Until next time, hope this helps. I've seen hundreds of students both succeed and fail, and these tips are what I think are most important. So hopefully you take them seriously!
Whether it is for college or AP Calculus in high school, many of you are just hitting the hard part of this year in calculus. So here are a few tips to get you off on the right foot.
- Practice the harder problems. Most homework assignments include a mix of easier, medium, and harder problems. Typically, it is best to take an assignment like "1-33 odd" and rework it as "1, 5, 9, 11-21 odd, and 22-33." If you are graded, then it just means that you should do the more difficult even problems as well.
- Read all of the problems. This means that you should spend a few minutes thinking about how you would approach each and every problem at the end of the chapter. Of course, you will actually write out the ones that are assigned, but take a moment to convince yourself that you at least know how to start all of them. If you have no idea how to start, get out your paper and pencil and give it a few minutes of dedicated thought. If you still can't get anywhere, ask your teacher or tutor.
- It isn't enough to get the right answer. Calculus is the first time where simply understanding the mechanics is not enough to get an A. Conceptual understanding is crucial. You have to understand why the derivative is what it is, how it relates graphically, etc., not just how to take derivatives.
- On tests, expect the unexpected. Related to all of the above points is the fact that tests will often throw new types of problems at you. This is why you have to practice problems outside of your homework assignments and understand the concepts behind your answers, not just the steps to get them.
- Know definitions and theorems precisely. You know those boxes in your textbook that have boldfaced or colored concepts in them? You have to know exactly what those say. Good examples are the limit laws and definitions of continuity. It's not enough to loosely understand them; you have to know exactly why they are exactly as they are. Know examples and counterexamples, meaning know when the definitions and theorems don't apply.
- Practice every day. Two days in a row, 45 minutes each, is better than every other day for two hours. This is especially important for those of you on odd/even type schedules, or for those of you in college, where you have MWF or TTh classes. Now, more than ever, you need a combination of understanding and performance. It's not enough to understand something, you have to know how to do it. Taking a whole day off sets you back too much. Weekends can erase several days worth of practice.
Should you take the SAT again?
College counselors, family members, peers, and anyone else you talk to will all try to downplay the SAT's importance. They think that they are doing you a favor by telling you to relax, and that it's not a big deal, and don't forget, you're the vice president of the glee club! There is some truth to all of that. I know a student that just got accepted to Berkeley with a score in the 1900's.
While a bad SAT score does not shut any doors, a great SAT score sure can open them. "But I can't get a great score!!" you say. For anyone who is taking precalculus or calculus during their junior year of high school, there is absolutely no reason why you should be getting less than a 700 on each of the math and writing portions of the SAT. You know how to do every single one of the math problems. Every. Single. One. (Notice I also slipped the writing portion in there. That's because it is very formulaic, and you are probably good with formulas, no?)
So what is keeping you from scoring well? You probably haven't done at least ten practice tests. What!?! Ten practice tests?? That's crazy!! Well, you don't take them all at once, but one section per day, that's 90 days, at around 20-30 minutes each, seems like a good way to spend a summer vacation, no?
"But I already took a class from Revolution!!!" you say. Let's have a Dr. Phil moment then - how's that working out for you? How did sitting through hours of stuff that you already knew, followed by doing homework that was neatly categorized and divided up into little concept nuggets actually prepare you for the test?
The SAT is by its very nature unpredictable. That means you practice for it, you don't study for it. Homework assignments that are divided up by subject matter artificially prepare you, because they put you in the frame of mind necessary to solve whatever type of problems they offer.
So back to my original suggestion - ten practice tests. One or two sections at a time. No studying, only practice, alternating with review of previous sections.
Then take the SAT again. I promise you won't be disappointed.
What should you look for in a tutor?
If you have read about our tutoring, you'll see that I (and we) firmly believe in private tutoring as a teaching method. That means that, regardless of whether you choose to go with a tutor from us or not, I think that hiring a private tutor is your best avenue for improved grades, raising test scores, and so on.
So how do you choose a tutor? In my experience, I have figured out a few things that really matter, and a few that don't. First, a brief list of what doesn't matter: score increase guarantees, the school your tutor graduated from, teaching certification, "proprietary methods," initial assessments, to name a few.
So, what does matter?
And, to be honest, that's about it. I'll finish with a few red flags that should send you running in the opposite direction. I've worked for companies of all sorts - some good and others great - and these red flags aren't my opinion, they are what I've gathered from observation.
- Promptness in communication - You might be surprised to find this seemingly irrelevant trait first on the list. In today's world, there is absolutely no reason not to be on the ball with electronic communication. Look for a tutor that responds promptly, and you'll find a tutor that cares about you or your child.
- Subject mastery - Especially in the world of mathematics, subject mastery is absolutely crucial to the ability to communicate effectively. It is not enough to have done well in the course. In my tutoring interviews, the first thing that I give is a test. If possible, you should do the same, preferably without warning. Ask your child's teacher or your professor for help in this arena.
- Rapport - All of the tutoring experience in the world is completely irrelevant to you or your child once the session is underway. What matters is whether the tutor gets along with and communicates well with the student. Ask for a half session meet & greet, and most good tutors will do this for free. After the tutor has left, gauge candidly your or your child's reaction.
- Long term view - Look for a tutor that mentions ideas that point to long term success, in tandem with the usual strategies to do well on an upcoming test. Building core intellectual skills and concept mastery are worth the investment in tutoring and require a commitment on the part of the tutor and the student. A lack of awareness of this fact will mean that you won't get any lasting results from the tutoring.
I'll probably think of some things to add to this, I always do. Until next time, happy schooling.
- Big companies - Don't get me started here. Places like K*plan, Pr*nceton R*view, R*volution, are marketing machines. They invest heavily, heavily, heavily in marketing. What they DON'T invest in is their tutors or their materials. Give me any of their books and I will find a mistake in five minutes. Their good tutors quit readily because their pay structure is so poor.
- Package requirements - Offering discounts is one thing (we do that), but requiring the purchase of a package? Use your imagination and you'll be right as to how this can go wrong.
- Initial questionnaire - If the very first thing that a tutoring company has you do is fill out a census style questionnaire, it means that the person assigning and managing the tutors is probably underqualified as an education professional.
New Year's (re)Solutions
Some of you are done with final exams, and some of you have them upcoming after break. Most of you, no doubt, are resolving to study harder, get better grades, all that stuff. So, I thought I would list a few promises that you can make to yourself and really keep. As always, you'll get no fluff from me. And this time, you might get more than a few objections if you actually follow through.
But this is good advice. Seriously. You can blame me if it goes wrong. But make sure you read the explanations that follow.
Until next year, happy holidays, and be sure to keep up with the YouTube videos.
- Stop copying everything the teacher writes. How many times have you missed something and then been completely lost during class? Rather than furiously trying to keep up with everything the teacher writes, how about trying instead to listen to everything and follow conceptually. It's okay if you miss the details, you can always look in your book. And it's way better than wasting a whole day trying to catch up to something you don't understand anyway.
- Stop doing your homework. Okay, not exactly. Stop doing exactly what you are assigned. Instead, start your homework by reading each and every problem at the end of the chapter. I'll repeat that, louder, because it is so important: READ EACH AND EVERY PROBLEM AT THE END OF THE CHAPTER. Think about how you would start each one. Work on the ones that you don't know how to do. Of course, you'll still do (most) of what you are assigned, but your real goal should be to prep yourself for the more difficult problems, the ones that might trip you up on the test.
- Stop working so hard on the same problem. Problems in textbooks and on tests are meant to be solved in an efficient manner. If you are encountering some impossible algebra, chances are that you made a mistake somewhere. Starting over is almost always better than continuing to push along what may be an impossible path.
- Stop writing between the lines. Don't write math the same way that you write everything else. Give active thought to how you organize symbols on the page: how you use parentheses, how you space the numerator and denominator of a fraction, how you carefully keep track of negatives. This piece of advice, if followed correctly, is probably worth more than any other, in terms of increased grades.
- Stop being so fancy with your graphs. With graphs, one of two things can go wrong. The first is that you try to use some fancy method for translating and reshaping a graph, and you forget to check that it goes through the right points. Seriously, PLUG IN POINTS!!! Never underestimate the method that a seventh grader uses for graphing. The second is that you take forever carefully labeling a graph, making sure it looks pretty. It's just a waste of time. Just label the key points, and get on with life.
- Stop checking over your work. There is a saying in business: "Build in quality control from the start." In other words, do something right the first time, and you don't have to pay an inspector to fix the mistakes. Incorporate this lesson into your test taking. Slow down and do it right the first time. Also, instead of checking over your work by rereading, try solving the problem a different way. Check your answer by plugging in some simple numbers. If you simply reread your solution, chances are you will simply make the same mistake(s) again.
- Stop reviewing your homework/notes. Instead, get out a blank sheet of paper, open up your book, and start working problems. Even if they are problems you have already done.
A New Toy
EDIT: I've realized that, in my efforts to go slowly in the videos, I sound totally boring and even might put you to sleep. I'll fix that.
EDIT: Okay now I sound less dull, at least in the first one below.
Just a quick post to point out that I just got a stylus for my iPad, and also got the app ScreenChomp, which allows me to make math videos, Khan Academy style. Right now I'm peppering the site with video content, so if you have a question, send it to me, and I'll answer it in a video! Check out the GRE problems below.
Word Problems, Part 1
At every level of abstract math, from algebra 1 to multivariable calculus, there is a common complaint from students: "I hate word problems." (After multivariable calculus, no one wants to look stupid, so they stop complaining.)
So there you are, stuck on your last homework problem, and it is a word problem. You aren't even confused by the math, or so it seems: you've mastered fractions, or factoring, or asymptotes, or the chain rule, or polar coordinates, or Green's theorem, or whatever it is that you happen to be studying between the ages of 12 and 65.
You ask your tutor, and the tutor begins "Well, you just have to break it down...write out what you know...draw a picture...blah blah blah." It makes perfect sense when the tutor works it out like that. You tell yourself that word problems are easy, and you kind of feel silly for not getting it. Then the next day there is a word problem on the quiz. And you stare at it and don't get it. What happened?
Math is a language. So, to be fair to math, let's call a word problem what it really is - a math problem written in a foreign language. And let's suppose that math is your native language. So, all you need to do is translate the word problem - which is written in a foreign language - into math, which is your native language. The trouble is, you don't really speak the foreign language that well.
What do you do when you have to translate a foreign language into your native language? You certainly don't stare at a whole passage and try to just "get it" all of a sudden. You go sentence by sentence, phrase by phrase, maybe even word by word. You break it up into whatever chunks are necessary.
For each chunk, you go to your dictionary. You don't think about the other chunks while you are focusing on each particular chunk, you rely on the dictionary. When you are done with every chunk, you will have an awkwardly written passage, but it will be in your native language. Now you can sort out what it means.
For now, I've got some more tutor stuff to do, but I'll work through an example in the next post. Until then...well, for most of you it is summer, and you probably aren't reading this, so you'll just see both posts at once.
The Math of Khan
Just a quick post to point you to one of the most valuable math tutoring sites on the net: Khan Academy. I guess it proves that I'm not much of a salesman if I show you a place where they are giving math instruction away for free! In all honesty, for those times when you are stuck and there isn't a tutor, teacher, or a classmate to turn to, try watching some videos of Sal Khan.
He's done something truly remarkable, and more power to him. In the fall, a number of schools are going to pilot for software bas
ed on the site, and my fingers are crossed for Renaissance Arts Academy
to get the nod. Of course, I'm biased, since I've been a math adviser there for the past few months, and I plan to continue into next year. Good luck on final exams, or on your summer courses, whatever the case may be.
Lately, whether it be prepping for the AP test that has come and gone, or prepping for California state testing which is about to start, I've been thinking and tutoring a lot with test prep strategies. I thought I would share a few. Once again, no filler material. And if you think you are too fancy for some of these, like you are in BC calc or something, then you have a lot to learn:
Until next time, always use parentheses, write big and clear, and always get your negatives right.
- The answers are part of the question on a multiple choice math test. Read the answers before you start working on the problem. It can save you work when you can immediately rule out wrong answers, and it can also save you frustration by preventing a trip down the wrong problem solving path.
- Read the question, read the answers, read the question. Do this routine every time.
- Plug in 0. Plug in 1. Plug in -1. Plug in 1/2. Plug in -1/2. Plug in 2. Plug in -2. In that order.
- Did you forget to plug in numbers to the answers as well? The answers are part of the question! Go back and reread the previous three bullet points.
- The first thing you should look at when you see a graph is the y-intercept. The next is the x-intercept. The next is the slope, or whether it is increasing or decreasing.
- Check your work in a different way than the way that you did the problem. Generally, that means testing out your answer by plugging in numbers if you found the answer algebraically or abstractly, and trying to do algebra or differentiate or whatever if you found the answer with numbers.
Anyone in AP Calculus knows that the test is coming up in May. So I thought I would share some tips. And this isn't going to be a useless "top 10 best ways to eat a good breakfast and make sure you study every night" list. You'll only get the actual useful stuff, and only serious readers need continue. Here goes:
- (1/x)' = -1/x2 and (x½)' = 1/2x½. These derivatives come up over and over, and you don't want to have to use the power rule every time. Memorize them and it will save you valuable minutes. The second one also tells you that if a square root is in the denominator of an integral, it will be in the numerator of the answer.
- (arctan(x))' = 1/(1+x2). This knowledge alone is worth one or two multiple choice questions, and you probably have forgotten it from first semester. While you are at it, go look up the derivatives of all of the trig and inverse trig functions. That means csc(x) and sec(x). These obviously will also show up in integrals.
- PRACTICE. Don't study. Do lots of problems. Only do problems. If you are tempted to lazily read over your notes, take a nap instead, until you are rested enough to do problems. Don't buy a book. If you are unsure how to find practice problems, you can get a step by step guide here.
- "Linearization" means using the tangent line. The formula for the tangent line is best remembered in point-slope form. That means f'(a) = [y-f(a)]/(x-a), or slope = rise/run.
- For those of you in BC calc, have your basic Taylor series memorized. That means knowing that sin(x) = x - x3/6 + ... WITHOUT going through the tedium of the Taylor series formula.
- There will be a numerical integration question among the open ended problems.
- Learn to find volumes geometrically; DON'T memorize the formula for "rotating around the y-axis" and so on. The AP test loves to ask you to find volumes involving cross sections that are triangles and rectangles, instead of circles.
That's enough for today. It's been a hectic month, and my posts are lagging a bit, in true blog fashion. But I promise more study tips soon.
So far, I've written about practice, and that's about it. The New York Times even wrote an article that let me off the hook for a whole blog entry, since they wrote about a study that "demonstrated" that practicing test taking was the most effective means to do better on tests.
The last angle in our approach we call "Motivation." It doesn't mean the desire to get up at 6:14am and set aside 3 hours and 32 minutes per weekday evening for homework, although those things are nice. What it means is that behind every mathematical concept, every problem and every problem solution, there is a motivation behind why anyone ever thought of it in the first place. And if you really want to understand an idea, to understand it to the point that it is a part of you, so that you can apply it in ways that you yourself haven't even considered yet, you absolutely must learn the motivation behind the idea.
One of my favorite pseudo-layman's books regarding this angle is Einstein's Clocks and Poincare's Maps. The theory of relativity is some pretty fancy stuff, but setting that aside for a moment, the author shows the reader the reason why Einstein was led to consider the ideas in the first place: he wanted to know how to coordinate the times between train stations that were far apart. This was during the turn of the previous century, before routine conveniences like telephones and wireless internet on transatlantic flights. Ultimately, this got Einstein thinking about relativistic time, a paradigm shift that led him to his now famous theory. To learn more, I invite you to check out the book, which is not for the mathematically faint at heart, but hopefully you have a better idea of how a practical consideration can motivate an idea.
For something more accessible, we can turn to high school geometry, when students must learn to make things like right angles and parallel lines using only a ruler and compass. In Euclid's time, this was the way that engineers, architects, and construction teams actually did things! When they were building the walls of some huge thing that we now know as ruins, they literally had huge compasses and rulers to guide them towards precise angles and measurements. When they had to construct congruent shapes, they had to get it right, or things literally collapsed. Euclid was a revolutionary in publishing his work on plane geometry, and it became an indispensable reference. (He also pioneered the domain of intellectual property rights, because the knowledge became carefully guarded and fought over, but that is a history lesson, and far from my domain of expertise.)
Whenever you have the question in math class of "Why should we have to know this?", you probably should rephrase it as "Why did anyone ever think of this in the first place?" Put that way, it is a fair and even crucial question to answer. We consider it part of learning mathematics, beyond interesting historical tidbits. So ask your teacher next time, and don't take "Because it's on the test!" as an answer.
The New York Times recently published an article about test taking as a method for learning, and the study goes as far as to claim it is better than "studying."
Far be it from me to gloat, but the timing of this article couldn't be better for me to say "I told you so." It also prompts me to clarify what constitutes practicing a math problem: it means starting from scratch, without your book or notes, and struggling through the points of confusion without giving in to the temptation of going to your book or notes.
So if you'll scroll a few entries down, you'll see that I once had no idea if our method works better than others. Well, at least now I've got a scientific study to back me up. Of course, those can always be wrought with logical fallacies and statistical errors, but that is a different discussion altogether. So, if you want to really learn, keep practicing. And you don't have to take my word for it.
"You Want to Know Geometrical Problems"
The title of this post is the title of a book on geometry that I came across in college. It had been translated from Japanese, which explains the weirdness. It does manage to highlight why we actually learn math and what it means to DO math: it means solving problems.
Although it is listed second on the tutoring approach, I believe practice is the most important component to improvement in math. (The reason that I list confidence first is because confidence makes people feel good, and I'd like potential customers to feel good while they read the site.) I mention the analogy between math and music. If you've ever been my personal tutoring client, you are probably sick of me and my analogies between math and music. The basics: songs ~ problems, performances ~ tests, chords/scales ~ concepts, song recordings ~ textbook.
Suppose that you were practicing for a piano recital. Would you rehearse a song enough times until you finally played it perfectly, all the way through, one time? If you did that, there would still be a good chance that your recital went fine. Let's be generous and say that 75% of the time you would perform just fine.
That rehearsal scheme is exactly how most students go about their math homework. They work each problem until they get it right, then they move on. If they do look back at the problem, then they read over it to "make sure that they still get it." Reading over the problem is about as useful as listening to a recording of yourself playing a song perfectly.
Supposing you screwed up the recital, would you try to explain it to the audience? "But I totally understood the song. I knew how to play every note and every chord perfectly. So you should be satisfied." (What's truly funny is, if you are familiar with American Idol and similar shows, you'll hear contestants try to do exactly that.) Of course, that sounds ridiculous, but the same justification is attempted with math tests all the time.
Hopefully by now you have a better idea of the role that practice plays in learning to DO math. So I'll get off my soapbox for now, until next time...
Practice vs. Study vs. Confidence vs. Motivation
On the left you'll see my explanation of our Tutoring Approach, and if you explore a bit further, you'll see that I've broken it down into four categories: confidence, practice, study, and motivation. For these first few blog entries, I'd like to explain that a bit further, if not to you, then to myself.
First off, a confession: for a long time, I had never actually written down my approach to tutoring. A few years back, there came a time when my tutoring requests exceeded what I could personally handle. So I began referring clients to colleagues of mine in the math department at Berkeley. Before I passed any torches, I had to tell each of them how I went about doing things. Lo and behold, my tutoring business was born, and with it, my tutoring method.
Later, when I taught a course to new graduate students on how to teach, I refined my methods further. (To this day I'm proud to say that I've personally influenced more than a few parts of the "how to teach" syllabus.) More recently, I launched this website, polished my methods a bit further, and laid them out for all to see. (I'll probably revise them further after I write this entry.)
Another confession: I have no idea how unique or groundbreaking my approach is. In fact, I used to call it unique, right here on this site, but I just revised that. But if you can forgive the shameless plug, I will tell you why I have an overwhelming hunch that I'm doing something right, something different from what other people are doing. Over and over my clients are shocked at the progress they make. Heck, I'm shocked. It's awesome; it's a thrill; it's why I do what I do. Okay, enough with the self-promotion.
Last confession: I have absolutely no formal training in education. What I do have is a lot, and I mean A LOT, of formal training in mathematics. My credentials are that of a professional mathematician, not a professional educator. That means that I know how to DO math. And that's what I do my best to teach clients: how to do math. Not how to understand it, not how to memorize it, how to DO it.