New Geometry Course for Talented Kids


September 27, 2017 Filed under: Curriculum Development,Gifted And Talented,Online Classes,STEM Education IMACS Staff Writer @ 5:00 pm

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.

EMF is a self-paced, online program designed for mathematically gifted middle school students.

Euclid of Alexandria (fl.300 BC)

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.

EMF is a self-paced, online program designed for mathematically gifted middle school students.

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.

To learn more about the Elements of Mathematics: Foundations (EMF) program and how to gauge whether it would be a good fit for your mathematically talented child, review detailed information on the main EMF website, have your child take the FREE online aptitude test, read blog posts here and here, read reviews, and watch our video.

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Educational Summer Camp Matters: Three Reasons Why Exercising the Mind is Important


March 23, 2017 Filed under: Computer Programming,Logical Reasoning,Robotics,STEM Education IMACS Staff Writer @ 1:00 am

A fun and educational summer camp featuring electronics, computer programming, virtual robotics, and logic puzzles.

The Institute for Mathematics and Computer Science (IMACS) believes in making sure kids exercise their minds this summer, in addition to their bodies, with fun yet educational summer camp experiences. Spending summers having fun and being outside is definitely important, but a small dose of educational summer camp is essential for all students too. Here are three reasons why:

1. Retain Knowledge: Summer vacation can often lead to forgetfulness and overall loss of learning. According to a study by the RAND Corporation, students lose, on average, one month of learning during the summer, all students lose some learning in math, and summer learning loss is cumulative over time. When kids aren't working certain parts of their brains during summer, they end up spending the first few weeks of the academic year refreshing what was lost. Educational summer camp activities keep those areas of the brain active so that children are ready to engage in new learning when school begins again.

2. Develop New Interests: Educational summer camps are a great way to explore new academic areas that kids don't have time for during the year because of school commitments and extracurricular activities. IMACS Hi-Tech Summer Camp, for example, often sparks the interest of kids who never considered math, electronics or virtual robotics as something they would enjoy. This is especially true for girls who often don't get enough exposure to these kinds of activities. At IMACS, we have had a number of students realize they want to study engineering after being immersed in hands-on projects and the logical and creative ways engineers think at our Hi-Tech Summer Camp.

3. Make New Friends: If your child enjoys fun, academic challenges in mathematics, computer programming and gaming already, they will not only have the opportunity to advance their skills, they will meet other kids their age with similar interests. Educational summer camp attendees are smart, fun, and enjoy a little friendly competition. It's a truly unique opportunity for your child to connect with other kids who appreciate their unique way of thinking. It's also a time for kids to be inspired by instructors who are genuinely passionate about their field.

Educational summer camp should be a part of every child's summer activities. It helps stop summer "brain drain", exposes kids to new ideas and pursuits, and leads to great friendships and memories. When choosing an educational summer camp, be sure to ask if the camp is staffed by high school and college kids or by highly-qualified and experienced instructors. After all, these kinds of camps should involve real thinking in addition to real fun!

IMACS Hi-Tech Summer Camp, with locations in South Florida and North Carolina, does for the mind what sports does for the body. Reserve your spot at one of our Open Houses and come for a chance to win a FREE week of camp! Not near an IMACS camp? Summer is the perfect time to try one of our outstanding online courses in mathematics and computer science.

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The Multiplied Foundation and IMACS Provide Over $10,000 in Scholarships for Hi-Tech Summer Camps


July 28, 2016 Filed under: Gifted And Talented,Meet IMACS,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
Peyton Robertson visits with one of the scholarship recipients
he is helping to sponsor at IMACS Hi-Tech Summer Camp.

The Institute For Mathematics and Computer Science (IMACS) has received a grant of $5,000 from the Multiplied Foundation Fund of the Community Foundation of Broward to provide full scholarships worth over $10,000 for 20 students to enroll in IMACS' 2016 Hi-Tech Summer Camp.

The Multiplied Foundation was founded by 14-year old IMACS student, Peyton Robertson, with the mission of supporting and expanding STEM (science, technology, engineering and math) education. Peyton has a deep appreciation for how early exposure to enriching STEM activities can motivate a young person. At 11 years old, he won the Discovery Education 3M Young Scientist Challenge. By 14, he was awarded three patents.

Peyton credits his academic accomplishments, in part, to the strong foundation in mathematical thinking that he developed while attending IMACS: "The early skills I developed at IMACS helped me to have a deeper understanding of the math and computer science classes that I have taken in school. My hope is that other students will benefit from the foundation that the IMACS program can provide."

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
Peyton with IMACS President Terry Kaufman.

"IMACS is honored to be working with the Multiplied Foundation to provide scholarships to 20 very deserving students," said IMACS President Terry Kaufman. "Bright and curious minds come from all backgrounds, and we all need to do more to identify and nurture these kids. We thank the Community Foundation of Broward for making this opportunity possible."

Camp scholarships were awarded to rising 4th through 9th graders who have a desire to build their math and logical reasoning abilities but who would otherwise not have the resources to attend. Recipients were selected from applicants at Piney Grove Boys Academy (PGBA) in Lauderdale Lakes and "I Have A Dream" Foundation in Miami.

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
Scholarship recipient James Wilson III enjoys the Computer
Programming & Virtual Robotics class at IMACS.

James Wilson III, a rising 5th grader at PGBA, is excited to attend the camp. "Every day we get to do a cool project and learn something new. I can't wait for tomorrow," exclaimed James. "It's great to see my son, who is very athletic and into sports, also be so intrigued and interested in technology thanks to his time with the program," observed James's mother, Melissa Mata. "The exposure he's getting at IMACS is definitely priceless."

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
Administrators from Piney Grove Boys Academy thanked Peyton
for providing this opportunity for their students.

Frances Bolden, Educational Administrator at PGBA, is also impressed with IMACS: "I could tell from meeting the staff and touring the facilities that IMACS is about challenging students through the latest technology to expand their knowledge to a new level." She added, "Everyone gave us a warm welcome, and I left knowing that our students were in good hands."

IMACS Hi-Tech Summer Camp program consists of logic puzzles, computer programming, virtual robotics, electronics, and an element of competition. Working solo and in teams, kids learn how to think logically and creatively while having fun.

About the Multiplied Foundation
The Multiplied Foundation's mission is to support and expand STEM (science, technology, engineering, and math) education. The Multiplied Foundation was founded by Peyton Robertson and seeded with the $100,000 he won during the 2015 Pebble Beach Pro Am's Chip Off Challenge. Each year, the Multiplied Foundation distributes 5% of its 12 quarter rated average value to organizations supporting STEM education. For more information, visit multipliedfoundation.org.

About IMACS
The Institute For Mathematics and Computer Science is an independent teaching and educational research institute focused on helping students reach their highest potential in math, computer science and logical reasoning. For more information, visit imacs.org.

About the Community Foundation of Broward
Founded in 1984, Community Foundation of Broward helps families, individuals, and corporations create personalized charitable Funds that deliver game-changing philanthropic impact. We provide leadership on community solutions, and foster philanthropy that connects people who care with causes that matter. Our 450 charitable Funds represent $173 million in assets and have distributed $89 million to create positive change. For Good. For Ever. For more information about Community Foundation of Broward, visit cfbroward.org or call
954.761.9503. Connect at #cfbroward @cfbroward

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Demonstrating Understanding in Mathematics


November 19, 2015 Filed under: Critical Thinking,Curriculum Development,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.

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!

IMACS students develop the creative problem-solving skills needed to put their knowledge to use in novel situations. Give your child the advantage of a genuine understanding of mathematics with our Mathematics Enrichment classes and online Elements of Mathematics: Foundations courses. Like IMACS on Facebook for the latest information about our local classes and online courses.
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IMACS Celebrates Ada Lovelace Day!


October 13, 2015 Filed under: Parenting,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
IMACS alumna Rachel Cuozzo (MIT), Rachel Gologorsky (Harvard), Sarah Thomas (Stanford) and Shakthi Shrima (Princeton) embody the spirit of Ada Lovelace Day.

October 13, 2015 is Ada Lovelace Day, a day to honor the achievements of women in science, technology, engineering and math. IMACS asks that you join us in celebrating this day by encouraging a girl to pursue her interests in the STEM subjects.

She may be your daughter, sister, student or friend. She may be enthusiastically expressive about her love for STEM, or she may be the quiet type who will share deep thoughts if you ask. Or she may be especially in need of your encouragement because she’s not yet received that message or, worse yet, has been actively discouraged from pursuing her passion for STEM.

What can you do to encourage a girl in STEM today or any day? If you have the knowledge and time to share, become a mentor to her. If you don’t have the time to commit to mentoring, help her find appropriate enrichment activities such as local events at the science museum or after-school programming classes that will keep her engaged, especially when social pressure can push her off track. Even something as simple as sharing stories about talented women in STEM who can serve as role models can make a difference.

IMACS is honored to have been a meaningful part of the education of numerous high-achieving girls who have gone on to amazing college and professional careers. They now serve as inspirational role models for our younger students. Who knows? The girl you encourage today may one day be an IMACS alumna studying STEM at a top university!

IMACS believes in the potential of bright young women to shape a better future for all people. That’s why IMACS created university-level courses that help talented girls develop the knowledge and skills prized by top universities and leading companies. Like IMACS on Facebook for the latest information about our local classes and online courses.
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Improving Math Education through History


December 18, 2014 Filed under: Curriculum Development,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.

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:

  • Respondents with low history scores
    • were more likely to indicate that they believed mathematics overall was like "cooking a meal" or "a tool for use in everyday life."
    • were more likely to believe that mathematics is a disjointed collection of facts, rules and skills than respondents with high history scores.
    • appeared to be more likely to agree with the statement that "the process of doing mathematics is predictable" than those with higher history scores.
  • Respondents with high history scores
    • exhibited more favorable views of mathematics.
    • were more likely to indicate that they believed mathematics overall is like "doing a dance" or "an art, a creative activity, the product of the imagination."
    • disagreed more often with the statement "everything important about mathematics is already known" than did their low-scoring counterparts.

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.

Give your talented child the benefit of learning mathematics in a way that allows him or her to experience the thrill of intellectual discovery and meaningful challenge. Enroll in the Elements of Mathematics: Foundations online program for bright secondary school students. EMF presents all of middle and high school math up to Calculus — and much, much more — as a cohesive body of knowledge made up of elegant and interconnected ideas. Visit www.elementsofmathematics.com to learn more.

Like IMACS on Facebook for the latest information about our local classes and online courses.

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Letters to a Young Math Teacher


January 2, 2014 Filed under: Math Enrichment,STEM Education IMACS Staff Writer @ 1:00 am

The following letter is extracted from the new book, Letters to a Young Math Teacher, by Gerald Rising and Ray Patenaude, which is available from Amazon.com and other sources. Gerry Rising is Distinguished Teaching Professor Emeritus at State University of New York at Buffalo where he co-founded the university’s Gifted Math Program for highly-qualified regional students in grades seven through twelve. Gerry Rising was for many years associated with IMACS activities and is a strong supporter of our work.


Letter Seventeen: A Bag of Tricks

Paul Rosenbloom enjoyed an international reputation as a senior mathematician when I joined his Minnemath Project at the University of Minnesota as his assistant director. His Elements of Mathematical Logic remains today a basic resource recognized worldwide to be of historical as well as academic significance.

In our very first conversation he told me that he considered teaching outside the classroom an important aspect of a mathematician’s life and he urged me to develop what he called “a bag of tricks” from which to draw math-related lessons for people of all ages. I would later see him draw from his own collection in a number of settings. He could captivate anyone from nursery school students to scientific colleagues and even senior political figures.

Within a month of that conversation I found myself seated on an airplane flying from Minneapolis to Denver. My seat partners were a bored nine-year-old and her mother. When the mother learned that I was a math teacher, she asked me if I would be willing to “show Marjorie some math.”

Fortunately, I had been reading David Silverman’s book, Your Move, and I had some pennies with me. I played a series of Nim games with Marjorie, finding her an interested opponent and a remarkably quick study.

Nim, for readers unfamiliar with the word, is German for “take” and in this context it applies to games with players taking one or more counters like coins from a collection following a given set of rules.

We started with One Pile Nim. I set out a line of pennies on Marjorie’s tray and explained the rules. We would take turns, at each turn removing one, two or three coins. The winner would be the player who could take the last penny or pennies.* When she quickly determined the strategy for playing, I suggested she challenge her mother. Her eyes lit up as she outplayed her mom.

Now she wanted more. I suggested the misère form of the game: same rules for play but this time the loser forced to take the last coin. Now I had her mother’s full attention as well, but again Marjorie, once she found the winning strategy, could outplay mom.

We went on to play the very simple Two Pile Nim game in which each player takes any number of coins from one of two piles with the winner taking the last coin, but then the less transparent game, Woolworth, which is isomorphic to Two Pile Nim. Woolworth is named for the five-and-ten-cent stores of the 1950s that have been reincarnated as today’s Dollar Stores. On a sheet of paper I drew the following diagram:

I placed nickels and dimes as shown in the figure. One player controls the coins on the left, the other the coins on the right. Players alternate moving either of their coins any number of squares to left or right but without joining or passing the opponent’s coins. The object of the game is to squeeze your opponent so that no moves remain.

By the time I showed Marjorie the connection between Two Pile Nim and Woolworth we were well on our descent into the Denver airport. I was rewarded for our hour together when Marjorie turned to her mother to tell her that she wanted to become a mathematician.

Where do you find lessons like these that require no background and yet are suitable to given age groups? I found mine from two sources: my reading of journals and books about mathematics and my observation of lessons taught by others. Appendix 2 lists a few of the many available book sources.

Now, because it is a good story, I will tell you about my use of a lesson copied from the remarkable teacher, Robert Wirtz.^

Late one school year when I was a math supervisor I was asked by a primary school teacher to demonstrate Stern blocks for her students. Stern blocks are inch cross-section blocks that come in lengths from one to ten inches. Cuisenaire rods are similar but with centimeter dimensions.

I took several sets of Stern blocks to this teacher’s classroom and showed the six-year-olds some of the relationships among them.

The students were excited by the opportunity to mix play with learning, and I was convinced that they were gaining from the lesson. I noticed, however, that their teacher was not happy with the children’s mix of play with learning. Several times she interrupted their activities to tell individuals to be quiet.

It was clear to me that the teacher was convinced that I wasn’t disciplining the children in the way she wished, so I suggested that I return the next day to teach another lesson. Despite her reservations about my conduct of this class, she jumped at the chance to have me return. It was clear that she was happy to pass responsibility for these irrepressible kids off onto anyone she could find.

I had seen Bob teach a lesson to a similar group and the next day followed his model. Before the class I had the teacher gather the students in front of the chalkboard. Saying nothing to them I drew a square on the board and within it marked two star shapes. What I had drawn looked like this:

I pointed to that little tail at the bottom of the figure and turned to the class, holding out the chalk.

Several hands waved and I offered the chalk to one of the wavers. The boy came up and carefully wrote a “2” below the diagram.

That got us started. I drew similar diagrams with different shapes and the children quickly caught on, competing to write the numbers below the figures. Then I began to complicate the requests by erasing their numbers and connecting two figures like this:

No problem. A volunteer wrote “5” below the boxes.

I continued in this fashion, soon replacing the boxes with numbers but using the same connecting segments. All of the children were eagerly participating, but not one word was said by me or by any of them.

After a half hour of this activity, I finally spoke up, calling the children’s attention to the fact that none of us had spoken until then. And I turned to their teacher to congratulate them on what we had accomplished.

Unfortunately, my lesson proved nothing to this young woman. Her response: “Yes, after you left yesterday I really scolded them for their behavior and you saw the result this morning.”

Even with the best of efforts, you cannot always win.

I have talked about these mathematical extras mostly in terms of their use outside of your classroom, but they can play a role in your classes as well. Here is an example described by Swedish math educator Barbro Grevholm:


One of the teachers worked with a smaller group of pupils that had difficulties with mathematics. It was Friday and the pupils were not concentrating. The teacher announced that if they worked well he would play a game with them for the final ten minutes. This motivated the pupils to pay attention to the lesson. When the teacher finally said that it was time for the game, the students were all alert and extremely concentrated. In the game the teacher threw a single die nine times and after each throw read the result aloud. The pupils drew on sheets of paper three by three grids and chose to put the die calls successively in any one of the nine squares. The winner was the one who, by adding up the resulting three three-digit numbers, produced to a sum closest to one thousand. The game was played several times and all students took part eagerly. There was almost complete silence during the game and everyone made the additions quickly. For some of the pupils obviously more calculations were made during those ten minutes than during the earlier part of the lesson.

Please take the examples I have used only as samples of my own and others’ collections of activities. You may or may not add them to your collection. What is important is that you accumulate such activities that work for you. They will serve you well.


Where Do You Find Such Activities

Sources of such activities are all around you. You just have to be alert to them and adapt them creatively.

Many such activities come from your reading. Mathematics and mathematics education journals describe topics and lessons that you can both enjoy and share with others. Of course, you should credit your source when you replicate such a lesson, but every writer I know would be delighted to know that his or her lesson was being duplicated.

The two best journal sources I know for such examples are the Mathematical Association of America’s Math Horizons and the British Mathematical Gazette.

Some non-standard texts are full of such lessons. Authors like Harold Jacobs and Sherman Stein are two who have gathered and present such useful topics. And some authors simply collect such presentations. Among these are Ian Stewart, Ross Honsberger and Howard Eves.

In a class by himself is Martin Gardner whose collections of such topics are unsurpassed. His range is amazing, covering everything from hexaflexagons to fractal music, almost all of them serving a perfect basis for well-planned demonstrations.

Even your daily newspaper can provide activities. Although many people are drawn to the Sudoku puzzles, I much prefer three other popular forms: Kakuro, KenKen and Numbrix. These puzzels are adaptable to students of all ages (I solve one or two each day) and can challenge them all. The simplest appearing is Marilyn vos Savant’s Numbrix. While this puzzle task is appropriate for primary grade students (all it involves is listing the integers from 1 to 81 in order in a 9×9 grid), individual Numbrix puzzles range widely in difficulty.

In addition to their intellectual challenge, there are two things about the Kakuro and KenKen puzzles that I find attractive: (1) they involve basic calculation facts that reinforce the solver’s skills and (2) they appear as free apps on mobile phones and tablets. This availability makes them perfect puzzles to introduce to fellow travelers.

* The strategies for playing these Nim games are included in Appendix 4. I encourage you to play the games before you look at those strategies.

^ Bob Wirtz and his wife adopted a number of children with severe learning problems and they developed math materials to teach them. With Mark Botel, then president of the International Reading Association, he gathered those materials into books for elementary school students. Sadly, they are difficult to find today for they include some very attractive activities.

Are you searching for inspired teachers who understand your bright and talented child? Look no further than IMACS! Distance learners in our university-level gifted math and computer science programs are each assigned a principal IMACS instructor who is available to answer questions. Like IMACS on Facebook to get FREE weekly logic puzzles.
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Driven to Abstraction: Strengthen Your Thinking Skills with Computer Science


December 5, 2013 Filed under: AP Computer Science,Computer Programming,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
IMACS is proud to be a Partner for CSEdWeek, December 9-15, 2013.
Have you registered for the Hour of Code? Do it now!

Abstract reasoning ability entered the national conversation this year as the Common Core State Standards in mathematics were broadly implemented in the United States. In particular, one of the eight Standards for Mathematical Practice is to “reason abstractly and quantitatively.” The so-called STEM subjects — science, technology, engineering and math — are well-known for emphasizing this skill. Given that STEM-related fields are where most high-skilled job growth is predicted, today’s students would do well to develop their ability to think abstractly.

So what is abstract reasoning, and why is it so important? Let’s break it down: To reason is to use logic in piecing together information, usually with the goal of forming an inference or conclusion. Abstract simply means that this process is a thought-based exercise of the mind as opposed to being based in concrete experience. For example, if you know that ice melts at temperatures above 32°F, you can reason abstractly that an ice cube placed on the counter of your room temperature kitchen will melt. You don’t have to take an actual ice cube out of your freezer and observe it for an hour to arrive at this conclusion.

Of the subjects that you could study in order to develop strong abstract reasoning skills, computer science is a natural and practical choice, as well as being a highly creative and exciting area in which to learn and work. The programming aspect of computer science is well-known and is one area where abstract thinking matters a great deal. Programming, after all, is the creation of a set of instructions that a computer can follow to perform a specific task. Such tasks typically involve the manipulation of digital information, decidedly not the kind of stuff you can grab hold of to see how it reacts in the tangible world.

Learning to program well involves developing the ability to think logically and abstractly so that you can anticipate how the computer will react to the instructions you give. Great programmers are actually capable of writing simple code without having to check it with a computer because they have the ability to analyze processes in their minds. If you cannot think abstractly, you may still be able to get your code to “work” with trial-and-error tinkering, but that approach lacks the robustness needed to solve meaningful problems that tend to be more complex.

The rich experience of learning computer science, however, is so much more than coding. When you study computer science, you engage in computational thinking, in which logic, abstraction and creativity come together to help solve intellectually interesting problems. As Professor Jeannette Wing of Carnegie Mellon University argues in her seminal article* on the topic, computational thinking is a skill set from which everyone would benefit no matter their career path.

Why so? Because when you study computer science, your mind learns to grapple with high-level questions such as: How can existing information be used to deduce further information that will help solve the problem? How should a complex system be designed in order to maximize simplicity and usability? How can a complex problem be broken down into smaller pieces that are easier to solve? Can a common approach be devised to efficiently handle similar problems?

If these questions seem like they would be applicable in a wide variety of fields, STEM and non-STEM, it’s because they are. In essence, when you study computer science you learn the valuable skill of thinking abstractly like a computer scientist even if you don’t plan on becoming one.

*Wing, Jeannette M. “Computational Thinking.” Communications of the ACM 49:3 (March 2006) 33-35.

Excited by an Hour of Code? Imagine a lifetime of intellectual advantage when you take online computer science courses from IMACS, including our College Board-approved AP Computer Science course. Like IMACS on Facebook to get FREE weekly logic puzzles.
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A Rocket Scientist Shares His Past and Muses on the Future of STEM


March 21, 2013 Filed under: Careers,Computer Programming,STEM Education IMACS Staff Writer @ 1:00 am

This month, IMACS chats with alumnus Daniel “Danny” Vidaud. Danny started taking IMACS math enrichment classes as an elementary school student and progressed through the introductory computer science class. He went on to earn his B.S.E. in Aerospace Engineering from the University of Michigan. Danny is currently in his third year as an engineer with Boeing.

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.
IMACS alumnus, Daniel Vidaud, soaks up the sun in Cartagena, Spain.

Tell us about your current position at Boeing. What exactly do you do?

I am an aerodynamics engineer working in technology and product development. In a nut shell, I am part of the team of architects for the external shaping of future commercial jet airplanes. We spend our time sketching up new, outside-the-box ideas and bringing them to life!

What were you like as a kid? What kinds of things interested you?

I was a very intuitive child. Very not normal. Constantly absorbing as much as I could about the world around me. I had a tendency to quickly gain a functional understanding of complex ideas. The downside was that this only applied to topics I found interesting. A repetitive spelling assignment, for example, was as interesting to me as watching paint dry in an empty room with no windows. I needed to actively seek ways to challenge my inspiration or I would inevitably fall into a state of no motivation.

I enjoyed music. The piano, I found, was quite versatile at conveying a variety of musical ideas, but I hated studying it. I couldn’t stand the classical books or the structured process. The musical expression was inspiring; the structured training was not. Instead, I decided that mimicking what I heard on the radio was inspiring enough to practice for hours on end. Free improvisation and jazz composition became the new method of study.

Computer games! Fun! Not so fun when they freeze and get choppy, right? So I decided it would be interesting to develop a theory on what made a computer “fast” or “slow” and subsequently exploit that theory to help others in creating new systems or maintaining their old systems to do what they needed them to do.

Physics and all other things I found interesting went along the same lines of thought. The approach was always the same: Take a complicated problem, gain a general intuitive understanding for how it works, then generate as many permutations or original ideas as possible.

Did you know from a young age that you wanted to be an engineer?

Always. I didn’t always know it was called engineering though. I just knew that I liked asking the “Why not?” question a lot. “Why can’t we do something like this?” Challenging the normal. Being weird. It just seemed like more fun to not do what everyone else was doing.

Given that, rocket science seemed like a viable candidate. No one was doing it, everyone said it was impossible, and it seemed like it might be a good place to start if I wanted to get involved in something really complicated that may have high demand and low supply. So I turned 15, applied with a pre-declared major of Aerospace Engineering to the University of Michigan, a few years later developed a powerful network of friends, and then came to work for Boeing in the heart of its commercial think-tank.

How did your IMACS classes prepare you for college? Your position at Boeing?

The teaching philosophy for computer programming at IMACS is not the classical piano book approach. You will not become an expert at solving any kind of existing, well-defined problem with one specific and popular language. You’ll spend a lot of time not learning the computer language that you will be taught in your first term in college.

Instead, you will gather an understanding of what you might call “computer linguistics”. The ability to communicate an idea through the assembly of conceptual components. The skill of decomposing a multipart task into a simple abstract algorithm. At which point you are then free to cut code in the language that would be most efficient for communicating that idea. IMACS computer science provides you with a different way of thinking, not just an add-on to your résumé about how you can write code in an industry favorite language.

At Boeing, we spend time studying new functional aerodynamic shapes to solve a variety of complex problems while keeping in mind the multidisciplinary nature of every component. With every new idea, you walk through the development process to show that it’s viable, or even patentable. Some of the skills I learned at IMACS allow me to draft up a few quick and dirty scripts in languages I had never coded in before. This allows me to save a significant amount of time repeating similar analytical tasks on multiple candidate solutions or parsing out test data in a useful way. After IMACS, you become more comfortable interacting with the machine and make the most of the computational power you have available at your disposal.

You also have some experience teaching. What do you think the US has to do as a nation to improve math, science and computer science education?

I was a substitute teacher at a high school and subsequently a graduate teaching assistant in a first-year programming course for future engineers at Purdue University.

Successful college students today are very aware of the concept of perceived economic value. Students today are more likely to seek out business-related, social science or history degrees rather than physical science or engineering degrees. It demonstrates a general sentiment that the technical degrees are no longer worth their perceived cost (years and/or intensity of training, accrued financial burden, etc.). Science is not the “cool” thing to do anymore as it once was when scientists were in the limelight of the 60’s. The perceived benefit of being in a technical field was much higher. Marketing happened by default on the news every time a rocket launched at the NASA Kennedy Space Center. Unfortunately, the need for technical degrees is inherently difficult to quantify and isn’t always obvious at a cultural or global economic level anymore.

Today, we’ve come to take for granted the engineering and scientific leaps that have been made in the recent past, such as leaps in wireless data transfer, functional nanotechnology and intuitive human/machine interfaces. Advances in biotechnology research (e.g. replacement organs, spray on skin) have unlocked a new approach for healing the infirm.

Unless the general culture regains an appreciation for scientific exploration by raising the perceived benefits and reducing the perceived cost (as was once shown in the 60’s during the Space Race, or during WWI and WWII in aircraft and military weaponry development, or personal computer development in the late 80’s), we will see a general stagnation in “technological advancement” as it has been traditionally defined.

Traditionally defined innovation and scientific exploration is a high-risk, high-expense endeavor. It will only happen when the global market demands it and demonstrates its true value. When the free market price of oil is allowed to inflate beyond the point of affordability without manipulation, the economy will require immediate and immense creativity in alternative energy and fuel technology. The need for scientists and engineers will be made immediately relevant and the market support will demonstrate the true benefit to all who depend on that which they take for granted.

The traditional ambition within transportation advances, for example, in the past century has repeatedly contained the adjectives “faster, farther, higher”. On the ground we went from conventional rail to high-speed rail (e.g. France’s TGV) to magnetically levitated trains (e.g. Japan’s MLX01). In the air we’ve seen US Air Force-funded demonstrators like the Boeing X-51 flying multiple times the speed of sound over the Pacific Ocean.

It seems the general population is now fairly uninterested in the traditional. We are no longer actively pursuing this long-established goal. In a modern culture that is approaching one of perfect information (made possible, in part, by economically accessible, internet-enabled, naturally intuitive smartphones), we have the ability to make more rational purchasing decisions. Now the market tends to instantly reward those who can make a substitute product for a cheaper price. Engineering and science is being redirected to the practical. For example, after you pick your destination, travel websites will automatically sort by price. The name of the game now isn’t “faster, farther, higher” anymore. The commercial business case is just not there for it.

Social awareness is starting to flow into the demand for science and engineering. Privatized venture philanthropy and private humanitarian and community foundation efforts have created a multi-billion dollar industry in the past 10 years. Modern innovation is making commodities such as digitally-based financial services for the poor or basic health services available to the masses that were previously prohibitively expensive.

Science, Technology, Engineering and Mathematics are here to make the world a better place. How we each define “better” will guide innovation and large private capital into the directions that have highest economical demand and true value. The only thing we can do now is try to show the world what we are capable of accomplishing and what they can do as the culture begins to appreciate, once again, how powerful ideas really are.

Go faster, farther, higher with gifted math and computer science online from IMACS! Middle school students can explore our Elements of Mathematics: Foundations online courses. Get weekly IMACS logic puzzles on Facebook.
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Three New Year’s Resolutions for Educators


December 27, 2012 Filed under: Computer Programming,STEM Education IMACS Staff Writer @ 1:00 am

After school, weekend and online programs in math and computer science for gifted children who enjoy fun, academic challenges.

It’s that time of year when we think about changes and improvements for the new year. Here are three that IMACS would be delighted to see on every educator’s list:

Teach computer science. The opportunity for American students to learn this increasingly important subject in school is woefully rare. If we want future generations to continue to have a high standard of living, we must prepare them for the jobs of tomorrow. While some of these jobs may not even have been invented yet, we are fairly certain that computational thinking learned through studying computer science will be a highly valued skill needed to succeed at them.

Be a guide, not an answer key. Give students time to puzzle through new concepts and problems and the opportunity to discover answers for themselves. Step in when needed to help avoid frustration. More learning will happen, more knowledge will stick, and more confidence will build for the next challenge.

Give children more unstructured time. This is such a challenging goal in today’s busy and competitive world, but it is well worth the effort. Our culture and history have long emphasized industriousness and productivity (for good reason), but we are now coming to understand that unstructured time is also a highly productive time for our brains. These are the moments when insight and unconstrained creativity lead to new ways of thinking and solving problems. Those are the seeds of progress.

Thank you all for a terrific 2012, and best wishes for a wonderful 2013!

Ring in the New Year with IMACS! Check out our online courses in gifted math and computer science. For talented middle school students, visit www.elementsofmathematics.com to register for our Operational Systems course that is FREE before January 1, 2013. Get weekly IMACS logic puzzles on Facebook.
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