Tuesday, April 30, 2024

Intervention based on science of reading and math boosts comprehension and word problem-solving skills

New research from the University of Kansas has found that an intervention based on the science of reading and math effectively helped English learners boost their comprehension, visualize and synthesize information, and make connections that significantly improved their math performance.

The intervention, performed for 30 minutes twice a week for 10 weeks with 66 third-grade English language learners who displayed math learning difficulties, improved students' performance when compared to students who received general instruction. This indicates that emphasizing cognitive concepts involved in the science of reading and math are key to helping students improve, according to researchers.

"Word problem-solving is influenced by both the science of reading and the science of math. Key components include number sense, decoding, language comprehension and working memory. Utilizing direct and explicit teaching methods enhances understanding and enables students to effectively connect these skills to solve math problems. This integrated approach ensures that students are equipped with necessary tools to navigate both the linguistic and numerical demands of word problems," said Michael Orosco, professor of educational psychology at KU and lead author of the study.

The intervention incorporates comprehension strategy instruction in both reading and math, focusing and decoding, phonological awareness, vocabulary development, inferential thinking, contextualized learning and numeracy.

"It is proving to be one of the most effective evidence-based practices available for this growing population," Orosco said.

The study, co-written with Deborah Reed of the University of Tennessee, was published in the journal Learning Disabilities Research and Practice.

For the research, trained tutors implemented the intervention, developed by Orosco and colleagues based on cognitive and culturally responsive research conducted over a span of 20 years. One example of an intervention session tested in the study included a script in which a tutor examined a word problem explaining that a person made a quesadilla for his friend Mario and gave him one-fourth of it, then asked students to determine how much remained.

The tutor first asked students if they remembered a class session in which they made quesadillas and what shape they were, and demonstrated concepts by drawing a circle on the board, dividing it into four equal pieces, having students repeat terms like numerator and denominator. The tutor explains that when a question asks how much is left, subtraction is required. The students also collaborated with peers to practice using important vocabulary in sentences. The approach both helps students learn and understand mathematical concepts while being culturally responsive.

"Word problems are complex because they require translating words into mathematical equations, and this involves integrating the science of reading and math through language concepts and differentiated instruction," Orosco said. "We have not extensively tested these approaches with this group of children. However, we are establishing an evidence-based framework that aids them in developing background knowledge and connecting it to their cultural contexts."

Orosco, director of KU's Center for Culturally Responsive Educational Neuroscience, emphasized the critical role of language in word problems, highlighting the importance of using culturally familiar terms. For instance, substituting "pastry" for "quesadilla" could significantly affect comprehension for students from diverse backgrounds. Failure to grasp the initial scenario could impede subsequent problem-solving efforts.

The study proved effective in improving students' problem-solving abilities, despite covariates including an individual's basic calculation skills, fluid intelligence and reading comprehension scores. That finding is key, as while ideally all students would begin on equal footing and there would be few variations in a classroom, in reality, covariates exist and are commonplace.

The study had trained tutors deliver the intervention, and its effectiveness should be further tested with working teachers, the authors wrote. Orosco said professional development to help teachers gain the skills is necessary, and it is vital for teacher preparation programs to train future teachers with such skills as well. And helping students at the elementary level is necessary to help ensure success in future higher-level math classes such as algebra.

The research builds on Orosco and colleagues' work in understanding and improving math instruction for English learners. Future work will continue to examine the role of cognitive functions such as working memory and brain science, as well as potential integration of artificial intelligence in teaching math.

"Comprehension strategy instruction helps students make connections, ask questions, visualize, synthesize and monitor their thinking about word problems," Orosco and Reed wrote. "Finally, applying comprehension strategy instruction supports ELs in integrating their reading, language and math cognition…. Focusing on relevant language in word problems and providing collaborative support significantly improved students' solution accuracy."

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A mathematical bridge between the huge and the tiny

A mathematical link between two key equations—one that deals with the very big and the other, the very small—has been developed by a young mathematician in China.

The mathematical discipline known as differential geometry is concerned with the geometry of smooth shapes and spaces. With roots going back to antiquity, the field flourished in the early 20th century, enabling Einstein to develop his general theory of relativity and other physicists to develop quantum field theory and the Standard Model of particle physics.

Gao Chen, a 29-year-old mathematician at the University of Science and Technology of China in Hefei, specializes in a branch known as complex differential geometry. Its complexity is not in dealing with complicated structures, but rather because it is based on complex numbers—a system of numbers that extends everyday numbers by including the square root of -1.

This area appeals to Chen because of its connections with other fields. "Complex differential geometry lies at the intersection of analysis, algebra, and mathematical physics," he says. "Many tools can be used to study this area."

Chen has now found a new link between two important equations in the field: the Kähler–Einstein equation, which describes how mass causes curvature in space–time in general relativity, and the Hermitian–Yang–Mills equation, which underpins the Standard Model of particle physics.

Chen was inspired by his Ph.D. supervisor Xiuxiong Chen of New York's Stony Brook University, to take on the problem. "Finding solutions to the Hermitian–Yang–Mills and the Kähler–Einstein equations are considered the most important advances in complex differential geometry in previous decades," says Gao Chen. "My results provide a connection between these two key results."

"The Kähler –Einstein equation describes very large things, as large as the universe, whereas the Hermitian–Yang–Mills equation describes tiny things, as small as quantum phenomena," explains Gao Chen. "I've built a bridge between these two equations." Gao Chen notes that other bridges existed previously, but that he has found a new one.

"This bridge provides a new key, a new tool for theoretical research in this field," Gao Chen adds. His paper describing this bridge was published in the journal Inventiones mathematicae in 2021.

In particular, the finding could find use in string theory—the leading contender of theories that researchers are developing in their quest to unite quantum physics and relativity. "The deformed Hermitian–Yang–Mills equation that I studied plays an important role in the study of string theory," notes Gao Chen.

Gao Chen now has his eyes set on other important problems, including one of the seven Millennium Prize Problems. These are considered the most challenging in the field by mathematicians and carry a $1 million prize for a correct solution. "In the future, I hope to tackle a generalization of the Kähler–Einstein equation," he says. "I also hope to work on other Millennium Prize problems, including the Hodge conjecture."

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Saturday, April 13, 2024

It’s common to ‘stream’ maths classes. But grouping students by ability can lead to ‘massive disadvantage

Two students work on maths problems at a whiteboard.

It is very common in Australian schools to “stream” students for subjects such as English, science and maths. This means students are grouped into different classes based on their previous academic attainment, or in some cases, just a perception of their level of ability.

Students can also be streamed as early as primary school. Yet there are no national or state policies on this. This means school principals are free to decide what will happen in their schools.

Why are students streamed in Australians schools? And is this a good idea? Our research on streaming maths classes shows we need to think much more carefully about this very common practice.

Why do schools stream?

At a maths teacher conference in Sydney in late 2023, I did a live survey about school approaches to streaming.

This survey was done via interactive software while I was giving a presentation. There were 338 responses from head teachers in maths in either high schools or schools that go all the way from Kindergarten to Year 12. Most of the teachers were from public schools.

In a sign of how widespread streaming is, 95% of head teachers said they streamed maths classes in their schools.

Respondents said one of the main reasons is to help high-achieving students and make sure they are appropriately challenged.

But almost half the respondents said they believed all students were benefiting from this system.

We also heard how streaming is seen as a way to cope with the teacher shortage and specific lack of qualified maths teachers. These qualifications include skills in both maths and maths teaching.

More than half (65%) of respondents said streaming can “aid differentiation [and] support targeted student learning interventions”. In other words, streaming is a way to cope with different levels of ability in the classrooms and make the job of teaching a class more straightforward.

The ‘glass ceiling effect’

But while many schools and teachers assume streaming is good for students, this is not what the research says.

Our 2020 study, on streaming was based on interviews with 85 students and 22 teachers from 11 government schools.

This found streaming creates a “glass ceiling effect” – in other words, students cannot progress out of the stream they are initially assigned to without significant remedial work to catch them up.

As one teacher told us, students in lower-ability classes were then placed at a “massive disadvantage”. This is because they can miss out on segments of the curriculum because the class may progress more slowly or is deliberately not taught certain sections deemed too complex.

Often students in our study were unaware of this missed content until Year 10 and thinking about their options for the final years of school and beyond. They may not be able to do higher-level maths in Year 11 and 12 because they are too far behind. As one teacher explained:

This comes as fewer students are completing advanced (calculus-based) maths.

If students do not study senior maths, they do not have the background for studying for engineering and other STEM careers, which we know are in very high demand.

On top of this, students may also be stigmatised as “low ability” in maths. While classes are not labelled as such, students are well aware of who is in the top classes and who is not. This can have an impact on students’ confidence about maths.

A man points to equations on a whiteboard.

What does other research say?

International research has also found streaming students is inequitable.

As a 2018 UK study showed, students from disadvantaged backgrounds are more likely to be put in lower streamed classes.

A 2009 review of research studies found that not streaming students was better for low-ability student achievement and had no effect on average and high-ability student achievement.

What should we do instead?

Amid concerns about Australian students’ maths performance in national and international tests, schools need to stop assuming streaming is the best approach for students.

The research indicates it would be better if students were taught in mixed-ability classes – as long as teachers are supported and class sizes are small enough.

This means all students have the opportunity to be taught all of the curriculum, giving them the option of doing senior maths if they want to in Year 11 and Year 12.

It also means students are not stigmatised as “poor at maths” from a young age.

But to do so, teachers and schools must be given more teaching resources and support. And some of this support needs to begin in primary school, rather than waiting until high school to try and catch students up.

Students also need adequate career advice, so they are aware of how maths could help future careers and what they need to do to get there.

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Tuesday, September 26, 2023

Math error: A new study overturns 100-year-old understanding of color perception

LOS ALAMOS, N.M., August 10, 2022—A new study corrects an important error in the 3D mathematical space developed by the Nobel Prize–winning physicist Erwin Schrödinger and others and used by scientists and industry for more than 100 years to describe how your eye distinguishes one color from another. The research has the potential to boost scientific data visualizations, improve TVs and recalibrate the textile and paint industries.

“The assumed shape of color space requires a paradigm shift,” said Roxana Bujack, a computer scientist with a background in mathematics who creates scientific visualizations at Los Alamos National Laboratory. Bujack is lead author of the paper by a Los Alamos team in the Proceedings of the National Academy of Sciences on the mathematics of color perception. "Our research shows that the current mathematical model of how the eye perceives color differences is incorrect. That model was suggested by Bernhard Riemann and developed by Hermann von Helmholtz and Erwin Schrödinger—all giants in mathematics and physics—and proving one of them wrong is pretty much the dream of a scientist.”

Modeling human color perception enables automation of image processing, computer graphics and visualization tasks.

“Our original idea was to develop algorithms to automatically improve color maps for data visualization, to make them easier to understand and interpret,” Bujack said. So the team was surprised when they discovered they were the first to determine that the longstanding application of Riemannian geometry, which allows generalizing straight lines to curved surfaces, didn’t work.

To create industry standards, a precise mathematical model of perceived color space is needed. First attempts used Euclidean spaces—the familiar geometry taught in many high schools; more advanced models used Riemannian geometry. The models plot red, green and blue in the 3D space. Those are the colors registered most strongly by light-detecting cones on our retinas, and—not surprisingly—the colors that blend to create all the images on your RGB computer screen.

In the study, which blends psychology, biology and mathematics, Bujack and her colleagues discovered that using Riemannian geometry overestimates the perception of large color differences. That’s because people perceive a big difference in color to be less than the sum you would get if you added up small differences in color that lie between two widely separated shades.

Riemannian geometry cannot account for this effect.

“We didn’t expect this, and we don’t know the exact geometry of this new color space yet,” Bujack said. “We might be able to think of it normally but with an added dampening or weighing function that pulls long distances in, making them shorter. But we can’t prove it yet.”

The Paper: The non-Riemannian nature of perceptual color space, in Proceedings of the National Academy of Sciences, by Roxana Bujack, Emily Teti, Jonah Miller, Elektra Caffrey, and Terece L. Turton. https://www.pnas.org/doi/10.1073/pnas.2119753119#sec-8

The Funding: Laboratory Directed Research and Development Program of Los Alamos National Laboratory.

8th Edition of International Conference on Mathematics and Optimization Method

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Maths problem: review finds shortage of teachers and researchers

There are significant challenges facing teaching and research in the mathematical sciences in Australia, according to a nation-wide review of the discipline.

The findings are part of a mid-term review of Australia’s 10-year plan for mathematical sciences published today, and overseen by the Australian Academy of Science’s National Committee for Mathematical Sciences (NCMS).

Issues raised included the continued long-term decline in the supply of qualified secondary mathematics teachers.

NCMS Chair and Academy Fellow Professor Alan Welsh said the impact of the COVID-19 pandemic on the university sector had also resulted in notable losses to the mathematical sciences research community.

“COVID-19 exacerbated gender imbalances in the mathematical sciences workforce,” Professor Welsh said.

“These impacts affect not just mathematical sciences research outputs but also the quality of mathematical and statistical education available at all levels in Australia.”

The mid-term review recommends:continuing to develop programs and addressing current issues in teaching to give all Australian school students access to outstanding mathematics teachers
urgently addressing the cuts to mathematical courses at universities, which have impacted the ability of university students in Australia to access a degree in which they can major in the mathematical sciences
emphasising the contributions of mathematical sciences in responding to national challenges and informing policy decisions.

National Committee member and Academy Fellow Professor Kerrie Mengersen said the opportunities provided by the mathematical sciences community can only be realised with appropriate recognition of its role in responding to developing areas of interest from the government and research sectors.

“This must be paired with infrastructure and resourcing to support excellent mathematical and statistical work for the research that underpins many solutions to contemporary challenges, and to ensure high-quality education to equip the next generation of Australians with the mathematical science knowledge needed for the future,” Professor Mengersen said.

8th Edition of International Conference on Mathematics and Optimization Method

AMU’s mathematics department ranked 1st in India by US News, World Report

NEW DELHI: The Aligarh Muslim University’s Department of Mathematics, has become number one in India as per the Best Global Universities for Mathematics in India ranking published by the latest US News and World Report 2023. The university secured the 137th position from the 175th in 2022.

ISI, Kolkata with a ranking of 342 secured the third position. IIT Kanpur with a rank of world 352 secured fourth position while IIT Madras and IISC Bangalore are at fifth and sixth positions with a ranking of 372 and 384 respectively. There are only six institutions in India on the list.

ALSO READ| DU, AMU, Visva Bharti University among 105 universities introducing FYUP from new session: UGC

Mohammad Ashraf, chairman of the department said the department's progress is evident from an overall subject score of 56.8, positioning it at the topmost on the national level followed by TIFR, whose ranking is 324 worldwide.

Ashraf said, “Our achievement in this important global ranking is a testimony to the several years of dedicated efforts from our faculty and research scholars to enhance the ranking of the department. It is an accomplishment that could truly change the trajectory of the Department’s future”.

“The department’s recognition also highlights the calibre of mathematics education and it is a testament to the impact of our teaching and research in India on the global stage”, he added. The US News Education monitors higher education data for the world's research-led institutions.

ALSO READ| CUET UG Result 2023: AMU offers admissions to 15 programmes; seats, fee, eligibility criteria

“ The subject rankings, according to the US News, are powered by Clarivate Analytics, which provided the data and metrics used in the rankings, and the bibliometric data are based on the Web of Science. The bibliometric indicators used in this US News ranking analysis have been drawn over the last five-year period from 2018-22. However, the citations to these papers came from all publications up to the most recent data available. The indicators include publications, total citations, books, conferences, number of publications that are among the 10 percent most cited, number of highly cited papers that are among the top 1 percent most cited in their respective fields, international collaborations, etc” an official statement from the university read.

8th Edition of International Conference on Mathematics and Optimization Method

Monday, September 4, 2023

Doctors with Math background have the upper hand in interdisciplinary research

Mumbai girl Aakriti Goel and Janhavi Ajit Rao from Bengaluru have something in common. Both women chose to become a doctor after investing years studying Engineering.
Aakriti who completed her BTech from BITS Pilani in 2015 and worked for several projects and startups, is now set to start her second innings as an MBBS student. Aakriti secured 1118 All-India rank in NEET 2021. Janhavi, an IT professional based in the US, chose to become a doctor after she was diagnosed with rheumatoid arthritis and realised that there is a lack of specialists to treat the disease. She was also touched by the service rendered by the doctors to the society. Janhavi completed her MBBS from MS Ramaiah Medical College, Bengaluru, in 2020. Currently, she is an internal medicine resident at Riverside University Health System, USA.

Dr Ajay Kumar Pal, associate professor, Department of Surgery, King George’s Medical University (KGMU), Lucknow, says that such kind of career switch is there but it is quite rare. “There are many students like me who qualify for both medical and Engineering entrance, but pick one,” says Dr Pal, who qualified for Medicine and Engineering in 2003.
However, Dr Pal, an MBBS graduate from Jawaharlal Nehru Medical College, AMU, believes that students with a Mathematics background have slightly better analytical skills as compared to those from pure Biological Science background.

There is no difference in skills as a doctor between Mathematics and non-Mathematics students. “Doctors who have studied Mathematics in class XII enjoy an upper hand in interdisciplinary research in areas such as Biomedical, Biotechnology and Biostatistics among others,” explains Dr Pal.
Tech intervention Technology has become a deciding factor in all disciplines. “Medicine is in a nascent stage in India and technology intervention is evident for better development of the infrastructure,” says Aakriti who wants to become a surgeon after completing her MBBS. “Candidates who come from Mathematics, Engineering or Technology background are better skilled with analytical and problem-solving and critical thinking abilities. Having an increased presence of these students in Medicine will definitely improve the efficiency and promote interdisciplinary research in this field,” says Aakriti.

8th Edition of International Conference on Mathematics and Optimization Method