Understanding and
Remembering Times
Tables Facts: What Works?
Richard Harvey-Swanston
University of Brighton
@RHarveySwanston
What is the current practice in your school?
To what extent is it
‘rote learning’?
To what extent does it
support children in
developing a conceptual
understanding?
Rote Learning: Memorisation
based on repetition.
What are your criteria for having
‘learned’ you x times tables?
• Recall x facts in approx. 3 seconds
• Recall x facts out of order
• Recall ÷ facts out of order
• To know all common factors and multiples
• Be able to use the knowledge to derive x facts
greater than 12 x 12
Fluency in
calculation
Rote
Learning
Conceptual
understanding
Problem Solving
Aims
• To critique rote learning strategies
• To unpick key ideas within multiplication
• To identify models and methods which can support
children in learning their x tables and
The problems with rote learning...
• Knowledge is not linked
to understanding of
what multiplication is.
• Retention is often poor.
• It deals only with
symbolic
representation.
• ‘Interference’ occurs
between remembered
facts.
Interference
• “errors occur in
remembering by
accidental association
with other similar
knowledge.”
• E.g. 54 as the answer to 7
x8
Frobisher, L. and Threlfall, J. (2004)“Teaching and Assessing
Patterns in Number in the Primary Years,” p58, in Orton, A.
(Ed.), Patterns in the teaching and learning of mathematics,
New York: Continuum
• Interference increases as
we learn more.
• Facts which share
similarities with other
facts are more likely to be
remembered wrongly
(e.g. 6x4 as 7x4 or 6x3).
• Facts remembered
wrongly are more likely
to be remembered
wrongly in future.
Rote learning...
• Lacks connection with a conceptual understanding
of what multiplication is.
• Does not support calculation fluency.
• Causes anxiety and avoidance amongst many
children, parents and carers.
• For many children, it is not an effective strategy.
Multiplication and division represent a significant
qualitative change in children’s thinking
Nunes and Bryant, (1996), Children doing mathematics, p 144, Oxford: Blackwell
The size of a set
e.g. The number of
oranges in a set.
The number of sets e.g. The
number of bags of oranges.
3x4
Addition and subtraction are unary operations representing the
same kind of element... We need to view multiplication as a
binary operation with two distinctive inputs.
Barmby, P., Harries, T., Higgins, S., et al. (2008) “The array representation and primary children’s
understanding and reasoning in multiplication,” Educational Studies in Mathematics [Online], 70(3), p 1
Children also need to understand...
Commutativity
5x3=3x5
Distributivity
8x7=
(4 x 7) + (4 x 7)
Associativity
(3 x 4) x 5 = 3 x (4 x 5)
12 x 5 = 3 x 20
Using Cuisennaire, multilink or
squared paper, can you show that:
3x5=5x3
Notice: What language are you
using?
8x7
= (4 x 7) + (4 x 7)
= (5 x 7) + (3 x 7)
(2 x 3) x 4 = 2 x (3 x 4)
3x5=5x3
Notice: What language are you
using?
3 rows of 5
=
5 rows of 3
8x6
= (4 x 6) + (4 x 6)
= (5 x 6) + (3 x 6)
(4 x 6) + (4 x 6)
(5 x 6) + (3 x 6)
(2 x 3) x 4 = 2 x (3 x 4)
(2 x 3) x 4
2 x (3 x 4)
What is your approach to solving
18 x 5?
https://www.youcubed.org/from-stanford-onlines-how-to-learn-math-forteachers-and-parents-number-talks/
http://thegriddle.
net/handouts/m
ult_color.pdf
Developing Multiplication Fact
Fluency, (2015) Brendefur et al.
• Intervention based on
physical models and
social interaction.
• In a 5 week period,
groups undertaking this
intervention gained an
extra 6 facts per minute
compared to rote
learning groups.
Fluency was
“approximately 3
seconds” in a
timed test.
Developing x fact fluency
1. Building arrays with physical models, finding
arrays in their environment and drawing diagrams
of arrays.
2. Making arrays on 12 x 12 square grids. Students
overlaid their derived strategies.
3. Without any materials, children had to discuss
how they might use related facts to solve a range
of unknown facts.
4. Pairs created sets of strategy cards, with a fact on
the front and the pair’s preferred derived fact
strategy on the reverse
Brendefur, J., Strother, S., Thiede, K., et al. (2015) “Developing Multiplication Fact Fluency,” Advances in Social
Sciences Research Journal [Online], 2(8), pp.142–154
https://www.scholastic.com/teachers/blog-posts/alycia-zimmerman/total-recall--helping-our-studentsmemorize-multiplication-facts/
http://tonyastreatsforteachers.blogspot
.co.uk/2012/10/i-cant-believe-ittheygot-it.html
How Close to 100?
You will need:
• two players
• two dice
• recording sheet
1. This game is played in partners. Two children share a blank 100 grid.
The first partner rolls two number dice. The numbers that come up
are the numbers the child uses to make an array on the 100 grid.
They can put the array anywhere on the grid, but the goal is to fll up
the grid to get it as full as possible. After the player draws the array
on the grid, she writes in the number sentence that describes the
grid.
2. The second player then rolls the dice, draws the number grid and
records their number sentence.
3. The game ends when both players have rolled the dice and cannot
put any more arrays on the grid.
4. How close to 100 can you get? How far way were you? How did you
calculate it?
So what can we do...
To develop conceptual
understanding
Connect x facts to
representations and the
repeated addition structure of
multiplication.
To develop fluency
Encourage children to share
the different ways they might
calculate a x table fact.
To develop fluency
Develop an understanding of
meaningful patterns.
To what extent to these
patterns/techniques support
development of:
•Fluency
•Understanding the concept of
multiplication
Meaningful Patterns
1 x 9 = 10 – 1
2 x 9 = 20 – 2
3 x 9 = 30 – 3
4 x 9 = 40 – 4
5 x 9 = 50 – 5
x 8 = 10 – 2
2 x 8 = 20 – 4
3 x 8 = 30 – 6
4 x 8 = 40 – 8
5 x 8 = 50 – 10
1
P 91. Frobisher, L. and Threlfall, J. (2004) “Teaching and Assessing Patterns in
Number in the Primary Years,” in Orton, A. (Ed.), Patterns in the teaching and
learning of mathematics, New York: Continuum International Publishing Group
Pattern in the 6 times table p2. Barclay, N. and Barnes, A. (2013) Big Ideas - An Idea With
Primary Potential?, [Online] . Available:
<https://www.atm.org.uk/write/mediauploads/resources/big_ideas.pdf>
Evaluation
• What was most useful to you?
• What could be improved?
Richard Harvey-Swanston
University of Brighton
@RHarveySwanston
[email protected]