You, Too, Are a Math Person; When Race Comes Into the Picture, That Has to Be Reiterated

Link to the article above.

Link to the article above.

Noah Smith's and my Quartz column "There's One Key Difference Between Kids Who Excel at Math and Those Who Don't" was also published in the Atlantic online as "The Myth of 'I'm Bad at Math.'" This is by more than an order of magnitude the most popular thing I have ever written. That might even be true for Noah. I followed up with the column "How to Turn Every Child into a 'Math Person'." Our message of "You, too, can learn math" should be clear enough, but such is the racism in our society, that when it comes to disadvantaged minorities, the message "You, too, can learn math" needs to be reiterated. Until we see the potential in everyone, regardless of race or household income, prejudice continues. 

The article linked above, "How Does Race Affect a Student's Math Education" talks about racial obstacles to everyone getting the message that the ability to do math is as nearly universal as the ability to read. Letting someone believe they can't learn to do math is like letting someone believe they can't learn to read. Either is cruel in the extreme. 

Sometimes people don't realize that math comes slowly to almost everyone. Those who spend a lot of time thinking about math get good at it. On this theme, in addition to "How to Turn Every Child into a 'Math Person'," see 

Thanks to Richard Watson for pointing me to this article. 

Miles: I wrote a follow-up column "How to Turn Every Child into a "Math Person" that gives links to some of the reactions to "There's One Key Difference Between Kids Who Excel at Math and Those Who Don't" and many resources for math learning. Here are some links to posts on math learning that didn't make it into that column:

Also, here are some Twitter discussions on math learning:

Calculus is Hard. Women Are More Likely to Think That Means They’re Not Smart Enough for Science, Technology, Engineering and Math

Believing that you can make it is an important ingredient in success. For women, that confidence is harder to come by when they are pursuing science, technology, engineering or math. A study by the Mathematical Association of America found that of those in Calculus I who initially intended to go on in science, technology, engineering or math, 

... if we restrict our analysis to just those who are earning an A or B in the course ...  18 percent of the women, but only 4 percent of the men, believed they did not understand calculus well enough to continue.

I’ll bet the phenomenon of women who are objectively doing equally well as men having much less confidence than men extends to many other moments in education in technical subjects--including training in economics. Professors and other instructors can do a lot of good by bolstering the confidence of female students who are in fact doing well in a class or in independent research. 

In my own experience, I have sometimes been quite surprised at how much my expressions of confidence in a female student in economics have meant to that student--as if that student were in a parched desert for such expressions of confidence, despite what seemed to me excellent skills. 

In addition to discouragement, low confidence in oneself also causes other people to underestimate one’s skills. It is quite difficulty to know how skilled someone is, but typically quite easy to tell how skilled they think themselves to be. So people use a job candidate’s opinion of herself or himself as a shortcut for judging her or his skills. 

For those who need to come across as more confident I highly recommend the weekend personal growth workshops conducted by Landmark Education Corporation, beginning with the Landmark Forum. In my view, almost everyone entering the dissertation writing and then job-hunting phases of getting a PhD in economics should do the Landmark Forum because of how much it will help the psychology of being able to focus on dissertation research and then the psychology of self-presentation for getting a job. I am sure that the same advice would apply to students in many other fields, at many stages of education. 

The Extensive Margin: How to Simultaneously Raise Quality and Lower Tuition at Elite Public Universities

Link to the article on the Michigan Daily website

Every year, tuition increases are in the news. What people don’t realize is that for elite public universities like the University of Michigan, there is an easy way to raise quality while lowering tuition. And it is a strategy that can make a particular university more elite while striking a blow against elitism. The strategy is to increase the size of the student body, with a careful plan that scales up everything from dorms to professors to academic advisors to socializing venues necessary to take care of the extra students. 

Increasing Quality: In talking of increasing quality, I mean in the first instance increasing the quality of the faculty, which over time has a big effect on the reputation of a university generally. A larger student body means more total tuition revenue that can pay for more professors. Even if the average quality of the professors in a department stayed the same, a larger department is likely to be perceived as stronger. (Whatever bit of research one thinks is important, it is more likely to have been done by someone in Department X if there are more total professors in Department X.) And it isn’t long before having a larger department makes it easier to recruit more impressive professors because there will be plenty of colleagues in a given field to work with in that department. 

Reducing Tuition Rates: Increasing the number of students by 50% should be accompanied by increasing the number of professors by 50% in departments that have strong enrollments, the number of professors doesn’t need to be increased that much in departments that have many low-enrollment classes to begin with. And having more students should make it possible to offer a wider range of summer classes which can have extra-low tuition because underused classrooms in the summer come at very low cost to the university. In addition, if the number of slots for out-of-state students is increased by a somewhat bigger percentage than the number of slots for in-state students, the same tuition rate for both in-state and out-of-state students will raise more money per student because of the higher fraction of out-of-state students. Note that this shift policy makes it easier, not harder, for in-state students to get admitted, because there are more slots for in-state students. 

Striking a Blow Against Elitism: The beauty of this policy is that while it makes the university more impressive academically, it strikes a blow against elitism since more total people can receive an excellent education. The imprimatur of that prestigious university is given to more total students. 

The biggest problem with the strategy is the mechanical aspect of rankings that gives credit to universities simply for turning away a lot of applicants. So an important aspect of this strategy is introducing measures of how much students know by the end of their college education. But that is something a university that cares about student learning should be doing anyway. 

To press the point, this strategy involves a genuine belief that the university actually teaches students a lot so that someone can arrive at the university not knowing much and leave it knowing a great deal as a result of what the university has done. If actually adding value is a central goal of a university rather than just choosing students who were already smart, then it needs to be measuring how much students have learned and especially how much knowledge and wisdom students have embodied in long-term memory. 

Measuring how much students have learned requires something like adding to the usual practice of student surveys at the end of a class some kind of test that students take two years or so after they have taken core classes to see if they still remember what they learned. To make it a good measure of long-term memory, students should be discouraged from cramming for that test two years later. An easy way to help avoid a temptation for the students to cram is to make their individual grades on that test two years later confidential. Of course, it is important that students try hard on that test. Some very low minimum score needed for passing should ensure that most students try reasonably hard.

Note that the student satisfaction surveys at the end of each class about their experience in the class should also be continued. Student learning may be the most important objective of university classes, but it is not the only one: it also matters how pleasant it was for students to be in the class and learn the amount that they learned. The point is to augment that measure of satisfaction with a measure of learning retained two years later. Having a measure of the success of a class based, say, 2/3 on the amount students learned and 1/3 on how much they enjoyed the experience would have a dramatic effect on how well that objective was achieved.  

Lydia Murray Interviews Me: Understanding where I am coming from, you might be interested in what Lydia Murray wrote based on her interviews with me and others in the article “Increasing tuition: a look at why costs continue to rise” for the University of Michigan’s student newspaper the Michigan Daily:

The University of California system saw a 15 percent cut in state funding between 2006 and 2013 and experienced a 52.8 percent increase in tuition revenue during that time frame. The UC system also maintained a three-year freeze in tuition between 2011 and 2014, when its Board of Regents voted for a five percent tuition increase per year for the next five years.

While it is clear that the University has increased its general fund at a much faster rate than other comparable institutions, according Miles Kimball — a University economics professor — increasing the budget is essential to maintaining a high quality of education. Kimball cited the increasing wage premium for highly skilled labor as a source for the expanding budget, as higher salaries are becoming necessary to maintain a high-quality faculty.  

Kimball said the UC system is now facing fiscal challenges because it has not adjusted its budget to rising costs.

“You can always ruin your University, and basically the UC system has been going in that direction,” Kimball said. “It is actually very troubled by the kind of budget straightjacket that it has been put into.”

Courant echoed these sentiments, citing the so-called “cost disease phenomenon” in which the cost of highly skilled labor has increased rapidly while the productivity of this group has not changed. According to Courant, the University needs to pay higher salaries to faculty members to retain them and the quality of education they provide for students.

“The relative price of University activity, of University faculty in particular, as well as other skilled labor at the University, tends to rise relative to the economy as a whole,” he said.

Alternatives to Raising Tuition

According to Kimball, the University does not need to raise tuition to increase its revenue and budget size. Instead, Kimball said the University could systematically increase enrollment so that there is a both a larger student population and a higher proportion of out-of-state students. The larger student body and the increase in the number of out-of-state students could then circumvent any need for tuition increases.

“There’s a missed opportunity where we could easily keep the tuition from going up simply by increasing the number of students,” he said. “I think that is a big missed opportunity, and, to the extent that, you miss that opportunity, you’re not doing an appropriate thing, because the appropriate thing would be to seize that opportunity.” …

“You’re going to have to increase the budget,” Kimball said. “But increasing the budget doesn’t mean that you have to increase tuition.”

One thing that didn’t make it into the article was my speculation that the University of California schools will show the damage from overly severe budget pressure more slowly than the University of Michigan would because they have better weather. I am very interested to know just how big a problem the budget pressure in the University of California system is and what strains that has caused. Perhaps it isn’t as bad as the picture I painted in my interview. If so, I hope you will correct me.  

Examining the Statistics in “Math at Home Adds Up to Achievement in School” by Talia Berkowitz, Marjorie Schaeffer, Erin Maloney, Lori Peterson, Courtney Gregor, Susan Levine and Sian Beilock

Patrick Warren’s questioning of the evidence for the value of Bedtime Math in helping kids with math (see last Thursday’s post) motivates me to state more clearly what the evidence is. The result the authors highlight most is certainly vulnerable to serious criticism: the result that among those students using the math app, math performance has a substantial correlation with how often the app is used. As Patrick points out, there is a serious problem here of omitted variable bias: how much a child likes math is likely to have a positive effect on how often the child uses the app and to have a positive effect on math performance regardless of whether the app does anything to augment performance or not. That is, kids who like math will tend to be good at math whether or not they have a math app because they will find some way to do math and thereby get better at it. (See my column “How to Turn Every Child into a ‘Math Person’”.) So that result offers very little proof of the efficacy of the app. 

However, the article has another piece of evidence that the authors should have led with. Parents were categorized into those who were anxious about math and those who weren’t. Among those parents categorized as anxious about math, the overall difference between kids’ math performance of those who received the Bedtime Math app and those who received a reading app was substantial: extra math achievement equivalent to what students get from 3 months of school. This was enough to be quite unlikely to be due to chance. The authors report a 4.8% probability of this result being due to chance, but that is for a two-tailed test that is only appropriate if one would have taken seriously a seeming finding that the kids did worse in math because of having a math app. With a more appropriate one-tailed test, there would be only a 2.4% probability of this result being due to chance. 

In other words, of the 4.8% chance of declaring a fluke a real result that the authors report, only 2.4% is the chance of declaring a fluke in the positive direction a real result. The other 2.4% is the chance of declaring a fluke in the negative direction a real result. If one is bound and determined from the beginning not to declare a seeming result in the negative direction a real result, then the overall probability of declaring a fluke a real result is only the 2.4% chance coming from a seeming result in the positive direction. (See the Wikipedia article on “One- and two-tailed tests.”)

Of course, this 2.4% p-value is truly correct only if this hypothesis of an effect on math anxious parents had been the one and only central hypothesis spelled out in advance–as could be true in a replication of the experiment

Why is this better evidence that the app does the job it was intended to do than the relationship between how much the app is used and math performance? The difference is that assignment to the math app group as opposed to the reading app group was random, and has no reasonable causal path to affect math performance than through the effects of the app itself. But amount of time spent with the app is nonrandom, and can easily reflect characteristics of a child or a child’s parents that could affect math performance in ways that don’t depend on the app at all.  

The authors genuinely don’t seem to realize the effects of the assignment to the math app or the reading app–in interaction with math anxiety on the part of the parents–represents their only piece of solid evidence for the efficacy of the app. Not only is this result not clearly described in the abstract, it is not featured in a figure. The figures are reserved for the result about usage that, as discussed above, provides very little proof of the efficacy of the app. Their language instead suggests that they are going the extra mile by doing this intent-to-treat analysis. It wasn’t the extra mile. It was the first mile! But it was a good mile. 

Here is the key passage:

We expected the math achievement of children with high-math-anxious parents to be more affected by use of the math (versus reading) app because these children would not generally be provided with high-quality math input at home (6). Therefore, we first separated parents on the basis of whether they were lower or higher in math anxiety (median split). We then performed an “intent-to-treat” analysis in which we looked at the effect of group (math versus reading app) on children’s end-of-year math achievement (controlling for beginning-of-year math achievement) independent of actual app usage. For children of high-math anxious parents, we found a significant effect of group, with children in the math group outperforming those in the reading group by almost 3 months in math achievement by school year’s end [beta-hat_21 = 5.25; t=1.99; P=0.048]. We did not find this same pattern for children of low-math anxious parents [beta-hat_31 = −0.61; t = −0.27; P = 0.79] (Model S3). An intent-to-treat analysis allows us to rule out factors possibly related to app usage—such as motivation or interest—as explaining our findings. 

This is the heart of the paper, unbeknownst to the authors. You don’t need to read anything else. 

The one thing that would make one be suspicious of this result is the possibility that the authors tinkered in various ways–including the split by math anxiety–to get the results they wanted. But that is easily remedied simply by having another group of researchers replicate the experiment. Although the substantive size of the effect is large, there is enough random variation that the statistical precision in the Berkowitz-Schaeffer-Maloney-Peterson-Gregor-Levine-Beilock experiment with 420 families is none too large. So it would be wise for someone undertaking a replication to involve at least 1000 families. The importance of the scientific question amply deserves that kind of care.

Note: Besides “How to Turn Every Child into a ‘Math Person’”, which I cited above, don’t miss the column I coauthored with Noah Smith: “There’s One Key Difference Between Kids Who Excel at Math and Those Who Don’t” and my column “The Coming Transformation of Education: Degrees Won’t Matter Anymore, Skills Will.” Also, on Bedtime Math specifically, don’t miss “Laura Overdeck: Math for Pleasure” and “Laura Overdeck: Street Math.”

DC Central Kitchen Brings Charter School Principles that Work to Adult Education that Turns People’s Lives Around

This is a heart-warming segment from the PBS newshour. But it is more than heartwarming. It is also very instructive. It suggests that the same principles that work in the “Knowledge is Power Program” or KIPP schools work for adults as well as children. As I wrote in “Magic Ingredient 1: More K-12 School” and quoted again in “The Coming Transformation of Education: Degrees Won’t Matter Anymore, Skills Will”:

“The secret recipe behind the “Knowledge is Power Program” or KIPP schools (which have been very successful even with highly disadvantaged kids) is this:

  1. They motivate students by convincing them they can succeed and have a better life through working hard in school.
  2. They keep order, so the students are not distracted from learning.
  3. They have the students study hard for many long hours, with a long school day, a long school week (some school on Saturdays), and a long school year (school during the Summer).”

Abstracting a bit from the specifics, the recipe is

  1. Give them hope
  2. Use tough love
  3. Make them work hard

By this recipe, they can gain skills that will turn their lives around.

John Stuart Mill’s Vigorous Advocacy of Education Vouchers

John Stuart Mill was a strong proponent of mandatory education. But he was a vigorous opponent of trying to push students into public schools. The key to his opposition is the key to many people’s support of pushing students into public schools: having students attend public schools is a way to get greater conformity.   

What did John suggest? Vouchers, at least for the poor. Many people think of Milton Friedman as the originator of the idea of education vouchers. But the idea is clear in paragraph 13 of On Liberty “Chapter V: Applications.”

Were the duty of enforcing universal education once admitted, there would be an end to the difficulties about what the State should teach, and how it should teach, which now convert the subject into a mere battle-field for sects and parties, causing the time and labour which should have been spent in educating, to be wasted in quarrelling about education. If the government would make up its mind to require for every child a good education, it might save itself the trouble of providing one. It might leave to parents to obtain the education where and how they pleased, and content itself with helping to pay the school fees of the poorer classes of children, and defraying the entire school expenses of those who have no one else to pay for them. The objections which are urged with reason against State education, do not apply to the enforcement of education by the State, but to the State’s taking upon itself to direct that education: which is a totally different thing. That the whole or any large part of the education of the people should be in State hands, I go as far as any one in deprecating. All that has been said of the importance of individuality of character, and diversity in opinions and modes of conduct, involves, as of the same unspeakable importance, diversity of education. A general State education is a mere contrivance for moulding people to be exactly like one another: and as the mould in which it casts them is that which pleases the predominant power in the government, whether this be a monarch, a priesthood, an aristocracy, or the majority of the existing generation, in proportion as it is efficient and successful, it establishes a despotism over the mind, leading by natural tendency to one over the body. An education established and controlled by the State should only exist, if it exist at all, as one among many competing experiments, carried on for the purpose of example and stimulus, to keep the others up to a certain standard of excellence. Unless, indeed, when society in general is in so backward a state that it could not or would not provide for itself any proper institutions of education, unless the government undertook the task: then, indeed, the government may, as the less of two great evils, take upon itself the business of schools and universities, as it may that of joint stock companies, when private enterprise, in a shape fitted for undertaking great works of industry, does not exist in the country. But in general, if the country contains a sufficient number of persons qualified to provide education under government auspices, the same persons would be able and willing to give an equally good education on the voluntary principle, under the assurance of remuneration afforded by a law rendering education compulsory, combined with State aid to those unable to defray the expense.

It is a mistake to see this issue primarily in the US context. Education in the US will probably muddle through, and may improve dramatically either because of technological progress (see my column “The Coming Transformation of Education: Degrees Won’t Matter Anymore, Skills Will”) or in a smaller way, because a Supreme Court ruling weakens the deadening hand of teachers’ unions. What is much more important is for the developing world to shift toward voucher-supported private education rather than trying to fix a public education system in which teachers very frequently fail to show up for work. Effective and inexpensive private education alternatives are springing up in many developing countries that should be encouraged by loosened regulation first and foremost, and by a share of public expenditure on education if possible. 

The August 1, 2015 edition of the Economist has a fascinating article on this:

The $1-a-week school: Private schools are booming in poor countries. Governments should either help them or get out of their way

Here are some key passages:

Although Mathare has virtually no services like paved streets or sanitation, it has a sizeable and growing number of classrooms. Not because of the state—the slum’s half-million people have just four public schools—but because the private sector has moved in. Mathare boasts 120 private schools.

This pattern is repeated across Africa, the Middle East and South Asia. The failure of the state to provide children with a decent education is leading to a burgeoning of private places, which can cost as little as $1 a week (see article).

The parents who send their children to these schools in their millions welcome this. But governments, teachers’ unions and NGOs tend to take the view that private education should be discouraged or heavily regulated. That must change.

… when public schools exist, they often fail. In a survey of rural Indian schools, a quarter of teachers were absent. In Africa the World Bank found teacher-absenteeism rates of 15-25%. Pakistan recently discovered that it had over 8,000 non-existent state schools, 17% of the total. Sierra Leone spotted 6,000 “ghost” teachers, nearly a fifth the number on the state payroll.

… private schools are innovative. Since technology has great (though as yet mostly unrealised) potential in education, this could be important. Bridge gives teachers tablets linked to a central system that provides teaching materials and monitors their work. Such robo-teaching may not be ideal, but it is better than lessons without either materials or monitoring.

Everything You Think You Know about Disciplining Kids is Wrong

Research is suggesting new approaches to helping kids behave. I would like to think that the claims here are true. But I think this research needs to be replicated by a skeptic.

Research in this area is crucial, since if behavior problems can be effectively dealt with, then a big argument people make to “Keep the Riffraff Out!” can be neutralized.