First Lesson Preview

Soon enough, I will be delivering my first lesson in the front of a class. The topic will be the first section of a chapter on exponential functions. This lesson will be a bit of an experiment and a good representation of what my philosophy explains. Many people look at the philosophies and theories of education as unpractical or even useless but what I think they don’t realize is that their applications are subtle at best. Good teachers, if one was to really analyze them, use these abstract concepts to create an effective learning environment. For example, a teacher that is authoritative but yet keeps a light atmosphere may do this because it’s his style but what he may not realize or ever think about is that his style creates an nonthreatening environment, which is a big idea in the theories. When I think about theories and philosophies I don’t think about how to create the ideal classroom; I try to look at my lesson and see the value in what I am doing. Outlining how your instructional strategies use theories allows you to see the value and then reflect on its benefit. To put it simply: We don’t take theories and philosophies and create a teaching technique. We reflect on our teaching technique and try to think about it in terms of the abstract concepts with a hope to improve.

I am not doing anything special or new in my first lesson but my explanation might be unique. Like I mentioned before, effective teachers seem to be able to pull mathematics out of the vacuum in which it is placed. This doesn’t mean saying, “I can figure out how much to tip a waiter by using this simple algebraic equation.” To me, this is a huge misconception that dominates mathematics education. I interpret “real world application” as taking a complicated concept like a function and presenting it to them in a different way than the text might explain it (e.g. drawing on the board of a machine that takes raw materials as its inputs and outputs something useful). This gives the abstract a more concrete nature which is easier understood by minds that haven’t developed certain capacities of thinking.

Exponential growth falls under this category.

So what I am going to try is the paper folding problem?

I will use an actual sheet of paper for this lesson.

How thick is a sheet of paper? t inches
If I fold it in half how thick? 2t inches
In half again? 4t inches
Again? 8t inches

I will then have students approximate the measurement. It should be around 1/8 of an inch.

How thick would it be if you could fold the paper 20 times in half? 50 times?

They will most likely guess a few inches where the answer is the size of the Liberty One building for 20 times and to the sun for 50 times.

This is a common activity but the value is extremely unappreciated.

One instruction strategy that I am interested in is Socratic questioning. This strategy is used to expose underlying assumptions of the target. Once they are exposed you then undermine what they think they know by proving their assumptions false. If done effectively, the surprise that occurs is enough to engage them in the concepts.

The sequence is the following: t 2t 4t 8t 16t 32t …
Factor out a t: 1 2 4 8 16 32 …
Some famous mathematician once said that mathematics is about pattern recognition. What is the pattern? The thickness doubles every time.

We could also say that the thickness t is doubled every time the paper is folded in half.

t x 2 x 2 x 2 x 2 x ….

If the paper is folded 5 times how thick is it?

t – no folds
2t – 1 fold – 2t
2(2t) – 2 folds – 2^2t
2(2(2t)) – 3 folds – 2^3t
2(2(2(2t))) – 4 folds – 2^4t
2(2(2(2(2t)))) – 5 folds – 2^5t – 32t – 32 sheets of paper

This introductory activity is about engagement and relationship. Some students may be able to look at a graph of an exponential function and understand that it grows very quickly but this activity illustrates it by using tangible examples.