The Associative & Commutative Properties | Schooling Terrence Howard
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Published 2024-06-08
This mini-lesson is about the Associative and Commutative properties of addition and multiplication, under the field of Real Numbers.
This is definitely NOT simply copy-and-pasted from old math videos I've already recorded in the past.
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All Comments (8)
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Yaaayyyy, a dozen random people on the internet now officially "Like" the associative and commutative properties!! A video lesson that I, OF COURSE, recorded in the present time and didn't just copy-and-paste together from a previous math lesson!
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Understanding trigonometry and algebra is one thing, but when you can't even understand your basic 1x table, that's about as stupid as it gets. Every child who has just learned multiplication knows the answer to that one, without even thinking. 😆
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I'd like to offer a small, but very important correction. It's easy to miss in such a quick and spoken presentation, but it is the underlying mistake in his whole spiel. 3:57 "If a and b are positive integers, a is to be added to itself as many times as it is indicated by units in B" (Terrence Howard) That is NOT the definition of muliplication. Using the language that Howard uses here, this definition would be "you add as many instances of a together, as is indicated by b". So, in the "correct" understanding, you take b instances of a, and add them up. In Howard's version, you take a, and add a to that b times. In his version the number b refers to the number of operations executed. In the "correct" version, b applies to the number of elements on which the operation is performed. It's easy to see now why his version is increased by 1 (or "a"). I'd like to see him give an answer to that "So 1 times 5 is?" with the commutative principle applied. If he applied his own explanation consistently, this would indeed by "6"... but the opposite version of "So what is 5 times 1?" would have to result in 10, using his version. It is not commutative.
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I can understand why he would say "you cannot make something smaller by multiplying". He is going off the literal dictionary English definition of the word "multiply", which means "to make more of". So the idea you could "make less of" by "making more of" sounds contradictory. What he failed to learn - and what school likely failed to adequately teach him - is that mathematical terms have specific and precise technical definitions wholly separate from their ordinary English counterparts. For example, one possible definition of multiplication on the natural numbers, i.e. { 0, 1, 2, 3, ... } numbers, is: 1. For any natural n, 0*n = 0. 2. For any naturals m and n, S(m)*n = (m*n) + n. where S(i) is the successor of i, in Peano arithmetic terms. The point here is that this is what "multiplication" means as a word in this context. It does not mean "make more of", it means "the binary function *: N x N -> N that satisfies the two properties above". It literally means that in this particular mathematical context. For the real numbers, it is somewhat more complicated, but we can still do the same thing, and then that becomes the meaning. Anything else simply is not what the word means in context, just as the word "president" means one thing when applied to a country, but a different thing when applied to a corporation.
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The communicative property means you can switch the order when you are operating on ANY TWO operands. The operation must also be associative to allow any order for any number of operands.I agree that this is not part of the definition of multiplication. It is a natural property of multiplication as it is defined. A ridiculous thing I heard Terrance Howard say while justifying his 1x1 = 2 "logic" is "You can't make something smaller by multiplying." He should ask a sharp 5th grader to explain fractions to him.
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I believe Mr. Howard thinks multiplication is the number of times a number is added to itself. Given 1x5 he thinks itself=1 and you add 1 5 times to get 6 because you are starting with the first number already being there. . I am unclear what "to itself" is supposed to refer to but I have always hated this definition.
