# U.S. Math Education Still in the Doldrums

(Photo: arjin j)

Every three years, the OECD, in the PISA assessment, studies 15-year-olds around the world to measure performance in reading, mathematics, and science. The results of the 2012 PISA assessment, which had a particular focus on mathematics, just came out and the United States does not fare well: “Among the 34 OECD countries, the United States performed below average in mathematics in 2012 and is ranked 26th.” I worry not so much about the rank, but about the low absolute level of proficiency to get this rank.

The U.S. students’ particular strengths and weaknesses are even more distressing:

Students in the United States have particular strengths in cognitively less-demanding mathematical skills and abilities, such as extracting single values from diagrams or handling well-structured formulae. They have particular weaknesses in items with higher cognitive demands, such as taking real-world situations, translating them into mathematical terms, and interpreting mathematical aspects in real-world problems.

Thirty-five years ago, the situation was similar. A large study of 13- and 17-year-old students across the United States, the second National Assessment of Education Progress (1977-1978), concluded: “Students appear to be learning many mathematical skills at a rote manipulation level and do not understand the concepts underlying the computation” [Carpenter, et al., Mathematics Teacher 73:329-338 (1980)]. One example was the question: “Estimate the answer to 12/13 + 7/8.” The choices were 1, 2, 19, 21, and “I don’t know.” Among 13-year-olds, 14 percent chose “I don’t know,” and only 24 percent chose 2. Among 17-year-olds, 16 percent chose “I don’t know,” and only 37 percent chose 2. The correct percentages are not much higher than the monkey line.

A frequent explanation for the U.S.’s poor PISA results is poverty — for example, by Daniel Wydo or (after the similar 2009 PISA results) by Stephen Krashen. That explanation is based on comparing wealthy U.S. schools (where fewer than 10 percent of students get free or reduce-price lunches) against the average score in countries with less poverty, such as Finland. The logic is superficially plausible, because Finland has only a few percent of students in poverty.

However, comparing wealthy students to averages of a country is a stacked comparison. The same authors who argue for the poverty explanation concede this point in other arguments, saying that it is unfair to let Shanghai’s PISA scores represent all of China, given that most of China is less wealthy than Shanghai. A fairer comparison is to compare comparable segments of the population.

That is what PISA did in its analysis of the effects of poverty. For example, see Figure II.2.6 (“Mean mathematics performance, by national quarter of socio-economic status”) on page 43 of Volume II of the PISA results (here is the data as a spreadsheet). Each country was divided into quartiles by socioeconomic status (SES), the average mathematics score computed by quartile, and then ranks computed for the top and bottom quartiles. The comparison is of top quartiles against each other, or of bottom quartiles against each other, rather than top quartiles against averages.

If poverty explained the U.S.’s poor performance, with scores of the bottom quartile dragging down and obscuring the alleged great performance of the U.S. top quartile, then the U.S. top quartile should do well internationally. However, when compared against the top quartiles in other countries, the U.S. rank drops from 26th to 32nd. Similarly, U.S. students with a SES at the OECD average do slightly worse than the U.S. average (Figure II.2.5 of the same report).

As a final example, look at the Finland and U.S. by-quartile comparison. In Finland, the bottom quartile has a SES of -0.68 (0.00 is the OECD average), and an average PISA math score of 488 (500 is the OECD average). In the United States, the second-highest quartile has an SES of 0.60 — they are much better off than the worst-off Finns — but almost the same average score, 494. Even though Finland has poor students, they do reasonably well.

In short, the “it’s poverty” explanation is not convincing. The usual U.S. math curriculum is simply weak, even in well-off schools. After decades of reform in the mathematics curriculum, students in the United States are still unprepared for full participation in society.

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1. caleb b says:

As a volunteer math tutor (and student tutor in college, and Big Brother), all of my anecdotal experience (including my own HS education) has confirmed that US students are simply not receiving the rigorous math necessary to truly understand the subject. My current 9th grade students are paralyzed by fractions and do not understand order of operations. Isolating a variable is a constant struggle as they cannot remember the single most basic principle of algebra (just do the same thing to both sides of the equation).

It is hard to know where to place the blame. I imagine that No Child Left Behind is a likely culprit, as well as school policies that overwelming favor leniency. Teachers cannot give zeros. They must ALWAYS allow work to be turned in, no matter how late. Teachers are required BY LAW to call parents twice in order to give a student a failing grade…despite the fact that parents might not have a working number or speak english.

I also imagine that part of the problem is the cultural incentive structure of acquiring good math teachers. If I were to win the lottery, I’d drop out of the corporate world and teach algebra to inner-city kids. But since I won’t win the lottery, I’m not willing to take a massive pay cut and lose future earnings just to go back and teach. So the field of math teachers is comprised of those that are smart enough to make solid corporate money, but choose to forgo all that money for the love of teaching. I can imagine that this pool is faily limited.

One final thought…although each generation believes the next is the ruin of society, I think this generation of kids really ARE the ruin of society. They can’t stay off their phones for even a few minutes and their attention spans and long-term memory are almost non-existent. The smart kids are still smart, but the average kid tries 1/3rd as hard as 20 years ago and expects nothing lower than a B in every class.

Well-loved. Like or Dislike: 44  9
• Taylor S. Marks says:

It’s been my observation that nearly nobody who owns a smart-phone has the ability to leave it down – there is no generational divide in this, except that fewer elderly citizens own smart phones than younger citizens.

It seems to me what we need are better teachers. Which I would guess means being quicker to fire teachers that perform poorly and increasing the salary we offer to teachers, so that we might lure in better ones.

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• Average.Random.Joe says:

I always find it ironic that teachers resist being graded.

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• nborlaug says:

Every time I hear someone of your generation complaining about the youth of today, I hope you remember that baby boomers are objectively a worse generation. Your generation inherited an incredible post war boom, a solid stock market, and yet you’ve managed to bankrupt the nation with a social security system that simply will not be available for us. Almost every dollar you take in social security will simply be stolen from young workers of today. But yeah, I guess we do use our phones a lot. Sorry about that.

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• caleb b says:

for the record: I’m 32 and my complaint is about 9th graders and the general decline of math skills I’ve personally witnessed over the last 10 years.

I’m not sure what a bankrupt SS system has to do with freshmen not being able to learn basic Algebra I over two academic semesters. I am sure it has something to do with their addiction to their phones though…

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• Average.Random.Joe says:

“although each generation believes the next is the ruin of society, I think this generation of kids really ARE the ruin of society.”

My dad has said this and then states that his grandfather used to say almost to the word this same thing. Written language was looked down on as people will not need to remember the oral history and their memory will dry up. I think we are confident and can agree that books have been a net benefit. Sure, people didn’t have as good as a long-term memory but they had more cognitive resources to spend on other things that are more productive. Same with the internet. Why memorize when I can look it up in 20 seconds? Learning context and how to look things up and using the tools should be focused on more than attention spans and memory capacity. Every older generation thinks the last tries less hard but only because tech made less work produce more so it looks like they are doing less. I assure you that I did significantly less homework in Kindergarten than my kids did. And every parent I have talked to feels the same way.

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2. josh says:

The SES lower quartile of students in Finland really can’t be compared to the SES lower quartile of students in the U.S. I don’t have data in front of me, but I’d hypothesize the variability of poverty is the largest in the world so it’s misleading to compare the countries this way. It’s better than just looking at the countries but it’s still flawed.

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• Chris Chan says:

I agree with what Josh said. There are several ways to measure poverty; what the OECD uses is the relative distance from a percentage of the median household income of a particular country. And there are several factors that contribute to well-being – and by extension, academic performance – that vary greatly among countries.

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• Enter your name... says:

If someone had said, let’s compare the bottom quartile of kids in Darfur against the bottom quartile of Americans, you can bet that everyone would say that this was invalid because it’s a different type of poverty: nearly all poor American kids, even at the lowest level, can count on eating some food every day, and getting some medical care, and so on—and Darfur orphans can’t. We’d all say not to trust that one, but instead to compare kids in similar situations: those who have stable homes and food security and medical care against other kids who also have stable homes and food security and medical care.

But that’s not really what happens with the Finland vs USA comparison. We’re comparing 25 Finns, two of which have serious problems, against 25 Americans, ten of which have serious problems. While the PISA study is better than nothing, you really do need more sophisticated measurements to do this properly.

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• steve says:

They don’t compare the bottom quartile of Fins to Americans – the article just explained why that doesn’t work. They compare the bottom quartile of Fins with the second quartile of Americans. Similarly, what disadvantage is there when the top quartile of Americans are compared against the top quartile of other countries? I’d imagine our top 25% are pretty well off compared to the rest of the world, and we still finished 32nd of 34.

Unfortunately, I don’t have a solution either. With math, I feel like the key lies in repetition. Start them early (multiplication and division at 8) and often. Also, in high school, there should also be a required course that stresses using math in real world situations, i.e. paying bills, tipping (though I know Freakonomics is anti), interest rates, etc. This won’t be enough, but they would be positive changes to the system.

Last note, maybe we could have a system like the lawyers in NY who need to work their first year pro-bono. Grads from engineering or related fields could teach math for a year before joining the workforce… (of course that invites a host of other problems)

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3. Dr. Bob says:

It’s amazing what folks will publish and not carefully look at the research on poverty and achievement. You would think they would have learned their lesson on data analysis after the last presidential election.

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4. James says:

I suggest that there may be a confusion of cause and effect: rather than poverty causing poor math scores,poor math scores cause poverty.

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• SC says:

Seems more like a “chicken or egg” situation to me…

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• James says:

Not really, because the ability to for instance comprehend simple* interest rate calculations can save you a whole bunch of money. Which, invested wisely, can make the difference between continued proverty and future prosperity.

*I mean that the calculations are simple, not that people shouldn’t understand compound interest as well

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• Enter your name... says:

SC, I’m betting that you’ve never worked in a place that sold lottery tickets. Poor math skills *cause* poverty (not the only cause, of course, but definitely one of them).

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• Lulu says:

So…getting a fleet of whiz-bang brilliant math teachers into every inner city school and turning all the kids into math WIZARDS is going to end poverty in our time. Brilliant! Let’s get right on that plan!

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• James says:

Hey, it worked for me! Not inner city, but being real good at math helped take this poor kid from the backwoods to jobs at world-class research institutes.

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• Linch says:

Your analysis would be much more convincing if this wasn’t a test of 15 year olds, most of which (shockingly) don’t have a full-time job.

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• Linch says:

Simply put, I fail to see how being bad at mathematics could cause children to be born into families with low income.

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• James says:

Linch, you have cause & effect reversed. The families were low income in part because the parents were bad at math. So the kids grow up in a culture in which they get the message “math is hard” dinned into them. (And where most other intellectual pursuits are likewise denigrated.) So if kids listen to those messages, they’ll likely grow up to be poor in turn.

In all honesty, I’ve come to think that the main reason I managed to escape this cycle is that I was just far enough out on the Asperger’s/autism spectrum to be pretty much oblivious to all the social cues telling me to stay dumb.

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5. Julien Couvreur says:

Russ Robert of EconTalk had a recent episode on education, where he made an interesting observation about the expectations and culture of parents.
In the US, how do most parents supplement their kids education? Extra-curricular sports, arts or maths?
That is an indication of what they feel their child can benefit most from.

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6. kip says:

a^4 + b^4 + c^4 = 37

Solved by an American!

Well-loved. Like or Dislike: 14  1
• Celine says:

How did you solve it? I’m less good at inequalities than I would like to admit.

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• Tristan says:

square a^2+b^2+c^2 and raise a+b+c to the fourth power. Then do algebra.

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• Gabe says:

Answer is right here, under the photo’s flickr account: http://www.flickr.com/photos/arjin/3272274144/in/photostream/

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• J Tepp says:

There are no inequalities in this problem

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• Enter your name... says:

Celine, he didn’t give you enough information to find out what a, b, or c could be. Without a couple of other equations, you can’t actually figure out what each of the variables stand for. Given just this one equation, there are multiple answers the fit. For example, a=1, b=0, and c=quad root of 36 fits; so does a=1, b=2, and c=quad root of 20.

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• Tristan says:

Can confirm answer. Also American.

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• Linch says:

a=0 b=0, c= fourth root of 37.

Do I get a prize?

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• kip says:

0 + 0 + 37^(0.25) = 37^(0.25), which does *not* equal 0. no prize!

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7. RGJ says:

We are probably the most culturally mixed country on that list. It is much easier to educate a homogenous population of children than it is one from a mixed base of ethic and cultural influences. Just is. Mutts don’t win the dog shows.

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• James says:

The only reason mutts don’t win dog shows is that the rules don’t allow them to be entered.

Well-loved. Like or Dislike: 5  0
• Average.Random.Joe says:

Yes, his quip was dumb as it doesn’t address the point but his point is valid. Diversity is going to be a factor in education learning styles. Language and culture have been shown to have natural tenancies to have greater proficiencies in different areas. Also, relating different concepts using things familiar to each child is a greater challenge the greater the diversity. And this is completely relevant to the topic because when we talk about the comparative ability among countries and the system structure, changing that structure may not help. Adopting other systems wouldn’t be appropriate and I am not sure that segregating people by ethnic or culture would be either. We can then address a solution that would be acceptable that would work with that issue.

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8. Paul Miller says:

I am a junior college mathematics instructor. It seems as though more and more colleges are adding remedial math courses to their curricula because fewer students are proficient enough to get into college level courses. One of the biggest discussions in community colleges is how we can increase retention without decreasing rigor. But as the skills of incoming freshmen seem to decline, that retention battle becomes harder and harder.

College enrollment data indicates that students who begin in remedial courses have a much lower graduation rate. Could it be that some colleges will go the other direction and cut remedial programs alltogether in favor of boosting retention numbers?

Many of my students ask why they have to learn math or comment that they won’t remember any of it in a few years. Maybe so…but will there come a day that basic arithmetic is a skill ONLY possessed by STEM’ers? What would that look like? Does the ability to add fractions translate into real life success & prosperity?

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• Enter your name... says:

> Does the ability to add fractions translate into real life success & prosperity?

It depends on what you think means success:

Will your students ever try to bake a cake from scratch, or to adjust a recipe up or down to fit their baking pan/expected number of guests? Will they need to figure out whether their mixing bowl is big enough to hold the cake batter?

Similarly, the practical reason to learn how to multiply fractions is so that you can figure out how much three and a half pounds of apples will cost you when they’re on sale, or how much carpet they need to buy if they want to carpet a hallway that is three and a half feet wide and eight and a half feet long. If they’re not seeing the everyday uses, then you should consider changing the topics in your story problems.

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• Paul Miller says:

I am speaking about something more fundamental than story problems. My question was more about the skill of adding fractions by itself. Students in remedial math won’t even touch story problems if they cannot perform the skill by itself.

I have noticed that students are getting really proficient at using technology to solve math problems. I think this should be celebrated. Maybe our focus for remedial math should be on teaching some common sense fundamentals and leave the heavy lifting to a calculator. Take that problem in the article for example, “Estimate the answer to 12/13 + 7/8.” Many of my incoming students would struggle because the first two things they remember from learning fractions is that they hate fractions and fractions are a lot of work. If they can successfully complete remedial math they should have no problem solving a problem like the above…but so many never finish. Could we focus the attention to the fundamental concepts of fractions so when they look at them they sense the size of the fractions, understand the operation, and can quickly calculate the EXACT sum on a calculator?

Five to ten years after graduation most 4-year college graduates (outside of STEM) would struggle with problems containing fractions if they have to work them out by hand. That is because it is a skill they do not have a chance to use and keep sharp. There are exceptions like construction and culinary jobs, but for the most part I rarely meet someone who uses them.

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