Weekly 7: My Vision for the Future/Second Book

Before my career begins as a teacher, I am filled with a mountain of dos and don'ts.  We study all the time of what a good teacher looks like, what type of qualities we should have, and how we should treat all of our students.  It is easy for there to be an overwhelming feeling of pressure for pre-service teachers.  What if I am not all those things?  Can I really be "perfect"?  The answer to these questions comes with prioritizing.  You can successfully choose different topics to focus on and slowly improve your teaching style.  Your career as a teacher should be dynamic and always improving.  We must learn to adapt our personal style to the group of students we have been given: hour by hour, grade by grade and year by year.

Throughout this course, I have learned how to appropriately educate all students.  Our semester has been filled with different levels of education majors and just strictly math majors.  There is a way to satisfy all of our thirsts for knowledge.  As an educator, it is important to build off of student interest.  When students are presented with choice and topics that spark their interest, then engagement will come automatically.  This aspect of learning needs to be acknowledged and emphasized.  When students care about learning then they will assume responsibility and be in charge of their own learning experience. When we have students driving their own learning, we have found success.


Book #1
I have read two books that enhance my knowledge about the mathematical journey for all students.  There are messages for teachers of all levels to follow.  Regardless of what grade you teach, the fundamentals stay the same, in my opinion.  The lessons that I learned from this book apply to all areas and levels of mathematics.  It is important to understand how to approach the subject to spark student interest and maintain engagement.

Here is my reflection and interpretation of the ideas presented in this piece.









Book #2

This book emphasized that the world is constantly changing.  Everything about it experiences some type of alteration with time.  Education is not different.  When it comes to math education, teachers must stay one step ahead.  We need to freshen up on our approaches, techniques, and lessons.  The one key idea that needs to remain the same is that we need to teach with understanding.  Some of the key ideas are:

• Different skills will be needed to support all types of students.  
• For any topic, the teaching demands flexible approaches for defining and solving problems.  
• Learning things with understanding can be used flexibly, adapted to new situations, and used to learn new things.

The learning experience needs to be fostered from many different angles.  This means that from a teaching standpoint, we have a lot of responsibility.  We have to understand the subject area as well as the troubles students might encounter and how to rebuild understanding.  I picture understanding the material is like climbing steps.  You have to climb the first step before the second and the second before the third.  Understanding is step one to the learning process and it is arguably the most important.

Another important topic that is presented in this piece are the five dimensions that shape learning experiences:

1. Nature of learning tasks.
2. The role of the teacher.
3. The social culture of the classroom
4. The kind of mathematical tools that are available.
5. The accessibility of mathematics for every student. 

Each of these aspects needs to be considered and be approached very delicately.  These are the five most important dimensions that basically define student learning.  The book goes into more detail about each, but I think that the interpretation goes beyond the textbook definition.  Each of these pieces have room for your own personal twist.  There is a way to personalize each one so that it works for your own teaching style.  For example, the social nature of the classroom will be varying from year to year and only you will know the dynamics of your classroom.  You are the only one with the knowledge of your students, just you. So, this list is universal, but changing from teacher to teacher.  

I would love to read the rest of this book in detail.  It seemed to really dive into a new understanding of how we must approach mathematics with students.  As intimidating as it may seem, teaching mathematics the right way can provide a beautiful experience for all.  It may take a "special" person to take on this task, but the world is full of special people, destined to do great things.  

Weekly 6: Gender Stereotypes in Education

Boys play with trucks and girls play with dolls.  Boys are tough and girls are emotional.  Boys grow up to be men that provide for their families.  This is the way it works.  No deviation from this pattern will be acceptable.  The amount of pressure that is produced from these gender stereotypes is revolting.  As a future educator, this topic is sickening.  You want your students to reach their highest potential, wherever that may be.  Regardless of gender, all students are given special gifts and talents.  It is up to us to find them   and encourage our students to pursue them.

The first initial questions that I have is:

• How soon are we exposed to gender stereotypes?
• Is the affect preventable?
• Does it affect all children? To what degree?
• What type of role does a teacher have in regard to gender stereotypes?
• How much does the classroom impact students' view of themselves?
• How did gender stereotypes affect me?

I believe that educators have the power to provide molds for their students.  The type of environment we provide can strongly influence ideas and behaviors.  This is why teachers need to be tuned in to what affect they have on their students.  It has been said that 

"Sitting in the same classroom, reading the same textbook, listening to the same teacher, boys and girls receive very different educations" ("Gender Bias," 2013).

Is the fate of our classroom predetermined?  I do not think so.  Teachers can be conscious of their behavior toward both genders.  So many times teachers are a product of their own type of conditioning as well.  Teachers already assume that girls are neat and organized, while boys might require more reprimanding.  Why is this?  This is due to the overwhelming influence of socialization.  Paul C. Gorski states that "socialization of gender within our schools assure that girls are made aware that they are unequal to boys" ("Gender Bias," 2013).  We assume that gender is the determining factor for our most dominate traits.  This idea might introduce the topic of nature vs. nurture, but that is for another time.  So many different factors affect the children that are put through the education system.  The students today are different than they were forty years ago.  There is no reason why the treatment of them should stay the same.  Our society is on its way to revolutionizing the role of genders.  Women are beginning to dominate in the work force and going beyond the normal domestic roles.  In order to foster this development, I want to know what I can do as an educator to instill the empowerment in both genders. My role as an educator goes beyond what I teach in my classroom.  It dips into the type of ideas I plant in my students' head.  What messages do I want to grow?  

This video explains the important effect of stereotypes in the classroom.  Throughout history, we have dealt with race and gender in education.  What happens when students and teachers see these stereotypes in a different light... 

"When you are not aware, you will let it happen."

Holding students to the same standards is the foundational layer to success for every person in your classroom.  Students who believe they can do well will achieve close to that expectation.  It is as simple as changing their mindset when it comes to understanding and performance.  This is the type of power educators have is underestimated.  In my opinion, teachers need to reassess the type of messages they are sending to their students.  Maybe the focus for improving education would shift from the need for more governmental funding and reducing budget cuts to providing the best role models in the classroom.  If student performance is directly correlated to the types of messages they receive, then it is not about money at all.  Women need to believe that they are equally capable to pursue the same careers as men and races need to believe they have the right to feel successful.  

Be a part of the solution and not the problem. 

Gender Bias in the Classroom - Boundless Open Textbook. (n.d.).Boundless. Retrieved June 17, 2014, from https://www.boundless.com/sociology/understanding-education/issues-in-the-u-s-education-system/gender-bias-in-the-classroom/

Gorski, P. C. (n.d.). Gender Bias in Education. Gender Bias in Education. Retrieved June 15, 2014, from http://www.edchange.org/multicultural/pap

Weekly 5: Accessible Mathematics

Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement

This book by Steven Leinward takes an inside look at what education is missing and what teachers can do to provide a top-notch education for students.  Now, there is a balance.  Teachers can provide the best education for students while keeping their sanity. For teachers who go about this in the wrong way, we see exhaustion and burn out.  He says that teachers and students have the ability to perform at their highest potential if the focus is on the correct areas.  The ten instructional shifts that are addressed in this book are (with explanation):

• Incorporate ongoing cumulative review into every day's lesson. 
• When students are given the opportunity to refresh ideas about what they have learned, the retention rate is high.  A quick review quickly pulls students into the right mindset and they get ready to learn.  This tool can also be used as a quick assessment tool.  
• Adapt what we know works in our reading programs and apply it to mathematics instruction.
• There is no reason why the basics for one content area cannot translate to another.  Reading is an essential and foundational piece to a child's education.  When we can incorporate the same ideas and fundamentals, students are given more opportunities to grow.  Literal questions will demand a higher level of thinking to get students used to responding with mature and well-developed answers.  
• Use multiple representations of mathematical entities.
• We know that there are multiple different types of learners.  As teachers we must be able to foster all types of learning.  This is why using multiple representations can guide students' learning and understanding.  What works for one student will not always work for another.  Using differentiated instruction can help bridge the gap.  
• Create language-rich classroom routines.
• With more and more classroom becoming non-English, understanding mathematical language becomes tougher and tougher.  So, when students can find common ground to relate to, the understanding goes beyond language.  There is a way to present all information in a less strict and uniform way.  When students see math as instruction, their success rate decreases.  
• Take every opportunity to support the development of number sense.
• Number sense is the comfort with numbers that include estimation, mental math, equivalence, and on understanding place value.  This is probably the most fundamental idea to mathematics.  In order to build these thoughts, teachers must ask rich questions and introduce intriguing ideas that demand this familiarity in its responses.  
• Build from graphs, charts and tables.
• When we work with graphs, charts, and tables, we will most likely be dealing with real life data.  This makes the information more interpretable and understandable.  Students can use mathematical language and rich ideas when they are trying to work through these ideas. 
• Tie the math to such questions as How big? How much? How far? to increase the natural use of measurement throughout the curriculum. 
• The biggest tragedy in math education is that teachers most often skip the lesson on measurement.  If it is not skipped, then it is significantly neglected.  Measurement is one of the lessons that will carry students further through life than anything.  So when we are teaching this lesson to students, we must ask rich questions that lay firm foundations.  
• Minimize what is no longer important, and teach what is important when it is appropriate to do so. 
• There is no way that a teacher can cover a whole text book in one year.  Teachers put too much pressure on themselves to do everything.  A realistic goal is to sufficiently cover the most important topics for students.  This way, a teacher can build and plan to teach students to his/her best ability.  Students will have gained more useful knowledge if the teacher makes that decision.  
• Embed the mathematics in realistic problems and real-world contexts.
• Real-world problems are the best way for students to learn.  The one question that teachers are tired of hearing is "When are we going to use this?"  When students are given a real-world problems, they will automatically use mathematical language and stumble into understanding themselves.  
• Make "Why?" "How do you know?" and "Can you explain?" classroom mantras.  
• When students are encouraged to initiate this sense of wonder with each other, the classroom environment changes.  Students are now becoming the teachers and students are brought to the same page as everyone else.  Explaining answers puts an emphasis on the need to explain/defend a position with evidence.  

For a future teacher, some questions that might arise are:  Which of these are obtainable?  What one can I work on the most?  How many of them seem impossible?  What can benefit my students the most?  What sacrifices will I have to make to assure one of these shifts is possible?

I believe that all of these "changes" are obtainable.  This is why the author emphasizes 'shifts' because it should not be detrimental.  Theses ideas take already existing instruction and makes it better.  Teachers should always be willing to grow and change within their career.  As we gain more knowledge about the way students learn, we are figuring out the ways we should teach.  This is exactly what is happening here.  Teachers can take little steps to change for the better without adding more stress or work to their already hectic lifestyle.  My biggest takeaway from this is that most of these shifts encourage students to have more fun with learning and for teachers to incorporate more useful learning.   There is a way for students to have fun and learn at the same time.  When learning and fun occurs at the same time, success has been reached!

The hardest shift for teachers might be creating more rich and real-world examples.  This would require more prep for teachers and a deeper understanding.  Teachers must be able to foster the learning for all students.  If a student was having trouble, the teacher must understand the topic so well that the instruction might have to take a different direction to attain understanding.  Teachers need to learn to be dynamic and versatile during every part of their lesson to make instruction purposeful.  Additionally, this shift might take teachers outside of their comfort zone.  When teachers go into their own unique planning, teachers never know how students will respond or if the message of the lesson will ever successfully reach students.  The only thing that I can say is that risks might be worth the reward.  There will be lessons that fail, that is inevitable.  A few failed attempts and some tweaking might make the perfect lesson.  You just have to ask yourself:  What lesson has more utility for students?  Which one seems to be more successful?  Did students enjoy it?  If the end result is met, then the lesson is perfect.  Grow and learn from your students.  Sometimes the lesson you felt failed the most leads to the most learning opportunities.  Just know that the students will always come back and you can constantly develop your passion for teaching.  

For future and current teachers, this book can be very helpful.  Regardless of if you are in you first or fortieth year, the lessons to be learned are universal.  Like I said before, teachers should always be learning and growing.  If these shifts are already integrated in your classroom, how can they be made better?  What is the weakest one?  These shifts are just meant to maximize mathematics education.  The main goal is to make math more approachable and beneficial for students.  Who can argue with that?

Improving teachers one 'shift' at a time. 

Weekly 4: Communicating Mathematics

Three.....Two....ONE....

These were the last three words our elephant heard before he took the fall to the first floor.  He put all of his trust in the hands of four mathematicians.  Now, judging from our performance, who knows if he would trust us again.  

From looking at Galileo's Thought Experiment, Galileo wanted to explore what is now gravitational pull on an object.  He wanted to know what exactly what would happen if two objects with different weight were dropped at the same height, which would reach the ground first.  Picturing this in his head, he could only do so much.  Today, we know that the gravitational pull is equal on both of these objects, so they will hit at the same time.  However, this was not the information available at the time.  The only way to support or disprove this prediction was to gather data.  This was our task as a class.  We wanted to put ourselves into the mindset of Galileo.  After selecting from an array of action figures, we were told to predict how many rubber bands would be needed to keep our elephant off the ground.  As a group, we needed to take into consideration that the weight of the object.  How would that affect the rubber bands?  Would it?  There were many ideas that were bounced around:

• What is the stretch capacity of a single rubber band?
• How much will be taken up by knotting the rubber bands together?
• Will each rubber band stretch to their fullest?
• Which ones would bear the most weight from the elephant?
• Are we trying to just keep off the ground or are we trying to be the closest?
• How accurate can our measurements be from one rubber band to two bands to ten?

These were questions that we had to find answers to.  To start, we measured one rubber band resting and then two, and three.  Then, we measured how the distance increased in rubber band stretch when the elephant was dropped.  

This is the distance the ten rubber bands stretched
when the elephant was hung.  This is not the maximum
distance the rubber bands can stretch because this
does not indicate peak distance.

We took the distance and created a ratio with the number of rubber bands used.  As the number of rubber bands increased, the decimals got bigger and bigger.  In order for us to make our prediction, we wanted to be on the safe side.  We took the smallest decimal and used that number to find out how many rubber bands we needed to go just shy of the 505cm to the ground.  When we found those calculations, we found that we would need 16.69 rubber bands.  Again, playing it safe, we put 16 rubber bands on the bungee.  We thought that our prediction was relatively reserved, but we had no idea to what degree.  After the first drop, we were under a meter from the ground.  Those were not the results we wanted to see, so we did a few more calculations.  Our second try used more of a trial and error method.  We knew that adding two more bands would give the elephant a significant further drop.    By adding two, the elephant was close.  Close was not good enough for us so we added another.  This drop proved to be too much.  Our friend went smashing trunk first into the ground.  We did not have to be bungee experts to know that we had gone too far.  Thankfully, death was not the result, but it did give us enough information to calculate the difference.  

Looking at our results, there were a few things that I noticed:

• Weight did affect the result of the drop.  Our elephant was one of the heaviest objects that could have been used and we had the least amount of rubber bands.  Barbie dolls had around 25 and we started out with 16.
• It is hard to predict how much each rubber band is going to stretch and which ones stretch the most.  I think that this hindered our calculations.  
• Guess and check is probably the most successful way to find results.  Once we went from 16 to 18 and then to 19, we could see which was best.  Thankfully, there are not real bungee jumpers involved because the calculations need to be as accurate as they can be. 
• Repeated trials is important too.  There are a lot of factors that should not be overlooked.  The way the object is dropped can change and the way the rubber bands fall could affect the results.  

Weekly 3: Nature of Mathematics

The Role of our Number System..in Education

Ever since the beginning of our time, numbers have been another lesson that is taught to us during our developing years.  Numbers are included in early development as a milestone of understanding.  First, we learn to verbalize numbers while slowly developing the meaning.  This order relates to how our brain develops.  The ability to speak will come before the ability to understand.  Where did this understanding come from?  There has to be teachers before there are learners. Where did this system originate and why has it become a norm for education?  This is the topic I want to explore.  I want to discover the purpose of the number system in education.  From a future teacher perspective, I learned the role of our number system through other teachers who insisted its importance.  What benefits does knowing the number system give to students?  Finally, what is important for me as a teacher to educate myself with?

The 'Number System' can included many different aspects.  In the most broad sense possible, it can be summed up in the ways that all people reason about numbers.  For each person, the definition will be different.  Not only does each person have a different definition, but regionally, culturally and historically, the definitions have changed.  Here in the United States, we commonly see numbers in our speech and learning as one, two, three....(1, 2, 3.....).  We have both a numerical representation as well as a visual representation.  So, in teaching, the introductory grades are just learning to incorporate this language into their vocabulary.  As their fine motor skills develop, they begin to write their visual representation.  Then, as students grow and develop, meaning starts to develop.  Students begin to incorporate numbers with meaning.  This knowledge is brought into everyday life.  The connection between learning and using in crucial.  Students must be able to make the connection between what they are learning and what type of benefit it provides them.  In education, is important for learning to be purposeful.  If your students cannot connect with the material presented, then it will not become an integrative part of their everyday life.  As a teacher, this learning needs to be meaningful and easily applicable.  So, making that connection for students is absolutely crucial.

However, our number system did not just appear.  It took many centuries from its development to become what it is today.  So, the historical aspect is not to be neglected.  Students need to understand the different steps it went through.  Knowing this aspect makes the content seem more deeply rooted.  Rich history can amplify a lesson for students.  We know that counting in its simplest form can start with tallying.  Students can think like the earliest counters by using this method.  Then from there, we can dig into how other cultures used their earliest counting methods.  This learning can create links to history and/or geography.  There are so many ways to integrate the learning of the number system into education.  The importance must be emphasized over and over so that students can deepen their comfort with these topics.  After all, counting, reasoning, and numbers are everywhere.  There is not an aspect of life that does not utilize our number system.  As teachers, we must educate our newest learners into number system geniuses.

All in all, teachers have already been educated about the importance of the number system.  We see its importance and the role it plays in society so we teach it to our students.  Then, students just see numbers.  I think that if we introduce the role of numbers in their most historical form we would see a deeper appreciation.  It should not take a college degree for us to see where numbers came from.  If we educate students in the proper order we might see a stronger response to the number system.  I understand that students cannot be so overwhelmed with the history of every aspect of math, but we can prioritize.  The number system is going to be a part of their lives forever.  Lets teach them right.

Weibull, Nikolai. "A Historical Survey of Number Systems." Math Chalmers. N.p., n.d. Web. 25 May 2014. <http://www.math.chalmers.se/Math/Grundutb/GU/MAN250/S04/Number_Systems.pdf>. 

Weekly 2: Doing Math

Working with Tessellations

Tessellations can be defined as a repeated pattern on a flat that uses geometric shapes with no spaces or gaps. When I think of ones that I see in everyday life, they are present on floors, wall tiling, and tapestry.  There are patterns present everywhere.  The more intricate the pattern, the more angles and shapes become a factor.  The way the shapes interact is absolutely fascinating.  Below I used GeoGebra to test these interactions.  For the most part, I used circles (midpoints) and lines to create grids/boxes. 

This was the starting shape to
the first tessellation.

When overlapped, this pattern is created.  

For the second pattern, I used the same size circles, but with different frequency and placement.
It's amazing how similar, yet different the two look. 

This is what one segment of the pattern looks like blown up.
 It appears to be composed of individual curved shapes
and not the product of circles and lines.  

Next, I put lines through every vertex in every direction.  The lines created a horizontal and vertical grid with their diagonals. The new shapes that were created make this pattern more dynamic and visually appealing.

Tessellations introduce and explore many parts of mathematics.  Students can work with these at many different levels.  An important lesson that students can learn is how shapes can interact.  In the examples above, we could see that overlapping circles can create ovals that make a floral pattern.  Then, moving to the next example, we can see that a slight shift in position can create a pattern completely unlike the rest.  Students can be creative and make discoveries on their own. They can answer questions like: Is this a linear pattern?  What happens if one more shape is added? How could lines be utilized?  What if only circles or squares were used?  Can you make different shapes from the beginning?  What types of ideas have you explored?

These patterns provide an inside look at the role of angles, lines, and circles.  When simple shapes are put together and overlapped in a repeating patterns, it appears to create a complex image.  This was the greatest mathematical lesson that I gained.  Not only can a simple object be made to appear complex, but knowing that complexity can always be made simpler. 

Weekly 1: History of Math

Euclid’s Fifth Postulate is one of the most historically discussed postulates in the history of mathematics.  Many of the most famous and recognized mathematicians have worked on the success of this postulate.  Some failed, some succeeded, and some work that aided in its discovery was not recognized until after the mathematician’s life ended.  The amount of effort and intelligence that went into this postulate is something to be recognized.

           At first, Euclid created five postulates.  The first four postulates describe the properties in exactly one single line.  It is not until the fifth postulate that parallel lines are introduced.  Mathematicians thought that the parallel (fifth) postulate could use all four previous postulates to help prove it.  The reason for so much discussion over the fifth postulate was because unlike the first four, there were many more discrepancies in the fifth. Mathematicians worked on the postulate until they found something to fall through in their work.

            The infamous Fifth Postulate is also known as the Parallel Postulate because it can draw conclusions about the characteristics of parallel lines.  This postulate was necessary to prove Proposition 29.  Euclid’s first 28 propositions could be proven with his first four.  Therefore, no further propositions, including parallel lines, could be proved without the understanding and correctness on the Fifth Postulate.  Interestingly, in Euclid’s original writings of the postulate, he assumes the creation of the parallel lines.  This did not sit well with other mathematicians, so the work began.

            There were many consequences that were created by the fifth postulate.  Many of these consequences looked to help conquer the ambiguity contained in the postulate.  They looked to observe the possibilities and behavior of the interaction between two lines.  It was not until the work of Saccheri and his approach to finding the answer.  Instead of trying to prove what Euclid had written, he wanted to prove that there existed a contradiction.  Although his work did not arrive at a concrete conclusion of contradiction, his method of using quadrilaterals was not completely a waste.  

            Following Saccheri were mathematicians looking for answers, but never found any.  The next big attempt came from Bernhard Riemann.  In Euclid’s Fifth Postulate, the length of the line is not defined and many people assumed that it meant infinite length.  Riemann found that an undefined length did not necessarily mean that the length was infinite.  This led to the discovery Elliptic Geometry.  Using a three dimensional plane, postulates 1, 2 and 5 were modified using circles.  In Elliptic Geometry, all the perpendiculars have equal length and the angles’ sum is determinable unlike plane geometry.  Riemann’s discoveries used a new and unique approach to geometry. 

            All of the failures and successes of mathematicians led to many discoveries.  Some work may have made accidental discoveries on the way, but without unique points of view, geometry would not be dynamic.  It took the work of Euclid, Riemann, Lobachevsky, Bolyai, and Gauss to challenge the work of others and make lasting discoveries.  Euclid laid most of the groundwork when it came to these discoveries.  If it was not for his work, he would have not paved the way for the other mathematicians.    Euclid’s Fifth Postulate created frustration for years, but without this frustration and determination, geometry would not have been redefined. 

The main message that I can take away from Euclid's work is that persistence is key.  He may not have been able to finish proving his work like he wanted to, but he paved the way for other mathematicians.  We need to understand that even mathematical tasks are done better by teams.  Math is not meant for one genius to try and conquer.  If you look at the other historical details of mathematics, you will see that multiple people have done their part to contribute to certain discoveries.  Yes, we know Euclid's name because of the work, successes, and troubles he had.  Experiencing difficulties only proves that he is human, which might have been a question I have had before.  

"Euclid as the father of geometry." Khan Academy. N.p., n.d. Web. 12 May 2014. <http://www.khanacademy.org/math/geometry/intro_euclid/v/euclid-as-the-father-of-geometry>.

"Parallel Postulate." - AoPSWiki. N.p., n.d. Web. 12 May 2014. http://www.artofproblemsolving.com/Wiki/index.php/Parallel_Postulate