Monday, October 26, 2015

Semantics of Elementary Arithmetic Expressions

Musing on the Semantics of Arithmetic Expressions

What is meant by the arithmetic expressions that we read and write? I certainly do not intend to present anything new, just report some observations made while working out what arithmetic can mean, with an eye toward the possibilities when numbers denote more than just themselves. I want to look at elementary mathematics as an active endeavor where you start by writing an expression for something you want in terms of what you know, and whose evaluation plays as a movie in your mind. 
For addition and multiplication what actions flow out from the form of their expression? In the infix notation used at this level, expressions are of the form: "(operand operator operand)"; the parentheses are just my notation to improve blog readability. It has long been noted that the semantics are ambiguous: do we read (2 + 3) as instruction to calculate, or is it just another way to write (5).  If the meaning is calculation, we use a process that takes two numbers and produces a third. If the expression is to just to be rewritten, then 
we look in a big table we carry in our heads. This is what most of us do for small numbers
These are real questions when you are trying to instruct a computer; they are also real questions whenever their answer is not clear from the context. It was clear in elementary education before we tested students on material long after it was taught.  For small numbers elementary math instruction first teaches a calculation process and later assumes a student has the big look-up table.

For our purposes, the arithmetic expression shall mean the operator denotes calculation with operands: the expression (2 + 3) means "add 3 to 2 " and produce the sum. How do we know it does not mean "add 2 to 3"? In a sense we do not care, the sum produced by any process that we know about produces the same result, and in fact we could have chosen the reverse order, but the two processes are different: they treat their two operands differently, and we must choose one. This is reflected in our usual interpretation of the expression for addition as "add N2(1) to N1(2)" and not "produce the sum of the two operands"; or for multiplication: "multiply N1 by N2" and not "produce the product of the two operands".
The classical words for the two operands in both addition and multiplication are different from each other and are suggestive of the calculation process:

addition: Augend + Addend = Sum
subtraction: Minuend - Subtrahend = Difference

multiplication: Multiplicand x Multiplier = Product
Division: Dividend / Divisor = Quotient

...contain a wealth of Latin. If you happen to know Latin, you will understand these meanings more deeply. For example "minuend" comes from a form meaning "about to be lessoned" and "subtrahend" comes from a form meaning "about to be taken away". In general, "-nd" will carry the meaning "about to be --ed".

Interesting! What appear to be exotic labels suggests a practical way to look at the calculation process. I choose to read "Augend" as the operand about to be augmented, with "Addend" about to be the one doing the augmenting. The "Multiplicand" is the operand about to be multiplied, and "Multiplier" is the one that will do the multiplying

This is consistent with the operational meaning: choose one operand to be acted on and the other to act on it. For addition one operand plays the passive role, waiting to be augmented by the second operand. In multiplication, again one operand plays the passive role, waiting to be multiplied ( stretched either lengthwise or crosswise in an amount determined) by the second operand. The classical nomenclature suggests actions and actors.

Any suggestions for different words for the operands? Surely you can improve these: ("abt" is "about to be", "at" is "about to")

Addition:             (atbAdded-to + atAdd) 
Multiplication:    (atbMult-ed + atMultiply)


Addition:            (atbAdd-edTo + atbAdd-ingTo)
Multiplication:   (atbMultipli-ed     x  atbMultipli-er) .

These position-specific "words"  for "operand" are all adjectives, each one now modifies an unvoiced "operand"; e.g.       "[the] atbAdd-edTo [operand]" in
"(the atbAdd-edTo 2 + the atbAdd-ingTo 3)" for the expression    (2 + 3).

A more practical notation might forgo the "about to be":

Addition:            (To 2 + Add 3)
Multiplication:   (Mult 2 x By 3)

and this suggests the more elegant

Addition:            (To 2 , Add 3) means "add 3 to 2", or               "with 3 augment 2".
Multiplication:    (Mult 2 , By 3) means "multiply 2 by 3",              or "with 3 multiply 2".

None of these feel quite right; but the real "problem" is now apparent. "add" and "multiply" are not symmetrical. "Add" refers to what is is going to be added, not to what is about to be added to; "multiply" refers to what is about to be multiplied, not to what it is going to be multiplied by:
  • (2 + 3) means add 3 to 2, 
  • (2 x 3) means multiply 2  by 3.
In short, one adds "to", while one multiplies "by".

Things might be less confusing if we could use symmetric terminology for (operand , operator, operand). For addition: 
  • the name of the operator "+" could be "augmentation":
  • the left operand is the "augment"-this-operand,
  • the right operator is the do-it-"with"-this operand; ie,
Addition expression: 
(augment-this-operand  +  by-this-operand)

Multiplication expression: 
(multiply-this-operand  x  by-this-operand).

Do you think the order of the operands should be reversed? In the order shown above the operands on the right are those most likely considered prior to those on the right.

For both addition and multiplication there is generally the connotation that after the calculation the left operand has been changed, and the right one did the changing; however, neither the sum nor the product depend on the operand order: addition and multiplication commute. To reduce confusion in subsequent work, however, I will try to keep the changed operand on the left.

There is a major difference between addition and multiplication: In multiplication we must first consider the changing operand because Multiplication distributes the change it specifies over the entire changed operand while, for addition, the augmentation can be effected by one "attachment".  In a sense multiplication is melding; and addition, an add-on. At this juncture all this is premature; perhaps you can see it. It will come up in later posts, and I may go back and change the first post where I think some of it appears more naturally. 

Tuesday, October 20, 2015

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.


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.      


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.


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.


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.

Elementary Thoughts