones 1 ten

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward ones

Counting Backward 77 0 zero

Go Noodle!

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

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.

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

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

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)

Enactive Iconic Symbolic

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?

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

Greg Tang

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 b c

Discuss a b c 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 ath/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 thinking

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 Adapted from Exemplars Classic Exemplars Rubric. Retrieved from: (Exemplar C) Adapted from Exemplars Standards-Based Math Rubric. Retrieved from: (Exemplar S)

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

Go Noodle

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

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 ). 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, 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,

References Romberg, T. A., & Kaput, J. J. (1999). Mathematics worth teaching, mathematics worth understanding. Mathematics classrooms that promote understanding, 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, Go Noodle

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