Digital Literacy and Oral Language

With working two jobs to pay child support, while trying to also get through college, my stress level has been on overload the past several weeks. I am happy to have found the emotional energy needed to prepare and write this blog entry. Month after month of sleep deprivation, from trying to do too much, eventually triggers the law of diminishing returns. Ambition is a good thing when tempered by wisdom.
Technology is ever changing. Graphing calculators and programs have been around for a while but they are getting better and more powerful to use. From a recent journal article (http://www.nctm.org/Publications/mathematics-teacher/2014/Vol108/Issue5/Graphing-Projects-with-Desmos/), I learned about a free graphing calculator program available at https://www.desmos.com/calculator. This program does more than graph equations, it also allows you to import images (such as water from a fountain) which can be used to plot the graph of parabolas or other equations corresponding with the image.  Art and beauty in the world around us contain shapes and patterns related to math. Modern entertainment, such as movies and computer games, runs on and is driven by math. Today’s world runs on ones and zeros. There is so much more to math than the often tedious, ordinary, and even boring regular math traditionally taught in school. Incorporating and making connections with things that kids love will help them see math as part of a much larger world. Without those connections, they are like Neo before waking up from the Matrix – unaware of the vastly larger and more useful world of math.
Another way students can make learning connections is through classroom discussion. Jeff Foxworthy famously said, “women don’t want to hear a man's opinion, they want to hear their own opinion in a deeper voice.” (https://youtu.be/-4EDhdAHrOg) Similarly, teachers sometimes seem as if they don’t really want to hear a student’s opinion (when asking questions for which the teacher already knows the expected answer). Higher level thinking is not promoted when the goal of questions is to provide validation, in the form of recitation, of the teacher’s already known answers relating to what is in the text.
An older article (Research Matters: Classroom Discussions of Literature by Rick VanDeWeghe http://www.ncte.org/journals/ej/issues/v93-1), outlines the findings of a study that was done which shows the positive impact on student achievement when there is less monologue on the part of the teacher and more genuine questions and dialogue from students. An “authentic question” is one that does not already have a predetermined answer. Students become more engaged in learning when they are the ones asking the questions since, “unlike teachers, students don’t (typically) ask questions when they already know the answer.” Student questions are the most important driver of dialogue style classroom discussions. Teachers can promote the chance for this type of discussion by validating what students say and incorporating student contribution (“uptake”) in comments designed to “initiate, nurture, guide, and sustain (discussion) momentum.” In other words, thought-provoking discussion is more likely to occur when teachers encourage independent thinking through student generated questions.
By relating things which matter to students (through the use of technology) and by encouraging original thought (through student generated discussion) teachers can engage students in collaborative learning. Students become active partners with the teacher rather than recitation repeaters of what the teacher already knows. The teacher becomes more a discussion leader than (“traditional”) teacher with the students as fellow teachers. In becoming teachers of themselves and each other, students while teaching learn.

Writing Instruction

The writing assignments I particularly enjoyed during high school were poetry during 12th grade English. I loved reading but hated reading assignments because they involved writing. I always did well in English but never enjoyed it until the class taught by Mrs. Potter. Our class was used as a broadcast class so we were televised and had to speak into microphones when making comments. I had never written poetry before and was very intimidated with the idea. As we began to learn structure and rules for various poetic forms I gradually started to think I might actually be able to do it. One of the poems I wrote won a poetry contest that year.
The writing assignments I disliked were doing mechanical things with language such as diagramming sentences. Had I paid more attention those exercises might have helped in learning a foreign language but I found them to be tedious. The motivation is different with analyzing structure of something you are learning instead of something you already know how to do. With learning to speak French and with writing poetry, they were things I did not yet know which were challenging to learn; knowing the mechanics was helpful in figuring out what to do. With diagramming sentences in English, the exercise did not seem helpful since I already knew how to use proper grammar. I did not see the need to learn the mechanics of something which was automatic.
This week's reading talked about doing composition in math classes at the same level as with other subjects. I had not thought of having a composition book or writing papers as part of learning math. I think doing so would be an excellent way to give students more of a voice in what they are learning. Too often students remain silent during lectures and then work on homework without much participation in discussion. One way to reinforce learning is to teach. Writing gives the opportunity to teach through written explanation. As students develop communication skills through writing, they will become able to explain math to someone who is a "non-math" person. This will serve them well and us one day when they are writing text books.

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.