School Teacher Who Helped The Space Program.

When we think of the rocket program, we think of Goddard, of the women in Hidden Figures, but did you know of the Russian School Teacher who formulated many of the principles used in the space program.  What was really impressive is that he never built a rocket on his own, or experimented with them but he turned Jules Verne speculation into reality.

Konstantin Tsiolkovsky was born in 1857 in the Siberia area of Russia.  His father was Polish and had been deported there.  When he was ten years old, he sickened with Scarlet Fever which caused him to lose most of his hearing.  He chose not to let it slow him down. 

Due to the hearing loss, he was educated at home and when he was old enough, his family sent him to attend college in Moscow.  While there, he obtained an solid education in science and mathematics but when his father discovered he was overworking himself, he was called home where he took exams to become a teacher and passed. At the same time as working as a teacher, he read Jules Verne stories of space travel and he even began to write science fiction.

From here, he transitioned to writing scientific papers on gyroscopes, rocket control, rocket fuel, escape velocities and how action and reaction works in space travel. In addition, he studied the Chinese rockets and used math and science to create rocket dynamics. This lead to him to publish the equations in a Russian aviation magazine and was called the Tsiolkovsky Formula.  This formula laid out the relationships between rocket speed, the speed of gas at its exit, the mass of the rocket and it's propellants. It is the Tsiolkovsky Formula which formed the basis of much of the modern space craft engineering.

Furthermore, he is responsible for building Russia's first wind tunnel so various aircraft designs could be tested to determine their aeronautical abilities. Since no one was willing to help finance the wind tunnel, he dipped into his family's monies to pay for it. He spent so much of his adult life exploring topics associated with aerodynamics, rockets, heat transfer, friction, and other such topics but the Aeronautics congress helped in St. Petersburg didn't think much of his research. 

In addition, he published the Investigations of Outer Space By Rocket Devices in 1911 and the Aims of Astronauts in 1914. Furthermore, he published information about his theory of multistage rockets based on his knowledge of propellants dynamics in 1929.  By 1921, the predecessor of the Academy of Science chose to elect him as a member and granted him a pension for all the contributions he made.  He died in 1935 but his knowledge was used in creating both the Russian and American Space problems.  He shares the honor of being the Father Of Modern Rocketry with Goddard and Oberth since all three of these people came up with much of the same information independently.  

I chose Tsiolkovsky since he majored in Math in college, taught school, and managed to develop foundational concepts that contributed to the development of the space program.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If you have music that is really, really slow at. 20 beats per minute, how many beats will there be played in one hour?

Warm-up

 

If the basic tango steps take 8 counts of music and 4 counts make a bar of music, hoe many steps will a couple take in 34 measures of music?

Black Friday

Happy Thanksgiving

 

Why Teach Statistics To Grades K to 12?

From my personal experience, the two topics that get ignored the most is probability and statistics.  I know I end up skipping over it in high school because I often need to finish the main mathematical concepts and I run out of town.  I know many teachers who do the same simply because the probability and statistics sections are incorporated into most math classes and don't seem to fit.  

There is this perception that statistics should only be taught as a separate class in high school and in some, it's an AP class for those who have finished all their required math classes.  Unfortunately, many do not think schools should be teaching statistics before then.

It has been pointed out that people are currently exposed to so much data via the internet and people need to be able to interpret it.  Consequently, studying statistics, even from kindergarten, is an important skill in today's society. Unfortunately, most people are not very good at statistics unless it is in a context they love, such as in sports or fantasy leagues.  If you speak to someone who is into basketball, they can give you all the stats for their favorite team or player or if they are involved in a fantasy league, they know the stats of every player so they can create their team.

So why start have students learn about probability and statistics so early?  It takes a while for students to develop sound statistical thinking and reasoning and for that it has to start in kindergarten.  Right now,  the common core standards introduces formal stats in the 6th grade and middle school but elementary grades can focus on collecting, organizing, and describing data in a variety of ways. It is also important for all students to make sense of graphs and data in subjects other than math, in subjects such as science and social studies. 

Furthermore, all statistical lessons should be sprinkled throughout the different units rather than treating the topic with one chapter out of the book. It has been suggested that we begin introducing probability and statistics  to kindergarteners, we ask students about their personal preferences such as type of music, television show, or ice cream flavor, or look at measurements like number of books, heights of students, or shoe size for measurements. Even in kindergarten, students can create their own surveys and begin to analyze the data they collected.  

When statistics is taught throughout the school year in math and other topics, it helps students develop critical thinking skills and to think more critically about the topic in general. In addition, statistics does not always work the same way as most math.  For instance, math is usually taught so students follow a series of steps to come up with the answer but in statistics is about context and the answer making sense. 

Furthermore, it is important to take most elementary school activities a bit further.  For instance, if you have students who count and chart shoe sizes for everyone in the class and then graph all the results, most times, the lesson stops there but it is important to have students analyze the data they collected. Analyzing the data helps students learn more about grouping and scales which help in Algebra and Geometry.  tHis helps build mathematical reasoning. 

When we start teaching statistics in elementary and we have students analyze the data, they develop a sense of data and statistics so by the time they reach middle school, they are ready for a proper introduction of the topic and have a solid foundation for the topic.  Let me know what you think, I'd love to hear.  Have a great day.

Math Explains Why Some Volcanic Bombs Do No Explode.

When ever a certain type of volcano erupts you end up with volcanic bombs being tossed out.  Some explode on impact and others don't but it hasn't been until recently that math was used to explain why some don't. First off, a volcanic bomb is mass of cooling lava that flies through the air during an eruption.  In order to be classified as a bomb, it has to be larger than 2.5 inches. Furthermore, it is usually red or brown that weathers to a yellowish brown.

Volcanic bombs tend to occur in volcanoes near or surrounded by water and the bombs absorb a lot of water  when they are launched through the water. Then the water is turned to steam causing the bomb to explode. However, some never explode and that has confused many a scientist since no one knew why this happened. The problem is that the ones that explode mid-air are fine but its the ones that do not because they can hurt people or animals, or damage houses, cars, and other things. To help satisfy scientists curiosity, a volcanologist worked with two mathematicians to try to find the answer as to why only some bombs explode.

One of the first things they did was to create a mathematical model designed to simulate the launch of volcanic bombs because its too hard to to study the fast moving objects in real life. They used data from real life data bombs to help establish the parameters of mathematical model. In the model, they varied the temperatures and pressures to see which effected the bombs.According to the results, it appears water both causes the bombs to explode and keeps them from exploding.

They's discovered the as the magma rises towards the surface, the surrounding pressure decreases while the trapped water is turned to vapor which escapes leaving bubble holes. The object is then pushed through the water and turned into a bomb. The key to the bombs is that some bombs are not solid yet so there are ways for the water to escape so the pressure buildup is eliminated and it doesn't explode. If the bomb does not develop the bubbles and doesn't have the escape routes, so the pressure inside the bomb increases and explodes.  Fortunately most bombs allow the gases to escape. 

The scientists who created the mathematical model love that this shows how mathematics can solve a non-abstract problem. Explaining to people that you are figuring out why some volcanic bombs explode and using math to do it, impresses people and they can relate to it.  Now if you'd like to share this with students,  the American Mathematics Society has provided a couple of algebra based activities for the classroom.

The first activity looks at how the ideal gas law applies to the situation and gives students the opportunity to use it.  They also provide a link to a simulator that allows students to play with various numbers in the ideal gas law.  The second activity has students practice linear equations in reference to lava flows.  It is actually an activity provided by Science Friday.  

Check it out and let me know what you think.  I'd love to hear.  Have a great week. 

Warm up

 

If a snail travels at the average speed of 1 mm per second, how many feet does it travel in one hour?

Warmup

If there are 20159 snails in one ton, what is the average weight of one? 

Hot Dogs And Number Theory

Who ever thought that number theory could be applied to hot dogs.  I certainly didn't until I read all about it.  This is also one of those wonderful real life applications of the Chinese remainder theorem and least common multiple.  

Think about it.  Hot dogs come in packs of 10 while the buns come in a packs of 8, so you end up figuring out how many of each you need. So the easiest way is to buy 8 packs of hot dogs and 10 packs of buns but who wants to deal with 80 hot dogs.

On the other hand, if you use the least common multiple methodology, you end up buying 4 bags of hot dogs and 5 bags of buns to get 40 hot dogs.  If you were my mother, you'd buy a bunch of hot dogs, freeze them, and then thaw the exact number needed and the same for buns but that's no fun mathematically. 

Now it turns out that the the factors of 4 and 5 are relatively prime because the two numbers have no factors in common and the lowest common multiple is their product.  If two numbers are not relatively prime such as 12 and 15, they will have a least common multiple that is less than their product. So what happens if you have one or two hot dogs left over. The left over hot dogs then take the whole problem into the Chinese Remainder Theorem which was identified by a Chinese mathematician over 2,000 years ago. 

The Chinese Remainder Theorem belongs to a field of study referred to as modular arithmetic which looks at the remainders left after a division problem. This particular area is used in a variety of applications from astronomy to cryptography.  Basically what the theorem says is that if you are dividing a number by relatively prime numbers, there will a unique solution that is greater than or equal to zero but less than the product of the two factors regardless of the remainder.

For instance, if you have a packs of 5 hot dogs and packages of buns with 8 and you have one hot dog left over, you start with 6 hot dogs and 8 buns.  If you add packs to each, you get a solution of 3 hot dog packs and two packages of buns which is 1 unique solution less than the product of 5 x 8 or 40.  Thus the Chinese Remainder Theorem tells us there exists a solution and provides us with a method to find the solution. 

As far as notation, it is written as X = remainder mod base.  The base is the number of objects. This notation, can be rewritten into algebraic equations such as base(a) + 2nd base (b) = remainder and from here it can be solved. It is possible the answer might be a negative number.  Interesting connection between hot dogs, lowest common multiple, and the Chinese Remainder Theorem.  Let me know what you think, I'd love to hear.  Have a great day.

Why Did The Stock Market Originally Use Fractions Rather Than Decimals.

 

When I stated teaching math all those years ago, we'd always list the stock market as one of the places in real life that used fractions.  We could pull examples out of the newspaper or off the internet to show that Apple might be 46 3/4 or Pepsi was 123 1/8.  We could have students pick a stock to follow daily and then have them calculate the percent increase or decrease, graph it, just do so many things with the fractions.  Now, even the stock market is listed using decimals. 

Prior to April 9, 2001 when the Securities and Exchange Commission ordered the stock market to change from using fractions to listing everything in decimals.  The stock market is over 200 years old and when it began, it was based on the the Spanish trading system dating back to the 1600's. This system was based on fractions rather than decimals. 

Back in 1792, 24 bankers, brokers, and merchants formed the New York Stock Exchange by signing the Buttonwood Agreement.  Since America was so new, they looked to Europe for a system to model their new exchange on.  After looking at different systems, they decided to base it on the system used by Spain.  

Originally, Spanish traders used gold doubloons either whole or cut into quarters, eighths, or half so that they could use their fingers and not the thumbs to count things so rather than being base ten, it was actually base eight. When the New York Stock Exchange, the eighth of a dollar or 12.5 cents was the smallest unit used for trades but eventually, it was changed to 6.25 cents or one sixteenth of a dollar for large trades to change the spread.  This was done to minimize loses.  Think about it, if you have 100,000 shares and each share does down 1/8th you'd lose that is about $12,500 but if the smallest amount is 6.25 cents per share, the loss is only $6,250.  Since then, they've made the smallest units thirty-secondths and sixty-fourths.

Unfortunately, having the stock market use fractions created some problems because we have a decimal based society.  It made it hard to determine how many shares you could buy if you had $5000 dollars to invest and share ran 43 1/4 per share.  One had to change the fraction based values into decimal values in order to do the actual calculations. 

So in 1997, they signed to Common Cents Stock Pricing Act which required the listing of stock to change from fractions to decimals and the process began in August 2000 when the New York Stock Exchange offered seven stocks in decimal form and by September, they offered 57 stocks in decimal form. The process was completed in 2001. 

This change has been beneficial to the investor because the all price increases were more precise and the losses became smaller. Furthermore, the United States stock market form matched the stock markets in the rest of the world.  So when the stock market changed their listings from fractions to decimals, I lost one of my more useful real life examples.  Let me know what you think, I'd love to hear.  Have a great day.

The Math Of Snow

 Snow is a wonderful thing. It can be light and airy or it can be quite dense and heavy.  You can ski or snowboard on it.  It can be used to make ice cream or a snowman but there is so much math associated with it.

For instance, we see ratios used when discussing how much snow has to fall in order to get one inch of water.  If it is an average snow which is neither too wet or too dry, you need ten inches of snow to equal one inch of water.  On the other hand, if your snow is wet, it only takes five inches of snow to equal one inch of water but if your snow is dry, you'll need 15 inches to get the same one inch of water.  Knowing these ratios means you can figure out how much water it equals.  For instance if you only get four inches of wet snow, you'd set up a proportion of 4/x = 15/1, so if you solve for x, you'll get 0.27 inches of water.

Furthermore, if your snow is such that you can create a proper snowman, one made of of three spheres decreasing in size, but the ratio is based on different ratios depending on who you talk to.  There is the  the 1:2:3 ratio, or the 3:5:8 (top to bottom) ratio for the spheres,  all suggestions based on the Fibonacci series. Thus, if you know the radius of the top sphere, you can calculate the radii of the other two so that your snowman has the proper proportions.  A mathematician in Poland actually created a Snowman calculator which allows people to determine the maximum sized snowman they can build based on the amount of snow in their backyard based on the golden ratio.

This article gives some interesting information on how much it cost the city of Boston to clear out 99 inches of snow or snow that was 8 feet 3 inches tall.  Back in 2015, Boston got hit with 99 inches of snow and the city had to send out every truck they could just to keep the snow from completely shutting everything down.  The workers spent 185,000 man hours and drove about 293,000 miles which is about 12 times around the earth. In addition, they had to use 76,000 tons of salt and the whole process cost them about $35 million dollars.

Now to figure out the best way to clear the snow, mathematicians have to apply the Chinese Postman problem in which the postman wants to deliver to every house, on every street, backtracking as little as possible.  It boils down to finding the best routes between intersections with an odd number of streets. This lead to a mathematician in the 1990's to design an algorithm focused on optimizing snowplowing. Since most cities use more than one snow plow and have to cover large areas, the best way is to break the down the network of streets into smaller units for the best results.

I hope you find these tidbits interesting.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If one and a half tablespoons of fresh coconut equals one tablespoon dried. What is the percentage decrease from fresh to dried?

Warm-up

 

If one cup of grated coconut weighs 3 ounces, how many cups of grated coconut are in one pound of grated coconut?

The Math Behind The Music That Makes You Feel Good.

It is well known that when you listen to certain types of music, your mood improves and you feel a lot better.  In fact, you end up feeling really good.  The type of music may differ a bit but it all fits the parameters of  a mathematical formula.

A Dutch neuroscientist did some serious research to find out how the brain translates music into emotion and he focused specifically on the music that makes us feel good. This gentleman figured out a mathematical equation to help him analyze the anatomy of the music that makes us feel warm and fuzzy.

He looked at a data base that contained 126 songs from the past 50 years  that made us feel really good.  He applied statistical analysis to identify the characteristics of the song responsible for making us feel good. This neuroscientist scoured scientific literature to see which keys and tempos were the most responsible for make us feel good.  Then he looked at the scores for the key and tempo before looking at the lyrics to see which ones had the identified characteristics. 

After identifying all of these things, he then applied a regression model to determine which songs could then be classified as feel good songs. At the end, he came up with the Feel Good Index (FGI) which contains all the identified lyrics that make us feel good, the tempo in beats per minute, and the key. Basically the higher the FGI a song has, the better it will make us feel. The ideal feel good song has happy lyrics with a tempo of 150 beats per minute and is in a major third musical key.  If a song has all of these characteristics, it will positively impact your mood, making you feel so much better.

The song that hit the top of the list is "Don't Stop Me Now" by Queen. This song meets all of the criteria needed to be classed as a proper feel good song. The thing is, even without lyrics, music can make us feel better, or impact us emotionally. It has been found that the beats in music without lyrics, triggers the motor areas of our brain, making us want to move around.  These fast tempo songs, feel more energetic, are more likely to make us get up and move or at least move parts of our bodies, and are linked to a more joyful state of being. 

It is not known by the brain associates major keys with positive emotion and minor keys with negative emotion. It is thought this association is a learned behavior although some people claim it is actually a biological reaction. There are other studies out there which link easy going music with the ability to sooth away road rage and with lowering your blood pressure. 

The one thing this neuroscientist did discover is that the mathematics works as long as someone does not have a personal association such as if "The Twist" was playing when their boyfriend broke up with them, they might not feel as good after listening to it but for most people it would cheer them up because it meets the criteria.  Let me know what you think, I'd love to hear.  Have a great day and a great weekend.

The Mathematics of Apples

There is nothing that I like more than having slices of a nice crisp apple at the end of any meal.  Apples have a very specific shape and if you cut them precisely the right way, you'll notice a star in the center. If you look at an apple, it is pretty much spherical except for the dimple at one end. 

Apples first developed in Central Asia thousands of years ago.  Over time, apples spread to Europe and the United States.  If an apple is grown from seed, it is often different from its parent so apples are mostly grown by grafting apple cultivators onto a rootstock so the trees grow faster and have the desired characteristics. 

Recently a paper was published on the shape of an apple.  Mathematicians and physicists used observation, experiments, theory, and calculations to figure out how apples grow and form. It started with a simple theory on how apples form and grow but it took off when they were able to connect observations of actual apples at various stages of their growth with experiments, theory, and calculations.

The first thing they did was to collect apples from quite early to ready to pick and map the measurements for each stage.  They focused on the growth of the dimple or cusp over time and how it related to the apple . Then they needed a theory to explain the growth so they focused on the singularity theory which is used to explain everything from black holes to the light patterns found at the bottom of the pool. Although the cusp of an apple has little in common with the light patterns at the bottom of the pool or a droplet coming off a column of water, it is the same shape as the others.  In addition, singularity theory is also responsible for explaining the slight deformation at the stalk end.

So once they had a theoretical framework, they began using numerical simulation also known as mathematical modeling to develop an understanding of the different growth rates of the fruit cortex and core cause the cusp to form. The cusp appears to develop due to the different rates of growth between the bulk of the apple and the place where the stem is. They used a gel to recreate a physical representation of the growth and by changing the growth and composition of the gel, they were able to mimic the development of the fruit.

This is just the first step in looking at a larger topic. They've explored a biological singularity but now they need to figure out how the molecular and cellular mechanisms work in regard to the formation of the cusp itself.  Eventually, they hope to develop a broader theory of biological shape.  I think this is absolutely fascinating.  Let me know what you think, I'd love to hear.  Have a great day.

Interpreting Graphs

 

Students often have difficulty with interpreting graphs, especially if they are asked to come up with a plausible story to explain the graph. Normally, we tell students to look at the title of the graph, look at how the axis are labeled, check for the units used.  Are they in feet, ounces, years? What scale are they in?  Are they in one tick is one foot, 10 miles, 100 pounds?  Finally, we ask students about the shape of the graph.

With this information, they are better able to interpret the graph but what if they are given a graph like the one to the left and asked to create a "story" about it.  This graph has none of the things we tell students to look for so they often flounder due to the missing information.  I've given students graphs with no information and asked them to create a story to go with it and they struggled so I had to break the process down into smaller parts.

I had to use questions like:

1.  Is the graph going up or down?  What does it mean if the graph is going up (increasing) or going down (decreasing). 

2.  What types of things either increase or decrease?  

3.  Why would it change from increasing to decreasing or vice versa?

4. What does it mean if you end up with a straight line parallel to the x-axis?

Did you know that when students are asked to create a story or narrative to go with the graph, it helps improve their understanding?  If you look back at the graph I posted at the top, there are so many possible stories.

1.  Each tick on the y-axis represents $10 while each tick on the x-axis represents a week.  So the first week, John deposits 8 dollars, the second week he puts in $4.00 so now he has a total of $12 but on the third week, he had to take out $3.00 to go to the movies but then he added $11.00 to make up for what he'd taken out........

2.  Different scenario, is that Jose is out driving, he is accelerating for the first two minutes, then he has to slow down between the second and third minutes when traffic cleared up and he was able to continue increasing his speed until he reached his cruising speed.

Lots of possibilities but if you aren't sure how to teach a section on interpreting graphs so students are able to practice, check out this site. This pdf is from open university in India has some very nice activities with the necessary resources to complete it. The first activity has students bringing in examples of graphs from newspapers, magazines, etc but if you are somewhere students do not have access to this type of media, you can search the internet and print some out. 

Divide the students into groups and have them look at all the graphs they came up with or you provided.  they want to look at the graphs and divide them into the easy group which are graphs that are easy to understand without much thought, or  the hard ones that you need to really examine to figure out.  

Students will take the hard ones and write down what it is about the graphs that make them difficult to understand.  Next they will look at the easy ones and write down what it is about these graphs that make them easy to understand.  The final step in this activity is to compare the two lists to see what is the same in each and what is different basically a compare and contrast.

The second activity has students working in groups of two or three people. Each group is given a bunch of cards that they match the story with the graph. They want to make sure the story they read matches the picture of the graph and they identify the characters or variables in the story.  The second part of this activity has them looking at the examples from activity one to identify the variables the graphs are telling the story about. The final step is to have the groups create their own cards with story and graphs that they give to another group to match up.

The third activity is again matching graphs with stories but they are based on distance and time.  After they match and decide if the graphs tell the story, they make their own sets of cards to tell more stories.  The fourth activity has students interpreting the data on a auto rickshaw race where the vehicles are going around a bend. The final activity has students being the writers of a movie who are creating an escape scene using a description and graph.

I like this because it has students matching scenarios with graphs and they are given the chance to create their own for others.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

Why would you need measuring implements showing different scales?

Warm-up

 Explain the difference in place value signified by the ths ending such as hundreds and hundredths or millions and millionths?

TeacherVision

 

I am always looking for sites which offer additional worksheets or lessons or resources I can use in my classroom when I want to spice things up a bit. Yes I said worksheets because they can form the basis of interactive activities designed to get students up and moving around.  In addition, it is nice to look for lesson plans that might provide new ways or ideas for teaching certain topics.

I found TeacherVision which has worksheets, lessons, and resources for teachers. The site allows you to do a search based on subject and grade.  For instance, if I look up math and 10th grade, I find I have access to access to 221 resources including activities, teaching resources, classroom tools, worksheets, games, graphic organizers, and lesson plans.

This is a paid site but they allow you to download three items free before they expect you to subscribe to their service at either a per month cost for one or two years for individuals or you school could sign up for all the teachers.  If you purchase the premium subscription, you can access everything and download what you want. 

The site does credit where the item came from.  For instance, I downloaded an activity on blood types which is more of a science worksheet but it does deal with real life applications of percentages so it can be used in math class. I like that it has students reading a chart on which blood type can accept blood from which group, and the percentage of the population who has that blood type.  After reading the chart, they are asked to interpret the data to answer questions. The second half has students asking people for their blood type and then using the information to create a bar graph. In addition, there are answers available so you can easily correct the work.  

It is easy to log in using your google or facebook account or you can set up an account from scratch but I don't like the way they count your free items.  If. you click on the item such as an activity, worksheets, lesson plans or graphic organizer to see what it entails, they count that  as one of your free ones and you don't have to actually download it.  I prefer to see what it looks like before I download it and would rather they count the free ones based on what is downloaded.  On the other hand, you can view teacher resources without having it count so you can read them to see what they are like.  The difference on  appears to be based on if it is something written out step by step versus a prepared worksheet, organizer, or activity that must be downloaded to be used.

This is not something I'd normally use since I cannot easily check out some of the material before downloading it because I want to make sure it does what I want it to do but at least you can get ideas for teaching topics in your math class. Check it to see how you feel about it.  Let me know what you think, I'd love to hear.  Have a great day.

Did She Help Crack The Enigma Code?

Although Alan Turing was known for creating the machine that broke the Enigma code, he was actually one of a few people who worked together and one of those people was a woman.  A woman who I would never have heard of except for the movie.  Her name was Joan Clarke and she was even engaged to Alan for a while during the war.

Joan Clarke attended Cambridge to study maths and even got a double first when she finished in 1939 but she was unable to receive the degree until 1948 because Cambridge didn't grant degrees to women in mathematics.

It is at this point that history differs from the film.  In the film, they show her answering a puzzle in the newspaper which leads to her being the only woman asked to crack a tougher puzzle in under 6 minutes which she does and she is hired to work on the project. Turing did do this but she was not involved.

In reality, she was approached in 1939 while at Cambridge to attend the Government Code and Cypher School before being assigned to Bletchley as a clerk earning only 2 pounds a week.  This was much less than men doing the same job.  Note that the Government Code and Cypher School was created to break the German Enigma code. While doing this job, her ability shone through and she was transferred to Hut 8 which is where Turning and the others were working on cracking the Enigma code. In order for her to be granted a raise to go with this promotion, she had to be classified as a linguist since she was the first senior cryptanalyst.

In 1940, Turing wrote down his version of the Enigma theory for all the incoming recruits for huts 6 and 8 but before it was given to them, he had Clarke read it to see if it was understandable.  Due to materials captured in February and May of 1941, Hut 8 was able to make advances because they now knew some of the keys used for the starting position of the Navel Enigma machines.  In August 1941, the cryptanalysts in Hut 8 were able to begin using a code breaking technique referred to as Banburismus. 

Banburismus was developed by Turing and used a Bayesian sequential procedure designed to produce and handle a probability network containing thousands of pieces of evidence.  Clarke was one of the first females to learn this technique and was extremely good at it. This technique helped break the Naval Enigma code and by 1943, the machine (the bombe) built by Turing was used to break codes. 

Most of her time was spent decrypting Naval messages and forwarding them to Naval command to be used. Her good work was recognized and in 1944, she was made the deputy head of Hut 8. Unfortunately, she was still paid less than a man doing the same job and she would not rise any higher due to being female. After the war, she continued working for the Government Communications Headquarters (GCH) until she met and married a coworker in 1952. Due to his health issues, she left but 10 years later she returned to the GCH and worked there until she retired in 1977.  Once retired, she helped others when writing about Bletchley Park and World War II codes.

Although she and Turing became engaged in 1941 and he told her he was homosexual, the engagement did not last long because he felt the marriage would not last.  In addition, she did help create certain techniques but she never received credit for them because she was a women. We do not know all of her contributions due to the secrecy associated with Bletchley Park and thus do not know if she actually participated in breaking the Enigma code as much as shown in the movie.  We do know, she made some extremely important contributions but didn't receive the credit she could.  She was also considered one of the  foremost female cryptanalysits there and her contributions saved lives and helped Britain win the war.  

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

Alan Turing

On the trip home, I ended up watching a movie based on the life of Alan Turing.  I'd heard the name but didn't really know much about him but by the end of the movie I knew quite a bit.   The movie was based on a book about his life and it intertwined three time lines of his life.  It alternated between his early life at school when his best and only friend introduced him to codes, his years during the war, and his life at the point when he was accused of the crime of being a homosexual. 

The most interesting part of the movie and the part which took up most of the film was that of his involvement at Bletchley Park in trying to break the code associated with the Enigma machine.  The code for the Enigma machine was changed every 24 hours at midnight and the code breakers had only one day to figure out the code but Alan decided he could build a machine that could decode the messages each day beginning with the 6 A.M. report. He worked with one female and several males over the course of the war but about two years into the war, they managed to "program" this early computer to break the code.  They knew the first report out every morning was the weather report and this formed the basis of the computers ability to break the daily code.  

Although others get the credit for creating the first computer in 1946, Alan Turing's work showed that he believed one machine could be made to do any well defined task as long as it was appropriately programmed. It was called the "Universal Turing machine" and relied on the stored program which is what most modern devices operate with now a days. Even the technology that came from creating the atom bomb in the Manhattan project was based on that same concept. 

After the war, he continued extending his ideas. Turing was quite fascinated with the interplay between the human thought processes and the development of computerized though and artificial intelligence.  Much of his work in the late 1940's focused on the emerging power of computers during the Cold War and the fight for nuclear supremacy.  In 1950, he published a paper explaining the "Turing test" which was designed to see if the computer could pass as human. In this test, a human would ask questions and based on the answers would determine if the unseen entity was human or a computer. 

It is also interesting that in school, he was more interested in Maths and Science than studying the classics such as Latin and Greek.  He studied mathematics at the University of Cambridge between 1931 and 1934 when he received his degree. Alan obtained a fellowship at Kings College based on his research in probability after he graduated. Soon afterwards, he headed to the United States to work on his PhD at Princeton which he received in 1938.

He came to the attention of others when he published a paper on the Entscheidungsproblem or decision problem which meant that there had to be a way to determine if a certain problem was solvable based on a given mathematical system.  Both he and Church (an American) came to the same conclusion that they are not solvable and Alan's paper lead to his being given the opportunity to work on his Phd under Church. It was his work on this problem that lead to his creation of the Turing machine and the basis of breaking the Enigma code.

Unfortunately, in the mid 1950's an investigation into his house being broken into lead to his being ousted as homosexual and he was convicted of the crime.  He ended up taking the chemical treatment to keep his choice under control and in 1954, he died.  The coroner determined it was suicide due to the cyanide found in his system.  He was only 41 years old and that is so sad because who knows what else he might have contributed if he'd been able to continue working.  Let me know what you think, I'd love to hear.  Have a great day.