Semantics in Elementary Math

Where I'm Coming From

1. Have not taught mathematics.
2. Did theoretical physics, and taught some.
3. Did engineering at Bell Labs. Retired from there.
4. Did research in the Dept of Civil and Env Eng, Princeton.
5. Started to wonder why most of us eventually have trouble applying the mathematics we learned in elementary and even secondary school; the initial exam results in my first undergraduate physics course indicated I did not not have a clue.
6. Read, talked, thought, and wrote over the past year.

Where I Am Now

I assert: 

1. The mathematics taught in elementary school is both too abstract and too concrete. It is largely based on teaching an arithmetic of abstract numbers which is then used in concrete algorithms whose utility is largely superseded by electronic computation.  
2. Common Core is more a matter of rationalizing the traditional core, less a matter of getting at a core of problems in it.
3. Within the framework of current instruction questions arise that can not be addressed because there is no explicit consideration of the semantics in the mathematics taught; what is the meaning of the numbers and expressions that appear? In the recent discussion of Multiplication is Addition, it was necessary to go way beyond elementary mathematics to argue that multiplication is not addition. Consideration of the semantics of addition and multiplication is a path all of us can follow to understand why it is not a good idea to teach that it is.
4. Semantics is the bridge between numerical calculation and word problem solving.
5. Those who teach the use of mathematics in subjects like engineering, physics, etc., should be in the discussion. Mathematics is too useful, too meaningful, to leave its explanation entirely in the hands of mathematicians. 
6. Manipulables in support of an active approach to simple arithmetic can be used to make a concrete semantics. I will discuss this in follow-on blogs.

My experience suggests it is more effective, and usually more efficient, to assert and apologize than to hedge and explain. Those for whom I am writing know the subject far better that I do; I can only hope you work the tension between cutting me some slack and giving your best shot. I also know I have a responsibility to argue for what I assert. That will follow in this or subsequent blogs.

Semantics of Addition and Multiplication

Outside the classroom and aside from pure mathematics the semantics of operands and expressions is the primary point of mathematics, or at the least understanding the semantics is the primary value added by us. Computers can take care of both numerical and algebraic calculation; i.e. reducing to a useful value or form the numerical and algebraic expressions that encode our understanding of the semantics of the problem.

Of course we also must be able to do simple calculation problems.  We must learn how to do simple and easy syntax because it can be easier to do it than it is to set it up for a computer to do; computers do the complex or onerous syntax.
In any case bringing consideration of meaning into the elementary learning of calculation by addition and multiplication shows how mathematics deals with real problems.

Addition

N apples + M oranges = (?). As apples and oranges, a member of the sum collection N+M is not defined; what is an (apple AND orange). If instead of apples and oranges we consider a member of each of the collections N, M, and (N+M) to be an (apple OR orange ), or perhaps just a fruit, then the addition is well defined: (?) is (N+M) fruits.The point is simple: only a common, additive, identifying characteristic that adheres to each element of the operands is summed by addition. Since it is common to all of the terms in the addition expression, it may be factored out and then the numerical expression alone, N+M= (N+M), describes the addition of the N individuals and M individuals. For this reason ignoring semantics in elementary addition education is not greatly problematical. After all, any rewriting of a numerical addition expression as another numerical addition expression does not change the meaning of the terms in the resulting expression. At some level the semantics of any addition problem is the same as that of any other: you might as well leave the semantics implicit. And finally, from what we have said, it is not possible to chose to draw an apple(orange) from the sum; the element drawn is either an apple or an orange. I think this is the case for a sum in Type Theory, but not for Category Semantics or Set Theory. Obviously this is not elementary issue, but I would appreciate some help here.      

Multiplication

Unlike addition, semantics is an active participant in multiplication. Generally the meaning of one operand differs from that of the other, and the meaning of the product is the result of the interaction of the meanings of the operands; in a real sense the meaning of the product is the product of the meanings of the operands Ignoring the semantics of multiplication and just relying on syntax is dangerous: for the whole number M = 3 consider

3 x N = (1+1+1) x N = (1 x N) + (1 x N) + (1 x N).

If one ignores any meaning of 1 (or 3)  beyond its property as a whole number and takes (1 x N) = N, then 

3 x N = N + N + N,

and one is led to infer multiplication by a whole number is equivalent to multiple addition. This can be true for an N that only carries the meaning of a whole number, but most of us have little truck with such problems, and it misleading to suggest equivalence in our problems where multiplication is not addition because what is being added is not the same as what is being multiplied. We illustrate the issue with exemplars of two broad classes of multiplication problems: rate/ratio and area calculations.

Rate as Speed

The distance D traveled in a Time T at a constant speed R for T = N hours and R = M miles/hour is given by the evaluation of the expression M miles/hour x N hours. For M =3 and N = 2, D follows from

3 miles/hour x 2 hours = 

(1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) =

2 miles + 2 miles + 2 miles.

The semantics is apparent in 1 mile/hour x 2 hours = 2 miles; it is not the case that three Ms =  M plus M plus M  because 1 x M does not equal M. In the real problem "1" is a speed that acts over time that results in a distance. ( I would go further and say it is not a good idea to decompose 3 miles/hour because it suggests there are 3 parallel trips at speed 1 mile/hour while the original problem concerned one trip at 3 miles/hour. If a problem in elementary education is the application of mathematics to word problems, then the mathematics should be taught as following from words not numbers. Your thoughts?)

Area of a Rectangle

For length L feet and width W feet, say L = 3 feet and W = 2 feet, the area A is

3 feet x 2 feet = 6 square feet.

Here each operand is a length measured in a unit of 1 lineal foot, think of a foot-long ruler; while the area is measured in a unit of 1 square foot, think of a square tile one foot on each of its sides. This is the common refrain in multiplication problems: the meaning of a product is the product the meanings of the operands.

Wrapup

Multiplication is not addition. Addition is not equivalent to multiplication: it has different semantics, and the addition process is not equivalent to the multiplication process. The product resulting from multiplication can always be rewritten as a multiple addition, but each of its terms the product in an already performed multiplication: the multiple addition is simply a rewrite of the product, a different way to express it, not an equivalent way to obtain it.

It is the case that virtually all algorithms for multiplication start by decomposing the operands into parts that can be easily multiplied with each other to produce numbers whose sum is the desired result. Now that computers can do the actual arithmetic, teaching the multiplication algorithm can be taken as an opportunity to show/see how big problems are broken up into smaller ones, and the results of the small ones are then swept up into the solution of the big one. Eureka! they have the keys to the kingdom.

Looking Ahead

Here I've written mostly about the semantics of problems using arithmetic. I want to turn to some ideas on making school mathematics a subject with a more active approach.  If semantics/meaning is to go anyplace, ways have to be found to make mathematics an active endeavor involving things you can touch, feel, and name while at the same time raising good mathematics questions. The useful heart of mathematics holds meaning not manipulation. I believe two strands of science, physics and computer science, have developed useful methods and concepts that can be suggest ways to go forward in the effort. Your nominations? Of course my dream is to entice you to join.

I want to discuss ideas for a hands-on coordinate system to make calculation on a number line self-contained, calculation by measurement if you will. It should be in a follow-on blog.

Addendum

I want to revisit the semantics of addition and multiplication to look for better, or at least other, ways to encapsulate their meanings. This discussion is intertwined with the one in the next post. It looks for visceral understanding of the difference between addition and multiplication.

Multiplication spreads the effect of one of its operands over the other to produce a third that generally differs qualitatively from each. In a process of multiplication, and here I picture the physical process: speed acting over time (or time over speed ) to produce distance: 1 mile/hour x 2 hour = 2 miles, extending along a line produced by velocity converting time into distance.
Multiplication of length by width produces area, it distributes width(length) along length(width) to produce area by extending the entire the one dimensional length (width) by the one dimensional width(length) to produce a two dimensional area. 1 foot-length-rectangle x 2 foot-width-rectangle = 2-square-foot-rectangle. Here the extension by one operand is transverse or crosswise to the other operand, sweeping out an area, not "extension along" as it was in the previous conversion calculation.

Addition  produces a sum with the same qualitative features as those of the operands whose addition produced it. There is no connotation of extension, rather addition augments or appends one operand to the other with no conversion of units or change in dimension.