September is here, and so is the school year for my US friends. Start a habit of doing some problem solving in your math classes this year! Here is the 1990 Calendar of Problems from 35 years ago f…| Reflections and Tangents
In a conference center near San Francisco International Airport on Monday, new ways to teach math were taking off, and with little time to spare. In less than two weeks, the city’s public schools will roll out a new curriculum for the first time in more than a decade. The San Francisco Unified School District’s […] The post How SFUSD’s Teaching Overhaul Aims to Revive Student Math Scores appeared first on The Frisc.| The Frisc
Wow the summer is flying by, and it is back-to-school time already in many parts of the US. If you’re about to start your academic year, here is the August 2010 Calendar of Problems from 15 y…| Reflections and Tangents
Joram’s seminar 2025 Here is my summary of the recent Joram’s seminar that took place on July 9 and 10 in Jerusalem. Much of the seminar was about the the paper Product Mixing in Compac…| Combinatorics and more
Note: this article| mathesis
Note: this article| mathesis
Note: this article| mathesis
Quanti venti conosci?| wok
A couple recent questions asked what constitutes “standard form” for a quadratic equation; that will lead us to some older questions about “standard form” for a linear equation. We’ll see that “standard” isn’t quite as standard as you might think. Standard form for a quadratic The first question is from Charliemagne, a teacher (apparently in …Is There More Than One Standard Form for an Equation? Read More »| The Math Doctors
Let’s look at a nice little challenge: to find a cubic function with maximum and minimum at given locations – without using calculus. We’ll explore how to solve it with graphing software, and using algebra in a couple ways, and finally with calculus. And, surprise! They all give the same answer, though the results look …A Cubic Challenge Read More »| The Math Doctors
Last time, we considered how to represent algebraically the division of a line segment in a given ratio. At the end, we touched on a subject I recalled discussing extensively almost four years ago: that such a “division” can be either internal (inside the segment, as you’d expect) or external (elsewhere on the line containing …Internal and External Division of a Segment Read More »| The Math Doctors
A series of recent questions dealt with proportional division of a line segment. The context was vectors, and we’ll use them a lot, though the main ideas can be understood using ordinary geometry. We’ll see a mistake so easy to make that AI did it just as humans do; and how textbooks can make it …Dividing a Segment: Get the Order Right! Read More »| The Math Doctors
The monoid of n×nn \times n matrices has an obvious nn-dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So its category of representations is generated by this one obvious representation, in some sense. And it’s almost freely generated: there’s just one special relation. What’s that, you ask? It’s a relation saying the obvious representation is nn-dimensional! | golem.ph.utexas.edu
Frank Cassano and Anya Sturm are math teachers at Marin Academy, an independent high school in San Rafael. They presented on “Integrating Argumentation and Problem-Solving” at the Calif…| Henri's Math Education Blog
Spring is here, and I’m sharing the May Calendar for some problem solving enjoyment in the last stretch of your school year. This is the May 2019 Calendar of Problems from 6 years ago for you…| Reflections and Tangents
Welcome to April, and I hope you’re ready to spring into some problem solving! Here is the April 2011 Calendar of Problems from 14 years ago for you and your students to try. I have the last …| Reflections and Tangents
On BlueSky, Bryan Meyer asks: Henri, in your experience, what are the pros and cons of using function diagrams with kids (in addition to the more standard Cartesian representation)? My BlueSk…| Henri's Math Education Blog
Update: Let me mention a ninth paper that just appeared on the arXive. IX. … and the optimal sofa for the moving sofa problem is … Gerver’s sofa. Optimality of Gerver’s Sofa…| Combinatorics and more
How can we tackle relational algebra of the sort seen in optimization models using tools from AlgebraicJulia?| blog.algebraicjulia.org
Acsets are great, but what if attributes could be variables?| blog.algebraicjulia.org
A follow-up to Algebraic Geometry for the Working Programmer, this post explains a category-theoretic approach to symbolic open dynamical systems.| blog.algebraicjulia.org
In this series of posts, we investigate the duality between algebra and geometry in order to develop new types of lenses. In this first post, we review some basic ideas about algebraic geometry that will be needed in the coming posts.| blog.algebraicjulia.org
Bless me Father for I have sinned. It’s been four and a half years since my last curriculum article. My entire teaching career has been spent navigating a student ability gap that got noticeably wider sometime before the pandemic. The strongest kids are getting better, the weaker ones spent middle school not learning a thing […]| educationrealist
Here’s the thing: I only have one prep this year. The whole year. I have a lot of credentials. I can teach any subject except science. (OK, language and PE as well, but they’re not real…| educationrealist
In early December, I attended the California Math Council Northern Section conference in Asilomar, as I’ve done almost every year since the mid-1980’s. In my last post, I discussed my session on fr…| Henri's Math Education Blog
At some point, maybe thirty years ago, it became fashionable to emphasize functions and their multiple representations in secondary school math. This was in part driven by the newly available elect…| Henri's Math Education Blog
If $latex V$ is a module of a Lie algebra $latex L$, there is one submodule that turns out to be rather interesting: the submodule $latex V^0$ of vectors $latex v\in V$ such that $latex x\cdot v=0$…| The Unapologetic Mathematician
There are a few constructions we can make, starting with the ones from last time and applying them in certain special cases. First off, if $latex V$ and $latex W$ are two finite-dimensional $latex …| The Unapologetic Mathematician
There are a few standard techniques we can use to generate new modules for a Lie algebra $latex L$ from old ones. We’ve seen direct sums already, but here are a few more. One way is to start …| The Unapologetic Mathematician
As might be surmised from irreducible modules, a reducible module $latex M$ for a Lie algebra $latex L$ is one that contains a nontrivial proper submodule — one other than $latex 0$ or $latex…| The Unapologetic Mathematician
Sorry for the delay; it’s getting crowded around here again. Anyway, an irreducible module for a Lie algebra $latex L$ is a pretty straightforward concept: it’s a module $latex M$ such …| The Unapologetic Mathematician
It should be little surprise that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra $latex L$ we define an $latex L$-module t…| The Unapologetic Mathematician
It turns out that all the derivations on a semisimple Lie algebra $latex L$ are inner derivations. That is, they’re all of the form $latex \mathrm{ad}(x)$ for some $latex x\in L$. We know tha…| The Unapologetic Mathematician
We say that a Lie algebra $latex L$ is the direct sum of a collection of ideals $latex L=I_1\oplus\dots\oplus I_n$ if it’s the direct sum as a vector space. In particular, this implies that $…| The Unapologetic Mathematician
Let’s go back to our explicit example of $latex L=\mathfrak{sl}(2,\mathbb{F})$ and look at its Killing form. We first recall our usual basis: $latex \displaystyle\begin{aligned}x&=\begin{…| The Unapologetic Mathematician
The first and most important structural result using the Killing form regards its “radical”. We never really defined this before, but it’s not hard: the radical of a binary form $…| The Unapologetic Mathematician