The Most Advanced Mathematics Curriculum Ever Devised
For Talented Secondary School Students
Is Now Available Online.
"This curriculum does a wonderful job of captivating young minds and provoking within them a lifelong thirst for
learning, producing graduates with unusually mature insight into mathematics."
Professor Emeritus Vincent Haag Franklin & Marshall College
"My son cannot stop talking about the mathematical ideas he is learning. He is choosing to do EMF over Minecraft!"
Nan Rosenberry EMF Parent
"I was taking a practice test for the timed AMC8 exam when I came across a problem involving non-standard mathematics. Thanks to EMF, I was already familiar with these ideas and solved the problem a lot faster."
Peyton Robertson EMF Student
"As a homeschooler, our son is an avid user of online educational resources. I recently asked him to pick his favorite online course. He answered, 'most definitely EMF, by a wide margin'."
Arvinder Oswal EMF Parent
"EMF does more than teach; it inspires. Using the ideas I learned from EMF, I was able to show the student I tutor that math is truly fun!"
Hossain Turjo EMF Student
"The way of thinking about a problem that I learned from your program is something I use every day. I think I am a better physician because of the years I spent in EM."
Cara O'Brien, M.D. and EM Alumna while a Resident at Duke University
"After many years of boredom in elementary school, I finally felt
challenged. EM was also wonderful preparation for my future schooling."
Tuni Kundu, Ph.D. and EM Alumna while a Mechanical Engineer at Fluor
"The content and the pedagogy are of the highest quality and I recommend it without reservation."
Professor Emeritus Gerald R. Rising State University of New York at Buffalo
"I still remember remarking to a friend in middle school that, after coming out of each EM class, I felt smarter! I still marvel that I was exposed to such a wealth of mathematics at such a young age."
Rochelle Pereira, Ph.D. and EM Alumna while Assistant Professor of Mathematics at The College of St. Catherine
"There is no doubt that the Elements of Mathematics curriculum gave me the logical thinking skills and mental framework that
inspired and enabled me to develop the Chocolate Fix puzzle system."
Mark Engelberg, EM Alumnus Inventor of ThinkFun's Chocolate Fix logic game
"I have since graduated from Yale College and Harvard Law, and I can honestly say that my EM classes were the most challenging academic experiences of my life."
Ellen Moskowitz, J.D. and EM Alumna while at Ropes & Gray
"Our program in mathematics is the strongest in the nation and for [an EM student] to be able to jump in with our best
students and perform at the highest level is ample testimony of the strength of the program."
Professor Andrew M. Gleason Harvard University
"EM was the greatest program ever imagined -- and there is no way I could have stayed in teaching 46 years had it not been for the EM program."
Elements of Mathematics: Foundations is a carefully planned
sequence of self-study courses for well-motivated secondary school students with superior verbal and analytical skills.
The Elements of Mathematics: Foundations courses
are being offered online by the Institute for Mathematics and Computer Science (IMACS), which specializes in courses for exceptionally talented middle and high school students.
Elements of Mathematics: Foundations
is not an accelerated version of the standard US mathematics curriculum. Instead it establishes
foundational concepts that allow the suitably talented child to progress quickly through material
for which others would require significant drill and practice. It then proceeds to cover concepts in a mathematically consistent
way, incidentally covering all of the standard high school curriculum — and much, much more.
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, and your child
will need to comprehend and accept this statement:
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 acing the test without studying, you should expect that you will need to study in EMF.
If you're used to getting 100% on all things math, you should definitely expect to get less than perfect scores in EMF.
If you're used to redoing the same question until you get a "perfect" score, you should expect that you will need to give more careful consideration to your answers in EMF before submitting them.
If you're used to getting help from your parents on "school" math when you need it, you should expect that they will be unfamiliar with 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
is a series of 15 interactive, self-study math courses designed specifically for bright secondary school students that goes well beyond the typical gifted math class offered in schools or online.
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 self-contained,
self-study course that allows the talented student to complete all of middle and high school mathematics with the exception of 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? Talented students who have completed all of elementary school math but have not yet completed algebra and geometry would gain the most from EMF. However, students who already have some experience of algebra and/or geometry may still find benefit because EMF introduces concepts that are not covered in standard high school mathematics classes.
The Elements of Mathematics: Foundations online courses are based on Book 0: Intuitive Background of the
Elements of Mathematics (EM) series of textbooks. 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 for talented middle and
high school students while maintaining a level of mathematical rigor found only at the university level. Aimed solely at talented students and unconstrained
by the need to follow a standards-based curriculum, EM focused on providing precocious students with a deep understanding
of mathematical structure.
Along the way, Book 0: Intuitive Background covered all of middle and high school mathematics
except calculus before the end of middle school. After completing the Intuitive Background series,
students would begin the EM formal logic 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 Urbana-Champaign
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
Edward C. Martin, Institute for Mathematics and Computer Science
EM Senior Editor
Lennart Råde, Chalmers Institute of Technology, Gothenburg, Sweden
Hans-Georg 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 Urbana-Champaign
The Elements of Mathematics: Foundations courses are offered by the
Institute of Mathematics and Computer Science (IMACS) through its distance learning division,
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 widely-acclaimed curricula for
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:
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.
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. Students who have completed all of elementary school math but have not yet completed algebra and geometry
would gain the most from EMF. However, students who already have some experience of algebra and/or geometry may still
find benefit because EMF introduces concepts that are not covered in standard high school mathematics classes.
Your child must be motivated, independent, and talented.
He or she must also have excellent reading skills as students will need to comprehend meaningful text explanations of mathematical concepts.
Before starting EMF, your child must have completed all of elementary school math, and
must have mastered addition, subtraction, multiplication and division of
multiple-digit whole numbers Learn more »
The whole numbers are the numbers 0, 1, 2, 3, ..., and so on.
Your child must be familiar with "long division" to obtain a quotient and a remainder.
For example, 96 divided by 29 yields a quotient of 3 and a remainder of 9:
must understand the following terms: sum, product, multiple, clockwise,
EMF courses are self-study courses that require a certain level of intellectual maturity. 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 HTML5-compliant 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 HTML5-compliant
and not suitable for use with EMF. Mobile devices such as the iPad are also not suitable for use with EMF.
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,
Advances in technology have allowed our developers to create this comprehensive series of self-study courses. Suitably talented students are able to work
independently and at their own pace while still developing a deep understanding of the material.
IMACS is working diligently to bring these courses online as quickly as possible. Currently,
the first four courses are available, and the fifth will be
available in Winter of 2013.
To experience EMF through interactive sample pages, click on the Sample Content box in the left margin of this page.
The study hours needed to complete each course varies considerably.
The first course consists of more than 70 engaging exercises. Talented, well motivated students can
complete this course with approximately 35 hours of study. Each enrollment is for a period of three months.
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 non-numeric 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.
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 balloon-popping car.
Students are exposed to various theorems about integer relationships and carefully guided through the first steps of how to make a
well-reasoned and logical argument in support thereof.
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 mind-stretching, 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.
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 rock-paper-scissors 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 n-tuples.
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
Decimals and an Application of the Rational Numbers
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
FAQ (Frequently Asked Questions)
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 self-study. 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 at
least 10 years old. He or she must have completed all of elementary school math and should understand terms such as sum, product, multiple,
clockwise, and counter-clockwise. Your child must be motivated, independent, and talented with superior analytical skills. Students must
also have excellent reading skills as they will need to comprehend text explanations of mathematical concepts.
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?
A talented high school student who has completed algebra and geometry courses may be eligible for the Advanced Mathematical Logic courses
offered by www.eimacs.com. Click here
for a free aptitude test for the AML program.
However, if a high school student wishes to complete the Elements of Mathematics: Foundations program,
he or she must start with the first course because EMF introduces concepts that are not covered in standard high school
mathematics classes. For example, the first course addresses properties of non-numeric 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 better-developed
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 "self-study" mean?
Self-study 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 courses are self-study, what should my child do if he or she needs help understanding the material?
When students are logged into the EMF site they may access the "EMF Help Forum."
In this forum, EMF students may post questions to be discussed and answered by other
students. They are able to rate the helpfulness of answers given and earn "Math Wizard" points
for contributing answers marked as helpful. Questions that cannot be answered satisfactorily by other students are filtered
through to experienced instructors and mathematicians who will respond appropriately.
To use the EMF Help Forum, students must obey strict rules including rules prohibiting the request or provision of answers
and the use of inappropriate language. Students who do not obey these rules will be prohibited from accessing the EMF Help Forum and
will not be permitted to enroll in subsequent courses. An EMF student is also free to discuss questions with
parents and other individuals.
If EMF is self-study, 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, and underdeveloped study habits. In order to excel in EMF most students will need to do some studying,
especially prior to taking the in-course 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 email@example.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 self-study.
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 one-hour 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.
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 and reading
skills to excel in EMF on a self-study 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 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 university-level 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 re-enroll 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 re-enrollment 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,
university-level mathematics AND the ability to relate these complex ideas to middle and high-school 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 world-class 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.
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
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
firstname.lastname@example.org 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 self-study.
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, teacher-led 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 email@example.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 self-study.
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 firstname.lastname@example.org 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 self-study.
Gifted and Talented Resources
The following are recommended resources for information on serving the needs of gifted and talented children: