In 1637 René Descartes gave us a geometrical model of multiplication.
The multiplication of two lines has long been associated with the 'area model' of multiplication. However, in 1637, Rene Descartes modified a diagram of Euclid, (Elements, Book VI, Proposition 12), to present the diagram 'La Multiplication' below, which states:
"For example, let AB be taken as unity, and let it be required to multiply BD by BC, then I have only to join the points A and C, and draw DE parallel to CA; and BE is the product of this Multiplication."
It's importance was profound, yet is largely lost on mathematics teachers today. It was the first time a major mathematician had multiplied one line by another, to produce, not an area, but another line! Just as importantly, the diagram reveals the proportional relationship 'as the Unit 1 is to the Multiplier, the Multiplicand is to the Product'. (The ratios 1 : Multiplier and Multiplicand : Product are equal.)
How to transform Descartes' multiplication diagram onto the Cartesian Plane
Mashing Devlin & Descartes...
As the Unit 1 scales to the Multiplier, the Multiplicand scales to the Product. Where a (Multiplicand) × b (Multiplier) = c (Product), as 1 is to the Multiplier b, the Multiplicand a is to the Product c. (The proportion 1 : b :: a : c is noted.)
Mathematicians like Stanford University math professor, Dr. Keith Devlin, insist multiplication is NOT repeated addition. Yet bloggers and elementary school teachers respond by saying multiplication and addition are connected via the Distributive Law. For example:
2 × +4 = 2 × (0 + 1 + 1 + 1 + 1) = 0 + 2 + 2 + 2 + 2 = 0 + 8 = +8
A fact too often ignored, is multiplication also distributes over subtraction, so:
2 × –4 = 2 × (0 – 1 – 1 – 1 – 1) = 0 – 2 – 2 – 2 – 2 = 0 – 8 = –8
By digging a little deeper, the simplistic MIRA myth, Multiplication Is Repeated Addition, is immediately seen to be just half the integer multiplication story. If we say MIRA, we must also say MIRS, Multiplication Is Repeated Subtraction! In reality, integral multiplication of a multiplied by b written as a × b may involve EITHER a added to zero b times in succession OR a subtracted from zero b times in succession, according to the sign of the multiplier. *
Whilst this clarification is important, it's still unwise to limit multiplication to 'either repeated addition or repeated subtraction according to the sign of the multiplier'. As ancient Chinese and Indian teachers would have known only too well, upon achieving mastery of 'numbers of positives' and 'numbers of negatives', where opposing units simply cancel each other out, as soon as fractions get introduced, the repeated addition|subtraction of the Multiplicand, (to|from nothing-China, zero-India), Multiplier times, no longer works as an algorithm. **
Strangely, the best understanding of Multiplication existed 500 years ago. By multiplication, whatever we do to the Unit to make the Multiplier, we do to the Multiplicand to make the Product! That is the ONLY meta-multiplication definition that spans the Natural Number System all the way to the Real Number System.***
The Unit was the missing link in Keith Devlin's hypothesis, that multiplication is a 'generalized notion of scaling'. Whatever gets done to the Unit to make the Multiplier, gets done to the Multiplicand to make the Product! There are two simultaneous variations or scaling operations in multiplication. These simultaneous scaling operations are made visible with similar triangles. Genius mathematicians like Euclid, Leonardo Pisano (Fibonacci) and René Descartes understood this. However, mathematics progressively got 'dumbed down' in the teaching over time, (as andragogies became pedagogies), and many deep insights such as this were lost!
To understand multiplication the way it was intended to be understood, all we need do is morph Euclid's and Descartes' diagrams onto our modern Cartesian Plane as seen above. Drag the Multiplicand sideways and the Multiplier up and down and watch the Product appear via the tandem instances of scaling or variation, I call Proportional Covariation or PCV!
Teachers in the lower grades are thus advised to introduce multiplication with physical manipulatives that maintain the proportional nature of multiplication. As 1 block placed three times makes three blocks, 2 blocks placed three times make six blocks. Here, as 1 is to 3, so 2 is to 6, which is written 1 : 3 :: 2 : 6 as a proportion. Whilst such a conversation may come much later, such (Euclidean-style) multiplication is both historically and pedagogically pure, as the video below reveals.
For a historical perspective and how Isaac Newton's insight about negative line segments (and numbers) should have led to the above, read on!
[embeddoc url="http://www.jonathancrabtree.com/mathematics/wp-content/uploads/2016/07/2016-JCRABTREE_A_New_Model_of_Multiplication_via_Euclid_Final.pdf" download="all"]
FOOTNOTE: If you would like to know more about my decades-long struggle to understand multiplication, you can watch my 'Lost Logic of Elementary Mathematics' slideshow, (online via Microsoft OneDrive), at:
My recent elementary math conference paper exploring multiplication, and how a London haberdasher trashed Euclid's multiplication definition, is at http://bit.ly/LostLogicOfMath
* My Lost Logic of Elementary Mathematics slideshow has an idea for teachers (using bottle caps) that explains multiplication by negative integers.)
** As an aside, the ancient Chinese and Indians both took a symmetrical approach to elementary mathematics. China had both red positive rod numerals and black negative rod numerals. When an equal number of each opposing color existed, they were simply removed from the counting board. In India, zero was defined as the sum of two equal yet opposite numbers. Today, after India, we accept 0 = +n + –n yet preach the doctrine +n > –n which would have made no sense in either ancient China or India. If n steps right|East were greater than n steps left|West our maps wouldn't work! Alas, western elementary mathematics was built on only half the integers, the positive ones. The primary reason is Euclidean geometry, (which the West rightly revered), had neither zero nor negative numbers. The secondary reason is western churches had a problem with concepts such as infinity and zero, which India did not. Western Churches controlled not only society, but education in society and only God was infinite and the void, or zero, was the realm of the devil!
To appreciate how simple eastern math was in China 2100 years ago compared to western math today, consider the western question, "What's negative seven minus negative four?" which many if not most adults get wrong. (No, the answer isn't negative eleven.) Now consider the same (hypothetical) question posed in China 2100 years ago. "What's seven negatives minus four negatives?" which many if not most Chinese children would have replied correctly, with "Three negatives!" (NOTE: Refer to LLEM slideshow for more.)
*** We then select whatever algorithm we want to calculate the answer. That algorithm might entail: repeated doubling as is the case with binary Egyptian methods, repeated subtraction as is the case when the integral multiplier is negative or proportional scaling, which is what Proportional Covariation (PCV) is all about.