Multiplication Operation I am a father of two young boys and I looks forward to exploring mathematics with them for as long as they will let me :-). I would really like for them to have a deeper understanding of mathematics than what I had when I was a young student. As I think about how I might approach some of the topics, there is one that remains particularly unclear to me to this day - the multiplication operation. Now I do not have a strong background in mathematics (e.g. never had a course in abstract algebra), so please forgive me if some things that I say are off - maybe  even way off. 
I have seen that there have been debates online as to what multiplication is, and how to teach it to students.  Often the discussion turns into interpretations of multiplication (e.g. repeated addition, scaling, etc.) but the discussions/debate from this approach seem to be fruitless. Other times properties of multiplication are discussed, but often the properties are the same as those found under different types of operations. Integer multiplication may be associative, but so is integer addition - leaving me no more informed about the unique and universal thread for the concept of multiplication. 
From my perspective, I am most confused by the many definitions for the multiplication operation depending on the type of objects of interest (real numbers, complex numbers, matrices, etc). I always think to myself, "why would mathematics allow the same name to be associated with multiple definitions?". It seems like there must be something that all the definitions must have in common. Surely, not just any binary operation on a set of objects can be labeled multiplication on a whim...or can it? So this is my question, is there a characterization of the multiplication operation that holds true for all operations labeled multiplication, that it is agreed on within the academic community, and is unique enough to be able to distinguish it from other operations (namely addition)? If so, please do share. And if not, how would you explain why the same term has various definitions in mathematics to students learning about operations like multiplication?
From my limited mathematical knowledge, it appears that the only thing in common with different definitions of multiplication on different objects is that that they all rely on the use of the addition operation in their construction. So perhaps the term addition is used to reference an operation for a set of objects that is considered to be the simplest method for combining/connecting two objects in a set, and multiplication is a more complex method for  doing so (perhaps based on the use of simpler operations, like addition, already defined for the set). But, I would prefer that my discussion with my sons not rely on my experience. Hence, the reason for the post. Many thanks for taking the time to review my write up and I look forward to any insight that may be offered.
 A: It is not multiplication that is particularly special. Rather, the special operation is addition. Multiplication need not be associative, commutative, or be associated with an addition operation over which it distributes. Multiplication of octonions is not associative or commutative, and groups, semigroups, and  monoids only have one operation in general, often called multiplication. There are even algebraic structures with binary operations that have no special properties at all, and these operations could conceivably be called multiplication. The structures in question are called magmas.
Addition, however, is almost always associative and commutative and every element usually has an inverse, in which case the inverse of $a$ would be written as $-a$. The operation in a commutative monoid is often called addition; it is associative and commutative but there need not be inverses.
In situations where an operation is called addition and is not associative or commutative, the author would often use different notation for it ($a+'b$, for example) or specify this explicitly to avoid confusion. I know of at least one example: in "Algebraic Topology" by Allen Hatcher the operation in higher homotopy groups is written as addition. It is always associative, and it is commutative except for the relative homotopy group $\pi_2(X,A)$.
Edit: I linked some Wikipedia articles but the links aren't working. This is because I'm on my phone and it's hard to get the links right. I'll fix it when I get on a computer, but if anyone is feeling generous feel free to fix them.
A: From the perspective of abstract algebra, ring theory in particular, a multiplication operation must have a few different properties.
Mulipication is associative, meaning $(a\times b)\times c=a\times(b\times c)$. There must be an identity, so $a\times1=1\times a= a$. Its relationship to addition comes from the distributive property, that $a\times(b+c)=(a\times b)+(a\times c)$ and $(a+b)\times c=(a\times c)+(b\times c)$.
Depending on the type of objects being used, it may also be required that multiplication is bilinear by scalars or modules.
Of course, these concepts in algebra were developed as a generalization of the expected behavior of multiplication over real numbers. However, there are a few properties that multiplication does not need to have. There does not need to be an inverse for every element (indeed, even in the real numbers, $\frac10$ is not defined), or in fact any element, as can be seen in the integers. What is probably more surprising is that multiplication does not need to be commutative. It is perfectly acceptable to have an algebra where, in general, $a\times b\ne b\times a$. In contrast, every element of a ring or group must have an additive inverse, and in rings, addition is commutative.
So, there is a set of rules that characterizes multiplication, and distinguishes it from addition; it is not just any binary operation.
A: I appreciate your question.
It is my experience - in abstract algebra especially - that most operations are simply called "multiplication" by default.  We dont say "times" or "multiply", but we might say "product" and refer to it as "multiplication".  Its not entirely uncommon to see addition, either, but in general theorists prefer what they call "multiplicative notation" over "additive notation", when dealing with abstract objects.  The terminology is simpler than using the ambiguous term "operate" or "operation", though it would be more accurate and far less misleading.  So I understand your confusion. In abstract algebra it is the intent of the teacher to convey abstract ideas in a way that is generalizable and doesnt lock you into one way of thinking, but that is often undermined for some students by the unfortunate use of vocabulary.  What seems normal and convenient and easy for the expert to default to is often a crutch to educating the layperson.
I dont believe there is a "common thread", no.  Integer multiplication can be defined as repeated addition, but you run into problems with matrix multiplication.  You might be able to add a matrix to itself an integer number of times (scalar multiplication of a matrix), but how do you add a matrix to itself a matrix number of times?  Indeed, matrix multiplication is derived from vector dot product multiplication (row and column vectors of a matrix).
The vector dot product just happens to be a special case of whats called the inner product (generalizeable to different kinds of objects).  Inner products (again notice the multiplicative implication, although it isnt actual multiplication), by definition, share in certain kinds of properties: conjugacy, linearity, positive-definiteness, zero-product property.
These properties are shared with real and integer multiplication, too. So this might just be the common thread youre looking for, but I cant be sure. I cant be sure it doesnt apply to integer addition, as well.
Getting back to regular multiplication. Has it occurred to you that defining multiplication as repeated addition is only valid if one of the two operands is a positive integer?  For example, $2\cdot 4$, $5 \cdot 2.3$, $ 1.7\cdot 8$ or even $-2.9 \cdot 8$.  But this definition loses its meaning when dealing with $1.5 \cdot 9.3$.  How do you add any kind of number to itself a fractional number of times?
Typically the problem is circumvented by introducing students to fractions.  As long as a real decimal number is rational it can be expressed in terms of integers.  And bare in mind, we arent really talking about division yet, just the abstract idea of fractions.  Although you could introduce students to integer division prior.
Multiplying by irrational numbers becomes tricky, too. Or imaginary.
One thing Ive found particularly useful in my mathematical education is to understand the development of mathematics in a historical context.  Why did things develop as they did, what incentivized the change, etc.  Its as useful as modern proofs and seeing practical application. All three are essential to motivate a desire to learn and a genuine comprehension.
I will agree with you on this. I far too frequently see one idea (not just one ambiguous term like "multiplication" applied to different objects), but Im talking about one specific idea applied to one kind of object, defined in multiple ways simultaneously.  Take the natural base $e$, Euler's Number, for example.  Ask any two mathematicians how its defined, you might get two different answers.  I prefer to default to the historical origination, and prove every other property from there, rather than interchangeably relying on a multitude of simultaneous definitions whose connection or equivalence is not obvious.  When I see mathematicians do this I question how well they know the material to not see the potential for circular reasoning in what they do. Im content to use any other equivalence, just so long as I recognize it as a proven theorem and dont treat it as a secondary definition. In general definitions are arbitrary and man-made, and no rational person can rely on two distinct definitions being equivalent, unless its been proven to be such, but a derived theorem from a definition is not the same as the definition.
