Vocabulary Instruction

Core vocabulary concepts were taught to me in grades K-12 differently than current research-based instruction. I remember writing lists of words and definitions by copying them from a text, handouts with word lists that required drawing a line from a word to its corresponding definition, crossword puzzles, and word searches. In the case of word searches, definitions were not provided. My math experience up through high school was that it was not necessary to understand the “why” of something as long as you could find the right answer on a test. Independent learning strategies required for text-based learning were not taught me prior to college.
It is interesting to note that puns and word play can result in higher level thinking. I attended a theatrical performance about puns; it was a play on words. There I learned that dragon milk comes from cows with short legs and a cow without legs is known as ground beef. Beef wasn't what's for dinner though, we ate venison instead. Oh deer!
In addition to incorporating the recommended dose of dry humor, I want to teach math differently.
I don’t think I really began to understand numbers and mathematics until I learned some of its history.  We aren’t allowed to divide by zero but why? None of something is nothing and you can’t have some of nothing yet the sum of nothing is nothing which sounds like something but is it? History gives context to answers and even paradox. This statement is unprovable.

(http://www.radiolab.org/story/161744-loops/)

Comprehension Instruction

During Bart Nelson’s calculus class at Snow College we encountered the following function:

f(x) = 1/x

(http://mathworld.wolfram.com/GabrielsHorn.html)

When rotating this function, where x ≥ 1, around the x-axis on a graph you get something which resembles a cornucopia or trumpet.

We were surprised when calculating the volume yielded a finite answer but when solving for the surface area the answer was infinite. He explained that this means you could never finish painting the inside of it but you could pour it full of paint.

(http://blog.plover.com/math/gabriels-horn.html)

This seeming paradox led to a discussion about what infinity means and to the question, “Just how big is a ‘point’?” Supposedly a point, by definition, is something which has no mass and takes up no space. Another way to think of them (very informally and non-mathematically of course) is that points are bits of nothing. Then, if you stack up enough bits of nothing, you somehow get a line which has infinite length. These infinite lengths of nothing are then connected side by side to make the plane represented by our graph. If the premise upon which our geometry is based seems paradoxical we should not be too surprised when our mathematics yields paradoxical results.

Additionally, the results are dependent on allowing the possibility of a function with infinite length. Just how big is infinity? If you take the biggest number in the world and add 1 to it then you have a larger number. So infinity as a number can’t happen but we want it to anyway so we are going to invent the largest possible number and call it infinity.

I did not realize it at the time but infinity is actually not a number.

(https://www.khanacademy.org/math/recreational-math/vi-hart/infinity/v/9-999-reasons-that-999-1)

At the beginning of the course, we talked about numbers and how they do not exist. Up until that point my experience was that math was logical and always made sense. Why would he say that numbers do not exist? He then asked, “Have you ever seen a 'two'?” We said that of course we had seen a 'two' and he asked for examples. Someone wrote the word "two" on the board and someone else wrote the number ‘2’. He asked which one was the 'two' and whether what we had written was a ‘two’ or just a representation of a 'two'? What about the roman numeral (II) or what if we write it in French (deux) or Spanish (dos)? Is what we have written a 'two'? What if we made a statue in the shape of a ‘2’? Would that be a 'two'? Is ‘two’ concrete or abstract? Is ‘two’ something tangible? Does ‘two’ actually exist or is it just a concept? We finally agreed that 'two' does not actually exist but is a concept to describe quantities. ‘Two’ may describe a property of a set of items which do exist but the number ‘two’ does not exist as a tangible thing.

This was the beginning of my understanding of how math was a language. Having my assumptions challenged helped me keep an open mind later when we encountered functions such as “f(x) = 1/x”.

If you think about language, words do not physically exist either as tangible entities separate from the things they describe. “Apple” is the name which we use for a particular fruit but the word “apple” is not itself an apple. Numbers are names for quantities of things. You might say there are two sheep in a field but the word “two” is not a sheep. Numbers are words. Math is a language.

Language, Literacy, and Learning in Math

A former professor, Bart Nelson, taught me that the so-called “real numbers” are poorly named because none of them are real; numbers do not exist and the term “imaginary number” is redundant. I was intrigued. What did he mean by that? As he explained his point of view I saw numbers in an entirely new context. Through a BBC documentary, 'The Story of 1' narrated by Terry Jones (Monty Python), I found out that zero was invented in India and is relatively new.  Mathemusician Vi Hart’s YouTube video series helped me realize that 0.999… equals 1 and that infinity is not a number. Something even more strange was learning about practical application for imaginary numbers when solving equations involving alternating current circuits; I figured that imaginary numbers were only for thought exercises or number games and it never occurred to me that they could have real world application. Much of my enthusiasm for math has come from sources outside the traditional classroom.
Prior to starting school again I used to enjoy chess, reading, movies, computer and board games, or even riding a unicycle. Now I work two jobs in order to afford to be a student so those pursuits are put on hold until some time in the distant future.
I plan to teach high school math. I will minor in chemistry or French. I want to teach higher level math since math is more interesting beyond algebra. I thought this would mean teaching calculus but, from what I can tell so far, it seems there is more focus on statistics rather than calculus when preparing for college with the common core.
In the past, I have been a substitute teacher and tutor for subjects such as English, French, math, chemistry, and physics. While working as a math lab tutor I was able to assist students learning material not understood from other tutors and professors. Several students were on the verge of changing majors because they had not been able to get through the math. I encouraged them not to give up on future career goals and helped them pass their classes. It was very rewarding. As I contemplated becoming a teacher I felt I could make the most difference by preparing students in high school for college.
Then I went through a divorce and my career plans got put on hold long enough that I no longer have the confidence I once did as a tutor but I have not lost the desire I had to make a positive difference.
There are elements of math in communication, entertainment, music, dance, martial arts, sports, games, and in nature but they are so interwoven that we do not typically think of them as math. We tend to think differently about everyday job and leisure activities that are math related and math which is taught in schools. For most of us, "math" is what we learn that might be required to obtain a degree or diploma but which we will never use again. That may be partially true for some but there is a great deal more we do end up using than we might realize. We often use it unaware or are surprised when we find ourselves commonly using math which we thought to never use again.
To me math is a language. I see myself as a linguist.
Literacy in math means learning to understand and communicate math words, graphs, pictures, or symbols the way you might with any other language.
This definition of literacy in relation to math is synonymous with fluency. I speak fluent French and hopefully also speak, read, and write fluent math.
As with any language, one becomes less proficient with lack of use. I am studying again to help regain some of the fluency I have lost. This experience will help me relate to other "language learners" who I will be teaching to become fluent (or in other words more literate) in the sometimes foreign language of math.