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 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.
> push fractions and long division (which, true, most elementary school teachers do not really understand)
Wow.. Really? Then how are they going to teach anything to the students except doing + - *? How about trigonometry, or even Tales Theorem? Hell, how about percentages? That's too hard and not usefull, too?
Is this generally accepted? For me it seems like the fact that parents would riot over?
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.
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.
"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.
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.
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?
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.
> 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."
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.
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.
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)
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"
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.
>> 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.
Looking at the US-flattering 4th grade scores, it appears that the Singapore average is 599, whereas the US Asian average is a pitiful 582. 8th grade sees the US suffer: Singapore declines to 593, and US Asians to 549. I don't know how to get a breakdown of US distributional information by race, but your language is, at best, highly misleading. The US 8th grade average all-included is 508, far below Singapore, but I think Singapore (and Taiwan!) are a little closer to 100% Asian.
I've heard nothing but great things about Singapore math, and on a gut level I'm shocked too that US 8th graders were barely able to approximate "almost one plus almost one", but in the big picture it's hard to conclude that our education is failing.
I don't know how to get a breakdown of US distributional information by race, but your language is, at best, highly misleading.
The last time I saw a comment of this nature in a thread on the subject of mathematics education in the United States and east Asia (both places I have lived), someone advised me not to feed the troll. But that was a different troll, and taking your comment, even where you incorrectly say "highly misleading" about my comment, as an attempt to advance the discussion, I'll invite onlookers to look at the evidence.
and anyone who takes a look at Exhibit 1.1 of that link (on pages 34 and 35 of the .PDF document), which is a good example of a comparative data distribution display, can see how the national median level of performance in the United States compares to the bottom quartile level for Singapore, and on the other hand where the top quartile line for the United States appears compared to the median line for Singapore. Q.E.D.
Unfortunately, once upon a time a blogger ignorant of the large body of research on textbook content and classroom practice in different countries for elementary mathematics in different countries of the world
took the lazy way out and said that if "race" is taken into account, then the United States is second to none in provision of public education, which is simply a lie. That meme has spread through some politically tendentious blog networks, but every serious professional researcher on comparative education policy can, and does, point to more meaningful differences between the United States and other countries. It would have helped that blogger also to be more familiar with the huge literature on "race" issues in countries all over the world,
but let me just disagree with the suggestion in your comment by pointing that nobody who makes the suggestion made by the blogger has actually gathered the data to show all the steps to prove that "race" as such makes any difference at all in educational attainment. Meanwhile I have taken care, in links already shown in my first comment above to document both the known inferiority of provision of primary education to some "race"-defined groups in the United States
and the degree to which other countries outperform the United States in providing primary education to the most disadvantaged groups in each of those countries.
the United States is conspicuous in how little it meets the educational needs of its strongest students in mathematics.
in the big picture it's hard to conclude that our education is failing
There is certainly room for semantic disagreement about how bad performance has to be before it is regarded as "failing" performance, but I note for the record that the United States has abundant resources devoted to K-12 schooling
but underperforms compared to what other countries do with less abundant resources. I didn't use the word "fail" or "failing" or "failure" in my comment, but I did suggest, and I think I suggested this with warrant, that United States schools could do a better job of teaching fraction arithmetic to the young people in their care.
...took the lazy way out and said that if "race" is taken into account, then the United States is second to none in provision of public education, which is simply a lie.
As is his norm, tokenadult ignores the data and vaguely appeals to authorities while attacking straw men.
Thaumasiotes didn't claim the US was #1, he merely pointed out that most of the gap Tokenadult cited is caused by student quality, not education policy. Nothing tokenadult has cited (here or elsewhere) addresses this point.
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"
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.
Or to answer that one you can just think "Hey, can I reuse the same calendar two years in a row?" and if the answer is no, then 365 isn't divisible by 7.
I don't see long division as some algorithm to be memorized. It is a fairly simple application of the theorem:
Dividend = Divisor * Quotient + Remainder. and the distributive law.
I was never taught long division as some algorithm to be followed and knowing what it accomplished and how has been enough to perform the operation with ease.
Contrasting that with stuff from later on in my life, I am not so confident with the math I picked up starting high-school. I guess I began following an approach that asked me to commit to memory - focusing on grades as opposed to concrete understanding.
I consider myself pretty decent at math and I would have to look it up if I needed to do long division. It wouldn't take me terribly long to do figure it out, but I don't currently "know" how to do it. Hasn't prevented me from being successful in mid- to upper-level college math yet.
Agreed. I never learned long division, and now I'm getting a PhD in engineering. Computers are good at arithmetic - let them do it! Teach your kids to code instead.
My SAT Math score is 660, ACT Math is 26, I'm in average-level math classes and usually wind up with an A- or B+.
I've done MIT Intro to Computer Science on OCW and am currently working on Stanford Algorithm Design and Analysis on Coursera. MergeSort took maybe 10 minutes to implement in Python. I have little trouble with recursion (now), and am decent at understanding and mentally modeling complex systems (i.e. AP Chemistry).
But The Powers That Be have decided that the rapid and accurate selection and application of procedures to small, contrived "problems" are the best measure of suitability for a STEM education, so I have little to no shot anywhere besides my state's JavaSchool. Maybe they're right.
But from experience, I will say this: when you teach or test sorting algorithms, you're teaching and testing ideas and how students think. When you teach and test fractions, you're teaching a procedure and testing a student's ability to execute it quickly and accurately without getting bored or transposing digits.
IMHO, the relative weights we put on those completely separate abilities are backwards.
(I have never had trouble with the ideas behind fractions, but make me find the common denominator ten times in 5 minutes and I'll probably screw up once or twice.)
I never said you shouldn't try implementing division yourself; everyone should try implementing everything themselves someday. I'm saying that anyone can, if they wish, do a lifetime of useful work (including writing "real-time 3D engines") without ever knowing or implementing a division algorithm. They will start with anything they want.
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.
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).
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.
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.
