There is a feedback aspect that this article does not seem to get.
> After all, the material hasn't changed (much), so if the calculators were good enough for us 10 or 15 years ago, they are still good enough to solve the math problems.
Technology changes the material: it changes what you think you need or want. You could almost say dialup internet was fine for viewing text web-pages -- why does it need to be faster? Broadband enabled web video -- which in turn spurred the need for broadband.
The calculator example looks a bit like a phone. If you took a phone from 2000 and one from 2007, they would look pretty similar. It is still a phone, is it not? We do not need something more, do we? And then the iPhone appeared.
> Here's the thing. Some technologies don't change all that quickly because we don't need them to. . . . Look at cars or power plants
Crikey! If you wanted to hit on two of the things we do most need to change, and have for decades, it might well be just those.
The whole article disturbs me a bit, actually, because it seems dangerously full of the anti/un-creative mindset. You will never invent anything if you just look at what you have and think of justifications for why everything is pretty much fine. You create by finding faults and imagining what you do not have. Look at those two calculator pics, and think of them as representing some part of the web now and in 2021. Scary? Well that is what it will be unless you get irritated and make some weird unexpected new stuff!
Excellent point! Commonly told anecdote, lady sees Faraday's inventions and asks "What is it good for", he answers "What good is a newborn baby!" (I just found out that this didn't exactly happen this way, see http://www.jstor.org/pss/986790, but you get the point). Some tech advances are pushed by necessity, some just happen (or may be pushed by some other necessity) and create the question or need that they then answer.
I loved the HP48 series in high school and college, and m48+ brings it to my iPhone. For the rare times I feel the need for something more powerful than the calculator built into Spotlight, and something less powerful than Clojure, I reach for it.
Never go into the HP50 series, though it does look nice. Probably would have started using it if I were still in school.
RPN FTW! I was never able to lend my calculator out because it took too long to explain how it works.
I got (and still have) a copy of the Advanced Programmer's Manual for the HP 48G. Great environment for someone to learn programming while you're bored at school.
Want to see a non-incremental innovation for multi-touch calculators?
A friend of a friend wrote this: http://mathtouchapp.com/ (He doesn't know I'm linking it here).
The app throws away the calculator metaphor and starts from scratch. Instead, the author uses the back-of-a-napkin as the metaphor.
You start with a blank page and add systems of equations. You can visually link variables together. You can insert values along with its unit of measurement and numeric precision. You can feed results to graphs.
At $10, this is cheaper than getting the Nspire if you already have an iOS device. But of course, you can't use it when taking the SAT. And it would be cool if you can export it to Wolfram's computational data format and trade formula libraries.
You need to actually understand the math instead of just punching the buttons. Then again, isn't that what Sal Khan's videos are for?
I'm suspicious that I won't find it useful. Pencils and paper are awfully nice. I bet that even a trained user of this software can't write an equation faster than I can.
Sometimes it is nice to be able to typeset equations, or feed them into Matlab or Mathematica, and I guess this might help with that. But I don't own Matlab or Mathematica anymore.
And, yes, the paper requires you to know that m is a mass with units of kilograms and q is a charge with units of coulombs, and that the m in one equation is the same as the m in another. But if you can't keep track of that yourself I'm not sure why you're bothering to write down equations.
I don't remember. I'm not very good at writing posts that get a lot of upvotes. You are more then welcomed to try -- this app can get all the marketing help it gets.
Thanks for the link. I just finished listening to the talk in its entirety. I would love to know what you thought about the actual content of the talk ( ie. not the presentation skills...Conrad Wolfram is charismatic and his analogies are compelling ). I personally think he has completely missed the whole point of math, why does one do math, math education, and problem solving using math.
Interesting. I thought the most compelling part of the talk was his interpretation of the purpose for doing math and how we've failed to educate toward that purpose.
I assume you're referring to his assertion that deep understanding of by-hand calculation is not necessary (the driving a car vs servicing a car analogy). I would agree that this may be a false analogy, and that a basic understanding of mathematical concepts (presently drilled into place through hand calculation) may be a fundamentally important step to being able to do math. Especially for those who continue to any advanced math study, fundamental understanding is almost certainly necessary.
But this talk is more about the common student who doesn't go on to college or post-grad math. Overall the broader point of the talk still rings true to me: because of an overemphasis on calculation, the majority of students are receiving insufficient math education to successfully make life decisions and contribute to their own fields of work/study. In my mind, Wolfram presents a viable solution in recommending refocusing math education towards logical thinking and correctly structuring problems, and in utilizing computers for calculation.
>the most compelling part of the talk was his interpretation of the purpose for doing math
Exactly. Except that he misinterprets that very purpose. He assumes the purpose is entirely utilitarian. He implicitly says math should be about solving common real-world problems, like his concocted life-insurance premium minimization, which helps the common student make life decisions (as if!). While 100% of pure math folks would actively disagree with him, even 99% of applied math folk would have a tough time agreeing with his utilitarian mindset. We simply don't do math to just pay the bills. People were trisecting angles and squaring the circle and computing surds for over thousands of years with no utilitarian aspect whatsoever. He also says math problems must be rephrased as CS programming problems!!! I'm all for algebraic computing tools, and Mathematica is a rather neat addition to that category, but aside from that, the rest of his talk is beyond hyperbolic.
Koblitz makes the case against computing here:
http://www.math.washington.edu/~koblitz/mi.html
But there's a very wide literature on why math & computers don't mix in the classroom - read Dr. Jeffery King on the art of math (http://www.amazon.com/Art-Mathematics-Jerry-P-King/dp/030644... ) - there's whole chapters in that book devoted to debunking what Conrad is talking about.
I don't think there's much competition, either. HP appears to have bowed out of the graphing calculator market years ago. (though a Google turns them up?)
Also, for high-end math wizardry, it's very easy to simply not use a calculator: Maxima [1] obseleted my calculator for tasks that weren't tests
I suspect as smartphones get cheaper and finish their takeover, the calculator firmware will get loaded into an in-app emulator and there they will live.
The HP graphing calculators are still alive and kicking.
the HP 50g is honestly a far more awesome tool than the TIs (and for those few who care, allows RPN entry too). (it also has a pretty outstanding distribution of reviews on amazon)
Yes portable ipod touches and the like are moving to partially supplant these tools, but there are a lot of use cases where a calculator with dedicated physical button and a far better battery life than any apple gadget, while still fitting in your pocket and being a general purpose CAS, is quite a nice baseline.
The HP 32SII (discontinued, now $139 on ebay; originally sold for less than half that price) and the HP 35S (redesigned version of the 33 which was the ugly successor to the 32SII) are non-graphing RPN machines that are very good for quick calculations. (My first and favorite HP, the 41, is much older than these, but it's too expensive to have one laying around everywhere I might need one.)
Even if I'm in front of a computer, typing in to a console window, it's faster to turn on the calculator than to enter bc, python, or matlab and type in the calculation. And with RPN you get to see the intermediate results so you know you're not forgetting something.
dc is the standard RPN console app that's been around forever (according to Wikipedia it predates the C language). It loads instantly but I always have a terminal window running it somewhere.
I used a TI-66 as an undergrad (Purdue - Computer Technology), but I graduated from college(the first time) in 1988. Was great that I could program in frequently used equations. Later as an MBA student I picked up an HP 19Bii Business Consultant Calculator for all the financial functionality.
They let you use a calculator on the SAT now? That is my "You kids get off my lawn" moment of the story.
Yes, because they are more interested in your understanding of math, not your ability to perform arithmetic. In my math and physics classes in college and high school, I typically only picked up my calculator once I had solved the problem. My real answer was an expression with variables in it, but to check my work with the back of the book, I had to plug numbers into the expression I derived.
Ideally, wouldn't math classes teach students how to solve the parts of problems that computers aren't as good at? I mean, I'm happy that I can use memorized multiplication tables to make my quick estimates of various quantities accurate to two orders of magnitude instead of one, but that's about all they're good for. Anything more, and I pull out my phone or head over to wolframalpha. And I'd much rather have spent those years of brute-force arithmetic practice advancing to higher, more interesting math.
The point of a math education is to understand what's going on. If you can't work problems, you don't really know what's happening, and you will have trouble applying the concepts outside of strictly defined problem spaces from classrooms.
Learning to use computing to do the heavy lifting is great, but I often see learning to use the sin() function used as a proxy for learning what sin() is, for example.
> If you can't work problems, you don't really know what's happening
On the contrary: If all your 'work' can be done even faster by a machine, you don't know any more than the machine does and your skills are worthless once your wages are more expensive than just buying and maintaining the machine.
Real mathematicians use software like R and Matlab and so on, because their real skill lies in knowing what to do with the software and how to interpret the results it gives.
Real mathematicians use software like R and Matlab and so on, because their real skill lies in knowing what to do with the software and how to interpret the results it gives.
Of course! But what I would contest is that (at least in my humble experience) knowing what to do with the software and how to interpret the results comes with learning about how those results are produced.
When I want to analyze a complicated circuit, I don't use a pencil and paper, I use pspice. But I would not be able to create a useful model, run useful tests, or make sense of the output if I was not able to (at least once upon a time) do it on paper.
This post seems to contradict what you said above:
> It seems like more and more, math classes are teaching students how to use a graphing calculator to avoid solving problems.
This is, simply, wrong: Calculators are banned for as long as students are learning to calculate. Once they are actually solving problems, as opposed to grinding through numbers (which isn't something they need to be doing anyway), they're thinking in ways calculators cannot match. A calculator can only do the most mechanical, least problem-solving-like parts of solving a problem.
I haven't changed the batteries in 8 months. I can do math on my TI-89 Titanium without even looking at it, that's not possible with any tablet. Real buttons are worth the extra money. Depending on the complexity of the problem, there's many I'd rather use a calculator than even Maple or Mathematica, both of which would be an absolute nightmare to do on my iPhone. Also, you can still program it, which isn't available on anything iOS related unless you used javascript somehow (Fucking up the semi-empirical mass formula in Nuclear Physics was practically a pastime on my homework before I wrote a small basic program). Furthermore, no teacher/professor in their right mind is going to let anyone use a device capable of wireless communications on a test.
Calculators haven't been allowed on the GRE either but with the new one (it either started this month or next month), they are offering an on-screen calculator built into the test.
Yup. You could make a decent argument that a graphing calculator has been marginalized into a $70+ aid for high school exams and standardized tests. That's the one thing that computers/phones can't encroach on.
Geogebra is open source and funded with EU grants. They're about to launch version 4, I think, but I don't care because v5 has been in stable beta for the last 6 months.
The fault lies with the college board. A useful standardized test would either:
* not require a calculator at all, testing understanding instead of computation or
* allow access to real world tools-- including free ones like http://wolframalpha.com and http://desmos.com/calculator (full disclosure-- I helped build the latter)
How much time is wasted teaching the unnecessary skills of how to use an antiquated, expensive device merely because tests require it? We should be teaching our students which resources are available, which to use in which situations, and how to plug in the gaps between them
People find it amazing that an analog oscilloscope from 1985 sells for $200 on the used market. Or that a milling machine from 1965 sells for $1,200. Tools, certainly solid tools, derive their value from their ability to meet the need, and once silicon density intersected with the needs of high school math they reached 'equilibrium.' From that point on, a solid calculator has an intrinsic value. What is more its pretty clear that the value is higher than the cost of manufacturing it, so building solid calculators is nearly always going to have some 'profit' associated with it.
Personally the TI-92+ was the pinnacle, it was basically a Sun-2 workstation with Macsyma installed in a handheld unit. I've still got mine :-)
The reason an analog 'scope from 1985 can fetch $200 is a combination of things, and you're forgetting at least two:
1) A quality scope is expensive to make. Those 'scopes, expensive as they may seem, are priced at a fraction of the cost to make one; it is not just their utility, but also the cost of making on that keeps their value above $5.
2) Modern, new quality 'scopes cost thousands of dollars, so $200 seems cheap in comparison.
True, both of these. Although the recent move to 'USB' type scopes has some people wondering if maybe moving all the 'display/ux' bits to a laptop warrants the price reduction.
I was trying to explain the difference between the Tektronix probe frontend (expensive american brand) and the Rigol (cheap chinese brand) scopes to a hobbyist and they couldn't really appreciate the flat frequency response from DC to the scope's rated frequency of the Tek front end, whereas there was a little more than 1dB dropoff in the frequency response on the Rigol as it approaches its rated frequency.
(Oh and the Tek actually 'works' to nearly double its rated frequency and the Rigol certainly does not do that)
I worry though that 'good enough' will make it hard to find quality test gear going forward.
Amazed at being expensive or cheap? With oscilloscopes and milling machines, the old adage goes, "they don't make them like they used to."
To get better performance that some of those old 'scopes would cost thousands of dollars today. Not everyone needs color graphics. Give me a green line and I'll be fine.
+1. I've got a 50mhz DT telequipment d83 which is older than me and built like a battleship with no integrated circuits at all. It is bomb proof, feature packed and still as good as new!
I still have my TI-83+ from my highschool days. Honestly, I couldn't survive without it. When I have to do hard core matrix algebra or graphing, and I need to be in front of my book, I can't use the PC or tablet -- it just doesn't fit in the zone. The physical nature of the calculator, the portability, and tactile feedback are all necessary.
Come back to teenagedom in 1985. You listened to the FM radio, because CD's hadn't been invented yet. You made calls on a landline, and yes, it was called "the phone". You didn't have a beeper. You didn't have "the internet" (it had been invented, but you had not heard of it yet). Instead, you had local dial up BBS's on a 1200 bps modem (if you were very lucky) or on a 300 bps modem (typical). You most likely had an Atari 2600 as your "game system" (that is, if you had a "game system" at all). And your calculator of choice was the HP-15C.
Well, maybe most didn't have a HP-15C, but that was the calculator I had. Still have it actually, and it still works as well today as it did then.
You're a bit off about CD's. They weren't popular yet, but they did exist. They first went on sale in the US in 1983, and 4% of albums sold in 1985 were on CD.
For portable music in 1985, there were also cassette tapes.
While I am geeky, I always felt that the graphing calculators did not contribute much to my math classes. And neither were they a significant gateway into technology - few people went beyond the simplest functionality.
Other people must be thinking this as well, but it seems likely/obvious that the graphing calculator will go extinct, replaced by an app on a touchscreen device (not an emulated ti-82, but a better more intuitive graphing calc). Sounds like a pretty promising start-up. That the TI-82 is still $70 is unbelievable.
You cannot use general purpose device like tablet/smartphone in tests. I think more realistic market would be a integrated teaching and exam environment for desktop computers (yeah, those things still exist), with necessary math functionality built-in, but intentionally limited enough for educational purposes.
Show me a $70 smartphone that can run for months on a single charge, looks fine without a backlight, and has instant response time, and I'll use it to replace my TI-83.
Maybe. I'm not sure I want my calculator showing an alert about what my friends are doing tonight when I'm trying to focus.
Actually, I started working on something like this last week, as a pet project to keep me busy. In essence, what you need is a basic programming language that's geared a little bit more towards mathematics (so you can do little things like 2log(n) instead of 2 * log(n) and actually use π as a constant instead of something like Math.PI), a REPL and a way to graph functions. That's feature equivalence, and that's easy.
The cool part is when you start thinking about how to innovate on top of that. Then you can add things like:
* the ability to display and remember the results of a whole range of calculations on a single screen instead of always having to write stuff down or bring back old calculations, e.g. http://stuff.stdout.be/graphcalc.png (think iPad app)
* the ability for teachers to follow along on their own screen as you're doing calculations, to help out, maybe even allow teachers to take over control (remotely) to show stuff
* save calculations and make little reports, which could serve as homework
* slowly learn general-purpose programming along the way, simply by having the thing based off a language like CoffeeScript instead of Texas Instruments' own half-assed TI-BASIC.
* extensions for physics and chemistry, so it's not just about math
I don't know if it'd really make sense as a startup (tablets are still too expensive and the nice thing about graphing calculators is that they're small and people can't browse the internet on them) but the whole "let's try to reinvent some ancient technology using modern tools" thing is pretty fun.
If anybody else is looking for a weekend hack project, I'd gladly open-source what I have thus far.
Frankly if you replicate the TI-83's layout in a GUI, and emulate its OS, and sell it for $10 on the App Store, you'd make millions. Any student carrying around an iPhone/iPod Touch/iPad would gladly pay that, rather than $100 for a new bulky TI-83, and use that in school instead. Makes their lives simpler by only having a single device.
Replicating the interface is key. From my experience, most teachers were horribly confused if someone brought in a TI-89 or, forbid, a TI-92. Why? Different UI that they don't know. You can't teach students how to use their calculator unless you know how to use it yourself.
After, you can add in all sorts of extra features like you mentioned. But "A TI-83 that runs on iOS" is probably the first feature that people will pay for.
It does bring up the issue of students cheating on tests, using their iOS "calculators", but last time I checked it's not too hard to hide notes on your TI-83 either...
It's an idea I've had for an iOS app for a while, but don't have the time to dedicate to it (working on something else).
The presence of a network connectivity on an iPhone (or even an iPod Touch) is a bigger problem than the ability to put notes in it. Notes are just a basic help, while the full answer is a different matter.
Have a 14 year old TI-89, that I still use to this day. It's small, simple, quick, and have all the functions and symbols I use under the custom menu. Amazing their haven't been any screen and function improvements (built in programs).
> After all, the material hasn't changed (much), so if the calculators were good enough for us 10 or 15 years ago, they are still good enough to solve the math problems.
Technology changes the material: it changes what you think you need or want. You could almost say dialup internet was fine for viewing text web-pages -- why does it need to be faster? Broadband enabled web video -- which in turn spurred the need for broadband.
The calculator example looks a bit like a phone. If you took a phone from 2000 and one from 2007, they would look pretty similar. It is still a phone, is it not? We do not need something more, do we? And then the iPhone appeared.
> Here's the thing. Some technologies don't change all that quickly because we don't need them to. . . . Look at cars or power plants
Crikey! If you wanted to hit on two of the things we do most need to change, and have for decades, it might well be just those.
The whole article disturbs me a bit, actually, because it seems dangerously full of the anti/un-creative mindset. You will never invent anything if you just look at what you have and think of justifications for why everything is pretty much fine. You create by finding faults and imagining what you do not have. Look at those two calculator pics, and think of them as representing some part of the web now and in 2021. Scary? Well that is what it will be unless you get irritated and make some weird unexpected new stuff!