I designed this for teachers who prefer a visual summary of the key ideas on pronoun use for transgender and nonbinary students. I also have longer text-based pieces for those who want more details: Non-Binary Students; Non-Binary “They” and Style Guides; Non-Binary Students and Pronouns

(Infographic last revised 1/16/21.)

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I designed this for teachers who prefer a visual summary of the key ideas on degendering language. I also have longer text-based pieces for those who want more details: Non-Binary Students; Non-Binary “They” and Style Guides; Non-Binary Students and Pronouns

A separate issue: “Parents” can be problematic because not all of your students will have or live with their parents. So please be mindful of that as well.

(Infographic last revised 1/16/21.)

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An open tool box showing some tools
An open tool box showing some tools
Photo by Tekton on Unsplash

This is a common criticism of Common Core (CCSS): It offers these strange new methods that students must use.

Except… only the first part of that is true. CCSS does offers some new strategies, but it doesn’t say that students have to use them.

This article isn’t a defense of CCSS, by the way. It’s far from perfect; it has plenty of problems. But one of its problems isn’t dogmatic adherence to specific methods.

So why do people think it does that? That’s been simmering in the back of my mind for a while, but it’s finally coming forward: Because…


Four quadratic functions on a coordinate plane.
Four quadratic functions on a coordinate plane.

When I was in school, I was taught the Quadratic Formula. I was taught that it was the most efficient, more reliable way to find the roots of a quadratic function.

This is what I was taught: Given a function in Standard Form, ax² +bx+ c, its roots can be found by evaluating:


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Photo by Jon Tyson on Unsplash

I’ve been thinking a lot lately about my name.

My legal name is “Paul”. It’s the name I’ve worn most of my life.

When I was too young to remember, I was Timmy. That was based on my middle name, which is Timothy. My older brother went by Mark, his middle name, and I went by Timmy.

At some point, he switched back to his first name, John, and I switched back to my first name, Paul. I was too young at the time to remember why.

When I was in school, I wanted to be a writer, and I…


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Photo by Brian Stalter on Unsplash

I have taught high school mathematics for nearly a decade. I have a BS in Mathematics. The Algebra II curriculum, which I largely built for my school, is based on “the story of functions”.

And yet, it was only the other day that I noticed something that was woefully wrong about the way that I’ve been presenting functions.

Mathematics education is filled with convenient half-truths. When we introduce division, we try to ignore what happens with the remainder. We call it the number “line” even before we introduce either negatives or non-integers (the positive real numbers are, technically speaking, a…


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Photo by Ross Findon on Unsplash

I’ll keep this one short. Also, it’s on calculus, for whatever that’s worth.

The Power Rule for Differentiation says that the derivative of a monomial ax^b is abx^(b-1). Last night I noticed a way to derive this for positive integers that I believe I’ve seen before (so I’m not claiming originality), but which is different from the standard way of doing so.

First of all, though, the context: I wasn’t even thinking about calculus! I was thinking about the standard linear equation, which is typically written in the United States as y = mx + b.

I have a longer…


A Guide for Public School Educators

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Photo by Sharon McCutcheon on Unsplash

In an earlier article, I discussed binarist language and provided some ideas for avoiding it with your students. In this article, I’m focusing specifically on pronouns.

First of all, though: Thank you for taking the time to read what follows. For many people, the increasing visibility of non-binary persons and the accompanying language can seem overwhelming. However, when you help create a climate that respects marginalized students, studies have shown that all students feel safer and respected.

If you’d like to learn more about creating a school environment for LGBTQIA students, or create a GSA (Gender and Sexuality Alliance) at…


Don’t get caught up on the concept of “fractions”.

A photo of a portion of a mathematics book defining “fraction”.
A photo of a portion of a mathematics book defining “fraction”.
Photo by the author of Milne’s 1893 “Elements of Arithmetic”

There is one topic students of mathematics consistently struggle with, to the point that it has become legendary: Fractions.

I teach Algebra II. Fractions don’t exist.

I’m not saying, of course, that 1/2 and 5/31 aren’t things that might occur. I mean that I encourage students to stop obsessing on “fractions” as an isolated concept.

The History of Division Notation

In “Elements of Arithmetic” (1893), William J. Milne writes, “(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.”

Let’s examine the origin of the ÷ sign itself. My primary…


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Photo by Arash Asghari on Unsplash

I’m going to build a simple set of functions. This set will wind up being very familiar when I’m done, but let’s pretend for a few minutes first.

Function and Operators

In case you’ve forgotten what a mathematical function is, there are a few quick ways to refresh your memory. A common way is to say that a function is a machine that takes some input and spits out a predictable output. The output may change with the input, but the key is: If you put a specific thing into the machine, you can always predict what you’ll get out.

More formally speaking…

Clio Hartzer

They/them and Mx. I think and write often about social equity. I teach high school mathematics. I have a bunch of articles on The Good Men Project.

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