## Wednesday, January 13, 2010

### Rainforest, Reform "Math" Update: Long Division Is No Longer Taught: It Stifles Students' Creativity

Scary: Only 57% of college freshman at the University of Washington could solve this problem below (231 / 7 = 33) without a calculator, using old-fashioned "long division." Here's a hint why - according to a math teacher quoted in the NY Times, "We don’t teach long division; it stifles students' creativity.”

From Professor Cliff Mass:

Last quarter I taught Atmospheric Sciences 101 at the University of Washington, a large lecture class with a mix of students, and gave them a math diagnostic test as I have done in the past. The results were stunning, in a very depressing way. This was an easy test, including elementary and middle school math problems. And these are students attending a science class at the State's flagship university--these should be the creme of the crop of our high school graduates with high GPAs. And yet most of them can't do essential basic math--operations needed for even the most essential problem solving.

PDF version of the full test and results, and here's a blank version to give your kids and friends.

Consider these embarrassing statistics from the exam:

43% did not know the formula for the area of a circle

86% could not do a simple algebra problem (problem 4b)

75% could not do a simple scientific notation problem (1e)

52% could not deal with a negative exponent (2 to the -2)

43% could not do a simple long division problem with no remainder (see above)!

47% did not know what a cosine was.

I could go on, but you get the message. If many of our state's best students are mathematically illiterate, as shown by this exam, can you imagine what is happening to the others--those going to community college or no college at all?

What explains this mathematical illiteracy?

Starting in the mid-90s colleges of education and "curriculum specialists" in districts become enamored with a new way of teaching math--called reform or discovery math. Instead of teaching the basics --followed by practice to mastery, the idea was that students could only learn math they "discovered" themselves. Working on problem sets was considered "drill and kill." Direct instruction by teachers and equations in books were out. Long division was out. Integrated math books where all topics were swirled together were in. Group learning and playing with objects (manipulatives) were in. Describing one's through process was considered more important than getting the right answer. Most of this proved to be a disaster, but those pushing it--professors in education schools and district curriculum types--were believers, even though there was no empirical proof that it worked.

See a video demonstration here of some "rainforest math":

HT: Ironman

See previous CD posts on rainforest math here, here and here.

At 1/13/2010 9:57 AM,  Andy said...

Long division isn't a great way to solve that problem.

Most division of small integers is easier to do using trial multiplication. So 3 * 7 = 21 so 30 * 7 = 210, 210 + 21 = 231. So 33 * 7 = 231.

Although long division is useful in other respects, for example for polynomials.

At 1/13/2010 9:58 AM,  geoih said...

I have my own long division horror story. I was taking a test for a science class in college that involved doing a lot of arithmetic (including division), only I forgot my calculator. I had to do all of the division by hand on the back of the test paper. If I hadn't known how to do long division, I would have failed the test. As it turned out, I did know how to do long division, and I did just fine.

At 1/13/2010 10:30 AM,  Anonymous said...

Everyone here is a constructivist," Gabriel Reich, a genial education professor at Virginia Commonwealth University, told me at a reception sponsored by the John Dewey Society. (Dewey, a pragmatist philosopher who died in 1952 and taught for years at Columbia Teachers College, is regarded, alongside the Swiss cognitive psychologist Jean Piaget, as one of the fathers of progressive education.) Reich was trying to explain to me why it was presumptuous for professional mathematicians (and many parents) to be up in arms about the currently fashionable constructivist idea that instead of explaining to youngsters, say, how to do long division, teachers should let them count, subtract, make an educated guess, or otherwise figure out their own ways to solve division problems. College math professors may complain that young people taught the constructivist way arrive in their classrooms unable to perform the basic operations necessary to move on to calculus, but so what? "Why should we privilege professional mathematicians?" Reich asked. Long division, multiplication--"those are just algorithms, and a calculator can do them faster than we can. Most of the people here at this meeting don't think of themselves as good at math, and they don't think math is creative. [The constructivist approach] is a way to make math creative for many people who never thought of it that way."

There are no wrong answers in constructivist theory, so Reich, pursuing his mathematical theme, had a tough sell the next day when he presented a paper to his fellow educators arguing that the principles of constructivism should be modified a bit in teaching arithmetic. "I know some constructivists might take issue with what I'm saying," was his delicate way of telling his audience that when a student says two and two equals five, there might be a problem, if only with the child's non-constructivist parents who might have "right-answer" concerns. Reich was suggesting that the youngster's incorrect (or "incorrect") answer be "vetted by the class" to see if it "works." That way, he explained, "the students are learning to act as members of a mathematical community--they are becoming mathematicians."

It might strike an outsider to the world of ed schools as absurd to spend multiple minutes of precious math-class time having other students "vet" answers to problems that a teacher could explain quickly using simple objects. But a sense of disconnect between the pedagogic theory taught to ed-school students (nowadays called "preservice teachers") and their lived classroom experience after graduation pervaded the AERA sessions.

Weekly Standard

At 1/13/2010 10:41 AM,  Anonymous said...

... but those pushing it--professors in education schools and district curriculum types--were believers, even though there was no empirical proof that it worked.

"Believers", that's the right word. They truly are "believers". They believe in marxist theory and little else. This isn't an accident, it's the end goal of the marxist left. They have spent nearly four decades on a Gramscian march through our institutions of higher learning and, for all practical purposes, have captured our teacher's education schools. People, like William Ayers and Howard Zinn, have great influence and have strived at every turn to undermine traditional education.

Education either reinforces or challenges the existing social order, and school is always a contested space – what should be taught? In what way? Toward what end? By and for whom?

- Self-professed marxist and professor of education, Bill Ayers, World Education Forum, November 2006 w/ Hugo Chavez in attendance.

Give me four years to teach the children and the seed I have sown will never be uprooted.

When an opponent declares, "I will not come over to your side," I calmly say, "Your child belongs to us already… What are you? You will pass on. Your descendants, however, now stand in the new camp. In a short time they will know nothing else but this new community."

At 1/13/2010 10:50 AM,  Cabodog said...

A long time ago, as an engineering student, I knew what a cosine was.

Now what's important to me is that I know that by using the internet I can easily contract a math problem to someone in India.

To build something of substance, to run a company, is it really that important that we know the details or just know it needs to get done?

What is more important to GDP?

At 1/13/2010 10:54 AM,  Anonymous said...

In my children’s elementary school, students in the early grades had no desks at all but instead sat in circles on a rug, hoping to re-create the “natural” environment that education progressives believed would facilitate learning. In the 1970s and 1980s, progressive education also absorbed the trendy new doctrines of multiculturalism, postmodernism (with its dogma that objective facts don’t exist), and social-justice teaching.

More powerfully than any previous critic, Hirsch showed how destructive these instructional approaches were. The idea that schools could starve children of factual knowledge, yet somehow encourage them to be “critical thinkers” and teach them to “learn how to learn,” defied common sense. But Hirsch also summoned irrefutable evidence from the hard sciences to eviscerate progressive-ed doctrines. Hirsch had spent the better part of the decade since Cultural Literacy mastering the findings of neurobiology, cognitive psychology, and psycholinguistics on which teaching methods best promote student learning. The scientific consensus showed that schools could not raise student achievement by letting students construct their own knowledge. The pedagogy that mainstream scientific research supported, Hirsch showed, was direct instruction by knowledgeable teachers who knew how to transmit their knowledge to students—the very opposite of what the progressives promoted.

The ed-school establishment has worked busily to discredit Hirsch. In 1997, the journal of the American Educational Research Association (AERA), the umbrella organization representing most education professors and researchers, launched an unprecedented 6,000-word dismissal of his work. Hirsch recounts, too, how he finally got the nod to teach one course on the black-white achievement gap—a hot topic—in Virginia’s education department, though not until he had won all of his university’s academic honors, written one best-selling book on education, and written another listed by the New York Times as a notable book of the year. But whereas his courses in the English department always overflowed with students, his education course drew only a handful for three straight years. Finally, one of the students broke the news: the education faculty had repeatedly warned them not to take the course.

[...]

But the ed schools’ closed “thoughtworld” (Hirsch’s term) has insulated itself from science. For that matter, future classroom teachers must search far in ed-school syllabi to find a single reference to any of Hirsch’s work—yet required readings by radical education thinkers such as Paulo Freire, Jonathan Kozol, and ex-Weatherman Bill Ayers are common. From these texts, prospective teachers will learn that the purpose of schooling in America isn’t to create knowledgeable, civic-minded citizens, loyal to the nation’s democratic institutions, as Jefferson dreamed, but rather to undermine those institutions and turn children into champions of “social justice” as defined by today’s America-hating far Left.

City Journal

At 1/13/2010 11:14 AM,  gettingrational said...

Why not issue multiplication flash cards to all grade school students? This would be a simple way to help form a base for further math learning. Memorizing times (x) up through 9x9 by the fifth grade is common sense.

Common sense and simplicity are dismissed early on in the educational bureaucracy of many school districts. The bureaucratic and highly paid hirearchy would stop growing.

This is what has happened with the Seattle School District. Memorization of times tables was thrown out for Integrated Math with calculators. The results on math tests were dismal. What was the solution? Evermore complexity with Discovery Math. More consultants, on-going committees with facilitators, more program directors, more funding from gov't and private grants, more paid tutors and teaching aides as well as more expensive and ever changing textbboks.

I am not saying math isn't difficult for most of us but why make it more difficult?

I think my best volunteering at my kid's private grade school was helping students learn their times tables. When my kids joined the Seattle School district in middle school the math learning was bewildering and downhill for them and me.

Andy, long division is the easier way to solve the problem -- if you have memorized basic times tables. Are you a Seattle School District Discovery Math Coordinator?

At 1/13/2010 11:18 AM,  reprise8 said...

Well, this expalains a few things (keeping in mind that as a land surveyor, I use math, algebra, and trigonometry daily. I also delve into occasional statistics/probability and calculus):

1. Why US students not only lag behind but are losing ground in math proficiency as compared to other developed nations - it's by design. It was intended.

This teaching method is designed (and succeeds) to equalize students at the bottom of the skills ladder by allowing the less math-inclined to achieve a higher grade while learning no more than they otherwise would have, and simultaneously ensuring that the more math-inclined learn nothing.

2. If this sort of teaching method is also used in other subjects, it also may explaing the occasional school shoot-em-ups perpetated by students.

At 1/13/2010 11:53 AM,  Bret said...

I think this is wildly overblown.

It is a pop quiz, no studying. Who knows when the last time the students did anything with a cosine?

At 1/13/2010 12:18 PM,  Patrick said...

I guess I learned math the "old" way by drill and kill and hated it. Who would have thought that I would get into algebra in high school and start to love math? I even went on to become an engineer. I would have to agree partly that the math you truly learn is the math you discover for yourself. I look back and wonder what exactly I was learning by doing basic math for so many years and seemingly learning nothing from it. I do believe you have to practice something to retain it but I wish I had been taught some creative ways of relating to math before high school. Some people are wired to learn by drill and kill but some people aren't. Too bad education isn't more flexible.

At 1/13/2010 12:42 PM,  Anonymous said...

My ears are bleeding. What in the name of all that is sensible is all this horse sh**? And it is horsesh**. What genius stated that it doesn't matter if I don't know how to do something because I can get someone else to do it(paraphrased)? Besides Tom Sawyer?

I guess those poor slobs who built the Roman roads, arches, and cities had to depend on "working out a cluster problem" by having someone else do the math for them. Not to mention the inventive boom that occured along with the industrial revolution, and the information age, and the medical advances since 1950, and the increasing efficiency of food production and distribution, etc. Someone else can take care of it; let them eat cake.

Can anyone fathom the kind of damned nonsense that is being taught? The ability to understand and reason, and to know what a cosine is, is necessary for continued development and evolution. Otherwise we would all still be in mud huts trying to cook a frog over a damned dung fire.

Of all the dumb sons of bit**es in the universe the fools in charge of education are only bested by the morons who want to redistribute wealth. Go ahead a**holes, put your best US public school student up against a mediocre German, Japanese, or Dutch student and see who has to solve the "cluster of problems" in their shorts afterward.

It is exactly this sort of poison that allows people to be led by the nose to the slaughterhouse, metaphorically and literally. Keep it up, stupid sh**s, and you will have a nation of 30-year old infants. We're close as it is.

Dan Patterson

At 1/13/2010 1:49 PM,  OA said...

This is why liberals oppose vouchers and support the public education monopoly on tax dollars. (Except at the college level where they support massive public support of mostly liberal private universities.)

This process oriented crap (vs. results oriented) can only stick around if everyone is similarly handicapped. By giving points for "explain your thought process" instead of getting credit for doing the math right they also bump up standardized test scores that would otherwise show the long term failings of the public education system.

I lucked out with my schools, although there were still some "teachers" who weren't even close to qualified.

After seeing this I understand better why my sister, who is a teacher, bought some math and science books to supplement her kid's schooling.

At 1/13/2010 2:11 PM,  Anonymous said...

Big problem with this :
College students generally don't care if you tell them a test doesn't count.

At 1/13/2010 2:29 PM,  richard said...

So you think you can calculate a bit, huh?
-------------------------
I choose 2 numbers smaller than a hundred.

I give Steve the Sum of these two numbers,

I give Paul the Product of these numbers.

Steve and Paul talk to each other:

- Paul: I don't know them
- Steve: I knew that!
- Paul: Now I know!
- Steve: Now I know too!

Which numbers did I choose?

At 1/13/2010 2:56 PM,  Lyle said...

Of course 40 years ago teenagers could not add either, back before computers it was an interesting game to go to McDonalds, and see what relationship to reality the counterpersons total was. This is why McD. introduced the picture menu box and computer. Speaking of this I recall when you took a book of log tables to a test, (because the slide rule was not accurate enough) and added and subtracted furiously to calculate. Maybe we should teach using log tables for long division it does work, it makes division into subtraction. A six digit log table makes a nice thick book.

At 1/13/2010 3:11 PM,  Jet Beagle said...

Lyle: "Of course 40 years ago teenagers could not add either"

Your experience was certainly different from mine. From 1969 through 1971 I was night manager in South Louisiana at two Burger Chefs (then a national chain). I trained 20 or more kids to count back change to customers. I don't remember a single one having a problem. They didn't have to add up anything, though. Our cash registers did that.

Twenty five years later, when I opened my first bookstore, not a single young worker could grasp the simple concept of counting back change. These college students could not function at the counter without calculators. I quickly installed a point of sale system to eliminate the requirement that they add, subtract, or count change.

At 1/13/2010 3:11 PM,  EJB said...

im reminded of a comment froms omeone regarding the quivelant in english/spelling. It went something to the effect of "despite years of advancment and countless billions spent of education advancement, a guy is still making millions selling video tapes teaching kids to spell using meathods that have been known for decades"

At 1/13/2010 3:23 PM,  randian said...

Long division? You should be able to do 231/7 in your head.

At 1/13/2010 3:39 PM,  Jet Beagle said...

Lyle: "Speaking of this I recall when you took a book of log tables to a test"

It's been a while. I had forgotten completely about using log tables.

At 1/13/2010 5:28 PM,  Lyle said...

Jet Beagle, this was in Pasadena, Ca so that likley there were differences in educational quality around the country at the time. (Although that was supposed to have been the golden age of CA education).

Then I visited Germany and the waiters did the sums in their heads and were correct!

Of course the 1960s/1970s were the begining of the new math, when arithmetic was depreciated.

At 1/13/2010 5:40 PM,  Stephen said...

While I'm not disputing that the level of math skill of college students isn't up to par, there is a clear incentive problem biasing the particular results posted downward. There is no accountability for getting the answer right/wrong. The student isn't even required to put their name on the top of the test.

At 1/13/2010 6:52 PM,  OA said...

Stephen said...

While I'm not disputing that the level of math skill of college students isn't up to par, there is a clear incentive problem biasing the particular results posted downward. There is no accountability for getting the answer right/wrong. The student isn't even required to put their name on the top of the test.

So only if you give them a clear incentive would they score up to potential? It's 13 questions on 1 page. I can understand 50 questions getting blown off, but not 13 relatively easy questions. Only a few of them would you maybe even write something down other than the answer.

That might be even a worse problem if they just care only when it "counts". I bet you drop that test into a school in Asia and they'd do 100% effort just because the teacher was asking them to take it.

But you could be right. I don't think Japan and China destroyed light manufacturing in the US as much as the US stuff really got shoddy. My last Sunbeam products just sucked. No quality in the construction. Same with my Zenith tv, and 2 GE clock radios.

At 1/13/2010 7:10 PM,  Anonymous said...

I'm a student at the University of Washington. Keep in mind that this class is a 101 class for non-science majors. Basically a very easy class for people who don't want anything to do with math (or much of a challenge of any kind).

At 1/13/2010 7:27 PM,  Anonymous said...

Cabodog said...
A long time ago, as an engineering student, I knew what a cosine was.

Now what's important to me is that I know that by using the internet I can easily contract a math problem to someone in India.

To build something of substance, to run a company, is it really that important that we know the details or just know it needs to get done?

What is more important to GDP?
1/13/2010 10:50 AM

The issue is that at some point the contractor in India won't need YOU to hire him, if he can do the math he can also figure out how to build something of substance. Thereby cutting YOU out of the deal.

How long will America remain a strong nation if our best jobs consist of doing each other's nails and dry cleaning?

At 1/13/2010 10:28 PM,  Anonymous said...

> They believe in marxist theory and little else. This isn't an accident, it's the end goal of the marxist left.

Let's hope you never get near a classroom. I guess it's only left that go into teaching, leaving the right to do the more important things like bring down the banking system.

At 1/13/2010 11:23 PM,  Blank said...

Here in Ecuador we work with problems this size:

Multiplication and Division Exercises

At 1/14/2010 12:13 AM,  Anonymous said...

Well, 40 years ago I was 17. My particular school district taught new vs. old math in alternating years. I'm not sure why that was the case - whether it the district couldn't afford the new math books for all grades at the same time, but the benefit was that you understood the math from both perspectives and for most of my classmates we could calculate accurately.

I was old enough to have a slide rule, but by the time I got to high school, electronic calculators were becoming common so I never learned how to use the slip stick except for multiplication. Someday I may brush up on my Dad's slide rule and give my young adult kids an history lesson.

I can still do long division and can do an amazing amount of arithmetic in my head because of new math and old math concepts, and that I try to do stuff in my head to keep it sharp. It's funny when my wife wants me to calculate her gas mileage and I can calculate in my head faster than my kids can punch it up on their cellphone calculators.

The problem with this mess is that in applied math and science, you need to get correct answers and you need to get them quickly. While I think it is helpful for students to understand multiple ways of getting correct answers, there needs to be emphasis on getting the right answers and in being confident that the method did indeed get the correct answer deliberately rather than just a lucky guess.

I think it is also very bad to rely on a calculator - there is value in being able to do math without one.

BTW, one of the most useful concepts is the 5-4-3 right triangle. You can check the squareness of a foundation, a window or a picture frame with it. The 4 and 3 are the legs of the right triangle and the 5 is the hypotenuse. You know it's perfectly square when the hypotenuse is 5. Doing a foundation? just scale the lengths up accordingly.

I'd bet that most high school grads couldn't figure out the formula for converting Fahrenheit to Celsius if their lives depended on it.

At 1/14/2010 1:18 AM,  Anonymous said...

I guess it's only left that go into teaching, leaving the right to do the more important things like bring down the banking system.

You can start your education HERE. There are a few big words, but I think that you can handle it.

Fannie and Freddie have been run by Democrat Party hacks since 1991 when Walter Mondale's campaign manager, Jim Johnson, was put in charge. As Edward Pinto, a former chief credit officer for Fannie Mae, points out in the article, the fraud started almost immediately. Throw in Bill Clinton, Andrew Cuomo and the CRA, and bringing down the banking system looks like leftist effort in community organizing.

But, hey, you can always teach the kids about "global warming", it's settled science.

At 1/14/2010 8:59 AM,  Max said...

You can't be right, that your students really didn't finish this test with 100 %. Perhaps we have different rules in Germany, but here you have to learn to solve these problems without calculators (starting from end of high school through the first 3 years of university).

I don't even understand how people could enter college/university without a 100 % score on this one Oo (except perhaps you are a non-science person)

At 1/14/2010 2:07 PM,  Mike S. said...

So you think you can calculate a bit, huh?
-------------------------
I choose 2 numbers smaller than a hundred.

I give Steve the Sum of these two numbers,

I give Paul the Product of these numbers.

Steve and Paul talk to each other:

- Paul: I don't know them
- Steve: I knew that!
- Paul: Now I know!
- Steve: Now I know too!

Which numbers did I choose?

There's more than one right answer.

Given X and Y with each between 1 and 99, inclusive, there are 1104 products which can be formed using at least two possible X,Y pairs. Paul's product must be one of these 1104, which is why he doesn't know.

There are also 24 sums for which all possible X,Y pairs that form them also result in products that are in the set of 1104 just mentioned. Steve's sum must be one of those 24, which is how he knew that Paul doesn't know.

For Paul to now know what the factors are, his product must only have one set of factors which add up to a sum in that set of 24. There are 117 products which fit the bill.

Now for this to let Steve know what the numbers are, his sum must only have one possible product of addends in that set of 117. Unfortunately, all of this is true for three different number pairs:

(1,8) - Paul's product 8 could be 1*8 or 2*4, so he doesn't know; Steve's sum 9 has possible addend products 8, 14, 18, and 20 which are all ambiguous, so he knew; Paul knows if Steve had had 2+4=6 he could not have said "I know that" because 1*5=5 is not ambiguous; and Steve knows Paul wouldn't know Steve's sum if his product were 14 (1+14 or 2+7 would work), 18 (1+18 or 2+9 or 3+6), or 20 (1+20 or 4+5).

(7,8) - Paul's product 56 could be 1*56, 2*28, 4*14, or 7*8, so he doesn't know; Steve's sum 15 has possible addend products 14, 26, 34, 44, 50, 54 and 56 which are all ambiguous, so he knew; Paul knows all Steve's addend products would not be ambiguous if he had 1+56=57 (4*53=212 is not ambiguous), 2+28=30 (1*29=29), or 4+14=18 (1*17=17), so he could not have said "I know that" with them; and Steve knows Paul wouldn't know Steve's sum if his product were 14 (1+14 or 2+7 would work), 26 (2+13 or 1+26), 34 (2+17 or 1+34), 44 (1+44 or 4+11), 50 (2+25 or 5+10), or 54 (2+27 or 6+9).

(8,11) - This comment is already too long for me to "show my work"

At 1/14/2010 3:21 PM,  OA said...

Anonymous said...
...I think it is also very bad to rely on a calculator - there is value in being able to do math without one.

Exactly. One key problem with the calculator crutch is you have to key it right. If you mess up, how do you know it's wrong?

The same goes for delegating or outsourcing the math as someone else suggested. If you can't review and catch errors, something is going to slip through eventually.

At 1/17/2010 5:04 PM,  Scott Kerr said...

"I'm a student at the University of Washington. Keep in mind that this class is a 101 class for non-science majors. Basically a very easy class for people who don't want anything to do with math (or much of a challenge of any kind)."

I agree with the above statement. The sampling population is very biased. I am also a former UW student, as soon as I read the post I was reminded of all of my buddies in school who were not majoring in science or math; they would ask which classes they could take to get easy science credit, Atmospheric Sciences 101 always came up.

It worked both ways though, I took Architecture 150 to get art credit so I wouldn't have to compete with real art students in drawing or painting.

At 1/19/2010 10:28 PM,  Mark said...

How, professor, do you explain the inherent inability to understand the chemical properties of CO2, and implications, there at AEI? There's math involved with that equation too. Some people are number dunces, but the good news is it's curable with education. What are the odds?