I always encourage students to get a feel for math.  Specifically, whenever possible they should learn to make accurate estimates and gain an intuitive understanding of whether the answer is a reasonable one.

When I was young I thought history mattered little.  After all, what good is history at getting anything done.  Instead, with math I could better understand how things work, and create new things.  Hence, I knew when I was quite young that I wanted to be an engineer.  And I followed through.

Recent research suggests that during summer break, most students lose 2-2.5 months of the math computational skills they learned through the year.  One of the reasons for this is that many are taught math only using procedures and steps rather than teaching concepts.

Recently I've been exploring 2 web sites that you may find quite interesting.

The solution to problems involving probabilities are not always intuitive and often require careful thought. Below is a famous one (commonly called the "Monty Hall" problem, after the famous host of the "Let's Make a Deal" game show).

Let's say there are three doors.  Between one of the doors is a new bike and behind each of the other two doors is a barrel of onions.  You get to choose any of the three doors and will receive the corresponding prize behind the door.

After choosing your door, the host opens one of the remaining doors to reveal a barrel of onions.  He then gives you a choice to keep the door you originally chose or switch to the remaining door.  Which would you choose?

Many would say "it doesn't matter.  There are two doors and one prize, so each has a 50% chance of being the bike.".  But this isn't accurate. In reality, you are better off switching doors EVERY time. 

The reason is as follows. You have a 1/3 chance of having the bike behind the chosen door and a 2/3 chance that it's behind one of the other doors. When one door is revealed to have onions, the probability remains that the chosen door has a 1/3 chance of the bike while the new information increases the probability of the remaining door having the bike to 2/3.  So the player should always switch.

If you have trouble understanding probabilities, it always helps to 'exaggerate' and then 'simplify'.  To exaggerate, let's say there are 100 doors with one bike prize and 99 corresponding barrels of onions. Let's say you pick door #1 and the host proceeds to reveal 98 barrels of onions from the other doors and gives you the opportunity to switch doors before the last door is opened.  Then would it make sense to switch?  Of course it would!

Now, to make the problem a bit harder, let's assume the host is completely clueless as to which door holds the real prize.  Then should you always switch to get a higher probability?

Again, to solve this problem you should exaggerate (e.g. choose 100 doors) and simplify.  It is HIGHLY unlikely in this scenario that as the host reveals door after door that he will not stumble upon the door holding the bike.  Under such conditions, each door maintains EQUAL probability since the host is randomly choosing doors.  Therefore, he is not increasing the probability that any one remaining door holds the real prize relative to other doors.  It is as if he were "flipping a coin" and each event (i.e. opening of a door) were independent of the others. 

So the real question is whether Monte Hall knew what was behind each door.  And any avid game show observer knows that at least some of the time he must have known what was behind each door to keep the audience engaged.  And therefore, odds are that it was indeed best to switch whenever given the option. :)

Every year there are a number of studies that rank the "best" and "worst" jobs.

Such job rankings typically take factors into account such as income, growth opportunity, challenge, physical demands, and safety.

On sample article is from The Wall Street Journal in early 2009.  This article highlighted the best 5 jobs as follows:

1. Mathematician   
2. Actuary   
3. Statistician   
4. Biologist   
5. Software Engineer

Of course, everyone is different and someone who grows up cutting wood in his/her backyard may dream of becoming a lumberjack (which was ranked last) and very must dislike one of the so called "top jobs".  So, the golden rule remains to "do what you love" as #1 priority and not to build your life dreams off any survey.   BUT, it's insightful that the 5 "best" jobs are all heavily dependent on math skills.

I encourage you to think of math not as "one of many school subjects"  Instead, recognize that math is useful in day to day life, required in excelling at a number of job roles, and teaches you how to think logically which will help you be able to better solve a wide variety of different problems you will encounter throughout life.  And arguably, problem solving is the most important skill that anyone can acquire.

There are certain numbers or sets of numbers that are "famous".  By famous, I mean that they are well known not only in mathematical circles, but also known by those learning more diverse subjects as natural occurrences or even popular culture.

One of those famous "sets of numbers" is the Fibonacci sequence of numbers.  These numbers are simple to generate but associated proofs & usages could be studied for many months.

To form the Fibonacci sequence of numbers, simply start with the numbers 0 and 1.  Then continue the expand the set with new entries that are the sum of the previous two entries.  So, the Fibonacci sequence would be:
{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ... }

This seems simple enough.  The interesting aspect to me is that this sequence of numbers has been used in one form or another for literally 1000s of years.  Its used in poetry, financial markets, computer programs, and the modeling of natural phenomenon such as population growth, shorelines, flowers and more.

In fact, I used a similar equation to the described Fibonacci recursion, in the form x^2 - x - 1 = 0 in my senior project (many years ago) to create a simple pseudo-random number generator for data encryption.  I gleaned the idea from a book called "Understanding Chaos."

I encourage you to learn more about the Fibonacci sequence and send me your thoughts on what you think are some of its most useful and interesting applications.  At a minimum, you'll get a better appreciation (and appear quite sophisticated to your friends/family) while experiencing books/movies that cite the sequence such as the "The Da Vinci Code".

A common rule is that a $1 today is worth more than a $1 tomorrow.  The reason is that it costs money (interest) to borrow money.

A corresponding rule is that if you have $1 now, and save it in an account with interest, the amount in your account will grow in time.

Borrowing and saving money is a vitally important subject that not enough people understand.  So, it is good to review.  The key math concepts you need to understand are: Additional and Multiplication.  Also, its good to understand exponents as a short-cut to writing long equations.

Let's say you start with an amount of money we call "P".  So, today you have P dollars.  Let's say you can find an account that will pay you an interest rate of 5% per year.  Then...

P = $$ this year
P(1+5%) = $$ you'll have next year

So, if P = $1, then next year you'll have 1(1+5%) = 1(1.05) = 1.05

And the following year?  Well, you will earn 5% more again, so...

P(1+.05)(1+.05) = 1.1025  But of course, dollars use only 2 decimals, so in Year 2 you'll have $1.10.

You can continue this for each ensuing year, and you'll quickly discover the following formula is true:

P is the principal (the initial amount you borrow or deposit)

r is the annual rate of interest (percentage)

n is the number of years the amount is deposited or borrowed for.

A is the amount of money accumulated after n years, including interest.

Assuming the interest is paid once a year:

A = P(1 + r)^n

So, lets say you say $1000 per year for 10 years and use an account that guarantees you 5% interest per year.  How much will you have after 10 years?  Using the above formula:

A = 1000(1.05)^10 = $16,288.95

Not bad!  Lets say your friend only has $5000, but he found someone to pay him 15% interest per year.  Let's see how much he would have after 10 years.
   
A = 5000(1.15)^10 = $20,227.79

WOW.   As you can see, the interest rate is VERY important.

This is why it so important to try and borrow as little as possible from others, and instead loan money to others by keeping money in accounts that earn interest. 

Of course, things can get a bit more complex if interest is calculated (or compounded) on a daily or monthly basis, rather than an annual basis, but you should be able to easily create a new formula using simple math.

There are lots of articles on this subject online, but feel free to let me know if you have any questions.
 

There are many ways to help impact intuition and understanding to any student.  But two of the key ways are:

1. Take Math "outside the classroom" and link "boring math" to something in everyday life or "in the news" 
2. Always encourage the use of estimation so math can be used informally and quickly to provide added insights.

It's hard to avoid today's news that discusses the $900B stimulus package.  But few can understand how large is $900B.  To make it more real, I encourage you to first do some simple math calculations using estimation. 

There are about 300M people in the US, so you can quickly calculate the size of the stimulus for each person in the US.  And if you assume 3 persons per family, you can just as easily calculate how much is being spent on each family.  This type of calculation could be done by anyone in 4th grade or higher.

For more advanced students, I encourage you to take this number, and/or the US debt and begin to see what happens when you compound interest.  For example, we have roughly $10.7 trillion dollar debt and last year (2008) we paid ~$412B in interest alone.  What is the interest rate?  How does our debt change each year based on both the deficit and interest? And what happens if we had another $1-2 Trillion dollars to the debt this year?  How long will this take to pay-off if we bring out debt under control each year and pay a given amount back each year to our debtors?

This will build valuable math skills in terms of estimation and reasoning, while also helping you learn more about our country's finances.