Addition and Subtraction within Twenty Monica Moore

Common Core Expectations Experience with addition/subtraction within 20 is a Grade 1 standard; fluency is a Grade 2 standard Standards distinguish strategies from algorithms Standard algorithm viewed as the culmination of a long progression of reasoning with quantities, base-ten system and the properties of operations

Incorporating Mathematical Practices: Which appropriate? CCSS Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

Research Based on CGI Cognitively Guided Instruction Not a curriculum; focuses on teacher knowledge Understanding student thinking allows teacher to help students build on the knowledge they already have Intent is students are make sense of what they are doing; challenged to think; and take responsibility Classroom becomes a learning community

Problem-based; Student-Centered Problems generated by a book, content from another unit the students are studying or something relevant to the students lives Teachers make instructional decisions based on the student’s thinking The research identifies progression through types of problems and strategies

Example of Problem Types There were some children at the park. Four more children showed up. Then there were 11 children at the park. How many children were at the park to start with? There were 7 children at the park. Then 4 more showed up. How many children were at the park all together? There were 7 children at the park. Some more showed up. Then there were 11 children in all. How many more children came? [Problems from Illustrativemathematics.org]

Prior Knowledge Students must have experience with counting and cardinality (Know number names/counting, able to compare numbers). Students must be able to represent numbers 1 to 20 using concrete objects, especially 2-digit numbers to enhance the concept of place value [Numeracy Project] Place value developed WHILE learning meaning of the operations. Concept of place value integrated into the problems solving as they explore and make use of base-ten blocks, counting frames and other base-ten materials

Symbolic representation Students will develop conceptual understanding of what an equal sign represents. [ Example of activities that promote this are balance [How many more fish]

Types of Addition Problems Result Unknown: Add one addend to another Robin had 4 toy cars. Sue gave her 7 more cars for her birthday. How many cars did she have then? Change Unknown: Second addend unknown Robin has 4 toy cars. How many more cars does she need to get for her birthday to have 11 cars all together? Start Unknown: First addend unknown Robin has some cars. She got 7 cars for her birthday. She now has 11 cars. How many did she have before her birthday?

Types of Subtraction Problems Separation From: Result Unknown Roger had 13 stickers. He gave 9 stickers to Colleen. How many stickers does Roger have left? Separating: Change Unknown Roger had 13 stickers. He gave some to Colleen. He has 4 stickers left. How many stickers did he give to Colleen? Separating: Start Unknown Roger had some stickers. He gave 9 to Colleen. He has 4 stickers left. How many did he have to start with?

Part-Part-Whole Problems Whole Unknown Tim has 3 red marbles and 4 blue marbles. How many marbles does he have? Part Unknown Tim has 7 marbles. 3 are red and the rest are blue. How many blue marbles does he have?

Compare Problems Difference Unknown Quantity Unknown Referent Unknown Ella has 9 pieces of candy. Trey has 4 pieces of candy. How many more pieces does Ella have than Trey? Quantity Unknown Ella has 9 pieces of candy. Trey has 5 pieces less than Ella. How many does Trey have? Referent Unknown Ella has 9 pieces of candy. She has 5 more than Trey. How many does Trey have?

Direct Modeling This is the first stage students go through when approaching addition or subtraction. Build understanding of operations as actions Students use concrete objects to physically model the action taking place within a story problem. Can become diagrams (pictures) such as bar graphs Easier when discrete objects vs. measurement (a unifix cubes represents a marble vs. an inch0

Matching (Comparison) Construction of one-to-one correspondence between two sets until one set is exhausted. Example: Mark has 6 mice. Joy has 11 mice. Joy has how many more mice than Mark? Trial and Error (Completion) The start is unknown. Example: Robin had some toy cars. Her friends gave her 5 more toy cars for her birthday. Then she had 11 toy cars. How many toy cars did Robin have before her birthday?

Counting Strategy Counting strategies are used when students recognize that they do not need to physically model the action(s) taking place within a story problem. Students may use fingers, counters, or tally marks to determine the sum or difference.

Types of Counting Strategies Counting on from first. A child begins counting forward from the first addend in the problem. Counting on from larger. A child begins counting with the larger of the two addends.

Derived Facts Based on the understanding of relationships between numbers. [different from memorized facts] Six frogs were sitting on lily pads. Eight more frogs joined them. How many frogs were there total? To solve this using derived facts, a child may use 6+6 (doubles) to reach twelve, and then add 2 more.

General Instructional Strategies Provide many experiences for students to construct strategies to solve the different problem types. Begin with concrete materials in order for direct modeling to take place, allowing the students to build their own understanding of the operations. Move through problem types. Opportunity for differentiated instruction in ways problems are worded: making action clear; sequencing with action Sue had some marbles. Emma gave her 7 more. Sue now has 15. How many did she start with? Emma gave Sue 7 marbles. She now has 15. How many did she have to start with?

Instructional Strategies Con’t Provide multiple and varied experiences that will help students develop a strong sense of numbers based on comprehension-not rules and procedures. Teachers should constantly be observing the students’ methods and strategies and make variations to the instruction in order to enhance their own approaches. Student share strategies; learning from each other

Misconceptions Students frequently have trouble in joining, start unknown problems when the computation involves completion-type subtraction, however, is perceived as addition. [teaching key words] Sandra has some silly bands, and Charlie gives her five more. She then has eleven total silly bands. How many silly bands does she start with? Sandra has 6 silly bands. Charlie has five silly bands. How many more does Sandra have then Charlie?

Misconceptions Students often mistakenly name the subtraction, or minus, sign for takeaway; however, takeaway is only one of four different subtraction situations. [importance of language] Many young children often mistake an equal sign as another operation, and do not fully understand that it is similar to a balancing act [language “is the same as” ]

CGI approach Correlation between teacher knowledge and student achievement Significantly higher in problem solving No loss on standardized achievements on fluency of number skills Allowing students to struggle

Activities Ten-frame Five-Frame Grouping and Grazing (Adding and Subtracting) How Many Under the Shell? Number Stories Balance Equations Dominoes 7 Up Game

Activities Dominoes Domino Math video 6 + 4 = 10

Activities Pan Balance Strengthen understanding and computation of numerical expressions of equality. Equality is a relationship, not an operation.

Literature Books 12 Ways to Get to 11 by Eve Merriam A Chair For My Mother by Vera Williams How Many Feel in a Bed? by Diane J. Hamm How Many Snails? by Paul Giganti Rooster’s Off to See the World by Eric Carle Ten Sly Piranhas: A Counting Story in Reverse by William Wise The Giraffe That Walked to Paris by Nancy Milton What Comes in 2’s, 3’s, & 4’s? By Suzanna Aker Equal Shmequal by Virginia Kroll