What Do You Do With A Broken Calculator? A bit of history In the 1970’s a graduate student of mine at MIT went into second grade classrooms with some hand calculators that had some, indeed most, of their keys taped over. Only the 0, the 1 the + and the – key were not covered. After distributing the calculators to the students she did the following exercise with the class: Ana: What year is it? Class: 1974 Ana: Can you get you calculator to display 1974 without removing the tape! ? Class: Hmm….. After a while, as the students figured out that they were
really not being asked to add 1 nearly two thousand times, some of them hit
upon the idea of adding 1000 + 100 + 100 + 100 + … + 10 + 10 +10 + 1 + 1 + 1 +
1, that is to say they discovered/invented the fundamental idea underlying place value. Shortly after that incident, in a fourth grade classroom, the problem Ana posed to second graders presented little difficulty. However, one student presented a new solution, i.e., 1111 + 111 + 111 + 111 +110 + 110 + 110 + 100 + 100 There was much discussion as to which way to get to 1974 was preferable. For what it is worth, the class decided that this way was preferable since the number of additions needed to reach the goal was equal to the largest digit (in this case 9), whereas the first method required a number of additions equal to the sum of the digits (in this case 21). This pair of classroom experiences and others like it prompted me to write a piece of software entitled “What Do You Do With A Broken Calculator?” which was published in the 1980’s by Sunburst Communications. The
idea underlying the design of the Broken Calculator Numbers and the operations we perform on them are tools that we as humans have devised to help us describe the world around us. One of the functions of schooling is to help youngsters develop some facility with the tools they are being taught to use. Master craftspeople have long used an effective strategy in training their apprentices to use the tools of their craft. They would confront the apprentice with the following challenge, “Can you perform this task, without using the tool that is normally used? That is to say, do you understand the nature of your tools well enough so that you can improvise an alternative way of getting the job done?” This perspective, applied to the teaching and learning of
place value and the arithmetic operations, is what drove me the write a small program originally entitled "What Do You Do With a Broken Calculator?" in the early 1980's. That version of the program, which was written in a language that is no longer supported, was adapted and improved by the Seeing Math project of the Concord Consortium. [The Seeing Math version of the software was written in Flash - a language that is increasingly unsupported by browsers. To avoid difficulties in the future, I have rewritten the program in GeoGebra which is continually updated to reflect software development] You can see this rewriting of the original Broken Calculator in GeoGebra by clicking here * The Broken Calculator is a
four function calculator in which the user can selectively disable keys,
either numbers or operations. After doing so, they may use the
remaining numbers and operations to try to perform a computation with
their now more limited toolkit. Here is a short list of sample problem types that we have used with the Broken Calculator -
Estimation with the Broken Calculator In the original version of the “…Broken Calculator” there was a mode called “leading digit” mode. In this mode, as soon as the user enters a non-zero digit, the keys for 1 through 9 are disabled. As a result the only numbers that can be entered into the calculator in this mode are numbers like 2, 40, 700, 3000, 0.1, 0.06, 0.001, etc. While this feature is not implemented in the current linked version it can be simulated easily. In doing so the user is forced to attend to place value, the order of magnitude of numbers and the results of computations. Are all calculators broken? Any calculator
(or digital computer, for that matter) is a rational number machine - it cannot
represent real but irrational numbers numerically. Moreover, there is a large collection of problems that calculators must produce INCORRECT answers to - e.g., 1 divided by 3. In this case one might suppose that the calculator might distinguish between .333...3 as a truncated decimal which is incorrect and .3 with a line over (or under) the 3 to indicate the repeating pattern. That would be a correct answer, but one has to consider how it is arrived at. [N.B. repeating decimals indicated with an underline, thus 1/6 = .16, 1/3 = .3, 1/7 = .142857 etc.] Does the calculator ‘know’ that a pattern repeats - -by calculating until the pattern repeats and then assumes it will continue to do so? or, -does the calculator ‘know’ the way we humans ‘know’? How do we humans
know that 1/7 = .142857 ? A closing remark Students and teachers who think that the only goal of learning and teaching the arithmetic operations is accuracy may think of such problems as wasted effort. My colleagues and I have found that working on such problems can lead to deep understanding of the standard (and even non-standard) computational algorithms. * You may still be able to use the version of the Broken Calculator adapted from the original by the Seeing Math project of the Concord Consortium. If your browser supports Flash (the language in which the Concord Consortium version was written). Please be aware that many browsers no longer support programs written in Flash. Broken Calculator - Seeing Math version
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