Beginning to Learn a New Language
Before I discuss the world of mathematical expression, let us look at how we tend to develop our first linguistic language. There is an increasing degree of abstraction and intellectual requirement as we proceed upward toward a formal written system that codifies the language. Here is a list which I think describes the stages we go through:
- Begin with babbling sounds separate the “Noise" from the "Signal"
- Combine the sounds into labels for things and ideas, words then sentences but also have Body expression, inflection, -- a multi-dimensional concept
- Learn that sounds and images on paper are related
- Break the strange code that freezes language onto a page. Permanence
- Read -- with increasing ability over time
- Write – highest level involves planning, translating and reviewing
Since translating the above probably gave you considerable difficulties, I provide a "Rosetta Stone".
I doubt you were very successful in quickly reading the passage. Did you get frustrated and decide to quit. How many of the characters did you think you had figured out, only to find that a moment later you forgot. And how often did you need to go back to recheck something you thought you knew. .
- Observe how you sound out the Greek characters, but you would not sound out the first two words (the English ones) as "fo-re-i-gin laa-ng-oo-ahg-ees"
- You easily recognize those as complete units and ignore the individual characters.
- Now observe what happens when you read the other words (I am assuming English is your primary language).
- I suggest that you try the same passages, except I put them in your “mother tongue” (native language)
Observe that the first line was in a script font, yet you probably had no problems reading it. The next line has no spaces between the words and yet most of us can readily make out the words. It may take a little more time than reading the words as they are written here, but nevertheless you are able to insert those breaks. In the third line there are few vowels and yet you can make out the words or the intent of the message. I would say that most who have developed reading skills are very robust readers in our primary language.
Consider Mathematical symbols.
I am going to refer primarily to some formulas that we encounter in statistics, but you should quickly see this is equally an issue with ordinary algebra.
Consider, for example:
Consider Mathematical symbols.
I am going to refer primarily to some formulas that we encounter in statistics, but you should quickly see this is equally an issue with ordinary algebra.
Consider, for example:
We expect our students to quickly realize that the symbol X is very different from that symbol X-Bar (the X with the line across the top) , In reality X-bar is the condensation of information that is virtually equivalent to an entire paragraph.
Those of us mathematically literate are also very familiar with the way the “word order” follows in this formula, since we do not read it character by character, or from left to right. We read the X – (X-Bar) as a complete unit, working from the inside of the parenthesis outward, and knowing that the summation is a final modifier, not an initial action.
It is incumbent on us to place ourselves in the same position as the neophyte to statistics. I suggest you picture such activities as using your TV remote control and the setting of a program to be recorded or finding the fast forward and pause buttons.
(Note: I apologize for the mixing of the X with the bar across the top and the word "X-Bar" but there is often an inherent problem of putting certain types of characters in web pages. A more complete copy of this material will be posted shortly.)
Those of us mathematically literate are also very familiar with the way the “word order” follows in this formula, since we do not read it character by character, or from left to right. We read the X – (X-Bar) as a complete unit, working from the inside of the parenthesis outward, and knowing that the summation is a final modifier, not an initial action.
It is incumbent on us to place ourselves in the same position as the neophyte to statistics. I suggest you picture such activities as using your TV remote control and the setting of a program to be recorded or finding the fast forward and pause buttons.
(Note: I apologize for the mixing of the X with the bar across the top and the word "X-Bar" but there is often an inherent problem of putting certain types of characters in web pages. A more complete copy of this material will be posted shortly.)