1. Building Foundations for Mathematics
Defining Numerical Fluency

2. Fluency versus Memorization
Are FLUENCY and MEMORIZATION the same thing? How are these ideas related/different?

3. Fluency in the CCSS
The word fluent is used in the standards to mean “reasonably fast and accurate.” Developing fluency in each grade may involve a mixture of:
1) just knowing some answers
2) knowing some answers from patterns, and
3) knowing some answers from the use of strategies.

4. What is Conceptual Understanding?

5. Conceptual Understanding and Procedural Fluency
When procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations (Fuson, Kalchman, and Bransford 2005). Martin (2009, p. 165) describes some of the reasons that fluency depends on and extends from conceptual understanding:
To use mathematics effectively, students must be able to do much more than carry out mathematical procedures. They must know which procedure is appropriate and most productive in a given situation, what a procedure accomplishes, and what kind of results to expect. Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results.
National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author. (p. 42)

6. Why Teach Facts using Strategies?
Straight memorization isn’t working!
Memorization is devoid of reasoning and sense- making.
These strategies are applicable to other contexts.

7. Memorizing provides a single path

8. Strategies provide a web of understanding

9. Table Talk
What is a take away you have about fluency and memorization?
How are you or will you altering the memorization/time test cycle?

10. Strategy Chart

11. Strategy Chart 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

12. Counting On (+1, +2) Activities/Games: Subitizing plates and dots
Shift from “counting all” to “counting on”
Activities/Games:
Subitizing plates and dots
Dot card addition
Count On Cup
Bump

13. Strategy Chart Counting On1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

14. Strategy Chart +0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

15. Plus Ten: Materials Useful Materials: Double Ten Frames Hundreds Chart
Base Ten Blocks
Unifix Cubes

16. Strategy Chart Plus Ten1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

17. Strategy Chart Doubles1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

18. Using Doubles (aka Doubles +1/-1)
4 + 5 = ? Think 4 + 4, and 1 more
Linking Cubes
Number Talks
Laminated cards
Games: Match ‘em Up

19. Strategy Chart Using Doubles (Doubles +1/-1)1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

20. Making Ten (combinations of 10)
Ten Frame Toss (Roll die, fill the frame)
Go Fish
Make 10 Solitaire (playing cards or 10-frames)
Bump
Moose Smack

21. Strategy Chart Making Ten (Combinations that make 10)2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

22. Using Tens
Decompose one number, thinking about how to “complete a ten”
Helpful materials: Double ten frames, Rekenrek
This thinking paves the way for other mental math and understanding regrouping.

23. Decomposition

24. Composing and Decomposing
Building and taking apart numbers
6+3=9
9: 1+8, 2+7, 3+6, 4+5
Looking for patterns/relationships between numbers
7 + 4 = 11 and = 11
any two-digit number can be broken up into tens and ones
(e.g., 35 = )
Unitizing numbers
Unitizing is the place value understanding that ten can be represented and thought of as one group of ten or ten individual units.
Using numbers as reference points
Benchmark or Friendly Numbers

25. Research
“…research evidence points in one direction: The best way to develop fluency with numbers is to develop number sense and to work with numbers in different ways, not to blindly memorize without number sense.”
Jo Boaler

26. Table Talk
What will you try this week, month, or trimester to help your students build fluency?

27. Resources