Everyone (yes, EVERYONE) who learns new math struggles at first and then slowly puts the pieces together. The only difference is that the people with more experience learning math know what to expect and how to push through it. This is the key to not only understanding math but really to doing anything tough in life.

Think about how it feels when you start a new topic in math. Can you identify these stages in your process? Do you let yourself get caught up in frustration before you push through?

• Stage 1: Bewilderment – I’m sorry. The fact that no one uses this word anymore made it all the more important to me that I manage to use it in a sentence today. Anyway, back to our topic: This is the stage where you have literally no idea what is going on. All you know is that someone is writing symbols on the board and they might as well be making them up. You try to look in the book and it looks more made up than the stuff you saw on the board! Pushing through this stage means asking tons of questions and truly reading your math book. When I say reading, I mean sitting with a pen and paper and trying the examples right along with the book.
• Stage 2: Stumbling – You know you have hit stage 2 when you can do problems in the book, but only if you have your notes. In other words, you get the first step or two and then get stuck. BUT — All it takes is a look at the notes or an example (and a few minutes of frustration) and you can figure out where to go. A lot of people get stuck here because they haven’t learned that the frustration they’re feeling is no different than “feeling the burn” when working out. Those little trip ups and times when you are trying to find you mistake? Those are exactly the moments your brain is building the connections it will need for the next problem.
• Stage 3: Robotic Understanding – After a while of working problems and stumbling about you will find yourself at this stage. Things are starting to feel much different now. While you may not always know WHY you do certain steps, you can do problems on your own and are getting the right answers most of the time. You’re seeing connections between this and things you learned before even though some of the ideas are still fuzzy. This is a good place to be and a place to be careful! Plenty of people figure they got it at this stage and stop practicing/studying. Don’t fall for that! Keeping the work up now will pay off with…
• Stage 4: True Understanding – Truly understanding a math concept or idea means that you can work problems that use it (even if you get stuff wrong here and there – that’s normal!), see how it is connected to other ideas, and even be creative with it (as in do a problem that is different from anything you have seen and apply this new idea to it). In my personal experience, there are probably 100 levels of “true understanding”. Even with all my experience, there have been times where I suddenly realize something about a topic I learned 10 years ago! If you are taking math class, you won’t necessarily reach this stage until you have had a few new topics that your mind can start to connect together. In my experience, the difference between accepting being in the “robotic understanding” stage and pushing to the “true understanding” stage is what separates the A/B’s from everyone else.

Your goal: Next time you are feeling overwhelmed and frustrated learning math, remember: It is part of the process. You will have to find your own way to push through it, but I am hoping just knowing this much will keep you motivated!

math anxiety, math success, studying tips, test-taking

While there are plenty of methods to get a good hand drawn graph of a linear equation, many plot more points than necessary and (if you aren’t careful) misrepresent the location of two important points: the x and y intercepts.

Using the intercepts to plot your line takes care of both of these problems and more importantly: its fast! In this video, I will use two examples to show you exactly how it’s done.

http://www.mathbootcamps.com/media/fastgraphsoflinearequations.mp4 graphing equations, linear equations

Its time to get the bootcamps rolling for 2012! You’ve been waiting much too long to see when the next pre-algebra, algebra, and stats bootcamps will be held. Well, here they are for your math reviewing needs!

Pre-Algebra Bootcamp
This bootcamp is designed for anyone who needs to review the basics like working with fractions, negative numbers, and variables before starting algebra.

Feb 6 – 12
Mar 5 – 11

Algebra Bootcamp
A little rusty on algebra? This bootcamp reviews important algebra skills that come up in courses like algebra 1 (or beginning algebra) and a few of the topics you may find in intermediate algebra / algebra 2. This is perfect for anyone who is preparing to take courses like college algebra or pre-calculus.

Feb 20 – 26
Mar 19 – 25

Statistics Bootcamp
Would you believe me if I said we go from making histograms to hypothesis testing in a week with this bootcamp? Covering much of the same material as an AP stats or college level introductory statistics course, this bootcamp can help you review all of the most important introductory statistics concepts.

Feb 13 – 19
Mar 12 – 18

Remember, these are all online and you get a full 6 months access to the materials after the bootcamp is over. It’s a great deal and always fun / interesting. Think about it: only the most serious of students sign up for a math bootcamp ! Hope to see you in one this spring!

courses, mathbootcamps news

Are you taking an online math class this spring? I could give you tons of advice but my #1 tip is:

Treat it like a real math class.

People feel silly writing notes from something they are reading online. That’s silly. You take notes in a face to face class right? Don’t let yourself become more passive simply because there isn’t the pressure of the professor looking at you funny while you just sit there doing nothing.

Write.

If there are online videos, take notes. If there are online notes, write your own notes from them and try to work the examples as you go. You should never be looking at any math without a notebook in front of you.

math success, online math classes, studying tips

If you thought you missed out on taking this round of the statistics bootcamp, you are in luck! The deadline has been extended and you can now sign up anytime before THIS MONDAY.

If you are enrolled in a stats class for the spring semester, this would be a great way to get ahead of your classmates and make the semester that much easier on yourself! Here are the details:

Start date: Monday Jan 9, 2012
Duration: 7 days – all online
More info: http://www.mathbootcamps.com/statistics_bootcamp.php

Hope to see you there!

mathbootcamps news, statistics, tutoring

At this point, I haven’t even written out any equation or expression and you already know what I’m talking about. There are problems that come up in algebra where it seems like you should be able to cancel a variable out and move on. Yet, in these very situations your professor keeps telling you it can’t happen!

It may not always seem like it, but there really is a pattern and a simple rule of when you can and can’t cancel a term or a variable. Let’s focus on rational expressions specifically. Those are expressions (like the one below) where both the numerator and the denominator are polynomials.

When can you cancel terms in a rational expression?

You can cancel any matching factors that occur in both the numerator and the denominator. Let’s use numbers to understand this a little better:

In the fraction above, 2 is a factor of 4 since 2 x 2 is 4. Similarly, 2 is a factor of 14 since 2 x 7 = 14. Based on my rule, I should be able to cancel the 2′s and get a fraction that is still the same thing! Let’s see:




If you check on a calculator you will find that 4 divided by 14 is the same as 2 divided by 7 – so this rule seems to be working here. What is really going on?

Why does this work?

Every number other than zero, when divided by itself is 1. Since a fraction simply represents division (in one sense), and we really just have .

Trying it with Variables

Now that everything makes some sense with numbers, let’s try variables. For us, the variables represent numbers so they will behave in the same way. Take for instance the expression

The numerator has a couple of factors which we can find by factoring out an x (doesn’t that phrase make a lot more sense now?!): . This shows us that and are factors of the numerator. Looking at the denominator, we see is also a factor since . Remember, when both the numerator and the denominator share a factor, you can cancel that factor since it is really just 1.


The only time you can’t cancel terms in the numerator and denominator is when they are both NOT factors.. That’s why you can’t cancel in . The x is not a factor of the numerator; its just a term being added. Cancelling the here would be like cancelling the in and saying that is 1. If you check it in a calculator it certainly isn’t!

algebra, factors, simplifying

Every semester I have a few students that have put themselves in a really tough situation. Dreading math and being convinced they’re no good at it, they put off taking any math requirement to their last couple of semesters. The moment I hear this, I imagine jumping in a time machine, finding them in their first semester – just as they are registering and saying:

Sign up for a math class now! Take your math courses in your first year! Please! Not only will you do better, you will be so happy you got it done!

This is what a time machine looks like.

Math is a skill and like any other skill, it gets rusty over time. Even if your major only requires one or two math courses, you want to do well right? The idea is – take it while the math from high school is still fresh. Even if it has been a couple of years, that will be easier than when it has been four or even six years!

If you already have a math phobia, then making it the one thing that is keeping you from graduating in your last semester isn’t exactly helpful. If you are taking it in your first year, you can be more relaxed (knowing that you can drop if you must). Being more relaxed will make learning easier and probably help you do better than if you had all kinds of “my last semester” stress.

If you are a college or high school student, do me a favor. Right now, look at your schedule for next semester. Have you taken your math requirements yet? If not, make sure you are signed up for a math class next semester. Get it done.

college planning, math success, studying tips

Sometimes easy and sometimes hard, our calculus problem of the week could come from any calculus topic. If you really want to get better at calculus, following these problems is a great way to make yourself practice! Past calculus problems of the week.

This week’s problem:
Earlier today, I managed to find a really amazing new (to me) math blog. Not only that, there was this (insanely) interesting and creative calculus problem! So, of course, I must share! The problem is: (quoted)

“You’re in a museum and you’re looking at a painting which is hung above eye level. (There is a specific painting which is hung high in the entrance room at the Brooklyn Museum that I think of with this problem.) You are standing some distance away from it. The question is: what is the largest angle x that you can get as you walk forwards and backwards? (See diagram below for setup.)”

Head over this way to see the answer and more!

calculus problem of the week

I’m traveling this weekend – so there won’t be a calculus problem of the week this time around. You’ll have to get your fix next week! In the mean time, you can find old problems here. Make sure you take the time to follow me on twitter while you’re at it!

If you want to go to graduate school (not counting business or law school), chances are you will have to take the GRE or Graduate Records Exam. This test is designed to measure how much you have managed to absorb in four years of college but in reality, the math portion tests heavily on high school level math! The only difference between GRE math questions and the type you saw in high school is the level of trickiness or problem solving you are expected to do. From the GRE’s website:

The content in these areas includes high school mathematics and statistics at a level that is generally no higher than a second course in algebra; it does not include trigonometry, calculus or other higher-level mathematics

Alright, so no calculus or trig – then what can you expect?

• Arithmetic – can you find percentages? add fractions? know how to work with absolute value and a number line? Can you simplify ?
  Specifically you should be familiar with:
  • the properties of integers: divisibility, prime numbers, prime factorizations, basic arithmetic, exponents, radicals, ratios and percents, using absolute value, the number line, and decimal representation.
• Algebra – You find the usual stuff here such as simplifying expression and solving linear equations but also a couple of surprises as well.
  • rules of exponents, factoring and simplifying, relations and functions, solving both linear and quadratic equations, solving systems of equations, word problems (big!), dealing with the graphs of functions and inequalities.



• Geometry – There are quite a few geometry questions on the GRE and since most people take this in high school, you should really take the time to brush up on this topic. However, you will NOT have to worry about constructing proofs – only problems such as:
  • congruence, similarity, special triangles, properties of parallel and perpendicular lines, polygons and quadrilaterals, area, perimeter, volume, the Pythagorean theorem, and angles.
• Statistics and Data Analysis – If you recently took statistics, don’t worry. There is no hypothesis testing or linear regression on the GRE. Instead it focuses on the basics such as:
  • mean, median, mode, range, standard deviation (not calculating it!), quartiles, percentiles, reading from graphs and tables of all types, probabilities including compound events (like “or”, “and”), independent events, random variables, probability distributions, counting methods including combinations and permutations, and Venn Diagrams.

  A high score on the quantitative section of the GRE can mean the difference between getting into graduate school and making other plans – even if you are planning on majoring in a non-quantitative field. Don’t neglect this section, and start studying early. You can get a great score even if you aren’t a math whiz!

  GRE, test prep