Representations, Models, Diagrams… Think about the following

What’s wrong with how and why? “The smarter kids' brains aren't accepting lines, dots, circles and squares as an acceptable means of learning division. Yes, our children have been reduced to using lines, dots, dashes, squares and circles to learn division.” Smarter kids aren’t able to use representations, area models, groups of objects circled, or number lines to see that, for instance, 38 ÷ 15 is 2 groups of 15 with 8 as a remainder? Many of those who ask about “this stuff” can’t recall understanding ANYTHING they did in math class?” Number lines, area models, and other representations for operations and both how and why they work have been used for a LONG time. What’s different is that the Common Core State Standards for Mathematics actually includes statements like…

Kindergarten K.OA – Operations and Algebraic Thinking A.Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. 3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1).

Grade 1 1.NBT – Number and Operations in Base Ten C. Use place value understanding and properties of operations to add and subtract. 4. Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

Grade 2 2.NBT - Number and Operations in Base Ten B. Use place value understanding and properties of operations to add and subtract. 5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 6. Add up to four two-digit numbers using strategies based on place value and properties of operations. 7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Grade 3 3.NF - Number and Operations – Fractions A. Develop Understanding of fractions as numbers. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

Grade 4 4.NBT Number and Operations in Base Ten B. Use place value understanding and properties of operations to perform multi-digit arithmetic. 6. Find whole number quotients and remainders…using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain calculation by using equations, rectangular arrays, and/or area models.

Grade 5 5.NF Number and Operations – Fractions B.Apply and extend previous understandings of multiplication and division to multiply and divide fractions. 7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

And now the assessments…

PARCC: Grade 3 - Art Teacher’s Rectangular Array

PARCC: Grade 4 – Three Friends’ Beads

PARCC: Grade 4 - Fraction Comparison

Issues/Challenges we face… Teacher background/comfort in using varied representations: – Visual models – typically within OA, NBT, NF – and to a lesser extent (more specific) in MD? Area models Bar diagrams Number lines

Issues/Challenges we all face…(cont.) Teacher background in understanding language like: – “1.OA – Use strategies such as counting on, making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction.” – Compose and decompose – Relationship between addition/subtraction; multiplication/division – Other language issue?

What about the use of models/representations and grading? Daily classroom work, assessments?

Getting teachers to know when to move on… This is the issue…