Henri Picciotto's

Math Education Newsletter

April 2021

Online Professional Development Workshops This Summer!

Hello everyone,

Only six weeks have passed since the previous issue of this newsletter. The reason I'm sending this one off so soon is to announce my summer workshops. Because of Covid-related uncertainty, the workshops will be online. Without the in-person time, this will be a different experience. But on the positive side, it will be less expensive, and you can attend from wherever you are!

Scroll down for more information, or go to the Summer Workshops page on my website.


New on my Blog and on

Before my 32 years in high school math, I was a K-5 teacher and math specialist. That background informed much of my subsequent work in the classroom and as a curriculum developer. It is also the source of many of the ideas in this newsletter.

Seeing is Believing?

I welcome the increased availability of beautiful computer animations of math concepts. They are often elegant and satisfying — the contemporary manifestation of so-called "proofs without words", which are popular among mathematicians and math teachers. However these very animations can set a pedagogical trap: they are most effective with an audience that already understands their content. For beginners, they work best if they are preceded with some preliminary work, and followed by discussion or writing to supply the missing words. "There is no royal road to geometry", and there is no shortcut to understanding: students must engage intellectually, not watch passively. I discuss all this in a blog post.

Fraction Rectangles

Fractions, of course, are difficult. When teaching 4th and 5th grade in the 1970's I struggled with this, and came up with a powerful learning tool: fraction rectangles. The idea is that it is much easier to think about (say) `2/3` and `4/7` when looking at a 3 by 7 rectangle.


I discussed this in great detail on my website in 2013: Fraction Arithmetic on Grid Paper. Last year, I added explanatory videos to that page, as I'm told that these days "people expect that". Finally, a couple of weeks ago, I created Fraction Rectangles, a bare-bones GeoGebra applet as a support for this approach. (I used it to make the above figure.) I was tempted to make the applet more powerful and easier to use, but that would have undermined the basic pedagogical concept here: this approach is intended to empower the student, not highlight my GeoGebra prowess. So I decided to facilitate the drawing of the figures, and not go beyond that. Read more about this on my blog.

Pentomino Puzzle Books


Also in the 1970's, I had a weekly "math lab" in my class, during which students would solve assorted geometric puzzles, including many pentomino puzzles I created. In the 1980's, this led to my first publications: a series of books, starting with three pentomino puzzle books. Those remained in print for decades. They are now available as free downloads on my website. A new edition of a companion book (Pentomino Lessons) was published by Didax in 2013 under the title Working with Pentominoes. That book makes the curricular connections explicit, and works well as a complement to the three pentomino puzzle books.

Popular Pages?

Every month, I get an email message from Google with an update on my site's "analytics", as they put it. One curious fact is that month after month after month, these two PDF's are among the "top performing pages", among the top three, in fact:
From Factored to Standard Form (quadratic functions in Algebra 2)
Skidding Distance (an application of the square root function)
Hundreds of people visit those every month. How could that be? Surely those are not the best items on the site! Could it be a Google Analytics bug?
In case you're curious, here are the other recent "top performers" according to Google:
January: V-Shaped Graphs (good one, especially the first three questions)
February: Henri Picciotto's Math Education Page (the site's front page)
March: Virtual Pentominoes (big fun!)

Online Summer Workshops

Symmetry: a unifying thread across grades and cultures.

with Henri Picciotto

Symmetric designs are found in virtually every culture, and are interesting to students across the grades. We will use that as a foundation for lessons about essential and enrichment math topics for grades 6-8. This approach will allow us to build on student creativity while furthering their visual sense and their mathematical growth.

When: July 19-23, 2021 — 1:00-3:00 pm EDT / 10:00-12:00 pm PDT
More info: This workshop is hosted by the Atrium Summer Math Institute: Flyer | Website
The Institute will offer two other workshops, both of which sound good! (Liz Caffrey: Designing Projects that Challenge and Contextualize. May Vang: Counting with Purpose: Teaching Social Justice and Math.)
Registration: (early bird rate until May 1) click here.

No Limits! (Algebra 2, Trigonometry, and Precalculus)

with Henri Picciotto and Rachel Chou

A challenge in teaching upper level high school classes is the limited pedagogical range of most curricula. This is particularly harmful to the students who find symbol manipulation difficult, but it is also cheating our stronger students of the multi-faceted understanding that would serve them best. We will present a number of activities to complement the corresponding lessons in any textbook with the intelligent use of electronic tools (Desmos and GeoGebra), creative alternate representations, and problem-solving throughout.

When: August 2-6, 2021 — 1:00-3:00 pm EDT / 10:00-12:00 pm PDT
More info: The workshop is hosted by Menlo School, where Rachel Chou chairs the Math Department. Flyer.
Registration: Opens on April 14. Pre-register.

Scholarships: 75% discount for a limited number of public school teachers at each location (first come, first served).

Questions? Go to my Summer Workshops page, and/or email me.

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