Maths: April 2023

Thursday, April 20, 2023

In his first speech of 2023, the Prime Minister will set out his priorities for the year ahead and ambition for a better future for Britain.

In his first speech of 2023, the Prime Minister will set out his priorities for the year ahead and ambition for a better future for Britain.

The PM will commit to taking the necessary action to deliver for the long term on issues such as low numeracy rates.

As part of this, he will set a new ambition of ensuring that all school pupils in England study some form of maths to the age of 18.

The Prime Minister is expected to say in a speech today [Wednesday 4 January]:

This is personal for me. Every opportunity I’ve had in life began with the education I was so fortunate to receive.

And it’s the single most important reason why I came into politics: to give every child the highest possible standard of education.

Thanks to the reforms we’ve introduced since 2010, and the hard work of so many excellent teachers, we’ve made incredible progress.

With the right plan – the right commitment to excellence – I see no reason why we cannot rival the best education systems in the world”.

Recognising the practical challenges involved, the PM will acknowledge that reform on this scale won’t be easy. He will commit to starting the work of introducing maths to 18 in this Parliament and finishing it in the next.

Around 8 million adults in England have the numeracy skills of primary school children. Currently only around half of 16-19 year olds study any maths at all and the problem is particularly acute for disadvantaged pupils, 60% of whom do not have basic maths skills at age 16.

Despite these poor standards, the UK remains one of the only countries in the world to not to require children to study some form of maths up to the age of 18. This includes the majority of OECD countries, including Australia, Canada, France, Germany, Finland, Japan, Norway and the USA.

The Prime Minister will commit to take action to reverse these trends by introducing maths to 18 for all pupils in England. He will say:

One of the biggest changes in mindset we need in education today is to reimagine our approach to numeracy.

Right now, just half of all 16–19-year-olds study any maths at all. Yet in a world where data is everywhere and statistics underpin every job, our children’s jobs will require more analytical skills than ever before.

And letting our children out into the world without those skills, is letting our children down”.

Maths to 18 will equip young people with the quantitative and statistical skills that they will need for the jobs of today and the future. This includes having the right skills to feel confident with finances in later life, including finding the best mortgage deal or savings rate.

The government’s focus on literacy since 2010, including phonics, has led to significant improvements in standards. In 2012, only 58% of 6-year-olds were able to read words fluently. By 2019, the figure had risen to 82%. Our renewed focus on numeracy will aim to match this achievement.

The government does not envisage making maths A-Level compulsory for all 16-year-olds. Further detail will be set out in due course but the government is exploring existing routes, such as the Core Maths qualifications and T-Levels, as well as more innovative options.

The ambition is the PM’s first major intervention on education since entering office and reflects his mission to ensure that more children leave school with the right skills in numeracy and literacy.

At the Autumn Statement, the government announced that it will invest an additional £2bn in schools next year and £2bn the year after, taking school funding to its highest ever level.

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How to improve math skills among American children

In the past two decades, researchers have made great strides in uncovering how children learn math, but little of that new knowledge has trickled down to teachers, according to a new book on math education.

The gap between research and practice is particularly unfortunate, given the current state of American students’ math skills, said Nancy Krasa, co-author of How Children Learn Math: The Science of Math Learning in Research and Practice.

“American children are not doing well at math,” said Krasa, who is an adjunct assistant professor of psychology at The Ohio State University.

“In 2019, only about one-fourth of high school seniors scored at or above the proficiency level in math. And all indications are that this has only gotten worse with the learning loss associated with the COVID-19 pandemic.”

But there is a way to meet this challenge, according to Krasa, who is a licensed psychologist specializing in children with learning difficulties, including those who have trouble with mathematics.

“The scientific research on how children learn math has exploded in the past 20 years, with thousands of new studies focused on how children come to understand numbers and various other aspects of math,” she said.

“The problem is that little of this work has been accessible to teachers on the front line.”

Krasa said she and her co-authors, Karen Tzanetopoulos and Colleen Maas, wrote How Children Learn Math to bring the latest discoveries in math learning to teachers and parents and give them research-based ways to teach the fundamentals to young students. The book focuses on toddlerhood through the learning of fractions.

A good example of one of the new discoveries is learning spatial skills. The importance of spatial skills in early math is one of those crucial findings that hasn’t made it to early education teachers, Krasa said.

Most people think of spatial skills in terms of geometry, but recent research suggests that a person’s spatial skills are associated with their math skills more generally.

“That’s something most teachers would have no idea about, but the results are remarkably consistent,” she said.

“What is not yet entirely clear is how they are related — why do people with good spatial skills have an easier time with math?”

One hypothesis is that humans think of numerical quantities along a mental number line, as if they existed in space. One real-life classroom application is that a physical number line in the classroom, if properly used, may help teach young children about numbers.

But research shows that children begin developing spatial skills even before they get to school. One way very young children learn spatial skills is by playing with blocks.

One study Krasa and her co-authors mention in the book found that when mothers and their 3-year-olds build with blocks together, the amount of spatial language, related gestures, and planning support the mothers provide predicts the children’s math skill in first grade.

The impact of playing with blocks and its effects on spatial skills goes well beyond the early grades.

Another study found that children’s preschool block-building skills predicted their high school math course selections, math grades and standardized math scores.

One implication of recent research is that children should be screened for spatial skills early in life, just as they are for reading skills, Krasa said. The good news is that “spatial skills are trainable, especially if we can identify those who need help early.”

Another important finding of recent research is the importance of language in learning mathematics, she said.

“Math language is very abstract. Students may understand math concepts better with familiar terms, such as ‘and’ instead of ‘plus’, for example," Krasa said.

“Also, math is not separate from reading. Research has shown that children with reading disabilities, particularly dyslexia, are at a great risk for math failure.”

One study found that of children who had been diagnosed with a developmental language disorder in kindergarten, 55% had serious math difficulties by the fifth grade – more than 10 times the rate found in the general population.

Despite the alarming statistics about math knowledge among American children, Krasa said working on the book has convinced her that the situation is not hopeless.

“I believe that with the proper supports, all children in the normal range of intelligence can learn math. Even with challenges like poverty, reading and language disability, weak spatial skills and attentional issues, they can learn and understand the fundamental concepts,” she said.

The key is that students have to begin early, or, if they don’t, they have to go back and begin with the fundamentals. Math skills and concepts that students learn in high school are built on those from elementary school – and those are built on skills learned in preschool and at home.

That means many of the problems that students face in high school find their roots in early math education.

“If we’re going to get it right, we have to start from the beginning,” Krasa said.

The U.S. failure in math education is not the fault of teachers, she said. They are doing the best they can given their training and the challenges they face.

“We want teachers to have the latest research on how children actually learn math so they can help turn things around. That’s why we wrote this book.”

The book offers activities that are easily understandable for teachers and parents, but that aren’t currently being used in most classrooms, she said.

These new approaches are desperately needed.

“Clearly, something is not working in math education in this country. We could be doing much, much better,” Krasa said.

New algorithm aces university math course questions

Multivariable calculus, differential equations, linear algebra — topics that many MIT students can ace without breaking a sweat — have consistently stumped machine learning models. The best models have only been able to answer elementary or high school-level math questions, and they don’t always find the correct solutions.

Now, a multidisciplinary team of researchers from MIT and elsewhere, led by Iddo Drori, a lecturer in the MIT Department of Electrical Engineering and Computer Science (EECS), has used a neural network model to solve university-level math problems in a few seconds at a human level.

he model also automatically explains solutions and rapidly generates new problems in university math subjects. When the researchers showed these machine-generated questions to university students, the students were unable to tell whether the questions were generated by an algorithm or a human.

This work could be used to streamline content generation for courses, which could be especially useful in large residential courses and massive open online courses (MOOCs) that have thousands of students. The system could also be used as an automated tutor that shows students the steps involved in solving undergraduate math problems.

“We think this will improve higher education,” says Drori, the work’s lead author who is also an adjunct associate professor in the Department of Computer Science at Columbia University, and who will join the faculty at Boston University this summer. “It will help students improve, and it will help teachers create new content, and it could help increase the level of difficulty in some courses. It also allows us to build a graph of questions and courses, which helps us understand the relationship between courses and their pre-requisites, not just by historically contemplating them, but based on data.”

The work is a collaboration including students, researchers, and faculty at MIT, Columbia University, Harvard University, and the University of Waterloo. The senior author is Gilbert Strang, a professor of mathematics at MIT. The research appears this week in the Proceedings of the National Academy of Sciences.

A “eureka” moment Drori and his students and colleagues have been working on this project for nearly two years. They were finding that models pretrained using text only could not do better than 8 percent accuracy on high school math problems, and those using graph neural networks could ace machine learning course questions but would take a week to train.

Then Drori had what he describes as a “eureka” moment: He decided to try taking questions from undergraduate math courses offered by MIT and one from Columbia University that had never been seen before by a model, turning them into programming tasks, and applying techniques known as program synthesis and few-shot learning. Turning a question into a programming task could be as simple as rewriting the question “find the distance between two points” as “write a program that finds the difference between two points,” or providing a few question-program pairs as examples.

Before feeding those programming tasks to a neural network, however, the researchers added a new step that enabled it to vastly outperform their previous attempts.

In the past, they and others who’ve approached this problem have used a neural network, such as GPT-3, that was pretrained on text only, meaning it was shown millions of examples of text to learn the patterns of natural language. This time, they used a neural network pretrained on text that was also “fine-tuned” on code. This network, called Codex, was produced by OpenAI. Fine-tuning is essentially another pretraining step that can improve the performance of a machine-learning model.

The pretrained model was shown millions of examples of code from online repositories. Because this model’s training data included millions of natural language words as well as millions of lines of code, it learns the relationships between pieces of text and pieces of code.

Many math problems can be solved using a computational graph or tree, but it is difficult to turn a problem written in text into this type of representation, Drori explains. Because this model has learned the relationships between text and code, however, it can turn a text question into code, given just a few question-code examples, and then run the code to answer the problem.

“When you just ask a question in text, it is hard for a machine-learning model to come up with an answer, even though the answer may be in the text,” he says. “This work fills in the that missing piece of using code and program synthesis.”

This work is the first to solve undergraduate math problems and moves the needle from 8 percent accuracy to over 80 percent, Drori adds.

Adding context Turning math questions into programming tasks is not always simple, Drori says. Some problems require researchers to add context so the neural network can process the question correctly. A student would pick up this context while taking the course, but a neural network doesn’t have this background knowledge unless the researchers specify it.

For instance, they might need to clarify that the “network” in a question’s text refers to “neural networks” rather than “communications networks.” Or they might need to tell the model which programming package to use. They may also need to provide certain definitions; in a question about poker hands, they may need to tell the model that each deck contains 52 cards.

They automatically feed these programming tasks, with the included context and examples, to the pretrained and fine-tuned neural network, which outputs a program that usually produces the correct answer. It was correct for more than 80 percent of the questions.

The researchers also used their model to generate questions by giving the neural network a series of math problems on a topic and then asking it to create a new one.

“In some topics, it surprised us. For example, there were questions about quantum detection of horizontal and vertical lines, and it generated new questions about quantum detection of diagonal lines. So, it is not just generating new questions by replacing values and variables in the existing questions,” Drori says.

Human-generated vs. machine-generated questions The researchers tested the machine-generated questions by showing them to university students. The researchers gave students 10 questions from each undergraduate math course in a random order; five were created by humans and five were machine-generated.

Students were unable to tell whether the machine-generated questions were produced by an algorithm or a human, and they gave human-generated and machine-generated questions similar marks for level of difficulty and appropriateness for the course.

Drori is quick to point out that this work is not intended to replace human professors.

“Automation is now at 80 percent, but automation will never be 100 percent accurate. Every time you solve something, someone will come up with a harder question. But this work opens the field for people to start solving harder and harder questions with machine learning. We think it will have a great impact on higher education,” he says.

The team is excited by the success of their approach, and have extended the work to handle math proofs, but there are some limitations they plan to tackle. Currently, the model isn’t able to answer questions with a visual component and cannot solve problems that are computationally intractable due to computational complexity.

In addition to overcoming these hurdles, they are working to scale the model up to hundreds of courses. With those hundreds of courses, they will generate more data that can enhance automation and provide insights into course design and curricula.

Monday, April 17, 2023

AI takes up garb of maths, science teacher as online education website launches ChatGPT rival

Chegg is an American company that provides services to students. These services include physical and digital textbook rentals, help with homework, tutoring and more. The NYSE-listed company is now launching a new study aid software called CheggMate, which will offer personalized study guides and practice tests to students, using the power of artificial intelligence (AI). The underlying technology of CheggMate is GPT-4 which has been developed by OpenAI and also powers the paid version of the ChatGPT.

The company claims that the software is designed to adapt to students by processing data on the classes they are taking and the exam questions they have missed, providing targeted support to help them achieve their goals.

The new CheggMate software will be focusing on math and science, according to a report by Reuters. In order to avoid being misused like ChatGPT, CheggMate creators claim that it has been designed to limit reviews of answers to current exam questions.

Chegg's CEO, Dan Rosensweig, has described CheggMate as a "tutor in your pocket," which will enable students to get the personalized support they need to succeed. The software will be available for free initially, and Chegg expects it to reduce content costs and boost profitability over time.

Also read: ChatGPT created its own language to break free from word limits; meet Shogtongue

On the other hand, the platform ChatGPT has been facing flak for unregulated use by students. Last year, the launch of ChatGPT led to some schools banning access to the chatbot, out of concern that it could be used to plagiarize coursework. CheggMate plans to tackle the issue by focusing on math and science.

CheggMate is not the only educator that is trying to incorporate the power of AI to boost their scale and efficiency. Khan Academy was also one of the first online education websites that included the latest GPT-4 technology to derive personalised experiences for students.

Also read: OpenAI ChatGPT VS Google Bard: Which AI chatbot is the real disruptor?

Monday, April 10, 2023

‘Math has become my art:’ Podcaster, mathematics professor and advocate speaks at Wichita State

Once planning on being an art teacher, Pamela Harris found a new art with mathematics.

“I think math has become my art,” Harris, now an associate professor at the University of Wisconsin-Milwaukee, said. “People ask me that, they’re like, ‘Oh, do you still paint, do you do like sculptures’ and I was like, ‘I don’t need to.’

“For me, my creative outlet has become math, right? I get to create mathematics.”

Harris visited Wichita State to present in the “Lecture Series in the Mathematical Sciences,” on Friday, and also on Sonia Kovalevsky Day, an all-women event geared toward empowering future generations of women in STEM.

“A lot of the math that I do is, it’s so easy to explain,” Harris said. “I can show (this function) to a kid and they can play with the numbers, even though they might not get to the formula of how many there are,” Harris said.

Harris said she’s talked at lecture events before and is able to see the participants engage with math in a way that’s not just ‘how fast can you multiply,’ or ‘how fast can you do this arithmetic problem?’

“And that part for me is so important,” Harris said. “To see a new generation of students not believe the math is just the stuff that’s in your textbook, like to see that math can be creative and artistic and then you can just have fun and play with it.”

Harris’s presentation was titled “Parking Functions: Choose your own adventure,” and followed the classic “choose your own adventure format,” where audience members help direct the course of the presentation.

“It’s kind of nice because there’s so many things that I could talk about that then I just put it all together and then you know, set the slide deck so that it selects whatever the audience kind of wants,” Harris said.

Harris has been working on this research since 2017, and while she saw parking functions in other contexts of work, it was never the sole investigative point — until working with a group of graduate students at a conference.

“I was like, no, these objects in and of themselves are really interesting,” Harris said. “And there’s more to be said here and discovered.”

Harris also is the co-host of the podcast, Mathematically Uncensored, which began in 2020. Harris and Aris Swinger started the podcast to make private conversations public about being a mathematician of color.

“There’s still a lot of negative experiences we have,” Harris said. “He’s a Black man, I’m Latina. And so as minorities in math, there’s many of us, but still, we’re very underrepresented in the STEM fields.”

Along with co-hosting the podcast, Harris is also the president and co-founder of Lathisms, Latinxs and Hispanics in the Mathematical Sciences.

The organization started with a conversation on Facebook seven years ago.

Harris tried to compile a list of Latino people in math and realized there wasn’t one, so the founders started a calendar for Hispanic Heritage Month, where a biography was released each day of that month.

As the calendar gained traffic, someone suggested starting a podcast, then a YouTube lecture series was created, along with the creation of a book.

“It’s so interesting to me that I think as a kid if somebody would have been like, ‘imagine yourself being old,’” Harris said, laughing. “I would have never thought that this could be a life. I didn’t know that you could be a mathematician and travel everywhere and talk about the thing that you love.”

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Does a Child’s Math Skills Have a Genetic Basis?

A recent study published in the journal Genes, Brain and Behavior has uncovered several genetic variations that could be linked with mathematical abilities in children.

The research involved conducting genome-wide association studies on 1,146 elementary school students from China, focusing on 11 categories of mathematical ability. The results revealed seven single nucleotide genetic variations in the genome that showed a strong correlation with mathematical and reasoning skills.

Additional analyses revealed significant associations of three mathematical ability categories with three genes. Variants in LINGO2 (leucine rich repeat and lg domain containing 2) were associated with subtraction ability, OAS1 (2’-5’-oligoadenylate synthetase 1) variants were associated with spatial conception ability, and HECTD1 (HECT domain E3 ubiquitin protein ligase 1) variants were associated with division ability.

“Results of our research provide evidence that different mathematical abilities may have a different genetic basis. This study not only refined genome-wide association studies of mathematical ability but also added some population diversity to the literature by testing Chinese children,” said corresponding author Jingjing Zhao, Ph.D., a professor in the School of Psychology at Shaanxi Normal University, China.

Reference: “A genome-wide association study identified new variants associated with mathematical abilities in Chinese children” by Liming Zhang, Zhengjun Wang, Zijian Zhu, Qing Yang, Chen Cheng, Shunan Zhao, Chunyu Liu and Jingjing Zhao, 22 February 2023, Genes, Brain & Behavior.
DOI: 10.1111/gbb.12843

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Researchers Just Developed a Math Formula For Achieving a Male Orgasm

For the first time, a team of mathematicians has developed a model that can map the best way for those with penises to reach sexual climax.

After combing through decades of data on both physiological and psychological arousal, the researchers say they've found the ideal conditions required to achieve orgasm.

"We have developed the first successful mathematical model of sexual performance," says mathematician Konstantin Blyuss, one of the lead researchers from the University of Sussex in the UK.

"Our results cover the physiological and psychological aspects required to reach climax. They reinforce, and mathematically prove, existing studies into the psychology of sex."

Of course, there is no one-size-fits-all when it comes to sex and sexual satisfaction, so take this with a grain of salt. The researchers aren't guaranteeing an orgasm for everyone every time.

Instead, they just hope to skew the odds a little more in one's favor based on data.

The formulae themselves are actually pretty intense mathematical stuff that won't be a great help in the heat of the moment. (The research was inspired by the use of mathematics to improve sports performance, but it does not involve thinking about baseball.)But the researchers did come up with one important takeaway message: Too much psychological arousal early on can make it harder to reach orgasm.

"A key finding is that too much psychological arousal early in the process can inhibit the chance of reaching climax," says Blyuss.

"Simply put, our findings can be summarized as 'Don't overthink it.'"

More importantly, the team has been able to do what others haven't – find a way to use mathematical models to predict whether someone will reach climax.

Of course, the elephant in the room here is why the researchers looked at human males, who usually find it much easier to climax than others.

But many of them also experience sexual dysfunction at some point in their lives, and they tend to have a much simpler arousal cycle, so it was the best starting point for researchers to create these equations.

"Our findings shed light on a socially taboo subject, which we believe could have useful applications for the clinical treatment of sexual dysfunction," said mathematician and co-researcher Yuliya Kyrychko, also from the University of Sussex.

"With what we have learned from this study, we intend to mathematically model the female sexual response, which is physiologically – and mathematically – more complex than the male response."

The researchers developed the equations by analyzing data around the four stages of the male arousal cycle: excitation, plateau, orgasm, and resolution.

One of the main studies they looked at was the iconic 1966 study behind the Masters-Johnson theory of sexual response cycle, which included data from 10,000 sexual acts performed in the lab between 382 women and 312 men.

The team then compared their results with research from the Netherlands that dates back to 2006.

In these studies, researchers put consenting participants in fMRI machines and monitored their neurological changes as they performed sexual acts and reached climax.

The model also took into account research on phenomena such as spontaneous arousal during the day, the responses of males with spinal cord injuries, and 'wet dreams'.

Looking at all of this data, researchers came up with two different mathematical equations – one dealing with the psychological factors involved and one dealing with the physiological side of things.

As mentioned above, these formulae aren't really something you can plug in and use in the bedroom. You can see some of the calculations in the paper below.

A screenshot from the paper. (Blyussa & Kyrychko, Chaos, 2023)

But they are there to map the conditions that would lead an arousal cycle to end in orgasm – or not. This could help researchers better understand sexual problems in the future.

Because every single person's sexual activity involves such different stimuli, the team had to use what's known as 'stochasticity'. Basically, it's the phenomena of randomness that can be statistically analyzed.

"We are able to find optimal stochastic escape paths that show how a sexual response progresses toward an orgasm under the influence of small stochastic perturbations," Blyussa and Kyrychko write.

Now that we've been able to quantify and model the elusive orgasm, we're looking forward to math helping all of us get a little closer to satisfaction.

The research has been published in Chaos: An Interdisciplinary Journal of Nonlinear Science.

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