This column originally appeared at RealClearEducation.com on July 29, 2014

Over the weekend the New York Times Magazine ran an article titled “Why do Americans Stink at Math?”  by Elizabeth Green. The article is as much an explanation of why it’s so hard not to stink as an explication of our problems. But I think in warning about the rough road of math improvement, the author may not have even gone far enough.

The nub of her argument is this. American stink at math because the methods used to teach it are rote, don’t lead to transfer to the real world, and lead to shallow understanding. There are pedagogical methods that lead to much deeper understanding. U.S. researchers pioneered these methods and Japanese student achievement took off when the Japanese educational system adopted them.

Green points to a particular pedagogical method as being vital to deeper understanding. Traditional classrooms are characterized by the phrase “I, we, you.” The teacher models a new mathematical procedure, the whole class practices it, and then individual students try it on their own. That’s the method that leads to rote, shallow knowledge. More desirable is “You, Y’all, We.” The teacher presents a problem which students try to solve on their own. Then they meet in small groups to compare and discuss the solutions they’ve devised. Finally, the groups share their ideas as a whole class.

Why don’t US teachers use this method? In the US, initiatives to promote them are adopted every thirty years or so—New Math in the 60’s, National Council of Teachers of Mathematics Standards in the late ‘80’s--but they never gain traction. (Green treats the Common Core as another effort to bring a different pedagogy to classrooms. It may be interpreted that way by some, but it’s a set of standards, not a pedagogical method or curriculum.)

Green says there are two main problems: lack of support for teachers, and the fact that teachers must understand math better to use these methods. I think both reasons are right, but there’s more to it than that.

For a teacher who has not used the “You, Y’all, We” method it’s this bound to be a radical departure from her experience. A few days of professional development is not remotely enough training, but that’s typical of what American school systems provide. As Green notes, Japanese teachers have significant time built into their week to observe one another teach, and to confer.



Green’s also right when she points out that teaching mathematics in a way that leads to deep understanding in children requires that teachers themselves understand math deeply. As products of the American system, most don’t.

Green’s take is that if you hand down a mandate from on high “teach this way” with little training, and hand it to people with a shaky grasp of the foundations of math, the result is predictable; you get the fuzzy crap in classroom that’s probably worse than the mindless memorization that characterizes the worst of the “I, we, you” method.

But I think there are other factors that make improving math even tougher than Green says.

First, the “You, Y’all, We” method is much harder, and not just because you need to understand math more deeply. It’s more difficult because you must make more decisions during class, in the moment. When a group comes up with a solution that is on the wrong track, what do you do? Do you try to get the class to see where it went wrong right away, or do you let them continue, and play out the consequences of the their solution? Once you’ve decided that, what exactly will you say to try to nudge them in that direction?

As a college instructor I’ve always thought that it’s a hell of a lot easier to lecture than to lead a discussion. I can only imagine that leading a classroom of younger students is that much harder.

There are also significant cultural obstacles to American adoption of this method. Green notes that Japanese teachers engage in “lesson study” together, in which one teacher presents a lesson, and the others discuss it in detail. This is a key solution to the problem I mentioned; teachers discuss how students commonly react during a particular lesson, and discuss the best way to respond. That way, they are not thinking in the moment, but know what to do.

The assumption is that teachers are finding, if not the one best way to get an idea across, then a damn good one. As Green notes, that often gets down to details such as which two digit numbers to use for particular example. An expectation goes with this method; that everyone will change their classroom practice according to the outcome of lesson study. This is a significant hit to teacher autonomy, and not one that American teachers are used to. It’s also noteworthy that there is no concept here of honoring or even considering differences among students. It’s assumed they will all do the same work at the same time.

The big picture Green offers is, I think, accurate (even if I might quibble with some details). Most students do not understand math well enough, and the Japanese have offered an example of one way to get there. As much as Green warns of the challenges in Americans broadly emulating this method, I think she may underestimate how hard it would be. It may be more productive to try to find some other technique to give students the math competence we aspire to.

"Stereotype threat" refers to a phenomenon in which people perform worse on tasks (especially mental tasks) in line with stereotypes, if they are are reminded of this stereotype.

Hence, the stereotype for women (in American culture) is that they are not as good at math as men; for older people, that they are more forgetful than the young; and for African-Americans, that they are less proficient at academic tasks. Members of each group do indeed perform worse at that type of task if the stereotype is made salient just before they undertake it (e.g. Appel & Kronberger, 2012).

Why does it happen? Most researchers have thought that the mechanism is via working memory. When the stereotype becomes active, people are concerned that they will verify the stereotype. These fears occupy working memory, thereby reducing task performance (e.g., Hutchison, Smith & Ferris, 2013).

But a new experiment offers an alternative account. Sarah Barber & Mara Mather (2013) suggests that stereotype threat might operate through a mechanism called regulatory fit. That's a theory of how people pursue goals. If the way you conceive of task goals matches the goal structure of the task, you're more likely to do well than if it's a poor fit.

Stereotype threat makes you focus on prevention; you don't want to make mistakes (and thus confirm the stereotype). But, Barber & Mather argue, most experiments emphasize doing well, not avoiding mistakes. Thus, you'd be better off with a promotion focus, not a prevention one.

To test this idea, Barber & Mather tested fifty-six older (around age 70) subjects on a combined memory/working memory task. Subjects read sentences, some of which made sense, others which were nonsensical either syntactically or semantically.

Subjects indicated with a button press whether the sentence made sense or not. In addition, they were told to remember the last word of the sentence for as many of the sentences as they could. Task performance was measured by a combined score: how many sentences were correctly identified (sensible/nonsensical) and how many final words were remembered.

Next, subjects read one of two fictitious news articles. The one meant to invoke stereotype threat described the loss of memory due to aging. The control article described preservation of memory with aging.

Then, subjects performed the sentence task again. We would expect that stereotype threat would lead to worse performance.

BUT the experimenters also varied the reward structure of the task. Some subjects were told they would get a monetary reward for good performance. Others were told they were starting with a set amount of money, and that each memory error would incur a penalty. 

The instructions made a big difference in the outcome. As shown in the graph, framing in terms of costs for errors didn't just remove stereotype threat; it actually lead to an improvement.

This outcome makes sense, according to the regulatory fit hypothesis. Subjects were worried about errors, and the task rewarded them for avoiding errors.

These data are the first to test this new hypothesis as to the mechanism of stereotype threat, and should not be seen as definitive.

But if this new explanation holds up (and if it applies to other groups) it should have significant implications for how threat can be avoided.

References:
Appel, M., & Kronberger, N. (2012). Stereotypes and the achievement gap: Stereotype threat prior to test taking. Educational Psychology Review, 24(4), 609-635.

Barber, S. J., & Mather, M. (2013). Stereotype Threat Can Both Enhance and Impair Older Adults’ Memory. Psychological science, published online Oct. 22, 2013. DOI: 10.1177/0956797613497023.
Hutchison, K. A., Smith, J. L., & Ferris, A. (2013). Goals Can Be Threatened to Extinction Using the Stroop Task to Clarify Working Memory Depletion Under Stereotype Threat. Social Psychological and Personality Science, 4(1), 74-81.

In Why Don't Students Like School? I pointed out that cognitive challenge is engaging if it's at the right level of difficulty, but boring if it's too easy or too hard. It sounds, then, like it would make sense to organize students into different classes based on their prior achievement.

It might make sense cognitively, but the literature shows that such a practice leads to bad outcomes for the kids in lower tracks. Those classes tend to have less demanding curricula and and lower expectations for achievement (e.g., Brunello & Checchi, 2007).

Further, assignment to tracks is often biased by race or social class (e.g., Maaz et al., 2007).

What tracking does to students self-perceptions has been less clear. A new international study (Chmielewski et al., 2013) examined data from the 2003 PISA data set to examine the association of different types of tracking and student self-perceptions of mathematics self-concept.

The authors compared systems with

• Between school streaming: in which students with different levels of achievement are sent to different schools.
• Within school streaming: in which students with different levels of achievement are put in different sequences of courses for all subjects.
• Course-by-course tracking: in which students are assigned to more or less advanced courses within a school, depending on their achievement within a particular subject.

Controlling for individual achievement and the average achievement of the track or stream, the researchers found that course tracking is associated with worse self-perceptions among low-achieving students, but streaming is associated with better self-perceptions.

This figure shows the difference between the self perceptions of higher and lower achieving students in individual countries, sorted by the type of tracking system.

The data suggest that when students are tracked for some but not all of their courses, they compare their achievement to other, more advanced students, perhaps because they see these students more often. Students who are streamed within or between schools, in contrast, compare their abilities to their fellow stream-mates.

But why is there self-concept higher than higher-achieving students? This effect may be comparable to a more general phenomenon that people are poorer judges of their competence for tasks that they perform poorly. If you're not very good, you're not good enough to realize what you lack.

The authors do not suggest that between school steaming is the way to go (since it's associated with higher confidence). They note that the association is just the reverse of that seen in achievement: kids who stream between schools seem to take the biggest hit to achievement.

References

Brunello, G., & Checchi, D. (2007). Does school tracking affect equality of opportunity? New international evidence. Economic Policy, 22, 781–861.

Chmielewski, A. K., Dumont, H. Trautwein, U. (2013). Tracking effects depend on tracking type: An international comparison of students' mathematics self-concept. American Educatioal Research Journal, 50,  925-957.

Maaz, K., Trautwein, U., Ludtke, O., & Baumert, J. (2008). Educational transitions and differential learning environments: How explicit between-school tracking contributes to social inequality in educational outcomes. Child Developmental Perspectives, 2, 99–106.

Part of the fun and ongoing fascination of science of science is "the effect that ought not to work, yet does."

The impact of values of affirmation on academic performance is such an effect.

Values-affirmation "undoes" the effect of stereotype threat (also called identity threat). Stereotype threat occurs when a person is concerned about confirming a negative stereotype about his or her group. In other words a boy is so consumed with thinking "Everyone expects me to do poorly on this test because I'm African-American" that his performance actually is compromised (see Walton & Spencer, 2009 for a review).

One way to combat stereotype threat is to give the student better resources to deal with the threat--make the student feel more confident, more able to control the things that matter in his or her life.

That's where values affirmation comes in.

In this procedure, students are provided a list of values (e.g., relationships with family members, being good at art) and are asked to pick three that are most important to them and to write about why they are so important. In the control condition, students pick three values they imagine might be important to someone else.

Randomized control trials show that this brief intervention boosts school grades (e.g., Cohen et al, 2006).

Why?

One theory is that values affirmation gives students a greater sense of belonging, of being more connected to other people.

(The importance of social connection is an emerging theme in  other research areas. For example, you may have heard about the studies showing that people are less anxious when anticipating a painful electric shock if they are holding the hand of a friend or loved one.)

A new study (Shnabel et al, 2013) directly tested the idea that writing about social belonging might be a vital element in making values affirmation work.

In Experiment 1 they tested 169 Black and 186 White seventh graders in a correlational study. They did the values-affirmation writing exercise, as described above. The dependent measure was change in GPA (pre-intervention vs. post-intervention.) The experimetners found that writing about social belonging in the writing assignment was associated with a greater increase in GPA for Black students (but not for White students, indicating that the effect is due to reduction in stereotype threat.)

In Experiment 2, they used an experimental design, testing 62 male and 55 female college undergraduates on a standardized math test. Some were specifically told to write about social belonging and others were given standard affirmation writing instructions. Female students in the former group outscored those in the latter group. (And there was no effect for male students.)

The brevity of the intervention relative to the apparent duration of the effect still surprise me. But this new study gives some insight into why it works in the first place.

References:

Cohen, G. L., Garcia, J., Apfel, N., & Master, A. (2006). Reducing
the racial achievement gap: A social-psychological interven-tion. Science, 313, 1307-1310.

Shnabel, N., Purdie-Vaughns, V., Cook, J. E., Garcia, J., & Cohen, G. L. (2013). Demystifying values-affirmation interventions: Writing about social belonging is a key to buffering against identity threat. Personality and Social Psychology Bulletin,

Walton, G. M., & Spencer, S. J. (2009). Latent ability: Grades and test
scores systematically underestimate the intellectual ability of negatively stereotyped students. Psychological Science, 20, 1132-1139.

Illiteracy and its costs to individuals and to society has long been a focus of concern in public policy. A corresponding lack of ability in mathematics--innumeracy--has received increasing attention in the last few decades. The ability to use basic math is more and more important as modern day society grows more complex.

Some children have a problem in learning to read that is disproportionate to any other academic challenge they face. Some children have a corresponding problem with math. For some reason, the ideas just don't come together for these students.

In a recent article, David Geary (2013) reviews evidence that one cause of the problem may be a fundamental deficit in the representation of numerosity.

Geary describes three possible sources of a problem in children's appreciation of number.

To appreciate where the problems may lie, you need to know about the approximate number system.  All children (and members of many other species) are born with an ability to appreciate numerosity. The approximate number system does not support precise counting, but allows for comparison judgements of "more than" or "less than." For example, in the figure below you can tell at a glance (and without counting) which cloud contains more dots.

This ability --making the comparison without counting--is supported by the approximate number system. (Formal experiments control for things like the total amount of "dot material" in each field, and so on.)

The ability depends on not on the absolute difference in number of dots, but on the ratio. Adults can discriminate ratios as low as 11:10. Infants can perform this task, but the ratio of the difference in dots must be much greater, closer to 2:1.

Many researchers believe that this approximate number system is the scaffold for an understanding of the cardinal values of number.

So the first possible source of problems in mathematics may be that the approximate number system does not develop at a typical pace, leaving the child slow to develop the cognitive representations of quantity that can support mathematics.

A second possibility is that the approximate number system works just find, but the problem lies in associating symbols (number names and arabic numerals) to the quantities represented there. Geary speculates that regulating attention may be particularly important to this ability.

Finally, It is possible for children to appreciate the cardinal value of numbers and yet not understand the logical relationships among those numbers, to appreciate the structure as a whole. That's the the third possible problem.

Geary suggests that there is at least suggestive evidence that each of these potential problems creates trouble for some students.

The analogy to dyslexia is irresistible, and not inappropriate. Math, like reading, is not a "natural" human activity. It is a cultural contrivance, and the cognitive apparatus to support it must be hijacked from mental systems meant to support other activities.

As such, it is fragile, meaning it lacks redundancy. If something goes wrong, the system as a whole functions very poorly. Thus, understanding how things might go wrong is essential to helping children who struggle early on.

Gear, D. (2013) Early foundations for mathematics learning and their relations to learning disabilities. Current Directions in Psychological Science, 22, 23-27.

How much help is provided to a teacher and student by the use of  manipulatives--that is, concrete objects meant to help illustrate a mathematical idea?

My sense is that most teachers and parents think that manipulatives help a lot. I could not locate any really representative data on this point, but the smaller scale studies I've seen support the impression that they are used frequently. In one study of two districts the average elementary school teacher reported using manipulatives nearly every day (Uribe-Florez & Wilkins, 2010).

Do manipulatives help kids learn? A recent meta-analysis (Carbonneua, Marley & Selif, 2012) offers a complicated picture. The short answer is "on average, manipulatives help. . . a little." But the more complete answer is that how much they help depends on (1) what outcome you measure and (2) how the manipulatives are used in instruction.

The authors analyzed the results of 55 studies that compared instruction with or without manipulatives. The overall effect size was d = .37--typically designated a "moderate" effect.

But there were big differences depending on content being taught: for example, the effect for fractions was considerable larger (d = .69) than the effect for arithmetic (d = .27) or algebra (d = .21).
More surprising to me, the effect was largest when the outcome of the experiment focused on retention (d = .59), and was relatively small for transfer (d = .13).

What are we to make of these results? I think we have to be terribly cautious about any firm take-aways. That's obvious from the complexity of the results (and I've only hinted at the number of interactions).

It seems self-evident that one source of variation is the quality of the manipulative. Some just may not do that great a job of representing what they are supposed to represent. Others may be so flashy and interesting that they draw attention to peripheral features at the expense of the features that are supposed to be salient.

It also seems obvious that manipulatives can be more or less useful depending on how effectively they are used. For example, some fine-grained experimental work indicates the effectiveness of using a pan-balance as an analogy for balancing equations depends on fairly subtle features of what to draw students’ attention to and when (Richland et al, 2007).

My hunch is that at least one important source of variability (and one that's seldom measured in these studies) is the quality and quantity of relevant knowledge students have when the manipulative is introduced. For example, we might expect that the student with a good grasp of the numerosity would be in a better position to appreciate a manipulative meant to illustrate place value than the student whose grasp is tenuous. Why?

David Uttal and his associates (Uttall, et al, 2009) emphasized this factor when they pointed out that the purpose of a manipulative is to help students understand an abstraction. But a manipulative itself is an abstraction—it’s not the thing-to-be-learned, it’s a representation of that thing—or rather, a feature of the manipulative is analogous to a feature of the thing-to-be-learned. So the student must simultaneously keep in mind the status of the manipulative as concrete object and as a representation of something more abstract. The challenge is that keeping this dual status in mind and coordinating them can be a significant load on working memory. This challenge is potentially easier to meet for those students who firmly understand concepts undergirding the new idea.

I’m generally a fan of meta-analyses. I think they offer a principled way to get a systematic big picture of a broad research literature. But the question “do manipulatives help?” may be too broad. It seems too difficult to develop an answer that won’t be mostly caveats.

So what’s the take-away message? (1) manipulatives typically help a little, but the range of effect (hurts a little to helps a lot) is huge; 2) researchers have some ideas as to why manipulatives work or don’t work. . .but not in a way that offers much help in classroom application.

This is an instance where a teacher’s experience is a better guide.

References

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (in press). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology. Advance online publication.

Richland, R. E.  Zur, O. Holyoak, K. J. (2007). Cognitive Supports for Analogies in the Mathematics Classroom, Science, 316, 1128–1129.

Uribe‐Flórez, L. J., & Wilkins, J. L. (2010). Elementary school teachers' manipulative use. School Science and Mathematics, 110, 363-371.

Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J.
(2009). Dual representation and the linking of concrete and symbolic
representations. Child Development Perspectives, 3, 156–159.

There is a lot of talk these days about STEM--science, technology, engineering, and math--and the teachers of STEM subjects. It would seem self-evident that these teachers, given their skill set, would be in demand in business and industry, and thus would be harder to keep in the classroom.

A new study (Ingersoll & May, 2012) offers some surprising data on this issue.

Using the national Schools and Staffing Survey and the Teacher Follow-Up Survey, they found that science and math teachers have NOT left the field at rates higher than that of other teachers. In this data set (1988-2005)  math teachers and science teachers left teaching at about the same rate as teachers in other subjects: about 6% each year.

Furthermore, when these teachers do leave a school, they are no more likely to take a non-education job than other teachers: about 8% of "leavers" took another job outside of education. Much more common reasons to leave the classroom were retirement (about 15%) or an education job other than teaching (about 17%).

The authors argue that teacher turnover, not teachers leaving the field, is the engine behind staffing problems for math and science classes.

So what prompts teacher turnover?

The authors argue that on this dimension math and science teachers differ. Both groups are, unsurprisingly, motivated by better working conditions and higher salaries, but the former matter more to math teachers, and science teachers care more about the latter.

But in both cases, the result is that math and science teachers tend to leave schools with large percentages of low-incomes kids in order to move to schools with wealthier kids.



Ingersoll, R. M., & May, H. (2012). The magnitude, destinations, and determinants of mathematics and science teacher turnover. Educational Evaluation and Policy Analysis, 34,  435-464.

Someone needs to tell Glen Whitney that algebra doesn't matter.

Poor, deluded Whitney has seen the negative attitude that most Americans have about mathematics--it's boring, it's confusing, it's unrelated to everyday life--and concluded that Americans need a mathematical awakening.

To prompt it, he's spearheading the creation of a Math Museum in New York City, the only one of its kind in North America. (There had been a small math museum on Long Island, the Goudreau Museum. It closed in 2006).

Whitney reports that he loved math in high school and college, but didn't think he was likely to make it as a pure researcher. He went to work for a hedge fund, creating statistical models for trading. When the Goudreau Museum closed, he organized a group to explore opening a math museum that would be more ambitious.

 A rendering of the plan is shown below.

The plan is for exhibits similar to those seen in science museums--plenty of interaction and movement on the part of visitors, and a focus on the fact that mathematics is all around us.

All around us to the point that Whitney currently gives math walking tours in New York City. As he notes in a recent interview in Nature, math is in "the algorithms used to control traffic lights, the mathematical issues involved in keeping the subway running, the symmetry of the mouldings on the sides of buildings and the unusual geometry that gives gingko trees their distinctive shape."

A traveling exhibition, Math Midway, has been making the rounds of science museums around the country, whetting appetites for the the grand opening (December 15th, 2012).

The most popular exhibit is a tricycle with square wheels which can be ridden smoothly on a track with inverted curves, calculated to keep the axles of the trike level. In the photo below it's ridden by Joel Klein (former New York chancellor of education and current leader of News Corporation's education venture).

Whitney says that the beauty of the tricycle exhibit is that it gives people the sense that math can make the impossible seem possible.

Next impossible challenge: persuade people who think that math is mostly irrelevant and should be dropped from public schooling for most kids that they are wrong.

The Math Museum looks like a long step toward making that goal seem possible.

More at MoMath.org.

Making a change to education that seems like a clear improvement is never easy. Or almost never.

Judith Harackiewicz and her colleagues have recently reported an intervention that is inexpensive, simple, and leads high school students to take more STEM courses.

The intervention had three parts, administered over 15-months when students were in the 10th and 11th grades. In October of 10th grade researchers mailed a brochure to each household titled Making Connections: Helping Your Teen Find Value in School. It described the connections between math, science, and daily life, and included ideas about how to discuss this topic with students.

In January of 11th grade a second brochure was sent. It covered similar ideas, but with different examples. Parents also received a letter that included the address of a password-protected website devised by researchers, which provided more information about STEM and daily life, as well as STEM careers.

In Spring of 11th grade, parents were asked to complete an online questionnaire about the website.

There were a total of 188 students in the study: half received this intervention, and the control group did not.

Students in the intervention group took more STEM courses during their last two years of high school (8.31 semesters) than control students (7.50) semesters.

This difference turned out to be entirely due to differences in elective, advanced courses, as shown in the figure below.

An important caveat about this study: all of the subjects are participating in the Wisconsin Study of Families and Work. This study began in 1990. when women were in their fifth month of pregnancy.

The first brochure that researchers sent to subjects included a letter thanking them for their ongoing participation in the longer study. Hence, subjects could reasonably conclude that the present study was part of the longer study.

That's worth bearing in mind because ordinary parents might not be so ready to read brochures mailed to them by strangers, nor to visit suggested websites.

But that's not a fatal flaw of the research. It just means that we can't necessarily count on random parents reading the materials with the same care.

To  me, the effect is still remarkable. To put it in perspective, researchers also measured the effect of parental education on taking STEM courses. As many other researchers have found, the kids of better-educated parents took more STEM courses. But the effect of the intervention was nearly as large as the effect of parental education!

Clearly, further work is necessary but this is an awfully promising start.

Harackiewicz, J. M, Rozek, C. S., Hulleman, C. S & Hyde, J. S. (in press). Helping parents to motivate adolescents in mathematics and science: An experimental test of a utility-value intervention. Psychological Science.

When I first saw yesterday's New York Times op-ed, I mistook it for a joke. The title, "Is algebra necessary?" had the ring of Thurber's classic essay "Is sex necessary?" a send-up of psychological sex manuals of the 1920s. 

Unfortunately, the author, Andrew Hacker, poses the question in earnest, and draws the conclusion that algebra should not be required of all students.

His arguments:

• A lot of students find math really hard, and that prompts them to give up on school altogether. Think of what these otherwise terrific students might have achieved if math hadn't chased them away from school.
• The math that's taught in school doesn't relate well to the mathematical reasoning people need outside of school.

His proposed solution is the teaching of quantitative skills that students can use, rather than a bunch of abstract formulas, and a better understanding of "where numbers actually come from and what they actually convey," e.g., how the consumer price index is calculated.

For most careers, Hacker believes that specialized training in the math necessary for that particular job will do the trick.

What's wrong with this vision?

The inability to cope with math is not the main reason that students drop out of high school. Yes, a low grade in math predicts dropping out, but no more so than a low grade in English. Furthermore, behavioral factors like motivation, self-regulation, social control (Casillas, Robbins, Allen & Kuo, 2012), as well as a feeling of connectedness and engagement at school (Archambault et al, 2009) are as important as GPA to dropout. So it's misleading to depict math as the chief villain in America's high dropout rate.

What of the other argument, that formal math mostly doesn't apply outside of the classroom anyway?

The difficulty students have in applying math to everyday problems they encounter is not particular to math. Transfer is hard. New learning tends to cling to the examples used to explain the concept. That's as true of literary forms, scientific method, and techniques of historical analysis as it is of mathematical formulas.

The problem is that if you try to meet this challenge by teaching the specific skills that people need, you had better be confident that you're going to cover all those skills. Because if you teach students the significance of the Consumer Price Index they are not going to know how to teach themselves the significance of projected inflation rates on their investment in CDs. Their practical knowledge will be specific to what you teach them, and won't transfer.

The best bet for knowledge that can apply to new situations is an abstract understanding--seeing that  apparently different problems have a similar underlying structure. And the best bet for students to gain this abstract understanding is to teach it explicitly. (For a discussion of this point as it applies to math education in particular, see Anderson, Reder, & Simon, 1996).

But the explicit teaching of abstractions is not enough. You also need practice in putting the abstractions into concrete situations.

Hacker overlooks the need for practice, even for the everyday math he wants students to know. One of the important side benefits of higher math is that it makes you proficient at the other math that you had learned earlier, because those topics are embedded in the new stuff. (e.g., Bahrick & Hall, 1991).

So I think there are excellent reasons to doubt that Hacker's solution to the transfer problem will work out as he expects.

What of the contention that math doesn't do most people much good anyway?

Economic data directly contradict that suggestion. Economists have shown that cognitive skills--especially math and science--are robust predictors of individual income, of a country's economic growth, and of the distribution of income within a country (e.g. Hanushek & Kimko, 2000; Hanushek & Woessmann, 2008).

Why would cognitive skills (as measured by international benchmark tests) be a predictor of economic growth? Economic productivity does not spring solely from the creativity of engineers and inventors. The well-educated worker is more likely to (1) see the potential for applying an innovation in a new context; (2) understand the explanation for applying an innovation that someone else has spotted.

In other words, Hacker overlooks the possibility that the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills. The on-the-job training in mathematics that Hacker envisions will go a whole lot better with an employee who gained a solid footing in math in school.

Finally, there is the question of income distribution; countries with a better educated populace show smaller income disparity, and suggesting that not everyone needs to math raises the question of who will learn it. Who will learn higher math in Hacker's ideal world? He's not clear on this point. He says he's against tracking, but notes that MIT and Cal Tech clearly need their students to be proficient in math. Does this mean that everyone gets the same vocational-type math education, and some of those going on to college will get access to higher math?

If that were actually implemented, how long before private vendors offer after school courses in formal mathematics, to give kids an edge for entrance to MIT? Private courses that cost, and to which the poor will not have access.

There are not many people who are satisfied with the mathematical competence of the average US student. We need to do better. Promising ideas include devoting more time to mathematics in early grades, more exposure to premathematical concepts in preschool, and perhaps specialized math instructors beginning in earlier grades.

Hacker's suggestions sound like surrender.

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