How to Begin

Imagine someone taught you how to play Brahm’s Lullaby on the piano but told you the name of the song was Go to Sleep, Baby.  You can completely master the song.   However, your ability to apply the song is limited to the request, “Play Go to Sleep, Baby.” 

This is analogous to what is often a best case scenario in math.  A student completely masters a skill that is only repeated under a particular contextualization.  The problem in learning math is exacerbated by students not mastering the skills they’ve been taught.

To get everybody on the same page a teacher gives a pre-test and a book has a prerequisite section.  Using our previous example, both methods seek to rename the song appropriately, and get students to become proficient playing it.  The problem is that the focus here is too narrow, too skill based.  It only addresses how to do something, not what it is that is being done, or why it works.

An example in math is,

Simplify the following expressions: 

1.      a³× a²

2.      (a³)²

If a student is stuck they will ask for help and someone will tell them, “If the bases are the same you add the exponents but when a power is raised to another power, you multiply them.”  Maybe, if the student has a diligent teacher, they will be presented with a list of rules for exponents.  If the student has previously been taught these things they perform the task and all involved are happy. 

Unfortunately the student has just been taught Lullaby by the wrong name, so to speak.   Even more unfortunately, this will likely earn both the student and teacher high marks, and a student will likely do well on standardized tests, and all without understanding. 

When the student takes a college entrance exam they’ll see a problem like:

Simplify the following (without a calculator)

4⁵ + 4⁵ + 4⁵ + 4⁵

With that, I will only assume that you know basic arithmetic facts.  That doesn’t mean that the pace will be slow, but the prerequisite knowledge will be covered in embedded contexts. 

One last note to be made is that some math is convention.  That is, we simply agree upon some things, it’s not that the nature of all things in math are discoverable.  For example, the features of a coordinate plane, x being the horizontal axis and y being vertical are so because they are defined to be so.   But the fact that there are infinitely many prime numbers is a discoverable fact.  (The discoverable facts are a lot more fun to learn.)