1. How to Help Struggling Student’s
Obtain Math Fact Fluency

2. Three Phases of Fact Fluency:
• Phase 1: Counting Strategies – using object
counting or verbal counting to determine an
answer
• Phase 2: Reasoning Strategies – using known
information to logically determine the answer
of an unknown combination
• Phase 3: Mastery – efficient (fast and
accurate) production of answers

3. Two Approaches to Fact Fluency:
• Conventional Wisdom: Mastery grows out of
memorizing individual facts by rote through
repeated practice and reinforcement.
• Number-Sense View: Mastery that underlies
computational fluency grows out of
discovering the numerous patterns and
relationships that interconnect the basic
combinations.

4. Conventional Wisdom:
• Believe that phase 1 (counting strategies) and
phase 2 (reasoning strategies) are not
necessary for achieving the storehouse of
facts that is the basis of combination mastery.

5. Number Sense View:
• Phase 1 and phase 2 play an integral and
necessary role in achieving phase 3 mastery of
basic number combinations is viewed as an
outgrowth or consequence of number
sense, which is defined as well-interconnected
knowledge about numbers and how they
operate or interact.

6. What does this mean for struggling
learners?
• According to conventional wisdom, learning
difficulties are due largely to defects in the
learner.
• According to the number-sense view, they are
due largely to inadequate or inappropriate
instruction.

7. Struggling Learners and
Conventional Wisdom:
• Belief that learning difficulties are attributed largely or
solely related to their cognitive limitations.
– Heavy reliance on counting strategies
– Have capacity to learn reasoning strategies but an
apparent inability to spontaneously invent such strategies
– Inability to learn or retain basic number
combinations, particularly those involving numbers greater
than 5
– A high error rate in recalling facts
*Struggling learners appear to get stuck in phase 1 (counting) of
number-combination development. These learners can sometimes
achieve phase 2 (reasoning), at least temporarily, if they are taught
reasoning strategies directly. In this approach many struggling
learners never achieve phase 3 (mastery).

8. Struggling Learners and
Number-Sense:
• Believe for two reasons struggling students have difficulty mastering the
basic combinations:
– They lack adequate informal knowledge, which is a critical basis for
understanding and successfully learning formal mathematical and devising
effective problem-solving reasoning strategies.
– The conventional approach makes learning the basic number combinations
unduly difficult and anxiety provoking. The focus on memorizing individual
combinations robs children of mathematical proficiency.
• Far more challenging than meaningful and many give up on learning the basic
combinations. (inefficiency)
• When children focus on memorizing facts by rote instead of making sense of
school mathematics or connecting it with their knowledge, they are more prone
to misapply this knowledge because they make no effort to check themselves or
they miss opportunities for applying what they do know (inappropriate
applications)
• When instruction does not help or encourage children to construct concepts or
look for patterns or relationships, they are less likely to spontaneously invent
reasoning strategies, and thus they continue to rely on counting strategies.
(inflexibility)

9. So . .How do we Help Struggling Learners:
• Conventional Wisdom:
– Recommend focusing on short-term, direct
approach
• Number-Sense:
– Recommend a long-term, indirect approach

10. Conventional Wisdom Suggestion:
• Extensive drill and practice
– Can be known as “over-learning”

11. Number-Sense Suggestion:
• Efforts to promote computational fluency be
intertwined with efforts to foster conceptual
understanding, strategic mathematical
thinking and a productive disposition.
– This includes four instructional implications :

12. Four Implications:
1. Patiently help children construct number sense by
encouraging them to invent, share, and refine informal
strategies.
• Help children gradually build up big ideas, such as composition
and decomposition
2. Promote meaningful memorization or mastery of basic
combinations by encouraging children to focus on looking
for patterns and relationships.
• Concentrate on Fact Families
• Encourage children to build on what they already know (build
on if they already know doubles)
• Use different reasoning strategies which may require different
approaches (may require more structured discovery learning
activities)

13. Four Implications Continued:
3. Practice is important, but use it wisely.
– Use practice as an opportunity to discover patterns and
relationships
– Focus on making reasoning strategies more automatic, not on
drilling isolated facts.
– The learning and practice of number combinations should be
done purposefully.
– Practice to ensure that efficiency not be done prematurely –
that is before children have constructed a conceptual
understanding of written math and had the chance to go
through the counting and reasoning phases.
4. Students should be encouraged to use a variety of
strategies including automatic or semi-automatic rules
and reasoning processes.

14. In Conclusion:
• Using the guidelines for meaningful inquiry-based, and
purposeful approach can also help students achieve
the other aspects of mathematical proficiency:
conceptual understanding, strategic mathematical
thinking, and a productive disposition toward learning
and using mathematics.
• Such an approach can help all children and may be
particularly helpful approach for children who have
been labeled as struggling learners, but who do not
exhibit hard signs of cognitive difficulties.

15. 10 Minute Math Protocol Ideas:
• After initial lessons in math fact strategies are introduced,
teachers need to guide students through phase 2 (reasoning
strategies) and phase 3 (mastery) working toward quick
recall in quick 10 minute activities where specific strategies
are developed for students.
– The activities are teacher-led, student-centered, and essential
for developing basic fact fluency and focus on:
• Build Number Sense
• Build Vocabulary
• Build Conceptual Knowledge
• Build Mathematical Modeling Skills
• Build Procedural Fluency
• Build Strategic Competence
• Build Strong Mathematical Disposition

16. Some Common Strategies to Use:
• Doubles
• Doubles Plus 1
• Half Facts
• Number Line It
• Ten More
• Ten Less
• Multiples
• Counting Up

17. How/When to Use:
• Whenever a small chunk of time exists:
– Morning Opening
– Part of Bell Ringer/Do Now
– During Calendar
– Before Lunch
– Before/After Recess
– During Bathroom Break/Snack
– End of the Day