Actually, the correct syntax for x squared is x**2. Well, at least in python.

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.

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.

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.

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.

Here is a response to Dan’s challenge:

http://db.tt/yMbwePEP

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

http://www.wolfram.com/cdf-player/

but the very expensive Mathematica to create:

http://www.wolfram.com/mathematica/

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.

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.

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.

http://mathquill.com/

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.

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.

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.

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?

Static version here:

http://www.carnegielearning.com/galleries/6/

Click on the link for the proof tool

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 http://webdemo.visionobjects.com/ (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 http://www.mathpen.co.uk/ , 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).
=====

Here’s my submission: http://www.wikiphys.org/images/similartriangles.py

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.

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.)

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 http://www.geogebratube.org/student/m45125. 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.)

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.

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. http://bit.ly/16vXu5V 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 http://bit.ly/11Ysm1y 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?

@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: https://share.ehs.uen.org/node/12530. That is not my vision of a modern, relevant course in geometry.

@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?

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.

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.

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.

Sorry meant that last one @Max

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.

Yup, looks like it.

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.

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:
http://blog.stephenwolfram.com/2010/11/the-free-form-linguistics-revolution-in-mathematica

Teachers can also reference Step-by-step math solutions (a feature of Pro) by visiting here:
http://www.wolframalpha.com/pro/step-by-step-math-solver.html

Best,

Amy Wolff
Wolfram Research

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.