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Mathematical Journeys: My Imaginary Friend

There’s no such thing as the square root of a negative number. Right?

Since squaring a number is defined as multiplying it by itself, and multiplying a negative times a negative gives a positive, all squares should be positive. Right?

So any number you want to take the square root of should be positive to begin with. Right?

So what if it’s not?

What do you do if you’re chugging through a problem and suddenly find yourself confronted with

x = √(-9)

It seems like to finish this problem we’ll need to take the square root of a negative number – but we can’t, so what do we do? Drop the sign and hope nobody notices? Mark it as ‘undefined’ like dividing by zero? Give up? Cry?

Well, actually, we don’t have to do any of that, because we’ve got an imaginary friend to help us.

Meet i.

i is a mathematical constant, whose sole definition is that i^2 = -1. Or, in other words, i = √(-1). i is an imaginary number – people used to think taking the square root of a negative number was impossible, so they called such results imaginary. i is known as the “imaginary unit” or the “unit” imaginary number, and he functions very similarly to the number 1 in the realm of real numbers.

Because he’s imaginary, i can be a bit difficult to wrap your head around. Just remember that he’s a constant, like 3 or 12 or even π. Unlike other special constants like pi or e, though, we have no real way to articulate his value. We can say that pi is roughly equivalent to 3.14…, and that e is roughly 2.718…, but what is i? i is just i. i is the square root of negative one, and that’s the only way we can really describe it, since he’s not a real number. We just have to accept that “the square root of negative one” would theoretically have a concrete value, and assign it a special symbol like we did with the other special constants.

So let’s put our imaginary friend i to work on our earlier problem of

x = √(-9)

i functions much as 1 does for real numbers, so we can rewrite that equation as

x = √(9 * -1)

Which we can split up into

x = √(9) * √(-1)

And now we know how to deal with that second term – we just use our imaginary friend! √(-1) = i, so

x = 3 * i

Which we can rewrite as simply x = 3i.

Imaginary numbers all include the symbol i, since they’re all essentially multiples of the imaginary unit. So when math hands you an impossibility, just grab your imaginary friend and jump into the realm of the unreal!

New Subjects Added!

Just a quick update: I’ve completed several more certification tests through WyzAnt and added new subjects to my list. New subjects include Pre-Calculus, ACT prep (Math, English, and Reading) and GRE prep (Math, Reading, and Essay Assistance). If you or someone you know is in need of tutoring in any of these subjects, look no further!
~Ellen

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WWTK: Summer Fun with Math!

Nobody likes doing homework in the summer. It’s just a fact of life. My advice to students who want to stay sharp during the summer is to inject fun into your work and work into your fun. Find a way to connect your personal fun time back to the subjects you’re learning in school. The best way to accomplish this, in my opinion, is to look for school skills in unusual contexts. If you’re interested in maintaining your English or writing, you can join a book club or arrange one with your friends. Take this summer as an opportunity to read that book you’ve been dying to get to, and while you’re reading think about it critically and talk about it with others. I’m part of a “Bring Your Own Book” club right now, where each month we are given a topic and have to find and read a book that fits the topic. BYOBook clubs are a great chance to see a broad range of interpretations of a given theme and think about your reading in a larger context (what does the topic “animals as main characters” mean to six different people?). If you’re organizing the club, start with various genre selections and move on from there (Sci-fi night, horror night, etc.).

Finding math in unusual contexts is a bit more difficult, but for me it’s a lot more fun. There are plenty of seemingly-unrelated skills and activities that actually involve a great deal of math, and if you look for it you can learn to think about math and its impact on the world in a broader context. Here are just a few examples of fun activities that involve math.

Dungeons & Dragons or other tabletop Role-Playing Games
Dungeons & Dragons is the most well-known of a genre of games known as Tabletop Role-Playing Games (RPGs). In a D&D game, each player creates a unique character using a complex chance-based generation system involving rolling dice to determine various statistics. The game is full of math, including but not limited to multiplying to find base stats or critical damage, or adding and subtracting various modifiers. Dice are rolled on the fly and numbers tallied up and called out, resulting in the need for quick mental math. Playing D&D can be a great way to practice your mental math skills without it seeming like drudgery. Plus the game itself is a great lesson in problem solving and algorithms, as you figure out which patterns of addition and multiplication are needed for which actions and very naturally arrive at your most optimal workflow. The storytelling aspect of the game can be helpful for creative writing practice as well, as you think about how best to phrase your statements or work together to figure out the solution to a puzzle.

Knitting and Crocheting
Knitting and crocheting are great for math practice. Learning the basic stitches is an act of problem solving and devising algorithms, as you figure out how to hold your work and where the various parts of the stitches go. Once you have the basics down, you’ll still need to count your stitches, follow patterns for lacework, and even use math to figure out how fast to increase or decrease. For a real math workout, though, you should go through the gauntlet of altering a pattern. Use a different weight of yarn than the pattern calls for; you’ll need to figure out your stitch ratio and then use lots of multiplication to figure out how all the numbers will change. It’s a complex process, but incredibly rewarding.

Mini-Golf
Want a fun afternoon of geometry- and physics-based fun? Head out for a round of mini-golf! Navigating around the obstacles requires planning and forethought – can you figure out how to predict where your shot will go before you start? Banking off of the sidewalls provides an exercise in angles of incidence – can you avoid the obstacles in the first place with a carefully-lined-up shot? How are you deciding which tee to use, and how do the bumps and hills affect your ball’s trajectory?

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Writing Rundown: Finding Your “However”

It takes practice to find your writing style, whether it be in fiction, research papers, or analytical essays. The best piece of writing is both grammatically correct and organized, but also contains the essence of the person who’s writing it. When I correct students’ papers, I try to avoid suggesting alternate sentences in their entirety, since a paper written by you shouldn’t sound like one written by me. Even if we are answering the exact same prompt in the exact same way, the tone and character of each paper will be distinct, unique to each of us. Finding your style is a slow process, and generally comes about organically as a result of experience. Write more papers and you will begin to zero in on what makes a paper sing for you.

This is not to say that there aren’t tips and techniques I can give to help you find your writing style. By far one of the most useful techniques in my own experience has been working with what I call “Finding your ‘however’.” The name comes from my sister, who always used to incorporate the word ‘however’ into practically every sentence in her paper, even when it didn’t make sense. It was just a quirk of the way her brain liked to formulate thoughts, so it ended up in her rough drafts a lot because it was in her head while trying to sort out what she wanted to say. I had a similar experience, but in my case it was not ‘however’ but ‘I believe that’. I used to start every sentence in an opinion essay with “I believe that,” since that’s how my brain formulated thoughts. In an opinion piece, though, the idea that these statements are your beliefs is implied – you don’t need to keep telling the reader that. It was hard for me to remember that I could just state my opinions as fact and with confidence, and the reader would understand implicitly.

We all have these little quirks in the way we write, and they’re different for each person. It’s a natural reflection of the way our brains process information. Usually these quirks end up detracting from the strength of the overall paper, throwing the difference between spoken and written English into stark contrast and sounding stilted or affected when read. Catching yourself in the process of falling into a quirk is difficult, though, so my strategy is simply to recognize and acknowledge your little ‘however’. In my case, I recognize the fact that I like to start every sentence with “I believe that,” and that this quirk will need to be dealt with before my paper can be considered finished.

But here’s the key – ignore it when writing your rough draft. It’s easiest to get the information out on paper if you’re not worrying about editing, so just get it all out and don’t censor yourself. On your first round of editing, after you have a completed draft, go through methodically and remove all of those little ‘howevers.’ In my case, when it’s time to edit I start by going through and cutting out every single instance of “I believe that,” adjusting sentences where necessary.

And don’t worry – if you work in this way long enough, your brain will eventually figure out that you don’t need those quirks and you’ll find they stop making their way into your paper in the first place. I hardly ever start sentences with “I believe that” anymore, but I still make a first pass through the paper and check for them anyway. In fact, far from being a detrimental quirk, my little ‘however’ has become a tool that I use to help me generate content. When I’m having trouble articulating a point, I’ll say it to myself starting with “I believe that” – and then simply write the rest of the sentence without those first three words. Find your ‘however’ and keep on top of it, and you’ll be well on your way to writing a great essay.

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Literature Spotlight: Psychological Punishment

Crime and Punishment by Fyodor Dostoyevsky is a novel about guilt, morality and emotion. Throughout the novel many characters espouse the idea of reason and willpower over emotion – that if you have sufficient mental faculties you can prevent emotion from getting in the way of your actions and behave truly rationally. The student Raskolnikov believes this with all his heart when he sets out to murder a pawnbroker for the good of the community. This concept is quickly proved to be fundamentally flawed, however, as his inner guilt throws him into emotional turmoil and his brain attempts to protect him from the ugly truth of his actions. Raskolnikov displays several textbook examples of psychological defense mechanisms throughout the course of the novel, proving that even the most thorough reasoning and intellect cannot prevent the emotional and psychological response to a crisis.

Psychological “defense mechanisms” are the brain’s way of protecting itself from full awareness of unpleasant thoughts or behaviors. In Raskolnikov’s case, they are his brain’s way of protecting him from the full reality of what he has done, and the guilt associated with that reality. Some are more primitive, living in the subconscious and relying on emotion, and some are more complex and rely on conscious thought. Raskolnikov displays many of them over the course of the novel, as his brain attempts to come to terms with what he has done. Here are just a few examples.

Dissociation
A relatively primitive defense mechanism, Dissociation involves losing track of time; the narrative of life becoming disjointed. It allows the person to disconnect from reality for a while, seeking psychological shelter by escaping to a different sense of time. At one point in the novel, Raskolnikov gets confused about whether he has been to the police station yet. He states, “Bah! I am mixing it up; that was then. I looked at my sock then, too, but now…” (P. 192) An action that he associates with guilt (looking at the sock) causes his brain to try to escape to a different time, a time before he had interacted with the police. Dissociation appears in many places in the novel, with Raskolnikov frequently getting mixed up with regard to who was present during a certain time, whether he had said something or just thought it, and even with regard to his dreams and their relationship to reality.

Displacement
Displacement is when a person redirects their thoughts or actions away from the actual source of their problem, sometimes referred to as “taking it out on someone else.” It allows the person to vent their frustrations when they cannot take them out on the actual source. Raskolnikov is angry at Luzhin and lashes out at him almost from the very beginning, and his anger isn’t necessarily warranted to the extent that he experiences it. Getting mad at Luzhin is a way for his brain to release the feelings of anger and guilt that he feels towards himself without hurting someone he cares about.

Acting Out
Acting Out is when a person performs extreme behaviors in an attempt to express thoughts or feelings that they can’t comfortably acknowledge. The classic example of acting out is a child throwing a temper tantrum. Raskolnikov acts out for much of the middle part of the novel; the most noteworthy example is when he runs into Zametov at a cafe and makes a speech explaining exactly how he committed the murder, under the guise of conjecture. He makes this speech in a creepy whisper, standing too close to Zametov, and tops it off with a strange laugh, so that Zametov is convinced he is a madman. This allows him the freedom to get the confession off his chest in a safe way, a way where he knows nobody will actually believe him. He continues to act out for the next few chapters, saying and doing things that offend and incense others and lend credence to their belief that he is delirious.

Rationalization
Rationalization involves providing a seemingly-plausible alternate explanation for why something is happening, an explanation that sounds more reasonable or is easier to deal with than the real one. Raskolnikov blames his guilty conscience and crazy dreams and thoughts on a physical illness. “ ‘It is because I am very ill,’ he decided grimly at last, ‘I have been worrying and fretting myself, and I don’t know what I am doing…I shall get well and I shall not worry…’ “ (P. 169) Blaming his guilty feelings on an illness allows him to distance himself from the reality of the situation, and also ties in to his belief that he can control his emotions – he comes up with a seemingly-logical reason for his emotional actions in order to devalue them.

Intellectualization
Intellectualization is the blaming of every emotional reaction on some kind of intellectual reason, finding an explanation for unwanted emotions that ignores the obvious source of them. While waiting in the police station Raskolnikov begins to get nervous, but intellectualizes the problem by saying “It’s a pity there’s no air here…it’s stifling…It makes one’s head dizzier than ever…and one’s mind too…” (P. 148)

The use of intellectualization ties in particularly strongly to the overall theme of reason over emotion and highlights the importance of these defense mechanisms to the text. Using textbook examples of defense mechanisms shows just how little control Raskolnikov really has over his own psychology. When setting out to commit the murder Raskolnikov is convinced he can get away with it, specifically because he will use his reason and not allow emotion to get in the way. Raskolnikov believes that the reason most crimes are badly concealed and easily detected “lay not so much in the material impossibility of concealing the crime, as in the criminal himself. Almost every criminal is subject to a failure of will and reasoning power by a childish and phenomenal heedlessness, at the very instant when prudence and caution are most essential.” (P. 112) Raskolnikov believes this “failure of will” is an avoidable circumstance, and further believes that he will not suffer the failure himself, because in his mind his actions are “not a crime.” His entire reasoning process relies on the idea that murdering the pawnbroker is not actually a crime, and that consciously remembering this fact will be enough to keep him sane and safe after the act is committed.

Right from the beginning, however, he runs into problems. His plan does not go as smoothly as he had anticipated, with one problem after another piling up and straining his nerves, culminating in the pawnbroker’s sister coming in at the wrong time and Raskolnikov being forced to murder her as well. No matter how thoroughly he had rationalized the murder of the pawnbroker, he did not anticipate the sister and so he has no justification for her death. This unjustified murder pokes holes in Raskolnikov’s carefully thought out reasoning process, and his own brain begins to betray him. Raskolnikov reacts psychologically to the murders, even as he endeavors to explain his actions rationally. By showing us these psychological defense mechanisms, Dostoyevsky argues that even the most well-reasoned justification is no match for the emotional turmoil that follows a crime. Raskolnikov’s guilt is his punishment, and all the defense mechanisms in existence cannot protect him from it.

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Ellen’s Choice: Tangent to the Classics

School’s almost out for the summer, and to me, summertime is a perfect excuse to try learning and growing in new, fun ways. When I tutor students over the summer, I make a concerted effort to inject some fun into our work, so that it doesn’t feel like homework. We read fun or unusual books, or we put a twist on a project. Write a creative, narrative response to a work instead of an analytical essay, or go on a little “field trip” to find learning in unexpected places. I’ve recently devised a new fun “field trip” type activity, and I’d like to share it today.

But first, some background. I participate in a monthly “Bring Your Own Book” club, where each month we are given a topic and we each choose a book that relates to the topic to read and bring in. We always end up with a really interesting mix of genres and types of stories, all revolving around a theme (such as “books with animals as main characters” or “books that have inspired music”). Since I tutor high school English, I tend to gravitate towards the classics, and I’ve brought in such books as Watership Down and Ragtime in previous meetings.

Last month, our theme was “alcoholic writers,” and after a quick search for a list of popular writers who were alcoholic, I settled on F. Scott Fitzgerald. My initial idea was to reread The Great Gatsby, a classic I hadn’t read since high school but remembered enjoying. I headed to the library and quickly found the F’s, then found about half a shelf of F. Scott Fitzgerald. Reaching for Gatsby, my eye was suddenly drawn to the book snuggled up next to it on the shelf, a volume slightly larger than Gatsby and also by Fitzgerald. I picked it up and looked at the blurb on the back, flipped through a few pages, and took it home with me instead of Gatsby. That book was “The Crack-Up,” an anthology of essays, letters, and notebook jottings from Fitzgerald’s life. It turned out to be a much more interesting and exciting experience than simply rereading an old classic.

Fitzgerald wrote essays about living in New York during the jazz age at a time when the jazz age was scarcely even over – an essay from 1931 had the same sort of wistful tone we’d expect from a period piece written today. But by far my favorite part of the collection was “the notebooks,” a sequence of random jottings, bits and pieces, collected into categories that started with each letter of the alphabet. “C” was for “Conversations and Things Overheard,” “D” was for “Descriptions,” “E” was for “Epigrams and Wisecracks.” It struck me as an intriguing glimpse into the inner workings of an author plying his craft, and my experience as a dancer and choreographer made this behind-the-scenes look into his process resonate with me even more strongly. I know the feeling of having an inspiration strike you and feeling that “I must write this down right now or I’ll forget it!” urge. Many of the little bits and pieces had Fitzgerald’s distinctive gorgeous wording, and I could almost see the reason behind putting each piece into his notebook. Lots of them were examples of “that’s a perfect way to describe this,” or just “I like that turn of phrase.” Some of my favorites were the ones completely devoid of context, such as:

Impersonating 46 presidents at once.

I absolutely loved this collection, and the experience gave me a great idea for a new “field trip” activity. I call this one “Tangent to the Classics.” The rules are simple:


  1. Head over to your nearest library and use the catalog to search for a classic novel you enjoy. It can be anything, really, so long as you enjoyed reading it.

  2. Jot down the call number and head into the stacks as if you were going to check it out.

  3. When you find the classic you searched for, don’t pick it up. Take a look at the books sitting to its immediate left and right. Read the blurbs on the back, flip through a few pages in the front, and choose one of those two books to check out.

  4. Take it home and read it. If it’s by the same author as your classic, think about how this book changes your conception of the author and his work. If it’s by a different author, think about why it was located right next to the classic. Libraries have an organizational structure that generally puts similar books near each other. Why is this book tangent to the one you searched for?

Try this one out and let me know how it goes! I’d love to see your experiences in the comments. What are your reactions to reading a book that was tangent to a classic?

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Mathematical Journeys: What Does the Function Look Like?

This week’s Math Journey builds on the material in The Function Machine. If you have not yet read that journey, I suggest you do so now.

In The Function Machine we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values.

But plotting point after point is a tedious and time-consuming process. Wouldn’t it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points?

Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a function’s graph from the equation itself – and it’s those clues that we’ll be talking about today. They come in four basic flavors: the power, the sign, the co-efficient, and the constant.

The Power

Let’s start with our old standby from the previous journey: y = x + 4. When we talk about “power” in this context, we’re referring specifically to the highest exponent on an x value. The highest power in this problem is one; there are no exponents so the x is simply raised to the first power. This means that for every value of y, there is exactly one corresponding value of x. If x is 1, y is 5. If x is 2, y is 6, and so on. For every given increase in x, there is a proportional increase in y, in this case it’s 1 to 1. And that means that this graph is a straight line. Easy enough, right?

Well, let’s throw a bit of a wrench into the works here, shall we? Your new function is y = x^2. Now, if I turned the machine around backwards and told you that y was 4, what would you give me for x? You might give me 2, right? 2 squared is 4. But hang on, there’s more than one thing you can square to get 4.

Not seeing it?

How about negative 2? When you square a negative number it goes positive, right? So your x value could just as easily have been – 2 as positive 2. And the same thing would have been true for any value of y, right – the corresponding x value could be either the square root of y or the negative square root of y. So in this case, there is more than one corresponding x value for any given value of y – in fact there’s exactly 2 corresponding x values for each y (with the exception of 0, of course). That means that this graph is NOT a straight line.

Turns out, it’s actually a parabola. All functions with x^2 as their highest power (known as quadratic functions) graph out as parabolas. The specific parameters of each parabola are determined by the other categories of clues, but the power tells us that this graph will be some kind of parabola. In the same way, the powers of higher-power functions also tell us the type of shape they will graph; third power functions (ones with a cube as their highest power) will form hyperbolas, and so on. This holds true with functions that include radicals as well; the type of power indicates the rough shape of the graph.

The Sign

Let’s take our quadratic function of y = x^2. When you plot some points it becomes clear that this is a parabola opening upwards; the larger the x values become, the exponentially larger the y value becomes. But what if I made one slight change to this equation?

Y = – (x^2)

Now I’m asking you, essentially, to take each of those y values and invert it. If x is 2 (or negative 2), y would now be negative 4. This holds true for every value of y, so if you plot a few of those sets of points it quickly appears that you’ve just flipped the parabola upside down. And indeed, the sign on the highest-power x value dictates which direction the graph will be facing (at least in terms of up-and-down; the side-to-side graphs are usually dictated by higher powers in the first place or by radicals or other more complex types of functions). If we were dealing with a straight line, the negative sign would indicate that the line travels downward as it moves to the right, rather than upward. Y = -x, for example, is a line with a negative slope, which means it moves down and to the right rather than up and to the right as y = x does. If you graphed both of those line functions, they’d come out to be mirror images of each other. So the sign on the highest-power x value dictates direction.

The Co-Efficient

When we talk about a co-efficient in math, we’re generally referring to the number that is multiplied by a variable. Take, for example, the function y = 3(x^2). How would this differ from our original y = x^2?

Well, let’s follow the problem through. With a co-efficient, each time we get the square we’ll need to multiply it by 3 before it becomes the y value. This will mean that each y value is quite a bit larger than the y value in our original problem. The curve will be quite a bit steeper, since using 2 for x will give us 12 for y instead of 4. So with a co-efficient above 1, the graph will show up steeper/skinnier/more closed. With a co-efficient that is a fraction, however, the graph will show up shallower or more open. Think about y = (1/3)(x^2). With 2 for x, you’d now end up with 4/3 for y; even less than with the original problem. So the co-efficient tells us how steep or sharp the progression of the curve is. Higher numbers mean sharper curves, while smaller fractions mean more gentle progressions.

The Constant

The constant is my favorite clue. A constant is a number that does not involve a variable. In our original y = x + 4, that +4 is the constant. That constant is the y-intercept – the value at which x is 0. If x were 0, all terms with x’s in them would become zeros and all you’d have left would be the constant. So with a quick look at the constant you can figure out one of your points with no work at all. But here’s the really fun part. Since it doesn’t involve a variable, the constant doesn’t actually change the shape of the curve itself. What it does do is move it around the plane. Take a look at y = x^2 versus y = x^2 + 4. That +4 on the end simply means that every y value you normally would have gotten is now 4 places higher on the graph. The whole curve has been lifted up four places on the graph. If it were a negative 4 – you guessed it – it would have moved down four places.

So the natural next question is: what if you want to move it right or left on the plane? Well, that involves getting a second co-efficient into play. Let’s change our equation to x^2 + 2x + 4. That 2x will shift the graph horizontally – but it’s a little bit more complicated than you might think. The signs here are actually reversed – adding 2x moves the graph to the left, and subtracting it moves the graph to the right. Also, it’s not a one-to-one ratio; in fact the ratio varies depending on the equation itself. Remember, too, that the constant is still the y-intercept, so if you get sideways transposition involved the center won’t necessarily be cleanly at an easily-discernible value anymore; but the curve will still cross the y-axis at 4. Combining those two pieces of information, along with the power, sign, and leading co-efficient to tell you the shape of the curve, will get you well on your way to knowing what the graph looks like.

Remember back at the beginning when I told you that using these clues would allow you to plot just a few points and sketch the graph more quickly? Well, here’s how we put it all together. Let’s take a new equation:

y = 3(x^2) + 5x – 2

What can we tell about the graph from the clues presented here?

First, the power. This is a quadratic function, which means we’re dealing with a parabola. The leading sign is positive, so it’ll open upward. The leading co-efficient is 3, which is greater than 1, so it’ll be a sharper, steeper curve, 3 times steeper than the basic parabola. We’re adding 5x, so the graph will be transposed to the left, and the y-intercept is at – 2. We’d still need to work out and plot a couple of points (personally, I’d factor the quadratic to find the x-intercepts and work from there – more on that next time), but now we have a better idea of what the graph would look like – and we can see all of that just from the equation alone!

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Ellen’s Choice: Teach the Concept, Not the Algorithm

I hear a lot about math teachers from my students, and while every teacher is unique, some comments are repeated over and over. By far the most common one I hear is that their teacher didn’t really explain something, or was incapable of elaborating when questioned and simply repeated the same lecture again. As a tutor, my first priority is to make sure the student understands the material, and if they’re still confused, to find another way to explain it so that it makes sense. In order to do that, I need to have a thorough understanding of the concepts myself, so that I am not simply reading from a textbook but actually explaining a concept. In my years of tutoring math, I’ve developed a point of view and approach to math that I refer to as “teaching the concept, not the algorithm.”

An algorithm is a step-by-step procedure for calculation. The term is used in math and computer science, but the concept of an algorithm is universal. I could tell you that I have an algorithm which consists of 1) close the windows; 2) put on a sweater; 3) check the thermostat; 4) turn it up 3 degrees if it displays lower than 68. This is pretty obviously an algorithm to solve the problem of “I am cold right now.” We have algorithms for everything in our life, and most of the time we don’t even think about it that way. We see a problem, we work out a set of steps to solve it, and we complete those steps and observe the result.

In math class, however, students frequently encounter teachers who simply teach the algorithm; handing them a formula for solving a problem without ever really teaching them the core concepts involved or why the formula is what it is. This results in a lot of rote memorization with no understanding of why the numbers are where they are in that formula. My golden question for math teaching is always “Why?” Why does this work? Why can I do that? What am I trying to accomplish here, in the grand scheme of things? If I can explain the concept to the student so that they understand what they are doing on a macro scale and why their actions work and make sense, then it doesn’t matter if they forget the formula itself, they should be able to figure it out organically by going through the conceptual process again.

I’ll give you an example from my favorite math teacher, Mr. Lazur. (I wrote a whole blog post about Mr. Lazur’s teaching style, which heavily influences the way I tutor.) I had Mr. Lazur for Geometry, a subject notorious for the amount of formulae it throws at its students. Every single type of shape has three or four formulas associated with it, and keeping them all straight can be a nightmare for students. Mr. Lazur got around this by showing us WHY the formulas look the way they do, ensuring that his students could always reverse-engineer the formulas from the concepts if they couldn’t remember them directly.

In this example, we’re learning about the volume of a cylinder. We’ve just spent the previous few days discussing volume of cubes and rectangular prisms, so Mr. Lazur starts us off by reminding us of exactly what volume means. It’s the amount of stuff required to fill up the shape; the amount of water that would be displaced if the shape were dropped into a bucket. Then he pulls something out that nobody was expecting: one of those CD spindles that you buy with blank, recordable CDs in them. He points out that a stack of CDs is a cylinder, taking them off the spindle and setting the stack on his desk. He asks us to imagine that each CD is actually a truly 2-dimensional object, ignoring the tiny thickness of the plastic. He tells us that the process works just the same with truly 2-dimensional objects as it will with these CDs. How could we figure out the amount of stuff required to fill up this shape, he asks. Assuming it was truly 2-dimensional, we wouldn’t be talking about volume anymore; it’d be area, right? He asks for the area of a circle, and we give it to him. We know this; it’s easy stuff we’ve known for months now.

A = πr^2

So that’s how much space this single, 2-dimensional CD takes up, right? He picks up another CD and places it against the first one, flat sides together. How much space would 2 of them take up? He separates them again, holding them side by side. It’d just be twice the amount of space the first one took up, right? 2 circles’ worth of area.

He writes on the board: 2πr^2

So how much space would 5 of them take up? 5πr^2

And how much space would a stack of them that was h CD’s high take up? hπr^2

Mr. Lazur then circles that last line and turns to us. “This is the formula for volume of a cylinder. It’s just the area of the flat face, multiplied by the height of the stack of those faces. πr^2(h).”

When I started writing this blog post I wasn’t thinking about the formula πr^2(h) – I was thinking about that stack of cylinders. The formula followed organically from thinking about the concept. And that’s the key – you can derive an algorithm easily from a concept, but if you never teach the concept all the algorithms in the world are just meaningless memorization.

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Writing Rundown: Prewriting Techniques

Prewriting often gets the short end of the stick with students rushing to get that paper written before its due date. Since many teachers don’t require prewriting to be turned in with the paper, many students feel that it’s a corner they can cut to save time and launch straight into writing a first draft. In reality, prewriting is actually a great time-saver, particularly when you don’t exactly know what you’re going to talk about. It helps you to organize your thoughts, as well as make sure your points are clear and your concept isn’t too broad or too narrow. Prewriting is especially helpful in situations where you’re given a very broad prompt – or even no prompt at all (as was the case with my IB World History term paper, whose prompt consisted of ‘Write a paper about something from 20th century world history’!)

Prewriting is usually defined broadly as anything you do before writing your paper, and can take many forms. This blog post will discuss a few of the most common forms and their pros and cons.

The Outline

By far the most common prewriting technique is the Outline. In an outline, you plot out the framework of your paper by first listing the most important or main ideas and then fleshing them out with supporting details. In a well-built, detailed outline, most of the information for the paper will be present, just in sentence fragments or keywords. Outlines are great for making absolutely sure you know where you’re going with your paper before you start, and for keeping you on task during the writing process itself. I generally write an outline before I start writing the draft of any paper. However, it’s not always the ONLY prewriting technique I use. Outlines work just fine on their own when the topic is relatively straightforward, for things like cause-and-effect relationships or comparison-contrast papers. Outlining is a very linear prewriting form, though, and for some people it’s difficult to generate ideas and plans of attack using a linear method (myself included). For us, there are alternate methods.

Clustering (aka “Word Cloud,” aka “Word Net”)

This next technique goes by many names, but the most common are the Clustering technique and the Word Cloud or Word Net. In this technique, you start with one word or concept which you want to be the central focus of your paper. You write that word or concept in the center of a blank piece of paper and draw a circle around it. From there, you begin to free associate, writing down words or concepts that relate to the main word in the areas around it. Each word or concept gets a circle drawn around it, and then gets a line linking it back to the main idea. From each of those related words, more sub-ideas are generated in the same way, written down, and linked to the sub-concept. What makes this technique a favorite of mine is that it works really well for complexly-interconnected concepts. Each time a concept is written down, lines are drawn linking it not just to the main concept, but also to any other related concepts anywhere on the page. This makes it easy to see common threads running through the concept, and to see alternate ways to organize the information. For example, writing my Literature Spotlight on Wuthering Heights I began with a word cloud surrounding the central idea of the title and the idea of “weathering a storm”. As I built the cloud out from that central idea, I found that all of my main points were connected through a single symbol – the lightning striking the house. That symbol started out pretty far down one of the branches of my cloud, but seeing how many of my other points related to it, I decided to make another cloud with that symbol at the center. That second cloud is what eventually became the outline for the essay. Word clouds make it easy to rearrange your information and look for connections beyond the ones you first noticed, and are extremely helpful for spatial/visual learners who think better in geometric or spatial reasoning than in a linear fashion.

Freewriting

Freewriting is the last prewriting technique I’m going to talk about today. Freewriting is not to be confused with launching straight into a first draft – this is a prewriting technique, not a drafting technique. In freewriting, sometimes called “stream-of-consciousness” writing, you put your pen down on a blank piece of paper and just start writing – and you don’t stop writing for at least ten or fifteen minutes. Jot down everything that comes to mind, trying to stay on topic but not worrying if you stray. The important thing is that the pen should never stop moving – just write down everything that comes into your head. This exercise attempts to remove the filter that normally exists in your head – and by giving you the freedom to stray off topic, you get around the brain’s tendency to self-censor and second-guess itself. When the fifteen minutes are up, go back and read over what you wrote. You’ll probably need to synthesize this information into another form of prewriting such as a word cloud or outline before you can use it to write your paper.

The advantage to this kind of prewriting is that you may find yourself writing about aspects of or angles on the topic that you didn’t expect. When done correctly, freewriting can get you very deep into your psyche and tell you things about yourself that you weren’t aware of. For this reason, I find freewriting to be very useful for a specific kind of writing assignment – one that asks for a deep and personal opinion from the writer. Assignments like the admissions essay required by so many colleges, which challenge the applicant to discuss their dreams and goals or their opinions and beliefs, are particularly well-suited to freewriting. That research paper for history class? Not so much.

So next time you sit down to write a paper, give a few of these a shot. It’s always better to be thoroughly organized before you open up that word document, and it’s always easier to write when you know what you’re writing about.

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Literature Spotlight: Nora Grows Up

Since I’ve been tutoring English literature students, I’ve noticed a pattern: every time we read a book that I remember reading in my high school classes, I enjoy it far more as an adult than I ever did as a teenager. Time and time again I pick up a book I remember hating in class, resigned to slog through it and discuss metaphor and symbolism with my student, only to find that I thoroughly enjoy it. Each time I come out of the unit with a fresh new appreciation for the work in question. As this happens more and more I’ve come to the conclusion that there are whole worlds of theme and subtext in many novels that are only apparent to a reader who has reached adulthood, because they require the reader to have experiences beyond those of an average high-school student. In today’s Literature Spotlight I’d like to illustrate this point using a recently-transformed work for me, A Doll’s House by Henrik Ibsen. One of the main themes in A Doll’s House is the idea of Nora’s reluctance to grow up. This theme and all of its associated points are much more clearly apparent if the reader has already had the experience that Nora is undergoing.

One main point associated with this theme of growing up is the idea that actions have consequences. Specifically, that borrowing money creates a responsibility to the lender. A healthy borrowing relationship relies on a level of respect for the lender. Early in the play, while talking obliquely about the idea of borrowing money, Nora and her husband Torvald have this exchange:

Torvald: Yes, but what about the people who had lent it?
Nora: They? Who would bother about them? I should not know who they were. (P. 6)

This idea that she does not even care who is lending her the money is troubling, and indicates that she is only concerned about being able to maintain her perfect home. If that means she needs more money for fancy clothing or Christmas presents, then she’ll simply borrow it from next month’s paycheck. When questioned about what she would do if he suddenly dropped dead and there was no next month’s paycheck, her response is simply “If that were to happen, I don’t suppose I should care whether I owed money or not.” (P. 6) The hypothetical becomes real when Nora borrows money from Krogstad. As expected, she has no respect for him at all, even though she knows exactly who is lending her the money. When he challenges her to think of his family, she waves the issue aside. This resistance to thinking about money realistically is just one illustration of Nora’s reluctance to accept the responsibilities of being an adult.

In many households, teenagers are given an allowance or allowed to use their parents’ credit card and very little consequences are present for wild spending. Teenagers do not have to pay bills or make mortgage or car payments, and so the idea that borrowing money is a responsibility is harder for them to understand. By and large, they are not as aware of the lender/borrower dynamic because their primary lender is the “bank of Dad” and he is likely to forgive them rather than insist on consequences. Compare Nora in Act 1 to the average teenager. When Nora begs Torvald for money at the beginning of the play, her dialogue is very reminiscent of a child begging for an advance on their allowance. The relationship between Torvald and Nora at this point is far more parent and child than it is husband and wife, which ties back in to the central idea that Nora is Torvald’s plaything. But this relationship is difficult for teenagers to recognize as unusual, since the play is primarily from Nora’s point of view and Nora’s part of the relationship is the part that teenagers tend to inhabit with their parents. Only with a bit of distance does it become apparent how strange it is that a married couple would have such a dynamic.

A second point associated with the main theme of growing up is the idea that life is complicated and messy. At the start of the play, Nora has been relatively successful at maintaining her idyllic life. She twitters about, singing like a skylark, and when problems begin to arise, she tries desperately to cling to her carefree life. Two symbols used in the expression of this theme are the Christmas tree and the fancy dress-ball. When confronted with impending problems Nora fusses over the tree, the dress, and the tarantella to distract her from dealing with them. She believes if she can just get the tree to look perfect, everything will be fine, and clings desperately to that belief even as her mountain of secrets and lies begins to crumble down around her.

Nora often talks about how wonderful it would be to have lots of money and not have to worry about anything. Clearly, she is aware on some level that she’s not approaching these issues in a useful way, but she resists changing because that would mean acknowledging the possibility that her life is not really as happy as she believes it to be. She wants everything to work itself out so she can go back to playing with the children and being carefree, but life doesn’t work that way, and sooner or later you have to confront the hard problems. This concept probably goes over most teenagers’ heads because they are still living at home with their parents. Most parents try very hard to give their children an idyllic life; they’ll conceal financial realities from their kids and try their hardest to give off a carefree appearance. Parents want to shelter their kids from the harsh reality of life – but those kids will be adults soon and they will have to learn the truth one way or another. At some point, everyone has to come to the realization that their life is not as perfect as they once thought it to be. For most high-school readers, however, this realization won’t happen for several more years, and the theme is more easily apparent to a reader who has already gone through the process.

In Act 3, Nora forces Torvald to have a hard conversation, and then decides that she needs to leave him, if only for now. This is Nora having that realization – life is messy, and marriage is more than just flitting around like a songbird dressing the Christmas tree and playing hide-and-seek with the children. Two statements in her conversation with Torvald illustrate the thoroughness of her realization. The first is when she states, “You have never loved me. You have only thought it pleasant to be in love with me.” (P. 191) Couldn’t that sum up an incredible number of high-school romances! This statement shows a maturity of thought and emotion on Nora’s part that is rare in the still-developing brain of a teenage high-school student. Loving someone is more than just being in love with them. It’s about accepting another person into your life, faults and all, and working continually to improve each other in an equal and fair relationship. Nora received none of this from Torvald, nor did she give it. Teenagers having little or no experience with deep and loving relationships may be confused by her statement, but to the married adult, who understands that marriage is more about standing in line at Home Depot than it is about grand and romantic gestures, her comment makes a great deal of sense.

The second statement comes when she tells Torvald, “No, I have never been happy…only merry.” (P. 192) She has realized that flitting around like a songbird is not the same as being truly content with her life. Real happiness is the conscious knowledge that you are being true to yourself and are in the place you would like to be, professionally and personally. Nora realizes that she’s been nothing but a doll to her husband, and her father before him, and that she needs to go off on her own and find out who she is. This is Ibsen’s brilliant depiction of the experience of waking up one morning realizing you’re unhappy with your life and you need to make a change – quit your job, travel the world, and figure out what things and people you need in your life to be happy. This sort of soul-searching happens a lot in college and the years soon after, so once again, the average high-school student would not have had this experience.

Nora through the course of this play displays a process of emotional maturing reminiscent of a young adult going through college and a first job, figuring out what they want to do with their life and who they are as a person. High-school students have not yet gone through that process, so it’s harder for them to see the significance of those moments in the context of the play. This is just one example; many novels have themes that share similar qualities of life experience, and are better appreciated as adults than as teenagers. We tend to forget that a teenager’s brain, physiologically, is not finished developing yet, and new connections and pathways will continue to emerge until well into adulthood. Other novels that have given me this experience upon rereading with students include The Good Earth and The Scarlet Letter. So a final note to all my adult readers: give those old high-school novels another shot. You just might find that you thoroughly enjoy them as adults.