Connecting my teaching and academic life! What do I want to Remember!

Recently I am becoming increasingly aware of the importance of seeing the connections between my professional and academic time at Grand Valley State University!  It has been an extremely demanding semester for me so far.  I am at school for 22+ hours a week, then daily I drive downtown to Grand Rapids for 15.5+ hours of class per week.  This sounds like a normal 40 hour work week, until I consider the 9+ hours I am traveling from place to place throughout each week.   I constantly struggle, finding that I have a hard time connecting my time teacher assisting and going to class.  I think part my challenge has to do with the psychological reset that occurs as I drive to Grand Rapids from my placement over 45 minutes away.  I really think a year from now when I look back on my experience I am gonna remember how I made a decision to transition.  What transition? That transition I will unfold right now!

Starting tomorrow, I am gonna utilize my driving time and after school time to carry out at least 15 minutes of active reflection and connection daily.  Instead of being overwhelmed by my separate feeling academic and teaching worlds I am going to do my best to connect them.  From time to time I will need to post here to do that, but most times the connections will involve concentration and perseverance to utilize the traveling moments in my day!

One year from now, I want to remember that I made this change.  I decided that I can learn, and apply my learning to my profession immediately.  The practice of reflection that I will develop through perseverance and focus could be one of the best decisions I ever make.

As I think further, some other things I want to remember one year from now are how scared I was when I first started teaching.  I was amazed that I am less confident in front of a room full of 7th graders than I am in front of 30 college students.  I want to remember how much I have grown from a timid by the book teacher, to a teacher that is not afraid to do difficult and different things to enhance student learning and engagement.  A last very important thing that I want to remember is how much I have struggled, sacrificed, and learned on my way to becoming a teacher!  I did all of that because I believe that each student has the potential to become an awesome individual and to grow from learning mathematics whether that be from the content or from the perseverance and abstract thinking skills they gain to defend their own arguments.  I am so excited for what the rest of this semester brings, but I am excited starting now to better connect my teaching and learning in college!

Wait What… Is Math a Science?

According to the Merriam Webster Dictionary, science is “a knowledge about or study of the natural world based on facts learned through experiments and observation”.  Last Wednesday my history of mathematics class carried out a neat experiment called barbie bungee.  The activity involves rubber bands, and a barbie or action figure.  Similar to a bungee jumper, we tied a string of rubber bands to the figures legs with the goal of seeing how we could drop our figure from a ledge so it ended up very close to the ground!  To carry out this experiment with accuracy we had to conduct a series of smaller trials before we dropped our figure off the bridge! 

The activity was a blast and we nearly won the competition between 6 groups of math students, but our hulk like figure barely nicked the ground because our calculations were just off!  He may not have died, but he would have been severely injured which was not our intent!  Following the activity my professor proposed a unique question, “Is math a science?”

At first my reaction was that yes mathematics is a science.  Mathematicians carry out trials in which they test conjectures and see if the conjectures hold.  Mathematicians look at patterns in the natural world, and seek explanations for their being.  Mathematics can be utilized to make sound hypothesis and we can test our predictions.  Lastly, I thought about the scientific method.  First we ask questions, then research and gather background information, construct a hypothesis, test our hypothesis by doing an experiment, analyze your data and draw a conclusion, communicate our results.  In mathematics it seems to me there are very similar processes, first we ask a big question, then we think about it and construct a conjecture.  Once mathematicians have a conjecture they form a proof and test their conjecture trying to think of counterexamples, then mathematicians build a proof or share their counterexample, and draw a conclusion about the validity of a conjecture.  Then they share out their idea by having a formal proof published on their conjecture.

On that last point I immediately detected a difference between math and the natural sciences. I could be wrong about this, but it seems to me that mathematicians only communicate their results when they have constructed a formal proof.  I mean they do not publish their counterexamples, or if they find a conjecture to be false.  They may discuss their results with others, but they will not end up in a paper.  In science a hypothesis could be wrong, and a conclusion can still be developed.  Another distinction between science and mathematics is that science deals with that which we can feel, smell, see, hear, and touch most times whereas mathematics is more abstract.  Mathematics deals with the concepts behind the manipulation and exploration of patterns.  Mathematics builds on previous axioms and provides explanations for patterns rather than drawing from experiments we can see.

A great example of math not being a science involves number theory.  Number theory was not created through experimentation and observation, but rather by abstract thinking and critiquing counting objects in the world.  Mathematics can be used to form great predictions in the sciences, but it is not a science itself because it does not rely on physical experiments for its growth and development as a field.  Mathematics and science are closely related, but I would say that math is not a science because it does not involve experimentation on the physical world, but rather abstract quantities and figures.

The reality behind this whole discussion is that math needs science, and science needs math in order to effectively grow and expand.  Without science, math is just a made up and distant group of ideas. Without math, science will progress extremely slowly because data cannot be organized clearly, and predictions made by scientists will be weak and faulty.  In our barbie bungee activity we saw the importance of utilizing mathematics in a scientific experiment.  Science and math are connected like peanut butter and jelly, they are separate entities, but they combine to make something wonderful!

My Reflection towards 100% classroom awareness and 95% student engagement!

Wednesday this past week I had the privilege of teaching about rational numbers.  I began the lesson with a basic question what are rational numbers?  

I had a good thought of trying to make the lesson an inquiry lesson.  My desire to have students explore to find the definition was inspired by my background in the GVSU math department.  The professors constantly encourage us to let ourselves find multiple ways to a solution, and to provide opportunities for student discovery.  As a new teacher I sometimes feel that I must teach this way or else my students will not develop a real deep conceptual understanding of the mathematical topics I am covering!  This is my internal struggle right now.  A fight between teaching conventionally: learning a procedure that a teacher models and then students begin to use that procedure.  Along with some conceptual explanation throughout the lesson that often gets lost in the shuffle of an adolescent mind.  Compared to the alternative which is a discovery based lesson: where students explore some numbers or aspect of life and find out on their own the patterns or models present by building from their prior knowledge.  This way they first get the concept, and then derive the subsequent procedure.  The great war in my teaching mind is between these two tactics.  With the first teaching conventionally, I find myself frustrated that students may never understand why they do what they do, and they will struggle to apply what they learn to real life.  With the alternative, I find that I feel like a bystander to student learning, and have difficulty determining how to effectively step in and focus student thinking without corrupting it to make it exactly like my own.  Currently, I am persevering in problem solving to determine how to best teach students, and the problem is an ever evolving one I feel.

The reality remains that I believe deep down I need to strike a balance between these two approaches to teaching.  I am pretty sure the more discovery sided lesson has a better long term impact for student memory and implementation, but the conventional approach has merit too in that humans have always learned already discovered concepts through modeling the behaviors of others.  I want to be a teacher that has students learn how to think deeply, but also explains concepts clearly and in a manner which students build a math toolbox.  Through the past four weeks, my first four weeks trying to teach 15+ students at a time, I have certainly learned that I have so much to learn and grow in!

Before I digress, back to the lesson.  I asked the big question, then displayed some examples and had the students guess whether the numbers were rational or irrational, then I told them whether they were rational or not.  I utilized numbers like, 3/2 , √2 , 5, -2/3, and 5/0.  Once students began to see which numbers were rational and which were irrational they developed some ideas about what could make a number rational.  The idea of having an inquiry lesson could have worked, but students had no definition to work from, so I was using examples of which they had no idea whether they were rational or irrational. The students had nothing to construct their own framework from as they had no experience with irrational or rational numbers.  Intriguingly, even with the flaws in my lesson structure the students provided great possible explanations for why certain numbers could be rational and some could not.  For 5/0 they said because the denominator is a zero the number is not rational.  It is not possible to have 0 groups given 5.

The point at which I had most overestimated my students prior knowledge was when I presented them with the square root of 2.  I did not realize that in 7th grade most students have hardly or never seen square roots.  The students had little idea of what square root even meant, yet I was expecting them to tell me whether it was rational or irrational.  I used this example, remembering my Sophomore year of college where we proved the square root of 2 was irrational.  My poor planning had led my students astray.  I wonder how often teachers cause their students more challenges than help like I did.  Still as I mentioned prior, the students had some solid ideas, one said a rational number must be one where there is a ratio, and another said any fraction is a rational number.  In the end, I stated the definition, and as much discussion as the search caused, in the end it seemed to have been wasted.

Later, I taught a bit more on converting fractions to decimals and vice versa, and throughout the class I noticed that when I spent too much time on individual student responses other students checked out and became disengaged.  I wonder how in full group discussions I can best keep all students engaged and learning?  When students work in smaller groups and engage in mathematical discourse I have noticed that they learn a lot!  Hopefully I will be able to develop better methods to assist struggling students while the lesson is occurring.

After being encouraged to talk to a few professors and teachers about my frustration regarding the gap in experience between students. I  have learned that I need to model how I think about solving problems, then give students time to try to think about the problem my way or their own way building on what they know.  If I verbalize how I approach problems, then perhaps my students will become more comfortable with asking questions throughout the problem solving process.  My teacher and professors have also encouraged me to set up more opportunities for students to teach each other.  In the past two days, I have tried to connect students with my coordinating teacher by having the less experienced students sit by the more experienced students in table groups.  This way the “lower” students at the beginning of the year will grow from the expertise of the “higher” students.  A further problem this week I discovered is that when shy and less experienced students sit further back, it is harder for me as a teacher to call on them as often as is necessary to help them to be engaged and learning.  Thus, there are many challenges in the classroom, but I think I am beginning to understand how to help struggling students learn.

To wrap things up, I am really enjoying my experience as a teacher assistant though it is very challenging to teach well!  I realize that it may take me many, many years to become the teacher I want to be, but I am looking forward to the growth I can and think I will experience.  In the coming weeks, I hope to develop a better wider view of the whole classroom as I teach.  By this I mean, that I want to have a full awareness of student behavior so that I can prevent problems and consistently address issues that detract from the learning environment of the classroom I teach in.  I want to become more proficient at utilizing yellow cards, which are warnings where students talk to me after class about their behavior.  Another thing I would like to figure out is how to plan lessons well so that they are engaging and interesting for at least 95% of my students.  I look forward to seeing what happens as I strive towards my goal of helping student’s best learn and being satisfied with my performance as I teacher.

0, Zero, Cero, Zilch, Nada, Null, and Yes it has a History!

Zero, so simple it seems, but the truth is there is more to zero than meets the eye!

When I think zero, I first think of a number with no positive or negative value.  In the 7th grade mathematics class I am teacher assisting in, some students had difficulty figuring out which number system zero should be a part of.  The answer to this question is much more challenging than it seems at first.

Zero is considered to be a rational number because zero is considered to be an integer.  Zero is an integer though it is the only integer that is neither positive or negative. and 0 divided by any other integer equals zero.  Since a rational number is the quotient of two integers, zero can be considered a rational number.

Zero is a whole number also if you consider the definition of a whole number to include all nonnegative integers.  There are raging debates throughout the web on whether zero is a whole number or not.  A few fascinating ones can be found at the math forum and a physics forum.

Zero is a natural number depending on which definition of natural numbers you utilize.

N∈{0,1,2,3,4,5,…}

or N∈{1,2,3,4,5,…}

So the answer to most questions related to zero is… that defining it is complicated.

Zero in my view is like a neutral number.  It is the break even point which separates above and below ground, above and below the sea.  The point at which a debt and a fortune are equal, so there is neither debt nor fortune.  Brahmagupta came up with the first rules regarding the concept of zero in the 7th century in India.  He said,

“When zero is added or subtracted from a number the number remains unchanged.”

He also established that anything multiplied by zero was equal to zero.

He viewed numbers as abstract entities, and this probably led to his contributions in defining the rules of zero.

Brahmagupta

There are many holes remaining in my definition of zero.  Zero is very difficult to define in one way.

The movie and book Holes.  Included a character named Zero.  It was said of Zero that he had nothing going on in his stupid little head.  That was the reason he was called Zero.  The reality was that the camp counselor that declared that did not fully understand zero.  Zero was a very smart kid, he just needed someone to believe in him!  Intriguingly the number zero is like this.  From a distant ignorant perspective the idea of zero seems pointless, but for those that view the bigger picture and get to understand zero the number is indispensable.

Zero from holes

Euler was certainly wiser than I, but there seems like a lot of mystery here!

“To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.” Leonhard Euler

Volume of a Sphere, and a Soccer Ball

When I think about spheres I first think of soccer balls.  Not just any soccer ball, but a jabulani world cup ball!

Ball used at the World Cup in South Africa

As a student four years removed from Calculus, I first think about the volume of a sphere in terms of the basic volume formula.  I am excited to use this math formula because I will likely be teaching about the formula in a 7th grade mathematics classroom I am assistant teaching at this year.  One key thing to note about formulas like the volume formula is that they were not always so readily available.  I was curious where the formula

V=4/3*pi*r^3

Actually came from: Archimedes is said to have discovered and proved the formula during the 3rd century B.C. in Greece.  Of all his works the work he is said to have been most proud of was the discovery of the relationship between a sphere’s volume and a cylinder circumscribing that sphere.  This is particularly impressive considering he also invented a claw that destroyed ships, a very close estimation of pie, and created a place value system of base 100,000 because he was disappointed with the greek number system.  He was so proud of this discovery that he even had a sculpture of the phenomena on his grave.  The relationship between the two is that the a sphere with the same diameter as the height and diameter of a cylinder has 2/3 of the volume of a cylinder.

Archimedes found that a sphere's volume is 2/3 that of a circumscribing cylinder of the same height and diameter.

Archimedes found that a sphere’s volume is 2/3 that of a circumscribing cylinder of the same height and diameter.

Moving past Archimedes, about 2000 years later Gottfried Leibniz and Isaac Newton worked separately to discover Calculus.  How can two men discover calculus at the same time in separate places? Check here

Calculus can be utilized to prove the formula for the volume of a sphere.  I could spend a lot of time in my own words explaining the proof for the volume of a sphere utilizing Calculus, but rather than waste your time I will point you to two helpful sites that explain the process.  The first is a webpage that shows how integration and the pythagorean theorem can be used to show that the Volume of a sphere is what Archimedes said it was.  The second is a video that walks learners through the proof.

sphere

What is incredible and fascinating about the proofs for the volume of a sphere is that Archimedes proved it without any Calculus! Calculus had been no where near invented in the 3rd century, but Archimedes still was able to form a geometric proof.  In fact, the proof is so challenging and complex that most textbooks have a simplified version.

Returning back to the beginning, a Jabulani soccer ball is a sphere.  I wondered what the volume of that sphere is?  It turns out the answer is not as straight forward as I first thought.  There is not an instant plug and chug as there so rarely is in real life.  As I searched for the specifications of a Jabulani match soccer ball.  I could find only the circumference which is considered to be between 68.5 cm and 69.5 cm.  So I knew there would be some error in my answer as the circumference was a range.  By first utilizing the formula for Circumference I had to find the radius of the sphere.  That formula is:

Circumference= 2*pi*radius

Then I had to carry out some algebra to solve for the radius.  So I divided 69 centimeters by 2*pi.  When I did this I found the radius of a jabulani soccer ball to be about 11 centimeters.     I inserted the radius into the formula for the volume of a sphere.  

Volume=4/3*pi*radius^3

and found that the volume of a jabulani soccer ball was 5546.44 cubic centimeters.  If I wanted to further challenge myself or my students I could ask them the range of the volume of all FIFA regulation jabulani soccer balls.  

chaff

So what should we take from this exploration through the history of finding the volume of a sphere, and the volume of a jabulani soccer ball?  

so what

First, the history of mathematics is fascinating and may be something that teachers can use to engage students in a lesson, and help them to realize why mathematics matters!  Another key discovery is that mathematical problems do not often give us the necessary information to solve them in one step.  We must seek out and find a way to figure out the information we are missing to find the solutions we are looking for.  Lastly, there are so many mathematical resources at our disposal today that educators need to carefully consider the problems that they are giving students.  A good question to sometimes ask students may be, “Are math problems in real life this simple? Why do you think that?” This practice will help students in their critical thinking skills and their ability to recognize the many variables that influence a real problem.  Math can be challenging, but students will benefit much more from our teaching if we challenge them and expect their best effort!

Landscaping in an Educational Complex: 3 Key Points of Consideration

A peculiar title, but one that teachers have to consider throughout every school year.  In my current 7th grade math classroom at a small charter school of about 60-70 7th graders, we have what I have imagined as a relatively standard classroom set up.  The teacher’s desk is in the front left of the classroom.  The students seats were organized in 6 rows that are staggered with 3 tables in each row.  Each row has space for six students in it as each table holds two students.  In the back of the room there is a storage closet and all of the classroom textbooks on the left wall.  For a better understanding of my coordinating teachers classroom layout…

Classroom Layout 7th Grade Math Class

Classroom Layout 7th Grade Math Class.  The door to enter the room is in the lower right and the storage closet is a bit more pushed back at the bottom.

When I discussed our classroom layout with my coordinating teacher, he explained that a few years back he was offered the option of having individual desks for students, or having tables.  He chose tables because he believed that tables would be easier to move in ways that promoted collaboration, and also they would make it easier for students to have discussions with their neighbors.  I believe that creating collaborative options is one of the keys to landscaping in a classroom for a teacher.  In the future I will definitely arrange my tables or desks in a way that promotes student collaboration and microteaching between students.

I could go on and on about the classroom landscape, but for the sake of time I want to share my present three most crucial considerations for the landscape.

1. Collaborative value-The layout promotes student discussion and teamwork.

2.Uniqueness-The layout is intriguing and interesting; is it just like all of there other classes, if it is you may want to tweak it.

3. Accessibility-The teacher can easily maneuver around the classroom to best meet student needs.

The classroom layout of a teacher is vital to their goals in creating a vibrant classroom culture, and a sound space for student learning.  I am excited to learn more about the secrets to a great classroom layout throughout my experiences observing teachers and seeing new schools.

So when I say Math I mean…

It has been too long since I last blogged, but I am entering a new chapter in my learning.  I am looking forward to discovering more about the history of math through an exciting capstone course!

Now you are probably thinking hold up, the history of mathematics how can that be fascinating?  Why does it matter what math is and where it came from I just need it to balance my checkbook or figure out how much change I should be receiving for my 20 dollar bill.

twenty dollars

Personally, I love history! My opinions aside, mathematics history is valuable to give people a base for understanding the importance of the beginning of various mathematical operations, concepts, and theories.  I believe that mathematics history is crucial to finding meaning in mathematical learning.  That is the real reason I am so excited to learn more about math history.

I currently do not know much about the history of mathematics, but from what I do know in my prior knowledge five of the most incredible discoveries in mathematics history in no particular order were…

The Ability to count

The Pythagorean theorem- vital to building structures and keeping right angles.

The discovery of Algebra by Al Khwarizmi(spelling may be questionable)- key to abstract reasoning in mathematics and simplifying complex problems and challenges.

The Nine Point Circle (By Euler)

The calculation of ballistics(long shot)

Clearly there are hundreds more discoveries that I have not listed and my list is very meager, but these discoveries are important.  Understanding why something was discovered adds a significant amount of value to a topic.  If your educational experience was like mine you remember growing up asking why do we need to learn this Alg-ee-bra?  The history of mathematics has a ton to say about the answer to questions like this!

The history of mathematics is important, yes, but what is math?  Is it counting, coming up with theories about numbers?  Defining math is complicated, so when I say Math I mean the study of patterns in the world and in our minds and how they connect to each other.  Math is utilized for a variety of reasons, but the study of patterns in order to apply models to patterns in the world around us is central to my definition of math.  As a future teacher I need to continue to develop my conviction on the definition and importance of mathematics, and be flexible to learn more from others throughout my career!

If you want to think further with me consider what it means to have a degree in mathematics? The diagram below could kick start your discussion or thoughts!

expertise profile math