An Online Middle School Curriculum
for Mathematically Talented Minds
EMF
Affordable. Scalable. Proven

Highly talented mathematics students need nurturing, and that does not mean just giving them the next curriculum faster! EMF is the solution.


EMF News: University professor visits EMF Math students.

About EMF

Elements of Mathematics: Foundations (EMF) is an online curriculum designed for mathematically talented students that covers Pre-Algebra, Algebra, Geometry, Precalculus and beyond in grades 6–8.

EMF Offers Proven Solutions That Work for All Stakeholders …

… Because Schools and Districts Need to Allocate Resources Wisely

  • Covers Pre-Algebra through Precalculus and beyond in grades 6-8.
  • Exceeds all national and state math standards.
  • Learning Management Systems integration.
  • Scalable for any sized district.
  • Affordable.

… Because Teachers Want All Students to Learn and Grow

  • Stretches the brightest students by challenging them at their ability level.
  • Effective in-class differentiation.
  • No additional teacher training required.
  • Automatic grading, online gradebook and weekly progress reports.
  • Free up time to work with non-EMF students.

Partner Schools, Districts and Organizations

Join the growing ranks of school, districts and organizations providing talented students with the most advanced secondary school mathematics curriculum available today!

  • Barrington 220 Community Unit School District, Illinois
  • Broward County Public Schools, Florida
  • Collier County Public Schools, Florida
  • Denver Public Schools, Colorado
  • Fairbanks North Star Borough School District, Alaska
  • Madison School District, Arizona
  • Pembroke Pines Charter School, Florida
  • Project MEGSSS, Missouri
  • San Antonio Academy, Texas
  • Scheck Hillel Community School, Florida
  • Somerset Academy Pemroke Pines, Florida
  • St. Mark's Episcopal School, Texas

Free Trial & Licensing

Licensing fees for institutions such as school districts and traditional schools vary according to the number of students enrolled in EMF. Please complete the form below to request more information or a free trial of EMF.

Homeschool? Visit www.EMFmath.com.

Frequently Asked Questions

Questions? schools@EMFmath.com
EMF Home Page
© 2002 – 2018 IMACS. All rights reserved.

Courses

EMF recognizes that certain fundamental concepts elegantly unify the various branches of mathematics. EMF first teaches these concepts and then builds on that foundation to explore modern mathematics while covering all of Pre-Algebra through Precalculus.

  1. Operational Systems
    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.
  2. The Integers
    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.
  3. Sets, Subsets and Set Operations
    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.
  4. Ordered n-Tuples
    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.
  5. Mappings
    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.
  6. The Rationals
    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 proof-builders, students also advance to the next step in learning how to construct well-reasoned and logical arguments.
  7. The Decimals
    This course builds upon knowledge of the rational numbers to introduce decimal numbers and their properties, arithmetic operations on decimals, and positional 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. Students then apply what they have learned about decimals, decimal approximations, and percentages in the study of rates (that is, ratios of measured quantities), the slopes of straight lines, a case study of a fictional world in which mathematics is necessary to analyze a social and political issue, and an extended introduction to elementary descriptive statistics.
  8. Probability
    This course provides an introduction to elementary probability theory and covers one-stage, two-stage, and multistage random experiments, the Product Rule, counting subsets, combinatorics, and random digit generators. This is one of EMF's most technologically ambitious courses with over half the pages containing an interactive device, narrated animation, or virtual classroom. Using these tools students learn about, replicate, or analyze the outcomes of a wide variety of random experiments online. The course concludes with an exploration of one of the most important methods in probability — Monte Carlo simulation — and two famous questions — the birthday problem and the Monty Hall puzzle.
  9. Number Theory
    An exploration of numbers for their own fascinating sake is a joy that every young person should experience. This course provides that opportunity by investigating some of the most intriguing and timeless questions in Number Theory. Along the way, students learn about prime and composite numbers, prime factorization, and number bases as well as examining elegant ideas such as Euclid's Lemma, the Sieve of Eratosthenes, and the Fundamental Theorem of Arithmetic. Students expand their logical reasoning skills with an introduction to the powerful proof technique of mathematical induction. Interactive exercises help students practice their proof-writing skills with simpler conclusions, while animated narrations enhance rigorous yet accessible proofs of more significant results such as the multiplicativeness of Euler's totient function.
  10. Algebra: Groups, Rings and Fields
    This course focuses on the study of algebra, in particular the kind of algebra that is usually learned by mathematics majors at university. As an incidental matter, students who complete the EMF algebra series will be able to solve any high school algebra problem with ease but, more importantly, will be well-prepared to study the high-level mathematics that is at the heart of important disciplines such as public-key cryptography and particle physics. Building on a solid understanding of operational systems, this course introduces groups, rings, and fields and their mathematical properties. While typical high school algebra students are limited to applying these rules mechanically to solve numeric equations, EMF students are guided to their own intuitive "discovery" of these behaviors through interactive exercises involving numeric and non-numeric mathematical structures. Students continue to sharpen their logical reasoning skills by proving several of these properties using EMF's proof-building technology.
  11. Algebra: Relations, Mappings, Equations
    This course continues the study of abstract algebra by introducing the fundamental notion of a relation and its properties. Whereas the previous course in the EMF algebra series focused on combining elements of a set with operations, this course investigates ways of relating elements of a set, one to another. Mappings, a key mathematical construct studied earlier in EMF, are then described as special types of relations. By expanding ideas and techniques introduced in the fifth and sixth EMF courses to the general context of any field, students learn to solve linear, quadratic and rational equations with mappings by calling upon the powerful properties of relations covered in the first part of this course. As with all EMF courses, students are exposed to mathematical ideas typically not seen until college and continue to gain experience building proofs of important results.
  12. Algebra: Relational and Ordered Operational Systems
    This course ties together the previous two courses and completes the study of abstract algebra in EMF. Students learn how groups, rings and fields can be combined with certain kinds of relations to form new structures, specifically relational systems and ordered systems. Within such systems, techniques for solving inequalities are given a much more in-depth treatment than has been possible in earlier courses. Skeleton proofs are provided of various properties of ordered groups, rings, and fields, and students are asked to complete them and then to apply these properties to easily solve inequalities over these systems.
  13. Real Functions I
    This course introduces the real number system as an extension of the ordered field of rational numbers and goes on to investigate certain fundamental classes of functions on the real numbers. After learning about the real numbers and their properties, students prove that the set of real functions with function addition and multiplication form a ring. Armed with this fact, various ring properties covered in earlier EMF courses, and powerful online graphing tools, students investigate systems of first degree equations and inequalities on the real numbers. The course then turns to analyzing quadratic equations and inequalities, first graphically and then by factorization and completing the square. The analysis culminates with an in-depth derivation of the quadratic formula — typical high school algebra students are merely asked to memorize this formula — after which EMF students will be well-versed in methods for solving first and second degree equations and inequalities, as well as understanding why these methods work. Real Functions I wraps up with second look at statistics, including the normal distribution, and an introduction to trigonometry. Throughout the course, students put their understanding and skills to work on realistic, complex problems that underscore algebra's usefulness in the real world.
  14. Real Functions II
    This course begins with an introduction to sequences and series, including in-depth coverage of arithmetic and geometric sequences and series. Students continue to expand their knowledge of real polynomial functions such as integer and rational power functions, exponential and logarithmic functions, the absolute value function, rational functions, and asymptotes. The course builds on material from Sets, Mappings, and Probability to explore the Binomial Theorem, a fundamental result with uses as varied as the distribution of IP addresses and financial modeling. Practical applications, including two extensive case studies in statistics, continue to demonstrate algebra's usefulness in the real world. The course culminates with an introduction to the field of complex numbers as a field extension of the real field and to polynomial functions over the complex field. Throughout, EMF students will prove many of the results about real and complex functions that typical high school students are only asked to memorize and apply. With respect to developing students' proof-writing skills, Real Functions II marks the completion of the transition from outline proofs to paragraph proofs.
  15. Geometry: Incidence and Transformations
    This course begins the formal study of Euclidean geometry in EMF. Working in the three-dimensional context of space, the two-dimensional context of planes, and the one-dimensional context of lines, students investigate familiar geometrical figures and a variety of ways of transforming them. In the first half of the course, real-world experiences with physical objects are used to inform the mathematically rigorous construction of analogous geometric objects and the properties they should have. The second half of the course builds on concepts from Sets and Mappings in order to explore transformations — reflections, translations, rotations, and magnifications — as mappings on sets of points. Students prove numerous results that are typically taken as given in high school geometry as they ascertain which of these transformations are permutations and which permutations preserve betweenness, collinearity, angle measure, distance, and circular orientation. As this is EMF, the proofs that students complete go well beyond the basic two-column high school proof in complexity, sophistication, and consequence.
  16. Geometry: Congruence and Similarity
    This course continues the study of Euclidean geometry in EMF. Students begin by investigating congruent figures in order to extend their knowledge of reflections, translations, rotations, and glide-reflections, together known as isometries. They then prove that these "rigid motion" mappings form a group. Equipped with their understanding of group properties from EMF's abstract algebra courses, students spend the first part of this course exploring congruence and symmetry, including properties of congruent triangles and a classification of polygons in terms of their symmetry groups. The second part of the course focuses on magnifications and their properties. Students explore what happens when magnifications are composed with themselves and with isometries and discover that these compositions, known as similitudes, also form a group. The third part of the course provides an in-depth study of circle geometry incorporating multiple applications from the first two parts. Because EMF teaches mathematics as a unified body of knowledge rather than a collection of unrelated topics, students come to realize a fundamental connection between algebra and geometry that most traditional school math students will never know.
  17. Geometry: Coordinates and Measurement
    This course concludes the study of Euclidean geometry in EMF, with a focus on measurement in two and three dimensions: of area, volume, distance, taxi-distance, conic sections; introduces vector spaces; and reexamines isometries in the context of coordinate systems.
  18. Precalculus Coda
    This course brings together various topics covered in earlier EMF courses and expands upon that material with an extensive study of vectors in the real number plane and space, linear transformations (including orthogonal mappings), and matrices and their application in solving systems of equations. Together these topics comprise a typical Precalculus course and provide a detailed understanding of the basic principles behind The Calculus.

Supplements

The following supplemental short courses become available to EMF students following the completion of certain specified EMF courses.

  • Pre-Algebra
    This short course explains topics, terminology, and notation that are taught in a typical school Pre-Algebra class by relating them to the more advanced language and concepts taught in EMF. Students who successfully complete the first seven EMF courses, this supplement, and recommended practice tests will be well-prepared to take any school Pre-Algebra test and succeed in a traditional school Algebra 1 course.
  • Abstract Algebra Readiness
    This short course tests a student's readiness for EMF courses 10-14, which focus on Abstract Algebra. Students who complete the first nine EMF courses should use this supplement to review the important ideas covered in those courses as these ideas provide the mathematical tools necessary to master the EMF Abstract Algebra courses.
  • Algebra 1
    This short course explains topics, terminology, and notation that are taught in a typical school Algebra 1 class by relating them to the more advanced language and concepts taught in EMF. Students who successfully complete the first 13 EMF courses, this supplement, and recommended practice tests will be well-prepared to take any school Algebra 1 test and succeed in a traditional school Algebra 2 course.
  • Algebra 2
    This short course explains topics, terminology, and notation that are taught in a typical school Algebra 2 class by relating them to the more advanced language and concepts taught in EMF. Students who successfully complete the first 14 EMF courses, this supplement, and recommended practice tests will be well-prepared to take any school Algebra 2 test and succeed in traditional advanced high school math courses.
  • Geometry
    This short course explains topics, terminology, and notation that are taught in a typical school Geometry class by relating them to the more advanced language and concepts taught in EMF. Students who successfully complete the first 17 EMF courses, this supplement, and recommended practice tests will be well-prepared to take any school Geometry test and succeed in traditional advanced high school math courses.
  • Precalculus Readiness
    This short course tests a students readiness for EMF's Precalculus Coda course. Students who complete the first 17 EMF may use this supplement to review the important ideas covered in those courses before advancing to the Precalculus Coda course.
  • Precalculus
    This short course will explain topics, terminology, and notation that are taught in a typical school Precalculus class by relating them to the more advanced language and concepts taught in EMF. Students who successfully complete the entire EMF program, this supplement, and recommended practice tests will be well-prepared to take any school Precalculus test and succeed in an Advanced Placement Calculus class.

Technology

EMF is supported on the free Chrome and Firefox browsers. In addition to laptops and desktops, EMF is compatible with iPads, large Android tablets, and touchscreen computers. However, some features are easier to use with a mouse or other pointing device. EMF is not compatible with smartphones.

Examples of how EMF uses the latest technology are shown below.

  • Virtual Classrooms

    In EMF's Virtual Classrooms, a clever cast of professors and students discuss current course topics. Here's a simulated classroom from EMF09 (Number Theory) that introduces proof by induction:

  • Puzzles and Games

    EMF uses puzzles and games to reinforce student understanding. In this case, the String Game helps students to understand set concepts such as union and intersection. Try your hand! Move the pieces from the right onto the diagram to obtain clues about the identity of the three labels. Click on a label to make your choice.

  • Experiments

    EMF uses modern technology to generate experiments such the Galton Box shown below, as well as many interesting Markov chain simulations.

  • Animations

    Hundreds of animations and interactive tools help students understand difficult concepts with relative ease. In this animation, the sun's rays make the ball's shadow appear as the wheel turns at a steady pace, resulting in a distance-time graph of a sine curve:

  • Explorations

    Today, computers and even many mobile devices have capabilities that teachers could only dream of a few years ago. EMF uses the very latest technology to allow students to explore mathematics in ways that are only now becoming possible. Consider, for example, the extrusion illustrated below. Press the buttons to extrude the plane object to a solid, then zoom, rotate and drag this 3D object to explore it.

Reviews

Broward County EMF Survey

The following is a sampling of findings from an anonymous survey given to EMF families in Broward County Public Schools, Florida in spring of 2016.

  • 100% of parents reported that their child's ability to analyze complex problems, self-study skills, and ability to focus for extended periods of time had improved since beginning EMF.
  • 92% of parents report that their child enjoys doing EMF and finds EMF challenging.
  • 92% of parents think that EMF has made their child more ready for high school and college mathematics than they would have been without it.

Read the full summary of the 2016 Broward County Public Schools EMF Survey.

Student Quotes

  • "EMF is a tough program, but it pays off. I learned how to prove concepts, and I learned to work harder and on a schedule." — Andrei Chekmasov, EMF Student
  • "EMF allows students to avoid the trap of believing that math is nothing but problems to solve with rules and a calculator." — Corbin Diaz, EMF Student
  • "EMF is definitely the course for students with a thirst for mathematical knowledge." — Luke Freed, EMF Student
  • "After doing EMF, I not only understand how to use formulas but I can actually prove them. This program changed me from a girl who loved math into a real mathematician." — Skylar Gamberg, EMF Student
  • "EMF is a way of looking at everything that you took for granted in earlier years, and proving them to make them more detailed and complete." — Ian Gonzalez, EMF Student
  • "EMF is challenging in a good way — like a puzzle. You're working hard, improving your mind, and still having fun!" — Grace Hancock, EMF Student
  • "EMF is way more exciting than other math programs and keeps you on your toes. I enjoy it even when it takes a really long time to solve the hardest problems." — Imaan Nanji, EMF Student
  • "I cannot even imagine what my life would have been like without having seen the beauty of mathematics through EMF." — Vievie Romanelli, EMF Student
  • "EMF has increased my ability to write proofs and my mathematical skill overall. It has some of the most interesting mathematical concepts I have ever seen." — Albert Wang, EMF Student
  • "Now that I have completed EMF, I can attest to their claim that their students will be ready to take calculus. I am currently about half way through AP Calculus BC and am having no problem at all with the material. — Alexander Yue, EMF Student
  • "If you're good at math but could sleep through math class because you find it boring, EMF is awesome and will actually keep you awake. — Isabella Yeung, EMF Student

A Brief History

In the mid 1960s, facing a shortage of home-grown mathematicians and scientists, the US Office of Education (later to become the US Department of Education) made a huge investment in mathematics curriculum development projects. So-called New Math was designed to replace the traditional mathematics curriculum for students across the board. The effort is generally regarded as having failed, not least because insufficient funds were allocated to support the necessary and substantial teacher retraining.

At the same time the more focused Elements of Mathematics (EM) project set itself the task of replacing the standard program for mathematically talented students. EM would take precocious ten-year-olds and prepare them to learn undergraduate-level mathematics by seventh grade and upper division mathematics as college freshmen. The middle school portion of this curriculum became known as Elements of Mathematics: Foundations (EMF).

Unfortunately, not only did EM require that teachers be retrained … they also needed to have mathematics degrees, greatly exacerbating implementation issues. Nevertheless, EM was taught and survived for many years at a handful of centers in the US to the benefit of mathematically talented children lucky enough to live in those locations.

In 2011, a mathematics educator and EM alumna was clearing out an attic when she came across her old Foundations textbooks. In light of the many recent technological advances, she wondered whether there might now be a way to teach it without the need to employ highly skilled mathematicians and to make it accessible to talented students anywhere. A successful pilot study ensued, and the following year, EMF was launched as an online, self-study, self-contained mathematics program for talented middle school students.

Recommended Reading

Calculation was the price we used to have to pay to do mathematics. Prof. Keith Devlin, Devlin's Angle, 2 May 2018.

A Mathematician's Lament. Paul Lockhart, 2002.

Additional Information

EMF Reveals the Beautiful and Creative Side of Math. The IMACS Blog, 11 Oct. 2019.

Beyond Math Competitions: Developing Math Talent to Its Fullest Potential. The IMACS Blog, 28 Jun. 2019.

University of Florida Professor Delivers Math Lectures to BCPS Top Middle and High School Students. Broward County Public Schools, 10 Jun. 2019.

Talented Middle School Math Students Forge Early Path to University Coursework. The IMACS Blog, 6 Mar. 2019.

EMF Math vs. The Traditional Approach to Learning Mathematics. The IMACS Blog, 20 Feb. 2019.

Online Gifted Math Program Allows Middle School Students in Illinois District to Study Advanced Math. District Administration, 12 Oct. 2018.

Gifted EMF Alumna Shares Advice for Current & Future Students. The IMACS Blog, 27 Aug. 2018.

EMF Math For Talented Middle School Students is Complete! The IMACS Blog, 23 Jul. 2018.

Challenging mathematically talented middle school students. District Administration, 8 May 2018.

Online Math Program for Talented Students Doubles Enrollment in Florida's Broward County Public Schools. District Administration, 23 Oct. 2017.

Gifted Math: A Better Alternative to Acceleration. The IMACS Blog, 17 Nov. 2016.

Broward County Schools to Expand Online Math Program for Gifted Students. THE Journal, 2 Aug. 2016.

Online Gifted Math Program to Grow to Several Hundred Students in Broward County Public Schools. PR Newswire, 2 Aug. 2016.

Not Your Father's Algebra: A New Online Course for Talented Students. The IMACS Blog, 12 Feb. 2015.

IMACS Introduces Elements of Mathematics: Foundations. The IMACS Blog, 25 Oct. 2012.

IMACS

EMF is developed by the Institute for Mathematics and Computer Science (IMACS), an independent teaching and educational research institute with 25 years' experience developing original curriculum for talented K-12 students. IMACS' curriculum has been used in prestigious programs such as Duke's Talent Identification Program and Johns Hopkins University's Center for Talented Youth. Over 4,500 students from across the US and around the world attend local IMACS classes or study its online courses.

EMF