# Category Archives: Calculators

## Rebuilding the Antikythera Mechanism out of Lego

Andrew Carol, an engineer at Apple, has rebuilt a model of the ancient Antikythera Mechanism entirely out of Lego blocks. Watch this amazing 3-minute video:

A fuller story behind all this is here. I feel like running out and buying out the entire stock of Lego from some unsuspecting toy store.

I was just talking with an older colleague of mine yesterday — he’s been teaching math at my college for over 50 years — about how technology has changed since he started, and I remarked that in many ways I’m more amazed by the mechanical calculator technology of the 50’s and 60’s than I am by modern digital computers. I remember my Dad bringing home an old mechanical calculator from his work and opening it up to reveal gears upon gears inside. Watching this video reminds me of that.

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## Student (mis)understanding of the equals sign

Interesting report here (via Reidar Mosvold) about American students’ misunderstanding of the “equals” sign and how that understanding might feed into a host of mathematical issues from elementary school all the way to calculus. According to researchers Robert M. Capraro and Mary Capraro at Texas A&M,

About 70 percent of middle grades students in the United States exhibit misconceptions, but nearly none of the international students in Korea and China have a misunderstanding about the equal sign, and Turkish students exhibited far less incidence of the misconception than the U.S. students.

Robert Capraro, in the video at the link above, makes an interesting point about the “=” sign being used as an operator. He makes a passing reference to calculators, and I wonder if calculators are partly to blame here. After all, if you want to calculate 3+5 on a typical modern calculator, what do you do? You hit “3”, then “+”, then “5”… and then hit the “=” button. The “=” key is performing an action — it’s an operator! In fact, I suspect that if you gave students that sequence of calculator keystrokes and asked them which one performs the mathematical operation, most would say “=” rather than the true operator, “+”. The technology they use, handheld calculators, seems to be training them to think in exactly the wrong way about “=”. What we have labelled as the “=” key on a calculator is really better labelled as “Enter” or “Execute”.

In fact, the old-school HP calculators, like this HP 33c, didn’t have “=” buttons at all:

That’s because these calculators used Reverse Polish Notation, in which the 3 + 5 calculation would have been entered “3”, then “5”, “+”, then “Enter” — and then you’d get an answer. What HP calculators label as “Enter”, on a typical modern calculator would be labelled “=”, and in that syntax lies a lot of the problem, it seems.

The biggest problem I seem to encounter with “=” sign use is that students use it to mark a transition between steps in a problem. For example, when solving the equation $3x - 2 = 10$ for x, you might see:

$3x - 2 = 10 = 12 = x = 4$

The thought process can be teased out of this atrocious syntax, but clearly this is not acceptable math — even though the last bit of that line (x=4) is a correct statement. If the student would just put spaces, tabs, or even a semicolon between the steps, it would be a big improvement. But many students are so trained to believe that the right answer — the ending “4” — is all that matters, they have little experience with crafting a good solution, or even realizing that a mathematical solution is supposed to be a form of communication at all.

What are some of the student misconceptions you’ve seen (or perpetrated!) with the “=” sign? If you’re a teacher, how have you approached mending those misconceptions?

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## Wolfram|Alpha and the shrinking future of the graphing calculator

Image via Wikipedia

By now, you’ve probably heard about Wolfram|Alpha, the “computational knowledge engine” that was recently rolled out by the makers of Mathematica. If you haven’t, here’s a good place to start. There is considerable debate among ed-tech people as to exactly what kind of impact Wolfram|Alpha, abbreviated W|A, is going to have in education. For me, W|A is still a little raw and gives back  too many “Wolfram|Alpha isn’t sure what to do with your input” responses when given mathematically legitimate (at least they seem so to me) queries. But the potential is there for W|A to be a game-changing technological advance, doing for quantitative information what Google did for text and web-based information back in the 90’s. (W|A is already its own verb.)

One thing that seems clear is that, with technology available that is free and powerful and hardware-agnostic, technology that previously has ruled the ed-tech roost can’t survive for much longer. I’m thinking particularly of the graphing calculator. These have been a fixture in math education, especially at the pre-college level, for the better part of 20 years. But now here is W|A, which can graph functions, perform symbolic algebra and calculus computations, even solve differential equations and do number theory and statistics and all manner of interesting stuff besides, including but very much not limited to mathematics. In short, it does everything a graphing calculator does. But, importantly: W|A is free, runs on any web-enabled device (including, as I can attest to by experience, an iPod touch), is fast, is portable (see the links I just shared?), and — perhaps most importantly of all —  has an army of developers who are constantly adding new features into the system.

You could spend \$150 to get the latest and greatest from Texas Instruments, a handheld device that does what a graphing calculator does — but no more. (Here’s my first-hand take on the NSpire and details on what I see as its demerits.) Or, you could spend a little more than twice that much and get a netbook computer that gives you access to W|A as well as a suite of office tools and more. Computing hardware has become so small and cheap, and online quantitative tools so functional and powerful, that it’s very hard to see how graphing calculators can survive the next 5 years.

If graphing calculators do survive, it will be for one main reason: The AP exams. I was talking with a local high school AP Calculus teacher this week who impressed on me that  she cannot afford to drop graphing calculators and move on to using netbooks or some other more sensible technology because, quite simply, there are questions on the AP Calculus exams that require the use of graphing calculators. Students have to have total fluency with graphing calculators — and not some other, calculator-like technology — in order to do as well as they possibly can on the exam, which is part of this teacher’s professional responsibility. The AP already succeeded in killing the TI-92 calculator — a really good technology for its time, when laptops still weighed 15 pounds and costs thousands of dollars — for no better reason than because it had a QWERTY keyboard. Today, the AP might succeed in keeping W|A and other similiarly useful, perhaps even transformative, technologies out of the hands of students pretty much for the same reasons, which is a real shame and quite backwards-looking.

But then again, I don’t know what the AP folks have in mind. Perhaps there are plans afoot to migrate the AP exams away from dependency on graphing calculators. It certainly wouldn’t take much for the AP folks to write their own lightweight graphing tool that does nothing more than plot functions, find intersection points, shade in areas, and do numerical integration (rarely are graphing calculators used on the AP free-response portion for more than these four things). Make it extremely basic, put it on the web, free for all to use, and provide it on specialized computers for students taking the exam. That way, students can learn how to use technology rather than learn how to use a graphing calculator, and both teachers and students can be freer to choose the extent and type of technology they want to use in their classes. And such a thing would probably have a longer shelf life than any TI calculator for sale or in production.

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## Calculator blasts from the past

One of the real treats of the ICTCM was the Saturday 8:00 AM session titled “Three Decades of Handheld Devices: How Mathematics Teaching Changed Along with Them” given by John Kenelly. Prof. Kenelly has a long history of involvement in the development of calculator technology, and he gave a fascinating talk full of good thoughts on the direction of handheld technologies today, war stories from the past, and good jokes. (Example of the latter: “Getting a spreadsheet to work on a calculator is like getting a dog to walk on its hind legs — it can be done, but it ain’t pretty!”)

I will try to say more about Prof. Kenelly’s ideas about the future of handheld technologies in a later post, but for now I wanted to share one of the really cool parts of his talk — the calculators themselves, some of which are now antiques. He had a bag full of these old-school devices (some of which are less than 10 years old but still old-school) which he generously let us paw over.

Here is a Hewlett-Packard HP 35, the world’s first handheld scientific calculator, from 1972. Check out that red LED display and, in contrast with the NSpire, the sheer paucity of keys on the keyboard:

Here’s a rare example of a Casio fx7000, from 1985 — the world’s first handheld graphing calculator.

I was downright startled to learn that sitting right across the aisle from me at this talk was Hideshi Fukaya, the lead engineering on the development team for the Casio fx7000 and the person rightfully considered to be the inventor of the graphing calculator.

Moving ahead up the timeline, here is a Casio Cassiopeia. More of a palmtop computer than a calculator, and it ran Windows CE. Anybody remember good old WinCE and why that abbreviation was particularly apt?

I guess I am just a sucker for old-fashioned calculators.

Aside: I’d love to do a spreadsheet in which one column has the year in which a calculator was made and another column has the number of buttons on the calculator, and run a regression analysis on it.

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## Encountering the NSpire; or, My calculator can beat up your calculator

One of the biggest conversation pieces here at the ICTCM is the Texas Instruments NSpire, their most recent entry in a long line of calculators. Here’s a firsthand look at it; click to enlarge, and then just take your time to look at the thing and think about it:

On the right there is a normal-sized TI-30-something scientific calculator. That should give you an idea of the scale. Here’s another shot with me holding it, which should also give an idea of the size of this thing; and another shot which gives a better view of the screen.

But let’s go back to that first photo. First of all, yes, the NSpire does actually have not one but two keyboards. They snap in and out; the one that’s un-snapped is just a duplicate of the TI-84’s keyboard. The one that’s snapped in is, well, let’s just say “busy”.  The first thing you notice is that there are buttons between the buttons. The little rounded buttons are a kind of alphanumeric keyboard. Well, really the first thing you notice is that this thing is big. Really big. It’s hard to get past the big-ness of the thing. How can the massive size not be a factor in getting kids to use the thing? Would you want to whip this out on the bus to do your homework, knowing that doing so clearly identifies you as the kid that needs to get beaten up?

From what I can tell, the NSpire is supposed to be a full-featured computer algebra system in a handheld device. If that’s so, then it certainly wouldn’t be the first time TI has tried to market such a thing. That honor would go to the TI-92 graphing calculator, which I owned about 10 years ago and, honestly, I really liked it, even though apparently I was the only person who did, because it was a marketing disaster and got banned by the AP Calculus exam to boot. (It was banned from the AP not because people didn’t like it but because it had a QWERTY keyboard.)

I am not sure what the NSpire brings to the table in terms of CAS functionality that isn’t already available in industry-standard CAS computer software like Maple or Mathematica. I overheard one person giving a rave review because it treats functions as geometric objects, whatever that might mean. I don’t think that a function is a geometric object — the graph of one certainly is — so I’m a little in the dark here.  I believe it means that you can enter in a function and view it dynamically in multiple representations, so if you have a graph of a function with the tangent line drawn at a point, for instance, you can go to a split-screen view and set up a spreadsheet that shows all this data, and then if you move the point of tangency the stuff in the spreadsheet changes as well. More here (complete with annoying music).

There is also a computer software-only version of the NSpire, so you can use the CAS without owning the calculator. That sounds more likely to be useful. The downside is that, according to the TI rep with whom I spoke at the vendor booth this morning, TI is ditching Derive — its simple and very serviceable CAS that has been around since forever — to focus solely on the NSpire line of products. They have already quit producing Derive and will cease tech support for it in 2010. I think this is a huge mistake, and TI will end up paying for it in the end. But that’s the subject of another post.

Isn’t the NSpire just really, really over the top here? I think so. After a certain point, you simply cannot cram more and more stuff onto a proprietary device. You will either make the device too expensive, too bulky, too confusing to use, or too proprietary in the sense that the device is trying to reinvent software applications that already exist in a simple, affordable, and ubiquitous way. (Think MS Excel, versus the proprietary spreadsheet on the NSpire.) I think TI crossed all four of those boundaries years ago, and the NSpire is just a step further — several steps further — in a direction that is really just a dead end.

The thing doesn’t even have a touch screen, for goodness sake, which is so easy and cheap to implement that it’s unfathomable why you wouldn’t build one into the calculator instead of having hot-swappable keyboards. Swapping keyboards, for gosh’s sake. What kind of user interface is that? Are students — who are used to iPhones and, at worst, the 12-15-button interface of a cell phone — supposed to see the NSpire as a device they will actually adopt and use?

The session I attended this morning went into this issue, regarding just how far can you possibly push the technology of the graphing calculator before you simply must abandon the format and move through a paradigm shift. More on that later, though.

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## Saturday agenda for the ICTCM

It was a full day yesterday here at the ICTCM, and the day was capped off with a very enjoyable dinner with Maria Andersen and Scott Franklin, along with two of Maria’s friends who (if I understood Maria right) are soon-to-be math bloggers. I have photos and a video forthcoming.

Today will be no less busy:

• 8:00-8:45: Session on handheld calculating devices over the last 30 years and how they have changed teaching. Very interested in this talk; I’ll have more to say about some of the handheld technology I’m seeing here.
• 9:00-9:45: Session on using Maple 11 in the advanced calculus and modern algebra classroom.
• 9:45–10:30: Exhibit hall surfing.
• 11:30-12:05: Session on labs in mathematics classes.
• 12:30-1:15: Session on using Geometers Sketchpad alongside computer algebra systems.
• 1:30-2:15: Session on Winplot.
• 2:30-3:15: Take a break!
• 3:30-4:15: Session on blogging with concept maps. Two of my favorite things put together, so this ought to be fun.
• 4:30-5:15: Haven’t made up my mind yet — either a session on CaluMath or a session on using Geometers Sketchpad in calculus courses.

Unfortunately the internet access I am paying \$10 a day for isn’t wireless — or at least, there is wireless but yesterday it didn’t play nice with me. So I won’t be blogging continuously. Which is probably a good thing because I need to pay attention at these sessions. Speaking of which, it’s time to head down to the first one.

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## 5 inside facts about calculators

Every semester before classes start, I get emails and phone calls from new students or their parents wanting to know about calculators. What calculator is required for your classes? What calculator should we get? I have a TI-**, can I use it in your class? And so on. A lot of these folks are trying to purchase technology, the uses of which are not entirely clear to them — and so they end up buying the wrong kind of device or spending a lot more money on a calculator than is really necessary.

From my vantage point as a college math prof — not beholden to any calculator company or big ed-tech advocacy group — here’s the scoop on what people really need to know about these things.

Fact 1: You probably don’t need a graphing calculator. Despite intense marketing for graphing calculators, the fact is that most settings do not require graphing capabilities from a calculator. In fact, the only situations I can think of where graphing calculators are required are AP Calculus courses (some of the AP exam questions require graphing calculators) and some college courses. And my experience is that the college courses that require graphing calculators are getting fewer in number — they spiked when the graphing calculator craze was in full bloom in the 90’s but are diminishing with the advent of cheap and useful computer software (see fact #2). So before you shop for a calculator, check with the school or the professor to see what you really need. You may need less than you think.

Fact 2: Most people in the real world use software, not calculators, for the heavy-duty stuff. Scientists and engineers don’t pull out their TI-84’s when they want to calculate numerical derivatives or find eigenvalues of matrices — they get out Mathematica or Matlab. Financial types don’t start typing in numbers to a micro-scaled, half-featured quasi-spreadsheet on a calculator — they get out Excel. Statisticians fire up SPSS or SAS. Although the little comparison charts on the backs of the calculator packages list what kinds of applications the calculator can be used for, the basic fact is that calculators are for small-scale number crunching done on the fly, and that’s about it. Any application beyond that level is relegated to a computer, which is much better suited for handling it. So it doesn’t make a lot of sense to me to drop a lot of money on a calculator, just to get functionality that you pretty much will never use in a real setting. Get a calculator that will do what you need it to — and no more.

What the great majority of students need instead of a fancy graphing calculator is a solid scientific calculator. There’s no single definition of “scientific calculator”, although I would define one as a calculator with the following abilities:

• Raising numbers to arbitrary powers (usually via a y^x button);
• Taking arbitrary roots;
• Calculator the log base 10 and the natural logarithm;
• Raising the number e to a power;
• Doing trigonometric and inverse trig functions;
• Doing basic statistical calculations (mean, standard deviation)
• Doing multi-level calculations using parentheses.

I’d even consider the statistical functionality to be optional. The rest of the list is basically everything you’d need in a standard algebra II, precalculus, or calculus course as well as entry-level quantitative courses in other discplines such as business. With that, we can move on to the remaining three facts:

Fact 3: The optimum price point for a good scientific calculator is about \$50. I check out the calculator offerings every time I go into an office supply store or the Target store up the road from me, and without collecting any data formally, I’d say that around the \$50 mark, one reaches a point of diminishing returns on the usefulness of a calculator. A \$100 calculator is not twice as good as a \$50 calculator, and a \$200 calculator (yes, those are out there) is a lot less than twice as good as a \$100 model. You do get more functionality the more you spend, and sometimes you get some useful technological features, such as USB ports for connecting to a PC. But those don’t necessarily make the calculator more useful (see fact #2) and you might better spend the money on good software.

Fact 4: You can get a good scientific calculator for under \$25. This TI-36X calculator is a good example, as is this Casio FX-115 (although I’m not a fan of Casio calculators; I think their displays are hard to read). Both of those models are excellent buys for the money. This doesn’t even consider sales at office supply stores or the possibility of picking up more advanced models on eBay for less. I tell my students: You should never spend more than \$100 on a calculator; you should try to spend less than \$50; and you can probably spend less than \$30.

Fact 5: The most important features of a calculator are usability and portability, not functionality. Let me explain this with an example. Around our house, we have (I think) four calculators. Two of these are a TI-83 and a TI-86 that I brought home from work. Another is a simple four-function calculator that has a flip-down stand and nice big comfortable rubber keys, and another is a slightly more advanced calculator that fits into a shirt pocket. When I’m grading papers at home, or when my wife is working out the finances for the day, which calculators do we go to the most? Hint: They aren’t made by TI. The calculator with the flip-down stand and rubber keys is comfortable on the hands, and we hit incorrect keys less frequently. The small calculator is easy to slip into my wife’s purse or my coat pocket. Both are solar, too, so we never have to deal with them not turning on in a critical moment (final exam, anyone?). If I need to do something more advanced than a square root, I might dig out the TI’s — although I’m more likely to fire up Maple 10 on my laptop.

If a calculator isn’t easy to use, both in terms of its user interface and — very importantly — human factors, you won’t use it unless forced to. Buy a calculator whose keys are easy to hit, whose display is crisp and clear, whose menu system (if it has one at all) doesn’t remind you of those hellish voice-mail menus you get when calling up the credit card companies. It doesn’t matter if the calculator will solve second-order differential equations if it makes your hand cramp up to type on it or gives you a headache when you have to look at it for a long time.

A good calculator is a tool that will prove itself useful for a long time to come. But it’s possible to spend too much and get a calculator that is so bulky or complicated that you won’t use it, just as it’s possible to spend too little and get one that doesn’t do what you need. But it’s very good when you get something that’s just right.

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