There has been a major and on-going back-and-forth about the need for learning (and teaching) single-digit × facts well-enough so that students access/know/recall/[whatever verb you like] them automatically. I think the automaticity is crucially important, but I strongly believe that if automaticity isn’t approached through understanding, then whatever semblance of fluency seems to have achieved is likely to be subject to erosion due to limitations of cognitive load, unthinking obedience, brute-force memorization, post-colonial whatnot, and other factors. (I still have the scars from a traumatic experience with flashcards in 3rd grade. I remember it like it was only yesterday even though it happened about 70 years ago, but I’ll save that story for a future post—or maybe just leave it as one more painful memory.)

Anyway, once I understood multiplication as repeated addition, and knowing that addition facts didn’t change, and having gotten familiar with the commutative property of addition and extending it to multiplication, I realized that to get faster at my times facts all I had to do was practice. With that in mind I made some powerpoint animations of multiples of whole numbers from 3 to 9. I turned those into videos so people could access them more easily. They’re all short: the longest is less than 3 minutes. And they’re in 3 formats (I intend to insert some screenshots here very soon):

Just the facts (products on a 10×10 grid)
nx3, nx4, nx5, nx6, nx7, nx8, nx9

Multiples step-by-step on the number line
nx3, nx4, nx5, nx6, nx7, nx8, nx9

Multiples making tens on a 10×10 grid
nx3, nx4, nx5, nx6, nx7, nx8, nx9

In case you are wondering, I like Holey Cards for the 100-question time tests. I have my students do these every 2 or 3 days and I have them graph their own progress until they achieve something like exit-velocity. The frequency of the time tests and students graphing their own progress seems to desensitize them to any potential anxiety (“Familiarity breeds contempt.”) and make it clear that the goal was to plot and see their own progress towards 100% accuracy in <3 minutes (so that they would become fluent enough in the nuts and bolts of arithmetic), not any kind of high-stakes.

Last week I posted an overview of what (for me) is the fundamental remediation activity to do with students K-10: race games. This week’s subject is Diffies, an extension of race games, whether they involve big or small whole numbers, fractions (common fractions or mixed numbers), decimals, every number to the right of zero on the number line. The link here is a fresh revision of my how-to instructions for Diffies, the paper-and-pencil extension to hands-on race games. Diffies will remind you of the 2nd half of a race game, the race back down to 0, where students have to take away the amount shown on the dice they roll, working their way down to zero. The graphic below shows a blank diffy (click here for a pdf). Once you get a number in each of the four corners, you put the differences in between, working your way down to zero as in a race game. You keep putting the differences in between the new corners until you work your way down to zero. Game over!

Diffies are typically done by a single student, although a second student could join or help. I like rolling dice to get the numbers in the corners, but you could use Excel to get a number at random from a range you specify.

Composition and decomposition (breaking a number into pieces or taking putting numbers together) is a key skill, whether the numbers are whole numbers or fractions, the bread and butter of addition and subtraction in grades K-5, and the prerequisites of 5-8 and beyond. It really helps students and teachers if the new skills and understandings we’re supposed to teach and learn are a lot like old ones, stuff we already know. (Math is often like that—or can be, when properly taught.) I’ve been playing race games with my students and with pre-service teachers for the last four decades or so, and I try to capture what it is we do that is so engaging and instructive for everybody.

Usually I introduce students (and their teachers, if I’m doing professional development) to race games in-person, by playing a game or two but I’ve realized that I need some kind of write-up that explains how to play a race game and what I thinking and saying to my students as the game progresses. Here’s a link to the latest version of that write-up in pdf format.

Adapted from Richard Lesh’s translation model and elaborated over the years to reflect what I think makes for effective teaching of K-8 mathematics and for thinking about and anchoring math instruction at any level.

Counting, naturally, is the foundation for the whole edifice, conceptually and chronologically. Teaching and learning, we build our way up—and keep in mind where we’ve been.

To find the difference between two numbers, you can add up from the little one or subtract down from the big one: your result will be the same. When kids do diffies, they get lots of practice in additive differences—exactly what students do when they play race games—especially in the 2nd half of a race game: the race from your target number back down to zero.

This morning I finally got around to sorting through, spiffing up, and posting these PowerPoint animations that show how to do diffies. I’ve got a brief overview of diffies here. I’ll add more details later.

(Click below to download these how-to diffy animations in PowerPoint pptx)

Diffy 1-20

Diffy-2-digits (00-99)

Diffy-3-digits (000-999)

Diffy-mixed numbers with halves, thirds, and sixths

Diffy-mixed numbers with halves, fourths, and eighths

Download a blank diffy in pdf format

Lazy me! I haven’t posted anything since last year. Instead of boring you with the details, I’ll just crow about my bikes4fish channel passing 300,000 total views, with the Root Spiral of Theodorus still in the lead. Here’s what I just now grabbed from youtube.

Spurred by springtime (or something!)—maybe invigorated by my long layoff—I’ll be adding more pages to this website in the coming days and weeks, especially (freely) downloadable pdfs and slideshows of grids and activities, that teachers, parents, or whoever is interested can use with their students, children, or whomever.

Also, I’m happy to say, my Root Spiral of Theodorus pops up right at the top of Google searches for “root spiral”, “root spiral of theodorus”, and similar phrases:

Is there an animation you would especially like to see? Let me know on the Contact page.

Last week I saw that my video “Root Spiral of Theodorus” had passed 200K total views. I posted it to YouTube a little more than 4 years ago after putting it together in PowerPoint and then saving it as a video. Most of the views have been from India (over 50%), followed by the USA (at a little less than 30%), then the United Arab Emirates. If you google “root spiral”, my video comes up right away, just below Wikipedia’s explanation. I’m glad it’s so popular, but I’m not sure why. If you have what you think is a good explanation for its popularity, please let me know.

Now I’m going to try to come up with something new. If you have any ideas for what I should work on, you can let me know by using the Contact form. If you’d like to download my original PowerPoint slideshow so you can modify it to suit your own needs or whimsy, or see how I put it together, just hit the Animations tab.

My bikes4fish youtube channel has 339 subscribers (as of this afternoon). I haven’t submitted any new content in awhile, but I’m working on a few more animations that I hope you will like when I “release them into the wild” sometime soon.

This morning when I checked the youtube stats for my youtube channel, I saw that the Root Spiral of Theodorus had hit 100K views.
Here’s a screenshot that shows the most popular videos:

Total views are over 160K.