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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.
Note: This story is adapted from our IMACS Math Enrichment Program for talented students in grades 1-8. The solution and analytical process are revealed at the end of this post, so don’t scroll down too far if you want to avoid the spoiler!
Max, The Lost Dog
Sofia, a bright IMACS student, has lost her dog, Max. Luckily, Max was wearing a collar when he was found by another IMACS student, Emily, who loves using her logical reasoning skills to solve mysteries. Tucked into the collar was a piece of paper with clues on where to return Max should he become lost. Follow the clues to help Emily find which house is Max’s so that Sofia can be reunited with her dog … and possibly make a new friend in Emily. (Since both girls take online math classes through eIMACS, they had not met each other yet.)
Clue 1: The number of the house where Max lives is the sum of two numbers, each of which is a power of 2 that is greater than 0 and less than 10. These two numbers that add up to equal Max’s house number can be the same, but they don’t have to be.
Clue 2: Max lives in Galois Gardens, a neighborhood represented by the following arrow diagram in which each dot represents a house.
The fork to the left is Leibniz Lane, the fork to the right is Abelian Avenue, and the house at the bottom is on Via Vertex. Here is how you interpret the arrows:
• The house number at the end of a red arrow is equal to the house number at the start of that red arrow plus 3.
• The house number at the end of a blue arrow is equal to the house number at the start of that blue arrow plus 4.
Clue 3: The biggest number for any house in the neighborhood is 15.
Clue 4: The number of the house where Max lives is a multiple of 3.
The Analytical Process of Solving the Mystery
Now that you’ve given yourself a chance to work through the problem, let’s go step by step to understand one way we can think about putting the clues together.
Clue 1: What are the powers of 2 that are greater than 0 and less that 10? They are 1=20, 2=21, 4=22, and 8=23. So we need to take one number from the set {1,2,4,8} and a second number from the same set {1,2,4,8} and add them together to come up with the list of possible house numbers. Remember that both numbers can be the same but do not have to be. We’ll leave the arithmetic to you, but we should all arrive at the following possibilities:
Clue 2: Although we don’t yet know any of the house numbers in the arrow diagram of the neighborhood, what can we figure out from the information given? One thing is the location of the house with the smallest number. It has to be the house at the bottom of the picture because the arrows tell us that every time you go from that house along either of the two roads, you add 3 or 4 each time. So you keep getting bigger and bigger house numbers, and they never fall back below the number you started with at the bottom, whatever that number is.
Likewise, is there anything we can say about the largest house number in the neighborhood? Well, it’s probably at the top of the diagram since the house numbers get bigger as you go up the roads. But is it the house on the top left or the one on the top right? Now we need some variables now to help us keep track of these houses.
Let’s call the house number at the bottom “A,” the one on the top left “B,” and the one on the top right “C.”
We know B is bigger than A, and C is bigger than A. Can we say anything about how much bigger B and C are than A? Well, to get from A to B, we add 3 and 3 and 4. That is, we add 10 = 3 + 3 + 4. So A + 10 = B. To get from A to C, we add 4 and 4 and 3. That is, we add 11 = 4 + 4 + 3. So A + 11 = C. We’re starting with at same number, A, and adding 11 to get to C but only 10 to get to B, so C must be bigger than B. Now we know that the house with the biggest number is on the top right.
Clue 3: This clue, along with what we figured out from Clue 2, tells us that C = 15. In other words, the address of the house at the top right is 15 Abelian Avenue. With a little arithmetic, we can now determine the rest of the addresses in the neighborhood as follows:
Now that we have all the addresses in the neighborhood, we can go back to our original list of possible house numbers from Clue 1 and cross off the ones that do not show up in the neighborhood. Here’s what our list should look like now:
Clue 4: We’re down to only three possible house numbers—4, 8, and 12. Clue 4 tells us that the house number is a multiple of 3. Which of these possibilities is a multiple of 3? Only 12 is a multiple of 3. Hooray! We figured out where Max lives—12 Abelian Avenue! Sofia will be so happy to have Max back at home, and Emily will be quite pleased with herself for having solved the mystery and done a good deed.
There’s no mystery as to who has the best online gifted math curriculum. IMACS! Register for our free aptitude test. Play along with our weekly IMACS logic puzzles on Facebook.
Show a child some tricks and he will survive this week’s math lesson. Teach a child to think critically and his mind will thrive for a lifetime.
Math word problems confound many students for a variety of reasons. Too often, well meaning parents and teachers attempt to help students struggling with these kinds of problems by offering them tricks or shortcuts for getting to the solution without necessarily understanding what’s going on. Just do an internet search for something like “how to solve word problems,” and you will get a slew of Web pages with tips, tricks and strategies like the following:
• Search for key words that will tell you which mathematical operation you should use (e.g., “lost” means subtraction).
• Cross out non-essential information so that you are not distracted by it.
• Draw a picture to illustrate each step of a problem.
Let’s consider these three examples for a moment. To suggest that looking for key words within the text of a math problem is somehow a “strategy” to solving it more easily seems a bit absurd. Basic reading comprehension skills are obviously essential to solving any problem, math or otherwise, where the information is communicated via natural language. No key word mapping is going to provide a fool-proof way of understanding what a word problem is asking. If you’re starting with a word problem that is well designed and written with clarity, there simply is no “trick” to understanding the words.
The other two examples sound reasonable on the surface, but a closer examination reveals problems with them too. To determine which pieces of information are relevant to the solution (as in Example #2), you need to be able to evaluate and analyze the given information. To correctly translate the words of a problem into a step-by-step illustration (as in Example #3), you need to be able to understand and prioritize the information. In other words, tricks like these seem to require the very skills needed to solve the problem without the tricks! Furthermore, a “tip” cannot tell you whether you have enough information to solve the problem, or how to find any missing information. Only critical thinking and logical reasoning skills can help you there.
This approach of relying on “clever strategies” to replace critical thinking runs counter to the IMACS philosophy of teaching children the fundamental skills they need to solve problems throughout their lives, whether they be elementary word problems or complex Calculus problems, whether it’s making correct change or correcting the flight path of an exoplanet-bound space craft.
Meaningful Strategies That Matter
While silly tricks may work for some students as a short-term fix to “just get through the problem,” there are practical and helpful techniques that should be part of any meaningful approach to understanding and solving word problems.
Pay attention to units of measurement. Word problems commonly feature units of measurement. When finding a solution, it’s important to pay close attention to how you are adding, subtracting, multiplying, or dividing amounts in the problem so that you don’t end up combining “apples and oranges.” If you carefully track the units of measurement through each step of your computation, you can compare the units of your answer to what it should be based on the question. This is a good technique to check the reasonableness of your answer rather than one that “magically” solves the problem for you. Here’s an example:
No one would think that the following approach is right
because adding children to cookies makes no sense! The answer requires a relationship in the form of “X cookies per child” – or, more algebraically,
With this in mind, what makes sense is the following:
While we’re on the topic of cookies, here’s another example:
Most people know to do something with ( 4,500 / 3 ) and 5, but what? Looking at the question in the problem, we know that our answer should be in the form of “X cookies.”
What if we set up the equation correctly
but then worked out our answer as follows?
Hmm … 300 cookies per minute per minute? The fact that our answer is not measured purely in cookies as expected tells us that we made a mistake in our calculation.
Use reverse word problems to practice abstract thinking. The process of solving word problems often requires us to connect a real-life situation described in language we know with the abstract version written with variables and equations. As with learning to speak a foreign language, thinking abstractly becomes more natural with practice and regular use, and the earlier you start, the easier it is.
When kids are younger, parents can help them to develop an intuition for abstract reasoning by doing “reverse” word problems in the guise of creative play. Start by writing down an equation such as 7 + ___ = 10. Next, ask your child to imagine a story that goes with this equation. He might say something like, “It takes 10 gold stars to get extra recess, and I already have 7. How many more stars do I need?” For 3 + (8 × 2) = 19, she might say, “When I started robotics, I took 3 introductory classes. Then, I took 2 classes per week for 8 weeks. So, I have taken a total of 19 classes.”
Be sure to keep the level of difficulty appropriate for your child’s math ability. Frustrating him with a computationally hard problem will only defeat the purpose of teaching him to think abstractly.
Word problems can be used effectively to teach important mathematical concepts and to help give real-world context and purpose to what may seem like useless hieroglyphics to some people. But without a firm foundation in critical thinking and logical reasoning skills, a student is sure to be at a disadvantage when trying to evaluate, analyze, prioritize, and synthesize the numerous pieces of information embedded in the narrative. Tips, tricks, and strategies may help at the margins, but there is simply no substitute for genuine understanding.
When your school’s gifted math classes are not challenging enough, it’s time to try IMACS. Register for our free aptitude test. Play along with our weekly IMACS logic puzzles on Facebook.
Brandi Parsell leads a discussion with IMACS students in a math enrichment class.
Former IMACS instructor, Brandi Parsell, offers advice on how to address the ultimate question in a way that stimulates logical reasoning and critical thinking skills.
It can be endearing, or at times downright frustrating – that eternal question, “why?”. When bright children discover that single word, they seem to grab onto it and won’t let go. Sometimes the answers are simple, and sometimes we find ourselves at a complete loss for words.
This innocent question, however, is a signal to parents that a child is ready to be challenged to think logically. The creativity is there – we can see it in their everyday play. It is how we encourage that creativity and shape it into critical thought that will form a solid basis for a child’s learning ability.
Critical thinking is one of the hardest subjects to teach older students; any schoolteacher will tell you so. But if you begin to give your children the necessary tools when they are as young as three or four years old, they can develop these skills more easily. When the question of “why” is put once again on the table, the best policy is to ask, don’t tell. Challenge your children; ask what they think the reason might be. Chances are you will be pleasantly surprised.
Often parents believe that when their child reaches school age, he or she will at last find satisfaction for that curiosity. Talented students, though, may become bored with traditional school curriculum. When such a student is not challenged to exceed our expectations, this frustration often takes the form of careless errors and lack of effort. If children begin to develop these kinds of bad habits, too often they give up quickly when faced with a truly challenging problem. It is important that bright students are encouraged to go beyond what is merely “expected” of them.
“Talented students owe it to themselves to stretch their minds as far as they can,” said Burt Kaufman, co-founder of IMACS. Burt spoke from experience. For over 40 years he worked closely with bright pre-college students and developed challenging mathematics curriculum materials to stimulate them to become true students – disciplined logical thinkers with an insatiable thirst for knowledge and understanding.
Parents are a child’s first teachers, and the best teachers don’t give away the answers. Turn your child into a detective, and yourself into their greatest source for clues.
An IMACS foundation in logical thinking sets students on a path to successful learning. Take our free aptitude test to begin your IMACS journey.
IMACS introduced Boosted Learning for Achievement on Standardized Tests (BLAST) in 2007 to help second graders who are struggling in math. After receiving a growing number of inquiries from parents searching for an effective remedial math program, we saw that it was time to make the benefits of our curriculum more widely available. IMACS is all about helping students reach their potential in math, and we firmly believe in this philosophy for all children regardless of natural ability.
BLAST, which is funded primarily through corporate sponsorship, is open to students who score below the 70th percentile in problem solving in first grade on the Stanford Achievement Test. Here are some highlights from our inaugural BLAST class:
• The average increase in math score across all students was 15 percentage points.
• Two-thirds of the students improved their math scores. The average increase among this group was 27 percentage points.
• Of the students who improved their math scores, more than half also improved their reading scores by an average of 19 percentage points.
• Every student in the bottom third of the class according to the first year’s math scores increased his or her math score. The average increase among this third was 29 percentage points.
• All but one of the students in the bottom third of the class according to the first year’s reading score increased their reading scores. The average increase among this third was 23 percentage points.
While our BLAST kids gained a lot from their IMACS experience, we were also able to take away some valuable lessons on how to help students who are having difficulty with math.
Learn to Think, THEN Think to Learn
Most children who come to us struggling with math have not sufficiently developed the ability to think logically and critically. And this is true whether they are below average or gifted and talented. Most have never been exposed to any kind of training in this area because the traditional approach to teaching is based on the assumption that logical reasoning and critical thinking skills will develop almost as a side benefit of studying the core subjects. It’s not quite putting the cart before the horse. Rather it’s more like having a stand-alone cart and hoping that a horse will trot on over to be attached. What we’ve found in both our BLAST program and in our advanced math program is that if you start children off by teaching them how to think logically, it makes a world of difference in their ability to learn whatever material comes after. And mathematical logic, like other subjects, can be taught and learned. It’s the intellectual equivalent of pumping iron in the gym when you’re training for competitive sports. Students first build up their critical thinking skills so that they are ready and, just as importantly, feel ready to be take on a challenging subject.
Individual Attention Encourages Risk-Taking
So how do we help students feel ready? It’s really just about rekindling and harnessing the natural curiosity they were born with. When a child is struggling with math, he or she may not feel comfortable speaking up in large group settings to answer questions or ask for help. Children with above average math abilities may feel the pressure of expectations. They may believe that they have to understand right away and cannot make mistakes. Those who are not naturally strong in math may fear being labeled as “dumb” if they give incorrect answers or admit that they don’t understand. They may also worry that others will think they are trying to “act smart” if they participate in class. At IMACS, we are able to break through this barrier of self-consciousness by working with children in small groups. When our students realize that they are in a place where it’s okay to take risks and make mistakes, they let down their guard and allow their natural curiosity take over. Simply put, they rediscover the joy of learning and leave the program feeling empowered to take on challenging work. A parent of one of our students summed it up beautifully: “BLAST went beyond my expectations. In a matter of 10 weeks, I have seen my child work with numbers on her free time and love math. Her self-esteem is soaring unbelievably.” That’s exactly the kind of feedback that makes us want to rise and shine each and every morning.
While we have our own in-house logic exercises that we use with our students, we’re also a big fan of logic games by ThinkFun. We like that ThinkFun games are affordable and constructed well to last a long time. And they usually have multiple levels of difficulty, which makes them accessible to a wider range of ages and abilities. Another great source are the weekly logic puzzles posted every Friday on our Facebook page. If you have a suggestion for other logic games or ways to help struggling students, please leave a comment.
If you are interested in being a corporate sponsor of BLAST, please send us an email at info @ eimacs.com.
IMACS online classes are designed to fit into your schedule. Take our free aptitude test to see if our program is right for you.
IMACS’ Senior Curriculum Developer, Edward Martin, recalls a lesson learned from his own father about the virtues of honing one’s mental agility with games, an exercise that is particularly effective for developing abstract thinking skills.
When he was in a reflective mood, my dad occasionally revealed interesting episodes from life as a younger man. He recalled, for example, for several years riding the train to work in company with two fellows who spent the hour-long journey either with eyes closed or staring into space while sporadically uttering cryptic phrases such as “D five to B six.” Puzzled by this behavior, after a few weeks my dad asked them what they were doing. It turns out that they were playing chess…without a board! The amazing thing was that they were usually able to agree upon the outcomes of these wholly-imagined contests.
Few of us are blessed with such powers of concentration. However, the ability to concentrate is something that can be trained. As with many such faculties, it improves with use. Even 8- or 9-year-old youngsters are capable of feats approaching that of my dad’s glassy-eyed chess-playing companions. Next time your energetic third or fourth grader is acting up in the car, challenge him or her to a game of tic-tac-toe…without the board. Brush aside any incredulity that such a thing is possible. Just jump in and say, “I’ll go first. I’ll be X and I’ll put my first X in the top left corner.” Before you know it, your recalcitrant fellow traveler will be caught up by the spirit of competition and will be totally focused upon the goal of putting you in your place for having the temerity to issue such a challenge.
In this quick and simple little exercise, you achieve: honing concentration skills, thinking about and communicating grid-based information, and peace and quiet. It’s a win-win situation!
What are some of the mind-stretching games that you play or played?
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Why do we bother teaching children their ABC’s when they are so young? It’s not as if they are going to write the next Gettysburg Address once they know the difference between b and d. Why do piano teachers spend so much time with their beginning students on proper finger placement? It’s not like they are going to compose a symphony once they know that a treble clef means play these notes with your right hand. So why does IMACS start students off with an introduction to logic course? It’s not as though they’re going to solve the twin prime conjecture once they know the difference between modus ponens and modus tollens.
Now before you leave a comment about why that’s not a perfect analogy, you’re right. The point was to push your thinking in that direction before you think about this: Why do we use an exaggerated voice when talking to a baby that doesn’t yet understand words? Why do we toss off aphorisms to small children who don’t have the life experience to know what they mean? (Here’s a fun list if you want to practice sounding clever and wise.) It’s because when a growing brain comes to understand the nuances of tone, the subtleties of language, the connotations beyond the denotations, then and only then can its potential be realized as the developed brain of a novelist, journalist, and yes, even blogger unleashing the full power of human communication. Now, it’s a long time between teething and typing away at the next great idea. But parents still talk to their kids in these ways because they understand, if only subconsciously, that it’s part of teaching children how to express themselves effectively in adulthood. That’s likely how their own parents spoke, and so it’s very familiar.
Formal logic, on the other hand, is not as familiar. So the question naturally arises, “Why spend so much time teaching it when all I’m looking for are advanced classes for gifted and talented students?” For those who go on to major in math or computer science at college, the benefits of taking logic courses are obvious. They will be expected to write complicated proofs or programs that are logically coherent. If they choose these fields as professions, their ability to make a living (and stop mooching off of you) will largely depend on their ability to do this really well. And just as with teaching the skills that lead to great story-writing, you don’t start when the kid is already in college. By then, you want your college student to already have these tools so he can blow away the curve and make you so proud that you will not stress when he goes on his first spring break trip where there will be no mischief whatsoever. You start earlier by first teaching them how to craft a well formed and logically consistent argument, and then you layer on the advanced courses that require this fundamental skill in order to be successful.
But what about students who are not going into math or CS? Well, are there any future philosophy majors out there? How about pre-law? Wouldn’t you know it, they too will need the ability to make logically coherent arguments in their coursework, whether it’s debating the forms of Plato or taking the opposite side of a Supreme Court decision. Speaking of debate, we’ve had a number of high school students tell us that their logic skills helped them to excel in debate class. Come to think of it, if your child plans to be in a position where she needs to think about ideas in an orderly fashion, connect pieces of information together, draw conclusions from her analysis, and present her argument for scrutiny, understanding logic would be a huge advantage. Someone stop me now before I break out into “I’d like to teach the world a proof…”
Do you have a story about a situation in which understanding logic helped you? Leave it in a comment below.
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