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FUNFORMS
A NEW MATH LEARNING SYSTEM
Words and symbols are the chief tools used in human thought.
What effects might new symbols have on our understandings of mathematics?
Funforms is a new mathematical numerical notation learning system. "Fun"
stands fo
r
fun
damental
fun
ctional and
fun
. "Forms" stands for
form
ulae.
Funforms are fundamental formulae that are fun to work with. They are
starkly simple. They allow math functions to take place mechanically on
paper. Thus, they are quite functional. A coll
eague of mine [Harold Larson]
and I developed the system more than 20 years ago.
HISTORY
: Before humankind began to use counting numbers, tally marks
were used. A tally mark simply consisted of a cut in a stick or a knot in a
piece of rope or a mark on a
piece of paper [or some other similar
representation]. Each knot or notch stood for a single object in a collection
on a one

to

one basis. It was not a true counting system, but it did allow for
a person to keep track of whether or not all members of a
group were still
present in the group when it was "recounted" [compared]. Later tally marks
probably were grouped for convenience in visualizing the total number of
members in a group.
Until Hindu

Arabic numerals were introduced to Europe in the 12
th
ce
ntury,
the numerical system in common usage in the western world was the Roman
numeral system. Most scholars agree that Roman numerals were suitable
only for writing down the results of calculations made on an abacus or by
using some other similar system.
Roman numerals were not easily
manipulable by the individual who was using them. Roman numerals served
well as a permanent record of the results of a calculation.
BENEFITS OF HINDU

ARABIC NUMERALS
: Hindu

Arabic numerals
were a major step forward. They
emphasized
place order
to indicate what
power of ten the particular
coefficient
of 10 being written or displayed stood
for. Many scholars have regarded the introduction of the 0, which kept the
place order if there were zero of that particular power of 1
0 in that particular
column as a step forward, but it also introduced problems such as dividing
by zero.
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Arabic numerals are arbitrary symbols, so their intrinsic meaning is not
immediately apparent. Their big advantage was/is that by learning a variety
of rules, they could be manipulated by the individual using them
and
they
still served the purpose of functioning as a peripheral memory/record
keeping system. They had all of the advantages of the Roman numerals
system plus the advantage of being manipul
able.
FUNFORMS
: Funforms are basically
binary
in design (although a trinary
system also exists). It is really a
place order, tally mark
system. The
place order is maintained by the structure of the glyph
, so zeroes are not
necessary. To our knowledge,
these two innovations make Funforms
unique
among numeral systems. Starkly simple in design, the idea of place
order is preserved, but
coefficients are unnecessary
except transiently
during manipulation. The meaning of the number is obvious to anyone with
passing familiarity. That is to say that the formulae or glyphs are
iconic
or
ideographic
. Most importantly, as the user applies Funforms,
mathematical operations become transparent
, and understanding the
nature of the mathematical transaction becomes a
pparent to the user.
Funforms is a
geometrically progressive
system as opposed to our own
number system, which is arithmetically progressive. In Funforms,
arithmetic progression is preserved if one looks at the exponents, however.
In the system in commo
n usage, we count 1, 2, 3, 4, etc. In Funforms we
count 1, 2, 4, 8, etc.
We believe that Funforms
can be taught to children as a game
. We
believe that there may be advantages for learners who have dyscalculia to
learn Funforms. We also believe that for
the general student and the gifted
Funforms will have benefits just like the benefit of learning a foreign
language. There is much to be learned about one's own language by
studying another language. Most scholars agree that words and symbols are
the ch
ief tools used for higher types of human thought. In the same way that
the available tools limit or enhance a carpenter’s ability to carry out chosen
activities, the tools available to have thoughts may limit or facilitate
thinking. That makes a new symbo
l system like Funforms potentially
enabling new and different conceptual formations to those who learn to use
it.
DEFINITIONS FOR USING FUNFORMS
:
In Funforms we use a vertical line called a
staff
.
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The staff has various possible positions ("points") at
which particular
number values can be indicated as present. These positions occur at
regular intervals. By using lined paper, each line can conveniently
serve as such a potential position.
A point is chosen to represent
unity point
.
A line perpendic
ular to the staff called a
flag
can be drawn at any of
the potential positions [points].
A flag drawn to the right of the staff at unity point has a numerical
value of one.
By convention
number values double
at each successive position
going down the s
taff.
Positive values
are drawn to the right of the staff.
Negative values
are drawn to the left of the staff.
All potential positions ("points") below unity point have a whole
number value that corresponds to a whole number power of 2.
All points
above unity point are fractional in nature and represent
whole number negative powers of 2.
Thus, going down successive points on the staff, the number values
would be:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, etc. [representing]:
2
0
, 2
1
, 2
2
,
2
3
, 2
4
, 2
5
, 2
6
, 2
7
, 2
8
, 2
9
, 2
10
, 2
11
, etc.
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Figure 0
This is how a group of Staffs look
SOME RULES NECESSARY TO MANIPULATE FUNFORMS
:
Recall that numerical values double each time a flag [or a series of
fl
ags] moves down one position [or set of positions] on the staff.
Numerical values halve each time a flag [or a series of flags] moves
up one position [or set of positions] on the staff.
No more than one flag can be at any one position (except temporarily
during manipulation). [That is, there is either
one
flag at any given
point (position),
or
there is
no
flag there.]
Any two flags at a position [point] are the equivalent of one flag at the
next position down.
(And conversely any single flag at a positio
n is the same as two flags
at the preceding position [point] going up.)
With that introduction, please let us begin counting:
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Figure 1
Note the repeating and expanding patterns. After getting to four, figures
from 1
to 3 are repeated [added on to four], before 8 is reached. Then all
figures up to 8 are repeated [added on to 8] before 16 is reached, and so on.
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Figure 2
TO REITERATE SEVERAL IMPORTANT POINTS FROM THE
DEFINITION AND RU
LES SECTIONS
:
Positive integers have flags to the right of the staff.
Negative integers have flags to the left of the staff.
By convention, the first position on the staff, which is marked by a
flag extending to the right, represents the number one. This
position
is called
unity point
.
Whole numbers are written at and below unity point.
Any flags written above unity point represent fractional numbers.
HOW TO USE FUNFORMS
BEYOND COUNTING
: Addition and
subtraction are easily carried out following the r
ules already learned:
ADDITION
is carried out by simply combining [or “coalescing”] whatever
number values are to be added from the individual figures or glyphs and then
simply "clearing" them by following the already learned rules that:
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No more than one
flag is allowed at any point [EXCEPT temporarily
during calculations].
Any two flags at one position [point] are the equivalent of one flag at
the next position down
Simple addition problems follow [See Figure 3, page 8.]:
Note the first [3] figure i
s
red
. It is made up of a flag at the 2 position
and a flag at the one position
The 5 figure is black. It is made up of a flag at the 1 and at the 4
positions.
In the third figure both the flags from 3
and
the flags from 5 appear.
[The figures have be
en combined or coalesced.] They appear in
exactly the
same positions
that they originally appeared in.
The third
figure is the result of
combining
the first two figures.
Color has
been
maintained.
[So there is be one red and one black flag at the
unity
position; one red flag at the two position and one black flag at
the 4 position.]
Then the task is to
clear the excessive flags
.
Remember that:
N
o more than one flag is allowed at any one position
[point] and
T
wo flags at any one
position are the same
as one flag at the next
position down
.
The two flags at the unity position [point] become one flag at the next
position down [the two position].
The other flags are simply re

copied to their proper positions.
Now there are two flags at the two posit
ion...but that is
not allowed
in
a final figure.
They become one flag at the next position down.
There is already a flag at that position [the four position]...but that is
not allowed in a final figure.
The two flags at the four position
become one flag
at the next position down, the eight position.
The FF figure is now in its simplest form and nothing further needs to
be or can be done.
That is the answer. [See Figure 3, page 8]
25 + 18
= and
14 + 34
= are shown using the same techniques in Figure 3
.
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Figure 3
Negative numbers are easily understood in terms of their positive
counterparts.
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SUBTRACTION
as an operation is intrinsically understood, and there is
no basic difference between adding a negative number and
subtracting a
positive number. The Funform figure to be subtracted is simply rewritten
with the flags extending to the left of the staff. Values that cancel one
another [a flag at each side of a given position] are dropped. If a left
pointing flag does
not have a suitable flag to the right, one is obtained by
moving a right pointing flag up to that level. [A flag at one level is the
same as two flags at the immediately preceding level up.] If there are no
flags available [no flag that can be moved up],
the resultant number after
clearing all possible flags has a negative value.
Pay particular attention to 32
–
1, please. [Fig. 4, page 10] Note that
whenever there is a long gap between a minus number value [in this case
–
1] and a positive one, exactly w
hat you see there happens. Each
intervening point becomes filled with a flag beginning one position up
from the lower positive flag. Each position extending to and including
the original negative flag ends up with a flag. You should only have to
see it
operate once to be able to apply it each time the situation presents
itself. I think of this like unzipping a zipper. Similarly, if you added 1 to
31, you would begin with 2 flags at the one position, which would
become 1 flag at the two position [where
there was already a flag], and
the flags would “tumble” down sequentially, leaving just one flag at the
thirty

two position. I think of those events mechanically like a zipper
closing or like a Jacob’s ladder.
Similarly, when you add one to 127, the resul
ts are to continuously find 2
flags at each succeeding [down] position as you cleared the problem.
This leads to a final flag one position after the last flag. There is no need
to keep doing it mechanically. It always happens that way.
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Figure
4
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Figure 5
MULTIPLICATION
simply amounts to serial addition. It is done by
writing the multiplicand at each position that a flag exists on the multiplier
[using the formulaic qualit
ies of Funforms], and then clearing the resultant
figure by the rules already learned. It helps to think about the multiplicand
in terms of its spaces and flags
beginning at unity point
. For example “10”
could be thought of as “space

flag

space

flag”. I
f 10 were to be multiplied
by 6, one would write space

flag

space

flag first at one of the two flags that
make up 6, and then by writing space

flag

space

flag at the level of the other
flag making up 6. [This use of the component parts of "6" has to do wit
h the
formulaic qualities of Funforms.] The two intermediate figures would then
be coalesced onto one staff and cleared according to the standard
procedures.
Another way to carry this out, which helps demonstrate the slide rule,
mechanical functions of Fu
nforms is to use a strip of paper. On the strip the
multiplicand is copied at one edge, beginning with attention to where unity
point is. The strip is then placed at the level of each flag of the multiplier on
a new staff (one new staff for each flag in
the multiplier). The multiplicand
is copied from the strip on to each new staff at each position of the flag on
the multiplier. The intermediate figures are then combined (coalesced) and
the figure is cleared. The reader should now try to carry out that
problem on
his/her own.
Now for some demonstrations of multiplication:
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“10” is read as “space

flag

space

flag”. It is first written at the position of
the two

flag of six [
red
], and then at the 4

flag of six [
blue
].
Fig
ure 7
I have attempted to “help” the reader by color coding the operations. That,
is unnecessary in real operations, but it may make the steps somewhat easier
to follow until the reader is familiar with the operations by having done
them.
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Figure 8
3 X 5=
and
6 X 7=
appear immediately above in Figure 8
In the first figure “5” is written at each flag of the formula for 3, namely 1
and 2. “5” reads “flag space flag” beginning at unity point. In the second
demonstratio
n in figure 8, “7” which reads flag, flag, flag is written first at
the 2

flag of “6” [red] and then at the 4

flag of “6” [blue]. The figures are
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combined, then cleared according to already learned rules. More difficult
multiplication problems are shown
below in Figure 9.
Figure
9
FRACTIONS
:
Consider the following Funform figures in Figure 10. In the first 24 is
repeatedly halved by moving it up one position at a time. As it passes unity
point, it becomes fractional,
at least in part. [24, 12, 6, 3,
3
/
2
,
¾
]
The same is true for "fiveness" written at various positions.
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Figure 10
Recall that numbers halve as they move up one space. Consider how “3”
and “5” are valued as they mov
e up and down the staff. [See Figure 10
above.]
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Now for some manipulation of
FRACTIONS
:
In
adding fractions
, there is no need to seek the lowest common
denominator.
Figure 11
5
/
8
[
red
] +
¾
[black] is added exactly lik
e you would add any Funform
figure. You do not need to know that they have fractional values to correctly
add them. [See Fig. 11 above.]
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Figure 12
7
/
8
+
1 ½
is added just like adding any other Funform figures.
In
mult
iplying fractions
, it is again wise to say to oneself how the
multiplicand looks, beginning at unity point. “
5
/
16
” for example would be
said aloud as “space

space

flag

space

flag” [again attending to unity point
and naming the features at each point ascend
ing the staff].
Multiplying fractions is done just like multiplying whole numbers, attending
to how the flags on the multiplicand relate to unity point. In Figure 13
below,
½
X
½
is shown.
½
can be thought of as space

flag. When space

flag is written a
t the
½
position, the result is
¼
.
The reader/learner is again reminded that during
multiplication of
fractions,
it helps to describe (in your mind or on a strip of paper) the multiplicand
beginning at unity point. “3/4” would be read “space, flag, flag
.”
[Ascending from unity point] That formula is then written at whatever
positions are dictated by the formulaic factors in the multiplier (the formula
of the multiplier). In figure 13, page 18,
¾
is written first at the
½
position
[red] and then at the
¼
position [blue]. The usual combining leads to the
next to the last figure, and by clearing the excess flags according to the rule
that 2 flags at any one position are the same as one flag at the next position
down, the final figure is found to be
9
/
16
.
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Figure 13
In
multiplying by fractions
, the only difference between fractions and
whole numbers is that somewhat more attention may need to be paid to the
location of unity point. It is unity point that determines where
the figure will
be written in the transitional figures. One says “space, space, flag, space,” or
whatever
from unity point
to describe the Funform figure to be multiplied.
DIVISION
like multiplication, for any whole power of an exponent of 2, is
done as
follows: the number to be divided is simply re

written at the position
the divisor is below unity point. If you intend to divide by 8, for example, 8
is written 3 spaces down from unity point … the figure to be divided is
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simply written 3 spaces
farther u
p
on the staff than the position at which it
was originally written.
In the case of division by a fraction that is a whole power of 2, the number to
be divided is written the number of positions
down from
unity point that the
divisor fraction is written a
t above unity point. {Thus, in the case of
½
as
the divisor, which is written one position above unity point, the answer will
begin being transcribed at one position below unity point. If the number
being divided by ½ is “10”, for example, the operator w
ill say to him or
herself “
space, flag, space flag
, describing “10” as it appears written
beginning the description at unity point. That figure will be written
beginning at the
space
that is one down from unity point. The resultant
Funform figure will re
ad from unity point
space,
space, flag, space flag
, or
20.)
DIVISION
for other numbers is just serial subtraction, while keeping score.
I believe that a student should be asked, “How many times can ‘
the divisor’
be subtracted from ‘
the number to be div
ided’
”. The results of the repeated
subtractions are simply recorded off to one side and cleared at the end of the
operation. How many times can 7 be subtracted from 35 is the way that I
would pose the problem depicted in Figure 14. I believe this questi
on makes
it much clearer to the learner what the challenge is.
Just as in fractions, where there is no need to seek the lowest common
denominator, there is no need in division to try at each step of the way to
find the largest number that can be subtracte
d from the number to be divided
(like the "lowest common denominator"). The divisor is turned into a
negative number and placed at any given position along the staff of the
number to be divided (just as it would be done in subtraction). On a staff at
som
e distance away that is reserved for keeping track of the developing
answer [the “recording” staff], a flag indicating how many positions down
the negative number was placed at, is written. The subtraction takes place
and the process is repeated until the
re is only a staff left, or the flag(s)
remaining on the staff to be divided is/are too small to subtract the divisor
from. In that case, what is left is the remainder. After the process is
complete, the recording staff is cleared in the usual way, and t
hat is the
answer.
Division, to summarize, for numbers that are not simply whole powers of
two is just serial subtraction, while
keeping score
.
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Figure 14
The reader needs to actually carry out these operations rather
than just scan
them. Some sample illustrative problems of multiplication and division with
numbers that include fractions follow:
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Figure 15
Figure 16*
*Note that the above multiplication
is actually carried out as 3 X 3½.
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Figure 17**
**This multiplication problem was carried out as 5 X 1
3
/
8
.
Figure 18
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Figure 19
DISCUSSION
: It s
hould be apparent to the reader at this point that it is
possible to use a complex Funform figure when doing a manipulation. For
example, if one wanted to add, divide or multiply by seven, one could use a
Funform figure of 8 minus 1. For smaller and simp
ler numerical values, this
concept is irrelevant.
For a much larger numerical value, however, one could easily simplify the
operation... for example consider 127. It is the same as 128 minus 1. Using
128 minus one would allow the person making the calc
ulations to use only
two flags in the manipulation. 7 flags would be involved if one chose to use
127 as it is ordinarily written.
No calculations are necessary to develop a simpler form of the Funform for
math operations. One can simply put a flag at
the highest value position
immediately after the last flag of the original Funform and put minus flags at
every open position
and
a minus flag at the lowest value position containing
a flag.
For example in 127 converted to this format there is a flag a
t the position
representing 128 and a negative flag at the position of 1. If one were doing
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the same operation with a numerical value of 126, there would be a flag at
128 and a negative flag at the two position (the lowest value position with a
flag in th
e original Funform with a value of 126).
IMPLICATIONS OF FUNFORMS
:
We believe that Funforms most important use will be helping children
[and interested adults] understand multibase arithmetic.
It will help them see the connection between whole numbers
and
fractions.
It will help them see that there is no mystery about negative numbers.
One of the most important aspects of Funforms is that it makes the
understanding of arithmetic calculations transparent. [For example, in
multiplying the multiplicand is
“multiplied” by each of the component
factors of the multiplier, and then these are added. Or that division is
just serial subtraction, while keeping score.]
Funforms familiarizes students with base 2 and its various
permutations, the system used in one
form or another, by all digital
machines.
By using a staff with 5 positions beginning at unity point, the student
would be dealing with base 16.
By using one with 6 positions, the student would be working in base
32. The implications for understanding f
undamental aspects of those
bases [or others], in the computer field seem obvious.
Funforms could easily be read by an optical scanner and by a human,
as contrasted with other artificial constructs like UPC codes.
Funforms does not use zero as a number or
as a placeholder. The
absence of a value at any point is made obvious by the absence of any
structure indicating a value at that point.
Funforms is ideographic. The meaning and component factors of any
number is its formula. It is a starkly simple struct
ure, so to speak.
We have not worked on a way to speak Funforms beyond space

flag

flag,
but it could possibly be done using a Morse code type convention or by
using the terms of the musical scale (do, re, me, etc.). It is possible that it
could even be
sung. Until a means is agreed on, we suggest using the space

space

flag

flag

space system, always beginning at unity point.
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ISSUES YET TO BE ADDRESSED
: We have considered using a 3D
form, as depicted below. Positive and nega
tive integers would still remain
180
o
apart [See first two flags in the figure immediately above.] That would
allow imaginary numbers to be depicted at a 90
o
angle from them [See
above.] Other interesting axes might include irrational numbers; numbers
ba
sed on natural logarithms; or transcendent numbers like pi.
Joel S. Steinberg
November 28, 2004
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