> **Please post a comment if any of these insights are > new to you or need clarification.** > > I am writing this post because of observations from > Peter Duveen and GSC Chandy in a post by Robert > Hansen titled 'An interesting point' at > http://mathforum.org/kb/message.jspa?messageID=7918003 > and > http://mathforum.org/kb/message.jspa?messageID=7919346 > > > In basic arithmetic, numbers are counts of objects OR > counts of actions. Just referring to both the > multiplicand and multiplier as factors that can be > swapped around dumbs down the logic of arithmetic. > > Many elementary math teachers fail to see, let alone > explain, that one of the numbers in 3 x 2 is > invisible. Yet which one is it when they are both > written down? > > As any child can tell you, it depends of the story > the expression summarises. > > Multiplication and division involving a single unit > both require a number you can't see. > > In the case of multiplication, the invisible number > is the multiplier. > > In the case of division, the invisible number is the > divisor. > > The reason is you cannot 'see' adverbs. > > Both a multiplicand and a dividend describe a number > of units that are operated upon. > > Therefore multiplicands and dividends are adjectives > describing the number of implicit nouns (units) in > the story. > > The product and quotient are the adjectives that > describe the final scene. ie final number of units > > 1/3 of a pizza can be seen. The adverbial multiplier > or divisor is unseen. The product or quotient is > seen. > > In the case of multiplicand x multiplicand you would > have > > 3 pizzas x 3 pizzas = 9 square pizzas ie nonsense > > 3 pizzas (seen) x 1/3 (unseen) = 1 pizza seen and > eaten! > > The fun rhyme by Ogden Nash... > > Minus times minus results in a plus, > The reason for this, we needn't discuss. > > ... is silly as this is saying > > verb times verb = verb > > Similarly negative times negative = positive is also > flawed logically as this is akin to saying... > > adjective times adjective = adjective > > In 3 bones x 2 the multiplicand is 3. > > Three takes on the role of an adjective. > > The verb is multiplication (iterative adding of 3 to > zero). > > Two is the adverb describing how many times you add > the multiplicand to zero. > > You can have 2 x 3 bones as all you have done is tell > the same story another way so the multiplier can come > before the multiplicand as the multiplicand still > carries the unit. > > minus (verb) times (verb) minus (verb) equates to > > take away times take away > > plus (verb) times (verb) plus (verb) equates to > > add times add > > Multiplication and division are both iterative > actions. > > With the expression 3 x 2 the story is about adding > the multiplicand 3 to zero twice (the adverbial). > > The implicit sign of the multiplier instructs us to > either repeatedly add or repeatedly subtract the > multiplicand from zero. > > The expression 3 x -2 means taking away the > multiplicand 3 from zero twice. The unseen multiplier > adverb now has another action associated with it! > > -3 x -2 is taking away the negative multiplicand 3 > from zero twice. Whether the child's model is taking > away cold (-ve) by adding heat (+ve) or taking away > debt by adding wealth, the taking away a negative > multiplicand can be shown to create the same result > as adding a positive multiplicand. > > So taking away -3 from zero twice is the same as > adding 3 to zero twice. Not as intuitive as my > 'natural' integer axioms* yet still capable of > demonstration via a zero based number line. > > When you see mathematicians writing about minus times > minus, or plus times plus, you know their students > may be on a path to cognitive conflict. Children have > a well established pre-school understanding that > adding (plus) and taking away (minus) are things you > DO and not numbers you SEE. > > Similarly negative times negative and positive times > positive are flawed constructs if their purpose is to > describe reality. So an adjective repeatedly > described with an adjective is nonsense too. > > Multiplicands and dividends count units and > multipliers and divisors count actions. > > You may be interested to explore my axioms via the > link further below. Alternatively if you teach > children, read on! > > Let me share a simple story... > > =============== > > Podo the Puppy was hungry yet didn't have any bones. > So she went hunting. Soon she came across a large > pile of bones. > > Podo picked up two bones in her mouth and ran back > home to chew them. Yet when she got back, she had a > visitor. So Podo ran off and fetched another two > bones so they could both share a meal together. > > However as luck would have it, when Podo returned, > another friend had arrived! So once more Podo ran off > and fetched two more bones. > > Then they all enjoyed eating together and telling > stories about their day. > > ================== > > In the above story, you SEE zero bones at the start. > > Then you see TWO bones. Then you see TWO more bones. > And again you see TWO more bones. > > At no stage do you ever SEE the number three. > > The reason is the 3 in 2 x 3 or 3 x 2 is an adverb. > It is the number of times Podo fetched 2 bones. > > The 2 is the adjective that is implicit in the noun > unit (bones). > > You can have multiplicand x multiplier as 2 bones > three times, an approach popular in India. > > You can have multiplier x multiplicand as three times > 2 bones which is common in the west. > > So Indians would likely read 2 x 3 as "Two three > times" meaning 2 2 2 and implicitly combine the > addends. It doesn't matter whether you put the > multiplicand before the multiplier or vice versa, > provide you GET which is which. The adverbial numbers > are the invisible multiplier. > > Following India's invention of zero as a number > around 1400 years ago AND even more explicitly since > the number line was 'invented' in 1685 by John > Wallis, the complete expression within the story is 0 > + 2 + 2 + 2. ie mk = m added to zero k times > > Model the story on a number line and you start at > zero, jump two to the right, then two more to the > right and then again two more to the right to arrive > at 6 bones and complete an equation. > > So in 2 (bones) x 3 or 3 x 2 (bones) you never see > the 3. It is the sum of identical events. These > events may be serial or simultaneous. > > You do of course SEE the PRODUCT which is six, before > of course, they get eaten and the story ends with > zero bones and a good sleep for all the puppies. > > Only after children fully understand the stories > behind expressions and vice versa and the different > roles multiplicands and multipliers have, should > children be exposed to abstract 'unitless' arrays and > area models. > > The term 'factors' should not be used just because > the 'times tables' are commutative. Far better that a > times table chart be labelled or color coded with the > terms multiplicand and multiplier with products > within. > > So what are your thoughts? Do you better appreciate > how multiplicands and multipliers are different and > why it's misleading to 'dumb down' explanations of > the elements of multiplication to interchangeable > factors and a product? > > And thank you for reading! > > Jonathan Crabtree > > TOWARDS A MODEL OF 'NATURAL' INTEGER LOGIC > * http://www.jonathancrabtree.com/about/?page_id=166 > > > Message was edited by: Jonathan Crabtree
Perhaps all of this shows that is it useful to make the units (or in pure mathematics, the underlying set of elements) the same, even if this means some sort of generalization.
3 apples + 2 apples = 5 apples.
But what do we put in the right side blank below?
3 apples + 2 oranges = 5 _____.
One way out:
3 pieces of fruit + 2 pieces of fruit = 5 pieces of fruit.
There can be an analogue in pure mathematics ("pure" simply meaning no real world units):
For 3 + 2 = 5, if we can explicitly address the question of an underlying set of elements:
3 elements in the set of rational numbers + 2 elements in the set of rational numbers = 5 elements in the set of rational numbers.
But what do we put in the right side blank below?
3 elements in the set of rational numbers + 2 elements in the set of irrational numbers = 5 elements in the set of _______.
One way out:
3 elements in the set of real numbers + 2 elements in the set of real numbers = 5 elements in the set of real numbers.
That is, as we can generalize apples and oranges to pieces of fruit, we can take the union of two sets each containing one type of element but different in type from set to set, and then we can define these two different types of element in this union more generally to be one type of element.