What’s Wrong with “Multiplication Is Repeated Addition”? (2008)(denisegaskins.com)
denisegaskins.com
What’s Wrong with “Multiplication Is Repeated Addition”? (2008)
https://denisegaskins.com/2008/07/28/whats-wrong-with-repeated-addition/
417 comments
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 gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
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 gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.
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 addition doesn't prevent you from looking at the special properties of this new operator.
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 addition doesn't prevent you from looking at the special properties of this new operator.
I think the author's argument becomes more clear when you consider multiplication in rings other than the integers, for example (square) matrices. The product of two matrices A*B does not correspond to repeated addition, and it is not commutative (A*B does not equal B*A in general).
I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning about mathematics. However, this is true for a lot of basic properties that hold over the integers and not in other domains, so I'm not really convinced that it's actually wrong to teach kids about multiplication this way.
I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning about mathematics. However, this is true for a lot of basic properties that hold over the integers and not in other domains, so I'm not really convinced that it's actually wrong to teach kids about multiplication this way.
I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition.
For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.
For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases and all of the more advanced concepts the author (and the related articles) discuss can be taught later.
I don’t think the aruments made here are compelling. Multiplication can be defined in this way. Saying that multiplication is repeated addition is not the same as saying that multiplication IS addition. Addition relates to multiplication as multiplication relates to powers.
>> 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.
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.
Learning isn't possible without unlearning whatever temporary construct we used as a crutch, but it doesn't mean using that temporary construct is wrong even if it's not technically correct.
More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.
More specifically in mathematics, the interplay between formalism and intuition, like a mental danse or gymnastics, is a powerful process in furthering our understanding of mathematical truths:
From a formal perspective, mathematical objects can be created in so many ways, some constructions being more intuitive and beautiful than others (axioms, groups, rings, fields). The formalism itself let us see what intuition can't.
From an intuitive perspective, it's useful to latch on whatever concept one have to learn the next level of abstraction, while acknowledging that the intuition might not be 100% correct. Like using addition to intuitively understand multiplication, or addition and multiplication to intuitively understand fields. The intuition let us familiarize with otherwise novel ideas.
Ironically, this article wants to be very normative about which mathematical intuition is better (which there isn't, I'm sure many don't think of "multiplicand" as something special), while disregarding any cues from any formalism.
(Unrelated to pedagogy, but related to when multiplication is not repeated addition.)
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:
What if it were that 2*2 = 5 ?
Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4
2*2 = (1+1) * (1+1) = 1+1+1+1 = 4
1. Associative
2. Commutative
3. Multiplicative Identity
4. Increasing ( If i,j > 1, then m(i,j) > i,j )
5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )
6. Uniqueness of Prime Factorization
An easy example is the odd numbers.
1,2,3,4,5, ... n
1,3,5,7,9, ... odd(n)
odd(n) = 2n-1
inverse_odd(x) = (x+1)/2
m(i,j) = inverse_odd( odd(i) * odd(j) ) = 2ij - i - j + 1
It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.
My favourite way of thinking about these operations comes from category theory:
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
* addition: the cardinality of the disjoint union of two sets
* multiplication: the cardinality of the direct product of two sets
* exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a
I like this article, but I feel like it misses the value of teaching analysis to students. It's true that it's not-quite-right to say multiplication is repeated addition, but it's also a really nice demonstration of the analytical/generalization approach anyone could use to "invent" multiplication on their own.
This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.
This same pattern can be seen driving discoveries in math. - Integrals are generalizations of Riemann sums. - Fractional exponents generalize taking square roots. - The gamma function sorta generalizes factorials. It's okay to start with something not-quite-right and explore how it generalizes.
I'll tell you what's wrong: the mentality of the author of this text
Multiplication has started as repeated addition. That's where the idea came from.
> It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication.
Nobody. Literally nobody said that
Of course, you'll need to forget a bit the idea of repeated multiplication when you get into the rationals/reals/complex numbers, but even there it kinda makes sense
So no, I think this is the kind of teacher that makes the students even more confused and prone to hating math
Multiplication has started as repeated addition. That's where the idea came from.
> It is as if there were two types of addition: regular, random, “wild” addition and the specially-bred variety of addition to which we give the name multiplication.
Nobody. Literally nobody said that
Of course, you'll need to forget a bit the idea of repeated multiplication when you get into the rationals/reals/complex numbers, but even there it kinda makes sense
So no, I think this is the kind of teacher that makes the students even more confused and prone to hating math
The HN crowd is way above me when it comes to math, but I'm going to chime in here w/ a question, come what may.
I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later).
I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)
I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.
With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.
It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?
I read about this a couple of years ago when I was doing basic arithmetic instruction at home to supplement my 5 y/o daughter's schooling. We were just doing basic numeracy activities at the time-- playing with physical representations of a number line, etc. I wanted to have some idea of how to approach multiplication when we got to that point (which we did about a year later).
I was flummoxed by the whole "multiplication isn't repeated addition" debate because of my own arithmetic education nearly 40 years before. The "light bulb" moment for me was reading an article that described multiplication of real numbers as scaling. I'm a embarrassed that, at >40 y/o, I had such an epiphany. (Then again, math has never been something I've had great intuition for.)
I knew that her teacher was likely going to introduce multiplication as repeated addition and I felt like I wanted to have the first word on the subject with her.
With that in mind when we got to multiplication in our home activities I pulled the physical number line analog out again (a board w/ some small nails in it) and used some elastic and rubber bands to illustrate "stretching" a number as multiplication of an integer by another integer (granted, for small numbers only-- I only had the patience to drive so many nails). I did a little bit of "shrinking" an integer by multiplying it by a fraction (because fractions are something we have covered extensively, if only as an excuse to have cake!) to show that multiplication is can scale a number to be both larger and smaller.
It seems intuitive to me, but I'm a rank amateur. Was this a reasonable strategy? Did I do more harm than good?
One of the best concise guides I've encountered to the relationship between the various algebraic operations -- and how they circle back to one another -- is chapter 22 "Algebra" of volume I of the Feynman Lectures on Physics. For some reason this tied everything together in my head in a way that made so much sense, when the pedagogy I head leading up to it simply did not supply me the same vantage point.
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
https://www.feynmanlectures.caltech.edu/I_22.html
From there I guess I've always thought as the various algebraic operations as functions which build upon one another. I think the article's point about not overlooking multiplication's Dimensionality as a key component is a good one, but neither should that focus avoid talking about the functional relationships of the various operations.
As someone with a pragmatic, visual mind, I rely entirely on this kind of reasoning in order to understand and apply math. For example:
- Multiplication is repeated addition.
- Power is repeated multiplication.
- The square of a number can be visualized as a geometric square when you duplicate a row of x items x number of times.
- The cube of a number can be visualized as a geometric cube when you duplicate a row of x items x number of times and then you take the resulting geometric square and duplicate it x times along the orthogonal axis.
- n to the power of p can be visualized as a tree with height p such that each branch splits up into n branches at each level.
- The logarithm base x of y is the height 'number of levels' of the tree when each branch splits into x branches at each level until the number of tips is equal to y.
- A factorial can be visualized as a tree whose branches split up in such a way that there is one fewer branch at each level until the branches cannot be split anymore.
- Multiplication is repeated addition.
- Power is repeated multiplication.
- The square of a number can be visualized as a geometric square when you duplicate a row of x items x number of times.
- The cube of a number can be visualized as a geometric cube when you duplicate a row of x items x number of times and then you take the resulting geometric square and duplicate it x times along the orthogonal axis.
- n to the power of p can be visualized as a tree with height p such that each branch splits up into n branches at each level.
- The logarithm base x of y is the height 'number of levels' of the tree when each branch splits into x branches at each level until the number of tips is equal to y.
- A factorial can be visualized as a tree whose branches split up in such a way that there is one fewer branch at each level until the branches cannot be split anymore.
I 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.
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.
The nicest formal definition of multiplication is "the operation that distributes over addition". When defining operations on abelian groups that might be called "multiplication", it's the distributive property that makes the term useful, and brings us into the categories of ring, semiring, [weird math prefix here]-ring, etc.
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".
I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)
Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?
> Is there really a difference between multiplication and repeated addition, or am I tilting at windmills here?
You are tilting at windmills
> Is it even necessary for teachers to define multiplication?
No, at least not in formal terms
> Or is the teacher’s job to provide plenty of examples of multiplication in action?
I think so
> Should we let the students intuit their own definition(s)?
Yes
> Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?
It will introduce new difficulties
> Or do we already teach this way, only in different words?
We don't, thankfully
> If you are an elementary teacher, how do you teach multiplication to your students?
I will let elementary teacher answer
> Are some students clueless because, no matter how we explain it, they just don’t pay attention?
While we can't always blame teachers when students don't pay attention, if it is systematic, maybe there is a problem with your teaching.
> Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?
I will let teachers respond
Ok, I hope you are still there and didn't just click the "downvote" button.
I think the author raises good points for at least high-school level students in STEM fields. But in reality, the important thing in elementary school is that we want people to know how to count. 5 apples at $0.50 each is $2.50, this kind of thing.
Dimensional analysis is very important in engineering, and I definitely think is should be taught, as well as everything in the article, but at college level, not in elementary schools. Teaching such abstract concepts too soon is a recipe for disaster, I know because it has been tried. The expectation was that it will make better engineers in the future, the reality is that while it may have helped a tiny minority get college degrees, it mostly produced kids who couldn't do simple arithmetic.
You are tilting at windmills
> Is it even necessary for teachers to define multiplication?
No, at least not in formal terms
> Or is the teacher’s job to provide plenty of examples of multiplication in action?
I think so
> Should we let the students intuit their own definition(s)?
Yes
> Will it help students if we change our focus from “how to get the answer” and teach them to identify the multiplicand, the “this per that” unit? Or will that introduce new difficulties I haven’t considered?
It will introduce new difficulties
> Or do we already teach this way, only in different words?
We don't, thankfully
> If you are an elementary teacher, how do you teach multiplication to your students?
I will let elementary teacher answer
> Are some students clueless because, no matter how we explain it, they just don’t pay attention?
While we can't always blame teachers when students don't pay attention, if it is systematic, maybe there is a problem with your teaching.
> Have you tried using bar diagrams to model elementary arithmetic situations? And if so, how did your students respond?
I will let teachers respond
Ok, I hope you are still there and didn't just click the "downvote" button.
I think the author raises good points for at least high-school level students in STEM fields. But in reality, the important thing in elementary school is that we want people to know how to count. 5 apples at $0.50 each is $2.50, this kind of thing.
Dimensional analysis is very important in engineering, and I definitely think is should be taught, as well as everything in the article, but at college level, not in elementary schools. Teaching such abstract concepts too soon is a recipe for disaster, I know because it has been tried. The expectation was that it will make better engineers in the future, the reality is that while it may have helped a tiny minority get college degrees, it mostly produced kids who couldn't do simple arithmetic.
In some settings, "repeated addition" is the only sensible way to think of multiplication. Such as multiplying an elliptic curve point with a scalar.
Imagine explaining the content of this post to a five year old instead of that multiplication is repeated addition.
The more methods one studies, the better one understands. There is no one true method.
I like how 3blue1brown presents it in terms of animated number line translations
https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
https://www.youtube.com/watch?v=mvmuCPvRoWQ&t=365
> Is that really how we want our students to think? Multiplication is not a mere sub-species of addition. Multiplication is its own animal, an independent operation.
Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.
Wtf, why not? Why wouldn't you want people to draw connections between different parts of math. The whole point of math is to find the patterns and interconnections.
I 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.
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.
I feel like people here have a weird definition of "truth". Platonism aside, there is no outside ground truth on what "addition" or "multiplication" is. It is an abstract symbol defined solely by what it does to numbers. As long as your definition describes its behaviour, your definition is no more or less right than any other definition.
If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.
Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.
If you want to define multiplication as repeated addition (and addition as repeated invocation of the succesor function), by all means do so. Its not wrong.
Sure you could say it doesn't work for complex numbers, but usually when you introduce complex numbers you say something like, multiplication works as normal except i×i=-1. I hardly see that as a problem with the original definition of multiplication. Its more about the definition of i than the definition of multiplication.
Why not just say:
There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.
There is the symbol '*'. And when there's two integers it does one thing, and you can think about it as repeated addition. But when there's something different, it works differently. But now we're just going to concentrate on integers.
I think what the author is missing is the word "scalar". I do understand the sentiment that in general multiplication is not repeated addition. But multiplication with a scalar is.
In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.
Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.
In the usual curriculum you do the unit analysis as a separate step, as it's usually in the physics lessons where units are used. In math it's unitless.
Of course, we should start teaching immediately that you can't add apples and pears. Maybe we should just do this explicitly there, that 5 times 3 apples is fine, as 5 is scalar. Explain that immediately and then add the unit analysis in physics.
I know how addition is done in hardware. I've programmed multiplication and division on computers that only had shift and add instructions. I've implemented IEEE 754 floating point packages, too.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
https://github.com/DigitalMars/dmc/blob/master/src/CORE16/FL...
Multiplication is repeated addition.
Yes, I know about roundoff errors, overflows, etc.
A few years ago, I was teaching a class and decided to record a short series on the fundamentals of mathematics, for complete "beginners" - whether they be kids or adults.
This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:
https://www.youtube.com/watch?v=Wd1lzqR3xW0
The other videos you'll find on that channel are:
https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
(PS: I am generalizing this approach in our upcoming app, https://teaching.app)
This is the first video in the series Thinking Mathematically (after the introductory video, "Why think Mathematically?") which I put on YouTube under a channel of the same name. It proceeds through the sets of numbers, N -> Z -> Q -> R -> C and yes it's for beginners. Would love some feedback:
https://www.youtube.com/watch?v=Wd1lzqR3xW0
The other videos you'll find on that channel are:
1. Why think mathematically?
2. Numbers and Algebra
3. Sets and Infinity
4. Logic and Probability
https://www.youtube.com/channel/UCuge8p-oYsKSU0rDMy7jJlA
(PS: I am generalizing this approach in our upcoming app, https://teaching.app)
I think the author is trying to say that _from a teaching point of view_, thinking of multiplication as a kind of addition is suboptimal.
Because you can indeed multiply through repeated addition (which she acknowledges), repeated addition is one valid way to look at multiplication. But it might not be the maximally helpful framework for students.
Because you can indeed multiply through repeated addition (which she acknowledges), repeated addition is one valid way to look at multiplication. But it might not be the maximally helpful framework for students.
You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be defined in any or all of these contexts if one enjoys doing so. There are far ranging context to define an object and less far ranging. I am quite sure that in grade school it is quite unhelpful to look for the furthest ranging context in which one could define multiplication. E.g, let us start in grade 1 with the definition of an algebra and derive everything from that.