Tag Archives: algebra

3D picture language for mathematics

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There’s a new, 3D picture language for mathematics called ‘quon’ according to a March 3, 2017 news item on phys.org,

Galileo called mathematics the “language with which God wrote the universe.” He described a picture-language, and now that language has a new dimension.

The Harvard trio of Arthur Jaffe, the Landon T. Clay Professor of Mathematics and Theoretical Science, postdoctoral fellow Zhengwei Liu, and researcher Alex Wozniakowski has developed a 3-D picture-language for mathematics with potential as a tool across a range of topics, from pure math to physics.

Though not the first pictorial language of mathematics, the new one, called quon, holds promise for being able to transmit not only complex concepts, but also vast amounts of detail in relatively simple images. …

A March 2, 2017 Harvard University news release by Peter Reuell, which originated the news item, provides more context for the research,

“It’s a big deal,” said Jacob Biamonte of the Quantum Complexity Science Initiative after reading the research. “The paper will set a new foundation for a vast topic.”

“This paper is the result of work we’ve been doing for the past year and a half, and we regard this as the start of something new and exciting,” Jaffe said. “It seems to be the tip of an iceberg. We invented our language to solve a problem in quantum information, but we have already found that this language led us to the discovery of new mathematical results in other areas of mathematics. We expect that it will also have interesting applications in physics.”

When it comes to the “language” of mathematics, humans start with the basics — by learning their numbers. As we get older, however, things become more complex.

“We learn to use algebra, and we use letters to represent variables or other values that might be altered,” Liu said. “Now, when we look at research work, we see fewer numbers and more letters and formulas. One of our aims is to replace ‘symbol proof’ by ‘picture proof.’”

The new language relies on images to convey the same information that is found in traditional algebraic equations — and in some cases, even more.

“An image can contain information that is very hard to describe algebraically,” Liu said. “It is very easy to transmit meaning through an image, and easy for people to understand what they see in an image, so we visualize these concepts and instead of words or letters can communicate via pictures.”

“So this pictorial language for mathematics can give you insights and a way of thinking that you don’t see in the usual, algebraic way of approaching mathematics,” Jaffe said. “For centuries there has been a great deal of interaction between mathematics and physics because people were thinking about the same things, but from different points of view. When we put the two subjects together, we found many new insights, and this new language can take that into another dimension.”

In their most recent work, the researchers moved their language into a more literal realm, creating 3-D images that, when manipulated, can trigger mathematical insights.

“Where before we had been working in two dimensions, we now see that it’s valuable to have a language that’s Lego-like, and in three dimensions,” Jaffe said. “By pushing these pictures around, or working with them like an object you can deform, the images can have different mathematical meanings, and in that way we can create equations.”

Among their pictorial feats, Jaffe said, are the complex equations used to describe quantum teleportation. The researchers have pictures for the Pauli matrices, which are fundamental components of quantum information protocols. This shows that the standard protocols are topological, and also leads to discovery of new protocols.

“It turns out one picture is worth 1,000 symbols,” Jaffe said.

“We could describe this algebraically, and it might require an entire page of equations,” Liu added. “But we can do that in one picture, so it can capture a lot of information.”

Having found a fit with quantum information, the researchers are now exploring how their language might also be useful in a number of other subjects in mathematics and physics.

“We don’t want to make claims at this point,” Jaffe said, “but we believe and are thinking about quite a few other areas where this picture-language could be important.”

Sadly, there are no artistic images illustrating quon but this is from the paper,

An n-quon is represented by n hemispheres. We call the flat disc on the boundary of each hemisphere a boundary disc. Each hemisphere contains a neutral diagram with four boundary points on its boundary disk. The dotted box designates the internal structure that specifies the quon vector. For example, the 3-quon is represented as

Courtesy: PNAS and Harvard University

I gather the term ‘quon’ is meant to suggest quantum particles.

Here’s a link and a citation for the paper,

Quon 3D language for quantum information by Zhengwei Liu, Alex Wozniakowski, and Arthur M. Jaffe. Proceedins of the National Academy of Sciences Published online before print February 6, 2017, doi: 10.1073/pnas.1621345114 PNAS March 7, 2017 vol. 114 no. 10

This paper appears to be open access.

Does education kill the ability to do algebra?

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Apparently, the ability to perform basic algebra is innate in humans, mice, fish, and others. Researchers at Johns Hopkins describe some of their findings about algebra and innate abilities in this video,

While the researchers don’t accuse the education system of destroying or damaging one’s ability to perform algebra, I will make the suggestion, the gut level instinct the researchers are describing is educated out of most of us. Here’s more from the March 6, 2014 news item on ScienceDaily describing the research,

Millions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don’t come easily, it gets worse: Most preschoolers and kindergarteners can do some algebra before even entering a math class.

In a just-published study in the journal Developmental Science, lead author and post-doctoral fellow Melissa Kibbe and Lisa Feigenson, associate professor of psychological and brain sciences at Johns Hopkins University’s Krieger School of Arts and Sciences, find that most preschoolers and kindergarteners, or children between 4 and 6, can do basic algebra naturally.

“These very young children, some of whom are just learning to count, and few of whom have even gone to school yet, are doing basic algebra and with little effort,” Kibbe said. “They do it by using what we call their ‘Approximate Number System:’ their gut-level, inborn sense of quantity and number.”

A Johns Hopkins University March 7, 2014 news piece by Latarsha Gatlin describes the research further,

The “Approximate Number System,” or ANS, is also called “number sense,” and describes humans’ and animals’ ability to quickly size up the quantity of objects in their everyday environments. We’re born with this ability, which is probably an evolutionary adaptation to help human and animal ancestors survive in the wild, scientists say.

Previous research has revealed some interesting facts about number sense, including that adolescents with better math abilities also had superior number sense when they were preschoolers, and that number sense peaks at age 35.

Kibbe, who works in Feigenson’s lab, wondered whether preschool-age children could harness that intuitive mathematical ability to solve for a hidden variable. In other words, could they do something akin to basic algebra before they ever received formal classroom mathematics instruction? The answer was “yes,” at least when the algebra problem was acted out by two furry stuffed animals—Gator and Cheetah—using “magic cups” filled with objects like buttons, plastic doll shoes, and pennies.

In the study, children sat down individually with an examiner who introduced them to the two characters, each of which had a cup filled with an unknown quantity of items. Children were told that each character’s cup would “magically” add more items to a pile of objects already sitting on a table. But children were not allowed to see the number of objects in either cup: they only saw the pile before it was added to, and after, so they had to infer approximately how many objects Gator’s cup and Cheetah’s cup contained.

At the end, the examiner pretended that she had mixed up the cups, and asked the children—after showing them what was in one of the cups—to help her figure out whose cup it was. The majority of the children knew whose cup it was, a finding that revealed for the researchers that the pint-sized participants had been solving for a missing quantity. In essence, this is the same as doing basic algebra.

“What was in the cup was the x and y variable, and children nailed it,” said Feigenson, director of the Johns Hopkins Laboratory for Child Development. “Gator’s cup was the x variable and Cheetah’s cup was the y variable. We found out that young children are very, very good at this. It appears that they are harnessing their gut level number sense to solve this task.”

If this kind of basic algebraic reasoning is so simple and natural for 4, 5, and 6-year-olds, then why it is so difficult for teens and others?

“One possibility is that formal algebra relies on memorized rules and symbols that seem to trip many people up,” Feigenson said. “So one of the exciting future directions for this research is to ask whether telling teachers that children have this gut level ability—long before they master the symbols—might help in encouraging students to harness these skills. Teachers may be able to help children master these kind of computations earlier, and more easily, giving them a wedge into the system.”

While number sense helps children in solving basic algebra, more sophisticated concepts and reasoning are needed to master the complex algebra problems that are taught later in the school age years.

Another finding from the research was that an ANS aptitude does not follow gender lines. Boys and girls answered questions correctly in equal proportions during the experiments, the researchers said. Although other research shows that even young children can be influenced by gender stereotypes about girls’ versus boys’ math prowess, “we see no evidence for gender differences in our work on basic number sense,” Feigenson said.

Parents with numerically challenged kids shouldn’t worry that their child will be bad at math. The psychologists say it’s more important to nurture and support young children’s use of their number sense in solving problems that will later be introduced more formally in school.

“We find links at all ages between the precision of people’s Approximate Number System and their formal math ability,” Feigenson said. “But this does not necessarily mean that children with poorer precision grow up to be bad at math. For example, children with poorer number sense may need to rely on other strategies, besides their gut sense of number, to solve math problems. But this is an area where much future research is needed.”

Here’s a link to and a citation for the paper,

Young children ‘solve for x’ using the Approximate Number System by Melissa M. Kibbe and Lisa Feigenson. Article first published online: 3 MAR 2014 DOI: 10.1111/desc.12177

© 2014 John Wiley & Sons Ltd

This paper is behind a paywall.

When twice as much (algebra) is good for you

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“We find positive and substantial longer-run impacts of double-dose algebra on college entrance exam scores, high school graduation rates and college enrollment rates, suggesting that the policy had significant benefits that were not easily observable in the first couple of years of its existence,” wrote the article’s authors.

The Mar. 21, 2013 news release on EurekAlert which includes the preceding quote recounts an extraordinary story about an approach to teaching algebra that was not enthusiastically adopted at first but first some reason administrators and teachers persisted with it. Chelsey Leu’s Mar. 21, 2013 article (which originated the news release) for UChicago (University of Chicago) News (Note: Links have been removed),

Martin Gartzman sat in his dentist’s waiting room last fall when he read a study in Education Next that nearly brought him to tears.

A decade ago, in his former position as chief math and science officer for Chicago Public Schools [CPS], Gartzman spearheaded an attempt to decrease ninth-grade algebra failure rates, an issue he calls “an incredibly vexing problem.” His idea was to provide extra time for struggling students by having them take two consecutive periods of algebra.

…

In high schools, ninth-grade algebra is typically the class with the highest failure rate. This presents a barrier to graduation, because high schools usually require three to four years of math to graduate.

Students have about a 20 percent chance of passing the next math level if they don’t first pass algebra, Gartzman said, versus 80 percent for those who do pass. The data are clear: If students fail ninth-grade algebra, the likelihood of passing later years of math, and ultimately of graduating, is slim

…

Gartzman’s work to decrease algebra failure rates at CPS was motivated by a study of Melissa Roderick, the Hermon Dunlap Smith Professor at UChicago’s School of Social Service Administration. The study emphasized the importance of keeping students academically on track in their freshman year to increase the graduation rate.

Some administrators and teachers resisted the new policy. Teachers called these sessions “double-period hell” because they gathered, in one class, the most unmotivated students who had the biggest problems with math.

Principals and counselors sometimes saw the double periods as punishment for the students, depriving them of courses they may have enjoyed taking and replacing them with courses they disliked.

It seemed to Gartzman that double-period students were learning more math, though he had no supporting data. He gauged students’ progress by class grades, not by standardized tests. The CPS educators had no way of fully assessing their double-period idea. All they knew was that failure rates didn’t budge.

Unfortunately, Leu does not explain why the administrators and teachers continued with the program but it’s a good thing they did (Note: Links have been removed),

“Double-dosing had an immediate impact on student performance in algebra, increasing the proportion of students earning at least a B by 9.4 percentage points, or more than 65 percent,” noted the Education Next article. Although ninth-grade algebra passing rates remained mostly unaffected, “The mean GPA across all math courses taken after freshman year increased by 0.14 grade points on a 4.0 scale.”

They also found significantly increased graduation rates. The researchers concluded on an encouraging note: “Although the intervention was not particularly effective for the average affected student, the fact that it improved high school graduation and college enrollment rates for even a subset of low-performing and at-risk students is extraordinarily promising when targeted at the appropriate students.” [emphasis mine]

Gartzman recalled that reading the article “was mind-blowing for me. I had no idea that the researchers were continuing to study these kids.”

The study had followed a set of students from eighth grade through graduation, while Gartzman’s team could only follow them for a year after the program began. The improvements appeared five years after launching double-dose algebra, hiding them from the CPS team, which had focused on short-term student performance. [emphasis mine]

Gartzman stressed the importance of education policy research. “Nomi and Allensworth did some really sophisticated modeling that only researchers could do, that school districts really can’t do. It validates school districts all over the country who had been investing in double-period strategies.”

I’m not sure I understand the numbers very well (maybe I need a double-dose of numbers). The 9.4% increase for students earning a B sounds good but a mean increase of 0.14 in grade points doesn’t sound as impressive. As for the bit about the program being “not particularly effective for the average affected student,” what kind of student is helped by this program? As for the improvements being seen five years after the program launch. does this mean that students in the program showed improvement five years later (in first year university) or that researchers weren’t able to effectively measure any impact in the grade nine classroom until five years after the program began?

Regardless, it seems there is an improvement and having suffered through my share algebra classes, I applaud the educators for finding a way to help some students, if not all.