To all.....
I feel badly that I have completely hijacked a perfectly nice thread. But now that I've done it. I don't know how to put the genie back into the bottle!
Apologies to all who might be reading this thread for its original topic!
@Socks said:
I've seen nothing to undermine the basic idea that multiplication is a form of addition, at least when it comes to integers, I need more than assertions, it's no good saying 'honest' or that it's hard to conceive'
You are not wrong. For integers, one way to determine the correct value of a multiplication problem is to use repeated addition. But pedagogically, that realization should have happened for you after you learned what multiplication was. It should have been a happy discovery that (for integers) you can simply use repeated addition to derive the correct answer to a multiplication problem.
You should have also discovered that another way to find the answer to a multiplication problem is to look up the value in a 'multiplication table'.
But we don't say that multiplication is, 'looking up a value in a multiplication table.' Even though it gives us the correct answer every time.
We also shouldn't say that multiplication is 'repeated addition'. Even if it gives us the correct answer every time (for integers).
I am afraid that I am running out ways to assert that the fundamental mathematical operation of multiplication is scaling (or rescaling) a numerical value. You want me to quit making assertions but that is what mathematics consists of (assertions). If I cant make a fundamental assertion then I can't do math!
@Socks said:
I'd ideally need to be shown, perhaps proof of some kind,
Does @BigDave's math problem help? How do you use repeated addition to get the value for:
1.1x1.1
@RThurman said:
You were taught as a child that multiplication is repeated addition was because a teacher did not understand the basic mathematical operation of multiplication.
Not at all, right or wrong, it's something I've arrived at myself.
Thats right! All boys and girls should be allowed the pleasure of discovering for themselves that one clever trick to solving a multiplication problem (for integers) is to use repeated addition. Its a fun little trick! But that's all it is. It is not multiplication.
@RThurman said:
To all.....
I feel badly that I have completely hijacked a perfectly nice thread. But now that I've done it. I don't know how to put the genie back into the bottle!
Apologies to all who might be reading this thread for its original topic!
I wouldn't apologise, it's interesting stuff, your posts have lots of useful insights, and the original topic has already been addressed !
@RThurman said:
You are not wrong. For integers, one way to determine the correct value of a multiplication problem is to use repeated addition. But pedagogically, that realization should have happened for you after you learned what multiplication was. It should have been a happy discovery that (for integers) you can simply use repeated addition to derive the correct answer to a multiplication problem.
I suppose I'm not coming from an education angle, or even from a practical angle, or even trying to suggest that multiplication-is-a-form-of-addition (MIAFOA) has any kind of utility (it may or may not, but that's not my point), I am saying "that is fundamentally what it is", or at the least that it's a valid interpretation, it could be the least practical way of thinking about multiplication, it could become impractical with negative numbers, fiendishly complicated with fractions and laborious with very large numbers, or it could even be such a complex way of thinking about a process that you need a supercomputer just to work out a restaurant bill . . . . whilst still being 'right' or a valid interpretation of what the process is actually doing.
The 1.1 * 1.1 example is a good one in that whilst 10x3 can be seen as ten threes to be added together, and is something most people can do in their head, 1.1 * 1.1 is much less intuitive . . . but the MIAFOA concept doesn't fail as a concept it just becomes impractical, and of course we don't judge the validity or truth of an idea on its practicality, MIAFOA as a method for multiplcation might be useless or even unworkable whilst still being the correct, or a valid, description of the underlying process.
@RThurman said:
You should have also discovered that another way to find the answer to a multiplication problem is to look up the value in a 'multiplication table'.
But we don't say that multiplication is, 'looking up a value in a multiplication table.' Even though it gives us the correct answer every time.
The argument here (from both the stretching/shrinking and the MIAFOA perspectives - and from the perspective of any other method) would be that the table is not the process, it's simply listing the results of the process, the process happened prior to the table been drawn up, and we are really discussing here is the process.
@RThurman said:
You want me to quit making assertions but that is what mathematics consists of (assertions). If I cant make a fundamental assertion then I can't do math!
Wouldn't you say assertions in maths are underpinned by proofs ?
@RThurman said:
Does @BigDave's math problem help? How do you use repeated addition to get the value for:
1.1x1.1
Yes, as mentioned above, it's a good example in that it teases apart the difference between a practicable method and the underlying process.
How would I use repeated addition to get the value for 1.1 x 1.1 ? I wouldn't ! I don't use repeated addition for anything as far as I am aware, perhaps some very simple multiplications, most maths seems to be memory, or some kind of kludging, a calculator and guesswork and occasionally a hammer, and to add to your 'number-line' and 'stretching' ideas a lot of maths gets done visually (for me at least, but I'm sure it's common) as shapes and lines and pattern of dots and so on.
But . . . . I would say fundamentally the question 'what is 1.1 x 1.1' is asking what is one-point-one one-point-ones combined or added together (or at the least it's a valid interpretation of the question).
@RThurman said:
Thats right! All boys and girls should be allowed the pleasure of discovering for themselves that one clever trick to solving a multiplication problem (for integers) is to use repeated addition. Its a fun little trick! But that's all it is. It is not multiplication.
Lol we are going to have to disagree on this one, again just to reiterate, not that I think anything you've said is wrong at all, and if I tried to teach maths to schoolchildren I would simply be increasing the chances that they would all go to prison and most of them would be adding up their parking fines with a compass . . . it's just that I think at the root of multiplication lies a dark secret, underpinning the whole concept, if you dig down to the foundations, you are adding values together . . . I think !
@Socks said:
I think at the root of multiplication lies a dark secret, underpinning the whole concept, if you dig down to the foundations, you are adding values together . . . I think !
OK -- since we are revealing deep dark thoughts. Here are the first four fundamental operations of math:
1) counting (making number values)
2) combining number values (addition/subtraction)
3) growing and shrinking number values (multiplication/division)
4) determining the rate that numbers can grow and shrink (exponentiation)
These are the first four things you can do with numbers. They are qualitatively different things you do with numbers. The don't rely on the previous operation they are simply different things you do.
I realize that these are assertions.
But so is your statement that its turtles (erm... I mean addition) all the way down!
Comments
To all.....
I feel badly that I have completely hijacked a perfectly nice thread. But now that I've done it. I don't know how to put the genie back into the bottle!
Apologies to all who might be reading this thread for its original topic!
You are not wrong. For integers, one way to determine the correct value of a multiplication problem is to use repeated addition. But pedagogically, that realization should have happened for you after you learned what multiplication was. It should have been a happy discovery that (for integers) you can simply use repeated addition to derive the correct answer to a multiplication problem.
You should have also discovered that another way to find the answer to a multiplication problem is to look up the value in a 'multiplication table'.
But we don't say that multiplication is, 'looking up a value in a multiplication table.' Even though it gives us the correct answer every time.
We also shouldn't say that multiplication is 'repeated addition'. Even if it gives us the correct answer every time (for integers).
I am afraid that I am running out ways to assert that the fundamental mathematical operation of multiplication is scaling (or rescaling) a numerical value. You want me to quit making assertions but that is what mathematics consists of (assertions). If I cant make a fundamental assertion then I can't do math!
Does @BigDave's math problem help? How do you use repeated addition to get the value for:
1.1x1.1
Thats right! All boys and girls should be allowed the pleasure of discovering for themselves that one clever trick to solving a multiplication problem (for integers) is to use repeated addition. Its a fun little trick! But that's all it is. It is not multiplication.
I wouldn't apologise, it's interesting stuff, your posts have lots of useful insights, and the original topic has already been addressed !
I suppose I'm not coming from an education angle, or even from a practical angle, or even trying to suggest that multiplication-is-a-form-of-addition (MIAFOA) has any kind of utility (it may or may not, but that's not my point), I am saying "that is fundamentally what it is", or at the least that it's a valid interpretation, it could be the least practical way of thinking about multiplication, it could become impractical with negative numbers, fiendishly complicated with fractions and laborious with very large numbers, or it could even be such a complex way of thinking about a process that you need a supercomputer just to work out a restaurant bill . . . . whilst still being 'right' or a valid interpretation of what the process is actually doing.
The 1.1 * 1.1 example is a good one in that whilst 10x3 can be seen as ten threes to be added together, and is something most people can do in their head, 1.1 * 1.1 is much less intuitive . . . but the MIAFOA concept doesn't fail as a concept it just becomes impractical, and of course we don't judge the validity or truth of an idea on its practicality, MIAFOA as a method for multiplcation might be useless or even unworkable whilst still being the correct, or a valid, description of the underlying process.
The argument here (from both the stretching/shrinking and the MIAFOA perspectives - and from the perspective of any other method) would be that the table is not the process, it's simply listing the results of the process, the process happened prior to the table been drawn up, and we are really discussing here is the process.
Wouldn't you say assertions in maths are underpinned by proofs ?
Yes, as mentioned above, it's a good example in that it teases apart the difference between a practicable method and the underlying process.
How would I use repeated addition to get the value for 1.1 x 1.1 ? I wouldn't !
I don't use repeated addition for anything as far as I am aware, perhaps some very simple multiplications, most maths seems to be memory, or some kind of kludging, a calculator and guesswork and occasionally a hammer, and to add to your 'number-line' and 'stretching' ideas a lot of maths gets done visually (for me at least, but I'm sure it's common) as shapes and lines and pattern of dots and so on.
But . . . . I would say fundamentally the question 'what is 1.1 x 1.1' is asking what is one-point-one one-point-ones combined or added together (or at the least it's a valid interpretation of the question).
Lol
we are going to have to disagree on this one, again just to reiterate, not that I think anything you've said is wrong at all, and if I tried to teach maths to schoolchildren I would simply be increasing the chances that they would all go to prison 
and most of them would be adding up their parking fines with a compass . . . it's just that I think at the root of multiplication lies a dark secret, underpinning the whole concept, if you dig down to the foundations, you are adding values together . . . I think ! 
OK -- since we are revealing deep dark thoughts. Here are the first four fundamental operations of math:
1) counting (making number values)
2) combining number values (addition/subtraction)
3) growing and shrinking number values (multiplication/division)
4) determining the rate that numbers can grow and shrink (exponentiation)
These are the first four things you can do with numbers. They are qualitatively different things you do with numbers. The don't rely on the previous operation they are simply different things you do.
I realize that these are assertions.
But so is your statement that its turtles (erm... I mean addition) all the way down!
Here are my four basic maths operations:
1) counting
2) turtles
3) cheating
4) guessing