mathematics

Metacognitive questioning and the use of worked examples

The use of worked examples

We're all familiar, I'm sure, with the use of worked-out examples in mathematics teaching. Worked-out examples are often used to demonstrate problem-solving processes. They generally specify the steps needed to solve a problem in some detail. After working through such examples, students are usually given the same kind of problems to work through on their own. The strategy is generally helpful in teaching students to solve problems that are the same as the examples.

Worked-out examples are also used in small-group settings, either by working on the example together, or by studying the example individually and then getting together to enable those who understood to explain to those who didn't. Explaining something to another person is well-established as an effective method of improving understanding (for the person doing the explaining -- and presumably the person receiving the explanation gets something out of it also!).

Metacognitive differences between high and low achievers

An interesting study comparing the behavior of high and low achieving students who studied worked-out examples cooperatively found important differences.

High achievers:

  • explained things to themselves as they worked through the examples
  • tried to construct relationships between the new process and what they already knew
  • tended to infer additional information that wasn't directly given

Low achievers on the other hand:

  • followed the examples step-by-step without relating it to anything they already knew
  • didn't try to construct any broader understanding of the procedure that would enable them to generalize it to new situations

Other studies have since demonstrated that students taught to ask questions that focus on relating new learning to old show greater understanding than students taught to ask different questions, and both do better than students who ask no questions at all.

Learning to ask the right questions

An instructional method for teaching mathematics that involves training students to ask metacognitive questions has been found to produce significant improvement in students' learning. The method is called IMPROVE -- an acronym for the teaching steps involved:

  • Introduce new concepts
  • Metacognitive questioning
  • Practise
  • Review
  • Obtain mastery on lower and higher cognitive processes
  • Verify
  • Enrich

There are four kinds of metacognitive questions the students are taught to ask:

  1. Comprehension questions (e.g., What is this problem all about?)
  2. Connection questions (e.g., How is this problem different from/ similar to problems that have already been solved?)
  3. Strategy questions (e.g., What strategies are appropriate for solving this problem and why?)
  4. Reflection questions (e.g., does this make sense? why am I stuck?)

A study that compared the effects of using worked-out examples or metacognitive questioning (both in a cooperative setting) found that students given metacognitive training performed significantly better than those who experienced worked-out examples (the participants were 8th grade Israeli students). Lower achievers benefited more from the metacognitive training (not surprising, because presumably the high achievers already used this strategy in the context of the worked-out examples).

More reading

Here are some papers by the creators of IMPROVE on their studies into the benefits of metacognitive instruction (in PDF format):

http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG8/TG8_Kramarski_cerme3.pdf

http://www.hbcse.tifr.res.in/episteme/allabs/zemira_abs.pdf (no longer available)

http://www.icme-organisers.dk/tsg18/S32MevarechKramarski.pdf (no longer available)

References: 
  • Mevarech, Z.R. & Kramarski, B. 2003. The effects of metacognitive training versus worked-out examples on students' mathematical reasoning. British Journal of Educational Psychology, 73, 449-471.

Gender Differences

In general, males are better at spatial tasks involving mental rotation.

In general, females have superior verbal skills.

Males are far more likely to pursue math or science careers, but gender differences in math are not consistent across nations or ages.

A number of imaging studies have demonstrated that the brains of males and females show different patterns of activity on various tasks.

Nicotine has been shown to differentially alter men's and women's brain activity patterns so that the differences disappear.

Both estrogen and testosterone have been shown to affect cognitive function.

Training has been shown to bring parity to differences in cognitive performance between the sexes.

Age also alters the differences between men and women.

Widely cited gender differences in cognition

It is clear that there are differences between the genders in terms of cognitive function; it is much less clear that there are differences in terms of cognitive abilities. Let me explain what I mean by that.

It's commonly understood that males have superior spatial ability, while females have superior verbal ability. Males are better at math; females at reading. There is some truth in these generalizations, but it's certainly not as simple as it is portrayed.

First of all, as regards spatial cognition, while males typically outperform females on tasks dealing with mental rotation and spatial navigation, females tend to outperform males on tasks dealing with object location, relational object location memory, and spatial working memory.

While the two sexes score the same on broad measures of mathematical ability, girls tend to do better at arithmetic, while boys do better at spatial tests that involve mental rotation.

Having said that, it does depend where you're looking. The Programme for International Student Assessment (PISA) is an internationally standardised assessment that is given to 15-year-olds in schools. In 2003, 41 countries participated. Given the constancy of the gender difference in math performance observed in the U.S., it is interesting to note what happens in other countries. There was no significant difference between the sexes in Australia, Austria, Belgium, Japan, the Netherlands, Norway, Poland, Hong Kong, Indonesia, Latvia, Serbia, and Thailand. There was a clear male superiority for all 4 content areas in Canada, Denmark, Greece, Ireland, Korea, Luxembourg, New Zealand, Portugal, the Slovak Republic, Liechtenstein, Macao and Tunisia. In Austria, Belgium, the United States and Latvia, males outperformed females only on the space and shape scale; in Japan, the Netherlands and Norway only on the uncertainty scale. And in Iceland, females always consistently do better than males!

Noone knows why, but it is surely obvious that these differences must lie in cultural and educational factors.

Interestingly, the IEA Third International Mathematics and Science Study (TIMSS) shows this developing -- while significant gender differences in mathematics were found only in 3 of the 16 participating OECD countries for fourth-grade students, gender differences were found in 6 countries at the grade-eight level, and in 14 countries at the last year of upper secondary schooling.

This inconsistency is not, however, mirrored in verbal skills -- girls outperform boys in reading in all countries.

Gender differences in language have been consistently found, and hardly need reiteration. However, here's an interesting study: it found gender differences in the emerging connectivity of neural networks associated with skills needed for beginning reading in preschoolers. It seems that boys favor vocabulary sub-skills needed for comprehension while girls favor fluency and phonic sub-skills needed for the mechanics of reading.The study points to the different advantages each gender brings to learning to read.

There's a lesson there.

There are other less well-known differences between the sexes. Women tend to do better at recognizing faces. But a study has found that this superiority applies only to female faces. There was no difference between men and women in the recognition of male faces.

Moreover, pre-pubertal boys and girls have been found to be equally good at recognizing faces and identifying expressions. However, they do seem to do it in different ways. Boys showed significantly greater activity in the right hemisphere, while the girls' brains were more active in the left hemisphere. It is speculated that boys tend to process faces at a global level (right hemisphere), while girls process faces at a more local level (left hemisphere).

It's also long been recognized that women are better at remembering emotional memories. Interestingly, an imaging study has revealed that the sexes tend to encode emotional experiences in different parts of the brain. In women, it seems that evaluation of emotional experience and encoding of the memory is much more tightly integrated.

But of course, noone denies that there are differences between men and women. The big question (one of the big questions) is how much, if any, is innate.

Studies of differences, even at the neural level, don't demonstrate that. It's increasingly clear that environmental factors affect all manner of thing at the neural level. However, one study of 1-day-old infants did find that boys tended to gaze at three-dimensional mobiles longer than girls did, while girls looked at human faces longer than boys did.

Of course, even a 1-day-old infant isn't entirely free of environmental influence. In this case, the most important environmental influence is probably hormones.

Hormones and chemistry

A lot of studies in recent years have demonstrated that estrogen is an important player in women's cognition. Spatial ability in particular seems vulnerable to hormonal effects. Women do vary in their spatial abilities according to where they are in the menstrual cycle, and there is some evidence that spatial abilities (in both males and females) may be affected by how much testosterone is received in the womb.

Another study has found children exposed to higher levels of testosterone in the womb also develop language later and have smaller vocabularies at 2 years of age.

Hormones aren't the only chemical affecting male and female brains differently. Significant differences have been found in the brain activity of men and women when engaged in a broad range of activities and behaviors. These differences are more acute during impulsive or hostile acts. But — here's the truly fascinating thing — nicotine causes these brain activity differences to disappear. A study has found that among both smokers and non-smokers on nicotine, during aggressive moments, there are virtually no differences in brain activity between the sexes. A finding that supports other studies that indicate men's and women's brains respond differently to the same stimuli — for example, alcohol.

What does all this mean? Well, let's look at the question that's behind the whole issue: are men smarter than women? (or alternately, are women smarter than men?)

Is one sex smarter than the other?

Here's a few interesting studies that demonstrate some more differences between male and female brains.

A study of some 600 Dutch men and women aged 85 years found that the women tended to have better cognitive speed and a better memory than the men, despite the fact that significantly more of the women had limited formal education compared to the men. This may be due to better health. On the other hand, there do appear to be differences in the way male and female brains develop, and the way they decline.

For example, women have up to 15% more brain cell density in the frontal lobe, which controls so-called higher mental processes, such as judgement, personality, planning and working memory. However, as they get older, women appear to shed cells more rapidly from this area than men. By old age, the density is similar for both sexes.

A study of male and female students (aged 18-25) has found that men's brain cells can transmit nerve impulses 4% faster than women's, probably due to the faster increase of white matter in the male brain during adolescence.

An imaging study of 48 men and women between 18 and 84 years old found that, compared with women, men had more than six times the amount of intelligence-related gray matter. On the other hand, women had about nine times more white matter involved in intelligence than men did. Women also had a large proportion of their IQ-related brain matter (86% of white and 84% of gray) concentrated in the frontal lobes, while men had 90% of their IQ-related gray matter distributed equally between the frontal lobes and the parietal lobes, and 82% of their IQ-related white matter in the temporal lobes. Despite these differences, men and women performed equally on the IQ tests.

It has, of course, long been suggested that women are intellectually inferior because their brains are smaller. A study involving the intelligence testing of 100 neurologically normal, terminally ill volunteers found that a bigger brain size is indeed correlated with higher intelligence — but only in certain areas, and with odd differences between women and men. Verbal intelligence was clearly correlated with brain size for women and — get this — right-handed men! But not for left-handed men. Spatial intelligence was also correlated with brain size in women, but much less strongly, while it was not related at all to brain size in men.

Also, brain size decreased with age in men over the age span of 25 to 80 years, suggesting that the well-documented decline in visuospatial intelligence with age is related, at least in right-handed men, to the decrease in cerebral volume with age. However age hardly affected brain size in women.

What is all this telling us?

Male and female brains are different: they develop differently; they do things differently; they respond to different stimuli in different ways.

None of this speaks to how well information is processed.

None of these differences mean that individual brains, of either sex, can't be trained to perform well in specific areas.

Here’s an experiment and a case study which bear on this.

It's all about training

The experiment concerns rhesus monkeys. The superiority of males in spatial memory that we're familiar with among humans also occurs in this population. But here's the interesting thing — the gender gap only occurred between young adult males and young untrained females. In other words, there was no difference between older adults (because performance deteriorated with age more sharply for males), and did not occur between male and female younger adults if they were given simple training. Apparently the training had little effect on the males, but the females improved dramatically.

The “case study” concerns Susan Polgar, a chess master. You can read about her in a recent article (http://www.opinionjournal.com/la/?id=110006356 ), which I noticed because the Polgar sisters are a well-known example of “hot-housing”. I cited them in my own article on the question of whether there is in fact such a thing as innate talent. Susan Polgar and her sisters are examples of how you can train “talent”; indeed, whether there is in fact such a thing as “talent” is a debatable question. Certainly you can argue for a predisposition towards certain activities, but after that … Well, even geniuses have to work at it, and while you may not be able to make a genius, you can certainly create experts.

This article was provoked, by the way, by comments by the President of Harvard University, Lawrence Summers, who recently stirred the pot by giving a speech arguing that boys outperform girls on high school science and math scores because of genetic differences between the genders, and that discrimination is no longer a career barrier for female academics. Apparently, during Dr Summers' presidency, the number of tenured jobs offered to women has fallen from 36% to 13%. Last year, only four of 32 tenured job openings were offered to women.

You can read a little more about what Dr Summers said at http://education.guardian.co.uk/gendergap/story/0,7348,1393079,00.html, and there's a rather good response by Simon Baron-Cohen (professor in the departments of psychology and psychiatry, Cambridge University, and author of The Essential Difference) at: http://education.guardian.co.uk/higher/research/story/0,9865,1399109,00.html

Parts of this article originally appeared in the January and February 2005 newsletters.

References: 
  • Canli, T., Desmond, J.E., Zhao, Z. & Gabrieli, J.D.E. 2002. Sex differences in the neural basis of emotional memories. Proceedings of the National Academy of Sciences, 99, 10789-10794.
  • Everhart, D.E., Shucard, J.L., Quatrin, T. & Shucard, D.W. 2001. Sex-related differences in event-related potentials, face recognition, and facial affect processing in prepubertal children. Neuropsychology, 15(3), 329-341.
  • Fallon, J.H., Keator, D.B., Mbogori, J., Taylor, D. & Potkin, S.G. 2005. Gender: a major determinant of brain response to nicotine. The International Journal of Neuropsychopharmacology, 8(1), 17-26. (see http://www.eurekalert.org/pub_releases/2005-02/uoc--bao021705.htm)
  • Geary, D.C. 1998. Male, Female: The Evolution of Human Sex Differences. Washington, D.C.: American Psychological Association.
  • Haier, R.J., Jung, R.E., Yeo, R.A., Head, K. & Alkire, M.T. 2005. The neuroanatomy of general intelligence: sex matters. NeuroImage, 25(1), 320-327.
  • Hanlon, H. 2001. Gender Differences Observed in Preschoolers’ Emerging Neural Networks. Paper presented at Genomes and Hormones: An Integrative Approach to Gender Differences in Physiology, an American Physiological Society (APS) conference held October 17-20 in Pittsburgh.
  • Kempel, P.. Gohlke, B., Klempau, J., Zinsberger, P., Reuter, M. & Hennig, J. 2005. Second-to-fourth digit length, testosterone and spatial ability. Intelligence, 33(3), 215-230.
  • Lacreuse, A., Kim, C.B., Rosene, D.L., Killiany, R.J., Moss, M.B., Moore, T.L., Chennareddi, L. & Herndon, J.G. 2005. Sex, age, and training modulate spatial memory in the Rhesus monkey (Macaca mulatta). Behavioral Neuroscience, 119 (1).
  • Levin, S.L., Mohamed, F.B. & Platek, S.M. 2005. Common ground for spatial cognition? A behavioral and fMRI study of sex differences in mental rotation and spatial working memory. Evolutionary Psychology, 3, 227-254.
  • Lewin, C. & Herlitz, A. 2002. Sex differences in face recognition-Women's faces make the difference, Brain and Cognition, 50 (1), 121-128.
  • OECD. Learning for Tomorrow's World –First Results from PISA 2003 http://www.oecd.org/document/0/0,2340,en_2649_201185_34010524_1_1_1_1,00.html
  • Reed, T.E., Vernon, P.A. & Johnson, A.M. 2005. Confirmation of correlation between brain nerve conduction velocity and intelligence level in normal adults. Intelligence, 32(6), 563-572.
  • van Exel, E., Gussekloo, J., de Craen, A.J.M, Bootsma-van der Wiel, A., Houx, P., Knook, D.L. & Westendorp, R.G.J. 2001. Cognitive function in the oldest old: women perform better than men. Journal of Neurology, Neurosurgery & Psychiatry, 71, 29-32.
  • Witelson, S.F., Beresh, H. & Kigar, D.L. 2006. Intelligence and brain size in 100 postmortem brains: sex, lateralization and age factors. Brain, 129, 386-398.
  • Witelson, S.F., Kigar, D.L. & Stoner-Beresh, H.J. 2001. Sex difference in the numerical density of neurons in the pyramidal layers of human prefrontal cortex: a stereologic study. Paper presented to the annual Society for Neuroscience meeting in San Diego, US.

Early development

Children’s understanding, and their use of memory and learning strategies, is a considerably more complex situation than most of us realize. To get some feeling for this complexity, let’s start by looking at a specific area of knowledge: mathematics.

Children's math understanding

Here’s a math problem:

Pete has 3 apples. Ann also has some apples. Pete and Ann have 9 apples altogether. How many apples does Ann have?

This seems pretty straightforward, right? How about this one:

Pete and Ann have 9 apples altogether. Three of these belong to Pete and the rest belong to Ann. How many apples does Ann have?

The same problem, phrased slightly differently. Would it surprise you to know that this version is more likely to be correctly answered by children than the first version?

Whether or not a child solves a math problem correctly is not simply a matter of whether he or she knows the math — the way the problem is worded plays a crucial part in determining whether the child understands the problem correctly. Slight (and to adult eyes, insignificant) differences in the wording of a problem have a striking effect on whether children can solve it.

Mathematics also provides a clear demonstration of the seemingly somewhat haphazard development in cognitive abilities. It’s not haphazard, of course, but it sometimes appears that way from the adult perspective. In math, understanding different properties of the same concept can take several years. For example, children’s understanding of addition and subtraction is not an all-or-none business; adding as combining is grasped by young children quite early, but it takes some 2 to 3 years at school to grasp the essential invariants of additive relations. Multiplicative relations are even harder, with children up to age 10 or so often having great difficulty with proportion, probability, area and division.

Neurological differences between children and adults

Part of the problems children have with math stems from developmental constraints — their brains simply aren’t ready for some concepts. A recent imaging study of young people (aged 8-19 years) engaged in mental arithmetic, found that on simple two-operand addition or subtraction problems (for which accuracy was comparable across age), older subjects showed greater activation in the left parietal cortex, along the supramarginal gyrus and adjoining anterior intra-parietal sulcus as well as the left lateral occipital temporal cortex. Younger subjects showed greater activation in the prefrontal cortex (including the dorsolateral and ventrolateral prefrontal cortex and the anterior cingulate cortex), suggesting that they require comparatively more working memory and attentional resources to achieve similar levels of performance, and greater activation of the hippocampus and dorsal basal ganglia, reflecting the greater demands placed on both declarative and procedural memory systems.

In other words, the evidence suggests that the left inferior parietal cortex becomes increasingly specialized for mental arithmetic with practice, and this process is accompanied by a reduced need for memory and attentional resources.

Not just a matter of brain maturation

But this isn't the whole story. As the earlier example indicated, difficulties in understanding some concepts are often caused by the way the concepts are explained. This is why it’s so important to keep re-phrasing problems and ideas until you find one that “clicks”. Other difficulties are caused by the preconceptions the child brings with them — cultural practices, for example, can sometimes help and sometimes hinder learning.

Other domains: neurological differences between children and adults

What's true of mathematics is also true of other learning areas. When we teach children, we do need to consider developmental constraints, but recent studies suggest we may have over-estimated the importance of development.

In an intriguing imaging study, brain activity in children aged 7-10 and adults (average age 25 years) while doing various language tasks was compared. Six sub-regions in the left frontal and the left extrastriate cortex were identified as being significant. Both these areas are known to play a key role in language processing and are believed to undergo substantial development between childhood and adulthood.

Now comes the interesting part. The researchers attempted to determine whether these differences between children and adults were due to brain maturation or simply the result of slower and less accurate performance by children. By using information regarding each individual's performance on various tasks, they ended up with only two of the six sub-regions (one in the frontal cortex, one in the extrastriate cortex) showing differences that were age-related rather than performance-related (with the extrastriate region being more active in children than adults, while the frontal region was active in adults and not in children).

The researchers concluded that, yes, children do appear to use their brains differently than adults when successfully performing identical language tasks; however, although multiple regions appeared to be differentially active when comparing adults and children, many of those differences were due to performance discrepancies, not age-related maturation.

Childhood amnesia

Let's talk about childhood amnesia for a moment. "Childhood amnesia" is a term for what we all know -- we have very few memories of our early years. This is so familiar, you may never have considered why this should be so. But the reason is not in fact obvious. Freud speculated that we repressed those early memories (but Freud was hung up on repression); modern cognitive psychologists have considered immature memory processing skills may be to blame. This is surely true for the first months -- very young babies have extremely limited abilities at remembering anything for long periods of time (months), and research suggests that the dramatic brain maturation that typically occurs between 8 and 12 months is vital for long-term memory.

But an intriguing study (carried out by researchers at my old stomping ground: the University of Otago in New Zealand) has provided evidence that an important stumbling block in our remembrance of our early years is the child's grasp of language. If you don't have the words to describe what has happened, it seems that it is very difficult to encode it as a memory -- or at least, that it is very difficult to retrieve (before you leap on me with examples, let me add that noone is saying that every memory is encoded in words -- this is palpably not true).

This finding is supported by a recent study that found that language, in the form of specific kinds of sentences spoken aloud, helped 4-year-old children remember mirror image visual patterns.

The role of social interaction in memory development

Another study from my favorite university looked at the role mothers played in developing memory in their young children. The study distinguished between reminiscing (discussing shared experiences) and recounting (discussing unshared experiences). Children 40 months old and 58 months old were studied as they talked about past events with their mothers. It was found that mothers who provided more memory information during reminiscing and requested more memory information during recounting had children who reported more unique information about the events.

In general, parents seldom try to teach memory strategies directly to children, but children do learn strategies by observing and imitating what their parents do and this may in fact be a more effective means of teaching a child rather than by direct instruction.

But parents not only provide models of behavior; they also guide their children's behavior. The way they do this is likely to be influenced by their own beliefs about their children’s mnemonic abilities. If you don't believe your child can possibly remember something, you are unlikely to ask them to make the effort. But when parents ask 2 – 4 year olds to remind them to do something in the future, even 2 year olds remember to remind their parents of promised treats 80% of the time.

By 3 yrs old, children whose mothers typically asked questions about past events performed better on memory tasks than those children whose mothers only questioned them about present events. Observation of mothers as they taught their 4 year olds to sort toys, copy etch-a-sketch designs, and respond to questions regarding hypothetical situations found 3 interaction styles found that related to the child’s performance:

  • imperative-normative, in which mother gave little justification for requests or demands;
  • subjective, in which mother encouraged child to see his own behaviour from another’s point of view;
  • cognitive-rational, in which mother offered logical justifications for requests and demands.

Children whose mothers used the last two styles were more verbal and performed better on cognitive tasks.

A study of kindergarten and elementary school teachers found that children from classes where teachers frequently made strategy suggestions were better able to verbalize aspects of memory training and task performance. Although this made no difference for high achieving children, average and low achievers were more likely to continue using the trained strategy if they had teachers who frequently made strategy suggestions.

Conclusion

What lessons can we learn from all this?

First, we must note that there are indeed developmental constraints on children's capabilities that are rooted in physical changes in the brain. Some of these are simply a matter of time, but others are changes that require appropriate stimulation and training.

Secondly, the importance of language in enabling the child cannot be overestimated.

And thirdly, for children as with older adults, expectations about memory performance can reduce their capabilities. Supportive, directed assistance in developing memory and reasoning strategies can be very effective in helping even very young children.

References: 
  • Best, D.L. 1992. The role of social interaction in memory improvement. In D. Herrmann, H. Weingartner, A. Searleman & C. McEvoy (eds.) Memory Improvement: Implications for Memory Theory. New York: Springer-Verlag. pp 122-49.
  • Liston, C. & Kagan, J. 2002. Brain development: Memory enhancement in early childhood. Nature, 419, 896-896.
  • Reese, E. & Brown, N. 2000. Reminiscing and recounting in the preschool years. Applied Cognitive Psychology, 14 (1), 1-17.
  • Rivera, S.M., Reiss, A.L., Eckert, M.A. & Menon, V. 2005. Developmental Changes in Mental Arithmetic: Evidence for Increased Functional Specialization in the Left Inferior Parietal Cortex. Cerebral Cortex, 15 (11), 1779-1790.
  • Schlaggar, B.L., Brown, T.T., Lugar, H.M., Visscher, K.M., Miezin, F.M. & Petersen, S.E. 2002. Functional neuroanatomical differences between adults and school-age children in the processing of single words. Science, 296, 1476-9.
  • Vergnaud, G. 1997. The Nature of Mathematical Concepts. In T. Nunes & P. Bryant (Eds.), Learning and Teaching Mathematics: An International Perspectives (pp. 5-28). Eastern Sussex: Psychology Press Ltd.

Math skill in 1st grade linked to jobs, wages

Knapsack problem

A study involving 180 13-year-olds who had been assessed every year since kindergarten has found that their understanding of the number system in first grade predicted functional numeracy more than six years later, but skill at using counting procedures to solve arithmetic problems did not. Researchers controlled for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors.

Math anxiety starts before school, impacts math achievement

Numbers, comic image

"The general consensus is that math anxiety doesn't affect children much before fourth grade.” New research contests that.

Study 1: found many first grade students do experience negative feelings and worry related to math. This math anxiety negatively affects their math performance when it comes to solving math problems in standard arithmetic notation.

Development of mathematics in children — a round-up of recent news

  • Fifth grade students' understanding of fractions and division predicted high school students' knowledge of algebra and overall math achievement.
  • School entrants’ spatial skills predicted later number sense and estimation skills.
  • Gender differences in math performance may rest in part on differences in retrieval practice.
  • ‘Math’ training for infants may be futile, given new findings that they’re unable to integrate two mechanisms for number estimation.

Grasp of fractions and long division predicts later math success

One possible approach to improving mathematics achievement comes from a recent study finding that fifth graders' understanding of fractions and division predicted high school students' knowledge of algebra and overall math achievement, even after statistically controlling for parents' education and income and for the children's own age, gender, I.Q., reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication.

The study compared two nationally representative data sets, one from the U.S. and one from the United Kingdom. The U.S. set included 599 children who were tested in 1997 as 10-12 year-olds and again in 2002 as 15-17-year-olds. The set from the U.K. included 3,677 children who were tested in 1980 as 10-year-olds and in 1986 as 16-year-olds.

You can watch a short video of Siegler discussing the study and its implications at http://youtu.be/7YSj0mmjwBM.

Spatial skills improve children’s number sense

More support for the idea that honing spatial skills leads to better mathematical ability comes from a new children’s study.

The study found that first- and second-graders with the strongest spatial skills at the beginning of the school year showed the most improvement in their number line sense over the course of the year. Similarly, in a second experiment, not only were those children with better spatial skills at 5 ½ better on a number-line test at age 6, but this number line knowledge predicted performance on a math estimation task at age 8.

Hasty answers may make boys better at math

A study following 311 children from first to sixth grade has revealed gender differences in their approach to math problems. The study used single-digit addition problems, and focused on the strategy of directly retrieving the answer from long-term memory.

Accurate retrieval in first grade was associated with working memory capacity and intelligence, and predicted a preference for direct retrieval in second grade. However, at later grades the relation reversed, such that preference in one grade predicted accuracy and speed in the next grade.

Unlike girls, boys consistently preferred to use direct retrieval, favoring speed over accuracy. In the first and second grades, this was seen in boys giving more answers in total, and more wrong answers. Girls, on the other hand, were right more often, but responded less often and more slowly. By sixth grade, however, the boys’ practice was paying off, and they were both answering more problems and getting more correct.

In other words, while ability was a factor in early skilled retrieval, the feedback loop of practice and skill leads to practice eventually being more important than ability — and the relative degrees of practice may underlie some of the gender differences in math performance.

The findings also add weight to the view being increasingly expressed, that mistakes are valuable and educational approaches that try to avoid mistakes (e.g., errorless learning) should be dropped.

Infants can’t compare big and small groups

Our brains process large and small numbers of objects using two different mechanisms, seen in the ability to estimate numbers of items at a glance and the ability to visually track small sets of objects. A new study indicates that at age one, infants can’t yet integrate those two processes. Accordingly, while they can choose the larger of two sets of items when both sets are larger or smaller than four, they can’t distinguish between a large (above four) and small (below four) set.

In the study, infants consistently chose two food items over one and eight items over four, but chose randomly when asked to compare two versus four and two versus eight.

The researchers suggest that educational programs that claim to give children an advantage by teaching them arithmetic at an early age are unlikely to be effective for this reason.

Gender differences in effects of anxiety on performance

Two studies indicate that, while anxiety is present in both sexes, it only impairs performance in females.

A British study looking at possible gender differences in the effects of math anxiety involved 433 secondary school children (11-16 years old) completing customized (year appropriate) mental mathematics tests as well as questionnaires designed to assess math anxiety and (separately) test anxiety. These sources of anxiety are often confounded in research studies (and in real life!), and while they are indeed related, reported correlations are moderate, ranging from .30 to .50.

Previous research has been inconsistent as regards gender differences in math anxiety. While many studies have found significantly greater levels of math anxiety in females, many studies have found no difference, and some have even found higher levels in males. These inconsistencies may stem from differences in how math anxiety is defined or measured.

The present study looked at a rather more subtle question: does the connection between math anxiety and math performance differ by gender? Again, previous research has produced inconsistent findings.

Findings in this study were very clear: while there was no difference between boys and girls in math performance, there were marked differences in both math and test anxiety. Girls showed significantly greater levels of both. Both boys and girls showed a positive correlation between math anxiety and test anxiety, and a negative correlation between math anxiety and math performance, and test anxiety and performance. However, these relationships between anxiety and performance were stronger for girls than boys, with the correlation between test anxiety and performance being only marginally significant for boys (p<0.07), and the correlation between math anxiety and performance disappearing once test anxiety was controlled for.

In other words, greater math anxiety was linked to poorer math performance, but it was significant only for girls. Moreover, anxiety experienced by boys may simply reflect test anxiety, rather than specific math anxiety.

It is worth emphasizing that there was no gender difference in performance — that is, despite laboring under the burden of greater levels of anxiety, the girls did just as well as boys. This suggests that girls might do better than boys if they were free of anxiety. It is possible, however, that levels of anxiety didn’t actually differ between boys and girls — that the apparent difference stems from girls feeling more free to express their anxiety.

However, the finding that anxiety is greater in girls than boys is in line with evidence that anxiety (and worry in particular) is twice as prevalent in women as men, and more support for the idea that the girls are under-performing because of their anxiety comes from another recent study.

In this study, 149 college students performed a relatively simple task while their brain activity was measured. Specifically, they had to identify the middle letter in a series of five-letter groups. Sometimes the middle letter was the same as the other four ("FFFFF") while sometimes it was different ("EEFEE"). Afterward the students completed questionnaires about their anxiety and how much they worry (Penn State Worry Questionnaire and the Anxious Arousal subscale of the Mood and Anxiety Symptom Questionnaire).

Anxiety scores were significantly negatively correlated with accuracy on the task; worry scores were unrelated to performance.

Only girls who identified themselves as particularly anxious or big worriers recorded high brain activity when they made mistakes during the task (reflecting greater performance-monitoring). Although these women performed about the same as others on simple portions of the task, their brains had to work harder at it. Then, as the test became more difficult, the anxious females performed worse, suggesting worrying got in the way of completing the task.

Greater performance monitoring was not evident among anxious men.

[A reminder: these are group differences, and don't mean that all men or all women react in these ways.]

Myths about gender and math performance

Two new reviews debunk several theories for the reasons for gender gaps in math performance.

Is there, or is there not, a gender gap in mathematics performance? And if there is, is it biological or cultural?

Although the presence of a gender gap in the U.S. tends to be regarded as an obvious truth, evidence is rather more equivocal. One meta-analysis of studies published between 1990 and 2007, for example, found no gender differences in mean performance and nearly equal variability within each gender. Another meta-analysis, using 30 years of SAT and ACT scores, found a very large 13:1 ratio of middle school boys to girls at the highest levels of performance in the early 1980s, which declined to around 4:1 by 1991, where it has remained. A large longitudinal study found that males were doing better in math, across all socioeconomic classes, by the 3rd grade, with the ratio of boys to girls in the top 5% rising to 3:1 by 5th grade.

Regardless of the extent of any gender differences in the U.S., the more fundamental question is whether such differences are biological or cultural. The historical changes mentioned above certainly point to a large cultural component. Happily, because so many more countries now participate in the Trends in International Mathematics and Science Study (TIMSS) and the Programme in International Student Assessment (PISA), much better data is now available to answer this question. In 2007, for example, 4th graders from 38 countries and 8th graders from 52 countries participated in TIMSS. In 2009, 65 countries participated in PISA.

So what does all this new data reveal about the gender gap? Overall, there was no significant gender gap in the 2003 and 2007 TIMSS, with the exception of the 2007 8th graders, where girls outperformed boys.

There were, of course, significant gender gaps on a country basis. Researchers looked at several theories for what might underlie these.

Contradicting one theory, gender gaps did not correlate reliably with gender equity. In fact, both boys and girls tended to do better in math when raised in countries where females have better equality. The primary contributor to this appears to be women’s income and rates of participation in the work force. This is in keeping with the idea that maternal education and employment opportunities have benefits for their children’s learning regardless of gender.

The researchers also looked at the more specific hypothesis put forward by Steven Levitt, that gender inequity doesn’t hurt girls' math performance in Muslim countries, where most students attend single-sex schools. This theory was not borne out by the evidence. There was no consistent link between school type and math performance across countries.

However, math performance in the 29 wealthier countries could be predicted to a very high degree by three factors: economic participation and opportunity; GDP per capita; membership of one of three clusters — Middle Eastern (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia); East Asian (Hong Kong, Japan, South Korea, Singapore, Taiwan); rest (Russia, Hungary, Czech Republic, England, Canada, US, Australia, Sweden, Norway, Scotland, Cyprus, Italy, Malta, Israel, Spain, Lithuania, Malaysia, Slovenia, Dubai). The Middle Eastern cluster scored lowest (note the exception of Dubai), and the East Asian the highest. While there are many cultural factors differentiating these clusters, it’s interesting to note that countries’ average performance tended to be higher when students attribute less importance to mastering math.

The investigators also looked at the male variability hypothesis — the idea that males are more variable in their performance, and their predominance at the top is balanced by their predominance at the bottom. The study found however that greater male variation in math achievement varies widely across countries, and is not found at all in some countries.

In sum, the cross-country variability in performance in regard to gender indicates that the most likely cause of any differences lies in country-specific social factors. These could include perception of abilities as fixed vs malleable, attitude toward math, gender beliefs.

Stereotype threat

A popular theory of women’s underachievement in math concerns stereotype threat (first proposed by Spencer, Steele, and Quinn in a 1999 paper). I have reported on this on several occasions. However, a recent review of this research claims that many of the studies were flawed in their methodology and statistical analysis.

Of the 141 studies that cited the original article and related to mathematics, only 23 met the criteria needed (in the reviewers’ opinion) to replicate the original study:

  • Both genders tested
  • Math test used
  • Subjects recruited regardless of preexisting beliefs about gender stereotypes
  • Subjects randomly assigned to experimental conditions

Of these 23, three involved younger participants (< 18 years) and were excluded. Of the remaining 20 studies, only 11 (55%) replicated the original effect (a significant interaction between gender and stereotype threat, and women performing significantly worse in the threat condition than in the threat condition compared to men).

Moreover, half the studies confounded the results by statistically adjusting preexisting math scores. That is, the researchers tried to adjust for any preexisting differences in math performance by using a previous math assessment measure such as SAT score to ‘tweak’ the baseline score. This practice has been the subject of some debate, and the reviewers come out firmly against it, arguing that “an important assumption of a covariate analysis is that the groups do not differ on the covariate. But that group difference is exactly what stereotype threat theory tries to explain!” Note, too, that the original study didn’t make such an adjustment.

So what happens if we exclude those studies that confounded the results? That leaves ten studies, of which only three found an effect (and one of these found the effect only in a subset of the math test). In other words, overwhelmingly, it was the studies that adjusted the scores that found an effect (8/10), while those that didn’t adjust them didn’t find the effect (7/10).

The power of the adjustment in producing the effect was confirmed in a meta-analysis.

Now these researchers aren’t saying that stereotype threat doesn’t exist, or that it doesn’t have an effect on women in this domain. Their point is that the size of the effect, and the evidence for the effect, has come to be regarded as greater and more robust than the research warrants.

At a practical level, this may have led to too much emphasis on tackling this problem at the expense of investigating other possible causes and designing other useful interventions.

Reference: 

Kane, J. M., & Mertz, J. E. (2012). Debunking Myths about Gender and Mathematics Performance. Notices of the AMS, 59(1), 10-21.

[2698] Stoet, G., & Geary D. C. (2012).  Can stereotype threat explain the gender gap in mathematics performance and achievement?. Review of General Psychology;Review of General Psychology. No Pagination Specified - No Pagination Specified.

Dealing with math anxiety

A new study shows that some math-anxious students can overcome performance deficits through their ability to control their negative responses. The finding indicates that interventions should focus on anticipatory cognitive control.

Math-anxiety can greatly lower performance on math problems, but just because you suffer from math-anxiety doesn’t mean you’re necessarily going to perform badly. A study involving 28 college students has found that some of the students anxious about math performed better than other math-anxious students, and such performance differences were associated with differences in brain activity.

Math-anxious students who performed well showed increased activity in fronto-parietal regions of the brain prior to doing math problems — that is, in preparation for it. Those students who activated these regions got an average 83% of the problems correct, compared to 88% for students with low math anxiety, and 68% for math-anxious students who didn’t activate these regions. (Students with low anxiety didn’t activate them either.)

The fronto-parietal regions activated included the inferior frontal junction, inferior parietal lobe, and left anterior inferior frontal gyrus — regions involved in cognitive control and reappraisal of negative emotional responses (e.g. task-shifting and inhibiting inappropriate responses). Such anticipatory activity in the fronto-parietal region correlated with activity in the dorsomedial caudate, nucleus accumbens, and left hippocampus during math activity. These sub-cortical regions (regions deep within the brain, beneath the cortex) are important for coordinating task demands and motivational factors during the execution of a task. In particular, the dorsomedial caudate and hippocampus are highly interconnected and thought to form a circuit important for flexible, on-line processing. In contrast, performance was not affected by activity in ‘emotional’ regions, such as the amygdala, insula, and hypothalamus.

In other words, what’s important is not your level of anxiety, but your ability to prepare yourself for it, and control your responses. What this suggests is that the best way of dealing with math anxiety is to learn how to control negative emotional responses to math, rather than trying to get rid of them.

Given that cognitive control and emotional regulation are slow to mature, it also suggests that these effects are greater among younger students.

The findings are consistent with a theory that anxiety hinders cognitive performance by limiting the ability to shift attention and inhibit irrelevant/distracting information.

Note that students in the two groups (high and low anxiety) did not differ in working memory capacity or in general levels of anxiety.

Reference: 

[2600] Lyons, I. M., & Beilock S. L. (2011).  Mathematics Anxiety: Separating the Math from the Anxiety. Cerebral Cortex.

You can download a copy of the study here: Math anxiety.pdf

News Topic dyscalculia


Older news items (pre-2010) brought over from the old website

Right parietal lobe implicated in dyscalculia

By temporarily knocking out an area in the right parietal lobe (the right intraparietal sulcus), researchers have induced dyscalculia in normal subjects, providing strong evidence that dyscalculia is caused by malfunction in this area. These findings were further validated by testing participants suffering from developmental dyscalculia. Although less well-known, dyscalculia is as prevalent as dyslexia and attention deficit hyperactivity disorder (around 5%).

Kadosh, R.C. et al. 2007. Virtual Dyscalculia Induced by Parietal-Lobe TMS Impairs Automatic Magnitude Processing. Current Biology, online ahead of print March 22

http://www.sciencedaily.com/releases/2007/03/070322132931.htm
http://www.eurekalert.org/pub_releases/2007-03/ucl-tro032107.php

Scientists find brain function most important to math ability

A finding that an area of the brain widely thought to be involved in processing number information generally, in fact has two very separate functions, may be the key to diagnosing dyscalculia. One function is responsible for counting 'how many' things are present and the other is responsible for knowing 'how much'. The brain activity specific to estimating numbers of things is thought to be the brain network that underlies arithmetic and may be abnormal in dyscalculics.

[1336] Castelli, F., Glaser D. E., & Butterworth B. (2006).  Discrete and analogue quantity processing in the parietal lobe: A functional MRI study. Proceedings of the National Academy of Sciences of the United States of America. 103(12), 4693 - 4698.

http://www.eurekalert.org/pub_releases/2006-03/ucl-sfb030606.php

Calculation difficulties in children of very low birthweight

Learning difficulties, including problems with numeracy, are common in Western populations. Many children with learning difficulty are survivors of preterm birth. Although some of these children have neurological disabilities, many are neurologically normal. A neuroimaging study of neurologically normal adolescent children who had been born preterm at 30 weeks gestation or less found an area in the left parietal lobe where children without a deficit in calculation ability have more grey matter than those who do have this deficit.

[1281] Isaacs, E. B., Edmonds C. J., Lucas A., & Gadian D. G. (2001).  Calculation difficulties in children of very low birthweight: A neural correlate. Brain. 124(9), 1701 - 1707.

http://brain.oupjournals.org/cgi/content/abstract/124/9/1701
http://news.bbc.co.uk/hi/english/sci/tech/newsid_1512000/1512664.stm
http://www.independent.co.uk/story.jsp?story=90945

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