An amazing new result
The asymptotics of r(4,t) - by Sam Mattheus and Jacques Verstraete
SACRAMENTO—The California State Board of Education today approved the 2023 Mathematics Framework for California Public Schools, instructional guidance for educators that affirms California’s commitment to ensuring equity and excellence in math learning for all students.
“I’m thankful for everyone who worked tirelessly to develop this framework to ensure California’s students have equitable access to rigorous and high-quality math instruction that will prepare them for the future. The framework has struck a great balance in new ways to engage students in developing a love for math while supporting those on an accelerated path,” said Mary Nicely, Chief Deputy Superintendent of Public Instruction. “Our State Superintendent is a champion of equity and excellence, and it is our core mission that every child—regardless of race, ZIP code, or background—has access to a quality education. The approval of the revised Math Framework is one more step forward to meeting the needs of all California’s students.”
The vote today concludes four years of work to update math instructional guidance aligned with the California Common Core State Standards for Mathematics (PDF), which are rigorous learning standards that detail what every student should know and be able to do at every grade level. The framework approved today is the third iteration and reflects revisions responsive to thousands of public comments fielded over two 60-day public comment periods and two public hearings.
The draft was presented by Dr. Mike Torres, Executive Director of the Instructional Quality Commission and a former high school math teacher. Others who participated in the presentation include Dr. Kyndall Brown, Executive Director of the California Mathematics Project at the University of California, Los Angeles; Omowale Moses, Founder and Chief Executive Officer of Math Talks; Dr. Adrian Mims, Founder of The Calculus Project; Ellen Barger, Chair of the Curricular and Improvement Support Committee of the California County Superintendents; and Dr. Linsey Gotanda, Vice Chair of the Instructional Quality Commission.
“This framework provides strategies to challenge, engage, and support all students in deep and relevant math learning by building on successful approaches used in nations that produce high and equitable achievement in math,” said State Board President Linda Darling-Hammond. “It also draws on the experiences of educators who have worked for a decade to develop successful strategies for teaching California’s rigorous standards, carrying those lessons to others across the state. This framework provides teachers and schools with a path to greater excellence with greater equity.”
The guidance includes strategies to:
More information is available on the California Department of Education's Mathematics Frameworks web page, which includes frequently asked questions, an overview, and a timeline of events in the framework’s development.
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Tony Thurmond —
State Superintendent of Public Instruction
Communications Division, Room 5602, 916-319-0818, Fax 916-319-0100
Session of Senior Capstone presentations today at ONU, featuring topics on Partial Differential Equations and Number Theory. It was definitely a success, with a lively attendance. Many thanks to these awesome students!
Today, the topic was the Euclidean algorithm...
Just before erasing this number theoretic Tibetan mandala :)
My first post-pandemic faculty-student paper and the 14-th overall (written with Rachael Harbaugh, ONU '23, a talented Mathematics Education major). Glad for this "restart". Students need confidence, need to be exposed to interesting math topics, and then we hope for the best in their future timelines.
Mathematics Capstone Colloquium - December 7, 2022 @ohionorthern (from L to R: Dr. Chowdhury, Joelena Brown, Rachael Harbaugh, and Dr. Caragiu)
Joelena Brown: "Rectangular Donut Numbers" (advisor Dr. Chowdhury)
Rachael Harbaugh: "Extending a Putnam Problem to Fields of Various Characteristics" (advisor Dr. Caragiu)
Random Fibonacci recursion
In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation , where the signs + or − are chosen at random with equal probability , independently for different . By a theorem of Harry Kesten and Hillel Furstenberg, random recurrent sequences of this kind grow at a certain exponential rate, but it is difficult to compute the rate explicitly. In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943...(sequence A078416 in the OEIS), a mathematical constant that was later named Viswanath's constant.
In my classes I often try to communicate the specific sense of wonder arising from mathematical experimentation, to the benefit of undergraduates. Here "phi" represents the Euler's totient function, in a serendipitous trigonometric (albeit nonlinear) mix.
Chaos evokes images of the dinosaurs running wild in Jurassic Park, or my friend’s toddler ravaging the living room.
In a chaotic world, you never know what to expect. Stuff is happening all the time, driven by any kind of random impulse.
But chaos has a deeper meaning in connection to physics and climate science, related to how certain systems – like the weather or the behavior of a toddler – are fundamentally unpredictable.
Scientists define chaos as the amplified effects of tiny changes in the present moment that lead to long-term unpredictability. Picture two almost identical storylines. In one version, two people bump into each other in a train station; but in the other, the train arrives 10 seconds earlier and the meeting never happens. From then on, the two plot lines might be totally different.
Usually those little details don’t matter, but sometimes tiny differences have consequences that keep compounding. And that compounding is what leads to chaos.
A shocking series of discoveries in the 1960s and ‘70s showed just how easy it is to create chaos. Nothing could be more predictable than the swinging pendulum of a grandfather clock. But if you separate a pendulum halfway down by adding another axle, the swinging becomes wildly unpredictable.
What’s the difference between a pack of cards and the weather?
You can’t predict your next poker hand – if you could, they’d throw you out of the casino – whereas you can probably guess tomorrow’s weather. But what about the weather two weeks from now? Or a year from now?
Randomness, like cards or dice, is unpredictable because we just don’t have the right information. Chaos is somewhere between random and predictable. A hallmark of chaotic systems is predictability in the short term that breaks down quickly over time, as in river rapids or ecosystems.
Isaac Newton envisioned physics as a set of rules governing a clockwork universe – rules that, once set in motion, would lead to a predetermined outcome. But chaos theory proves that even the strictest rules and nearly perfect information can lead to unpredictable outcomes.
This realization has practical applications for deciding what kinds of things are predictable at all. Chaos is why no weather app can tell you the weather two weeks from now – it’s just impossible to know.
On the other hand, broader predictions can still be possible. We can’t forecast the weather a year from now, but we still know what the weather is like this time of year. That’s how climate can be predictable even when the weather isn’t. Theories of chaos and randomness help scientists sort out which kinds of predictions make sense and which don’t.
Read other short accessible explanations of newsworthy subjects written by academics in their areas of expertise for The Conversation U.S. here.
See https://www.fq.math.ca/Scanned/11-5/trigg.pdf - Fibonacci Quarterly Vol. 11 No. 5 (Dec. 1973)
This course of 25 lectures, filmed at Cornell University in Spring 2014, is intended for newcomers to nonlinear dynamics and chaos. It closely follows Prof. Strogatz's book, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering."
AN ELEMENTARY NOTE ON THE GREATEST PRIME FACTORS OF LINEARLY RELATED INTEGERS
JP Journal of Algebra, Number Theory and Applications
Volume 52, Issue 1, Pages 95 - 100 (October 2021)
Sequential Experiments with Primes got into the 100 Best Number Theory Books of All Time @ https://bookauthority.org/books/best-number-theory-books As an undergraduate college faculty member, I am happy. Thank you! :-)
I believe the attractiveness of the book lies not only on the novelty of certain ideas, but also in the style in which said novelty is attained. It's a sort of "jazz" with numbers (unfolding as a sustained creative piece not unlike the free development of a jazz gig). A jazz with no particular rigid/studied reverence to other established theoretical
approaches. Just free self-sustained jazz discovering new facts. In its way, it's structured as a sort of "dessins d'enfants" leading to a different look on the mystery of prime numbers.
Quanta Magazine: Computer Scientists Attempt to Corner the Collatz Conjecture
"A powerful technique called SAT solving could work on the notorious Collatz conjecture. But it’s a long shot."
“Those of us who do math for a living actually are drawn to math often because of this exploratory nature, and we love the creativity,” Su said. “The argument that I make is that these virtues carry over to other areas of your life.”
The friendship paradox in real and model networks
George T Cantwell, Alec Kirkley, and M E J Newman
Journal of Complex Networks, Volume 9, Issue 2, April 2021
Published: 27 May 2021
student co-author(s) in italics