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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ E-Math News Volume 3, Number 1 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Contents ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Everyday Life - Hand Counting vs. Machine Counting Schedule of Public Classes Math in Industry - Design of Experiments in 13 Steps Family Math - Logic Book Review - The StatSoft Electronic Statistics Manual and The NIST/SEMATECH Manual Ask Statman - Tests for Non-Bell-Shaped Data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "In the country of the blind, the one-eyed man is king." H.G. Wells ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Everyday Life ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Hand Counting vs. Machine Counting The Presidential election of 2001 gave us all something more than headaches it gave us a real-life example of a simple, but not widely understood, mathematical concept - bias. The word "bias" means that one side is favored over another in an argument, a group, an election, etc. The mathematical meaning is nearly the same - distortion in an estimate that is systematic and not random. Thus a vote count smaller than the number of votes cast (or larger) is biased - both mathematically and colloquially. In the 2001 election everyone agreed that an accurate, unbiased vote count was the goal. However, people disagreed strongly about how to achieve such a count. Some felt that machine counting was the best way while others felt that hand counting would be more accurate. Some suggested that simple hand counting was not enough, but that the "voter's intention" should be determined before counting a vote. In order to obtain an accurate vote count, bias must be eliminated or, at the very least, minimized. This article will examine sources of bias in each of these techniques and how they can be eliminated. Let's start with machine counting. Machine counting offers a good way to avoid human sources of bias, but it has its own sources of bias. The sources of bias will depend on the design of the machine and the care taken in its manufacture. Let's just discuss a machine like those used in the Florida election counting. In the Florida election count, holes were punched in ballot cards to indicate a vote. To count votes, machines passed a beam of light through the hole. If the light made it through to the other side, a detector indicated a vote. If the light didn't make it to the detector, no vote was counted. Several potential sources of bias exist using this counting machine. If the light source and the sensor are not carefully aligned to the position of the hole in the card a clear, legal vote might not be counted. If any path for stray light to reach the detector exists a vote could be counted that did not exist. If the light source and detector are too small, slight variations in the position of the holes due to card manufacturing tolerances might cause some votes to be counted and others not. Careful manufacturing and calibration can eliminate these sources of bias. The machines must be calibrated before the candidates are assigned to hole positions to avoid bias being introduced by the people performing the calibration. Before a machine is used to count votes, its ability to count accurately must be tested. As a matter of fact, Florida law requires that vote counting machines be tested before use. Several methods exist to test machines before use, a Gage Repeatability and Reproducibility study being one possible choice. Now let's take a look at hand counting. Hand counting can work well when a clear criterion for a legal vote is provided (e.g. the hole is cleanly punched through) before any votes are cast and all of the counters are honest. The difficulty comes in insuring that the counters are all honest. A few dishonest counters can bias the vote count and, in a close race, change the outcome of an election. If the vote counters do not know which hole corresponds to which candidate, the dishonest vote counter problem is eliminated. Unfortunately, keeping vote counters (who are also US citizens and, therefore, legal voters) ignorant of the connection between hole position and candidate would be a practical impossibility. Now let's take a look at the method of determining a "voter's intention" before hand counting. If the vote has been unambiguously cast, (e.g. the hole has been cleanly punched for only one candidate), there is no difficulty determining the voter's intention. If the vote has not been clearly indicated, (e.g. the hole has not been punched clean), then there is doubt about the voter's intention. Since the ballot is secret, we cannot ask the voter what was intended. Any method of determining the voter's intention when the vote is in doubt short of asking the voter is highly subject to bias, both in the standard and mathematical senses. The person or group making the determination is very likely to favor one candidate over the other. Dishonest members can follow their bias without compunction. Honest members may still be influenced by their bias in subtle ways that cause them to favor one candidate over another. Bias can be overcome when determining a "voter's intention." Everyone counting votes must not know which punched hole corresponds to which candidate. If you don't know if hole A or hole B indicates a vote for your preferred candidate, you cannot be swayed to count one hole more than the other. But as we have already discussed, this is a practical impossibility. Machine counting offers the most practical solution to the problem of bias. Most states have adopted some form of machine counting to provide more accurate counts than could be obtained by hand counting. If more people understood the concept of bias, we would have better election laws (e.g. laws that don't permit the determination of voter intention) and fewer people willing to resort to an inferior counting method in close races. What can you do to help prevent vote count bias? First, be absolutely certain that your vote is unambiguous. If your precinct uses a punched card ballot, be absolutely certain the correct holes are cleanly punched. If you make a mistake, ask for a new ballot. Second, write your state representatives and ask them to review the vote counting policies in your state. Ask them to consider using counting methods that are demonstrably low in bias. Whether you are a Democrat, a Republican, or, like me, a member of a third party, it is in your best interest to have a fair, unbiased vote counting system. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ "How to Lie with Statistics" provides a clear, witty explanation of many simple mathematical facts that are misunderstood and misused in daily life. You can purchase a copy form Amazon.com at the following link http//www.amazon.com/exec/obidos/ASIN/0393310728/mathoptionsinc ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Schedule of Public Classes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Date Class Location ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ February 15 & 16, 2001 Objective Experiments San Francisco for Mixtures and Bay Area, CA Discrete Factors Expand your knowledge of Design of Experiments to include advanced topics April 18-20, 2001 Performing Objective Experiments Atlanta, GA Learn practical Design of Experiments and become a top performer. May 16-18, 2001 Performing Objective Experiments Anacortes, WA Learn practical Design of Experiments and become a top performer. This class includes a free whale watching trip!
SAVE $300 BY REGISTERING EARLY! Visit http//www.mathoptions.com/class_registration.htm for details. You can learn more about these classes and register to attend at http//www.mathoptions.com/public.htm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Math in Industry ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Design of Experiments in 13 Steps By Robert Johnson Introduction This document is intended for anyone who will use Experimental Design/Response Surface Methodology to explore the behavior of one or more responses when several factors are varied in a systematic way, based on an experimental design. Approach 1. Formulate a clear statement of the question or the problem. What are the goals, and how will the results be used? Use this information to guide the steps below. 2. Select the properties to be measured. These are called responses. 3. Select the quantities to be varied. These are called factors, or independent variables. Distinguish between continuous factors, which can be set at any level within a range, and discrete factors, which can take on only certain values. An example of a discrete factor is a choice to use either process A or process B. You can't choose a setting halfway in between the two. At this time, you may also want to select factors called "blocks". These are factors which are not of direct interest, but which must be taken into account in order to reduce variability. An example is the choice to use oven C or oven D to treat samples. Next, you will need to consider whether there are constraints on the choice of factor levels. For example, if the factors are percentages in a mixture and each component is a factor, then there is a constraint that all factor settings have to add up to 100%. If there is a mixture constraint, you will have to use a design appropriate to the situation. Finally, if there are several constraints, you may have to purchase a custom-made design from someone who has the specialized software for generating such designs. 4. Choose a mathematical model. This is the type of equation that will be used to fit the data points. The most common choice is the quadratic model, including all second-order terms. 5. Choose an experimental design. The choice will be determined by the type of model, the number and type of factors, and the need (if present) to reduce the number of runs. I-Optimal designs provide greater economy of runs than classical designs such as the central composite design. At this stage, decide how many replicate runs will be performed. The decision will be based on how much prior knowledge is on hand concerning the random variation of each response. More replicate runs give more precise knowledge of the standard deviation of the response. The significance of a response result is always measured against the amount of response variation present. 6. Run the experiments. Make sure that they are done in the exact way specified. Scramble the run order if the design does not provide the runs in scrambled order. ("Scrambling" the run order is an improvement over simple randomization.) 7. Perform the regression analysis. There are several software packages that will do this, like STATISTICA or STRATEGY. 8. Perform diagnostic tests on the regression results. Here we look at how well the model fits the data, and check on the basic regression assumptions of independence, Normal distribution of residuals (errors), and constancy of variance across the factor ranges. It is also the time to look for outlier data points in the response. If some of these quality checks fail, it may be possible to remedy the situation by transforming the data, for example changing a response from a linear to a logarithmic scale. 9. Generate response surface plots. These can be either contour plots in two dimensions, or 3-D plots. Only two factors and one response can be selected at a time. All other factors are fixed at some level for each plot. Example Suppose you have three factors, called A, B, and C. Generate three plots of the response vs. A and B, with the third factor C set at low, medium, and high levels. If necessary, continue by plotting the response vs. A and C, with Factor B at low, medium, and high levels. The nature of the response surfaces may indicate which type of plot is most revealing of the behavior. Repeat the plotting for each response. Note that discrete factors cannot be used as one of the axes of these plots. However, one can generate plots for the continuous factors, one for each setting of a discrete factor. If there are no continuous factors, response surface methodology does not apply, and one is forced to work in the mode of "factorial" or "fractional factorial" designs. 10. Find the sweet spot(s). A sweet spot is a set of factor values which gives you the best possible result for all responses simultaneously. Actually it is a region in factor space, not a point, which means that there are typically several sets of factor settings which will accomplish the desired result. In simple situations, it may be possible to find optimum factor settings merely by looking at one or a few response surface plots. In many if not most cases, the situation is too complex to search visually through a stack of response surface plots. In such cases, use a search feature like the GRIDSEARCH program within STRATEGY or "Prediction Profiles and Response Desirability" in the Experimental Design module of STATISTICA to find the sweet spot(s). 11. Run experiments to confirm the sweet spot(s). This validates the whole analysis process up to this point. This step is necessary for gaining confidence for the next two steps. If confirmation is found, you may find it useful to estimate the robustness of the settings. Are the responses overly sensitive to small variations in the settings of the factors? Use the plots and the regression equation to estimate robustness. 12. Write a detailed report. The purpose is mainly to inform the decision-makers, but is also for the record, so that related future work can build on the work just completed. 13. Make decisions based on the results. A decision based on good planning, good data, and a valid procedure for analyzing the data is reasonably safe from being revised or rescinded. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Robert Johnson is a Statistician at DSM Desotech. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Are you having trouble finding an experiment design that fits your specific experimental conditions? Are the designs available to you too costly in time and resources? Do you need to study factors that only have specific, restricted levels available for study? Please give me a call at (888) 764-3958. I can help you get the design you need. Or visit http//www.mathoptions.com/Custom%20Experiment%20Design%20Service.htm for more information. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Family Math ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Logic by Beth Heffernan Long awaited by many parents is the arrival of logic into the minds of their children. Somewhere in the chaos of two year old tantrums you will be told "Don't worry, at age seven children become reasonable and logical". Actually, beginning with the most elementary sensorial exploration, children are taking themselves into logical thought patterns in a step by step manner. With the proper atmosphere and support from adults, especially parents, the overwhelming nature of the world is sorted, categorized and eventually open to abstract manipulation. Your child is forced to handle information this way because he knows so little of the "real world". Information is attained by physical exploration, and repeated over and over till the body "knows" it. Mental pathways are created in the brain. With increasing age your child's attention span grows longer, and the narrow trail of initial information widens. In this way your child readies himself or herself for math and science. Parents are wonderfully able to facilitate the process by providing opportunities for physical accomplishments, playing labeling and later sorting games, pointing out details of the world with precise language, and, most importantly, sharing and praising the exciting acquisition of skills and prowess. You don't need anything fancy to do this. You do need information on age appropriate abilities, and lots of time and attention. You probably already know that!
All basic mental skills are built on physical accomplishments. Grasping a toy and getting it into his mouth , learning to balance through crawling, walking, and jumping, catching the rolling ball, and myriads of other tasks we take for granted are the foundation for your child's mental growth. Until your child has good control of his body and a reasonable grasp of the order of the world, no abstract thought is possible. However, beginning around age one, your child will usually be willing to consider concepts if they are translated into concrete examples in his world. Three is tangible if there are three steps to his front door. Weaving math and scientific observation into your child's life should begin at a very early age. Bearing in mind that some children are more logical and precise than others, make the study of situations exciting. Math is a way of looking at the world to make sense of it. It is everywhere. It is counting your family members and pets. It is estimating the number of fleas on a little dog versus a big dog. It is the names of all the shapes in their toy box. Who gets a bigger piece of birthday cake if the cake is cut in different ways? Zero is real to all children, and more respectful of their minds than "all gone". Montessori preschoolers manipulate binomial and trinomial equations using fun puzzles (they don't know this, but many report memories of the feel of the equation when it is presented to them as adolescents). And money, ah, money can be the start of so many discussions. Your child knows what is important to you. If math and science are real and exciting to you, so it will be for them. However, if you dislike math or downright fear it, all is not lost. This could be one of the golden opportunities we get with our children to relearn the world ourselves. Math is everywhere for you too, and you handle many mathematical functions each day easily. Which interest rate pulls less money out of your pocket? How do you divide dinner for four into dinner for six with unexpected company? Good deals, checking accounts, and "when are we going to get there?" estimates happen all the time. Take heart and dive in. Next issue Math activities for ages one through six. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Beth Heffernan is Vice President of Math Options. You can reach her at mailtoBeth@MathOptions.com ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Book Review ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The StatSoft Electronic Statistics Manual and The NIST/SEMATECH Manual by John Raffaldi In the past, book reviews have focused on printed material. This time however, the focus is on two Internet based electronic statistics manuals. The first considered is by StatSoft, makers of STATISTICA, a general purpose statistics package. The second, an almost complete manual, is a joint effort between the National Institute of Standards and Technology (NIST) and SEMATECH, a consortium of semiconductor manufacturers based in the United States. The Statsoft manual is a broad, in-depth collection of statistical knowledge that provides information on nearly any aspect of statistics. The manual's "Basic Statistics" hyperlink contains a good background for those unfamiliar with basic statistics to build a foundation of knowledge for understanding the other sections in the manual. Despite this good introduction, those less familiar with statistics may have a difficult time understanding the rest of the manual due primarily to the information density. Nonetheless, we are fortunate that Statsoft has taken the initiative to create the manual and made it available to the general public without charge. The NIST / SEMATECH manual is an in-progress effort to provide a useable statistics resource to the semiconductor industry. The manual, nearing completion, provides eight general topic categories Explore Measure Characterize Model Improve Monitor Compare Reliability Written by NIST statisticians and industry experts, the intent of the authors is to create an applied statistics tool to help those less familiar with statistics. Although the manual contains some mathematical notation for those people liking that sort of thing, the authors are creating a useable resource for any industry. The authors are still soliciting comments from the general public if you would like to make comments to them. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ John Raffadli is is a software engineer at SHS.com and the Math Options expert on Gage Repeatability and Reproducibility Studies. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ StatSoft's Electronic Statistics Manual can be found at http//www.statsoft.com/textbook/stathome.html The NIST/SEMATECH Manual can be found at http//www.itl.nist.gov/div898/handbook/ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Ask Statman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Written by Dr. Charles Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Dear Statman- It seems that many of the articles you write assume the data we examine are bell-shaped. Very often I look at data that are definitely NOT bell-shaped. How can I compare the results of two experiments without this assumption? Signed, No-bell Dear No-bell- As you might have guessed, statisticians have come up with ways of dealing with data when you can't assume the data are bell-shaped (normal). The bell curve (normal distribution) needs two numbers, or parameters, to describe it the mean and the standard deviation. In fact, many distributions we deal with, like the exponential, Weibull, chi-squared, F, etc., use parameters. The tests we use to compare data that come from these kinds of distributions are called parametric tests. The t-test and F-test are examples of parametric tests (previous issues of E-Math News have covered these subjects). There are different statistical procedures where you don't have to assume the data come from a particular distribution. For this reason, they are called non-parametric tests. Sometimes we would like to know if a treatment [a particular situation] has an effect, but we don't know how the data pile up (the distribution). To find out if one treatment is different than another, we have several different non-parametric tests to choose from. Note that while these tests don't require the usual assumption of bell-shaped data, we are not completely free of assumptions. We still have to be sure that the observations are independent. From a practical standpoint, this means making sure that the data are randomly gathered and come from samples that have been randomly assigned to the different treatments. Let's consider the case of a cook who wants to know if adding a particular spice to his chili will improve the flavor. He makes up a big pot of it and transfers half to a second pot to which he adds his spice. He then makes up 24 bowls of chili, 12 from each pot, marking the bowls underneath so that he knows which bowls contain the extra ingredient. From the 24 bowls, he sets out 12 pairs of bowls, each pair having one bowl with and the other without the added spice. His friend consents to act as judge and pick a winner from each pair. Let's look at the results of the taste test. Pair # Winner Sign 1 Spice + 2 Spice + 3 No spice - 4 Spice + 5 No spice - 6 No spice - 7 No spice - 8 Spice + 9 Spice + 10 No spice - 11 Spice + 12 Spice +
From the results, it looks like there is no clear winner. But how do we show that statistically? Let's look at the last column. If the judge had a difficult time telling the difference between the two types of chili, then we would expect the spiced to win about half the time, and the un-spiced to win the other half. The third column has a plus sign each time the spice won, and a minus sign each time it didn't. There are a total of seven +'s and five -'s which seems pretty even. This taste test is a little like tossing a fair coin 12 times and noting the number of heads and tails. In order to determine the probability of getting at least five -'s (or five "heads" on a fair coin) we use the binomial distribution. While you may not be familiar with the formula, the results of using it are available in tables. In this test, we must consider the smaller of the total number of pluses and minuses. Here, the smaller value is five. From the binomial table we find that the probability of getting five tails or less out of 12 tosses using a fair coin is ~0.387 (see http//www.mathoptions.com/binomial.htm for the table). Since we are not sure that the spice will improve the flavor, it turns out that we have to double this probability. (For those who are interested, this is called a two-tailed test. It is called two-tailed because we have to find the "area under the curve" of the probability distribution in both tails of the distribution. If we were very sure that the spice would improve the taste, then we could use what is called a one-tailed test and the p-value would have been only 0.387. Two-tailed tests are more conservative and hence safer. I recommend always using a two-tailed test unless you are absolutely certain a one-tailed test is indicated. ) So, 2 x 0.387 = 0.774. This last quantity is called the p-value and indicates if the result is significant. Usually, we want a small p-value of less than 0.1. A p-value of 0.774 means we can't tell the difference between treatments. Had there been only one "-" in the last column, then the p-value would have been 0.0064 (2 x 0.0032 from the table), indicating a statistically significant difference. But since we really got five -'s, it looks like our cook has to go looking for some other ingredients to improve the taste of his chili. Now, there are a few points I need to make. First, note that the data were gathered in pairs. That is, the cook took a single pot of chili and performed his "treatment" on half of it. Say the cook had made several different pots of chili on different days and then added the spice randomly to half of them. In this case, the data might not be considered "paired" and so we could not use the above test. The test used above is called the sign test (hence the columns of +'s and -'s). It assumes the data come in pairs. A good time to use this test is when you have a sample that is measured both before and after treatment. That way, the data are naturally paired. If the judge couldn't tell the difference between a pair, we would have had to remove that pair completely from the analysis. Also, when the sample size is >25, you can use an approximation to get p-values. The formula is z = (|2r - n| - 1) / square root(n) where z is called the "z-score", r is the number (please see note 1) of +'s or -'s, n is the number of pairs, and the symbol "| |" means absolute value. In this example, we have r = 7, and n = 12 producing a z-score of 0.289. From the standard normal table, this corresponds to a probability of 0.386 (see http//www.MathOptions.com/normal.htm for a normal probability table). Since we have to double the probability, the p-value is then 2 x 0.386 = 0.772, very close to the exact p-value of 0.774 from the binomial table. As you might be aware, the higher the z-score, the more significant the difference. A z-score of 1.96 means the p-value is 0.05. Said another way, when z=1.96 you are 95% confident that the two treatments are different. Lastly, this test can also be used with numerical data, where the one with the higher score gets the +. However, since this test reduces the differences between treatments to +'s and -'s, some information is lost. This test doesn't take into account by how much one treatment beat the other. Therefore, the sign test won't be as powerful at detecting treatment differences as other tests that make use of the size of the difference between pairs. Next month I will explain a non-parametric alternative to the t-test called the Mann-Whitney test. Thanks, Statman. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Notes (1) In this case, we don't have to use the less frequent of the number of +'s and -'s. That's because the absolute value takes it into account. We would have gotten the same z-score with r = 5. (2) As in the case for U and U', this formula for z assumes no ties. The presence of ties requires a small correction. This correction is not important in most situations. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you have a question for Statman, please send it to mailtoStatman@MathOptions.com. Statman will answer questions about basic statistics that are of general interest to people working in industry. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Copyright 2000 by William D. Kappele, Beth Heffernan, John Raffaldi and Charles S. Whitman ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ If you like E-Math News, please forward it to a friend. You may copy and redistribute E-Math News in its entirety freely. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A free newsletter published every other month by Math Options Inc. http//www.MathOptions.com 4702 Camano Place FAX (503) 218-6587 Anacortes, WA, 98221 Toll Free (888) 764-3958 William D. Kappele, Editor Bill@MathOptions.com To subscribe to or unsubscribe from E-Math News please visit http//www.mathoptions.com/e-math.htm. If you don't have access to the World Wide Web, please send E-Mail to mailtoBill@MathOptions.com with either "Subscribe E-Math News" or "Unsubscribe E-Math News" as the subject. |
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Vol. 3, No. 1