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# How to explain multiplication and division like René Descartes and Isaac Newton

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Addition and subtraction are inverse operations. So, it is no surprise primary school teachers often explain multiplication via repeated addition and division via repeated subtraction. Yet such approaches become tricky as soon as fractions and negative numbers get introduced. In this presentation, I reveal how long lost ideas of René Descartes and Isaac Newton can be used with children as young as Grade 2. Play a game. Sing a song! The new ideas you will discover explain multiplication and division, from Naturals to Reals, more deeply and simply than ever before, in ways currently missing from western F–8 curriculums.—In 1968 at age 7 in Grade 2, Jonathan Crabtree noticed the definition of multiplication taught for centuries made no sense. Undeterred by failure, he explored hundreds of original source mathematics books and manuscripts spanning 16 languages. Strangely, Euclid’s definition of multiplication, broken upon translation into English in 1570, was never fixed! Thus, the writings of brilliant mathematicians were ignored and the foundations of mathematics education in the West are worse today than in ancient India and China. Having hunted for the lost logic of mathematical minds for decades, Jonathan’s new research-based ideas make mathematics simpler than ever before.

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### How to explain multiplication and division like René Descartes and Isaac Newton

1. 1. Thanks Jonathan and Podo. What’s on the agenda? How to explain multiplication and division like René Descartes and Isaac Newton. Jonathan Crabtree 26th Biennial Conference of the Australian Association of Mathematics Teachers Inc. 11 July 2017
2. 2. René Descartes 1596 – 1650 Isaac Newton 1643 – 1727
3. 3. Thanks Jonathan and Podo. What’s on the agenda? Our Two Goals Today 2. To explore how children might see, sing and play along with DesCartesian multiplication and division. 1. To explore and modernise Descartes and Newton’s (DaN’s) ideas. i.e. to convert DaN’s andragogy to pedagogy!
5. 5. Two multiplied by three is two added to itself three times and that equals six. But that’s crazy! Two added to itself three times is eight. Grade 2C 1968 Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved
6. 6. Of course 2 multiplied by 3 equals 2 added to itself 3 times! It’s Euclid’s multiplication definition from 300 BCE. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved
7. 7. …to multiply a by integral b is to add a to itself b times Collins Dictionary of Mathematics Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved In book VII, Euclid defines multiplication as ‘when that which is multiplied is added to itself as many times as there are units in the other’ The Development of Multiplicative Reasoning in the Learning of Mathematics
8. 8. How can 2 added to 1 three times be seven? 1 + 2 + 2 + 2 Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved and 2 added to itself or 2 three times be six? 2 + 2 + 2 + 2
9. 9. With 2 multiplied by 3, the three hops of 2 drawn on the number line start at zero. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved
10. 10. So 2 multiplied by 3 equals two added to zero three times, not itself. 0 + 2 + 2 + 2 Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved
11. 11. …to multiply 2 by 3 is to add 2 to itself ZERO 3 times …to multiply a by integral b is to add a to itself ZERO b times Jonathan, Age 7. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Copyright © 2017 Jonathan Crabtree All Rights Reserved In book VII, Euclid defines multiplication as ‘when that which is multiplied is added to itself PLACED TOGETHER as many times as there are units in the other’ Jonathan, Age 55.
13. 13. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Click for Paper
14. 14. Thanks Jonathan and Podo. What’s on the agenda? Q. Why should we explain multiplication in the manner of Descartes and Newton? … 3 answers. Background | Why DaN? | How DaN? | Pedagogies? | Questions?
15. 15. Thanks Jonathan and Podo. What’s on the agenda? A1. The definition of multiplication attributed to Euclid (since 1570) is wrong! (In all his Book VII multiplicative propositions and proofs, Euclid never mentioned addition.) Background | Why DaN? | How DaN? | Pedagogies? | Questions?
16. 16. Thanks Jonathan and Podo. What’s on the agenda? A2. The andragogy of multiplication in 1500 was totally different to the pedagogy of multiplication in 1600. Background | Why DaN? | How DaN? | Pedagogies? | Questions?
17. 17. Thanks Jonathan and Podo. What’s on the agenda? A3. Descartes and Newton (and others) improved maths andragogies, yet we’re inadvertently following the dumbed down & over-simplified maths pedagogies of 1600! Background | Why DaN? | How DaN? | Pedagogies? | Questions?
18. 18. Thanks Jonathan and Podo. What’s on the agenda? FYI, the additive identity element 0 and multiplicative identity element 1, have NOT been fully incorporated into western elementary maths pedagogies! Background | Why DaN? | How DaN? | Pedagogies? | Questions? In Europe, around 1600, 0 was still just a ‘placeholder’ and 1 was NOT a number – it was a ‘unit’ from which numbers were formed. Bank robbery anecdote.
19. 19. Thanks Jonathan and Podo. What’s on the agenda? a × +3 = 0 + a + a + a a × –3 = 0 – a – a – a Background | Why DaN? | How DaN? | Pedagogies? | Questions? a × b equals a, either added to zero b times, or a subtracted from zero b times, according to the sign of b.
20. 20. Thanks Jonathan and Podo. What’s on the agenda? ab equals 1, either multiplied by a b times, or 1 divided by a b times, according to the sign of b. Background | Why DaN? | How DaN? | Pedagogies? | Questions? a+3 = 1 × a × a × a a–3 = 1 ÷ a ÷ a ÷ a
21. 21. Returning the identity element zero reveals patterns not often seen. a × +4 = 0 + a + a + a + a a × +3 = 0 + a + a + a a × +2 = 0 + a + a a × +1 = 0 + a a × 0 = 0 a × –1 = 0 – a a × –2 = 0 – a – a a × –3 = 0 – a – a – a a × –4 = 0 – a – a – a – a So, integral multiplication involves either repeated addition or repeated subtraction, from zero, depending on the sign of the multiplier. Copyright © 2017 Jonathan Crabtree All Rights Reserved Background | Why DaN? | How DaN? | Pedagogies? | Questions?
22. 22. Returning the identity element one into the pattern of exponentiation. a+4 = 1 × a × a × a × a a+3 = 1 × a × a × a a+2 = 1 × a × a a+1 = 1 × a a 0 = 1 a–1 = 1 ÷ a a–2 = 1 ÷ a ÷ a a–3 = 1 ÷ a ÷ a ÷ a a–4 = 1 ÷ a ÷ a ÷ a ÷ a So, integral exponentiation involves either repeated multiplication or repeated division, from one, depending on the sign of the exponent. Idea extension via Disquisitiones Arithmeticae, Carl F. Gauss, 1801. Copyright © 2017 Jonathan Crabtree All Rights Reserved Background | Why DaN? | How DaN? | Pedagogies? | Questions?
23. 23. Thanks Jonathan and Podo. What’s on the agenda? As we will see, the ideas of Descartes and Newton extend from the Naturals to the Reals. Our current approaches do not. Background | Why DaN? | How DaN? | Pedagogies? | Questions?
24. 24. Thanks Jonathan and Podo. What’s on the agenda? Q. Why should we explain division in the manner of Descartes and Newton? Background | Why DaN? | How DaN? | Pedagogies? | Questions?
26. 26. Thanks Jonathan and Podo. What’s on the agenda? A. The models of division taught in 2017 fail to work with negative divisors! (e.g. 12 ÷ –4) Background | Why DaN? | How DaN? | Pedagogies? | Questions?
27. 27. Copyright © 2016 Jonathan Crabtree All Rights Reserved 2. You can't have negative four groups! (Partitive model.) 12 ÷ –4 Background | Why DaN? | How DaN? | Pedagogies? | Questions? 1. There aren't any negative fours in twelve! 3. You can’t repeatedly subtract –4 from 12 until you get to zero! (Quotative model) (i.e. You have to convert the ÷ to × and apply LOS!)
28. 28. Thanks Jonathan and Podo. What’s on the agenda? Our Two Goals Today 1.To explore and modernise Descartes and Newton’s ideas. 2. To explore how children might see, sing and play along with DesCartesian multiplication and division.
29. 29. Copyright © 2016 Jonathan Crabtree All Rights Reserved Descartes “For example, let AB be taken as unity, (1), and let it be required to multiply BD (the multiplicand) by BC (the multiplier), I have only to join the points A and C, and draw DE parallel to AC; and BE is the product of this Multiplication.” Background | Why DaN? | How DaN? | Pedagogies? | Questions?
30. 30. https://www.geogebra.org/m/je3SEyQr BLUE GREEN RED Background | Why DaN? | How DaN? | Pedagogies? | Questions?
31. 31. https://www.geogebra.org/m/je3SEyQr Background | Why DaN? | How DaN? | Pedagogies? | Questions? https://www.geogebra.org/m/x90Ylkv8
32. 32. Background | Why DaN? | How DaN? | Pedagogies? | Questions? https://www.geogebra.org/m/v62CqVEN
33. 33. Thanks Jonathan and Podo. What’s on the agenda? Our Two Goals Today 1.To explore and modernise Descartes and Newton’s ideas. 2. To explore how children might see, sing and play along with DesCartesian multiplication and division.
34. 34. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Johann Huswirt on Multiplication (modernised) “As the Unit is to the Multiplier, the Multiplicand is to the Product.” i.e. Whatever we do to the Unit to make the Multiplier we do to the Multiplicand to make the Product Huswirt, J. (1501). Enchiridion Algorismi, (Handbook of Algorithms) Cologne, Germany.
36. 36. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Multiplication is Proportional Covariation (PCV) “As the Unit 1 is to the Multiplier b, the Multiplicand a is to the Product c.” Within a × b = c the proportion is 1 : b :: a : c
37. 37. Background | Why DaN? | How DaN? | Pedagogies? | Questions? The proportion here is 1 : 3 = 4 : 12 a × b = c 4 × 3 = ?
38. 38. Background | Why DaN? | How DaN? | Pedagogies? | Questions? The proportion also ‘commutes’ as 1 : 4 = 3 : 12 a × b = c 3 × 4 = ?
39. 39. Background | Why DaN? | How DaN? | Pedagogies? | Questions? Johann Huswirt on Division (modernised) “As the Divisor is to the Unit, the Dividend is to the Quotient.” i.e. Whatever we do to the Divisor to make the Unit we do to the Dividend to make the Quotient Huswirt, J. (1501). Enchiridion Algorismi, (Handbook of Algorithms) Cologne, Germany.
41. 41. Background | Why DaN? | How DaN? | Pedagogies? | Questions? 1 is placed together 3 times, so 4 is placed together 3 times a × b = c 4 × 3 = ?
42. 42. Background | Why DaN? | How DaN? | Pedagogies? | Questions? 1 of 3 equal parts is taken, so 1 of 3 equal parts is taken a ÷ b = c 12 ÷ 3 = ?
43. 43. Background | Why DaN? | How DaN? | Pedagogies? | Questions? × and ÷ are ‘inverse’ operations so the ‘Done That, Do This’ arrows are inverted!
44. 44. Background | Why DaN? | How DaN? | Pedagogies? | Questions?
45. 45. Background | Why DaN? | How DaN? | Pedagogies? | Questions?