Date: Nov 8, 2012 12:35 PM
Author: Jonathan Crabtree
Subject: How teaching factors rather than multiplicand & multiplier confuses kids!
**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!
TOWARDS A MODEL OF 'NATURAL' INTEGER LOGIC
Message was edited by: Jonathan Crabtree