Why Does Exploding Dots Work?

A few months from now, if James Tanton and his Global Math Project co-conspirators have their way, ten million schoolchildren will take a huge mathematical step from the twenty-first century all the way back to the Bronze Age: instead of using a gadget with a state-of-the-art interface (say, a telepathic smartphone that tells you the answer to an arithmetic problem when you merely think the question), these kids will solve arithmetic problems by moving counters around on boards, the way people did thousands of years ago.¹

But if you think Tanton is a back-to-basics reactionary, you’ve got him all wrong: he’s a math-Ph.D.-turned-math-educator possessed by the conviction that math can be made understandable to, and exciting for, everyone. Tanton’s “Exploding Dots” approach to precollege math is designed to bring illumination and joy to a subject that students all too often associate with mystery and misery, and the Global Math Project’s aim is to carve out one week each year (“Global Math Week”) from the grade-K-through-12 academic calendar, in which every student gets a chance, if only for an hour, to experience that illumination and joy.

What is Exploding Dots? One answer is, it’s a dynamic way of doing calculations using representations that have immediate visual appeal, in such a way that all the intermediate stages in a calculation have meaning. Another answer is, Exploding Dots is a perfectly poised halfway-point between kids’ very tangible native ideas of number (counting sheep, fish, cookies, etc.) and the algorithms students are taught to use when computing with numbers.

But that’s all just a lot of words. If you don’t know what Exploding Dots looks like, stop reading right now and watch this short (wordless!) teaser video:

 

 

(For a longer intro, with actual words, you can view the first video at the Exploding Dots homepage. Or, check out the book “The Magic and Joy of Exploding Dots”.)

My essay this month will be about why Exploding Dots works, interpreting the question as both a mathematical one (why does it give correct answers?) and as a pedagogical one (why does it succeed in the classroom?). And if you don’t know what Exploding Dots is, that’s okay; I’ll lead you up to it by degrees.

MONEY

Let’s start by considering a student who has two dimes (10-cent pieces) and twelve pennies (1-cent pieces). She wants to know how much money she has in total, so she trades ten of her pennies for a dime. Now she has three dimes and two pennies, and she announces that she has thirty-two cents.

Would a classmate furrow his brow in puzzlement and wonder why that “method” gave the right answer? Probably not. Money has value, and he knows that the value of ten pennies is the same as the value of one dime. So two dimes plus twelve pennies has the same value as three dimes plus two pennies.

Now let’s make the scenario just a little more complicated. Suppose the first student instead starts with twelve dimes and twelve pennies. There are two different procedures she could follow. She could first trade ten pennies for a dime (yes, I know they should be drawn different sizes, but work with me here), and then trade ten dimes for a one-dollar bill (or, as shown in the picture for graphical uniformity, a schematic one-hundred-cent coin), ending up with one dollar, three dimes, and two pennies:

Or: she could first trade ten dimes for a one-dollar bill or coin, and then trade ten pennies for a dime, ending up (again) with one dollar, three dimes, and two pennies:

Schematically, here’s what the two calculations look like, visually depicted as alternative paths:

We’ve got a “question” (represented by twelve dimes and twelve pennies; “How much is it worth?”) and an “answer” (represented by a dollar bill, three dimes, and two pennies; “That’s how much!”), and two different paths that lead from the question to the answer, with each path involving an intermediate step. If you agree that trading ten pennies for a dime, or trading ten dimes for a dollar bill, is a fair trade, then you shouldn’t find the steps mysterious: each involves a fair trade that doesn’t change the total value of the coins.

Nor should you find it surprising that the two paths converge on the same final answer. Or should you? If you try a few more examples of a similar kind, using pennies, dimes, dollar bills, and ten-dollar bills, you’ll observe that in every case the process of trading in a bunch of coins for a single coin (or bill) of the same total value exhibits confluence: the process always ends, and the end-state doesn’t depend on the choices you make along the way. After one has had enough experiences like this, one’s intuition becomes so solid that it seems impossible to doubt that confluence must occur. At the end of the article, I’ll try to show how, even if one can’t actually manage to doubt this fact (true facts are the hardest ones to doubt!), there are ways to challenge one’s intuition, to interrogate it, and to learn something useful in the process. (Hint: What if we add 25-cent pieces to the game?)

DIFFERENT NAMES

Instead of telling our story of the two students in terms of different collections of coins that have the same total value, we can tell the story in terms of different English words and phrases that describe the same number. We might say that the student with twelve dimes and twelve pennies starts with “twelvety-twelve” cents, and that she figures out that twelvety-twelve is another name for one hundred and thirty-two. (This may strike you as whimsical, but “twelvety” as an alternate name for 120 isn’t something I just made up; the term appears in a manual of arithmetic published in 1854.²)

The idea that different names can refer to the same number can be hard for people to accept. This difficulty makes psychological sense; after all, the very first math skill kids pick up is counting, and the very first thing they learn about counting, even before they master the art of doing it right, is that there is one and only one right way to do it.

But getting comfortable with multiple representations of a number, and learning to choose the most convenient one for the problem at hand, is a valuable skill. An important part of anyone’s mathematical training is the principle “Switch between representations as needed”, and the follow-up principle “If the representation you need doesn’t exist, invent it!” For instance, here’s a trick I often use when managing my calendar. Say I have to figure out what day is three weeks before the first of May. I choose to think of the 1st of May as the 31st of April. Then, to go back three weeks, I subtract 21 (the number of days in three weeks) from 31, giving me the correct answer, the 10th of April. This trick works despite the fact that “the 31st of April” strictly speaking doesn’t exist.

(Do any of you have other, maybe better, examples of problems that can be solved by using nonstandard representations?)

Once we accept that numbers can have multiple representations, a new paradigm for solving arithmetic problems become conceivable: the question and the answer are just different names for some number, and we can morph the question into the answer by making a sequence of small changes, each of which transparently leaves the value of the number alone.

What’s more, under this paradigm (as opposed to the algorithmic paradigm most frequently taught in schools), there isn’t one and only one path to the answer. Just as in the “twelvety-twelve cents” example, where we saw two paths from 12 dimes + 12 pennies to 1 dollar + 3 dimes + 2 pennies, Exploding Dots has lots of ways to get from a question to the answer. (Perhaps we should call such a nondeterministic scheme a “polyrithm”?)

I think this flexibility is one reason Exploding Dots has been so warmly received in so many classrooms. Calculation, instead of being an exercise in rule-following, becomes an improvisation, a sequence of choices made within a framework that encourages trust and rewards play.

A SENSE OF PLACE

What I’ve described so far is related to Exploding Dots, but it isn’t Exploding Dots. Tanton’s approach takes us one crucial step further by bringing the concept of place value to center stage. Instead of 12 pennies plus 12 dimes, Tanton shows us 24 dots divided between two different boxes; the 12 dots in the rightmost box represent pennies (1’s), the 12 dots in the next box to to the left represent dimes (10’s), and box to the left of that is empty because there are no dollar bills (100’s), or at least not yet.

Trading ten pennies for a dime looks like this:

And trading ten dimes for a one-dollar bill looks like this:

The two ways to convert 12 dimes + 12 pennies into 1 dollar + 3 dimes + 2 pennies look like this:In Tanton’s terms, what happens is that when ten or more dots occupy the same box, we are permitted (though not required) to “explode” them, meaning, to replace them by a single dot in the box immediately to the left. Thus, a single visually simple operation, “exploding”, represents a variety of different change-making substitutions (trading ten pennies for a dime, trading ten dimes for a dollar, etc.) depending only on the positions of the boxes in question. This multivalence is helpfully reminiscent of the multivalence of digits in numerals (the two 9’s in “929” mean two different things because of their different positions in the numeral).

Exploding Dots answers questions many kids actually have but are afraid to ask, such as: Why do we add and subtract and multiply from right to left, but divide from left to right? The liberating answer is: We don’t have to do it that way! One of my favorite moments in the Exploding Dots narrative is the passage where we add numbers from left to right and still get the correct answer. If you find Dots unfamiliar and unintuitive, here, in a more traditional format, is an example of adding from left to right and getting the correct answer:

 
 1999
+2998
-----
 3
 18
  18
   17
-----
 4997

After an initial transgressive thrill, students who see this get some real understanding of why, regardless of whether you add from left to right or right to left, you’re bound to get the correct answer as long as you do the adding properly. And then, for many students, a second insight dawns: they’ve been taught to add from right to left not because one has to, but because it saves space.

The goal of saving space collides with the goal of fostering understanding. If you want to understand why the traditional algorithms work, you’ll want to bring to light steps that are normally hidden from view; after you’ve excavated those hidden intermediate results, Exploding Dots is the perfect armature on which to hang them.

I should mention here that Exploding Dots owes a lot to earlier practices in math education, such as chip trading; for a fuller discussion see my earlier essay on “The Global Roots of Exploding Dots”, or watch the video of my presentation at the Global Math Week 2017 kickoff.

OFF-BASE?

One fan of Exploding Dots, Keith Devlin, writes: “Exploding Dots does nothing for the expert. But for the learner, it can be huge. It does nothing for the expert because it represents on a page what the expert has in their mind. But that is why it can be so effective in assisting a learner arrive at that level of understanding!”

By “experts”, I’m assuming Devlin means not just mathematicians, but all sorts of people who work with numbers. However, if (misreading Devlin for argument’s sake) we take “expert” to mean “mathematician”, then Devlin’s encomium become reminiscent of some of the words of praise that were initially heaped on the New Math of the 1960s by well-meaning educators who relished the chance to teach kids to think the way mathematicians think. And since the New Math crashed and burned, we ought to think about how Exploding Dots is similar, and how it’s different.

One much-parodied innovation of New Math was the distinction between numbers and numerals. Another was the idea of writing numbers in different bases. Regardless of which side you take in the “math wars”, you’ve got to admit that Tom Lehrer’s parody of New Math is funny:

 

And in fact Exploding Dots does (implicitly) highlight the difference between numbers and symbolic representations of numbers, and it does traffic in base two, base three, and so on. So skeptics may wonder “Haven’t we been down this road before?”

In the problem that Lehrer put to music (taken from a real New Math text, I believe³) the student is asked to subtract 173 from 342, where both numerals are to be interpreted not in the ordinary base-ten way, but rather in base eight (“just like base ten if you’re missing two fingers”, Lehrer jokes). This change in the meaning of numerals threatens to overturn much of a student’s hard-won expertise in the ways of arithmetic. No wonder many students confronting such a problem feel frustrated. Exploding Dots does an end-run around that frustration by initially downplaying the fact that base eight is a way to represent numbers on a par with familiar base ten. Instead, the students start by playing a game with pictures of dots in boxes and with rules like “You can trade eight dots in one box for a single dot in the box to the left”. Students have no experience with games like this, and no expectations of themselves; it’s a level playing field, and nobody gets “Exploding Dots anxiety” because it doesn’t look like math yet — it looks like fun and games with machines and secret code-names for numbers. Only later do they realize that these Dots are covert bearers of numerical information, and that in a sense, many of the algorithmic processes they’ve learned in school are just Exploding Dots processes without the Dots.

But still, some parents might say: “My child will never need to compute a tip at a restaurant in base eight, so what’s the point? Why confuse kids about what ‘173’ and ‘342’ mean?”

There are, analogously, parents who don’t want their toddlers to learn two languages, because they fear the kids will grow up confused. But the fact is that kids who grow up learning two languages have a big advantage that monolingual kids don’t have: it’s easier for them to pick up other languages later on. Likewise, I suspect (though I don’t know if anyone’s studied this) that kids who learn multiple representations for numbers have an easier time learning more abstract math later on. Certainly there’s a conceptual link between writing integers in multiple bases in middle school and working with polynomials in high school (where kids learn that one can think of the decimal notation “929” as signifying the number obtained by evaluating the polynomial 9x²+2x+9 at x=10). The seemingly unmotivated use of different bases in the early years can help prepare the student for other concepts years down the road.

DOES EXPLODING DOTS WORK?

At this point (better late than never!) I feel compelled to admit that the title of this essay, interpreted in its pedagogical sense, makes an assumption that hasn’t been proved according to prevailing standards for evaluating teaching practice. Exploding Dots is still too new to have been tested for classroom efficacy, and in some ways I’m reluctant to see it tested — not just because I’m nervous that it will not be found to be pedagogically effective, but also because I’m nervous that it will be found to be pedagogically effective.

I’ll explain what I mean by that in a minute.

But first, let me mention my own experience teaching Exploding Dots in K-12 classrooms (hastening to say that this is just anecdote, not data). This past year, I visited both my son’s 5th grade classroom and my daughter’s 3rd grade classroom, and gave each of them “a dose of the Dots”, using examples that related to the sorts of arithmetic they were studying. I thought at the time that both classes went well; many students were engaged, calling out ideas, suggesting variations, urging me to give them harder examples.

The kids seemed to love it. But did they? My son later told me, “You know, Dad, lots of kids hated Exploding Dots. They didn’t get it, and it made them feel stupid.” I’d committed a mistake that I fear is typical of academics who dabble in K-12 teaching: I didn’t read the room. Maybe puzzled faces weren’t as numerous as my son would have me think, but maybe the puzzled kids hid their puzzlement because they didn’t want to disappoint me, or to rain on the parade of their classmates who seemed to be having so much fun. I should have taken active steps to assess how well my lesson was succeeding. Or maybe the puzzled faces were out there in plain sight and I just didn’t want to see them, because that would have required me to take stock of what was going on and slow the heck down.

My daughter’s verdict initially seemed much more like what I’d hoped for. “People loved Exploding Dots, Dad. They really liked you.” “Yeah,” I persisted, “but do you think it helped them understand addition and subtraction better?” “Oh no,” she said; “it was just fun.”

Measuring how much benefit students derive from an experience can be a tricky matter, but there are methodologies for assessing teaching practices, and now that the Global Math Project is setting its sights on reaching ten million children per year, it’s inevitable that people will try to measure how good a job Exploding Dots does at fulfilling its mission.

That’s where I get nervous. If exposure to Exploding Dots is found to have no correlation with subsequent performance in math, teachers may conclude that there are better ways for them to use class time. On the other hand, if Exploding Dots seems to be a win, Big Ed is going to try to turn it into something more like a curriculum, complete with worksheets, homework, tests, and all that.

Maybe that’ll be a good thing overall, but I can’t help feeling that something essential will be lost in the process. I think that part of what makes kids excited about Exploding Dots is precisely that it’s a mathematical side-trip; it’s fun to hear James Tanton say “Kaboom!” and “Kapow!” as he explodes all those dots, when you know that the question “What different sorts of sounds do exploding dots make?” will never, ever be on an exam. I’ve often felt that even something as exciting as a class field-trip to Oz or Narnia could have all the joy sucked out of it if educators were to devise a suitably stultifying and rigidly regimented suite of worksheets. (“Find a Talking Beast and an ordinary beast of the same species. What kind of beasts did you choose? Write down three ways in which they are similar. Write down three ways in which they are different.”) Part of the charm of Exploding Dots comes from its status as “indie” math; can it be absorbed into the mainstream without losing some of its soul?

Cartoon by Ben Orlin, @mathwithbaddrawings.

WHEN PERPLEXITY IS PERPLEXING

If you’re teaching Exploding Dots and a kid says “I don’t get it”, the student’s perplexity might mean many things. It could be that the student just needs you to repeat something that you’ve already explained that somehow didn’t get absorbed on first hearing. Maybe the student wants to see more examples. Maybe the student just wants to slow you down, to get some more time to assimilate something that actually did get absorbed but hasn’t been digested yet. Or maybe, most challengingly, what’s bothering the student is something that the student has no ability to articulate, because it hearkens forward to a level of sophistication that the student can only glimpse through a glass darkly.

I like what Ben Blum-Smith writes about this sort of situation: “When a student ‘isn’t getting’ something, we often have very little info about what’s actually happening in their mind. Often they are reasoning totally coherently but about something different than we think we’re talking about.” And I would add that even if the thought process lacks coherence, it may still possess some validity.

Here’s an analogy: You might say that a student’s question (or a question disguised as a statement, such as “I don’t get it”) indicates the presence of an itch inside the brain. With a bodily itch, we can name the body-part that itches, or at least point to it. But we often lack the words to specify the locations of the itches that affect our brains. When you’re a student in this situation, it takes bravery to admit to an itch you can’t point to. And when you’re a teacher responding to such a student, it takes a different kind of bravery to admit that you don’t know how to scratch it. In such a case, the only thing a teacher can do (but what a big thing!) is to respect the itch.

I’ll wrap up this essay with a discussion of one particular deep-brain itch that, for some students, might manifest itself as a professed confusion about why Exploding Dots works. I’ll take us back to the story about pennies, dimes, and dollars. It seems pretty intuitive that when we “simplify” a pile of twelve dimes and twelve pennies by repeatedly trading a bunch of low-value coins for a single high-value coin (whose value is the sum of the values of the low-value coins), the end result doesn’t depend on the choices we make. As I wrote above, it might seem impossible to question, doubt, or wonder about this.

DOTS MADE DIFFICULT

So now, let’s revisit that question, except this time, I’ll use nickels (5-cent pieces), dimes (10-cent pieces), and quarters (25-cent pieces). As before, the rule is, you’re allowed to replace any bunch of coins with a single coin whose value equals the sum of the values of the replaced coins. Is it obvious that the choices we make won’t effect the final outcome when all possible simplifications have been made?

Stop and think about this for a moment before reading on.

Now consider a pile containing two 10-cent pieces and two 5-cent pieces. You could trade two 10-cent pieces and one 5-cent piece for a 25-cent piece, ending up with a 25-cent piece and a 5-cent piece, which can’t be combined further.

10 + 10 + 5 + 5 = (10 + 10 + 5) + 5 → 25 + 5; done!

Or, you could trade the two 5-cent pieces for a 10-cent piece, ending up with three 10-cent pieces, which can’t be combined further.

10 + 10 + 5 + 5 = 10 + 10 + (5 + 5) → 10 + 10 + 10; done!

So here we have a situation in which different routes to “simplifying” a collection of coins
can lead to two distinct “answers”, depending on the choices we make along the way.

Now that we’ve seen that this sort of thing can occur when our coins have denominations of 5, 10, and 25, we can ask ourselves, more pointedly than before, why doesn’t the sort of failure-of-confluence happen when we use denominations of 1, 10, and 100?

One answer we can give is that the 10+10+5 → 25 process and the 5+5 → 10 process, viewed as “chemical reactions”, use some of the same “reactants”. Specifically, both of them “use up” nickels: the more nickels you use up in “making” quarters, the fewer nickels will be available for “making” dimes. This doesn’t happen when the two reactions are 1+1+1+1+1+1+1+1+1+1 → 10 and 10+10+10+10+10+10+10+10+10+10 → 100; the former reaction, far from depleting the reactants in the second reaction, increases their number.

There’s more that can be said along these lines⁴ (for instance I haven’t discussed the role of divisibility and the fact that 10 does not go into 25 evenly) but I hope you can see that there are depths lurking beneath the surface of seemingly shallow change-making games. In a sense, when you first read the “MONEY” section of this essay, you latently knew that trading ten pennies for a dime doesn’t interfere with later trading ten dimes for a dollar (and in fact facilitates it), but did you know that you knew it? And did you know that this noninterference was relevant to understanding why the procedure always leads to a final outcome that’s independent of the choices that are made?

Sometimes doing math isn’t about building upward, but about excavating downward, to see what underlies things that seem obvious. Sometimes we need to doubt things that seem indubitable, or at least trick ourselves into believing that we are doubting them (which can be just as effective). We do this because we want to find out why these things are really true.

A DIFFERENT DIFFICULTY

I didn’t mean to imply that the chemical reactions analogy (or rather the mathematical version of it, which I’m leaving unstated) is the only way to think rigorously about confluence. A different approach is to make use of the fact that each counting number has one and only representation using the digits 0 through 9.⁵ This fact is embedded in the way we learn numbers through the process of counting, so it can be hard to see what’s at stake, just as it can be hard for fish to see the water they swim in. But if we use algebra to turn the matter around and phrase it differently, suddenly what seemed obvious can seem much less clear. If we’re told that a, b, c, a‘, b‘, and c‘ are whole numbers between 0 and 9 inclusive, and we’re told that 100a + 10b + 1c is equal to 100a‘ + 10b‘ + 1c‘, how do we know that we must have a=a‘, b=b‘, and c=c‘? It’s not hard to prove this from simpler principles,⁶ but you’re unlikely to do it until you recognize that there’s something to be proved!

It’s time for more honesty here: “simpler” is a relative term. For many students, numbers essentially are numerals. The idea that the properties of numbers might be considered in a context that doesn’t start with their decimal representations, but is built upon some more abstract foundation that’s independent of how we represent numbers — Peano’s axioms, for instance — is a strange idea that won’t make sense to them until much later in their mathematical educations, if ever. This inversion of what’s “simple” and what’s “complicated”, when it happens, usually takes place in college or grad school, as a response to the changed nature of the problems a student is tackling. The inversion typically doesn’t take place in K-12, nor should it, and it would be a pedagogical mistake to make students worry about subtleties that they’re not ready for. But my point is that some of these students may already, in some fashion, be intuiting those subtleties, even if they don’t say so (because they don’t know it).

I’m often unsure what to tell a student who says “I don’t get it”, whether the “it” is Exploding Dots or something else, especially after I’ve given an explanation that seemed to satisfy everyone else. But I do know that we teachers need to recognize that a student’s dissatisfaction with a standard explanation may stem not from a mental block that needs to be removed, but from a probing nature that needs to be encouraged, and intuition that needs to be nurtured.

Thanks to Ben Blum-Smith, Sandi Gubin, Brian Hayes, Ben Orlin, Henri Picciotto, and James Tanton.

Next month (July 17): A pair of shorts.

ENDNOTES

#1: Actually, in some classes the kids will use slates, chalk, and chalk-erasers, or mini-whiteboards, markers, and marker-erasers, but the principle is the same.

#2: Likewise, you can find “eleventy” in J. R. R. Tolkien’s “The Hobbit”, though it’s unclear to me whether Tolkien the philologist was resurrecting an archaic name for 110 or importing a number-word from Old English that never got into modern English to begin with. See https://en.wiktionary.org/wiki/eleventy; and the links for discussion of twelfty/twelvety (=120) and eleventeen (=21).

#3: Does anyone know which textbook Lehrer got the problem from?

#4: The end of the story has not been written, but if you’re interested in finding out more about confluence, you might look into the theory of rewriting systems, and in particular the Diamond Lemma (aka Newman’s Lemma), which is a powerful tool for recognizing situations in which different ways of making changes are guaranteed to lead to the same final outcome.

#5: To see how this claim about uniqueness of decimal representation of numbers relates to the claim about coins, notice that if we keep trading in collections of ten coins of the same denomination for a single coin of the next power-of-ten denomination, the number of coins keeps going down, and it can’t go down indefinitely, so at some point the process must stop. That is, at some point, there will be nine or fewer coins of each denomination. If there were two different terminal states for such a process of repeated change-making, there’d have to be two different decimal numerals corresponding to the same positive integer.

#6. I know a left-to-right proof and a right-to-left proof.

The right-to-left proof starts with the observation that when you divide 100a + 10b + 1c by 10, you get remainder c, whereas if you divide 100a’ + 10b‘ + 1c‘ by 10, you get remainder c‘; so if 100a + 10b + 1c and 100a’ + 10b‘ + 1c‘ are the same number, we need to have c = c‘. What’s more, the two quotients must be equal; that is, we need 10a + 1b to equal 10a’ + 1b‘. Now we’re in the same situation we were before, but with smaller numbers. The same reasoning applied to the two new numbers lets us deduce that b = b‘, and one more turn of the screw reveals that a = a‘.

The left-to-right proof starts with the observation that if a < a‘, then (since b and c are at most 9), 100a + 10b + 1c is at most 100a + 90 + 9, which is strictly less than 100a + 100, which equals 100(a+1), which is less than or equal to 100a‘ (since a+1 ≤ a‘), which is less than or equal to 100a’ + 10b‘ + 1c‘; that is, a < a‘ implies 100a + 10b + 1c < 100a’ + 10b‘ + 1c‘. Similarly, a > a‘ implies 100a + 10b + 1c > 100a’ + 10b‘ + 1c‘. So the only way for 100a + 10b + 1c and 100a’ + 10b‘ + 1c‘ to be equal is if a = a‘. Summarizing: we’ve shown that if a, b, c, a‘, b‘, c‘ are digits between 0 and 9, and 100a + 10b + 1c = 100a’ + 10b‘ + 1c‘, then a = a‘. Subtracting 100a from the left-hand side and the equal quantity 100a‘ from the right-hand side of 100a + 10b + 1c = 100a’ + 10b‘ + 1c‘, we get the simpler equation 10b + 1c = 10b‘ + 1c‘. Now repeat the procedure, successively deducing that b = b‘ and c = c‘.

REFERENCES

Kiran Bacche, James Tanton, and James Propp, The Magic and Joy of Exploding Dots: A revolutionary concept that changes the way we learn and teach mathematics. (The essay you just finished reading appears as a preface to this book.)

Keith Devlin, “The Power of Simple Representations” (Devlin’s Angle, July 13, 2017).