Wednesday, September 19, 2018

A Few More

Learn, Mathematics, Child, Girl, Formula

Most children have a person they look up to, someone they want to be like.  Sometimes its a sports celebrity.  Sometimes,  its a model or actress but how many times do you hear someone say "I want to be just like Katherine Johnson when I grow up." When I was young, I didn't know of any female mathematicians.  My high school Algebra teacher inspired me to go into Math.

I think its time we introduce students to some of these wonderful women who contributed to the field of mathematics.  The movie Hidden Figures introduced us to some fantastic ones but unless we see a movie or find something on the internet, most are unknown compared to Descartes, Galileo, or others.

1. Sophie Germain who lived during the French Revolution. When the revolution began, she shut herself in her father's study to read and learn.  It was upon learning about Archimedes that she developed an interest in mathematics. She taught herself Greek and Latin so she could study some of the mathematics in their original languages. 

Although she could not study at the local university, she managed to obtain study notes so she could learn.  Eventually one of the professors discovered she was submitting papers on a false name, so he became her mentor.  By the time she died, she'd become the first woman published by the French Academy of Sciences, proved Fermat's Last Theorem, and her work on the theory of elastisity.

2. Sophia Kovalevskaya born in to a Russia where women were not allowed to attend University so she married a paleontologist and they moved to Germany.  She was privately tutored until she received her Doctorate in Mathematics.  She was known for her papers on partial differential equations, Abelian equations, and Saturn's rings.

After her husband's death, she was appointed as lecturer at the University of Sweden, before becoming the first woman to be granted a full professorship in the region.   She won prizes from both the French Academy of Sciences and the Swedish Academy of Sciences before she died in 1891 at the age of 41.

3. Emmy Noether was lauded by Albert Einstein as the most brilliant and creative mathematicians produced since the higher education of women started.  She grew up in Germany where there were rules against women matriculating to higher at Universities.  Finally she received her PhD, when she wrote on a a topic in Abstract Algebra but she was unable to secure a university position for many years until she was granted an "unofficial associate professor" at the University of Gottingen but she lost it in 1933 because she was Jewish.

Due to the discrimination, she moved to America to teach and conduct research at Bryn Mawr College and the Advanced Institute for Advanced Studies.  Over time, she developed the mathematical foundations for Albert Einsteins general theory of relativity and advances in the field of algebra.

There are other examples I can share and will in the future.  I chose to omit Hypatia and Ada Lovelace because they are fairly well known but these three are not as well known.  If we want to convince women to going into mathematics, we need to show them some role models.

Let me know what you think, I'd love to hear.  Have a great day.


Tuesday, September 18, 2018

The Marquise Du Châtelet


Paris France Eiffel Tower Night Night Pari  If you are like me, you read the name in the title and went "Who?"  This lady was listed as a woman who should be known in science but its another aspect of her that interested me.

The Marquise Du Châtelet born
Gabrielle Emilie le Tonnelier de Breteuil in Paris on December 17, 1706.  This is a time when wealthier women were expected to marry, run the house and bear heirs.  Its amazing that Gabrielle did this and so much more in her time.

She grew up in a traditional household of her time.  When she was 18, she married the Marquis Du Chatelet, a man with a title but little money.  Within a few years she'd produced two sons and a daughter while running the household and pursued traditional entertainments like the opera, etc.  But it was while she was pregnant with her second son that she picked up a mathematics book and began her road to studying and becoming fluent in the subject. 

Eventually, she studied Descartes Analytic Geometry with two of the leading mathematicians of the time.  All through the 1730's and 40's she continued to raise her children, run the house hold, read, study, and publish a few works.  She was even published by the Royal Academy of Sciences which for her time was extraordinary.   

She became pregnant again late in 1748 or early 1749, she worked hard to finish translating Isaac Newton's Principa into French because pregnancies late in life tended to result in death. She did not just translate Newton's work, she either corrected or completed many of his ideas so they are what we know today.

This work was published ten years after her death in 1759 and in time for Haley's comet.  Imagine, a woman in 18th century France who educated herself enough in mathematics that she could read, translate and understand all the concepts and mathematics involved in Newton's work.  That awed me so much.  I know how hard it is to translate from one language to another and get it so it means the same thing in both languages. Even in today's world, her translation is the only full version of Newton's work.

In many ways, she was typical of her time including having lovers outside of marriage.  One of her lovers - Voltaire - she was with for 15 years while the other fathered her last child.  Still, through it all, she wrote many papers, translated, and continued running her house for her husband. When she died in 1749, she was only 43 years old but she left her mark in a male focused world.

Thank you for letting me share this wonderful lady with everyone.  Let me know what you think, I"d love to hear.  Have a great day.

 

Monday, September 17, 2018

Bringing Math into the Classroom.

Kitten Veterinarian Feline Doctor Fur Cat
The other day I read about a third grade teacher who organized her classroom to resemble a veterinarian clinic with several math stations.  Each station provided a different aspect of the math involved in the field of veterinary clinic.

I realized there is no reason, high or middle school teachers should not invite representatives from the bank, a store, the sewer department, the police, or any business to come in and speak to students on how math is used in their jobs.

We have a man who is in charge of the water and sewer department who is always willing to come in to explain the math involved in his job.  It's amazing on how much they have to monitor in the sewer department as they move waste and process it.  Out here, most of the wastes are moved using suction through pipes on the ground.  I don't think we have any buried pipes because the ground freezes.  They also have to keep something running through the pipes to keep them from freezing.  

He is always willing to talk about all of that.  He brings manuals to show the information they use to keep the system running.  He is also great about discussing how he has to keep passing certain math tests to move up the management ladder.  The more classes he passes, the more responsibility he gets.  Its cool.

We don't have any banks out here but I could have the manager of either of the stores come in to discuss the math they use.  They have to order supplies in and if they want the bigger items like washers, dryers, refrigerators, and freezers, they have to figure out shipping via the barge.  They can explain how they calculate the price mark-ups so they make a profit.

There is a local engine repair place who charge a per hour fee plus supplies.  Someone could explain how they arrive at an hourly cost for their labor and how they price the parts.  The business also offers to help people purchase boats and four wheelers (ATV's) and the representative could explain how that works mathematically.  

The only other business of any consequence is the health clinic manned only by health aides.  It would be possible for one to visit and explain how they use mathematics in their jobs.  I know its used when they calculate how much medicine to administer medicines among other things.

In most places, there are so many more businesses to call to see if anyone would be willing to visit your classroom to discuss the math they use.  Most places have access to Insurance companies, larger stores, gas stations, movie theaters, doctors, lawyers, temporary agencies, etc.  

This would add a touch of reality to the mathematics we teach in school.  It can help them see how mathematics is found in the world around them and exists everywhere.  Let me know what you think.  I'd love to hear.  Have a great day.

Friday, September 14, 2018

Mortgage Math.

House, Cemetery, Haunted House  In many small schools, consumer math is no longer being taught.  The school only offers math classes designed for college bound students.  Unfortunately, many of the consumer math classes do not take time to look at mortgages and how a simple eighth of a percentage can change the over amount paid.

When I was growing up, my mother always told me to pay extra every month so the mortgage would be paid off sooner.  This does work because you end up making at least one additional payment per year.

Furthermore, most students don't realize that the larger the down payment, the lower the amount of money they have to borrow which is important to planning ahead.

Lets look at the down part of buying a house.  Most down payments are either 3%, 10%, and 20% of the agreed upon price.  It is good to point out that with a 20% down payment, the purchaser does not have to purchase PMI or private mortgage insurance to protect the lender in case you cannot make payments.  PMI costs between 1/2% and 1 % per year added to the mortgage payment.  Once the loan balance amount reaches 78 percent of the original amount, the PMI is removed.

When securing the mortgage, there are several different types such as a conventional loan for a flat percent over a 15, 20, or 30 year term.  Rates change frequently so its important when choosing a rate, to choose one of the going rates.  In addition, to conventional there are ARM or adjustable rate mortgages which are often guaranteed to remain fixed for a set period of time such as 5 or 10 years before the interest rate is recalculated every year.  This type of mortgages is often appealing in a time when mortgage rates are rising frequently.  Another type is the interest only where the borrower pays for the interest only and often requires the borrower to pay the whole borrowed amount after 2 to 5 years. The idea is the borrower will secure a proper loan before the balloon payment is due.

The last element to look at are interest rates.  Every lender offers a slightly different rate based on your credit scores but many online real estate places such as Realtor.com and Zillow.com offer current interest from several sources so students get an idea of the different rates.

So here are three elements students can explore using a spreadsheet while using math formulas for interest, payment, and down payments to decide which deal is the best.  Use this information to provide a spreadsheet based project where they:

1.  Find a house they would like.
2.  Determine the down payment for 3, 10, and 20 percents.  How much is each.
3.  Find the amount being borrowed.
4.  Calculate the monthly payment for each balance based on mathematical formulas using a low, medium, and high interest rate.
5.  Use the figures for taxes, etc from the websites to be included in the monthly payment.
6.  Calculate any PMI payments at 1 percent of the loan per year.
7.  Once the spread sheet is set up, let them play with paying off the loan by adding $50, $100, or $200 per month to see how it changes the payoff time. 

This is real world application of Math which prepares students for the real world in which they may consider buying a home.  The more they know, they better job they will do planning for buying the house.

Let me know what you think, I'd love to hear.  Have a great day.

Thursday, September 13, 2018

Gambling and Probability

Play Card Game Poker Poker Chips Chips Car Although most of our students will not grow up to be professional gamblers, the study of probability involved in gambling is worth looking at since it is one way to peak student interest.

I know a guy who used to live in Los Angeles before moving up to Alaska.  At that point, he'd pop over to Las Vegas to earn extra money gambling.  He said the reason he never became a professional was he needed to spend time with his kids. 

Its interesting what things a gambler has to keep in mind while gambling.  I'm not talking about the old folks who hit the casinos once a month, or those who hit a gambling establishment occasionally.  I'm talking about those people who make a regular income with the game of chance.

There are three things both the casinos and the gambler must consider about the game.  First, they are dealing with definite possibilities.  Second, the expected value or the amount of money one can expect to get from the game.  Finally, the volitility index or standard deviation when the game is played.

Lets look at these in more detail.  As stated there are definite probabilities within the game.  A gambler knows the probabilities depend on the number of outcomes or sample space.  When you roll a six sided die you have a one in six chance of landing on a specific number but when you are talking poker which uses multiple decks, the probabilities are quite small.  In poker, trying to draw a four in five card is only 0.00024 while drawing a flush is even smaller.  Knowing these odds, helps guide a professional's betting choices.

A second factor in gambling is the expected return per game.  In other words, if the game were based on flipping a coin where you get $1.00 every time the coin comes up heads or  you lose $1.00 every time you get a tail, you would expect to end up with nothing because the probability is 50/50.  Mathematically, it is EV = (.5(1.00) + .5(-1.00)) = 0. Because the odds are equal, this situation is considered fair because no one has the advantage. 

If on the other hand the dealer gave you $1.50 every time a head came up and you lost $1.00 every time a tail came up the odds would change to EV = (.5(1.50) + .5(-1.00)) = .25 or you'd be 25 cents richer per game on average.  This means that for every 100 games played, you'd expect to be $25 ahead.

Usually, gambling casinos have negative EV's so they have the advantage.  They want to have enough money to pay all their bills.  Even thought the EV is negative overall, professional gamblers still come to play because the actual amount they win is often different than the theoretical or EV. 

The volatility index or the standard deviation is what gamblers are concerned with.  This tells them whether they can win or loose a bit more than what is normally expected. They use the deviation for a specific number of rounds to help them decide when to continue or when to stop playing.

Furthermore, its not just mathematical odds, its also their ability to read the body language of the other players to determine if they should stay in the game or fold.  Its not a straight calculation because there is a human element.

Let me know what you think, I'd love to know.  Have a great day.