### 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.