My early forays as a curriculum developer date back to my days as a K-5 math specialist in the 1970’s. A key insight of my young self was that activities intended for students were that much more worthwhile if they were also interesting to me. I learned to view with suspicion activities that were boring to me and other adults, but supposedly good for children to slog through. Thus started a career-long search for low threshold, high ceiling problems and puzzles. (That way of describing such activities originated in the 1980’s Logo movement, but that is another story.)

Among my first creations were pentomino puzzles whose difficulty spanned the whole range from Kindergarten to adult. (If you don’t know what pentominoes are, visit Geometric Puzzles in the Classroom before reading further.) Those puzzles were collected in books and cards that remained in print for an astounding 30 years or so. Alas, those are no longer available, as they have fallen victims to test prep obsession, to the gobbling up of smaller publishers by mega-corporations, and to the latter’s lack of interest in actual children’s education. I have now made those puzzles available online on my Web site, as I did for the not-unrelated SuperTangram puzzles*.

Anyway, what prompted this post was this message from Meghan Lee, a teacher in Santa Cruz:

I’ve been using your website like crazy this year. I’m teaching Geometry and used your transformation materials, and mixed in a little bit from your Space class. I used many of your puzzle problems to create a presentation project: each group had a different spatial/area puzzle including the polyarcs. One particularly artistic student drew these, which I thought you’d appreciate.

Polyarcs were my own invention, which you can read about here. Here are Aviva’s polyarc creations:


Fun stuff! I’d love to see other student polyarc art! Start them off with this worksheet. (Prerequisites: they need to know how to find the area and perimeter of a circle.)

Meghan’s project assignment also included these options, among others: Tiling Rectangles with Polyominoes, and various SuperTangram puzzles*.

Back in the day, I used geometric puzzles in my elementary school classes as activities in a weekly “math lab” session, and as menu options for students who finished other work early. The logistical key was that each student had a folder which listed all the options, and where they checked off the puzzles they had successfully solved.


* Alas, I am the only source of plastic supertangrams. If you sell math manipulatives and are interested in selling them, get in touch! Likewise if you want to buy some sets from me.

Follow up: see Meghan’s 2022 guest blog post about polyarcs.

7 thoughts on “Polyarcs”

  1. Here is a short version:Logo was a computer language that swept through elementary schools in the early days of educational computing. Seymour Papert, the MIT professor who was the founder and leader of the Logo movement, was a prophet of educational transformation through technological change. His utopian vision has failed to materialize, to say the least, but for some of us it triggered a fundamental shift in perspective. The concept of “objects to think with” (in his book Mindstorms) readily expands to “objects to talk and write about”, and is in a lot of ways foundational to my tool-rich educational vision.Logo was an environment that empowered students in multiple ways: it offered students opportunities to experiment, to explore and solve problems their own way, to set their own goals and ask their own questions. Starting on day one, all students were able to achieve interesting results. That was dubbed “low threshold”. But Logo was a full programming language, meaning that just about any project a student could conceive, they could pursue. “High ceiling”! (Or even “no ceiling”!)The fact that the environment was highly visual and geometric was a plus, and it opened up opportunities to learn a lot of geometry, from very basic ideas about angles, to college-level math. (See the amazing book _Turtle Geometry_, from MIT Press, by Abelson and DiSessa.)Logo has many descendant languages. Scratch is probably the best known. It added tremendous possibilities for animation and more, but unlike Logo, it is not (nor is it intended to be) a full computer language. Snap (from UC Berkeley) is directly inspired by Scratch. I can do everything Scratch can do, but it makes it possible to program anything — no ceiling!Anyway, when the “low floor, high ceiling” language took off with Jo Boaler's help, the elders among us know where the phrase came from!


  2. This is fascinating. I knew about Logo, but I didn't know that 'low threshold/high ceiling' originated there.It doesn't need to be you, but someone should write a history of math manipulatives. I'll read anything about how you invented all the tools you invented, their relation to what came before or after, etc. Seriously: anything. It's fascinating stuff. Someone should just get you and Marilyn Burns in a room with a voice recorder, some coffee, and lock the door until all this history has come out.


  3. I've written a little bit on the history of algebra manipulatives:http://www.mathedpage.org/manipulatives/alg-manip.html(Scroll to the end.)Other than the Lab Gear, I've been mostly a user, not an inventor, though as you know, I did come up with many clever uses for the geoboard, the circle geoboard, and pattern blocks, in addition to the puzzles mentioned in this post. Some of my ideas were ground-breaking, e.g. geoboard squares leading to the Pythagorean theorem. Also: I helped create great activities for Zome, under George Hart's leadership. (Zome history can be looked up online.)EDC invented pattern blocks, and created tremendous sets of mirror puzzles, and tangram puzzles, all of which really shaped my curricular aesthetic back in the 1970's. And yes, Marilyn knows a lot of the early history! I was already a fan of hers in 1975!


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