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!

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.

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 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.

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.

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 always willing to adjust my rate for a student with a special circumstance, so if you’re in need of help, don’t hesitate to contact me!)

As for the specific changes they’re making to the test itself, I have more worries there. Again, I think their intentions are good, but I suspect some of these changes will have unbalancing ramifications that the College Board is unaware of. I’ll detail a few of my biggest worries below.

Passages About School Subjects

In the current SAT, passages for the reading comprehension are deliberately selected to be obscure and deal with topics the students wouldn’t have heard about before. This is done to ensure that students are forced to draw their information exclusively from the passage, and to prevent students from having an unfair advantage due to prior knowledge of the topic. In the 2016 redesign, passages will now deal with topics from school subjects such as science, history and social studies. The College Board says that this will make the test more relevant to the information that students are learning in school – but I was under the impression that the whole point of the obscure passages was that they weren’t relevant to the information from school – you couldn’t get lucky and get a passage you knew a lot about already.

Here’s an example: suppose two students, Student A and Student B, are given the same booklet on the same test date. This booklet, by luck of the draw, contains a majority of history-based passages, including one about the Haitian Revolution. Student A loves history, and actually wrote a term paper about the Haitian Revolution the previous year. Student B, on the other hand, hates history and has an awful time reading about it – the names and dates just jumble together into a big mess in her head. Obviously, Student A has an unfair advantage overall on this test – her love for history means she will likely be better able to find the information needed in the passage. Student B will be at a disadvantage, and will probably struggle to make sense out of the information that Student A found so enthralling. On the other hand, of course, Student A’s extensive knowledge of the Haitian Revolution from her term paper might lead to her relying on information not present in the passage, which could cause her to miss some questions. Or she could still get the right answers, but not even need to use the critical reasoning skills that the test is supposed to be evaluating.

Overall, I’m reserving judgement on this topic until we hear more from the College Board about the process of creating the new test. I do see ways in which they could balance the test – make sure there are exactly the same number of passages and questions about each subject, for example, or make the topics discussed come from higher-level topics than the students would have covered. However, I would want to know that the test structure has been heavily beta-tested before it is unleashed on the general student population.

No Penalty For Guessing

The concept of “being penalized for guessing” versus “not being penalized for guessing” is actually a gross oversimplification. It’s not that the current SAT can somehow tell whether you knew that answer was right when you picked it – it’s all about probability. There are 5 answer choices for each question of the SAT, and a wrong answer gets you a ¼ – point penalty. This means that if you can narrow it down to 3 choices or less, you’re actually better off guessing, statistically-speaking. So the SAT doesn’t penalize guessing, it penalizes blind guessing. It actually rewards educated guessing. By removing the penalty for wrong answers, they are no longer offering an incentive to avoid random guessing. The College Board says this change is to encourage students to offer “the best answer they have to every question” – but that’s what I already teach my students to do with the current system! The only real difference, if taught appropriately, is that in the last minute or so of each section students should now go back through and randomly pick answers for any questions they were completely clueless on, since leaving a question blank is now, statistically, a bad idea.

So this change will have very little effect for students who are already learning smart test-taking strategies – so why am I worried? Well, it could artificially inflate the scores of students who are less prepared. Unprepared students who are guessing blindly could now conceivably get lucky and happen to pick the right answers enough of the time to raise their scores, which would give an inaccurate assessment of their abilities and knowledge. Without the discouragement of wrong-answer penalties, you just encourage blind guessing. Most of the changes in the SAT over the years have been in response to either substantially decreasing or substantially rising test scores, and I worry that eliminating the penalty for guessing will cause scores to artificially rise, possibly resulting in an increasing number of perfect scores such as the ones that prompted the 2005 redesign.

The “Great Global Conversation”

My final worry is a somewhat different one. The introduction of the new “Great Global Conversation” sounds as though it is intended to encourage students to learn about and participate in a nation-wide discussion of the principles our nation was founded on and how they have developed over time. While this is an admirable sentiment in general, I can’t help feeling it is really not the business of a standardized test to encourage it. If we want our students to participate more in the conversation about our nation and where we should be headed, that’s fine – but why is the College Board worrying about it? Shouldn’t that ball be in the court of the country’s social studies teachers? I’m not convinced that putting the Gettysburg Address or Martin Luther King’s “I Have A Dream” speech on the SAT and asking questions about it would really affect the “conversation” that much. Especially since they’d be treated like any other passage on the SAT – that is, the test-taker would not be supposed to bring outside information into their process, but simply look for evidence in the text itself. It’s not as if the questions about the Gettysburg Address would be asking for the test-taker’s opinion on the topic; even a passage written by another notable figure about the Gettysburg Address would be followed with questions asking about that author’s opinion, not the student’s. It is certainly an admirable goal to have for students’ futures, but I don’t think the College Board should be trying to achieve it through a standardized test.