Seven months have passed since my last newsletter! This is the longest gap between issues in the seven-year run of the newsletter. The main reason is that summer travels (both work and vacation) severely limited my ability to blog and to add materials to my Web site.

Anyway, I'm back in the Bay Area. If you live around here, I may see you this weekend!

## Blog Posts

Here are links to posts on my Math Education Blog that you might find interesting.

If you are so moved, you may comment on the posts, and/or subscribe to the blog.

### Understanding "understanding"

As you probably know, I'm an advocate of "teaching for understanding". But what does that mean? "I know it when I see it", you might reply, and I believe you, but nevertheless, it is useful to be more specific. I list some easily observed components of understanding in this post. The post quickly became one of the top three most-visited on my blog in the past year. (Which is not saying much, admittedly, but still.) I'm hoping the list can help you improve your teaching and assessment strategies. It certainly has helped me develop worthwhile activities as a curriculum developer.

### More on extending exposure

A recurring theme of this newsletter (and my blog) is how to address the fact that students learn at different rates. The question, of course, is fundamental to teaching math effectively at any level. Unfortunately, many of the standard responses have problematic side effects. If schools heed the "detracking" advice in NCTM's *Catalyzing Change* document, the question will gain even more urgency. A full answer is of necessity multidimensional, but one component I've been promoting is *extending exposure*, a set of practices that rearrange how things are sequenced without taking more time. This includes lagging homework, separating related topics, and as I suggest in this new post, pursuing two units at the same time.

### Catchphrases

I have come up with various slogans over the years, mostly aimed at teachers. I decided to list them in a blog post. They include the ever-cheerful "Nothing Works!", the cryptic "Fast is slow, and slow is fast!", and one so long that it barely qualifies as a catchphrase: "Formulas and tricks should encapsulate understanding, not substitute for it." See the whole annotated list here. (A day or two later, I remembered some slogans of mine that I hadn't included. Maybe in a future post!)

## MathEducation.page

Some articles and curriculum materials, free downloads on my Web site.

### Letters and Postcards

An introduction to linear inequalities in two variables, and linear programming. Two versions: middle school, and high school. Start here. If you create a Desmos version of this, let me know!

### New GeoGebra Applets

Creating a sine curve is an original animation suggested by Rachel Chou, along with some questions to trigger reflection, discussion, or writing.

Representations of a trinomial (symbols, graph, Lab Gear — manipulable) This is a conversation starter or writing prompt to consolidate what was learned by other means in middle school or in Algebra 1.

Iterating Linear Functions. What happens when you use the output of a linear function as its next input? This supports some curriculum materials available here. (Algebra 1, Algebra 2, Precalculus)

More advanced: iterating f(x) = rx(1 – x), in two representations. Chaos lurks! (Part of my Infinity course.)

### Referenced On Twitter

Here are links to various items on my Web site that were referenced on Twitter in the past few months (in reverse chronological order). I'm guessing that if they were useful to the tweeter, they may be useful to some of you!

- Manipulatives for middle school and high school
- Pattern block dodecagons
- Teaching ("philosophical" articles)
- Function Diagrams: a parallel axes representation
- Geometric Construction: a sequence of puzzles to support geometric understandings.
- Kinesthetic activities for algebra and geometry
- Lab Gear: the best algebra manipulatives!
- Difference of Squares (animation to help visualize the algebra)
- Make These Designs (probably my most popular activity, at one time)
- Geometry Labs (frequent mentions)
- The perils of hyper-acceleration: much sooner is not usually much better
- π for regular polygons (though the slides don't seem to work on all browsers)
- Taxicab Geometry: a fun exploration for teachers.

As if you had time for all this! I'll stop now.