What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters.

After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2.

To return to TFA I think the author is talking from a pedagogical standpoint, that teaching that multiplication is just a bunch of additions under a trench coat is not the best way to go. I'm not sure that I agree personally.

In particular this bit regarding multiplier/multiplicand makes zero sense to me:

>Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

Of course 3 rows and 2 rows aren't the same, but what does it have to do with the order of the multiplication? Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.

Multiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.

>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.

Education is probably a special case, at least from my educational experience. There's often multiple ways to abstractly represent a problem in mathematics and find the desired solution. The example you pointed out is such a case.

When teaching mathematics, I think the goal is to introduce a lot of forms of mathematical thinking and approaches to solving a problem, to make you realize there are often multiple approaches and to take a peek at the insight of some of these approaches and how they often connect or think about different 'branches' of mathematics.

Teaching math needs to be explicit with this though: solve this using method X. They should also explain this to kids as to why they're doing it. One of the biggest mistakes I see in mathematics teaching is the perception there's "only one right answer." Well, yes and no. Under certain condition a specific answer exists, sometimes it's a set of answers and sometimes you they're not really assessing so much that you can get the correct answer, but instead that you understand a specific method.

The 'answer' is a means to an end to force you to step through, internalize a process, and hopefully at some point understand the deeper insight of that clever process you internalized and apply it to other problems you may encounter in the future. You may never use the exact process or insight behind it as-is, then again you may use facets of the reasoning you internalized later. If you don't work in professions that require this sort of abstract thinking I can see how the entire dance is quite silly but if you work in research, you often appreciate all these nuggets of deep insight and wisdom you've gained you can cobble together, remold, or lead to new insights. As a kid you probably have no idea where you'll work as an adult so maybe a lot of that effort is wasted.

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 multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?

> 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.