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## Helping Children Make Sense of Numbers

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**Helping Children Make Sense of Numbers**Building Understanding, Increasing Flexibility, and Using Operations in K-2 Face-to-Face Session – October 8, 2015 Coweta Committed to Student Success**Agenda**• Sense making • Meaning of numbers • Operations in the early grades • Kindergarten • First Grade • Second Grade • Reflection Our vision is to ensure the success of each student.**Sense making**Young children show a natural interest in and enjoyment of mathematics. Research evidence indicates that long before entering school children spontaneously explore and use mathematics – at least the intuitive beginnings – and their mathematical knowledge can be quite complex and sophisticated. In play and daily activities, children often explore mathematical ideas and processes; for example, they sort and classify, compare quantities, and notice shapes and patterns. NAEYC & NCTM. (2002). We believe as a learning community, we must continually improve.**Meaning of numbers**Sequential order Learning outcomes Fluency with the counting sequence One-to-one correspondence between words and objects What does How many? mean? Start counting mid-stream • Counting words • Counting objects • Counting on We believe we must see students as volunteers in their learning.**Adding and subtracting**• Traditional Sense • Join or put together and find a total • Separate or take apart and find the difference We believe we must provide challenging, interesting, and satisfying work for students.**Operations in the early grades**K 2 1 K 2 1 K 1 K 1 1 1 1 2 2 We believe that we are responsible for the success of each student.**Kindergarten mastery**Type of problem Example A + B = ? C – B = ? A + B = ? C = ? + ? • Add to, total unknown • Take from, difference unknown • Put together/take apart, total unknown • Put together/take apart, both addends unknown We will provide high-level, engaging work for all learners and leaders.**Result unknown: Add to**A bunny was on the grass. Two more hopped over. 1 + 2 = How many bunnies are on the grass? 3 Coweta Committed to Student Success**Result unknown: Take from**There were four apples in the bowl. I ate one. 4 – 1 = How many apples are left in the bowl? 3 Our vision is to ensure the success of each student.**Total unknown: Put together**The bowl has two red apples and two green apples. 2 + 2 = How many apples are in the bowl? 4 We believe as a learning community, we must continually improve.**Put together/take apart: Both addends unknown**Grandma has six flowers, some red and some yellow. How many are red and how many are yellow? 6 = 3 + 3 6 = 2 + 4 We believe we must see students as volunteers in their learning.**Building place value foundation**We believe we must provide challenging, interesting, and satisfying work for students.**First grade mastery**Type of problem Example A + ? = C C – ? = A A + ? = C or C – A = ? A + ? = C or C – A = ? A + B = ? C – B = ? or ? + B = C • Add to, change unknown • Take from, change unknown • Put together/take apart, addend unknown • Compare, difference unknown • Compare, bigger unknown • Compare, smaller unknown We believe that we are responsible for the success of each student.**Change unknown: Add to**Carlos has two cars. He received some more cars for his birthday. Now he has five cars. 2 + = 5 How many cars did Carlos get for his birthday? 3 What happened? We will provide high-level, engaging work for all learners and leaders.**Change unknown: Take from**Katelyn has four dolls. She gave some to a friend. Now she has two dolls. 4 – = 2 How many dolls did Katelyn give her friend? 2 What happened? Coweta Committed to Student Success**Addend unknown: Put together**Grandma has ten flowers. Six are red and the rest are yellow. 6 + = 10 or 10 – 6 = . How many flowers are yellow? 4 4 Our vision is to ensure the success of each student.**Compare: Difference unknown**Abby has two books and Trevor has four books. How many more? How many more books does Trevor have than Abby? How many fewer? How many fewer books does Abby have than Trevor? Abby’s books Trevor’s books 2 + = 4 2 4 – 2 = 2 We believe as a learning community, we must continually improve.**Compare: Difference unknown**Abby has two books and Trevor has four books. How many more books does Trevor have than Abby? Abby’s books Trevor’s books 2 + = 4 2 This is an example of using tape diagrams (or bar models) which is helpful in future grades. We believe we must see students as volunteers in their learning.**Compare: Bigger unknown**Trevor has two more books than Abby. Abby has two books. How many books does Trevor have? Trevor’s books Abby’s books 4 2 + 2 = We believe we must provide challenging, interesting, and satisfying work for students.**Compare: Smaller unknown**Trevor has two more books than Abby. Trevor has four books. How many books does Abby have? Trevor’s books Abby’s books 2 4 - 2 = We believe we must provide challenging, interesting, and satisfying work for students.**Second grade mastery**Type of problem Example ? +B= C ? – B = A A + B = ? C – B = ? or ? + B = C • Add to, start unknown • Take from, start unknown • Compare, bigger unknown “How many fewer? • Compare, smaller unknown “How many more?” We believe that we are responsible for the success of each student.**Start unknown: Add to**Jackson had some books. He received three books as presents. Now he has seven books. How many books did Jackson have to start? ? books + 3 more books = 7 books altogether ? 3 7 = + We will provide high-level, engaging work for all learners and leaders.**Compare: Bigger unknown**How many fewer? Abby has two fewer books than Trevor. Abby has two books. How many books does Trevor have? Abby’s books Trevor’s books 2 + 2 = ? Coweta Committed to Student Success**Compare: Smaller unknown**How many more? Trevor has two more books than Abby. Trevor has four books. How many books does Abby have? Trevor’s books Abby’s books 4 - 2 = ? Our vision is to ensure the success of each student.**Summary**• Item types here focus on mastery grade levels. You can introduce them earlier when the context is appropriate. • Sequence moves the position of the unknown from simplest to more complex: • Use friendly numbers so the math is clearer. A + B = ? A + ? = C ? + B = C C – B = ? C – ? = A ? – B = A We believe as a learning community, we must continually improve.**Reflection**• Think about your instruction. Is there enough variety of problem types? • Are you working in an appropriate developmental sequence? We believe we must see students as volunteers in their learning.**References**Common Core State Standards Initiative (CCSSI). 2010. Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards.pdf Copple, C., & S. Bredekamp. 2009. Developmentally appropriate practice in early childhood programs serving children birth through age 8. 3d ed. Washington, DC: NAEYC. Retrieved from https://www.naeyc.org/store/files/store/TOC/375_0.pdf Falkner, K.P., Levi, L., & Carpenter, T.P. (1999). Children’s understanding of equality: A foundation for algebra. Teaching Children Mathematics, 5(6), 56-60. Georgia Department of Education. (2015). Georgia Standards of Excellence in Mathematics. Retrieved from https://www.georgiastandards.org/Georgia-Standards/Pages/Math-K-5.aspx National Association for the Education of Young Children [NAEYC], & National Council of Teachers of Mathematics [NCTM]. (2002). Early childhood mathematics: Promoting Good Beginnings. Joint position statement of NAEYC and NCTM. Retrieved from https://www.naeyc.org/files/naeyc/file/positions/psmath.pdf The Common Core Standards Writing Team. (2011). Draft K-5 progression on Counting and Cardinality and Operations and Algebraic Thinking. Progressions Documents for the Common Core State Standards.Retrieved from https://commoncoretools.files.wordpress.com/2011/05/ccss_progression_cc_oa_k5_2011_05_302.pdf Van de Walle. J.A., & Lovin, L.H. ( 2006). Teaching student-centered mathematics, grades K-3. Boston, MA: Pearson Education. We believe we must provide challenging, interesting, and satisfying work for students.