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Making Struggle Productive – Henri's Math Education Blog
At the start of the opening chapter of our new book (There Is No One Way to Teach Math), Robin Pemantle and I argue that teachers should learn to embrace contraries. The idea is that instead of choosing one or another option in a binary pedagogical choice, and instead of seeking a supposedly happy medium…
Another Use for the Lab Gear – Henri's Math Education Blog
I am the creator of the Lab Gear, a manipulative environment for learning algebra. It is more expensive than the better-known algebra tiles, in their various versions, but it is much more effective in the classroom. At least that is the opinion of those teachers I know who have used both. As one of them…
Tangram Puzzles – Henri's Math Education Blog
When I was a beginning K-5 teacher in the 1970’s, my father-in-law introduced me to pentominoes. It was clear to me that there had to be some way to incorporate those in my teaching. It wasn’t long before I instituted a weekly "math lab" session for my students: every Friday, they had an opportunity to…
Can Bots Teach Math? – Henri's Math Education Blog
For many years, Dan Meyer has been challenging Sal Khan’s attempts to improve math education through technological innovation. In the latest episode of this ongoing saga, Dan concedes that bots may be helpful with very small items, but he correctly points out that they cannot support students' very big questions. I encourage you to read…
NCTM on Data Science – Henri's Math Education Blog
The National Council of Teachers of Math issued an official statement about data science. It is a Joint Position of NCTM, the National Science Teachers Association, the American Statistical Association, the National Council for the Social Studies, and the Computer Science Teachers Association. It is a short document, and you should read the whole thing,…
Proof in High School – Henri's Math Education Blog
Proof in Geometry Many years ago, when I was still teaching high school, I added a Teaching Proof page to my website, which included a bit of philosophizing and links to the relevant parts of the site. If you’re looking for ideas and materials on this topic, you should definitely check out that page. I’ve…
Order! – Henri's Math Education Blog
You can now order my new book, There Is No One Way to Teach Math (co-authored with Robin Pemantle). I wrote about it in my previous post, where I forgot to mention that I link to many resources throughout the book. If readers want to know more about a certain tool or technique, they will have…
The Three Triangles – Henri's Math Education Blog
I recently added a guided inquiry worksheet on my website’s Pythagorean theorem page. It leads students along a proof of the theorem based on similarity. It is called The Three Triangles. This is a worksheet I used and honed over the years when I taught geometry. Upon rereading it, I thought it would make a…
There Is No One Way! – Henri's Math Education Blog
This is the cover of my upcoming book about math pedagogy. Art teacher / artist Briana Loewinsohn created the image, based on this photo I came across ages ago: “There is no one way” has long been the slogan for my website and blog. It was inspired by the story of the zen master who…
Working with Pentominoes – Henri's Math Education Blog
This is an update of a post from 2013, when Didax published my book Working with Pentominoes. You can still buy the book. It is geared to grades 4-8, though I used some of the content in high school. Pentominoes are a geometric puzzle, a staple of recreational mathematics. This is what they look like:…
A Construction Unit – Henri's Math Education Blog
I have written about geometric construction a number of times on this blog, and on my website. I outlined my philosophical outlook on this topic here. Today I summarize some construction activities, a subset of the unit you can find on my website. —————————————————————————————————— Geometric construction with straightedge and compass is a standard topic in…
Teacher-Created Materials – Henri's Math Education Blog
In this post I would like to discuss the place of teacher-created materials in the big picture of math curriculum. In some ways, the topic is of limited importance. Teachers in the US do not have a lot of time and energy for this once they are done with lesson planning, paper grading, and assorted…
Functions from Tables – 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 electronic graphing technology, and in part by the realization that important curricular ideas should be approached in more than one way. As is often the…
Order of Operations – Henri's Math Education Blog
A guest post by Rachel Chou I recently saw this in my Facebook feed: I had many questions: Why is this posted in a facebook group called “Grandma’s recipes?!” 462,000 people cared enough to leave a comment? 6,900 people cared enough to re-share it? Most importantly: I hope no one is posing this question to…
Simplify? – Henri's Math Education Blog
We ask students to simplify expressions in various arenas: fractions, order of operations, radicals, and no doubt other topics I’m not recalling right now. What is the purpose of this? When is it appropriate? How much is too much? In my in-depth analysis of the Common Core Standards for Mathematics (CCSSM), I wrote approvingly of…
Geometry Labs Notes – Henri's Math Education Blog
In this post, I share thoughts about my Geometry Labs (free download), and complement some of Mimi Yang's notes about it. (In fact, this post was suggested by the existence of Mimi’s notes. You may find those useful, as she lists which topics are supported by various labs.) First, big-picture thoughts. The book includes obviously…
Manipulatives – Henri's Math Education Blog
Liz Caffrey is a master middle school math teacher in the Boston area. She is the author of last year’s Lab Gear, the Great Connector, a guest post on this blog. Today, she shares her big-picture thoughts about manipulatives. I inserted some links to pages on my website — you can find many more links…
Fractions – Henri's Math Education Blog
Last weekend, I shared my thoughts about teaching fractions with teachers of grades 3-5 at the Asilomar meeting of the California Math Council. After decades of work in high school, and hundreds of presentations to teachers of grades 7-12, this was a bit of a departure from my normal routine, and somewhat anxiety-provoking. The reason…
Factoring Trinomials – Henri's Math Education Blog
A recent online conversation got me thinking about the factoring of trinomials. To start with, I would like to step back, and think about why this topic is prominent in the teaching of algebra. In the age of computer algebra systems (CAS), factoring trinomials is not an important skill, except of course for the purposes…
What belongs in Algebra 2 – Henri's Math Education Blog
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 fractions. For various reasons, I only attended two other sessions. The first was by Eric Muller, who works at the Exploratorium, San Francisco’s…
Rate of Change – Henri's Math Education Blog
I often promote the idea that if a concept is important, we should teach it more than once, and preferably in more than one way. Rate of change is one such concept. It can be approached in various ways from middle school to calculus. Is there anything to add to the oft-repeated “rise over run”…
PCMI – Henri's Math Education Blog
In 2006, I attended the Park City Mathematics Institute. I shared some of the work I did there on my website. Here is some information about this summer’s program, from the amazing Peg Cagle. -- Henri —————————————————————————————————— An immersive, three-week residential program, the Park City Mathematics Institute / Teacher Leadership Program provides participating secondary math…
Seen Elsewhere – Henri's Math Education Blog
Today, I share some materials not from my website — though not unrelated! Robert Wirtz As a young elementary school teacher and math specialist in the 1970’s I came across Robert Wirtz’s materials, and was inspired by them. Here is an example, the one I call “Wirtz’s flags": The idea is to draw vertical lines,…
Classroom Basics – Henri's Math Education Blog
Last January, I addressed a blog post to a “young teacher”. Actually, it applied to beginners — whether young or not so young. The post was largely about big-picture thoughts on becoming a math teacher. It generated a thoughtful and heartfelt response from another veteran educator. If you’re a beginner, I encourage you to check…
Course Evaluations – Henri's Math Education Blog
I’ve written at length about assessment — the assessment of students by teachers. Today, I turn around and focus on how to maximize the benefits of student feedback in course evaluations. Early in my high school teaching career, I was given a form that my students were supposed to fill out. It had a long…
Lab Gear, the Great Connector – Henri's Math Education Blog
I had the good fortune of mentoring Liz Caffrey for five years, at the very beginning of her teaching career, when she worked at the high school where I chaired the math department. She is now a veteran and talented middle school teacher in the Boston area, but we’ve stayed in touch. In this guest…
Project SEED – Henri's Math Education Blog
Like most teachers, I learned the craft on the job, with the help of colleagues. The formal training I received in college was not that useful, with one exception: my short stint as a Community Teaching Fellow (CTF) when I was a math graduate student: I got paid as a Teaching Assistant, but my teaching…
Mentoring Young Teachers – Henri's Math Education Blog
In a recent post I addressed young teachers. Today I write to their mentors. (If you’re a mentor, you should probably read both posts.) I chaired the math department in a small private school for almost 30 years. When we had an opening for a math teacher, the head of school made the final decision, but I was asked to…
Dear Young Teacher – Henri's Math Education Blog
Dear young teacher, I am retired from schools, after 42 years in the classroom, grades K-12. In the second half of my career, I ended up mentoring a number of young first-year teachers. In general, that went very well, so I thought I would share some ideas about what made it work. Also, I participated…
Making a GeoGebra Slide Show – Henri's Math Education Blog
I’m a long-time user of interactive geometry software, of which the dominant instance these days is GeoGebra. Here are some ways it’s enhanced my teaching over the years. Most obviously, it provides an environment for students to explore geometry and geometric construction. I’ve written much about it on this blog, and shared some curriculum on…
We need to review! – Henri's Math Education Blog
It is a serious mistake to present important concepts only once, and move on. Most students need extended and repeated exposure to challenging ideas before those sink in. As I became more and more aware of this in the course of my decades in the classroom, I started to think in terms of a “preview…
Technology in Math Education – Henri's Math Education Blog
Technology influences both the content and the methods of math education, but the impact is slow and gradual, not sudden and dramatic. This is in part because it takes time for technology to reach the classroom, but it is especially because school and societal culture develops unevenly. In this post, I think about some specific…
For Algebra – Henri's Math Education Blog
The Atlantic published an excerpt from Temple Grandin’s latest book (Visual Thinking). They titled it “Against Algebra”, which puts it in a tradition of anti-algebra pieces in various magazines and radio programs (!). Alas, anti-algebra ideas are also present among some math educators. I have written about this repeatedly: Technology in Math Education (2022) My…
Polyarcs in the Classroom – Henri's Math Education Blog
You may be familiar with polyominoes, the figures that can be made by connecting unit squares, edge to edge. For example, these are the tetrominoes, each one made of four unit squares: Polyominoes are an example of polyforms. I discussed polyforms in this article: Geometric Puzzles in the Classroom. In particular, this is where I…
No Best Way – Henri's Math Education Blog
In 2020, I wrote No One Way, a blog post which I used to explain my website’s motto (“There is no one way.”) I argued that it is the math itself that demands that we approach important topics in multiple ways. As it turns out, this is a favorite topic of mine: in 2016, I…