261–270 of 425 posts
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Well, that would be true if all children were as smart and analytically adept as you.
I think this viewpoint is pernicious. A child doesn't have to be "smart" in order to deserve being told the truth. Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)
I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…
I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition. I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”. I also happen to have an 8-year-old going through third grade right now, and when we were talking through…
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The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.
There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.
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I really don't get the point you're making. If we're going to pull out random examples, monoids aren't guaranteed to be abelian; strings and concatenation form a monoid that's not abelian. If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue. The original article was contrasting addition and multiplicatio…
My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion. But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math. Personally I'm of the…
I agree not calling it multiplication would probably help, perhaps something like "linear transformation composition" might encourage students to keep it separated from real multiplication, I jsut found the argument in the original article kind of ridiculous to be honest.
These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…
With my six year old I told them to say "groups of" instead of "times". So 3 X 2 isn't read as "three times two" it is "three groups of two" which I think has been helpful.
>> Multiplication requires different units. >> 2 baskets × 3 apples per basket = 6 apples No it doesn't. In that example the "baskets" and "per baskets" (ie x/baskets) are cancelling each other. 2 baskets x 3 apples/baskets = 2 x 3 apples = 6 apples. Once you remove the unnecessary terms the actual multiplication is still basic repeated addition. 2 hours x 3 miles per hour = 2 x 3 miles = 3miles+3miles = 6 miles.
I think the real issue is about how to interpret the adjective "Repeated" when we turn to more general domains than the natural numbers. But that's not a huge issue, it's mostly a matter of terminology.
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There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.
But you can define multiplication based on abstracting repeated addition. That how we did it my analysis class, although I forget some of the details. So if you say multiplication is repeated addition, it really isn't a lie.
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The algebraic structures multiplication brings is different from the algebraic structures from addition. That's the point of rings (as opposed to groups). Nonetheless, in a ring (and all fields are rings), multiplication must and always is related to addition, through the distributed property (which I argue, the distributed property IS the mathematical term for "repeated addition"). Without the distributed property,…
Without the distributive property you have two magmas in general for the same set that don't see each other. I'm not sure what your point is. When you have two binary operations you need some sort of distributive property to build a structure. All this is trivial, if you consider a ring, you get a·0=0 as a property, if your starting point is the Peano axioms for the arithmetic of natural numbers that's one of them, f…
Consider the following polynomial: x0 * b^0 + x1 * b^1 + x2 * b^2 ... xn * b^n, where "n" goes to both positive infinity and negative infinity.
When "b = 10" and when "x" can be numbers from [0-9], we have the so called base-10 set of real numbers, do we not? IIRC, if b = sqrt(-1) * 10, we then have the set of complex numbers (a non-intuitive result. I may have made a mistake somewhere, but I assure you there's a surprising property along those lines).
That's the funny thing about real numbers and complex-numbers. Real numbers and even complex-numbers ARE polynomials, and therefore a polynomial ring. 3.1415926 == 3 * 10^0 + 1 * 10 ^-1 + 4 * 10 ^-2 ...
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I'm using a lot of words here. But all I'm saying is once again: Pi * 3 == 3.14... * 3 == 3 * 3 + 0.1 * 3 + 0.04 * 3 + ... == 9.42...
We can evaluate 3 * Pi by splitting Pi up into a set of additions (3 + 0.1 + 0.04 + 0.001...), even if that set of additions is infinite. Then evaluate 3*(each component). This is possible because Pi is easily represented as a polynomial X0 * 10^0 + X1 * 10^-1 + ... Xn * 10^-n.
There's a reason why polynomial multiplication is usually called "Carry-free multiplication". Because Real-numbers are just polynomials where you have that annoying "carry the one" property to keep track of. Remove the "carry the one" property (in say: Galois extension fields), and all the math still works.
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Hmmmm... I probably could have said all that in fewer words. TL;DR: "Real numbers ARE a polynomial ring". (And complex numbers are probably a polynomial ring, I just forgot how to prove that factoid)
> I'd like to see how "repeated addition" works in polynomial rings.
EDIT: Just thought of a cute and simple retort. You ever do a CRC32 check? There ya go.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
Earlier quoted context omitted.
The general concept is known as https://en.wikipedia.org/wiki/Lie-to-children and I think you'd be hard pressed to teach/learn/apply anything of significance without this "well actually it's more complicated than that" approach and recognizing how far you need to go with it to aptly do some task.
There's no need to lie to children. Telling children "multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with" is not a lie.