Milgram & Klein links:

• Why Long Division? speech by James Milgram, Stanford University
  
• The Role of Long Division in the K-12 Curriculum
  by David Klein, Department of Mathematics California State University, Northridge and R. James Milgram, Department of Mathematics, Stanford University
  

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Comments

"Secondly, if the estimate is too small, the student can continue with the procedure. This greatly alleviates the frustration of having to erase, and to some extent, allows one to get around a forgotten multiplication fact."

This is a plus? It's OK not to know your multiplication facts? What if the estimate is too high? One of the purposes of multi-digit multiplication and long division is to continue to practice and master your basic facts and to be able to do things like multiply 3 * 26 in your head. This doesn't happen with Forgiving Division.

"Research has shown that low-ability students show better retention and understanding when taught division with this method and become better estimators of quotients."

Better estimators? No, I don't think so. Forgiving Division is all about not having to be a good estimator. In exchange for a slower, forgiving process, you do not need to know your multiplication facts and you do not need to be a good estimator. I wonder what "research" shows when average or above ability students use this technique?

"Students who were taught the forgiving method were better at solving unfamiliar problems and were better able to explain the meaning of the steps (van Engen and Gibb, 1956)."

Not believable.

"The forgiving division method also gives students the opportunity to practice mental math. (page 166, grade 5)"

And just what are the kids doing when they have to mentally get the greatest quotient at each step with the traditional approach?

It takes less knowledge, skill, and understanding to use the "Forgiving Division" approach. Otherwise, they would call it something like the "Super Deluxe Division" approach.

Of course, all of this says nothing about how much practice the student has to do. And, it changes the focus away from lower expectations and no full algebra by 8th grade.

-- SteveH - 15 Nov 2005

"The forgiving division method also gives students the opportunity to practice mental math. (page 166, grade 5)"

And just what are the kids doing when they have to mentally get the greatest quotient at each step with the traditional approach?

I liked that part, too.

-- CatherineJohnson - 15 Nov 2005

One of the basic themes of constructivism is that "facts are not gatekeepers."

As far as I can tell, lots of people think this is important.

My neighbor, who has tutored kids for many years, was one of the first people I heard say this.

"Facts should not be gatekeepers."

-- CatherineJohnson - 15 Nov 2005

This a good summary of the NSF curricula philosophy (from the same document):

Our approach is based on educational research and is supported by the National Council of Teachers of Mathematics’ Principles and Standards for School Mathematics (2000). It is characterized by the following:

• Emphasis on problem solving. Students will learn the basic facts if they are encouraged to use a problem-solving approach. Students can invent their own strategies, learn from their peers, or learn from the teacher through class discussions. Students will discover the need to learn the facts as they encounter them in labs, activities, and games.

• De-emphasis of rote work. Students learn their math facts, but we de-emphasize the use of rote memorization and in first grade we do not administer timed tests. These can produce an undesirable result—students perceive that doing math is memorizing facts and rules which “you either get or you don’t.” Instead, students should feel confident that they can find answers through the use of strategies they understand.

Throughout first grade the focus remains on the use of strategies that are meaningful to students. Beginning in Unit 11, a systematic organization of the math facts is introduced via the Daily Practice and Problems. Facts are grouped together in ways that may help students think of them in a more efficient manner. However, students are still free to solve the problems using whatever strategies they wish. By the end of second grade, students in Math Trailblazers are expected to demonstrate fluency with the addition and subtraction facts. The first grade curriculum enables them to build to this fluency through experience with and understanding of the concepts of addition and subtraction.

• Facts will not act as gatekeepers. Students are not prevented from learning more complex mathematics based on their fluency with the math facts.

A more detailed introduction to our approaches for learning math facts is in the TIMS Tutor: Math Facts found in the Teacher Implementation Guide. This tutor discusses different strategies students often use while doing addition and subtraction problems. Additional background on problem solving can be found in the TIMS Tutor: Word Problems, also found in the Teacher Implementation Guide.

-- CatherineJohnson - 15 Nov 2005

A couple of interesting points:

Catherine asked the question: can you do decimal division using forgiving division? BTW, this is the same method that EM teaches.

The answer: no unless you cobble on the option of bringing down the next digit. Even so, the only way to get an exact answer is if you have a terminating decimal. If you have a repeating decimal, you will always have a remainder in forgiving division.

BTW, this is not answered anywhere in the EM curriculum at all. As a matter of fact, forgiving division is even repeated in the 6th grade curriculum. All over EM, even though they don't believe in rote memorization, students are required to memorize the fraction/decimal pair because they have never been taught to convert one to another.

If you read David Klein's paper in defense of long division (found on his website or NYC HOLD) you will find that long division and the ability to find out if a problem gives a terminating decimal, repeating decimal or non repeating decimal is central to understanding rational and irrational numbers in middle school.

I just don't understand the claim that long division is too hard to learn. Even educators who claim that math was hard for them will never claim that they couldn't learn long division and can't do it. Does anyone actually know a student who could not learn long division? My special needs son has no problem doing long division problems.

As for facts will not act as gatekeeper: I just finished The Myth of Ability by John Mighton. I read through his explantion of teaching fractions to children who do not know their times tables. I think the motto should be: facts don't necessarily act as gatekeeper, but DI does. He is successful because first he teaches the students the times tables they need by counting on their fingers (2, 3 and 5 times tables). With these facts established, he teaches addition and subtraction of fractions with the same and different denominators. All of his instruction is direct instruction in small steps. He makes sure that every student can do every step before he moves on.

-- AnneDwyer - 15 Nov 2005

Mabye it's a great program as long as it's supplemented with kumon or singapore curriculum at home? :-)

-- AlexStrifflerHernandez - 15 Nov 2005

no unless you cobble on the option of bringing down the next digit. Even so, the only way to get an exact answer is if you have a terminating decimal. If you have a repeating decimal, you will always have a remainder in forgiving division

So, to do decimal division, you would have to revert back to classic long division.

You'd have to bring the 0 down.

Yes?

-- CatherineJohnson - 15 Nov 2005

All over EM, even though they don't believe in rote memorization, students are required to memorize the fraction/decimal pair because they have never been taught to convert one to another.

They actually have the kids memorize it?

-- CatherineJohnson - 15 Nov 2005

TRAILBLAZERS stops at grade 5 (goes through grade 5, I should say).

So they can pretty much just transfer responsibility over to middle school.

I've read the first half of David's paper, but have put the second half on hold for a bit.

I think at this point I can follow it, but the effort to do so is so huge that it's a better use of my time to keep re-learning math & come back to Klein & Milgrim's paper when it won't be such a struggle.

That's another thing.

I am a math student.

Period.

Every day, I work on math.

What's more, I'm working on math just slightly above the level of TRAILBLAZERS & EVERYDAY MATH; I'm working through a pre-algebra text (Saxon 8/7).

As Steve said earlier, there are plenty of math concepts I have to 'discover' for myself, simply because no matter how clearly they're explained by a text or by people here, I still don't get them.

So I have to work them through, and find the 'hook' or the mode of explanation (either someone else's or my own) that finally gets it across.

If I had to discover everything through a series of, to my eye, haphazardly designed hands-on activities, I'd never make it out of pre-algebra.

-- CatherineJohnson - 15 Nov 2005

Oh!

That reminds me.

Remember when Carolyn said that knowing something can be done is half the battle?

This happens to me fairly often.

Yesterday Paul Miller left an explanation of the 'string orchestra' problem on Christopher's test.

I didn't follow his explanation (although it's possible I would have if I hadn't been quite tired by then. We are NOT getting enough sleep around here; not even close).

However, reading through Paul's explanation I saw the 'key,' and I knew the problem was possible to answer.

After that I was able to do it myself very quickly, even in my dead-on-my-feet state.

-- CatherineJohnson - 15 Nov 2005

'Discovery' happens in many, many ways; it's far more complex than a simple inductive process.

-- CatherineJohnson - 15 Nov 2005

I've always wondered - how do computers do long division?

Both long division algorithms there require looking at the two numbers and drawing from memory what multiplication factor would be a good start. (4 in the first method, 20 in the second). But how does a computer do that?

-- TracyW - 15 Nov 2005

All this talk of 'passive' students.....it's not like math is a breeze when people just tell it to you—

-- CatherineJohnson - 15 Nov 2005

Hey, you guys!

Tell me some math!

I want to hear some math!

-- CatherineJohnson - 15 Nov 2005

"Reality is that which, when you stop believing in it, doesn't go away." Philip K. Dick, Do Androids Dream of Electric Sheep?

"Facts should not be gatekeepers."

One of these makes sense, the other doesn't. Let's just say that I don't think it's facts that are barring the gates of knowledge. Teaching math without "facts" makes as much sense as teaching sociology without history.

At some point, to be useful, the theory has to match reality. It's awfully hard to tell when that happens if you don't actually know what is real.

-- DougSundseth - 15 Nov 2005

Reality is that which, when you stop believing in it, doesn't go away.

Oh, that's wonderful.

-- CatherineJohnson - 15 Nov 2005

Let's just say that I don't think it's facts that are barring the gates of knowledge.

Ditto.

-- CatherineJohnson - 15 Nov 2005

Philip Dick was a brilliant writer (and by the reports I've read, a pretty broken human being), and is currently very popular in Hollywood as a source of material to butcher. For example, Blade Runner was taken from Do Androids Dream of Electric Sheep.

-- DougSundseth - 15 Nov 2005

I think there's even less here than meets the eye. I believe the other name for Forgiving Division is Division by Partial Products. I think it might have been Instructivist who linked us to a particular school's web site demonstrating this method. Before attempting the actual division, the students for a list of partial products. First, they list 1x the divisor; then, they add it to itself to get 2x the divisor; then, they add that to itself to get 4x the divisor. Then they append a 0 to the divisor to get 10x the divisor; then, they add that to itself to get 20x the divisor.

So, there isn't any real multiplication in Forgiving Division at all. I don't think math should be about avoiding learning how to do things.

-- DanK - 15 Nov 2005

Yeah, I've read....MEMORY LOSS.....

What's his famous book?

The alternative history one?

sigh

-- CatherineJohnson - 15 Nov 2005

Dan

Yes, partial products.

-- CatherineJohnson - 15 Nov 2005

oh my gosh—I have to find that post

-- CatherineJohnson - 15 Nov 2005

The way computers do division varies depending how the numbers are represented. Integers get dealt with differently depending on whether they're considered "signed integers" or "unsigned integers". Signed and unsigned integers are essentially the same, except that signed integers reserve the first bit of the number to indicate the sign.

Ignoring these details, though, for integers, one way to do it is essentially... forgiving division! (No, I'm not kidding. Computers have to do this, though, because they are dumb. Kids don't because they are smart.) I'll do a specific example.

Say we want to divide 1739 by 43.

Guess: 2. 2*43 = too small.

Guess: 4. 4*43 = too small.

Guess: 8. 8*43 = too small.

Guess: 16. 16*43 = too small.

Guess: 32. 32 * 43 = too small.

Guess: 64. 64 * 43 = too big!

Now, what we do is essentially split the difference between 64 and 32, by calculating (64 + 32)/2 = 48. (What's that divide by 2 there? Cheating you say? Nope. Dividing by 2 in binary is as easy as dividing by 10 in decimal. All modern computers implement this. One thing to remember, though, is if you're dealing with integers, you have to drop off everything after the decimal point. Or, should I say, the binary point?)

Ok, next guess

Guess: 48. 48 * 43 = too big

Guess: 40. 40 * 43 = too small.

Guess: 44. 44 * 43 = too big.

Guess: 42. 42 * 43 = too big.

Guess: 41. 41 * 43 = too big.

At this point, the computer declares that the answer is 40. Why? It's the largest number that when multiplied by 43 leaves a positive remainder. That's essentially the definition of division.

Notice that we found the answer after 11 steps. 1739 requires 11 bits to represent in binary. This is no coincidence, since what the computer was doing is essentially a binary search for the answer.

There are faster (for the computer) ways to do it, but they require more explaination.

Edit: corrected answer. Should be 40, not 41.

-- PaulMiller - 16 Nov 2005

Interesting—

-- CatherineJohnson - 16 Nov 2005

"Facts should not be gatekeepers."

Apparently there are no gatekeepers for Ed Schools. The gates are wide open.

How about: "A math class should be a funnel and not a filter." I read something like that one time. They referred to traditional math as a filter.

How about the philosophy that says that kids shouldn't be held back a grade because research shows that these kids have a higher probability of dropping out.

-- SteveH - 16 Nov 2005

From: Algorithms and Everyday Mathematics by Andrew C. Isaacs

http://everydaymath.uchicago.edu/educators/Algorithms_final.pdf

"Today, being able to mimic a $5 calculator is not enough: Employers want workers who can think mathematically. How the school mathematics curriculum should adapt to this new reality is an open question, but it is clear that proficiency at complicated paper-and-pencil computations is far less important outside of school today than in the past. It is also clear that the time saved by reducing attention to such computations in school can be put to better use on such topics as problem solving, estimation, mental arithmetic, geometry, and data analysis (NCTM, 1989)."

Thinking mathematically is a new business reality? Apparently, the traditional approach didn't do the job. If you read this paper, you will see that much of the justification revolves around very low expectations of kids. Instead of trying to find ways to bring kids up to a higher level, they use them as justifications for lowering expectations.

... ...

"Reducing the emphasis on complicated paper-and-pencil computations does not mean that paper-and-pencil arithmetic should be eliminated from the school curriculum. Paper-and-pencil skills are practical in certain situations, are not necessarily hard to acquire, and are widely expected as an outcome of elementary education. If taught properly, with understanding but without demands for "mastery" by all students by some fixed time, paper-and-pencil algorithms can reinforce students’ understanding of our number system and of the operations themselves. Exploring algorithms can also build estimation and mental arithmetic skills and help students see mathematics as a meaningful and creative subject."

"... are not necessarily hard to acquire, and are widely expected as an outcome of elementary education."

Not hard to acquire, but apparently it saves so much time for ...

"... problem solving, estimation, mental arithmetic, geometry, and data analysis..."

You know, those things they don't get with a traditional approach.

Let's see:

1. estimation - see long division

2. mental arithmetic - see long division

3. geometry - "I am a polygon. I have two right angles. Only two sides are parallel. What am I? (4th grade, EM)

4. data analysis - How about 90 percent data collection and 10 percent data analysis.

One of the main justifications is that calculators save time which can be used for all sorts of other good things. Hogwash!

They wanted to do what they wanted to do. This paper is just an after-the-fact rationalization.

-- SteveH - 16 Nov 2005

"Both long division algorithms there require looking at the two numbers and drawing from memory what multiplication factor would be a good start. (4 in the first method, 20 in the second). But how does a computer do that?" -- TracyW^?

If you work in base 2 you only have two possibilities for the factor. It is 0 or it is 1. You can determine which one to pick by simply checking whether the divisor is larger than the portion of the dividend you are considering. There are no tables to remember here. This method is very slow.

Computers can use the division algorithm in larger bases, but then they need some sophisticated method of searching for the factor. You can perform this search by guessing what the factor is and applying some iterative algorithm that converges to your answer. This can be done quite fast

There are other methods that don't use the division algorithm at all. PaulMiller^? showed one way to do this above. Another common method is to find the reciprocal of the dividend and multiply by the divisor. This is an attractive method because we have nice fast methods of computing reciprocals and products on the computer.

-- KtmGuest - 16 Nov 2005

If taught properly, with understanding but without demands for "mastery" by all students by some fixed time, paper-and-pencil algorithms can reinforce students’ understanding of our number system and of the operations themselves.

I love the quotation marks around the word mastery.

-- CatherineJohnson - 16 Nov 2005

I've started reading Engelmann, and he has numerous rants about constructivism removing all pressure from educators to produce results.

-- CatherineJohnson - 16 Nov 2005

Thanks for the descriptions on computer division. PaulMiller's is "guess and check" done right!

(And I am trying to remember the formal name for that procedure - know I did it at school).

-- TracyW - 17 Nov 2005

"PaulMiller's is "guess and check" done right! (And I am trying to remember the formal name for that procedure - know I did it at school)."

There is a lot of guessing and checking techniques in engineering computer solutions, except that they are called search techniques. These have absolutely nothing to do with the no-previous-knowledge, semi-random or brute-force, guess and check used in K-8 math.

For formal search techniques (computer floating point processor division is another beast), there are generally two steps:

1. Find a bracket of the solution. 2. Find the solution

Paul's first stage reminds me of the Davis-Swann-Campy technique that searches for a bracket by doubling the step size during the search. The guesses go 2, 4, 8, 16, 32, and so forth until you have gone past the solution point. If the problem is set up to find the zero or root of an equation, then you stop when the answer changes sign. If you have a non-linear equation where you are trying to find a (local) minimum, then you look for when the result changes direction. The key goal is to find a bracket of the answer. (For non-linear equations, this doubling of step sizes can jump right over a solution when the step size get large, which it does very quickly.)

The simplest approach (binary search) to the second phase (find the solution) is to take that bracket and test the midpoint of the bracket. The midpoint is the answer or it allows you to throw away one half of the bracket. Repeat this process with the remaining bracket until the bracket gets small enough to meet some accuracy tolerances.

Bracketing and binary searching is something that can be easily taught and understood. The classic trick is to tell someone to pick a number between 1 and 64, but don't tell you. You will tell him/her what number it is by asking only 5 questions. For example, is your number equal to, less than, or greater than 32? You, of course, use the binary search technique.

If you know something about the equation that you are searching, then you might be able to speed up the search. If the equation is linear, then two points define the equation and you can solve for the answer. If it is non-linear, then you can assume that its shape is quadratic locally and use three points to estimate a solution that can be refined repeatedly.

A Newton search technique is fast, but assumes that you have a starting point "close" to the answer. (That can be a problem.) As long as you are near the solution, you don't need a bracket. Some versions of the Newton search technique require the analytic derivative of the equation, while others (Secant method) determine the derivative numerically. For many of the problems I do, finding a bracket or finding a point near the solution is harder than finding the answer. If the initial guess is not smart or close enough, then the search can wander off. If, however, you have a true bracket of a solution, then the binary search technique will guarantee a solution.

Do they teach any kind of search methodology to kids in K-8? No. It's just brute-force guess and check. If a student stumbles upon some semi-methodical search process, this is great, but don't expect the teachers to codify or formalize the process for the rest of the class. And, of course, forget about practice.

-- SteveH - 18 Nov 2005

Actually, for the particular case of division, Newton's works just fine from any starting point, because the function 1/x is convex.

-- PaulMiller - 18 Nov 2005

Actually, I should say, convex on the open interval from 0 to infinity.

-- PaulMiller - 18 Nov 2005