Math competitions are a popular way to enrich the lives of mathematically talented students.

Contests introduce ideas not covered in school, bring together like-minded peers, and celebrate academic achievement.

But math competitions have their shortcomings.

An education in advanced mathematics that starts with foundational topics, so-called “discrete math,” offers a more complete and inclusive way to develop math talent.

This approach allows students to grow beyond the limits of competitions and competition-inspired curricula and also gives those children who thrive in non-competitive environments a pathway to success.

## Math Competitions Are Great … For Some

I wasn’t the fastest guy in the world. I wouldn’t have done well in an Olympiad or a math contest. But I like to ponder. And pondering things, just sort of thinking about it and thinking about it, turns out to be a pretty good approach. -Jim Simons, Founder of the National Museum of Mathematics.

By design, most math contests incorporate time limits. This leads many people to wrongly believe that speed indicates talent.

In this way, competitions perpetuate harmful myths about what it means to be “good” at math. Contests end up deterring many creative thinkers from pursuing the subject.

Professionals in the field know that excellence in mathematics is more often achieved through slow, careful, and deep thinking.

There is also something very unusual about the kinds of problems that the IMO presents. … They have to depend on seeing a particular insight or trick. -Keith Devlin, Director of Stanford Mathematics Outreach Project

Math contests also elevate the importance of recognizing carefully designed situations over the ability to think through novel and open-ended problems.

Solving problems by quickly ascertaining the right approach is still valuable in many situations. However, making that a hallmark of an activity for advanced students is outdated.

Schools already teach traditional math using this method, and a rapidly-changing world requires flexible thinkers who are prepared to solve unrecognizable problems.

In the days of Math Olympiad, we were just seeing the tips of an iceberg. Now we were diving into the water and seeing the whole foundations of mathematics. And it was much more interesting than what was above water. -Zhiwei Yun, MIT Professor, IMO Gold Medalist

Competitions offer a wonderful introduction to higher math for those students who enjoy the competitive environment. But competitions only scratch the surface of what is accessible to bright, young minds in this wide-ranging and deeply complex field of knowledge and exploration.

Furthermore, students who prefer to ponder deserve a way to learn advanced mathematics at an early age.

Math competitions should always be among the options available to mathematically talented students, but parents and teachers must look beyond competitions and competition-inspired curricula in order help their students fully realize their potential.

## The Power of Discrete Mathematics

With discrete mathematics, students will be thinking flexibly and creatively right out of the box. There are relatively few formulas to memorize; rather, there are a number of fundamental concepts to be mastered and applied in many different ways. -David Patrick, AoPS CFO, MIT PhD, Alumnus of the SUNY Buffalo Gifted Math Program

Mathematics is filled with mind-expanding topics that are fantastically fun to learn. To name a few:

- Mathematical logic.
- Set theory.
- Relations and functions.
- Combinatorics.
- Probability.
- Number theory.
- Induction.
- Group theory.

They fall under the category of “discrete mathematics.” While a few discrete math topics that fit the competition mold do appear in math contests, several which are most important for a strong foundation in advanced mathematics are excluded.

Discrete math concepts are extraordinarily powerful because of their fundamental quality; they appear across mathematics as essentially the same ideas expressed in different ways depending on the specific branch of mathematics.

Arguably most powerful are the proof techniques from mathematical logic and set theory. They allow students to connect knowledge in new ways that are useful in reasoning through solutions to unfamiliar problems.

EMF Math starts by teaching concepts from discrete mathematics, such as the powerful proof technique of mathematical induction.

If students first master these fundamentals, it is no exaggeration to say that their understanding of much of the rest of mathematics will fall into place with significantly greater ease and depth.

High school math is simplified down to elementary cases of generalized principles, and undergraduate courses become sophisticated extensions of core ideas.

In 1979, Dr. Gerald R. Rising put this pedagogy into practice when he co-founded the Gifted Math Program at the State University of New York at Buffalo to serve talented middle and high school students. He selected the *Elements of Mathematics* (EM) textbook series, a research-based curriculum designed for gifted students that put discrete mathematics at the start of the learning process.

As you might expect, the SUNY Buffalo Gifted Math Program turned out many alumni who went on to distinguished careers in mathematics, math education, and more.

## The Rise of EMF Math

From 2012 to 2014, the EM textbooks covering discrete math topics and their manifestations in prealgebra were reborn as the first nine online courses of the Elements of Mathematics: Foundations (EMF) program.

The tenth and final foundational course covering groups, rings, and fields was released in 2015, marking the first time that an undergraduate-level course designed for talented children became always accessible.

With 10 courses covering discrete math fundamentals in place, EMF Math followed with courses on the algebra of real numbers (Algebra 1 and Algebra 2 in school math terminology), geometry, and precalculus.

The promise of teaching traditional high school math as elementary applications of fundamental ideas was fulfilled again, this time online and with global access. Bright and motivated students could now complete prealgebra through precalculus in three years without loss of depth. In fact, the depth would be *greater* than any other widely available math program for talented youth.

Since Betty Krist and I founded the University at Buffalo Gifted Math Program for bright grade 6-12 students of western New York in 1979, we have used books from the Elements of Mathematics series as a central aspect of our instructional program. Many aspects of these books are unique. In particular the well-thought-out mathematical concept presentations provide deep insights into the structure of mathematics and applications of mathematical thinking. I’m delighted to see that EMF Math has transformed these books into an online curriculum that opens up access to bright students anywhere in the world. -Gerald Rising, Director Emeritus of the SUNY Buffalo Gifted Math Program

## Ready for Calculus and More

Unlike in schools, acceleration in EMF Math is *not* about working through standard material faster. Rather, acceleration is a happy side-effect.

Mastering fundamentals first makes future learning easier and thus faster because all new concepts are just special cases of foundational ideas. There are no tricks or traps, just the transformation of precocious youngsters into formidable mathematical thinkers.

The mathematical maturity of these [EMF] students is astonishing. While only in middle school they have a grasp of abstract topics that most undergraduates do not see until their second or third year at the university. -Kevin Knudson, University of Florida Math Department Chair

Because EMF Math enables students to accelerate through traditional school math topics as part of their deep, foundational training in mathematics, they are prepared to excel in calculus, real analysis, and other undergraduate-level courses at an early age.

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, 8th Grade EMF Math Student

Where competition emphasizes speed, EMF Math emphasizes depth.

Where competition rewards recognizing tricks, EMF Math rewards open-ended thinking.

And where competition has limited scope, EMF Math unlocks the fascinating world of modern mathematics.

**Parents and teachers:** If you want to give talented students the best and deepest understanding of mathematics available to young minds, include EMF Math as a core part of their mathematics education and enrichment.

To go beyond what competitions and competition-inspired curricula can offer, enroll at EMFmath.com (individuals, including homeschool) or EMFmath.com/schools (school licensing).

## More on EMF Math

University of Florida Professor Delivers Math Lectures to BCPS Top Middle and High School Students

EMF Math vs. The Traditional Approach to Learning Mathematics

Gifted EMF Alumna Shares Advice for Current & Future Students

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