The IMACS Blog reconnects with alumnus Mark Engelberg who recently released three new coding games through ThinkFun and Target. (Win an autographed copy of one of Mark's new games!) Mark's passion for computer science, math and logic led him to an award-winning career in puzzle and game design. He is also an active speaker in the Clojure programming language community. In this post, Mark talks with IMACS about what it was like to be a homeschool dad to two amazingly talented children who are now accomplished young adults in their own right.
First, tell us about your new ThinkFun games.
Two years ago, I invented a programming puzzle game for ThinkFun called Code Master. Code Master was a commercial and award-winning success, so ThinkFun asked me to develop a new line of puzzle games, where each game would highlight a specific underlying principle of computer science. The new line of games is called the //CODE Programming Game Series. To design the games, I started from a list of concepts I wanted players to encounter, and kept trying out ideas and tinkering with the rules until I felt I had a combination of mechanisms that covered those concepts and was fun to play.
The first game in the series, On the Brink, is an introduction to procedural abstraction — the idea that we're not just writing programs to solve a one-off task, we're building components that solve multiple instances of an underlying problem, and then using those components to build higher-level components, and so on. The second game, Rover Control, focuses on the essential skill of stepping through programs in your head and visualizing the outcome, particularly with flow control constructs like if-then-else statements, while loops and for loops. The third game, Robot Repair, is all about Boolean logic. When developing Robot Repair, I leaned heavily on the material that I learned as a child in IMACS' Advanced Mathematical Logic courses. I think students in your Logic courses are really going to enjoy Robot Repair. Like their predecessor, Code Master, all three of the new games are "unplugged" computer science board games that come with a book of puzzles for solo or cooperative gameplay.
You are also a parent of two gifted children who are now ages 19 and 17. About how old were Alex and Molly when you first suspected that they were gifted? What were the signs?
I was a stay-at-home dad, and when Alex and Molly were toddlers, I taught them to read and write and gave them daily math lessons. Honestly, I mostly did those things to keep myself from getting bored; it was fun for me to feed their intellectual curiosity. My wife did the same with the kids when she was home from work. That was just our parenting style. We wanted the kids to have the opportunity to learn these core skills as soon as they were capable, and the easiest way to achieve that was to throw a lot of educational activities at them early, just to see what would stick. As it turned out, more of it stuck than we expected!
I didn't really have much to compare their development to until the day I went to Alex's kindergarten orientation. As the teacher explained what the students would be doing in the coming year, it dawned on me that my kids had already done those things. I realized that I had inadvertently given my kids a big jumpstart, and what I had been doing was working, so I might as well keep doing it. And that's how I ended up deciding to homeschool my kids.
Because I homeschooled my kids, the label "gifted" didn't really come into play. I never needed to get them tested, or prove anything to a school official. But it was clear to me that they had the capacity to learn faster than what our school system provides, so I set out to do my best to keep them challenged and allow them to fully develop their individual talents.
How did you feel when you knew for certain that they were gifted?
It felt like a tremendous responsibility to help them achieve their full potential, even more so because I was choosing to take on that responsibility myself rather than relying on the school system. If things didn't work out, I would have no one to blame but myself. It was a lot of pressure. I didn't really doubt my capability to teach them, but I knew it would be an enormous undertaking, one that would require the bulk of my time and attention for nearly 20 years. It was well worth it — I'm so proud of the young adults they've become — but I'm relieved to have reached the point where that responsibility is behind me rather than in front of me.
As you looked into ways to meet their educational needs, what did you discover?
One pleasant surprise was realizing that it isn't necessary to be an expert in everything in order to meet your kids' educational needs. In many cases, it's more about playing the role of an advocate, connecting them up with the right resources. I would track down private teachers, public school classes, homeschool co-op classes, other parents, books, videos, and websites for the areas I was less equipped to handle. I did my part for the homeschooling community by serving as a teacher of math and computer science for many other talented kids whose parents were not as adept at those particular subjects.
What were some of the programs, opportunities or approaches that you found helpful in making the most of their talents and interests?
One of the things that was really important to me was to make sure my kids had a strong foundation in math and computer science. Fortunately, I have the advantage that I learned the IMACS curriculum when I was in middle school and high school. This made it easy for me to teach these subjects because I'd already experienced first-hand what a top-tier math and CS education looks like. To duplicate that experience for my kids, I simply taught them using the same materials and techniques that I had learned at IMACS and in college. Those resources played a big role in giving me the confidence to homeschool.
What were some of the benefits and challenges of homeschooling? How did you deal with the challenges?
One of the biggest challenges of homeschooling is that it takes effort to find a community to be a part of. We are fortunate that here, where we live in Seattle, there is a robust homeschooling community and we were able to find a group that fit our personalities and make many dear, lifelong friends. Once we found the right community for us, those rich social interactions became one of the biggest benefits of homeschooling. Our weekly park days remain some of my most cherished memories.
Another big challenge is that, even though my kids are close together in age, they were far enough apart that I couldn't do the same lessons with them. I always felt like I was struggling to give each of them the level of individual attention I wanted to give them. My solution was to make certain parts of the day relatively structured and leave the other parts of the day flexible so that I only needed to schedule the structured parts. For example, I might be doing structured math lessons with one child while the other was doing creative writing or watching an educational video, and then switch.
What were some other challenges you faced in parenting gifted children, and how did you deal with those?
Gifted children can be wildly asymmetric in their abilities, and wildly different from one another in terms of their talents and interests. It was a constant challenge to avoid getting impatient when one of my children would struggle with a concept that came naturally to the other. I had to keep reminding myself that it was totally normal for them to have different strengths and weaknesses.
How did being a gifted adult who was once a gifted child affect your educational and non-educational parenting choices?
This was especially helpful when my kids' interests dovetailed with my own. My son, for instance, developed a keen interest in programming, and thanks to my own background, I knew exactly what to do to support him in that interest. When my kids' interests and abilities diverged from mine, it was a lot tougher, and I ended up doing what any parent would do … seeking out people and resources that could help my kids develop their talents.
What are your fondest memories of raising Alex and Molly?
Probably my fondest memories are the many times we played board games together over the years. For those moments, instead of having a parent/child relationship, or a teacher/student relationship, we were just people playing games together and enjoying each other's company. I feel like at those moments, I really gained some insight into their character, and got a glimpse of what they would be like as adults.
What are your hopes for them in the future?
My hope has always been for them to be good, caring, compassionate, independent, happy adults. The good news is … they already are. I'm done! Woohoo!
Robot Repair was inspired by Mark's study of the IMACS’ Advanced Mathematical Logic curriculum.
Residents of the continental United States are eligible for the Robot Repair Raffle. The autographed copy of Robot Repair will be shipped only to an address in the continental US. To enter the raffle, complete the following steps:
- Parents must register their child to take either the EMF Aptitude Test (ages 10-14, approximately) or the eIMACS Aptitude Test (ages 14-18, approximately).
- On the aptitude test online registration form, type "IMACS Blog" in the box for "How did you hear about us?" or "How did you hear about IMACS?".
- Have your child take the aptitude test for which he or she is registered. The test must be taken during the period October 1-30, 2017. (No minimum test score is required to qualify for the raffle.)
RJ was an IMACS student from kindergarten through 12th grade and took every class IMACS offered, including all levels of Math Enrichment, Computer Enrichment, Hi-Tech Summer Camp, University-Level Computer Science, AP Computer Science: Java Programming, and University-Level Logic for Mathematics.
With a 5.9 GPA and 2360 on the SAT, RJ graduated as co-valedictorian and was honored as a US Presidential Scholar candidate, National AP Scholar and National Merit Scholar. He and his teammates were also four-time Science Olympiad state champions.
RJ was accepted to Caltech, Georgia Tech, Carnegie Mellon, and the University of Florida. He chose Caltech where he took junior-level Computer Science classes as a freshman and served as a Teaching Assistant for those classes as a sophomore. RJ conducted summer research in computer vision and was selected for an internship at Northrup Grumman. He will graduate with a double major in Computer Science and Philosophy and plans to attend graduate school to specialize in Machine Learning.
“IMACS taught me an organic approach to problem-solving, a way of thinking that demonstrates the necessary rigor required to solve genuinely challenging problems. The introduction to proofs and logic that I got at IMACS was absolutely critical to my success at Caltech.”
As a young child, Mason excelled in IMACS Math Enrichment classes and, in subsequent years, attended several weeks of IMACS summer camps. He later completed IMACS’ university-level courses in Computer Science and Logic for Mathematics. Mason was honored by the College Board’s National Hispanic Recognition Program for his achievement on the PSAT/NMSQT, and his high school mathematical modeling team was recognized by the Consortium for Mathematics and its Applications. Mason graduated high school with a perfect 4.0 GPA and 34 on the ACT.
A natural at sharing knowledge, Mason enjoys tutoring students in science and math at a local homeschool co-op, co-taught a robotics class for two years, and was a Teaching Assistant for an Honors Calculus seminar. He accomplished this and more while being an all-county varsity and travel soccer player and a dual-enrolled student at a regional 4-year university.
Mason chose Rice University where, at 16 years old, he was accepted early decision into the George R. Brown School of Engineering. He plans to major in Materials Science and NanoEngineering.
“My early exposure to IMACS provided me with a unique foundation in logic and mathematics that helped me succeed in advanced STEM classes at a very early age. I understand numbers intuitively and am able to solve complex problems in my head thanks to my IMACS training.”
This month's IMACS blog post is by guest author and eIMACS student, Naomi Spargo. Naomi recently sent us the following letter and kindly agreed to let us share it with our blog readers. Learn more about the eIMACS online courses that helped Naomi develop a passion for math and computer science and be prepared for a summer program in Artificial Intelligence at Stanford.
I recently got back from the Stanford Pre-Collegiate Summer Institutes program and wanted to let you know that it was absolutely amazing! It was very intense, but I also learned a ton. I learned about all the different techniques that fall under the umbrella of artificial intelligence such as machine learning, neural networks, and search algorithms. (I found the search algorithms that I learned in eIMACS very helpful.) I also learned answer set programming, which had a very steep learning curve for most of the class because they had never encountered formal logic before. I was at a big advantage because of the logic classes I took with eIMACS.
Other than a more sophisticated approach to coding, what I took away from Stanford was a better understanding of how sprawling of a field computer science is. I've always approached coding in a very mathematical manner because mathematics has always been my strongest area of expertise. At this camp, I encountered many other coders that aren't particularly strong in math but have a heavy background in something else, like robotics.
The way we approach problems is extremely different. Depending on the problem, my approach works much better than theirs or vice versa. There are so many different fields within computer science; each field lends itself to a different type of mind. I now see the need to specialize within computer science, perhaps on data analysis or cybersecurity (both math-heavy fields).
I also wanted to take this opportunity to thank you for making advanced coursework available and enjoyable for young people like me. Your courses have been a godsend in three primary areas:
1. They helped me develop a passion for math/computers.
I had a strong dislike for math classes before I took your courses because the way math is taught in schools is very dull. My biggest frustration with school math classes was how shallow the content was; we were essentially being taught to memorize symbols. Only occasionally were we offered a real understanding as to how the symbols connected. I hungered for deeper meaning in mathematics, and was doing my best to figure it out myself.
The only math I enjoyed was the work I did myself with a store-bought textbook and a calculator. I didn't associate the math I did with any structured "class" because every class I had taken was slow, boring, and surface level. I was excited to learn, but my approach to learning was unorganized and inefficient. There is only so much an eleven year old can do on their own.
eIMACS was a huge assist in this. Your courses explain everything in approachable language so I was not intimidated away as a young child, but gave me the deep understanding that I wanted. The content is presented in an interesting manner. The material was clearly written by someone who loves what they are doing, and the instructors who helped me with my questions share this attitude. This passion translated to me; I no longer felt alone when it came to my love of math. eIMACS courses pull off a rare trick in education; they explain complicated ideas to 11 year olds without sacrificing complexity. Your courses are comprehensive; you leave no questions unanswered. I enjoyed them more than anything I have taken in high school.
2. They gave me a competitive edge in nearly every subject.
It is self evident that the three computer science courses I took with you helped me with coding. The diversity of languages you offer is a huge benefit, because certain languages lend themselves to certain problems. At Stanford, the AI class teamed up with the cryptology class to crack some numerical-based codes. Haskell, which I knew thanks to you, particularly lends itself to interpreting numerical data, so I could crack the codes much faster than students who were confined to object-oriented languages like Java. You also gave me experience with complex data structures, which most coders my age have not encountered.
Similarly, the logic I have taken with eIMACS helped me with math. I often reference my tautology template when proving theorems. It gives me an organized way to express my theories and postulates. I see logic as a honing device for mathematical thought. However, the reach of these courses goes far beyond math and science. My English essays (especially the persuasive ones) often employ tautologies written out in prose form. Logic has made my writing more coherent and streamlined; it has improved my word economy because I can organize my thoughts more efficiently.
My strong foundation in logic has also made me a more persuasive speaker. I do a form of debate called extemporaneous speaking, where I have thirty minutes to prepare a seven minute persuasive speech about a current political topic. Because my time allotment is so small, I need to break down complicated economic and political concepts quickly so that the average person can understand them. Using rules of logic is an excellent way to do this. My debate coach has told me I have "logos for days"; I directly attribute this to my eIMACS courses.
3. They allowed me to specialize on what I truly loved earlier than most students.
eIMACS courses are so much more advanced than what most high schoolers are exposed to. I know much more about the fields I'm passionate about and can specialize sooner as a result. Most kids my age have no idea what they want to major in. Because of IMACS, this is not the case for me.
I feel very fortunate to have taken eIMACS courses and am thankful to all my IMACS instructors for inspiring and encouraging me. I know that my eIMACS background has prepared me well for college and whatever I decide to do after, and I feel confident that the way of thinking logically that I learned in eIMACS will help me face any challenge.
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!
As most people know, Santa Clause generates an annual list of children who have been sufficiently nice to warrant a Christmas present (and, of course, a complementary list of children who have been too naughty to deserve a present). Last December, while checking his list twice, Santa discovered that the status of four children — Amy, Bobby, Charlie, and Donna — remained in doubt. He decided to talk to the four of them in order to rank them from nicest to naughtiest. In his conversations, the children made the following claims:
Amy said, "I am nicer than Charlie."
Bobby said, "I am naughtier than Donna."
Charlie said, "I am neither the nicest nor the naughtiest of the four of us."
Donna said something as well, but Santa couldn't hear her over the noise of eight hungry and overworked reindeer.
As was, perhaps, to have been expected, the two nicer children both told the truth and the two naughtier children both lied. Can you order the children from nicest to naughtiest?
Try your best before looking at the solution to this year’s IMACS Holiday Logic Puzzle.
Acceleration Helps But Gifted Kids Need More
For mathematically talented children, acceleration is a commonly used approach to address their need to be challenged. While acceleration can save gifted math students from unnecessary repetition, by definition acceleration cannot go deeper than the standard curriculum, and deeper is what many of these students desperately need. Mathematically talented students deserve to be inspired by engaging curriculum, not merely less bored. Educators know this and are seeking better solutions that offer gifted math students the opportunity to study genuine mathematics with the depth and sophistication that match how their gifted minds work.
Common Characteristics of Gifted Children
The National Association for Gifted Children (NAGC) lists common characteristics of gifted children. Twelve that relate particularly to the study of mathematics are:
- Rapid learner; puts thoughts together quickly *
- Excellent memory *
- Advanced comprehension of abstract ideas
- Enjoys solving problems, especially with numbers and puzzles
- Thinking is abstract, complex, logical, and insightful
- Longer attention span and intense concentration
- Learn basic skills quickly and with little practice *
- Asks probing questions
- Highly developed curiosity
- Interest in experimenting and doing things differently
- Puts idea or things together that are not typical
- Vivid imaginations
Of these twelve characteristics, acceleration can make maximum use out of only 1, 2 and 7 to deliver a learning experience that is somewhat better than no accommodation at all. The other nine characteristics require an approach that is fundamentally different from the standard curriculum, which is designed to work with typical students.
EMF Understands Gifted Thinkers
Elements of Mathematics: Foundations (EMF) provides that fundamentally different approach. Developed by a team of mathematicians who teach gifted children, EMF was designed from scratch to leverage all 12 characteristics. The result is a curriculum that allows talented students to immerse themselves in the study of mathematics where their unique way of thinking fits naturally and is an enormous advantage to learning at the highest levels.
One of the most important aspects of EMF is that it allows students to experience the excitement and satisfaction of intellectual discovery. The standard curriculum, accelerated or not, tends to present the final result as a given to be accepted blindly. By contrast EMF emphasizes the path leading up to the result and teaches some basic techniques of logic used to get there.
EMF students are guided through thought-provoking exercises that lead them to keen observations. Then using logical and creative thinking to analyze and synthesize this information, students are able to arrive at the result themselves. This approach creates a much deeper, intuitive understanding of mathematics and is strongly aligned with how gifted children learn, especially the nine characteristics that acceleration ignores.
Gifted Math 2.0
Acceleration has been an acceptable compromise solution for gifted math students and their educators for a long time. Many gifted students, however, have long needed and deserve better. They need to learn mathematics in a way that embraces their endless curiosity and unique way of looking at the world. They need a curriculum designed specifically for the way their minds work. That curriculum is EMF.
IMACS Staff Writer @ 1:00 am
Fiona first attended IMACS Hi-Tech Summer Camp when she was ten. Over the next eight years, she went on to complete IMACS’ university-level courses in University Computer Science, AP Computer Science: Java Programming, and Logic for Mathematics. Homeschooled since fifth grade, Fiona was awarded a National Merit Scholarship, named an AP Scholar with Distinction, and scored 2360 on the SAT. She is also a second degree black belt in Taekwondo and won a bronze medal at the 2013 AAU Junior Olympic Games.
As a high school student, Fiona studied four years of undergraduate, advanced undergraduate, and graduate classes in mathematics and physics at Northwestern University where she is the only student to have won the Outstanding Achievement in Advanced Mathematics Classes by a High School Student prize four years in a row. During that time, Fiona also served as a Teaching Assistant for the University of Chicago’s Young Scholars Program, a position usually reserved for undergraduates.
Fiona chose the University of Chicago, where she intends to major in Mathematics and Physics. She plans to pursue a career as a university professor of mathematics. Fiona directly credits IMACS for helping her to develop a deeper appreciation for and a sense of ownership of the mathematics that she learns.
“Logic and computer programming classes at IMACS taught me to how to organize my thinking and break problems into solvable pieces. The natural structure of logic and computer programming allowed me to see the structure of a mathematical proof as I was writing or reading it. When I understand it at that level, I feel I can reconstruct it and use it creatively as well.”
Toolboxes are handy things to have around, especially when they are filled with useful tools. A maker might include a hack saw, hammer and soldering iron in her toolbox. A musician might include a metronome, pitch pipe and recording device in his toolbox. What would you find in the toolbox of a problem solver? Many things, to be sure, but among them is likely to be an array of logical reasoning skills.
Logic promotes clarity of thought in understanding ideas. As the complexity of our world increases, each of us will face a growing number of unfamiliar and challenging situations. The shift to a knowledge-based economy keeps progressing. The significant decisions we must make keep multiplying. People are needing to think critically more and more but with less and less time and information. In a world such as this, logical reasoning is a powerful skill that can help a person sort through the noise and think clearly. When faced with new and difficult questions, the experience of having solved hard problems with logical thinking can be the difference between a panicked "Where do I even begin?!" and a calm "Let's see where to begin."
For example, you might start by looking at the problem from different angles to see what's inside. Investigate further as you anticipate obstacles and how they might affect your analysis. Then build a mental map to help you navigate your way. What are the different paths you can take? As you develop a feel for the problem, divide it into more manageable pieces. Working your way through the pieces will likely require sideways thinking with creative approaches and capturing those tricky insights that flutter in and out of focus. Connect the dots as you build a way forward. When you've got all your pieces solved, glue them together for a complete solution!
People who learn to think logically are simply better equipped to analyze complex problems. From Brexit to antibiotic-resistant bacteria, unanticipated developments often lead to unprecedented situations that leave many smart, experienced people wondering what to do. Solving problems does not stop at the schoolhouse door. It is when you step into the "real world" that, if you are fortunate, the most interesting and difficult problems present you with opportunities throughout your life to make a meaningful difference. What will be in your child's toolbox when those opportunities arise?
Logic is the systematic study of reasoned argument. Why study logic? For students planning to take university-level math courses, which typically involve reading and writing proofs, the benefits are evident. But even for students heading in a different direction, developing strong logical reasoning skills has benefits too.
Logic Supports Learning Mathematics
Few would dispute that arithmetic is a fundamental base upon which the successful learning of math is built. Consequently, the development of arithmetic skills receives enormous attention in elementary school. Logic is equally fundamental to learning all other areas of math but is rarely taught in elementary school. Were students to develop a strong foundation in logic at a young age, their capacity to understand middle and high school math would be greatly enhanced.
In arithmetic, we learn how to combine or break apart numbers in order to proceed from the information given to the answer sought. If we bake three dozen cookies, multiplication tells us how many cookies we have in all. If 17 cookies are left, subtraction tells us how many were eaten. Honing arithmetic skills in elementary school allows students to quickly and accurately call upon them later, thereby making it is easier to learn more advanced math in the future.
Logic is a similarly powerful and essential tool for learning more advanced math. Whereas arithmetic is used in constructing and deconstructing information that is numeric in nature, logic is essential in constructing and deconstructing assertions that are mathematical in nature.
Students typically encounter a weak version of proofs and logic for the first time in high school geometry. But by then, pre-algebra and elementary algebra have already been taught as mostly procedural rules and algorithms to accept at face value. As a result students do not develop a genuine understanding of the derivation of those rules and algorithms and why they work. By the time students reach high school, their natural curiosity about mathematical ideas has been reduced to a hard-to-break habit of memorizing and applying rules blindly.
A worthy goal of the Common Core math standards is to develop mathematical understanding in addition to procedural skills. True mathematical understanding comes with seeing how knowledge is pieced together step-by-step on the path to deriving an important result. This is similar to what we know about developing scientific understanding in young minds — hands-on projects make learning stick better than memorizing facts. Math’s version of "hands-on" is your brain working its way through the reasoning that mathematicians followed in order to provide us with neat and tidy rules.
With a strong foundation in logical reasoning, students would have the key analytical tool needed to work their way successfully through the reasoned arguments that lead to the rules. Some may even wish to achieve the deepest understanding by using logic to independently derive the mathematical truths they were once asked only to memorize.
Logic Supports Complex Problem Solving Beyond Mathematics
The amazing thing about logic is that it makes us better thinkers and problem solvers well beyond mathematics. Computer science could not exist without logic. If you want to understand the underlying principles and algorithms and be able to apply them in any programming environment, you must understand logic. Engineering disciplines rely on logic too, especially electrical engineering as any Minecrafter worth her redstone will tell you. The entire practice of law is founded upon reasoned argument, which is why the LSAT has sections on logical reasoning and analytical reasoning.
People who learn to think logically are better at analyzing complex problems. Mathematics, with its diversity of complex problems, is a natural thought playground for developing logical thinking skills. The first steps in solving any math problem are assessing the situation and coming up with a plan of attack. Can you view the larger problem as a combination of more manageable cases? Can you take what you know and derive what you need to know? Would additional information lead you to certain conclusions? Answering these types of questions requires logical thinking.
People who learn to think logically are also better at assembling solutions. Once you have assessed the problem, broken it down, and gathered the necessary information, how do you put it all together to arrive at a solution? When faced with a novel situation, can you devise an approach where there was not one before? As the pace of change in our world accelerates, every field increasingly needs people who can strongly grasp core concepts and combine them sensibly to solve any problem, especially unfamiliar ones. Learning to apply logical thinking to mathematical situations is excellent training for handling complex problems, no matter which college major or career you choose.
Logic has a way of promoting clarity of thought in understanding ideas that were previously unfocused or muddled. Whether you are planning on becoming a professional mathematician or just trying to get through math homework on your way to a different intellectually demanding career, there is no doubt that strong logical reasoning skills would ease your path.
This blog article is part of the Hoagies' Gifted Education Page March 2016 Blog Hop on Math Education Enrichment. Please click on the Blog Hop image above to learn more about Math Education Enrichment from other Hoagies' Blog Hop participants.
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