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Note: This story is adapted from our IMACS Math Enrichment Program for talented students in grades 1-8. The solution and analytical process are revealed at the end of this post, so don’t scroll down too far if you want to avoid the spoiler!
Max, The Lost Dog
Sofia, a bright IMACS student, has lost her dog, Max. Luckily, Max was wearing a collar when he was found by another IMACS student, Emily, who loves using her logical reasoning skills to solve mysteries. Tucked into the collar was a piece of paper with clues on where to return Max should he become lost. Follow the clues to help Emily find which house is Max’s so that Sofia can be reunited with her dog … and possibly make a new friend in Emily. (Since both girls take online math classes through eIMACS, they had not met each other yet.)
Clue 1: The number of the house where Max lives is the sum of two numbers, each of which is a power of 2 that is greater than 0 and less than 10. These two numbers that add up to equal Max’s house number can be the same, but they don’t have to be.
Clue 2: Max lives in Galois Gardens, a neighborhood represented by the following arrow diagram in which each dot represents a house.
The fork to the left is Leibniz Lane, the fork to the right is Abelian Avenue, and the house at the bottom is on Via Vertex. Here is how you interpret the arrows:
• The house number at the end of a red arrow is equal to the house number at the start of that red arrow plus 3.
• The house number at the end of a blue arrow is equal to the house number at the start of that blue arrow plus 4.
Clue 3: The biggest number for any house in the neighborhood is 15.
Clue 4: The number of the house where Max lives is a multiple of 3.
The Analytical Process of Solving the Mystery
Now that you’ve given yourself a chance to work through the problem, let’s go step by step to understand one way we can think about putting the clues together.
Clue 1: What are the powers of 2 that are greater than 0 and less that 10? They are 1=20, 2=21, 4=22, and 8=23. So we need to take one number from the set {1,2,4,8} and a second number from the same set {1,2,4,8} and add them together to come up with the list of possible house numbers. Remember that both numbers can be the same but do not have to be. We’ll leave the arithmetic to you, but we should all arrive at the following possibilities:
Clue 2: Although we don’t yet know any of the house numbers in the arrow diagram of the neighborhood, what can we figure out from the information given? One thing is the location of the house with the smallest number. It has to be the house at the bottom of the picture because the arrows tell us that every time you go from that house along either of the two roads, you add 3 or 4 each time. So you keep getting bigger and bigger house numbers, and they never fall back below the number you started with at the bottom, whatever that number is.
Likewise, is there anything we can say about the largest house number in the neighborhood? Well, it’s probably at the top of the diagram since the house numbers get bigger as you go up the roads. But is it the house on the top left or the one on the top right? Now we need some variables now to help us keep track of these houses.
Let’s call the house number at the bottom “A,” the one on the top left “B,” and the one on the top right “C.”
We know B is bigger than A, and C is bigger than A. Can we say anything about how much bigger B and C are than A? Well, to get from A to B, we add 3 and 3 and 4. That is, we add 10 = 3 + 3 + 4. So A + 10 = B. To get from A to C, we add 4 and 4 and 3. That is, we add 11 = 4 + 4 + 3. So A + 11 = C. We’re starting with at same number, A, and adding 11 to get to C but only 10 to get to B, so C must be bigger than B. Now we know that the house with the biggest number is on the top right.
Clue 3: This clue, along with what we figured out from Clue 2, tells us that C = 15. In other words, the address of the house at the top right is 15 Abelian Avenue. With a little arithmetic, we can now determine the rest of the addresses in the neighborhood as follows:
Now that we have all the addresses in the neighborhood, we can go back to our original list of possible house numbers from Clue 1 and cross off the ones that do not show up in the neighborhood. Here’s what our list should look like now:
Clue 4: We’re down to only three possible house numbers—4, 8, and 12. Clue 4 tells us that the house number is a multiple of 3. Which of these possibilities is a multiple of 3? Only 12 is a multiple of 3. Hooray! We figured out where Max lives—12 Abelian Avenue! Sofia will be so happy to have Max back at home, and Emily will be quite pleased with herself for having solved the mystery and done a good deed.
There’s no mystery as to who has the best online gifted math curriculum. IMACS! Register for our free aptitude test. Play along with our weekly IMACS logic puzzles on Facebook.
Show a child some tricks and he will survive this week’s math lesson. Teach a child to think critically and his mind will thrive for a lifetime.
Math word problems confound many students for a variety of reasons. Too often, well meaning parents and teachers attempt to help students struggling with these kinds of problems by offering them tricks or shortcuts for getting to the solution without necessarily understanding what’s going on. Just do an internet search for something like “how to solve word problems,” and you will get a slew of Web pages with tips, tricks and strategies like the following:
• Search for key words that will tell you which mathematical operation you should use (e.g., “lost” means subtraction).
• Cross out non-essential information so that you are not distracted by it.
• Draw a picture to illustrate each step of a problem.
Let’s consider these three examples for a moment. To suggest that looking for key words within the text of a math problem is somehow a “strategy” to solving it more easily seems a bit absurd. Basic reading comprehension skills are obviously essential to solving any problem, math or otherwise, where the information is communicated via natural language. No key word mapping is going to provide a fool-proof way of understanding what a word problem is asking. If you’re starting with a word problem that is well designed and written with clarity, there simply is no “trick” to understanding the words.
The other two examples sound reasonable on the surface, but a closer examination reveals problems with them too. To determine which pieces of information are relevant to the solution (as in Example #2), you need to be able to evaluate and analyze the given information. To correctly translate the words of a problem into a step-by-step illustration (as in Example #3), you need to be able to understand and prioritize the information. In other words, tricks like these seem to require the very skills needed to solve the problem without the tricks! Furthermore, a “tip” cannot tell you whether you have enough information to solve the problem, or how to find any missing information. Only critical thinking and logical reasoning skills can help you there.
This approach of relying on “clever strategies” to replace critical thinking runs counter to the IMACS philosophy of teaching children the fundamental skills they need to solve problems throughout their lives, whether they be elementary word problems or complex Calculus problems, whether it’s making correct change or correcting the flight path of an exoplanet-bound space craft.
Meaningful Strategies That Matter
While silly tricks may work for some students as a short-term fix to “just get through the problem,” there are practical and helpful techniques that should be part of any meaningful approach to understanding and solving word problems.
Pay attention to units of measurement. Word problems commonly feature units of measurement. When finding a solution, it’s important to pay close attention to how you are adding, subtracting, multiplying, or dividing amounts in the problem so that you don’t end up combining “apples and oranges.” If you carefully track the units of measurement through each step of your computation, you can compare the units of your answer to what it should be based on the question. This is a good technique to check the reasonableness of your answer rather than one that “magically” solves the problem for you. Here’s an example:
No one would think that the following approach is right
because adding children to cookies makes no sense! The answer requires a relationship in the form of “X cookies per child” – or, more algebraically,
With this in mind, what makes sense is the following:
While we’re on the topic of cookies, here’s another example:
Most people know to do something with ( 4,500 / 3 ) and 5, but what? Looking at the question in the problem, we know that our answer should be in the form of “X cookies.”
What if we set up the equation correctly
but then worked out our answer as follows?
Hmm … 300 cookies per minute per minute? The fact that our answer is not measured purely in cookies as expected tells us that we made a mistake in our calculation.
Use reverse word problems to practice abstract thinking. The process of solving word problems often requires us to connect a real-life situation described in language we know with the abstract version written with variables and equations. As with learning to speak a foreign language, thinking abstractly becomes more natural with practice and regular use, and the earlier you start, the easier it is.
When kids are younger, parents can help them to develop an intuition for abstract reasoning by doing “reverse” word problems in the guise of creative play. Start by writing down an equation such as 7 + ___ = 10. Next, ask your child to imagine a story that goes with this equation. He might say something like, “It takes 10 gold stars to get extra recess, and I already have 7. How many more stars do I need?” For 3 + (8 × 2) = 19, she might say, “When I started robotics, I took 3 introductory classes. Then, I took 2 classes per week for 8 weeks. So, I have taken a total of 19 classes.”
Be sure to keep the level of difficulty appropriate for your child’s math ability. Frustrating him with a computationally hard problem will only defeat the purpose of teaching him to think abstractly.
Word problems can be used effectively to teach important mathematical concepts and to help give real-world context and purpose to what may seem like useless hieroglyphics to some people. But without a firm foundation in critical thinking and logical reasoning skills, a student is sure to be at a disadvantage when trying to evaluate, analyze, prioritize, and synthesize the numerous pieces of information embedded in the narrative. Tips, tricks, and strategies may help at the margins, but there is simply no substitute for genuine understanding.
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Editor’s note: The material below is adapted from ‘Book 0 Chapter 1: Operational Systems’ of the Elements of Mathematics series. The EM series of secondary school mathematics textbooks is a 27-volume collection written and published by the IMACS Curriculum Development Group and serves as a basis for the high-level online math courses available through eIMACS, the distance-learning division of IMACS.
In our previous IMACS blog post on modular arithmetic, we introduced the Clock Game as a fun way to teach children about modular addition. Now, we’ll introduce a few variations on the Clock Game that make it more interesting. If you’re not familiar with the Clock Game, we suggest that you first read our introduction to modular addition before adding on with this post. Have fun!
The Double Game
In the double game, there are two pieces at each number at which to play instead of just one.
The game described in our introduction to modular addition can also be played as a “double game.” The only difference is that when playing double games, a player has more choices. For example, if, as a result of a particular move, a player has to place a checker at a number, one of whose circles is unoccupied and one of whose circles is occupied by the opponent, the player has the choice of either taking the opponent’s checker by replacing it with one of his or her own, or placing a checker on the unoccupied circle and leaving the opponent’s checker on the math. Double games, therefore, last longer, since all the available places must be occupied before the game is over.
Kings and Double Kings
Another variation of the games which can make them more interesting involves the use of kings and double kings. Only one place at each number is used in these games. In this variation, if the move requires it, you may place a second checker on top of one of yours already occupying a particular position, thus creating a “king checker.” If you have a king checker already occupying a certain position, then, if the move requires it, you may place a third checker on top of the king, thus making a “double king.” If a double king occupies a particular position, no further checkers may be added to the pile and so the play continues as in the simplest version of the game.
If your opponent would normally land on a place where you have a double king, then that double king cannot be removed, and the opponent cannot place a second checker. The hour hand, however, should still be moved so as to point at the position occupied by the double king even though your opponent could not place a checker there. If, however, your opponent lands on a place where you have a king, then both your checkers may be removed and replaced by one of your opponent’s checkers. The game ends when all of the positions are occupied by at least one checker upon the completion of a turn. This type of game is scored by counting 3 points for each double king, 2 points for each king, and 1 point for each single checker.
Kings, Double Kings and the Double Game
This variation utilizes the rules of the last two sections simultaneously. The first checker of each move must be placed on an unoccupied position, but there may be several choices for the second checker. For example, suppose your second checker has to be placed at “3,” your opponent has a king on one of the positions labeled “3,” and you have a checker (or a king) on the other position labeled “3.” You may either remove your opponent’s king and replace it with one of your own checkers, or you may form a king (or a double king) of your own. In such a situation, you may always choose either one of the two possibilities offered you by the two positions corresponding to the number at which your second checker is to be placed.
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Mathematics educator, Georges Papy, in 2009.
IMACS pays tribute to Belgian mathematician and mathematics educator, Georges Papy, who died in Brussels this past November at the age of 91. Georges and his wife, Frédérique, also a noted mathematics educator, were two of the seminal figures in the development of what eventually became the IMACS Mathematics Enrichment curriculum. We are deeply grateful to the Papys for their life’s work on behalf of mathematics students and their teachers.
The Early Years
Georges Papy was born in Anderlecht, a municipality in Brussels, Belgium of not quite seven square miles, on November 4, 1920. World War I had ended just two years before, and the country, which had suffered immensely under four years of German occupation, was still recovering. Against this backdrop and coming of age during the rise of Hitler, it is no wonder that young Georges was a member of the armed underground resistance forces in Belgium during 1941 – 45, serving in particular in the areas of intelligence and action. An educator from the start, he taught clandestine courses to students at the University of Brussels during 1941 – 42, and taught in a clandestine school for Jews during 1942 – 43 in the town of Méan, 50 miles to the southeast of Brussels.
After the end of World War II, Papy earned his doctorate in mathematics (with the highest distinction) from the University of Brussels in 1945, and was granted his advanced teaching diploma by the Science Faculty of the University of Brussels in 1951. After serving from 1949 first as Reader and then as Senior Lecturer at the University of Brussels, he became Professor of Algebra in the Science Faculty in 1956. The year before, he became a member of the prestigious Institute for Advanced Study in Princeton, New Jersey.
Finding His Purpose: Mathematics Education
Papy in 1970.
During the mid 1950s, influential academic mathematicians were leading efforts to improve the quality of mathematics education in France. In parallel with these developments, Georges became deeply interested in improving the quality of mathematics education at the secondary school level, and assumed a position of leadership in Belgium of what became known as the New Math movement. [This was inspired in part by a colloquium organized in 1959 in Royaumont, France, by the agency that in 1963 would become the OECD. A follow-up meeting held in Yugoslavia under the chairmanship of the American mathematician Marshall Stone gave rise to a secondary school mathematics curriculum that was published in Paris in 1961 under the name “Mathématiques nouvelles” (“New Math”).]
Papy’s work in mathematics education would accelerate quickly from that point. In 1961, he founded the Centre Belge de Pédagogie de la Mathématique (Belgian Center for Mathematics Pedagogy). From 1962 onward, Georges was called upon as an expert in mathematics education by several international groups, including UNESCO, IBM, and the OECD. During 1960 – 70, he served as president of the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (International Commission for the Study and Improvement of Mathematics Teaching), and was founding president of the Groupe International de Recherche en Pédagogie de la Mathématique (International Research Group in Mathematics Pedagogy), starting in 1970. Amid this flurry of activity, Papy even found time to serve as Senator in the Belgian government during 1963 – 64.
Modern Mathematics: A Crowning Achievement
Papy with a copy of Mathématique Moderne, Vol. 1, 1963.
During 1963 – 66, with the collaboration of his wife, Frédérique, Georges published the groundbreaking six-volume series entitled Mathématique Moderne (Modern Mathematics), which represented a fundamental reformation of the secondary school mathematics curriculum based upon the unifying themes of sets, relations, functions, and algebraic structures (such as groups).
His mathematical educational interests expanded to include elementary school education. Starting in 1967, under the auspices of the CPBM, Frédérique and an associate taught experimental mathematics classes to six-year-olds. In the years that followed, Papy and Frédérique published several volumes comprising what amount to annotated accounts of these experimental classes and those that succeeded them. The mathematical underpinnings of this elementary school work remained the same as those of Papy’s earlier secondary school work, prominently featuring the use of multicolor arrow diagrams to represent relations and functions.
The Papy Minicomputer
As part of this work at the elementary school level, Papy developed what became known as the Papy Minicomputer and published a related text, Minicomputer, in 1968. This is a two-dimensional, mixed binary/decimal abacus made of square boards subdivided into four squares, each color-coded with Cuisenaire rod colors. He attributed the design of the Minicomputer to some work by the renowned Belgian cosmologist Msgr. Georges Lemaître (who was the first to propose what he called “The Primeval Atom” but which Fred Hoyle disparagingly labeled “The Big Bang Theory”). In the mid-1950s, Lemaître had proposed the introduction of new digits to represent numbers. The digits were formed from lines and curves that revealed an underlying binary structure. Papy adapted this idea and transformed it into a two-dimensional board on which checkers may be placed.
The IMACS Connection
In 1969, Georges and Frédérique met IMACS founder Burt Kaufman at the first conference run by the International Commission on Mathematics Education in Lyon, France. At the time, Burt was the director of a federally-funded mathematics curriculum research and development project called the Comprehensive School Mathematics Program. He managed to recruit Frédérique as his Director of Research, a role that she fulfilled during 1973 – 78. Of course, this meant that Georges paid frequent visits to the United States. Consequently, he and his wife had a very significant influence in setting the direction for the development of what has now become the IMACS Mathematics Enrichment curriculum.
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IMACS alumna Natasha Chen on her way to visit the Institute for Advanced Study in Princeton, NJ.
Originally published in the May 1992 issue of The Computing Teacher, this essay by IMACS alumna Natasha Chen brought much needed attention within computer science education circles to the debate over teaching computer programming for beginners. Amazingly, almost 20 years later, the key issue raised by the article – how to teach fundamental computing concepts in a way that is effective, engaging and empowering – remain unresolved. Many students still find themselves in syntax-driven classes that focus on a short-term ability to “make apps now!” in the language du jour rather than being taught strong fundamentals that enable success throughout college and career, regardless of which programming language is used. At a time when the US faces a huge gap between technical job positions and qualified individuals to fill them and the UK is grappling with a need for true computer science in its national curriculum, IMACS revisits this issue with a renewed urgency.
A Bad Beginning In BASIC
My first experience in computer programming classes was an elective course in BASIC back in sixth grade. I chose that class because I thought that computers were powerful and capable of doing many interesting things. Electives usually have a reputation for being fun, but my classmates and I heard stories about the difficulty of this course and how only one or two kids ever got A’s. I thought to myself, “Maybe they just weren’t interested in computers. But I am, so how bad could it really be?”
Pretty bad! Forget learning anything that encouraged us to think, wonder and explore. We were asked to study the history of computers, memorize the names of hardware, and master the rules of syntax. We were sixth graders. We weren’t about to enter the high-tech world of programming. All we wanted was to see what neat things we could do with the computer. The class wasn’t difficult at all; memorization is hardly a challenge if you take the time to do it. There were so few A’s because no one cared to do busy work, and that’s all that was offered.
Swamped By Syntax
After my sixth grade BASIC experience, I never wanted to take another computer science course again. Of course, when you are eleven years old, G.P.A. and class rank don’t mean much to you. But by the time I was about to enter my junior year in high school, I started thinking about those things … and college … and the classes I needed to take. To avoid another BASIC nightmare, I decided to bypass Computer Programming I (BASIC) and go straight into Computer Programming II (Pascal). Pascal was different enough from BASIC to make me think that it had to be better. I found out that the improvement was far less than I had hoped. We jumped right into the syntax of Pascal: program (input, output), begin-end, etc. Even after two years of studying Pascal, I still can’t remember all the rules.
It’s like trying to interest small children in reading. You try to show how much there is to discover in books by reading with or to them. Maybe they will pick up some books on their own and then some more, and pretty soon they will have built up a library. But if you dump the library on top of them, ask them to memorize the Dewey Decimal System and then put the books back in order, by the time they finish – assuming they do – they won’t care to look at another book, much less read one.
One of the students from that CP2 had a very hard time with the syntax of writing information-processing programs, but when it came to the one graphics program we were assigned he was an absolute genius. I can’t begin to describe all the amazing things he could make the cursor do. But he was never recommended for the AP Computer Science class. By having no place for such a student, the computer education system may very well be tossing aside some of the best computing minds, simply because they don’t fit the mold of the curriculum whose rigidity derives from that of the languages used.
AP Computer Science: As Good As It Gets?
Five of us made it to the AP CS class and four more came from the CP3 (Fortran) class. We spent the first two days refreshing our memories of the syntactic rules that had evaporated over the summer. It seemed as if the purpose of CP2 was to teach us the rules and now, we had to remember them so that we could play the game – someone else’s game.
The last year was one of frustration. The most commonly heard outburst in our classroom was undoubtedly, “Stupid computer!” No matter what, it was always the computer’s fault. It seemed that my classmates’ programming strategy was to let the computer find their errors in the hope that this would somehow help them solve the problem. It never occurred to them that the computer is simply a testing ground for a well-thought-out idea. A human could conceivably go through an algorithm manually; it would just take an intolerably long time. For most of my classmates, this was their fourth year in the computer science education system, and all it had taught them was to rely too much on the computer and not enough on themselves.
The root of this problem lies in what is currently thought to be important for students to learn in computer programming courses, namely, syntax. Whoever is designing the high school computer science curriculum seems to think that, once students learn the rules of the language they are studying, having them write programs that demonstrate how those rules are applied will teach them what is important in computer science. As a student who has been through the system, who has had to waste the majority of my patience, concentration, and effort on keeping the syntax of my programs straight, leaving barely enough of these qualities to devote to solving the problem, I can tell you that such a belief could not be more wrong. This method of teaching is brainwashing. It is not like brainwashing or similar to brainwashing; it is brainwashing. It is a danger to computer science, sending out trained hackers instead of enthusiastic visionaries. Fortunately for me, I was given the opportunity to recover.
The author's original copy of "The Schemer's Guide."
Saved By Scheme
In ninth grade, my math class began an introductory computer programming course in Logo. My classmates and I looked forward to every other Friday when we studied Logo, not only because it meant a break from math, but also because working in Logo was a lot of fun, and it was easy. Unfortunately, that course was ‘squeezed out’ as far as my class was concerned by the pressure of our math courses over the next three years. During that time, however, the course had been rewritten using the programming language Scheme.
As a senior, I had a study hall period that I sometimes spent in my math classroom doing homework. It was on one of these days that I happened to overhear my math teachers talking about Scheme. I was already tearing my hair out in my Pascal class trying to learn something for the upcoming AP Computer Science exam – in fact, all I was learning was the page number of the reference section in our textbook, which I frequently consulted to see whether type declarations or variable declarations came first or to re-check how to declare a record for a linked list. Enticed by what I heard, I willingly gave up my study hall to come in four days of every week to learn Scheme on my own for no credit at all, using The Schemer’s Guide.* My reward was that I regained the enthusiasm and interest I thought I had lost six years earlier.
In the four months it took me to complete my course in Scheme, I learned more about computer programming than I had in my two years of Pascal. In less than five minutes after I began reading the text, almost everything I learned more than three years previously in our aborted Logo course came back to me. Five minutes, not the two days it took to recover from just one summer away from Pascal. There were hardly any rules of syntax to remember. Furthermore, throughout the entire four months, I never touched a computer. The ease of learning and using Scheme gave me such confidence in the programs I wrote that I didn’t feel the need for the security of a compiler to check my work.
Simple Language Makes Learning Complex Concepts Easier
The thing I liked most about taking this course in Scheme was that I knew that I was learning something. Every concept I had ever tried and failed to understand comprehensively in my Pascal class – searching and sorting procedures, recursion, processing binary trees – was made clear when I studied them in Scheme. These things occur so naturally in Scheme that I couldn’t help but understand. After mastering the concept, I could then go back into my Pascal class and easily master the code. The point here is that concepts like these are universal in computer science. After you understand them, then you can learn the rules of any language in order to encode them. But it doesn’t matter how well you have mastered the syntax of a language if you don’t understand the meaning of what you are typing or the reason why it works.
Recursion serves as a prime example. My understanding was very vague; I knew that something was done over and over again. But after seeing the first recursive program in the Scheme text, I understood what it was all about. When we took the test on recursion in AP CS, my whole class seemed to choke, with grades in the 60s and 70s, and the second highest score in the 80s. Thanks to my background in Scheme, I aced the test. I couldn’t quite believe it myself! For the first time, I fully realized that Scheme was not only easy to learn; it was also easy to learn from.
It is a myth to think that length and complexity make a program impressive – a prevalent idea among my AP classmates. Scheme code is clear and easy to understand. There is no need for pseudo-code. Thoughts go straight from your head to clear, simple code. The strange thing is that Pascal sets you up to fall into the trap of complexity and to disobey the laws of top-down design. As a procedural language, it enticed us into trying to do too much in one procedure simply because it could be done. With Scheme, I couldn’t write a function that had more than one purpose. It is as inherently top-down as it is inherently recursive.
Computer Programming Is Fun Again!
In my four months of studying Scheme, I not only covered and understood everything that had been presented in the AP Computer Science course, but went beyond that to study functional programming, data and functional abstraction, objected-oriented programming, and artificial intelligence. I still can’t believe all the amazing things I have learned in this short time. This is how I wanted to learn when I was in that BASIC class in sixth grade. I had to wait six years to do it, but it was well worth it.
I wrote this article because I don’t want another kid to have to go through the frustration that I did and not get anything out of it. I don’t want another kid’s enthusiasm snuffed out by a pile of library books. I want students who study computer science to be inspired to create their own game. Kids never liked rules anyway, and that’s all we are – kids.
*The Schemer’s Guide serves as the basis for IMACS‘ series of University Computer Science courses.
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It’s winter at IMACS headquarters in South Florida. What does that mean? Sunshine. Temperature in the 70s and 80s. New Yorkers at the mall. Gotta love it! With the holidays around the corner, you might be wondering what to get for the math or science lover in your family. So we asked members of our IMACS family to share some of their favorite books or movies for our Winter 2011 Staff Picks list. Instructors Guy Barmoha, Frances Keiper and Jeff Piskun offered some terrific recommendations. If you’re still in need of ideas, check out more favorites from our Summer 2011 Staff Picks blog post.
Guy Barmoha, Senior IMACS Instructor
Guy has a BS in Mathematics from Florida State University and a Master of Science in Teaching Mathematics from Florida Atlantic University, where he continues to serve as a Teaching Assistant. He has been a part-time instructor with IMACS since 1995 and has taught all levels of our Mathematics Enrichment courses as well as the summer Logic Puzzles course, a Computer Enrichment course, and an Electronic course.
Guy is an award-winning teacher who taught middle school math for 11 years, including the Great Exploration in Mathematics gifted program. He received the prestigious Edyth May Sliffe Award for Excellence in Teaching in 1997. In 2001, he was a finalist for Broward County Teacher of the Year and was named Broward County Middle School Teacher of the Year in 2004. For the past four years, Guy has been a Mathematics Curriculum Specialist for Broward County Public Schools, and before that, he spent two years teaching math to distance-learning students. As you can tell, Guy is one busy, well, guy! But he always manages to find time to teach at IMACS. As he puts it, “Teaching here is always a bright spot in my week – the interactions with the students, parents, and staff always make it feel like home. Not to mention the great curriculum!”
Guy’s survey response is below:
“There are quite a few books that I enjoyed reading, not all related to mathematics. Even though we are mathematics educators, we all understand the importance of literacy. It is hard for me to pick a favorite book, so I will choose two to discuss. One that has some sentimental value, especially since we are discussing IMACS, is called What Is the Name of This Book? by Raymond Smullyan. This is more a compilation of logic puzzles than a novel; however, there are surely stories told within the book. You may ask why I chose this book. Well, the answer is simple – this is the book that reminds me of learning to teach at IMACS. In this book, students have to solve logic puzzles to find their way around the Island of Knights and Knaves, where Knights always tell the truth and Knaves always lie. Many of our logic puzzles that we use at IMACS are based on the problems in this book and other Raymond Smullyan books. I can still see our late founder, Burt Kaufman, teaching from this book and showing all the cases and subcases that students have to consider to solve the puzzles.
The second book is named Innumeracy by John Allen Paulos. Since we mentioned how important literacy is, we should not overlook the importance of numeracy. This book discusses consequences of innumeracy. A story from the book that sticks with me is the one about the stockbroker. The story goes something like this …
… Suppose I told you that I could predict the movement of the stock market correctly for the next five weeks in a row. Would you trust me enough to be one of my clients? Most people would. Well what you may not know is that I gave the same deal to 2,048 people. I then told half of them that the stock market will drop, and I told the other half that the stock market will rise. After the first week, there are 1,024 people who believe I predicted the movement of the market. Of those 1,024 people, I tell half that the market will drop and half that it will rise over the second week. By the end of this cycle, I will have 64 people who will believe that I am knowledgeable enough about the stock market to be able to predict its movement for five weeks in a row. …
This book was full of interesting situations like the one mentioned above. This is why I enjoyed reading it and refer to it often.”
Frances Keiper, IMACS Instructor
Frances has a BS in Mathematics from Stetson University and an MS in Applied Math from the University of Central Florida. She has been teaching part-time for IMACS since 2003 and previously taught math at Broward College. Frances worked for IBM Federal Systems Division at Cape Canaveral, Vandenberg AFB, and in Houston on Space Shuttle ground support software and on an upgrade to Mission Control flight information systems. She’s also had a few international assignments with IBM in Melbourne, Australia, where she created software and hardware upgrades for various banks, and in Kuala Lumpur, Malaysia, where she managed a project for the state-owned telephone company.
Frances’s survey response is below:
“I loved Dune by Frank Herbert – the book, not the movie. I got totally lost in the fantastic but somehow believable world Herbert created and just hated it when I finished the book. (I recommend skipping the sequel.) Herbert created a physical world full of sand containing creatures perfectly suited to that environment such as giant worms that travel rapidly over and through the sand. Then he populated the world with people who had their own elaborate social order and customs, again perfectly suited to that world but unlike anything we know on earth. The magic of the book is that it is so rich in detail and written so vividly that it becomes very real to the reader. You are drawn into this made-up universe. I don’t believe, even amidst the mountains of details regarding the physical and the social systems of Dune, that there is a single bit that is illogical. It all fits together so beautifully. You just have to believe.
For a more recent favorite, I really liked The Disappearing Spoon by Sam Kean. It’s full of quirky tales from the Periodic Table.
I loved the play Proof, which I saw at American Heritage Schools’ Mosaic Theater. And I liked the movie, too. It’s about a famous mathematician who is working incredibly hard to develop a proof, something that will be acclaimed the world over. He is an old man in the story and dies. His daughter, a mathematician in her own right, cares for him and grieves at his death. At first you’re not sure if the old mathematician was a genius or crazy. His behavior was bizarre. After a while, you’re convinced he was crazy.
Then a young math student shows up and trolls through the old man’s notebooks looking to find or maybe steal his brilliant but maybe non-existent proof. The twist toward the end is that the student finds the brilliant proof, but the person who developed it is the old mathematician’s daughter! I especially loved the fact that the hero is a FEMALE mathematician!
I almost never miss checking in to see what topics NPR’s Science Friday covered during it’s most recent broadcast. And I also really like an Australian call-in science show from Triple J Youth Radio. It’s called Dr. Karl. Dr. Karl himself is an enthusiastic and engaging science guy who can explain anything in simple terms and will readily admit that he’s not qualified to answer when he isn’t!”
Jeff Piskun, Senior IMACS Instructor
Jeff has a BS in Mathematics from Villanova University and an MS in Sports Administration from St. Thomas University. He started with IMACS in 2003 and is currently one of our part-time instructors. Jeff also teaches middle school math. A big sports enthusiast, he worked for over six years in sports and entertainment venue management for several professional teams, including the Florida Marlins, Miami Heat, and Florida Panthers.
Jeff’s survey response is below:
“Two books I would recommend are Freakonomics and Super Freakonomics by Steven Levitt and Stephen Dubner. These two economists use basic economic principals and basic statistics to look at real-life situations and investigate the hidden side of everything. The books are written in a fun but eye-opening manner and explain things like: why charging parents late pickup fees will actually increase late pickups not deter them; why your realtor actually does not have your best interest at heart; how walking drunk is more dangerous than driving drunk (not endorsed!); that many government programs actually have the opposite effect than intended (shocker). Parents should note that the books address some adult topics, so beware, but they are really an eye-opening read.
I enjoy movies that are similar to the original Stargate with Kurt Russell and James Spader. I like anything with codes and ancient secrets and the science/adventure/historical fiction genre overall. I’m intrigued by the idea that even though certain events are part of “history,” we still do not know everything about what occurred and why. Many many years later there are still mysteries and puzzles that need to be solved and theories to be investigated. Codes, puzzles, and mysteries based on historical events and real data are exponentially more fascinating than fictional ones.”
Guy, Frances and Jeff – Thank you for the awesome recommendations!
To our readers – Thank you for making our blog a part of your online experience. We sincerely appreciate your time, comments and feedback. Our next post on December 22nd will be a classic from the IMACS vault. All the best to you and your families for a wonderful holiday season and happy and healthy New Year!
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Milo Gardner at US Army code breaking school, Fort Devens, 1957.
Milo Gardner may be retired, but he’s not letting his mind rest idle. This amateur code breaker chatted with IMACS about his mathematics background and how it influenced his life. After serving in the United States Army as a cryptanalyst, Milo went on to earn his B.A. in Mathematics with a minor in History of Economic Thought at California State University – Chico. What followed was a career in aerospace engineering and an M.B.A. from CSU – Fullerton. Years later in retirement, Milo’s curious mind, a letter to the editor, and a meeting at a coffee shop led him to his current passion – decoding ancient Egyptian math texts. IMACS blog readers can review his extensive list of online publications on the subject. Now let’s hear from Milo in his own words.
Please tell us about your mathematics background, including what interested you as a child.
Baseball and sports of all types filled my youth. In the 8th grade I was sports editor of our school newspaper. This meant computing box scores and other game statistics for the various teams and writing up short narratives. Several sports teams were enjoyed during high school. Basketball was my favorite. Math was a special high school interest, as well as Spanish, chess and the sciences. Algebra I, Algebra II, Geometry and Trigonometry were studied in the context of set theory. That is, math students determined in advance which number system(s) would solve particular problems.
Right out of high school, the US Army was joined with hopes of spending time in Europe. My high school language skills were tested first. Two entrance exams for the Defense Language Institute at Presidio of Monterey, California, were taken. Luckily, the second exam score was low. The Army’s code-breaking tests followed, which were passed. In a five-month code breaking school, language and number-based patterns were studied. Techniques identified one-to-one, two-for-one, and other substitution systems. Finishing third in a class of 20, Germany was selected as my two-year assignment. Lower ranked students were sent to Africa, Turkey, Korea and Japan. In Germany, ad hoc Russian language projects were assigned to cryptanalyst and linguist teams. A two-month side trip to Lebanon widened my foreign language exposure to include Arabic.
Upon discharge from the Army, a number of my co-workers went to work for the National Security Agency (NSA) in Washington, D.C. I went to college in Northern California and earned a Mathematics degree with a minor in History of Economic Thought, with an intention of applying to the NSA. Computer programming courses were taken, as I was preparing to be a high school math teacher as a back-up career. Upon graduation, wishing to be married and raise a family, a southern California aerospace engineer accepted. The pay was better than teaching. Equally important the location kept me and my wife-to-be close to both of our families.
What kind of work did you do in the aerospace industry?
The first aerospace position was at Vandenberg AFB as a systems analyst. Western missile test range issues input computer-generated data to hand drafted range safety charts. The charts factored in daily wind measurements and drag aspects of missile parts that allowed three-second delays for the missile safety officer to blow up errant missiles. Nearby populations were protected. Subsequently, college programming skills were applied working with a team that automated the hand drafted charts.
The second aerospace job was at Rockwell International in Fullerton, California. Failure patterns of Minuteman I, II and other avionics guidance systems were studied in terms of maintainability issues. While at Rockwell, evening business classes were taken towards an MBA degree. Upon graduation, a career change allowed our family move to northern California to be near our respective families.
You raised a daughter who went on to become a civil engineer. What advice can you offer to parents of mathematically talented girls on how to nurture their talent?
Missy, my CE daughter, was self motivated by the second grade. All three of my children played musical instruments. Parental guidance consisted of coaching youth softball and baseball teams up to age 12 and supporting each child’s active social life. Thereafter, tryouts for competitive teams were arranged. Missy made a traveling team at age 14 in an organization that her older sister played. My wife and I attended most of Missy and her sister’s weekend tournaments. Our children knew their parents were their biggest supporters.
Missy was strong willed and stood up to basketball, softball and volleyball coaches in high school. Her sports experience gave her confidence to confront one engineering professor in college who refused to call on women in the class. After receiving a failing grade at mid-term, she called a conference with the professor and asked why he had not taken even a single question from one woman in the class. Her strong position was understood and respected. Thereafter, in-class questions were answered sufficiently for her and other women to earn passing grades. Today, she is an engineer working for a private firm. Incidentally, she told us this story weeks after the situation had occurred. Missy had the confidence to handle it herself.
Milo with his wife Bertha and grandson Chris, 2005.
In retirement, you now pursue code breaking as a hobby with a specialization in ancient Egyptian math texts. How did you become involved in these pursuits?
In 1962, an upper division college history of math class told a fuzzy history of zero story. Zero, as we know it today, did not reach Germany until 1200 AD, in time for the birth of our base 10 decimal system in 1585 AD (that defined n^0 = 1), an unbelievable assertion. “Some day I’ll research that topic,” I told myself. In 1988, six months were spent studying the topic at two local university libraries. Medieval and older Near East numeration systems including Classical Greeks used zero, a round figure topped by two dots in clear ways. Babylonians and Egyptians used zero as a limit 1,500-2,000 years earlier. Zero was also the value of empty sets in an Egyptian double entry accounting system. The older uses of zero did not use placeholders. Care had to be taken to read the context of mathematical documents and inventories –- issues that I knew well as a military code breaker.
After completing the study of the longer history of zero, an unexpected event took place. Acting on the dehumanization of classroom math topics (e.g., omitting personal stories like how the pre-teen Gauss summed the addition of 1 to 100 by a formula), I wrote a letter to the editor of the Sacramento Bee on the weaknesses of the 1990 California Math Framework. The day after the letter was published a phone call from a retired electrical engineer was received. Noel Braymer asked that we meet at local coffee shop. I said yes, and my retirement world changed for the better.
Noel had worked on 1650 BCE Egyptian text called the Rhind Mathematical Papyrus (RMP) for 15 years. A 50 member RMP 2/n table was encoded. The table took up 1/3 of a papyrus that contained 87 other problems. Noel offered a modern number theory solution to the 2/n table. Number theory stresses prime numbers in ways that ancient and modern mathematicians parse divisors of composite numbers into primes. The ancient 2/7 was recorded as 1/4 + 1/28. What set of ancient rules were used in the entire 2/n table?
The 2/n table encoded 2/3, 2/5, …, 2/101 to concise unit fraction series in ways that scholars hotly debated during the 20th century. Robin-Shute published the Rhind Mathematical Papyrus text in 1987 and suggested one incomplete solution. Noel gifted the book to me, and asked if I would assist in publishing his work. I said yes, provided the ancient scribal methods were also decoded and published.
Ten years later, working with a linguist, a sister document to the RMP, the Egyptian Mathematical Leather Roll (EMLR), was decoded with modern number theory. Attempting to explain connections between the two documents, aspects of the RMP 2/n table construction methods slowly emerged. Another seven years passed. Finally, by considering Egyptian wages paid in commodities, a 2011 paper included a complete solution to the 2/n table problem as an appendix.
In your opinion, what mix of interests and skills makes a person well suited for a career in cryptography? What should kids who are interested in code breaking be doing now to prepare themselves for a career in this field?
Students should enjoy solving all types of puzzles. I have loved crossword and other puzzles since high school. Learn a foreign language and learn about new and old foreign mathematical issues. Enjoy competitive individual and team games. Chess and bowling took up much of my free time. Choose your games and puzzles wisely. To pursue a puzzle solving career, a student should learn to contribute to all sorts of teams. Team membership is an important skill in many aspects of the adult world.
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Editor’s note: The material below is adapted from ‘Book 0 Chapter 1: Operational Systems’ of the Elements of Mathematics series. The EM series of secondary school mathematics textbooks is a 27-volume collection written and published by the IMACS Curriculum Development Group and serves as a basis for the high-level online math courses available through eIMACS, the distance-learning division of IMACS.
Consider a clock face such as the one shown below:
If the clock shows 10 o’clock, what time will it show 11 hours later? If the clock shows 5 o’clock, what time will it show 12 hours later? If the clock shows 6 o’clock, what time will it show 6 hours later? If you answered 9 o’clock, 5 o’clock and 12 o’clock, respectively, then you’d be right. This exercise suggests a new kind of arithmetic. Let’s call it “clock addition” or, because this kind of addition involves just the numbers 1 through 12, “Clock 12 addition” and denote it by +12. Then the following statements correspond to the three questions above:
10 +12 11 = 9
5 +12 12 = 5
6 +12 6 = 12
Notice that the number 12 plays a very special role in Clock 12 addition. When you add 12 to any of the numbers from 1 to 12, the number remains unaffected. For example, 5 +12 12 = 5, 12 +12 12 = 12, etc.
In ordinary addition the number 0 has this property. This suggests that it might be convenient to replace the “12″ on our clock face by a “0″ so that our clock face looks like this:
With this slight change from the usual clock face, we can now summarize the situation as follows: Clock 12 addition involves the twelve numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11. We denote this set of the numbers by Z12 (read, “Z sub 12″). Here is the rule for computing x +12 y for any numbers x and y in Z12.
For all numbers x and y in Z12, x +12 y is equal to:
x + y, if x + y is less than 12, or
x + y – 12, if x + y is greater than or equal to 12
In plain English, that means: To find the Clock 12 sum of x and y, first find the ordinary sum of x and y. If this ordinary sum is less than 12, you are finished – the Clock 12 sum is just the ordinary sum. On the other hand, if the ordinary sum is greater than or equal to 12, then subtract 12 from the ordinary sum to get the Clock 12 sum.
Notice that in any case, as long as x and y are in Z12, x +12 y will also be in Z12. (Can you prove that?) Since for any numbers x and y in Z12, the Clock 12 sum, x +12 y, is again in Z12, we call +12 an “operation” on Z12. Similarly, ordinary addition is an operation on the set of all natural numbers, N, which consists of the numbers 1, 2, 3, … . On the other hand, ordinary addition is not an operation on Z12, and ordinary subtraction is not an operation on N. (What are some examples that support this statement?)
We have been calling +12 “Clock 12 addition” because it’s very closely related to the kind of adding you do on a real clock. The operation +12 is also called “addition modulo 12″ or “addition mod 12″ for short.
Clock 12 Addition Game
Now we’ll describe a game called “Clock 12″ that two players, A and B, can play on a circular board. You’ll need a movable arrow for the hour hand so that your game board looks like the figure below.
Player A begins the game by placing a checker on one of the twelve numbers around the edge of the clock face. The hour hand is placed so that it points at the number on which the checker has been placed. The corresponding time is called “the hour.” For example, if Player A puts a checker on 10, the hour hand is placed so that it points at 10. Player A then says “The hour is 10 o’clock.”
Player B makes a move in two stages: first by placing a checker of a second color on one of the unoccupied numbers, and then by placing a second checker of the same color in such a way that the following is true: The starting hour plus (mod 12) the number of hours corresponding to Player B’s first checker is equal to the time corresponding to Player B’s second checker. The arrow is then made to point at the number on which Player B’s second checker was placed, thus indicating the new hour. Player B then declares the new hour.
Example 1
Suppose Player A begins the game by placing a checker on 10. Then the hour is 10 o’clock. If Player B chooses 4, then his or her first checker is placed on 4, and the second checker is placed on 2 because 10 +12 4 = 2. The hour hand is then moved to 2 o’clock.
Example 2
Suppose Player A chooses 0. If Player B chooses 4, the first checker is placed on 4. In this case, 0 +12 4 = 4, and Player B already has a checker on 4. Therefore, there is no need to place a second checker on 4, and so Player B’s turn ends with the hour being 4 o’clock.
Example 3
Suppose Player A chooses 10 and Player B chooses 0. Then, since 10 o’clock plus 0 hours is 10 o’clock, Player B’s second checker ought to be placed on 10. However, Player A’s checker is already on 10. In this situation, Player B takes Player A’s checker and replaces it with one of his or her own. The hour hand, which was pointing to 10 before Player B’s move began, will not be changed. The hour is still 10 o’clock.
The players continue to alternate turns using rules described in the examples above. If, after completing his or her turn, a player leaves the board so that there are no more unoccupied numbers, then the game is over and the player with the most checkers on the clock wins!
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The benefits of math competitions are well known: focusing on goals, dealing with pressure, learning teamwork, and building friendships are among those typically mentioned. Math competitions also allow for the much needed celebration of intellectual achievement the way athletic achievement has always been celebrated. At IMACS, many of our math enrichment students enjoy competing in the American Mathematics Competitions, International Mathematical Olympiad, and MATHCOUNTS among other contests, so we have a very positive view of math competitions, particularly for the kids who thrive in that environment. We’re extremely proud of our numerous students over the years who have performed well in these prestigious contests. We’re equally proud of our numerous students over the years who have thrived in quiet contemplation.
This brings to mind an article published last week by The Wall Street Journal on the challenges encountered by Type A parents raising Type B kids. The article talked about different ways that ambitious, competitive, and hard-driving parents modify their interactions with or expectations of their dreamy, mellow, and seemingly laid-back children to foster healthy parent-child relationships. Whether Type A or not, most parents are cognizant of today’s ultra-competitive global environment, and many feel a sense of urgency to nudge, push, or even pressure their kids to achieve. It’s not a stretch to imagine the parents of a mathematically talented child thinking, “If all the other kids are involved in math and science, then my child should be doing even more.” Here’s the thing: What makes an activity suitable for even more depends on what works for your child and is not necessarily the same activity enjoyed by all the other kids but with more time dedicated to it or with better results.
For parents considering activities for their mathematically talented child, it is important to understand how innate personality factors into the mix that determines whether math contests provide a net positive experience for that child. Just as there are natural-born competitors among mathematically talented students, there are also natural-born dreamers. These kids used to get a bad rap for being unfocused, undisciplined, and even lazy. There was no observable productivity associated with daydreaming so, of course, it had to be a waste of time. Not so fast. In 2009, researchers from the University of British Columbia published a study in the Proceedings of the National Academy of Sciences detailing that brain activity increases when our minds wander. Reporting on this finding, ScienceDaily.com put it well: “[B]rain areas associated with complex problem-solving – previously thought to go dormant when we daydream – are in fact highly active during these episodes.” And from The Wall Street Journal: “These sudden insights … are the culmination of an intense and complex series of brain states that require more neural resources than methodical reasoning.”
If your child is more of a dreamer (or just not drawn to competition), activities that would cultivate his or her talent in and appreciation for mathematics may differ from what is offered through a typical regional, national, or international math contest. Students of this personality type often find more success and satisfaction with math enrichment programs that focus on deep problem-solving over computational prowess. This is not to say that your child shouldn’t at least try participating in some type of competitive math. The experience just might open up a different side of his or her mathematical personality. In fact, our math enrichment classes use game-playing and mini-competitions as teaching tools, and most of our students really enjoy this aspect of the class the best. But if it’s clear that competition does not bring out the best in your child, we encourage you to explore other options including letting your child have more free and unstructured time to let his or her mind ruminate about the wonders of mathematics.
For parents of talented children, the secrets of mathematical success are really not that different from general advice on positive parenting. They include understanding what kind of child you have, knowing what motivates him or her, and fostering an environment that includes the kind of math activities or, quite possibly, freedom from structured activities that align best with that motivation. And if having a dreamer for a child still makes you worry, just think about the great mathematical and scientific discoveries we owe to dreamers of the past. You never know what grand ideas are simmering behind those eyes staring off into the distance.
Dreamer Matching Puzzle
We made a simple matching game out of the Wall Street Journal article that reported on the aforementioned University of British Columbia study. See if you or your kids can match the great thinker with his profound idea and what he was reportedly doing at the time moment of insight. Answers may be found in the article.
Great Thinker: (A) Archimedes, (B) Newton, (C) Einstein, (D) Descartes, and (E) Tesla.
Profound Idea: (1) special relativity, (2) coordinate geometry, (3) alternating electrical currents, (4) calculating the volume of an irregularly shaped object, and (5) law of universal gravitation.
Dreamy Activity: (i) lying in bed watching flies on the ceiling, (ii) taking a bath, (iii) watching an apple fall from a tree in an orchard, (iv) taking a walk, and (v) imagining trains and lightening.
Editor’s note: Regular readers of this blog will notice a slight change going forward. IMACS will be switching to a bi-weekly publishing schedule with our next post appearing on November 10, 2011.
Whether a dreamer or competitor, IMACS has a place for you. Sign up for our free aptitude test. Play along with our weekly IMACS logic puzzles on Facebook.
Chess champion and IMACS student, Rachel Gologorsky, enjoys tournament competition.
My earliest memory of chess was from when I was around three years old. I remember sitting on the edge of a rug playing with colorful LEGO blocks. When I looked up, I saw my father and my oldest brother playing chess. The board was illuminated so brightly that it looked as if the pieces were shining. I looked back down at my blocks, and they seemed so boring compared to the multitude of pieces and squares and interesting stuff happening on the board above. That was when I was first aware of my desire to play on that board and with those pieces. I would crawl over to where my father and brother were playing, take a few game pieces, and put them on the board too. After “playing” with them a few times in this way, I was henceforth banished from the vicinity of their games. This continued for two years until my mother determined that I had the capacity to understand what was going on. However, nobody believed her, so she taught me the basics herself. It took me about a week to grasp those rudiments of chess, after which I was handed over to my father for further instruction.
My father and I would take a chess set and chess book from our library at home and bike to one of our favorite places. There, my father would set up a position from the book on the board, and we would puzzle it out together. In the beginning, we would actually move the pieces, but after a few weeks my father urged me to work out the entire solution and move the pieces only in my head. We studied tactics, strategy, endgames, etc. I gradually improved to a stage where I could have a “reasonable” game with him (“reasonable” in the sense that I wasn’t losing after the fourth move). A few years later, when I was eight or so, I learned that there was a chess club at a local elementary school, so naturally I went to check it out. There I found good competition with kids my age and with one boy in particular. We were about evenly matched, so when he went to a tournament and won a big trophy, I wanted one too! That’s how I got started with tournaments.
For a few years though, competition took a back seat to other important family events, which had the strange habit of conflicting with tournament schedules. School was also a factor: It would have been difficult to actively study chess and play in tournaments given all the classwork and homework I would have had to make up for the days I missed. Also, tournaments usually run through Sunday, and my Sundays were taken up with IMACS. I loved IMACS so much that I wasn’t willing to skip even one class if I could help it. When I was in third grade, my parents withdrew me from the school I was attending for a variety of reasons, including lack of a challenging curriculum that kept me perpetually bored even though I skipped a grade. I was homeschooled while my parents searched for another school, and I recognized this as an excellent opportunity to have a say in my education. I was on my best behavior for months before I convinced my parents to homeschool me forever! Now that I was homeschooled, I had loads of free time. Also, my IMACS classes were moved to Thursday. So, now that I had the time and was not sacrificing IMACS, I got interested in competitive chess again.
From my friends at tournaments, I found out about the Internet Chess Club (ICC). With my new ICC membership, I played for hours after my studies were done (which at 3rd grade took a grand total of an hour and a half). My parents always encouraged me to play up (i.e., at a higher level), so already in elementary school I was regularly playing up in K-12 events. This wasn’t very good for my rating as I usually lost many games, but as a consolation, from age 8 on, I was Florida’s Top Girl at the K-12 level for several years in a row. At that time, Women’s World Champion Grandmaster, Alexandra Kosteniuk, was handing out the Florida’s Top Girls prize. After a few years of shaking hands with her, my mom asked her to coach me. I improved dramatically under Alexandra’s guidance – I developed an opening repertoire, whereas before I usually made up my own openings (that sometimes didn’t turn out well). As my rating shot up, I made the Susan Polgar’s National Team for Girls and was invited as the representative from Florida to play in her 2009 National Invitational for Girls event. That was my introduction to girls-only events. I took second place, and made new friends. My mom is always checking the Web site for FIDE (the World Chess Federation, or Federation Internationale des Echecs). There she found out about the 2010 North American Youth tournament. The fact that one of my new friends was going to attend helped seal the deal. We were both US representatives, and it felt awesome to tell people that I was representing the United States of America. A series of wins and draws took me to the final round where I ultimately won! And so in my first international tournament I got my first gold medal for the US. As I hadn’t been having great results before that tournament, the win was welcome and kept me studying chess even with my increasing workload. (Studies at my age now aren’t as easy as in 3rd grade!)
My coach, Alexandra, started a family, which meant that she rarely traveled from her home in Russia to the US anymore. Thus, my parents decided to choose a different coach to fill in the gaps – four-time US Champion and Grandmaster Alexander Shabalov. Under his guidance I won the 2011 US Girls Junior (U21) Chess Championship this past August with an undefeated score. Whenever I’m feeling low, there always comes a win to keep me motivated. Chess is an amazing game, and I’m very competitive, so that also always brings me back to competitions. I love playing and seeing how I stand against some of the big names in chess (or more often, those who say they’ve played them). My favorite competitions are the one day tournaments, because I get to go home quickly. The longer 6-7 day tournaments, like the Susan Polgar, North American Youth, and the US Junior Girls leave me homesick toward the end. (For how long can one eat hotel food?) Now I’m in a lull between tournaments, so I practice by playing on ICC and chesscube.com for about an hour each day. My next big tournament will be the World Youth Chess Championship in Caldas Novas, Brazil, this November where I will represent the United States.
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