How to Learn math without fear, Stanford expert says

Stanford Prof. Boaler finds that children who excel in math learn to develop "number sense," which is much different from the memorization that is often stressed in school.
Image Credit: THEPLANETWALL STOCK

Students learn math best when they approach the subject as something they enjoy, according to a Stanford education expert. Speed pressure, timed testing and blind memorization pose high hurdles in the youthful pursuit of math.

"There is a common and damaging misconception in mathematics – the idea that strong math students are fast math students," said Jo Boaler, a Stanford professor of mathematics education and the lead author on a new working paper. Boaler's co-authors are Cathy Williams, cofounder of Stanford'sYouCubed, and Amanda Confer, a Stanford graduate student in education. 

Curriculum timely

Fortunately, said Boaler, the new national curriculum standards known as the Common Core Standards for K-12 schools de-emphasize the rote memorization of math facts. Maths facts are fundamental assumptions about math, such as the times tables (2 x 2 = 4), for example. Still, the expectation of rote memorization continues in classrooms and households across the United States.

While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added.

On the other hand, people with "number sense" are those who can use numbers flexibly, she said. For example, when asked to solve the problem of 7 x 8, someone with number sense may have memorized 56, but they would also be able to use a strategy such as working out 10 x 7 and subtracting two 7s (70-14).

"They would not have to rely on a distant memory," Boaler wrote.

In fact, in one research project the investigators found that the high-achieving students actually used number sense, rather than rote memory, and the low-achieving students did not.

The conclusion was that the low achievers are often low achievers not because they know less but because they don't use numbers flexibly.

"They have been set on the wrong path, often from an early age, of trying to memorize methods instead of interacting with numbers flexibly," she wrote. Number sense is the foundation for all higher-level mathematics, she noted. 

Role of the brain

Boaler said that some students will be slower when memorizing, but still possess exceptional mathematics potential.

"Math facts are a very small part of mathematics, but unfortunately students who don't memorize math facts well often come to believe that they can never be successful with math and turn away from the subject," she said.

Prior research found that students who memorized more easily were not higher achieving – in fact, they did not have what the researchers described as more "math ability" or higher IQ scores. Using an MRI scanner, the only brain differences the researchers found were in a brain region called the hippocampus, which is the area in the brain responsible for memorizing facts – the working memory section.

But according to Boaler, when students are stressed – such as when they are solving math questions under time pressure – the working memory becomes blocked and the students cannot as easily recall the math facts they had previously studied. This particularly occurs among higher achieving students and female students, she said.

Some estimates suggest that at least a third of students experience extreme stress or "math anxiety" when they take a timed test, no matter their level of achievement. "When we put students through this anxiety-provoking experience, we lose students from mathematics," she said.

Boaler contrasts the common approach to teaching math with that of teaching English. In English, a student reads and understands novels or poetry, without needing to memorize the meanings of words through testing. They learn words by using them in many different situations – talking, reading and writing.

"No English student would say or think that learning about English is about the fast memorization and fast recall of words," she added.

Strategies, activities 

In her paper, "Fluency without Fear," Boaler provides activities for teachers and parents that help students learn math facts at the same time as developing number sense. These include number talks, addition and multiplication activities, and math cards.

Importantly, she said, these activities include a focus on the visual representation of number facts. When students connect visual and symbolic representations of numbers, they are using different pathways in the brain, which deepens their learning, as shown by recent brain research.

"Math fluency" is often misinterpreted, with an over-emphasis on speed and memorization, she said. "I work with a lot of mathematicians, and one thing I notice about them is that they are not particularly fast with numbers; in fact some of them are rather slow. This is not a bad thing; they are slow because they think deeply and carefully about mathematics."

She refers to the famous French mathematician, Laurent Schwartz, who wrote in his autobiography that he often felt stupid in school, as he was one of the slowest math thinkers in class.
Math anxiety and fear play a big role in students dropping out of mathematics, said Boaler.

"When we emphasize memorization and testing in the name of fluency we are harming children, we are risking the future of our ever-quantitative society and we are threatening the discipline of mathematics. We have the research knowledge we need to change this and to enable all children to be powerful mathematics learners. Now is the time to use it," she said.

Source: Standford Unversity

New technology may identify tiny strains in body tissues before injuries occur

The top image shows how the new algorithm is able to identify an area (in red) where stress has created a weak spot in a small piece of plastic wrap. The older method (shown in the bottom half of the picture) is unable to pinpoint the place where the plastic wrap is weakening.
Credit: John Boyle, © The Royal Society (used with permission)

Researchers at Washington University in St. Louis have developed algorithms to identify weak spots in tendons, muscles and bones prone to tearing or breaking. The technology, which needs to be refined before it is used in patients, one day may help pinpoint minor strains and tiny injuries in the body's tissues long before bigger problems occur.

The research is available online Aug. 27 in the Journal of the Royal Society Interface, which publishes research at the nexus of the physical and life sciences.

"Tendons are constantly stretching as muscles pull on them, and bones also bend or compress as we carry out everyday activities," said senior investigator Stavros Thomopoulos, PhD, professor of orthopaedic surgery. "Small cracks or tears can result from these loads and lead to major injuries. Understanding how these tears and cracks develop over time therefore is important for diagnosing and tracking injuries."

To that end, Thomopoulos and his colleagues developed a way to visualize and even predict spots where tissues are weakened. To accomplish this, they stretched tissues and tracked what happened as their shapes changed or became distorted.

The paper's first author, John J. Boyle, a graduate student in biomedical engineering, combined mechanical engineering fundamentals with image-analysis techniques to create the algorithms, which were tested in different materials and in animal models.

"If you imagine stretching Silly Putty or a swimming cap with a picture on it, as you pull, the picture becomes distorted," Boyle said. "This allows us to track how the material responds to an external force."

In one of the experiments described in the paper, Boyle sprayed a pattern of dots on plastic wrap, stretched it and tracked the dots.

"As you pull and stretch the plastic wrap, eventually tears begin to emerge," he explained. 

"The new algorithm allowed us to find the places where the tears were beginning to form and to track them as they extended. Older algorithms are not as good at finding and tracking localized strains as the material stretches."

In fact, one of the two new algorithms is 1,000 times more accurate than older methods at quantifying very large stretches near tiny cracks and tears, the research showed. And a second algorithm has the ability to predict where cracks and failures are likely to form.

"This extra accuracy is critical for quantifying large strains," said Guy Genin, PhD, professor of mechanical engineering and co-senior investigator on the study. "Commercial algorithms that estimate strain often are much less sensitive, and they are prone to detecting noise that can arise from the algorithm itself rather than from the material being examined. The new algorithms can distinguish the noise from true regions of large strains."

Thomopoulos, who also is a professor of biomedical engineering and of mechanical engineering, works with Genin to study the shoulder's rotator cuff, a group of tendons and muscles that connect the upper arm to the shoulder blade. They want to learn why some surgeries to repair rotator cuff injuries ultimately fail. Their goal is to increase the odds that the tissue in the shoulder will heal following surgery, and they believe the new algorithms could help them get closer to that goal.

How soon the new algorithms could be used in patients depends on getting better images of the body's tissues. Current imaging techniques, such as MRI and ultrasound, lack the required clarity and resolution.

Genin also explained that although the goal of the current study is to better understand how forces at work on human tissue cause injury and stress, the algorithms also could help engineers identify vulnerable parts of buildings and other structures. Our muscles and bones, he said, are influenced by the same strains that affect those structures.

"Whether it's a bridge or a tendon, it's vital to understand the ways that physical forces cause structures and tissues to deform so that we can identify the onset of failures and eventually predict them," he said.

In the long run, they want to use the algorithms to prevent additional injuries following surgery to repair knees, shoulders and other tissues. They also said it may be possible some day to predict problems before they occur.

The group, which applied for a provisional patent earlier this year, hopes the algorithms will be useful to researchers in the medical and engineering fields.

As a piece of plastic wrap is stretched, the new algorithms identify the location (in red) where it is weakening, which is where the material eventually breaks.

Source: Washington University in St. Louis