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Mental Math Part 1:
Fact Learning
Mental mathematics (that is, calculating mentally without the use of external
memory aids) is a combination of cognitive strategies that enhances flexible
thinking and number sense. It improves computational fluency by developing
efficiency, accuracy, and flexibility.
Mental Computation Components
Mental Computation
May 2016
Fact
Learning
Mental
Calculations
Computational
Estimation
Fact Learning
1, 2, 3 –
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Fact learning refers to the acquisition of addition, subtraction, multiplication, and
division number facts (limited to single digits 0–9).
Learning math facts is a developmental process where the focus of instruction
is on thinking of and building number relationships. Recall of facts becomes
automatic for students through repeated exposure and practice.
Arthur Baroody identifies three stages through which students typically progress
in acquiring basic facts:
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Counting Strategies: using object counting (e.g., blocks, counters, fingers) or
verbal counting to determine the answer
Reasoning Strategies: using known information (i.e., known facts and
relationships) to logically determine the answer of an unknown fact
Automaticity or Mastery: efficient (fast and accurate) production of answers
(Baroody, 2006, p. 22)
Teaching the Facts
The following are a few principles for teaching the facts that were provided
in A Guide to Effective Instruction in Mathematics by the Ontario Ministry of
Education:
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Most students can learn the basic facts accurately, although their
speed may vary considerably. Students should have many experiences
modelling facts using concrete and pictorial representations.
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7
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2
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4
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3
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2
8
0
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6
4
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7
5
4
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2
6
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4
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9
6
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6
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7
5
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6
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5
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8
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6
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2
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3
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Students should be encouraged to look for patterns and relationships
between the operations and numbers in the facts.
Students need strategies that help them reason their way to the solutions
for the facts, rather than strategies for memorizing the facts.
Students should not be compelled to memorize facts if they have limited
strategies for solving facts. (It can be a waste of time, and it limits the
opportunity for students to learn the whole range of fact strategies they
will use throughout their elementary-school years.) Students who have a
repertoire of strategies will be able to find an accurate answer, and over
time their speed will naturally increase. (Ontario Ministry of Education,
2006, p. 13)
Assessing the Facts
“Evidence strongly suggests that timed tests cause the early onset of math
anxiety for students across the achievement range.” (Boaler, 2014)
Basic facts should be assessed through observation, interviews, games, selfassessment, and strategy-focused paper/pencil practice. Although the goal is to
have students recall facts in a reasonable time frame, researchers strongly caution
against the use of timed tests. “Children who tended to use more sophisticated
mathematical strategies experienced the most negative impact on achievement
due to math anxiety. Thus, it appears that some of our best mathematical thinkers
are often those most negatively affected by timed testing.” (Ramirez et al., 2013)
Reflection and Discussion
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How are basic facts currently being taught and assessed?
Can we develop a robust set of assessment tools and strategies (beyond timed
tests) in order to determine student progress toward fact recall?
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What supports do you need related to basic facts?
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What difficulties have you experienced teaching basic facts?
Bibliography
Baroody, Arthur J. Why Children Have Difficulties
and Division: Strategies, Activities &
Mastering the Basic Number Combinations
Interventions to Move Students Beyond
and How to Help Them. Reston, VA: National
Memorization. Portsmouth, NH: Heinemann,
Council of Teachers of Mathematics, 2006.
2014
Available online at http://webmedia.jcu.edu/ ———. Mastering the Basic Math Facts in
cmsett/files/2014/06/baroody-facts-article.pdf
Addition and Subtraction: Strategies, Activities,
Boaler, Jo (2014). “Research suggests timed
and Interventions to Move Students Beyond
tests cause math anxiety.” Teaching Children
Memorization. Portsmouth, NH: Heinemann,
Mathematics 20.8: 469–473.
2015
Ramirez, Gerardo, Elizabeth A. Gunderson,
Susan C. Levine, and Sian L. Beilock (2013).
“Math Anxiety, Working Memory, and Math
Achievement in Early Elementary School.”
Journal of Cognition and Development, 14(2):
187–202. Available online at http://dx.doi.org/
10.1080/15248372.2012.664593
O’Connell, Susan, and John SanGiovanni.
Mastering the Basic Math Facts in Multiplication
Ontario Ministry of Education. A Guide to Effective
Instruction in Mathematics, Kindergarten to
Grade 6, Volume Five: Teaching Basic Facts and
Multidigit Computations. Toronto, ON: Province
of Ontario, 2006. Available online at http://
eworkshop.on.ca/edu/resources/guides/
Guide_Math_K_6_Volume_5.pdf (English)
and www.atelier.on.ca/edu/resources/guides/
GEE_math_M_6_fasc5.pdf (French)