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IMACS celebrates influential career of co-founder Edward C. Martin

It is with bittersweet emotion that the co-founders of the Institute for Mathematics and Computer Science (IMACS) announce the retirement of partner, colleague, teacher and friend, Edward C. Martin. Ed has enjoyed a prolific career of nearly 50 years in mathematics education. What follows is a modest attempt by two of his students to share highlights of a life dedicated to inspiring young people through his exceptional writing and teaching to study mathematics.

For Ed, the most rewarding part of his career has been “witnessing students come to the realization that mathematics is a big, wonderful, logically-connected, all-encompassing, intellectually satisfying Garden of Earthly Delights.” If you were fortunate to have been a student or colleague of Ed’s, please take some time to leave a comment with your memories and wishes for retirement. He will enjoy reading every word!

From Cornwall to Cambridge

If there were ever a child destined to become a magical mathematician and teacher, it would be Edward C. Martin. The first auspicious sign of Ed’s future career came in 1948 when he was born in St. Ives, Cornwall, England, a town that is the subject of the famous arithmetical riddle As I Was Going to St. Ives:

As a young student at St. Ives County Primary School, Ed found math to be “very routine.” Nonetheless, it was important to do well because in England at that time, a compulsory, nationwide exam determined whether a child would attend an academic secondary school with an emphasis on university preparation or a school akin to a trade school. Ed’s family actually moved to Brighton around this time, but he stayed in St. Ives with his grandparents to minimize any disruption in the year before the all-important exam.

After the exam, Ed joined his family in Brighton. His scores earned him a spot at the Varndean Grammar School for Boys where his father was a math teacher. Four years later, his father began teaching math at Worthing High School for Boys, and this is where Ed completed high school.

Ed credits his father, who was his math teacher during his last two years of high school, as the person from childhood who most influenced him to study mathematics. During this time is when the subject started becoming really interesting to him. “We were preparing for the nationwide final high school exams and the special entrance exams to Oxford and Cambridge,” Ed recalls. “It was the mental challenge I enjoyed rather than anything else.”

Interestingly, Ed’s father never addressed him by name in class, opting to call him “you” instead. But their bond was clearly a special one. “We would laugh ourselves silly over some incident that had occurred in math class that day, much to the bemusement of the rest of the family, who could understand neither what we were talking about nor why it was so funny,” remembers Ed.

Cambridge University

After high school Ed attended Fitzwilliam College, one of the colleges that form Cambridge University. Unlike in the United States, a college student in Britain concentrates on one area of study and takes courses related only to that subject. Of course, Ed studied all courses in mathematics! At the time, the Department of Pure Mathematics and Mathematical Statistics was housed in an old multi-story building surrounding a quadrangle near the center of town in Cambridge. “It always amused me that, as you left the complex and went out onto the street outside, there was a notice on the archway that said ”You are now entering the real world’,” recalls Ed with a smile.

Throughout his years at Cambridge, Ed studied under several famous mathematicians, including John Horton Conway, Peter Swinnerton-Dyer and John Willis. Ed fondly recalls his weekly one-on-one tutoring sessions with Professor Conway. Ed was Conway’s student at the time Conway was working on one of his most famous discoveries, the classification of simple groups.

“On one occasion when I arrived, he was obviously very excited,” Ed recalls. “So I asked him what was going on.” Conway hadn’t gotten any sleep the previous night because his bath had been interrupted by a phone call from Fields Medal winner Professor John G. Thompson, then of the University of Chicago, who wanted help in analyzing a new discovery. So Conway jumped out of the bath, sat down soaking wet at his home office desk, and worked right through the night!

“I asked him what was involved in this work, and he explained that it could be thought of in terms of packing spheres in 24 dimensions,” Ed continues. “Of course, I was none the wiser! Incidentally, I subsequently learned that there is great significance in the fact that 24 dimensions are involved. It’s an amazing fact that the only natural number n such that 1² + 2² + 3² + … + n² is itself the square of a natural number is 24.” (The sum is in fact equal to 70².)

The Making of a World Class Teacher

After graduating with his bachelor’s degree in May 1969, Ed did something completely unexpected but probably a lot of fun! He and two friends bought a Land Rover and spent three months driving through southern Europe. They traveled far enough to roam around northwest Africa and the northern limits of the Sahara Desert.

Upon returning home from this adventure, Ed learned that his father had enrolled him in a one-year postgraduate teaching course at Cambridge, so Ed headed back to school to pursue his teaching degree. There he met Alan Bishop who led the teaching course. Ed credits Bishop as someone who was “very influential in molding me into the kind of teacher that I am, emphasizing that there’s more to it than simply imparting knowledge.”

During the decades to follow, Ed would go on to become a respected and beloved teacher of over 1,700 students, not counting the many more who benefited from the textbooks and online courses he wrote and edited. Whether in a classroom or online, Ed’s students know him to be the kind of teacher who brings clarity to complex ideas without watering down the rigor. He can instantly see the flaws in a student’s thinking and knows just what and how much to say to get a student back on track. In fact, one of those lucky students, Terry Kaufman, would one day co-found IMACS with Ed.

Terry remembers when Ed was his high school teacher for a college-level linear algebra course. “What I remember so vividly was that he would give us class time to work on our assignments, and he would sit at the front of the class with a loose-leaf notebook on his teacher’s desk. Whenever one of us would need help with a challenging problem, he would open his notebook, which contained copious notes he had taken in his college courses. The notes were so meticulously taken and in such small print that today it would appear to look like they had been printed at about an 8-point font. It was amazing how he always found the information needed to help us with our questions from within his notebook.”

Fate and the First Job

As Ed completed his postgraduate year at Cambridge, he applied to various American universities, seeking a graduate assistantship while simultaneously studying for his PhD in mathematics. The chairman of the math department at Southern Illinois University (SIU) received one of Ed’s applications and noted that he was qualified in both mathematics and education. As fate would have it, the SIU math department did not have the funds to hire Ed, so the chairman forwarded Ed’s application to Burt Kaufman.

At that time, Burt was director of the Comprehensive School Mathematics Program (CSMP), a federally-funded development project tasked with writing mathematically rigorous textbooks for talented middle and high school students. These books, which covered most of an undergraduate degree in mathematics, would eventually become known as the Elements of Mathematics (EM) series.

Burt was looking for an editor who could bring consistency to the EM books. Up to that point, each year a different visiting professor would spend a sabbatical year working with Burt’s team to write and then test out the material with actual students. “These professors all had different writing styles and didn’t have much, if any, experience working with secondary school students,” explains Terry, who was also Burt’s son.

As Terry recalls, his father decided to hire Ed sight unseen. In fact, the day that Ed received the reply from SIU explaining that there were no funds to hire him was the very same day that he received the offer letter from Burt. “The letter indicated that one of the CSMP advisors was on sabbatical in London, and I could go and interview him if I wanted to,” remembers Ed. “This I did, and we got on very well together.” Ed accepted Burt’s offer, which was unconditional, and began working as a Teacher/Writer in August 1970.

Comprehensive School Mathematics Program

The work of writing the original EM books had begun in 1966. So by the time Ed joined the effort in 1970, he had the daunting task of taking four years of material written by multiple authors and smoothing everything out from a stylistic point of view, introducing cross-references between the books, and ensuring that they all had complete answer keys.

“At one point, it was decided that the approach to set theory used in the advanced college-level books should be changed,” recalls Ed. “So we commissioned a new author to rewrite the book in question, and I then had the task of editing his work into what was by then EM style as well as chasing down every place in later books that referenced axiomatic set theory and editing them to match the revised approach.”

By the time Ed was promoted to Senior Editor of the EM series in 1974, math teachers across the US had heard about these uniquely engaging textbooks. CSMP offered summer workshops to train teachers on the EM materials, and among Ed’s responsibilities were helping to run the workshops. After attending these workshops, a number of teachers started using the EM books in their classrooms. Ed then provided ongoing professional development support during the school year.

Without a doubt, Ed’s profound influence on math education has rippled throughout the decades as the teachers he trained went on to bring the EM approach to students across the US, a number of whom became math educators themselves.

High School of Glasgow

Ed’s exceptional teacher-training abilities did not go unnoticed. In 1976 he was recruited by the High School of Glasgow to come in and disrupt the way mathematics was being taught at the school. “The headmaster, who also had a background in math, realized that his math department was not keeping up with current thinking in mathematics education,” explains Ed. “So I was charged with retraining the math teachers and improving the quality of math teaching at the school.”

Teachers who disagreed with the change either left or tried to preserve the status quo by complaining that Ed’s approach would not work with underperforming students. Undeterred, Ed volunteered to teach all the weaker students and showed that these concerns were unfounded. “Things turned around markedly during the six years I was there,” he is pleased to say.

In his spare time and on weekends, Ed continued to work remotely as Senior Editor for CSMP. During summers he would return to the US to focus on editing the EM books and training teachers on how to use them.

University of Bath

Ed’s next big challenge came in 1982 when he was asked to lead the development of a new mathematics curriculum for British secondary schools. This project with national implications was funded by textbook publisher Addison-Wesley and hosted at the University of Bath. As head writer, Ed’s job was to train a team of eight other writers in the art of textbook writing, to do some of the writing himself, and to do all the final editing and preparation for publication. “I’m proud of the math curriculum we produced,” says Ed, “because it represented a fresh and exciting deviation from the curriculums that were common at that time.”

While at the University of Bath, renovations caused Ed to have to share an office with computer scientist Iain Ferguson who was working on a different research project hosted at the university. This was surely a serendipitous allocation of space because the professional collaboration between these two scholars would come to last more than 30 years and continues today.

“In my first memory of collaborating with Ed, I was working on ways of displaying demographic and other geographic information on early desktop computers while he was editing a mathematics textbook,” recalls Iain. “I needed an algorithm for choosing which pixels to light up to draw a line across a screen.” Ed came to work the next day with such an algorithm, but Iain noticed immediately that something wasn’t quite right. After Ed gave it a few tweaks, it worked like a charm!

“Maybe that story typifies our collaboration over the years, with each of us feeding off of ideas from the other,” says Iain, “although I unequivocally attribute all the best ideas to Ed.”

Project MEGSSS of Broward County, Florida

In 1985 while at the University of Bath, Ed was again recruited by Burt Kaufman who was then heading up an effort in Broward County, Florida to teach the EM curriculum in a large, diverse school district.

Ed mentioned to Iain that he was thinking of accepting Burt’s offer to move to Fort Lauderdale and teach advanced mathematics to talented middle and high school students. “He gave me a couple of the old EM books to peruse,” recalls Iain. “I was astounded that anyone would try to teach mathematics of that type to students so young.” Iain was so intrigued that he convinced his then-fiancé-now-wife to emigrate to the US so that he could join Ed and Burt in this ambitious program known as Project MEGSSS (Mathematics Education for Gifted Secondary School Students).

Ed, as well as Iain, found the work to be very demanding but ultimately rewarding. While much of their time was spent planning lessons, teaching classes or grading papers, every spare minute was spent developing new material. “Burt insisted that what Iain and I produced should be good enough to teach him everything he needed to know to be able to teach these subjects to his students,” says Ed. “This was no small task because Burt’s lesson planning was meticulous and voluminous.”

The MEGSSS students certainly benefited from what was then a rare opportunity to study modern mathematics with some of the best teachers in the world. On one memorable occasion, some of the students came up with original solutions to puzzles from the book To Mock a Mockingbird by mathematician, philosopher and author Raymond Smullyan. They sent their solutions to Smullyan who later telephoned Burt to offer his congratulations!

For Ed, the best memories of MEGSSS are of the great kids that he got to work with, a good number of whom still keep in touch with him. Many went on to attend some of the most prestigious universities in the country followed by PhDs or accomplished careers in a wide variety of industries. Some are even parents of a new generation of EM students. In one of those “small world” moments, one of Ed’s star MEGSSS students ended up collaborating on research during his post-doctoral studies with a professor whose high school math teacher had attended Ed’s CSMP training workshops!

An official accolade was in the mix too. In 1993 Ed was one of 24 teachers nationwide to be honored with the Mathematical Association of America’s Edyth May Sliffe Award for Distinguished High School Mathematics Teaching.

Despite its successful track record for turning out some of the brightest math students in the country, MEGSSS was eliminated due to Florida’s statewide budget crisis in 1992. The project wound down at the end of the 1992-93 school year.

Institute for Mathematics and Computer Science

With such highly valued knowledge and skills, Burt, Ed and Iain could have easily pursued lucrative careers in the nascent technology industry that was just then witnessing the rise of America Online. Instead, they joined with Terry to establish IMACS in July 1993 with the goal of making the Project MEGSSS curriculum available as an after-school enrichment program.

Their complementary skill sets made them an effective team. “Ed is an amazing mathematician, Iain is a computer science wizard, and my father was an outstanding teacher with the vision for what was possible,” explains Terry. “I was the only one with any business experience, so I handled all operations and financial aspects of IMACS while learning from my father how to teach various aspects of our curriculum.”

What started with just 37 students in Plantation grew steadily to over 1,000 local students at four dedicated teaching centers and seven on-site school locations across South Florida. IMACS is also among the pioneers of distance-learning programs for precocious youth, having launched its first online course in 1998. Since then, over 30,000 students across the world have benefited from studying IMACS’ online curriculum.

Looking back at the company’s steady growth over the past 25 years, one might think that success was a foregone conclusion. Not so! “Founding IMACS was a fairly heart-stopping venture,” recalls Ed. The four partners took no salary for two years while recruiting a small part-time workforce and trying to attract local students. It was during those early days that they developed the IMACS Mathematics Enrichment curriculum based on books that Ed and Burt (along with two CSMP advisors) wrote back in the 1970s.

To create curriculum that is original, unique and engaging for bright and curious students requires tremendous labor and ingenuity. Luckily IMACS had Ed. “He was the key contributor to so many aspects of our curriculum development. We always knew we could count on him to solve any challenge,” recollects Terry. “As I learned working with Ed at IMACS, he could explain almost anything on any topic. In fact, I often refer to him as our Google before there was Google because if we needed to know anything, we would just ask Ed.”

Ed’s mathematical brilliance and all-around problem-solving skills would prove to be invaluable once again as IMACS pursued its most recent curriculum development project — the ambitious Elements of Mathematics: Foundations (EMF) online program for talented middle school students.

Elements of Mathematics: Foundations

Ed’s focus these past four years has been on developing the EMF online curriculum in modern mathematics for highly advanced middle school students. Launched in 2012, EMF leverages advanced web technology to make EM’s Book 0: Intuitive Background material available anywhere in the world as an online, self-study program. This development is particularly important in locations where mathematically talented students have little or no access to appropriately challenging material or instruction that makes them have to think harder instead of merely work faster as with acceleration.

Change can be uncomfortable, as Ed knows from his time at the High School of Glasgow. “The primary difficulty was and continues to be the inertia that characterizes mathematics teaching in the US,” laments Ed. Because EMF treats the subject as a unified whole built upon solid logical foundations, teachers would need to know how everything fits together and to be equipped for laying groundwork that allows for later developments to connect back. “It was partly because of the difficulty of finding suitably qualified teachers that we created EMF in the form of dynamic web-based courses supported by an online help forum,” he explains.

EMF is currently licensed by a growing number of partner schools, school districts and after-school organizations that are looking for a cost-effective way to keep their most advanced math students engaged beyond what acceleration can offer. Having seen math education trends come and go over his career, Ed is all too familiar with how they squander the potential of bright students by failing to challenge them at their ability level. So it should come as no surprise that he would like to see the EMF program adopted by school systems all over the United States and around the world.

How can EMF students learn modern mathematics, much of it college-level, with no teacher? The secret is in the painstakingly drafted text, all of which was written or edited by Ed. Whether through timeless passages from the original textbooks or in newly written material, Ed’s unparalleled ability to anticipate students’ thought processes and to write for a young audience without watering down content are why talented and motivated students can learn from EMF as if a master were there teaching it.

It’s amazing to think that the EMF curriculum, which started with mathematicians from around the world traveling to the US to work with Burt, is now online and traveling out through the internet to students in over 20 countries! Needless to say, Ed is very proud of the role he played in bringing to fruition the vision that Burt had back in the 1960s of a math curriculum that is worthy of the nation’s and now the world’s brightest young minds and that will prepare them for the needs of tomorrow.

Back Across the Pond

Following the completion of EMF’s main development work this past summer, Ed and his wife returned to England where most of their families live. While he will continue to make occasional contributions to IMACS’ curriculum development efforts and teach the most advanced online IMACS students, Ed is looking forward to devoting more time to musical composition, which gives him almost as much pleasure as his math education activities.

Of course, Ed is already missed at IMACS for many reasons, not the least of which is his British sense of humor. He’s got some wickedly clever jokes! “Like the one about the three types of mathematicians — those who can count and those who can’t,” says Iain with an impish grin.

When someone accomplishes so much for so long, it’s hard to imagine them retiring. But that time has come for Ed and deservedly so. “My hope is that he continues to be healthy and lives for many years to come,” says Terry, “and that he continues contributing to IMACS for as long as he is able.”

What has mathematics taught Ed about life? “A mathematical approach can often help to make sense of many of life’s mysteries,” he explains. “Thinking mathematically gives you an expectation that there will be linkages and underlying structure that will make it easier to see your way through whatever you might be facing.”

Dearest Ed, thank you for this wonderful parting wisdom and for so much more than words can fully express. May your rest be anything but routine!

About the Co-authors

Skylar Gamberg is a 7th grade student from Weston, Florida where she lives with her parents and younger sister. Skylar attends Falcon Cove Middle School and is part of Broward County Public School’s EMF program. She loves learning math and foreign languages. Skylar is grateful for the opportunity to work with Mr. Martin who has taught her much about mathematics, its history, and the lives of mathematicians, especially John Conway. Skylar credits Mr. Martin with influencing her future career choice and goals. She hopes to become a math teacher and a mathematician.

Natasha Chen was Ed’s student during her 10th, 11th and 12th grade years in Broward’s Project MEGSSS program. She went on to earn her BS in Mathematics as a Trustee Scholar at the University of Southern California followed by an MBA from New York University and career in finance. Natasha now balances her duties as Communications Director for IMACS with being the homeschool parent of an EMF and online IMACS student. She has fond memories of Ed’s spellbinding lessons, especially on group theory, and his infinite patience.

Acknowledgements

The co-authors are immensely grateful to Ed, Iain and Terry for taking the time to answer their interview questions in detail. Special thanks go to Ed for organizing, editing and writing the foreword to Burt Kaufman: An Appreciation, which provided essential background material for this article.

The Institute for Mathematics and Computer Science (IMACS) is thrilled to announce that the Elements of Mathematics: Foundations (EMF) online program for talented middle school students is now complete with the release of the 18th and final EMF course, Precalculus Coda.

Precalculus Coda brings together topics from earlier EMF courses and expands on them 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.

Upon successful completion of the EMF program, students will have covered all national and state math standards for pre-algebra through precalculus but from an entirely modern mathematical approach. In addition, they will have enjoyed thorough introductions to college-level topics such as Abstract Algebra, Logic, Set Theory, Number Theory and Topology, and be well-prepared to excel in a college-level Calculus course.

Why Modern Mathematics
Mathematically talented students typically master school math with little effort. As a result, they tend to become bored with and disengaged from math if limited to traditional curriclum. They may also fail to develop the mental resiliency and study skills that come with having to work hard at math problems designed for their ability level.

The EMF curriculum is designed specifically to engage young, gifted thinkers in ways typically reserved for university math majors. It is useful to distinguish the two ways in which EMF is college-level: (1) the sophisticated use of logic and reasoning via proofs, and (2) the introduction of modern areas of mathematics such as Topology.

EMF incorporates all the concepts from a traditional curriculum but uses a logic and proof-based approach not usually entrusted to students until college. Talented middle school students who are intellectually mature and motivated can benefit from this more sophisticated approach; it fosters the development of creative thinking skills that are not easily replaced by computer processing power as algebraic manipulation skills already have. In other words, EMF prepares students with skills for the future, not skills of the past.

EMF's incorporation of exciting topics in modern mathematics—topics ignored by traditional school math—makes it far more interesting and intellectually engaging. These topics represent areas of active research as opposed to traditional school math subjects that were understood completely by the late 17th century. The idea of cutting off a child’s education in literature, history or science at the 17th century is ludicrous. The same applies to mathematics.

Access Matters
For children in many parts of the US and around the world, EMF now fulfills the promise of opening up access to a comprehensive, unified, proof-based approach to modern mathematics. IMACS certainly believes in the value of talented and dedicated teachers, but we also believe that lack of access to such teachers has unfairly limited for too many bright students the right to be challenged at their ability level on the way to achieving their full potential in mathematics. We aim to change that with EMF.

Course 1 < $20
In celebration of EMF's completion, tuition for the first course has been dropped to $19.95 (regularly $59.95) for a limited time. Through August 31, 2018, new families save 67% on the first EMF course or 25% on a bundle of all 18 courses. Visit EMFmath.com to learn more and enroll.

EMF in Schools
Did you know that numerous districts, schools and after-school programs license EMF? In fact, the first cohort of EMF students in Broward County Public Schools, Florida completed the program last year. As a result, many of them will be enrolling in Calculus as 9th graders this fall. If you would like to see EMF offered in your school, share the EMFmath.com/schools website with the relevant decision-makers in your district.

The following is an excerpt from Geometry: Incidence and Transformations, the first of three Geometry courses in the self-paced Elements of Mathematics: Foundations (EMF) series. EMF covers Pre-Algebra through Precalculus, plus several university-level topics, with the depth and sophisitication needed to challenge and mathematically talented children. Limited Time Offer: The first EMF course is available at 25% off regular tuition for students who enroll on or before October 31, 2017. Learn more at www.EMFmath.com.

In this course and the next our main focus is on Euclidean geometry. As you have undoubtedly noticed, in the preceding courses we have quite frequently talked about, considered, and even worked with a wide variety of geometrical concepts. But we have been relying on common knowledge and your intuitive understanding of such things; we have only rarely hinted at the formal underpinnings of this very important and pervasive area of mathematics.

The history of the study of geometry is very long and honorable, springing out of humanity’s attempts to describe mathematically the forms, shapes, and patterns seen in the real world.

Roughly speaking, our intended plan of study in this course is as follows: Each of us has a store of experiences with real objects, their forms, and their shapes. These experiences will be refined as we draw pictures of objects, observe specially constructed models of objects, and think about experiments that could be performed with these objects. As we sharpen our experiences with physical objects, we will describe corresponding geometric objects (such as points, lines, and planes), and decide what properties these geometric objects should have if they are to be mathematical replicas of the real objects. Of course, geometric objects, like all other mathematical objects, are abstractions, so the only role that can be played by our drawings and experiments with physical points, lines, and planes is that of serving as a guide to the properties that should be ascribed to their abstract geometric counterparts.

The properties we ascribe to the geometric objects will be called agreements. That is, we will agree to accept a certain property about geometric points and lines, say, because our experience strongly suggests that physical points and lines have that property. Of course, agreements in geometry, as in real life, have consequences. This then is the activity of geometry: to find out what the consequences of the basic agreements are.

In other words, we shall agree that the geometric objects we wish to study have certain basic properties. We shall then deduce that geometric objects with these properties must also have certain other properties. We will discover these new properties, not by looking at the objects (remember, geometric objects are abstractions!) but by thinking about them. We will express our thoughts in arguments of the following general form: Since we have accepted our basic agreements, then we must also accept that such and such is the case. But then we are constrained to accept that thus and so, etc. Finally we examine these new properties in order to decide whether they too are compatible with our experience of physical objects. In this way, Euclidean geometry becomes the mathematics of the shapes and forms of the world around us.

A typical pre-algebra curriculum is designed to promote computational or procedural fluency. Unfortunately, that is rarely enough to engage gifted minds. Pre-Algebra Plus, an online course within the Elements of Mathematics: Foundations program, offers gifted students a better option.

Five Reasons Gifted Students Prefer Pre-Algebra Plus

1. Pre-Algebra Plus was created specifically for the way gifted minds work. Course material invites students to play with numbers, patterns, and puzzles. Unnecessary repetition is traded away for deep exploration of complex ideas. This approach appeals to both traditionally "mathy" kids and those with untapped mathematical talent.
2. Instead of telling students what the rules are, Pre-Algebra Plus takes students on the journey of discovering the rules through observation, conjecture, and reasoning. As one parent put it, Pre-Algebra Plus teaches math from the inside out.
3. Pre-Algebra Plus introduces basic logic and proof techniques. Logic and creativity complement each other naturally in the study of genuine mathematics. Creative thinking helps students leverage what they know and observe into new (to them) ideas about mathematical concepts. Logical thinking helps order and cement those ideas into cohesive arguments and conclusions.
4. Pre-Algebra Plus is hard! The course explores several university-level topics not normally studied until college. For a mathematically talented child, being challenged is a joy. After all, what's the point of a Formula One racecar if you only get to drive it on a straightaway, even at accelerated speeds?
5. Pre-Algebra Plus is individually-paced and available online 24/7, allowing each student to work on the course when convenient and go at the pace that works best for his or her learning.

Is your mathematically talented child ready for pre-algebra? Does he or she need a flexible, online course designed for the gifted thinkers? Pre-Algebra Plus is that course. Visit the Pre-Algebra Plus website to learn more about course content and enrollment.

Special Offer: Pre-Algebra Plus is offered to new students at a 33% discount to regular price. Visit www.EMFmath.com for details.

Last week, The Atlantic published an excellent article questioning the trend toward requiring convoluted explanations of mathematical thinking in Common Core-aligned math classes. The authors rightly pointed out that verbal explanations are hardly the only way of determining whether a student understands a concept and that many of the brightest mathematical minds are verbally challenged.

The theory that if you cannot explain, you do not understand does logically lead to the conclusion that if you understand, you can explain. (IMACS Mathematical Logic students will recognize this as an example of contrapositive inference.) That theory, however, is completely false, yet it continues to drive the misguided practices of Common Core-aligned pedagogy to the detriment of another generation of students who, we worry, will be irreparably damaged in their understanding and appreciation of math.

Visual explanations are often a natural way to demonstrate mathematical understanding when designed thoughtfully and taught well, but even they can be gamed. There is another way, however, that is an effective measure of true understanding — demonstrating how well you can apply your knowledge to a novel situation. This approach does not call specifically for a verbal, visual or symbolic explanation, but it does require that teachers have the mathematical depth to recognize understanding when presented in a variety of explanatory modes, not just how a scoring rubric of model answers dictates.

Consider for a moment the art and science of cooking. Some people can only follow a recipe, and many have compared this to when students can only "plug 'n chug" math formulas and algorithms. Some people read cooking magazines or watch the Food Network and then impress with the right vocabulary at parties, the same way verbally skilled students will learn the right Common Core-friendly phrases to use in answering certain types of problems. Then there are the chefs, the ones who understand why certain ingredients and/or cooking methods work well together and what the fancy foodie talk actually means. What can they do with their genuine understanding?

If you've never watched the television show Chopped, take a moment. Competing against the clock and other chefs, each contestant must use everything in a basket of mystery ingredients to prepare an appetizer, entrée or dessert. When the ingredients are unveiled, it is not unusual for them to include such oddities as grasshoppers, gummy bears and leftover pizza. Talk about a novel situation!

It's hard to imagine a single traditional recipe that calls for such ingredients, or what erudite words you could utter about the culinary characteristics of a grasshopper. But leave it to the chefs on Chopped to shred the pizza crust, melt the gummy bears and use them with other ingredients to make breaded and glazed gourmet grasshoppers with a pepperoni pâté!

Mathematics is a lot like cooking. When you have a genuine understanding of mathematical concepts, you know what to do when faced with a problem that is unlike any you’ve seen before but that requires putting your knowledge together in a new way. Whether you can impress the Pulitzer Prize people as well is beside the point. Bon appétit!

The Institute for Mathematics and Computer Science (IMACS) recently released its first online algebra course, Algebra: Groups, Rings and Fields. This is the tenth course in the Elements of Mathematics: Foundations (EMF) program for talented secondary school students. Our latest self-paced offering has generated a fair amount of inquiries from parents seeking options for their mathematically advanced child. The answers to some of those questions can be found in the FAQ at EMFmath.com. IMACS responds to others, which have been consolidated by topic, in this week’s blog post.

Q: My child aced pre-algebra and is ready for algebra. Why can’t I just enroll her in the first EMF algebra course?

When people use the term “algebra,” they’re usually talking about high school algebra or what mathematicians call “elementary algebra.” By contrast, EMF teaches the kind of algebra that a mathematics major at university learns called “abstract algebra.” A student who has mastered pre-algebra is, no doubt, ready for high school algebra. However, this same student, no matter how talented, is simply not ready to jump directly into abstract algebra.

Why? Just as success in high school algebra is built on the foundation laid by elementary school math up through pre-algebra, success in abstract algebra requires a strong foundation in various mathematical structures and reasoning techniques that are rarely taught outside of a university setting. This important foundation is built up through the first nine courses of the EMF program. In fact, one might think of these early courses as constituting “pre-abstract algebra.” As such, they are an integral part of the EMF program and essential to success in the later courses.

In case you’re wondering, students who complete the EMF algebra courses will have learned all of elementary algebra and be able to solve any high school algebra problem with ease. But they will also have learned a great deal more and be well-prepared to study the high-level mathematics that is at the heart of important disciplines such as particle physics and cryptography.

Q: What if my child already took high school algebra and geometry? Is there anything left for him to learn in EMF?

As IMACS principal founder and EMF co-author, Burt Kaufman, once wrote, “It is surely a sad state of affairs that in the traditional high school curricula, the student encounters very few, if any, mathematical ideas that postdate the seventeenth century. … It would be ludicrous if an English curriculum for the high school never contemplated confronting the student with a piece of literature written after Shakespeare.”* That’s one key reason why the EMF curriculum was created—to teach modern mathematics to talented, young students who are capable of benefiting from advanced material that goes beyond the outdated math curricula used in schools.

Naturally, more experienced students will find some of the EMF material familiar, but EMF approaches these topics from a far more sophisticated standpoint. Between the new mathematical structures and techniques for reasoned argument that they will be learning, there is much for these students to gain in EMF if they are motivated to learn real mathematics as opposed to just school math.

Q: We tried other math programs for advanced kids, but they just seemed to be about going faster or preparing for competitions. That’s not working for our son who’s more of a deep thinker. I’ve heard that EMF takes a different approach. Can you explain?

First, a pair of quotes:

The competition-inspired approach to math has its merits. But it’s hardly the only approach worthy of mathematically talented kids. The gifted population is filled with individuals who have exceptional talent and prefer to take their time. EMF is an ideal option for these students because the program is self-paced and encourages patience in coming to a deeper understanding of complex and beautiful ideas. At the same time, we’ve had numerous EMF students who also enjoy and excel at competition math. In fact, in situations where speed is of the essence, the non-standard mathematics to which EMF students are exposed gives them a distinct advantage over others who are seeing these ideas for the first time during a competition.

Q: You say that EMF is “mathematician math,” and that it’s taught the way a math major at university would be taught. That’s nice, but what are the benefits for a talented child who has no interest in being a math major, let alone a mathematician?

Before your child writes off being a math major completely, especially if that choice is based on experiences with school math, consider the following:

Whether your child decides to pursue a math major or not, there are several important skills that the EMF approach to mathematics teaches. As with all IMACS programs, EMF fosters the development of logical reasoning skills in young students. While some people believe that logic is cold and inhibits creativity (think: Star Trek’s Spock!), our experience teaching the EMF curriculum over the past 30+ years suggests otherwise. To the contrary, we have found that when IMACS students are equipped with the logic skills to construct their own well-reasoned arguments and critique those of others, the clarity of thought that this produces unleashes creative and innovative ideas that were previously unfocused or muddled.

Which brings us to how EMF encourages creative thinking. As Professor Devlin reminds us, mathematics is a creative discipline. The fact that what passes as “math” in schools is devoid of creativity should not be taken as evidence that true mathematics is indifferent to creative thinking. School math tends to take a “tell-then-drill” approach where the teacher states a rule and then students apply the rule to umpteen haphazard problem sets. By contrast, EMF uses carefully constructed exercises and interactive technology to guide students to their own “discovery” of mathematical results. To be successful in EMF, a student simply must think creatively in order to cross the bridge from keen mathematical observations to the “A-ha!” moments of intuitive understanding. And when these moments happen, the joy and pride of having climbed the intellectual mountain are profound.

Another skill that EMF promotes is abstract thinking. Imagine a world in which most jobs involve people interacting with tangible objects in the present. Perhaps you envisioned the Industrial Revolution, a time when mechanical inventions led to an explosion in manufacturing. Today, we find ourselves amidst a Knowledge Revolution wherein technological inventions mean that well-paying jobs require abstract thinking about intangible ideas such as code. This is obviously true in tech, but because tech touches every industry now, it’s also true for wide-ranging fields from medicine to music to law to film. It’s the question on many people’s minds: Are you going to design and program the robot, or will you be replaced by the robot? Whatever career your child pursues, he or she will almost certainly need to think abstractly. Abstract thinking is fundamental to the study of genuine mathematics, which is what EMF teaches.

Is EMF right for your child? IMACS created a 30-minute, online Aptitude Test to help prospective parents and students answer this question. Register to take the FREE test at EMFmath.com.

* Kaufman, Burt, Jack Fitzgerald, and Jim Harpel. MEGSSS in Action. St. Louis: CEMREL, Inc., 1981.

A recent study published in the Philosophy of Mathematics Education Journal confirms that teachers’ images of mathematics and their mathematics history knowledge are interlinked. According to the study’s lead author, Danielle Goodwin of the Institute for Mathematics and Computer Science (IMACS), "By and large, the teachers with low history scores in this study were the teachers who exhibited narrow, negative views of mathematics."

Key findings from the study include:

Attitudes Influence Decisions that Affect Students

Why does this matter? Because educators’ views of mathematics affect student learning experiences in a variety of ways, from daily classroom instruction to curriculum selection and development to far-reaching proposals for national math education reform.

Teachers’ images of math are typically based on their own limited experiences as young students, and so teacher education programs should incorporate mathematics history into their curriculum as a way of reshaping attitudes, the study suggests. Doing so would help future teachers develop an appreciation for and understanding of math as a subject that is alive and fundamentally creative. Fostering this viewpoint could help teachers help their students understand that mathematics is a natural place for inventive problem-solving where questioning and investigating are highly valued.

"Teachers who have rule-oriented images of mathematics can weaken student learning by representing mathematics in misleading ways," says Goodwin. Instead of conveying as healthy the struggle of intellectual discovery that naturally takes place in mathematics when new ideas are explored, "struggle" in US K-12 math classrooms has come to mean being "bad at math." This unfortunate association has left generations of Americans hating math and believing in the myth that they are not "math people."

Current teachers and pre-service teachers who want to improve their ability to teach math don’t have to wait for curriculum changes at schools of education. There are wonderful and accessible resources that provide a willing and curious mind with a deeper understanding of mathematics in the context of its rich history.

Recommended Reading and Viewing

If you’re still looking for a holiday gift for your child’s math teacher, perhaps one of the recommended books below would be appreciated. For the visually-inclined,
the videos and movies that follow provide many hours of awe-inspiring and sometimes humorous enlightenment.

UPDATE, July 28, 2014: IMACS has completed the update of our AP^® Computer Science: Java Programming course to include eight fully-elaborated labs that far exceed the minimum requirements of the College Board. IMACS’ Be Prepared for the AP^® Computer Science Exam online course has been updated as well. Students who are enrolled directly through eIMACS in our AP^® Computer Science: Java Programming online course receive free access to the Be Prepared course.

Following a recent review of the AP^® Computer Science A course and exam, the College Board has decided to replace its case study requirement with a requirement to complete a minimum of 20 hours of hands-on lab experiences. This change, which will take effect for the 2014-2015 school year, is being implemented to more effectively support student learning of core concepts in computer science. IMACS continues to follow closely all communications from the College Board, as well as discussions within the APCS community, on the forthcoming changes and will act accordingly.

From the beginning, IMACS’s philosophy has been to emphasize computational thinking and mastery of foundational ideas in computer science. This approach is reflected in how our Curriculum Development Group has meticulously designed our CS courses and, more importantly, in the success our CS graduates find in college, graduate school and at top tech companies. As such, IMACS fully expects that our AP^® Computer Science: Java Programming course will continue to exceed, as it always has, all of the College Board’s requirements and remain College Board-approved.

GridWorld Case Study

Since the 2007-2008 school year, AP^® Computer Science A has used the GridWorld Case Study to reinforce lessons on object-oriented programming.* GridWorld provides Java code designed to simulate the behavior of objects (Rock, Flower, Bug and Critter) in a grid. Ground rules such as Rocks cannot move, Critters eat Flowers and Bugs move forward and turn 45 degrees if blocked are part of the initial set-up. Given these starting parameters, students then write additional code that extends these various classes of objects. A student’s understanding of computer science concepts in the context of the GridWorld code is then tested on the AP exam with one free-response question and a handful of multiple choice questions.

College Curriculum Study

In 2011, the College Board undertook a College Curriculum Study in which institutions of higher education were surveyed about the AP^® Computer Science A course case study.^ Of the 117 institutions that responded, 91% said they were not likely to change their credit/placement policy for AP^® CS A if questions on the case study were not included in the exam. About two-thirds of respondents rated the inclusion of a case study as not important or only somewhat important.

Clearly, GridWorld is now past its prime. As the College Board noted on its website, the case study requirement in AP^® Computer Science A needed updating “to stay aligned with the most recent practices in the continually changing field of computer science.”

Labs, Labs, and More Labs

This March, the College Board plans to release details of three sample AP^® Computer Science A labs as examples of how the new lab experience requirement may be implemented. One expectation is that their shorter length will make the labs easier to integrate into the course curriculum throughout the school year. Teachers and curriculum developers will have the flexibility to include sample labs or other comparable labs at points they feel are most relevant and pedagogically effective. It is also expected that the sample labs will be more connected to real-world situations, perhaps increasing student interest in taking the course and studying computer science.

Most importantly, labs are expected to support student learning of fundamental ideas in computer science. Whereas the case study questions on the current exam are tied heavily to the context of the GridWorld code, the 2015 AP^® Computer Science A Exam will test a student’s understanding of core concepts that are reinforced by hands-on lab experience, not knowledge specific to any particular lab. As an educational institution that has always emphasized foundational concepts in CS over code manipulation skills in the programming language du jour, IMACS is pleased to see the College Board take this important step.

Learn how you can give your child an unfair advantage in computer science. To find an IMACS teaching center near you, visit www.imacs.org. Talented middle and high school students can take university-level computer science online through our eIMACS distance-learning division.

*For readers who may be unfamiliar with object-oriented programming, it’s an approach in which the programmer creates “objects” with specified attributes and behaviors as modular, reusable code.
^https://tinyurl.com/LabsAPAC2013

The new school year is now a month old. By this time, most children who attend a public K-12 school in the US will have experienced the new Common Core State Standards in Mathematics (CCSSM). On the one hand, IMACS is pleased to see that key elements of the teaching philosophy we have lived by for more than 20 years are reflected in the Common Core Standards for Mathematical Practice. For a variety of reasons, however, we maintain a healthy amount of skepticism about whether the implementation of the CCSSM will lead to meaningful, positive change in mathematics education, particularly for our most talented youth.

Common Core Was Not Designed for Gifted Kids

First, the CCSSM was not designed with exceptional kids in mind. The official Common Core Web site states plainly that:

The Web site further acknowledges that Common Core, like its predecessors, cannot adequately address the unique needs of individual learners:

As to what educators should do about serving the diverse needs of a student body, Common Core guidance leaves them with unresolved internal conflict, offering both:

and

[Note that Common Core does not include gifted children as an example of “students with special education needs.”]

But children in the right-hand tail of the distribution do have special education needs. Whether due to a failure to understand this fact, budgetary pressure, or some other constraint, some school districts seem to be latching on to (ii) above, using the arrival of Common Core as a reason to reduce or eliminate services or accommodations for gifted students. Should this become a national trend in education policy, our country will surely suffer as the majority of gifted children who rely on public education are left without appropriate alternatives.

What About Creative Problem Solvers?

Notwithstanding the potential for improving the thinking skills of typical students, the CCSSM are simply not built to inspire or nurture the creative problem solver. The unfortunate embracing of computerized testing as a cheap means of measuring “learning” — consequently resulting in a culture of teaching to the test — has made the K-12 classroom a place to dread for many unique thinkers. The plan to continue use of computerized testing under the new standards suggests that the non-standard thinker may still be out of place in the Common Core classroom.

IMACS recently asked Gerald R. Rising, SUNY Distinguished Teaching Professor Emeritus at the University at Buffalo, how he thought the CCSSM would affect mathematics education for bright children, to which he replied, “Any imposed curriculum can have a depressing effect on special programs for gifted students.” He also shared the following anecdote about the limits of standardized testing:

IMACS has been delivering courses and administering tests online to bright and creative children for over 15 years, so we know a thing or two about designing effective computerized assessments of high-level thinking skills. Let’s just say that it takes tremendous creativity, foresight, and a deep understanding of how to leverage the power of technology. If high-stakes testing is here to stay, as it appears to be, we sincerely hope that the consortia working on Common Core-aligned assessments will find ways to reward (or at least not penalize) creative problem solvers.

Inadequate Investment in Training

Common Core marks a major change in teaching philosophy for math education in the US. The intent is to move away from just teaching procedural skills by giving equal weight to conceptual understanding. Teaching math with an emphasis on thinking and understanding, however, is not something one becomes proficient in after a few hours of training, which is all that many districts have provided to their teachers.

Such a radical shift in mindset can be especially challenging for some who have taught math with a completely different focus for many, many years. This is not to say that teachers are incapable of learning to teach a new way — quite the contrary. But, as with any field undergoing fundamental change, extensive training and professional development are necessary if districts and schools want a successful implementation of Common Core. So far, the evidence suggests that they cannot or will not be making that investment.

“Common Core” Textbooks In Name Only

Many of the textbooks currently on the market that say they are aligned to the Common Core standards were developed before the creation of the CCSSM. Note the example below of pages from old and new versions of a math text currently being used in California. The pages on the left were from the edition published in 2009, the year before states began adopting the CCSSM. The pages on the right are from the current edition that proclaims “Common Core” on the cover. (Click on an image to enlarge.)

Furthermore, such textbooks often only align to the specific content skills listed in the CCSSM rather than subscribing to the overall philosophy of the CCSSM. Many that claim to be aligned to the CCSSM do not include problems or tasks that involve the higher-level thinking skills that are supposed to be measured by the new Common Core standardized tests being developed.

Awareness and Advocacy Are More Important Now Than Ever

What does all this mean if you are the parent of a talented child? Probably more work for you. Just over a month ago, nearly two-thirds of respondents to a poll on education said they had never even heard of Common Core! So, if you’re thinking that someone else will speak up first, don’t count on it. Advocacy for a gifted student has never been easy given the lack of awareness and amount of misinformation about their unique educational needs. With the potential for Common Core to bring more harm than good to the education of exceptionally bright kids, it is more important now than ever to be heard.

This year IMACS celebrates 20 years of educating talented, young students in mathematics and computer science. In all this time, we have never wavered in our philosophy that providing children with a deep and strong foundation in logical reasoning would enable them to take on virtually any intellectual pursuit with ease and confidence.

In mathematics, we continue to receive regular confirmation of our approach. Recent IMACS graduates often write to tell us of how advanced they are compared to their college math classmates, even at elite universities. Non-IMACS students who were so deftly skilled at applying formulas and algorithms in high school suddenly found themselves in college turning to our graduates for help in proving why these formulas and algorithms worked. It seems this phenomenon is steadily growing in computer science.

As strong advocates of K-12 computer science education, we are heartened by the broad realization that teaching children about this amazing and empowering field is of great importance. At the same time, IMACS urges parents, educators, and policy makers to understand the difference between coding and computational thinking, as well as the consequences of promoting one path over the other. As CS education decisions are made, we must not repeat the ruinous mistakes of math education policy lest we end up with computationally illiterate generation after generation as well.

Learning to Code Isn’t Enough

In a recent article titled “Learning to Code Isn’t Enough,” computer scientist Shuchi Grover offered the most articulate and convincing argument we’ve read on the shortcomings of the “learn to code” craze. In particular, Ms. Grover notes that the cognitive benefits gained through the process of good programming often fail to develop in online coding academies:

The IMACS Approach

At IMACS, we have taken a considerably different approach to teaching computer science than the trendy, new organizations. Most importantly, we focus on universal thinking and problem-solving skills. That’s really what any programming exercise comes down to: thinking clearly about how to solve a particular problem. As Ms. Grover points out:

In our introductory computer science classes, IMACS deliberately uses programming languages that have trivial amounts of easily-mastered syntax. As a result, our students are able to concentrate their mental energy on learning the core concepts in computer science instead of on memorizing rules of syntax. Rather than focusing narrowly on ideas that only apply to a specific environment, IMACS classes develop computational thinking skills that can be applied to any programming situation.

Learning to Think with Logo

Children may begin taking IMACS Computer Enrichment classes as early as 3rd grade. Computer Enrichment uses Logo, an easy-to-learn language with a strong graphical component, to introduce students to programming ideas. Using a language with graphical components allows even our youngest students to understand and master advanced programming and problem-solving techniques.

IMACS Computer Enrichment places a heavy emphasis on computational thinking — thinking about logic, thinking about processes, thinking about good design. (All this takes place in a fun-filled class that incorporates interesting puzzles and problems.) A working program is not the main goal; rather, it is understanding how and why a program works or doesn’t. With a firm foundation rooted in computational thinking, IMACS students as young as 11 or 12 are well-prepared to move up to our university-level classes in computer science.

University-Level Computer Science

The IMACS curriculum continues with our Modern Computer Science track comprised of three university-level classes. The first course, UCS1, teaches the fundamental principles of computer science using Scheme. Scheme’s expressive yet simple syntax allows students to focus on learning universal concepts applicable in any programming language, even future languages not yet invented.

The second course, UCS2, begins in Scheme, but by the end students are programming in Haskell and Python. One reason that we introduced these additional languages into UCS2 was to show our students just how easy it is for them to learn new languages given their solid foundation.

The third course is our College Board-approved Advanced Placement Computer Science course in Java. This summer IMACS will be updating our APCS course with a new section on how to write Android phone apps. Although app development is not part of the AP Computer Science curriculum, the new component will allow IMACS students to gain experience in developing real applications.

The IMACS Advantage

While it sounds impressive to say that students who complete the entire IMACS computer science curriculum will graduate with significant experience in five diverse programming languages, what matters is that they leave us with something even more highly-prized: the ability to succeed in virtually any coding environment. Incidentally, whether or not IMACS graduates go on to study or pursue careers in computer-related fields, they gain an unfair advantage over their peers throughout their lifetimes thanks to their unmatched ability to dissect problems and articulate solutions. IMACS CS alumni, we look forward to receiving your emails.