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Exhibit A:

The simplest thing, “Take a picture of one of the proofs you just wrote and email it to me.” turns into twenty minutes of troubleshooting cameras that don’t work, and picture files we can’t find in order to attach them, and how to login to your school email account.

This isn’t an exhibit of doing mathematics or of technology enabling a classroom. This is an exhibit of an entire classroom spending time and administrative capital accommodating the limitations of computers, of technology disabling a classroom.

The tools need to get out of the way. When I use the Internet to communicate these words across time and space, I’m not consciously aware of all the technologies that facilitate that communication. They are out of my way. Computers are a natural medium for communicating words. In Kate Nowak’s class, the tools are consciously in the way.

Featured Comments

Dave Major:

Over the past couple of months I’ve heard “yeah, that’s cool, but I can do the same using x, combined with y and converted using z, backing onto Dropbox” far too many times.

Paul Topping:

With plain text, we go to a computer first to type it. Many of us have noted how he hardly ever handwrite anything longer than a phone number or address these days. The same can’t be said for math notation. Some can write math using LaTeX but that is far from ideal. Even mathematicians who are LaTeX experts do not handwrite it on paper or a whiteboard. They use standard math notation.

42 Responses to “Computers Are Not A Natural Medium For Doing Mathematics”

  1. on 05 Nov 2012 at 8:13 amChris Robinson

    The first person to find the line between “effective instructional tools” and “consciously in the way” and implement it into a digital math curriculum is going to be a very popular (and probably rich) person.

  2. on 05 Nov 2012 at 8:16 amjosh g.

    a) Yes. Anyone who thinks kids will automatically be able to handle these things because they’re “digital natives” is welcome to come to my class and help reset the passwords for my classes of 11-yr-olds who have no idea how to manage a username and password.

    b) Is *any* medium natural for doing math for these kids, or is it all learned? Would a room full of kids who’d never used a compass before take just as much hand-holding?

  3. on 05 Nov 2012 at 8:24 amBen K.

    There are solutions. What about a Livescribe pen? It will record the page as you or a student write and transfer it to the internet where anyone can see it. You could even have the student talk through the proof as they are writing it so that you can hear their thought process.

    Or a document camera. Have the student write out the proof, come up to your desk and snap a picture with the document camera. This will save right to your desktop. You could immediately name it with the students name.

    These cost money but would work.

    Or, have student pull out their cell phones. I’m guessing that every student in your class that has a cell phone, knows exactly how to take a picture and send it (email or text) to someone. Just post your google voice/text number or email address on the wall.

    Tools will get out of the way if you are using the right tool.

  4. on 05 Nov 2012 at 8:42 amKate Nowak

    Ben, agreed that those are all potential solutions to this particular “project student work in front of the class” task. (The document camera probably being the simplest, however, using a document camera to take a photo of the work of a train of 18 kids would not be a quick process, either. Livescribe pens in my experience are buggy as hell, and not every 14 year old is carrying around a cell phone, as hard as some might want to wish it into being so.)

    However this is not the only thing we need to do. It was just one example of the annoyances that crop up with one particular task. The idea with the tablets is to get the same robust, multi-purpose tech in the hands of every kid, and they are supposed to be the medium for lots of potential classroom tasks: seamless access to Geogebra/Desmos/Internet, sharing files, taking photos, annotating work, etc etc.

  5. on 05 Nov 2012 at 9:09 amDave Major

    Take that Prensky.

    The problem as I see it is we are all looking for a solution that does one specific thing amazingly well with an amazing experience (take a photo and email it to Kate without disrupting learning), but expecting it in a device that we also demand does everything this way.

    Over the past couple of months I’ve heard “yeah, that’s cool, but I can do the same using x, combined with y and converted using z, backing onto Dropbox” far too many times.

  6. on 05 Nov 2012 at 9:26 amJonathan

    I think part of the problem is the kind of tablets they dumped on you. Windows 8 being new and these tablet being built for rock bottom prices some of the issues could be the wrong tool for the job. Depending on student logins is another thing. In my setup we have these junk net books that require logins. We have iPads that don’t. Rather than have kids log in to a website for email, each iPad is tied to the same photo stream (via iCloud) and has a Dropbox as a fail safe. Uploading to Dropbox is simple, no logins, no nothing, and I can see everything quickly.

    I think the problem is the way in which the people buying the hardware think it will be used. Email, logins, and all that are clunky clunky.

    But I will say that the better, faster, 100% no problem way to deal with this is pencil and paper. Analog methods are usually the best. There are ways to get the tools out of the way, but too many schools are buying the wrong tools.

  7. on 05 Nov 2012 at 10:21 amDavid Wees

    Is a computer not a natural medium for mathematics, or is not a natural medium for the mathematics that we teach in schools?

    Mathematics is such a broad subject. I strongly suspect that there are many areas of mathematics which are naturally suited to computers, and which are relevant, useful things for our students to know.

    Could it be that our dissonance between what we can share via a computer, and what we can share via paper is because the mathematics we use has been largely designed and developed with the medium of paper in mind?

    I’d love to see a strong k-12 mathematics curriculum designed with the assumption that every child always has access to a computer, and the computer becomes the medium of communication, rather than pencil and paper. Of course our current curriculum doesn’t qualify.

  8. on 05 Nov 2012 at 11:11 amMike T (@gfrblxt)

    @David – If the goal is to communicate mathematics, then I agree with Dan – the computer is not a natural mechanism to do that. We teachers could generate worksheets (!) using computers, but it’s the discussion aspect of math education – which I think we all agree is lacking in general – that either requires low-tech (pencil, paper, board) or really fiddly tech solutions (I put smartboards, interactive pens, tablets, iPads, and practically any BYOD/1:1 type solution in that category). Right now, I vote for low-tech: I get much more useful feedback and dialogue by having kids at a regular whiteboard answering and asking questions, or writing answers on miniature whiteboards, or writing solutions and having students trade them with one another, than I do trying to use clickers, Socrative, or any other tech mechanism.

    The real problem, as I see it, is one of ease of communication. When an English teacher or a history teacher or a foreign language teacher wishes to communicate with their students or to implement technology that encourages dialogue, they have options – e-mail, blogs, Twitter, video chats, etc. These work because of the purely verbal nature of their dialogues. My wife, a biology teacher, holds review sessions via Moodle chat for her students in the evenings before tests and quizzes. She can respond to questions, give feedback, and so on, also because her subject is largely verbal (any images are common to her and her students, either in files she’s given them already or by references from the textbook). I watch her chats with great envy, because I see no equivalent way of doing this in mathematics.

    I say these things with regret, actually, because there are interactive tools that are wonderful for teaching math – GeoGebra, Wolfram Alpha, the SAGE project, Desmos’s calculator. But they are the math equivalent of Bunsen burners and flasks. Not everyone has access to them (because, as @Kate said, not everyone brings a smartphone to school, even if they did the school might not have the bandwidth, and not every school has a dedicated math computer laboratory that any student can use at any time). The other problem with them is that they are also tools useful for specific purposes – which often only have meaning once a particular base of knowledge has been achieved.

    I guess I’m still not convinced that the math that everyone needs to learn will require computers to do so. Computers are good tools for what they are good for, and it’s a lovely dream for every student to have access to them (they will, someday, and we should be talking about what it will mean for math ed when that happens), but I remain skeptical that the math we wish 5-10-year-olds to learn will require their presence.

  9. on 05 Nov 2012 at 11:27 amDavid Wees

    @Mike T

    I agree, the mathematics we want very young students does not necessary require the presence of computers, but perhaps it should at least be designed remembering that they will some day have access to computers, and that some of the tasks we emphasize at a young age might be different if we expect students to be using pencil & paper to communicate the math they know. Perhaps we need a bit more emphasis at a young age on formal logic & logical systems, for example?

    I also think we are looking at a product in process and complaining that it isn’t done yet. Well, we aren’t there yet. We don’t know exactly what computers are going to do to our society and the systems we have in place, but we should start consciously thinking about it, since it is unlikely they are going away.

    What choices could we make to ensure that our now dominant form of communication allows people to easily share mathematical knowledge with each other? I think if we abdicate ourselves from making these choices, we will find one of two things will happen.

    1. People will develop adhoc systems for sharing mathematics ideas and we will end up spending years & years trying to combine them into a single useful system (so we can be back where we are now, with basically a single language with which we share mathematical ideas).

    2. We purposefully create and use systems as mathematicians and mathematics educators which are easy for an ordinary mortal to understand and can be easily implemented.

  10. on 05 Nov 2012 at 3:32 pmColleen

    I agree with David Wees that we need to reevaluate what math we are teaching, at least at the high school level, given our access to computing. But currently I am still tied to standards that require both pencil-and-paper calculations as well as use of computer algebra systems. To carry this out the single most useful tech tool that I have is my class set of networked, document-based calculators. I pads and chrome books look fun, but they cannot offer me the same functionality.

  11. on 05 Nov 2012 at 4:48 pmPaul Topping

    With plain text, we go to a computer first to type it. Many of us have noted how he hardly ever handwrite anything longer than a phone number or address these days. The same can’t be said for math notation. Some can write math using LaTeX but that is far from ideal. Even mathematicians who are LaTeX experts do not handwrite it on paper or a whiteboard. They use standard math notation.

    The world has not yet settled on a standard user interface for typing and working with math notation for a number of reasons. However, many are working on it, my company included. I believe we will get there eventually.

    A user interface for working with math will probably have to be adopted by college professors and professionals before it can be confidently and universally used in K-12 education.

  12. on 05 Nov 2012 at 6:07 pmCarl Malartre

    We made progress since Mac OS Classic or Windows 95, but it’s not good enough.

  13. on 05 Nov 2012 at 6:45 pmlouise

    What do Chinese do when they use the computer? Our problem is the graphics used for math.
    I’m sure that in another 20 years there’ll be something else, but right now I’m spending the time to teach the “digital natives” Excel for graphing, and Word with Microsoft Equation for math writing. If I did not have those I would use open office and the free version of equationeditor. You could use google docs or whatever they’ve decided to call it.
    Sending via email may be clunky, but it’s better than having 150files all labeled “book1″ arrive in my dropbox. Plus this skill, together with the photo-taking skill, are necessary skills. What do you think college students do?
    Someone’s got to teach them. We’re having a nightmare because students have to take a photo against a plain background, and upload it to get an ACT ticket. Kids have to learn to do that. Yep, we’re using my personal camera and a wire.
    Last year I had kids take photos and flip-cam videos of their solved puzzles. They put their names on paper in block print, so I could quickly scan through. I did not have a single student who could not take and attach a photo much quicker than I can…

  14. on 05 Nov 2012 at 6:54 pmDan Meyer

    Comments from Dave Major and Paul Topping featured in the post proper.

    I take exception to the idea that we should start with the medium (computers) and then work backwards and define math accordingly. Silicon Valley has been in the middle of that project for the last decade and the results have been grim.

    It’s a give-and-take relationship with any medium, but computers take more than they give, at this point. Kate and her class are bending over backwards to accommodate that, and for what?

    Paul’s remark about LaTeX is at the center of the matter. Try to communicate the square root of five ninths using a computer. Now try using a paper and pencil. Now situate that root in the context of a larger proof or argument. Now add a diagram.

    Is there any contest here?

    I’m optimistic that computers will eventually have our students doing and communicating more math and more interesting math, but right now they aren’t the natural medium for doing mathematics.

  15. on 06 Nov 2012 at 4:19 amClimeguy

    I would rephrase your conclusion. Right now they aren’t the natural medium for doing most traditonal mathematics. For me computer technology allows me to do mathematics in creative and interesting ways. For example, doing probability experiments/simulations. You can always toss toothpicks to do the Buffon’s needle experiment which I do with younger students, but you can’t beat the computer for doing this: http://www.metablake.com/pi.swf

  16. on 06 Nov 2012 at 5:02 amWilliam

    Climeguy – that’s a pretty cool demonstration, but the computer is actually concealing most of the math. How are the angles determined? What’s their distribution look like? Does that matter? After 2500 iterations on my computer the estimate for pi was 3.24. How many iterations do you need to get an estimate accurate to the tenths place? How long will that take? It’s cool that the computer can compute things quickly, but it’s not explaining or arguing or communicating reasoning.

    I’d suggest that the *actual math* is the reasoning that convinces you that the probablility of a randombly positioned stick intersecting one of the lines is \frac{2}{\pi}. That’s all expressed in code that the reader doesn’t have access to.

    The fault, dear teachers, is not in our computers, to paraphrase Caesar, but in our keyboards. We have these awesome machines that compute things quickly, but we talk to them about math using smaller machines designed for a very different purpose. Keyboard layouts are designed for a particular language; keyboards for English are very different from keyboards for French (much less Greek or Russian). Why isn’t there already a widely adopted keyboard for typing math?[1]

    One possible reason is that the math we type isn’t one language like English [2]. The mathematical language we use to communicate is actually a collection of partially overlapping dialects: we use one dialect to write arithmetic, another for algebra, a third for euclidean geometry, a fourth for cartesian geometry, and more once we leave high school. We include all sorts of diagrams and pictures, and, as Dan so effectively advocates, video.

    We often combine those dialects in a single work, so whatever input method garners wide adoption for communicating math will have to allow people to quickly and easily write in any of those dialects and include all those media. It turns out that none of those dialects is naturally convertible into a linear stream of characters, so the input method that wins won’t be something that ties you to a linear stream.

    A computer input method that lets us declaratively and non-linearly specify relationships between quantities is definitely a step in the right direction. Brett Victor’s Tangle is a good start. An input method that let you do that without requiring you to learn a programming language would be great guns.

    [1] The way you shape the byte-stream on the computer is a red-herring. Programmers have (mostly) awesome languages for building abstractions on top of byte-streams.

    [2] It turns out that English isn’t one language either. British keyboards are different from American.

  17. on 06 Nov 2012 at 5:10 amDon Abrams

    Yeah, I’ve had the hardest time with this lately. I’m creating a quick math skill assessment app for a Principal in Baltimore. The idea behind it is to take as little time out of the student’s and teacher’s time as possible while still providing an accurate assessment of K-5 math skills (since state/city testing is near useless).

    But how are teachers and kids to enter solutions and problems with math notation? Having kids or teachers learn LaTeX is just not a viable solution to me. I would make it a freehand touchscreen-only app, but then the process of manual grading greatly increases the overhead of the assessment. As it stands, a manual paper assessment is still superior (which is what is being used now).

    Any ideas?

  18. on 06 Nov 2012 at 5:11 amClimeguy

    @william the wikipedia page does a pretty good job of explaining the problem. But there are many more good ones out there. Cut-the-knot.org and George Reese’s is a favorite. Google can come up with many others. I love the math behind this thing. It still amazes me that it works. BTW – Ive been running the simulation for about 20 mins now and I’m getting 3.14 etc.

  19. on 06 Nov 2012 at 5:20 amWilliam

    No doubt there are good explanations. The wikipedia page is a good read if you already know a lot of mathematics. If you’re learning, though, it’s a harder read.

    Happily, if I don’t know what a random variable is, I can click on the wikipedia link and find out. Wikipedia’s discoverability for english words and terms is awesome.

    Vexingly, if I don’t know what the $\int$ symbol on the wikipedia page means, there’s no link to find out. How would I ever discover what that notation means? Because the mathematical notation is TeX compiled to png images (!) they’re not discoverable at all.

    I wonder how hard it would be to modify the program that produces the png images to add image-map links to the mathematical notation that go to the appropriate page on the wikipedia? Time to go digging!

  20. on 06 Nov 2012 at 9:10 amPaul Topping

    Since people are talking about Wikipedia’s math, I should point out that Wikipedia now support MathJax (www.mathjax.org) for rendering math. However, that is not the default. The user must log into Wikipedia and change a setting to see it.

    Once math is displayed using MathJax, the math can be copied to the clipboard as MathML which can be pasted into many applications and retain its mathematical meaning and structure. This all works even if the source for the math is TeX. Such math can also be accessible to students with disabilities via screen readers and other AT software, including my company’s MathPlayer product. Such math is also potentially searchable.

    MathJax is an open source JavaScript engine that displays math represented by MathML or LaTeX in all modern browsers. It is supported by Design Science (my company), American Mathematical Society, SIAM, and several other societies, publishers, and elearning companies.

  21. on 06 Nov 2012 at 9:15 amChuck

    I dream of a school where students come in with a simple little device (think smart phone with wifi connectivity) that takes all of the administrative work off the plate for teachers.

    I don’t have to take attendance, I don’t make seating charts, and my grading time is reduced. Then I get more time to do the part of my job I like – teach.
    And I imagine I still teach through some form of talking to my students and writing on the board.

    I think the one thing we can all learn from the Khan Phenomenon is how engaging it is to watch writing appear with the speaker removed from the picture.

    I’m hoping there will be a shift in the digital learning sphere away from trying to make kids learn better into trying to allow teachers more time to teach.

  22. on 06 Nov 2012 at 2:21 pmmr bombastic

    @Don, Why do they need an app? I think this gets back to the original point of the computer getting in the way. Hand out paper and grade it. Neither assessing nor grading routine skill oriented problems takes a lot of time. This app is a solution for a probem we don’t have. I already know what skills they are missing. I need an app that helps the students learn the skills (that I already know they don’t have).

  23. on 06 Nov 2012 at 7:04 pmAndrew Price

    Tools will consciously be in the way only as long as math is represented in a non-visual manner. But when it’s represented without latex, the generalization doesn’t hold. It looks like this…

    http://www.mindresearch.net/cont/programs/demo/tours/SolvingLinearEquations/progTour.php

  24. on 07 Nov 2012 at 4:48 amDon Abrams

    @bombastic The structure of the assessment is non-traditional– the students keep answering questions until they are unable to answer two questions in a row. The final one they get right is their ‘level’.

    This is possible because the school has already linearized certain paths in K-5 math (which I slightly disagree with, but I can collect data to prove if this is a good model or not and convert it to a graph model if linearization is a bad model). So during this assessment, after a student answers a question, a teacher immediately determines the correctness of the answer. This doesn’t scale very well in a classroom setting (volunteers often proctor these currently), which is why the app makes more sense. Right now it is being done with paper. I was asked to do this by the principal to help solve a problem he definitely has (and to help spread this out to the rest of the city).

    That being said, when you get a new student or a new class of 30 students, do you really know what skills each and every student is already missing? In Baltimore City, a 4 grade level swing within a classroom is not outside the norm.

    I think I am going to go the tablet only route. I’ll create some screen where a proctor (or proctors) can assess correctness for a classroom sized number of people taking the assessment (maybe even remotely). This will probably work way better than any automated grading, but still allow it to scale. We’ll see.

  25. on 07 Nov 2012 at 10:59 amcb1601ej

    This sounds like instrumentation theory (French School, names Trouche, Verillard, Rabardel, Chevalard, Lagrange, Artigue) which states that ‘instrumental genesis’ has to take place i.e. bridging the gap between mental schemes and tool use.

    Actually, there’s still a lot to do, but there are more elements that are ready than is in this post. Just think of implementing http://webdemo.visionobjects.com/equation.html?locale=default (which has been mentioned earlier in a comment to a post) or win8 Math Input Panel in an HTML5 or other tool.

  26. on 18 Nov 2012 at 3:00 pmDanijel

    There are software tools specifically designed for rapid math typing (the Math-o-mir notepad software, for example). Skilled students are able to copy math notes from blackboard directly to their computers at suficient pace.

    Still, typing math on a standard keyboard is unnatural because the keyboard is designed for text typing, not for math. The math notation, on the other hand, is designed to be written with a pencil.

    But math wants to be written. Therefore I think the math notation will change so that ‘digital natives’ can write it. Eventually the math will be written on machines, but it will look somewhat different than it is now.

  27. on 19 Nov 2012 at 3:35 amGnostradamus

    “I think the one thing we can all learn from the Khan Phenomenon is how engaging it is to watch writing appear with the speaker removed from the picture.”

    My head imploded here.

    Am I the only one who finds the idea that students who struggle with instructors in the room can learn from a video player utterly ridiculous?

    Anyone who thinks Khan is the model to emulate thinks teaching is about lecturing to a screen.

    Too much time and money is wasted on technological solutions to problems that arise only when you try to use computers to do things that work fine on paper. That includes voting in elections, and teaching math.

  28. on 19 Nov 2012 at 8:27 amjosh g.

    “Paul’s remark about LaTeX is at the center of the matter. Try to communicate the square root of five ninths using a computer. Now try using a paper and pencil. Now situate that root in the context of a larger proof or argument. Now add a diagram.”

    Okay, this is a bit old already but it still bugs me, because I don’t see this as being that big of a challenge *if* you’re still talking about me the teacher, standing beside the student talking about it together.

    Here are the steps:

    1) type “sqrt(5/9)”

    2) discuss

    3) open Geogebra, start drawing geometry (which conveniently doubles as a way to compute values related to that geometry)

    Is “sqrt(5/9)” cheating? I don’t know. There *is* a way to type mathematical formulae into computers in an easy-to-type way, by necessity because programming languages use them all the time. The only major flaw is that there isn’t a universal notation for every function – different languages have some slight variations. But honestly, anything Algol-like will be close enough, and if you type “squareRoot(5/9)” instead of “Math.sqrt(5/9)” a human programmer will know what you mean anyway.

    So the *real* problem is that the version of mathematical notation that’s easy to type is substantially different than the version that’s easy to write. In my mind, though, that doesn’t make one superior or more intuitive than the other. This doesn’t make the computer version less “math” than the paper version.

    Now I’m just talking about the typing issue here – I will never disagree with how big a pain having kids deal with digital handins can be, or other technical meltdowns. (Although once I finally wrestle kids into handing files in onto my course moodle site, the “keeping stacks of handins organized” problem is taken care of for me, which for me is freakin’ huge.)

  29. on 19 Nov 2012 at 9:29 amPaul Topping

    I have to disagree with josh g. As he points out, many programming languages support mathematical expressions but they are mostly all different in various details. And “sqrt” is not a radical sign. It all gets much worse when you start getting into accents (primes, hats, etc.), superscripts, and subscripts. Add units, piecewise functions, intervals, set operations, and your simple programming language gets very complex. Most importantly, it doesn’t look at all like the math in your math textbook.

    I do believe that this kind of programming language math IS less math than the paper version. As proof, mathematicians have used TeX for over 30 years and they still use standard math notation, not TeX itself, to communicate with each other. The two-dimensional aspect of standard math notation is very important to understanding because it models the underlying mathematical relationships.

  30. on 19 Nov 2012 at 12:53 pmDan Meyer

    Paul:

    And “sqrt” is not a radical sign.

    Agreed. And to the argument that the square root symbol is just as much a convention as “sqrt,” we could make a strong case for tradition (ie. the challenger needs to prove itself more than the incumbent) but also a case that we’d be replacing a ubiquitous medium with a scarcer one. Which is to say I have a pencil and paper at my ready much more often than I have a MathML parser.

  31. on 19 Nov 2012 at 1:10 pmPaul Topping

    No one is proposing that an end-user knowingly work with a MathML parser. Though, since you mention it, there are probably already several applications on your computer that contain MathML parsers.

    In fact, every Windows computer with Windows 7 or 8 contains Math Input Panel, an application that lets one handwrite math using a tablet and paste it into apps that can handle it. This happens behind the scenes using MathML, though there’s no mention of it in the user interface. There are other apps available elsewhere that also do this: VisionObjects and Enventra.

    I am not suggesting that math handwriting recognition is the answer to our dreams. IMHO, we can do much better with direct manipulation, WYSIWYG interfaces. After all, virtually no one uses handwritten input to their word processor. Typing is much faster. We just need to get this right for typing math.

  32. on 19 Nov 2012 at 1:20 pmjosh g.

    Arg arg arg, so much I agree with and yet so many nitpicking buttons of mine being pushed all at once! *bleargh*

    – I totally agree that a world in which kids couldn’t parse written rational expressions properly is a sad, scary world.

    – The “proof” Paul mentions doesn’t hold water with me at all. TeX is a convoluted mess meant to capture every detail of how to *format math notation for publishing*, and as such is twice as complex as most programming languages’ methods of doing math.

    – Most of how we type math into computers is basically function notation, which is something kids should be learning anyway.

    – I totally agree that I don’t want to push math education into a tech-dependant corner where kids can’t operate outside of Mathematica or Python or whatever. But why is this an either-or situation? If we agree that function notation is worth learning anyway, it doesn’t seem impossible for a kid to know that “sqrt(2)” means the same thing as what I write down as radical-sign-with-a-two-inside.

    – Given that most applications of math lean heavily on computing, isn’t it in our students best interests to know how to translate pencil-and-paper notation into something a computer can actually do computations with? (Again, that does not include TeX or MathML, which are used to format something for publishing, not do computations with.)

  33. on 19 Nov 2012 at 1:38 pmPaul Topping

    By all means, let’s teach kids programming. Progamming is a useful skill in the modern world but learning it also teaches them more fundamental skills such as divide-and-conquer, logic, abstraction, binding, etc.

    However, programming and functional expressions are not really mathematics. Most programming languages (and the ones mentioned in this thread) are algorithmic and imperative whereas standard math notation is declarative. For example, = in most programming languages is an assignment operation. This is not the same as = in mathematical notation which declares both sides as having a certain relationship, usually numerical equality but not always.

    Programming languages and math notation are very different kinds of languages because they describe quite different things.

  34. on 19 Nov 2012 at 7:17 pmDan Meyer

    josh g:

    But why is this an either-or situation? If we agree that function notation is worth learning anyway, it doesn’t seem impossible for a kid to know that “sqrt(2)” means the same thing as what I write down as radical-sign-with-a-two-inside.

    I was in the faculty lounge at Stonybrook last week. Two members of the faculty were conferring and they were drawing on the chalkboard not because computers weren’t readily available (they were) and not because the profs didn’t know how to use them (I’m sure they did) but because it wasn’t the easiest, most natural medium for expression.

    I agree that knowing more ways to express oneself (either in math or in language) is a good thing. It’s also an investment, though. There are costs and benefits. I’m open to arguments that the benefits outweigh the costs but I don’t see it in this thread.

  35. on 20 Nov 2012 at 8:40 amJames Key

    I think that we can make the same argument about texting. When all we had was the 10-key pad, it was really cumbersome to send even a simple text message. (viz. “texting is not a natural medium for communicating your thoughts)

    Then we got “T9 Word” and things got a little better. Then we got QWERTY keypads and things got easier still.

    Can’t we be optimistic that things will get better for us math nerds in re: using computers to communicate?

  36. on 28 Nov 2012 at 10:09 amSurf Movies

    I think that we can make the same argument about texting. When all we had was the 10-key pad

  37. on 02 Dec 2012 at 12:25 amMRE8

    Agreed! I am special education teacher in training, math is not traditionally my strong subject, but my current student teaching stint has forced me to quasi-master high school level geometry very quickly…and to acquire a new nemesis: Carnegie Math online. Really? While trying to accomplish the already impossible seeming task of massaging geometric theorems/formulas AND the prerequisite basic linear equation solving skills (that should have been learned last year) into reluctant teens with specific learning disabilities, emotional disturbances, and a hot text buzzing in their pocket, I have to also teach them the separate, non-transferable skill of using Carnegie Math’s clumsy “solver” to solve for the radius of a circle given the area?!!!! That’s why I have dry erase markers and laminated white paper always handy. I better stop here before this becomes a rant.

    I will admit that for the right student, higher level math on the computer can be rewarding. But, often, pen and paper is just more efficient and authentic – especially as geometry goes (we don’t measure ANYTHING on a COMPUTER with a protractor…) I believe at least one part of the story is that computer based learning often gets support because it makes data collection and assessment easier for TEACHERS…

    -end rant here-

  38. on 02 Dec 2012 at 7:13 amKevin Hall

    MRE8, I think those of us who are enthusiastic about these systems just believe the problems you’re describing are solvable. It’s pretty easy to imagine, 10 years from now, students doing Carnegie Learning work with “pencil and paper” (stylus and tablet), as long as the software has decent handwriting-recognition.

  39. on 02 Dec 2012 at 8:46 amMRE8

    Kevin, I am VERY enthusiastic about leveraging technology in general. I have all faith that we/they will find a solution. Many companies are a few ideas away from making digital “paper” a practicle reality (think sheet of paper that functions like an ipad). I guess I’m just disappointed with the current state of what I’ve observed in my – admittedly – narrow sample.

  40. on 03 Dec 2012 at 2:51 pmPaul Topping

    I am not a believer in “digital paper”. As long as we are looking for something that electronically simulates paper and pencil, we will be limiting ourselves to only what can be done with paper and pencil. Our digital simulation will only approach, not meet or exceed, the positive qualities of the medium it simulates.

    While TeX has its merits and its fans, in many ways its presence has slowed progress toward a better math UI. Its low price (free) and ubiquity has aided its adoption, but its nature is such that it is difficult to improve. To make matters worse, its creator, Don Knuth, dictated that the TeX name can only be applied to systems that process TeX as defined when it was invented. In other words, any attempts to improve TeX can’t be called TeX!

    Unfortunately, most of the market’s attention on new technologies, user interfaces, mobile devices, etc. does not include mathematics. I do believe this will happen in time. When it does, its capabilities will be far beyond those of paper and pencil. Our digital math facilities just need time to mature.

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