Oct 30, 2012
4:30

6:30
Specially Designed Instruction in
Math PDU Session Two
Text
Chapter Two: Learning to Calculate
Outcomes for Session One
Participants will review the foundational knowledge of the
psychological processes of math through the development of
number sense
Participants will deepen their understanding between natural
number sense and invented calculation tasks to solve
problems that begins with multiplication
Review the math psychological
processes
“The human brain has serious problems
with calculations. Nothing in its evolution
prepared it for the task of memorizing
dozens of multiplication facts or for carrying
out the multiple step operations required for
two

digit subtraction. Our ability to
approximate number quantities may be
imbedded in our genes, but dealing with
exact symbolic calculation can be in error
prone ordeal.”

Sousa
Example of CRA with Multiplication
Rita’s class is having a bake sale. Each
student in Rita’s class will bring 12 treats to
sell. There are 25 students in Rita’s class.
How many treats will the class bring in
altogether?
Concrete
Materials needed

Base Ten Blocks

Masking Tape
Concrete
Fill in the intersection of each
row and column using a block
that as the same dimensions
Representation
Abstract
By age 4, children have created, refined, and selected
algorithms for basic arithmetic (Griffin 2002)
Every two years the
child’s number sense
is reorganized based
on experience to
understand more
complex tasks.
This progression is predictable in
about 80% of the population. 20%
are slower and 20% are faster
.
4 year olds Operational Sense
Global Quantity Schema
Initial Counting Schema
more than
less than
1
2
3
4
5
Requires Subitizing
Requires one

on

one
Correspondence
6 year olds Operational Sense
Internal Number line has been developed
This developmental stage is a major turning point because
children come to understand that mathematics is not just
something that occurs out in the environment but can also occur
inside their own
heads.
1 10 20 30 40 50
a little
a lot
8 year olds Operational Sense
Double internal number line has been
loosley
developed to
allow for two digit operational problem solving
Loosely coordinated number line is developed to allow for understanding of place value
and solving double digit additional problems.
1 10 20 30 40 50
a little
a lot
1 10 20 30 40 50
a little
a lot
8 year old can now…
place value
mentally solve double

digit addition
know which of the double

digit numbers
are larger or smaller
read hours and minutes on a clock
solve money problems
solve balance

beam problems
10 year olds Operational Sense
Double internal number line has been well developed to
allow for two digit operational problem
solving
effeciently
These two well developed number lines allow for the capability of doing two digit
addition calculations mentally.
1 10 20 30 40 50
a little
a lot
1 10 20 30 40 50
a little
a lot
A 10 year old can …
Mental computation with double

digit
that involve borrowing and carrying
solve triple

digit calculations
translate hours to minutes
translate one monetary dimension to
another (quarters to nickels and dimes)
Multiplication: Natural or Invented?
What is multiplication?
Imaging studies show that the brain
recruits more neural networks during
multiplication than during subtraction.
3+3= 9
3X3= 12
Natural
vs
Invented
+

x
The big idea is that addition and subtraction are
somewhat natural skills that are accused for survival,
however multiplication is an invented tool and requires
additional work to conquer the skill.
Multiplication and Mistakes
•
The average adult
makes multiplication
mistakes about 10%
of the time.
•
Some multiplication
facts such as 8x7 and
9x7 can take up to 2
seconds longer to
solve and has an error
rate of 25%.
Why? Three factors …
•
Memory
•
Pattern and associations
•
Language
Memory
Memory and Multiplication
In a study in 1978 by Ashcroft they determined that
memory plays a critical factor when doing
calculations. It took less than a second to
determine the results of 2+3 or 2x3, but about 1.3
second two solve 8+7 or 8x7. Why?
Because…
•
The accuracy of our mental representations of
numerocity
drops quickly with increasing number size.
•
We remember best what we learned first. When we
begin learning our arithmetic fact, we started was
simple problems containing small digits.
•
Because small digits appear more frequently than
larger ones, we most likely receive much less practice
with multiplication problems involving larger numbers.
“Children in the primary
grades encounter a sudden
shift from their intuitive
understanding of numerical
quantities and counting
strategies to the rote learning
of arithmetic facts.
Unfortunately, most children
lose their intuition about
arithmetic in the process.”

Sousa
What is the tip on this bill?
20% tip
on
76.10
Memorization of multiplication
facts; Is this intuitive?
It still doesn’t solve the problem…
are easiest because it aligns with our intuitive understanding of number; base ten
system.
Memorization of multiplication
facts; Is this intuitive?
It still doesn’t solve the problem…
HOW?
Memorization of multiplication
facts; Is this intuitive?
It still doesn’t solve the problem…
This cuts the number to only 32 to
memorize.
This still doesn’t solve the problem.
So what does this mean for your
teaching?
Pattern Recognition and
Associations
Associative Memory
Associative Memory
Associative Memory
?
?
?
Associations interfere with
multiplication mastery
Listen to a
partner say
these facts out
loud.
What kind of
linguistic
information do
you hear?
6x9is54
7x8is56
8x8is64
The brains strong pattern

seeking ability detects the
rhythm of these entities
when said aloud, thus
making it difficult to keep
these three expressions
separate.
Carl Dennis lives on Allen Brian Avenue
Carl Gary lives on Brian Allen Avenue
Gary Edwards lives on Carl Edward Avenue
Who lives on Allen Brian
Avenue?
Where does Gary Edwards
live?
Name
Number
Allen
1
Brian
2
Carol
3
Dennis
4
Edward
5
Frank
6
Gary
7
Lives On
=
3x4=12
Carol Dennis lives on Allen Brian
Avenue
3x7=21
Carol Gary lives on Brian Allen
Avenue
7x5= 35
Gary Edwards lives on Carol
Edward Avenue
The brain’s ability to recognize patterns interferes with the child’s ability
to learn their multiplication facts. Learning multiplication actually
interferes with understanding addition.
In 1998 a study by Miller discovered that students in third grade took
more time to perform addition when they started learning the
multiplication tables, and errors like 2+3= 6 began to occur.
So what does this mean for your
teaching?
Language
Russian
English
Bilinguals
Language and Multiplication
25 x 30=
Exact
Approximate
This does not mean that students who struggle with language
processing will automatically struggle with mathematics, however
language plays a critical role in learning multiplication.
There is strong evidence to
suggest that the power of
language and verbal
memory can greatly
enhance the child’s
mastery of multiplication;
using poems and music are
examples of this power of
language
.
So what does this mean for your
teaching?
Next time we meet… 11

27

12
Chapter 7 How the Brain Learns Mathematics; Assessing math
learning disability; Best practice for remediation; CRA strategies
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