# Mathematics

Also see

Dyscalculia

## Latest Research News

I've reported before on studies showing how gesturing can help children with mathematics and problem-solving. A new Australian study involving children aged 9-13 has found that finger-tracing has a similar effect.

Students who used their finger to trace over practice examples while simultaneously reading geometry or arithmetic material were able to complete the problems more quickly and correctly than those who didn't use the same technique.

In the first experiment, involving 52 students aged 11-13, some students were instructed to use their index fingers to trace elements of worked examples in triangle geometry, involving two angle relationships (Vertical angles are equal; Any exterior angle equals the sum of the two interior opposite angles.). Students were given two minutes to study a short instructional text on the relationships and how they can be used to solve particular problems. They were then given two minutes to study two worked examples. The tracing group were given additional instruction in how to use their index finger to trace out highlighted elements. The non-tracing group were told to keep their hands in their lap. Testing consisted of six questions, two of which were the same as the acquisition problems but with different numbers, and four of which were transfer questions, requiring more thoughtful responses.

A ceiling effect meant there was no difference between the two groups on the first two test questions. The tracing group answered significantly more transfer questions, although the difference wasn't great. There was no difference in how difficult the groups rated the test items.

In the second experiment, involving 54 Year 4 students, the instruction and problems concerned the fundamental order of operations. The tracing group were told to trace the operation symbols. The tracing group did significantly better, although again, the difference wasn't great, and again, there was no difference in assessment of problem difficulty.

In another experiment, involving 42 Year 5 students (10-11 years), students were given 5 minutes to study three angle relationships involving parallel lines (vertical angles are equal; corresponding angles are equal; the sum of co- interior angles is 180°). While answers to the 'basic' test questions failed to show significant differences, on the advanced transfer problems, the tracing group solved significantly more test questions than the non-tracing group, solved them more quickly, made fewer errors, and reported lower levels of test difficulty.

In the final experiment, involving 72 Year 5 students, on the advanced test problems, students who traced on the paper outperformed those who traced above the paper, who in turn outperformed those who simply read the material.

The researchers claim the findings support the view that tracing out elements of worked examples helps students construct good mental schemas, making it easier for them to solve new problems, and reducing cognitive demand.

As with gesturing, the benefits of tracing are not dramatic, but I believe the pattern of these results support the view that, when cognitive load is high (something that depends on the individual student as well as the task and its context), tracing key elements of worked examples might be a useful strategy.

Further research looking at individual differences would be helpful. I think greater benefits would be shown for students with low working memory capacity.

[4046] . Getting the point: Tracing worked examples enhances learning. Learning and Instruction [Internet]. 2015 ;35:85 - 93. Available from: http://www.sciencedirect.com/science/article/pii/S0959475214000929

[4043] . Learning By Tracing Worked Examples. Applied Cognitive Psychology [Internet]. 2015 :n/a - n/a. Available from: http://onlinelibrary.wiley.com/doi/10.1002/acp.3171/abstract

A study of 438 first- and second-grade students and their primary caregivers has revealed that parents' math anxiety affects their children's math performance — but (and this is the surprising bit) only when they frequently help them with their math homework.

The study builds on previous research showing that students learn less math when their teachers are anxious about math. This is not particularly surprising, and it wouldn't have been surprising if this study had found that math-anxious parents had math-anxious children. But the story wasn't that simple.

Children were assessed in reading achievement, math achievement and math anxiety at both the beginning and end of the school year. Children of math-anxious parents learned significantly less math over the school year and had more math anxiety by the year end—but only if math-anxious parents reported providing help every day with math homework. When parents reported helping with math homework once a week or less often, children’s math achievement and attitudes were not related to parents’ math anxiety. Reading achievement (included as a control) was not related to parents' math anxiety.

Interestingly, the parents' level of math knowledge didn't change this effect (although this is less surprising when you consider the basic-level of math taught in the 1st and 2nd grade).

Sadly, the effect still held even when the teacher was strong in math.

It's suggested that math-anxious parents may be less effective in explaining math concepts, and may also respond less helpfully when children make a mistake or solve the problem in a non-standard way. People with high math anxiety tend to have poor attitudes toward math, and also a high fear of failing at math. It's also possible (likely even) that they will have inflexible attitudes to how a math problem “should” be done. All of these are likely to demotivate the child.

Analysis also indicated that it is not that parents induced math anxiety in their children, who thus did badly, but that their homework help caused the child to do poorly, thus increasing their math anxiety.

Information about parental anxiety and how often parents helped their children with math homework was collected by questionnaire. Math anxiety was assessed using the short (25-item) Math Anxiety Rating Scale. The question, “How often do you help your child with their math homework?” was answered on a 7-point scale (1 = never, 2 = once a month, 3 = less than once a week, 4 = once a week, 5 = 2–3 times a week, 6 = every day, 7 = more than once a day). The mean was 5.3.

The finding points to the need for interventions focused on both decreasing parents' math anxiety and scaffolding their skills in how to help with math homework. It also suggests that, in the absence of such support, math-anxious parents are better not to help!

http://www.futurity.org/parents-math-anxiety-979472/

A new study claims to provide ‘some of the strongest evidence yet’ for the benefits of gesturing to help students learn.

The study involved 184 children aged 7-10, of whom half were shown videos of an instructor teaching math problems using only speech, while the rest were shown videos of the instructor teaching the same problems using both speech and gestures. The problem involved mathematical equivalence (i.e., 4+5+7=__+7), which is known to be critical to later algebraic learning.

Students who learned from the gesture videos performed substantially better on a test given immediately afterward than those who learned from the speech-only video (average proportion correct around 42% vs 31% — approximations because I’m eyeballing the graph), and, unlike the speech-only group, showed further improvement on a test 24 hours later (around 46% vs 30%). They also showed stronger transfer to different problem types (35% vs 22%).

http://www.futurity.org/society-culture/to-teach-kids-math-keep-hands-mo...

[3377] . Consolidation and Transfer of Learning After Observing Hand Gesture. Child Development [Internet]. 2013 :n/a - n/a. Available from: http://onlinelibrary.wiley.com/doi/10.1111/cdev.12097/abstract

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.

Number system knowledge includes an understanding of the relative magnitude of numerals, their ordering, and the ability to combine and decompose them into smaller and larger numerals and to use this knowledge to solve arithmetic problems.

Tests for functional numeracy were chosen based on labor economic studies of employability, wages, and related outcomes in adulthood. One in five adults in the U.S. lacks the math skills expected of an eighth grader.

[3270] . Adolescents’ Functional Numeracy Is Predicted by Their School Entry Number System Knowledge. PLoS ONE [Internet]. 2013 ;8(1). Available from: http://dx.doi.org/10.1371/journal.pone.0054651

"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.

Study 2: found that second grade math anxiety affected second grade computations and math applications. Additionally, children with higher levels of math anxiety in second grade learned less math in third grade.

"Students are walking into classrooms at five and six-years-old saying that they aren't good at math before they've even stepped into a math classroom.”

Vukovic, R.K. et al. (in press) Mathematics Anxiety in Young Children. Journal of Experimental Education

[3322] . Mathematics anxiety in young children: Concurrent and longitudinal associations with mathematical performance. Contemporary Educational Psychology [Internet]. 2013 ;38(1):1 - 10. Available from: http://www.sciencedirect.com/science/article/pii/S0361476X12000471

### 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.

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.]

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.

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

[2698] . Can stereotype threat explain the gender gap in mathematics performance and achievement?. Review of General Psychology;Review of General Psychology. 2012 :No Pagination Specified - No Pagination Specified.

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 lobule, 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.

Working memory capacity and level of math anxiety were assessed in 73 undergraduate students, and their level of salivary cortisol was measured both before and after they took a stressful math test.

For those students with low working memory capacity, neither cortisol levels nor math anxiety made much difference to their performance on the test. However, for those with higher WMC, the interaction of cortisol level and math anxiety was critical. For those unafraid of math, the more their cortisol increased during the test, the better they performed; but for those anxious about math, rising cortisol meant poorer performance.

It’s assumed that low-WMC individuals were less affected because their performance is lower to start with (this shouldn’t be taken as an inevitability! Low-WMC students are disadvantaged in a domain like math, but they can learn strategies that compensate for that problem). But the effect on high-WMC students demonstrates how our attitude and beliefs interact with the effects of stress. We may all have the same physiological responses, but we interpret them in different ways, and this interpretation is crucial when it comes to ‘higher-order’ cognitive functions.

Another study investigated two theories as why people choke under pressure: (a) they’re distracted by worries about the situation, which clog up their working memory; (b) the stress makes them pay too much attention to their performance and become self-conscious. Both theories have research backing from different domains — clearly the former theory applies more to the academic testing environment, and the latter to situations involving procedural skill, where explicit attention to the process can disrupt motor sequences that are largely automatic.

But it’s not as simple as one effect applying to the cognitive domain, and one to the domain of motor skills, and it’s a little mysterious why pressure could have too such opposite effects (drawing attention away, or toward). This new study carried out four experiments in order to define more precisely the characteristics of the environment that lead to these different effects, and suggest solutions to the problem.

In the first experiment, participants were given a category learning task, in which some categories had only one relevant dimension and could be distinguished according to one easily articulated rule, and others involved three relevant dimensions and one irrelevant one. Categorization in this case was based on a complex rule that would be difficult to verbalize, and so participants were expected to integrate the information unconsciously.

Rule-based category learning was significantly worse when participants were also engaged in a secondary task requiring them to monitor briefly appearing letters. However it was not affected when their secondary task involved them explicitly monitoring the categorization task and making a confidence judgment. On the other hand, the implicit category learning task was not disrupted by the letter-monitoring task, but was impaired by the confidence-judgment task. Further analysis revealed that participants who had to do the confidence-judgment task were less likely to use the best strategy, but instead persisted in trying to verbalize a one- or two-dimension rule.

In the second experiment, the same tasks were learned in a low-pressure baseline condition followed by either a low-pressure control condition or one of two high-pressure conditions. One of these revolved around outcome — participants would receive money for achieving a certain level of improvement in their performance. The other put pressure on the participants through monitoring — they were watched and videotaped, and told their performance would be viewed by other students and researchers.

Rule-based category learning was slower when the pressure came from outcomes, but not when the pressure came from monitoring. Implicit category learning was unaffected by outcome pressure, but worsened by monitoring pressure.

Both high-pressure groups reported the same levels of pressure.

Experiment 3 focused on the detrimental combinations — rule-based learning under outcome pressure; implicit learning under monitoring pressure — and added the secondary tasks from the first experiment.

As predicted, rule-based categories were learned more slowly during conditions of both outcome pressure and the distracting letter-monitoring task, but when the secondary task was confidence-judgment, the negative effect of outcome pressure was counteracted and no impairment occurred. Similarly, implicit category learning was slowed when both monitoring pressure and the confidence-judgment distraction were applied, but was unaffected when monitoring pressure was counterbalanced by the letter task.

The final experiment extended the finding of the second experiment to another domain — procedural learning. As expected, the motor task was significantly affected by monitoring pressure, but not by outcome pressure.

These findings suggest two different strategies for dealing with choking, depending on the situation and the task. In the case of test-taking, good test preparation and a writing exercise can boost performance by reducing anxiety and freeing up working memory. If you're worried about doing well in a game or giving a memorized speech in front of others, you instead want to distract yourself so you don't become focused on the details of what you're doing.

[2357] . Choking under pressure: Multiple routes to skill failure. Journal of Experimental Psychology: General [Internet]. 2011 ;140(3):390 - 406. Available from: http://psycnet.apa.org/journals/xge/140/3/390/

Mathematics is a complex cognitive skill, requiring years of formal study. But of course some math is much simpler than others. Counting is fairly basic; calculus is not. To what degree does ability at the simpler tasks predict ability at the more complex? None at all, it was assumed, but research with adolescents has found an association between math ability and simple number sense (or as it’s called more formally, the "Approximate Number System" or ANS).

A new study extends the finding to preschool children. The study involved 200 3- to 5-year-old children, who were tested on their number sense, mathematical ability and verbal ability. The number sense task required children to estimate which group had more dots, when seeing briefly presented groups of blue and yellow dots on a computer screen. The standardized test of early mathematics ability required them to verbally count items on a page, to tell which of two spoken number words was greater or lesser, to read Arabic numbers, as well as demonstrate their knowledge of number facts (such as addition or multiplication), calculation skills (solving written addition and subtraction problems) and number concepts (such as answering how many sets of 10 are in 100). The verbal assessment was carried out by parents and caregivers of the children.

The study found that those who could successfully tell when the difference between the groups was only one dot, also knew the most about Arabic numerals and arithmetic. In other words, the findings confirm that number sense is linked to math ability.

Because these preschoolers have not yet had formal math instruction, the conclusion being drawn is that this number sense is inborn. I have to say that seems to me rather a leap. Certainly number sense is seen in human infants and some non-human animals, and in that sense the ANS is assuredly innate. However what we’re talking about here is the differences in number sense — the degree to which it has been developed. I’d remind you of my recent report that preschoolers whose parents engage in the right number-talk develop an understanding of number earlier, and that such understanding affects later math achievement. So I think it’s decidedly premature to assume that some infants are born with a better number sense, as opposed to having the benefit of informal instruction that develops their number sense.

I think, rather, that the finding adds to the evidence that preschoolers’ experiences and environment have long-lasting effects on academic achievement.

At every level, later math learning depends on earlier understanding. Previous research has found that the knowledge children have of number before they start school predicts their achievement throughout elementary school.

One critical aspect of mathematical development is cardinal-number knowledge (e.g. knowing that the word ‘three’ refers to sets of three things). But being able to count doesn’t mean the child understands this principle. Children who enter kindergarten with a good understanding of the cardinal principle have been found to do better in mathematics.

Following research indicating an association between children’s knowledge of number and the amount of number talk their parents engage in, a new study recorded parental interactions for 44 young children aged 14-30 months. Five 90-minute sessions, four months apart, were recorded in the children’s home, and each instance in which parents talked about numbers with their children was noted and coded. The children were then (at nearly four years) tested on their understanding of the cardinal principle.

The study found that parents’ number talk involving counting or labeling sets of visible objects related to children’s later cardinal-number knowledge, whereas other types of parent number talk were not. Talk of larger sets, containing more than 3 objects, was particularly important. This is probably because children can recognize number sets of three or less in a holistic way.

A study involving 171 sedentary, overweight 7- to 11-year-old children has found that those who participated in an exercise program improved both executive function and math achievement. The children were randomly selected either to a group that got 20 minutes of aerobic exercise in an after-school program, one that got 40 minutes of exercise in a similar program, or a group that had no exercise program. Those who got the greater amount of exercise improved more. Brain scans also revealed increased activity in the prefrontal cortex and reduced activity in the posterior parietal cortex, for those in the exercise group.

The program lasted around 13 weeks. The researchers are now investigating the effects of continuing the program for a full year. Gender, race, socioeconomic factors or parental education did not change the impact of the exercise program.

The effects are consistent with other studies involving older adults. It should be emphasized that these were sedentary, overweight children. These findings are telling us what the lack of exercise is doing to young minds. I note the report just previous, about counteracting what we have regarded as “normal” brain atrophy in older adults by the simple action of walking for 40 minutes three times a week. Children and older adults might be regarded as our canaries in the coal mine, more vulnerable to many factors that can affect the brain. We should take heed.

A meta-analysis of 242 articles assessing the math skills of 1,286,350 people found no difference between the two sexes. This was confirmed in an analysis of the data from several large surveys of American adolescents (the National Longitudinal Surveys of Youth, the National Education Longitudinal Study of 1988, the Longitudinal Study of American Youth, and the National Assessment of Educational Progress).

[1924] . New trends in gender and mathematics performance: A meta-analysis. Psychological Bulletin [Internet]. 2010 ;136(6):1123 - 1135. Available from: http://psycnet.apa.org/journals/bul/136/6/1123/

When children learn to count, they do so by rote. Understanding what the numbers really mean comes later. This is reflected in the way children draw a number line. In the beginning, they typically put more space between the smaller numbers, with the larger numbers all scrunched up at the end (a logarithmic number line). Eventually they progress to a number line where the numbers are evenly spaced (linear number line).

Now a series of experiments with preschoolers and second graders has revealed that the more linear the child's magnitude representations (as seen on the number line as well as in other tasks), the better the child was at remembering numbers (for example, from a story with some numbers included).

This was true for preschoolers for numbers from 1-20 and for elementary school children for numbers from 1-1000, and for four different number tasks measuring numerical-magnitude representations (categorization and number-line, measurement, and numerosity estimation). Other types of numerical knowledge—numeral identification and counting—were unrelated to remembering numbers.

[1894] . Linear Numerical-Magnitude Representations Aid Children’s Memory for Numbers. Psychological Science [Internet]. 2010 ;21(9):1274 - 1281. Available from: http://pss.sagepub.com/content/21/9/1274.abstract

A number of studies have demonstrated that negative stereotypes (such as “women are bad at math”) can impair performance in tests. Now a new study shows that this effect extends to learning. The study involved learning to recognize target Chinese characters among sets of two or four. Women who were reminded of the negative stereotypes involving women's math and visual processing ability failed to improve at this search task, while women who were not reminded of the stereotype got faster with practice. When participants were later asked to choose which of two colored squares, imprinted with irrelevant Chinese characters, was more saturated, those in the control group were slower to respond when one of the characters had been a target. However, those trained under stereotype threat showed no such effect, indicating that they had not learned to automatically attend to a target. It’s suggested that the women in the stereotype threat group tried too hard to overcome the negative stereotype, expending more effort but in an unproductive manner.

There are two problems here, it seems. The first is that people under stereotype threat have more invested in disproving the stereotype, and their efforts may be counterproductive. The second, that they are distracted by the stereotype (which uses up some of their precious working memory).

[1686] . Stereotype threat prevents perceptual learning. Proceedings of the National Academy of Sciences [Internet]. 2010 . Available from: http://www.pnas.org/content/early/2010/07/20/1002815107.abstract

Consistent with studies showing that gender stereotypes can worsen math performance in females, a year-long study involving 17 first- and second-grade teachers and their 52 boy and 65 girl students has found that boys' math performance was not related to their (female) teacher's math anxiety while girls' math achievement was. Early elementary school teachers in the United States are almost exclusively female. Math achievement was unrelated to teacher math anxiety in both boys and girls at the beginning of the school year. Moreover, achievement was negatively associated with belief in gender stereotypes. Girls who confirmed a belief that boys are better in math than girls scored six points lower in math achievement than did boys or girls who had not developed a belief in the stereotype (102 versus 108). Research has found that elementary education majors have the highest rate of mathematics anxiety of any college major.

[1450] . Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences [Internet]. 2010 ;107(5):1860 - 1863. Available from: http://www.pnas.org/content/107/5/1860.abstract

Data from North Carolina's mandated End-of-Grade tests (2000-2005), which includes student reports on how frequently they use a home computer for schoolwork, watch TV or read for pleasure, reveals that students in grades five through eight (c.10-13), particularly those from disadvantaged families, tended to have lower reading and math scores after they got a home computer. The researchers suggest that the greater negative effect in disadvantaged households may reflect less parental monitoring.

[1635] . Scaling the Digital Divide: Home Computer Technology and Student Achievement. National Bureau of Economic Research Working Paper Series [Internet]. 2010 ;No. 16078. Available from: http://www.nber.org/papers/w16078

Analysis of 30 years of SAT and ACT tests administered to the top 5% of U.S. 7th graders has found that the ratio of 7th graders scoring 700 or above on the SAT-math has dropped from about 13 boys to 1 girl to about 4 boys to 1 girl. The ratio dropped dramatically between 1981 and 1995, and has remained relatively stable since then. The top scores on scientific reasoning, a relatively new section of the ACT that was not included in the original study, show a similar ratio of boys to girls.

## Factors influencing math performance

### Early math skills best predict school success

A review of data from six studies of close to 36,000 preschoolers has revealed that the single most important factor in predicting later academic achievement is that children begin school with a mastery of early math and literacy concepts. This was true even if they have various social and emotional problems. Children's attention-related skills also mattered. The very strongest predictor of future academic success was beginning school with a knowledge of numbers, number order and other rudimentary math concepts. The study controlled for IQ, family income, gender, temperament, type of previous educational experience, and whether children came from single or two parent families. Mastery of early math skills predicted future reading achievement as well as future math achievement. The opposite was not true.

Duncan, G.J. et al. 2007. School Readiness and Later Achievement. Developmental Psychology, 43 (6), 1428–1446.

http://www.sciencedaily.com/releases/2007/11/071112182442.htm

### Gesturing helps grade-schoolers solve math problems

Two studies of children in late third and early fourth grade, who made mistakes in solving math problems, have found that children told to move their hands when explaining how they’d solve a problem were four times as likely as kids given no instructions to manually express correct new ways to solve problems. Even though they didn’t give the right answer, their gestures revealed an implicit knowledge of mathematical ideas, and the second study showed that gesturing set them up to benefit from subsequent instruction. The findings extend previous research that body movement not only helps people to express things they may not be able to verbally articulate, but actually to think better.

Broaders, S.C., Cook, S.W., Mitchell, Z. & Goldin-Meadow, S. 2007. Making Children Gesture Brings Out Implicit Knowledge and Leads to Learning. Journal of Experimental Psychology: General, 136 (4).

### Young children can add and subtract without arithmetic

We knew infants can judge simple mathematical relationships, such as being able to tell when there are more objects in one group compared to another. Now a new study shows that children can apply that ability to Arabic numerals after learning to count but before they learned to add and subtract. When given such problems as, "Sarah has 15 candies and she gets 19 more; John has 51 candies. Who has more?", five- and six-year-old children answered correctly 64—73% of the time. The research suggests ways to improve children’s engagement with formal arithmetic.

Gilmore, C.K., McCarthy, S.E. & Spelke, E.S. 2007. Symbolic arithmetic knowledge without instruction. Nature, 447, 589-591.

### Executive function as important as IQ for math success

A study of 141 preschoolers from low-income homes has found that a child whose IQ and executive functioning were both above average was three times more likely to succeed in math than a child who simply had a high IQ. The parts of executive function that appear to be particularly linked to math ability in preschoolers are working memory and inhibitory control. In this context, working memory may be thought of as the ability to keep information or rules in mind while performing mental tasks. Inhibitory control is the ability to halt automatic impulses and focus on the problem at hand. Inhibitory control was also important for reading ability. The finding offers the hope that training to improve executive function will improve academic performance

Blair, C. & Razza, R.P. 2007. Relating Effortful Control, Executive Function, and False Belief Understanding to Emerging Math and Literacy Ability in Kindergarten. Child Development, 78 (2), 647–663.

### Language affects how math is done?

A comparison of activity in the brains of Chinese and English participants doing simple arithmetic using Arabic numbers has found that, although both groups utilised the inferior parietal cortex (an area connected to quantity representation and reading), English speakers displayed more activity in the language processing area of the brain, while Chinese speakers used the area of the brain that deals with processing visual information. There was no significant difference in the reaction time and accuracy of the Chinese and English-speaking volunteers. However, an earlier study comparing Canadian and Chinese students found that the latter were better at complex maths. The findings suggest that our native language, or different teaching methods, may influence the way we solve equations.

Tang, Y. et al. 2006. Arithmetic processing in the brain shaped by cultures. Proc. Natl. Acad. Sci. USA, Published online before print June 30, 2006.

### Preschool storytelling ability linked to later mathematical ability

A new study suggests that preschool children's early storytelling abilities are predictive of their mathematical ability two years later. In the study, three-and four-year-old children were shown a book that contained only pictures and were asked to tell the story to a puppet. Their abilities were measured in a variety of ways. Two years later, the children were given a number of tests of academic achievement, including a test of mathematical achievement. It was found was that those children who scored highly on the mathematics test had also scored highly on certain measures of their storytelling ability two years earlier. "Most strongly predictive of children's mathematical performance was their ability to relate all the different events in the story, to shift clearly from the actions of one character to another, and to adopt the perspective of different characters and talk about what they were feeling or thinking." This study suggests that building strong storytelling skills early in the preschool years may be helpful in preparing children for learning mathematics when they enter school.

O’Neill, D.K. et al. 2004. Preschool children's narratives and performance on the Peabody Individualized Achievement Test - Revised: Evidence of a relation between early narrative and later mathematical ability. First Language, 24 (2), 149-184.

## Factors impairing math ability

### Gender gap in math is culture-based

Data from the Trends in International Mathematics and Science Study and the Programme for International Student Assessment, representing 493,495 students ages 14-16 from 69 countries, have revealed only very small gender differences overall, but marked variation when nations are compared. For example, there are more girls in the top tier in countries such as Iceland, Thailand, and the United Kingdom–and even in certain U.S. populations, such as Asian-Americans. However, despite overall similarities in math skills, boys felt significantly more confident in their abilities than girls did and were more motivated to do well. Furthermore, although some studies have found more males than females scoring above the 95th or 99th percentile, this gender gap has significantly narrowed over time in the U.S. and is not found among some ethnic groups and in some nations. Greater male variability with respect to mathematics, where it exists, correlates with several measures of gender inequality.

Hyde, J. S., & Mertz, J. E. (2009). Gender, culture, and mathematics performance. Proceedings of the National Academy of Sciences, 106(22), 8801-8807.

Else-Quest, N.M., Hyde, J.S. & Linn, M.C. 2010. Cross-national patterns of gender differences in mathematics: A meta-analysis. Psychological Bulletin, 136(1), 103-127.

### Iron deficiency may affect maths achievement in children and teens

A U.S. national study of 5,398 children aged 6 to 16 found iron deficiency in 3% of the children overall, and 8.7% of girls aged 12 to 16 (7% without anemia). Average math scores for iron-deficient children with or without anemia were about six points lower than those with normal iron levels. Among adolescent girls, the difference in scores was more than eight points. Previous research has linked iron-deficiency anemia with lower developmental test scores in young children, but there is less information on older children and on iron deficiency without anemia. It is suggested that this finding may help explain why the female superiority in maths at younger ages reverses itself in adolescence.

Halterman, J.S., Kaczorowski, J.M., Aligne, C.A., Auinger, P. & Szilagyi, P.G. 2001. Iron Deficiency and Cognitive Achievement Among School-Aged Children and Adolescents in the United States. Pediatrics, 107 (6), 1381-1386.

## Math Anxiety

### Positive stereotypes can offset negative stereotype effect

A number of studies have now shown that negative stereotypes can impair cognitive performance, mainly through adding to working memory load. A new study has now shown that this effect can be mitigated by the activation of a positive stereotype. The research takes advantage of the fact that we all belong to several social groups. In this case, the relevant groups were ‘female’ and ‘college student’. As usual, when (subtly) reminded of negative stereotypes for women and math, women performed worse. The interesting thing was that this didn’t happen if women were also made aware that college students performed better at math than non-college students. Moreover, this was reflected in working memory capacity. It seems that, when both a positive and a negative stereotype are offered, people will tend to choose the positive stereotype, and the effects of this will cancel out the negative stereotype. It’s also worth noting how easily these stereotypes are activated: effects could be manipulated simply by subtly changing demographic questions asked before the test (and it is not uncommon that test-takers are first required to answer some demographic questions).

Rydell, R.J., McConnell, A.R. & Beilock, S.L. 2009. Multiple social identities and stereotype threat: Imbalance, accessibility, and working memory. Journal of Personality and Social Psychology, 96(5), 949-966.

### Stereotype-induced math anxiety robs women's working memory

Another study finds evidence that being told men are better at mathematics undermines women's math performance, and extends it by demonstrating that the anxiety induced by the stereotype mainly reduced the verbal part of working memory, and that this carried over to subsequent (non-math-related) tasks. The accuracy of women exposed to the stereotype was reduced from nearly 90% in a pretest to about 80% after being told men do better in mathematics.

Beilock, S.L., Rydell, R.J. & McConnell, A.R. 2007. Stereotype threat and working memory: Mechanisms, alleviation, and spillover. Journal of Experimental Psychology: General, 136(2), 256-276.

### Implicit stereotypes and gender identification may affect female math performance

Relatedly, another study has come out showing that women enrolled in an introductory calculus course who possessed strong implicit gender stereotypes, (for example, automatically associating "male" more than "female" with math ability and math professions) and were likely to identify themselves as feminine, performed worse relative to their female counterparts who did not possess such stereotypes and who were less likely to identify with traditionally female characteristics. Strikingly, a majority of the women participating in the study explicitly expressed disagreement with the idea that men have superior math ability, suggesting that even when consciously disavowing stereotypes, female math students are still susceptible to negative perceptions of their ability.

Kiefer, A.K., & Sekaquaptewa, D. 2007. Implicit stereotypes, gender identification, and math performance: a prospective study of female math students. Psychological Science, 18(1), 13-18.

### Women's math performance affected by theories on sex differences

In a salutary reminder to all researchers into gender and race differences, researchers found that women who received a genetic explanation for female underachievement in math or were reminded of the stereotype about female math underachievement, performed more poorly on math tests than those who received an experiential explanation (such as math teachers treating boys preferentially during the first years of math education) or were led to believe there are no sex differences in math.

Dar-Nimrod, I. & Heine, S.J. 2006. Exposure to Scientific Theories Affects Women's Math Performance. Science, 314 (5798), 435.

### Anxiety over maths blocks learning

The so-called "maths block" is notorious - why do we have such a term? Do we talk about a "geography block", or a "physics block"? But we do talk of a reading block. Perhaps the reason for both is the same.
The amount of information you can work with at one time has clear limits, defined by your working memory capacity. When we are anxious, part of our working memory is taken up with our awareness of these fears and worries, leaving less capacity available for processing (which is why students who are very anxious during exams usually perform well below their capabilities). Processes such as reading and working with numbers are very sensitive to working memory capacity because they place such demands on it.
A recently reported study by Mark H. Ashcraft and Elizabeth P. Kirk, both psychologists at Cleveland (Ohio) State University, provides the first solid evidence that, indeed, math-anxious people have working memory problems as they do maths.

Ashcraft, M. H., & Kirk, E. P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130(2), 224–237. doi:10.1037/0096-3445.130.2.224

## Neural substrate of mathematics

### Where math takes place normally and in children with fetal alcohol spectrum disorder

An imaging study involving 21 children with fetal alcohol spectrum disorder confirms the importance of the left parietal area for mathematical tasks. Children with FASD are particularly impaired in mathematical ability. Brain activity patterns also revealed that the involvement of regions in the left cerebellum and the brainstem in math processing may be specific to children with FASD.

Lebel, C., Rasmussen, C., Wyper, K., Andrew, G., & Beaulieu, C. (2009). Brain Microstructure Is Related to Math Ability in Children With Fetal Alcohol Spectrum Disorder. Alcoholism: Clinical and Experimental Research, 9999(9999). doi: 10.1111/j.1530-0277.2009.01097.x.