That morning, I met with Oleg in the house in the valley at the bottom of the steep road that nobody with any sense would ever try to drive up, the one carved straight up the side of the hill because people in that part of the world had apparently never heard of switchbacks. I…
Author: Clio
A Sad Pantoum, Mine
There was something about the trees: Their burning leaves, Curling up on the ground In a black-gray fog. Their burning leaves A dark sadness, full of regret In a black-gray fog. Dead fruit rotting on the earth: A dark sadness, full of regret, As our cold tongues inspired to speak like Dead fruit rotting on…
Expressions as Names
One basic concept in mathematics that I see students struggle with, and with which I struggled myself, is the notion of expressions. However, when we remove the mathematical component, we can see that expressions behave much like a concept that natural language speakers deal with fairly easily. So let’s start there, in natural language. You…
Sums of Positive Consecutive Integers: Proof
In my previous post, I tackled this problem: Try to express positive integers in terms of the sum of two or more consecutive positive integers. For instance, 3 = 1+2, 9 = 2+3+4, and so on. For which numbers 1 to 25 is it possible to do this? I incorrectly concluded that there were no…
Sums of Consecutive Positive Integers
Edit: The bit about the larger prime numbers was due to an error in my VBA programming, but it lead to a better understanding of the problem. Don’t take this article as “final”, is the point. This week’s puzzle in my Mathematics Reasoning class: Try to express positive integers in terms of the sum of…
The Difference of Squares
In our Mathematical Reasoning class tonight, we discussed this problem: Can your age in years be written in terms of the difference of two square numbers? If so, what two numbers? There are at least three mathematical problems contained here: Given a specific whole number (0 or a positive integer), attempt to find two square…
Logarithmic notation: Mathematics vs. computer programming
One of my concerns as an educator is the way in which peccadilloes of mathematical notation can get in the way of understanding. In the case of logarithms, this has become more troublesome as general education about numeric bases at the secondary level has apparently evolved primarily into an interest of computer programmers. The notation…
Minimal Eulero
Eulero is a grid-based logic puzzle. Like Sudoku, it involves Latin squares; however, while Sudoku relies on a single Latin square, Eulero consists of two overlaid squares, one of numerals and one of letters. An additional constraint is that each letter-number combination has to appear exactly once. This got me thinking: What are the fewest…
Trigonometry as the Study of Circles
I recently read John Derbyshire’s book, Unknown Quantity: A Real and Imaginary History of Algebra (Plume 2007 edition). I recommend it overall, although the second half becomes increasingly inaccessible to the layperson. One bit that particular stuck in my head, because of the way it caused me to rethink a mathematical concept, was this passage…
The Prefix Dia-
A good teacher is always aware that there are things they can learn. One of the ways in which I encourage students to connect mathematics to other topics is by showing how words and morphemes used in mathematics are used elsewhere. For instance, “median” in mathematics refers to a number in the middle of a…