Artifact for Standard #7: Planning Instruction

Jacob Choi

In preparing for this artifact, I was inspired from my supervisor’s comments that indeed, some evidence of my consideration in planning for different audiences could be shown to prove that indeed, I base the instruction on the type of audience and community the subject is being taught to. Thus for this artifact, I look at a familiar topic in a new light.

Combat Evolved was my Honors Capstone presentation. It compiles a milestone that is worth more than a year’s worth of work. In the following pages, we will dissect the process in which the capstone was transformed from 8 months of research into a 10 minute presentation to mathematicians, until it finally became a 12 minute presentation at the Honors Capstone Conference, 2009, where it won an Honorable Mention—one of 12 awards.

Of course, this paper was not presented to high-schoolers, but it demonstrates the necessity for watering down a dense mathematical subject into smaller parts that are more understandable by the average college student. This skill of presenting to a non-mathematical audience was advised by Dr. Enayat, whose children also attend Bethesda-Chevy Chase High School, and following the “dry run” at the CAS Research Conference, the following presentation was finalized for a most streamlined and informative 10 minutes.

As you go through this presentation, note that the anticipated audience is a panel of judges who are academic faculty and staff from American University, some members of the general public, and also college students. Most have little or no background in either military science or differential equations, or both, so I tried to cater this presentation to everybody based on the assumption that they would need some introduction to differential equations in the first place. This required the first two minutes of the presentation to be entirely set aside for the introduction of some core ideas: two from military science (power ratios and historical artifacts) and another three from mathematics (linear, square, and inverse square relationships and differential equations).

Slide 1Combat Evolved

Lanchester’s Laws in Modern Warfare

Presented by Jacob Choi

Advised by Dr. Ali Enayat

The opening slide was designed with three things in mind:

1. An absence of mathematical language to ease the audience

2. A plain illustration of the military background of the paper, and

3. Clearly stating the title, subtitle, and contributors to the project

Slide 2

Generally, an Artful Strategy must be supported withA thousand swift four-horse vehicles,

A thousand armored four-horse vehicles,A hundred thousand armored troops,

And provisions transported for a thousand miles.Sun Tzu, The Art of War, written between 480-221 B.C.

However, the second slide dived headlong into the topic, making the most out of the limited time. A quote from Sun Tzu was incorporated because it mentions quantitative numbers of four types of military forces. A background illustration gives context, as well as a time period in the bottom right corner.

Slide 3Our Age of Combat

• Fourth Generation Warfare (4GW)

�

• Asymmetric elements

• Unconventional warfare

• Unmanned vehicles

• Satellites, real time information

• Direct soldier-to-command communication

Still lacking in mathematical information, we plunge the audience back to today’s reality of combat, having just discussed Sun Tzu’s perception of it almost two millennia ago. All audience members should recognize and understand most, if not all, of the vocabulary described on the slide. This gives some confidence to the audience that the application demands some mathematics, not vice versa. All the bullet points are explained in full sentences with connections to current events such as the recent North Korean missile launch.

Slide 4 Combat ModelsThe Theory Behind Wargames

•Combat models rely on differential equations (DE’s, or DiffEQ’s)•Differential Equations model how the world works and reflect rates of change•Unlike algebra, where we find solutions, DE’s are used to model known scenarios with various inputs/influences

Maintaining the same silhouetted background for some consistency, we now focus on describing differential equations, a field of mathematics that most people are unfamiliar with. As advised by Dr. Enayat, I went about this using an example and fully describing how DiffEQ’s contrast with algebraic mathematics. This plants the seed of curiosity in audience members as they ponder how DiffEQ’s impact their world, and I offer some examples.

Slide 5Lanchester’s Laws

Lanchester’s Linear Law (Unaimed Fire)

• rR-bB=k, where rR is total number of blue troops slain by red forces, and bB is the converse.

Unique Case: If k=0, then rR=bB

•If k>0, then rR>bB, so red forces win

•If k<0, then rR<bB, so blue forces win

As the background grows more complicated, so too does the material. At this stage, the audience is not expected to read and comprehend all the information, so I pick apart and dissect every single variable and what it represents. For the “unique case” when both forces are wiped out, I quote three words that most people in the room would recall from the Cold War: “Mutually Assured Destruction” and many heads nod in agreement.

Slide 6Lanchester’s Laws

Lanchester’s Square Law

•The power of such a force is proportional not to the number of units it has, but to the square of the number of units

•What is a unit?

The linear law on the previous slide was simple to describe without words, but now the square law is presented with only words so that those in the audience who are less math-proficient have an opportunity to get the general idea behind the square law. Before moving on to the next slide, we propose a simple question, “What is a unit?” for audience members to think on.

Slide 7 Conditions for Lanchester’s Square Law

•Aimed Fire•Combat tactics,

not strategies•Unit sizes may vary, but

need to stay “small”—company sized

•Combat units are comparable types

•For example: infantryman vs. infantryman, or main battle tank vs. main battle tank

Naturally, this leads to conditions that must be present to allow Lanchester’s square law to work, even though we have not yet presented the law! To further illustrate what scale of the battlefield this can relate to, a large scale diagram of a company-sized element visually represents the structure and quantity of what one “unit” can potentially be. In purple/blue, other conditions are listed succinctly.

Slide 8Lanchester’s Square Law

bB2 - rR2 = k, where k is a constant

bBrR

dRdB

bBrR

dRdB

dtdRdtdB

=⇒−−==

∫ ∫ +=⇒= krRbBrRdRbBdB 22

Finally, the gibberish appears and only those who have completed basic calculus will catch most of these operations. Although differential equations is the level of math necessary to comprehend this slide, it is greatly streamlined and many steps in between are cropped out so that only the important facts show. In the change of background, we keep it simple but with a distinct taste of the military.

Slide 9

222

222

1

21

41

81

1612

41

41:

11...1

=≤=

=

+

≤

++

∑

Ex

Rrr n

“The other method of turning the enemy…entails the risk of attending a division of our own force, whilst the

enemy…retains his forces united and therefore has the power of acting with superior numbers against one of our divisions.”

General Carl von Clausewitz, Vom Krieg (On War)

Application: Division of Forces

nrrrR +++= ...21Where R represents the total number of Red units, composed of r1, r2, …., rn

Having explained where the law is derived from and how we can understand it from an application, we offer an example from a military maxim: Don’t divide your forces! The audience is also told how we knew it was good advice, but could not prove it until we use Lanchester’s square law. Even though we explain what the symbols represent, a numerical example is given on the bottom line.

The presenter does not read the quote—this is left for the audience to ponder on as we move through the theory.

Slide 10Application: Division of Forces

•Suppose red is an inferior force, and blue units are three times as effective as red units b=3r

•Red forces are two-fold the size of blue forces,

•rR2-bB2 = r(2B0)2 – 3rB02 = 4rB0

2 – 3rB02 = rB0

2 > 0

•Thus, red forces defeat blue forces, but in this situation, both groups are still unified

00 2BR =

Returning from a heavy math slide, we give a very concrete example. Instead of orally presenting the math, I describe it using common words: “Blue force is half as plenty as red force, but it is also three times more effective at fighting than red force.” A very plain slide has nothing to do with the topic, except a split arrow on the right to illustrate the “division of forces”

Slide 11Application: Division of Forces

•Now suppose that the blue units can divide the red forces into two approximately equal sized groups, and blue fights them in two sequential battles

•Blue units are still three times more effective than red unitsIn the first battle:

• blue versus reds leads to

•About 82% of the original blue units remain

In the second battle, red forces have more troops to begin with, but blue still wins with 57% of its original units remaining

0B 0R 01 32BB =

To compensate for the previous two heavy-math slides, this one returns the math problem to a real world problem, offering an alternate scenario and then showing the result at the end. The intermediate math sequences are eliminated to keep the page more attractive to non-math readers.

Slide 12

β=br

β Effectiveness

N (Battle Sequence)

1 2 3 4 5

1 0% 80%

2 71% 0% 60%

3 82% 58% 0% 40%

4 87% 71% 50% 0% 20%

5 89% 77% 63% 45% 0%

Even math majors will have trouble interpreting this slide so I depend on my oral descriptions to capture the audience. Color coordination saves the day, when I can say, “Yellow represents where both forces reach a stalemate, while blue squares show where blue wins and with how much...” A general rule is given on the left, and I must also affirm that both b and r are real positive numbers, so β is also a real positive number, although the N battle sequences can be counting numbers. Finally, we point to an example, “So with a β of 4, blue forces can beat red forces with 71% of its forces left in only two battles…”

Slide 13How does the β-index apply to warfare?•β-index draws from multiple input variables according

to a desired depth of accuracy•Generalized formula for Bn

01 BnBn β−=

•The β-index is only a ratio, so we are only interested in keeping it high!

The background returns us to the real world context as we explore the question of relevance. Using a red box to outline a general formula, we set up dialogue for exploring the factors that make up β. At this stage, we reiterate that Lanchester’s laws are almost a century old, but we want to see if they are still relevant in the 21st

century. The audience is also reminded that we are only interested in keeping the β-ratio as high as possible.

Slide 14

•US Army extends Basic Combat Training to ten weeks long, instead of eight weeks, as of October 2008, to emphasize “every soldier a warrior”

From here on, the slides have less written information and more visual information. This first example is something most people can associate with: more training equals better soldiers. Though it is common sense, a quantitative quote from the US Army validates this point. The background directly illustrates the point.

Slide 15

•Sea and air mobility allows the long-range capabilities of bringing heavy or large units to the battlefield.•An increase in force projection power corresponds to an increase in the β-index

Visual learners continue to benefit, for they need not read to understand this idea of increasing mobility. The idea of expanding flexible logistics is one that most of the audience can affiliate with, having probably been customers of the USPS, UPS, FedEx, or DHL. We bank on that link to drive the point home, while it is clear in the picture that there are multiple aircraft that are not only many in flight, but also large, signifying massive capabilities that metaphorically relate to the first large word on the slide: “Increasing…”

Slide 16

Having more troops creates a target-rich environment for the enemy…

As with the last slide, the picture is worth a thousand words. We also depend on the audience having a good understanding of “target rich environment” and its concept, so that rather than using mathematics to prove this point, we need not go further than recalling a concept that audience members already have learned.

Slide 17

…while small teams of covert forces are ideal for infiltration and have a much higher β-index thanks to rigorous training and many capabilities•Small vs. Large forces results in Lanchester’s Linear Law!

This is the only slide that is coupled with one before it, eventually concluding the suspense of a sentence. The visual contrast between many objects previously, to only a single aircraft and boat strikes the polar opposite of our headline about “sizes”.

To demonstrate a mathematically proven point, an anecdote is shared about how small vs. large forces will eventually result in Lanchester’s Linear Law.

Slide 18

“There is Local Intelligence;There is Inside Intelligence;

There is Counter-intelligence;There is Deadly Intelligence;There is Secure Intelligence”

Sun-Tzu

Kinesthetic learners will like this image best, as they are able to sense the vivid movement in this slide. However, the topic is about intelligence, usually a “quieter” subject. However, recalling Sun Tzu and another of his quotes, Tzu’s words form a tidy list that we dissect during discourse. Finally the audience learns that the aircraft pictured is used for electronic warfare (counter intelligence) and is being launched from a 21st century aircraft carrier. This relates it back to both the military science topic and mathematical implications that are associated with the logic and operations behind intelligence.

Slide 19 To close, we point the audience towards the horizon and have one more question for them to ponder. It is at this point that they may cease reading and paying attention to the screen as I verbally convey how powerful Lanchester’s Laws are. Anecdotes about how they have fuelled operations research and intelligence are shared, finishing with how this math prevents collateral damage and led to the successful finding of Saddam Hussein.

Slide 20

“You may fly over a land forever; you may bomb it, atomize it, pulverize it and wipe it clean of life—but if you desire to defend it, protect it, and keep it for civilization,

you must do this on the ground, the way the Roman legions did, by putting your young men into the mud.”

T. R Fehrenbach, This Kind of War, 1963

In an answer to the previous slide, I read this quote slowly and deliberately so that it seeps in to the audience. The image is designed to mimic the “fog of war” as a closing scene. At this point, it is reiterated that the mathematics used is still being developed today, and its applications minimizes waste and saves lives. Following my final words, I leave this slide on while inviting audience members to ask questions that they may have.

In reflection, the overall goal of the capstone presentation was met when the enthusiastic audience started asking questions. They were loaded and I responded accordingly as most of the questions were non-mathematical and not in need of clarification, but in search of exploration. This, I feel, demonstrated that I reached the non-mathematical crowd while still presenting the research as necessary. Moreover, some judges inquired as to what my aspirations were for this topic, and how I came about to choose it. My answer, though honest, did not fully represent the reason as being a need to investigate mathematics, but rather, a curiosity of military science that uses mathematics.

Such a work has taken eight months of preparation, forty-four pages of compiled information, and hundreds of hours spent between the Library of Congress, AU Library, and online in the world of journals and other resources. However, all of it resolved into a tight 12 minutes on stage. Much alike the varied research techniques and information sources, I feel that this conclusion reflected just that—teaching that same information with a variety of diverse techniques for multiple learning styles and audience members’ preferences.

Although I was not given the specifics of the audience’s backgrounds in academia, I could guess that most of them, being at American University, were humanities based majors. Given that assumption, I worked hard to stimulate their minds with less meat from mathematics, and rather gave them illustrations of how this math could be applied to their world.

Such an application that is clear to the audience validates the mathematics behind it, even if the audience members do not understand the mathematics. The objective of the Capstone was to show that this mathematics is still relevant to today’s world, having been conceived a century ago for other real world problems. After all, it is the highest tier on Maslow’s pyramid to problem-solve and it is not the mathematical understanding of differential equations that I had wished to convey to my audience, but rather, its potential to solve problems in the 21st century.