1 Mar 2002
Math Tutoring With Barry Goldman
BASIC TUTORING RULES:
You must give 24 hour notice if you need to cancell a session, and if your session is 5pm or later (these slots are in GREAT DEMAND) then call BEFOR NOON the day before. there are usually other students who would like these slots.
If you don't do the homework i assign and we have to do the work in the next session, your progress will be very slow! Do lots of homework and bring it with you so that i can see how you are doing.
ALGEBRA:
this course as specified is an entire semester of spelling, or a semester of piano lessons consisting entirely of finger exersizes and scales and you never get to play a song, and not even get a chance to hear what a song is like. BORING! Actually, impossible to really learn. (possibly, you can memorize it for the test, but what a waste of time, might as well retain a valuble skill, and knowlege base).
I wish we had the time to do songs... I will try here and there to bring in some of this material. what would be the songs in algebra? well, working out some of the patterns I have been trying to show you, some of the historical development, explaining or proving some of the rules and assertions. i.e. where the square root of two comes from, why it was so important for the greeks and in fact most early civilizations to express numbers as fractions, and why the square root of two cannot be expressed as a fraction. Or how mathematicians filled in the hole in the pattern and defined what to do with division by zero ( it also involves the square roots of negative numbers). How some of this math is used in science. etc...
what is its utility? will it make you more capable citizens?
I wish i could teach you how to grasp the world with quantitative skills, pattern recognition, but...
Algebraic statement of concepts and then algebraic manipulation is a major tool of science, engineering, economics, business... The use of the same rules for neg, 0 and positive numbers, the use of neg numbers, fractions etc for exponents and using the same rules for all, is another powerful abstraction. algebra and the concepts of functions, operators, abstract algebraic systems, allows mathematicians to build layer upon layer, where eventually they can manipulate huge complexes of problems with the movement of a few symbols.
plotting geometric problems on the x-y axes and then using algebra, is a powerful tool for solving geometric problems for which insight is hard to come by.
In general, it is amazing that we have two alternate tools to solve problems, geometrical, and algebraic. some problems will succumb to one approach or the other, or in the solution of a single problem we have at our disposal both approaches to complement one another.
The process of working out algebraic and numerical problems, builds concentration, attention to details whose meanings are not readily aparent, techniques for maintaining attention on a long sequence of steps and how to complete them without losing track. It Trains us in the technique of breaking up a conceptually difficult problem into a long train of conceptually simple steps; perhaps not the most spiritually advanced, or artistic way to solve a problem, but the development of quantitative measuring, mathematical manipulation, scientific experiment and the resulting technology has made western civilization the current dominant force on this planet, wether fortunate or unfortunate.
USEFULNESS OF: functions, graphing, solving systems of linear equations,
PRECALCULUS AND CALCULUS:
The whole point of these courses is that we can make both algebraic, and geometric models of the same process, or structure. Or not even that we can make models, but that we can engage both visual/kinesthetic and linguistic/logical abilities to help us solve and understand problems. Oddly enough almost every piece of algebra goes with a picture (even if it is 4 or 11 dimensional.) and almost every picture as some algebraic description. Very odd that so much of the crazy algebra cooked up inside mathematicians heads like square root of minus one, and 4x4 matrices etc... are actually found by the physicists to describe our real world in some way.
So don't forget to use graphs to help solve your problems or check them.
SEE IT IN OTHER CONTEXTS!!
I will try to show you math in the contexts of other subjects, and in the functioning of civilization in general. or suggest some light reading about math problems, history of math and science, famous mathematicians, scientists. do math puzzles! Perhaps learn some fun games, like GO, conway life, computer programming.
TECHNIQUE:
work slowly and methodicaly
tell story of math team, break problem into many simple steps
writing practice
always be writing at first, don't let yourself stop and think for too long especially on a test or quizz! If you don't know how to proceed just start sketching out something, this may jog your memory and then you will see how to do it.
when first learning how to perform new problems, don't take too much time agonizing about wether you are doing them right. Work through them quickly and see what the results are.
take alot of space, have clean page, like well lit, well organized spacious kitchen to cook a major project in. cramping of any kind causes tension, causes messups. So: DON'T WORK AT THE BOTTOM OF THE PAGE!!
work row by row, going down the page ONE STEP AT A TIME, consequently: don't begin a problem at the bottom of the page. further, leave enough room to complete the whole problem on ONE PAGE, having to flip pages or copy from one page to another lets in chance of error.
Try to do only one operation per line, copying the rest of the line out each time; tedious perhaps, but by concentrating on only one operation at a time, things won't get jammed or confused and you won't make stupid mistakes even when you know what you are doing. Also by writing out each step neatly doing only one operation at a time, if you get distracted, you can easily come back to the page and see what the next step is.
always write the original problem statement first.
write neatly and openly so it don't look like a mess of spiders wiggling all over the page.
the goal is to have a sequence of easily readable steps, so that you can look down your work and spot mistakes and left out negative signs etc..
Which brings us to negative signs. watch the negative signs!! write them boldly as if you mean them so that they don't disappear a few steps down. treat them with respect, as if they were angry ferrets with sharp teeth. When you get to one, be on the alert, something confusing is bound to happen. The negative sign is relatively new in history.
NO CALCULATORS!!
mathematics is about seeing patterns, getting the patterns laid down in your gut. The use of calculators and computers in schools is largely a result of large industries bullying over and colonizing the educational establishment. In no way should all highschool students be purchasing those stupid $100 texas instrument calculators! They are actually sophisticated graphic programmable computers that you could design moon flights on! They do not lead to more effective learning, at least not used in the way they usually are.
The calculator is a useful tool, once you have thouroughly mastered the understanding of the concepts and a developed a gut feel for what the results should be. No machine has yet been built by us that comes close to the intuition of a trained craftsman. It is your job to become a trained craftsman in the use of mathematics. you should definitely not rely on the calculator to be the arbitor of wether your answers are correct. It is our jobs to be constantly checking the machines around us for their correct function, after all we don't want them running the show. Neither do we want to become dependent on tycoons like Bill Gates for our abilities to solve our own problems.
once the craft is learned, the calculator becomes an excellent tool for performing repetitve dull sequences of calculations. but beware that even in the process of performing long sequences of calculations, if we do them by hand we may gain insight into some patterns and ultimately learn to streamline the process or gain real insight into the problem at hand, something that might not happen if a calculator or worse, a computer were set to the task mindlessly.
at anyrate, even though the homework seems like a boring sequence of problems, it is actually the occasion for you to embody skills. If you let the calculator do the work, you will learn nothing. when you are ready to build bridges and rocket ships or design vast marketing strategies, you can go ahead and use a calculator.
So keep trying to learn the multiplication table. If it is hard to memorize, you will just have to develop your own tricks of how to calculate or see the patterns to help you memorize. That's what mathematics is.
STRATEGY:
first step is to have a goal in mind. what do you think the answer is going to be like? How are you going to get there? keep this goal in mind, keep in mind THE PURPOSE of each step while you are working and then it will be harder to get lost.
on your way, if you've guessed what the answer is going to be like, you can check, does it look like you are going to end up there?
when you are done, check your result!
WAYS TOO CHECK YOUR RESULT:
do the problem again
do the problem later, when you might see something different do the problem in a different way
i.e. a different algebraic approach, with decimals instead of
fractions, geometrically instead of algebraicaly.
check it numerically:
if it is an arithmetic problem, reason by way of the size of the answer you ought to get. approximate.
if it is solving an equation for a numerical answer, substitute the answer ALL THE WAY BACK IN THE ORIGINAL PROBLEM STATEMENT.
if it is transforming one algebraic expression into another, substitute a value for the variable in the original problem and your final answer. now evaluate the two expressions, they should come out the same. Just in case you picked a special value, pick another one and try it.
This particular tool is important to use all the time. Suppose you are about to use what you think is a correct rule: (a+b)^2 = a^2+b^2. Try it on some numbers first:
(3+4)^2 = 3^2 + 4^2 ? 7^2 = 9+16 ? 49 = 25? nope! that wasn't the right algebra rule.
by developing these skills, you can become confident that you know what you are doing, that you are mastering algebra and not just following allong in a fog.
Learn to approximate!!
Keep practicing. It's a physical skill. It is said of musicians, dancers... that if you miss one day of practice each week you will notice difficulty with your skill, if you miss two days practice, others will notice...
so, i suggest that you do a little bit of math EVERY DAY as apposed to alot of math at a shot each week. Unless you are on a good roll, take a break once an hour, get the blood flowing, do handstands, cartweels play tennis, get a fresh view of what you are working on. Try doing math at different times of the day, to find the time that you concentrate the best.
as we progress in the semester i will continue to give review from the previous weeks work to keep in practice. In math all skills are built on the ones that come before. You have to keep them in practice.
HOMEWORK:
Work on ALL problems assigned even if you do not think you have the right answer. Write your name, the date assigned, and the textbook exercise section number at the beginning of each homework assignment. Write the problem number and the original problem statement for each assigned problem. Write legibly and show all work!
If you don't know how to do the problem, DO NOT JUST COPY DOWN THE ANSWER FROM THE BACK OF THE BOOK, DO NOT JUST COPY IT OUT FROM YOUR FRIEND.
Instead of checking your answers with the ones at the back of the book, try to work on some of the ways i've described to check your answers on your own.
more important for you to trudge ahead and try stuff that don't work so i can see how you are thinking than to write down the correct answer. then, come test time, there wont be any surprises. When i look at your work i can discover what is holding you back, and help you move forward.
[on the other hand that is another technique i should use, just have them watching me solve problems step by step correctly and copying, build the muscles. hmmm...]
