
Welcome!
Elements of Mathematics: Foundations is a carefully planned
sequence of selfstudy courses for wellmotivated secondary school students with superior verbal and analytical skills.
The Elements of Mathematics: Foundations curriculum allows talented students to complete all of middle and high school mathematics up to calculus, in approximately three years.
Each course in the Elements of Mathematics: Foundations
series introduces mathematical ideas not found in the standard US curriculum and allows students to explore these ideas in depth. Each course presents the beautiful and fun side of mathematics. Completing just one course helps students develop logical reasoning and abstract thinking skills that they may not otherwise, and each additional course has a cumulative effect.
As with many serious intellectual pursuits, succeeding in
Elements of Mathematics: Foundations is
as much about persevering through challenge as it is about learning new ideas. Parents of a prospective student should read the following with their child:
EMF will challenge you as never before. Keep the following in mind as you rise to the challenge ...
 If you're used to understanding ideas without much effort, you should expect to put forth real effort in EMF.
 If you're used to getting high grades without studying, you should expect that you will need to study in EMF.
 If you're used to redoing the same problem to get a higher score, you should expect to give more careful consideration to your first attempt in EMF.
 If you're used to getting help from your parents on "school" math, you should expect to work more independently on the "mathematician" math in EMF.
 If you're used to being taught mathematics through a standard approach, even in other programs for gifted and talented students, you should simply expect the unexpected in EMF because there is no other course like it.
Elements of Mathematics: Foundations
Elements of Mathematics: Foundations
is a series of interactive, selfstudy math courses designed specifically for bright secondary school students that goes well beyond the typical gifted math class offered in schools or online. It is not an accelerated version of the standard US mathematics curriculum.
Based on the Intuitive Background books of the Elements of Mathematics (EM)
series developed over many years by an international team of mathematicians,
Elements of Mathematics: Foundations is a selfcontained,
selfstudy program that allows the talented student to complete all of middle and high school mathematics up to calculus before
leaving middle school.
The Elements of Mathematics: Foundations curriculum embodies the idea that mathematics is a comprehensive body of knowledge rather than a collection of unrelated
topics. Running through this deep discipline are certain fundamental concepts that elegantly unify the various branches.
Students who develop an intuitive understanding of these core concepts and an appreciation for how they are woven throughout mathematics
have a nearly insurmountable advantage in advanced classes over peers who do not.
Elements of Mathematics: Foundations
provides deep and lasting insight, and students who have completed the EM courses have progressed to
study college level courses in high school and graduate level classes upon arrival at college.
Is Elements of Mathematics: Foundations right for your child?
EMF takes a sophisticated approach to mathematics that students would not normally encounter until college.
Regardless of whether your child has just finished elementary school or has already completed algebra and geometry courses,
the answer depends on the individual student. The free aptitude test is a valuable tool that parents can use to help them
make this determination.
The Elements of Mathematics: Foundations online courses are based on the Intuitive Background books of the
Elements of Mathematics (EM) series. EM is the result of a collaborative effort of an international team of
eminent mathematicians and mathematics educators.
Through more than a decade of research and development, these scholars created an original curriculum that is fun and engaging while maintaining a level of mathematical rigor found only at the university level. Aimed solely at talented middle and high school students and unconstrained by the need to follow a standardsbased curriculum, EM focused on providing precocious students with a deep understanding
of mathematical structure.
Along the way, the Intuitive Background curriculum covered all of middle and high school mathematics
up to calculus before the end of middle school. After completing the Intuitive Background books,
students would continue in the EM series, which covered a significant portion
of a college undergraduate mathematics degree by the end of high school. The formal logic segment of the EM curriculum is already
available online at www.eimacs.com as part
of the Advanced Mathematical Logic track.
The following are the principle authors of the Elements of Mathematics series.
 Robert Exner, Syracuse University
EM Senior Author
 Peter Braunfeld, University of Illinois at UrbanaChampaign
 Lowell Carmony, Lake Forest (IL) College
 W.E. Deskins, University of Pittsburgh
 Arthur Engel, University of Frankfurt, Germany
 Vincent Haag, Franklin and Marshall College
 Burt Kaufman, Institute for Mathematics and Computer Science
EM Director
 Edward C. Martin, Institute for Mathematics and Computer Science
EM Senior Editor
 Lennart Råde, Chalmers Institute of Technology, Gothenburg, Sweden
 HansGeorg Steiner, Institut für Didaktik der Mathematik, Universität Bielefeld, Germany
 Nicholas Sterling, Binghamton (NY) University
 Robert Troyer, Lake Forest (IL) College
 Wilson Zaring, University of Illinois at UrbanaChampaign
The Elements of Mathematics: Foundations courses are offered by the
Institute of Mathematics and Computer Science (IMACS) through its distance learning division,
eIMACS.
Our online math and computer science courses are developed by the
IMACS Curriculum Development Group, which draws on an average of over 25 years' experience that includes extensive time
teaching gifted children in a classroom setting.
Based in Plantation, Florida, IMACS was established in 1993 and maintains local teaching centers in Florida, Connecticut, Missouri,
and North Carolina. Over 2,800 students from across the US and around the world study our widelyacclaimed curricula for
precocious students.
EMF Aptitude Test
If you are unsure whether your child is ready to begin the first EMF course, IMACS recommends that you encourage him or her to take the EMF Aptitude Test. This test is designed to help you and your child decide if he or she is ready for and would enjoy the EMF courses.
The EMF Aptitude Test is not like the tests typically given in schools. First, it deals with a subject that your child has almost certainly never studied before. Second, it is designed to gauge thinking skills and intellectual maturity, not knowledge; thus the questions are actually a series of puzzles. Third, some of the later questions will be challenging, and prospective students should not be discouraged if they are unable to solve all the puzzles. In fact, an excellent score can be achieved without answering all the questions.
The EMF Aptitude Test can be taken at your convenience with no fee and no obligation. All you need is:
 internet access;
 compatible Web browser (see Getting Started on left menu bar);
 desktop or laptop computer;
 paper and pencil; and
 30 minutes of uninterrupted time when your child can give the test his or her best effort.
Click here to sign up for the EMF Aptitude Test.
Getting Started

Step 1: Prerequisites
The prerequisites are as follows:
 Your child should be of middle school age. Learn more »
As with any program for gifted and talented students, it is difficult to offer exact guidelines for the ideal age at which
to study EMF. At a minimum, students must have mastered all of elementary school math. Students who have completed
algebra and/or geometry courses will find some of the material familiar although approached from a
very different, more sophisticated standpoint.
 Your child must be motivated, independent, and talented.
 He or she must also have excellent reading skills as students will need to comprehend text explanations of mathematical concepts.
 Before starting EMF, your child must have completed all of elementary school math, and
must be fluent in arithmetic operations of multipledigit numbers, including long division.
EMF courses are designed for students with a high level of intellectual maturity. To complete EMF in a selfstudy manner, students will need solid executive functioning skills as well. To help parents determine their child's level
of readiness, IMACS created a free online aptitude test for prospective students to take prior to enrolling in the first course.
We encourage parents to take advantage of this test because a child's experience with EMF will be far more positive and effective
if courses are taken when he or she is ready.
See Step 3 below for more information on the aptitude test.

Step 2: Browser
The EMF courses use "HTML5" to effect video, audio and various types of animation, and require an HTML5compliant browser. The EMF courses
have been tested on the browsers listed below at the version numbers given. Note that Microsoft Internet Explorer prior to IE10 is not HTML5compliant
and not suitable for use with EMF. Mobile devices such as the iPad are also not suitable for use with EMF.
 Firefox v14, available here.
 Chrome v21, available here.
 Opera v12, available here.
 Safari v6.

Step 3: Sign up!
Sign up securely by clicking here. After completing
the registration process, you will receive an email with instructions on how to take the aptitude test as well as how to enroll in the first course,
Operational Systems.
Courses
Each course in the Elements of Mathematics: Foundations series introduces important ideas not covered by the standard US math curriculum. Each course contains carefully crafted exercises that guide students through the process of mathematical discovery. Challenging problems encourage creative thinking and a healthy level of intellectual struggle.
In this way, each course avoids the "tellthendrill" approach and allows students to experience the joy of finding things out for themselves. Completing just one course helps students develop critical thinking skills and a passion for mathematical ideas that they may not otherwise.
 Operational Systems Learn more »
This course covers modular arithmetic using secret codes and online games. Learn about operational systems and their properties (commutativity, associativity, neutral elements, invertibility) by building interactive machines and evaluating nonnumeric operations. Get a solid introduction to the concepts of least common multiple and greatest common divisor, as well as to the geometric notions of midpoint and reflection.
 The Integers Learn more »
This course introduces positive and negative integers with an unusual elevator and mysteriously disappearing nuts. Learn about adding, subtracting, multiplying and ordering integers by building interactive number lines and driving a balloonpopping car. Students are exposed to various theorems about integer relationships and carefully guided through the first steps of how to make a wellreasoned and logical argument in support thereof.
 Sets, Subsets and Set Operations Learn more »
This course introduces the building blocks of set theory, which provides the basic language in which most mathematical texts are written. "Set" is what mathematicians call a collection of objects; it's a tiny word but a powerful concept as you will see. Learn about set properties and relationships involving sets with whimsical videos and the increasingly challenging String Game. Through mindstretching, interactive exercises, students cover fundamental concepts such as elements, roster names, the empty set, subsets, Venn diagrams, intersection, union, set difference and complement, and the Pascal Formula.
 Ordered nTuples Learn more »
This course considers what happens when order is imposed upon a collection of objects. Building on concepts introduced in Operational Systems, interactive features such as taxi driver navigation and the rockpaperscissors game give context to the properties of Cartesian product sets. An intuitive study of open sentences in two variables and the graphing of their solutions plant the seeds for future courses in Algebra. Students deepen their understanding of various mathematical operations as these concepts are extended componentwise to sets of ordered ntuples.
 Mappings Learn more »
This course examines relationships between the elements of two sets. Students explore various types of mappings, including permutations, with interactive ball sorters, slide rules and clever animations. Elementary combinatorial exercises lay the foundation for advanced concepts in Algebra, Geometry and Probability. Applying the properties of function composition, students delve into fractions and percentages in a mathematically rigorous and intuitive way.
 The Rational Numbers Learn more »
This course builds on topics covered in previous courses in order to examine the rational numbers, commonly referred to as fractions, as a natural extension of the integers. The course uses detailed narrations and interactive number lines and coordinate planes to cover the absolute value function, the properties of operations on rational numbers, how they are ordered, and that they are densely packed, not evenly spaced as the integers are. Students learn how to solve equations of rational numbers through the foundational concept of mappings instead of with rote algebraic manipulation. Using interactive proofbuilders, students also advance to the next step in learning how to construct wellreasoned and logical arguments.
 Decimals and an Application of the Rational Numbers Coming soon! Learn more »
This course builds upon knowledge of the rational numbers to introduces decimal numbers and their properties, arithmetic operations on decimals, and position notation for decimals. Students learn to compute various decimal approximations of rational numbers and to evaluate errors in approximation. The course revisits percentages in relation to decimals and arithmetic operations on percentages. Armed with a deeper understanding of decimals, decimal approximations, and percentages, students conclude the course with a case study of a fictional world in which mathematics is necessary to analyze a social and political issue.
 Probability I Learn more »
 Number Theory I Learn more »
 Algebra of Operational Systems Learn more »
 Algebra of Real Functions Learn more »
 Geometry: Incidence and Isometries Learn more »
 Geometry: Similitudes, Coordinates and Trigonometry Learn more »
 Probability II and Statistics Learn more »
 Number Theory II Learn more »
To experience EMF through interactive sample pages, click on the Sample Content box in the left margin of this page.
Each enrollment is for a period of 90 days.
 Operational Systems $59.95
 The Integers $59.95
 Sets, Subsets and Set Operations $59.95
 Ordered nTuples $59.95
 Mappings $59.95
 The Rational Numbers $59.95
 Decimals and an Application of the Rational Numbers Coming soon!
 Probability I
 Number Theory I
 Algebra of Operational Systems
 Algebra of Real Functions
 Geometry: Incidence and Isometries
 Geometry: Similitudes, Coordinates and Trigonometry
 Probability II and Statistics
 Number Theory II
Enrollment Options
Options for Completing EMF Courses 16 
Total Tuition 
Enroll in EMF Courses 16 individually at $59.95 per course 
$359.70 
Enroll in EMF Course Pack 6 
$305.95 
FAQ (Frequently Asked Questions)
Recent updates to the FAQ
What is IMACS?
IMACS is the Institute for Mathematics and Computer Science, an independent teaching and educational research institute that specializes in
courses for mathematically talented students.
How do I know if my child is ready for EMF?
EMF courses are intended to be selfstudy. To help parents determine their child's level of readiness, IMACS created a free online
aptitude test for prospective students to take prior to enrolling in the first course. While the aptitude test is not required for enrollment, parents are
strongly encouraged to have their child complete the test because it is designed to help gauge whether EMF is likely to be a good fit for a prospective student.
Whether you decide to have your child take the aptitude test or not, please keep the following prerequisites in mind: Your child should be of middle school age with a high level of intellectual maturity. He or she must have completed all of elementary school math and be fluent in arithmetic operations. Your child must be motivated, independent, and talented with superior analytical skills and excellent reading skills as they will need to comprehend text explanations of mathematical concepts. To complete EMF in a selfstudy manner, students will need solid executive functioning skills as well.
My talented high school student has already studied algebra and geometry. Is my child too advanced for EMF? Can my child start with a later course?
Your child is not too advanced for EMF. The EMF
courses were written by mathematicians to teach modern mathematics to those talented middle and high school students who
are capable of benefiting from a sophisticated approach. At times the subject matter will be familiar to
older students, but the approach is very different.
All students must begin with the first course — operational systems — because it is unlikely that your child has
any experience with these structures (even though they are fundamental to modern algebra).
The first course also addresses properties of nonnumeric operational systems, and very few high
school students will have experienced these systems. However, a talented high school student would be expected to move through
the material at a much faster rate than most younger students because of the higher level of maturity and betterdeveloped
study habits.
A very talented high school student who has successfully completed algebra and geometry courses may also be eligible for the Advanced Mathematical Logic courses
offered by www.eimacs.com. Click here
for a free aptitude test for the AML program.
How do I register my child to take the aptitude test? Is the test required?
After you complete the registration
process for EMF, you will receive an email with instructions on how your child can take the aptitude test as well as how to enroll
in the first course, Operational Systems. Taking the test is not required in order enroll in the first course. However, if you are unsure of
whether your child is ready for EMF and would find these courses interesting, we strongly encourage you to have your child take the
test. The test is designed to measure intellectual maturity, and the information you will gain from the test results may help you make your decision.
What exactly does "selfstudy" mean?
Selfstudy means that an EMF student is expected to complete these courses without
assistance from an instructor. The aptitude test and prerequisites outlined above and on the
Getting Started page are
designed to help parents determine if their child is ready to begin taking the first EMF course,
Operational Systems. Parents are encouraged to give careful consideration to these guidelines before
enrolling their child in order to avoid unnecessary frustration from starting the EMF program too soon.

If EMF is selfstudy, why has my child not been able to do the work on his own?
Not every child will be able to study the EMF curriculum independently. This can be for a variety of reasons including,
but not limited to, starting at too young an age, needing stronger reading skills, lack
of intrinsic motivation, underdeveloped study habits, and needing stronger executive functioning skills. In order to excel in EMF most students will need to do some studying,
especially prior to taking the incourse tests. Some children who have natural mathematical talent and have done well in standard math
courses may still need small to substantial amounts of adult support to succeed in EMF, depending on their maturity. IMACS encourages all parents to
regularly monitor their child's progress in EMF and to follow up with their child whenever there are signs of difficulties.
What should I do if my child is having technical difficulties with the course website?
Please send an email to techsupport@elementsofmathematics.com.
Provide a detailed description of the technical problem your child is experiencing. IMACS will work to
resolve your issue as quickly as possible. Please note that EMF students should not use this email
address to request help with their coursework. EMF courses are intended to be selfstudy.
What is the appropriate pace for working through an EMF course?
The appropriate pace is whatever the student is comfortable with. However, it is best not to go too many days without completing some coursework.
A student needs to keep the material fresh in his or her mind since EMF courses move along steadily with little to no repetition.
A minimum of at least two or three onehour sessions spread out over a week is recommended.
How will I know how my child is doing in the course?
A report card will be emailed each Monday morning. If a student has not made progress in two weeks, report cards are no
longer sent until the student answers another exercise.
How are EMF course scores calculated?
For each course element (exercise, review quiz problem and test problem), IMACS analyzes the available raw scores to
determine a level of difficulty. This information is then used to adjust the grading scale for that course element. Think of it as a "curve" if you like.
For example, if a student earns 12 out of 16 points on an exercise, this equates to a raw score of 75%, which might result in an adjusted score of
90% for a difficult exercise.
The adjusted scores for exercises and review quiz problems are then accumulated to form an assignment average. Similarly, the adjusted scores for the
problems of a given test are accumulated to form an average for that test. If a course has more than one test, the individual test averages are
themselves averaged with equal weighting to form a single test average. The course score is then calculated as a weighted average of the assignment
average (60% weight) and the test average (40% weight). In view of the demands of this course, any student who obtains a course score of 85% or better
should be very proud of his or her achievement.
As more students complete a course that my child completed already, will my child's course score change?
No. If your child has already completed a course, your child's course score will not be affected by the scores of students who complete
that course after your child. IMACS will update the adjusted score algorithm and grade mapping as more students complete a course,
but this will affect only students who subsequently complete the course.
Why does IMACS calculate grades in this way?
As you know, the EMF curriculum is quite different from a standard mathematics course and far more challenging. Some exercise and test questions
are such that obtaining a standard "A" grade of 90% is challenging and rarely achieved. For such questions, a "B" grade of 80% is regarded as exemplary.
As such, it is difficult to properly reflect EMF student achievement within a standard K12 grading model using raw EMF scores.
Nevertheless, parents and students may find it useful to consider performance visàvis a familiar framework, and so the IMACS grading system
uses scores that are adjusted to reflect the varying levels of difficulty of exercises and test questions.
My child completed an EMF course and received a lower average score than he is used to getting. How should we interpret this result?
First, your child is to be commended for completing the EMF course. Remember that this rigorous and demanding program challenges young students
to learn and think deeply about complex ideas in mathematics that are rarely introduced outside of a university setting. If
a student's final average score is lower than 75%, however, IMACS recommends waiting at least a year before enrolling in the next course.
With time, a student who has the natural talent to learn the material may also develop the intellectual maturity, motivation, reading
skills and executive functioning skills to excel in EMF on a selfstudy basis.
Does a student have to take the courses in order, or can some courses be skipped or taken out of order?
The EMF courses must be completed in order and without skipping. These courses are designed to build an intuitive foundation for
mathematics through carefully planned steps.
While EMF is new to the online environment, the curriculum itself has been in use for over 20 years
and is reviewed regularly to determine if updating or reordering sections will improve its effectiveness.
What if my child completes the EMF series of courses?
He or she will be ready to take calculus classes at the AP level. Students who complete the EMF series with an overall average of 80% or higher may also be
eligible for the Advanced Mathematical Logic series of courses. An aptitude test for these universitylevel courses is
available at www.eimacs.com.
Are courses scheduled at specific dates and times?
No. For a given student, a course begins when that student enrolls in that course. Courses may begin at any time during the year.
As long as a student is enrolled in an EMF course, he or she will be able to access that course's material 24 hours a day,
seven days a week. In addition, a current student will have access to all previous EMF courses in which that student was enrolled.
A student may complete coursework at his or her convenience within the enrollment period.
What happens if my child does not finish a course within the enrollment period?
Parents may reenroll a student in a course that he or she did not complete within the original enrollment period. Once the course is completed,
if the student is enrolled in the next course, any time remaining from the reenrollment period will be added to the new enrollment period for
the next course in the sequence at no additional cost.
If my child completes a course before the enrollment period ends, can he or she start the next course?
Once a student completes a course, he or she may enroll in and begin the next course in the sequence. Any time remaining from the previous course
will be added to the enrollment period for the next course at no additional cost.
To enroll in the next course, click here
and follow the registration process.
Is there a way for parents or teachers to receive curriculum support from IMACS other than through the Help Forum?
The EMF curriculum is "mathematician" math as opposed to standard "school" math. Through the work of professional mathematicians
and mathematics educators, this advanced material has been made accessible for extremely bright and motivated young students.
Individuals capable of providing support to EMF students would need a rare combination of skills: the ability to understand abstract,
universitylevel mathematics AND the ability to relate these complex ideas to middle and highschool aged children.
Because very few middle and high school teachers have any experience teaching this level of mathematics, it would be challenging to find
appropriate curriculum support staff for EMF. Furthermore, the cost to hire such uniquely skilled support staff would necessarily raise the tuition for
EMF courses substantially. The goal of IMACS in offering EMF is to provide wide access to our worldclass curriculum in mathematics in
a way that is still relatively affordable.
Will IMACS provide any kind of documentation to show that my child has completed a course?
When a student completes an EMF course, IMACS provides a Student Transcript and a Completion Certificate
that may be downloaded through the student's account.
Is EMF the same as New Math?
While the origins of EMF date back to the heyday of the New Math, the EMF developers both then and now have taken an entirely different approach. They could afford to do so because the target audience for EMF is highly gifted students who have already mastered standard elementary school mathematics and beyond, quickly and effortlessly. Furthermore, New Math was implemented with woefully inadequate teacher training. By contrast, EMF's selfstudy curriculum is written by professional mathematicians who average over 25 years' experience educating gifted children.
Are you an accredited institution?
While IMACS is not an accredited institution, we have been operating local teaching centers since 1993.
During the 10 years prior to that, we taught the EMF curriculum in the Broward County
public school district as part of the acclaimed Project MEGSSS program. Regardless of accreditation status,
our students find that the most meaningful and enduring benefit from having been in our program is the advantage it
gives them in college and in their future careers.
What is your refund policy?
IMACS is not able to offer refunds on EMF courses due to their low cost.
What is your "vacation" policy?
Enrollment periods are fixed and may not be paused. However, the length of an enrollment period is
designed to give students a reasonable amount of flexibility in consideration of their often
active schedules.
I've read all the material available on the EMF website, but I still have questions that I need
answered before enrolling my child. How can I get more information?
Parents interested in enrolling a child in EMF may email specific questions to IMACS at
info@elementsofmathematics.com with the subject line "EMF enrollment question". Please note that EMF students
should not use this email address to request help with their coursework. EMF courses are intended to be selfstudy.
I am an educator interested in licensing EMF. How can I get more information?
EMF allows talented students to work through the curriculum on their own, at their own pace, without the direct support of a teacher. As such, educators interested in licensing EMF should be aware that EMF is not designed for traditional, teacherled classroom instruction. EMF is most suitable as a resource to license when a school does not have the curriculum or staff to fully meet the needs of their most talented students.
Educators interested in licensing EMF may email a request for information
to IMACS at info@elementsofmathematics.com with the subject line "EMF licensing". Please note
that EMF students should not use this email address to request help with their coursework.
EMF courses are intended to be selfstudy.
How will I know when future EMF courses come online?
IMACS will email parents of current EMF students when new courses come online.
You may also "Like" us on Facebook at www.facebook.com/eimacs to receive
course updates as well as free weekly logic puzzles.
I think I found a mistake in a course. What should I do?
First use the EMF Help Forum to verify that there is a mistake in the course and not in your understanding.
If you really did find a mistake, please email IMACS at info@elementsofmathematics.com with the subject line
"EMF mistake". Please note that EMF students should not use this email address to request help with their coursework.
EMF courses are intended to be selfstudy.
Gifted and Talented Resources
The following are recommended resources for information on serving the needs of gifted and talented children:
Organizations
Web Sites
Books

FREE APTITUDE TEST
Click to find out if the EMF program is a match for your talented child.
SPECIAL OFFER
15% discount
when you purchase EMF Course Pack 6.
Click to enroll now!
Imagine ...
 that the entire
world of science and technology is open to your talented child ...
 that your child has completed all of middle and high school mathematics except calculus by
the time he or she finishes middle school ...
 that by the time he or she
reaches high school, your child is ready for undergraduate work ...
 that your child will one day
make important contributions to science and mathematics ...
... and
 that a wellestablished curriculum
designed to make this happen is now available online!
