Warren Esty | Montana State University - Bozeman (original) (raw)

Warren Esty

Professor Warren Esty is an Emeritus Full Professor of Mathematics at Montana State University, Bozeman, MT. He has a Ph.D. and M.A. from the University of Wisconsin, and a B.A. from Oberlin College.

Prof. Esty has written three mathematics textbooks, The Language of Mathematics, Proof: Introduction to Higher Mathematics (with Norah C, Esty), and Precalculus.

He has won several minor and most of Montana State University's major teaching awards and been a featured speaker at Math Ed and Numismatic conferences.

Professor Esty's research interests are mathematics education and statistics for the study of ancient coins. He formerly did research in probability theory. He has published research in widely varying journals such as Annals of Statistics, American Statistician, Annals of Probability, Journal of Applied Probability, Theoretical Population Biology, Journal of the American Statistical Association, Mathematical Biosciences, Stochastic Processes and their Applications, Annals of the Institute of Statistical Mathematics, Mathematical Scientist, Numismatic Chronicle, American Journal of Numismatics, Arithmetic Teacher, Mathematics Teacher, FOCUS on Learning Problems in Mathematics, Psychology in Mathematics Education, the 1996 NCTM Yearbook on Communication in Mathematics, AMATYC Review, and numerous issues of the Proceedings of the International Conference on Technology in Collegiate Mathematics.

Now, in retirement, he writes webpages about ancient coins:
http://augustuscoins.com/ed/

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Talks by Warren Esty

Talk to a numismatic group (much revised April 2024), 2022

Percentile plots can help readers understand the weights in a group of coins. They are an alterna... more Percentile plots can help readers understand the weights in a group of coins. They are an alternative to histograms. (They work well to display any group of quantities, but this talk was about coin weights.) Compared to other methods, they have the major advantage that percentile plots show ALL the data and don't hide things (e.g. the type of distribution, skew, outliers, and sample size) that summary statistics like the mean and standard deviation do not reveal.

A simple method for creating them is illustrated.

ANS , 2022

“Statistics for Numismatics”. A PDF of the slides for an ANS talk given in May, 2022 , on the pre... more “Statistics for Numismatics”. A PDF of the slides for an ANS talk given in May, 2022 , on the presentation and analysis of numismatic data so it can be used to help answer questions in ancient numismatics.

Also, a youtube video of a reprise of the talk, very slightly reorganized, here:
https://www.youtube.com/watch?v=mmOT4Iqmsmw&t=4s

A very elementary talk on how RSA Cryptography works. The necessary math is illustrated with ext... more A very elementary talk on how RSA Cryptography works. The necessary math is illustrated with extremely simple examples. You need not know a lot of math to be able to follow it.

Money Talks, 2020

These are the (slightly revised) PowerPoint slides from my "Money Talks" talk given October 3, 20... more These are the (slightly revised) PowerPoint slides from my "Money Talks" talk given October 3, 2020, for the ANS. The talk is on youtube:
https://www.youtube.com/watch?v=VoBUVOUuvRA&feature=youtu.be
This talk discusses what we can learn from die studies in ancient numismatics and the many caveats there are to die analyses.

When I retired I had things I wanted to say about what was important to teach, so I gave a "retir... more When I retired I had things I wanted to say about what was important to teach, so I gave a "retirement talk" about teaching math and being a math professor. Here are slides in pdf format

The whole talk was videoed and put on youtube,
https://www.youtube.com/watch?v=6V3rEUTg9lc
If you watch it, start at 9 minutes 30 seconds in -- the beginning is local to Montana State University. You will see the camera did a good job of following me, but often failed to show the slides I was commenting on. Here are some (not all). This is really a separate talk on what we can teach that adds value to students.

My Academia.edu page has more about the book, "The Language of Mathematics." Also, here is a link to a web page about it: http://estymath.com/

Books by Warren Esty

Algebra is written in a symbolic language that is designed to express mathematical thoughts. This... more Algebra is written in a symbolic language that is designed to express mathematical thoughts. This is a text for a course that emphasizes language skills. Most math courses concentrate on computational skills; this one concentrates on how mathematical methods and facts are expressed so you can understand what mathematical symbolism says. This course emphasizes all the ways that mathematics is used to express thoughts in algebra and higher-level classes.

This file is a manual for instructors to teach them some of what they would know about the course... more This file is a manual for instructors to teach them some of what they would know about the course if they had already taught it. "The Language of Mathematics" is a course that emphasizes how mathematical thoughts, methods, and facts are expressed in symbolism. That is, it emphasizes how to read and write mathematics. Furthermore, it emphasizes how mathematical reasoning works.

This is a text for a course that introduces math majors and math-education majors to the basic co... more This is a text for a course that introduces math majors and math-education majors to the basic concepts, reasoning patterns, and language skills that are fundamental to higher mathematics. The skills include the ability to

read with comprehension,
express mathematical thoughts clearly,
reason logically,
recognize and employ common patterns of mathematical thought, and
read and write proofs. To see another section, see Section 2.2 on "Existence Statements and Negation" on Academia.edu.

The text is amazingly inexpensive. See about that at
http://estymath.com/Proof.html

This is Section 2.2 on "Existence Statements and Negation" in the text "Proof: Introduction to Hi... more This is Section 2.2 on "Existence Statements and Negation" in the text "Proof: Introduction to Higher Mathematics," by Esty and Esty. You can find out more about the text at http://estymath.com/Proof.html

A text designed to produce a deep understanding of algebra and trigonometry so that students will... more A text designed to produce a deep understanding of algebra and trigonometry so that students will be comfortable with their next math. Students will be well-prepared for calculus. This text is designed to be appropriate for self-study, as well as classroom use. You can find out more at http://estymath.com/precalc.html The book is sometimes inexpensive used on Amazon.com. Write me if you want to see some old exams or supplements.

Papers by Warren Esty

This is a page of notes on the differences between four articles on how to estimate the size of a... more This is a page of notes on the differences between four articles on how to estimate the size of a coinage. The method in the 2011 article, Formula (4), is preferred. Confidence intervals come from the 2006 article, Formula (4).

This article gives a statistical method for estimating the number of dies used to strike issues o... more This article gives a statistical method for estimating the number of dies used to strike issues of ancient coins and a demonstration that is it justified with real ancient-coin data. Its mathematical justification is also noted. This method is shown to be better than previous methods, including my own methods published previously.

Linkage can help arrange dies in chronological order and help determine how the mint was organize... more Linkage can help arrange dies in chronological order and help determine how the mint was organized. However, there are many caveats and typical deductions are often unwarranted unless the model of mint operation is remarkably simple. What can be deduced *if* the assumptions hold is discussed and the numismatic considerations that bear on *whether* the assumptions hold are outlined.

Numismatic Chronicle, 1984

... | Ayuda. Estimating the size of a coinage. Autores: Warren Esty; Localización: Numismatic Chr... more ... | Ayuda. Estimating the size of a coinage. Autores: Warren Esty; Localización: Numismatic Chronicle, ISSN 0078-2696, Nº. 144, 1984 , pags. 180-183. © 2001-2010 Universidad de La Rioja · Todos los derechos reservados. XHTML 1.0; UTF‑8.

Numismatic Chronicle, 1989

Look on Academia.edu for slides my 2024 talk on percentile plots. It tells you explicitly how to ... more Look on Academia.edu for slides my 2024 talk on percentile plots. It tells you explicitly how to easily make them with modern software you already have. In 1989 when this article was published they were not so easy to make! They are a good way to show all the data (weights) visually without hiding the type of distribution, skew, outliers, and sample size like summary statistics do.

See beginning at 6 minutes 20 seconds into this youtube presentation:
https://www.youtube.com/watch?v=mmOT4Iqmsmw
(or search for "Statistics in Ancient Numismatics, by Warren Esty")

Focus on Learning Problems in Mathematics, 1994

Observations about a college mathematics core course that emphasizes learning the language of mat... more Observations about a college mathematics core course that emphasizes learning the language of mathematics (which is basically the language of algebra) in order to develop students' abilities to read with comprehension, to express mathematical thought clearly, to reason logically, to recognize and employ common patterns of mathematical thought, and to grasp the nature of proof.
A qualitative study of the course's effectiveness.

Advances in statistics in numismatics over the previous six years.

dialnet.unirioja.es

... | Ayuda. Statical Analysis of hoard data in ancient numismatics. Autores: Warren Esty; Locali... more ... | Ayuda. Statical Analysis of hoard data in ancient numismatics. Autores: Warren Esty; Localización: XIII Congreso Internacional de Numismática, Madrid, 2003: actas-proceedings-actes / coord. por Carmen Alfaro Asins, Carmen Marcos Alonso, Paloma Otero Morán, Vol. ...

Numismatic Chronicle, 1991

... | Ayuda. The contribution of surface-silvering to silver content. Autores: Warren Esty; Local... more ... | Ayuda. The contribution of surface-silvering to silver content. Autores: Warren Esty; Localización: Numismatic Chronicle, ISSN 0078-2696, Nº. 151, 1991 , pags. 226-228. © 2001-2010 Universidad de La Rioja · Todos los derechos reservados. XHTML 1.0; UTF‑8.

Talk to a numismatic group (much revised April 2024), 2022

A simple method for creating them is illustrated.

ANS , 2022

Also, a youtube video of a reprise of the talk, very slightly reorganized, here:
https://www.youtube.com/watch?v=mmOT4Iqmsmw&t=4s

Money Talks, 2020

My Academia.edu page has more about the book, "The Language of Mathematics." Also, here is a link to a web page about it: http://estymath.com/

The text is amazingly inexpensive. See about that at
http://estymath.com/Proof.html

Numismatic Chronicle, 1984

Numismatic Chronicle, 1989

See beginning at 6 minutes 20 seconds into this youtube presentation:
https://www.youtube.com/watch?v=mmOT4Iqmsmw
(or search for "Statistics in Ancient Numismatics, by Warren Esty")

Focus on Learning Problems in Mathematics, 1994

Advances in statistics in numismatics over the previous six years.

dialnet.unirioja.es

Numismatic Chronicle, 1991

Numismatic Chronicle, 1986

How to estimate the size of a coinage Warren Esty has written several articles about estimatin... more How to estimate the size of a coinage

Warren Esty has written several articles about estimating the number of dies used to stoke a coinage, the last word being published in the 2011 article, “The Geometric Model for estimating the number of dies.” The earlier articles have useful discussion, but their statistical methods have been superseded by formula (4) with p = 1 in the 2011 paper which demonstrates it is a good estimator.]

1986 “Estimation of the size of a coinage: A survey and comparison of methods.” [This article has useful discussion of coverage and non-randomness, but mostly it considers and rejects statistical methods that don’t work well. The discovery of the methods that does work well is in the 2011 paper.]

1991-2 “The distribution of the number of coins struck by dies” in the American Journal of Numismatics, joint with Giles Carter. [The so-called “negative binomial” statistical model has a parameter, p, for the the distribution of the number of coins stuck by a die. Carter had used p = 2 for his “simplified method” and this paper used real data to reconsider the value of p. It concluded p = 2 was too high and lower values fit better. However, there was not enough data to decide on a particular value, later determined to be p = 1.]

2006 “How to estimate the original number of dies” in Numismatic Chronicle 2006. [Formula (1) is unchanged, but formula (2) is no longer thought to be the best. Use formula (4) in the 2011 article instead. Formula (4) is still the best way to give a confidence interval given the 2011 point estimate for the original number of dies.]

2011 “The Geometric Model for estimating the number of dies.” in Quantifying Monetary Supplies in Greco-Roman Times, edited by François de Callataÿ. [Formula (4) with p = 1 is the best way to estimate the original number of dies. Formula (1) will give very similar results if the sample is random, but it never is. For a confidence interval, use formula (4) in the 2006 article. ]

Low-silver Roman coins from the late third and early fourth centuries AD originally had a thin ap... more Low-silver Roman coins from the late third and early fourth centuries AD originally had a thin applied surface silvering. The contribution of that silver to the total silver content is discussed.

Numismatic Chronicle, 1993

Acceso de usuarios registrados. Acceso de usuarios registrados Usuario Contraseña. ...

AMATYC Review, 2005

In their sections on inverses most precalculus texts emphasize an algorithm for finding f [supers... more In their sections on inverses most precalculus texts emphasize an algorithm for finding f [superscript -1] given f. However, inspection of precalculus and calculus texts shows that students will never again use the algorithm, which suggests the textbook emphasis may be misplaced. ...

The Mathematics Teacher, 1992

The NCTM's Curriculum and Evaluation Standards for School Mathematics states, “Evaluation is ... more The NCTM's Curriculum and Evaluation Standards for School Mathematics states, “Evaluation is a tool for implementing the Standards and effecting change systematically” (1989, 189). Tests are one facet of evaluation, and we maintain that mathematics classes are strongly affected by the way in which test scores are used to generate final course grades. In the traditional secondary school mathematics class, current grading practices tend to drive instruction by putting constraints on specific course content and its organization. In turn, content and its organization affect testing and therefore grading. The interaction of these factors is an aspect of assessment that is not specifically discussed by the NCTM's evaluation standards. The purpose of this article is to examine the impact of grading on mathematics instruction and on the implementation of the curriculum and evaluations standards.

Arithmetic Teacher, Dec 1991

Many people have difficulty adding fractions. I think the emphasis on finding a *least* common de... more Many people have difficulty adding fractions. I think the emphasis on finding a *least* common denominator is at fault. The product of the two denominators serves equally well, is conceptually simpler, and is important in algebra.

Communication in Mathematics, K12 and beyond, 1996

Formulas for estimating the original number of dies from die-count data are given. The reasons th... more Formulas for estimating the original number of dies from die-count data are given. The reasons they might not yield good estimates are discussed.
Note: Although they were the best formulas known in 2006, additional data suggests a method which is better. See my 2011 article, "The geometric model for estimating the number of dies." However, the "Discussion" section of this older article is still relevant as are the confidence interval formulas.

Estimating the original number of dies from a sample with few or no duplicates is problematic. We... more Estimating the original number of dies from a sample with few or no duplicates is problematic. We cannot expect the "point estimate" to be at all precise (If there are no duplicates it would be infinity!), so confidence intervals are important. This paper discusses a formula for the confidence intervals. [The numerical example is mathematically correct given the data, but there is some doubt whether the data with 662 coins and only 2 die-duplicates is correct.]

The very rare radiates of Tacitus (AD 275-6) distinguished by XI or IA in the mint mark have more... more The very rare radiates of Tacitus (AD 275-6) distinguished by XI or IA in the mint mark have more silver than other radiates.