Earlier quoted context omitted.
Not defending the article, but how would you compute: \pi*\pi using repeated addition?
3.14159 + 3.14159 + 3.14159 + (3.14159*0.14159)
red pen time
What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
141–150 of 425 posts
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#142Literally nothing. It is even in the language - multiple times. Multiply n times. And even square makes perfect sense - whatever side you choice, which is commutativity.
Disconnected from reality concepts is the root of all evil.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#143Earlier quoted context omitted.
Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.
We're talking about matrix/matrix multiplication here, not matrix/vector, right?
I was talking about matrix/matrix composition (rather than multiplication) at first. Then I talked about matrix application.
A matrix is a function. Not all functions can be represented by matrices (although all smooth functions can be represented by a Taylor series that sums over matrices (and "the sum C=A+B" really means "the function C such that Cx = Ax+Bx for all x in the range of both A and B"))
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#144The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?
> but is 2 cm × 3 cm the same thing as 2 × 3? This made me think too, in particular the teacher's example. So; here's what I think it should be calculated. 3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm). What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm". However 3 cm + 3 cm != 2 cm X 3 cm as the author stat…
The author is confusing math with physics.
Math: 3x3 = 3^2 physics: cmxcm = cm^2
3cm x 2cm = (3x2)(cm x cm) = (math)(physics)
The context gives the meaning.
Or in computers, how many bits would the representation of a matrix of 2b x 2b occupy? 2b x 2b = 4b (not 4b^2 since memory is linear)
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#145These 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…
Anyone that teaches this way should find something else to do. Memorizing a formula is easy but if you want to do well later you must be able to complete the square.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#146I think the author mixes variables with numbers. If you add units in to the mix then you have to think like it variables so for example that’s taken “2 baskets × 3 apples per basket = 6 apples” would correspond to 2x3y/x=6y removing the numbers and the equation becomes clear xy/x=y or we the numbers for them selves 23=6, That is because numbers are like their own variable but with connection to other numbers(variables). So if we would define 1 as x then 2 would be 1+x or we could call it y but y would still be 1+x in relation to 1.
And also”3 cm + 3 cm = 6 cm” as 3x+3x=6x and removing the numbers it become 3x/3+3x/3=6x/3 => x+x=2x it makes more sense. And”2 cm × 3 cm = 6 cm^2″ can be then seen as 2x*3x=6x^2.
Multiplication doesn’t change anything. We must see what we add, in this case the unit and calculate accordingly both for addition as for multiplication.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#147These 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…
> You can definitely see multiplication as repeated addition. Only for rational numbers. Doesn't work for real and complex numbers. > It's even a useful thing to do, how else would you define multiplication? Axiomatically, not algorithmically.
>Only for rational numbers. Doesn't work for real and complex numbers.
Complex numbers aren't really relevant, in my opinion, because they are usually introduced as an extension of the rules for reals and polynomials. To multiply two complex numbers, you can totally forget that i is imaginary, do the multiplication as if it's just an ordinary variable, then substitute "i" back in. But that relies on being able to multiply polynomials, which would be difficult to define in terms of repeated multiplication.
To some extent all of mathematics is a lie. We can do multiplication on the reals because we have decided that it's allowed. It is reasonable to define multiplication at first as repeated addition and then define a way to extend that to the reals that is consistent with the first definition.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#148Earlier quoted context omitted.
Now that's an interesting point, did it start as repeated addition? It is just as conceivable that people were faced with a problem like 'each person needs 2 apples, we have 5 people, so we need 10 apples'? In this case repeated addition is a perfectly fine algorithm to calculate the product but the product itself is not defined as repeated addition, it's the solution to a particular type of problem.
Your problem just reinforces the notion that multiplication is repeated addition. "So we need 2 for him and 2 for her and 2 for him and 2 for him and 2 for her. 2+2+2+2+2 = 5 x 2 = 10"
I disagree, translating from "2 for (him + her + him + him + her)" to "2 for him + 2 for her + 2 for him + 2 for him + 2 for her" is using the distributive property of multiplication.
The sentence "2 for him and 2 for her and 2 for him and 2 for him and 2 for her" is unnatural and not the way people generally think (unless they're just tallying up but then we're not even talking about multiplication anymore, a tally rarely consists of all equal numbers)
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#149Earlier quoted context omitted.
No, it's not. It depends a bit on your teachers but also on which "level" of math you're in (in the US). Lower level but still algebra/geometry classes tend to teach facts, not derivations from foundational concepts. Those are the classes aimed at non-Honors and maybe non-College Prep students (2 of the 3 typical "tracks" students end up in the US, names may vary by state and decade).
The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.
The US education system varies tremendously. I don't think you can credibly claim that this teaching method is "standard" in the US. At least my experience was much better than you described. Indeed it wasn't until university-level mathematics that I started to get into the "stop trying to understand how they work and just memorize these formulae". Fortunately my engineering classes provided a through-line for understanding how those formulae worked, and I was much stronger for it than my math-major contemporaries.
Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
#150I strongly believe that helping younger students gain strong intuition for these operators pays dividends towards their later success in maths.
I've always run into the following problem: I try to motivate multiplication as repeated addition, which does help with intuition, but then things totally fall apart when we move on from integers into fractional values.
1/2 * 1/2 -> 1/4.
Sure you can teach someone to simply multiple the numerator and denominator, but it doesn't necessarily help them make clear sense of what's going on.