Exponents Make Sense!

Many teachers of algebra and pre-algebra courses will agree that students have a difficult time understanding how exponents work.  As usual, due to standardized test pressure, and various levels of holding educators responsible for student results on these tests, exponents are a classic example of a topic in math that gets taught in a "memorize these rules" sort of way.  This, of course, in the author's opinion, leads students down the context-free road of mathematics content, and inevitably to their demise as mathematical thinkers.

Bad Idea: Exponents are about a set of rules

First off, it is worthwhile to point out that algebra, philosophically speaking, really is a "set of rules", when dramatically oversimplified.  It gives us symbolic representations of various notions, and predictable ways for manipulating those representations.  This act alone is good practice for the mind, and development of skills along these lines is certainly worthwhile.  Bearing this in mind, however, it is important to note that algebra as a rule-set is an incomplete picture.

In a good enough pre-algebra classroom, students sometimes see the following definition:

2² = 2 x 2

Or something equivalent.  This is a very good starting point.  Unfortunately, the context often gets dropped from here on in.  Next, we present to students the rules that go along with exponents.  They learn that when multiplying powers of the same base, the powers add:

2² x 2³ = 2⁵

They are given no explanation for this, (or perhaps only a very brief one), and are simply told to memorize it as a rule.  "Algebra is about rules", they are told, "follow them, and pass the course," is the promise.

A similar approach is taken to dealing with composition of exponents:

(2³)² = 2⁶

This is again presented as a rule that algebra students simply need to memorize, more than likely because to spend time investigating the content of this rule seems wasteful to teachers who know such an investigation won't lead to more correct answers on assessments for their students.

Then, of course, are the remaining exponent rules, which again are presented to students to be memorized:

2⁰ = 1
2¹ = 2
2^-1 = 1 / 2

And so on.  The way it appears in an algebra and pre-algebra classroom, each of these facts is a separate entity, and collectively, they all give us the rules for how exponents work.  Algebra, after all, is only about rules.

Good Idea: Exponents are repeated multiplication, nothing more

The original presentation of exponents, i.e:

2² = 2 x 2

is in fact, totally correct.  By definition, when one puts an exponent on a number, it literally means "multiply that number by itself that many times".  While this may be the initial presentation of the idea, many math teachers ignore the fact that all of the exponent rules and properties inherit directly from this one fact.

Consider how to deal with multiplication of powers of the same base:

2² x 2³ = ?

(2 x 2) x (2 x 2 x 2) = ?

(2 x 2 x 2 x 2 x 2) = 2⁵

As can clearly be seen, using the original definition of exponents as repeated multiplication makes this "rule" not only obvious, but necessary.  There is absolutely no other way to position this "rule", and upon viewing this elementary algebraic "proof", (a dirty word in our high school math classrooms for absolutely no reason, in the author's opinion) the fact is inescapable.  The inevitability of this result, and triviality of its justification will actually carry with it the benefit that students are more likely to have better recall of it.

A very similar justification for the composition of powers can also be demonstrated:

(2³)² = 2³ x 2³ = (2 x 2 x 2) x (2 x 2 x 2) = 2⁶

Both of the above proofs can also lead to a discussion of the associative property and commutative property of multiplication, if the teacher so chooses.  (Again, though, since there is only a small chance that there's a standardized test question on these properties, it won't happen).

Zero and negative exponents can be a bit trickier, but not for a skilled teacher who truly has a grasp of the underlying content.  Please note the following:

2² = 2 x 2
2³ = 2 x 2 x 2
2⁴ = 2 x 2 x 2 x 2
2⁵ = 2 x 2 x 2 x 2 x 2

And so on.  It is instructive to notice that as the power of 2 goes up, an extra "x 2" is added to the right-hand side.  Clearly to advance down this chart means to multiply by 2.  Students can then develop some mathematical inquiry by being guided to ask the question "What happens if we go backward?"  

It is easy to see that if going down the chart means multiply by 2, then going up the chart means divide by 2.  Once this is realized, since 4 / 2 = 2, 2¹ simply must be equaled to 2.  There is no other possibility, nothing to get it confused with.  In context, this is not a rule, but another inevitable fact about exponents.  2⁰ similarly must be equaled to 1, since 2 / 2 = 1.  Negative exponents then clearly yield fractions, as we divide 1 by 2, and so on.  If we keep dividing by 2, we clearly get the reciprocals we would expect.




It is extraordinarily important that exponents be taught in this way.  Students will see that there is a rule to understand, and that algebra is a rule-governed system, but it is important that they then discover for themselves all the inherent necessary formulas which inherit directly from that initial definition.  This is not only important to nurture early-developed mathematical inquiry and curiosity, but it will also instill these algebraic notions into the minds of students for years to come, making the furthering of their mathematics education possible.

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.

Adding and Subtracting Integers

Any mathematics educator will agree that for many students, the arithmetic of positive and negative integers is a challenge.  Often students get confused when they have to subtract numbers, especially once dealing with negative numbers, since their initial instinct to count it out on their fingers no longer seems viable.

It is also true that this frustration that many students have ultimately leads to their disenfranchisement in the math classroom, and is largely responsible for many people deciding they don't like math at all.  Thus, in comes the current approach to reconcile this problem:

-2 + -3 = ?

3 - +2 = ?

5 - -3 = ?

Bad Idea: Keep Change Change

Students initially learned what addition was in a very natural way.  Its simple to say that if one started with 2 of something, and added 3 more, then clearly they would have 5 things.  This is where math plays on preexisting human intuition (interestingly enough, this is where mathematics begins in the first place).

But as soon as the problem is subtraction, it gets a little bit harder.  This can be remedied by thinking of subtraction as a reverse kind of addition.  In other words, the problem:

5 - 3 = ?

can simply be seen as asking "What number do you add 3 to, in order for the total to be 5?"  This very natural approach to subtraction does have its foundations in underlying mathematical rigor, and is thus very conceptually acceptable.  Of course, things get much messier when the numbers involved can include positive or negative numbers, mainly because it is much harder to visualize the difference between the two.

Thus, the current pedagogy proceeds as follows:

5 - -3 = ?

• Keep the first number the same
• Change the operation from subtraction to addition
• Change the sign of the second number
• Then perform the operation as though it were addition

So the student is expected to rewrite the problem as 5 + +3, which is clearly 5 + 3, which he/she can easily and intuitively deduce is 8.  Problem solved.

Since the students are now equipped with a procedural tool to make their lives easier, it is sometimes easy to ignore situations where this method utterly and totally fails.  Consider the following:

-5 - -3 = ?

The only thing I changed is I made the initial number negative to begin with.  Now the student has to do the same KCC procedure, which yields:

-5 + +3 = ?

Is this any better?  Is it any clearer what the answer is?  Now we need to come up with procedural rules to help student add numbers possibly of opposite signs.  So then we have something to the effect of:

When adding numbers with signs:

• If the signs are different, subtract them, if the same, add them
• Take the sign of the larger of the two numbers

I hardly know where to begin here.  We've run into a procedural nightmare here, and all we've been trying to do is add and subtract integers.  We've given them so much procedure to consider and look at that we've divorced it entirely from the context in which we began.  And its not actually any easier for students, I think. Instead, math has become a list of steps to memorize.

But the proceduralist has a solution to this problem as well:  When in doubt, have the students use their calculators.

This final step invalidates any and all previous steps, because the calculator is never wrong (as long as you use it correctly, which is then another whole giant thing sometimes).

I hope I have adequately illustrated the giant mess that is this current pedagogical practice.  Now, onto the solution.

Good Idea: The Number Line

First, a note on mathematical foundations.  In geometry, a very well-known axiom of continuity was decided upon by Richard Dedekind, named Dedekind's axiom in his honor.  What it basically states is that lines (from geometry) and the set of real numbers (of algebraic and analytic fame) are actually the same thing.  So numbers correspond to distances or locations on lines, in a very tangible and practical way.  This axiom is noteworthy in higher mathematics because it bridges the gap between many sophisticated mathematical fields.

But why is it noteworthy for middle-school and high-school-aged students?  While its not necessary to present it to them specifically and by name, it is important for math educators to realize that strong, foundational mathematics gives us the answer to helping students with a very simple problem.  The real numbers are points on a line.  Points on a line are real numbers.  This fact is inescapable, and has a tremendously powerful application to help solve our addition/subtraction problem.

Here is a number line:

Its very simple, really.  You start with a line.  You pick some point in the middle of it, and call it "0".  As you move to the right, the numbers get bigger, and to the left, they get smaller.  Clear so far?  Good.

As can be seen and verified quite easily, any number can be found on this number line.  Its always possible to draw a quick line, and label some point on it "-2", or "56", or "-24,563" if necessary.  So here is how I propose the addition/subtraction problem be solved:

-5 - -3 = ?

-5 is a real number, its a real location on this line.  Draw the line on paper if you need to (or visualize it in your mind if you can do that easily enough, which of course comes with practice).  Find "-5", wherever you want it.

Now, ask yourself, "what am I doing, adding or subtracting?"  If adding, then you wanna think "go to the right, that's where numbers are bigger", and to the left if you're subtracting.  Finally ask "is the number I'm adding/subtracting positive or negative?"  If its positive, your initial "direction" is correct.  If negative, reverse it, because again, numbers go down as you go to the left, up as you go to the right.

I'll admit, it can be argued that this method is at least as procedural as KCC.  But inherent in that procedure is an underlying concept.  KCC just feels like a "set of rules to follow, arbitrarily set down".  Here, I think the student understands what's really happening, and is in touch with the actual mathematics of the situation.

The advantages of this method can be summed up as follows:

• If a student is struggling with the idea of numbers themselves, thinking of them as distances and points on a line gives them a visual interpretation.  This visualization adds a dimension to their ability to comprehend, and thus can help many students whose visualizing skills are stronger than their algebraic abstraction skills.
• This is real math going on here, not a long-winded, arbitrary list of rules.
• When students see the coordinate plane for the first time, being familiar with the number line will make their lives much easier.  After all, the coordinate plane is just 2 number lines.

Welcome and Disclaimer

Welcome to my blog.  I have decided to use this medium to help me express my opinions on current classroom practices in mathematics education.  I realize that there is plenty to be said on education itself; from a political, cultural, or pedagogical standpoint.  However, my main focus here is on content.  It is my opinion that the current methods used to present mathematical content to students is in fact creating more problems than it is solving.  In my opinion, mathematics should be about problem solving, abstraction, and training the students' mind so that they may be disciplined, careful, calculating thinkers in their lives.  I feel that math educators have let students down in this regard.

Now for the disclaimer.  Please note, I do not purport to have all the brilliant answers and/or solutions.  I am not speaking from a point of view of many years of fruitful experience, just my humble beginnings.  I also understand better than anybody how important assessment is, and the fact that helping students pass tests has (whether we like it or not) become a large part of our jobs.  But those issues I reserve for others to discuss (and they certainly do so).  The scope for this blog is to be content, and more specifically, the right and wrong ways of teaching specific concepts in mathematics.  Most of my experience thus far is in Algebra and Geometry classrooms, so this is the space in which I will mostly be working.

Thank you for your interest, and I welcome your comments and opinions!

Glenn