Ubiquitous Numbers

One and Zero are always everywhere

The more you tutor math, the more skilled you become in finding good, clear ways to explain all types of math concepts to students. However, some concepts are more elusive than others. The difficulty of grasping a concept depends not only on the concept itself but also on the student who is assimilating it. One person, for example, can easily understand polynomial multiplication, and struggle with percentages, while someone else can find percentages very easy but have trouble with polynomials. There are also some concepts or topics that seem to be hard for a significant majority of students, like word problems for instance.
Substitution is a very powerful problem-solving technique, and it is widely used in a variety of situations. Students who find substitution easy have a clear advantage over students who have trouble understanding it. Substitution comes up in many different ways, some more complex than others. Some students understand the more basic forms of substitution but have problems applying the same techniques when dealing with more complex expressions. In fact, a very consistent general trend is the host of negative reactions students tend to show in varying degrees when facing bigger, longer, more complicated expressions. The more complex the expression, the more likely that students will get confused, or feel overwhelmed by it.
There is a particular way of using substitution to which most students react with a strong resistance: It involves transforming a given expression into another one that is equivalent in value but looks more complicated. This is done with the ultimate goal of simplifying the expression but it starts out by complicating it a little more. It is like climbing up a hill to find a way down the mountain.
The most common way of using this technique is by introducing a representation of the numbers Zero or One into the given expression.
Zero and One are very special numbers. They implicitly are everywhere in any given algebraic expression, even when we do not see them written out.
Zero is called the additive identity because zero plus any number is that same number ( x + 0 = x ).
One is called the multiplicative identity because one times any number equals that same number ( x∙1 = x ).
These two properties make Zero and One algebraically omnipresent in an implicit way.
Further, we have the following properties:
A number subtracted from itself equals zero ( x – x = 0 ).
A number different from zero, when divided by itself gives us one as the result ( x/x = 1 ).
These last two properties give Zero and One an infinite number of representations (“disguises” if you will) to show up in a formula. So, not only are Zero and One ubiquitous, they can come in a dizzying multiplicity of seemingly different forms.
The above properties of Zero and One, and their consequences, make them extremely useful in solving equations, and in manipulating algebraic expressions in general.
However, as I mentioned before, many students present a strong resistance to the idea of making an expression more complicated to be able to reduce it later. This is partly because they do not see the point of multiplying a number times one, or adding zero to it; partly because doing so seems to increase the problem in size and complexity; partly because they feel we are working backwards into some uncalled-for calculation, and finally because they believe they would not know what particular form of Zero or One they are supposed to introduce into the expression if they were doing the problem on their own.
Sometimes, when working with a student on a given problem, it is very easy for me to see a path to the solution using these types of techniques but I have learned to make sure the student does not feel like I was expecting him or her to be able to solve the problem in the same way. I only use these techniques when the student is completely stuck in the problem, not making any progress at all. When they see one possible solution, it gives them some perspective on the different factors playing a role in the problem. After showing them one possible solution method, if they do not feel comfortable that they would be able to successfully use the same method on their own, we focus on finding an alternate method that works better for them.

Integration by Parts in Calculus 2

It is really neat when students make their own discoveries.

This morning I had a Calculus 2 tutoring session about Integration by Parts.
My student was having some trouble with those integrals that are calculated by applying the integration-by-parts formula multiple times.
In this type of problem you get somewhat complicated expressions with parenthesis nested inside parenthesis, often with multiplying coefficients and/or negative signs in front of each parenthesis.
Students usually get confused when doing these integrals for the first time. The main reason is because they do not expect so much complexity. They believe the problems are going to be shorter than they actually turn out. So one key for getting these integrals right is to keep track of every step separately, identifying each new integral, and labeling it with a new variable of its own (using I, I1, I2, I3, ... works well). Then you keep working all the way down until you finally reach an integral you can actually solve, without any new “left-over” integrals. Then you re-trace your steps back one at a time, substituting each (ever longer) expression into the corresponding place for the previous partially solved integral, until you get to the original one.
We were working with the function (x^3)(e^x), where you start by deriving x^3, and integrating e^x.
My student understood everything but he wanted to make sure he would remember it later, so he started all over again. I suggested this time to start from the bottom up, so he integrated x(e^x), then (x^2)(e^x), and then (x^3)(e^x). I suggested for him to keep going so he integrated (x^4)(e^x).
At this point I said he could even memorize all the resulting formulas just for the test but it did not seem a good idea to him. Then I asked: “Well, maybe there is a pattern here. Do you see after factoring out e^x the last number (the constant term in the polynomial factor) is a factorial? And the polynomial always starts with the power of x we have in the original integral, does it not?”
I started looking for a way to factor out the polynomials. It turned out to be not always possible (the second-degree one already had complex roots) but all of a sudden my student exclaimed: “They are all derivatives!” So the pattern was not apparently multiplicative, but one more closely related to calculus, because it involves successive derivatives.
The best part here is my student saw it himself without my help. He realized what the pattern was before me.
That really made my day. I enjoy watching students making their own discoveries. This was not a textbook exercise, but a question I came up with after we went beyond the examples in the book, and my student was able to come up with the answer faster than me, showing he really understood the question. And now he has a very effective reference point to remember all those integrals if they show up in his exam! He won’t forget it because he discovered it on his own.