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Welcome to Cosmic Gnomon's Sundial Web Page.
Here you will find the lesson plan for a Geometry enrichment lesson that utilizes sundial research on the internet, links to other sundial sites, some pictures, a book list, and an opportunity to provide feedback.

Sundial Lesson Sundials and the Internet -- This is the lesson plan (with handouts) for a high school Geometry enrichment lesson which introduces students to sundials as they utilize the research benefits of the internet. Using simple geometric concepts, students then construct and set up their own working sundial.
Web Links Web Links -- This is a partial (and by no means inclusive) list of some interesting internet sundial sites.

Sundial Pictures

Sundial Pictures -- Here are some interesting pictures of sundials. Sundial pictures (photos, scanned images, drawings, etc.) you would like to have displayed on this site, would be gratefully considered and, of course, full credit given if displayed. Students' sundial constructions are especially welcome.
Sundial Books Books -- These books were found in local libraries. Most give some history on sundials and describe the variety of dial types and designs. Many have excellent photographs and illustrations of sundials and provide detailed instructions on dial construction using geometric and/or trigonometric approaches.
Feedback Form Feedback Form -- Please take the time to fill out this form to let us know what you think of our site. If you are a teacher and used the "Sundials and the Internet" lesson with your students, please use this form to let us know about any lesson improvements you found that would benefit other instructors.
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Welcome to Cosmic Gnomon's Curmudgeon Web Page.

Here you will find incoherent ramblings of a cantankerous mind.


In my capacity as a mathematics instructor at Moorpark College (a community college in Southern California), I do not allow students to use calculators in most of the courses I typically teach. It is the intent of this exposition to indicate why I have instituted this policy in my classes, and to offer advice and encouragement to others who have become frustrated with the mounting pressure to incorporate technology into EVERY aspect of mathematics instruction, whether appropriate or not. Let me point out immediately that, despite their protestations to the contrary, calculator manufacturers are not necessarily interested in the enhancement of mathematics instruction. I believe any actions taken by calculator manufacturers in this regard are best interpreted as being primarily for the purpose of maintaining and expanding an extremely lucrative market for their product. That is: CALCULATOR MANUFACTURERS ARE IN THE BUSINESS OF SELLING CALCULATORS; they are NOT in the business of mathematics instruction. Of course, similar comments could be made concerning textbook publishers, and many of the "mundane" objections to machines outlined below could be applied equally well to textbooks. I submit that there are essential differences between the use of textbooks and the use of machines in the context of mathematical instruction. One example of such differences can be illustrated by considering the reaction of a class being told that the pending test will be of the "closed book" variety, versus the reaction of a class, wherein calculators have been allowed, being told that the test will be of the "no calculator" variety. And this brings us to perhaps one of the biggest problems attendant to the use of calculators/computers/technology in the context of instruction/education: Student perception of and reaction to such use. As a classroom instructor, I am concerned with the way things "really are" in the classroom. I deal with typical students in typical classrooms, not the "learners" in "model classrooms" so often referenced directly in contemporary education courses and indirectly in studies designed to promote the latest pedagogical innovation. I believe the one overriding difference between these two settings is that typical students, no matter how highly motivated to excel, tend to seek the easiest way to get it done. To what extent do we allow machines to supplant "basic knowledge"? Since calculators can perform arithmetic operations, does this mean that knowledge of addition and multiplication tables need no longer be required, or emphasized? Perhaps the line should be drawn at knowledge of the algorithm for performing long division. Or perhaps with the algorithm for computing square roots. And wherever the line is drawn, how do we assess essential knowledge in the context of pervasive calculator availability? My own interests in language and in mathematical logic lead me to draw analogies between the English language and the mathematical language. In the context under consideration, while I find it annoying that Math teachers are victimized by such technology at a somewhat greater level than English teachers, I note that we share common concerns in this regard. Does the spell checker function of my word processing program preclude me from the requirement of any knowledge of proper spelling? Does the syntax checker allow me to ignore proper sentence structure? Does the availability of a printer free me from the necessity of legible handwriting? "AH HA!" you say, "He uses the very technology he would deny his students!" Yes, I make extensive use of the spell checker, and other features, available on my word processor, and I use a calculator for various mathematical computations ranging from administrative tasks to the solving of those same problems I require of my students. "So what's the difference? Is this just another example of 'Do as I say, not as I do'?" One difference is that I passed all those spelling tests in grammar school, and I satisfied the Subject A requirement for graduation from the University of California. I know my multiplication tables, and I can graph basic equations by hand. Which is to say that I've earned the right to use machines by proving that I can function without them. To borrow a phrase from my father: I'VE PAID MY DUES. The Mathematics Department at Moorpark College has no departmental policies governing calculator use. That is, calculator use in any particular section of any course is solely at the discretion of the instructor for that section. As such, it can happen that in some sections of a course, graphing calculators are required, and in other sections of the same course (and not just mine!), all calculators are strictly prohibited. Attitudes toward calculator use at Moorpark tend to fall into four general categories, depending on both the instructor and the particular course that instructor is teaching: 1) Graphing calculators required; 2) Calculators expected (perhaps with certain requirements or restrictions); 3) Calculators allowed at the discretion of the student (perhaps with certain restrictions); 4) Calculators prohibited. Courses in which graphing calculators are required are so designated in the schedule of classes. While no such mention of such is made, instructors of these courses tend to require, either directly or indirectly, a particular band and model of machine. I use the phrase "calculators expected" to indicate that some kind of calculator is deemed essential by the instructor for the course. An individual instructor's reasons for "expecting" a calculator vary, but, typically, they are "expected" in either computationally intensive courses such as Elementary Statistics, or for those courses in which the instructor feels it is necessary to be able to have numeric/decimal representations of irrational numbers and/or values of transcendental functions. Prior to adopting an adamant "category 4)" stance in almost all my classes - - that is, in my (good-old) "calculators at your own risk" days - - I expected calculators in certain courses, allowed calculators in others, and prohibited them in still others. In those situations where calculators were allowed but not essential, I tried to hold students to the maxim: "You may use a machine in conjunction with those things for which you, at least in principle, don't need a machine". (Obviously,) I have come to the conclusion that this approach is unacceptable, for a variety of reasons, both mundane and pedagogical. The "mundane" issues which concern me center primarily around money. Calculators can be expensive. To require a calculator in a class increases the cost of the class and thereby inhibits access to the class. Of course, in this day and age, it is not at all unreasonable to assume that every college student already owns a calculator. But is it the right kind of calculator (whatever that means)? "Last semester, I was required to have a graphing calculator, and now you tell me that I can only use a scientific calculator that DOESN'T do graphing?" Different calculator manufacturers configure their products differently. If calculators are to be required in a course, it seems only fair that some instruction on their use be presented. Rather than attempting to give instruction in both "algebraic" and RPN logic, a more effective approach would be to require a specific brand of calculator. "You mean I can't use my HP? I have to use a Casio?" Different calculators are designed for different purposes, and are constantly subject to design enhancements. "You're requiring a TI-42? My last instructor required a TI-21. My sister-in-law works at TI and she tells me they're coming out with the TI-43 in a couple of months." And for all of it, there are two issues implicit in the above three paragraphs which raise significant ethical concerns. It seems to me that requiring a specific type or brand of calculator, either directly or indirectly, turns the instructor into a calculator salesperson. It seems to me that NOT requiring a specific type or brand of calculator allows those able to obtain a highly sophisticated machine a potential academic advantage based on something other than academic acumen. And there is one other "mundane" issue that does more to sour me on machines than any of the objections indicated above. In the spirit of the previously mentioned maxim regarding calculator use, a first semester calculus student would, for example, be allowed to use a calculator for "routine arithmetic computations". But I see no viable way of preventing that student from also using the machine for everything from the evaluation of definite integrals to (at least massive hints concerning) the sketching of graphs of functions - - and to the storage in memory of the exact wording of that definition he just knows I'm going to be asking about on the test. Moreover, I have had students tell me of some rather innovative methods of exploiting the line-of-sight infrared communication capabilities of certain calculators in "do your own work" settings. And what about pedagogical issues? What is being taught, how is it being taught, and why is it being taught? And, in this day and age, the question of to whom is it being taught arises as well. At this juncture, it would seem appropriate to consider arguments presented by proponents of calculator use. Let me point out that in my own research into this area, I am constantly impressed with the fact that such proponents almost invariably have calculator manufacturers lurking in their background. Consider, for example, the bulk of the advertising to be found in various publications of the National Council of Teachers of Mathematics. For an excellent summary of the usual arguments advocating calculator use, I recommend the paper The Role of Calculators in Math Education, which is comprised of research compiled by Heidi Pomerantz of Rice University. This paper may be found on the Texas Instruments web site,, and is copyrighted by Texas Instruments, 1997. The basic idea is that calculators can eliminate the drudge work of long, tedious, boring, monotonous arithmetic and algebraic paper-and-pencil computations and allow students and teachers to get to the REAL mathematical ideas. In my research, however, I have been unable to determine what, in any particular course, constitutes drudge work, as opposed to REAL mathematics, in the eyes of calculator proponents. While a calculator is supposed to eliminate the drudge work of mathematical computations not necessarily germane to the problem at hand, I believe there is a difference between the drudge work after you know how to do it, and the drill work necessary to learn how to do it. It seems to me that in this day and age in which the proper role of math teachers is deemed to be the coaching of each of their learner mathematicians to nurture her/his own inner-Bernoulli, these two types of work may tend to be confused. Those courses in which I prohibit calculator use are Elementary and Intermediate Algebra and First and Second semester Calculus. (It should be noted that at Moorpark, there is a basic arithmetic course which serves as the prerequisite to Elementary Algebra. Were I ever to teach this course, I would prohibit calculator use in it, as well.) These courses serve as the introductory courses of their fields, and, as such, students should emerge from these courses knowing the basics of that field. (Tedious, monotonous, boring) drill is the time-honored method of mastering basics, both in mathematics, and in other fields, as I am so often reminded by my (music teacher) father, and my (athletic coach) brother-in-law. To borrow a phrase from Euclid: THERE IS NO ROYAL ROAD TO GEOMETRY. Of course, the real pedagogical issue for math teachers is WHY are we teaching what we're teaching, and to what extent do we allow machines to impact that. Certainly one of our goals is to prepare students for the post-academic real world, and the use of machines in conjunction with problem solving is an important aspect of that world. But the essential issue is whether calculator training should be, and in fact whether it can be, incorporated into EVERY math course in the context of other mathematical and pedagogical goals of those courses. Some proponents of calculator use tell us it is necessary to reorganize mathematics curriculum in such a way so as to fully exploit the power of the machines. This stance leads to a serious pedagogical concern, the question of who is driving course content these days. To what extent are we to change what we are teaching and how we are teaching it in accommodation of the machines? What I see is that control over these questions may well be in the process of being wrested away from the academic community by sales/profit oriented calculator manufacturers. (Consider, for example, HP's efforts with respect to Horner's Method for representing polynomials.) And, on the darker side, I am constantly forced to listen to the haranguing of my (high school teacher) wife, who is able to rattle off a litany of ways in which the use of calculators can be abused, ranging from unqualified teachers who view them as a panacea for inadequate instructional ability to school districts who illegally allow their use on state assessment tests. In that I do not allow calculators in my classroom, I am forced to address head-on, on a day-to-day basis, the issue of whether or not a machine is necessary for, or would be of any real benefit in, what I am teaching. The recalcitrant student in my mind is constantly asking "Why won't you let me use a calculator for this?" And almost invariably, my answer is "You're supposed to (be learning how to) be able to do this without a machine." Even in those instances where something may be lost by not having a calculator available, and I'm not convinced that such is ever the case, I certainly believe the gain in recovering "what is lost" is not worth the price of the recovery. As far as I can tell, students (at least grudgingly) accept my edict concerning calculators outside the classroom as well as in, and some (again grudgingly) even claimed to agree with the rationale behind the edict. The other side of it is that after the announcement of this edict on the first day in one of my classes, I was informed by one girl in no uncertain terms that since she could not use a calculator, she would not be able to take my class. (What's scary here is that the class was Elementary Algebra...) And this raises the issue, which I believe does more to prove my point than any of the above haranguing: What of the student who is so immured of the machine that he cannot be successful in a subsequent non-calculator-based course? And in there as well, perhaps just as a whisper in the background, I believe we math teachers should be talking to our students about those fundamental mathematical ideas underlying the symbol manipulation, the word problems, and even the proofs - - the ideas that caught us and made us math majors in the first place. Knowledge of what's happening "in theory", I believe, ought be more our goal than knowledge of the mechanics of punching out the right answer on a machine. I submit that calculators have no place in this. And in the context of these background whisperings, we secondary and lower division math teachers should never forget that most math courses at the junior level and beyond typically require dealing with implications of the Axiom of Infinity. As anyone who calls themself a math teacher should know, such is beyond the capability of any machine.


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This web site was first published on 1 Nov 1999 and last updated on 18 Apr 2003.
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