VOLUME 20 ISSUE 2 FALL 2011

Mathematica Militaris

THE BULLETIN OF THE

MATHEMATICAL SCIENCES DEPARTMENTS

OF THE FEDERAL SERVICE ACADEMIES

LTC Brian Lunday, USMA

LTC Doug McInvale, USMA

ASSOCIATE EDITORS:

Dr. Kurt Herzinger, USAFA

Dr. Jan McLeavey, USCGA

LCDR Tyree Clark, USNA

Prof. Tom Mahar, USNA

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Editor’s Note 1

Assessing and Improving Students’ Fundamental Mathematical Skills 2

Major Lee Evans and Major Chris Weld, USMA

Effectiveness of Repeated Fundamentals Testing in Collegiate Calculus 10

Major Benjamin Thirey and Major Phil LaCasse, USMA

Fundamental Skills: To learn or not to learn? 21

Dr. Chris Arney, USMA

noteworthy, I encourage you to direct questions and comments on these topics to the

authors and/or consider submitting a follow-up or response for the Spring 2012 issue.

Calling all Academies!! We value and welcome insights from all of our service

academies’ mathematics departments for publication Mathematica Militaris in order to

grow from shared perspectives.

The upcoming Spring 2012 issue will emphasize to the topic-- Beyond the Core Math

Program: Math Electives that Service Engineering Disciplines. At your service

academy, how many more math courses must an engineering major take, compared to a

2012 is the deadline for submissions for the Spring issue. We look forward to hearing

from you!

Brian J. Lunday

Co-Editor-in-Chief

brian.lunday@usma.edu

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Introduction

strength and versatility, that peculiar vigor and rapidity of comparison necessary for

However, the committee cautioned that “[the mind] should be taught gradually to develop

when students receive a letter of acceptance to the United States Military Academy

(USMA) and continues into the STEM disciplines. Approximately half of the cadets at

West Point will be STEM majors, but every cadet will have a four-course mathematics

sequence and a three-course engineering sequence in order to fulfill the requirements of a

Bachelor of Science degree. Considering students enter West Point with different

backgrounds in mathematics, it is important to establish a common baseline of

mathematical knowledge from which to build the curriculum.

concepts in which cadets must demonstrate proficiency as this baseline of mathematical

knowledge. Identifying those concepts begins with an assessment of West Point’s ten

The Engineering and Technology Goal Team and the Math and Science Goal Team,

assisted the Department of Mathematical Sciences to identify the 13 concept areas.

series of Fundamental Concept Exams (FCEs). In the subsequent sections, we will

discuss the sequence of exams and the standards for validating the exam. Then, we will

explain the numerous resources available to students. Finally, we will discuss ways to

track which questions should appear on the exams and ways to track student usage of the

website and other resources.

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fundamental pre-calculus skills and concepts that we have identified, they will most

likely not perform to their full potential in subsequent math, science, and engineering

courses. The 13 fundamental concept areas are:

Each of these areas is further divided into specific objectives. For example, under the

Logarithmic and Exponential Functions concept is the objective “Solve simple

Academy, a cadet candidate’s journey through the core mathematics sequence begins.

The core mathematics sequence is a four-course sequence consisting of mathematical

modeling and introduction to calculus, differential calculus, integral calculus, and

probability and statistics. A cadet candidate’s first contact regarding the Fundamental

Concepts Exam (FCE) requirements is through information mailed from the West Point

Admissions Office. They are led to the fundamental concepts website

, which informs

them of scheduled evaluations in fundamental mathematics skills following their arrival

and the standards required to achieve a passing mark. We will discuss this in more detail

in the Resources section of this article.

provides them an actionable outlet to do so prior to their arrival. Additionally, most cadet

candidates recognize their period at West Point will be rich with experience, but short on

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free time. Permitting them the opportunity to prepare for forthcoming academic

first summer at the academy, while developing a more thorough foundation for their

collegiate mathematics careers.

are conveyed to cadet candidates prior to their arrival at West Point. An 80% or better is

required of all students on an FCE in order to validate. This standard remains consistent

throughout a series of opportunities available throughout their Plebe year. A pencil and

scratch paper are the only authorized resources for the exams. Algebra, analytic

geometry, and trigonometry skills tested often require formulas committed to memory.

Memorization of this material is included among the basic building blocks to the

foundation of mathematical knowledge a cadet will need to successfully complete his/her

choice questions and 5 questions with work shown – a suitable number to gauge each

student’s proficiency in the fundamental concepts.

academic year begins to assist sectioning cadets by ability level. Those who struggled

significantly on the exam may be enrolled in MA100 (Pre-calculus Mathematics) where

they will continue to develop their fundamental skills prior to entering their USMA core

math sequence. Conversely, demonstrated proficiency on the Summer Gateway Exam –

while not the only indicator – is a required prerequisite for selection into the advanced

track of core mathematics beginning with MA153 (Advanced Multivariable Calculus).

All cadets in the advanced mathematics sequence validate differential calculus and

therefore take a three-course sequence.

subsequent FCEs do impact their MA103 course average. They account for 100 of the

2000 total available MA103 course points (5%), with only the highest of all FCE grades

achieved recorded per student. Although cadets scoring at least an 80% on the Summer

Gateway Exam are not required to take the FCE, if enrolled in MA103 they have the

option to take it for score to improve their course grade. Additionally, any cadet who

already achieved a passing mark of 80% or higher can continue to take the exams in an

effort to improve his/her grade.

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is at risk of failing MA103. These students are addressed on a case-by-case basis. If

allowed to proceed with their core math sequence, a deliberate remediation plan is agreed

upon as terms for their continuation. Students failing MA103 fall semester will join

those selected to enroll in MA100/101 to take MA103 in the spring. Completing the core

mathematics sequence for that population will likely require a summer semester

enrollment in order to catch up to their peers.

Resources

fundamental concepts exam. Shortly after they are offered admission to West Point, high

In addition to the fundamental concepts website, students who fail to validate the

Summer Gateway Exam have online practice problems and practice exams for the first 12

lessons of the semester. These practice problems are administered via a commercial

online assignment that provides real time feedback to the students. If students do not

understand how to work a problem, they can access links that allow them to read sections

of a textbook that explain the concepts, watch similar problems being worked, and even

chat with a tutor 24 hours a day.

of the semester, some instructors will spend class time reviewing an “FCE minute”

problem of the day, or short practice problem focused on a required FCE skill. In

addition to this classroom instruction, students have the ability to schedule additional

help them with this requirement.

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Concept Exam statistics are utilized

before and after each exam. Prior to the

exam these statistics assist in designing

questions. Each of the 13 fundamental

concept areas has multiple objectives,

making it infeasible to test all on every

exam. Therefore it is important to

deliberately compile exams with a

representative cross-section of questions

each exam is not necessarily the

objective. Fundamental skills have

varying levels of importance, as well as

levels of difficulty, and all these factors

are taken into account when preparing an

exam. Certain skills appear on the

majority of exams. For example, as seen

in Table 1, Skill 2b – to manipulate

algebraic expressions that contain integer

and rational exponents – was tested on

every FCE iteration in the Fall 2010 and

Fall 2011 semesters because it continues

to be a weakness of students and is a key

the 13 required fundamental skills. Recording scores by question on non-multiple choice

questions can be time consuming. However, this allows us to identify where positive and

negative trends exists in understanding these concepts. Any identifiable trends are shared

with the 20+ instructors of the course, and addressed when appropriate. Remediation

can be addressed at the course level by incorporating a fundamental concept that scored

poorly on the FCE into a subsequent course exam (labeled “WPR 2” and “WPR 3” in

Table 1) or course wide project, or alternatively left to instructors to address

independently as they deem appropriate. Remediation results using Written Partial

Review (WPR) exams are seen in Table 1 for the Fall 2011 semester.

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remediating weak areas, analyzing student involvement in the learning experience is

equally as important. With such a large number of resources available to the students, we

are constantly trying to determine whether or not students are using them and whether or

not they are helping students master the fundamental concepts. In order to answer the

question of whether or not students are using the resources, we use a web analytics

program to track website usage. There are many free, easy-to-use analytics programs

available online. With very little experience in website programming, it is very easy to

embed these trackers into the html coding of any website. The functionality of these

trackers can range from complex to extremely simple. For the purposes of our analysis,

we were looking to obtain data on how many people were visiting the website, when

those individuals were visiting the website, and where the visitors were coming from.

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students that performs relatively poorly on both the FCE and TEE. Since we are curious

improvement in order to see if that provided any insight into TEE performance. Out of

930 cadets in the Fall 2010 semester, 468 improved from FCE I to FCE II. Those 468

cadets had an average TEE score of 82.37%. The 462 cadets whose score decreased

between FCE I and FCE II had an average TEE score of 83.95%. If we further condition

on cadets who passed both versions of the FCE, 277 out of 567 cadets improved. These

277 cadets had an average TEE score of 87.74%. The 290 out of 567 cadets who did not

improve have an average TEE score of 87.50%. Finally, we looked at the subset of

cadets who scored less than 80% on both FCE tests. Of that group of 106 cadets, 70

improved from FCE I to FCE II. These 70 cadets earned an average score of 70.07% on

the TEE while the remaining 36 cadets scored an average of 73.03% on the TEE.

looking at the particular subset of the student population that passed one FCE and failed

the other. When examining the subsets of the population that passed both FCEs and

failed both FCEs, there is insufficient evidence to reject our null hypothesis.

lower than those who passed the first FCE but failed the second. This hints that

superficial learning is being used to achieve a test score of 80%, but a long term

understanding of the fundamentals is lacking.

Further Work and Conclusions

We observed noticeable improvement between FCE I and FCE II among the

population of students who initially failed to achieve a passing grade on FCE I. This

group of students received remedial training and as many opportunities to pass an FCE as

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needed before the final administration of FCE II. This suggests that retraining and

Additionally, this improvement occurred among a student population that is on the lower

end of the grade spectrum.

Regarding the predictive ability of FCE performance on TEE performance, our data

did not support any appreciable differences between the populations that we compared,

specifically those cadets who improved on FCE II versus those who did not. There does

seem to be a connection between students who score well on the FCE and performance

on the final exam, though we are unsure whether this performance reflects underlying

study patterns or inherent skill which makes individuals proficient at fundamental skills.

In the fall semester of 2011, we adjusted the FCE administration by only allowing two

Repeated focus on fundamental skills is important since an improvement was seen in

subsequent testing by a large portion of individuals who took multiple administrations of

the fundamentals exam. Repeated testing of fundamental skills from the start of the

semester until attaining a demonstrable level of proficiency helped a sizable number of

the course population with the administration of FCE II. Finally, the performance of

those students who failed both FCEs seems to mimic that of superficial learning, with no

discernable pattern. Again we find ourselves with the proverbial question of how to

encourage deep learning and self-learning on the part of our students.

References

[1] N. Jourdan, P. Cretchley and T. Passmore, “Secondary-tertiary transition: what

[3] K. S. Sofronas, T. C. DeFranco, C. Vinsonhaler, N. Gorgievski, L. Schroeder, C.

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Have you ever watched a basketball team warm-up before the game? What can you tell

about their abilities as players and as a team, if anything? Sometimes you can see very

clearly their advanced skill level -- crisp passes, quick-release shooting, sky-high

rebounding, slam dunks, and fancy moves. Sometimes you can see basic skills being

displayed and reinforced -- concentrated proper-form foul shots, perfect lay-ups,

excellent positioning, and organized movement with a purpose. And other times, you

demonstrate and perform the advanced skills and coordinated teamwork, even if they do

not possess the basic skills. It is surprising to me that some players are able to make

highly athletic and amazing plays, but are poor shooters (especially foul shots) and poor

passers or defenders. I am sure the coaches of players like this are frustrated, just as I am

with the students in my class, when they do not possess, develop, or refine their basic

fundamental skills. But, like in basketball, all is not lost -- these basic skills just might

not be the prerequisite to success that I once thought they were. Great basketball players,

especially many in this modern, post-Jordan era, have poor basic and fundamental skills.

And I suspect that many good students, in this information age, are able to perform high-

level modeling and problem solving despite not having all the basic algebra-trig skills we

desire they possess. Likewise sometimes a player doesn't appear to have the same

athletic level (poorer basic and/or advanced skills) as the others and yet when the game is

played turns out to be the star of the team. The player organizes the other players,

mathematics skills. And the big difference between sports coaching and our classroom

teaching is that we never really see our students play their "game". Education is just the

practice, warm-up mode for their real-life games that occur long after students leave our

classrooms. And the substantial difference is that one trains to play sports, but

mathematics is education that is meant to stay with students forever and enable them to

think in entirely new and higher-levels than we are able to teach them.

So what is my point? Just like a good, John Wooden-like coach, I believe as

conscientious math educators we need to educate students at all levels of mathematics

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(for lack of better words -- basic skills, advanced skills, and teamwork). We need

need to drill some of the skills into our students, but don't drill and kill or think of these

as prerequisite skills. I think we need to develop higher-level thinking, modeling,

problem-solving skills, but don't think these are all you need to do or that they come

easily to students -- this is difficult stuff. And finally, we need to develop their ability to

communicate, cooperate and collaborate. After all, a player with all the natural skills in

the world, but who can't run the plays or fit into the team will never help a team win

games. However, in basketball, a player may still entertain the crowd --- a factor in

sports that we don't need to develop in our mathematics students.

How do students learn? I am not an educational psychologist, but I am aware of

several different perspectives or theories of learning -- behaviorism, cognitivism,

students use their memory structures and their traits, beliefs, motivations, and emotions to

determine how information is perceived, processed, and stored in a form of instructional

scaffolding. This theory supports a depth perspective where there is a mental structure

being built so that later information builds upon earlier learning. I think many

mathematics educators are constructivists who see a mental pyramid being built in our

students' minds that allows more advanced mental attributes lying on top of the basic

skills until the zenith of the pyramid is their deepest, most complex, reasoning about

mathematics. Of course, everyone has holes and weak parts in their pyramid, but if there

is a good enough foundation from the earlier courses, we try to get our students to build a

higher, wider, and sounder structure to their pyramid. Unfortunately, if there are too

many holes or weaknesses, we have trouble knowing how to teach our students the higher

level topics and ideas until they shore up their foundation. Many times, I have found

bright students who had gone into short-term memorization mode in their math courses. I

constructed in our students' minds, then we have the obligation to give them the

opportunity to shore up their weaker foundation areas by giving them fundamental skills

and the equivalent obligation to build onto the top layers through more advanced topics

so they have a richer, larger, stronger, better structure called "their own mathematics."

What are our students learning? This is an easy one to answer -- calculus. Well, we

also need to add calculus' close relatives -- differential equations, analysis, numerical

computing, optimization, and linear algebra. In many ways, the K-12 math curriculum is

based on enabling students to reach calculus during senior year in high school or first

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year in college. Although I am happy to say at USMA, there is a bit more than the

mathematics, and lots of modeling for all our cadets, and options for graph theory and

more algebra for students who study math in depth. But much to my chagrin, there is

very little geometry in school or college these days. But why is calculus such a big

player in 21

century American math education? Frankly, I haven't a clue. I understand

that calculus was important in the 19

infrastructure -- engineering advancement was essential and we needed to use math and

the physical sciences to advance our fledging technologies. But by the time we entered

the information age in the late 20

physical science as we needed to advance the information, computing, and social sciences

that dominated our highest priority issues and problems. Yet, we haven't changed our

math education very much. Such a change is difficult -- we are set in our ways. But by

necessary for our undergraduate math majors -- just a nice branch of mathematics to see

continuity and dynamics. Yet we have an entire K-16 curriculum designed for the 1% of

the people in our country who will be engineers or physical scientists who really need

calculus. At USMA, perhaps the percent rises to 10%. By the way, at USMA, we do

have several fantastic math/interdisciplinary application courses in cryptology, chaos,

fractals, complexity, energy management, climate change, health care policy, operations

research, information science, network science, and computing that really excite students

and enrich our cadets' education. I believe we need more of our students (I would like to

say all of them) to experience this kind of math excitement.

I will end with a suggestion you probably expect by now -- balance. Let's balance the