- Department of Mathematics (retired)
Schoolcraft College
Livonia, Michigan
USA
Randy K Schwartz
Schoolcraft College, Mathematics, Faculty Member
- History of Mathematics, Islamic Science, History of Maghrib and Andalus, Multiculturalism, Multicultural Education, Middle East History, and 14 moreAnthropology of Food, Food History, Culinary History, History Of Food Consumption, History of Food, International Relations, Mediterranean Studies, History of Math, The History of Math, The history of mathematics, Arab world, The History of Mathematics (history), Al-Andalus, and Non-European roots of mathematicsedit
This page contains a link to the full text of the published article.
Research Interests: Mathematics, History of Mathematics, Middle East History, History of Mathematics (History), Islamic Science, and 8 moreMaths, Non-European roots of mathematics, The history of mathematics, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Research Interests: Mathematics, History of Mathematics, Middle East History, History of Mathematics (History), Islamic Science, and 8 moreMaths, Non-European roots of mathematics, The history of mathematics, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Islamic Science, and 8 moreNon-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
A course module that uses concepts of modeling, trigonometry, differential equations, and features of the graphing calculator.
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A course module that uses concepts of linear algebra, calculus, and differential equations, and features of the graphing calculator.
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A course exercise in business optimization using concepts of calculus and features of the graphing calculator.
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Research Interests: Mathematics, Multiculturalism, History of Mathematics, Middle East History, Mathematics Education, and 11 moreHistory of Mathematics (History), Islamic Science, Maths, Non-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Page 1. Have you ever noticed that "circle" and "circus" sound very much alike? Or that "radius" and "radio" differ only a little? Word Wise Circles Around, About, Across, & Through Rheta N.... more
Page 1. Have you ever noticed that "circle" and "circus" sound very much alike? Or that "radius" and "radio" differ only a little? Word Wise Circles Around, About, Across, & Through Rheta N. Rubenstein University of Michigan-Dearborn Randy K. Schwartz Schoolcraft College ...
This brief article recalls the life of the English weaver-turned-mathematician Thomas Simpson (Aug. 20, 1710 – May 14, 1761), whose name became attached to Simpson’s Rule for the numerical approximation of integrals. The article takes... more
This brief article recalls the life of the English weaver-turned-mathematician Thomas Simpson (Aug. 20, 1710 – May 14, 1761), whose name became attached to Simpson’s Rule for the numerical approximation of integrals. The article takes special note of the fact that during this era, mathematics had special appeal among certain circles of working-class men and upper-middle class women.
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This article shows how the possible roots of a polynomial with integer coefficients are constrained by combinatorial principles. Knowledge of properties of Pascal’s Triangle can enable one to infer information about the roots from a few... more
This article shows how the possible roots of a polynomial with integer coefficients are constrained by combinatorial principles. Knowledge of properties of Pascal’s Triangle can enable one to infer information about the roots from a few known facts about the polynomial.
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This article, with accompanying exercises for student readers, explores the mathematics of Mendel’s Laws of Inheritance, and shows how such calculations are used to solve practical problems arising in biology, horticulture, and agriculture.
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This article, with accompanying exercises for student readers, explores the Binomial Theorem and its generalization to arbitrary exponents discovered by Isaac Newton.
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This article, with accompanying exercises for student readers, describes the efforts carried out by Leonardo da Vinci and other early-modern scientists to rectify the area of a circle by pizza-like dissection, and to use circle area... more
This article, with accompanying exercises for student readers, describes the efforts carried out by Leonardo da Vinci and other early-modern scientists to rectify the area of a circle by pizza-like dissection, and to use circle area calculations to understand the geometry of tree-branching in nature.
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This report summarizes some representative talks presented at the 12th Maghreb Colloquium on the History of Arab Mathematics (COMHISMA12), held on May 26-28, 2016, at Cadi Ayyad University in Marrakesh, Morocco. This rich and diverse... more
This report summarizes some representative talks presented at the 12th Maghreb Colloquium on the History of Arab Mathematics (COMHISMA12), held on May 26-28, 2016, at Cadi Ayyad University in Marrakesh, Morocco. This rich and diverse colloquium series, the longest-running and still leading international conference on the overall history of Arab mathematics, is part of the effort to uncover and disseminate the mathematical contributions of Arab and other Islamic cultures.
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Malaria has been humanity's worst public health problem throughout recorded history. Mathematical methods are needed to understand which factors are relevant to the disease and to develop countermeasures against it. This article and the... more
Malaria has been humanity's worst public health problem throughout recorded history. Mathematical methods are needed to understand which factors are relevant to the disease and to develop countermeasures against it. This article and the accompanying exercises provide examples of those methods for use in lower-or upper-level courses dealing with probability, statistics, or population modeling. These can be used to illustrate such concepts as correlation, causation, conditional probability, and independence. The article explains how the apparent link between sickle cell trait and resistance to malaria was first verified in Uganda using the chi-squared probability distribution. It goes on to explain that the incidence of sickle cell within a given population is an example of an asymptotically stable equilibrium determined by the selective pressure of malaria. It summarizes the impact of malaria on human history in order to explain why this equilibrium has varied over time and space. Finally, the article summarizes how linkage analysis and other statistical modeling techniques are being used as an important step in developing genomic pharmaceuticals to combat such diseases.
Research Interests: Mathematics, Mathematics Education, Applied Statistics, Medical Education, Statistics Education, and 12 moreSickle Cell Anemia, Malaria, Education (Mathematics), Statistics education (Education), Maths, Sickle Cell Disease, Mathematics-Education, Maths Education, Genomic Medicine, Mathematic Education, Sickle Cell Trait, and PUBLIC HEALTH EDUCATION
سيدي عمر عسالي، جامعة عمار ثليجي بالأغواط – الجزائر : الميادين التطبيقية لهندسة الإسقاطات العربية: مثال تسطيح الكرة على سطح الأسطرلاب BEBBOUCHI Rachid : Astronomie –Astrologie : complémentarité ou symbiose ? CALVO Emilia : Un... more
سيدي عمر عسالي، جامعة عمار ثليجي بالأغواط – الجزائر :
الميادين التطبيقية لهندسة الإسقاطات العربية: مثال تسطيح الكرة على سطح الأسطرلاب
BEBBOUCHI Rachid : Astronomie –Astrologie : complémentarité ou symbiose ?
CALVO Emilia : Un exemple de l’application des mathématiques à l’astronomie: Les instruments trigonométriques et leurs applications aux calculs astronomiques.
DÍAZ - FAJARDO Montse : Les techniques dans l'astrologie: la table de force d'une planète d'Ibn Azzuz al-Qusantini.
MARTOS QUESADA Juan : Le mathématicien andalous Ibn Mu'adh.
SCHWARTZ Randy, Al-qibla and the New Spherical Trigonometry: The Examples of al-Bīrūnī and al-Marrākushī
الميادين التطبيقية لهندسة الإسقاطات العربية: مثال تسطيح الكرة على سطح الأسطرلاب
BEBBOUCHI Rachid : Astronomie –Astrologie : complémentarité ou symbiose ?
CALVO Emilia : Un exemple de l’application des mathématiques à l’astronomie: Les instruments trigonométriques et leurs applications aux calculs astronomiques.
DÍAZ - FAJARDO Montse : Les techniques dans l'astrologie: la table de force d'une planète d'Ibn Azzuz al-Qusantini.
MARTOS QUESADA Juan : Le mathématicien andalous Ibn Mu'adh.
SCHWARTZ Randy, Al-qibla and the New Spherical Trigonometry: The Examples of al-Bīrūnī and al-Marrākushī
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Research Interests: Mathematics, Multiculturalism, History of Mathematics, Middle East History, Mathematics Education, and 12 moreMulticultural Education, Islamic Science, Maths, Non-European roots of mathematics, The history of mathematics, Maths Education, History of Maghrib and Andalus, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Research Interests: Mathematics, Multiculturalism, History of Mathematics, Mathematics Education, Multicultural Education, and 9 moreIslamic Science, Maths, Non-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Research Interests: Mathematics, Multiculturalism, History of Mathematics, Mathematics Education, Multicultural Education, and 12 moreHistory of Mathematics (History), Islamic Science, Maths, Non-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Arabic Science, History of Math, The History of Math, and The History of Mathematics (history)
Review of the state of scholarship in the history of mathematics in the Arab and Islamic world, based on COMHISMA10, a 2010 conference in Tunisia.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Education (Mathematics), and 10 moreIslamic Science, Non-European roots of mathematics, The history of mathematics, Mathematics-Education, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Review of the state of scholarship in the history of mathematics in the Arab and Islamic world, based on a 2004 conference in Tunisia.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Islamic Science, and 8 moreNon-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Review of the state of scholarship in the history of mathematics in the Arab and Islamic world, based on a 2002 conference in Morocco.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Islamic Science, and 8 moreNon-European roots of mathematics, The history of mathematics, Maths Education, History of Arabic Mathematics, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Traces the origins of certain mathematical terminology to the Arab world.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Islamic Science, and 7 moreNon-European roots of mathematics, The history of mathematics, Maths Education, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
Traces the origins of terminology for trigonometric functions to the study of mathematics and astronomy in ancient India.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Non-European roots of mathematics, and 7 moreThe history of mathematics, Maths Education, History of medieval mathematics, History of Math, South Asian History of Sciences, The History of Math, and The History of Mathematics (history)
A summary of the ancient mathematical text "Jiuzhang Suanshu (“Nine Chapters on the Art of Calculation”), with accompanying challenge questions for students.
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A summary of the medieval mathematical text by Bhāskara II known as Lilāvātī, with accompanying challenge questions for students.
Research Interests: History of Mathematics, Mathematics Education, Multicultural Education, History of Mathematics (History), Non-European roots of mathematics, and 6 moreThe history of mathematics, Maths Education, History of medieval mathematics, History of Math, South Asian History of Sciences, and The History of Math
Citation for the Trevor Evans Award: "In this outstanding article, Randy K. Schwartz gives a historical perspective of the development of the meter as a precise form of measurement beginning his account in the wake of the French... more
Citation for the Trevor Evans Award:
"In this outstanding article, Randy K. Schwartz gives a historical perspective of the development of the meter as a precise form of measurement beginning his account in the wake of the French Revolution. The birth of this unit of measurement was not without its pains. Amidst the political turmoil of the 1790s, a heated debate ensued from the decision to replace over 250,000 different weights and measures with a system of measurement that would not be arbitrary. The author verses readers in the mathematical arguments as to how a meter should be defined – as a unit related to the swing of a pendulum or proportional to the size of the Earth. Readers see how each idea was implemented in its turn and learn the mathematical challenges inherent in each. The author explains the motion of a pendulum clock and why it does not lead to the desired universal measurement. Readers also learn about the French surveying project, the complications associated with it, and its ultimate success. In each step of the article, readers are exposed to wonderful descriptions of how mathematical tools, ranging from trigonometry to differential equations, were used to define the meter."
"In this outstanding article, Randy K. Schwartz gives a historical perspective of the development of the meter as a precise form of measurement beginning his account in the wake of the French Revolution. The birth of this unit of measurement was not without its pains. Amidst the political turmoil of the 1790s, a heated debate ensued from the decision to replace over 250,000 different weights and measures with a system of measurement that would not be arbitrary. The author verses readers in the mathematical arguments as to how a meter should be defined – as a unit related to the swing of a pendulum or proportional to the size of the Earth. Readers see how each idea was implemented in its turn and learn the mathematical challenges inherent in each. The author explains the motion of a pendulum clock and why it does not lead to the desired universal measurement. Readers also learn about the French surveying project, the complications associated with it, and its ultimate success. In each step of the article, readers are exposed to wonderful descriptions of how mathematical tools, ranging from trigonometry to differential equations, were used to define the meter."
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Review of Mary Terrall’s "The Man Who Flattened the Earth: Maupertuis and the Sciences in the Enlightenment", with accompanying challenge questions for students.
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“Located on the Ariege River in the south of France, the town of Pamiers is an important part of the attractive rural setting in these parts. Some may find it surprising that the town was where a key text in the history of mathematics was... more
“Located on the Ariege River in the south of France, the town of Pamiers is an important part of the attractive rural setting in these parts. Some may find it surprising that the town was where a key text in the history of mathematics was written almost 600 years ago. This wonderful feature from the Mathematical Association of America’s 'Loci' feature describes the Pamiers Manuscript and its importance within the world of mathematics. Authored by Randy K. Schwartz of Schoolcraft College, this feature is divided into 15 parts, including Barter Transactions, Prices and Purchases, and First Shoots of Capitalism. Some of the highlights here include a narrative history of the manuscript (which was intended to teach young men commercial arithmetic) and a discussion of its novel use of negative numbers.”— Internet Scout Report 18:31 (Aug. 3, 2012)
Research Interests: Economic History, Mathematics, History of Mathematics, Mathematics Education, Mediterranean Studies, and 8 moreHistory of Mathematics (History), History of Economics, The history of mathematics, Maths Education, History of medieval mathematics, History of Math, The History of Math, and The History of Mathematics (history)
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An essay marking the 250th anniversary of the first publication of Bayes’s Theorem, with accompanying challenge questions for students.
Research Interests: History of Mathematics, Mathematics Education, History of Mathematics (History), History of Statistics and Statistical Agencies, The history of mathematics, and 6 moreHistory of statistics, Maths Education, History of Math, History of Statistics and Probability, The History of Math, and The History of Mathematics (history)
An essay marking the approximate 2300th anniversary of the birth of Archimedes, with accompanying challenge questions to introduce students to his Method of Exhaustion.
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This paper examines Giuseppe Peano’s axiomatization of numbers and arithmetic. The motivation and goals of Peano are summarized, and his project is placed in the context of the historical development of mathematics. Details as to how his... more
This paper examines Giuseppe Peano’s axiomatization of numbers and arithmetic. The motivation and goals of Peano are summarized, and his project is placed in the context of the historical development of mathematics. Details as to how his system was formulated and developed are provided. The subsequent evolution of the foundationist project, with its split into different schools and its ultimate failure to achieve its aims, are also briefly examined.
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In this paper, we examine Abel’s 1824 proof of the insolubility (in radicals) of the general polynomial of degree five. We give a precise description of what his proof establishes and what is does or does not imply, and we give a sense of... more
In this paper, we examine Abel’s 1824 proof of the insolubility (in radicals) of the general polynomial of degree five. We give a precise description of what his proof establishes and what is does or does not imply, and we give a sense of the place that this work had in the overall development of algebra. The paper details the overall strategy, and some of the technical machinery, that were used in the proof. It also focuses on one key section of the proof so as to provide a basic sense of the ingenious methods that Abel brought to bear on this problem.
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The proof that pi is a transcendental number, first provided by Carl Louis Ferdinand von Lindemann in 1882, was and remains one of the most celebrated results of modern mathematics. The proof was of interest in its own right, and it also... more
The proof that pi is a transcendental number, first provided by Carl Louis Ferdinand von Lindemann in 1882, was and remains one of the most celebrated results of modern mathematics. The proof was of interest in its own right, and it also resolved a host of questions that had focused the attention of mathematicians since ancient times, including classical problems involving constructions with straightedge and compass. Here, von Lindemann’s result is conveyed in a form that will be accessible to the average undergraduate mathematics major.
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Explains the usefulness of, and strategies for, discussion of word etymologies in the mathematics classroom.
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These three related articles (1) explain why it is in society's interests to adopt a policy of inclusion toward learning-disabled students, and (2) summarize some research-based strategies for teaching mathematics to them.
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The wars launched by the U.S. against al-Qā’ida in Afghanistan (Oct. 2001) and against Saddam Hussein in Iraq (Mar. 2003) were in fact closely linked, but the nature of that link was intentionally obscured by government pronouncements.... more
The wars launched by the U.S. against al-Qā’ida in Afghanistan (Oct. 2001) and against Saddam Hussein in Iraq (Mar. 2003) were in fact closely linked, but the nature of that link was intentionally obscured by government pronouncements. The attacks of 9/11/01 convinced U.S. authorities that keeping their forces stationed in Saudi Arabia in order to prop up the Gulf oil states had created an entirely unstable and dangerous situation and that the troops would have to be pulled back. But this would require first overthrowing Saddam, who was still strong enough to threaten Kuwait and Saudi Arabia if the overt U.S. military presence in those countries were brought to an end. The U.S. government couldn’t openly explain its reason for invading Iraq; instead, its officials cobbled together a web of stories about Saddam’s nuclear capabilities, germ stockpiles, ties to al-Qā’ida, etc. Saddam’s overthrow is linked to 9/11 because he stood in the way of America’s need to politically stabilize its oil-rich Gulf allies, a need made dramatically clear by the attacks of 9/11.
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A report prepared for the Canadian Labour Congress, Feb. 2000
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This article identifies lessons about sustainable agriculture and its cultural underpinnings as derived from three decades of fieldwork carried out by botanist George F. Estabrook (1942-2011) in the countryside of northern interior... more
This article identifies lessons about sustainable agriculture and its cultural underpinnings as derived from three decades of fieldwork carried out by botanist George F. Estabrook (1942-2011) in the countryside of northern interior Portugal. There, he found remnants of pre-industrial methods being practiced for both subsistence and commercial farming. Some of the principles that Estabrook discovered are applicable to a particular type of land, such as methods of terracing and irrigating steep hillsides, and methods to fertilize shale soils or granitic soils. Other principles have broader applicability, such as the importance of maintaining species diversity, or the use of energy inputs that rely ultimately on the sun or other renewable resources. Estabrook was not arguing that we can or should go backwards in time and adopt pre-industrial agricultural practices wholesale. Rather, his point was that by studying these practices, we can discover certain principles that could be creatively adapted as guides toward solving some modern problems of sustainability.
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This brief article uses history to argue against the common belief that the intellect provides only misdirection when it comes to gastronomic matters, and that philosophical outlooks are powerless in reforming humans’ natural tastes.... more
This brief article uses history to argue against the common belief that the intellect provides only misdirection when it comes to gastronomic matters, and that philosophical outlooks are powerless in reforming humans’ natural tastes. Examples cited include Christian ascetic vegetarianism in Russia and England; Islamic notions of pureness and balance in cookery; the influence of conscious nationalism and modernism on European cuisine; and the philosophies of Sylvester Graham in the U.S. and Rudolf Steiner in Austria.
Research Interests: Food History, History Of Food Consumption, History of Cuisine, Culinary History, History of Food, and 7 moreFood History, Food Culture, History of food and eating, Food History, Gastronomy, Culinary Arts, History of Food and Consumption, History of Food and Drink, Food and Philosophy, and History of Food and Gastronomy
This article reviews several published sources for recovering the culinary legacy of the earliest Dutch settlers in what is now the United States. Some of their characteristic foods include fried-batter cakes such as oliekoecken and... more
This article reviews several published sources for recovering the culinary legacy of the earliest Dutch settlers in what is now the United States. Some of their characteristic foods include fried-batter cakes such as oliekoecken and pannekoeken; pon haus (pork scrapple); vegetables warm-dressed in a salad (sla) or pickled (gepekelde); and baked goods leavened with pearl-ash, such as koekjes (cookies).
Research Interests: Food History, History Of Food Consumption, History of Cuisine, Culinary History, Foodways, and 18 moreHistory of Food, Dutch American History, American Foodways, Food History, Food Culture, Colonial New York, Dutch-American History, Dutch American Studies, Dutch-American cuisine and foodways, History of food and eating, Dutch Americans, Food History, Gastronomy, Culinary Arts, History of Food and Consumption, History of Cooking and Food Culture, Dutch--Americans, Dutch-Americans, Dutch In America, History of Food and Drink, Dutch American Foodways, and History of Food and Gastronomy
This article retraces the legacy of Dutch-settler food customs in Upper Midwestern America since 1847, when the first immigrants arrived there from the Netherlands. It shows that the Dutch heritage has subdivided, running on two separate... more
This article retraces the legacy of Dutch-settler food customs in Upper Midwestern America since 1847, when the first immigrants arrived there from the Netherlands. It shows that the Dutch heritage has subdivided, running on two separate tracks. The more traditional dishes of European peasant origin, such as hutspot and boerenkool, have undergone some modern Western ingredient substitutions but are still enjoyed by people of Dutch extraction, not only as tasty foods but as symbols of identity. Meanwhile, other foods, such as coleslaws and cookies, have been thoroughly assimilated into the mainstream; if they are noticed as Dutch, it is only because they are named or “packaged” as such.
Research Interests: Food History, History Of Food Consumption, History of Cuisine, Culinary History, Foodways, and 16 moreHistory of Food, Dutch American History, American Foodways, Food History, Food Culture, Dutch American Studies, Dutch-American cuisine and foodways, History of food and eating, Dutch Americans, History of Food and Consumption, History of Cooking and Food Culture, Dutch--Americans, Dutch-Americans, Dutch In America, History of Food and Drink, Dutch American Foodways, and History of Food and Gastronomy
This pair of articles retraces the legacy of early Florida foodways, such as the production of starch from koonti (Florida arrowroot), learned from Seminole and other natives; tropical fruits such as papaw and guava; hush puppies; and sea... more
This pair of articles retraces the legacy of early Florida foodways, such as the production of starch from koonti (Florida arrowroot), learned from Seminole and other natives; tropical fruits such as papaw and guava; hush puppies; and sea creatures like crab, mullet, conch, and marine turtles.
Research Interests: Food History, History Of Food Consumption, History of Cuisine, Culinary History, Foodways, and 12 moreHistory of Food, Florida history, American Foodways, Food History, Food Culture, Southern Foodways, History of food and eating, Cultural history of plants and food, History of Food and Consumption, History of Cooking and Food Culture, History of Food and Drink, History of Food and Gastronomy, and History of Florida
This essay analyzes the transformation of foodways in the medieval Maghreb, focusing on the interplay between material and ideological factors. The transformation left a deep and permanent imprint on Europe via the cultural bridge between... more
This essay analyzes the transformation of foodways in the medieval Maghreb, focusing on the interplay between material and ideological factors. The transformation left a deep and permanent imprint on Europe via the cultural bridge between what are now Morocco and Spain. The rise and spread of Islamic culture across the continents of Asia, Africa, and Europe unleashed economic and technological forces so powerful as to turn formerly conservative social, medical and religious doctrines into actual “workhorses” for modernization. For example, the theory of humoral balance was used to justify an endless cascade of spices and other ingredients. The image of Bedouin deprivation and self-reliance, once a proud theme of Arab poetry, became an accusation against the nomad’s paltry diet and means of life. Islam itself, which had banned the consumption of alcohol, adopted sweets and other stimulants as symbols of paradise. Increasingly, people living under Islam viewed alimentation from a perspective that went beyond sustenance to stimulation, pleasure, even luxury— the essence of the modern outlook on food.