Posts

Why We Should Never Tell Students What They Can Tell Us A student-centered approach to deeper mathematical thinking There is a deceptively simple idea in teaching that can fundamentally reshape classroom practice: never say anything a student can say. This principle comes from Steve Reinhart’s article Never Say Anything a Kid Can Say! (2000), and it challenges teachers to resist the urge to explain, clarify, or summarize when students themselves are capable of doing that intellectual work. When we connect Reinhart’s insight with Robert Kaplinsky’s instructional reflection and Dan Finkel’s TEDx talk, Five Principles of Extraordinary Math Teaching , a coherent vision of student-centered learning emerges. In that vision, students are active sense-makers and communicators, and the teacher’s primary job is to design experiences that make student thinking visible. Lead with a question Dan Finkel argues that extraordinary math teaching begins with questions worth thinking about, not ...
Post a Comment
Read more

The Queen of Mathematics: A Gentle Journey Through Number Theory

Carl Friedrich Gauss once wrote that “Mathematics is the queen of the sciences and number theory is the queen of mathematics.” Every time I teach or revisit introductory number theory, I feel the truth of that line settle in. There is something dignified, almost royal, about working with the integers—these simple, familiar numbers that somehow hide endless surprises and patterns. In this post, I want to walk through the major ideas you would encounter in a first number theory course for math majors, and share a bit of the joy behind each topic. After all, number theory isn’t just a subject to study; it’s a subject to savor. Number theory begins with **divisibility**, and even the most basic idea that “a divides b” if b can be written as a times some integer. It feels small at first, but suddenly this tiny definition opens up entire landscapes. The Euclidean Algorithm, for example, takes two integers and dances backward through divisions to reveal the greatest common divisor. There is ...
Post a Comment
Read more

Remember, Remember the Fifth of November: From Bonfire Prayer to Symbol of Revolution

Every November 5th, the night sky glows with bonfires and fireworks across the United Kingdom. Children recite the familiar words: Remember, remember the Fifth of November, Gunpowder, treason, and plot. I see no reason why gunpowder treason Should ever be forgot. This short rhyme—known as the Bonfire Prayer —dates back over four centuries to the failed Gunpowder Plot of 1605 , when a group of English Catholics, including Guy Fawkes , attempted to blow up the Houses of Parliament and assassinate King James I. Their goal was to end Protestant rule and restore a Catholic monarchy. The plan failed, and Fawkes was captured, tortured, and executed. From Thanksgiving to Tradition In the years that followed, November 5th became known as Guy Fawkes Night or Bonfire Night . The government declared it a national day of thanksgiving for the King’s survival. Communities lit bonfires, rang church bells, and later burned effigies of Guy Fawkes in public squares. The annual celebrati...
Post a Comment
Read more

Rebuilding curves (and physical laws) from their tangents: a gentle introduction to the the Legendre dual

In this post, we will explore an elegant mathematical idea that links geometry, calculus, and physics: the Legendre transform (sometimes called the Legendre dual ). The basic concept is that a smooth curve or function can be re-described entirely in terms of the slopes of its tangent lines. Even more remarkably, this process can be reversed to recover the original function. The same idea appears in classical mechanics and thermodynamics as a tool for switching between different physical variables, such as velocity and momentum or entropy and temperature. 1. Geometric Intuition with Tangent Lines Suppose we have a smooth, upward-curving function f(x) . At any point x = a , we can draw a tangent line. The tangent line can be written in slope-intercept form: y = p·x + b Here, p = f′(a) is the slope of the tangent line, and b is its y-intercept, which can be found by substituting x = 0 : b = f(a) − a·f′(a) So, every point a on the curve corresponds to a line with slope p a...
Post a Comment
Read more

The Hidden Connections Between the Totient, Sigma, Tau, Möbius, and Dirichlet Convolution

The Hidden Connections Between Totient, Sigma, Tau, Möbius, and Dirichlet Convolution In number theory, certain functions show how integers interact with their divisors. This post introduces five of them — the Euler totient function phi(n), the sum of divisors sigma(n), the number of divisors tau(n), the Möbius function mu(n), and the Dirichlet convolution — using one example number: Example number: n = 12 1) Euler’s Totient Function φ(n) Idea: phi(n) counts how many numbers from 1 to n are coprime to n (that is, share no common factors with n except 1). Formula: If n = p1^a1 * p2^a2 * ... * pk^ak, then phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk) Example with n = 12: 12 = 2^2 * 3^1 phi(12) = 12 * (1 - 1/2) * (1 - 1/3) = 12 * (1/2) * (2/3) = 4 The numbers 1, 5, 7, and 11 are coprime to 12, so phi(12) = 4. 2) The Sigma Function σ(n) Idea: sigma(n) is the sum of all positive divisors of n. Formula: If n = p1^a1 * p2^a2 * ... * pk^a...
Post a Comment
Read more

Understanding Markov Chains — A Journey Through Probability and Prediction

Have you ever wondered how computers make predictions—like when your music app seems to know the next song to play, or how weather forecasts estimate tomorrow’s conditions based on today’s patterns? Behind these everyday tools lies a mathematical idea called a Markov Chain , named after the Russian mathematician Andrey Markov , who developed the concept in the early 1900s. What Is a Markov Chain? A Markov Chain models systems that evolve step-by-step over time where the next state depends only on the current state, not on the full history. This principle is called the Markov Property : “The future is independent of the past, given the present.” To specify a Markov Chain, you need: A set of possible states (for example, Sunny or Rainy). Transition probabilities that describe how likely it is to move from one state to another in one step. Transition Matrix (2×2 Case) We often collect the transition probabilities into a transition matrix . For two states A and...
Post a Comment
Read more

What is Mathematical Fluency?

What does it really mean for students to be mathematically fluent? If you’ve been in any math PD over the past few years, you’ve likely heard the phrase everywhere. We talk about fluency as something students should develop, strengthen, and demonstrate, but it can still feel abstract when we try to describe it in observable, classroom-ready terms. This post breaks down mathematical fluency into the two simplest frames we can use as teachers: what it looks like and what it sounds like . These descriptions can guide instruction, assessment, student goal-setting, and even walkthrough conversations with colleagues or administrators. What Mathematical Fluency Looks Like In a classroom where students are developing mathematical fluency, you see students making choices about strategies rather than following steps robotically. They use representations—number lines, diagrams, tables, graphs, manipulatives, symbolic expressions—and switch between them to make sense of a problem. They move ...
Post a Comment
Read more