Month: August 2024

Credit: Unsplash/CC0 Public Domain

In just a few months, voters across America will head to the polls to decide who will be the next U.S. president. A new study draws on mathematics to break down how humans make decisions like this one.

The researchers, including Zachary Kilpatrick, an applied mathematician at CU Boulder, developed mathematical tools known as models to simulate the deliberation process of groups of people with various biases. They found that decision-makers with strong, initial biases were typically the first ones to make a choice.

“If I want good quality feedback, maybe I should look to people who are a little bit more deliberate in their decision making,” said Kilpatrick, a co-author of the new study and associate professor in the Department of Applied Mathematics. “I know they’ve taken their due diligence in deciding.”

The researchers, led by Samatha Linn of the University of Utah, published their findings August 12 in the journal Physical Review E.

In the team’s models, mathematical decision-makers, or “agents,” gather information from the outside world until, ultimately, they make a choice between two options. That might include getting pizza or Thai food for dinner or coloring in the bubble for one candidate versus the other.

The team discovered that when agents started off with a big bias (say, they really wanted pizza), they also made their decisions really quickly—even if those decisions turned out to run contrary to the available evidence (the Thai restaurant got much better reviews). Those with smaller biases, in contrast, often took so long to deliberate that their initial preconceptions were washed away entirely.

The results are perhaps not surprising, depending on your thoughts about human nature. But they can help to reveal the mathematics behind how the brain works when it needs to make a quick choice in the heat of the moment—and maybe even more complicated decisions like who to vote for.

“It’s like standing on a street corner and deciding in a split second whether you should cross,” he said. “Simulating decision making gets a little harder when it’s something like, ‘Which college should I go to?'”

Pouring water

To understand how the team’s mathematical agents work, it helps to picture buckets. Kilpatrick and his colleagues typically begin their decision-making experiments by feeding their agents information over time, a bit like pouring water into a mop pail. In some cases, that evidence favours one decision (getting pizza for dinner), and in others, the opposite choice (Thai food). When the buckets fill to the brim, they tip over, and the agent makes its decision.

In their experiment, the researchers added a twist to that set up: They filled some of their buckets part way before the simulations began. Those agents, like many humans, were biased.

The team ran millions of simulations including anywhere from 10 to thousands of agents. The researchers were also able to predict the behaviour of the most and least biased agents by hand using pen, paper and some clever approximations.

A pattern began to emerge: The agents that started off with the biggest bias, or were mostly full of water to begin with, were the first to tip over—even when the preponderance of evidence suggested they should have chosen differently. Those agents who began with only small biases, in contrast, seemed to take time to weigh all of the available evidence, then make the best decision available.

“The slowest agent to make a decision tended to make decisions in a way very similar to a completely unbiased agent,” Kilpatrick said. “They pretty much behaved as if they started from scratch.”

Neighbourhood choices

He noted that the study had some limitations. In the team’s experiments, for example, none of the agents knew what the others were doing. Kilpatrick compared it to neighbours staying inside their homes during an election year, not talking about their choices or putting up yard signs. In reality, humans often change their decisions based on the actions of their friends and neighbours.

Kilpatrick hopes to run a similar set of experiences in which the agents can influence each other’s behaviours.

“You might speculate that if you had a large group coupled together, the first agent to make a decision could kick off a cascade of potentially wrong decisions,” he said.

Still, political pollsters may want to take note of the team’s results.

“The study could also be applied to group decision making in human organizations where there’s democratic voting, or even when people give their input in surveys,” Kilpatrick said. “You might want to look at folks carefully if they give fast responses.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Daniel Strain, University of Colorado at Boulder

Credit: Unsplash/CC0 Public Domain

In past years, elections in the U.S. have been marked by stories of long waiting lines at the voting polls. Add other barriers, like long commutes and inadequate transportation, and voting can become inaccessible. But these voting deserts are difficult to quantify.

In a paper, “Persistent Homology for Resource Coverage: A Case Study of Access to Polling Sites” in SIAM Review, SFI External Professor Mason Porter (UCLA) and his students applied topological data analysis, which gives a set of mathematical tools that can quantify shape and structure in data, to the problem of quantifying voting deserts in LA County, Chicago, Atlanta, Jacksonville, New York City, and Salt Lake City.

Using a type of topological data analysis called persistent homology, Porter and his co-authors used estimates of average waiting times and commute times to examine where the voting deserts are located.

Applying persistent homology to a data set can reveal clusters and holes in that data, and it offers a way to measure how long those holes persist. The combination of waiting times and commute times in the data creates a pattern, with holes filling in as time passes.

The longer the hole takes to fill, the more inaccessible voting is to people in that area. “We are basically playing connect-the-dots in a more sophisticated way, trying to fill in what’s there,” says Porter.

Moving forward, Porter hopes to use this strategy to more accurately determine voting deserts. Finding voting deserts will hopefully be used to make voting more accessible, but it requires better-quality data than what was available to him and his students.

“This is a proof of concept,” Porter said. “We had to make some very severe approximations, in terms of what data we had access to.”

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article to be given  Santa Fe Institute

From the Fibonacci sequence to the Bell numbers, there is more overlap between mathematics and poetry than you might think, says Peter Rowlett, who has found his inner poet.

People like to position maths as cold, hard logic, quite distinct from creative pursuits. Actually, maths often involves a great deal of creativity. As mathematician Sofya Kovalevskaya wrote, “It is impossible to be a mathematician without being a poet in soul.” Poetry is often constrained by rules, and these add to, rather than detract from, its creativity.

Rhyming poems generally follow a scheme formed by giving each line a letter, so that lines with matching letters rhyme. This verse from a poem by A. A. Milne uses an ABAB scheme:

What shall I call
My dear little dormouse?
His eyes are small,
But his tail is e-nor-mouse.

In poetry, as in maths, it is important to understand the rules well enough to know when it is okay to break them. “Enormous” doesn’t rhyme with “dormouse”, but using a nonsense word preserves the rhyme while enhancing the playfulness.

There are lots of rhyme schemes. We can count up all the possibilities for any number of lines using what are known as the Bell numbers. These count the ways of dividing up a set of objects into smaller groupings. Two lines can either rhyme or not, so AA and AB are the only two possibilities. With three lines, we have five: AAA, ABB, ABA, AAB, ABC. With four, there are 15 schemes. And for five lines there are 52 possible rhyme schemes!

Maths is also at play in Sanskrit poetry, in which syllables have different weights. “Laghu” (light) syllables take one unit of metre to pronounce, and “guru” (heavy) syllables take two units. There are two ways to arrange a line of two units: laghu-laghu, or guru. There are three ways for a line of three units: laghu-laghu-laghu; laghu-guru; and guru-laghu. For a line of four units, we can add guru to all the ways to arrange two units or add laghu to all the ways to arrange three units, yielding five possibilities in total. As the number of arrangements for each length is counted by adding those of the previous two, these schemes correspond with Fibonacci numbers.

Not all poetry rhymes, and there are many ways to constrain writing. The haiku is a poem of three lines with five, seven and five syllables, respectively – as seen in an innovative street safety campaign in New York City, above.

Some creative mathematicians have come up with the idea of a π-ku (pi-ku) based on π, which can be approximated as 3.14. This is a three-line poem with three syllables on the first line, one on the second and four on the third. Perhaps you can come up with your own π-ku – here is my attempt, dreamt up in the garden:

White seeds float,
dance,
spinning around.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Peter Rowlett

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10%—a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, we advocate for expanding what equity means in mathematics education. We believe policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power—and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom We interviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counsellor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life—whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, We collaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of colour and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrolment in advanced courses, we believe true equity will remain elusive. Mathematical power—the ability and confidence to use math to make smart personal and professional decisions—needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, we believe schools, universities and state leaders can create a more expansive and equitable math education system.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Megan Staples, The Conversation

Statisticians from the National University of Singapore (NUS) have introduced a new technique that accurately describes high-dimensional data using lower-dimensional smooth structures. This innovation marks a significant step forward in addressing the challenges of complex nonlinear dimension reduction.

Traditional data analysis methods often rely on Euclidean (linear) dependencies among features. While this approach simplifies data representation, it struggles to capture the underlying complex patterns in high-dimensional data, typically located close to low-dimensional manifolds.

To bridge this gap, manifold-learning techniques have emerged as a promising solution. However, existing methods, such as manifold embedding and denoising, have been limited by a lack of detailed geometric understanding and robust theoretical underpinnings.

The team, led by Associate Professor Zhigang Yao from the Department of Statistics and Data Science, NUS with his Ph.D. student Jiaji Su pioneered a novel method for effectively estimating low-dimensional manifolds hidden within high-dimensional data. This approach not only achieves cutting-edge estimation accuracy and convergence rates but also enhances computational efficiency through the utilization of deep Generative Adversarial Networks (GANs).

This work was conducted in collaboration with Professor Shing-Tung Yau from the Yau Mathematical Sciences Center (YMSC) at Tsinghua University. Part of the work comes from Prof. Yao’s collaboration with Prof. Yau during his sabbatical visit to the Center of Mathematical Sciences and Applications (CMSA) at Harvard University.

Their findings have been published as a methodology paper in the Proceedings of the National Academy of Sciences.

Prof. Yao delivered a 45-minute invited lecture on this research at the recent International Congress of Chinese Mathematicians (ICCM) held in Shanghai, Jan. 2–5, 2024.

Highlighting the significance of the work, Prof. Yao said, “By accurately fitting manifolds, we can reduce data dimensionality while preserving crucial information, including the underlying geometric structure. This represents a major leap in data analysis, enhancing both accuracy and efficiency. By providing a solution that overcomes the limitations of previous methods, our research paves the way for enhanced data analysis and offers valuable insights for diverse applications in the scientific community.”

Looking ahead, Yao’s research team is developing a new framework to process even more complex data, such as single-cell RNA sequence data, while continuing to collaborate with the YMSC team. This ongoing work promises to revolutionize the approach for the reduction and processing of complex datasets, potentially offering new insights into a range of scientific fields.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to National University of Singapore

Math education outcomes in the United States have been unequal for decades. Learners in the top 10% socioeconomically tend to be about four grade levels ahead of learners in the bottom 10% – a statistic that has remained stubbornly persistent for 50 years.

To advance equity, policymakers and educators often focus on boosting test scores and grades and making advanced courses more widely available. Through this lens, equity means all students earn similar grades and progress to similar levels of math.

With more than three decades of experience as a researcher, math teacher and teacher educator, WEadvocate for expanding what equity means in mathematics education. WEbelieve policymakers and educators should focus less on test scores and grades and more on developing students’ confidence and ability to use math to make smart personal and professional decisions. This is mathematical power – and true equity.

What is ‘equity’ in math?

To understand the limitations of thinking about equity solely in terms of academic achievements, consider a student whom WEinterviewed during her freshman year of college.

Jasmine took Algebra 1 in ninth grade, followed by a summer online geometry course. This put her on a pathway to study calculus during her senior year in an AP class in which she earned an A. She graduated high school in the top 20% of her class and went to a highly selective liberal arts college. Now in her first year, she plans to study psychology.

Did Jasmine receive an equitable mathematics education? From an equity-as-achievement perspective, yes. But let’s take a closer look.

Jasmine experienced anxiety in her math classes during her junior and senior years in high school. Despite strong grades, she found herself “in a little bit of a panic” when faced with situations that require mathematical analysis. This included deciding the best loan options.

In college, Jasmine’s major required statistics. Her counselor and family encouraged her to take calculus over statistics in high school because calculus “looked better” for college applications. She wishes now she had studied statistics as a foundation for her major and for its usefulness outside of school. In her psychology classes, knowledge of statistics helps her better understand the landscape of disorders and to ask questions like, “How does gender impact this disorder?”

These outcomes suggest Jasmine did not receive an equitable mathematics education, because she did not develop mathematical power. Mathematical power is the know-how and confidence to use math to inform decisions and navigate the demands of daily life – whether personal, professional or civic. An equitable education would help her develop the confidence to use mathematics to make decisions in her personal life and realize her professional goals. Jasmine deserved more from her mathematics education.

The prevalence of inequitable math education

Experiences like Jasmine’s are unfortunately common. According to one large-scale study, only 37% of U.S. adults have mathematical skills that are useful for making routine financial and medical decisions.

A National Council on Education and the Economy report found that coursework for nine common majors, including nursing, required relatively few of the mainstream math topics taught in most high schools. A recent study found that teachers and parents perceive math education as “unengaging, outdated and disconnected from the real world.”

Looking at student experiences, national survey results show that large proportions of students experience anxiety about math class, low levels of confidence in math, or both. Students from historically marginalized groups experience this anxiety at higher rates than their peers. This can frustrate their postsecondary pursuits and negatively affect their lives.

How to make math education more equitable

In 2023, WEcollaborated with other educators from Connecticut’s professional math education associations to author an equity position statement. The position statement, which was endorsed by the Connecticut State Board of Education, outlines three commitments to transform mathematics education.

1. Foster positive math identities: The first commitment is to foster positive math identities, which includes students’ confidence levels and their beliefs about math and their ability to learn it. Many students have a very negative relationship with mathematics. This commitment is particularly important for students of color and language learners to counteract the impact of stereotypes about who can be successful in mathematics.

A growing body of material exists to help teachers and schools promote positive math identities. For example, writing a math autobiography can help students see the role of math in their lives. They can also reflect on their identity as a “math person.” Teachers should also acknowledge students’ strengths and encourage them to share their own ideas as a way to empower them.

2. Modernize math content: The second commitment is to modernize the mathematical content that school districts offer to students. For example, a high school mathematics pathway for students interested in health care professions might include algebra, math for medical professionals and advanced statistics. With these skills, students will be better prepared to calculate drug dosages, communicate results and risk factors to patients, interpret reports and research, and catch potentially life-threatening errors.
3. Align state policies and requirements:The third commitment is to align state policies and school districts in their definition of mathematical proficiency and the requirements for achieving it. In 2018, for instance, eight states had a high school math graduation requirement insufficient for admission to the public universities in the same state. Other states’ requirements exceed the admission requirements. Aligning state and district definitions of math proficiency clears up confusion for students and eliminates unnecessary barriers.

What’s next?

As long as educators and policymakers focus solely on equalizing test scores and enrollment in advanced courses, WEbelieve true equity will remain elusive. Mathematical power – the ability and confidence to use math to make smart personal and professional decisions – needs to be the goal.

No one adjustment to the U.S. math education system will immediately result in students gaining mathematical power. But by focusing on students’ identities and designing math courses that align with their career and life goals, WEbelieve schools, universities and state leaders can create a more expansive and equitable math education system.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Megan Staples, The Conversation

A simple experiment and mathematical model suggest that when you snack on pistachios, you may need a surprisingly large bowl to accommodate the discarded shells.

Shelling your favourite snack nuts just got a lot easier: physicists have worked out the exact size of bowl to best fit discarded pistachio shells.

Ruben Zakine and Michael Benzaquen at École Polytechnique in Paris often find themselves discussing science in the cafeteria while eating pistachios. Naturally, they began wondering about the mathematics behind storing their snack refuse.

The researchers stuffed 613 pistachios into a cylindrical container to determine “packing density”, or the fraction of space taken up by whole nuts in their shells. Separately, they measured the packing density of the shells alone. In one experiment setup, the researchers poured the shells into a container and let them fall as they may, and in another they shook them into a denser, more efficient configuration.

Without shaking, the shells had about 73 per cent of the original packing density. Shaking decreased this number to 57 per cent. This suggests that, with any pistachio container, an additional half-sized container will hold shell refuse as long you occasionally shake the container while eating.

Zakine and Benzaquen backed up their findings by modelling pistachios as ellipsoids – three-dimensional shapes resembling squashed spheres – and their shells as hollow half-spheres and calculated their packing densities based on mathematical rules. These results confirmed the real-life experiments and suggested that the same ratios would work for other container shapes.

Despite these similarities, the researchers found about a 10 per cent discrepancy between the calculations and the real-life measurements. Zakine says that this is not surprising because pistachios are not perfect ellipsoids and have natural variations in shape. More broadly, it is tricky to calculate how best to pack objects into containers. So far, mathematics researchers have only had luck with doing calculations for spheres, like marbles, and uniform shapes like M&M’s, he says.

Going forward, the researchers want to run more complex calculations on a computer. But for now, they are looking forward to fielding mathematical questions whenever they serve pistachios at dinner parties.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Karmela Padavic-Callaghan*

Catering for students’ learning needs is something we all aim to do. But it can be challenging. Is it just about differentiation? What is the best way to differentiate? How do we put it into practice? Let’s explore some ideas, strategies and tips.

Differentiation

When you hear the word differentiation, what do you think of? Ability groupings? Open-ended tasks? Educational consultant Jennifer Bowden from the Mathematical Association of Victoria believes differentiation involves teachers considering “a whole range of different pedagogies … and making choices about pedagogical approaches based on the students that they teach”. In a nutshell it comes down to knowing your students and how they learn, so you can cater for their needs.

Find out what students know

Assessment is key to discovering what your students know – and don’t know! You can assess students to find out what knowledge they have, the concepts they understand and the skills they can apply to tasks.

Data from this assessment can then be used as a starting point to plan what you will teach.

Find out how students learn

You can go further than just understanding what your students know. Delve deeper and think about; what are your students’ learning behaviours? What are their attitudes towards learning maths? How do they learn best?

It’s important to note that this Is not about learning styles. It’s about knowing how a student:

• thinks and feels about maths
• becomes engaged in a topic, or problem
• responds to certain scaffolds
• makes connections between concepts
• applies what they have learnt.

When you understand your students on this level you have a greater insight into knowing how to best build their knowledge and skills.

Putting it into practice

Once you know your students well you are better prepared to meet their learning needs, but there are still many aspects to think about. Let’s unpack this further.

Planning for instruction

Maths expert Jennifer Bowden promotes the use of the instructional model known as launch, explore, summarise.

• Launch– begin with a question or a task for students to complete or explore.
• Explore– during this stage the teacher supports students at their different levels. Students can work on the same task, but it can be differentiated to extend or give extra support where needed by scaffolding. You can plan for the learning to be done independently, or in small groups.
• Summarise– upon competition of the lesson or task the students come together to share what they have learnt.

In an excellent podcast on the Maths Hub, Jennifer explains this model in greater detail.

Open-ended tasks

These rich tasks provide differentiation by output. Essentially all students are working on the same, or similar task, and students reach various outcomes, according to their individual knowledge and skill application.

Grouping students

There are times when you can best meet students’ needs through grouping them in certain ways. When doing so, consider the purpose of the groupings, and ensure the groups are flexible.

• You should be clear about the specific purpose of your groupings. What needs are you addressing by grouping students together? Are you extending them? Providing consolidation? Are you supporting them to ‘catch up’ on learning they have missed? Or providing intervention?
• Student groupings should beflexible and change according to their purpose. Sometimes groups are ability based, so students can complete different tasks, at different levels. Sometimes groups have mixed abilities so that students can use their various skills and levels of knowledge to problem solve and use their reasoning skills.

Student agency

Giving students a voice by encouraging them to discuss their learning can help you to understand their individual needs. Ask students about their learning; what they know and want to know, if they are feeling challenged and what helps them to learn. This feedback can help you plan and deliver lessons that cater for all student needs.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to The Mathematics Hub