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

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Strange materials cropping up in condensed matter laboratories

A conformal field theory in condensed matter (labelled CFT) gets a hand from string theory – the string theory hand has an extra dimension and a black hole in it. Credit: Image courtesy of Perimeter Institute for Theoretical Physics

Subir Sachdev, William Witczak-Krempa, and Erik Sørensen are condensed matter physicists. They study exotic but tangible systems, such as superfluids. And their latest paper about one such system has a black hole in it.

How did a black hole get into a condensed matter paper? "Well, it's a long story," says Sachdev, who is a professor at Harvard and a Distinguished Visiting Research Chair at Perimeter Institute.

It's a long story, he might add, that in a way starts with him: he was one of the first condensed matter physicists to venture into the strange land of string theory, where the black holes live. But that is getting ahead of the tale.

"Let's start here," Sachdev says. "Condensed matter physicists study the behaviour of electrons in many materials -- semiconductors, metals, and exotic materials like superconductors."

Normally, these physicists can model the behaviour of a material as if electrons were moving freely around inside it. Even if that's not what's actually happening, because of complex interactions, it makes the model easy to understand and the calculations easier to do. Electrons (and occasionally other particles) used in this kind of short-hand model are called quasi-particles.

However, there are a handful of systems that cannot be described by considering electrons (or any other kind of quasi-particle) moving around.

"What we try to do is understand a quantum system where you have electricity without electrons," says Sachdev. "Of course, the system does have electrons in it, but the behaviour of the system doesn't look like electrons moving at all. What you see is not even particles, but some lumps of quantum excitations that are doing strange quantum things."

"Without quasi-particles, it's a mess," says William Witczak-Krempa. Witczak-Krempa, a Perimeter postdoctoral fellow, is also a condensed matter theorist who collaborated with Sachdev on the paper. "It's this quantum fuzzball of stuff."

Describing such a fuzzball system is a challenge -- but it's crucial to understanding many modern materials, including superfluids and high-temperature superconductors. The broad problem of how to model systems without quasi-particles has been stumping condensed matter theorists for decades.

"What we decided to do was look at a simple case of such an electricity-without-electrons system," says Witczak-Krempa. "That turns out to be a quantum phase transition between a superfluid and an insulator."

A fair amount of work had been done on such systems, such that the team was able to make progress modelling the system using the traditional mathematical tools of condensed matter. Sachdev and Witczak-Krempa worked with Erik Sørensen of McMaster University on this aspect of their paper. Sørensen used a computer simulation -- specifically, a quantum Monte Carlo simulation -- to predict how conductivity should change with temperature and frequency as a superfluid turns into an insulator.

"This frequency dependence tells us how the quantum fluid behaves in time. This dynamic behaviour is notoriously hard to study using standard methods, including quantum Monte Carlo simulations," says Witczak-Krempa. "Erik's work was a huge computational achievement. It took months of processing time. And, in the end, the results still needed to be converted into a form that can be compared with experiments. This is where we tried something new."

To perform this conversion, Sachdev and Witczak-Krempa tackled the same system from a different angle: string theory. (Here, they build on Sachdev's previous work with Perimeter Faculty member Robert Myers and one of his graduate students, Ajay Singh.) One of the pillars of string theory is that certain quantum field theories (technically known as conformal field theories) can be translated into a theory of gravity with one extra dimension.

Sachdev explains where the extra dimension comes in. Wiggling his fingers above the tabletop, he conjures strings moving through the air.

"In certain configurations, the strings all end on a kind of membrane," he says, tapping his fingertips on the table's surface. "You might ask yourself: if you were living on the membrane [the table surface] -- and you didn't know about the extra dimensions where the strings were, what would you see?"

He answers himself: "Only the ends of the strings. They would look like particles. What's amazing is that string theorists found that the theories that you'd use to define the ends of the strings on the membrane are remarkably like the theory we want to use to describe our system."

The quantum field theory describing Sachdev and Witczak-Krempa's "fuzzball" system shares many fundamental properties with the conformal field theories associated with string theory -- so many that the researchers were able to map the two-dimensional field theory into a three-dimensional theory of gravity.

"We ended up studying the physics of this alternate reality," says Witczak-Krempa. "Using this technique, we were able to translate a very hard problem into a fairly easy one." Albeit a fairly easy problem involving a black hole.

"We wanted to look at the physics of the boundary -- the physics at the table top," says Witczak-Krempa. "But we wanted to heat it up a bit -- give it a finite temperature. It turns out that the natural way of doing this is to invoke a black hole." Really?

"There are various ways of developing an intuition about that," he says. "For instance, you can remember that the black hole will release Hawking radiation. The Hawking radiation escapes and eventually hits the boundary where the system lives, and heats it up."

Witczak-Krempa admits it's unorthodox: "Most condensed matter people would go: 'Why is there a black hole in this paper?' It's crazy. But what's even crazier is that this mathematical machinery works quite well. It gives you answers that make a lot of sense. You can compare them directly with Erik's Monte Carlo results, and they check out."

It's the first time results from a traditional large-scale condensed matter simulation have been compared to results from the new string theory approach.

Sachdev is cautiously thrilled: "There are a couple of issues we don't fully understand and one discrepancy we wish we understood better, but in general it's worked extremely well. It's progress on something I've been thinking about for more than 20 years. And now we finally have a theory that is perhaps not complete, but is encouragingly successful."

What's more, string theory has finally produced a set of physical predictions that experimentalists can go check. Sachdev and Witczak-Krempa are hoping that an experimental team will try soon.

"Let's see what happens," says Sachdev. "We're pushing string theory to a new regime. Whatever happens, we will learn more."

Source: Perimeter Institute for Theoretical Physics

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New tool for exploring cells in 3D created

The new software can generate editable models of mid-size biological structures such as this one of HIV. Credit: Image created by Graham Johnson and Ludovic Autin of The Scripps Research Institute

Researchers can now explore viruses, bacteria and components of the human body in more detail than ever before with software developed at The Scripps Research Institute (TSRI).

In a study published online ahead of print December 1 by the journal Nature Methods, the researchers demonstrated how the software, called cellPACK, can be used to model viruses such as HIV.

"We hope to ultimately increase scientists' ability to target any disease," said Art Olson, professor and Anderson Research Chair at TSRI who is senior author of the new study.

Putting cellPACK to the Test

The cellPACK software solves a major problem in structural biology. Although scientists have developed techniques to study relatively large structures, such as cells, and very small structures, such as proteins, it has been harder to visualize structures in the medium "mesoscale" range.

With cellPACK, researchers can quickly and efficiently process the data they've collected on smaller structures to assemble models in this mid-size range. Previously, researchers had to create these models by hand, which took weeks or months compared with just hours in cellPACK.

As a demonstration of the software's power, the authors of the new study created a model of HIV showing how outer "spike" proteins are distributed on the surface of the immature virus.

The new model put to the test a conclusion made by HIV researchers from super-resolution microscopic studies -- that the distribution of the spike proteins on the surface of the immature virus is random. But by using cellPACK to generate thousands of models, testing alternative hypotheses, the researchers found that the distribution was not random. "We demonstrated that their interpretation of the distribution did not match that hypothesis," said Olson.

A Team Effort

The cellPACK software began as the thesis project of a TSRI graduate student, Graham Johnson, now a QB3 faculty fellow at the University of California, San Francisco (UCSF) who continues to contribute to the project. Johnson had more 15 years' experience as a medical illustrator, and he wanted to create an easy way to visualize mesoscale structures. cellPACK is an expansion of Johnson's autoPACK software, which maps out the density of materials -- from concrete in a building to red blood cells in an artery.

The researchers see cellPACK as a community effort, and they have made the autoPACK and cellPACK software free and open source. Thousands of people have already downloaded the software from http://www.autopack.org.

"With the creation of cellPACK, Dr. Olson and his colleagues have addressed the challenge of integrating biological data from different sources and across multiple scales into virtual models that can simulate biologically relevant molecular interactions within a cell," said Veersamy Ravichandran, PhD, of the National Institutes of Health's National Institute of General Medical Sciences, which partially funded the research. "This user-friendly tool provides a new platform for data analysis and simulation in a collaborative manner between laboratories."

As new information comes in from the scientific community, researchers will tweak the software so it can model new shapes. "Making it open source makes it more powerful," said Olson. "The software right now is usable and very useful, but it's really a tool for the future."

Source: Scripps Research Institute

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Software models more detailed evolutionary networks from genetic data

Phylogenetic networks depict the movement of genetic sequences from one species to another as a means of showing where horizontal gene transfer may have taken place. Software by scientists at Rice University aims to reveal far more about species’ evolutionary histories than traditional tree models are able to. Credit: Luay Nakhleh/Rice University

The tree has been an effective model of evolution for 150 years, but a Rice University computer scientist believes it's far too simple to illustrate the breadth of current knowledge.

Rice researcher Luay Nakhleh and his group have developed PhyloNet, an open-source software package that accounts for horizontal as well as vertical inheritance of genetic material among genomes. His "maximum likelihood" method, detailed this month in the Proceedings of the National Academy of Sciences, allows PhyloNet to infer network models that better describe the evolution of certain groups of species than do tree models.

"Inferring" in this case means analyzing genes to determine their evolutionary history with the highest probability -- the maximum likelihood -- of connections between species. Nakhleh and Rice colleague Christopher Jermaine recently won a $1.1 million National Science Foundation grant to analyze evolutionary patterns using Bayesian inference, a statistics-based technique to estimate probabilities based on a data set.

To build networks that account for all of the genetic connections between species, the software infers the probability of variations that phylogenetic trees can't illustrate, such as horizontal gene transfers. These transfers circumvent simple parent-to-offspring evolution and allow genetic variations to move from one species to another by means other than reproduction.

Biologists want to know when and how these transfers happened, but tree structures conceal such information. "When horizontal transfer occurs, as with the hybridization of two species, the tree model becomes inadequate to describe the evolutionary history, and networks that incorporate horizontal gene transfer become the more appropriate model," Nakhleh said.

Nakhleh's Java-based software accounts for incomplete lineage sorting, in which clues to gene evolution that don't match the established lineage of species appear in the genetic record.

"We are the first group to develop a general model that will allow biologists to estimate hybridization while accounting for all these complexities in evolution," Nakhleh said.

Most existing programs for phylogenetics (the study of evolutionary relationships) ignore such complexities. "They end up overestimating the amount of hybridization," Nakhleh said. 

"They start seeing lots of complexities in the data and say, 'Oh, it's complex here; it must be hybridization,' and end up inferring too much. Our method acknowledges that part of the complexity has nothing to do with hybridization; it has to do with other random processes that happened during evolution."

The Rice researchers used two data sets to test the new program. One, a computer-generated set of data that mimics a realistic model of evolution, allowed them to evaluate the accuracy of the program. The second involved multiple genomes of mice found across Europe and Asia. "There have been stories about mice hybridizing," Nakhleh said. "Now that we have the first method to allow for systematic analysis, we ran it on a very large amount of data from five mouse samples and we detected hybridization" -- most notably in the presence of a genetic signal from a mouse in Kazakhstan that found its way to mice in France and Germany, he said.

Nakhleh hopes evolutionary biologists will use PhyloNet to take a fresh look at the massive amount of genomic data collected over the past few decades. "The exciting thing for me about this is that biologists can now systematically go through lots of data they have generated and check to see if there has been hybridization."

Source: Rice University

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Astronomers bring the third dimension to a doomed star's outburst

A new shape model of the Homunculus Nebula reveals protrusions, trenches, holes and irregularities in its molecular hydrogen emission. The protrusions appear near a dust skirt seen at the nebula's center in visible light (inset) but not found in this study, so they constitute different structures. Credit: NASA Goddard (inset: NASA, ESA, Hubble SM4 ERO Team)

In the middle of the 19th century, the massive binary system Eta Carinae underwent an eruption that ejected at least 10 times the sun's mass and made it the second-brightest star in the sky. Now, a team of astronomers has used extensive new observations to create the first high-resolution 3-D model of the expanding cloud produced by this outburst.

"Our model indicates that this vast shell of gas and dust has a more complex origin than is generally assumed," said Thomas Madura, a NASA Postdoctoral Program fellow at NASA's Goddard Space Flight Center in Greenbelt, Maryland, and a member of the study team. "For the first time, we see evidence suggesting that intense interactions between the stars in the central binary played a significant role in sculpting the nebula we see today."

Eta Carinae lies about 7,500 light-years away in the southern constellation of Carina and is one of the most massive binary systems astronomers can study in detail. The smaller star is about 30 times the mass of the sun and may be as much as a million times more luminous. The primary star contains about 90 solar masses and emits 5 million times the sun's energy output. Both stars are fated to end their lives in spectacular supernova explosions.

Between 1838 and 1845, Eta Carinae underwent a period of unusual variability during which it briefly outshone Canopus, normally the second-brightest star. As a part of this event, which astronomers call the Great Eruption, a gaseous shell containing at least 10 and perhaps as much as 40 times the sun's mass was shot into space. This material forms a twin-lobed dust-filled cloud known as the Homunculus Nebula, which is now about a light-year long and continues to expand at more than 1.3 million mph (2.1 million km/h).

Using the European Southern Observatory's Very Large Telescope and its X-Shooter spectrograph over two nights in March 2012, the team imaged near-infrared, visible and ultraviolet wavelengths along 92 separate swaths across the nebula, making the most complete spectral map to date. The researchers have used the spatial and velocity information provided by this data to create the first high-resolution, fully 3-D model of the Homunculus Nebula. The new model contains none of the assumptions about the cloud's symmetry found in previous studies.

The shape model, which is now published by the journal Monthly Notices of the Royal Astronomical Society, was developed using only a single emission line of near-infrared light emitted by molecular hydrogen gas. The characteristic 2.12-micron light shifts in wavelength slightly depending on the speed and direction of the expanding gas, allowing the team to probe even dust-obscured portions of the Homunculus that face away from Earth.

"Our next step was to process all of this using 3-D modeling software I developed in collaboration with Nico Koning from the University of Calgary in Canada. The program is simply called 'Shape,' and it analyzes and models the three-dimensional motions and structure of nebulae in a way that can be compared directly with observations," said lead researcher Wolfgang Steffen, an astrophysicist at the Ensenada campus of the National Autonomous University of Mexico.

The new shape model confirms several features identified by previous studies, including pronounced holes located at the ends of each lobe and the absence of any extended molecular hydrogen emission from a dust skirt apparent in visible light near the center of the nebula. New features include curious arm-like protrusions emanating from each lobe near the dust skirt; vast, deep trenches curving along each lobe; and irregular divots on the side facing away from Earth.

"One of the questions we set out to answer with this study is whether the Homunculus contains any imprint of the star's binary nature, since previous efforts to explain its shape have assumed that both lobes were more or less identical and symmetric around their long axis," explained team member Jose Groh, an astronomer at Geneva University in Switzerland. "The new features strongly suggest that interactions between Eta Carinae's stars helped mold the Homunculus."

Every 5.5 years, when their orbits carry them to their closest approach, called periastron, the immense and brilliant stars of Eta Carinae are only as far apart as the average distance between Mars and the sun. Both stars possess powerful gaseous outflows called stellar winds, which constantly interact but do so most dramatically during periastron, when the faster wind from the smaller star carves a tunnel through the denser wind of its companion. The opening angle of this cavity closely matches the length of the trenches (130 degrees) and the angle between the arm-like protrusions (110 degrees), indicating that the Homunculus likely continues to carry an impression from a periastron interaction around the time of the Great Eruption.

Once the researchers had developed their Homunculus model, they took things one step further. They converted it to a format that can be used by 3-D printers and made the file available along with the published paper.

"Now anyone with access to a 3-D printer can produce their own version of this incredible object," said Goddard astrophysicist Theodore Gull, who is also a co-author of the paper. 

"While 3-D-printed models will make a terrific visualization tool for anyone interested in astronomy, I see them as particularly valuable for the blind, who now will be able to compare embossed astronomical images with a scientifically accurate representation of the real thing."

Source: NASA

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