Knowledge of fractions and long division predicts long-term math success

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.

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

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.

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.

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.