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. Anderson, J. R., Reder, L. M., & Simon, H. A. (1996). Situated learning and education. Educational Researcher, 25, 5-11 Archambault, I., Janosz, M, Fallu, J.-S., & Pagani, L. S. (2009). Student engagement and its relationship with early high school dropout. Journal of Adolescence, 32, 651-670. Bahrick, H. P. & Hall, L. K. (1991). Lifetime maintenance of high school mathematics content. Journal of Experimental Psychology: General, 120, 20-33.Hanushek, E. A. & Kimko D. D. (2000). Schooling, labor-force quality, and the growth of nations. The American Economic Review, 90, 1184-1208. Hanushek, E. A. & Woessmann, (2008). The role of cognitive skills in economic development. Journal of Economic Literature. 46, 607-668.
There is a great deal of attention paid to and controversy about, the promise of training working memory to improve academic skills, a topic I wrote about here. But working memory is not the only cognitive process that might be a candidate for training. Spatial skills are a good predictor of success in science, mathematics, and engineering. Now on the basis of a new meta-analysis (Uttal, Meadow, Tipton, Hand, Alden, Warren & Newcombe, in press) researchers claim that spatial skills are eminently trainable. In fact they claim a quite respectable average effect size of 0.47 (Hedge's g) after training (that's across 217 studies). Training tasks across these many studies included things like visualizing 2D and 3D objects in a CAD program, acrobatic sports training, and learning to use a laparascope (an angled device used by surgeons). Outcome measures were equally varied, and included standard psychometric measures (like a paper-folding test), tests that demanded imagining oneself in a landscape, and tests that required mentally rotating objects. Even more impressive: 1) researchers found robust transfer to new tasks 2) researchers found little, if any effect of delay between training and test--the skills don't seem to fade with time, at least for several weeks. (Only four studies included delays of greater than one month.) This is a long, complex analysis and I won't try to do it justice in a brief blog post. But the marquee finding is big news. What we'd love to see is an intervention that is relatively brief, not terribly difficult to implement, reliably leads to improvement, and transfers to new academic tasks. That's a tall order, but spatial skills may fill all the requirements. The figure below (from the paper) is a conjecture--if spatial training were widely implemented, and once scaled up we got the average improvement we see in these studies, how many more people could be trained as engineers? The paper is not publicly available, but there is a nice summary here from the collaborative laboratory responsible for the work. I also recommend this excellent article from American Educator on the relationship of spatial thinking to math and science, with suggestions for parents and teachers. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N.S. (2012, June 4). The Malleability of Spatial Skills: A Meta-Analysis of Training Studies. Psychological Bulletin. Advance online publication. doi: 10.1037/a0028446 Newcombe, N. S. (2010) Picture this: Increasing math and science learning by improving spatial thinking. American Educator, Summer, 29-35, 43.
One strategy for thinking about interventions to boost kids success in school is to conduct the following sort of study. Step one: measure lots of factors early in life, i.e., before kids start school. Step two: measure academic success after kids have been in school awhile (say, fourth grade). Then see which factors you measured early in life are associated with school success measured later. Some factors are well-known, e.g., socio-economic status of the parents, and so you’d statistically remove those “usual suspects” first. In 2007 Duncan and colleagues introduced a new method of analyzing this type of data, and they applied it to six sizable international data sets that followed kids from as early as birth to 3rd grade, focusing especially on reading and math achievement. They concluded that early measures of math and reading, and measures of attention were significant predictors of later math and reading skills, but early social skills were not. Curiously, early math scores predicted later reading scores as well as early reading scores did. Their conclusions, while not startling, attracted a lot of attention because the new method was deemed quite useful, and because it was applied meticulously to several large-scale datasets. In 2010, another article was published using the same methodology, but with a startling result. David Grissmer and his colleagues noted that three of the data sets had early measures of fine motor skills. They found that, after they statistically accounted for all of the factors that Duncan et al had examined, fine motor skills was and additional, strong predictor of student achievement. I have to note that what the tests called “fine motor skills” strikes me as a bit odd. Cognitive psychologists think of that as being tasks like buttoning a button, or picking something up with tweezers—i.e., requiring precise movements, usually of the fingers. But in these data sets it was tested with tasks like copying simple designs, or drawing a human figure. These are not solely motor tasks. The fuzziness of exactly what the tasks mean may cloud the interpretation, but it doesn’t cloud the size of the effect—these tasks are a robust predictor of later math and reading achievement. There’s plenty of speculation as to why this effect might work. Perhaps the measure of “fine motor skills” is really another way of measuring some aspect of attention. Perhaps it’s another way of measuring how well kids can understand and use space. Or the effect may be more direct; it’s commonly thought that the motor and cognitive domains are intertwined, and so practicing motor tasks may aid cognition. The big question: does this mean that practice of fine motor skills will boost academic achievement? Those studies are ongoing, and I hope to report on the results here before long. Duncan, G. J., Dowsett, C. J., Claessens, A., Magnuson, K., Huston, A. C.,Klebanov, P., . . . Japel, C. (2007). School readiness and later achievement. Developmental Psychology, 43, 1428–1446. Grissmer, D., Grimm, K., J., Aiyer, S. M., Murrah, W. M., & Steele, J. S. (2010). Fine Motor Skills and Attention: Primary Developmental Predictors of Later Achievement. Developmental Psychology, 46, 1008-1017.
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