1. edTPA Task 4 Boot Camp Spring 2016

2. What is required for students to be mathematically proficient? According to The National Research Council (2001), proficiency includes the following strands: Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition

3. Conceptual understanding comprehension of mathematical concepts, operations, and relations – Students should be able to: recognize, label, and generate examples of concepts use and interrelate models, diagrams, manipulatives, and varied representations of concepts identify and apply principles; know and apply facts and definitions compare, contrast, and integrate related concepts and principles; recognize, interpret, and apply the signs, symbols, and terms used to represent concepts

4. What does this look like in the classroom?

5. For grades 3 through 5, the use of zeros with place value problems is simple, but critical for understanding. "What is 20 + 70?" A student who can effectively explain the mathematics might say, "20 is 2 tens and 70 is 7 tens. So, 2 tens and 7 tens is 9 tens. 9 tens is the same as 90."

6. In grades 5 through 6, operations with decimals are common topics. "6.345 x 5.28 = 335.016" NOT CORRECT! One factor is greater than 6 and less than 7, while the second factor is greater than 5 and less than 6; therefore, the product must be between 30 and 42.

7. For grades 1 through 4, basic facts for all four operations are major parts of the mathematics curriculum. "What is 6 + 7?" Although we eventually want computational fluency by our students, an initial explanation might be "I know that 6 + 6 = 12; since 7 is 1 more than 6, then 6 + 7 must be 1 more than 12, or 13."

8. "What is 6 x 9?" "I know that 6 x 8 = 48. Therefore, the product 6 x 9 must be 6 more than 48, or 54."

9. Students should be able to use manipulatives or make drawings to show and explain their reasoning. Why is 5 an odd number? "5 is an odd number because I can't make pairs with all of the cubes (squares). 8 is an even number because I can make pairs with all of the cubes

10. Procedural fluency skill in carrying out procedures flexibly, accurately, efficiently, and appropriately – Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving (NGA Center & CCSSO, 2010; NCTM, 2000, 2014). – Effective teaching practices provide experiences that help students to connect procedures with the underlying concepts and provide students with opportunities to rehearse or practice strategies and to justify their procedures. Practice should be brief, engaging, purposeful, and distributed (Rohrer, 2009).

11. Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

12. How would students solve this problem?

13. Why don’t students choose the most efficient method? They don't really choose any method. They just apply an algorithm without thinking. They know how to use procedures but not always when to use them. They perform procedures accurately but not always flexibly or efficiently. In other words, they lack procedural fluency.

14. How can you help students increase procedural fluency? Model for students (and/or allow them to discover on their own and from each other) multiple solution strategies Establish connections between new topics such as proportions (and cross-multiplying) and previous topics such as equivalent fractions. Build procedural understanding through conceptual understanding whenever possible

15. As students work toward addition and subtraction fluency they may also use such mental math strategies as: Counting on: 8 + 4 = □ (8 …9, 10,11,12) Counting back: 12 – 4 = □ (12…11, 10, 9, 8) Making tens: 5 + 7 = □ (5 = 2 + 3 so 3 + 7 = 10 therefore 10 + 2 = 12) Doubles: 6 + 6 = □ Etc…

16. As they work toward fluency in multiplication and division they may use such mental strategies as: Doubles (2 x 2 = 2 + 2) Double and double again (4 x 2 = (2 x 2) x 2) Halve, then double (6 x 8 = (3 x 8) + (3 x 8)) Doubles plus one more set (3 x 7 = (2 x 7) + 7) Add one more set (6 x 7 = (5 x 7) + 7) Etc…

17. Mathematical Reasoning and/or Problem Solving Reasoning enables children to make use of all their other mathematical skills and so reasoning could be thought of as the 'glue' which helps mathematics makes sense. – One uses it to navigate through the many facts, procedures, concepts, and solution methods and to see that they all fit together in some way, that they make sense. – Mathematical reasoning provides opportunities for students to develop and express insights about the mathematical competencies that they are developing. Problem solving allows students to draw on the competencies that they are developing to engage in a task for which they do not know the solution.

18. Resources http://blogs.edweek.org/teachers/coach_gs_teaching_tips/ 2012/07/procedural_fluency_more_than_memorization.ht ml http://blogs.edweek.org/teachers/coach_gs_teaching_tips/ 2012/07/procedural_fluency_more_than_memorization.ht ml Hiebert, J. (1999). Relationships between research and the NCTM standards. Journal for Research in Mathematics Education, 30(1), 3–19. Rohrer, D. (2009). The effects of spacing and mixed practice problems. Journal for Research in Mathematics Education, 40(1), 4–17. http://www.mathleadership.com/sitebuildercontent/sitebu ilderfiles/conceptualUnderstanding.pdf http://www.mathleadership.com/sitebuildercontent/sitebu ilderfiles/conceptualUnderstanding.pdf http://nrich.maths.org/10990