What is Mathematics for STEM? What is a Mathematics of Solution?

Why all the Fuss about Multiplication

        We only consider multiplication of non-negative integers. (multiplication in the natural                        number system). When we say "multiplication" the operands and the product are natural                      numbers.

I think it's because multiplication is the first encounter with arithmetic that is more than just elaborated counting. Students do not bring applicable intuition: multiplication must be taught and learned in an academic setting. The traditional approach skirts its difficulty one, or both, of two ways:

• it says multiplication is (multiple) addition so there is really nothing new, and/or
• it says its too difficult to understand it, remember the multiplication tables.

I assume each of these approaches contributes to an efficient route to worthy goals, but their pursuit short circuits the opportunity to see what multiplication really is. In the past it could be argued that students only had a need for vocational math, hence it wasn't helpful to know how arithmetic worked; it was just a tool, learn how to do it fast. That was probably a reasonable argument 50 years ago. It's a lousy argument for the world where today's student will work. Understanding how things work is the way to be effective with and around computers/processors, and that is where today's students will be. One doesn't compute with computers, one tells computers how to compute. To do that you need to know how computation works

Why is it Hard to Understand Multiplication?

I think there is a simple explanation: the definition of multiplication involves multiplication itself; it also involves addition. We all know what is being defined can not also appear in its definition; but it turns out there is an escape clause. We are interested in the definition of the multiplication operation, and, although the multiplication operation does appear in its own definition, one of its operands is different, and that allows the operation to work itself out as a process. This is best seen in an example. Consider 6 x 4 with the natural number multiplication definition:  

• 6 x 4 = 6 x (3 + 1) + 6 = 6 x 3 + 6 =

Read this as "(six times four) equals/is  (six times three) plus six". The multiplication operation being defined does appear on both sides of the equals sign, but what happens when we substitute the same definition for 6 x 3, and continue in this manner:

• 6 x 4  =

1. 6 x (3+1) = 6 x 3  +  6 x 1 =  6 x 3 + 6
2. 6 x (2+1) + 6 = (6 x 2 + 6 x 1)  + 6 = (6 x 2  + 6) + 6
3. (6 x (1+1)) +6)  + 6 = ((6 x 1 + 6 x 1) + 6) + 6 =   (( 6 + 6 ) + 6 ) + 6
4. (12 + 6) + 6 = 
5. 18 + 6 = 

• 24

The definition specifies a process for multiplication. When you expand the definition for an actual multiplication, you write down a process:

• multiplication is a process. 

In fact it is a concatenation of two processes, one for multiplication and the other for addition. Each step in a multiplication(addition) process is followed by another multiplication(addition) step until the process terminates; each of these processes is recursive. 

And finally we have spoken the name of a process that refers to itself.  The idea of a recursive process can be hard to grasp; the best way to get a feeling for recursion is to do recursions. Laying out the process as the one above is, reveals its dynamic as a pattern: it first lays out the decomposition of the multiplication one sub-multiplication at a time and then it collects these terms into one sum by one addition at a time. It first goes out and then it comes back:
      multiplication is an process executed as

• a recursive multiplication process followed by
• a recursive addition process. 

A two step process is a general architecture for problem solution:

1. Formulate an expression for the solution that can be evaluated.
2. Evaluate the expression.

The first step is often neglected in school mathematics. Multiplication is an example of that neglect; it is often defined as the multiple addition produced by the first step, and then it is said "multiplication is multiple addition". This approach takes for the solution an expression where the results of the multiplication in the first step have already been incorporated; it is easy to overlook in natural number multiplication where n x 1 = n. In word problems, problems that involve quantities in the real world, the product is not the same thing as either of the operands: the product is not just a different number of what an operand is for natural numbers.

And Your Point Is?

Its slogan could be "multiplication is multiplication plus addition":

• Multiplication is a process, 
• the process terminates when it produces a product.
• To understand multiplication is to understand that process.
• It is a recursive process:"multiplication is multiplication plus addition" 

Multiplication as a solution is dynamic. It is not just the inert answer to a problem. In this, it is an exemplar of the solutions that productive members of society construct. 

For the most part school mathematics is now taught as if the problems of mathematics education itself had been solved. Success is just figuring out how to get students, teachers, and the rest of us to buy in to a program that says: we lay it all out for you, all you have to do is retrieve it. It is a long way from here to "the use of this stuff is in the understanding of it". I want to participate in that journey.