Month: February 2012

Total 10 Posts

Okay, This Is Exactly What I’m Talking About

I don’t remember sending half these tweets at the Sal Khan live show last night but, being fair with myself, there were a lot of people in the crowd passing around a lot of different things. Luckily, somebody … nearby me … bootlegged the show so I could go back this morning and relive it.

A highlight:

We’re hoping Khan Academy turns into a platform for cognitive education research. You have two million kids doing problems every day. If you have a better way for people to conceptualize fractions [using video-based lecture] and you have a good way of measuring it [using machine-readable tasks] — we’re already doing A/B testing — you put five percent of the audience in front of that versus the control. In one day, you have the data for your Ph.D.

Khan didn’t say those bold-formatted words but they were deafening just the same.

The technologies that allow you to conduct A/B tests on mathematics education constrain the mathematics you can A/B test. There are methods for teaching fractions effectively that cannot be effectively A/B tested using the Khan Academy platform or any existing technology, for that matter. Yet the prevailing attitude in Silicon Valley is, “The limits of my sandbox are the limits of the known universe.” (Khan extends that to say, “You could spend a day in my sandbox and get all the data you need for a doctorate.”)

That sandbox might be really, really great. I’m willing to tolerate way less than the best fraction instruction if it’s accessible to everybody in the world with an Internet connection. But confusing the sandbox with the known universe isn’t doing anybody any favors here.

The rest of the set was classic Khan, but he played some new material I hadn’t heard:

  • “We are highly influenced by constructivist thinking.” [link]
  • “A lot of what we’re talking about is maybe the missing link in Montessori.” [link]

Comments closed, obv.

[3ACTS] Joulies

Watch this one minute clip and if you find yourself wondering, “Do Joulies really work?” ask yourself, “What would that kind of temperature graph look like?” and then click on through to the third act of the lesson plan to find out.

A few release notes:

  1. The Goods. My favorite part of the task is how much work the students have to do to translate the inventor’s claims to mathematics. He says, “When coffee is poured into a travel mug with Joulies inside, the coffee cools down to a perfect temperature three times faster than normal. Then, when the coffee would normally cool off, the heat that was captured is released actually keeping your coffee pleasantly warm for twice as long.” ¶ So students have to make an assumption about “perfect temperature.” Is that a range? Is it a single temperature? Then they have to make an assumption about the initial temperature of the liquid and its final temperature. Then they have to create their initial graph. ¶ They get to decide and own all of that. We don’t care. We care about their transformation of their initial graph and whether or not it fits the inventor’s claims.
  2. Formative Assessment. Why did I close the first act video with this frame and not this frame?
  3. The Competition. Sometimes I wonder why we should bother with that level of precision, why we should analyze these videos on a frame-by-frame basis when our competition in the video-based math curriculum space is basically drooling all over itself.
  4. Citation. Marco Arment performed a similar experiment with Joulies. I got the idea to use rocks in the sequel from Jeff Ammons.
  5. Feedback From Pearson. They told me I should consider changing the domain of the temperature graph from six hours to one, because it’s rare to drink the same cup of coffee for six hours, and to be a little kinder to ELL students by using “joulies / no joulies” rather than “joulies / plain.” Other than that, they get what I’m trying to do here and they support it.

Featured Comment

Criticism from Bowen Kerins is one of the big reasons why I bother posting this stuff. Here’s his entire comment:

I think the one-hour timeframe is better than six hours. More importantly, though, you’re running across limitations of video technology by having to make this decision at all.

To me the “best” solution would be to let students decide what their axes limits should be, then see the graph populated. A tablet-PC environment could make this happen. A static video takes this decision out of the hands of students because you’re forced to select this in advance, or to set up a limited number of options. The same is true for the vertical axis — my first reaction to the presentation was “Why does the vertical start at zero degrees Fahrenheit??” And what led to the maximum being 160 degrees?

I’d want students making those choices as well, ideally in an environment where a quick change doesn’t cost them anything. Even when students are asked in advance to create the initial graph, leave the axes totally unlabeled and let them make all the decisions.

I also think this flexibility would lead to students coming to different conclusions about the effectiveness of the Joulies. A one-hour or thirty-minute graph makes it look like the Joulies are doing a pretty good job, while the six-hour graph makes it look like they do nothing most of the time. It could even lead to a cool “how to lie with data” conversation, or at least an important conversation about the nonlinearity of the graph (to meet 8.F.5). Often students think all graphs and functions are linear. The short-term graph of the “no Joulies” seems linear enough… then boom it ain’t!

I’m also a little confused by your student work example — the graphs show that the Joulies version stays in the “perfect” zone for more than twice as long (75 minutes versus 30). So I would not agree with the student’s assessment that they “stay perfect for almost exactly the same amount of time”. The six-hour versus one-hour makes a big difference here, I suppose.

Last, two nitpicks: the video talks of coffee but then presents tea (no big deal but why use tea and not coffee?). And please show me an actual eighth grader with the quality handwriting exhibited in the “student work” ;)

Panel Remarks: Why Algebra Matters And How Technology Can Help

2012 Feb 21. They uploaded video of the event. My remarks begin at 30:26 in this video.

I was a panelist at Middle Grades Math: Why Algebra Matters and How Technology Can Help, a conference at Stanford co-sponsored by Policy Analysis for California Education, NewSchools Venture Fund and Silicon Valley Education Fund. (I know. Awkward, right.)

The following are my introductory remarks and responses to two questions.

Introductory Remarks

I could ask you how tall do you think that lamppost is. Just give a guess. I could ask you for a guess you know is too low. Too high. A wrong answer, if you will.

I could then ask you what triangles do you see on this image now. How many? What types? Where are they?

And once you point out certain triangles that you see, I could ask you, “Do you notice anything special about those triangles? Are they just arbitrary, random triangles drawn here or there?”

“No, there’s something special about them,” you tell me. “They’re mathematically similar-looking.”

And I could ask you, “How do you know that? Could you help me prove that?” And we know that if you have two angles that are the same in two different triangles, they are mathematically similar. “What two angles in each of those triangles are the same? Can you help me with that?”

And you might point out to me that all of these angles are the same. In fact they’re all ninety degrees. Because, you tell me, we’re all standing perpendicular to the ground. Even the lamppost is.

You could point out that these angles are the same.


“Because the sun’s rays all strike us at the same angle because we’re all in more or less in the same spot.”

So we have these similar triangles. And I could ask you, “What parts of these triangles are easier to measure than others?”

And we could have that conversation. You might point out that our heights are all known, except the lamppost’s. And our shadows are all fairly easily measured. And at this point, you could solve for that unknown height.

We’ve just gone from the concrete —

— to the very, very abstract.

— you and me — in this process called mathematical abstraction where we formalize the informal. It’s a process that is invaluable to our students, something that mathematicians do all the time. It’s engaging to students and it’s accessible to students at every level.

If I could think of one way to restrict access to math to the already-haves and close it off from the have-nots, I could do no better than to rush as quickly as possible to the highest level of abstraction as possible on this scene. And that’s, of course, how we see this problem in even highly regarded textbooks.

Like right here. I can’t ask you, “What triangles do you see?” Those triangles have already been abstracted.

I can’t ask you, “What information is easiest to gather here?” It’s already there.

The problem, as I perceive it, is print. The process that we went through, stepping that out gradually, required ten extra slides in a slidedeck. That cost me a few bits and bytes on a hard drive. It’s nothing. But ten extra printed pages in a textbook. That’s very expensive.

So I’m here today very optimistic about digital curricula and its ability to open that process of abstraction up to all of our students.

That’s not to say this process couldn’t go horribly, horribly wrong.

So I just want to point out here, to close it up and turn it over to you guys, that print is a medium. Same as digital photos. Same as a teacher’s voice. Same as a YouTube video. Same as a podcast. These are all different media. And as we know, the medium is the message. The medium defines and constrains and sometimes distorts the message. The math that can be conveyed in a YouTube video is not the same math that can be conveyed in a digital photo or a podcast or a print textbook.

We’re so enthusiastic here in the Silicon Valley and in this group about technology that disrupts and scales but I think it’s really important to point out here the fundamental misapprehension of this whole process of technology that we have is that there is one monolithic “mathematics” and we are all just innovating around “mathematics.” But those innovations distort what mathematics is. That’s the ball that I urge us all to keep our eye on today. I’m really excited to be here and tease apart those issues with you and take some questions. Thank you.

How will the Common Core Standards paired with the computer assisted adaptive assessments that are envisioned by the Smarter Balanced Consortia change or disrupt middle grades mathematics?

Yeah, I like the Smarter Balanced assessment items, particularly their printed items. And that’s kind of controversial to say, I guess, at a tech conference. But the stuff you can assess online is just different. Someone I admire says, “the computer is not the natural medium for mathematics.” Not yet. There’s no natural language processing. You can’t easily grade automatically and adapt to a written argument the student makes about some figures he says are similar, for instance. So that’s stuff that I would feel sad to lose in our hurry to get to computer adaptive testing.

What kind of support are teachers and schools going to need to transition to Common Core, to make the changes in middle school math? Should the state be providing some support? Should county offices or foundations be providing that support?

Most of the PD that I underwent as a teacher stayed very close to content. It was one or two degrees from how you teach content and, now that we have these practices in the Common Core, that opens up an entirely new PD challenge. The most effective PD I’ve facilitated — and I’ve facilitated some very ineffective PD, I’ll admit that — the effective stuff has always included a large component where we do the practices. Because I think if you’ve taught for thirty years under a particular style of teaching, it has to distort what your perception is of math and how it should be taught. It’s unavoidable, to be steeped in that for so long. So to realign yourself, I imagine, is a very difficult thing. So PD that involves problem solving, involves reasoning, argumentation, that’ll be essential going forward.


  1. Key Curriculum Press asks, “Where were the women?
  2. Jason Buell follows the money.

Photo by Jason Buell

What Silicon Valley Gets Wrong About Math Education Again And Again

The medium is the message. The medium defines, changes, and distorts the message. The words “I love you” mean one hundred different things spoken by one hundred different people. Those words convey different meanings spoken on the phone, written on a fogged-over bathroom mirror, and whispered bedside in a hospital.

YouTube videos, digital photos, MP3s, PDFs, blog posts, spoken words, and printed text are all different media and they are all suited for different messages. When you attempt to distribute mathematics through any of these media, it changes the definition of mathematics.

Silicon Valley’s entrepreneurs, venture capitalists, and big thinkers assume a shared definition of “mathematics.” They innovate around the delivery of that mathematics. CK-12 has PDFs. Khan Academy has YouTube videos. Apple has iPad apps. ALEKS and Junyo have computer adaptive tests. Very few of them understand that each of those delivery media changes the definition of mathematics.

Even worse, at this moment in history, computers are not a natural working medium for mathematics.

For instance: think of a fraction in your head.

Say it out loud. That’s simple.

Write it on paper. Still simple.

Now communicate that fraction so a computer can understand and grade it. Click open the tools palette. Click the fraction button. Click in the numerator. Press the “4” key. Click in the denominator. Press the “9” key.

That’s bad, but if you aren’t convinced the difference is important, try to communicate the square root of that fraction. If it were this hard to post a tweet or update your status, Twitter and Facebook would be empty office space on Folsom Street and Page Mill Road.

It gets worse when you ask students to do anything meaningful with fractions. Like: “Explain whether 4/3 or 3/4 is closer to 1, and how you know.”

It’s simple enough to write down an explanation. It’s also simple to speak that explanation out loud so that somebody can assess its meaning. In 2012, it is impossible for a computer to assess that argument at anywhere near the same level of meaning. Those meaningful problems are then defined out of “mathematics.”

Do you want to know where this post became useless to Silicon Valley’s entrepreneurs, venture capitalists, and big thinkers? Right where I said, “Computers are not a natural working medium for mathematics.” They understand computers and they understand how to turn computers into money so they are understandably interested in problems whose solutions require computers. Sometimes a problem comes along that doesn’t naturally require computers. Like mathematics. They may then define, change, and distort the definition of the problem until it does require computers.

Some companies pretend those different definitions don’t exist. They pretend that we all mean the same thing when we talk about “mathematics.” Khan Academy acknowledges the difference, though, and attempts to split it by saying, in effect, “We’ll handle the math that plays to our medium’s strength. Teachers can handle the other math.” So Khan lectures about things that are easy to lecture about with computers and his platform assesses procedures that are easy to assess with computers. Teachers are told to handle the things for which teachers are a good medium: conversation, dialogue, reasoning, and open questions.

That delegation only works to the extent that teachers and computers convey complementary definitions of mathematics. But the message from Silicon Valley and the message from our best math classrooms contradict one another more often than they agree. On the one hand, Silicon Valley tells students, “Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself.” Our best classrooms tell students, “Math is something that requires the best of your senses and reasoning, something that requires you to make meaning of tasks that aren’t always clearly defined, something that can make sense whether or not anyone is there to explain it to you.”

I won’t waste any effort complaining that my preferred definition of mathematics has been marginalized. That effort can be better spent. Anyway, in every way that affects Silicon Valley’s bottom line, the Common Core State Standards have settled that debate. Mathematics, as defined by the CCSS, isn’t just a series of discrete content standards. It contains practice standards, too: modeling, critiquing arguments, using tools strategically, reasoning abstractly, and others. The work of mathematicians. Any medium that tries to delegate one set of standards to computers and the other to teachers should prepare for a migraine.

Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.

Has your ed-tech startup been struggling to demonstrate statistically significant gains on the California Standards Test, which features tasks like this:

That’s your home-turf. Simple, machine-readable assessments. It will never get any easier for you than that. How much worse will your results look when we assess the same standard in 2014 with tasks that connect mathematical content to mathematical practices:

The medium is the mathematics. How does your medium define mathematics and is that definition anything that will be worth talking about in two years?

Full Disclosure: I’m a doctoral student at Stanford University in math education. I was a high school math teacher. I consult with ed-tech startups as time allows. I also develop digital math curricula that I sell to publishers and give away online.

Comment Policy: My usual policy is to close comments on posts that mention Khan Academy because they get silly almost instantly. But Khan Academy is only a symptom of a sickness that’s gripped this valley for as long as I’ve lived here. That sickness interests me and your thoughts on that sickness interest me. I’m leaving comments open but I’ll trashcan anything that doesn’t enhance our understanding of that sickness. That includes “Attaboys,” etc.

2011 Feb 7. Neeru Khosla, founder of CK-12, responds in the comments.

2011 Feb 7. Josh Giesbrecht posts a useful reply at his own blog, focusing on technology as an assessment, not technology as a medium.

2011 Feb 7. Web Equation from VisionObjects does a fantastic job translating scribbles on the screen to LaTeX.

2011 Feb 7. Silicon Valley’s (unofficial) rebuttal to my post is at Hacker News. Let me excerpt a few responses.

jfarmer poses a very productive alternative to my thesis, “What is technology good for?” rather than “How does technology change mathematics?”:

In design, a skeuomorph is a derivative object that retains some feature of the original object which is no longer necessary. For example, iCal in OS X Lion looks like a physical calendar, even though there’s no reason for a digital calendar to look (or behave) like a physical calendar. The same goes for the address book.

This is what I see happening in online education. I don’t think it’s a case of “lol, Silicon Valley only trusts computers,” but rather starting off by doing the most literal thing.

Textbooks? Let’s publish some PDFs online. Lectures? Let’s publish videos online. Homework and tests? Let’s make a website that works like a multiple-choice or fill-in-the-blank test.

These are skeumorphs. There’s no reason for the online equivalent of a textbook to be a PDF, it’s just the most obvious thing.

For me it’s 1000x more interesting to ask “On the web, what’s the best way to do what a lecture does offline?” than to say “Khan Academy videos are the wrong way of doing it.”

Arun2009 offers a common view, that mathematics is many different things to many different people:

The trouble with trying to arrive at any single definition of Mathematics is that Mathematics is different things to different people. A research level Mathematician might see it differently (finding patterns, abstraction, theory – axioms and proofs) from an Engineer who has a purely practical interest in it (cookie cutter methods and formulas). For everyday use Mathematics is a set of algorithms for doing stuff with percentages, fractions, basic arithmetic etc.

He’s absolutely right, but if we’re pragmatic in the least, we’ll ask “which of those definitions does the most good for students?” and we’ll look at the Common Core State Standards, which is the de facto definition of mathematics for those students. (My appeal to the CCSS was an attempt to reduce exactly this subjectivity.)

Various hackers took me to task for claiming it’s difficult to represent mathematical notation using computers. ie. the square root of 9/4.


Square root ? “(3/4)^(1/2)” or maybe “sqrt(3/4)”. There’s no complexity in parsing that. I do agree it is not as natural as on paper but maybe tablets will find a way to improve that. Thats what innovation is here for after all.

japhyr rebuts convincingly, in my opinion:

He is writing about math education. The characters (3/4)^(1/2) make sense to all of us who have already learned math and know some programming languages, but that syntax is pretty confusing to students who are just developing a real understanding of exponents.


One could make the argument that any mathematical syntax is equally confusing for the novice–so why not start them on something they’ll be using later anyways?

That’s an interesting idea but unless we’re also positing a universe with utterly ubiquitous computing, we’d be better off preparing students to communicate in media that are readily available. What if a computer isn’t available for our students to code some LaTeX to express themselves?

davidwees illustrates my overall point well. Does this seem natural to anybody?

\png \definecolor{blueblack}{RGB}{0,0,135} \color{blueblack} \begin{picture}(4,1.75) \thicklines \put(2,0.01){\arc{3}{3.53588}{5.8888}} \put(.375,.575){\line(1,0){3.25}} \put(1.22,1.375){\makebox(0,0){\footnotesize$ds$}} \put(.6,.5){\makebox(0,0){\footnotesize$x=0$}} \put(3.36,.5){\makebox(0,0){\footnotesize$x=\ell$}} \dottedline{.05}(1.0,.575)(1.0,1.10) \put(1.0,.5){\makebox(0,0){\footnotesize$x$}} \dottedline{.05}(1.5,.575)(1.5,1.40) \put(1.5,.5){\makebox(0,0){\footnotesize$x+dx$}} \put(1.22,.65){\makebox(0,0){\footnotesize$dx$}} \dottedline{.04}(0.6,1.12)(1.25,1.12) \put(1.0,1.14){\vector(-1,-1){.45}} \put(.58,0.83){\makebox(0,0){\footnotesize$T$}} \put(.77,1.05){\makebox(0,0){\scriptsize$\theta(x)$}} \put(1.18,1.16){\makebox(0,0){\scriptsize$\theta(x)$}} \dottedline{.04}(1.5,1.41)(2.1,1.41) \put(1.5,1.44){\vector(4,1){.67}} \put(2.22,1.59){\makebox(0,0){\footnotesize$T$}} \put(1.95,1.45){\makebox(0,0){\scriptsize$\theta(x+dx)$}} \end{picture}

Symmetry, takes the conversation to Khan Academy:

As to why many people might want to defend Khan Academy, well, its because I think I would have been much happier with Khan Academy than the math education I actually had, and I would very much like it to be available to children like myself. I was bored stiff in math class in middle and high school, and being able to work at my own base, not bound by the slowest person in the class, would have been amazing.

This same sentiment crops up in the comments thread here and I think it’s utterly on point. Teachers are a great medium for lots of things that a YouTube video isn’t. “Conversation, dialogue, reasoning, and open questions,” as I put it in my post. If you, as a teacher, aren’t taking advantage of your medium, if you’re functionally equivalent to a YouTube video, you should be replaced by a YouTube video.

Sudarshan summarizes that elegantly:

Incompetent/bored math teacher < khan academy < better online learning platform < Good math teacher.

FWIW, I stopped by that thread and summarized my argument in three lines:

  1. There are different ways of defining mathematics, and some of them contradict each other.
  2. Silicon Valley companies wrongly assume their platforms are agnostic on those definitions.
  3. For better or worse, if you’re trying to make money in math education, the Common Core State Standards are the definition that trumps your or my preference for recursion, computational algebra, etc, and those standards include a lot of practices for which, at this point in history, computers aren’t just unhelpful, but also counterproductive.

2011 Feb 10. Zack Miller e-mailed. Zack graduated from Stanford’s teacher training program. He now teaches at the charter school where I’ve been consulting with my adviser and two other graduate students. He’s a proponent of blended learning, individualized instruction, and Khan Academy, in particular. He gave me permission to excerpt his e-mail. I’m going to post the entire thing, bold one line, and make a comment on that line at the end.

Students’ mathematical learning and sense-making has a much higher ceiling if the material they encounter follows a coherent mathematical narrative that is relevant to and keeps pace with their current individual mathematical understandings.

The above is not a reality in most math classrooms. Students’ math understandings are Swiss cheese, largely because time, not competency, was fixed in their math education. What’s standing in the way of the above becoming a reality? Resources and good data. 1 teacher must provide 100 unique learners with learning experiences, daily. Can our country’s median teacher (or best teacher) provide a learning experience that is in each student’s zone of proximal development every day (and also be an expert in classroom management, curriculum, parent communication, data analysis, etc.)? No. So we’ve looked for ways over the years to ameliorate the situation (tracking, heterogenous classes, differentiation, etc.).

Silicon Valley is trying to innovate because students’ mathematical learning has a much higher ceiling if curriculum always meets their ZPD, and perhaps now there are ways we can offer that. THIS IS BLENDED (not to be confused with computers replacing traditional classrooms with traditional video lectures). Yes, the medium matters – without question – but breaking from the ratio of “30 students:1 pace” matters more. Any tool that gathers relevant data and/or provides good learning experiences helps me individualize, and (like many of your commenters) I disagree that there is no computer-based tool that does this (Geogebra, for one, was great for today’s lesson that you and your team designed!).

I agree that, at present, Silicon Valley’s current tools are limited. Pedagogy is weak. But I disagree that doing this:

“We’ll handle the math that plays to our medium’s strength. Teachers can handle the other math.”

Implies also doing this:

Silicon Valley tells students, “Math is a series of simple, machine-readable tasks you watch someone else explain and then perform yourself.”

Isn’t it possible to frame computer-based tools in a way that supports your definition of math (a definition that I happen to closely align with)? Doesn’t procedural fluency (which will still be a large portion of Common Core assessments, I believe) support reasoning, problem solving, and sense-making?

And lucky for our students, people like you will help Silicon Valley with their pedagogical shortcomings and misguided definitions of math. As the quality and quantity of “learning experiences offered” improve, students can have a more individualized experience, and that high ceiling can be realized.

My point is that the tools that allow you to individualize math instruction are not neutral on the question, “What is mathematics?” My second point is that not every answer to the question, “What is mathematics?” harmonizes with every other answer. Sometimes they clang around against each other awkwardly.

2011 Feb 15: EdSurge offers a summary of this post and its responses over at Fast Company.

2011 Feb 15: Just noticed (way late) that Jacob Klein, one of the founders of Motion Math, responded here. Klein is hip deep in the culture I’m describing here and he makes games that help students practice mathematical skills I value, so give him a look.

Adventures In Plagiarism At #GSDMC12

I was sitting in a morning session with six other people at the Greater San Diego Math Council’s annual conference when I saw a slide that looked familiar.

At first I figured I had read that language in some paper or another in grad school. (“Aversion” seems like the sort of language an academic might use to paper over a lack of insight.) Then I remembered I had used that language. When I wrote it in a presentation I gave two years ago.

The presenter went on to describe each of those bullet points using exactly the same words I did in that presentation, in exactly the same tone of voice. Fella had rehearsed.

Four things:

  1. The good news is that we’ve apparently solved the “perseverance” problem in the last two years.
  2. Don Whiteside storified my livetweeting and added his commentary.
  3. I edited his slide next to mine and described the rest of his plagiarism under the heading “This is not okay.” I’ll send that e-mail off to him, the president of his council, and the conference head later today.
  4. How should this have gone? Against the judgement of some folk on Twitter, I’m choosing not to name his name. I figure it isn’t sporting to slug someone beneath your weight class. (Define “weight class” however you want, okay.) I figure he’ll have to explain himself the next time he wants to give a talk in San Diego (or he should, if GSDMC has any self-respect) and that’s enough. I’m open to counter-proposals.

2011 Feb 3. Since you’re all no doubt dying for updates, my plagiarist attended my session this afternoon. (I don’t know why that surprised me. Of course he attended my session. Where else is he going to get fresh material?) This was before I e-mailed him. He came up to introduce himself as I was setting up. Before he got too far into his introduction, I said, “Yeah, I know who you are. I was in your session this morning.”

“You were?” he said, and very obviously winced. “I may have borrowed some of your material.”

“‘Borrowed’ isn’t the word I’d use,” I said.

He apologized several times. I thanked him. He left. I sent the e-mail. We’re square.

Featured Comment:


I’m seeing many commenters who think that frowning upon plagiarism is all about credit, but that’s a very shallow view. In this case, the plagiarist completely disconnected the audience from Dan Meyer’s extensive work in this area. That means they can’t read the extended discussions, they can’t draw on other related resources, they can’t contact the real author and ask their own questions. They can’t connect to the author to follow future developments.

The plagiarist is standing at the top of a well, with the audience at the bottom. He throws down a loaf of bread and a few pages from a book, and leads the audience to believe that basically that’s all that exists outside the hole, and he’s the only one who can provide it to them.

I really want to give the plagiarist the benefit of the doubt, that it was a stupid mistake that hasn’t happened before or since. But I can’t get past the idea that some of us spend 40 hours or more putting together the best 1 hour presentation we can. The plagiarist took the easy way out. It’s disrespectful to the original author, to the subject, and to the audience. I tend to think that disrespect like that doesn’t just randomly rear its head once.