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September 29th, 2017, 04:54 PM   #11
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Originally Posted by jonah View Post
Beer soaked recollection follows.
From an author's preface:

Learning to use mathematics could be compared to learning to drive. In either case, the quote from the Chinese philosopher Lao Tse is appropriate:

You read and you forget;
you see and you remember;
you do and you learn.

At the outset the learner-driver is presented with a bewildering set of rules and tasks, some
of which must be performed simultaneously, some sequentially. There are sound, sensible
reasons for each of these rules, as learners will discover on their first outing on a public road.
Mastering driving skills and gaining a sense of how to control the car only comes about by
following closely the routines demonstrated by the instructor, then practising them over and
over again, sometimes patiently, sometimes not! In the end, the new driver will be able to
handle a car easily and effortlessly, as if it were second nature. With these newly acquired
skills life is enhanced with previously unavailable choices. The new driver (with a car!) can choose where to go, who to go with, what route to take, what time is convenient, etc. And so it is with maths.
This makes some sense. Take driving a stick, for example. After awhile it becomes second nature and you don't even think about it. But if you try to explain in words how to do it you are reminded of how many things you are doing simultaneously.
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September 29th, 2017, 05:00 PM   #12
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$f(x) = x^2$ makes me uncomfortable. I keep thinking, "What's x?", even though I know x doesn't always have to be defined.
The expression $x^2$ represents a machine. What is a machine? It's something that has an input and an output. You input 5 and it outputs 25. You input -3 and it outputs 9.

So the expression $x^2$ can be thought of as a process or recipe. Put in one number, get out another.



If the rule is $x^2$ then this machine outputs the square of its input. Whatever the rule is, that's what the machine does.

Another feature of the function machine idea is that the machine is a black box. We do not know or care how the machine works, whether there are gears inside perhaps elves with calculators. All we can ever know about our machine is what it outputs for a given input. The only aspect of a function machine that we can observe are its inputs and outputs.

The most important feature of a function machine is that it always produces the same output for a given input. So if you put in 3 and it outputs 47 today, it will do exactly the same thing tomorrow and the day after. That property defines a function.

Last edited by Maschke; September 29th, 2017 at 05:03 PM.
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September 29th, 2017, 07:04 PM   #13
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We use letters in algebra as convenient symbols to represent numbers that are either not yet known or else are not yet specified. Any letter will do, minuscule or majuscule, Greek or Roman. Students have trouble with this: they want to know what number x is, but the whole point is that we don't know yet. When the student finally begins to get that x is just an abbreviation for "a particular but as yet unspecified number", we introduce function notation where letters are used as abbreviations for formulas (more precisely, rules). Using letters in two different ways causes further confusion among students.

In terms of teaching elementary algebra, it would have been better to use majuscule letters for functions and minuscule letters for numbers. When we see a formula like

$d = r * t$,

it allows us to capture a potentially infinite number of numeric examples in a single abbreviated formula.

$f(x) = x^2$ simply means square x once you know what number x represents.

Frequently, using function notation is simply a convenient abbreviation.

I write $f(x) = \dfrac{\sin(x)}{x}$ and

can thereafter just say $f(x)$ instead of writing down that fraction.

Last edited by skipjack; September 29th, 2017 at 11:45 PM.
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September 30th, 2017, 02:16 PM   #14
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In terms of teaching elementary algebra, it would have been better to use majuscule letters for functions and minuscule letters for numbers.
That would be helpful.
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September 30th, 2017, 03:06 PM   #15
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Can be looked at as a li'l puzzle.

Teacher preparing questions writes down:
3 + 5 = 8
Then erases the 5 and replaces it with an x:
3 + x = 8
Writes above on blackboard:
Ok class, what number does x replace?
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October 1st, 2017, 11:30 AM   #16
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Quote:
Originally Posted by Denis View Post
Can be looked at as a li'l puzzle.

Teacher preparing questions writes down:
3 + 5 = 8
Then erases the 5 and replaces it with an x:
3 + x = 8
Writes above on blackboard:
Ok class, what number does x replace?
That is very simple, of course, and I would have no problem with that.

One instructor early on asked if I ever did sudoku puzzles? I said that I do not. She said that they work the mind in a similar way.
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October 1st, 2017, 11:33 AM   #17
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That is very simple, of course, and I would have no problem with that.

One instructor early on asked if I ever did sudoku puzzles? I said that I do not. She said that they work the mind in a similar way.
What denis means by his example is that algebra is merely following up the consequences of that little puzzle.
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October 1st, 2017, 11:38 AM   #18
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Originally Posted by JeffM1 View Post
What denis means...
Hey, it's "D"enis; plus be careful not to use "P" !!
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October 2nd, 2017, 11:01 AM   #19
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Beer soaked ramblings follow.
Quote:
Originally Posted by ArthurDeco View Post
This makes some sense. Take driving a stick, for example. After awhile it becomes second nature and you don't even think about it. But if you try to explain in words how to do it you are reminded of how many things you are doing simultaneously.
From the same preface/introduction of an author:

Worked Examples
At an introductory level, you will not become proficient at using mathematics by simply
'reading' a mathematics book, cover to cover. A better approach is to read just one section at
a time, then put pen to paper and follow the methods which are demonstrated in the Worked
Examples, line by line. An index of Worked Examples is provided at the end of the book for
easy reference.

Graphs
Can you imagine someone who has never seen a car before, attempting to understand what
the controls, gears, steering, etc. look like from a verbal description? Understanding would be
enhanced enormously by the provision of some well-labelled sketches and diagrams.
Visualising mathematical functions/equations is not easy, especially for beginners. When
the function is graphed, much of the vagueness and abstractness is removed. In fact, many of the properties of the function are revealed. In Worked Examples, whenever appropriate,
graphs are plotted over an interval by calculating a table of points, then drawing the graph.
Unfortunately, in many economic applications, this process is often a very time-consuming
exercise and errors are all too easily made in calculations. In this text, the use of ...etc.
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October 4th, 2017, 02:31 AM   #20
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Beer soaked ramblings follow.
Quote:
Originally Posted by ArthurDeco View Post
Hello. I'm hoping I can get some guidance or help, or something.

Background. I am a 53 yr old who has been working in civil engineering doing design and drafting for 29 years. I do not have a degree or a professional license, however, and I want both. This is very important to me. I am very good at what I do, but I am very weak at math. Sounds strange for someone in engineering, I know, but I do know what I do on a daily basis.

Algebra completely baffles me, and always has. What the deuce is X? It's like hieroglyphics to me. Yet everything builds from there. I started basic algebra at the community college level a few years ago and struggled every step of the way, but have managed to get by. I am now at a university (online program) and in calculus, and I'm kind of freaking out about it.

My issue is that I cannot retain what I learn. I get concept fine. I can do visual things like graphing fairly easily. But I should be remembering what I learned last semester in trig that leads to calculus, and I'm struggling and feel like I'm attempting to reteach myself on the side while trying to keep up with the class.

I thought maybe I had ADHD, as I fit so many of the symptoms, so I had myself tested by a professional a couple months ago and I definitely do not have it. Which actually kind of frustrated me because I'm back at square one with the same retention and concentration issues and nothing to point to that can be identified and worked with.

Any thoughts or help or suggestions would be most appreciated. Please feel free to ask questions if you need more information to make a suggestion or provide a thought.
A few years ago, I saw a bunch of blind guys playing chess against those who can see.
I was very impressed. A few years later, I saw this news about a young blind girl who took on mathematics in college at a regular university. That was inspiring. As inspiring when Paul Schaefer commented on how inspiring it was to hear about a news where this 80 something woman was peeping at her neighbor for her erotic needs.
Then I tried solving rubik's cube using just touch markers as if I was blind. I realized just how blessed those who can see.
I say you sir are blessed with sight and mobility. Your perceived obstacles are only obstacles until you move them out of the way.

From another author:
To recognize a conic section, you often need to pay close attention to its graph.
Graphs powerfully enhance our understanding of algebra and trigonometry. How-
ever, it is not possible for people who are blind-or sometimes, visually impaired-
to see a graph. Creating informative materials for the blind and visually impaired is
a challenge for instructors and mathematicians. Many people who are visually
impaired "see" a graph by touching a three-dimensional representation of that
graph, perhaps while it is described verbally.
Is it possible to identify conic sections in nonvisual ways? The answer is yes,
and the methods for doing so are related to the coefficients in their equations. As we
present these methods, think about how you learn them. How would your approach
to studying mathematics change if we removed all graphs and replaced them with
verbal descriptions?
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Last edited by jonah; October 4th, 2017 at 02:36 AM.
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