tbeale

Diverse Text Set EDC 448 - Spring 2009 March 1st, 2009**
 * Thomas R. Beall

**Mathematical Modeling of Military Conflict **


 * Context for using the text set: ** What is the purpose of school? This is a question that politicians, educators, parents, and citizens argue about endlessly. Those who embrace the findings of “A Nation at Risk” argue that the primary purpose of public schools is to instruct the young in “basic skills” that they need to become productive members of society. Given that these “basic skills” are “reading, writing, and ‘rithmatic'”, some argue that the “A Nation at Risk” adherents are more interested in producing “followers” who will be the hewers of wood and haulers of water for the leading elites of society. Some would go on to argue that this philosophy is most evident in high school (and now middle school) JROTC programs. Teaching young men and women the services’ core values, dressing them in military-style uniforms, and introducing them to military culture without exposing them to the downside of military service are seen simply as means to recruit young men and women of limited means to be the nation’s “centurions”, not the nation’s “patricians”.

John Dewey, a psychologist, philosopher, and educational reformer, had a different vision. To him, the mission of the public schools was to educate the young to prepare them to be informed, contributing citizens in a democratic society – citizens who will make informed decisions about community and national policy.

While social studies and literature courses seem to be the best venue for this kind of education, mathematics can also be a powerful tool to help any citizen make informed decisions about national policy. The following texts demonstrate how mathematical models can be used to aid in making decisions about the use and employment of American military forces. They will be used in a series of lessons taught in an advanced high school mathematics curriculum, lessons that will help students grasp the power of mathematics and mathematical modeling as tools to gain insights, answer questions, and make informed and responsible decisions in the "real world."  · Citation: Wood, D. (2008) Afghanistan: Not enough troops. //Baltimore Sun Online//, July 15th, 2008. Available at: []. · Text Summary: This short news analysis from a military correspondent highlights the need for additional United States troops in Afghanistan. · Rationale: This text will serve as an introduction to the topic of using mathematical models to gain insights into real-world problems and solutions. The lesson will introduce simple Lanchester-type differential equations and demonstrate how they can be used to gain insights into questions such as, ”How many troops do we need in Iraq or Afghanistan?” · Use of text: I would use this text to introduce the topic, asking the question, “How do we decide how many troops are enough?” · Attachment: Go to above link. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Mathematical model (2009). //Encyclopædia Britannica Online//. Available at: []. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This article provides a definition of a mathematical model and gives examples of use in real-world applications. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: Having asked the question posed by “Text 1”, this text provides a definition of the concept we will develop and employ in the lesson. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I would ask students to focus on the statement, “Essentially, anything in the physical or biological world, whether natural or involving technology and human intervention, is subject to analysis by mathematical models if it can be described in terms of mathematical expressions” and, using the examples provided in the text, explore what mathematical expressions they know might be useful. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. <span style="font-family: 'Verdana','sans-serif'; mso-spacerun: yes; msobidifontsize: 12.0pt; msospacerun: yes;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Paulos, John Allen (2003). Lanchester’s law: Too few American soldiers?. //ABC News Commentary//, March 30th, 2003. Retrieved online February 23rd, 2009 from [] <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This commentary, published soon after the start of the Iraq War, essentially poses the same question as Text #1 – do we have enough troops deployed for the war we want to fight. In this case, the question was posed soon after the start of the Iraq War in 2003 (as we know, the answer turned out to be “No”). The commentator introduces the Lanchester model and suggests that it can provide insights into the answer to that question. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This article is an excellent introduction to the lesson because it suggests the utility of the mathematical model we will study, the Lanchester model, in a real-world, important application. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: This text is a natural follow-on to the first two. Having posed the basic question in Text #1 and then defining a mathematical model in Text #2, this text suggests that a mathematical model can be used to help answer questions like that posed in Text #1. The text also introduces some of the vocabulary that will make its appearance in more complicated texts to follow. By introducing the vocabulary in a fairly straightforward news commentary, students can begin to develop understanding of the terminology we will be using before embarking on the more difficult texts. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Lanchester, F. W. (1916). The principle of concentration, The N-square law. Chapter V in //Aircraft in warfare: The dawn of the fourth arm// (pp. 39 – 53). London: Constable and Company, Limited. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This is the text in which Frederick W. Lanchester, a British engineer, introduced his N-square law which became the cornerstone of combat modeling from his time to the present day. This is a challenging text but it was written not for the mathematician or scientist but for the educated layman. The mathematics is relatively simple, requiring an understanding of functions, basic algebra, and arithmetic – all within the reach of the upper level high school student. The mathematics is not challenging but Lanchester’s application of the mathematics in the development of a model portraying the dynamics of battle can be. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This is the foundational text of all Lanchester theory. It is one of the most challenging texts the students will read but it is the one that must be mastered in order to understand and apply the theory to models of military conflict. Students will not achieve the objectives of the lesson unless they master this text. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: We would spend some time on this text, perhaps several class sessions. As we read the text, we would use a number of the vocabulary comprehension strategies discussed during this course including techniques such as use of the Frayer model and “Previewing Content Vocabulary.” Important words to master include: //concentrate, validity, efficiency, strategic, tactical.// <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment:. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Hughes, W. P. (2000). A salvo model of modern missile combat. In //Fleet tactics and coastal combat// (pp. 268 – 274). Annapolis: Naval Institute Press. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This text comes from the U. S. Navy’s primary text on naval tactics. Captain Hughes (who was my thesis advisor at the Naval Postgraduate School) was the first naval officer since Rear Admiral Bradley Fiske (a contemporary of Lanchester) to apply mathematical modeling to naval combat. Hughes’ audience is the junior naval officer who may have a liberal arts or engineering degree but who may be seriously math averse. He presents a very simple, Lanchester-type model that has naval application. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: I selected this text as a secondary text to further demonstrate the utility of mathematical models in tactical decision-making. The vocabulary and grammar are more contemporary than Lanchester’s and may be more easily grasped by students. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I would offer this text as an alternative reading if students have inordinate difficulty with the Lanchester text. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Found at Google Books: [] <span style="font-family: 'Verdana','sans-serif'; mso-spacerun: yes; msobidifontsize: 12.0pt; msospacerun: yes;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: The History Channel (2007). Attack on an outpost. Available at: [] <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This short video portrays a reporter’s visit to a U.S. Army unit guarding an outpost in an Iraqi town. The outpost comes under attack by insurgents. The video shows the confusion and lack of glamor of modern military operations in Iraq. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: I selected this video because it portrays modern warfare in a manner that is more realistic than the Army’s recruiting video game or other games. This video speaks directly to the question, “Do we have enough troops in Iraq?” by portraying the real-life challenges of our soldiers and marines in places such as Iraq. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I would use this video in conjunction with Text #1 to introduce students to the question. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: The History Channel (2007). Medics treat battlefield wounded. Available at: [] <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This short video portrays medics performing battlefield first aid in an Iraqi town. Again, the video shows the confusion and lack of glamor of modern military operations in Iraq. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Just as with Text #6, I selected this video because it portrays modern warfare in a manner that is more realistic than the Army’s recruiting video game or other games. This video speaks directly to the question, “Do we have enough troops in Iraq?” by portraying the real-life challenges of our soldiers and marines in places such as Iraq. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I would use this video in conjunction with Text #1 to introduce students to the question. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: The History Channel (2004). Line of fire. Available at: [] <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This series of interactive maps portrays the arrangement of opposing forces in a number of historic battles. It gives meaning to the “force on force” interaction that Lanchester attempted to model. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: I selected this text to portray the real-life interaction of military forces throughout history. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I will use this text in conjunction with an Excel spreadsheet that portrays the Lanchester model to help students to connect the model to analysis of real-life military operations. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. <span style="font-family: 'Verdana','sans-serif'; mso-spacerun: yes; msobidifontsize: 12.0pt; msospacerun: yes;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Wattenberg, Frank (1997). One variable models: Equilibrium points and stability. Available at: [|http://www.math.montana.edu/frankw//ccp/modeling/continuous/onevar/learn.htm]. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: The Lanchester model is made up of differential equations. This text provides instruction, including interactive Java applets, on simple differential equations such as Lanchester’s. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This web page provides the necessary mathematical background to understanding Lanchester’s model. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: I will use this text in conjunction with Text #2 to help students to understand the mathematical foundation of the models we are discussing. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: Go to above link. **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #10: ** <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Beall, Thomas R. (2009). Interactive Lanchester model. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This is a portrayal of the basic Lanchester model on an Excel spreadsheet. Users enter the size of each opposing force and the effectiveness of their fire. A graph is produced similar to that found in Text #4 portraying the outcome of the engagement. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: I developed this model to permit students to use the Lanchester model to gain insights into battle dynamics in order to help them answer the basic question. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: Once students understand Lanchester equations and their potential use, we will use this spreadsheet model to conduct some basic analysis and gain some insights. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">D. Instructional Resources ** <span style="font-family: 'Verdana','sans-serif'; mso-spacerun: yes; msobidifontsize: 12.0pt; msospacerun: yes;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Kohler, Angela D. A. (2002). The dangers of mathematical modeling. //Mathematics Teacher//, 95(2), 140–145. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This article uses an example of a high school class conducting analysis on U.S. population growth data to demonstrate that three equally plausible model developed from the data can produce widely divergent results. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This article helps me to remain mindful of the fact that models derived from data analysis must be tested thoroughly for reliability and validity before they are accepted and used to help one derive insights into possible answers to questions. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: By me prior to teaching the lesson. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: This is a copyrighted article which I will provide on request. **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #12: **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Mathews, Susann M. (2004). Mathematical modeling: Convoying merchant ships. //Mathematics Teaching in Middle School,// 9(7), 382-391. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This article describes a workshop in which middle school mathematics teachers used recursive models (much like Lanchester’s) to understand why convoying of merchant ships was a critical element in the Allied victory in World War II. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This text demonstrates the potential use of models to educate both teacher and students. Because military history is the context, the article serves to answer skeptics that military analysis might not be appropriate or useful in a public school classroom. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: Again I would use this article to prepare myself for the lesson as well as to demonstrate the lesson’s potential to colleagues and parents. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: This is a copyrighted article which I will provide on request. **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #13: **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;"> <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Citation: Munakata, Mika (2006). Holding a mathematical modeling contest in the classroom. //Mathematics Teacher,// 100(1), 30-39. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text Summary: This article discusses how a high school mathematics teacher helped her students to develop different mathematical models to address real-life problems and answer questions. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Rationale: This article demonstrates that high school students can develop and use mathematical models in a meaningful and useful way. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Use of text: Again I would use this article to prepare myself for the lesson as well as to demonstrate the lesson’s potential to colleagues and parents. <span style="font-family: Symbol; mso-list: Ignore; msofareastfontfamily: Symbol; msobidifontfamily: Symbol; msolist: Ignore; msobidifontsize: 12.0pt;">· <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Attachment: This is a copyrighted article which I will provide on request. **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Using these texts together in the classroom: ** <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;"> As I have stated above, I would begin by asking the students to read Texts #1, #2, and #3 to set forth the basic inquiries we will be making and the basic definition of what we will be studying – mathematical models. Concurrently, I will ask the students to look at #6, #7, and #8 to provide background and context. We would then read Text #4 in which Lanchester sets forth his N-square law and the equations to support it. We would take as much time as necessary to understand this text, using various comprehension strategies. We would use Text #5, Hughes’ model, as necessary, to promote understanding. Concurrently, we would use Text #9 to ensure students understand the mathematics (differential equations) underpinning Lanchester theory. Finally, we would use Text #10 to conduct some basic analysis. <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;"> <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore; msobidifontsize: 12.0pt;">I. <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">National Council of Teachers in Mathematics (NCTM) Principles and Standards for School Mathematics. <span style="font-family: 'Verdana','sans-serif';">A. Use mathematical models to represent and understand quantitative relationships. <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">1. <span style="font-family: 'Verdana','sans-serif';">Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships; <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">2. <span style="font-family: 'Verdana','sans-serif';">Use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts; <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">3. <span style="font-family: 'Verdana','sans-serif';">Draw reasonable conclusions about a situation being modeled. <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">II. <span style="font-family: 'Verdana','sans-serif';">Rhode Island High School Grade Span Expectations for Mathematics (GSE’s). <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">A. __<span style="font-family: 'Verdana','sans-serif';">M(F&A)–12-2 __<span style="font-family: 'Verdana','sans-serif';">: Demonstrates conceptual understanding of linear and nonlinear functions and relations by representing and analyzing functions in several ways; recognizing properties of functions and characteristics properties of families of functions; applying knowledge of functions to interpret, model, and solve problems; analyzing characteristics of classes of functions (polynomial, rational, and exponential) to include domain, range, intercepts, increasing and decreasing intervals and rates of change; representing functions numerically, algebraically, graphically, and verbally (i.e. in written words), recognizing properties of a function from these representations, and transfers information from one representation to another; graphing polynomial, rational and exponential functions, including vertical and horizontal shifts, stretches, and compressions as well as reflections across vertical and horizontal axes; applying knowledge of functions to interpret and understand situations, design mathematical models, and solve problems in mathematics as well as in natural and social sciences. <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">III. <span style="font-family: 'Verdana','sans-serif';">GSE's for reading: R–12–8: Analyze and interpret informational text (which may include technical writing), citing evidence as appropriate by… <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">A. <span style="font-family: 'Verdana','sans-serif';">R–12–8.1 Explaining connections among ideas across multiple texts. <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">B. <span style="font-family: 'Verdana','sans-serif';">R–12–8.2 Synthesizing and evaluating information within or across text(s) (e.g., constructing appropriate titles; or formulating assertions or controlling ideas). <span style="font-family: 'Verdana','sans-serif'; mso-list: Ignore; msofareastfontfamily: Verdana; msobidifontfamily: Verdana; msolist: Ignore;">C. <span style="font-family: 'Verdana','sans-serif';">R–12–8.3 Drawing inferences about text, including author’s purpose (e.g., to inform, explain, entertain, persuade) or message; or explaining how purpose may affect the interpretation of the text; or using supporting evidence to form or evaluate opinions/judgments and assertions about central ideas that are relevant.
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">A. Print Resources: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #1: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #2: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #3: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #4: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #5: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">B. Media Resources **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #6: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #7: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #8: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">C. Online interactive resources **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #9: **
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Text #11: **<span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">
 * <span style="font-family: 'Verdana','sans-serif'; mso-bidi-font-size: 12.0pt;">Learning Objectives: **

Tom, I liked the diversity of your text set, in that you showed the numerous ways army personnel were needed (for example, medical professionals) rather than just infantry troops to fight. Now I'm not a huge fan of math, and in the past I have found that word problems from textbooks try to create real life problems, but don't do it very well. (Like who wants to measure the shadow from a tree? Right?) But I think this is a great way to connect a real life event and problem and the mathematical forces behind it. Your texts about the Lanchester model and the current articles and commentaries on the war made me think about things that never occurred to me. Also, just the idea of being able to find how many troops are needed, and how it worked in the past, as in text 8, were very engaging ideas for a lesson. Well done, sir. -Jay Whalen

Tom, I enjoyed your text set, considering I am not as interested in math. I thought that the commentary and article used to introduce the lesson were great ways to hook students into being intrested in the topic of math. The real life aspect of your text set would also continue to hold your students intrest in the topic of the Lancaster model. You did a great job making math interesting for me and am sure you would have yoour students interest as well. <span style="font-family: 'Verdana','sans-serif';">-Erin O'Toole