Re, your: " 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?"
You are declaring that children learn arithmetic most naturally via the mathematics of *quantities" which have whole-number numerators. In that much you surely are correct. But much of your rhetoric and logic will be simplified if you rework it all through the context of quantities.
A quantity is a pair, nD, where n is a number (the "numerator" in the quantity), and D (that quantity's "denomination") is a *kind* of things (or a name for all things of that kind) ... as in 3A or 3(apples) or 3(2s) or 3(8ths) or 3(negatives) or 3(bones) ... often called "monomials."
Your distinction between multiplicands and multipliers actually is the distinction between quantities and numbers. What you are calling "invisible numbers" are more commonly called *scalars* or "coefficients". Indeed, the "combos" of such quantities ... as in 3A(pples)+2B(ananas) ... constitute a primitive kind of *vector space*. Your "multipliers" are the scalar numbers, and your multiplicands are quantities ... as with 3*2B(ones)= 6B: quantities 2B and 6B, multiplier,3.
It is a bit foolish to argue that "3x2" always means a scalar multiple of a unit-quantity. In the case of rectangular area, for example, 3S(quares)-by- 2S(quares) yields 6S(quares) ... and in mechanics, 3F(eet) @ 2P(ounds) is 6W(orks), in ft-lbs ... and in Podo's case, 3T(rips)@2B(ones) = 6B. NOT foolish is your recognition that children's *natural* entry into "multiplication" is NOT strictly numerical ... but very quantitative because it is strongly linguistic, and it heavily relies on counting things.
The point that you trying to make is well founded: along each whole-scalars quantity-scale ...[ 0A, 1A, 2A,...] its scalar additions/subtractions, and its scalar multiplications and remainder-divisions, are natural consequents of using adjectives within a spoken language ... although each kind of "multiplications" among quantities requires its own "definition."
In one example, above, SxS=S is a "multiplication" of quantities along a single Q-scale. In another, above, FxP=W uses a cross-scale "multiplication." Cross-denomination multiplications are essential to arithmetic ... but are not nearly as natural as is scalar multiplication of quantities. S, all cross-denomination "multiplications" must be seap[arately developed/defined.
Children can "multiply" pizzas x prices to get costs ... but for pizzas x pizzas, the meaning would have to come from an unusual context (or from creating an unnatural table). For example, by using rows and columns of pizzas, 3P x 3P = 9P (or a 9P square) ... and they might all be square.
As for signs ... -3(4negs) ... or -(3(4negs)) ... is not illogical. When "negs" are regarded as a quantity's denomination, and 4 as its numerator, 3 of 4N(egs) = 12N and the negative of that is -12N.
In Podo's 3@2B(ones), the quantity, 2B, is scalar-multiplied by 3, to get the quantity, 6B.
In English (wherein adjectives are before their subjects), 3 of the 2N(egs) quantity gives the quantity, 6N(egs) ... while -2 of the 3P(os)) = -(2(3P)) = -6P. That leaves a conceptual gap to be closed. When it comes to scalar additions/multiples of quantities, commutatively does not apply, but a kind of "associativity" does, as does a kind of "distribution" across the vector combos.
- -------------------------------------------------- From: "Jonathan Crabtree" <firstname.lastname@example.org> Sent: Thursday, November 08, 2012 11:35 AM To: <email@example.com> 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! > > Jonathan Crabtree > > TOWARDS A MODEL OF 'NATURAL' INTEGER LOGIC > * http://www.jonathancrabtree.com/about/?page_id=166