Place value systems represent numbers using positional notation, with the value of each digit dependent on its place. However, place value is very abstract for young students. Seeing each place as a shape provides a visual foundation for number sense. Representing numbers as combinations of cubes, squares, and other shapes for each place makes place value more concrete and intuitive to understand. This place shape method works for any number system, not just base ten, and can help students more easily grasp formulas, algebraic manipulation, and advanced math concepts.

1. Place Shape vs. Place Value
A visual foundation for math
A
Dream Realizations
Presentation
with
Cognitive Engineering
Research Laboratories

2. http://www.brothersoft.com/free-rainbow-screensaver-76363.html
Dialog modified from the movie Kate & Leopold (2001)
climate.met.psu.edu

3. CEO of Dream Realizations - Rebecca Reiniger
Alexsis (Anna-Marie), Cooper, and Ute (Rebecca) in Second Life.

4. Creative Director of CERLabs - Cooper Patterson
What is cognitive engineering?
www.osdata.com/system/physical/basics.htm
Functional brain map from Science
Daily by Daniel Oldjira Fufa.

5. The Dual Nature
of Numbers:
Cardinality &
Ordinality
The Impact of
Number
Language
Our Myopia with the
Decimal System
Counting, an
ordinal concept,
dominates our
early focus with
numbers.
The patterns in our
number words are
cognitively
inconsistent with
our minds’
number sense.
Our myopia with base ten interferes with
the discovery of place shapes and
subQuan meta patterns required for
formula derivation and algebraic
manipulations.
The
Three
Blind
Mice of
Number
Sense
Open
Your
Eyes
and
See!

6. Where is the world headed with numbers?

7. Single digit numbers of popular number systems
binary 0 g
number system 1 h
2 i
3 j
4 k
5 l
6 m
7 n
decimal 8 o
number system 9 p
a q
b r
c s
d t
hexadecimal e u
number system f v
duotridecimal*
number system is the
entire single digit set
shown.
* also called duotrigesimal

17. Place Value...
Very abstract relationship for youngsters.

18. 119
128
146
205
10A
224
1013
10102
137
Ten

19. nine
eight
six
five
ten four
three
two
seven
One-Ro

20. One Hundred 2D100A
1219
1448
2027
2446
4005
12104

21. 100A
1219
1448
2027
2446
4005
12104
One Hundred 3D

22. The subQuan four three one two is composed of:
4 Cubes, 3 Squares, 1 Seg, and 2 Ones

23. Cognitive engineering perspective
H
O
W
M
A
N
Y
?

24. Each Place Value also shows Shape.
The best part is that it works in ALL bases, not just base ten.

25. Place Shape!

26. If we verbalize the subQuan Algebraically:
4 ten Cubes, 3 ten Squares, 1 ten Seg, and 2 Ones

27. Compare the previous visual image to this excerpt from http://
en.wikipedia.org/wiki/Positional_notation
Try describing this to an elementary student!

28. Excerpt from http://en.wikipedia.org/wiki/Positional_notation

29. The Dual Nature
of Numbers:
Cardinality &
Ordinality
The Impact of
Number
Language
Our Myopia with the
Decimal System
Counting, an
ordinal concept,
dominates our
early focus with
numbers.
The patterns in our
number words are
cognitively
inconsistent with
our minds’
number sense.
Our myopia with base ten interferes with
the discovery of place shapes and
subQuan meta patterns required for
formula derivation and algebraic
manipulations.
The
Three
Blind
Mice of
Number
Sense
Open
Your
Eyes
and
See!

30. Great. We have a
foundation based on
Place Shapes.
What can one build
on place shapes?

31. "We are here"
Visit us in Second
Life to experience
the full effect of
3D visualization.
You can help us
discover all
that changes
when you start with
Place Shapes.