Mathematical Operations are a Lie

In any mathematics curriculum, students have to deal with operations.  The ones they see most often are the "Big Four", i.e. addition, subtraction, multiplication and division.  Students learn them at first in the elementary grades, most often being taught by "teachers" who don't have a strong grasp of mathematical operations themselves.  As a result, they are simply told to memorize how they work, and also taught that when in doubt, they have their calculators to rely upon.

In actuality, there is a great deal of context for these cardinal operations, and it begins with the squelching of a horrendous lie.

Bad Idea: There are 4 main numerical operations

Addition is usually a good place to start.  Learning how to add for students tends to be easy, especially since addition has an intuitive aspect to it.  Everyone can easily ascertain that if they had 3 objects, and had 4 additional objects, then altogether they would have 7 objects.  At least educators have this component right.  But then things start to go haywire when subtraction comes into the picture, mainly because it is probably introduced to students as "the opposite" of subtraction, but with very little attention paid to this aspect of how it works.

Multiplication makes things even worse.  Where addition and subtraction had to some extent a context, multiplication is often presented without one.  The average to weaker students are lost here, and will for the rest of their math careers confuse addition with multiplication.  This is because they have no idea what multiplication is for.

At this point, it is hardly even necessary to talk about division.  Since it is taught as the "opposite" of multiplication, which is already devoid of context, division cannot have any context either.

The goal of elementary math education seems to place more priority on students being able to do it, whether they understand what they are doing at all.  Then, of course, as standards drop, educators are happy if the students can press the right keys on the calculator.

Good Idea: There is really only 1 numerical operation

As I mentioned above, addition seems to be the right place to start, since it is so intuitive to students even at a very early age.  And yes, subtraction is in fact just the "opposite" of addition, and this is the correct way of looking at it.  But where math educators at present drop this notion very early on, (usually before grade 4), it is fundamental to understanding how subtraction works.  Since students are told to resort to the calculator, and to think about subtraction divorced from its context, they need to resort to crazy, nonsensical rule tables to help them manage it.

So, if we keep the context for subtraction alive, it won't be such a challenge for students in the long run.  When answering a subtraction question such as the one below...

5 - 3 = ?

students need to be asking themselves the question "What number do I add 3 to in order to get 5?"  This is truly the only contextual way of thinking about this operation.  No tricks, no rules to memorize, no nonsense; just subtraction exactly as it is in the formal sense.

Kindly note how with the above two operations, we still have really only one, since to answer the subtraction problem, we simply had to ask an addition question.  Questions of the above type also serve a great purpose in education: the notion of working backwards, and logically reorganizing one's thoughts in a problem.  This is a fundamental problem-solving concept, not just in mathematics and education, but in real life.

Now, surely one could argue that multiplication and division need to be thought of separately.  However, multiplication can be very easily understood simply as the answer to a trivial problem:

If you like, one may define multiplication as a way of counting objects in a rectangular grid.  The answer to the problem "4 x 3 = ?" is quite equivalent to saying "How many objects will I have if I have 4 groups of 3 objects (or 3 groups of 4 objects)?"

This view of multiplication has several advantages:

• Students will very quickly realize that multiplication is simply repeated addition.  4 x 3 is simply 4 + 4 + 4 or 3 + 3 + 3 + 3.
• Commutativity now becomes extraordinarily obvious, seeing as how the same number of squares exist in the rectangle, no matter how it is arranged.  A good teacher will point this out.
• This serves as a phenomenal introduction to ways for computing area.  The area of a rectangle and multiplication are the same exact thing*.  No one has an excuse for not remembering that "formula".  Its not a formula.  Its a definition, intrinsically based on real, sense-making intuition: exactly how math should be.

So now we have multiplication added to our arsenal, and we've still not had to do anything fundamentally different from adding.  How should we tackle the problem of division?

Once the contextual relationship of addition and subtraction is well-laid in the minds of students, and they understand multiplication, it is no longer a stretch for them to ask an analogous question about division.  In other words, asking "What is 10 divided by 5?" is the same question as asking "What number do I multiply by 5 in order to to get 10?"  Long division, once understood by students, is just a decimal place-by-place way of arriving at precisely this answer.  (I invite the reader to perform some long division, and notice how they proceed using primarily multiplication.)

Finally, I'd like to close by pointing out that the cardinal operations in their proper context allow students to very easily understand the "rules" that go along with them.  Note that each of these rules is no longer something one has to memorize:

• Any number plus 0 is itself - Now, since students rely on their intuition about addition, this fact seems trivial and difficult to confuse.  Students tend to confuse this rule with multiplication, and think that a number plus 0 is 0.
• Any number times 0 is 0 - Clearly, this leads to an empty rectangular array of boxes, since one of the rows is 0.
• Any number minus itself is 0 - Again, intuitive.  Not so much when students rely on finger-counting and/or the calculator.
• Any number divided by itself is 1 - This fact is identical to the notion that a number times 1 is itself.  When division is properly considered in context, this makes a lot more sense.

The above are not rules to be memorized, but thoughts to inevitably arrive at, once a student understands the context of these operations.

*I cannot tell you how many times I've encountered students in high school who don't know the area "formula" for a rectangle.  Its quite alarming, considering the above.