Balls: Scott Walker has 'em

[Tellos Athenaios]

ONLY after the reasoning behind is fully understood.

But this is kind of interesting: it is easy to see what you mean with multiplication (even though the example is actually quite flawed because once you move on from positive integers the logic breaks down, and it ought to crash hard in your first year of any technical study worth its salt); however how do you explain this with division?

Multiplication can be defined inductively (which I guess you are doing: letting the kids consider it repeated addition) even if it is really not, but division offers no such explanation that stands up to even casual scrutiny.

Children should never memorize the multiplication table, for example, until they have understood what multiplication is. But after you understand it, yeah, there's no harm in memorizing it.

Nah they should just look up the logs and do the addition.

Seriously though, from a “understanding” point of view if you understand what you are doing when fiddling with logs you inherently understand multiplication. So I do think the log method is better than just summing values from memorised tables, occasionally forgetting to “carry over” (or whatever it is called in English?) those extra values and getting the wrong answer out.

[HoreTore]

But this is kind of interesting: it is easy to see what you mean with multiplication (even though the example is actually quite flawed because once you move on from positive integers the logic breaks down, and it ought to crash hard in your first year of any technical study worth its salt); however how do you explain this with division?

Multiplication can be defined inductively (which I guess you are doing: letting the kids consider it repeated addition) even if it is really not, but division offers no such explanation that stands up to even casual scrutiny.



Nah they should just look up the logs and do the addition.

Seriously though, from a “understanding” point of view if you understand what you are doing when fiddling with logs you inherently understand multiplication. So I do think the log method is better than just summing values from memorised tables, occasionally forgetting to “carry over” (or whatever it is called in English?) those extra values and getting the wrong answer out.

Multiplication as a serial addition is one kind of multiplication, multiplication represent other things too. A student will have to understand all of them.

I don't see how division is any different though, except perhaps that it isn't common to memorize 4/2 like people memorize 2x2...

[Tellos Athenaios]

Maybe I am simply not imagining properly how you treat or explain multiplication (or how much you expect your students to work from the concept rather than they just doing what appears to be working). You are saying you expect your student to think in kinds of multiplication, however multiplication is to the Real numbers what the AND operation is to Boolean values. It gets really intriguing once you start considering matrices & graphs, set theory and relations: multiplication is how you compute the result of a query on sets involving multiple relations. Back to the “simple” examples: how do you work with 0.5 and other such factors: they are in fact a division.

As for division: typically explained as partitioning. But that explanation works for 3 people sharing 6 items, but it can be written off after the first time an adventurous soul asks how you would divide 6 items among 0 people?

[HoreTore]

Maybe I am simply not imagining properly how you treat or explain multiplication (or how much you expect your students to work from the concept rather than they just doing what appears to be working). You are saying you expect your student to think in kinds of multiplication, however multiplication is to the Real numbers what the AND operation is to Boolean values. It gets really intriguing once you start considering matrices & graphs, set theory and relations: multiplication is how you compute the result of a query on sets involving multiple relations. Back to the “simple” examples: how do you work with 0.5 and other such factors: they are in fact a division.

As for division: typically explained as partitioning. But that explanation works for 3 people sharing 6 items, but it can be written off after the first time an adventurous soul asks how you would divide 6 items among 0 people?

Division by zero is impossible.

It might seem like a difficult concept for a 12-year old to undestand, but it's actually surprisingly easy to explain. As soon as students learn about multiplication and division, they learn about the relationship between the two, just like they did with maths. A student learns that if 6+3=9, you can turn it around and get a subtraction that reads 9-3=6. The same goes for multiplication and division, the division 4/2 is explained by showing that 2x2=4.

Thus, explaining why division by zero cannot be done is as simple as showing them the multiplication 2x0=0, and ask them to turn it into a division. Or, if one insists that 2/0 must be 0, ask them to swap their division around and see if the multiplication works.

As for multiplication with 0.5, that is indeed a division, and as such it is explained by swapping the 0.5 for the much more accurate number 1/2.

[Tellos Athenaios]

Division by zero is impossible.

Trick question.

It might seem like a difficult concept for a 12-year old to undestand, but it's actually surprisingly easy to explain. As soon as students learn about multiplication and division, they learn about the relationship between the two, just like they did with maths. A student learns that if 6+3=9, you can turn it around and get a subtraction that reads 9-3=6. The same goes for multiplication and division, the division 4/2 is explained by showing that 2x2=4.

But then you are not explaining multiplication at all. You explain how you compute multiplication, but that is not at all the same thing, and in fact is little more than pure memorisation of a symbol and a set of actions to arrive at output.

Thus, explaining why division by zero cannot be done is as simple as showing them the multiplication 2x0=0, and ask them to turn it into a division. Or, if one insists that 2/0 must be 0, ask them to swap their division around and see if the multiplication works.

But I can do that using such logic and the given example: 0/0 = 2. Perfectly valid according to the reverse logic since 0 * 2 = 0.

As for multiplication with 0.5, that is indeed a division, and as such it is explained by swapping the 0.5 for the much more accurate number 1/2.

Yes but here is the interesting thing: multiplication is scaling in a lot of applications. Remember how you said kind of?

The real problem is that if you want to teach concepts of operations like addition and multiplication definitions based on “reverse identities” do not hold in the general case. For instance with set addition and substraction.

{ 5, 6, 7, 8 } (+) {1 , 2, 3, 4, 5 } = { 1, 2, 3, 4, 5, 6, 7, 8}.
Now let's try our reversal technique:
{1, 2,3,4,5,6,7,8 } - {5,6,7,8} = { 1,2,3,4 }.

[HoreTore]

But then you are not explaining multiplication at all. You explain how you compute multiplication, but that is not at all the same thing, and in fact is little more than pure memorisation of a symbol and a set of actions to arrive at output.

But I can do that using such logic and the given example: 0/0 = 2. Perfectly valid according to the reverse logic since 0 * 2 = 0.[/QUOTE]

First of all, the product of any factor multiplied by zero is zero by definition, which happens to be one of the earliest concepts a child will learn when it comes to multiplication. So, it is highly unlikely that someone would argue that way.

Secondly, I'm using numbers here. I would of course supplement with reality when explaining it to a student. Ie. "if you have zero baskets with zero apples in them, how many apples do you have?" Anyhoo, a 12-year old thinks that the higher the number the higher the difficulty, so a strong kid would focus on 09324850932809423x94386724395 rather than focusing on the 0 or other low numbers...

Another way would be to say "can I have two slices of pizza, if I don't have a pizza?" (2/0)

Yes but here is the interesting thing: multiplication is scaling in a lot of applications. Remember how you said kind of?

The real problem is that if you want to teach concepts of operations like addition and multiplication definitions based on “reverse identities” do not hold in the general case. For instance with set addition and substraction.

{ 5, 6, 7, 8 } (+) {1 , 2, 3, 4, 5 } = { 1, 2, 3, 4, 5, 6, 7, 8}.
Now let's try our reversal technique:
{1, 2,3,4,5,6,7,8 } - {5,6,7,8} = { 1,2,3,4 }.

No idea what you're talking about here....

And I have to add that my knowledge of computer 1337 skillz are limited at best.

[Tellos Athenaios]

No idea what you're talking about here....

Well I was getting a bit far out in the whole concept thing. Summary: that “addition” and “substraction” operations are well defined outside the domain of real numbers, and sometimes you can't say x + y = z, therefore z - y = x. I chose an example with sets to elaborate both at the same time.