Teaching for Mathematical Understanding

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.

This is what a blank diffy looks like. Or you can do one from scratch on a sheet of paper. Here’s an example of one done from scratch on the back of an oversize envelope:

(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

Over 300K total views!

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.

Root Spiral of Theodorus clears 200,000 views!

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.

145K views & other news

This morning I checked in on my youtube videos at www.youtube.com/bikes4fish and was happy to see that the total views had cleared 145,000. Over 50% of the views have been from India—this year over 70% of the views have been from India. The most popular video is still the “Root Spiral of Theodorus“; in 2nd place is what I meant to be its companion, “Constructing Square Roots on the Number Line“. Formerly, the big months were October and November; now the big months are April and May. I’m wondering about the reasons driving all this, and about the fact that most of my more than 250 subscribers are from India. My best guess is that it all has to do with who’s trying to learn—or teach—what and when.

Since I mentioned wondering, a few days ago Hung-Hsi Wu told me about an excellent and thought-provoking speech by Dean James Ryan of the Harvard Graduate School of Education at HGSE’s 2016 Commencement. It’s worth sharing with you here, so here goes:

Dean James Ryan, HGSE

Ryan claims there are 5 essential questions that we all should keep in mind—plus a bonus question at the end. I’ll list them here (except for the bonus question), and include a transcript so you can read through the text and pore over it; but for starters, you should really watch and listen to his whole speech—the excerpt, actually—it’s not even 7 minutes long. Anyway, here are Dean Ryan’s 5 essential questions and his claims about them:

  1. “Wait, what?” is at the root of all understanding.
  2. “I wonder” is at the heart of all curiosity.
  3. “Couldn’t we at least…?” is at the beginning of all progress.
  4. “How can I help?” is at the base of all good relationships. And,
  5. “What really matters?” gets you to the heart of life.

Back to wondering about my videos’ popularity in India, I’m stuck on #2. If you’ve got something with better explanatory power than my rather obvious “who wants to learn/teach what/when curricular explanation, I’d love to hear it. I suppose #3, “Couldn’t we at least…?” was my motivation for making the videos in the first place: I thought it would be good to have a slide presentation (that later became a video) guiding students (i.e., James-Ryan-HGSE-5-questionspeople) slowly and surely through Theodorus’ square root spiral and another one that showed step-by-step how you could construct/locate some of the most commonly encountered real numbers—the square roots of 1…9 on the real line, all in the comfort and privacy of your own home/library/study space.

Now over 105,000 views on YouTube!

My geometry animations cleared 100K views and are closing in on 110K! Half of those views are from the insanely popular Root Spiral of Theodorus:

And about a sixth of the 100,000 + views are from its companion video, Constructing Square Roots on the Number Line:

As always, you can download the PowerPoint slideshows these animations were made from by clicking on the Animations tab. That way, you can modify them and pace them to suit your students and your style.

new adaptation of Lesh’s translation model

Here’s a new adaptation I just finished making of Richard Lesh’s translation model. (Google something like “Richard Lesh translation model” and you should have plenty to look at, including an adaptation I put together in MS Word about 10 years ago—(which you once could have found at www.soesd.k12.or.us/math, but that link is now dead.) The source I used for Lesh’s original model was Kathleen Cramer’s “Using a Translation Model for Curriculum Development and Classroom Instruction”, which appeared in a book edited by Lesh himself (2003).

I departed from Lesh’s model for a variety of reasons. I think that understanding any given thing in mathematics—whether you’re understanding it yourself, explaining it to other people, or checking to see if they understand whatever it was—is built up through an interaction of a lot of different things. If students can’t build up a concept explicitly from these various aspects but can only fall back on rote-learning or some mystical grok-it-and-rock-it incantations, it ain’t understanding: it’s a sign that they don’t understand a concept well enough to learn the next thing and may very well not be able even to remember the original jingles well enough later on to solve the problems they were made for.

I wanted algorithms sitting at the top because I believe algorithms represent great human achievements and serve us as compact “knowledge packets” (to use Liping Ma’s great phrase), which, if we actually understand what’s going on in the workings of algorithm and are not just going through the motions, can readily serve as the solid foundation for the next step in our knowledge of math.

At the base we’ve got concrete objects to work with. This is typical in early grades; later on we may tell students something I heard Hung-Hsi Wu say many times in his lectures, “Let’s try some numbers”, treating specific numbers as if they were concrete objects we could experiment on mathematically.

As the model shows, we can record what we’ve done or what we are doing by making sketches (the representations arrow going up and around the left side). We can describe what we’re up to (the descriptions arrow going up and around the right side) in natural language, i.e., everyday speech, but as our understanding advances, we need to formalize our descriptions with definitions and conventions. As we progress, representations and descriptions get more formal and iconic, leading us through tables and graphs and through symbolic expressions to an algorithm which becomes our standard operating procedure for addressing the problems that arise with this concept.

Students who are only able to do the algorithm necessary to get a correct answer, but, if questioned, can’t explain what’s they did or why, clearly don’t have a complete picture. It’s as if the pentagon and their understanding got flattened to a line segment underlining the word “algorithm”. I would suspect students showing this behavior had been trained to a degree of perfection in some little dance without meaning, maybe even accompanied by a catchy jingle, but I would expect most of them would have trouble with more advanced topics that always follow.

My solution to this—I hope you’ve guessed by now—would not be more jingles: garbled jingles are a rich source of students’ errors, especially in high school, when they may thrash cluelessly, vainly trying to remember how the catchy jingle went so they can get an answer.

Geometry animations clear 85,000 views!

Led by the Root Spiral of Theodorus with over 40,000 views, my bikes4fish channel of youtube geometry animations passed 85,000 total views on the 24th of February!

 I’m happy to add that Constructing Square Roots on the Number Line, the companion to the Root Spiral of Theodorus, passed 10,000 views one day later. I’m particularly fond of “Constructing Square Roots…” because for many students, square roots live in their calculators and they access them there, but they don’t know how to construct where square roots live on the (real) number line.

In other news, Dr. Pat Kenschaft has given me permission to start posting mp3 audio files from her Math Medley radio shows (1998–2004) featuring hour-long interviews with eminent mathematicians and math educators talking about K-12 math education and what needs to be done to do it right. Unfortunately, the links to those 300+ shows don’t work anymore, so, with Dr. Kenschaft’s permission, I’m making the ones I can find available here. Eventually, they’ll get a folder of their own in the menu bar, but in the meantime, here are interviews with Liping Ma and Richard Askey to get us started. (Dr. Kenschaft’s books are available from her Amazon Author page. I’m reading one of them, Change is Possible: Stories of Women and Minorities in Mathematics right now.)