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# Fixing Flaws in the Teaching of Elementary Mathematics

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Fixing Flaws in the Teaching of Elementary Mathematics

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### Fixing Flaws in the Teaching of Elementary Mathematics

1. 1. 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
2. 2. “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.
3. 3. • 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
4. 4. 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).
5. 5. 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
6. 6. CHINA | India | NCIS | Podo’s Paddock Pedagogies | Q&A From integer arithmetic with rods... ...to integer arithmetic with squares. Negatives Positives
7. 7. 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?’
8. 8. 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).
9. 9. Which numbers are greater? or or or or China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A
10. 10. Which numbers are greater? or or or or China | INDIA | NCS | Podo’s Paddock Pedagogies | Demo | Q&A xx xx x x x x x xxx x xxx xxxx xx xxx xxxx x
11. 11. Which numbers are greater? or or or or China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
12. 12. China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A - 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.
13. 13. China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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.
14. 14. China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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!
15. 15. China | INDIA | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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.
16. 16. 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
17. 17. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
18. 18. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
19. 19. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A Table Mat... (+5 multiplied by +4) i.e. 0 + 5 + 5 + 5 + 5 = 20
20. 20. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A Division Under Table Multi-Mat (+20 divided by +5) i.e. 20 – 5 – 5 – 5 – 5 = 0
21. 21. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
22. 22. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
23. 23. China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A 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
24. 24. 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
25. 25. 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 China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A
34. 34. 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 China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A
35. 35. –2 x –3 (Via Descartes, Newton & Crabtree) Copyright © 2016 Jonathan Crabtree All Rights Reserved China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A 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.
36. 36. –2 x –3 (Via Descartes, Newton & Crabtree) Copyright © 2016 Jonathan Crabtree All Rights Reserved China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A 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 !
37. 37. –2 x –3 (Via Descartes, Newton & Crabtree) Copyright © 2016 Jonathan Crabtree All Rights Reserved China | India | NCIS | Podo’s Paddock Pedagogies| Demo | Q&A As +1 is to –3 so –2 is to +6
38. 38. Questions? research@jonathancrabtree.com More papers and presentations @ www.jonathancrabtree.com Thank you! I look forward to reading your feedback! China | India | NCIS | Podo’s Paddock Pedagogies | Demo | Q&A