1. Math Facts through Conceptual Understanding
Emilie Schiff
K-5 Math Specialist and Instructional Coach.
Lake Orion Community Schools

2. Making the Case… Focus on number sense!
Research indicates that early number sense predicts school success more than other measures of cognition like verbal, spatial or memory skills or reading ability.
Jordoan, Kaplan, Locuniak, and Ramineni, 2007
Why are we talking about fluency and number sense

3. Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts By Jo Boaler Each take a paragraph. What stood out. Highlight, underline, mark up…Be prepared to share your M.I.P. Professor of Mathematics Education, co-founder youcubed with the help of Cathy Williams, co-founder youcubed, & Amanda Confer Stanford UNIVERSITY
Mark It Up.

4. What is Mastery of Basic Facts:
Addition and subtraction by the end of 2nd grade
Multiplication and division by 3rd grade
Fact fluency leads to other math fluent skills…
Imbedded in the Common Core-Standards for all students-
Quick response (3 seconds) w/o resorting to inefficient methods such as counting

5. Guided Intervention Fact mastery will not magically happen.
“Time is a poor intervention.”
Parents can help, by showing your children your active strategies, and your thinking aloud.
Teachers who relied heavily on textbooks that focused on memorizing basic fact strategies had students with lower number-sense proficiency.

6. Developmental Nature of Basic Fact Mastery
Counting
Reasoning
Mastery
Teaching basic facts requires essential understanding that student’s progress through stages that eventually result in “just knowing”. Listed are the three phases:
Phase 1: counting strategies
Phase 2: reasoning strategies-using known information to logically determine an unknown combination , student knows 3 +7 = 10 so just adds on 1 more.
Phase 3: mastery producing answers efficiently, just knows it, = 11.

7. Phase 1: Counting Strategies Counting strategies-
using concrete objects
verbal counting
Example includes:
5 + 1 = count on starting at 5, 6
5 + 2 = 5, 6, 7
4 +7 = 7 then 8, 9, 10, 11

8. Phase 2: Reasoning Strategy Reasoning Strategy:
using known information to logically determine an unknown combination
4 + 7, student knows 3 +7 = 10 so just adds on 1 more.
10 x 8 is 80 half of that is 5 x 8 is 40

9. Mastery Phase 3: Producing answers efficiently
just knows it
4 + 7 = 11
Retrieval within 3 seconds
Grade appropriate strategy
Efficient
In first grade knowing , (5 plus 1, plus 1) more would be seven would not be grade appropriate at 5th grade
Counting by 5’s is good to get to 20, but not 2000

10. Knowing Facts from Memory- “Passive Storage View”
Thinking that students will learn their facts if they just practice enough
What does this mean-
100 isolated addition facts
100 isolated multiplication facts
Also having to memorize subtraction and division- well over 300 pieces of information to remember
Not only having to memorize, but to keep practicing as well.
Strong evidence that this method does not work well! Think of how many students simply don’t have their facts mastered. People may say, well I did it, but early research shows that those who mastered facts used some strategy to remember those facts.
Inefficiency. There are too many facts to remember.

11. Effects of not knowing facts fluently…
Lack of basic fact automaticity has been shown:
Limit participation in math class discussions
Impede successful problem solving
Severely impair the development of the standard algorithms for multiple-digit addition and subtraction, long division and fractions
Misapply facts and not seeing reasonableness of answers
Inflexible thinking skills
If students don’t make the correct relationships by themselves they will often use incorrect relationships. An example is and 1 x 0
Inappropriate applications. Students misapply the facts and don’t check their work.
Inflexibly. Students don’t learn flexible strategies for finding the sums (or products) and therefore continue to count by ones.

12. Struggling Learners and Students with Disabilities:
Have difficulty memorizing so many isolated facts (but can be successful with strategies)
Drill creates, in a majority of students, unnecessary anxiety
Undermine student interest and confidence in mathematics
Connecting to what students already know allows students to learn basic facts for life.

13. Parent Supports
Scaffold the Language, but not the mathematic strategies. Explain the vocabulary they don’t know, in simpler terms, but don’t “dumb down” the mathematics.
Use Realia and Models
Use Graphic Organizers
Use gestures for together, take apart, groups etc.

14. Explicit Strategy Instruction:
Strategies can be effective to learning math facts.
Supports students thinking rather than give the students something new to remember.
Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting.
Strong evidence indicates that these strategies can be effective to learning math facts.
Supports students thinking rather than give the students something new to remember.
Teachers who relied heavily on textbooks that focused on memorizing basic fact strategies had students with lower number-sense proficiency.
Key: Help students see possibilities and let them choose strategies that help them get to a solution without counting.

15. Addition FACT Strategies
Use Doubles
Double
Double Plus 1
Double Plus 2
Bridge to 10
All Facts
Count On
Count On 1
Count On Turnarounds
Count On 2
Count On 3
Count On 0
Research shows that the most effective way for students to learn the basic facts is to arrange the facts into clusters. Each cluster is based on a thinking strategy that students can use to help them learn all the facts in that cluster. (Fuson, 2003, Thorton, 1990)
For example use doubles is on of the addition clusters. Within that cluster are double facts and facts that involve doubling then adding 1 or 2.

16. Prepare… Before teaching strategies students need to know…
Subitizing
The ability to instantly recognize the total quantity of objects in a group without counting
Small infants can subitize..
Trajectory state is developing states that by age 1-2 they should know 1, 2 and more
Many of our children come to school without any formal experience that allow them to build on this subiizing idea.

17. 1. Introduce Concept Through Real Objects Transitioning to Model
Counting bottles
Cubes in a cup
Addition stories
Count on 1 Cards

18. 2. Reinforce Through Models
Count on Cards
Cube Trains
Count on Cards with Numeral Cards
Does This Make Sense?
Moving from physical models to semi-abstract models to symbolic models (with numbers)
These are models that will continue to work with kids throughout their math cubes
Cube trains starts the work around arrays
Does this make sense starts students on the track of CCSSM college/career ready by having to justify their answers
“there is not a job out there that doesn’t require thinking! If it is automated, it is automated-we build machines to do the work…

19. 3: Practice through Games
Spin, Count On 1, Record Fact
Reinforce with Count On Flash Cards
Count On 1 Bingo
Total and Expression
Subitizing games/Other resources: Youcubed.org
sense/subitizing/
Send home as homework for family engagement

20. Multiplication Strategies
Use a Rule
One’s Facts
Zero’s Facts
Build Down and Build Up
Nine’s Facts
Six’s Facts
Last Facts
Use Tens
Five Facts
Doubling
Two’s Facts
Four’s Facts
Eight’s Facts

21. Multiplication Strategies
Use Tens
Five Facts
Doubling
Two’s Facts
Four’s Facts
Eight’s Facts
Use a Rule
One’s Facts
Zero’s Facts
Build Down and Build Up
Nine’s Facts
Six’s Facts
Last Facts

22. What to Do When Teaching Basic Math Facts
Ask students to self-monitor
Focus on self-improvement
Drill in short time segments
Work on facts over time
Involve families
Make drill enjoyable
Use technology
Emphasize the importance of quick recall

23. What Not to Do When Teaching Basic Facts
Don’t use lengthy timed test
Don’t use public comparisons of mastery
Don’t proceed through the facts in order from 0-9
Don’t work on all the facts at once
Don’t move to memorization too soon
Don’t use facts as a barrier to good mathematics
Don’t use fact mastery as prerequisite for calculator use.

24. Did we? Identify fact fluency is a developmental process.
Identify research based strategies that help develop fact fluency
Understand strategies that are ineffective for fact fluency development
Learn specific strategies that help develop fact fluency
Hand out games and additional resources from Boalers article.

25. Emilie Schiff: Elementary Math Specialist
Contact Information
Emilie Schiff: Elementary Math Specialist
Office (Main Office Hallway) is at Orion Oaks
Phone: (x 4010)