Do they not teach basic math anymore.

[snowman]

Originally posted by K.B.
......
There is a limit somewhere - it is different for each person, We all need to reach for our limit and then go beyond. If a calculator or computer helps go beyond then we can solve even more complex problems.

To clarify my earlier comment, the BS only applies to the last paragraph. Sorry for any confusion but I still get dizzy reading it.

THe rest of the post is right on.

[K.B.]

I'm mostly with you guys on the calculator use. I've had to work with school administration to have a middle school teacher not let all the kids use calculators all of the time. They become addicted. They get to high school and can only do fractions with the calculator. As I said before, "in moderation". The teacher in question now teaches something other than math. Its hard to get real math teachers in elememtary or middle school unless the district is big enough to hire curriculum specialists.

On the other hand I've also worked with special ed students in the high school that test at the 4th or 5th grade level (or even lower) in math. They've been doing "drill to kill" for several years and still don't get it. We have tried for decades to pound the basics into them with limited success. "General Math" class no longer exixts at our high school. We are having greater success now with other methods.

Sometimes we have them do 1 digit math without a calculator.
Like : 2x + 3 = 9

Sometimes we give them a multiplication chart for the next level.
22x + 3 = 47

Then a word problem might lead them to write
22.7x + 17 = 257.3
which they can only sove with a calculator with their limited skills. They understand the concepts of addition, multiplication and order of operations because they have used calculators. We need not punish them just because they never got the algorithims for long division or multiplication. (Most people don't really "get them" anyway - they just do them by memory.) Farther down the road they might be able to use a TI-83 to understand the change in slope by comparing the graphs. Only with a calculator will some of them ever get to see the beauty of the pythagorean theorem.

Way back in my day we used trig tables(and slide rules) to do trig. Calculators have changed that method a lot. Who remembers how to do a square root by hand? How would you do e^1.3 without a calculator? Calculators are a useful tool to help us think, not a replacement for thinking, just about like Bicycles are a tool to help us travel, not a replacment for walking/running. It's not "no calculators", nor is it calculators "all the time" but rather appropriate use at the appropriate time!

[snowman]

Teachers who teach math really need to know the subject for a lot or reasons. In the third grade I constantly argued with my teacher about being able to divide a small number by a larger number, ie a fraction. She said it was impossible. I said that if I have a single item I can always cut it up into smaller pieces. This problem persisted until college, where I learned the truth, The teachers didn't know what they were talking about. They didn't understand math. The teachers that I had, never understood the concepts. I am not a marvelous math student, so how do teachers cope with someone like one of my sisters, at 8th grade level or above in second grade. Math teachers really NEED to understand math to teach it. I suspect many do not.

In any case, I don't believe anyone should use a calculator until the 12 th grade and only then if they are proficient at that grade level. That’s the same grade slide rules used to be introduced.

As to "bone head" math, please excuse the term, but it is most descriptive, yes use calculators.

[onewhippedpuppy]

You have to learn the basics before introducing calculators. After you can do it by hand, a calculator just increases the speed by which calculations can be done. Who would want to pay an engineer an hourly rate to do calculations by slide rule? That said, I worked construction for years before returning to school to get my aerospace engineering degree, and have noticed that my talent for doing math in my head has slipped.

One question, why so much blame on the teachers? Parents, if your children are slipping in school, you need to look in the mirror, because that's where the buck stops. If they're gifted, you need to nurture it. If they're slow, assist them. At the end of the day, it's your responsibility that your kids succeed.

[K.B.]

Originally posted by snowman

In any case, I don't believe anyone should use a calculator until the 12 th grade and only then if they are proficient at that grade level. That’s the same grade slide rules used to be introduced.

I used to believe the same thing (back when I taught slide rules). Things are different now. Kids are different, society is different, our expectations are different, tools at our disposal are different. I can't imagine doing simple trig without a calculator. There's no point in wasting time multiplying .8666 by 1.414 by hand calculations. What does that prove if you get it right or wrong? That you are good or bad at arithmetic? You probably never heard of William Shanks, an English mathematician. His contributions to the body of knowledge of mathematics were significant back in his day. He spent over 15 years calculating the first 707 places of pi. He published his results in 1873. In 1944 it was finally found, that he had made a mistake in the 528th place, and all the following digits were wrong. Does that mean he was a bad mathematician? Would we expect someone to repeat his calculation by hand today? Would that make you a great mathematician? Time marches on. We are in a different world.

We all know Albert Einstein. Einstein was considered a bonehead by his math teachers. They told him to give up, go back to the basics, he would never get it, too many mistakes on the simple stuff. Einstein was put in the slot of being a lowly patent clerk and he transcended the limits (limitations) people imposed on him. He didn't waste his genius on simple arithmetic. He stepped beyond.

In an earlier post I used the term limits. I suppose a better term would have been limitations. Do a find and replace.

Man had a limit (limitation) for thousands of years of not being able to fly. After the Wright brothers, it took us less than a century to make it to the moon. Once we stepped beyond the limitation, we went way beyond. Of course kids need to learn the basics first. But being able to do basic arithmetic is not and end in itself. Basic arithmetic is not necessarily related to kids ability to understand higher levels of mathematical/logical thinking and reasoning. A lot of people with good memories can do arithmetic. Our world needs people to think logicaly, critically, and solve new problems in new ways. Kids need to learn logical thought processes to step beyond the limitations around them. Rote repetition of simple arithmetic isn't going to get them there. Exposure to spreadsheets, graphing calculators, and a variety of mathematical thought processes and logical reasoning will give them tools to confront the coming challenges.

[snowman]

Yeh, yeh, yeh, but how do you tell the difference between the boneheads and he ones that really understand what they are doing? Einstein is a one of a kind, and he made it in spite of the limits imposed on him. I kind of suspect any new Einstein would somehow do the same, so not to worry about overlooking him. For the rest of us mortals, you need some kind of standards, something to sort the wheat from the chaff.

It is necessary to have a cold, hard, mastery of arithmetic and multiplication to solve higher math problems like differential equations, where the educated guess is the rule. You can't make educated guesses without lots and lots of practice on mundane problems. Those guesses rely on learning all the "tricks" of math, especially arithmetic, which rely on lots and lots of practice doing simple, stupid multiplication and addition, subtraction problems that a calculator is useful for. It is impossible to use a calculator to solve differential equations (in general) Differential equations are math 101 in the real world, yet many of the people talking about this don't even know what a differential equation is.

Then we get to matrixes, partial diff eq, curls, vector math, spars matrixes, chaos theory and other stuff that is the meat of math and science. Much of this stuff, at this level relys on educated guesses, not systematic methods. THis is the math that makes modern car engines, nuclear reactors, bombs, electronics, cell phones, High def tv, the internet, high speed dsl connections, preditor drones, gps, and all the star wars stuff possible. Thats why calculators should not be taught until 12th grade, even in todays world. Computers are NECESSARY to actually utilize the results of this high math as the calculations would take years to do by hand, but it is NOT a chicken and egg problem, the basic math comes first, the computers, second. The computer is nothing more than a simple tool, like a power saw is to a carpenter.

[K.B.]

“George Boole (1815 - 1864) The original Working Class Boy Made Good, Boole was born in the wrong time, in the wrong place, and definitely in the wrong class - he didn't have a hope of growing up to be a mathematical genius, but he did it anyway.”
George Boole, invented Boolean algebra, the fundamental basis that electronics (computers) AND, OR, and NOT gates come from, 100 years before the transistor was invented.

History is filled with cases of ignoring and ridiculing those “different” people that society considered boneheads, and many of these were mathematicians, the ones who didn’t follow the norm, the odd, quirky, some suffering schizophrenia (like the movie “A Beautiful Mind”) or savants (like the “Rain Man”) Our inability to recognize genius has historically been present for centuries.

In some countries educational system, only the best move on. We could do that. Our public school system gives the opportunity for all to learn. We consider them all to be wheat, there is no chaff. We could give tests at 4th 8th and 12th grade and assign futures, jobs and lives, to people based on their test results. This premature specialization would lead us to recognize future plumbers, auto mechanics at an early age and prevent wasting time teaching those students subjects they don’t really care about anyway. It could also trap kids into an occupation they hate and miss out on one they might love. It could also lead people to do exactly what their parents do for a living as the learning would be more limited.

Kids in 4th grade are currently doing some simple robotics with computers. The current grade level content expectations for 6th , 7th and 8th grade allow students to experience mathematics concepts of probability, similarity, quadratic equations at an earlier time in their educational experience. We used to bore them with repetition, drill to kill, etc till they got to the 9th grade where they finally took Algebra. Today’s kids heads are wired differently than ours were. Kids that are interested in math need exposure at a young age as to what the possibilities are. Exposure to computers, calculators, robotics, gives them a chance to see what future careers in mathematics might be like. It shouldn’t take 12 years to master arithmetic. To borrow your carpenter example, We wouldn’t have many carpenters if we told them they had to have 11 years of experience with hand saws before they could use power saws. The tools are there, just don’t turn them into crutches you need for life, use them when appropriate. Yes, there are a lot of teachers who do not know what appropriate use of technology is. Kids need to learn the basics but they don’t need it drilled into their head till they hate math. They need constant exposure to basics as well as introduction to higher levels of thought and introduction to the use of modern technology.

[snowman]

TOTAL BS. Thats exactly why we lag behind all those other nations in math. The people that made the US famuous were brought up under the exact same system you say is bad. What we are doing now is a disservice to everyone involved. Forget the super math types, they will suceed in spite of whatever system we use, its the majority that need help. One of my close relatives, Luitzen Egbertus Jan Brouwer, was a math wiz, I suspect he would disagree with you. Todays kids are wired exactly like the kids of yesterday were. When you understand the math of Newton, Brouwer, Einstein, Bohr,and the like you are just coming up to the last part of the 19th century. Once you have mastered that you can talk about the 2oth century. Brouwer is considered to be the father, or one of the fathers, of modern math.

[K.B.]

The 4 most popular major philosophies of mathematics are Formalism, Inventionism, Intuitionism, and Platonism.

Brower was an intuitionist (as it appears you are.) He was also Dutch (which I am) He made many important discoveries using intuitionism. I certainly bow to his mathematical prowess. However Errett Bishop produced a systematic treatment showing that almost all of the important intuitionist material can be dealt with by methods of classical approach. In other words, there are different ways to reach the same mathematical conclusions.

I see why you are not supportive of using Boolean logic, true false, binary, computers. Brower likely supported 3 state logic, fuzzy logic, yes no maybe.

"Is pure mathematics simply a game which we play or do our theorems describe an external reality?"

Did we discover counting or did we invent it?

The slow development of artificial intelligence shows how complex the human brain is. We can’t duplicate it with computers very well. Computers have a difficult time understanding simple language, something that comes easy to us. Computers have an easy time solving math problems; something that often times seems difficult for us. Math is not difficult, it is merely contrary to our normal way of thinking .

If we better understood how the brain handles math, we could find approaches to teach math more effectively. I agree with your statement that the smart kids will make it no matter what. Even if they have a few bad teachers they seem to bounce back in no time. The cream rises to the top.

I did not say the educational system that brought us to where we are, was bad. I am saying our society is changing and so are our educational needs. Education has been changing continuously anyway. If it didn’t change it would become stagnant. Current educational literature and brain research contains a body of knowledge (or opinion) that differs from your opinion. It's not about math its about the human brain. I grew up in a “Leave it to Beaver” household. That doesn’t exist for the average kid any more. The undiscovered math that is out there will need new and different ways to discover it. It is not that math is changing; it’s our world, our people, our knowledge, and the tools at our disposal.

Our educational system appears to lag behind other countries systems for many reasons.
A. We educate everybody. When we compare tests we include the low along with the high. Other countries only promote the best. We are comparing our average with their top (see the Timms report)
B. Other countries have a much higher respect for education and teachers. It is valued more when it is not free for everybody.

The days are gone when a single person could understand all of math. For hundreds of years a person could study all of the known body of math. This body has become too large for any single person to learn. People will need to work collectively in hives to conquer larger problems. A collective conscious can solve more than a single mind. The famous mathematicians of the past are going to be replaced by groups, corporations, communities, working together to solve larger problems. We will need new methods to solve new problems. Our educational system is adapting to new knowledge of the human brain. If this is BS to you then so be it. I agree to disagree. If you don't think kids are different today then visit your local schools.

Snowman, this conversation has probably stretched beyond OT capacity. Thanks for the oportunity to discuss. You are welcome to have the last word (or last 2 letters if you wish)

[snowman]

ok