Drawing on the Right Side of the Brain is a drawing-instruction book, first published in 1979. In her book, Betty Edwards, Professor of Arts at California State University, builds on the then novel theory that the two halves of the brain function differently. The left hemisphere is responsible for verbal, abstract, symbolic activities. The right hemisphere serves for synthetic, holistic, intuitive perception and information processing. Under normal circumstances, the left hemisphere is the more active of the two. The book offers a series of exercises designed to subdue the rational, left side of the brain while firing up its right, imaginative part. The book carries the subtitle of A Course in Enhancing Creativity and Artistic Confidence. In the preface to the second edition (1989), the author describes how surprised she was to discover that, in the 10 year period following publication of the book,
… individuals and groups working in fields not remotely connected with drawing have found ways to use the ideas in my book. A few examples will indicate the diversity: nursing schools, drama workshops, corporate training seminars, sports-coaching schools, real-estate marketing associations, psychologists, counselors of delinquent youths, writers, hair stylists, even a school for training private investigators.
Conspicuously absent from this list are school teachers, and math teachers in particular. I do not believe the above passage is the result of a purposeful statistical study and do not intend to draw farfetched conclusions from it. It just made me ponder whether, holistically speaking, a similar approach may work for math instruction. Drawing is an R-mode (R for right) activity. Now, what can be said about mathematics? Mathematics is verbal for it’s a language, and it is abstract for in its heart one finds proofs and axiomatizations. It is symbolic, rational, logical, and, nowadays, it is very digital. These are all indications that mathematics is firmly entrenched in the analytic domain of the L-mode (L for left) way of thinking and perception.
On second thought, if the L-mode prevails for most people, why is innumeracy so pervading? After all, the left, analytic hemisphere of the brain is dominant more often than not. So it must be that mathematical thinking is R-mode. It’s math instruction that has come to depend on the L-mode. If this discrepancy is the source of widespread innumeracy, should math instruction not exploit more of the R-mode facilities?
I take at the face value the fact that Edwards’ book has been used in the circles far removed from the art of or the need for drawing (nursing schools, corporate seminars). When you teach creativity as a stepping stone for drawing, you teach a skill that can be applied elsewhere, even outside the classroom. This leads to a criteria for evaluation of the current math instruction methods. Does anything taught in the math classroom prepare students to face aspects of reality not directly related to math? Students who have taken B. Edwards’ course have also learned to see the world differently and to better use their mental powers. Is this true of math instruction? Unfortunately, I think, not. Even more so when math instruction emphasizes the “real world problems”: most of these problems are dull and have limited utility anyway. It appears possible to learn drawing for drawing’s sake and acquire a more universal skill along the way. It should be possible to perform the same feat in math classrooms. I assume math instructors would be proud to be nurturing their students’ imagination.
One of the arguments against New Mathematics was that its heavy formalism had little to do with how mathematicians really work. Humanistic Mathematics presents it in a humane way with a human context. One aspect of which is how the real mathematicians do it. No, not every one was born to become a mathematician, but that is besides the point. Future biologists dissect worms and frogs in biology labs as do future engineers and literary agents. Without creativity and imagination mathematics would not be possible. As Edwards’ experience demonstrates (see also books by E. de Bono), these skills can be taught. I suspect that good math instruction should foster students’ creativity in a deliberate manner and as a part of curriculum.

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The abacus has been around in various forms for over 2300 years. It was used for various counting and operational tasks. One might even call it the original math manipulative (unless you count fingers and stones). In my younger years, abaci were relegated to the bottom shelf or used as a toy for the kinesthetic kids. These days, abaci can meet the same fate that the abaci of my youth did. The first known abacus, the Salamis tablet, collected dust for over 2100 years. For all those lonely and banished abaci on dusty shelves everywhere, I dedicate this article on how to represent, add and subtract whole and decimal numbers.

As most teachers know, the use of manipulatives by younger elementary students helps them to understand the concepts of place value and operations later on. In my search for a variety of manipulatives to teach number sense, addition and subtraction, I came across a convenient tool in the abacus. I’m sure it was no coincidence that each row on the abacus included exactly ten beads, but there was no operators manual with the abacus I found. When I found an instruction manual several years later, I found that the manufacturer of the abacus saw it as no more than a counting device and had no idea of the place value power inherent in the design.

Representing Numbers With a Dusty Abacus
When I first started using an abacus as a manipulative in math class, I was teaching grade six. In the grade six curriculum, students were supposed to represent whole numbers greater that one million and decimal numbers to thousandths. If you count the number of places from one million down to thousandths, you get ten places. Coincidentally, the abacus had ten rods of ten beads each. I’m sure what I discovered was discovered long ago, and some manufacturers probably even send out better instruction manuals that make note of this, but at the time, it was a completely new discovery.
To make a long story short, I assigned each row a specific place value starting with millions at the top, and thousandths at the bottom. One could use a strip of tape or an indelible marker to label the rows. To represent a number, a student would simply move the number of beads for the value of each place in the number they were given. For example, the number 325,729 was represented by moving three of the hundred thousands beads, two of the ten thousands beads, five of the thousands beads, seven of the hundreds beads, two of the tens beads and nine of the ones beads.
I didn’t have a class set of abaci, so I made up little sketches of an abacus (six or so per page) and students showed representations of numbers using these.
Adding and Subtracting Numbers With a Polished Abacus
Once students are familiar with representing numbers using an abacus, they can move onto adding and subtracting numbers. The idea of adding using an abacus and place value is quite a simple process. Begin by representing the first number. Add the value of each place value in the second and subsequent numbers one at a time beginning with the lowest place value and regroup as necessary.
Consider this simple example, 178 + 255. The student would represent 178 on the abacus to begin. She would then add five to the ones row. Since there aren’t five more beads to add, this first move would also involve regrouping. The student would move the two remaining ones, then regroup by sliding all ten ones back and replacing them with a ten. She would then move three more beads since she already moved two of them for a total of five. Since there was some regrouping, there would now be eight tens. The students needs to add five more, so there would be another regrouping, this time of ten tens to make a hundred. Finally, the student moves two additional hundred beads; this time regrouping isn’t necessary. If everything was done correctly, the student would end up with four hundreds beads, three tens beads and three ones beads.
A variation on addition is to add the second and subsequent numbers from the highest place value to the lowest place value.
Subtracting is much the same as addition, but it involves “removing” beads. The procedure for subtracting is to represent the first number then to subtract the value of each place value in the second and subsequent numbers beginning with the highest place value.
Consider this example, 3.252 – 1.986. The student would first represent 3.252 using the abacus. He would begin by subtracting one one. This is fairly straight forward because there are enough ones available. In the next step, though, the student has to subtract nine tenths from two tenths. He begins by subtracting two of the nine tenths, but he then has to regroup one of the remaining ones into ten tenths. Once he has ten more tenths, he can subtract the remaining seven tenths. He continues by subtracting eight hundredths from five hundredths, and again, he has to regroup, this time, one of the tenths into ten hundredths. The final step also involves regrouping since six thousandths must be subtracted from two thousandths. In the end, the student hopefully ends up with one one, two tenths, six hundredths, and six thousandths (1.266).
Subtraction could also be accomplished by subtracting the lowest place value first, but this sometimes means more manipulations of the beads which means more chance for error.
Conclusion
The use of the abacus takes a little bit of time to master. It is important that the teacher and the students use the correct place value terminology (e.g. “regroup ten hundreds to make one thousand” instead of “turn ten green beads into one blue bead”), so the concepts of place value, addition, and subtraction can be transfered to mental strategies and paper/pencil algorithms.

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So you think you can’t do math, huh? Well, what if I told you that without even an algebra background I could teach you some calculus right here and now? You don’t believe me. I thought so. This unbelief proves how programmed you’ve been from early on to reject mathematical teachings. Whoever laid this negativity on you really did a number on you. Well, let’s get rid of that negative programming and feed you some really positive stuff.
In this short article, I am going to prove to you that you can do calculus. In fact, by the end of your reading this, you will know how to perform a fundamental operation which forms one of the key branches of this most important mathematical discipline. This fundamental operation is known as differentiation and as such forms the branch known as differential calculus. When we perform differentiation, we are finding derivatives, and this is the process we will learn more about in this article.
Differentiation is a procedure which allows us to do things like find the highest point of travel by a projectile, or to figure the maximum profit given a specific rule for the revenue of a particular business or enterprise. Differentiation also allows us to approximate, without a calculator, the square, cube, or even fourth roots of numbers (see my Why Study Calculus series of articles). How’s that for an important procedure! And to think I’m going to show you how to do some of this right here, right now, regardless of your previous training or education. It just doesn’t get better than this.
In calculus we work with things called functions, which are nothing more than rules which relate different letters. The letters are known as variables and the most common letters used are X and Y. For example, the rule which says that Y is always equal to three times whatever X is, is expressed as Y = 3*X, where the asterisk symbol “*” represents the operation of multiplication. We can also write this as Y = 3X, in which no space between the number 3 and the X means that we multiply these two together. The rule which says that Y is equal to X times itself is expressed as Y = X*X,. We also write this same function as Y = X^2, where the “^” symbol followed by the 2 means we multiply X by itself, also known as “X-squared.” The function that says that Y is equal to four times X-squared is written as Y = 4*X^2, or 4X^2.
Now bringing a little calculus into play, I will show you how to perform the operation of differentiation on each of the above functions, or rules. Remember. This procedure will allow us to determine more specific information about each of these functions and thus allow us to calculate certain important values. First take Y = 3X. Anytime we have a number times X, we can find the derivative by taking the number in front of X. Thus the derivative of 3X is—you got it—3. For 4X, the derivative is 4, and for 100X, the derivative is 100. What is the derivative of 18X?
Whenever we have a function like X^2, we find the derivative by taking the 2 and multiplying by the number in front of X and then X. In the function X^2, the number in front of X is a 1 understood. So the derivative of X^2 is 2*1*X or just 2X. In the function 4X^2, the derivative is 2*4*X or 8X. In the function 16X^2, the derivative is 2*16*X or 32X. What is the derivative of 9X^2?
Yes, folks. I’ve just taught you some calculus in less than a seven-hundred-word article. And you thought you couldn’t do math. Remember that the next time the naysayers start flocking around you. Oh, and by the way. If you answered 18 to the first question above, and 18X to the second, then I’ve proved my point.

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Under more pressure than ever to achieve, high school students often find themselves overwhelmed with increasing amounts of homework. While some high schools are guilty of assigning too much homework, there are ways of approaching the task that allow you to become more productive and efficient.
Organizing your homework assignments saves a lot of time and wasted effort. Make sure you record your assignments for each class as they are given, and then make a checklist – before you leave school for the day if possible – to make sure you have all the tools and supplies you need for each assignment, including textbooks, class notes, additional reading material from the library, etc. If you have a study hall near the end of the day, use at least a few minutes of that time to organize yourself.
Are there assignments that you believe are going to give you more trouble than others? Use your study hall time to think about those particular assignments, and either tackle them in study hall or think about some resources you can access before you leave school for the day. Maybe you need to clarify something with the teacher, or get information from one of your classmates about the assignment. Get a library pass and use your library time to gather resources for your evening’s homework.
You should arrive home with everything you need to complete the evening’s homework assignments. Again, organization will save you time and trouble. Wherever you do your homework, whether it’s at your own desk on stretched out on the bed in your room, have all the books, notebooks, and supplies you need within easy reach.
There are a few standard tricks to approaching reading assignments. Before you start to fully read the assigned passages, glance through them. Notice the chapter titles and the interior headings and subheadings, if any. Look at all the illustrations, charts, and tables, and be sure to read all the captions. Go to the back of the chapter or subchapter to see if there are any questions; if there are, read them. Then start reading the assignment in full, keeping a notebook handy. When you run across information which looks like it will help you answer the questions at the end of the passage, take notes, or at least jot down the page number as you come across relevant information. Even if you haven’t been assigned the questions as homework, try to answer them, at least by jotting down a few notes on each question (chances are your teacher will use these questions in class the next day, or as the basis for future tests). Make a list of your own questions about the subject to ask your teacher.
At the beginning of each course, consider going through your assigned texts and creating an outline of each textbook. Create a three-ring binder and provide sections for each chapter and subchapter, and file your class notes and completed assignments as they come up in the appropriate section. If a class – for instance, algebra or physics – includes formulas which you need to work with and commit to memory, write the formula and one or two examples of the calculations on a separate page, and file the homework on succeeding pages. Do the same with theories – the commutative and associative principles of math should each have a separate page in your algebra notebook, for example. Design these particular pages in a way which will stand out and be easily found as you page through the notebook. What you create by imposing an organization at the beginning of your course is a template for studying as you move through the course and submit to quizzes and tests.
Don’t forget to review your homework when you get it back from your teacher. If you’ve missed a few math problems, make sure you go over your work and figure out what you did wrong, and recalculate until you’re sure you’ve come up with the right solution and the reason for it. If your homework involves creative writing, which you enjoy, file it in its own binder. You may not have time to revise assigned writings now, but sometime in the future you may want to do some additional work on the subject, and you’ll have your writing assignments ready and organized, available for creative inspiration.
One neat trick to keep you on task: Keep track of your time. Create a log which records the assignment, including page numbers of reading material, and record a start time and a finish time for each task. After a while you will develop an understanding of just how much time it takes to complete various assignments, and will help you plan your day and evening. It’ll also help conquer the worst problem many of us have with any task – procrastination – because you’ll know that a certain assignment is only going to take fifteen minutes, and then it’ll be over, and you can move on to another part of your life, like watching your newest DVD. Also, in the event that you really do have way too much homework – some schools and teachers are notorious for piling on very demanding assignments, day after day – you’ll have evidence! If you’re consistently burning the midnight oil and then arriving at school the next day bleary-eyed and sleep-deprived, talk to your parents. Armed with specific information, they may be able to intervene, either directly or through your local PTA.

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How much time do teenagers really have for practice? Since they spend hours on the internet and TV, they sleep in on the weekends and they don’t appear to get paid big bucks by the hour – it may seem that they have all the time in the world just for practicing algebra. Well that might be the case for countries like Japan, Korea or Singapore, but nowhere near it in the US. The average teenage would spend a few hours a week at best on the entire school duties. The only way to get them to spend more time on homework would be to include watching reality TV shows on the weekly assignments… The problem with algebra, comparing to other school duties – is that it’s not simply memorizing stuff, but performing tasks that are more complex than almost anything else they need to do. Let’s look for example at the following math problem:
“Jane, Paul and Peter can finish painting the fence in 2 hours. If Jane does the job alone she can finish it in 5 hours. If Paul does the job alone he can finish it in 6 hours. How long will it take for Peter to finish the job alone?”
Solving this kind of a problem can take something like 10-20 stages: understanding the data, assigning variables, using formulas, constructing the equations, combining like terms, finding a common denominator, opening parentheses and so on. Now consider what happens when someone makes a mistake on one of the steps? Obviously that person will end up with the wrong answer. However the biggest problem is that they have a slim chance of finding the error. Most likely the kid will be frustrated and angry about the wasted time, and will move on to the next question. If that one won’t work either – that will be the end of math practice for that day.
So what can be done to avoid frustration and wasting time? With new subjects or ones of difficulties, real-time feedback is imperative (see my article on real time feedback). They can get it by asking for help from the teacher, other kids, family, friends or online – but no one should be left alone in this battle. A frustrated teenage will start avoiding the subject and develop an “algebra phobia”. It is much better to move on to the next subject and continue only with someone that can help and provide feedback. So here are the top 10 tips for effective practice:
1. Do not spend too much time and energy on a single exercise. Make a fair effort to solve the exercises, but if it doesn’t work – move on and get back to it with some help.
2. Allocate enough time. Don’t start Algebra practice if you have less than 30 minutes. You don’t have to use it all, but just the thought of what comes next can kill the practice.
3. Don’t allow interruptions. No emails, IM, Cell phone, parents or anything else. There is no better excuse to stop the practice then welcoming an interruption.
4. Set clear reachable goals before you begin – and meet those goals.
5. Practice as soon as you can after the class ended. Like paying bills and dentist appointments, practice is usually done at the latest possible time (and for the same reasons). However doing it while the material is still fresh in the mind can save a lot of time.
6. Always start a new subject with the easy ones. Do not skip them even if they seem too easy – they will help getting the principles memorized
7. Make sure you understand before you start. If the explanations are unclear, ask someone or use the internet to find out. Don’t waist too much time trying to solve an exercise while you still didn’t figure out what is it that need to be done.
8. Clear your head. If you have urgent things to do – do them first, if they’re not urgent write them down and get them out of your consciousness.
9. Use a calculator (not the one on the computer). It will save you time and “stupid” mistakes, and besides, you need to get use to it since it will be your only friend at the test.
10. Play background music (if it doesn’t disturb you). Not too loud and not to exciting so it won’t form an obstruction. Music can serve as a white noise blanket that masks all other interruption (phone rings, opening doors, calling someone, etc).

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