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## Sunday, October 7, 2012

### In defense of long division: Pro-reform professor capably shows why reform math doesn't work

By Laurie H. Rogers

How do you divide one number by a different number? I was taught to do it using long division. Long division is considered to be a “traditional” algorithm (or methodology). It looks like this:

Our children, however, aren’t being taught long division (or aren’t being taught it to mastery). They’ve been taught various “reform math” approaches, such as this one:
This reform approach is alleged to increase understanding of division, supposedly showing students WHAT division is. Reformers claim that traditional long division isn’t “intuitive” and that children struggle to learn it. (Never mind that an entire country was built on long division, that children around the world learn long division, that homeschoolers and private schools tend to teach long division, and that long division is a more-efficient method. People who are pro-reform math do not like long division and generally won’t teach it.)

What’s the big deal about that? Sure, the reform method is clunky, eking its way to the answer rather than getting there efficiently. So what? The field of public education appears to love it, so why not teach it to the children and not worry about it? Here’s why.

The reform method doesn’t work well with larger numbers. It isn’t efficient. It doesn’t include decimals. It adds too many steps, allowing for more possibility of error. It’s difficult to later switch students from that method to the more-efficient algorithm. And – most important – the reform method often leads to wrong answers. Despite what reformers seem to think – a correct answer is the entire point of a mathematics calculation. Math is a tool that we use to get a job done.

Not long ago, I was explaining this reform approach to some college folks. A young math professor was listening in, nodding his head and saying, “Uh, huh. Uh huh,” in that encouraging way people do when they’re in agreement. “You don’t sound shocked,” I said to him. And he wasn’t. He praised the reform approach and appeared to prefer it. I was surprised, but I thought perhaps he wasn’t familiar with its complications; he claimed to be seeing it for the first time. So, I asked him to do a problem for me.

Dividing 396.3 by 16 is simple for those who were taught long division. However, for those using the reform approach, the decimal poses a problem. Using long division, this is how the problem is done:

Long division efficiently provides a complete answer to the problem. On the white board that day, however, the young math professor used the reform method. His answer: 247 remainder 9. Whoops.

Not only was this additional information confusing, it was still incorrect. Later, he fixed his initial subtraction error (43 – 32 = 11, not 9), but didn’t complete the problem. His final answer was this:

Someone in the room (not me) wrote a huge question mark next to his work.

Math advocates will not be surprised to know that this person never budged on his assessment of which approach was better. He stubbornly maintained that the reform approach is easier, leads to more understanding, and should be the method taught to small children. He said the traditional approach is not “intuitive,” that children can’t learn it, and that division should be done with calculators anyway.

We stared at his garbled work on the board. When I asked him – politely, I swear – how the reform approach is “simpler,” he responded defensively: “I’m not the enemy.” Every time I reminded him that his answer was not complete, he would say, “I’m not advocating for this. I’ve just seen this for the first time today.” But then he would continue to advocate for it. He never wavered in his support for the method, despite his incorrect work and unfinished solutions. It was stunning. But this is reform math, and this is typical of reformers.

Division isn’t the only problem in reform math. I tutor students who come to me not understanding negatives and positives because – instead of being taught the number line – they were taught reform methods such as the building model and the balloon model (going up is positive, and going down is negative). Where in these models is zero? Where is the concept of infinity? Where are fractions and decimals? The number line contains all of it, but reformers purposefully avoid using the most-efficient methods such as the number line. Who loses? The children.

Many children also haven’t been properly taught fractions, they don’t know how to convert from fractions to decimals to percentages, they don’t know basic formulas such as the Pythagorean Theorem, the point-slope formula or the quadratic formula. They don’t know speed/distance/time ratios. Many don’t even know their basic multiplication facts, how to read an analog clock or a ruler, or how many days are in each month (which is critical to calculating calendar time).

Young children do NOT need to know why an algorithm works. They need to know the most efficient ways to get correct answers. Young children should NOT be forced to struggle and get things wrong initially. It causes them to become frustrated and to lose heart – whereupon reformers blame them for not being motivated. Who could possibly be motivated when faced with garbage like this?

Math is a tool to get a job done – unless, as my daughter noted wryly – 4th graders are beginning a PhD in number theory.

The dearth of basic skills is bad enough, but graduates also continue to struggle. Without long division, how do they divide a polynomial? Many math programs have deleted it from the curriculum. Not necessary, they say. Long division, not necessary. Basic math facts, not necessary. Fractions, not necessary. What’s important to reformers are fuzzy concepts for which they can’t be held accountable: “Deeper conceptual understanding,” “critical thinking,” “collaboration,” “real-world application,” and “self-discovery.” When you see those terms in your child’s math program, grab your babies and run.

This is what the children in many public systems face – stupid approaches from clueless people. Simple problems are made to be clunky, inefficient, and incomprehensible. These approaches will damage their futures forever. You would think that 30 years of absolute failure would have killed off reform math, but the fervor of reformers for fuzzy math is nearly cult-like, and their opposition to the efficiency and effectiveness of traditional math borders on hysteria. Reformers have no real support for their approach, no scientifically collected data to support it as being better than traditional math, but they will not let go of it, claiming that reform math would work if teachers would just do it properly.

Sadly for the children, reform math is now seeing a resurgence, due to the federal imposition of the Common Core initiatives. Many pro-reform-math decision-makers are using the Common Core to implement more reform math. It boggles the mind.

I stood next to this professor, listening to him claim something akin to “the moon is made of green cheese and pigs fly.” His process was inefficient and nearly incomprehensible, and his answers were incorrect, yet he preferred it. His face had the same blank look I’ve seen before on reformers – stubborn and closed -- in denial of what was right there in front of him and obvious to everyone else.

I gave up trying to talk with him. I know from experience that the indoctrinated will not listen. Weak outcomes, angry parents, frustrated community members, a nationwide mathematical Dark Ages, and millions of suffering children will not break through their certainty. Alas, this professor is not the only math professional to have accepted the Kool-Aid. Spokane Falls Community College uses reform approaches in some of its remedial classes, and there are others.

The traditional approaches to mathematics were developed by brilliant adults over thousands of years. It’s astonishing that all of that work is being tossed out for methods that are proved to be flawed – in proper studies and in student outcomes. It’s alarming that a dogmatic commitment to reform math appears to be worming its way into departments of mathematics.

Without intervention, there soon will come a day when few Americans know any math at all. At that point, American dependency on foreign professionals will be complete. What will happen then to our children?

Please note: The information in this post is copyrighted. The proper citation is:
Rogers, L. (October 2012). “In defense of long division: Pro-reform professor capably shows why reform math doesn't work." Retrieved (date) from the Betrayed Web site: http://betrayed-whyeducationisfailing.blogspot.com /.

Anonymous said...

Thanks for pointing out the differences between traditional long-division and reform division.

My older daughter never learned long-division (she is dependent on a calculator) and my younger daughter had great teachers (that's what the advanced kids get even though the middle of the road kids could benefit from those teachers too) and her teachers knew she needed it for math classes she would take down the road.

My younger daughter has many career options available to her because she was taught math in a way she could learn it (traditionally). My younger daughter wanted to be a dental hygienist (she goes to EWU) but figured out her first year of college that she didn't have the math skills necessary to get an A in chemistry. The program is very competitive and requires straight As on the prerequisites. Her elementary teachers taught her math using reform methods (didn't have to learn her math facts, never taught long-division, etc.). Even though her middle and high school teachers tried to fill in the gaps, there are still some holes which ended up making her switch majors.

It is pretty sad that we have an education system that allows some students to graduate with the skills needed to be anything they want to be in the future and leaves other kids with a limited number of career choices because they weren't taught math properly.

Too bad the professor didn't have an open mind and could admit that reform math doesn't work. Reform math is going to continue handicapping our kids (yes, it DOES handicap them) unless the reformers wake up and realize their math is actually harming students, not making math easier.

Winifred Daisley, Ph.D. said...

I am a neuropsychologist. part of my practice is assessing for learning disabilities. I see very bright kids making ridiculous error in math exactly because of this. They memorize a pattern of button-pushes on the calculator and have no idea what they are doing conceptually. They cannot reduce complex equations because of this. They do not recognize PATTERNS in numbers. And they have no earthly idea of estimation to see if the answer they got remotely makes sense. Thank-you for explaining this!!!!

Bruce Price said...

At the end of a new article called "Why American Kids Can't Do Math," I tried to sum up where all my work has pointed:

"CODA: You may be thinking that Reform Math is the dumbest thing in history. Perhaps not. Look-say (also known as Whole Word) was a method used to teach reading that virtually guaranteed children wouldn’t learn to read. These two programs operated in tandem to destroy the 3 R's. If only one of them existed, you might think it had to be some sort of grotesque accident. That there is both a math retardation program and a reading retardation program tells us that somebody was trying to eradicate both of them."

Michael Paul Goldenberg said...

This conversation would be more meaningful if two things were true: 1) you didn't insist upon reducing "all reform math" to whether the standard long division algorithm is taught or not; and 2) if you (and those who buy into your viewpoint) could demonstrate not that you can USE the standard approach, but that you can EXPLAIN why it works. I see nothing in your column that convinces me that you do, but I have no doubt that you can USE it.

But the idea behind exploring other approaches, if you were honest, is not to "do away with long division" (though some people favor that and have reasons for it that aren't quite as insane as you make them out to be) or because progressive mathematics educators "hate" long-division (that has to be one of the bigger whoppers of the Math Wars I've read lately.

I won't claim to know what EVERY educator believes (you seem to have no compunctions about mind-reading, but that's your prerogative). But I know that I won't teach "black box" arithmetic or anything else in mathematics. If you are happy with such an approach, and you employ it consistently, and you claim that many students don't wind up confused and full of fear and hatred when it comes to mathematics, please account for all the Americans who were in school long before the progressive reform movement that began to gain some small traction in the early 1990s. I graduated high school in 1968. There was no such thing as "math for understanding," and everyone was taught all the traditional approaches. If they got it, great. If not, too bad. And we have a nation that has LONG despised math, been deeply ignorant of it, and not at all ashamed of that fact. Further, because of people like you, most Americans haven't the slightest clue what mathematics actually is. I know this will come as a deep shock, but it's not about calculation or doing donkey arithmetic by hand.

I hope that didn't cause you too much stress. Read A MATHEMATICIAN'S LAMENT by Paul Lockhart, or any of Keith Devlin's books.

There is nothing new about kids graduating with limited math skills, by the way. Most folks do. And the reason predates the NCTM Standards, the curricula you revile, and the constructivist philosophy you equate with "bad" math teaching. When you can explain the history of US math education from, say, 1850 to 1989, then I'll be interested in your claim that public education is failing, that reform math doesn't work, etc.

Arezoo said...

Thank you for this blog. Very interesting and sad too. To me the most important issue is that math is really about logic. It's about teaching a child how to use the brain in a logical way to find the right answer- which is why math is so fundamental regardless of the field of work a person chooses later on. I can see the benefit of the reform method in understanding the concept of division however It seems that instead of using and learning logic they use a memorized way of getting to the answer - but apparently it doesnt always work for that purpose either.

Anonymous said...

A few thoughts -

1. I recall my first grade math teacher who, the very first thing, introduced us to sets - the players were simple - a whole set equals various combinations of subordinate subsets. While reading, I considered which method is subordinated to whom. While the method in question clearly provides solutions that are accurate in a subset of all possible problems, traditional long hand provides an answer to all possible problems, excepting one - the finite calculation of pi. Hmm. Do curricula definers really subscribe and support limiting the possibilities of students learning by the use of methods that appeal to children and seem to fit neatly in a box for sale?

And a sidebar to the intelligence that offered up an opportunity to discuss how best to explain how long division works, consider set theory. Perhaps the graphical nature of progressive set division would help you understand what most of us got in first grade, then elevated to division by grade 3.

By the time this reform division method is completed by a student, it appears you wind up with a fraction. By the time that fraction is addressed, you may be out of time for your exam or, worse, your astronauts in apollo 13 would be dead because all the brains in the room got a variety of answers using a method that in its design provides too many logical branch possibilities, or, otherwise creates more branches that need solving.

In medicine a "silver bullet" is the mythical pill for an ill, but not available. However, in mathmetics long division has been a reliable silver bullet. But I suppose in a capitalist society, plenty of snake oil salesmen love to sell a new and improved silver bullet. And instructors who prefer them are probably the very ones who missed out on the long division logic in the first place, but whip out e snake oil and hide behind it.

2. In the Art of War, a distinction is clearly drawn between tactics and strategy. Is the best winning strategy one in which the tactics are limited? Sometimes. But often poor leadership is often found amongst the troops who enjoy this choice. While there may be many tactics available, if you only have one to use, it goes without saying (but obviously some need it said again), the most efficient that provides for the greatest possibility of a positive outcome is better strategy.

3. In the matter of the absent minded professor who displayed a valid effort in the attempt to solve the experimental problem via the method, and upon failing continued in its support, I was reminded of some simple logic that may go something like this -

"Son, if I told you dung tasted good, would you go out and have some? And furthermore, would you then go out in public and continue to promote it after you had some?"

3. Or worse, "Do your investments include educational curriculum holdings?"

Michael Paul Goldenberg said...

I wonder why the parents who are so absolutely sure that any alternate examination of division (e.g., through examination partial difference/quotients) is the primrose path to hell, and who believe that what and how they learned arithmetic is the One True Way don't explain something like long division to said children?

What would be fascinating would be if they could do so in a way that would actually explain why the algorithm is sensible and efficient (under reasonable conditions: it gets ridiculous past a certain point to insist on eschewing the use of a calculator or computer, not because one is - ahem - dependent upon the latter, but because one is far too intelligent to waste one's limited time doing donkey arithmetic that calculators and computers do faster and more accurately.

But I digress. My deeper question is why so many of the people equate calculation and repetitive computation with mathematics and why they want to deny their kids a chance to think mathematically. This need not be an either/or situation, in that it's certainly feasible to learn basic arithmetic, become facile with basic computation, and to develop mental mathematics, estimation, and number sense (don't hear much about those last three from this particular pulpit, for some reason) and STILL be learning mathematical reasoning, too, even before middle school.

There are no jobs in arithmetic. There are no jobs in worksheets. There are no jobs in rapid computation/calculation by hand.

There certainly are jobs for people who can intelligently use computational devices and software, though if that's all they can do, those jobs aren't all that challenging or rewarding.

But people who can reason mathematically, solve problems that were hitherto unsolved or solve previously-solved problems more efficiently? Who can use mathematics to model the world in ways that lead to something new and useful? Yes, lots of jobs for those folks. Why anyone would mistake worksheet skills for the latter baffles me.

Anonymous said...

I have a high school student who has struggled with math since elementary. He would bring his math homework home for help, and even though I’m a college graduate and very good at math, I couldn’t help; I couldn’t even understand the question, much less show him how to get there. Eventually his self-confidence suffered, and it has continued to rear its ugly head because of feeling inferior as a small child. BRING BACK BASIC MATH. It’s clear, it’s direct, and it works!

R. Craigen said...

Hello Laurie. I'm also deeply involved in this issue (see our blog, wisemath.org ). I recognize the "type" you were dealing with here and let me say that I have difficulty believing this was a "math professor". I am a math professor myself, and I even know a few genuine math professors who are advocates of this fuzzy math. And what you describe doesn't ring true of them.

It DOES, however, ring true of several math EDUCATION professors I have met. These people, generally, have no background in mathematics and so really don't understand how a mathematician looks at the subject. That appeared evident in this professor's words and actions in your story.

So, regardless of how they described themselves to you, I'm quite certain you were dealing with a professor of education who claims an expertise in math education -- not a "math professor" with a PhD in mathematics. I'd be interested in any further evidence you can provde otherwise.

Assuming I'm right, I'd like to reinforce to anyone involved in this issue: Math EDUCATION professors are generally not "mathematicians". I have met many, and most of them have essentially no qualifications in mathematics per se. They think because they have taught it that they know the subject. But their knowledge is superficial and coloured by entirely non-mathematical ideas.

Michael Paul Goldenberg said...

Lots of anonymous posting here. I wonder why. . .

Bring back "basic math"? What makes you think that it's gone anywhere? If something is mathematically sound, then it is consistent with all other sound mathematics. The fact that, for example, the lattice method is not what you learned in school doesn't make it wrong, and in fact not only is it grounded in exactly the same mathematical ideas as the current standard algorithm, but it was the way people did multiplication until the printing press made it too difficult to reproduce in books (today's printing methods have no such limitations). If you'd grown up learning lattice multiplication and people were now pushing the current standard approach, you'd be screaming bloody blue murder and "bring back basic math," assuming you'd been taught lattice multiplication in the same mindless way you no doubt were taught multiplication as a kid.

Still waiting for someone here to explain why long division works mathematically. I know and can explain it in detail. And I have no problems with any approach that is mathematically sound. . .

Laurie H. Rogers said...

Three decades of reform math have produced a population that is largely math-phobic and math-illiterate. Besides having damaged the future of millions of children, these 30-odd years of failure also have put the welfare and security of this country at serious risk.

I have found that when people are defensive about reform math, or they go on the offensive and begin challenging others to prove this or that, there is some underlying reason for the tactic -- even if only to divert the discussion away from something in which they have a vested interest.

A lot of adults have made a lucrative living off of reform math, even as reform math completely failed the children.

It's unconscionable to close one's eyes to the suffering of children -- or to be unwilling to acknowledge that the cause of that suffering is one's own massive error.

Anonymous said...

I'm shocked that this is all you (Laurie) could come up with in response to Michael Paul's comments and questions. Please try re-reading his comments and replying as intelligently as you can muster.
Also please consider this question: You said kids do NOT need to know why an algorithm works as the purpose for doing mathematics is to get the correct answer. You also have repeatedly bashed the use of calculators/computers/technology in our math classrooms. This seems to contradict. If kids should do math without needing to understand why it works, but simply for computating correct answers, why not just teach them all how to use a calculator very efficiently? Similarly, we don't teach kids how to read an analog clock at school because everyone in the modern day just pulls their smartphone from their pocket to see it says 8:07.

Barry Garelick said...

Anonymous (the latest one):

I think Laurie addressed Michael adequately. An examination of Iowa Test of Basic Skills (ITBS) scores from the 40's through the mid 60's in Iowa, Indiana and Minnesota shows The scores (in all subject areas, not just math)increasingly steadily from the 40’s to about 1965, and then a dramatic decline from 1965 to the mid-70’s. One conclusion that can be drawn from these test scores is that the method of education in effect during that period appeared to be working. And by definition, whatever was working during that time period was not failing. That the math could have been made more challenging and covered more topics in the early grades does not negate the fact that the method was effective. While some may argue that standardized tests scores do not measure true knowledge or “authentic” problem solving skills, the rise of the ITBS scores during this period has been of considerable interest to various researchers for some time (including Dan Koretz who wrote about it extensively in a study he wrote for the Congressional Budget Office )

Over the last twenty years, the mathematical performance of students has been bad. High school math teachers see students who have not mastered basic math facts and who do not know how to do basic procedural operations with whole numbers and with fractions. The lack of such proficiency is alarming and was not seen in past eras when supposedly, according to many math reformers, traditional math was failing thousands of students.

Many of today's students are dependent on the calculator for doing routine calculations. It is not unusual in an algebra class in which the teacher does not allow calculators, to see students counting on their fingers when adding or subtracting.

students need procedural fluency. Sometimes that comes before understanding and sometimes understanding comes before procedure. To insist that understanding MUST come first before students earn the right to learn how to use the standard algorithms does them a grave dis-service. As Michael points out, lattice multiplication is a valid approach as are other approaches. Supposedly, students in various reform math programs are taught the alternative algorithms first in an effort to instill understanding of what is happening when we do basic math operations, and to better understand how the standard algorithm then works.

The result in many cases, however, is confusion. Why not teach the standard efficient algorithms first, and use the alternatives as a means for explanation. Some students will understand, some will not. But procedural fluency will help them solve problems and move forward in math. As for understanding, do it. And in fact, procedure and understanding work together. In solving problems, students learn what subtraction is, what it represents, and when and how it is used. Same for the other operations. With calculators, however, it is neither procedure nor understanding. Students often learn to press keys to get the answer, and forget what it is they are actually doing.

As for understanding always having to precede procedural fluency, sometimes understanding does occur first, and sometimes it doesn't. The insistence on understanding is a deterrant. It is possible to instill the notion that math is more than just calculations while still teaching for procedural fluency.

Anonymous said...

Anonymous said...

"Young children do NOT need to know why an algorithm works. They need to know the most efficient ways to get correct answers."

While many of your comments were shocking (and so wrong), this is by in large, the most alarming. You truly believe students should not understand why they are using an algorithm or how it works? If this is your belief, science would be so much easier for a science teacher. Just tell your students all of the facts, and give them no opportunity for self-discovery or investigation. You believe this would adequately prepare our science community?
I would love for you to survey 1000 random adults from your era taught long division and have them solve a complicated long division problem for you. Can you guess the outcome? Have you looked at what high achieving countries are doing in math? My guess is no. These countries are focusing on specific, developmentally appropriate standards. The students that are the products of these countries are able to apply their deep understanding of concepts and apply them to problems they have never seen before. Also, have you really looked at common core? There is still a point where common core asks for students to master long division. It's not being thrown out the window. Hundreds of math expects, more versed in math than you and I, know that if our students can truly have a deep understanding of math, they will be more successful when taught your most prized long division method.

SteveH said...

"Without intervention, there soon will come a day when few Americans know any math at all."

And long division is the key indicator of this?

"Young children do NOT need to know why an algorithm works."

They need to know something about why it works, but there are different levels of understanding. There is the Partial Quotients understanding level, and there could be an understanding of why the standard algorithm works.

How about an understanding of how division works for octal numbers. How about any base? The assumption is that everyone knows what understanding means for each topic and grade.

Mastery versus understanding issues in math revolve around defining the goal and which direction you approach the goal; top down or bottom up. If you don't go far enough from the bottom up, you will have what many people call rote knowledge. It is unflexible and if tests are done correctly, you will fail.

If you approach the the goal from the top down, you can get stuck at a conceptual understanding level and not develop the flexible skills that indicate whether you have a full understanding of the material - at that grade level.

You can get really good at long division, but not really understand much. But what would it mean to get really good at fractions? I would assume that "really good" would imply understanding. Others might not. Some like to see little linkage between skills and understanding. They see only speed.

When my son was very young, I remember thinking about all of the things I didn't like about my traditional math education. They didn't go far enough with understanding. Then I found out that our school was using the top-down MathLand approach. This was the wrong solution. It was the wrong direction and set low expectations of mastery. K-8 math became worse than when I was growing up. It still is worse.

Anonymous said...

Laurie, do you agree with following statement: If students do not know how to do long division, then they will not be able to use this method to as a pathway to understanding why 33 1/3 equals 33.33333333..... and similar problems.

I still remember when I learned this lesson in elementary school, using long division. Will students understand the meaning of repeated numbers so well, and relationships such as 1/3 = 0.333... if they can only use calculators to make this connection?

Mr. Goldenburg:

What was your pathway to becoming a math teacher? If you became a math teacher through studying within a Department of Education, then this does not give me confidence that you know better than Laurie Rogers or any of the rest of us lay people what is the best pedagogy for math.

From what I understand, Higher Ed Depts of Education indoctrinate their students in Reform Math. An elementary teacher so indoctrinated will tell parents, "Research shows that children learn math best when they are socially engaged and having fun. Reform math teaches deep conceptual understanding." These teachers, perhaps due to their indoctrination during their training, refer to non-reform curricula as "Drill and Kill" and "Parrot Math."

I prefer the term "Drill and Skill."

I have found that students who know their math facts are able to acquire deep conceptual understanding with much less effort than the peers who lack fact fluency. Algorithmic fluency has the same benefit.

I can't blame the teachers for being gullible. I blame the faculties at the colleges of education. It should not be the teacher's responsibility to pour through the research and identify genuine best practices and genuinely best curricula.

Anonymous said...

Here's another good one: when multiplying numbers with decimals such as 67.4 x 2.3 using the "traditional" algorithm (my 5th grade daughter is working on a whole page of this right now for homework) I was taught to count up the number of decimal places total in the problem - in this case two - then move your decimal that many spaces in your answer. Poof! Correct answer!
WHY?

(I know why and just explained it to my daughter, but hey, I'm a math teacher. I certainly didn't know why when I was a kid as I was taught Math using traditional methods and curriculum. I wasn't taught math to be mathematical, I was taught arithmetic.)

Anonymous said...

"The fact that, for example, the lattice method is not what you learned in school doesn't make it wrong..."

No, what makes it wrong is people like you - Confusing children with it BEFORE they learn the more efficient and fundamental algorithm.

I'm not anti-Lattice Method, I'm pro-Law of Primacy in learning. Teach the most efficient method FIRST. Use other methods when that approach fails for AN student. Show other methods AFTER students MASTER the most efficient method.

The traditional algorithm works in every case. EVERY case. Reform algorithms do not work in every case, actually, they don’t work well beyond rudimentary examples. They don’t work well with fractions, decimals, or large numbers. That’s when a reformer will stick a calculator in a child’s hand and say, “Do that.” (Pity the child.)

"Still waiting for someone here to explain why long division works mathematically. I know and can explain it in detail."

Keep waiting, right there by your keyboard, PLEASE! The more time you spend waiting here for an answer your question - the less time you have (I hope) to spend crippling some unwitting child with your "higher understanding" drivel...

Can you explain to me why human fecal material smells, physiologically? I know and can explain it in detail.

Of course, you don't have to be able to explain why it smells to know that it does smell, to know you would prefer to not roll in it, or to know when someone else is trying to sell you a load of odiferous bovine scat...

I'm catching that scent here with you Msr. Goldenberg. Your rabid rantings and pet theories scrawled across the Web (Mathematically Sane come to mind) make me weep for the unfortunates damaged by your brand of instruction and "coaching". Your financial stake in the promotion of a "reform" pedagogy are obvious to the most casual web researcher.

SteveH said...

MPG wrote:

"But I know that I won't teach "black box" arithmetic or anything else in mathematics."

Skills with little understanding can be fixed. Conceptual understandings with few skills are nowhere. In the old "traditional" days, there was a certain amount of sink or swim involved, but with reform math, kids are pumped (spiraled) along and the sink or swim filter happens in high school, when it's too late to do anything about it. At that point, it's easier to blame the kids, parents, society, and poverty. Even the kids will believe it. I'm seeing more people lately spouting off about how it's an IQ issue. They can't or won't separate the variables.

Some think that if these kids were more engaged, they would be motivated and then be able to achieve mastery. They place the onus on the student. This is not what I did with my son at home. I ensured (!) that he had mastery of basic skills at each step along the way. Long division was one of those skills, but it wasn't the most important one. Conceptual understandings are always nice, but they don't drive the mastery process. Real understanding is driven by mastery of skills. Conceptual understandings can only take you a few steps along that path.

For the traditional approach, black box learning is never the goal. It's what you get if you don't go far enough. For reform (top down) approaches, the assumption is that conceptual understandings and engagement will generate the motivation for students to master basic skills. They spiral around the material and expect that pie chart understandings of fractions will magically translate into understandings of how to handle rational expressions.

With such a focus on understanding, you would expect that reform math supporters would be pushing formal proofs at every step of the learning process. No. Reform math is recognized by simple conceptual understandings. They "trust the spiral" and hope that real world hands-on engagement will get the mastery job done. The onus is on the child. Knowledgable parents get the job done at home. In the old days, bright kids with no help at home could end up on a STEM career path. I was one of them. With reform math, this is virtually impossible.

abellia said...

I think something is missing from this discussion. First, these days anyway who wants an exact answer to a problem like the original one posed is going to use a calculator. It's fast and it's accurate. There is nothing wrong with using a calculator for doing such computations. That being said, there are many computations for which you want an exact answer where it is much faster to compute in your head than to use a calculator, or even locating a calculator. All of the basic integral facts people should know by memory because it is the easiest an fastest way to deal with such problems. Math isn't the same as computation, and long division is algorithmic computation. When I went to school, I learned an algorithm for exact computation of square roots. I've used it exactly 0 times since then. I do plenty of division in my "real life" job. I use a calculator.

One legitimate issue that reformers want to address is that some kids, when faced with some real-world problem, don't understand the operation or combination of operations that they need to use to come up with a solution. Harping on whether kids know or don't know long division doesn't address this at all.

Assuming kids can translate a problem to some math operations, most of the time people don't need an exact answer. What I think is often missing, and someone mentioned earlier, is number sense that comes with doing problems without algorithms OR calculators. If one has number sense, one would realize that 1000 / 26 isn't 3.84 when they type 100 / 26 into the calculator instead of the intended problem. But really, to solve a problem like this, one can round 26 to 25, realize there are 4 25's in 100 and 1000 is 10 * 100, so the answer is about 40. All with no calculator. Sure, it's nice to be able to get an exact answer by long division if you don't have a calculator and you need an exact answer, but this doesn't occur that often these days. Being able to deal with place value and the basic facts can get you to success most of the time.

Unfortunately, schools now teach algorithms for "estimation," which makes it seem algorithmic. Kids need tools that they can put in their bag and pull them out to solve problems as they come up. Adults need these too.

Number sense comes from dealing with numbers in your head, and that means non-exact answers when you get past the very simple, which means you can't test it easily. Maybe that's the real problem.

abellia said...

One additional thing. While long division may be much more effective in many cases than whatever the method the reform math professor was pushing, it's no panacea. Try 7845629735 / 64839 with long division. Not easy. You need lots of tools in your toolbox, and long division is a valuable one, but it's not about tools vs. concepts. It's about tools AND concepts. Both are important. Both reformers and traditionalists need to understand this.

Laurie H. Rogers said...

To Abellia: You appear to be dismissing the very notion of correct answers as being important. I do see that you are advocating for estimating in our head and using calculators whenever we want an exact answer. But mathematics isn’t a “social science” – it’s a tool to get a job done. Correct answers are the point. Dependency on calculators has not and will not provide students with sufficient math skills that will consistently lead them to correct answers.

Perhaps in your field, estimating is all you need. But in most “real-world” applications of mathematics, correct answers are essential. Estimating (without following through to the exact, correct answer) is how bridges collapse, machines explode, and people die from incorrect doses of medication. Unfortunately, a casual attitude toward good process, basic math skills, and correct answers permeates the math program in many school districts.

To parents and community members who are wondering why the children don’t know much math: The pro-reform responses here should be enlightening. This is what we math advocates butt our head against. Some advocates have been fighting reform math since its most recent incarnation in the 1980s. Most reformers simply cannot or will not understand what is required for the children to learn math. They are obstinately closed to it.

If you see that the people running your district are closed to the need for teachers to efficiently and effectively directly teach the students sufficient mathematics to mastery (i.e. to proficiency) – it’s probably time to grab the babies and run.

abellia said...

Laurie,

I don't dismiss precision. But if you want precision, you're going to use a calculator or other computer because it is fast and accurate. I work in an engineering environment. We use computers. People don't design bridges or machines these days without using computers. Does this mean that learning to do long division isn't useful? No. But it is a TOOL. If it happens to be the best tool for the job, use it. Most people in their daily lives don't build bridges. They need to figure out about how much something costs when it is 20% off. Neither long division nor a calculator is the most efficient way to deal with a problem like this.

And what's special about division? When was the last time you used a paper and pencil algorithm to calculate a cosine or a logarithm? I don't see a backlash against using computers/calculators for those computations.

SteveH said...

Calculators are great. How they are used in education is not. Calculators should be used to expand the complexity and interest of problems that are solved. In reality, they are used as avoidance tools. Why bother to learn long division when you have a calculator.

There are lots of reasons to learn long division, especially for number sense and quick estimation. There are also arguments against having students solve hundreds of problems this way. So what does one do with the extra free time and a powerful calculator? Um, nothing? Where are the interesting real world assignments that require calculator manipulation of large data sets? I never saw them. I never saw a single assignment where my son had to really exercise his calculator. I was in college when calculators took over for slide rules. More complex theories could be used that required 20 pages of calculations. Calculators were used to do more, not less.

The unfortunate problem is that many educators think this calculator trade-off applies to all parts of math. They see two clearly split worlds; calculation and understanding. Why master the skill when the calculator can do it. This leads to thinking that there is little linkage between mastery and understanding; that all you need is conceptual understanding. However, it's never carried to it's proper conclusion. Students don't use the power of calculators to solve more complex problems. It's a tool to avoid the issues of mastery, not a tool to improve complex problem solving. This lead me long ago to realize that reform math is really about talking a good game while lowering expectations and putting the onus of mastery on the students. Apparently, if students still don't do well in math, then either they have a low IQ or they need to work on engagement and motivation. How can you question "trust the spiral" if you assume it works by definition?

SteveH said...

K-8 reform math educators claim the higher ground of understanding and problem solving, but they don't define it. Parents should watch out for the word "conceptual". While it's nice to motivate the learning of each unit of material (not incompatible with traditional methods), real mathematical understanding is built on mastery of basic skills. Pie chart understandings of fractions don't magically translate into the ability to manipulate rational expressions.

The key distinction of reform math is a top-down approach to math. Educators hope that engagement and motivation will ensure mastery of basic skills. Schools don't accept that responsibility. Curricula like Everyday Math tell teachers to keep moving through the material and "trust the spiral". The assumption is that a wide mix of students can be in the same classroom but still devlop to their own level. I call it repeated partial learning, and it doesn't work. Mastery doesn't happen. My son's fifth grade Everyday Math teacher had to NOT trust the spiral because bright kids didn't know the times table. This is clearly not an IQ issue. Mastery was not getting done. The additional problem was that students were not expected to show mastery "in any one year". Problems and gaps in skills multiply until it's too late to fix them. Students typically get sorted by 7th grade and a STEM career is over for many of them. No amount of Project Lead The Way in high school will motivate away those gaps.

Parents need to watch out for the warning signs of missing skills in the earliest grades. They need to watch for notes being sent home to all parents about practicing "math facts". Parents need to ask their schools when they start to track in math. If your child is not on the top math track, she/he could be on the track to nowhere. It shouldn't be that way, but it is. Missing skills are very difficult to diagnose and repair. Parents need to find out exactly what is on the test used to do the tracking. I've noticed that many parents are caught completely off-guard when the tracking gets done, usually at the end of 6th grade. It's often too late to do anything about it.

Also, for 7th and 8th grades, parents have to look for rigorous math textbooks rather than curricula (like CMP) which do not prepare students for STEM-career high school math. You can often tell the proper math textbooks because they have simple names, like "Pre-Algebra" and "Algebra I". Most publishers also offer lower level math series, but they include subtitles like "Tools for a Changing World". The goal is to get your child through a proper course in Algebra I with good grades (80+) by the start of 10th grade. Algebra I in eighth grade is required if you want to keep all STEM career doors open. Parents should watch out for K-8 teachers who talk about how most adults don't need or use algebra. That will almost guarantee that your child will become one of those adults. If children say things like "I'm just not good in math", the real answer might be that they are just not taught well. They are unhappy because nobody is ensuring that they master basic skills.

SteveH said...

CCSS does not solve the problem of low math expectations in K-8. Although it sets a goal of pseudo-Algebra II by the end of high school, that goal could leave your child at a mountain top of math, not a base camp for rigorous college work. Count the number of times the word "fluent" is used in the CCSS math standard. The term is not calibrated. One has to wait for tests, like PARCC, to see what it means. We have to wait to see what level is required by each state to achieve "proficiency". Parents have to watch out for that word. "Proficiency" means barely acceptable. Even when our state uses terms like "exceeds expectations", one has to look elsewhere to see whether the student is prepared for rigorous math in high school. Current state standards are meaningless for those in high school preparing for college. CCSS won't change that. CCSS can't even guarantee that proficiency will mean no remedial courses in college. David Coleman, who was picked to head the College Board, claims that he wants to align CCSS with the SAT. However, it seems more likely that the SAT will change to align with CCSS. The former might be good, but the latter will ruin the SAT and increase the exodus to the ACT. CCSS can't pretend that "proficiency" means college-ready, and "exceeds expectations" is no guarantee of a good ACT or SAT score.

Laurie H. Rogers said...

Abellia, what you use in the work environment is dictated by the work you do. I am talking about teaching mathematics to children. What they learn should not be dictated by what YOU do in your single work environment, but by what are good principles for learning mathematics.

The last time I used pencil and paper to calculate a cosine was last year when I was teaching trig to my daughter.

The last time I used pencil and paper for long division was two days ago, when I modified a recipe to make 3/4 of what the recipe said to make.

To parents and community members: The pro-reform arguments being made here appear to me to be driven by the interests of those making them. It's a huge problem in many districts. The choice of curriculum is driven by the interests of the adults and not by the actual academic needs of the students. In Spokane, it's been an ongoing Christmas in the Department of Teaching & Learning, and yet the 56 pages of titles of reform and supplementary materials (many of them also reform) haven't produced properly educated students.

Please notice that the pro-reform arguments are not dealing with the absolute devastation in math skills that 30 years of reform math curricula have brought us.

abellia said...

Laurie,

You make my point. I can't imagine doing long division to calculate 3/4 of anything when cooking. People can get exact numbers more efficiently by applying some number logic. And you're cooking. Estimation is fine unless you're doing something very unusual in the kitchen. If I can't do such a calculation in my head and I need an exact answer, I'll pull my cell phone out of my pocket and let IT do the calculation.

I have a degree in math and have been doing math in various professional contexts for almost 30 years. Unless you're going to implement a computer algorithm to calculate a cosine, I can't imagine the value in knowing the process, unless you just find it interesting. Furthermore, an algorithm that a computer might use for this process is in all likelihood different than one used by human beings. Even before calculators, people used pre-calculated tables to solve problems involving trig. Understanding cosines is important to people doing math. Computing cosines isn't.

Being able to do division by some efficient algorithm is worthwhile, but it's also not the most important thing in the world. I do feel very strongly (and I think I said this at the beginning) that students should know their basic facts cold. This should extend to understand of operations on fractions AS NUMBERS. I'm also not saying that math is well taught in most schools. It probably isn't. But if inability of students to do long division were our biggest problem, we'd be in pretty good shape.

Many math teachers over-estimate the value of the DETAILS of what they teach. I find math interesting, so for me it's something I want to understand. But many people in their daily lives don't need math beyond fractions, ratios and some basic geometry and algebra. People don't factor quadratics in real life. Only about 6% of the population work in STEM careers. Those people like math, pursue it, and probably don't have much trouble learning it. But we shouldn't kid ourselves about the general applicability of much high school mathematics.

When you speak of principles for learning math, what is the purpose to the student of the math you want learned? Is it general understanding of the breadth and beauty of mathematics? Is it to provide practical tools that people can use in their daily lives? Is it to check boxes on some test so that we can say that students know the same things that their parents learned in school? Is it to allow EVERY student to do math in a STEM career (not happening). Is it all of these? I'm not saying one philosophy is superior to another, but until you answer this, you can't determine what should be taught.

Laurie H. Rogers said...

Abellia, your comments deny the experience of millions of children in this country.

I don't know you, and I don't know your actual math background or what you actually do with math. Your comments don't marry up with what I know about people who use math. They do marry up quite well with those who come out of education departments and who love reform math.

Math matters. Correct answers matter. Efficiency and effectiveness matter. This country cannot survive on estimation and a dependence on calculators.

You seem convinced that your casual attitude toward structure and content should drive what K-12 students learn. It's a false premise.

You said: "But if inability of students to do long division were our biggest problem, we'd be in pretty good shape."

Long division is NOT our biggest problem in math, and we are not in pretty good shape.

SteveH said...

"I do feel very strongly (and I think I said this at the beginning) that students should know their basic facts cold."

This isn't getting done and it's the biggest problem in K-6 math. You're quibbling about division and assuming that this argument reflects what's going on. Schools don't ensure the basics for much of anything. Division is just a small symptom of this issue. Reform math sprials (circles) through the material and leaves mastery up to the student. They think this process works by definition. It's partly a curriculum issue and partly a competence issue. Many don't even do what they say they are going to do.

"I'm also not saying that math is well taught in most schools. It probably isn't."

It isn't, but the teaching of division is not the key indicator of what's going on. Just because you don't buy the division argument, you can't build a counter argument based on it, especially when you appear to not know what's going on.

"But many people in their daily lives don't need math beyond fractions, ratios and some basic geometry and algebra. ... Only about 6% of the population work in STEM careers."

This is a specious argument for deciding what (and how well) kids should learn math in K-8. The material is not difficult and the goal is to keep all doors open. How do you know which kids are going to be those 6%? If schools teach math badly, why wouldn't you expect this percent to be larger? You should define what "cold" means and check to see if that is being done with reform math. It isn't.

"But we shouldn't kid ourselves about the general applicability of much high school mathematics."

The applicability to individuals can be huge. It can mean finding something they love to do. It can provide a good job. Right now, many affluent parents ensure mastery of the basics at home or with tutors. They keep the doors open. Your "shouldn't kid ourselves" attitude glosses over real problems in math curricula and condemns many kids from poor families to being in your 94% group. By 7th grade.

"Is it to allow EVERY student to do math in a STEM career (not happening)."

"Every?" The goal is to keep STEM career doors open for most all students in K-8. You don't see how bad the problem is. My high school son just went through the process and I saw it first hand. Even though he loves math, I had to use Singapore Math with him at home. I had to ensure that he knew his basic facts "cold". Long division is not a good example of what's going on. Reform math supporters love to jump on that argument to distract people from all of the other fundamental flaws in reform math. It's not just an implementation issue. The philosophy is flawed. They don't believe in knowing basic math facts "cold".

R. Craigen said...

Anonymous said: "I still remember when I learned this lesson in elementary school, using long division. Will students understand the meaning of repeated numbers so well, and relationships such as 1/3 = 0.333... if they can only use calculators to make this connection?"

Actually, 1/3 can be done in the most awkward, laborious way. It is illustrated using base 10 blocks in most fuzzy texts. And it is ridiculous, because doing 1/3 gives no insight for 1/7 -- so that must be done separately, and even though 1/3 is bad, 1/7 is a MONSTER by this method.

You could do a million of these and still not prove the underlying theorem which is necessary for the development of the modern number concept: that if m and n are positive integers, then m/n corresponds to a repeating decimal.

And calculators are no help either for this. Try it with 1/17. In general, calculus give us 11-digit (or so) APPROXIMATIONS. But the theorem sought here deals with EXACT VALUE. For this kind of analytic understanding calculators are worse than useless.

.r2. said...

To All Those Who Think A Calculator Is An Answer:
What exactly would you do, and where would your civilization be, if the power runs out? The true genius behind the calculator was that it was made without a calculator. .r2

abellia said...

It's kind of funny to have you guys pile on, here. In a room with an average bunch of people who know something about this stuff, I'd be put in the corner with the traditionalists.

This post started with a criticism of alternative division algorithms. I tried to point out that in the scheme of things, this isn't that big a deal, and real grownups did indeed use calculators/computers for real division problems because it was fast and accurate. I also admitted that it's probably good to know long division, but as far as useful mathematical tools go, my contention is that it's not that high on the list for most people these days. Nobody has given me a good reason to change my mind.

Kids who want to learn math can learn math, and I think the screams coming from some of you are unwarranted. I think it's unfortunate if kids aren't asked to learn their basic facts inside and out because anything other than immediate recall is inefficient. One might suggest, though I'm not sure anybody here has, that learning long-form multiplication and division is good to help kids get comfortable with place value, which is very difficult for many. I think the arguments beyond this for learning algorithms is much less strong, though knowing how to do things without a calculator is still valuable.

Some reform is crap. But there is a valid reason why people were concerned with the state of things in the past. Focusing too heavily on tools without understanding leads to people who can't apply the tools or, really, know any math. Is it really any worse when someone want to use the tool discussed above in the problem 2000 - 867 than it is when students write 72 / 9 as a long division problem (yes, they do that) and come to you confused? Or what about if someone can't function if the computer stops working AND they can't find a pencil and you need change for a \$20 after buying something that cost \$17.32?

You guys have a point. But there is also reason why people were wanting reform. The problem lies in the people peddling junk -- and there are plenty.

SteveH said...

Abellia,

I'm glad you're slowly getting up to speed on this issue, but you should reread my last post where I already addressed the issues you now raise. Many of us critical of these teaching techniques have advanced technical degrees, have had kids go through this process, and are quite capable of seeing differences between theory and implementation. Besides, you specifically avoided critical low expectations issues I raised about your post.

"But there is also reason why people were wanting reform. The problem lies in the people peddling junk -- and there are plenty."

You still have more to learn about the problem. This is not just about poor implementations of reform math. It's about how it's used for low expectations. I could come up with a top-down, discovery, hands-on, real world, Socratic method for learning math, but it would require much more effort from students. In reality, high minded talk of critical thinking and understanding is only an excuse for low expectations. Rote learning is only an excuse to completely change the direction of learning and allow educators to load on all of their own pet ideas of teaching pedagogy. Instead of understanding the issues of rote learning, they use it as an excuse to do what they want. It sounds good in general, but falls apart when the details are examined.

Specific reform curricula, like Everyday Math, are intrinsically flawed because they "trust the spiral" to make sure that mastery of the basics gets done. It doesn't happen. I saw it first hand. It's not just poor implementation. The problem is not that people are peddling this junk. The problem is the educator market demand that created it.

John Threlkeld said...

Don't have time to read everything here -- but this blog came up in some other research I was doing. So if this question has been answered, I apologize.

But I'm curious when the last time any of the posters (on either side of the argument) had to divide, for example, 86478 by 321 by hand for any purpose other than to demonstrate that you knew the standard algorithm?

John Threlkeld said...

To the poster who was horrified that her daughter solved 2000-867 by coming up with the simpler and equivalent problem 1999-866, you should be thrilled that your daughter can THINK this way. I would suggest that she already understands more about arithmetic than you do.

Anonymous said...

In order for education to be truly reformed, it will be necessary to re-start measuring whether or not kids have attained measurable fluency in learned curricular material.

For example, in K-1, if kids can write the whole alphabet at a minimum rate of 40 letters per minute, there will be no subsequent reading failures.

Anonymous said...

"Despite what reformers seem to think – a correct answer is the entire point of a mathematics calculation. Math is a tool that we use to get a job done."

Isn't that an argument for a calculator?

Laurie H. Rogers said...

Anonymous, 8:43, June 18:

No, my statement isn't an argument for calculator use.

Calculator use inhibits the gaining of proficiency in arithmetic skills. With calculator use, students become dependent on calculators, and they tend to not know whether the answers they get from a calculator are correct or even in the ballpark.

Calculators are fine to use in college and in the workplace, AFTER skills have been learned. They're also fine to use for certain skills in more-advanced math - such as with logarithms and vectors.

Otherwise, calculator use is counterproductive, not helpful.

With a solid curriculum, students can easily get to trigonometry without the use of a calculator. I know that to be true. My own daughter did it, and the students I tutor also are doing it.

Jason Tangent said...

The math professor in this article sucks. I did the method they described in under a minute, and it was the first time I had ever seen it. 396.3 divided by 16. Using what I know about the connection between division and fractions (gasp, math understanding!) I write the question as 396.3/16, which can be written as an equivalent fraction 3963/160 by multiplying the numerator and denominator by 10. To do this, I look at 3963, and notice that 20 groups of 160 is 3200, and 3963-3200=763, so I need more group of 160. 4 more groups are 640, and 763-640=123, which cannot be divided wholly into 160. This leave 24 groups of 160 (totaling 3840) and a remainder of 123, of 24 + 123/160. An answer! It may not be as a decimal, but fractions are cooler anyways :).

In my opinion, both are necessary. Long division if efficient, but intuition and understanding help when students forget the rules to the algorithm (which tends to happen). Also, it builds understanding of what division is. So many students mix up rules because they memorize them and without understanding they don't know what else to do. Memorization without Understanding = Foolish. Understanding without some Memorization = Inefficient. Understanding + Memorization = Lethal Combo. Why can't school teach both and have kids decide which they want to use? As a tutor, I ask "What does division mean?" and most don't know, because basics teachers don't put much emphasis on this. It means dividing a number into equal sized groups, each group the same size, and the answer is the group size. Is that difficult to teach?

Christy Wilkinson said...

I totally agree with this post! Why on earth would anyone even need to explain long division? It is simple and self explanatory! One can explain it with pebbles or sticks. I had a specific "common core" math meltdown tonight over the ridiculous sideways approaches that are being inflicted onto my 4th grader. Ugh! Frivolous and illogical. I have been teaching her traditional mathematics since she started school because the silly common core ways stressed her and consumed more time.

magicact said...

I don't want my kids bogged down with long Drill and Kill homework. I would rather they have extra time to watch Youtube videos, listen to music, and experiment with drugs and sex.
Besides all this repetition is silly as we all know that "real life" involves little to no repetition.
Why should we waste time developing young brains anyhow? A brain not used will atrophy over time and as that is similar to trophy it must mean it is a good thing (just thought I would toss in some Reform English while we are on the subject of Common Core).
We must remember the words of our Master Bill Gates....a smaller brain is easier to manipulate and control.
Submission now trumps education in the language of the reformers.

DanielAjoy said...

For those who DO want they kids to learn long division, and want them to practice the procedure. Here at the bottom (Long Division Practice Worksheets) are the worksheets I use with my kids. The thing I like about them is that they are partially completed already so they have a way to determine if they are still on the right track.