What is an Axiom?

In mathematics, we often come upon terms such as axiom, theorems, postulates, etc., but what are the differences between them, and how are they used in mathematics? After taking a look into Euclidean Geometry again, I thought it would probably be a nice refresher for myself to look more closely into the definitions. 

Axioms and postulates are two words I hear interchangeably used in mathematics, but I have always been confused whether they are the same thing. Sometimes a statement is an axiom and sometimes the same statement can be referred to as a postulate, depending on who you are talking to. While looking further into this, I found that axioms are often referred to as postulates because like axioms, postulates are statements that do not need to be proven as true because it is self-evident that they just are true. For example, since we recently have looked at Euclidean Geometry, Euclid's Fifth Postulate is "If a straight like falling on two straight lines make the interior angles on the same side less that two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than." While reading, we can see that this is obviously a true statement because if we have to lines that intersect another line, and the interior angles created are less than 90 degrees, the two lines will have to intersect eventually. We don't need a proof to show that this is true, so it can be considered an axiom or postulate by definition.

Axioms and postulates are different from theorems because of the fact that they don't need to be proven, some even say that they cannot be proven because they are so self-evident there is no need for them to be proven because it is just logical. Theorems tend to be challenged more often then axioms since they are subject to more interpretations and because there are many various methods of deriving theorems. Axioms and postulates are often used to help prove theorems.

Axioms and postulates are evident throughout all mathematics, but one area where there are many famous ones are in Euclidean Geometry.  The historic mathematician Euclid came up with many postulates and theorems in Euclidean Geometry, that we still use today. Euclid's most famous postulates are the first five postulates:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight lines segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All Right Angles are congruent.

5. If a straight like falling on two straight lines make the interior angles on the same side less that two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than.

His Fifth Postulate, which I mentioned above, is one of the most famous postulates in mathematics. It caused a lot of controversy among mathematicians for many years that resulted in many proofs trying to prove and disprove the postulate. 

Euclid came up with his axioms and postulates to establish initial assumptions to help him prove other theorems throughout his studies. This, to me, is why axioms and postulates are so important to mathematics. Without axioms, we wouldn't have any set assumptions about mathematics that we can all agree are true without any controversy. Axioms make proving theorems so much easier. They give us a foundation of true assumptions, which then are used to prove theorems. Without them, we would have try to prove these self-evident statements, but with them we just have to acknowledge that it is a true assumption. 

Now that I have clarified the differences between axioms, postulates, and theorems, and I have refreshed my understanding of how helpful axioms and postulates really are, I have realized that I don't think I would want to try proving much without them! 

Second Book Review

The second book I chose to read was "Accessible Mathematics: 10 Instructional Shifts That Raise Student Achievement" by Steven Leinward. I chose to look at this book for the second book choice because I was interested at looking at some of these shifts for myself. After listening to Joe and Karen talk about the book, I thought it might offer some good insight on what I could do in my teaching methods to possibly be more effective, especially with my student teaching coming up in the fall.

Now onto my review. I really liked what I got from the book when I skimmed/read the book. I think the author really picked out the obstacle of teaching mathematics, the fact that there is a "lack of direction and will to make powerful instructional shifts in how we plan, implement, and assess daily mathematics instruction." This to me is so true! We all were taught math as certain way, mostly through the teacher packing us with information on the lesson for the day, finishing the lesson with a couple of examples, and sending us home with problems 1-30 even (or something like that) for "practice." We are so aware that this is not effective for children to learn mathematics and lacks a deep understanding in the mathematics behind the math that we are teaching. Yet, I still see and hear  that this is still the way a lot of teachers teach mathematics. There has to be an answer to better was of teaching, and Leinward tells us that "the answer lies between the place of what we know and what we actually do."

The author offers 10 instructional shifts that math teachers can make to help students better understand mathematics. All of these shifts that he suggests are solid and are something that I would definitely want to try out and can see myself trying. The author explains that shifts 1, 2, and 10 all deal with review, discourse, and stimulation in-depth understanding of mathematics. Shifts 3, 4, and 5 target the use of representations, communication, and number sense, which all are important for students to demonstrate understanding. Shifts 6, 7, and 9 ground the mathematics in the world around us and build understanding from data measurement and apply key mathematical ideas. Lastly, shift 8 is a reminder that we can't do it all and we need to focus on what is important. All of these shifts are focused on understanding, something that is lacking in mathematics curriculum today and needs to be focused on if we want students to actually learn mathematics.

Lastly, I would like to comment on the set up of this book. It is very easy to read, understand, and apply everything that the author has in this book! The chapters are set up nicely so that each shift is addressed in its own chapter. The reader is given an explanation of the reasoning of the shift, the benefits it can have, was to incorporate it, examples of direct dialogue and activities, and at the end of each chapter it gives a "So What Should We See in an Effective Mathematics Classroom" box offering goals for classrooms to have that are incorporating these shifts. This is so awesome for a teacher because the author is making it almost effortless to incorporate these shifts! They also have 5 appendices at the end of the book that offers more support and examples of how you could incorporate these into your teaching. Overall, this book is really setting teachers of mathematics for success in making these shifts. Shifts in teaching are easy to make, rather than whole method changes, and the book really wants you to try them!

A Math Teacher-to-be's Lament on the "Mathematician's Lament" (1st book review)

Where do I start?

This book gave me a whole slew of emotions as I read through it. In fact after I read it, I am still unsure of what kind of star rating that I would give the book, because I agreed and disagreed with the author about so many things that were addressed in the book. 

So...I guess I will start with this. I read the book "Mathematician's Lament" by Paul Lockhart and forwarded by Keith Devlin. I first read the forward, and it gives a nice summary of the purpose behind the book itself and the publishing of the book. Lockhart had originally wrote this Lament as a way to just lament about his frustrations of the ways mathematics is taught and presented nowadays. He had no intention of publishing it for public viewing, but had shared it with some fellow mathematicians. Devlin had came across a copy of it, read it, and thought that there was such good information in the piece that he shared it with some other fellow math teachers. They all agreed and felt mutually about the frustrations Lockhart emphasizes in the book. So, naturally, Devlin decided it needed to be published, and got it published so that it could be shared with the world. Whether that was a good decision or not, I am unsure based on how I felt after reading the book.

I can see why Lockhart had never intended for this to be published for public display. The way that the book is written is almost like one of those letters that your mom, dad, or friend had encouraged you to write to someone who you were really upset with, but then never send it. You basically write it to get your frustration out, yell at the other person(or in this case, object), and give them a piece of your mind, in hopes that it settles the anger that you have about whatever had happened, and never send it to the person. If you were to read this letter aloud to someone else, parts might not make sense, you could contradict yourself at different points in the letter, and it would probably be more like word vomit expressing anger and frustration, but never offer a solution to make your frustration into something that is productive.

I would say that this is a pretty accurate explanation of the layout and context of this book.

I first started reading this book, and I agreed with just about everything he said. He explained the problems with the way that math is taught in school, how there is no understanding, and students are forced to memorize and mindlessly go through the process of finding an answer in math. They don't get to see the mind-blowing components because there is a lack of discovery and exploration of mathematics in classrooms today. He explains that math is a creative topic, and we are teaching it without the creativeness. I really enjoyed the metaphor he gives for the way that math is taught in school and how much is lost by teaching it this way. He compares the way we teach math to the way we teach art. We all know art is a subject solely based on creativity. So imagine if we taught art by only allowing students to do paint by number paintings, and that is what they do the whole time they are taught in school. They are never giving the freedom to paint what they want, try out different medias and ways of creating art, or shown cool art that has been created by artist in history. So, they got through school thinking that art is this boring subject where all they do is think "number 3...that's purple. I need to paint all the 3's purple, and then I am done and can move on to the next number to do it all over again. Yay." How is this fun?! Well, it's not. We know that. So, Lockhart poses this idea that if we are teaching one style, one way, and never allowing ourselves to teach students math outside of that structure, how are we expecting them to be interested and see deeper understanding behind mathematics? You wouldn't have interest in art without creativity, or any subject involving creativity for that matter, so why do we expect students to be interested and understand math when we are teaching it without its creative components?

Personally, if the book would have stopped after this section, I would have been able to give the book an awesome review. This is an awesome point! and the point that needs to be seen by all teachers who are teaching mathematics at some point. Can you imagine if we made math fun, meaningful, creative, and purposeful how much more excitement students might have for math?!

However, the book continues, and he continues on his rant of anger and frustration. 

He continues with other points in the book about this lack of creativity when teaching mathematics. At the end of each chapter, he provides this little summary/argument/discussion between two people, Simplicio and Salviati, where one person, Simplicio, is questioning some of the statements that Lockhart has made in the chapter, and the other person, Salviati, is explaining his reasoning (so Lockhart's actual reasoning for his statements). I know, it sounds weird and confusing, but it is also nice to have this bit of dialogue at the end to clarify some of his thoughts. At one point he brings up that there should be a curriculum for math, and that we should simply allow students to work with it, explore, and discover the concepts themselves without the teacher teaching them any of it. At one point, Simplicio asks what he suggests we do with students in younger grades as far as math then, and Salviati responds with have them play games and explore. To me, I am agree with Simplicio here. This solution doesn't make sense. What do you think would happen if we gave a bunch of kindergartners games during a math lesson and told them to figure out how to count? Can you say "Chaos"?!! These students have no sense of numbers, what counting is, or how to even do it, and you expect them to figure it out on their own through games? I would like to see these games he would have them playing, because I don't think it would happen. In fact, when he made this argument, my first thought was that this guy must not have an experience in teaching mathematics and must be a pure mathematician. Throughout the a majority of the rest of his points I continued to think the same thing based on what he said, until I was later reminded in our class discussion that he was indeed a math teacher for many years. 

Students need structure. I agree with the fact that we need to implement more creativity and discovery into our current mathematics curriculum, and that unless something is changed, students are never going to have the understanding we desire for them to have, but that can't be done without actually teaching them things and having the teacher structure the learning. The teacher needs to be the mediator of the learning. It is our responsibility as a teacher to give the students the tools they need for learning, and help them successfully grasp a concept. For Lockhart to basically claim that we need to just step back as a teacher and let them discover alone, doesn't make sense to me. We need to be there to help push them in the right direction when they are going astray. Also, if we don't teach them the basics of math, like counting, place value, etc, how are they supposed to discover things in math?

This pattern of agreement with points of frustration, disagreement of how to handle it or how we really need to do things, and what he is suggesting just doesn't make sense, continued for me throughout the first part of the book. I won't even start with the second part of the book, because it is just a continuation of him complaining, not making sense, and never offering a solution to anything he was complaining about. 

Overall, my feelings towards the book are, if you want to listen to a lot of venting and get riled up about problems in mathematics, this is a good book for you. But, if you are like me and are looking for a book that points out issues in mathematics curriculum and teaching and offers ways to fix these problems or suggestions to teaching math with more creativity, then you might want to pass because you will probably get frustrated with the authors points and get sick of him complaining and not doing anything to solve the problem he's complaining about.  

Hippopotamus Bungee Jumping: 101

This week, we did a really interesting experiment. After looking into Galileo's experiments dealing with the gravitational pull on an object, we took his experiments and put them into action by thinking about forces an object can have based on the weight of the object. In other words... we chose an action figure/barbie/animal, and tried to give them the best bungee jump of their lives off the second floor staircase in Mackinac Hall, using only rubber bands. How does this seem like a valid experiment that we all learned from, you ask? Well, let me explain our process of the experiment and our results a little better, and then maybe you will understand. 

Alright, so here was our task. Our group decided to pick the hippo as our fearless jumper. We were then shown where we would be bungee jumping the hippo from, which was in the stairwell of Mackinac Hall from the second floor to the first floor. We measured the distance from the top of the railing, where we would be starting, to the floor of the first floor. This gave us the distance we wanted the hippo to get as close as possible to, so that he wouldn't hit the floor. The distance we measured was 199 inches, which we converted to centimeters and got 505.46 centimeters. After, we were given 10 rubber bands and some questions to answer. 

Ultimate Goal: How many rubber bands do we need to get the hippo as close to the floor, without touching, as we can? 

Along with the goal came many other questions we had to answer to make an educated guess. How far does the hippo travel using one rubber band? How about two? Or three? Is the distance he travels as we add rubber bands consistent? Or does it change? How do we want to go about collecting this data? How are we going to make sure the experiment is precise? These were all running through my table mates and I's heads as we thought about how to conduct the experiment. 

We attached the first rubber band securely around the back ankles of the hippo. We actually used a full rubber band just to make sure he was secure, and then attached the second rubber band, as if it was the first, so we didn't have the hippo's force on one and a half rubber bands. We held a meter stick on top of our desks, and a folder at a certain measurement to start. Sarah held the hippo with the bottom of its feet perpendicular to the top of the meter stick and we counted down, three..two..one, and she dropped him. If the hippo hit the folder we would lower it just a little, and try it again just to make sure he could travel any further. If he didn't hit the folder, we moved it up just a little to see if it could hit the folder a little higher up. When we found that the hippo was consistently reaching the same measurement at least twice, we recorded that height for the for one rubber band. We added another rubber band and did the same steps, until we found a measurement for two rubber bands. We continued this pattern to find the data shown in the table below. 

# of Rubber Bands	Distance Traveled (cm)	Difference of Jumps
1	27 cm	27 cm
2	55 cm	28 cm
3	75 cm	20 cm
4	99 cm	24 cm
5	117.5 cm	18.5 cm
7	156.5 cm	39.5 cm

We wanted to start keeping track of the difference of the distance of a jump and the jump previous to see if there was any pattern how far the hippo traveled each time. We got to the fifth jump and found that the average between the differences so far were about 22 centimeters. For the sake of time (since we were running short on it) we knew that 7 went into 21 evenly, so we rounded our average down to 21 centimeters between jumps. We then added 2 rubber bands, at once, to see if they reached about 42 more centimeters then the measurement with 5 rubber bands. We found that the rhino traveled 156.5 centimeters, making a difference between 5 and 7 rubber bands 39.5 centimeters. 

We needed to use the data we found to make a prediction about how many rubber bands we needed to use for the hippo to travel 505.46 centimeters. So, we multiplied the distance the hippo traveled with 7 rubber bands by 3, giving us 21 rubber bands. From this we found the hippo would travel about 469.5 centimeters. However, there would still be about 35.96 centimeters left for the rhino to travel to hit the ground, based on our averages we found from our data. So we decided to add another 22 centimeters, which was our average differences of the distance traveled using 7 rubber bands, to the estimated distance traveled using 21 rubber bands. This gave us a distance of 491.5 centimeters. we knew there would only be about 14 centimeters between 491.5 centimeters and 505.46 centimeters, so we decided to estimate that we would need 22 rubber bands. The work is shown in the picture below.

After we got all the rubber bands attached, we walked with the rhino to the stairwell, hoping that we made a good estimate and the hippo wouldn't "die" from hitting the floor. Here is the video of our first attempt. 

We were pretty happy with only the first attempt, and I think we were about 30 centimeters from the floor, which we pretty close to what we estimated. However, It is hard to see in the video, but when Sarah released the hippo, the rubber bands got stuck on the railing a little bit and altered how much the hippo was going to travel. So, we tried one more time, and here is the second attempt close up and from a distance. 

As you can see we got much closer on this try, I think the judges put us at 17 centimeters from the floor. So we predicted we would be about 14 centimeters from the floor and we were 17, which is very close to our prediction! Yay! We were happy that the way we decided to estimate worked so well. 

Overall we found that in doing an experiment like this, it is very important to try to be as accurate as possible. This will hopefully give you a more accurate estimation. Also, it is more difficult to bungee jump a hippo than you think! If we were to change anything about this experiment, It would have been nice to have a way to calculate the force the hippo had using a scale and calculating the force. This would have made it easier to estimate the number of rubber bands, and it would also probably be more accurate too. Another thought is, it might have been fun to have done about 5 trials or so of the actual bungee jumping. Then we could have found an average of about how close the hippo got. I just wonder if we could have gotten him any closer. I would have also liked to try adding one more rubber band just to see if he would have hit the floor or we could have gotten him extremely close. 

A Different Model For Khyyam's Cubic Solution

While working on Khyyam's Cubic solution in class, my table distributed each piece of the cube to a different person so we would all have something to work on. I was given the triangular pyramid, the piece that was 1/6 of the whole cube. I really struggled while trying to visualize the whole cube put together with three individual parts. I looked multiple times at the model at the front of the class, but I still couldn't make very much sense of it. I was extremely frustrated by the time that class was done, and I didn't get anywhere! It seemed like I had a different vision in my head as to how the pieces should go together. I decided to work on it for my daily work. I continued to try to understand the model that we did in class, but every time I tried to sketch the net of the piece I confused myself all over again. So, I decided to try something I knew I could visualize. I didn't know if it was going to work, but I thought I would try it. This is what I came up with:

This made more sense to me than the model we had in class, and it still had a triangular prism and a triangular pyramid, but it had a rectangular pyramid instead of a square pyramid. From here, as a continuation of my daily work, I decided to draw the nets of each piece and actually construct the figure and see if it would work.

I started with the net of the triangular prism (the one in the top left corner in the image above). I first started by drawing smaller nets that were to-scale of the actual figure I was going to create, I used the scale were 6cm = 1 4/10cm, and (72)^(1/2)cm = 2cm. I visually pulled the each triangular base flat first, and the I knew the square their bases were attached to was the 6cm x 6cm square face of the cube. I then visually unraveled the sides of the prism and laid it out flat. So my smaller scale version the net of the triangular prism looked like this:

 Similarly, I did the same thing to create the net of the rectangular pyramid. I knew that the base of the rectangular pyramid was the was the same dimensions as the rectangle in the net of the triangular prism because that was were the base of the rectangular pyramid would rest when I put the cube together. So, I visually just pulled down each of the sides to create the net. I knew that the smaller triangular side in the back was going to have the two 6cm sides on the outside because hypotenuse of the triangle was the measurement of the side of the base. It was a little tricky making this one, because I had to make sure that the lines were not only 6cm but also that they formed a 90 degree angle. the other two triangles on the smaller end of rectangular base I just laid flat, but then the last triangle was a little more tricky.I knew when I put the other 3 triangles together that the last triangle would be made with two of the hypotenuses and the last side of the rectangular base. So, I knew it was an equilateral triangle. This is the net of the rectangular pyramid that I came up with:

Lastly, I had to make the smallest and trickiest piece. I knew that the base of the triangular pyramid would lay on the equilateral triangle from the rectangular pyramid, so the base was an equilateral triangle with side lengths of (72)^(1/2)cm. It made it simple from there because I knew that each side of the base was going to be the hypotenuse of the triangles creating the sides. However, it was very tricky because again I had to make sure that were the 6cm sides met was a 90 degree angle. To make sure, I erected a perpendicular line from the midpoint of each side of the equilateral triangle. This created a line for the two 6cm side lengths to meet to ensure that they created a 90 degree angle. This is what I came up with for the net of the triangular pyramid:

Once I got the nets sketched that were to-scale, I made the bigger nets and folded each piece of the cube. Here are the 3 separate piece after I folded and taped them:

They all turned out fairly well and fit together! Here is the succession of adding each piece:

And the final product!

It kind of looks like the pieces didn't all fit perfectly in the picture, but that is because of my taping abilities. They really did fit snugly when I held them. 

Overall, from this, I felt like this reinforced the idea that there isn't only one way to do things. I think it was challenging trying to come up with a different way to make the pieces that still works and fits together. I am also very satisfied that I did do this because I became a lot less frustrated knowing that the way I was trying to construct it and was thinking about it in class really did work!

Creating a Tessellation

In class I was not having much luck creating a tessellation, so I decided to work on that for my weekly work this week. I went to the "Islamic Tessellations" link and chose to look at a YouTube video on "How to Make an Islamic Tessellation." I went through the video, and thought it might be easiest if I just do it on dot paper. However, it was a lot harder than I thought and I decided to switch to Geogebra shortly after, which make it a lot easier since I was using so many circles. 

I started out by making a central point with two lines perpendicular to each other, and where the lines intersected I made the center of my first circle (this is shown in my screenshot below).

I then began overlapping circles in the way shown in the video I watched. I went all way across the screen and up and down the screen.

I did this by making the center of the first circle a point on my next circle. This created the flower-like pattern in the center of the circle, which I really liked. It doesn't really show to continue to repeat the flower pattern in the video I watched, but I chose to continue you the flower pattern to kind of make it my own design. 

I then added lines that ran through the center of flower petals. these lines were at a diagonal, and were parallel to each other, and then I added another set of parallel lines that were perpendicular to the first set and also went through the center of petals. I also made a square with the polygon tool to enclose the flower inside of it.

After I titled with the smaller circles, I decided to overlay larger circles of different colors in the same pattern. I decided to have a little fun and made different color patterns as well, and it ended up looking pretty cool.

I added more circles and scrolled out a little so that I could add more lines to intersect the petals like I did before. I also messed with the colors again.

Lastly, I wanted to make polygons around my bigger flowers like I did with the smaller flowers. I also hid all the points when I finished just for looks.

I was struggling to come up with an idea for a tessellation in class, but watching the video on how to make tessellations really helped me to understand how to make the tessellations work in Geogebra. The video gave me an idea of how and where to start, and then I was able to come up with my own creation just by having a little direction. I am really happy with the way my tessellation ended up looking, and I am glad I had fun creating it, rather than getting frustrated. 

What is Math?

So what exactly is math? Often, math is seen as a subject that not many people have interest in. Its complexity can be overwhelming, and one bad impression with math can often turn people away. When I think about what math is, it is hard for me to give a formal definition because it a very broad concept. There are so many elements and components of math, so it can be taken in many different directions. It is also subject of interpretation of the person defining it because it is so broad. Some have much different ideas as far as what is math and what isn't math.

It is so hard for me to try to even think of a formal definition of what math is. However, if I was to have to give some kind of definition, I would say that math is the discovery of patterns of change that help us to explain things that happen in the world around us. It can use a range of levels of thinking, from logic and reasoning to simple calculations and calculations using algorithms. There are so many parts that I know I did not talk about, but we could be here forever discussing the amount of information that math encompasses. So, to me, math is a very broad topic containing many larger topics within it, and it focuses on the patterns among the topic of study. From this, we can explore the change in things all around us. 

The top milestones in the history of mathematics that I feel like effect me the most, based on the fact that I am going to be a teacher, is kind of hard to narrow down. There are so many important milestones in the history of mathematics that will effect me as a teacher, and have effected what I am going to eventually teach, but I think there are definitely some that I can say are going to make the explanation of a pattern much easier and things that will aide my students in their discoveries with math. So, if I was going to give a top 5, I would say they would be...

1. Algorithms: These are an important tool to teaching lower elementary and middle school math. It is obviously most important that when using the algorithms with students that they actually understand why and how the algorithm works, but once they do, they can make computing in mathematics that much easier and less time consuming. 

2. Modeling in Mathematics: I don't know if this tool is really considered an important milestone in the history of mathematics, but I definitely think that it is an important tool for understanding mathematics and that is why I mention it. Modeling in math gives a picture or imaging explanation of a problem. In teaching math, I cannot express how important it is for students to be able to do this! It helps with their reasoning and understanding so much, and benefits the teacher as well, by giving us a look at how they were thinking and reasoning with the problem. 

3. Pythagorean Theorem: This is important in my mind, because this is one of the discoveries that I am sure I will be teaching a lot, when it comes to geometry. This makes it easier for students to work with triangles while learning geometry. 

4. Euclid's Theorems: These are important, again, because they help students reason with shapes in geometry. It gives students this idea of reasoning with shapes through components that they have. It helps to see congruence in shapes, and gives a concrete reason why they are or are not congruent. 

5. Calculator: This seems like a silly one to put on, but calculators are such a useful tool to us while doing math. Sometimes they are overused for simple calculations, but they can be so helpful for making math more efficient, especially time wise. There is so much that we can do with them that is helpful to us, and therefore, is an important milestone in math history.