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Introducing Middle School Students to Chaos

By which I of course mean mathematical chaos--sensitive dependence on initial conditions. With the right groundwork, it can be easier than you think! We often teach middle school students about linear and exponential functions using the concept of recursive sequences, sequences in which the next term is generated from the previous one: In a linear sequence, you add a constant to the previous number; in an exponential sequence, you multiply the previous number by a constant instead. This is a great, intuitive framing for illustrating the difference between these two basic models--and curiously, a great way to introduce chaos too!

The logistic map is a simple population model from mathematical biology that pares down the famous logistic equation into a single recursive definition:

$$x_n=r x_0 (1-x_{n-1})$$

\\(x_n\\) is, roughly speaking, the population as a fraction of carrying capacity, and \\( r \\) is an abstract growth-rate parameter between 0 and 4. This is a more complicated definition than what they’ve seen before, but it really only introduces one twist on the exponential sequence: Multiplying by \\( 1-x_{n-1} \\) . We can think of this as a force of decay--the trouble a species encounters as it nears carrying capacity and runs out of resources.

During the 1970s, the logistic map attracted a great deal of attention as a system that became chaotic through doubling bifurcation--limiting behavior that split into ever-more complex loops with increasing frequency until the behavior at any given time was, practically speaking, unpredictable. And it did so surprisingly photogenically as \\( r \\) increased--but not quite quickly enough that students could be expected to discover this behavior themselves working by hand. So, I built this simple Google Sheets document for my class so they could experiment with changing the value of \\( r \\) (cell A1 in the sheet). After allowing them some time to experiment, I asked them for their observations--what did they notice happened at various values of \\( r \\) ? The behavior was clear enough that they could identify many of the well-studied characteristics of the map--the limiting behavior, the period-doubling, the islands of stability.

Side note: I seeded my students with values to try--ones that would produce good-looking values. If I did this again, I don’t think I’d do this! They had little enough trouble experimenting that I think more of their interest could have been captured with a simple “try any numbers between 0 and 4, go!”

I then showed them these two diagrams, asking them to try and make sense of them:

(Bifurcation diagram by Geoff Boeing.)

The first image they took to quickly, connecting the points on the graph with the limiting behavior of the map. The second took more explanation on my part--I’ll certainly be reflecting on ways to bring the punchline of the lesson--sensitive dependence on initial conditions--more to the fore in the structure of the lesson. All told, this lesson took a little over half an hour to deliver, and I was pretty pleased with the results! I don’t think most of their experiences with chaos here were particularly revelatory--only a few students seemed to be bowled over by how strange the behavior of the map is--but rather were something they could take in stride, a new sort of behavior that math might sometimes exhibit.

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Chalkdust Magazine just published a little article I wrote on the CORDIC algorithm that some calculators use to compute sine, cosine, and a whole boatload of other stuff! Go read it!

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Geometry of Circles - Philip Glass Music on Sesame Street (1979)

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The introduction to 1997′s Astro Algebra from Edmar and Harcourt Brace & Co., for Macintosh System 7.

If, after consulting this document and the Astro Algebra User’s Guide, you still experience difficulties, please contact Edmark’s Macintosh Technical Support department.

Telephone: (425) 556-8480 Fax: (425) 556-8940 24 hours a day Internet e-mail: mactech@edmark.com Or visit our World-Wide Web site at http://www.edmark.com America Online e-mail: Edmark Mac America Online Forum: Keyword “Edmark” Compuserve: 73252,3441

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An educational comic with an astonishingly broad remit, reading each Learnt strip feels like drawing yourself out of a long Wiki-wandering session and reflecting on what you’ve learned. Learnt, though, manages to be more thorough and directed in its brief 1-2 page spreads than those sessions ever end up being. Inventive visuals pull your attention to the novel ideas of the topic, while Nelson’s self-caricature smoothly leads a narrative educational monologue around the page. These narratives range from simple storytelling to some deft play with structure of comics dialogue, switching between narrators and tonally echoing Nelson’s discoveries about the comic’s subjects. As narrator, Nelson takes something like the role of a science presenter, adapting the style of a Cosmos or similar to make these visual centerpieces approachable with humor, experience, and feeling—all learning is, in the end, a personal journey.

Those following the particular interests of this blog might be interested in:

Homework -- New Math wasn’t the only education shift the ‘50s saw!

Counting Systems

Zeno’s Dichotomy

The Zeroth Law of Thermodynamics

Global Maps -- Ah, projection!

SAT/ACT -- On the troubled history of American testing (and an interesting stylistic experiment!)

You can support Kelli Nelson in making more Learnt on Patreon: https://www.patreon.com/kellinelson.

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This is, bar none, the coolest post-exam project I’ve heard of an AP Calc class doing! “Poetry Out Loud,” a national poetry contest, has quite the following in Vermont—and adapting its format to mathematical ends seems to have been quite a success! 

“I was jealous of Poetry Out Loud,” O’Donnell said. “It was such a full spectrum of participants, from those students who clearly enjoy poetry and performing to those who don’t really, but for reasons of their own, go through with it anyway.

“I look for every excuse I can find to bring poetry and math together,” he continued, “so last year after the AP exam in early May, when I was looking for a project for the class, kind of spur-of-the-moment, I put ‘Theorems Out Loud’ on the course calendar. I’d had those students for three years in a row, and they were very willing to go along with the idea.”

Click through for a description of the poems and a walk-through of how O’Donnell put the event together!

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Every Joke Printed in the Margins of Masterton and Slowinski’s Chemical Principles, 3rd Edition

Chapter 1: Basic Concepts (p. 21) If you don’t know the symbols for the elements, you should learn them, now

Chapter 2: Atoms, Molecules, and Ions (p. 25) Philosophical speculation seems to be easier than doing convincing experiments

Chapter 5: The Physical Behavior of Gases (p. 89) The H2 molecule would encounter the same difficulties as a commuter trying to get on a New York city subway at rush hour

Chapter 6: The Electronic Structure of Atoms (p. 122) A flying flea obeys classical mechanics

Chapter 9: Liquids and Solids; Change in State (p. 252) We lose a lot of our snow that way in Minnesota

Chapter 11: Water, Pure and Otherwise (p. 301) Thanks, but I think I’ll go swimming somewhere else

(p. 318) Distilled water, however, has the advantage that it doesn’t contain any bacteria

Chapter 12: Spontaneity of Reactions; ∆G and ∆S (p. 334) This is one of the few truly profound relations in all of science

Chapter 14: Rates of Reaction (p. 371) It’s somewhat sobering to realize people are thermodynamically unstable

Chapter 20: Oxidation and Reduction: Electrochemical Cells (p. 547) Some graduate students are more productive than others

Chapter 21: Oxidation-Reduction Reactions (p. 580) Unfortunately, Cl2 doesn’t taste good either

Chapter 22: Nuclear Reactions (p. 609) Tourists, maybe?

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Frederick the great, Konigsberg, bridges, pokemons and Euler.

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Pepys and the Parallelogram

This day come home the instrument I have so long longed for, the Parallelogram.

—Diary of Samuel Pepys, 13 January 1669

So ends that day’s entry in Samuel Pepys’ diary, sounding for all the world a man on the verge of Euclidean revelation. Pepys, though, was not a scholar; his excitement had another source. A minor official in the British government, he could afford to try the cutting-edge gadgets of near-industrial Britain. The Parallelogram (or pantograph) was one such high-tech toy, a mechanical device for scaling images up and down:

[A]nd there I had most infinite pleasure, not only with [Mr. Spong’s] ingenuity in general, but in particular with his shewing me the use of the Parallelogram, by which he drew in a quarter of an hour before me, in little, from a great, a most neat map of England — that is, all the outlines, which gives me infinite pleasure, and foresight of pleasure, I shall have with it; and therefore desire to have that which I have bespoke, made. Many other pretty things he showed us, and did give me a glass bubble, to try the strength of liquors with. —9 December 1668

The Parallelogram fascinated him for the next several months, with him showing it around so much as to begin wearing it out:

At noon home with my people to dinner, and then after dinner comes Mr. Spong to see me, and brings me my Parallelogram, in better order than before, and two or three draughts of the port of Brest, to my great content, and I did call Mr. Gibson to take notice of it, who is very much pleased therewith; and it seems this Parallelogram is not, as Mr. Sheres would, the other day, have persuaded me, the same as a Protractor, which do so much the more make me value it, but of itself it is a most usefull instrument. Thence out with my wife and him, and carried him to an instrument-maker’s shop in Chancery Lane, that was once a ‘Prentice of Greatorex’s, but the master was not within, and there he [Gibson] shewed me a Parallelogram in brass, which I like so well that I will buy, and therefore bid it be made clean and fit for me. —4 February 1669

He even purchased one as a gift:

I did give him [Captain Deane] a Parallelogram, which he is mightily taken with; and so after dinner to the Office, where all the afternoon till night late, and then home. —22 April 1669

With all this build-up, it may be surprising to learn how simple this device is—just four rods which together form a parallelogram with variable angles and two adjacent sides extending out of the figure.

By Inigolv on Wikimedia

To use a parallelogram, the end of one of the long rods was fixed in a position on the drafting table. Then, depending on whether the aim was to enlarge or to shrink, a pen would either be placed at the end of the other long rod or at the vertex of the parallelogram between them. Then, the point without the pen would be traced over the image to be reproduced—and the pen would necessarily scale it up or down appropriately. Note as well that these three points are aligned. The same tool can be applied as well to a number of uses—type making, miniature work, early mass-production techniques, and wood-working:

From Woodgears.ca

In order to answer how the Parallelogram works, we must first nail down what we mean when we say that we have scaled an image. If I add one unit to each side of a square, it will get scaled up—but it I do the same to a thin rectangle, I’ll be making it comparatively more wide than I am making it long. One figure is a scaled version of another if the changes are in proportion to the lengths. This is called similarity in geometry: When the ratios of corresponding sides in two figures are all equal, the two figures are similar.

What keeps those ratios the same? The distances between the fixed point and the other two crucial points on the Parallelogram will always be in proportion—try using the properties of parallelograms to find similar triangles embodied in the Parallelogram’s functioning. They’re there—remember your ASA similarity postulates!

From there, there is only one more question: If I know I have proportional distances to corresponding points from some fixed point, how do I know that this will give me similar figures? An answer involves ASA similarity with triangles drawn of corresponding sides of the figures and that point. Is there another?

This whole line of reasoning—using proportional distances from a fixed point—is so useful it has its own name: A dilation about that fixed point.

Here are some good questions: If the three crucial points of the Parallelogram are not in a line, are the figures still similar? Is it still a dilation? What if we shrank the parallelogram all the way down, moving that vertex point to the intersection of the two long rods? Why prefer this form of the Parallelogram over others?

Using this same principle, you can build your own crude Parallelogram out of standard school supplies. Take two rubber bands of the same make and tie them together. At one end of the band chain, loop in your pen or pencil; fix the other end to a spot with a tack or even just your thumb. To enlarge a drawing, move your pen so that the knot between the two bands traces over the lines in that figure. Observe the pattern your pen traces. Why does it act like a Parallelogram?

Don’t worry if your figures come out decidedly wonky—after all, even enthusiastic Parallelogrammer Pepys struggled with his tools:

After dinner, Mr. Spong and I to my closet, there to try my instrument Parallelogram, which do mighty well, to my full content; but only a little stiff, as being new. —17 January 1669

It’s tempting to see Pepys’ enthusiasm for his new gizmo as childish, the fancy of a wealthy dilettante with no real appreciation for the labor-saving device. Yet Pepys, for all the of strange ebullience of his diaries, was no stranger to the power of mathematics. He recognized the use of even simple arithmetic so much that he took on a tutor in the subject for a number of years, working to improve his analyses of the functioning of the Royal Navy. We can never know how much effort he put into understanding the Parallelogram, of course, but if he put in even the slightest—or, for that matter, moved one of his friends to think on the it—as a teacher, I can hardly begrudge him his toy.

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Finally, A Tolerable Kline

Turns out, I still have some credentials that get me access to the Sage dissertation clearinghouse! I've been reading through Howell's work on teacher preparation for the New Math, and—while it's not directly in the lane of my research—it looks to be very useful. Howell synthesizes the existing literature on New Math with a great deal document-based research, sleuthing out what preparatory structures existed at the time as well as what precursors these programs had. This aligns nicely with some of the doubts around teacher competence expressed in my interviews—I'm sure I'll be citing it there.

Howell covers a curious series of episodes I don't think have received much attention in the other literature: Televised promotion of New Math content and pedagogy. One series, Mathematics for Teachers, was produced by the State of New York—one of the first major efforts to come out of the state public broadcasting system—but by far the most influential was Continental Classroom, produced by NBC (with, of course, a whole passel of funders from Rockefeller to the NSF). The NCTM reports that these series were popular (as was an elementary-aimed sequel to Mathematics for Teachers they promoted), with teachers requesting broadcasts and reels. Reels! I rarely think that the presence or absence of technology in a school makes too much of a difference, but I have to wonder at what a series of similar ambition could do with modern distribution channels. Was its popularity, in fact, a product of novelty and scarcity—or are modern attempts too narrow and targeted?

Her literature synthesis provides the invaluable service of sorting through Kline's voluminous writing on the New Math and condensing it down. Much of Kline is really good work, but his distaste for the New Mathematics overflows in every piece he writes. It's not enough to dismiss his work, especially in light of how tremendously influential a contemporary writer he was, but his arguments do tend to the historical and aesthetic, making direct use of him as a scholar difficult. By placing Kline in a historical context himself, Howell finds a good way of incorporating him into modern work.

As everyone does—and I imagine I must—Howell closes with a discussion of what the New Math means for the Common Core. The section is generally strong, but has, I think, the flaw of missing how similar Common Core and the NCTM standards are in terms of content. This is not to say that all teachers were prepared for Common Core implementation, of course, but rather to note that the backlash has (and has always had) substantial political motivation that can elevate voices of objection that would have previously languished.

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Some Resources for Education from GDC

The Game Developers Conference (GDC) released slide decks and recordings for quite a number of its talks this year. While I first listened to these as an interested player, I noticed that several had good take-home lessons for education in general, and math education in particular:

Slide Decks:

How Twitch Made Me a Better Teacher by Sean Bouchard

Bouchard lays out some interesting parallels between streaming video games for analysis (rather than comedy, say, or esport spectating) and his work in the classroom. It's an interesting analogy, particularly for how it suggests that lectures could become enlivened by interaction. (As someone who's taught math, it's hard not to take his "let people know what you're playing" advice as an argument for the necessity of stating a lesson's objective.)

DONNA: Gender Inclusive Game Education in Practice by Jenny Brusk

Like mathematics, the games industry faces a distressing gender imbalance in its classrooms and workplaces. Brusk summarizes what was effective in the University of Skövde's DONNA effort to recruit—and support—more women in its game design and study programs. Check out the recommendation to pilot models for gender-inclusive programs!

Videos:

Continuous World Generation in 'No Man's Sky' by Innes McKendrick and Building Worlds Using Math(s) by Sean Murray

From a mathematical perspective, these two talks are fascinating examples of how various mathematical ideas can be pulled into games, both opportunistically and rigorously. Particularly fascinating is Murray's discussion of how realistically-modeled terrain proved to be too realistic—it was less frequently exciting than they had hoped. Students might be provoked into thought by McKinnes' mentioning how the team rejected techniques from cartography for reasons of speed. When is accuracy important, and when is efficiency? How can we quantify that trade-off?

Education Soapbox by Bonnie Ruberg, Karen Schrier, Christopher Totten, Marcelo Viana Neto, Emma Westecott

A theme runs through several of these talks—Totten, Schrier, Ruberg—of how easy it is to neglect the history of the past and complexity of the present by hewing close to familiar models of content and teaching. Committing to an egalitarian practice is not enough when the default programs of study devalue the artistic component of games (and math); such a practice may seem disingenuous when teaching a curriculum from which women’s contributions have been omitted. Neto then provides a really excellent primer on liberatory education per Freire and hooks—hopefully, those lessons won’t be wasted on a captive audience!

Put a Face on It: The Aesthetics of Cute by Jenny Jiao Hsia

Though it's rarely what got us into the field, teachers ultimately have to become designers if they ever want to go off-manual. There's good material available on best practices in terms of typography, graphs, imagery, usually centered on minimizing reading difficulty and predictably drawing the eye. Put a Face on It suggests that we can go further, using 'cuteness'—appealing, clear emotional indicators—to communicate what we, the designers, think about what they're reading and what they're doing. I can imagine this being particularly useful as we move into using more interactive activities, where how to use them is not pre-informed by a decade of school experience.

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Really Communicating

Math doesn't have billion-dollar telescopes to plunk down on sacred mountain peaks. Nobody has non-consensually sampled cell lineages to forever research in its name. New researchers don't make their names embedding with other cultures, finding ways to condense their lives for the consumption of other math professors.

But they do embed, don't they?

For all but a beneficently-fellowshipped few, surviving as a mathematician is predicated on the reverse relationship, exporting the field to students often from drastically different cultures from those who produced the subject list, presentation, and teaching model that we call a math class. Math as it is taught has a culture of its own, one which we reflexively insist must be approached wholly on its terms. Something very much like colonialism can evidence itself here—a sort of colonialism that Dr. Chris Emdin seeks to dismantle in his For White Folks Who Teach in the Hood.

The core of this effort centers on the analogy of "neo-indigeneity"—that for students from minoritized cultures, schools can be a mechanism that require them to discard their own cultural traits in order to simply integrate into the school system, and even more so in order to find scholastic success. The replacement values Emdin charges schools with propagating are white and upper-class, even for many schools established explicitly for self-advancement; the history of American education combined with the reality of contemporary power make alternatives difficult even to imagine.

Emdin is very open with how difficult it has been for him, as well; he recounts his failures along the way, and gives credit to the sources—traditional and neo-indigenous—that informed his clade of practices. Yet, For White Folks avoids the usual errors of giving into orderly memoir (or the hyper-organized thesis), instead proceeding through his practice recommendations with practicality; building in order of pre- and co-requisite practices. To drive for anti-colonialism, these practices engage with the cultures of his students—for example, his description of "Pentecostal Pedagogy" drawn from observation of how church-going experiences work as teaching experiences. Three of its core tenets—call-and-response, processing emotions, and catching the spirit (sharing leadership)—draw on the need for deeper teacher-student communication, a theme which repeats (Alongside the ceding of full control) in the book. (They also allow for the notion that student who is not all right—pedagogically or emotionally—ought to be afforded space in the class for this, running somewhat counter to my traditional instincts on the matter.)

Through communication, Emdin sees a path forward—one more consonant with his Pentacostal Pedagogy than traditional stances—for resolving the fact that a pedagogy best-received by most students in a class will not be well-received by all. After all, cultures are staggering in their variety, both as a whole and each internally—and I prefer majoritarianism to colonialism, but I'd rather avoid both. Cogenerative dialogues—student discussion circles that meet with the teacher—serve as a space for student-led communication about classroom teaching. By hosting such groups with a purposefully diverse set of students from the class, the teacher may hope to both address such flaws as their students will consider obvious and will move on unanimously, and as well pinpoint errors that are more particularly felt—all the while building bridges between the student cultures present in the room. More radically, he suggests co-teaching not by other professionals, but by the students themselves, allowing for a pedagogy that comes very near to directly from their own culture while allowing students to pursue a given subject in-depth; an extension of peer-teaching to the full classroom. His novel arguments for even more accepted models like peer-teaching make his suggestions for incentives feel a bit automatic by comparison—the peer-teaching recommendations in particular resemble nothing else more than a value-added model.

Emdin largely concentrates on finding remedies rather than critiquing the state of education—but when he does critique, it is the model of devastating brevity. As someone who has co-taught, his observation that co-teaching privileges volume of teacher-teacher communication over student-teacher communication hit hard. These do occasionally misfire—one can easily imagine, for example, that there are more reasons than hideboundness that a female teacher might have for misgivings about dropping in on jump-rope or pickup basketball that a male teacher would not.

His best, though, is saved for the close—and is a defense of his method as a whole. The White reader of his work might well be hesitant to engage with the cultures of their students for fear of co-option; indeed, Emdin himself is careful to frame his practices as opportunities for students to share their own culture, rather than his attempt to lead them in it. Yet, there is no neutral position outside of culture, there is no space without participation—and as the sociologists will (now) tell you, an insistence on pure observation can be a form of—in Emdin's words—subhumanization.

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The Pedagogical is Political

By far the most prominent piece of New Math scholarship of the past decade is Christopher J. Phillips The New Math: A Political History (review by Alex Bellos for Nature here), an efficient book on the often complex internal and external political struggles that initiated, shaped, and ended the New Math movement. So influential was it, I became aware of it entirely by accident while I was on a sabbatical from even thinking about my New Math research project--it was just leaping up those scholarly-publisher best-seller lists! I finally read it during my time in Boston, where I found it a good and informative read (if, admittedly, not too useful to my own research). The book can, at times, delve heavily into establishing the philosophical notions that it sees describing the struggles that unfolded over those two decades. Phillips, though, is a strong enough academic writer to keep those sections moving, and setting us these priors helps him a great deal in building his case. As Bellos notes, the purely descriptive history is strong as well; even someone without much knowledge of the era would be able to access this story.

The most interesting response, though, must certainly be the one from Jeremy Kilpatrick in Science--himself not just a renown math education researcher, but one of the few living curriculum experts who worked on New Math programs. He rightly notes that the dramatic differences between the various New Math programs like the UICSM and the SMSG makes Phillips’ decision to concentration more an act of myopia than simplification--particularly in that it makes the key failure of the SMSG not to require--or even offer--implementation training for teachers an incidental fact rather than a factor that substantially determined the course of the entire Movement. (This is a running theme in Kilpatrick’s accounts of the era.) This criticism also points toward a perception that the narrative of this book is heightened somewhat by an undue level of speculation on personal motivations--and while I think Kilpatrick is perhaps underplaying how much documentation is now available to speak to the New Math-er’s thought processes, he is right in such analysis being not terribly useful in a book of political history. One is hard-pressed in pulling much modern implication from such speculation.

Most of the other work from the past few years on the New Math is also operating in that sort of program-history vein--and indeed could be put in the general category of “centennials.” Several pieces from the Mathematical Association of America explore that organization’s role in shaping various New Math programs.

There are two major recent works I haven’t tracked down yet that also show some promise. The first is The Fight for Local Control, a political history(!) on the suburban capture of rural districts and the resulting policy agenda. My research indicates that--in amongst conflicts over desegregation and the Titles--the book has some substantial discussion on the New Math. (Also, published by Cornell U. Press--go Big Red!) The other is what promises to be a fascinating dissertation: That Much-Maligned Monster: An Examination of Teacher Preparedness and Training in the Era of New Math, 1950 to 1975. What a title! Unfortunately, it appears to have suffered the same fate as so many dissertations: Paywalled in the Sage database! Was it ever thus?

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Endpapers

Four years. I started my research project back in 2013, and—after years of only occasional work on it—it’s time I brought it to a close. (Also, I did promise several of the participants I would send them the results, and I feel not so great about that!)

THE RESEARCH QUESTION: How did the New Math impact the future mathematicians and math educators taught by those programs?

Perhaps, in retrospect, a less-broad question would have helped me jump quicker into the analysis! But little has been researched on the New Math past about 1980; indeed, the research involving its students seems to end upon their graduation! Some initial summation is necessary.

AS IT STANDS: I have

A bibliography in need of paring,

Multiple-choice answers to questions about what showed up in my participants’ curricula,

A collection of thirty-odd responses to short-answer questions on how they felt and feel about those choices,

A few hours of transcribed in-depth interviews with preliminary coding,

AND:

An analysis specifically of how problem-solving was folded into New-Math–style programs, based on the interviews and free responses.

My plan, to start with, is to work on building similar analyses on the other emergent themes of my data. First, though, I need to do the standard database trawl to make sure no-one’s snagged this topic yet!

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How rock-climber’s rucksacks came to be the basic kit of every school-child!

My editor, Steve Drummond, isn't that old of a guy. He's from Michigan — Wayne Memorial High School, class of '79.

But when he starts talking about backpacks, he dips into a "back in my day" tone that makes you think of a creaky rocking chair and suspenders: "You know, Lee, when I was in school, no one had a backpack!

"You just carried your books in your arms." He says it like he's talking about sending a telegram with Morse code. "No one really thought about it, that's just what you did."

It seems there really was a period where kids had too many books for the old book-strap method, but before the popularity--and standardization--of the backpack. “The book-strap of yesteryear could never go round today’s bulky curriculum,” says a 1966 edition of Newsweek. And the book-strap had other problems, too:

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Stop Having Fun

UNCF and the Education Post have a survey report out on how big-picture questions in education are seen in Black communities that I think is worth a read. An incisive variety of questions—and subjects! I do wish, though, that advocacy organizations didn’t feel the need to use “fun” diagram designs like this one:

The practice of starting a bar chart past 0—in this case, at 70%—is dubious enough when the bars are abstract rectangles. With those percentages represented as cartoon pencils, their comparative size is stressed, and is definitively given by the decision to indicate that pencils start with their erasers at the base of the chart. Consequently, 96% looks to have more than twice the magnitude of 80%! At the other end, the pencils are sharpened—and beyond objections as to muddying area and length, the sharp points obscure what actual value each bar reaches, as does the lack of interval tics. One could argue that this uncertainty has the intentional effect of making the data less immediately comparable—which is sensible, as the four items are not all from the same survey or on the same scale. But why, then, present them in a format designed for comparison? Either make the case or unlink the infographic.

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This looks like a good use of Damon’s platform! I appreciate his conviction on issues of school equity, but I worry that direct speech by celebrities often has the narrow effect of buoying activist’ spirits (and perhaps swinging a few fans), while being easily dismissed as inexpert, inflated, or angry by most who see it scroll through their feed. Finding instead a few documentarians with expertise in the field to collaborate on the project, melding expertise with star-power—I’m very interested to see what comes of this!