The Math You Don't Learn is Harder Still

[twisting_edge]

I had no idea things had gotten nearly this bad. Does anyone here have any experience of this? It is evidently only recently the increase in this "teaching" philosophy has ceased expanding.

From NYT:http://www.nytimes.com/2006/11/14/education/14math.html"

At the same time, parental unease has prompted ever more families to pay for tutoring, even for young children. Shalimar Backman, who put pressure on officials here by starting a parents group called Where’s the Math?, remembers the moment she became concerned.

“When my oldest child, an A-plus stellar student, was in sixth grade, I realized he had no idea, no idea at all, how to do long division,” Ms. Backman said, “so I went to school and talked to the teacher, who said, ‘We don’t teach long division; it stifles their creativity.’” [emphasis added]

Note that the NYT prevents you from accessing articles more than 14 days old. I've saved a copy in case the debate continues past that point.

 

[Ronnin]

My have been martialed through high school with the help of TI calculators. They do not issue textbooks or any meaningfull homework. All I get from them when I ask them how they figure certain things is, "Let me borrow your calculator and I can tell you." I love calculators and computers, don't get me wrong, but at their level they should be doing as much by hand calculation as possible. Here in Texas every is geared toward the TAKS test. So if it isn't tested for on that, you just don't need to learn it I guess.

 

[Rach3]

Here's a working link:

http://www.nytimes.com/2006/11/14/education/14math.html?_r=1&ref=education&oref=slogin"

__________________
"If we cannot end now our differences, at least we can make the world safe for diversity." -John F. Kennedy

"I know that the human being and the fish can coexist peacefully." -George W. Bush

 

[theCandyman]

:rofl:

How does long division stifle creativity?

 

[Rach3]

‘We don’t teach long division; it stifles their creativity.’”

My initial thought is that it's a reasonable complaint. These are young students in elementary schools, they have no inkling of algebraic manipulation and presumably have no clue how these "algorithms" work. They must take them for granted, memorizing a procedure without any understanding of what it means. I'd think that blind faith and memorization are not appropriate introductions to math. Why can't long division be held off until these kids are doing algebra, and can actually prove and comprehend what they're doing?

It's not like there's an urgent pragmatic need for long division, either - not even in the hard sciences. It's on the same level as say, digit-by-digit algorithm for computation of square roots (which if I'm right dates back to the medieval ages.) Does anyone complain about that algorithm not being taught in grade school?
__________________
"If we cannot end now our differences, at least we can make the world safe for diversity." -John F. Kennedy

"I know that the human being and the fish can coexist peacefully." -George W. Bush

 

[russ_watters]

I don't know about the creativity argument, but there apparently is some legitimate debate over this issue: http://www.doc.ic.ac.uk/~ar9/LDApaper2.html

Suppose a student arrives at college knowing those aspects of division discussed above but not having ever learned LDA. Will s/he have any problems in mathematics or engineering courses without the ability to do LDA? Of course not. One doesn't know whether to laugh or cry when a college or university mathematician says that s/he knows what mathematics students arriving at university need to know and that this includes LDA. Not only can this not possibly be correct but, more generally, as I wrote some years ago, it would be better if students entering college "knew even less than they do now in terms of factual knowledge if only they knew something about the mathematical enterprise – what it means to prove something, why certain approaches tend to be effective in solving certain kinds of problems, what it means to do mathematics instead of just pushing symbols around on a piece of paper" [5].

I can certainly see the logic of it being more important to know what math is than how to do certain pieces of it. The logical thought processes learned by doing word problems, algebra, and geometry are a foundation for everything that comes later. We had someone in here recently who complained that a kid in his class who was getting an "A" in a calculus class didn't have a clue as to what a derivative actually was. To me, knowing that is just as important as being able to memorize the steps in an algorithm to solve one.

Lets face it:

When was the last time you added/subtracted numbers by hand? (for me: this morning)
When was the last time you used algebra? (for me: this morning)
When was the last time you used calculus? (for me: this morning)
When was the last time you did long division? (for me - its been years)

On the other hand, to be perfectly honest, I can't figure out what kids would be doing in math class were they not learning things like manual arithmetic. It is tough looking back from here, but it doesn't seem like there is much to math before you get to algebra...

 

[Ronnin]

I think long division reinforce some simple concepts such as place value. If nothing else by hand calculation does teach some attention to details. It may also help in understanding fractions and mixed numbers in general later on.

 

[FrogPad]

I am a product of integrated math (a new way of teaching math).

Before high school I loved math. My first day of my first year of high school we were all handed out Ti calculators. We checked them with the teacher and were told that we were able to hold onto them for the year. This was the start of a new way of teaching math. Integrated math (in whatever form I was exposed to) was tested on our class, we were the guinni pigs for the program. I'm sure it really started on us due to some type of funding, but that is merely speculation.

Integrated math, is all the typical math programs integrated into one. I never took an algebra class, a trig course, or calculus. In fact I had no idea what I was doing, we were just shown new ideas and expected to use them to solve problems. Our in class assignments were done in groups. We were required to keep a journal that would count as 10% of our grade, and was encouraged to use on exams.

The journal was a place for us to write down "math notes". We were told that we would never have to memorize all these formulas in the workplace, so why learn that way? We worked in groups, and this mainly consisted of the "smartest" (whoever understood integrated math) doing all of the work.

I started at the university in pre-college algebra. Some people actually started in basic math courses when they went to university. Classes that would teach multiplication and division.

One teacher wrote a book at our school that was title (I don't have the exact title) along the lines "everything you need to know in high school math, that integrated math doesn't teach". It was very thick. He ended up resigning because he hated the program.

Anything good out of all this?
- I can type blindfolded on my calculator. That's a good skill right?
- The program is no longer taught at the school.

Anyone a lawyer? I'd love a class action lawsuit against the school. A lot of students had to take college courses that cost money, just to be at an introductory university level. I shouldn't have had to pay for those courses.

If you ever meet someone who has gone through integrated math, please give them a hug.

 

[FrogPad]

I think the use of a calculator is important in moderation. Since I became so proficient in using one, I can do some things that take a long time to do by hand. Some students have NO idea how to use one at all. I'd rather be those students, but still... the use of the calculator should probably still be taught.

My suggestion would be all math courses (in high school) should be taught without a calculator. They could reserve time at the end to use a calculator to check results, or something along those lines. I know there are more politics here, such as student failure rate, etc... but seriously, I wouldn't mind punching whoever thought this was a good idea in the eye socket. Not to mention all those mathematicians who had their work referenced as as (1) instead of Euler's identity (for example).

 

[arildno]

Long division is an excellent example of how we ought to sub-group a given number in an intelligent manner so that the result of a particular arithmetical process can be rephrased as "simply" as possible.

For example, if we are to perform the division 23456:12 (i.e, get that number's decimal representation), to develop the understanding that we first might divide the "thousands" by twelve, then the "hundreds" by 12 and so on, then this, indeed, is very relevant for understanding arithmetics as such.
But this is, essentially, what "long division" is about..

 

[twisting_edge]

My initial thought is that it's a reasonable complaint. These are young students in elementary schools, they have no inkling of algebraic manipulation and presumably have no clue how these "algorithms" work. They must take them for granted, memorizing a procedure without any understanding of what it means. I'd think that blind faith and memorization are not appropriate introductions to math. Why can't long division be held off until these kids are doing algebra, and can actually prove and comprehend what they're doing?

The alternative is for them to never have any sort of skill they can see how to apply during the years between. How often do you actually need to divide 27 remaining gumdrops between three kids? Probably not very often. When you do, you just count them out. But there are plenty of cases where you could use math, if only you knew what it was. Even if you do not use the math to actually perform the calculation, just thinking that you could if you were willing to put in the effort leads to a much more positive attitude when those rare occaisions arise that being able to do so is a real benefit, and becomes worth the effort.

Then there is the parallels to language. No one would try to argue that younger children learn different languages and thought structures more slowly than adults.

Then there is the issue of putting the cart before the horse. What is algebra? How does algebra make any sense at all if you have no uderstanding of the underlying operations? Why doesn't addition associate since multiplcation does? If you actually knew what addition and multiplication were, and had practice using them, you'd see that right off the bat. I know I did. I didn't have to think about what the meanings of additional and multiplication were, since I simply knew, having had a great deal of practice using them by the time anyone got around to telling me about the associative rule.

It's a lot like entry level calculus. Simply being able to solve the problem with sufficient thought isn't adequate. Being able to solve it almost without thinking is vital if you want to move beyond it. Yes, everyone knows you can derive the division rule from the multiplcation rule. But, as one of my professors said, "I don't care if you've taken five qualudes and passed out in a puddle of beer. If I roll you over and shout, 'Division!', I want you to tell me this," as he pointed at the board.

He was right, and for two reasons. When you get 100 questions on a test, you aren't going to have time to run through the derivation. But, far more importantly, you're going to miss some solutions completely for the simple reason you did not recognize a pattern you ought to have been intimately familiar with.

 

[physics girl phd]

"it stiffles their creativity"

Wow -- I thought math gave my creavitity a new set of tools.

Also -- I am even still known to do a bit of long division from time to time. Sometimes my calculator (or my dad's slide rule!) are lost in the lab, and I don't want to turn on the computer for a simple calculation. Sometimes I do it in my head to figure out best price/quantity (not always listed) at the grocery store etc... although I probably now more commonly use scientific notation to make the division easier, not all answers are whole numbers...

 

[arildno]

The main fallacies in modern pedagogical approaches to maths are the following:

1. Kids can't understand logic.
This is utterly false, but is, if not explicitly stated, a main tenet in modern pedagogics (mainly because it is the pedagogical gurus themselves who don't understand logic).
To teach in accordance to the principle that kids actually DO have a sense of logic means that you believe logical rigour, to SOME extent, is actually beneficial to the development of the kids' understanding of maths, rather than detrimental.

For example, demanding of kids to actually define their terms, which quantities are "known" and which are "unknown" and so on, and REQUIRE that they actually write this down as a preliminary list before they start the actual problem solving is an example of logical rigour that would immensely benefit many kids in structuring their thought processes in a constructive manner.
This, however, in modern pedagogics, would be termed an "abstract" requirement; the important thing is that the kids get their answers right, right?
WRONG!

2. Maths needs to be practically applicable in the kid's day-to-day life.

The simple, unavoidable fact is that the maths you learn beyond the calculation of percentages will only marginally affect the "ordinary person's" life. (And even the maths before that can usually be neglected without adverse effects.

Thus, you get silly, utterly fake applicability problems filling up the maths books, rather than problems designed primarily to develop mathematical SKILL and mathematical UNDERSTANDING, irrespective of its "practical" value.

 

[Gokul43201]

Lets face it:

When was the last time you added/subtracted numbers by hand? (for me: this morning)
When was the last time you used algebra? (for me: this morning)
When was the last time you used calculus? (for me: this morning)
When was the last time you did long division? (for me - its been years)

With a little rewording...

How often do you add subtract numbers, by hand or in your head? many times a day (almost entirely in my head)
How often do you use algebra, on paper or in your head? many times a day, in my head; much less on paper
How often do you use calculus, on paper or in your head? maybe a few times a day, on average
How often do you do long division, on paper or in your head? almost never on paper, but in my head, many times a day

The results will probably vary from person to person...so I don't think there's much of to take away from this.

In my opinion, the seeing of a dichotomy between learning basic skills and learning concepts is itself the problem.

 

[turbo]

In my opinion, the seeing of a dichotomy between learning basic skills and learning concepts is itself the problem.

True. As soon as we learned long division in our tiny grade school, the teacher gave us some problems with the divisor or dividend missing. She was a pretty sharp old bird because if a kid did not understand the relationship between multiplication and division they would struggle with the problems and she hadn't gotten the concepts across well enough. She not only taught the operations, she tied the new concepts back to things we already knew and demanded that we understand the relationships. Doesn't this happen in schools anymore?

 

[Quaoar]

I've found long division to be very useful when certain unscrupulous professors ban the use of calculators on tests that require extensive numerical manipulation...

Besides, is long division really THAT hard? Took me a day or two to "get" it.

 

[BobG]

... Sometimes my calculator (or my dad's slide rule!) are lost in the lab ...

If it were your slide rule, you'd be in contention for second sexiest female PF member. Oh, well, I guess you could always take out his knee with a billy club and steal it.

I bought a nice calculator recently-but the manual is so thick that I think it will take me the summer just learning how to use it properly. I do not use 99 percent of the functions, and now I wonder why I bought it at all. But I still would like to learn how to make good use of it.

It's also embarrassing if you're given a problem that uses a function you don't know how to use. Then there's the poor guy that bought a new calculator for around $200 and had to give up and go buy a simpler calculator for around $150. I'm always amazed at how many students don't realize a lot of the constants they need are already in their calculator and that they can store any additional constants they need (Even slide rules have built in constants, and, technically, you can store any additional constants you need with a trusty pocket knife, but - aauuugh! - it might be better to just remember them.)

 

[Bystander]

(snip) Why can't long division be held off until these kids are doing algebra, and can actually prove and comprehend what they're doing?(snip)

"Algebra?" Don't say "algebra" out loud in public --- take the Pharmaceutical Technician Training approach --- teach algorithms for mixing two different stock concentrations to prepare a prescribed concentration; call it algebra, and the tech preparing your prescription is going to panic --- call it "Pharm Tech," and they'll learn it, not understand it, but prepare your medication "correctly" more often than if they had to do the "algebra."

"New math?" "Interactive math?" Garbage. The damage was done fifty years ago. "Reversible?" Gonna take a lotta work.

 

[Mk]

if only they knew something about the mathematical enterprise – what it means to prove something, why certain approaches tend to be effective in solving certain kinds of problems, what it means to do mathematics instead of just pushing symbols around on a piece of paper."

When wa the last time you added/subtracted numbers by hand? (for me: this morning)
When was the last time you used algebra? (for me: this morning)
When was the last time you used calculus? (for me: this morning)
When was the last time you did long division? (for me - its been years)

Right. Then again when I need long division I usually have a calcultor.

Integrated math, is all the typical math programs integrated into one. I never took an algebra class, a trig course, or calculus. In fact I had no idea what I was doing, we were just shown new ideas and expected to use them to solve problems. Our in class assignments were done in groups. We were required to keep a journal that would count as 10% of our grade, and was encouraged to use on exams.

The journal was a place for us to write down "math notes". We were told that we would never have to memorize all these formulas in the workplace, so why learn that way? We worked in groups, and this mainly consisted of the "smartest" (whoever understood integrated math) doing all of the work.

Sounds like my kind of class if it was taught well and with very good students (unlikely)! Just smash everything together. I bet you can see connections much better and can learn more.

 

[turbo]

Even slide rules have built in constants, and, technically, you can store any additional constants you need with a trusty pocket knife, but - aauuugh! - it might be better to just remember them.

Surely you have an old Dietzgen Polymath kicking around that you could scribe some constants on!

 

[Moonbear]

When was the last time you added/subtracted numbers by hand? (for me: this morning)

Sometime in the wee hours of this morning.

When was the last time you used algebra? (for me: this morning)

Probably a week or so ago.

When was the last time you used calculus? (for me: this morning)

I couldn't tell you, unless you want to count the "trapezoid method" for getting area under a curve on a graph.

When was the last time you did long division? (for me - its been years)

This morning. It takes me longer to open the calculator on the computer, or find the handheld calculator buried on my desk somewhere (and I'm still fighting with a budget form that if I minimize the window, it freezes up). Been multiplying by hand too. "Too many sheep, times too much money per day to house them, times...oh geez, how long is this experiment going to take?!"

On the other hand, to be perfectly honest, I can't figure out what kids would be doing in math class were they not learning things like manual arithmetic. It is tough looking back from here, but it doesn't seem like there is much to math before you get to algebra...

Fractions. I still don't know why some kids have so much trouble with them. I guess I could relate better to slices of pie and cake than other kids did.

 

[twisting_edge]

When was the last time you added/subtracted numbers by hand? (for me: this morning)
When was the last time you used algebra? (for me: this morning)
When was the last time you used calculus? (for me: this morning)
When was the last time you did long division? (for me - its been years)

The correct answer to each of those problems, and this probably applies to almost everyone here, is "dozens of times every day."

The trick is that most of the time you don't even notice you are doing it. "Do I really want to buy a new calculator now, instead of waiting for my old one (missing for a week or two) to reappear?" Most people wouldn't bother to work out their monthly budget for a purchase likely to be less than $100 or so (we're talking a basic fourbanger with some extras). You know approximately what your expenses are, approximately how often you'll get paid, and about how much money you can spare. You don't need to lay out the budget because you have a good feel for the numbers, and a little error one way or the other isn't a big issue.

There are people for whom that simply isn't true. They are hopelessly incapable of realizing how the numbers add up without actually sitting down and adding them up all the time. I bet there's a far higher ratio of such people from school systems based on this sort of math than there are from those that practice actually doing arithmetic.

 

[russ_watters]

With a little rewording...

How often do you add subtract numbers, by hand or in your head? many times a day (almost entirely in my head)
How often do you use algebra, on paper or in your head? many times a day, in my head; much less on paper
How often do you use calculus, on paper or in your head? maybe a few times a day, on average
How often do you do long division, on paper or in your head? almost never on paper, but in my head, many times a day

The results will probably vary from person to person...so I don't think there's much of to take away from this.

Does the thought process by which you divide things in your head really count as long-division?

Ie, what is 27 into 1479?

My thought process rounds 27 to 30 and 1479 to 1500, calls the answer 50, then adds a few because I knew I was low due to rounding -- answer: 52. Can you actually come up with the exact answer (to a few decimal places)? Doing long-division requires holding partial solutions in your head, subtracting remainders, etc. I can't/don't do that - I have trouble holding on to a 7-digit phone number if I the area code and exchange aren't ones I am familiar with. Besides, while adding two two-digit numbers takes about 2 seconds, long division requires considerably more with all the steps. I can't see how it could possibly be done in someone's head faster than using a calculator.

That's different from addition, in which I can certainly add two two (or more) digit numbers in my head and get the exact answer, in just a couple of seconds.

 

[tehfrr]

I use long division perhaps once a month. That seems frequent enough to make it worthwhile. I think it is terrible that someone could even get to high school without knowing long division. *two thumbs down*

 

[moose]

When was the last time you added/subtracted numbers by hand? (for me: this morning)

If by hand you mean actually writing something down, then Monday while taking the average score of 20 tests. I didn't have my calculator with me.

When was the last time you used algebra? (for me: this morning)

About two hours ago

When was the last time you used calculus? (for me: this morning)

About two hours ago

When was the last time you did long division? (for me - its been years)

Well, when taking the average of those tests, I was dividing by 20, so nothing complicated there. Umm, I usually use an approximation to a decimal point or two using various methods... I haven't actually done long division for years.A surprising amount of people don't go through the thought process Russ mentioned.

My thought process rounds 27 to 30 and 1479 to 1500, calls the answer 50, then adds a few because I knew I was low due to rounding -- answer: 52.

It makes things so much easier on yourself.

 

[Hurkyl]

My thought process rounds 27 to 30 and 1479 to 1500, calls the answer 50, then adds a few because I knew I was low due to rounding -- answer: 52.

Once you've figured it's about 52, and want more detail, is it really that hard to multiply 27 by 50 (1350) and subtract from 1479 (129), at which point you know it's more like 54.5? :tongue:

 

[mathwonk]

one of the professors quoted in these articles, sybilla beckmann of uga, is a friend and colleague of mine, and she was largely misquoted.

the new emphasis is not a "back to basics" in the sense purely of memorization of rote stuff. It involves a much deeper approach to basics, like understanding the basic ideas.

she has devoted about 10 years of her life to this effort, and I have volunteered once to teach these new courses she has developed for new teachers in training, emphasizing thinking and problem solving.

her books for elementary teachers are excellent and are written by a research mathematician in arithmetic geometry, who cares about education. unfortunaTELY IT IS VERY HaRD TO GET A COHERENT OR ACCURATE JOURNaLISTIC ACCOUNT OF WHAT IS HApPENING. THEY JUST LIKE CONTROVERSY.

 

[mathwonk]

for example the focus on decimals is in understanding positional notation, not just memorizing how to carry and borrow, but one what that means, and why it works.

i taught my class for example why .99999...= 1.000000, a topic of frequent interest on this forum, and one student wrote me of her struggles to get her superiors in her new job to believe her, when she explained this.

 

[0rthodontist]

I have a notion that all learning is rote at first. It works in Japan, why not here? If the first, or at least a crucial, step to becoming an expert at a subject is developing a specialized memory for the subject, as appears to be true, why not go at it directly?

Students should be forced to solve calculational problems in their heads without paper. If they can remember the symbols, that is the first step to becoming comfortable with the symbols, which is the first step to understanding the symbols. Teach long division but force students to do it mentally.

 

[mathwonk]

i aGREE. as in science:first you observe, then you reflect. without rote learning of basic math facts there is nothing to reflect upon.

that is why we cannot teach homomorphism laws in abstratc algebra to people who do not know the common rules for exponents from precalculus.

on the other hand, rote test prep books like saxon's math books have proven very harmful in their almost idiotic aPPROACH TO TEACHING Math as a list of rules.

also, books entitled " quantum cohomology and toric decompositions of orbifolds, for idiots" DRIVE ME ABSOLUTELY UP THE WALL. ONLY IN AMERICA WOULD A BUYER embrace the idea of being caled an idiot by his teacher.

 

[Moonbear]

We also have to remember that the vast majority of high school students are not going to go on to advanced math courses. They don't need to learn calculus or how to do proofs; they need to know how to balance their checkbook and figure out how many cans of paint they need to paint the house.

 

[physics girl phd]

If it were your slide rule...

Well, I inherited it. Had to fight Mom for it.

But everyone in the sisterhood rules. Smart women are ALL sexy.

 

[Bystander]

The "Dummies" and "Idiots" series don't assume a "complete" background of the reader --- they review/summarize the necessary background (when properly done --- the half dozen or so I've skimmed), and tie a subject into an integrated picture for the reader rather than treating it in isolation.

 

[Evo]

In my job, if I didn't have a calculator, I'd have to do long division several times a day. Luckily I do have a calculator, but knowing that if I didn't I could actually do the long division is a nice thing. Sometimes if it isn't that long, I'll do it on paper. The calculator saves me time. How can anyone think that knowing how to do long division can be a bad thing?

I have to create legal documents that have to be 100% accurate. I cannot guess at the numbers I quote. A wrong answer can result in a lawsuit. No one checks my numbers for accuracy when it goes to legal. Now there's a scary thought.

 

[whitay]

We also have to remember that the vast majority of high school students are not going to go on to advanced math courses. They don't need to learn calculus or how to do proofs; they need to know how to balance their checkbook and figure out how many cans of paint they need to paint the house.

School has taught me a minimum amount of calculus. I wouldn't mind learning proofs. I can balance check books, I can do quite a variety of accountancy work as I've done a fair bit of voluntary work experience at 2 accounting firms. However I don't know how many cans of paint i'd need to paint a house, mainly because I am not into manual work.