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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.

The IMACS Blog reconnects with alumnus Mark Engelberg who recently released three new coding games through ThinkFun and Target. (Win an autographed copy of one of Mark's new games!) Mark's passion for computer science, math and logic led him to an award-winning career in puzzle and game design. He is also an active speaker in the Clojure programming language community. In this post, Mark talks with IMACS about what it was like to be a homeschool dad to two amazingly talented children who are now accomplished young adults in their own right.

First, tell us about your new ThinkFun games.

Two years ago, I invented a programming puzzle game for ThinkFun called Code Master. Code Master was a commercial and award-winning success, so ThinkFun asked me to develop a new line of puzzle games, where each game would highlight a specific underlying principle of computer science. The new line of games is called the //CODE Programming Game Series. To design the games, I started from a list of concepts I wanted players to encounter, and kept trying out ideas and tinkering with the rules until I felt I had a combination of mechanisms that covered those concepts and was fun to play.

The first game in the series, On the Brink, is an introduction to procedural abstraction — the idea that we're not just writing programs to solve a one-off task, we're building components that solve multiple instances of an underlying problem, and then using those components to build higher-level components, and so on. The second game, Rover Control, focuses on the essential skill of stepping through programs in your head and visualizing the outcome, particularly with flow control constructs like if-then-else statements, while loops and for loops. The third game, Robot Repair, is all about Boolean logic. When developing Robot Repair, I leaned heavily on the material that I learned as a child in IMACS' Advanced Mathematical Logic courses. I think students in your Logic courses are really going to enjoy Robot Repair. Like their predecessor, Code Master, all three of the new games are "unplugged" computer science board games that come with a book of puzzles for solo or cooperative gameplay.

You are also a parent of two gifted children who are now ages 19 and 17. About how old were Alex and Molly when you first suspected that they were gifted? What were the signs?

I was a stay-at-home dad, and when Alex and Molly were toddlers, I taught them to read and write and gave them daily math lessons. Honestly, I mostly did those things to keep myself from getting bored; it was fun for me to feed their intellectual curiosity. My wife did the same with the kids when she was home from work. That was just our parenting style. We wanted the kids to have the opportunity to learn these core skills as soon as they were capable, and the easiest way to achieve that was to throw a lot of educational activities at them early, just to see what would stick. As it turned out, more of it stuck than we expected!

I didn't really have much to compare their development to until the day I went to Alex's kindergarten orientation. As the teacher explained what the students would be doing in the coming year, it dawned on me that my kids had already done those things. I realized that I had inadvertently given my kids a big jumpstart, and what I had been doing was working, so I might as well keep doing it. And that's how I ended up deciding to homeschool my kids.

Because I homeschooled my kids, the label "gifted" didn't really come into play. I never needed to get them tested, or prove anything to a school official. But it was clear to me that they had the capacity to learn faster than what our school system provides, so I set out to do my best to keep them challenged and allow them to fully develop their individual talents.

How did you feel when you knew for certain that they were gifted?

It felt like a tremendous responsibility to help them achieve their full potential, even more so because I was choosing to take on that responsibility myself rather than relying on the school system. If things didn't work out, I would have no one to blame but myself. It was a lot of pressure. I didn't really doubt my capability to teach them, but I knew it would be an enormous undertaking, one that would require the bulk of my time and attention for nearly 20 years. It was well worth it — I'm so proud of the young adults they've become — but I'm relieved to have reached the point where that responsibility is behind me rather than in front of me.

As you looked into ways to meet their educational needs, what did you discover?

One pleasant surprise was realizing that it isn't necessary to be an expert in everything in order to meet your kids' educational needs. In many cases, it's more about playing the role of an advocate, connecting them up with the right resources. I would track down private teachers, public school classes, homeschool co-op classes, other parents, books, videos, and websites for the areas I was less equipped to handle. I did my part for the homeschooling community by serving as a teacher of math and computer science for many other talented kids whose parents were not as adept at those particular subjects.

What were some of the programs, opportunities or approaches that you found helpful in making the most of their talents and interests?

One of the things that was really important to me was to make sure my kids had a strong foundation in math and computer science. Fortunately, I have the advantage that I learned the IMACS curriculum when I was in middle school and high school. This made it easy for me to teach these subjects because I'd already experienced first-hand what a top-tier math and CS education looks like. To duplicate that experience for my kids, I simply taught them using the same materials and techniques that I had learned at IMACS and in college. Those resources played a big role in giving me the confidence to homeschool.

What were some of the benefits and challenges of homeschooling? How did you deal with the challenges?

One of the biggest challenges of homeschooling is that it takes effort to find a community to be a part of. We are fortunate that here, where we live in Seattle, there is a robust homeschooling community and we were able to find a group that fit our personalities and make many dear, lifelong friends. Once we found the right community for us, those rich social interactions became one of the biggest benefits of homeschooling. Our weekly park days remain some of my most cherished memories.

Another big challenge is that, even though my kids are close together in age, they were far enough apart that I couldn't do the same lessons with them. I always felt like I was struggling to give each of them the level of individual attention I wanted to give them. My solution was to make certain parts of the day relatively structured and leave the other parts of the day flexible so that I only needed to schedule the structured parts. For example, I might be doing structured math lessons with one child while the other was doing creative writing or watching an educational video, and then switch.

What were some other challenges you faced in parenting gifted children, and how did you deal with those?

Gifted children can be wildly asymmetric in their abilities, and wildly different from one another in terms of their talents and interests. It was a constant challenge to avoid getting impatient when one of my children would struggle with a concept that came naturally to the other. I had to keep reminding myself that it was totally normal for them to have different strengths and weaknesses.

How did being a gifted adult who was once a gifted child affect your educational and non-educational parenting choices?

This was especially helpful when my kids' interests dovetailed with my own. My son, for instance, developed a keen interest in programming, and thanks to my own background, I knew exactly what to do to support him in that interest. When my kids' interests and abilities diverged from mine, it was a lot tougher, and I ended up doing what any parent would do … seeking out people and resources that could help my kids develop their talents.

What are your fondest memories of raising Alex and Molly?

Probably my fondest memories are the many times we played board games together over the years. For those moments, instead of having a parent/child relationship, or a teacher/student relationship, we were just people playing games together and enjoying each other's company. I feel like at those moments, I really gained some insight into their character, and got a glimpse of what they would be like as adults.

What are your hopes for them in the future?

My hope has always been for them to be good, caring, compassionate, independent, happy adults. The good news is … they already are. I'm done! Woohoo!

Robot Repair Raffle

Robot Repair was inspired by Mark's study of the IMACS’ Advanced Mathematical Logic curriculum.

Residents of the continental United States are eligible for the Robot Repair Raffle. The autographed copy of Robot Repair will be shipped only to an address in the continental US. To enter the raffle, complete the following steps:

1. Parents must register their child to take either the EMF Aptitude Test (ages 10-14, approximately) or the eIMACS Aptitude Test (ages 14-18, approximately).
2. On the aptitude test online registration form, type "IMACS Blog" in the box for "How did you hear about us?" or "How did you hear about IMACS?".
3. Have your child take the aptitude test for which he or she is registered. The test must be taken during the period October 1-30, 2017. (No minimum test score is required to qualify for the raffle.)

RJ was an IMACS student from kindergarten through 12th grade and took every class IMACS offered, including all levels of Math Enrichment, Computer Enrichment, Hi-Tech Summer Camp, University-Level Computer Science, AP Computer Science: Java Programming, and University-Level Logic for Mathematics.

With a 5.9 GPA and 2360 on the SAT, RJ graduated as co-valedictorian and was honored as a US Presidential Scholar candidate, National AP Scholar and National Merit Scholar. He and his teammates were also four-time Science Olympiad state champions.

RJ was accepted to Caltech, Georgia Tech, Carnegie Mellon, and the University of Florida. He chose Caltech where he took junior-level Computer Science classes as a freshman and served as a Teaching Assistant for those classes as a sophomore. RJ conducted summer research in computer vision and was selected for an internship at Northrup Grumman. He will graduate with a double major in Computer Science and Philosophy and plans to attend graduate school to specialize in Machine Learning.

Congratulations, RJ!

Cori wanted to understand the fundamental concepts that make up the sophisticated theories of Computer Science. She started IMACS during the summer before tenth grade and went on to complete IMACS’ University Computer Science track and AP Computer Science: Java Programming course. Cori scored a 5 on the AP Computer Science A exam, 800 on the math portion of the SAT, and was named a National Merit Scholarship Finalist.

Cori’s keen interest in using her knowledge to understand real-world problems inspired her to write a Naive Bayes machine learning algorithm to detect political bias in online news articles. Her paper on it was selected for presentation at the Georgetown Junior Science and Humanities Symposium, and she was honored as an affiliate winner for the National Center for Women and Information Technology’s Aspirations Award.

Cori chose Brown University where she was accepted early decision and plans to study Computer Science and Economics. She hopes to pursue a career applying data science within education or a service-based field.

Congratulations, Cori!

As a young child, Mason excelled in IMACS Math Enrichment classes and, in subsequent years, attended several weeks of IMACS summer camps. He later completed IMACS’ university-level courses in Computer Science and Logic for Mathematics. Mason was honored by the College Board’s National Hispanic Recognition Program for his achievement on the PSAT/NMSQT, and his high school mathematical modeling team was recognized by the Consortium for Mathematics and its Applications. Mason graduated high school with a perfect 4.0 GPA and 34 on the ACT.

A natural at sharing knowledge, Mason enjoys tutoring students in science and math at a local homeschool co-op, co-taught a robotics class for two years, and was a Teaching Assistant for an Honors Calculus seminar. He accomplished this and more while being an all-county varsity and travel soccer player and a dual-enrolled student at a regional 4-year university.

Mason chose Rice University where, at 16 years old, he was accepted early decision into the George R. Brown School of Engineering. He plans to major in Materials Science and NanoEngineering.

Congratulations, Mason!

Snow began IMACS in 1st grade and completed all levels of IMACS Math Enrichment. She returned during high school to complete the IMACS University Computer Science track. Snow’s talents and keen interest in programming, civics, and service propelled her to 1st place finishes at FIU’s Google Startup Competition, Ft. Lauderdale’s Code for America Civic Hackathon, JusticeHack Miami, and Yale’s Bulldog Invitational Mock Trial Competition.

In addition to being selected as a National Merit Finalist, Snow was named Cadet of the Year by the Florida Wing of the Civil Air Patrol and won 1st Place at the National Cadet Special Acts Competition. Snow credits IMACS for the way she thinks and approaches every aspect of her life.

With a 5.5 GPA, 1580 out of 1600 on the SAT, and 800 on the SAT Math II exam, Snow was accepted at Dartmouth, UC Berkeley, NYU, Vanderbilt, Wellesley, and the University of Miami. She chose Dartmouth where she plans to double major in Computer Science and Economics and eventually pursue a career in cyberlaw or cybersecurity.

Congratulations, Snow!