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This month the IMACS Blog caught up with Azzara Nincevic, who has been a star student at IMACS for seven years now. Azzara enjoys reading, drawing, and classical ballet. Although she dances at least 12 hours per week and performs throughout the year, she always finds time for IMACS.

“When I began IMACS in first grade, I immediately loved it.” Azzara says. “Having taken an interest in math, I quickly learned the traditional material and was looking for more challenging enrichment. When I attended class at IMACS, all of the problems were thought-provoking.”

As a member of her school’s math team, Azzara attends competitions such as MATHCOUNTS and Mu Alpha Theta where her IMACS background has been an invaluable asset. As Azzara describes it, “The IMACS curriculum helped me to develop logical thinking skills and the ability to quickly solve math problems, which are key to succeeding at math competitions.”

While Azzara’s achievements in mathematics and ballet, by themselves, are enough to impress anyone, it’s her recent performance on the AP^® Computer Science A exam that readers will recognize as a rare feat. Soon after starting IMACS Math Enrichment program, Azzara enrolled in our Computer Enrichment & Virtual Robotics class where she developed a great interest in programming. Over the years, she continued with IMACS University Computer Science (UCS) track, which culminates in our AP^® Computer Science: Java Programming course.

AP^® exams are typically administered to high school students, but at the time that Azzara was ready for APCS, she was only just entering seventh grade. That didn’t deter her. “After inquiring, my mom and I found out that there is no minimum age requirement for an AP^® exam, so I registered. With the preparation that IMACS gave me, I was able to score a 5 on the AP^® Computer Science exam as a seventh grader.”

With such a busy schedule, Azzara appreciates that one of the greatest benefits of IMACS is that the computer science and logic programs are accessible online and self-paced. “I was able to excel at my own pace and access the IMACS curriculum anytime and anywhere.”

What does the future hold for Azzara? “I am entering the eighth grade with a greater passion for and interest in math and computer science. IMACS made me realize that I would like to pursue computer science in college and after. The fundamental skills that I have learned in the UCS courses and the logical thinking skills I have learned in the Math Enrichment and Mathematical Logic courses give me the advantage I need to be successful. As such, I plan to continue with IMACS in the upcoming years.”

Have you seen the latest video of a young child reciting multiplication tables or the digits of pi? Or maybe you know a kid who has always gotten straight A’s. Pretty neat, but does it equate to being genuinely good at math? No. Bright students often do well in school with little or no effort. And an airtight memory facilitates excellent grades, especially when those grades depend on regurgitating information that’s already been provided. Being genuinely good at math is more about having a deep understanding of how and why things work. It also means being able to take that understanding and apply it in novel situations. This is where the ability to reason logically and abstractly separates skilled thinkers from those who only learned how to go through the motions. As IMACS graduate Zachary Kaufman put it, “Graduate school at Stanford was so much easier with the logical thinking skills I learned at IMACS. While classmates tried to memorize each type of problem, I was able to strongly grasp core concepts and use them to solve any problem, even if it was different from those I had seen.” Zachary is a skilled thinker who is genuinely good at math. Will your child be?

This month the IMACS Blog speaks with IMACS student, Fiona Brady. According to Fiona’s mom, Susan, “IMACS was the first time Fiona had encountered a community of teachers and learners who were excited to hear her ideas and creative ways of problem solving.” After the Brady family moved out of the area, Fiona continued taking courses through our distance-learning program, eIMACS.

Having studied University Computer Science and AP® Computer Science through eIMACS (and scoring a 5 on the AP® exam), Fiona was able to pick up the Python programming language* when she encountered it at a summer mathematics camp at the University of Chicago with students several years older than she.

For students as talented as Fiona, homeschooling and early college courses often make the most sense as they and their families seek educational options that provide enough challenge, flexibility and inspiration to help them reach their highest potential. Let’s hear what Fiona has to say about pursuing this path:

Please tell our readers a little about yourself.

I’m turning sixteen this fall and I’m in tenth grade. I’m a second degree black belt in Tae Kwon Do, and I enjoy figure skating and horseback riding. I don’t feel like this gives a real image of me, but there it is. I enjoy making things with cardboard and duct tape, but definitely not wallets. I’ve made my Halloween costumes for the last few years. The year before last, I was Medusa. I wore a snake hat that I built in my bedroom and needed to turn sideways to get out. I have since learned that on some occasions it is important to get dressed outside of your room. When I’m not doing math, I love reading.

You are homeschooled and also taking college classes at Northwestern University. What is it like to do both? How do you balance the academic workload, extracurriculars and time with friends and family?

Homeschooling is not like regular school because there is no large division between having fun and learning. So I don’t balance it. However, when I have a large assignment due, my mom probably doesn’t see me for two days. My extracurricular activities — skating, horseback riding and Tae Kwon Do — force me to do something active. I also enjoy volunteering at the barn where I ride because they work with children with special needs. Our three dogs keep me pretty busy too, especially my own puppy, Mole (named because the white fur around his nose made him resemble a star-nosed mole when I first got him).

What circumstances led you to take university classes?

I have always liked math, so I started taking more than one math class a year. In my eighth grade year I took five. After that, I sort of ran out of other options. I participate in the University of Chicago’s Young Scholars Program, which is led by Professor Paul Sally. He and others at Chicago gave me advice and helped to set up a meeting with the head of the Northwestern Math Department, Professor Mike Stein. Professor Stein gave me permission to sit in on the courses, and introduced me to the professors.

Which classes are you taking at Northwestern? How did your IMACS courses prepare you for those classes?

Last year I took a course on Abstract Algebra and one on Multivariable Calculus and Linear Algebra. This year I am taking Physics and Analysis. IMACS was the first place where I encountered the idea that to learn something you have to own it; that is, you have to be able to form a picture of it in your head, and you need to be able to construct it from basic principles. In the IMACS computer science classes I took, you really needed to do that, otherwise you would get lost in the middle of writing a program and forget what you were doing. IMACS Logic for Mathematics is a continuation of that because it is constructing the basic principles of mathematics, which are skipped over in most high school classes (but assumed to be known in college courses).

[Editor’s Note: Three years running, Fiona has received the award from the Northwestern Mathematics Department for outstanding achievement in mathematics by a high school student.]

What advice would you give to young students who are thinking about taking university classes before they officially enter college?

Ask your teacher questions. I’ve had people in classes ask me questions, instead of asking the teacher. That’s a really big mistake, and it’s an even worse mistake to make in college because the professors are amazing. One of the things I most admire about the professors I’ve had at Northwestern is the unshakably solid understanding they have of the material. Also, if your professor asks the class a question and you think you know the answer, you should raise your hand. Even if your answer is not correct, that just gives you the opportunity to ask a question and figure out what you don’t understand before you try to learn something that builds on it or have a test.

What do you see yourself doing in the future?

I have three more years before I go to college and I want to keep taking classes and learning more. Being a professor sounds like an interesting career. (Being a stuntman does, too, but I don’t think I’ll pursue that.)

*IMACS added Python to University Computer Science II in November 2012 after Fiona had completed the course.

UPDATE, May 27, 2014: IMACS student, Peyton Robertson, impressed President Barack Obama at today’s fourth annual White House Science Fair. Watch Peyton with President Obama starting around the 4:35 mark of this video.

UPDATE, October 9, 2013: Congratulations to IMACS student, Peyton Robertson, on winning the title of “America’s Top Young Scientist,” $25,000 and a trip to Costa Rica! Watch Peyton’s winning moment.

From the moment you meet IMACS student, Peyton Robertson, you can’t help feeling that he is one of those bright, young people who is going to leave his mark on the world in a big way. His creative ideas and energy seem boundless. His smile and enthusiasm are infectious. And he’s just 11 years old!

Most recently, Peyton entered the 2013 Discovery Education 3M Young Scientist Challenge and is one of only 10 finalists (and the youngest) in this prestigious national science competition for 5th through 8th graders. The Young Scientist Challenge encourages students to explore science and innovation during the pre-teen years when interest in math and science typically starts to decline.

Peyton’s project, SOS: Sandless Operational Sandbags, focuses on developing a more effective and less costly sandbag design to protect against damage from saltwater flooding. As Peyton notes in his finalist video, 80% of the $43 billion dollars worth of flood damage caused since 2005 has been from saltwater flooding. Peyton’s native Florida is at risk for hurricane-driven saltwater flooding every year. So he’s decided to do something about it by re-engineering the basic tool of flood control: the sandbag.

The design of traditional sand-filled sandbags means that they are heavy, difficult to transport and, when stacked, leave gaps through which water readily flows. Peyton’s design uses a thin, expandable polymer to keep his bags light and easy to transport. He also pre-fills his bags with enough salt so that when it dissolves, the salt content of the solution inside is higher than that of seawater, helping to keep the seawater from penetrating the bags. Finally, Peyton uses an ingenious interlocking design to minimize the gaps between the bags when they swell. When the water recedes and the bags dry out, they return to their thin, easily transported and stored form, ready for use during the next storm.

IMACS is so proud of Peyton and delighted to count him as one of our many exceptional students. When we asked Peyton’s mom, Shannon Robertson, to describe the influence that IMACS has had on his education, this is what she had to say:

Congratulations, Peyton! Your friends at IMACS wish you the best in the finals of the 2013 Young Scientist Challenge.

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

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

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

Learning to Code Isn’t Enough

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

The IMACS Approach

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

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

Learning to Think with Logo

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

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

University-Level Computer Science

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

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

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

The IMACS Advantage

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

Can you link all nine dots using four or fewer straight lines, without lifting your finger and without tracing the same line more than once?

We all understand what is meant when a person exclaims, “That’s so creative!” This is a typical reaction to a new and often pleasing, useful or clever idea. It might range from a musical composition to bionic limbs to proof of a longstanding conjecture in mathematics. Whatever the form, praise for the idea is rooted in its originality. Designed to be ever-learning, the human mind relishes new “food for thought,” and so as a society, we place a high value on creativity.

Books and blogs offer up a variety of tips on how to foster creativity. One piece of advice that is at odds with our experience at IMACS is the claim that logic stifles creativity. There is even a meme of Albert Einstein with the quote* “Logic will get you from A to B — Imagination will take you everywhere” posted frequently in support this misconception.

To the contrary, many students who studied our curriculum in mathematical logic have gone on to thrive in fields — puzzle and game design, teaching and medical research, just to name a few — that require creativity and logic. They succeed because they have a fundamentally superior understanding of what it means to think with logical creativity.

With respect to the cliché “think outside the box,” some consider logic as a villain that keeps creativity bound and gagged inside the box. As the argument goes, illogical ideas are either withheld or discarded too soon when they might have led to winning ideas, thereby hindering the creative process. But consider this quote on creativity by iconic creative force, Steve Jobs:

“Creativity is just connecting things.” In our view, logic is one of the most powerful facilitators, if not the most powerful facilitator, of connectivity. Yes, logic will get you from A to B … and then to C … and to D and to E and so on, and that’s a good thing. Logic helps you think clearly about your knowledge and experiences, make connections among them and synthesize new ideas.

When learned well, logical reasoning has the power to unleash creativity. Build enough connections and you can follow a more direct path to the winning idea. You needn’t worry that the unstable bridge of an illogical idea was the only way you were going to get there.

*This attribution to Einstein has not been documented on Wikiquotes, not even as disputed or misattributed. Perhaps the meme is the result of creativity without logic.

By drawing lines that extend outside the box formed by the outermost dots, one can find a solution to the puzzle at the top of this post.

This month’s guest blog post comes from Christopher Tiwald. Christopher studied the Elements of Mathematics curriculum from 6th to 9th grade as a student in Lincoln Public Schools in Lincoln, Nebraska. He attended the University of Nebraska-Lincoln where he majored in Political Science. After exploring a diverse series of careers, Christopher became a self-taught software engineer and now works at Conductor, Inc. as a Technical Operations Engineer.

The most common complaint you hear from students in math class is “We won’t use this in the real world.” I’m ten years out of high school and six years out of college. I’ve made sandwiches professionally, waxed floors full-time, went to grad school, dropped out of grad school, and worked in professional politics. I can honestly report, “We were wrong.”

It turns out the real world doesn’t reward education as much as passion, self-starting, and the ability to solve problems. The Elements of Mathematics curriculum, or “Elements” as we called it in school, taught me how to learn. For four years I studied problems that bent my brain and stretched my imagination. They required every last modicum of creativity I could muster and, in truth, I wasn’t terribly good at them. I didn’t graduate top of my class — nowhere close. When I did graduate I vowed, “Never again”, and studied liberal arts instead of science in college.

But I couldn’t escape the Elements of Mathematics. Freshman Philosophy is a breeze when you’ve tackled deductive number theory. Supply and demand are trivial when you recognize their curves as simultaneous equations. College grades are driven less by raw IQ than they are by pattern recognition. Once you know how to solve one class of problems, it’s easier to solve again and again. It’s like the compound interest of education — compound interest, of course, being a concept I first encountered in Elements.

I thought it would stop when I finished college. Surely the real world, crammed full of unqualified positions, internships, and the entry-level, would not echo the Elements of Mathematics?

Did you know that if you have five ingredients to make a sandwich, the number of sandwiches you can make with two ingredients is exactly equal to the number you can make with three? Have you ever canvassed door-to-door for a campaign? The most efficient way to knock doors is often an Euler circuit — a route that takes you down every street exactly once. Here’s a fun one from the economist Thomas Schelling: Ski resorts are a closed mathematical system. The amount of time it takes to sit on the lift and wait in its line are inversely dependent. If you make the lift faster, you make its lines longer, because the time people take to ski down the mountain or warm up in the lodge won’t change.

You don’t need to know these things to run a sub shop, win an election, or enjoy a winter vacation, but knowing them gives you a more thorough, practical understanding of the world, one that is immediately useful to employers and in daily life. You learn new concepts quickly because you learned how to learn. You solve problems more efficiently because you’ve solved thousands of similar problems before. The Elements of Mathematics is a competitive edge without qualification. Bosses love it as much as professors.

As for me, I wouldn’t be who I am without the program. Elements changed the way I interact with the world. After wandering from career to career, I landed comfortably in software engineering, something I taught myself. That says less about me than it does my education. All jobs are strings of solvable problems. Once you’ve learned how to think, the rest is easy.

Andrew Hacker, emeritus professor of political science at Queens College, City University of New York, recently wrote an op-ed piece in The New York Times entitled “Is Algebra Necessary?” His opinions caused quite a stir in the ongoing debate over mathematics education in the US. IMACS sees value in some of his ideas, such as teaching quantitative reasoning starting in kindergarten, and we agree that schools should not subject students to the “ordeal” of struggling with algebra. However, IMACS believes that education professionals should focus on changing the way mathematics is taught, not on eliminating the requirement to study algebra.

First, consider the attendant consequences of the author’s proposal, which is to create alternative math courses that “familiarize students with the kinds of numbers that describe and delineate our personal and public lives.” Perhaps the large yet still minority percentage of students who cannot pass traditional algebra would be allowed to satisfy their high school math requirement with these alternative classes. As the author sees it, this would limit “misdirecting precious resources” presumably by redirecting them to the new classes. Does that mean we abandon the majority of students who can pass traditional algebra to the ineffective mores of a failing system (now with even fewer resources) because, hey, at least they’re not struggling? Never mind the fact that neither are they soaring as high as they could and will need to as the influence of technology on their world grows!

Teaching mathematics effectively to all students is the outcome we should be striving toward. Make no mistake—we understand the consequences of the prolonged economic stress on families, school districts, and public higher education. It is natural in such times to direct limited resources to activities most likely to lead to gainful employment. So let’s consider a key element of the author’s argument—that high schools are not even teaching students the math skills they will need in the workplace. He writes:

The author seems to miss the point that the mathematical reasoning skills needed to succeed in the workplace are the same ones needed to succeed in algebra. In both cases, you must be able recognize a problem or challenge, gather information relevant to finding a solution, analyze and synthesize the information to derive a solution, and effectively apply the solution. This takes critical thinking and logical reasoning abilities, and the current pedagogical approach is to try to impart these skills through the process of teaching algebra, almost as a side-effect.

Were the US education system to focus more in elementary school on building these fundamental skills, not only would students find learning high school algebra (and learning in general) easier, they would also be better equipped to succeed in the workplace where problem solvers are highly valued. Furthermore, as learning algebra becomes easier, it becomes less time-consuming, thereby freeing up instructional time to add topics from skill-training to more advanced math as desired. It may sound like an idealistic vision to the millions of people who have come to believe they are “bad at math” when they are more likely the product of bad math curricula, but we have seen this approach work for IMACS students for over 20 years.

Of course considerable training occurs after hiring in STEM fields! The need for accuracy and the complexity of the body of knowledge are so much greater in these fields than in others that are more subjective in nature (e.g., the arts) or that rely more on so-called “soft skills” (e.g., politics). We need only to look to the history of the Space Shuttle to understand the consequences of errors in judgment. On a happier note, consider that the successful entry, descent, and landing of the Mars Science Laboratory was almost 10 years in the making. On-the-job training is necessary in many non-STEM fields from law to portfolio management to journalism to costume design. Why should we expect STEM be any different?

As for the Toyota-sponsored Machine Tool Mathematics class, the course catalog description is “An applied mathematics course designed for machinists which includes instruction and practice in algebraic and trigonometric operations. (2 hour lecture, 2 hour lab).” That doesn’t sound like the math is different from what is taught in high school. Rather it sounds more like the way the math is taught is different. With a hands-on lab, it actually sounds like fun! That is a huge distinction and goes back to our main point that we need to significantly change our approach to how we teach mathematics in the US.

At IMACS, we support the idea of schools using math curricula that accurately incorporate real-world examples that students care about. This approach helps put abstract concepts in context but, more importantly, gets students interested in learning and helps them understand how mathematics shapes our world. Those who appreciate mathematics in its pure and abstract form are pretty special people, but math should and can be accessible to the majority as well. There also need to be more options for visual-spatial learners who may not “get” math concepts when presented on a bland white board. Rather than redirecting limited resources to creating new alternative classes, the focus should be on redesigning the algebra curriculum (as well as those for elementary, middle, and the rest of high school math) to present mathematics for what it truly is, a deep discipline centered around simple but beautiful ideas, rather than a bunch of numbers, funny symbols, and boring algorithms.

Unless you’ve been hiding under a rock for the past year, you’ll have noticed that the campaign to teach kids (and adults) how to code is everywhere you turn. As parents, politicians, and educators debate how to produce more graduates in technology fields, the push to introduce computing at an earlier age gets stronger. For example, MIT’s Lifelong Kindergarten group is collaborating on programming software aimed at kids in preschool to second grade. There are even board books for babies on HTML and CSS! We suspect that such novelties are more for tech parents’ enjoyment.

IMACS believes these efforts are well-intentioned and some, when implemented, will be well-designed. But before you click over to Amazon to buy your little drool monster a book on Web design, IMACS can offer you a few examples of how to introduce computational thinking to children through easy activities that are familiar to you, even if you think you’re computationally challenged.

Computational Thinking vs. Coding

This short and informative paper by Jeannette Wing, head of the Computer Science Department at Carnegie Mellon University, explains clearly what computational thinking is and is not. The following excerpted quote is a good summary of the focus of this blog post:

Learning to think abstractly is an essential skill if you want to succeed in computer science. It makes solving problems easier, which in turn makes working on those problems more fun. Notably, students can learn to think like a computer scientist without entering a single line of code into a computer. In fact, our experience in teaching CS is that writing computer programs is trivial for students who first develop computational thinking skills. Let’s see where in our daily lives we can show kids relatable examples to help them make the transition to abstract thinking.

Stacks and Queues

In computing, a stack is an object in which data expressions are stored and retrieved in such a way that the first data expression to be stored is always the last data expression to be retrieved. It is an example of a so-called Last-In-First-Out (LIFO) object. The same idea applies to various real-life constructs that young kids encounter, even babies who love board books.

Obviously, you can’t explain LIFO with words to a baby and expect the baby to understand, but you can certainly demonstrate the concept with your actions. Take the classic Fisher-Price Rock-a-Stack, for example. Start with the rings off the cone and then load them on in the intended manner with the blue ring going on first. Try to get just the blue ring off. Can you do it while the other rings are still on the cone? No, you have to take the rings off one by one with the blue ring coming off last.

Older kids can appreciate the same concept with examples they come across in their lives: unloading plates from the dishwasher into the cupboard, setting the table with said plates the next day, selecting a product such as cosmetics from a store shelf, putting said product back if you decide not to buy it. You get the picture.

A queue is similar to a stack in that it is an object used to store data expressions. In the case of a queue, however, the first data expression to be stored is always the first to be retrieved. Queues are examples of First-In-First-Out (FIFO) objects. Kids encounter them every time they go through a checkout line or a drive-thru. Switch that Rock-a-Stack cone for an empty paper towel roll, and you’ve got yourself a baby-friendly queue.

Sorting Algorithms

Sorting is one of the oldest problems in computer science. Although the end goal (an ordered list) is conceptually easy to understand, getting there can be complex. Add to that the need for sorting algorithms to be computationally efficient and you’ve got yourself an interesting abstract puzzle.

If your kids are old enough to know or learn how to put words in alphabetical order, then make a project out of sorting the books on their bookshelf. Decide on a sorting key such as title or author’s name. For this example, we’ll use title. For the first shelf, ask your child to try a simple bubble sort. Traverse the shelf from left to right, compare the titles of two books, and swap them if they are in the wrong order. Repeat this process until all books on this shelf are in the correct order.

For the next shelf, you can use a simple insertion sort. Take all the books off that shelf and put them in order one by one in a pile on the floor. Each time that you add a book to the ordered pile, be sure to put it in the right place relative to the books that were previously added.

Now that you have two properly sorted sets of books, you and your child can work together to sort all books. Sounds like a good time to use a merge sort. Move the sorted books off the first shelf into another pile on the floor while keeping them properly ordered and separate from the pile of books from the second shelf. Reshelve the books as follows: repeatedly compare the titles of the two books that are atop the two piles, selecting the one that goes first, and continuing until both piles are exhausted.

Object Oriented Programming

An “object” in computer programming is a complex structure containing data fields and instructions. These objects interact with each other to create even more complex computer programs. The beauty of object oriented programming is that you can reuse objects to do common computing tasks without having to reinvent them each time. Over time, programmers can build up a “library” of useful objects.

The following analogy certainly isn’t perfect, but it will help get the point across about these seemingly mysterious objects. If you’re planning an outing with kids, you’ll need a few things to help keep your sanity: nourishment, entertainment, and possibly a change of clothes. So grab three bags and make some objects! In the nourishment object, you’ll probably need fruits, carb snacks, a protein, and beverages. For the entertainment object, how about art supplies, books, sporting equipment, and portable gaming device? Kids are made to get dirty, and the weather may change, so pack a top, bottom, and outerwear in the clothing object. Throw those “objects” in your huge tote “library” and you’re ready to go!

Think Like a Computer Scientist

Planting the seeds of computational thinking, especially the ability to think abstractly, is really a matter of recognizing the examples in your life that can be used to foster discussion with your children. Like any new endeavor, remembering to look at events in a computational light takes practice. You might just find yourself thinking like a computer scientist when it comes to solving the data problems in your own adult life.

Enhance your computational thinking skills with online computer science courses from IMACS! Register for our free aptitude test. Solve 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.