1. Importance of Procedural Fluency & Conceptual Understanding
USU STARS! GEAR UP
Sheryl J. Rushton
Weber State University

2. Mathematics Teaching Practice 6
Mathematics Teaching Practice 6. Build Procedural Fluency from Conceptual Understanding
Effective teaching of mathematics build fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems.

3. If a small gear has 8 teeth and the big gear has 12 teeth and the small gear turns 96 times, how many times will the big gear turn?
In a gear: Ratio of turns is inversely proportional to ratio of number of teeth
Turns of output gear/Turns of inout gear = number of teeth on input gear/number of teeth on output gear

4. Not just “Cross Multiply and Divide”!
Difficult for students who don’t understand what is meant by a particular proportional situation or why a given solution strategy works
Students often use more sophisticated reasoning when not doing “cross multiply and divide”

5. Define
Conceptual understanding: comprehension of mathematical concepts, operations, and relations.
Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

6. NCTM Position
To develop procedural fluency, students need experience in integrating concepts and procedures and building on familiar procedures as they create their own informal strategies and procedures. Students need opportunities to justify both informal strategies and commonly used procedures mathematically, to support and justify their choices of appropriate procedures, and to strengthen their understanding and skill through distributed practice. 
NCTM Standards & Positions, Statements/Procedural-Fluency-in-Mathematics/

7. Procedural fluency is more than memorizing facts or procedures, and it is more than understanding and being able to use one procedure for a given situation. Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving (NGA Center & CCSSO, 2010; NCTM, 2000, 2014).

8. 7.RP.1
7. RP.2
7.EE.4
8.EE.5

9. Which fish tank has more goldfish?
Tank A
Tank B
Find and defend two different answers to this problem.

10. Proportional Relationship
Essential Understanding 6:
A proportion is a relationship of equality between two ratios. In a proportion, the ratio of two quantities remains constant as the corresponding values of the quantities change.
Example
A clown walks 100 cm in 4 seconds. How far will the frog hop in 8 seconds if the frog travels at the same speed as the clown?

11. Student Solution Strategies
A clown walks 100 cm in 4 seconds. How far will the frog hop in 8 seconds if the frog travels at the same speed as the clown?
Student Solution Strategies
Unit Rates: 100 𝑐𝑚 4 𝑠𝑒𝑐. =25 𝑐𝑚/1 𝑠𝑒𝑐.;𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑦 𝑏𝑦 8 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Factor of Change: Travel twice as long, go twice as far.
Fraction Strategy: = 𝑥 8 → 𝑥 2 2 = 200 8
Cramer, Post, & Currier

12. Teachers need to step outside the textbook and provide hands-on experiences with ratio and proportional situations.
Initial activities should focus on the development of meaning, postponing efficient procedures (cross multiply and divide) until such understandings are internalized by students. 

13. Students need to see examples of proportional and nonproportional situations so they can determine that it is appropriate to use a multiplicative solution strategy.
If side issues are not raised, how can students be expected to be aware of and understand them?

14. Activity 1:
Ms. Adams, Mr. Barton, Ms. Crane, and Mr. Dahl

15. Activity 2:
Classy Limousine Company

16. Math Race
The kids work on practice problems together, and once a group has an answer, a member brings the work to the teacher for review.
If the answer is correct, the group gets a sticky note to place on a number board.
In addition, the group gets to shoot one “trashketball” shot.
A made shot results in an extra sticky note.
At the end of the practice, a randomizer is used to pick the winning number, and the corresponding group wins a prize.
Orr, 2014

18. Another great example:
“Concept First, Notation Last” – Leah Alcala

19. References
Cramer, K., Post, T., & Currier, S. (1993). Learning and teaching ratio and proportion: Research implications. In D. Owens (Ed.), Research Ideas For the Classroom (pp ). NY: Macmillan Publishing Company.
NGA Center & CCSSO, (2010). Common core state standards. Washington D. C.: NGA Center & CCSSO.
NCTM. (2010). Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics in Grades 6 –8. Reston, VA: NCTM.
NCTM (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
NCTM. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: NCTM.
Orr, J. (2014, April 15). Games in math. Retrieved from isageek.com/games-in-math/