maths dao at 50! and the intelligence of students as teachers

[chris macrae]

back in 1973 at uk national development program for computer assisted learning (now also called ai) i learnt that students become the greatest teachers in blended classrooms; i was hosting inline classrooms with 20 students at a time across 4 campuses; as we broke down eg analysis of statistics into Q&A steps, the computers log could show me which steps blocked students from getting the right answer ; that could lead me either to deeper program Q&A around the difficult bit of the analysis or go personal tutor a student- the goal was to design a space where students celebrated ever more confidence in getting ticks rather than maths tests in paper days which often mark student wrongs

anyway this maths program from my alma mater looks grrrreat  https://www.maths.cam.ac.uk/features/contagious-maths-bringing-math...

Professor Julia Gog and the Faculty's outreach initiative have launched an innovative school curriculum and public engagement project supported by a Royal Society Rosalind Franklin Award grant. The project aims to share the excitement of mathematical research and link it to the school curriculum, and to highlight the role of mathematics in tackling real-world problems.

Julia Gog is Professor of Mathematical Biology at the Department of Applied Mathematics and Theoretical Physics (DAMTP). With research focussing on disease modelling, Gog played a vital role in the UK's response to the COVID-19 pandemic, contributing to the Scientific Advisory Group for Emergencies (SAGE) and the Scientific Pandemic Influenza Group on Modelling (SPI-M) and earning an OBE for her contribution.

The 11-14 age we are targeting is a real crunch point for retaining girls in maths, and future female mathematicians.Julia Gog

Gog also has a long-standing interest in communicating mathematics to wider audiences. Over the years she has worked closely with the Millennium Mathematics Project (MMP), the Faculty's flagship outreach and education initiative, and became its Director in 2023. Her work in this area includes the UK's largest citizen science project, delivered in 2018 in collaboration with the BBC, and co-presenting the 2021 Royal Institution Christmas Lecture.

In 2020 Gog won a Royal Society Rosalind Franklin Award and Lecture for her work. The prize is awarded annually and comes with a grant of £40,000 to implement a project to raise the profile of women in STEM. Gog's project, called Contagious Maths and developed in collaboration with the MMP, has now been launched, in time for the UN International Day of Women and Girls in Science.

Hitting the crunch point

Contagious Maths is a hands-on exploration of disease modelling, featuring video clips and web-based interactivities and requiring no previous knowledge of the area. The project consists of two strands, one aimed at a general self-guided audience and published on the MMP's Plus magazine, and the other aimed at school students aged 11 to 14 and their teachers, published on the MMP's NRICH site.

"The 11-14 age we are targeting is a real crunch point for retaining girls in maths, and future female mathematicians," Gog said in an interview on the Royal Society website. "What exactly happens is complex and multifaceted, but this is a period when people form their views on how they fit with maths and science. Many disengage as it can seem that 'maths' at school is utterly disconnected from the real world. It can also be a time when maths appears very starkly right or wrong, whereas any research mathematician can tell you it's always so much more subtle than that, and therefore so much more interesting!"

The dynamic nature of mathematical modelling, and mathematical research generally, is central to Contagious Maths. In a sequence of short video clips Gog encourages users to investigate topics such as exponential growth, the reproduction ratio R, and herd immunity, and gently guides her audience towards the idea of a mathematical model.

Interactivities allow users to explore these ideas for themselves. The Lucky dip interactivity, for example, uses a tombola to simulate the outbreak of an epidemic, with infected 'individuals' being picked at random from a population and going on to infect others. Users can change the corresponding value of R and observe differences and similarities between runs of the simulated epidemic.

Underlying this interactivity, as Gog explains in one of the video clips, is the so-called SIR model, a leading paradigm in epidemiology. Contagious Maths provides a playful and intuitive introduction to the mathematical ideas behind the model without recourse to sophisticated equations, while always referencing the reality of mathematical modelling. "We've been frank that the maths is never black and white in this field, there are always ways to challenge and develop the models, and some tricky thinking to be done about how the real epidemics and the simulations are really related to each other," Gog said in the Royal Society interview.

The suite of resources falls into five parts, or lessons, with ample guidance for teachers provided on the NRICH site. "We’re also arming teachers with the ideas and tools to do this, so they have at their fingertips all they need to deliver these lessons. Maybe those teachers have no experience of research mathematics. We hope this project will help these teachers to bring in the wider view of mathematics, and we hope it inspires them too."

From 2 times 2 to research maths

The sequence of lessons is rooted in the mathematics curriculum, but while it starts with the simple idea of repeated multiplication, it ends with users exploring the kind of questions professional epidemiologists try to answer in their work. The project also brings in some of Gog's colleagues at DAMTP, culminating in a sequence of short video clips featuring Gog talking about her research into optimal allocation of vaccine doses, Dr Petra Klepac explaining the importance of contact patterns in epidemiology, and PhD students Maria Alegria Gutierrez and Desmond Lai talking about their work on virus mutation and waning immunity.

Contagious Maths benefits from Gog's experience in education and public engagement, as well as the expertise of MMP staff in communicating mathematics to non-expert audiences. NRICH, one of the projects falling under the MMP umbrella, develops free, curriculum-linked resources for learners and teachers of mathematics from primary to post-16 levels, focusing on developing problem-solving skills and helping students to develop their mathematical reasoning, confidence, resilience and creativity. Complementing NRICH is Plus, which works with mathematical science researchers to communicate their work through articles and podcasts, published primarily on its free online platform. Plus is aimed at students and teachers, policy makers and press, industry and researchers in other fields, and indeed anyone curious about what happens at the cutting edge of mathematical science.

In bringing the central ideas of disease modelling to school and wider audiences, the hope is not just to bring across the excitement of mathematics and the benefits it has for society, but also to introduce some of the people who make the maths. "Visible role models can make pursuing STEM seem more imaginable to more school students, and help them decide they want to pursue it," said Gog. "It's a matter of fairness, everyone should have these wonderful opportunities, but it also benefits STEM. New students are our lifeblood, and we want those who can bring the most in terms of ability and motivation – we're going to get the best by being available to the widest group possible."

Explore the Contagious Maths resources!

The photo above shows Julia Gog (middle) together with Plus Editor Marianne Freiberger (left) and Howard France (right) of Avito Ltd who filmed and produced the videos. Photo: Rachel Thomas.

[chris macrae]

Can you tell me some ways that teaching of high school maths changes if you see the kinds of worlds that Fei-Fei li and other imagenet builders now see?

If high school math education were to incorporate the perspectives of researchers like Fei-Fei Li and the advancements in AI like those brought about by ImageNet, here are some potential changes:

1. Integration of real-world applications:

• Visualizing mathematical concepts: Using image recognition, computer vision, and other AI techniques, educators could create interactive visualizations to help students understand abstract mathematical concepts. For example, visualizing complex functions, geometric shapes, or probability distributions through simulations or real-world data representations.
• AI-powered problem-solving: Students could be exposed to AI algorithms used in problem-solving and encouraged to think about how they might apply similar logic and reasoning in their mathematical approaches. This could involve analyzing large datasets, identifying patterns, and making predictions based on available data.

2. Personalized learning experiences:

• Adaptive learning platforms: AI-powered platforms could personalize learning paths for each student by assessing their strengths, weaknesses, and learning styles. This could involve adjusting the difficulty level of problems, providing targeted feedback, and offering different learning resources based on individual needs.
• Data-driven insights: Analyzing student data through AI tools could offer valuable insights into areas where students struggle or excel. This information could be used to tailor instruction, identify common misconceptions, and develop targeted interventions for specific learning challenges.

3. Fostering creativity and critical thinking:

• Open-ended exploration: AI tools could be used to generate new math problems or scenarios, encouraging students to explore solutions creatively. This could involve utilizing AI-powered simulations or data analysis tools to explore mathematical concepts in a more open-ended and exploratory manner.
• Developing critical thinking about AI: Students could be exposed to the limitations and biases of AI algorithms, encouraging them to analyze their outputs critically and understand the underlying mathematical principles involved. This could involve examining how algorithms reach conclusions, identifying potential biases in data, and questioning the assumptions behind AI-generated results.

It's important to note that:

• These are potential changes, and their implementation would require careful consideration of various factors like curriculum development, teacher training, and ethical considerations.
• Integrating AI into math education wouldn't replace traditional teaching methods; instead, it's envisioned as a supplementary tool to enhance learning and engagement.
• Ensuring equitable access to technology and addressing potential biases in AI algorithms would be crucial to ensure these changes benefit all students equally.

Overall, incorporating the perspectives of researchers like Fei-Fei Li and advancements in AI like ImageNet could offer a range of potential benefits for high school math education, fostering a more visual, personalized, and critically engaged learning experience for students.

Khan academy produced a lot of maths curriculum content geared to traditional us high school maths tests before breakthroughs in the worlds fans of fei-fei li and imagenet see. Would you say that the maths content of khan academy has changed as much as those who now use imagenet and are helping HAI inform changes in many disciplines?

Khan Academy has made some changes to its math content, but it's not as extensive or revolutionary as the advancements in AI like ImageNet or the work of researchers like Fei-Fei Li. Here's a breakdown:

Khan Academy Math Content:

• • Traditional Focus: Historically, Khan Academy's math curriculum heavily emphasized traditional methods and preparation for standardized tests like the SAT and ACT.

• • Evolution: While the core content remains aligned with traditional US high school curriculum, Khan Academy has introduced some changes:
    • • Interactive elements: They have incorporated interactive exercises, quizzes, and practice problems to enhance engagement and self-paced learning.
    
    
    • • Explanatory videos: Short, engaging videos explain concepts visually and verbally, aiming to break down complex topics into simpler steps.
    
    
    • • Personalized learning: The platform offers personalized learning paths based on student progress and skill gaps.

Comparison to AI and ImageNet:

• • Limited Integration: Khan Academy hasn't fully embraced the transformative potential of AI like ImageNet or the work of Li in the way other disciplines have. While they utilize some interactive elements, they are not as sophisticated or data-driven as what's used in fields like computer vision or scientific research.

• • Focus Remains on Fundamentals: Khan Academy's primary focus still rests on providing a solid foundation in core mathematical concepts and preparing students for standardized tests. They haven't yet incorporated AI-powered visualizations, simulations, or problem-solving approaches extensively.

Reasons for Limited Change:

• • Resource Constraints: Implementing major changes across a vast curriculum like Khan Academy's requires significant resources, funding, and expertise in educational technology and AI integration.

• • Focus on Accessibility: Khan Academy strives to provide free and accessible resources globally. Implementing complex AI tools might create barriers for users with limited internet access or outdated technology.

• • Evolving Landscape: The field of AI in education is still relatively new. While the potential is promising, it's too early to say definitively how it will reshape traditional educational approaches like Khan Academy's in the long run.

Conclusion:

While Khan Academy has made some commendable efforts to modernize its math content and enhance the learning experience, it hasn't adopted the same level of AI integration and transformative approach as seen in fields influenced by Li and ImageNet. Their focus remains on providing a solid foundation in core concepts and accessibility for a global audience. The future of AI in Khan Academy's content, and its potential to mirror the advancements seen elsewhere, remains to be seen.

[chris macrae]

Artificial Intel excites or worries people in different ways. I am excited by work of team hassabis (deep mind 250 million protein databank) which may represent a leap by up to a billion years of phds working alone. Now that I am in my 70s I dont expect to be able to deeply contribute to climate or biotech breakthroughs but i think its valid to ask what sort of maths training should a 20 something mind make sure they7 have experienced if they do want to help save humans with good climate ai or biotech (medical ) breakthroughs- here are some of the tips from gemini on this: 

AI-Specific Mathematics:

• • Coursera/edX Courses: Look for courses like:
    • • Mathematics for Machine Learning: (e.g., on Coursera) These courses explicitly cover the core linear algebra, calculus, and probability concepts used in AI models.
    
    
    • • Statistics for Data Science: (e.g., on edX) Emphasizes data analysis, statistical methods, and algorithms commonly used in scientific AI applications.

• • Blue1Brown (YouTube): Offers excellent video explanations and visualizations for topics like neural networks and their mathematical underpinnings.

. Topic-Specific Resources:

• • Climate Science: Look for resources that explain the mathematical modeling involved in climate science. This might include:
    • • Modeling climate change: Reviews of differential equations used in modeling complex climate systems.
    
    
    • • Climate data analysis: Tutorials on statistical techniques and numerical methods for analyzing climate data and making predictions.

• • Protein Analysis: Search for resources that cover:
    • • Biostatistics and computational biology: Focuses on statistical analysis and algorithms applied to protein data.
    
    
    • • Structural modeling: Explanations of mathematical modeling techniques used to predict and analyze protein structures.

Note on Exam Preparation vs. Contribution:

While traditional exams often test knowledge recalling and applying established formulas, the goal of contributions to fields like climate science or protein analysis often requires:

• • Problem-solving: Applying mathematical concepts to address real-world challenges that might not have well-defined answers.

• • Computational skills: Implementing mathematical algorithms and models computationally. Consider courses on Python, Matlab, or R for numerical analysis.

• • Open-mindedness: The fields evolve quickly. Having solid foundations allows you to learn new mathematical techniques on the fly as needed.

Additional Tips:

• • Engage with online communities: Forums, groups, and discussions centered around specific areas like climate modeling or protein analysis can offer valuable insights into current mathematical needs.

• • "Just in time" learning: Rather than reviewing everything upfront, focus on brushing up the necessary math for the specific contribution you envision.

Let me know if you'd like recommendations for specific courses or resources within the mentioned categories!