**Division problems:**

I. A 4th grade TERC/Investigations sheet, assigned in early November [click to enlarge]:

II. From a similar point in the 4th grade Singapore Math curriculum [click to enlarge]:

III. Extra Credit:

a. Why do American math students need more clues than their Singaporean counterparts do?

b. Is 4th grade too early for the long division algorithm?

## 19 comments:

In answer to is fourth grade too early, the Primary Mathematics teaches the division algorithm with 3-digit numbers in 3rd grade. 4th grade is just a review and extending to 4-digit numbers. There is no reason not to teach it in 3rd grade. When it is first taught, the lesson in the textbook uses pictures of place-value discs. You are supposed to teach it concretely, and make groups out of the hundreds, rename left-over hundreds as tens, and so on. Do it with base-10 material until they understand the process. Then show how the process relates to the written record. So by the time they get to doing just on paper, they no longer need all the steps spelled out. But this is in third grade. In fourth grade you might review the process briefly with base-10 material. Guide the students in doing it with base-10 material. But because they have learned the process with 3-digit numbers already, there is not much time spent on it in 4th grade. There is more time spent on learning the steps in the textbook in 3rd grade. Though even the workbook at that level does not spell out the steps over and over again. US workbooks seem to expect students not to know the steps by the time they do the workbook. That they have not been taught well. Or need reminding over and over again.

It's always troubling to see how much less our students are expected to learn compared to their peers in other countries.

These assignments are completely different concepts. They are both useful mental math strategies but the first is a more complicated procedure. It is also far more sophisticated. It is basically teaching the Distributive Property of Multiplication of Multiplication over Addition. A useful pre-algebra skill.

The second is compensation, a mental math strategy for subtraction. The sheet is also showing as many "hints" but using only symbols, no words. The Investigations is more left-brain.

IMO, third grade is too soon to teach long division. Having taught it in fourth grade many years ago (at least before the constructivist movement), it is appropriately challenging in fourth.

In terms of traditional algorithms, the question here is why does the Singapore not use regrouping?

"the first is a more complicated procedure"

More complicated than the long division algorithm?

"It is basically teaching the Distributive Property of Multiplication of Multiplication over Addition"

The distributive law is also embedded in the long division algorithm.

"The sheet is also showing as many "hints" but using only symbols, no words."

There are no hints here that show how to solve the long division problems.

"The Investigations is more left-brain."

?

"the question here is why does the Singapore not use regrouping?"

What makes you think it doesn't?

How else to teach division but with regrouping. Singapore does that in grade 3. And grade 4. It does it with objects, as I previously posted. Then pictures. Then abstract. Not words. By the time they get to the page posted, they know all about regrouping. 4th grade is too early only if you just teach it with words, rather than using base-10 material. My son certainly had no trouble learning it in 3rd grade, using the Primary Mathematics, which is more than just textbook page (you have to teach it with objects first, that is not in the textbook, it is in the guides) though he used base-10 blocks to do the regrouping for quite a while. He learned it thoroughly that way, and never forgot the concept. So by the time he got to dividing by 2 digits, it was a breeze.

Long division does NOT teach the Distributive Property in an recognizable form. If you think about the important concepts of math, long division is not one of them.

The Singapore does not teach regrouping in THIS example. It uses compensation.

Are you kidding me about the hints to solve the long division problem? The only reason the strategy works is because the divisor fits so neatly into the dividend. The exercise reinforces math facts and flexibility. It also teaches students how math works.

If you pick two random pages from two programs to hold up as examples, the same qualifications for judgement should be applied to both.

You bring up one, long division. I point out the two pages are not even comparable and you decide to focus on the semantics of my statements. When I asked why does the Singapore not use the traditional algorithm, I MEANT on this page? Compensation is not a big time saver but Distribution has very important applications for algebra. a(b+c)=ab+ac

So, you are saying long division is a useful skill? The first six years of math versus the rest of your math education. Arithmetic is merely preparation for algebraic thinking. Long division is pointless.

Is your blog just a venue to complain or are is your mind open to consider alternatives? If you just want to complain and you are set in your way, that's fine with me and I won't bother reading it anymore.

"Long division does NOT teach the Distributive Property in an recognizable form."

Yes, but one can teach why the long division algorithm works by showing how it uses the distributive law.

"The Singapore does not teach regrouping in THIS example. It uses compensation."

In this example, nothing is taught explicitly. What students get out of this problem set is (1) practice with estimation (2) practice with long division and (3) a window into how estimation can help them see if their long division answers make sense.

My problems of the week don't generally compare teaching methods; they compare tasks. That's why they're called "problems of the week" rather than, say, "lessons of the week."

"The only reason the strategy works is because the divisor fits so neatly into the dividend. "

? In half of these problems there are remainders. And in half of these problems, the divisor does not divide evenly into the first two digits.

"Arithmetic is merely preparation for algebraic thinking. Long division is pointless."

Is polynomial long division also pointless?

"Is your blog just a venue to complain or are is your mind open to consider alternatives?"

You should feel free to share your alternatives here, and to be open to the comments made by myself and other commenters.

I like both of these problem sets. You could combine them together in a lesson to teach kids when do use the long division algorithm and when to look at the numbers in front of you and see if you can apply mental math strategies. If so, that should be quicker. Unless you spend 3 minutes trying to break a number into two multiples of 8 that are each easier to work with. Getting kids to switch between mental math an cranking on algorithms is not trivial but it is worthwhile. Being able to do that is a sign of real mathematical competence/confidence.

The assertion that arithmetic is only a stepping stone to algebra is untrue. Arithmetic is useful in itself for many people who may never use algebra. The calculator is not the solution to their mathematical needs since people seem to need actual hands-on computational experience to understand arithmetic and develop number sense. Also, one could continue the argument by saying algebra is not important, it is just a stepping stone to calculus, which is a stepping stone to differential equations, which is a stepping stone to functional analysis. And so the end of math education is unatainable for almost everyone.

"The Singapore does not teach regrouping in THIS example. It uses compensation."

? The example is not teaching anything about compensation. It is just a practice exercise on estimation and long division. Why are you assuming the approach used in teaching the knowledge needed to do the exercise?

1) You originally asked, "Why do American math students need more clues than their Singaporean counterparts do?"

My original point was that this is not a good example to use when comparing the number of "hints" needed. I reiterate they are not practicing the same skill.

2) "Yes, but one can teach why the long division algorithm works by showing how it uses the distributive law."

In your own words, "My problems of the week don't generally compare teaching methods; they compare tasks. That's why they're called "problems of the week" rather than, say, "lessons of the week." "

3) "The only reason the strategy works is because the divisor fits so neatly into the dividend. "

When I said "the strategy" I was referring to the TERC task.

4) "? In half of these problems there are remainders. And in half of these problems, the divisor does not divide evenly into the first two digits."

I beg to differ. There are no remainders in any of these problems and the fact that the divisor does not fit into the first two digits is irrelevant.

5) Allow me to clarify how the first "task" presumes understanding of a more sophisticated skill than long division. In the first set of tasks, the hints remind students to look at the dividend as a sum of parts that are visibly compatible with the divisor. 162÷6 is composed of 120÷6 + 42÷6. It becomes obvious that 20 + 7 is the answer.

The second task asks students to estimate and then divide. Rounding 2475 to 2500 show them that 500 a good estimate. Then they practice long division to find out that the answer is 485. 485 is close to 500, so my actual answer is reasonable. This skill is not as effective, useful, nor foundational as recognizing that 2500÷5-75÷5 = 500-15.

If a student were to apply the first strategy to the second set of problems (again - the first being more sophisticated and yes, IMO left-brain using your own definition "logical, systematic, analytical and one-at-a-time" and not right-brain "intuitive and holistic,"), he or she would arrive at the correct answer without using long division.

3594÷6 is the same as 3600÷6-6÷6 or 600-1

4214÷7 is the same as 4200÷7+14÷7 or 600+2

6480÷9 is the same as 6300÷9+180÷9 or 700+20

6) In response to another reader, "The assertion that arithmetic is only a stepping stone to algebra is untrue. Arithmetic is useful in itself for many people who may never use algebra."

When students understand the laws of arithmetic, and not just how to perform computations, algebra BECOMES accessible to all.

BTW, I posted anonymously because I was not at my own computer but it was I: Aly V your long-time open-minded reader who enjoys a good debate provided my opponent is intellectually honest.

I know I have written a lot here so I will address the uselessness of long division in a different post.

Both problem sets have students dividing numbers. In one case, the steps are spelled out to them. In another, they are expected to use the long division algorithm (whose steps *are not* spelled out to them) after first estimating the quotient. One of the problems I have with Investigations is that it often forces students, including those who already *get it* right away, to do things in a highly prescriptive fashion, with clues that spell out the logic for you.

I've blogged about the many virtues of the long division algorithm earlier, and so has bky (see here: http://oilf.blogspot.com/search/label/long%20division) In particular, how do you appreciate the mechanics of how fractions convert to repeating decimals without the long division algorithm?

The recursive character of the long division algorithm has many other virtues, including an opportunity to see the distributive law in action. The more one works with the distributive law, the better one understands it. IIn terms of "left-brain" vs. "holistic," I can't think of a better example of the former.

As for *teaching* the distributive law, here's you quoting me:

" "Yes, but one can teach why the long division algorithm works by showing how it uses the distributive law."

In your own words, "My problems of the week don't generally compare teaching methods; they compare tasks. That's why they're called "problems of the week" rather than, say, "lessons of the week." " "

The two statements you quote are, indeed, nonsequitors. They are taken from different points in our discussion. They're also both true.

"When students understand the laws of arithmetic, and not just how to perform computations, algebra BECOMES accessible to all."

Without experience with numerical long division, you'll be at a total loss when trying to do polynomial long division. Do you have an alternative to polynomial long division?

Aly V, I look forward to your open-minded examination of the pro-long-division posts I linked to above, and your intellectually honest reactions to them.

Both problem sets have students dividing numbers. In one case, the steps are spelled out to them.

Please acknowledge that the steps that are spelled out for them in the first sheet are not the same steps that are not pointed out to them in the second sheet. The first does not cue, remind, hint, guide the students to use long division. It reinforces an important skill, applying the Distributive Property.

You do not address my point that they are not comparable activities.

You counter my saying Singapore does not "teach" by saying these are problems not lessons and then you say but one could teach the Distributive Property with long division. I am not taking your quotes out of context anymore than any quote is taken out of contest besides I am not misrepresenting the context you used. Mine was a logical conclusion to reach given what you said and therefore it does follow (not Non sequitur).

You say one of the problems you have with Investigations is that it "forces students to do things in a highly prescriptive fashion." What is long division if not highly prescriptive? In your post of Feb. 25, 2009 you state in reference to LD, "As bky points out, unlike the partial sums algorithm, the steps are precise and predetermined." Prescriptive means related to the enforcement of rules or methods.

You are remiss in not acknowledging your error concerning the divisibility of these numbers. At first glance, I agree it would be easy to miss. Being adept at applying the Distributive Law, I was quickly able to divide these numbers mentally.

In my original post, I clearly state that I do advocate teaching the LD algorithm. I also agreed to address the degree to which it is useless in a future post. By bringing it up again, you are changing the subject.

"Without experience with numerical long division, you'll be at a total loss when trying to do polynomial long division."

I think a total loss is an exaggeration.

Alternatives to polynomial long division? Synthetic division? The rational root theorem? Graphing and evaluating the zeroes?

Although I did say LD is useless, I meant relative to its merits and long-term applicability. I am not anti-long-division.

As far as my "open-minded examination of the pro-long-division posts linked to above," and my "intellectually honest reactions to them," I have given you no reason to speculate that I would do anything other than that.

"Please acknowledge that the steps that are spelled out for them in the first sheet are not the same steps."

Happily acknowledged. I thought that was obvious.

"It reinforces an important skill, applying the Distributive Property."

So does doing long division.

"You do not address my point that they are not comparable activities."

I thought that was obvious, too. The Singapore Math students are further along. They know the long division algorithm, and why it works, and don't need hints about how to find the quotients in these problems. Do you think Investigations 4th graders could sit down do the Singapore math problem set as is?

"one could teach the Distributive Property with long division. "

And, having accomplished this, one's students will learn more about the applicability and power of distributive law while applying the long division algorithm.

"Mine was a logical conclusion to reach given what you said and therefore it does follow (not Non sequitur)."

You'll see this yourself if you reread my previous comment, but I used the term "non sequitor" not in reference to your conclusion (I'm not even sure which conclusion you are referring to), but in reference to the second of the two statements you quoted of mine. That second statement is a nonsequitor with respect to the first atatement.

"You say one of the problems you have with Investigations is that it "forces students to do things in a highly prescriptive fashion.""

The full text of the sentence you cite here was:

"One of the problems I have with Investigations is that it often forces students, including those who already *get it* right away, to do things in a highly prescriptive fashion, with clues that spell out the logic for you."

Yes, the long division algorithm involves precise steps (it's an algorithm, after all); but in the Singapore problems there are no clues that spell these steps out for you. It is assumed that you know when and what to divide into what, when and what to multiply, when and what to subtract. And why.

"You are remiss in not acknowledging your error concerning the divisibility of these numbers."

Happy acknowledged. I've never been good at mental arithmetic. That's one thing I like about the standard algorithms, and had I actually sat down and done the long division here, I would have realized my mistake.

"I think a total loss is an exaggeration."

An empirical question that could be tested by attempting to teach polynomial long division to someone who has never done long division before.

I'm not sure it would be a good idea to use anyone as a guinea pig, however.

"Alternatives to polynomial long division? Synthetic division? The rational root theorem? Graphing and evaluating the zeroes?"

I find synthetic division far less transparent; graphing is imprecise; and how is the rational root theorem related to polynomial division?

Ali V, I'm hoping we can move beyond what has turned into a comment thread "post mortem" that I'm guessing is boring everyone else, if not ourselves, and that we can hear your objections to the arguments given by myself and others that long division isn't useless "relative to its merits and long-term applicability."

I find synthetic division far less transparent; graphing is imprecise; and how is the rational root theorem related to polynomial division?

How is graphing imprecise? What is it that you think the purpose of polynomial long division? The rational root theorem reveals the factors of a polynomial and is an alternative to polynomial long division.

Your use of Non sequitur (as it is spelled), still makes no sense to me in your previous post. Do you mean the two statements are not related? If so, you are neither using the expression correctly, nor understanding my point.

Okay, fine. I will drop it. I am done. I am in favor of teaching the standard algorithm for long division and all arithmetic operations. It was never my point to argue against it.

Your previous entries add little to this discussion. I think I have addressed long division sufficiently and will bore your readers no more with my point by point commentary. Your mind is made up and that is fine. It is your blog, after all.

As a teacher and a student, I am disappointed in your response to this left-brained thinker.

Not so boring. It is always interesting to see how these arguments denigrate into a he said/she said thing and I don't agree with your interpretation of what you said because I interpreted it different. It is fascinating. Maybe it has something to do with trying to communicate through messages. I wonder if this method of communication, not face to face, contributes to potential ill-will and frustration that arises between people of different opinions? I wonder what makes a good debate under these circumstances?

The Primary Mathematics does teach all the stuff that is in the Investigations example. Just earlier (3rd grade) and in a different manner (concrete to visual). So by fourth grade students are not at the same place in Investigations vs Primary Mathematics.

I wonder if the Investigations goes as far, if they are still guiding the students so carefully by this level?

It is important to note that students should take a pre-test before beginning any Singapore to make sure they are on the right place. The Singapore teacher's guide suggests that the most advanced third graders will place into 3rd grade.

I am glad you found this interesting but I found it incredibly frustrating, perhaps for the reasons you stated. Nevertheless, I have unsubscribed from this blog. I have no particular bias. I prefer to challenge students where they are and not be restricted to a set curriculum.

Everyone is so fascinated with Singapore because it is in English. I spent some time in Japanese math classes. It is not the materials, it is the way the content is taught.

I respect Katherine Beals and her perspective but I wonder when the last time she stepped foot in a classroom. I do not feel comfortable when anyone is so rigidly dogmatic. I have taught, coached, and tutored gifted math students for the last 22 years. I think this stuff may have come from out in left field but now it has bounced off the wall and jumped the shark.

"So by fourth grade students are not at the same place in Investigations vs Primary Mathematics.

I wonder if the Investigations goes as far, if they are still guiding the students so carefully by this level?"

Thanks, Anonymous, for bringing the discussion back to these points. They were precisely what I had intended to convey with this problem set all along--nothing more, nothing less.

I don't think the point is that "everyone is fascinated with Singapore Math" in so much as the difference in expectations between some curricula, whether from Singapore or earlier time periods in the U.S., and current US curricula.

And math from Singapore is staring to vary too, becoming "a Singapore approach" in the U.S., with possibly lowered expectations as U.S. publishers start adapting it.

I am not sure which teacher's guide the poster is referring to, no teacher's guide of Primary Mathematics says which grade level a student should do. Likely advanced third graders in the U.S. would have to start with the beginning of third grade, or possibly sooner, having used the current math in the U.S. previously with its low expectations. If a student starts the Primary Mathematics, they should of course start where previous knowledge or lack thereof places them as with any curriculum that is more advanced than the usual fare. For example, some of the material on the website for The Art of Problem Solving. Or a curriculum from many years ago in the U.S. And gifted students would hopefully be able to proceed at their own pace, as well as use other material, as each and every curriculum has its own flaws, even, gasp, Primary Mathematics.

It seems to me the purpose of comparing curricula here is to point out how low the expectations are for U.S. students, as opposed to pushing a particular curriculum and Primary Mathematics does happen to be available and in English. I think there were some French material posted at some point here. I went to school briefly in Europe, then switched to a U.S. school, and easily skipped a grade.

Education really is not important in the U.S., despite lip service to it. Thus funds are cut to schools, and especially universities, sports are more important, celebrities more important, CEO'S and rich peoples' continued profit more important, even not offending someone's spirituality more important than factual information.

Math books that are easy enough and prescriptive enough to give an illusion of success abound. So a student can get a good grade on a test from an easy curriculum and feel successful with little understanding of math.

This article puts it pretty well, I think

http://pelham.patch.com/articles/singing-the-blues-about-pelhams-elementary-math-program

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