
Knowledge of fractions and long division predicts long-term math success - ColinWright
http://www.sciencedaily.com/releases/2012/06/120615114057.htm
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Mvandenbergh
The way mathematics is generally taught, once you fall behind you're basically
done. If you get really lost in 5th grade math, you'll spend the next 5-7
years (depending on whether your school system requires you to take a math
class every year of high school) barely scraping by and getting promoted every
year to a new math class while falling ever further behind. Finally, you'll
reach the happy point where despite 10-12 of mathematical instruction, you
basically don't know any mathematics past basic operations. Much better to
adapt to the level of the learner and the speed they can work at - I'd rather
have someone genuinely understand all the material that they've nominally
learned than get barely passing grades on what looks like more material on
paper. Unfortunately, a system of discrete math classes broken up by grade
doesn't work any better for slow math learners than for gifted math learners.

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droithomme
I've mentioned before here that I have taught college, high school, and
elementary school. Also adult remedial education. The findings here are well
known and have been for a long time, they come up all the time in the
placement exams used in community colleges. Most people can do addition,
subtraction and some multiplication. Fractions, long division, and everything
that follows is a complete void for the majority of american adults.

Their conclusions are wrong though. It's not that they weren't taught
fractions well enough. It's that the educational system is broken. Kids start
doing badly in math at fifth grade. The reason is that's the age where they
are old enough to start to realize on some level their time is being wasted,
the schools are exhausting and the materials probably useless. Too much time
is spent in school pandering to the slowest student. Classes are no longer
segregated by ability. Fast students become bored, slow students never catch
up, and the rest just misbehave or check out mentally.

To fix the problem, you don't need to push fractions and long division (which,
true, most elementary school teachers do not really understand) harder, you
need to prevent the students from burning out and losing interest by age 10.

We don't see these results in adults in other countries because they don't
burn out their students at such an early age.

The US schools are unlikely to reform for bureaucratic and political reasons.
Some argument will be contrived to link fractions to pay, or require higher
salaries and funding in order to teach fractions properly. Perhaps yet more
computers in the classroom will be proposed as the answer: truckloads of iPads
for everyone! This will not solve the problem though.

If you want your kids to learn math properly, tools like Khan Academy are
good, especially their web based hierarchical exercise software. This allows
students to proceed at exactly the pace they need, independently of every
other student.

If you are starting from the beginning, use a sensible curriculum like the
Singapore Math series. By fourth grade though you should have a teacher who
understands math better than the average american to assist students with it
as needed. This is not possible to have in most school districts and is not
going to change since people are unwilling to basically burn the schools to
the ground and start over (fire everyone and requiring teachers to pass
competency exams before rehiring, also eliminate 90% of bureaucracy and rules
that impose restrictions on teaching). So things will continue as they are in
the schools.

~~~
yuushi
I disagree heavily with your argument. If the student is struggling with
fractions and long division, I don't think you can blame schools "pandering to
the slowest student" for such a problem. If they are struggling so much with
basic concepts, I'd have to argue that they are the slow students.

> We don't see these results in adults in other countries because they don't
> burn out their students at such an early age.

If you've looked at education in other countries, I don't understand how you
can say this. Many places push their students far harder than the US (for
example, Singapore, Japan, South Korea). Rote memorization is pushed far
harder as well. How is the material that is taught any less "useless" in the
eyes of students in these countries?

I think culture definitely comes into play here. I'm not going to say
incompetent teachers don't exist or that there isn't too much bureaucracy in
schools; there clearly is, however, laying the blame entirely at their feet is
being dishonest. I think culture and parental expectations play a larger part
than many people want to admit.

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kjhughes
Performance on fractions and long division predicts math "success" because
those tasks embody math education's emphasis on procedural skills over
conceptual understanding.

Tokenadult's comprehensive comment includes the 12/13 + 7/8 gem that sums up
the problem. Kids see 12/13 + 7/8 and, if they're "good" at math, start
turning the procedural crank they've been taught. (If they're "bad" at math,
they freeze with a defeated feeling that there are too many procedural steps
to remember.)

Get kids to see 12/13 + 7/8 primarily as "almost one plus almost one" (as
thaumasiotes describes), and only secondarily as the beginning of a tedious
procedure, to make substantive progress in math education.

~~~
ajuc
A few years ago there was a long article about math education in USA by
Russian math professor that immigrated. He blamed lack of "word problems"
solved by intuition and guessing and substituting values at first, only then
using algebra.

I can't find it now, but it showed exactly the same problems - math was just
pattern mathing problem for most students - no real understanding.

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yk
"We need better teaching of fractions" sound like a dangerous cargo-cult
solution to the problem of poor understanding of math. If the teaching of
fractions would be better, than the understanding of fractions would simply
become a worse predictor of long term math success.

The main problem of math teaching is not, that students do not understand
fractions (they could easily learn them later), but that they do not
understand that math is a (sometimes really helpful) way to think about
problems.

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Someone
I haven't read the paper itself, but that text, to me, does not give a _"clear
message [...] that we need to improve instruction in long division and
fractions"_.

The text talks about correlation between the two, but does not give any
arguments for causation. Judging from the title of the paper ("Early
Predictors of High School Mathematics Achievement"), neither does that paper.

~~~
SudarshanP
People who derailed in 5th grade are probably unlikely to get back on track
because education in the higher classes stands on the shoulders of the
previous classes.

If it not a causation, this may also mean that "non motivated students" never
care about math. Either in 5th grade or in high school or later.

Or maybe it is a combination of both. The article only shows correlation. But
how can one study causation without destroying the futures of students?

~~~
Someone
Make two groups; give one of them some extra and/or supposedly improved math
schooling. Check their math grades and, years later, their success in higher
education. See how much they correlate.

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thesteamboat
> The clear message is that we need to improve instruction > in long division
> and fractions

I guess it's my turn to be the guy who says `correlation does not imply
causation'. It's clear that something in math education is broken, but I feel
like the more important point lies in the next sentence:

> At present, many teachers lack this understanding [of rudimentary
> mathematics].

I feel like this is much more important than a renewed emphasis on
fractions/long division.

Full quote for context:

> "The clear message is that we need to improve instruction in long division
> and fractions, which will require helping teachers to gain a deeper
> understanding of the concepts that underlie these mathematical operations.
> At present, many teachers lack this understanding. Because mastery of
> fractions, ratios and proportions is necessary in a high percentage of
> contemporary occupations, we need to start making these improvements now."

------
tokenadult
Poor teaching of fraction arithmetic in elementary schools has been a pet
issue of mathematics education reformers in the United States for a long time.
Professor Hung-hsi Wu of the University of California Berkeley has been
writing about this issue for more than a decade.

<http://math.berkeley.edu/~wu/>

In one of Professor Wu's recent lectures,

<http://math.berkeley.edu/~wu/Lisbon2010_4.pdf>

he points out a problem of fraction addition from the federal National
Assessment of Educational Progress (NAEP) survey project. On page 39 of his
presentation handout (numbered in the .PDF of his lecture notes as page 38),
he shows the fraction addition problem

12/13 + 7/8

for which eighth grade students were not even required to give a numerically
exact answer, but only an estimate of the correct answer to the nearest
natural number from five answer choices, which were

(a) 1

(b) 19

(c) 21

(d) I don't know

(e) 2

The statistics from the federal test revealed that for their best estimate of
the sum of 12/13 + 7/8,

7 percent of eighth-graders chose answer choice a, that is 1;

28 percent of eighth-graders chose answer choice b, that is 19;

27 percent of eighth-graders chose answer choice c, that is 21;

14 percent of eighth-graders chose answer choice d, that is "I don't know";

while

24 percent of eighth-graders chose answer choice e, that is 2 (the best
estimate of the sum).

I told Richard Rusczyk of the Art of Problem Solving about Professor Wu's
document by email, and he later commented to me that Professor Wu "buried the
lead" (underemphasized the most interesting point) in his lecture by not
starting out the lecture with that shocking fact. Rusczyk commented that that
basically means roughly three-fourths of American young people have no chance
of success in a science or technology career with that weak an understanding
of fraction arithmetic.

Plenty of other mathematics teachers in the United States have noticed adult-
age students who have trouble with elementary fraction arithmetic.

<http://www.brianrude.com/fractionsquiz2.htm>

Professor Wu has written other important articles about what needs to be
reformed in United States mathematics education.

<http://math.berkeley.edu/~wu/Lisbon2010_2.pdf>

<http://math.berkeley.edu/~wu/NCTM2010.pdf>

<http://math.berkeley.edu/~wu/NoticesAMS2011.pdf>

<http://math.berkeley.edu/~wu/CommonCoreIV.pdf>

Other mathematicians who have written interesting articles about mathematics
education reform in the United States include Richard Askey,

<http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf>

<http://www.math.wisc.edu/~askey/ask-gian.pdf>

Roger E. Howe,

<http://www.ams.org/notices/199908/rev-howe.pdf>

Patricia Kenschaft,

<http://www.ams.org/notices/200502/fea-kenschaft.pdf>

and

James Milgram.

ftp://math.stanford.edu/pub/papers/milgram/milgram-msri.pdf

ftp://math.stanford.edu/pub/papers/milgram/report-on-cmp.html

All those mathematicians think that the United States could do much better
than it does in teaching elementary mathematics in the public school system.
Several of them identify lapses in teaching fraction arithmetic as a major
issue. I think so too after living in Taiwan twice in my adult life (January
1982 through February 1985, and December 1998 through July 2001). Taiwan is
not the only place where elementary mathematics instruction is better than it
is in the United States. Chapter 1: "International Student Achievement in
Mathematics" from the TIMSS 2007 study of mathematics achievement in many
different countries includes, in Exhibit 1.1 (pages 34 and 35)

<http://pirls.bc.edu/timss2007/PDF/T07_M_IR_Chapter1.pdf>

a chart of mathematics achievement levels in various countries. Although the
United States is above the international average score among the countries
surveyed, as we would expect from the level of economic development in the
United States, the United States is well below the top country listed, which
is Singapore. An average United States student is at the bottom quartile level
for Singapore, or from another point of view, a top quartile student in the
United States is only at the level of an average student in Singapore. I've
been curious about mathematics education in Singapore ever since I heard of
these results from an earlier TIMSS sample in the 1990s. I have seen the
textbooks used in Singapore (and have used those to teach my own children,
including a grown child who is now a hacker) and own many of the textbooks
used in Taiwan and China (as I read Chinese). The United States could plainly
be doing better in elementary mathematics education.

The article "The Singaporean Mathematics Curriculum: Connections to TIMSS"

<http://www.merga.net.au/documents/RP182006.pdf>

by a Singaporean author explains some of the background to the Singapore
mathematics materials and how they approach topics that are foundational for
later mathematics study. I am amazed that persons from Singapore in my
generation (born in the late 1950s) grew up in a country that was extremely
poor (it's hard to remember that about Singapore, but until the 1970s
Singapore was definitely part of the Third World) and were educated in a
foreign language (the language of schooling in Singapore has long been
English, but the home languages of most Singaporeans are south Chinese
languages like my wife's native Hokkien or Austronesian languages like Malay
or Indian languages like Tamil) and yet received very thorough instruction in
mathematics. It would be good for the United States to take advantage of its
greater degree of linguistic unity and childhood wealth to reach the
educational standard of the top-performing countries in other parts of the
world.

<http://www.pisa.oecd.org/dataoecd/50/9/49685503.pdf>

<http://www.pisa.oecd.org/dataoecd/17/26/48165173.pdf>

~~~
simonbrown
It seems strange that "2" was listed after "I don't know".

~~~
jccc
Most likely it wasn't that way on the test. They just saved it for last in the
report.

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raverbashing
Let me call BS on this one

Does anyone remember how to do long division by hand today? I certainly can't

But math is much more than your "grocery shopping math"

You don't need a calculator for most of math. And I really find it difficult
to correlate "how to do long division" in success in areas like statistics,
number theory and even calculus, because it's mostly concepts, not "1+1"

~~~
wcarss
I am not a mathematician and I do not do long division on any regular basis. I
certainly haven't written it out in over a year.

But occasionally I have to think about whether a number like 365 is divisible
by a number like 7, and I don't have a calculator or computer beside me. I can
wait until I'm around electronics, or I can think:

"How many times does 7 go into 36? 5 times." "That makes 35, so now, there's
15 left." "How many times does 7 go into 15? 2 times." "That makes 14, so I'm
at 364 -- one day is left over." "7 went into 365 52 times, with one left
over."

That is a useful algorithm. You can use that to answer questions about the
world. It seems reasonable that if you're not capable of accomplishing that,
then you're not going to do well in an environment that requires you to solve
problems.

~~~
raverbashing
I agree, and of course can do this 'approximation' by hand

(even though I may do it backwards sometimes, like, 365/7 ok, 365 looks like
700/2 so that's a start, or 7x5 = 35 so 7x50 = 350 so there goes)

What I meant to say is that math is not only about juggling numbers

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jvdh
I wonder if this has to do with the spatial aspect. Americans use that awkward
imperial system, which makes it really hard to easily chop up distances.
Almost all other countries use the metric system, which makes it very easy to
understand dividing lengths.

Remember that for ordering and numbers many people use a 2-dimensional line to
represent things in their head.

~~~
maxerickson
The splitting out over more units is a little awkward, but it is usually
possible to quickly divide imperial units by 2,3 and 4 (of course it is easy
to divide decimals by 2 and 4, my point is that imperial doesn't lose this).

For people frequently working with medium distances, all they have to do is
learn that a mile is 80 chain (for longer distances, decimal miles are about
equivalent to decimal km).

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Paul_S
I'm unfair judging this by the press article rather than the study itself but
isn't that like saying kids who are smarter do better at math later in life?
IQ tests of small kids are wildly inaccurate because of how kids develop at
different speeds so how can you control for it?

But whatever argument they need to get to teach better maths to kids I'm all
for it.

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boboblong
What we need to do is throw away the idea that smaller class sizes are always
better, at least for mathematics. I have nothing against primary- and
secondary-school teachers--god bless 'em--but we are letting their egos get in
the way of educating our children. If they don't really understand the
material themselves--and many of them don't--they need to admit it and let
their kids go to someone who does understand the material when it comes time
to teach math for the day.

I don't care if there are so few people who both understand math and can teach
it to children (and are willing to take the job) that all of our math classes
consist of 500 students. If that's the case, bring in bright highschoolers
(who also understand the material) either as volunteers or part-time workers
to act as "floaters". Let the class consist of one highly-paid individual with
a math degree and passable teaching ability and half a dozen low-paid
teenagers who understand fractions and are good with kids.

Of course, this would require many adult schoolteachers to admit that, when it
comes to elementary math instruction, they are totally unqualified. That's a
hard thing for a teacher--or, for that matter, an adult--to admit.

