Computers Are Not A Natural Medium For Doing Mathematics, Ctd.

You may have heard that San Jose State University’s recent partnership with Udacity ended with MOOC-enrolled students passing courses at much lower rates than their on-campus cohorts. Lots has been said about these results (Phil Hill has a good round-up of the coverage) but there’s one line that deserves more coverage:

When students did get to the online programs, even navigating the computer systems could be daunting. One of the questions that tutors were frequently asked was how to do exponential notation on a computer.

Again we find computers are not a natural medium for doing mathematics. There’s nothing intuitive about pressing Shift + 6 to write an exponent, no inherent connection between the idea and the action. This isn’t true for computer science, where the medium is perfectly suited for the course. Or even for English composition, where typing words is only one intuitive abstraction away from writing them with a pen.

I’d wager 90% of people reading this already know how to type an exponent on a computer. They believe it’s easy enough to teach and I don’t think they’re wrong. But this is only one instance of a problem with a lot of reach. Notation makes math difficult on a computer. But notation also makes math more powerful and interesting. That tension will be very difficult to resolve and, so far, online math providers have generally resolved it in favor of the computer at the expense of math’s interest and power.

In our relentless transition from classroom-based math to computer-based math, these SJSU-Udacity results offer us a chance to pause and ask ourselves, “What’s now missing?”


Computers Are Not A Natural Medium For Doing Mathematics

2013 Jul 26. Okay, taking friendly fire on Twitter, I posed this challenge:

Use a computer to compose a clear proof that a triangle’s midsegments create similar triangles and send it to me for assessment.

My guess is you’ll find the process a lot less annoying and a lot more clear when you pick up a pencil and some paper.

Featured Comments

David Lippman:

But on the other hand, the reality is that if our students use math any time later in their life, there’s a really good likelihood it will involve a computer, whether it’s using Mathematica to solve complex problems, doing computer programming, or just entering formulas in Excel. There is value in learning the notation for entering formulas in a computer, and it provides an valuable side benefit of reinforcing proper syntax to ensure proper order of operation.

Mr. K:

I’ve slowed down on the Euler Project problems, but last I checked, I was just short of 200. Those problems require a computer to do the math, but you don’t do the math on the computer. I have a notebook (ink on paper) dedicated to that that has a couple hundred pages filled with notes and drawings — that’s the math.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. It could be worse – you could be looking at something like the SBAC pages, which are “designed” to be functional even without a keyboard, and therefore require you to use drag and drop for just about everything.

    @Rhett: COBOL and Fortran too. Now we just need to install the source code compatibility patch in our students, and we’ll be good to go.

    I suspect part of the problem is that education is still revisiting the evils of software development from 20 years ago, where engineers expected people using the computers to adapt to the software. Trying to develop interfaces that don’t get in the way is less than 10 years old as a common practice, and still really in its infancy.

  2. I think your definition of “computer” is a little too narrow. Multitouch computing is a game charger for certain topics and subjects, also the first generation of windows tablets with the precision of the stylus seemed to work well in math classes. The tools are improving, but they are still just tools whether a pencil or a laptop.

  3. I have a love/hate relationship with entry notation on a computer. Sure, a palette-based entry would make things easier, and sure, paper-and-pencil is more natural. For initial exposure and practice in the classroom, and for exploring new concepts in an fluid way, paper works great. And sure, learning the notation adds a layer of complexity that risks detracting from the primary lesson.

    But on the other hand, the reality is that if our students use math any time later in their life, there’s a really good likelihood it will involve a computer, whether it’s using Mathematica to solve complex problems, doing computer programming, or just entering formulas in Excel. There is value in learning the notation for entering formulas in a computer, and it provides an valuable side benefit of reinforcing proper syntax to ensure proper order of operation.

  4. A note on computers and math:

    I’ve slowed down on the Euler Project problems, but last I checked, I was just short of 200. Those problems require a computer to do the math, but you don’t do the math *on* the computer. I have a notebook (ink on paper) dedicated to that that has a couple hundred pages filled with notes and drawings – that’s the math.

  5. Here is a question: What is a more powerful way to teach/introduce kids variables? Using pencil and paper or having them add a score keeping functionality to a game they created in Scratch or Etoys?

    Also define “doing mathematics.” Is it defined as being able to “write” the arbitrary symbols that are efficient when using pencil and paper? Are those symbols “intuitive”?

    Instead of asking what are computers, not good for, why not ask what are they good for?

  6. Apparently, I missed the day they taught us why writing a tiny number up and to the right of another number is an intuitive, natural expression of exponentiation. Big fractions definitely are harder to read in a linear programming language-type notation, but exponents? This is what the calculators show now, and it’s how spreadsheets and programming languages work, so I don’t see how we can avoid teaching this. But that’s a minor point.

    The problem is that computers are good enough communication tools for things that are more common than communicating about math. The people who built early computers already knew basic math and can deal with programming language syntax, so we have programming languages and TeX, which doesn’t help someone who is still developing mathematical literacy. Then normal people took over the development of computers as a communication medium and found ways to do word processing, audio and video over computer networks, but they weren’t interested in doing K-12 math, either. And there were some serious constraints on displays, memory, storage and processing that no longer exist but which still cast a shadow over what we are stuck with.

    Today’s computers and user interfaces barely respect the cognitive limits of adults. Given that history, there is no reason to think they should be good tools for educating children. But kids and teachers are more adaptable than technology, and there are no philanthropists who made a lot of money off of teaching who want to spend it promoting teachers.

  7. Math is traditionally done with pen or pencil on paper, chalk on chalkboard, marker on dry-erase board, or marker on an overhead-projector transparency.

    They’re all roughly similar: allowing free-form writing and drawing on a blank canvas.

    Why can’t that be done on a computer?

    It can! You might use a Wacom pen and tablet, or you might use a touchscreen like on an iPad, or whatever. The notes could be collected using an app like Note Taker HD.

    Those are still not common, but they’re more common than they used to be. Many students take notes with an iPad and a special pen (or similar tablet). I tried it and found the lag to be too high. Microsoft Research has prototype touchscreens with much lower lag.

    There are even apps that can recognize handwritten mathematical notation and convert it to LaTeX or whatever. But that’s not needed to make notes for oneself, or to make notes on a screen for a class, or to do homework to be handed in – so long as it’s legible-enough.

  8. Go to:

    and watch 13:45 to 15:13

    In the video Alan Kay (one of those early Computer Scientists who actually did think about how kids could use computers to learn) talks about how the how “Jacques Hadamard a famous French Mathematician decided to poll his 99 buddies, who together made up the 100 great mathematicians and scientists in the world and ask them how they did their thing. Only a few out of the hundred said they used mathematical symbology at all. Most said they did it in imagery and figurative terms, and an amazing ~30 percent (included Einstein) were down in the mud pies. Einstein said ‘I have sensations of a kinesthetic or muscular type’. The sad part is that every child in the US is taught Mathematics and Physic through this channel (using mathematical symbols). They (mathemeticians/scientists) use this channel to communicate, not to do their thing.”

    So why do we force them to go through a layer of symbols to understand concepts?

    Why do we not use kids visual/kinesthetic/spatial sense to help them learn?

    Seymour Papert did this when he came up with Logo which uses Turtle Geometry.

  9. I think Carnegie Learning’s interface for drawing proofs with flowcharts is excellent.

  10. Here is a response to Dan’s challenge:

    This is a Computational Document Format file, which requires the free CDF Player to view:

    but the very expensive Mathematica to create:

    So I freely acknowledge that this might be cheating, since Mathematica is a commercial package beyond the means of most K12 schools, but that is meant to make a computer a better place to do mathematics. Still, this “proof” (really a proof without words) that the medians of a triangle form a similar triangle seems to me to be accessible and useful for a student learning elementary geometry. I await assessment of this proof by Dan and/or other readers.

    Not sure how much this vitiates Dan’s argument, even if folks find this proof useful and clear. The proof itself carefully avoids the use of notation (Even the triangles vertices aren’t labeled!), which seems to be Dan’s gripe, and it isn’t exactly news that computers are useful places for manipulating graphics. But this does get to Mr. Steve’s point, I guess: maybe we shouldn’t get too caught up in notation–introduce it as needed to conveniently convey the ideas at hand–instead work on developing reasoning and modeling capabilities, and for this purpose, visualization and other sensory apparition is primary.

  11. Cal Armstrong

    July 27, 2013 - 6:45 am -

    As a 1:1 school, the math department found the laptops next to useless for reasons mentioned above… but 10 years ago we switched to pen-based tablets. Pen-based tablet computers provide the open canvas that paper provides and helps students tie in the graphical/image/dynamic aspects that paper can’t give. Troy, you should definitely check out Windows tablet computers — they’ve been lagless for a decade. The Microsoft Surface (uses a pen) using Microsoft OneNote is an excellent mathematical space/tool.
    The tablet computers also allow students to use audio and video in addition to (or replacing) the mathematical notation they may have issues with writing down.
    Two challenges: (1) screen size. You can set sheets of paper side-by-side to see things in their entirety without scrolling. Until later high school this isn’t typically an issue. (2) permanence — digital ink is too easily erased and the discussion of error has to be a managed and practiced process in a space where there can be no record of what occurred.

  12. I have always seen the use of computers as a means of enhancing the understanding and learning processes in mathematics. Paper and pencil are often the most appropriate media for writing, and if a computer copy is required, use a scanner!
    Regarding the exponent or power notation, there are various reasonable possibilities, a^3, e^x, exp(x+1), but for writing the superscript takes some beating, and given that the students hopefully have got used to + – and even / they really should accept superscripts for powers (it is doubtful that a better written notation exists).
    Regarding the writing of formulae in computer programs, you just have to do it their way, you have no option.

  13. Is there a platform out there with a stylus based interface that incorporates OCR technology essentially programming math notation?

  14. The idea that math is less natural on a computer (writing math was never natural to begin with) is an artifact of not learning it on a computer first – if computers are integral to the learning process through out there is no reason that it should be unnatural.

  15. The general criticism here seems to be, “Computers are the best medium for certain mathematics,” like David Wees with his Mandelbrot sets or David Lippman with his Mathematica for complex problems. That tempers my post title somewhat without fully contradicting it.

    Please keep in mind the terms here. We aren’t talking about an online elective course teaching the ten concepts for which computers are the best medium possible. We’re talking about thousands of people taking required, introductory-level courses where the syllabi are set.

    At any other time, I’d be very interested in Mr. Steve’s reconceptualization of Algebra instruction without any notation or JamesN’s reconceptualization of all of math instruction on computers.

    For now, though, the rules of the game are fairly clear. Students need help with remedial algebra, which includes a lot of notation that computers struggle to accommodate.

    I’m obliged to Rob Root and Mike Lawler (on Twitter) for taking on the challenge of a (relatively) simple Geometry proof using digital media only.

    Rob’s rendering required proprietary software and a player that weighs in at 234MB on OS X.

    Mike’s version uses an entire iPad screen to write a proof that would’ve taken a fraction of the area to write more legibly on paper. Information resolution took a nose dive.

    Do you see what I’m saying? I won’t bet long money against computers in this space but at the moment, for these particular MOOCs, computers are getting in the way of students expressing their mathematical understanding.

    @Kevin Hall, got a link?

  16. I’m still working on the notation thing. Why is exponentiation notation any different using paper and pencil than say, MathType, CAS, or even superscript. None really embodies he meaning of exponentiation. They’re just symbols for a concept. Or is it the certain online math programs can’t easily do the symbols? Which are difficult in a post.

  17. You should take a look at the Mathspace web interface or even better, the app (currently iOS, android on the way). No problem with writing exponents, checking work, ‘OCR’ type functionality and key – showing working out and reasoning. The app also recognises handwriting, so it is as though you are writing on paper. It also has some ‘inner loop learning’ (nod here to Michael Feldstein).

    Yes, I do work for Mathspace, but as it is so new, I reckon few if any of you will have seen how it may help resolve the issue you are discussing.

    Tim (unapologetically a dude!)

  18. Is there really a natural medium for doing mathematics? I know computers may not always be the most transparent way of doing things, but I don’t think any kind of notation really conveys the meaning of something like exponentiation. The ideas don’t exist on paper or a line of code – and I really think that we do carry a lot of baggage about what is “natural” based on simply how we were taught, because those symbols are what we have attached the meaning to.

  19. Computers have not been good at communicating the conceptual understanding that mathematics requires. That is why so many students are failing. Students need a live person to communicate the concepts. Computers are more suited for the number crunching. I received numerous phone calls this summer almost every morning in fact from students who were taking their math classes by computer. The students needed to complete problems that they had no understanding of. The computers were showing them procedures to follow. Procedures in isolation no less. The computer can not teach a student to understand the math. The most it can do is show students a procedure. On Webb’s depth of knowledge the students would only be at level 1 with a rare level 2 after the computer course.

  20. Agreed, computers are not (yet) a natural medium for learning or doing math. I love computers and technology. I used to be a programmer. I am sad that we haven’t yet reached a point where computers ARE the natural medium, since they are so mathematically based. But we’re not there yet.

    I think, for me, the problem lies in the differences between the inputs and outputs. It’s easy enough to say that if you want students to know that a certain notation means multiply a number by itself so many times you should teach that to them. But the fact is, regardless of the input method you give them, they still have to be able to read the standard superscript notation and apply it. Then you’re dealing with two different abstractions of the same concept. If a student already has facility with that concept then an additional abstraction is no big deal. But when one hasn’t achieved automaticity with one abstraction yet, having more than one for the same concept makes it too challenging.

    I also think there is a kinesthetic component to writing that is helpful for the human brain. I’ve used iPads as writing tools before, and while the lag and line fatness are no biggies, I find some students simply don’t process as well without additional tactile feedback. I’ve even had students who seem to learn better when they write on construction paper rather than notebook paper just because of the feedback. Until tablets can get better localized tactile feedback I think that will hinder their adoption as math doing devices. I do know someone who is working on that very problem, but the solution is years away.

    And I have to agree with Ben Hicks. Is there really a natural medium? Perhaps paper is so far the best, but it doesn’t mean it will always be so.

  21. Thanks for weighing in here, Amy and Patti.

    Along with Patti, I’m somewhat sympathetic to Ben’s argument that paper simply has first-mover advantage. Except that paper is also more ubiquitous than computers. Certainly it pays to know how to communicate mathematically in multiple media, but until you can find a computer and communicate mathematically with it as easily and quickly as you can with paper, that argument will have limited appeal.

  22. timteachesmath

    July 27, 2013 - 7:41 pm -

    How could the edX student get his symbol? Draw with “Web Equation” and his symbol appears ready for copy-paste, click icons (and hopefully learn syntax) withSciweavers or Codecogs , take a cell phone picture, Fluidmath?, ask Google how to write exponents or steal a screenshot from a Google images search, . . .

    This is perhaps one more step for students, but I think the tradeoff is worth the effort. Now, they can ask legible mathematical questions via email, blog, or stackexchange, add course notes to a class wiki, and know how to produce symbols for a Powerpoint or paper in Chemistry or wherever these come up for them. These are all skills I’d love my students to have, especially as we dream of a Conrad Wolfram/Art Benjamin conception of math class. I think it is true that a typewritten solution is a different product, just like a typed paper takes more effort and once faced a similar fight, but might the extra effort, attention, and focus have positive learning outcomes as well?

  23. Three distinct (but obviously not unrelated) things are being discussed here: ‘doing math,’ ‘learning math,’ and ‘communicating math,’ and the role of computers is different in each. Computers are amazing tools for ‘doing math.’ When confronted with difficult problems in nearly any domain I almost always turn to a computer. Dynamic geometry programs are great for exploring geometry problems, Desmos and/or Grapher make nearly any problem regarding functions more concrete and approachable, and if I have an algebra or combinatorics problem worth its salt, I almost always use Wolframalpha. Why risk a careless error that could derail an otherwise successful solution strategy? Even spreadsheets provide a great way to examine systems in a dynamic way. I could go on and on, but that’s not really the point is it? Despite the title, this article isn’t really about ‘doing math,’ it’s about ‘learning math’ and ‘communicating math.’

    With regard to ‘learning math,’ I would say that the issue is not that the computer is not a natural medium, but that the instructional approach described is not a natural approach to learning math on a computer (well, really anywhere, but that’s a different conversation). If the learning was centered on ‘doing,’ then MOOCs might be really successful by emphasizing the tools I described above, but if the salient issues of the courses are that students don’t know how to typeset superscripts, then the courses are not focusing on doing.

    Notation, however, is not something we can overlook just because we can do math with some great digital tools. Patti’s point about levels of abstraction is a great one. There really is nothing intuitive about writing an exponent as a superscript to the right, but for better or worse, that is the standard notation and students need to connect the operation with notation. Matching the action with the notation is arguably one of the most difficult aspects of learning math, but precise notation can be a powerful tool for a mathematician. Asking students to learn two or more sets of notation for different contexts all at the same time they are learning the concept only adds to their cognitive difficulties. In that regard, Dan, I think you are absolutely correct that computers are not yet ready.

    But now we are talking about the overlap between learning math and communicating math, and I think (hope) that soon the technology will rise to the challenge. I’m sure that at first many authors preferred typewriters to early word processors for lots of good reasons, but modern word processors have far surpassed their analog ancestors. They are now superior both in terms of actual typesetting and final product, but they also facilitate creativity with the ability to visualize, edit, cut and paste, annotate, version, etc. Why can’t ‘math processors’ one day do the same? Perhaps it will take some awesome blend of typing, speaking, and writing along with software that suggests and predicts symbols, notation, and formating, but looking at how we interact with our smartphones I can’t help but feel that it isn’t that far off.

  24. While I agree that many of those tools (proof tools, computer algebra systems, dynamic geometry, including my own programs) may distract, steal time and actually hinder the proper teaching of math thinking, we must not be too unimaginative. If you extrapolate IT and home computing from 10 years ago to 10 years in the future, it should be obvious that computers will become a central part in education, like it or not. This is a great blog to help to go into the right direction.

    When I do math I prefer the blackboard. But then I find myself copying what I wrote to paper, and then putting my notes somewhere so that they do not get lost. Moreover, I often test my results on a computer with a math software, which scatters things to still another place. The blackboard lessons I do are only kept by my students in their notes. They have to be reinvented the next time I do the same class. When I prepare something on my computer, I have to bring the device to the class. In the end, the only place that keeps all this together is my brain.

    Wouldn’t it be nice to have a tablet computer, which can run my software, photograph blackboards or load the current content of the computer controlled whiteboard, take notes just like on paper, project everything to a beamer, take stuff from mail, add links to papers or books, and collect all this in a single surface of one program? And why shouldn’t students want to do the same?

  25. A couple of comments I would make here:
    – I totally agree with Aran (and others) that a precise and standard notation is important. Seeing that the computer world has difficulties with getting a standard (OpenMath, MathML, Epub vs iBooks) format out there, we certainly should embrace a notational system, in my opinion.
    – This system, imo, is more intuitive than some say. Linear notations sometimes need many brackets, where traditional notations sometimes show structures ‘at a glance’.
    – Furthermore, assessments are and will remain to be on paper. In my opinion this means that we must keep thinking about both computer and digital media and their relation wrt transfer etc.
    – There are developments like those from visionobjects, mathspace (thanks dude Tim will certainly check it out) and some GPL stuff under development. It is possible to WRITE on tablets and I feel it’s one of the few ‘mistakes’ Jobs made when dismissing the stylus. It could combine both worlds, when OCR recognition finally merges. For Dean and completeness sake I paste a recent ALT-messageboard post with some relevant links.
    In my opinion MathML (and OpenMath) never really got adopted that well. There are three ways for input:
    1. One is use normal handwriting on paper and scan. However, scanning can also be done like apps like Camscanner, just photograph the work and e-mail/share it as pdf or image.
    2. Input editors. Like word, but Wiris, dragmath (in Moodle). There are numerous options. Can be quite tedious.
    3. Formula recognition. State of the art is by visionobjects (some may know them from the handwriting MyScript calculator on iOS) at (Equation, link with LateX, MathML, Wolframalpha). As mentioned before Microsoft has a Math Input Panel, and there are some more of them. I must flag up , a project one of my PhD students is working on. Will take some time before it’s done but can keep you posted (or you contact her yourself of course).

  26. One aspect I think it’s usually overlooked is that almost everyone who has learned math the old way has been able therefore to learn computing and even programming by their own means. Learning the basics by intuition and trial/error.
    And Dan, it’s not only the exponential on computers, haven’t you been asked about it on calculators? Take a look at the most common model we use (Spain). See the exponent key? Would it be so difficult to guess? And taking that knowledge into the computer environment?

  27. Here’s my submission:

    This is a proof that the midsegments of _a_ triangle make similar triangles, using 19 lines of open source vpython.

    It is not, however, what you were really looking for – a proof that it works for all triangles.

    Even though it didn’t really nail the assignment, as I was writing it, I thought about how it helped with a different set of skills than the pencil/paper proof did. In particular, finding the locations of the midpoints in a general fashion is the type of thing that I see kids struggling with. Incorporating the dot product (instead of using the VPython built-in anglediff) is another thing that you can get with this approach that you don’t get with the manual approach.

    I use VPython for my AP Physics kids, and I think that the same thing applies – the computer makes some impossible tasks possible, provides a different outlook or requires different skills (in a good way) than the pencil/paper on some tasks, and makes some tasks that are easy with pencil/paper very hard or impossible. For me, it opens up new avenues of exploration, but that doesn’t mean closing the old ones.

  28. nobody ever seems to mention *money*
    in suchlike threads. which is strange.
    because these “computer” things *ain’t*

    in money, or time, or… i don’t know
    quite how to say this, but it’s the dealbreaker
    for me a *lot* of the time… faith-in-the-system.

    because once i’d been messed around
    by corporate telecommunications entities
    a few times, i *despaired* of ever getting
    acceptable services *even if* my time
    and money were somehow unlimited:
    without the sponsorship of some *other*
    corporate entity, i’m their *victim* and
    must be made to understand this by
    being put on hold, while being billed,
    *forever* (in some kafka-was-right
    “before the law” type deal).

    the power goes out routinely right around here.
    but the sun keeps shining and i’ve laid in a supply
    of paper and pencils.

    probably *very* few of our students will have
    “fun, free, easy” access to high-tech
    tools for much longer… *even if* they’ve
    got such access now. so we’d serve ’em well
    to *at least* make sure they’re up on
    the paper-and-pencil stuff…

    thanks for kicking off yet another great thread, dan.

  29. Used to be, not long ago, hardly anyone was expected to communicate mathematics with a computer until…college, and typically only if studying science-y things. (Sure we used graphing calculators and computers to /do/ math before that, but not to express ourselves.) I didn’t have to learn TeX, Maple, Matlab, etc until engineering school. Anyone else not with me? At that point in her education, a student has a certain wherewithal to learn how to type a notation they already understand. We weren’t trying to learn two difficult things at the same time.

    Teaching high school kids, I became sensitive to when technology was going to cause too much friction, and often avoided it for that reason. Learning about the idea of variables, or how precisely an exponential function differs from a linear one, or what a similar triangle /is/ much less how it /proves/ anything, is a heavy enough lift for a 14 year old. Can you imagine having to learn to use a pencil at the same time? When the tech really adds something useful, you have to be very careful to predict when to explicitly be like, “when you want to type an exponent, use a ^”.

    There seems to be a little bit curse of knowledge running through this comment thread. Maybe there is some on the part of the MOOC instructors. There’s necessarily less immediacy about the consequences of assuming a student knows that ^ means exponent when you don’t have to see the panic on the kid’s face.

    Now let’s take kids who haven’t had much success in math, and ask them to learn not just new ideas, but a non-intuitive way of communicating at the same time that they /also have to be taught/? Let’s recognize how significant that is.

    Related: Not that it solves the fundamental problem, and lots of you already know this, but the best work in the area of making the interface frictionless and intuitive and not-scary, even for a user who doesn’t think they’re mathematical, is being done by Han and Jay at Desmos. Hands down. (MathQuill, linked in an earlier comment, is the thing behind the magical typing that happens on the Desmos calculator.)

  30. The friction comes from trying to retrofit an old approach into a new technology. If all you’re trying to do is use a 21st century tool to convey a 3rd century (BCE!) idea, well, bullocks! Employ a modern method with a modern tool, i. e. prove similarity through transformation, namely dilation. If you are interested, see If you feel that this is missing classical “rigor,” please explain to me and your fourteen-year-old students why this is not a compelling proof. (On a computer or with pencil and paper, your choice.)

  31. Sure it’s compelling but you have to admit there is a-LOT going on there. I kind of doubt that would prove anything to a 14 year old because of the amount of information coming at them all at once. Though I could see a kid constructing the figure and giving a verbal argument along with it. I’ve taught kids to use Geogebra for inductive reasoning, to test out conjectures, and to construct arguments, but you actually kind of prove my point. I had to be very intentional about teaching them Geogebra. It took a huge amount of instructional time. It was worth it, but we’re kidding ourselves if we assume learning Geogebra as part of a course isn’t a big decision with implications to instruction that most teachers don’t understand very well.

  32. Point taken, but this would not be an isolated use of GeoGebra, rather a natural application within a curriculum with transformations as its foundation. And I am not kidding myself, it is a sea change in pedagogy and content, but one whose time has come. We have the tools to let students explore the beauty and connectedness of geometry – we need to show them how to use those tools. Just because Euclid laid out what was known in one fashion, does not mean we are forever bound to that axiomatic system. New tools -> new approaches.

  33. I’d encourage everyone here to read Aran Glancy’s comment in full (and Kate’s — both bumped to the main post) particularly the parts that call me out for overreaching in my post title.


    But now we are talking about the overlap between learning math and communicating math, and I think (hope) that soon the technology will rise to the challenge. I’m sure that at first many authors preferred typewriters to early word processors for lots of good reasons, but modern word processors have far surpassed their analog ancestors.

    I’m certainly open to (and hopeful for!) that future. Your view acknowledges that the current slate of digital tools has costs along with their benefits. I don’t disagree. My intention here is to call out the technologists and MOOC-ophiles who only see benefits.

    @Kevin, looks promising!

    Kate Nowak:

    There seems to be a little bit curse of knowledge running through this comment thread.

    The upsides of these tools are so prominent and the downsides so obscure when you already know how to use them. But we’re talking about kids who often need a tutorial for copy and paste.

    Jennifer Silverman:

    If you are interested, see If you feel that this is missing classical “rigor,” please explain to me and your fourteen-year-old students why this is not a compelling proof. (On a computer or with pencil and paper, your choice.)

    After two browsers and a 30% hit to my laptop’s battery, I downloaded the file and opened it locally. (I’m sure you can imagine my smug, self-satisfied expression throughout. “Jennifer’s totally proving my point!” I said to myself.)

    I guess I’m curious where you see this proof fitting in the Van Hiele levels, or if you see those levels as outmoded.

    I’m curious if someone “reading” the proof will understand why the numbers aren’t changing. (Assuming she understands, first, what the numbers are describing.) Will they realize that corresponding sides are proportional, and that that fact alone isn’t enough for similarity, but the fact that one triangle is “nestled” into the other also fixes their angles. Will they have the opportunity to see that, by proving similarity, we’re also one step away from proving that the midsegment is parallel?

    Is any of this important mathematics or has Geogebra outmoded all of it?

  34. I’m afraid that GeoGebra’s html5 display did not give you all you needed to see. Here is a screen shot that includes the construction protocol as well as the algebra panel. I think it’s pretty easy to follow and explain when you see how it was constructed.

    Since this would not be an isolated proof in the curriculum I’m working on, students would get that the midsegment is parallel. (See for a bit more on that.)

    As for whether or not GeoGebra has outmoded the “important mathematics,” I’m sure you know the answer. (“Trolling” was explained to me earlier today…)

    There has to be more than one approach if we are expected to teach this in a meaningful way to EVERY kid in America. Why can’t we use the tools of our era to answer the questions of antiquity?

  35. @Jennifer:

    If you feel that this is missing classical “rigor,” please explain to me and your fourteen-year-old students why this is not a compelling proof.

    How about… producing a bunch of examples — which is what the program can do — is not the same thing as a proof? That the program does not eliminate possible pathological situations that might break the circumstances? That this sort of worry occurs all the time in mathematics, and one of the primary goals of mathematical education is to convey they difference between examples and proof?

    While I’ve worked with computer results plenty of times in proof production, and will even accept a proof where all cases were checked by computer rather than by hand, I don’t see how the GeoGebra program is exhaustive in this case. Also, the program is to an important extent lying. Here is one of the examples produced:

    3.67/1.83 = 2
    3.47/1.73 = 2
    3.08/1.54 = 2

    these values are incorrect. 3.67/1.83 is approximately 2.00546 and 3.47/1.73 is approximately 2.00578. Only 3.08/1.54 is exactly 2.

    There is of course rounding going on, but given an entire branch of mathematics is dedicated to handling computer-rounding issues (hence avoiding critical errors in actual engineering problems), that’s hardly something to be hand-waved over.

  36. @Jason:

    Yes, I know all of that. I am advocating for acceptance of a “proof” that makes sense to reluctant learners, that engages them because it is visual and comprehensible. I certainly want those students who are motivated to learn and to appreciate the beauty of an elegant proof in the classical style to do those proofs. However, much of that is lost on other students, who then think the whole thing is stupid and shut down. A very quick search led me to: That is not my vision of a modern, relevant course in geometry.

  37. Jennifer:

    As for whether or not GeoGebra has outmoded the “important mathematics,” I’m sure you know the answer. (“Trolling” was explained to me earlier today…)

    My point is I’m not sure where the important mathematics is in the Geogebra proof.

  38. @Dan, @ Jason

    It is true that manipulating the Geogebra sketch only allows one to verify that for all the vertices one chooses the ratio of sides is uniformly 2. No one should mistake that observation for a proof, and Jason is quite right to point out that we should aspire to conveying the power of abstraction and generalization to our geometry students. To that end, This construction begs for the rigorous reasoning that I offered in my cdf: The triangles AB’C’, A’BC’, & A’ B’ C are all similar to ABC with the same scaling of length, 1/2, and so the sides of A’B’C’ are uniformly scaled sides of ABC, and thus these are similar triangles. Need to grant Dan that, as it stands, this is at a van Heile level of about 1: analysis, but the diagram can, with the right prompting (and the right students), take us all the way to level 4: deduction. (I’d say level 5: rigor only comes from extended experience at level 4, and so this could be a step on the way there, but not the whole thing.)

    I’m hearing Jennifer say that the offered examples are as much as she can expect of her students, and until we’ve done some time teaching in her classes, who are we to argue?

  39. Max Ray (@maxmathforum)

    July 29, 2013 - 2:27 pm -

    What I’m wondering, from Jennifer and others, is if the Geogebra sketch is one she shows her students, or one her students showed her.

    If I asked students, “what can you tell me about the triangle formed by the midpoint of two sides and the vertex where they meet?” and they showed me that sketch, I’d be pretty thrilled. Especially if they explained that they dilated Triangle ABC to show it always landed on the midpoint triangle, essentially asserting that there exists a dilation from Triangle ABC to the midpoint triangle and that therefore Triangle ABC is similar to the midpoint triangle.

    On the other hand, showing students the sketch doesn’t seem like it would be helping them understand the many things it could mean to prove that the two triangles are similar.

    Stepping students through “creating” the sketch so that I’m doing the thinking and they’re doing the clicking also would not be helping students understand what it could mean to prove two triangles are similar.

    I’m willing to accept finding and explaining (justifying?) a transformation that maps one figure onto another as a decent rigorous proof for high-school students, and one for which computers would be a lot more intuitive than paper and pencil. But ONLY if it is the students who are doing the thinking, planning, constructing, trying, erring, explaining, and justifying. And the explaining and justifying is probably best done orally (technologically linked to the constructing using a screencapture podcast), second best in writing, and only third best, if at all, in typing.

    Also, while we’re trolling, I’m going to say that Sketchpad is worth paying for because not only is it better designed to be closer to intuitive than Geogebra, but it was also built incredibly thoughtfully to do many math-y things better than Geogebra. And because I’ve heard that Geogebra was stolen intellectual property which is NOT the same as thinking something up for yourself and deciding to make it free.

    Disclosure: despite the origins, the Math Forum has no current relationship financial or otherwise to Geometer’s Sketchpad.

  40. The approach I am taking to Geometry is to begin with a study of transformations. The algebra required to demonstrate the mapping of one figure onto another through coordinate geometry would be adaptable by the individual teacher to the particular group of students. (I write for a program in 12 public schools; I keep expectations flexible by scaffolding and providing editable resources that teachers can strip down.)

    I would expect that students could construct this proof (or demonstration, justification, etc.) on their own, since they will have already verified the conjectures that translations, rotations, and reflections produce congruent figures and dilations produce similar figures, in an earlier unit.

    I am very sorry to hear that GeoGebra may be “stolen intellectual property,” and I’m wondering if you might be able to point me toward a reliable source to corroborate what you’ve heard. At this point, I am committed by my organization to open-source software, but that allegation leaves me feeling very unsettled.

  41. @Dan

    “My intention here is to call out the technologists and MOOC-ophiles who only see benefits.”

    That is a noble intention, and you are right to point out that many people only look at the benefits. The natural response to this is to point out the weakness of which it is clear there are many. My question is, however, how do we steer the conversation away from arguing which list (the list of pros or the list of cons) is bigger, toward a conversation about how to strike a balance that is beneficial to students?

    I would agree with those challenging Jennifer that a sketch in GeoGebra is not a rigorous proof, however, I would counter that the need for rigor is often lost on students. To them it comes across as something we make them do just to make geometry harder for them. But what if it went like this:

    T-What can you tell me about the triangles formed by the midsegments of a triangle?
    S-[after creating GeoGebra sketch] They are are similar to the big triangle?
    T-Can you find an example where there aren’t?
    S-[After moving points around] No.
    T-How can you be sure?
    Enter rigorous proof.

    @Max: I’m a big proponent of GeoGebra and actually think it is superior for many more reason than just because it is free, but I value intellectual property much more than I value GeoGebra. Please share your reasons for believing that it is stolen intellectual property!

  42. Max Ray (@maxmathforum)

    July 29, 2013 - 7:31 pm -

    I probably shouldn’t have posted about a rumor that I can’t back up with anything more than hear-say. It’s pretty clear from some internet searches that Key Curriculum has never made a public statement or any legal statements against Markus Hohenwater. All I can say is that I’ve heard professors and friends of mine say that’s why they won’t use Geogebra. But if no one is on record saying that, I was remiss in passing along hear-say.

    Further internet searching also led me to notice a lot of impassioned defensiveness on both sides of the Sketchpad/Geogebra debate, and since I’m opposed to internet defensiveness I’m going to back way off and say that having used both tools I love dynamic geometry, period. I love that not only can it become an intuitive tool, it can actually change your intuitions about shape. What it means for a rectangle to sometimes be able to be square but always still be a rectangle feels different when you’re used to rectangles that you can drag vigorously. So people should use the dynamic geometry software that they have available to them, a lot, because it’s awesome. And they should appreciate the user communities that have made Sketch Exchange and GeoGebra Tube wonderful places, and they should figure out which one has the math behavior and the user interface and the features they most want. And while I’d assert that an individual should be willing to pay if the one they like is the one that costs money, I understand that there are lots of situations that’s not feasible in. But if I had a class set of laptops, I’d prioritize buying Sketchpad over buying graphing calculators, for example. I might even prioritize it over whiteboards, and I’d definitely prioritize it over a Smart Board. Might even prioritize it over a daily trip to Starbucks, but that’s easy for me to say since I’m allergic to caffeine.

    Finally, colleagues have reminded me that while for a long time the Math Forum and Key Curriculum had no financial relationship, at the moment there is some kind of deal where we are offering courses for graduate or continuing education credit that are cross listed with Sketchpad training courses. So while I didn’t remember that while I was initially defending Sketchpad, you may now consider me extra biased towards Sketchpad. Sorry guys. I’ll go back to my policy of not trolling because it’s never the right thing to do.

  43. I slept on this; here’s some more thoughts.

    The midsegment problem is a particular case of this:

    If you dilate two sides of a triangle with the same scale factor, leaving the angle between them constant, you will get a similar triangle.

    What the Geogebra needs to handle Dan’s objection is some way to slide the scale factor to values other than 2. This also dovetails nicely into Jennifer’s transformational geometry curriculum.

    What’s more, this general version of the concept connects up with other mathematics; for example, the definition of the tangent function (in such a way that even in a Conrad Wolfram style all-computer curriculum I’d say the concept is essential), or this standard from Common Core (8.EE.6):

    Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting the vertical axis at b.

  44. @Jason

    Thanks for your honesty. Like I said, GeoGebra works best for me, but you are right that Sketchpad is a great program with many strengths.

  45. I agree a lot with Aran Glancy’s distinction between doing, learning, and communicating math (post #24). Computers are so good with the doing part. Students are supposed to be doing the learning part. The communicating part is, I think, the part you’re looking at, Dan.

    I’m in a similar boat. While I don’t usually teach math, I do teach chemistry. Computers are my bane when it comes to writing formulas and compounds. Besides superscripts, subscripts are also necessary, and if you’re doing nuclear chem, then all four “corners” can be used. My students are still figuring out where to put which number and what’s capitalized. If they also have to figure out exactly which keyboard strokes to do it, they’ll be awfully frustrated.

    I think the bottom line is, computers aren’t much of a tool for learning if the students don’t know how to use them.

  46. Back to the OP, like Cal above I’ve been doing almost all of my math for the last eight years on various (mostly pc/Windows) tablets. As a high school teacher, I wanted to save, edit, project and publish/share my solutions and lessons. You can see hundreds of these in Journal files and PDFs at our school math dept wiki As many contributors said above though, typing algebra is a such a pain, without stylus input to Windows Journal I just wouldn’t have stuck with it.

    But tablet computers have changed how I //do// math (and how I teach.)

    I’m much more likely to haul up Wolfram|Alpha to check a result (or get a hint, now it will show some methods); I’ll verify and illustrate a result in Geogebra or Cabri since I’m already at the computer; I’ll use animation, video or photography to explain including to myself. I might even do a simple bit of programming to search for particular results or counter examples. I’ve used various online tools to rehearse routine skills to gain fluency and get rapid feedback.

    Above all, as I am doing now, I’m more likely to collaborate.

    With pencil and paper, I think I was slower, more routine in my solution methods and relied more on following standard examples and doing standard problems.

    I wonder if anyone else has found using computers to do math changed how they work?

  47. Mr. Steve:

    Only a few out of the hundred said they used mathematical symbology at all. Most said they did it in imagery and figurative terms, and an amazing ~30 percent (included Einstein) were down in the mud pies. Einstein said ‘I have sensations of a kinesthetic or muscular type’. The sad part is that every child in the US is taught Mathematics and Physic through this channel (using mathematical symbols). They (mathemeticians/scientists) use this channel to communicate, not to do their thing.

    So why do we force them to go through a layer of symbols to understand concepts?”

    This hits on several of the themes here — experts vs. beginners and doing vs. learning vs. communicating.

    Experts are saying this as experts — however, were they taught mathematics using “muscular” methods or without symbology? I’m guessing that these sensations of math are WHY they are experts now — it was a subject that to them is multi-faceted, fully engaging, in their brains and bodies in pictures and feelings. What percentage of K-12 students have that level of engagement?

    Was this the way that they themselves were taught K-12 level math? Extremely unlikely.

    Are these the ways that they communicate math to one another?

    The points made about learning basic skills of notation and not expecting students to learn two or more notations for one concept they have not yet grasped are key.

  48. George Bigham

    August 5, 2013 - 8:02 am -

    I recently took the Machine Learning class at Stanford through Coursera which is the kind of class I hope to prepare my HS students for. In that class the prof always introduces ideas, explains, and ‘prototypes’ in written or pre-typed math notation. Written math notation or simple 2-d graph sketches are very useful to learn from at the theoretical level. The work on the computer is only done later for what you would expect a computer to do: compute.

    I guess the moral of the story is that doing hand written math to get an understanding of concepts is best, even for the most tech advanced. The power of computers come into play when we apply these concepts to particular data. Unfortunately many people confuse math with computation, but computation is just the often messy and tedious by-product of applied math and it can now be passed on to the computers.

    So I think it is best to leave hand written math as the first medium, especially at lower grades. Once students get to a level where they can apply the abstract concepts to big numbers or excessive computations, they should turn to the computers. Basically I think computers should just be used as glorified calculators because if computer science professors still prefer hand written math to introduce and justify concepts, math teachers can too.

  49. Hi, Dan,

    I wanted to point out that although it is not intuitive to use Shift + 6, for example, to write an exponent, Wolfram|Alpha with its free-form linguistic capabilities eliminates this problem and even helps teach users the proper inputs. Also, since version 8, Mathematica has incorporated Wolfram|Alpha’s free-form linguistics in its programming
    environment, as noted in the blog post below:

    Teachers can also reference Step-by-step math solutions (a feature of Pro) by visiting here:


    Amy Wolff
    Wolfram Research

  50. There seems to be a lot of confusion about the meaning of “doing math”. I have been “doing math” for at least 50 yrs now, as pupil, student, teacher of math, statistics, control engineering and more, and math is “done”in the head. Then it get written down somehow. A large amount of which is screwed up and thrown away. A computer with suitable software is able to do some of the algebra and a lot of the geometry, but in the end the useful bits have to be collected together for presentation purposes. I have been writing computer programs for math and engineering for many years, and my conclusion is that the goal posts keep moving!
    Have a look at , my site, with three programs that you may find useful.

  51. While ultimately math is done in the head, physical and other representations can be an important part of the process, whether with diagrams in game theory or pattern blocks used in the early grades or exploring functions with a graphing calculator. Fractal geometry grew out of the representations possible on computers.
    Doing math includes thinking about and exploring representations.
    That’s where I see technology, computers, as a natural part of doing math.