# Should I include games in my math class?

My initial thoughts when I have heard about mathematics teachers implementing games in their classrooms was “Great, I would definitely include games in my regular task selection, but I do not see how they can be operated effectively to encourage student thinking in a mathematical mindset.” Time and time again I have played games without giving much thought to the mathematics behind them.  My thought as a teacher was that games are fun and exciting, but rarely can be used to effectively promote student learning in mathematics class.  That all shifted on Tuesday April 19th during a teacher named Mrs. Soto’s senior statistics course.  As I observed her teaching I began to see the tremendous potential for mathematical growth locked within games rather than focusing solely the game itself.

Horse racing was the game of the day!

My long time beliefs on the nature of games in a mathematics classroom were influenced by a lack of experience considering the mathematics behind the games I played as a child.  Also my negative beliefs stemmed from a misunderstanding of the love students have for games.  I saw games as pointless at times, and also a source of competition which is often based on luck.  Before observing Mrs. Soto’s class I likely would not have used many games in my first year of my teaching because I perceived games as too relaxed for effective mathematical learning to occur.  In my experience, I have rarely made sense of the problems in games, critiqued the reasoning behind their various methods or persevered in solving problems that occur within games.  Now I realize that mathematics by its very nature is meant to be a subject of discovery and exploration.  Games are an awesome opportunity for students to expand their critical thinking and to gain a deeper understanding of seemingly simple processes!  I often forget that mathematics provides the “why” for much of the phenomena that occurs in games and in gambling as well!

### So how did all this change in my thinking come about?

Mrs. Soto began her class with a game where students placed bets on horses based on their projected odds of winning the race.  Then, she ran horse race simulations from a site to see which bets won.  Each student received a payout when their horse won a race.  There were only 5 races, so students needed to pick the horses they thought were going to win carefully.  Students picked based on the odds for each horse winning in a single race which were set at 7:1, 5:1, 7:2, 13:5, and 18:1.

Some horses have a much higher likelihood of winning than others according to the experts.

Students played the game and did not win much because their understanding of the odds of winning was skewed because they had not worked with this kind of probability much.  Eventually, students were able to ask questions and discover while playing the game that the horse that had the best odds of winning was the horse with 13:5 odds.  Students discovered after a while that picking the horse with the best odds each time paid the biggest dividends.  I asked Mrs. Soto after class about how she goes about running games in the classroom and if she runs them regularly.

Mrs. Soto explained that the key to teaching with games is to demonstrate the game as a group first with a few students.  Then answer any big questions about the game before starting.  Mrs. Soto said that games are now a part of her routine in her statistics classes because she found that students thought they were the best part about her class from year to year.  She regularly has about 2 games per week.  She said that she used to avoid games because she did not think that they were very good for math teaching, but now she has found that students love them and learn a lot from them, so she tries to use them regularly.

After my observation with Mrs. Soto I would say that I have a much greater understanding of just how engaging mathematical games can be, and how students can easily be encouraged to actively think about what is happening mathematically within a game.  Student thinking can be probed as a teacher can see how the student is developing while playing the game.  Mathematical games seem to me like a tremendous opportunity for students to think critically, make sense of problems, and to learn how to reason abstractly.  There is so much more to games than meets the eye.  I hope to reveal that to students in my future classroom by playing content related games from the real world on a regular basis. Also I can relate them to what students are currently learning while encouraging students to be themselves and think critically about the outcomes in games.

I am very thankful to Mrs. Soto for the opportunity to observe her class.  One observation between teachers can change your thinking, and I am so excited about what I have learned from Mrs. Soto’s classroom.  I look forward to implementing my observations in the future!

# Teaching Tornado Year in Review

irThe whirlwind that was my first two semesters of teaching daily in a classroom is almost over.  Just one week remains in what has been a dynamic, bouncy, and transformative journey!  I am looking forward to a calm summer after what has felt like a tornado for me personally.  With all that said, I truly believe that I have learned a lot about the nature of mathematics teaching, and about how much I still have to learn.

Teacher Assisting and Student Teaching came and went so fast.

From my time at a small charter school at the middle school level on the north side of Holland, to today finishing up at a large high school on the Northwest side of Holland, I have seen a great shift in my development as an educator.  I began the final semester of my senior year thinking that I could model a mathematical process and students would just understand.  Like osmosis or diffusion the high concentration of mathematical thinking in my mind would diffuse into the lower concentration of their minds.  I quickly learned that whether at the middle school level or the high school level students develop most mathematically by working on problems and developing their thinking.  Don’t get me wrong,  some students will pick up perfectly well when I show a problem and present a procedure, but a majority need something more.  My thinking is not good enough for most students because mathematics has its own language.  Often times I fail to translate math well enough, and a great number of students are not translating the language of math on their own.  Over hours and hours of examples, and seeing students watch the method, and attempt to repeat the method taught. I have come to the realization that this is often times not the best way to go.  I am excited to share my reflections from my year teacher assisting at a middle school and student teaching at the high school level.

Teaching about similar polygons with students watching my method and repeating.

A few key things I learned I believe will begin to combat the issues I faced in my teacher assisting and student teaching semester.  Number one…

## Engaging Student’s Minds

I must activate and encourage student thinking about mathematics because students are not made like robots to have the information that I know directly passed on to them.

Psychologically students learn best when they are actively associating, making connections to previous things they have learned and have experience with.  The process of making memories begins with encoding where students construct memories from their perceptions of events.  A mathematics teacher lecturing at a board may seem fascinating to some students and thus it will be perceived as interesting and encoding will occur on an acoustic and visual level, but many students crave something more and their attention is just not gained through direct instruction.  Better than visual or acoustic encoding, semantic encoding is said to be most effective for long term storage most of the time.  So what is semantic encoding and why should a math teacher care about it?

Semantic encoding is the process of converting to a construct something that has a particular meaning or can be applied to a specific context.  I care about semantic encoding because in today’s mathematical landscape student’s understanding of mathematical concepts depends on applying meaning to mathematical symbols and numbers.

Therefore, I want to focus on giving students rich open ended tasks which as much as possible have a real context.  These tasks will engage students and allow them to make their thinking visible to their classmates through small group collaboration and sharing out to the class.  One activity where I successfully engaged student’s minds was in a Clinometer Activity where they measured the height of very tall objects using trigonometry.  To do this I need to create a classroom culture where students feel safe to share and are willing to make mistakes.  This is not an easy task, but through persistence and clear classroom norms I believe that this is possible.

Giving students opportunities to apply what they are learning is key! Pictured: Clinometer Activity for utilizing trigonometric functions.

## Make Sense of Problems & Possible Solutions

Second, I learned that in math class it is very important to emphasize making sense of a problem before diving into solving it.  If students have no clue what they are doing they give up or come out with inaccurate answers.  Making sense of a problem eliminates answers that are way off in many problems.  Instilling student confidence in making sense of problems expands their critical thinking skills and prepares them for the real world.

## Valuing Student Thinking

Lastly, I have learned that even struggling students appreciate it when teachers really are interested in what they are thinking and respond with thankfulness.  I can value student thinking by creating a classroom culture which sees mistakes as an opportunity to grow and participation as a necessity to personal learning.  Also, I can provide a variety of means for students to share their thinking through questioning, writing to explain their reasoning, having respectful arguments, and encouraging students to share their thoughts with their classmates on a consistent basis!  Talking out your thoughts cannot be merely something that is asked, but something that is expected from every student from early on in the year.

In the end, I had an outstanding year of learning as a mathematics teacher that will stick with me and serve me well for the rest of my teaching career.  I have been able to build a foundation for future success.  Day by day through reflection and gradual transformation I have the opportunity to become an excelling teacher!

4. ##### Also, I have seen the value of developing relationships beyond mathematics curriculum by talking to students about their lives on a regular basis.  I try to keep this focus in order to encourage students to grow in character and integrity before growing more as mathematicians.

What a year a tornado indeed, but I have learned so much!

# Mathematics, so much more than numbers and simple figures!

When I began the semester in my History of Mathematics course I thought math meant, “The study of patterns in the world and in our minds and how they connect to each other”.  I even created a blog post called “So when I say Math I mean…”  sharing my thinking and my scarce, but meaningful prior knowledge about the most important discoveries in mathematics.  My definition of mathematics and my understanding of where the field is at today has expanded by at least five times the amount of ideas that I thought mathematics contained before the semester.

I would say that the way I define mathematics has been refined by the course as the discovery of patterns in the world, in the minds of men and women, and how these patterns connect to each other and extend beyond the point of our human understanding.  One update to my definition lies in the point that I now believe that mathematics is discovered while notation is invented to explain our discoveries in mathematics.

Additionally, I would say that mathematics extends beyond our understanding because as of now I think forever there will be quantities, shapes, and ideas that connect the way the world works that we cannot comprehend as finite beings.  Through our discussions in the math capstone course, I have come to have a view that mathematics is so much bigger than I first thought because mathematics can explain social, physical, natural, and celestial phenomena to me it can be viewed as a seperate, but connected to science.  For a more in depth discussion on how many sciences there are I read a neat post on the number of sciences.  I think for me personally throughout my time in the history of mathematics, I have come to realize that I do not really care whether mathematics is a science or not? What is more important is that I understand that the field of mathematics when explained well and utilized to its fullest potential, makes science easier to understand.  Mathematics allows for amazing calculations, thoughts, and ideas that inform the concepts of science, future scientific discoveries and research.

Throughout this semester I have had a number of my preconceptions about matheamtics challenged, and have been reminded of the beauty of mathematics.  Of my misconceptions the one that has been most challenged was that mathematical figures rarely look really exciting and interesting.  Also, I did not realize that mathematical objects could be so mobile.  Tesselations in particular stood out to me as I remember briefly learning about them at some point in high school.  I came to see tesselations value in helping students to recognize the awesomeness of mathematics. Students can see how we can discover tesselations that no one else has ever created, but were possible all along by using some deep thought and drawing a repeating pattern that students must make reasoning choices and possibly calculations to ensure that the shapes or figures they use connect perfectly.  Students may blindly fit patterns together like putting together legos, and it is the role of a teacher to help students to make their thinking visible in regard to shape selection for their tesselations.  Students in everyday life make choices which require mathematical reasoning without even noticing that they are mathematical.  Teachers like myself, need to help students realize that doing mathematics is normal, and that they already use mathematical reasoning in their everyday lives anyways, so doing mathematics at school will serve to improve their reasoning and ability to make use of structures to solve everyday problems.

Neat basic tesselations

3D Tesselation

A tesselation I created

Another perception of mine that was further challenged was the idea that mathematicians sit at a desk and work at problems for hours on end, rarely getting up to do anything in the real world.  Archimedes was a prime example of the falsehood behind my thought.  Archimedes built a claw as seen below which picked ships up out of the water, and destroyed them by forcing them to capsize.  This invention was considered one of the greatest defensive war mechanisms of his time between 200 and 300 B.C.  He also created a death ray from mirrors that was said to set fire to ships.  Archimedes made many other vital contributions to mathematics like the calculation of pie to many more digits than it had previously been discovered to, and the creation of His wheel.  Some other amazing mathematician works that impacted their day greatly were Newton’s discovery of the laws of gravity, Gauss’s invention of the electromagnetic telegraph a huge step in long distance communication, and Mandelbrott’s work with Fractals which contributed to the concept of space filling curves now being used when considering traffic flow in cities.

The Claw of Archimedes

Benoit Meandelbrot’s Fractal

The aspect of mathematics that most astounded me throughout the course and had a big impact on my view of the purpose of mathematics was the concept of topology.  Topology is the the study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures(google).  Topology allows mathematics to extend far beyond the obvious visible world, to allow people to find new solutions and study the world in unique ways because topology shows us that we can change shapes and figures to form brand new objects that do not seem possible from an initial view.  A great example of this is in Topology playdough  what appears to be playdough with two holes and one hole locked around a pole can be molded so that two holes can be around the pole.  Another example of the application of topology can be in untying knots as topology shows us that there are many ways to untie a knot beyond just reversing the steps by which the knot was tied.  Topology may lead to many of the next great discoveries in mathematics.

To wrap up, Mathematics is so much more than countable patterns, and stagnant shapes, but a language and framework by which we can better understand the world.  Mathematics is exciting, meaningful, and necessary for unlocking the sciences.  Mathematics is beautiful, and I did a lot of mathematics throughout the Math Capstone course, and came to the realization that mathematics is not always obvious to us when we are first forming patterns, or manipulating numbers, but that is why mathematics is discovered. Following the discovery of mathematics people invent notation to describe their discoveries.  The end quote says it all,I have thoroughly enjoyed the history of mathematics and I hope that you can enjoy it too!  Some great places to begin your search are…

The story of mathematics

Mactutor Math History

The Mathematical Association of America

Weebly Mathematicians of History

# Aristotle’s Wheel, An Investigation!

This past week I was shown something that peaked my interest! That something was Aristotle’s wheel.  To begin I will show you what Aristotle’s wheel looks like!

Can you notice the issue in this picture?  Think Mathematically…

Maybe a video will help? Aristotle’s wheel

Or if you prefer a Geogebra animation

I will give you some time to let that sit in your head, maybe you have already guessed what seems to violate what you know about circles, but maybe you have not.  I often times enjoy checking into history, so perhaps that will lead us to the issue, but also the solution!

To begin there is much argument over who thought of this problem first, but we are more concerned with the mathematics behind it rather than who found the problem.  The story must begin with Aristotle( unless you have found better historical journals about this topic, please comment and post the link to the comment).  The problem first appeared in Aristotle’s the Mechanica.  Aristotle’s wheel stumped him, and Galileo along with many mathematicians throughout history as well.  James Cardan was one of the greats to take a shot at Aristotle’s wheel, but all he did was identify a point to point correspondence, but did little to develop that idea (Drabkin, 1950).  It took over one thousand years from the time the problem was conceived to the time that people could explain the paradox clearly.

The Issue

Aristotle’s wheel is a paradox because there are two wheels one within another that are spinning.  The issue comes in that the smaller wheel covers the same distance as the larger wheel when they are both rotated together by one full revolution, but the smaller wheel has a different circumference than the larger wheel right? So how can this be?  Here’s the wheel again.

Aristotle’s wheel

Check the wheel again and think about cars, friction, and any other concepts of rotation that you know of.

The Solution or what I understand of it!

The solution has come about in two major approaches.  The physics approach contends that the inner wheel has the same line length because there is much greater friction on a sliding object than a rolling circle.  Thus, the outer wheel appears to have the same length as the inner wheel because the inner rim has more friction acting upon it, so the inner wheel turns in smaller turns than the outer wheel, but overall turns the same length.

The point to point correspondence approach contends that regardless of the rotation rate, the inner part of the wheel’s points on the line can connect to the outer line with a one to one correspondence.  Thus, I believe an appropriate connection would be to say that the points on the inner wheel are an injection with the points on the outer wheel of Aristotle’s wheel as they rotate forward.  So though, the outer wheel appears to have a much longer circumference the inner wheel can still form a one to one correspondence with the points of the outer wheel.  Thus, the outer wheel appears to match the length of the circumference of the inner wheel when they are rolling from one space to another.

So either by a point to point correspondence and an understanding of the slipping of the inner wheel we see that Aristotle’s wheel is a paradox which is true and does not violate mathematical laws though it appears to do so.  Aristotle’s wheel has vexed mathematicians through the ages, but it has great implications for students and scholars today.  The problem shows us that what we see is not always what is.  By unraveling the deeper context and possible ways to view problems, students and scholars alike can solve problems that seem to contradict the laws of mathematics or the elements set in place in other subjects.  Aristotle’s wheel is a wonderful problem to help students think abstractly and to help them persevere in problem solving via working at a problem that initially makes no sense.  I gave this problem to eight grade students this week and with support the students were able to grasp the paradox, and begin working toward a solution.   I am looking forward to making a lesson plan on the paradox in the future because this problem challenges students to develop flexibility and apply many mathematical practices!

“The curious paradox is that when I accept myself just as I am, then I can change.”
Carl Rogers

References

Drabkin, I. E.. (1950). Aristotle’s Wheel: Notes on the History of a Paradox. Osiris, 9, 162–198. Retrieved from http://www.jstor.org/stable/301848

# Gauss… Establishing a solution method the Chinese had created over 1800 years prior…

Systems of Linear equations have always intrigued me! I am not certain of exactly why, but I found there relative simplicity in high school yet complex problem solving processes fascinating!

Graphing to solve systems of linear equations

## Beginnings of Solving Systems of Linear Equations

The Babylonians around the 19th century BC were said to be the first to work at solving systems of linear equations.  The Babylonians looked at only 2 variable equations though and they became proficient at solving these 2×2 systems.  They called the two dimensions or unknowns of these systems length and breadth (Texas A&M mathematics department).  Thus, Babylonian mathematics is largely considered the beginning of looking at systems of linear equations in mathematics.

Despite the fact that the period of time the Babylonians were in predated the library of Alexandria these discoveries were made and utilized throughout Babylon.  Though the Babylonians started this field, there were no further discoveries for over 3000 years…  but not so fast, a dim view of history could see the mathematical history this way. In reality, long before Carl Friedrich Gauss came up with the modern matrix method of solving systems of linear equations, the Chinese had utilized matrices to solve 3×3 and higher systems of linear equations (Smoller, 2001).  The solution examples using matrices can be seen in the Nine Chapters of the Mathematical Art by Chiu Chang Suan Shu.  During the Han dynasty (206 BCE-220 CE ), the Chinese came up with a solution method for systems of linear equations with an unlimited number of unknowns or variables.  Later there discoveries became what we now know as Gaussian elimination.  When we teach elimination and solving systems of equations by matrices as completely separate in math class we are doing students a disservice by not connecting their learning.  Gaussian elimination is connected to solving a system of equations through the use of matrices.  I believe the reason teachers skip over this connection is because they do not believe they will have time to connect these ideas in clear detail because the connection relies on an understanding of linear algebra.

Do you have any other ideas for reasons why teachers rarely connect the elimination method and matrices method for solving systems of linear equations?  As a student in high school I remember never really understanding what the rows in a matrix stood for, so I wish my understanding of matrices could have been made more explicit somehow.

In relation to my previous post on Mathematics as a science (Gauss’s vote is that math is a science)!

## Modern Day Matrices

James Joseph Sylvester is said to be the first person to name the arrangements of coefficients and constants involved in a system of linear equations as a matrix.  Sylvester has a unique story beyond that he was the chair of mathematics at Virginia University and arguably the most esteemed mathematician of his time in the United States from (1820-1840).  Prior to his naming of the matrix and finding of the discriminant of a cubic equation in London, he fled the United States because he believed that he killed a student with his sword stick when he whacked a disrespectful student in the head(more full story).  Regardless of his blunder, Sylvester went on to make some large discoveries with respect to matrices and their name came from him.

In the twentieth century, the more recent developments of matrices from a tool to a part of mathematical theory were driven by female mathematician Olga Tausky Todd(1906-1995).  The advances she made contributed to matrix theory’s creation and the measurement of vibrations on airplanes during World War II.  She was a brilliant mathematician and explorer of mathematics.

“I did not look for matrix theory. It somehow looked for me.” –Olga Taussky Todd in American Mathematical Monthly

## Bringing it all together

I am really excited to incorporate all of this history into my teaching on systems of linear equations in the future to help my students to understand the value of solving systems of linear equations and its place in history!  I think I have developed a strong base to build a lesson from by exploring the interesting history behind systems of linear equations.  History matters in the learning of mathematics because it gives students a context to understand the importance of mathematical discovery!  Students often struggle to find reason to see the value of mathematics because it is taught as though it is basic and dull, but many mathematical discoveries took many years to be established and spread around the world!  As an educator I have the responsibility to build up the problem solving and history behind the mathematical operations and theory my students are learning!  As a result of this endeavor I truly believe my students can find mathematics more valuable and more beneficial when they know how something came about, and why it could be important to them today!

Olga Taussky Todd

# Reflection on my Professional Practice Teaching on Expressions and Equations

This past Wednesday, I taught a lesson on equations and solving equations using addition and subtraction.  My professor Jon Hasenbank from the math education team was on hand, and I greatly benefited from the experience as I have every time that I have been able to teach and effectively reflect on my teaching!  Here are some of my thoughts after the experience, and how I believe I have developed and grown as a teacher!

The stages of development! We grow in almost everything we do consistently, and I am growing as a teacher like those in the picture. I would say I am a sitting teacher that wants to play at this point. Sometimes I walk, but I fall a lot!

## How did it go?

I thought the lesson I taught went fairly well overall.  There certainly could have been more student discussion and participation, but I think the lesson went well because students began to understand the main points in analyzing an equation.  Additionally, I was able to show my students the relevance of equations in the real world as a problem solving device!  Students really seemed to appreciate that equations are involved in the making of roller coasters!  My goal was to have a good attitude and be enthusiastic while teaching and I believe that I succeeded in that goal!  A second goal I had was to engage students in conceptual mathematical thinking and to clearly state the mathematical concepts in language my students could understand and relate to their lives.  Overall, I was pleased with my progress in my teaching though I could have used more preparatory time for the lesson!

## How do you know?

I know that I succeeded in having an improved attitude and more confidence because my professor said that I seemed genuine and excited about helping the students learn!  Beyond that I was able to provide students an improved amount of time to discuss their ideas about equations.  Also, my CT told me that my pacing was improving and that helped my confidence seem to be higher.  I tend to have a lot of trouble with keeping my pacing high enough to finish a lesson in a helpful amount of time for students to process what they have learned.  I was by no means perfect, but I think I took some great strides to break down concepts for students.

## Why is that so?

I think that is so because I got the students interested and more engaged in the lesson from the beginning than usual.  Students were interested in developing problem solving skills in mathematics because they could see the practical value it had for their lives!  I think that I effectively built student interest and promoted conceptual student understanding utilizing the ability to relate problems to students!  The more I was able to connect in the information to their lives the more interested they generally become!

## How did you grow?

I grew tenfold in my realization of the vitality of rich mathematical tasks that peak student interest through that class period!  The more students are engaged in education typically the more they will focus and learn, so I want to keep learning about more ways to motivate students!  I definitely grew by realizing that I need to focus on making student thinking more visible overall!  I grew by learning more areas that I need to grow in, but I found that the more I prepare the better I do as a teacher.  Another area I grew a lot in was my ability to make sense of mathematics problems for students.  I did this by relating the problems to students’ lives.

The cognitive coaching time and my reflection helped me to know that I was growing because they allowed me time to process my thoughts!  Prof. Hasenbank asked me about the steps I thought I could take to improve my ability to make real world connections explicit for students!  He suggested some great resources for rich mathematical activities.  Prof. Hasenbank suggested that I work on presenting as the expert because a lot of times I showed growth in expressing mathematical concepts, but could not mathematize challenging problems efficiently on the fly!   I know how I am growing the more I reflect on where I was when I started to become a better teacher to now where I feel I could teach alright!  I think that the more I reflect the more I consolidate and maintain my learning effectively!  That way I can continue to build on the last brick the next time I teach rather than starting from scratch over and over again!

## What’s Next for me?

Next I will work on finding more rich activities for utilizing in class. I will consistently spend time thinking about how I can relate stories to students to keep them more engaged!  Also, I would love to keep up my great attitude and honest enthusiasm I had at the end of the week!  I hope to continue growing in my practice of making mathematical thinking visible for both me and my students!  Education is an open door! I am loving the ride and excited to keep developing my student friendly language!

students need to talk in class to develop a great classroom culture! Lecture just is not good enough!

# Journey Through Genius=Awesome book for more than just Math Majors!

Throughout my reading and exploration of Journey Through Genius by William Dunham I was blown away by the interesting stories behind every mathematicians’ contributions to mathematics.  I really enjoyed how he set up each Genius with plenty of background and explanation of where they came from to the point of their written theorem and proof.  Dunham fused mathematical explanation with outstanding history to form what may be the most interesting math book I have ever read.  Though I am not really one to talk because I have not read a whole lot of mathematics books, I found the book enjoyable, and think it could be enjoyed by high school and even middle school students as well.  The theorems are very dense at times, and require much thought to follow, but Dunham makes each and every Genius’s story fascinating through his story telling and brilliant selection of detail.  Additionally Dunham did a wonderful job explaining the mathematical problems that have yet to be solved, and sparking the intrigue of the reader by laying them out clearly and showing that some problems carry on for hundreds to thousands of years.  These thoughts made me wonder if I could solve one of the great problems of history if I sat in front of it long enough!  I could go on and on, but I will highlight a few questions that I wondered about as I was reading, and my favorite math genius’s story and work!

First to the questions,

I wondered what the greatest known prime number currently is?  I know that computers now are programmed to find greater prime numbers, but how many digits is the current greatest prime? Over 17 million according to the science daily!  Are there infinite primes?  Yes Euclid answered that question within the book with his proof of the infinitude of primes.

Will anyone ever write a better or more influential set of math books than Euclid’s Elements? Dunham seems to suggest that will never happen, what do you think?

Is playing with numbers and proofs the same as experimenting? I think that observations clearly occur, but I am not sure I am comfortable calling working with numbers in proofs experimenting.  One explanation I favor says that logical deduction is the method mathematicians use to create theorems and prove their results.  Recently, some mathematicians have launched simulations which seem to have control and are more experiment like.  Does this make math experimentation a reality?  I thought experimentation needed control?

I wonder if working mathematics in abstract means will continue to bring us back to concrete real world discoveries?

George Cantor

I really enjoyed learning about Cantor, and found his work to be perhaps the most interesting.  Something about infinity just makes me get excited and want to discover more about mathematics! Despite my intrigue in Cantor, I found Isaac Newton to be the Genius I most enjoyed learning about because of his interest in persevering in problem solving.  Not only did Newton connect science and mathematics through physics and his work creating and working with calculus, but Newton never backed down from a challenge.  He declared that he solved challenging problems by thinking on them continuously.  John Maynard Keynes an economist at Cambridge in the twentieth century said, ” I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret.” (Dunham,p.164, 1991)  One of my main goals for my students is that they learn how to persevere in problem solving like that.  I want them to not give up even when problems are very difficult.  Newton exemplified what a great problem solver and hard worker is.  He certainly had a growth mindset and was always up for discovering new ideas through his creativity and flexibility in his problem solving approach!

Sir Isaac Newton

Anyone can learn and benefit a lot from the discoveries and stories outlined in a Journey Through Genius!  I really enjoyed this book and found it to have many great insights that scope well beyond mathematics class.  Never Give up! Think in different ways! Never be afraid to question the establishment!  These practices lead to growth and amazing discoveries like those outlined throughout Dunham’s masterpiece!

Final thought, “Genius is patience.”

Isaac Newton