Five Success Secrets for Mathematics Teachers via Ancient China & India Jonathan J. Crabtree

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1 Five Success Secrets for Mathematics Teachers via Ancient China & India Jonathan J. Crabtree Mathematics Historian Mathematical Association of Victoria (MAV) Annual Conference 2017 La Trobe University Melbourne Campus, Australia 7th December

2 Our Two Goals Today 1 st Explore China and India s ancient mathematical ideas on positives and negatives and reveal five success secrets. 2 nd Blend Chindian integer ideas with ratios and proportion and play multiplication and division games in which the answers appear in (simpler) ways. Eg. 2 x 3, 12 4 and if time (2 10 ) x (3 10)

3 Consider the ~150 BCE rod numerals of China... Positives Negatives The units and hundreds places had vertical rods while the tens and thousands places had horizontal rods. So, 5342 negatives appeared as.

4 The set of vertical and horizontal rod numerals.

5 Zero as a number was neither known nor needed.

6 Rod numerals were used with counting boards. Rod Numerals were also used in Japan as depicted here

7 Success Secret #1 The Grammar Flip (from abstract to concrete)

8 What s negative seven minus negative four? Most adults reply, Negative eleven. What s seven negatives minus four negatives? Most children reply Three negatives.

9 In negative seven we have adjective noun. In seven negatives we have adjective noun. Seven is a simpler adjective than negative and Chinese positives and negatives were nouns!

10 What s positive seven plus negative three? Is there a simpler approach to introduce positives and negatives than the number line?

11 Ground Level Zero No Positives (Bumps) and No Negatives (Holes)

12 Podo the Puppy, loves to dig!

What s positive seven plus negative three?

24 Positive numbers just count or measure things (units). Negative numbers just count or measure opposite things! DIRECTION FINANCIAL POSITION POPULATION TEMPERATURE SUFFICIENCY SEA LEVEL TIME North/South, East/West, Right/Left, Up/Down Assets/Debts, Profit/Loss Births/Deaths, Immigration/Emigration Hot/Cold, Above Zero/Below Zero More Than Enough/Less Than Enough Above/Below Past the hour/to the hour Q. How + ve result bad & ve result good? A. Your cancer test! Copyright 2016 Jonathan Crabtree All Rights Reserved

25 Success Secret #2 Defining, Splitting and Adding Zero

26 Defining zero. Now Nothing, nil, none, nought, or... Any number subtracted from itself. e.g. 5 5 = CE, Brahmagupta (India). Zero = The sum of a positive number and negative number of equal magnitude. सम-ऐक यम खम (Brāhma Sphuta-siddhānta, Chapter 18:30a).

27 NOTE: Negatives were NOT considered less than zero. - China had positives and negatives in the 2 nd 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 negatives for ~1400 years without any concept of zero. The Chinese obviously did not think negatives were less than a non-existent idea! Negatives and positives of the same size are equal in number yet describes units opposite in nature.

28 Splitting zero into equal numbers of opposites.

29 When you don t have enough for subtraction, just add zero as equal numbers of opposites. For 3 positives minus 7 positives, just add zero! + 3 minus + 7 (we re 4 short.) After adding zero in the form of + 4 and 4, we have 7 positives minus 7 positives, and 4 negatives left over, so = 4

30 The following integers sum to zero, as per Brahmagupta s definition of zero by addition, (red/pos and black/neg). 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

31 The following integers sum to zero, as per Brahmagupta s definition of zero by addition, (red/pos and black/neg). 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

32 Success Secret #3 How Integer Multiplication (REALLY) Works

33 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.)

2 x 3 Two negatives subtracted three times!

42 + 1 1 Take away + 1 from 0 and 1 remains. 1 Take away 1 from 0 and + 1 remains. + 1

SIDE OF NEGATIVE MULTIPLICANDS SIDE OF POSITIVE MULTIPLICANDS Addition of Integers to Zero 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

43 SIDE OF NEGATIVE MULTIPLICANDS SIDE OF POSITIVE MULTIPLICANDS Addition of Integers to Zero Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero Subtraction of Integers from Zero

44 SIDE OF NEGATIVE MULTIPLICANDS SIDE OF POSITIVE MULTIPLICANDS Addition of Integers to Zero Negatives Added N Times to Zero Multiplier Positives Added N Times to Zero Negatives Subtracted N Times from Zero Multiplier Positives Subtracted N Times from Zero Subtraction of Integers from Zero 2 x 3? Two negatives taken away three times produces six positives in Q3 so 2 x 3 = 6 +

45 Success Secret #4 Multiplication and Division are (ALWAYS) Proportional Concepts!

Done That Do This Multiplication! https://www.youtube.com/watch?

48 From repeated addition and repeated subtraction to Proportion 2 x 3 =? Whatever we do to the Unit (1) to make the Multiplier ( 3), we do to the Multiplicand ( 2) to make the Product. We placed three Units and changed their sign to make the Multiplier 3. Copyright 2017 Jonathan Crabtree All Rights Reserved

49 2 x 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! Copyright 2017 Jonathan Crabtree All Rights Reserved

52 Success Secret #5 Proportional Covariation Extends from the Integers to the Real Number System.

53 a 2 x a x b = c As 1 is to b, so a is to c a b = c As b is to 1, so a is to c C C b

54 Proportional Covariation Applets

Q1. + 2 (0 + 3) 2 Positives Added 3 Times onto Zero makes 6 Red Positives Q2. 10 (0 + 3) 10 Negatives Added 3 Times onto Zero makes 30 Black Negatives Q3.

55 Q (0 + 3) 2 Positives Added 3 Times onto Zero makes 6 Red Positives Q2. 10 (0 + 3) 10 Negatives Added 3 Times onto Zero makes 30 Black Negatives Q3. 10 (0 10) 10 Negatives Subtracted 10 Times from Zero makes 100 Red Positives Q (0 10) 2 Positives Subtracted 10 Times from Zero makes 20 Black Negatives Altogether, we have 106 red positives and 50 black negatives. After cancellation, 56 red positives or + 56 remains.

56 Questions? More papers and Thank you! I look forward to reading your feedback!

Hydrostatics and Stability Prof. Dr. Hari V Warrior Department of Ocean Engineering and Naval Architecture Indian Institute of Technology, Kharagpur

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