Fixing Flaws in the Teaching of Elementary Mathematics

Fixing Flaws in the Teaching of
Elementary Mathematics
(and Having Fun in Podo’s Paddock!)
Jonathan J. Crabtree
Mathematics Historian
Challenges in Mathematics Education for the Next Decade
14th International Conference of the Mathematics Education for the
Future Project. Balatonfüred, Hungary, 11 September 2017.
Conference paper @
http://directorymathsed.net/public/HungaryConferenceDocuments/LongPapers/CrabtreeLong.doc

“I would rather have
questions that can't be
answered than answers
that can't be questioned.”
Richard Feynman, 1918 – 1988, Nobel Prize winning physicist.

• What is zero?
• What is division?
• What are negative numbers?
• What is greater, –7 or +3?
• What is multiplication?
FAQs: Frequently asked questions
A number subtracted from itself 5 – 5
Repeated addition
Repeated subtraction
Less than zero
+3
Infrequently questioned answers: IQAs

Our Two Goals Today
2nd to reveal integer operations via a ‘Chindian’
multiplication table (Podo’s Paddock), that fix
foundational flaws relating to integer multiplication
and division. Then we demonstrate what we learned!
1st to blend China and India’s original ‘Chindian’
ideas on negative and positive integers with Newton’s
3rd Law, in a Closed Integer System (the NCIS).

CHINA | India | NCIS | Podo’s Paddock Pedagogies | Q&A
Consider the ~150 BCE rod numerals of China...
Negatives
Positives
NOTE: The units and hundreds places had vertical rods while
the tens and thousands places had horizontal rods.
E.g.–5342 appeared as not

CHINA | India | NCIS | Podo’s Paddock Pedagogies | Q&A
From integer arithmetic with rods...
...to integer arithmetic with squares.
Negatives
Positives

CHINA | India | NCS | Podo’s Paddock Pedagogies | Demo | Q&A
Today, most children reply, ‘Three negatives.’ even though most don’t know what negatives are!
Simple substitute the relevant noun for ‘negatives’. E.g. What is seven debts minus four debts?
Now, Grades 7 & 8 level.
‘What’s negative seven minus negative four?’
Contrast the relative complexity.
Most adults tentatively reply, ‘Negative eleven?’
150 BCE, Grade 2 level.
‘What’s seven negatives minus four negatives?’

China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A
Contrast the explanations of zero.
Now
Nothing, nil, none, nought, or ...
‘Any number subtracted from itself.’ e.g. 5 – 5 = 0
628 CE, Brahmagupta.
Zero = ‘The sum of a positive number and
negative number of equal magnitude.’
सम-ऐक्यम्खम्(Brāhma Sphuta-siddhānta, Chapter 18:30a).

Which numbers are greater?
or
or
or
or

or
or
or
or
xx xx x x
xx xx x xxx
x x
xx xx x xxx xx
5+ < 7–
9+ > 4–
1+ < 3–
5+ ≹ 5–
xx xx

- Neither China nor India ever defined negative
numbers as being ‘less than nothing/zero’.
- From a central origin, West 7 is NOT less than East 3.

Regarding subtraction, Brahmagupta wrote:
“From the greater should be subtracted the smaller; (the final
result is) positive, if positive from positive, (e.g. +7 – +4 = +3)
and negative, if negative from negative (e.g. –7 – –4 = –3).”
Prakash, Satya. (1968). A critical study of Brahmagupta and his works: A most
distinguished Indian astronomer and mathematician of the sixth century A.D.
New Delhi: Indian Institute of Astronomical & Sanskrit Research.
So for China and India –7 > –4, which is NOT taught today.

CHINDIAN SUMMARY & CONCLUSIONS
Negative numbers cannot (by definition) be less than zero.
- China had positives and negatives in the 2nd Century BCE, yet
their first mathematics text with ‘zero’, appeared in 1247 CE,
The Mathematical Treatise in Nine Sections, by Qin Jiushao.
- So, China used positive and negative integers for ~1400 years
without any concept of zero. The Chinese obviously did not think
negative numbers were less than a non-existent idea!

CHINDIAN SUMMARY & CONCLUSIONS
Negatives and positives of the same size are equal and
opposite in nature.
- Integer ordering such as –7 < –4 is a flaw, contradicted by an
elementary Indian law of mathematics.
Having questioned answers, and found flaws in core mathematical
foundations, we now evolve and fix multiplication and division pedagogies.

China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A
Isaac Newton’s 3rd Law of Motion
The ‘Closed Integer System’
- “For every action, there is an equal and opposite reaction.”
- A law of integer quantity conservation states, for a system
closed to all transfers of integers out of the system, integer
quantity within the system must remain constant over time,
before, during and after integer operations.
The Newtonian Closed Integer System NCIS

Where did the socks go? (Part 1)
The Closed Integer System (i.e. Drawer & Floor)
Q. From six socks in a drawer, two are taken and placed on
the floor. How many socks remain?
A. Six socks remain, four in the drawer and two on the floor!
SOCK COUNT DRAWER FLOOR TOTAL
START 6 0 6
OPERATION Minus 2 Plus 2
END 4 2 6

Where did the socks go? (Part 2)
Q. From two socks on the floor, two are taken and placed
in a drawer with four. How many socks remain?
SOCK COUNT FLOOR DRAWER TOTAL
START 2 (Minuend) 4 (Augend) 6
OPERATION Minus 2 (Subtrahend) Plus 2 (Addend)
END 0 (Difference) 6 (Sum) 6

Table Mat... (+5 multiplied by +4)
i.e. 0 + 5 + 5 + 5 + 5 = 20

Division Under Table Multi-Mat (+20 divided by +5)
i.e. 20 – 5 – 5 – 5 – 5 = 0

With equal and opposite actions in a Newtonian Closed Integer
System (NCIS), multiplication and division occur simultaneously.
INTEGER COUNT ÷UNDER TABLE ×ON MAT TOTAL
START 20 0 20
OPERATION –5, –5, –5, –5 +5, +5, +5, +5
END 0 20 20
INTEGER COUNT 20 ÷ 5 UNDER TABLE ×ON MAT TOTAL
START Dividend 20 0 20
OPERATION
Quotients of 5
subtracted 4 times
Multiplicands of 5
added 4 times
END 0 Product 20 20

In a Newtonian Closed Integer System (NCIS),
pairs of operational terms can be identified.
ZERO Start Stop ZERO
MULTIPLIER
(Times multiplicand added to zero)
DIVISOR
(Times quotient subtracted until zero)
MULTIPLICAND QUOTIENT
PRODUCT Stop Start DIVIDEND
DIVISION
MULTIPLICATION

The following integers sum to zero, as per Brahmagupta’s
definition of zero by addition, (red/pos and black/neg).
–1+1
These integers, being equal
in size, yet opposite in nature,
cancel each other out as zero.
Take away +1 from 0 and –1 remains.
Take away –1 from 0 and +1 remains.
–1+1
–1+1

Now, forget Euclid’s definition of multiplication as it has
been quoted in English since 1570. It’s wrong! *
a multiplied by b is NOT a added to itself b times.
a multiplied by b is EITHER a added to zero b times, OR, a
subtracted from zero b times, according to the sign of b.
(Where a can be either positive or negative.)
* See the 2016 math education conference paper @ www.bit.ly/LostLogicOfMath
We now use a Chindian multiplication table, called Podo’s Paddock in the
extended Crabtree paper Fun in Podo’s Paddock, available at:
http://directorymathsed.net/public/HungaryConferenceDocuments/
China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A

Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero
81 72 63 54 45 36 27 18 9 +9 9 18 27 36 45 54 63 72 81
72 64 56 48 40 32 24 16 8 +8 8 16 24 32 40 48 56 64 72
63 56 49 42 35 28 21 14 7 +7 7 14 21 28 35 42 49 56 63
54 48 42 36 30 24 18 12 6 +6 6 12 18 24 30 36 42 48 54
45 40 35 30 25 20 15 10 5 +5 5 10 15 20 25 30 35 40 45
36 32 28 24 20 16 12 8 4 +4 4 8 12 16 20 24 28 32 36
27 24 21 18 15 12 9 6 3 +3 3 6 9 12 15 18 21 24 27
18 16 14 12 10 8 6 4 2 +2 2 4 6 8 10 12 14 16 18
9 8 7 6 5 4 3 2 1 +1 1 2 3 4 5 6 7 8 9
–
9 –
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 0
+
1 +
2 +
3 +
4 +
5 +
6 +
7 +
8 +
9
9 8 7 6 5 4 3 2 1 -1 1 2 3 4 5 6 7 8 9
18 16 14 12 10 8 6 4 2 -2 2 4 6 8 10 12 14 16 18
27 24 21 18 15 12 9 6 3 -3 3 6 9 12 15 18 21 24 27
36 32 28 24 20 16 12 8 4 -4 4 8 12 16 20 24 28 32 36
45 40 35 30 25 20 15 10 5 -5 5 10 15 20 25 30 35 40 45
54 48 42 36 30 24 18 12 6 -6 6 12 18 24 30 36 42 48 54
63 56 49 42 35 28 21 14 7 -7 7 14 21 28 35 42 49 56 63
72 64 56 48 40 32 24 16 8 -8 8 16 24 32 40 48 56 64 72
81 72 63 54 45 36 27 18 9 -9 9 18 27 36 45 54 63 72 81
Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero
Addition of Integers to Zero
SIDEOFPOSITIVEMULTIPLICANDS
SIDEOFNEGATIVEMULTIPLICANDS
Subtraction of Integers from Zero

+2 x –3 (Via China and India)
Copyright © 2016 Jonathan Crabtree All Rights Reserved

+2 x –3
Two positives subtracted three times!

+2 x –3
Cancel opposites (China) or subtract the zeros (India).

+2 x –3= –6
See the answer!

–2 x –3

–2 x –3
Two negatives subtracted three times!

–2 x –3
Cancel opposites (China) or subtract the zeros (India).

–2 x –3 = +6
See the answer!

Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero
81 72 63 54 45 36 27 18 9 +9 9 18 27 36 45 54 63 72 81
72 64 56 48 40 32 24 16 8 +8 8 16 24 32 40 48 56 64 72
63 56 49 42 35 28 21 14 7 +7 7 14 21 28 35 42 49 56 63
54 48 42 36 30 24 18 12 6 +6 6 12 18 24 30 36 42 48 54
45 40 35 30 25 20 15 10 5 +5 5 10 15 20 25 30 35 40 45
36 32 28 24 20 16 12 8 4 +4 4 8 12 16 20 24 28 32 36
27 24 21 18 15 12 9 6 3 +3 3 6 9 12 15 18 21 24 27
18 16 14 12 10 8 6 4 2 +2 2 4 6 8 10 12 14 16 18
9 8 7 6 5 4 3 2 1 +1 1 2 3 4 5 6 7 8 9
–
9 –
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 0
+
1 +
2 +
3 +
4 +
5 +
6 +
7 +
8 +
9
9 8 7 6 5 4 3 2 1 -1 1 2 3 4 5 6 7 8 9
18 16 14 12 10 8 6 4 2 -2 2 4 6 8 10 12 14 16 18
27 24 21 18 15 12 9 6 3 -3 3 6 9 12 15 18 21 24 27
36 32 28 24 20 16 12 8 4 -4 4 8 12 16 20 24 28 32 36
45 40 35 30 25 20 15 10 5 -5 5 10 15 20 25 30 35 40 45
54 48 42 36 30 24 18 12 6 -6 6 12 18 24 30 36 42 48 54
63 56 49 42 35 28 21 14 7 -7 7 14 21 28 35 42 49 56 63
72 64 56 48 40 32 24 16 8 -8 8 16 24 32 40 48 56 64 72
81 72 63 54 45 36 27 18 9 -9 9 18 27 36 45 54 63 72 81
Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero
Addition of Integers to Zero
SIDEOFPOSITIVEMULTIPLICANDS
SIDEOFNEGATIVEMULTIPLICANDS
Subtraction of Integers from Zero
–2 x –3?
Taking away
two negatives
three times
produces six
positives in Q3
so –2 x –3 = +6

–2 x –3 (Via Descartes, Newton & Crabtree)
Whatever we do to the Unit to make
the Multiplier, we do to the
Multiplicand to make the Product.
We placed three Units and changed
their sign to make the Multiplier – 3.

Previously we placed three Units
and changed their sign to make the
Multiplier.
So, you place three Multiplicands
And change their sign to make the
Product !

As +1 is to –3 so –2 is to +6

Questions?
research@jonathancrabtree.com
More papers and presentations @
www.jonathancrabtree.com
Thank you!
I look forward to reading your feedback!

Fixing Flaws in the Teaching of Elementary Mathematics

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Fixing Flaws in the Teaching of Elementary Mathematics