My New Calculator

I bought a new handheld calculator!

Actually, it is not new; it was made about 1965. It is a Curta Type I.

Curta calculators were invented by Curt Herzstark (1902 – 1988) who, according to Wikipedia:

… was born in Vienna, the son of Marie and Samuel Jakob Herzstark. His father was Jewish and his mother, born a Catholic, converted to Lutheranism and raised Herzstark Lutheran.[1][2] In 1938, while he was technical manager of his father’s company Rechenmaschinenwerk AUSTRIA Herzstark & Co., Herzstark had already completed the design, but could not manufacture it due to the Nazi German annexation of Austria. … perhaps influenced by the fact that his father was a liberal Jew, the Nazis arrested him for “helping Jews and subversive elements” and “indecent contacts with Aryan women” and sent him to the Buchenwald concentration camp. However, the reports of the army about the precision-production of the firm AUSTRIA and especially about the technical expertise of Herzstark led the Nazis to treat him as an “intelligence-slave”.… he was called to work in the factory linked to the camp…. There he was ordered to make a drawing of the construction of his calculator, so that the Nazis could ultimately give the machine to the Führer as a gift after the successful end of the war. The preferential treatment this allowed ensured that he survived his stay at Buchenwald until the camp’s liberation in 1945, by which time he had redrawn the complete construction from memory.[3]

After the war he moved to Liechtenstein. In 1947 they started building and selling the calculators. Production continued until 1971 when handheld electronic calculators became widely available. In that time 79,572 Type I and 61,660 Type II machines were built. The Type II calculators have an additional two figures accuracy.

The calculator, which has about 600 parts, is based in the stepped cylinder (also called the Leibniz wheel) invented by Gottfried Wilhelm Leibniz who, as I recall, also did some work in calculus. The stepped cylinder has been used in calculating machines since Leibniz’s time.

Here is an excellent video animation explaining how the calculator works.

Of course all it does is add. Subtraction is accomplished using nines complement arithmetic, Since multiplication is repeated addition and division is repeated subtraction, it can also do those operations. But then things get interesting. There are lots of other arithmetic operations that can be done on a Curta calculator. They make use of various numerical algorithms for engineering, and business applications. If you are interested in more about this click here. For a quick example, the video below shows how to calculate square roots on a Curta using the idea that the square of any positive integer, n, is the sum of the first n odd integers, ${{n}^{2}}=1+3+5+\cdots +\left( 2n-1 \right)$  There are actually several algorithms for square roots and others for cube roots.

There is a Curta Calculator website. Here you will find lots of information about the Curta calculators including technical information and drawings, historical material, pictures, articles, simulators, cleaning and repair instructions, photos and videos.

Now, if I could just figure out where the batteries go….

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Graphing Calculators

I was asked to pass the following information along to you. I decided to do so because you may want to know about one of the newest CAS calculator models and because their follow-up offer for attending the summer institute is so generous. If your school and students need help getting calculators and/or you want to keep up with the latest trends, you may be interested in looking into this.

The calculator is the new Hewlett-Packard HP Prime. It is a CAS calculator with really good graphing features. HP is offering the HP Prime AP Summer Institute Program, a 3-Day Institute this summer in either Statistics or Calculus with expert teacher trainers to introduce their new Mathematics Solution, the HP Prime Wireless Classroom. Following the Summer Institute, teachers who attended will receive a donated HP Prime Wireless Classroom Kit for their school with 30 HP Prime Graphing Calculators and the HP Prime Wireless Kit (a $5,000 value!). The institute is an opportunity to improve your knowledge of teaching mathematics with a technology that makes learning intuitive for students and receive the technology to keep for classroom use. For more information click the link https://h30602.www3.hp.com/assets/hpmath/web1.html Some Graphing Calculator History Graphing calculators first came on the market around 1989. In the early 1990s after it was announced that graphing calculators would be required on the AP Calculus exams starting in 1995, there were a series of workshops following the AP calculus reading, then at Clemson University. They were called the Technology Intensive Calculus for Advanced Placement conference or TICAP. Half the readers were invited to stay after the reading for the conference. The next year the other half were invited, and others the third year. Casio, Texas Instruments, Hewlett-Packard, and Sharpe all contributed and provided their calculators to the participants. The Texas Instrument calculators (then the TI-81 and TI-82) emerged the most popular and have since been the most popular in the United States. TI to their credit makes a good product and provided, and still provides, lots of help for teachers in the form of print material, programs for the calculators, workshops, and meetings. They have developed ways to connect the classroom’s calculators to the teacher’s computer. Other manufacturers have done the same, but not on TI’s scale. TI has made improvements in their calculators and other manufacturers have made newer and improved machines as well. While similar in functionality, I think the Casio PRIZM to be a bit better than the latest TI-84 model; it is also a bit less expensive. TI’s ‘Nspire line is an excellent CAS machine. Casio also has several CAS calculators and HP has now come out with the new HP Prime model (mentioned above). There are others. TI has a whopping 92% market share with Casio far behind at 7%. While their machines are excellent, Casio and HP are playing catch-up and have a long way to go. If you are just starting out or have limited funds for your class, you might consider a different brand. I often think the main problem is that the buttons are all in the wrong place! That is, the keyboards are different than the TIs we all learned on. They are different for you, but students who have never learned the old way will have no trouble with the keyboards. You won’t either – just sit down with the guidebook for a couple of hours and you’ll become an expert (or let the kids help you!). You can also use the manufacturer’s online instructions or go to a summer institute such as HP’s mentioned above. Some Graphing Calculator Opinion Now comes the real heresy. Graphing calculators don’t graph all that well. Their screens are small and often crowded. Tablets such as an iPad, or computers do a much better job. (For example, TI-Nspire’s operating system is available as an iPad app that is easy to use and much easier to see (okay maybe I’m getting old and my eyes are not what they once were). Still the many other graphing and mathematics related apps available are fabulous. Graphing, statistics and geometry apps abound and will only increase in number and functionality. This is the future. The reason graphing calculators are still here is because the Educational Testing Service, for good reason, will not let students use any device with a QWERTY keyboard on their exams including the Advanced Placement exams. The primary reason is that they are afraid that students will copy the secure questions using the QWERTY keyboards on iPads and computers. For some reason, students apparently cannot figure out how to do this with the alphabetic keyboards on graphing calculators. (Or as Dan Kennedy once quipped, there is nothing wrong with the old method of writing them on your cuff.) Other more important reasons tablets are not allowed include being able to photograph the questions, get information and help through the internet during the exams, or communicating with others in or out of class with tweets and instant messages. These are real problems that need to be considered, but I cannot believe a work-around is not possible. It must be possible to make an app that will allow only the use of approved apps during exams. After all they have done that for graphing calculators. If technology helps students learn mathematics – and I believe it does – then students should have the best available technology. End of sermon. Take a moment or more to consider the new and improved calculators. Update – iPad’s “Educational Standardized Testing” Option I wrote the paragraphs above a few days ago. This morning the new operating system 8.1.3 for iPad became available. They have a new feature for “educational standardized testing.” You can turn it on under Settings > Accessibility > Guided Access. Once turned on you open any app, triple-click the home button, and the controls for that one app appear on the screen. The individual settings are slightly different for each app. You may turn off the keyboard, turn off the touch screen, or disable the dictionary. On apps with their own buttons you can turn off any or all of the buttons by circling them. A time limit may also be set. It appears for now that each iPad must have these features adjusted individually. Unfortunately, to change or turn the restricted features on again all a student needs is his or her passcode or fingerprint. In addition, there should be a way to turn off all the other apps where students may quite legitimately have notes or homework saved. (Most of my students last year in one-to-one classrooms took most of their notes and did their homework on their iPads.) This is a good start, but it has a long way to go before it can be used in group settings. Stay tuned for updates. Calculators First some history and then an opinion I remember buying my first electronic calculator in the late 1960s. It did addition, subtraction, multiplication, and division, and could remember one number. It displayed 8 digits and had a special button that displayed the next eight digits. I remember using those next eight digits never. To buy it I had to drive 40 minutes and spend$70 – expensive even today.

The square root of 743 computed using the algorithm discussed in the post. The third iteration (fourth answer) is correct to 10 digits.

With it I learned an iterative algorithm for finding square roots: guess the root, divide the guess into the number, average the quotient and the guess, repeat using the average as the new guess.  You could do it all without writing anything down. (See the illustration on a modern calculator – accurate to 8 decimals in only 3 iterations (fourth answer), but then I could find the next 8 with the special button.)

Since then, I’ve had lots of calculators of all sorts.

Graphing calculators hit the general market around 1989 or 1990. This was the same time as the “reform calculus” movement. The College Board announced that the AP calculus exams would require graphing calculators in 1995 – five years to get the country ready.

The College Board held intensive training immediately following the reading. These were the TICAP conferences (Technology Intensive Calculus for Advanced Placement). Half the readers were invited for the first year and the other half for the second, then more for the third year.

Casio, Hewlett-Packard, Texas Instruments all gave participants calculators to use take home. Sharpe lent them calculators (and we haven’t heard of Sharpe since). Sample lessons were taught using Hewlett-Packard CAS calculators and then the same lesson was taught using TI-81s. The HP computer algebra system calculators, with far more features but using the far more complicated reverse Polish notation entry system, lost in the completion to the simpler to use, but less sophisticated TI-81s.

The teachers were not all happy. A friend of mine, due to retire in 2-3 years gave up his AP calculus classes early so he would not need to learn the calculators. Others embraced technology. The AP program forced the graphing calculator into high schools where they were used to improve learning and instruction. Yet even today not all high schools have embraced technology.

The calculator makers, especially Texas Instruments, provided print materials, software, workshops and conferences that helped teachers learn how to use graphing calculators in their classes at all levels.

Technology, as a way to teach, learn, and most importantly, do mathematics, caught on big time. And that was and is a good thing.

I think graphing calculators are very quickly becoming obsolete and should be phased out.

Technology has bypassed graphing calculators. Tablet computers, PCs, Macs, iPads, and the like, even smart phones, can do everything graphing calculators can do. They are more versatile. The larger screens are easier to see and can show more information without crowding.

The initial investment may be more than for a graphing calculator, but once purchased the apps are relatively cheap. There are many free apps that not only do computations and graphing, but CAS operations as well. Interactive geometry and statistics apps are also available.

These, along with online textbooks and internet access, put everything students need to learn math literally at their fingertips. Graphs and other results can be easily copied and printed, or pasted into note-taking apps.

One disadvantage is the initial cost for the hardware (but of course many students already have the hardware). The other disadvantage is the ability to communicate and find help both in the room and around the world during tests. Photographing the questions for later use by others is another concern.  I think (hope) it is just a matter of time before this problem can be overcome perhaps with an app that allows access only to the apps the teachers allow for tests.

Technology, like time, marches on.

Calculator Use on the AP Exams

In my final post on reviewing for the AP Calculus exams I return to calculator use.

I hope everyone has been using calculators all year long. (My opinion is that students should be given a CAS calculator, or CAS computer program, or now even an iPad CAS app on the first day of Algebra 1 before they get their books – or earlier. But we’ll save that for another post.)

The exam will not instruct students when or if to use a calculator on a particular question. Sometimes the multiple-choice answers will provide a big hint (if the choices are decimals), other times not. On the free-response questions if the problem can be done by calculator it is expected that students will use their calculator – they don’t have to, but there is no credit for finding an antiderivative or a derivative by hand. It is the numerical answer that counts.

Here again is what students are allowed to do on the exams with their calculator without showing any additional work.

• Graph a function in a window. They may have to determine the window themselves.
• Solve an equation. This is usually best done by graphing both sides and finding the point(s) of intersection, but any equation solving routine in the calculator may be used.
• Find the numerical value of a derivative at a point.
• Evaluate a definite integral.

For the last three items students should write what they are doing on their paper. That is, write the equation, the definite integral or ${v}'\left( 3.5 \right)=$ followed by the number from their calculator. Use standard notation not calculator syntax.

For anything else, the work must be shown. Calculators can find extreme values, but readers will look for the appropriate “calculus” and not just the answer. In fact, a correct answer with no supporting work may not receive any credit.

Another handy skill is to be able to store and recall the numbers found without retyping them on their calculator. This saves time, students are less likely to make a typing mistake, and it avoids round off error.

Answers from calculators should be correct to three places after the decimal point. This does not mean they must be rounded or truncated; more places may be given as long as the first three are correct.

Remember that arithmetic simplification is not required. Only answers from calculators need to be given as decimals. If the computation gives 1 + 1, or $6\pi$ or ½ + cos(3) or $\sin \left( \tfrac{\pi }{6} \right)$ then the answer may be left that way. This also applies to a long expression resulting from a Riemann sum. Once the answer consists of all numbers, stop! You cannot make the answer any better and if you make a mistake or type the wrong key, then the final answer will be wrong and the point will be lost.

Be careful not to round too soon. If, for example, a student find the limits of integration and rounds them to three places to write on their paper, they will have earned the point given for limits of integration. BUT if these rounded values are then used to compute the integral and the rounding causes the final answer to not be correct to three places then they will lose the answer point.

Finally, get a copy of the directions for the two parts of the exam and go over them with your students before the test. Be sure students understand them.

Good luck to your students on the exam. Nah, luck has nothing to do with it. You prepared them well, they will do well.