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Mathematical Journeys: Carl Gauss and the Sum of an Arithmetic Series

There's a famous (and probably apocryphal) story about the mathematician Carl Friedrich Gauss that goes something like this: Gauss was 9 years old, and sitting in his math class. He was a genius even at this young age, and as such was incredibly bored in his class and would always goof off and get into trouble. One day his teacher wanted to punish him for goofing off, and told him that if he was so smart, why didn't he go sit in the corner and add up all the integers from 1 to 100? Gauss went and sat in the corner, but didn't pick up his pencil. The teacher confronted him, saying “Carl! Why aren't you working? I suppose you've figured it out already, have you?” Gauss responded with “Yes – it's 5,050.” The teacher didn't believe him and spent the next ten minutes or so adding everything up by hand, only to find that Gauss was right! So how did Gauss find the answer so fast? What did he see that his teacher didn't? The answer is simple, really – it's all about... read more

Literature Spotlight: A First-Person Ensemble

Bram Stoker's Dracula is a novel told in epistolary form – meaning the story is told entirely through documents, in this case journal entries and newspaper clippings. Epistolary is a very effective technique for writing certain types of stories, and one that I feel is generally under-appreciated. In Dracula the epistolary form is used brilliantly to enhance the sense of mystery and suspense in the novel, and to add to the overall chilling effect of the story. One of the ways in which epistolary form enhances the suspense is through the use of first person narration – from multiple sources. In a traditional first-person narration the reader follows a single protagonist, knowing only what they know and seeing only what they see. This can be a welcome insight into a character's psyche, but can also be restrictive to the author since they cannot add outside information to the story. In epistolary form many characters can contribute first-person narratives to the novel... read more

Ellen's Choice: From Page to Screen

This past weekend I went to see the long-awaited movie adaptation of John Green's bestselling novel “The Fault in Our Stars.” I'm a big fan of alternate-medium depictions of various art forms (movies based on books, theater, or games, books that expand upon a movie or TV show, etc.) and I love to think about the ways in which a story is adapted for a new medium. Movies, TV, books, and live theater all have their own distinct methods of storytelling, and it's an enlightening exercise to think about how the source material has to change to fit the new style. The Fault in Our Stars movie is one of the most faithful, and I think successful, adaptations I've seen in a long time. I'd like to take a moment to discuss a few of the ways in which I felt they most successfully navigated the transition from book to movie. I'll refrain from spoilers in case any readers have not read the book or seen the movie yet. Visual Effects The Fault in Our Stars, in book form, contains... read more

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 i2 = -1. Or, in other words, i = √(-1). i is an imaginary number... read more

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”... read more

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... read more

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... read more

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... read more

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... read more

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... read more

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... read more

Ellen's Choice: How I Feel About the SAT Redesign

A few weeks ago I posted an article about the impending SAT redesign and the changes that have been announced. I mentioned at the end of that article that I'd be posting another one soon with my thoughts on the redesign, once I'd had time to think more about them. Well, this is that article. Overall, I think the motivation for the redesign is good – that the College Board's heart is in the right place and they're acknowledging some of the very real problems that the current SAT has. I'm very happy with their partnership with Khan Academy as well. I'm happy to hear that they acknowledge that students really do need some kind of prep help for the SAT, and that if they're going to force every student who wants to apply to college to take it, they should be offering free prep help for everyone who wants it. Not everyone can afford a private tutor, and money should not be a limiting factor in every student's ability to thoroughly prepare for the test. (That said, I am... read more

Mathematical Journeys: Undoing the Unknown Exponent

This journey is heavily inspired by the youtube mathematician Vi Hart, whose videos describing mathematical concepts through doodling in a notebook were the inspiration for much of my mathematical journeys series. I'll put a link to her video on this topic at the end of the journey, and I highly encourage everyone to go check her out. Let's talk exponents. But to do that, first we should talk about multiplication. Multiplication is a shortcut for adding a bunch of the same number together. If I gave you: 5 + 5 + 5 + 5 + 5 + 5 = ? You could just add them normally, treating each of those 5's as a size-5 step along the number line. But since each of these addition steps is the same size, a faster way to figure out the result would be to determine two things: the size of the step, and how many steps we have. Then we can multiply the size of step (in this case, 5) by the number of steps. In this case, we have a total of 6 size-5... read more

WWTK: Online Resources

I've recently discovered several online resources that I find very helpful for the various subjects I tutor. Since my tutoring subjects break down into three broad categories (Math, English, and SAT Prep), I'll choose one from each category to discuss today.     SAT Prep For SAT preparation, you can't beat the College Board website ( There's no better way to prepare than to hear it directly from the test makers. In addition, twitter users can follow @SATQuestion to receive the official SAT Question of the Day on their feed each morning. Particularly now given the announcement of the impending redesign, staying connected to the College Board will keep you up to date on all the changes. There's a place on their website to sign up for email updates, so you'll never miss a thing!   Math Having recently started working with middle-school students, I found a sudden need for worksheets to practice with... read more

Literature Spotlight: The Lightning, The Tree, and The Chimney-Stones

Title choice is an often-overlooked aspect of literature. What the author chooses to call his or her work can serve as a window into their intentions, showing in a subtle way the aspects of the novel to which they wish to draw the reader's attention. As an example, take Emily Brontë's classic novel Wuthering Heights. According to the dictionary, “wuthering” means “blowing strongly with a roaring sound” when describing a wind, and “characterized by such a sound” when describing a place. The word also has close associations with the more common “weathering,” implying enduring harsh weather or coming through a storm. Throughout Brontë's novel are references to this idea of weathering out a storm or withstanding howling winds. Most of the major plot developments take place during thunderstorms, and the various characters are likened to different aspects of a storm. This theme comes to a head during Heathcliff's disappearance midway through the novel – not coincidentally in the... read more

The Redesigned SAT: What You Need To Know

The news broke recently that the College Board is once again changing the SAT. These new changes, scheduled to be implemented in spring 2016, represent a pretty large departure from the SAT of the past. The College Board states that this new SAT will “ask students to apply a deep understanding of the few things shown by current research to matter most for college readiness and success.” Here are the changes that will have the biggest effect on test preparation, as I see them: An Increased Focus on Evidence-Based Analysis The new SAT will place a higher priority on analysis based on evidence. In the critical reading and writing sections students will now be asked to support their answers with evidence, including citing portions of the passages. In effect, the new SAT will require students not only to know the correct answer, but to be able to explain why the answer is correct, and point to specific evidence in the passage that supports their choice. The essay... read more

Mathematical Journeys: The Unperformed Operation

Come with me on a journey of division. I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of piles I'd made, which in this case is 4. You'd probably write that as: 32 ÷ 4 = 8 So there are 8 candies in each pile. Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4 instead... read more

Good Luck SAT Students!

Good luck to all students taking the SAT this morning!  Remember: they're trying to trip you up, so watch your feet!   Don't feel like you did your best?  Anxious about how many questions you skipped?  Don't worry, there are more test dates this year.  Many people take the SAT multiple times, and if you get some tutoring in between (from yours truly!), you can dramatically increase your scores on the second time through.   The remaining test dates for the current school year are:   January 25 March 8 May 3 June 7 I recommend you start studying for the SAT at least one month in advance, longer if you plan on going it without a tutor. If you'd like to work with me for the January or March test cycle, send me an email ASAP. The sooner we can get to work, the higher your scores will be!

Mr. Lazur and the Volume of a Cone

I will never forget my favorite math teacher. Mr. Lazur taught ninth grade CAS Geometry (my school's version of AP) and also twelfth grade IB Calculus, so I was fortunate enough to have him as a high school freshman and then again as a senior. I'm incredibly thankful for Mr. Lazur because his fun and informal teaching style got even the most anxious students to actually enjoy math. In his classes I learned to think about math on a more “macro” scale, thinking about the concepts and how they related to each other rather than getting bogged down in numbers. He also knew exactly when and how to give a practical demonstration of a confusing concept so that none of us would ever forget it again. One of these demonstrations has stuck with me ever since, and I don't think I'll ever lose the knowledge it provided. We were in Geometry, working on volumes of solids. The previous day we'd learned the formulas for volume for cubes and cylinders, and today we were supposed to... read more

A Few Thoughts on Why We Learn Algebra

I recently responded to a question on WyzAnt's “Answers” page from a very frustrated student asking why he should bother learning algebra. He wanted to know when he would ever need to use it in the “real world” because it was frustrating him to tears and “I'm tired of trying to find your x algebra, and I don't care y either!!!” Now, despite that being a pretty awesome joke, I really felt for this kid. I hear this sort of complaint a lot from students who desperately want to just throw in the towel and skip math completely. But what bothered me even more were the responses already given by three or four other tutors. They were all valid points talking about life skills that require math, such as paying bills, applying for loans, etc., or else career fields that involve math such as computer science and physics. I hear these responses a lot too, and what bothers me is that those answers are clearly not what this poor student needed to hear. When you're that frustrated... read more

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