Practicing Teaching Analysis
During my clinical observations at a couple of different schools, I had the opportunity to watch a couple of teachers really making an intentional effort to help their students with learning vocabulary.
In a middle school class I observed, I watched as the teacher and students worked through a problem of factoring a quadratic and then finding the zeros. After finding them, however, he told the class, that these were called the "zeros" or they can also be called "solutions" or "roots."
There are so many times that we toss around different words for things, without even realizing it, but without letting the students know that we may be using more than one word to mean the same thing, and thus we lead them to be confused. Simply doing as this teacher did and instructing them right up front that these words can be used interchangeably I am sure helped them avoid a lot of confusion. (And in fact, I'm sure my deduction is correct as I hear that his students consistently are the high scorers on the end of level tests they are required to take--and one thing that tends to be difficult on those tests is when differently terminology is used from the stuff you are familiar with, even if you understand the concept.)
I also sat in on a geometry class. I sat in on a lesson introducing tangent lines to circles and was VERY impressed by what I saw. The idea was not introduced by displaying an abstract picture of a circle with a line tangent to it, or with a lecture. Instead, the teacher had a real picture demonstrating circles with lines tangent to them (I think it was a picture of something with pipes, if I remember right). Then someone came up and pointed out what they thought the tangent was. Then there was a discussion, along with more real life pictures demonstrating circles and tangents to them. I was actually kind of blown away by how good this was! This of course made it so much more real to students. After discussing the idea in terms of real life pictures, some abstract ones were brought in, but still with real life pictures that showed the same idea as the abstract drawings. I was impressed by what I saw, and all the different ways that the teacher had found where circles and their tangent lines were found in real life things we encounter. It made the lesson he was teaching seem so much more relevant and important, even to me, so I'm sure to the students! I loved the way this was taught. There are many skills in math that students need to learn in order to be successful in higher math classes, but which they may not see any immediate use for at all. Getting students thinking about things they have encountered already in their lives, that demonstrate this idea, before abstracting the concept was so helpful to both making it seem important, and to making the idea easy to grasp.

Critical Literacy

I came into this class with the understanding that "critical literacy" meant being able to "read and write the world." Through this literacy class and others I am taking simultaneously, however, I have become more aware of how students become literate, different practices to help them, achievement gaps in schools, etc. With all of these, my idea of critical literacy and what good teaching looks like have become more refined.
Good teaching now looks more like this: specific and intentional questioning, scaffolding, lack of assumptions, not simply giving information but facilitating the discovery of that information, not merely teaching facts but teaching thinking.
I would say my definition of Critical Literacy has expanded from simple being able to "read and write the world" to being able to read, write, listen to and understand, and speak about the world. And more specific to my classroom, being able to critically think about and question, and then use the tools discovered my class to do so.

Text Set o' Joy!!

Novels:

• Lund, Gerald N. (1983). The alliance. Salt Lake, UT: Deseret Book Company
  
  This book has always been one of my very favorites. It is a fiction book that takes place in the future, about 18 years after "World War III" has happened. This book is a relatively simple read, so it is accessible to those not reading at high grade levels, yet the characters are incredibly well developed and the plot is so intriguing that  those of high reading levels will also love it. The underlying moral issue being fought about in this book is the ability to choose how to act. This book doesn't have anything to do with mathematics but it is a book that encourages critical thinking, so I would strongly recommend this book to any student or teacher.

Trade Books:

• Devlin, Keith (2000). The Language of Mathematics. New York, NY: Holt Paperbacks.
  
  This was my first experience with a math trade book. I was surprised at just how fascinating this was. In this book, Devlin explains in a way interesting to both mathematicians and non-mathematicians much of the history and development of math we use today. I would love to use this in a class, but depending on the grade level I am teaching. This would be something great to use in an advanced level class in a high school. There might be some words unfamiliar to them, but with a bit of vocabulary discussion, this would be very accessible to that group. I have been thinking that it might be a good idea to assign this book to be read during the school year or semester, but set mile markers to be hit along the way. I would require them to read to a certain point by a certain time, and there would be something, maybe a small writing response quiz for each mile marker, to keep them accountable for the reading. When first assigning this, I would probably read some aloud to them, before having them read it on their own so they can see just how interesting and accessible the book is. Then throughout the semester, here and there I would want to read some more out loud to them, or have their peers do so, during class. This would then be good material for in class, small group discussions every so often. (I would contemplate using this in a different high school class also, but I would want to be careful and use something else if I have students whose reading levels are significantly below where they should be.)

• Livio, M. (2002). The golden ratio: The story of phi, the world's most astonishing number. New York, NY: Broadway Books.
  
  This book is another that I have yet to read. From the reviews I have read, however, I am really interested in it. It is a book that discusses, of course, the Golden Ratio. It is supposed to be engaging for both mathematician and non-mathematician alike. It discusses not only math, but goes through art, architecture, botany, biology, physics, and mathematics, so it should be a good cross-curricular book.

• Seife, Charles (2000). Zero: The biography of a dangerous idea. New York, NY: Penguin Books.
  
  I have not yet read this book, but I have heard time and time again good reports of this book. From what I have heard, this is very accessible, informative, and interesting. This is a book I would probably like to keep on my shelf for students to be able read. I might possibly consider having students read a selection from the book as part of their homework.

Additional Textbook:

• Burton, D.M. (2007). The history of mathematics (6 ed.). New York: McGraw-Hill.
  
  This is a textbook that I had for the History of Mathematics and Number Theory class I took from Jim Cangelosi and Utah State University. This textbook has so much that is so intriguing in it. I have not read the whole thing, but I have enjoyed much of what I have read, and furthermore it was recommended to our class, by Jim, as a book we should definitely have on our shelves. This will be an excellent resource for history on the math I will be teaching in my classes, to make the class more interesting.

Websites:

• Gillespie, M. J. (2011). Aggiemail DUMMY https://sites.google.com/a/aggiemail.usu.edu/dummy/home website.
  
  This is my own website that I have been putting together. (I probably will want to figure out how to change the name from "Dummy" before using it in my classroom...) This website has a collection of explanations of a few interesting problems in Probability Theory, as well as a few links to further information/applets/videos on the subject. I would probably put a link of this to my class website or blog so students could go to it for something interesting. I could also forsee showing this to my students, and having them work together in groups to find an interesting problem in math/probability and then work together to come up with an explanation of the problem and solution, such that it could be an addition to the website. This could be fun and also help them learn how to explain math ideas in a way understandable to others.

• Isaacson, Michael (2012). USUMATH4500.weebly.com. Utah State University Math Department. Website.
  
  This is a website that was created for the Math 4500 class I am in. On this website we are able to upload lesson plans and such that we have done, for the rest of the class to be able to have. This text has many well written lesson plans that both I and future colleagues may find useful in our classrooms.

• Bogomolny, A. (2012). Interactive mathematics miscellany and puzzles.   http://cut-the-knot.org/ (accessed April 2012).
  
  This website is awesome! I have used it several times trying to find explanations of interesting math problems. It has explanations and applets and all kinds of good stuff. It has math games and puzzles, explanations of a large range of mathematical things from geometry to algebra to calculus, probability, and more. I would use this website probably both in finding intersting things to present to my students and also possibly as a resource for them to use should I assign them to present something.

Simulations/Applets: 

• Chang, K., Graham, S., Koen, V., Lindsay, M./The New York Times (8 April, 2008). Interactive feature: The Monty Hall problem. http://www.nytimes.com/2008/04/08/science/08monty.html (retrieved 10 April 2012).
  
  This simulation of the Monty-Hall Problem is really fun! I love this problem and will definitely present this problem in my class, and then hopefully have a class activity, where we do the game in class, and figure out the reason for the results. This online simulation would be good to give to them if we don't end up having time to do it in class, I could assign it as a way fun homework assignment, or as for those who might miss the in-class activity.

Children's Books:

• Ellis, Julie (2010). Pythagorus and the ratios: A math adventure. Watertown, MA: Charlesbridge Pub Inc.
  
  This is a book I have not yet read, but I have read another by this author and just loved it! This would be something fun to have to help explain ratios in a non-threatening way. Probably better for younger kids, but honestly, I think probably high school students could have fun with it too.

• Ellis, Julie (2004). What's your angle, Pythagorus? A math adventure. Watertown, MA: Charlesbridge Pub Inc.
  
  I love this book! I read it to one of my roommates as a bed time story! This just fascinated me because it teaches the Pythagorean Theorem in a very understandable way. Though it might be more difficult for them to read, I think it could be understood by even third graders. I think probably junior high school is where I would be able to get the most use out of this. I'd probably just have students sit down on the ground (how didn't love story time like that in the younger grades?!) and read it to them. Then we'd do the things in the book that the young Pythagoras did. 

• Neuschwander, C. (1999). Sir Cumference and the dragon of Pi (a math adventure). Watertown, MA: Charlesbridge Pub Inc.
  
  I have not yet read this one, but it was recommended with the books of Julie Ellis, which I was already very impressed with. This book I think would be great for teaching in a junior high school, which is where I would love to teach for at least some amount of time. Sometimes students find math in general to be an unapproachable concept. But kids can do knights and dragons!!
• Neuschwander, C. (1997). Sir Cumference and the first round table (a math adventure). Watertown, MA: Charlesbridge Pub Inc.
  
  Similar to what I have said about the other children's books, I would use these as a way of helping make math less scary and daunting, and make it seem more understandable and accessible to my students who will be at various levels of ability and understanding.

• Neuschwander, C. (2001). Sir Cumference and the great knight of Angleland (a math adventure). Watertown, MA: Charlesbridge Pub Inc.
  
  As said with the others, I would use this as a tool to help teach math to students who have a hard time grasping the abstract ideas thrown to them with math.

Articles:

• Stripp, A. (9 September 1999). How the enigma works http://www.pbs.org/wgbh/nova/military/how-enigma-works.html (retrieved April 2012). Article.
  
  This is an article describing how the Enigma (a ciphering/deciphering machine used by the Germans during WWII) works. I include this text because cryptography is done using prime numbers, and this is a real world example of how prime numbers are important in the "real world," a disputing a claim so many students make that the math they are learning bears no importance to their real lives. I also like this because the Enigma is something that can be discussed tying mathematics and history together. *I would love to use this in conjunction with the Nova video of cracking the Enigma, if I can find it.

Videos:

• Devlin, Keith (2008). Authors@Google: Keith Devlin retrieved from http://www.youtube.com/watch?v=3pRM4v0O29o.
  
  I love this video. It is just over an hour long and is really good. It discusses the origin of probability theory, discussing the problem of points. It is really interesting as he discusses how this is something that we now take for granted, but was a hard idea for people to accept at first. With probability we are able to essentially see into the future. This is a really interesting video and I would probably use this as something for my students to watch on a day when I am not there and have a substitute. There is a lot they can learn from this, so the time won't have to be wasted, but I also won't have to worry about the math ability of the substitute. 
• K50aker, (12 Aug 2007). "Abbott and Costello 13 X 7 = 28." Online Video Clip, http://www.youtube.com/watch?v=rLprXHbn19I (retrieved April 2012).
  
  This is a funny clip that would be great to use as a start to a discussion on mathematical reasoning, and what is sound and what is not. In the clip, one of the men explains in several ways that seem reasonable why 13 times 7 is 28. There is, of course, a definite flaw in each one, but it seems so reasonable, that it is really funny, and I think could be a great conversation starter.
• Mandlebrot, B., (February 2010). Benoit Mandelbrot: Fractals and the art of roughness http://www.ted.com/talks/lang/en/benoit_mandelbrot_fractals_the_art_of_roughness.html (retrieved 9 April, 2010).
  
  This is a Ted Talk where an elderly gentleman, Benoit Mandlebrot, discusses roughness, and fractals. He begins by showing a closeup picture of califlower, a vegetable we see all the time, and shows how this is a demonstration of a fractal. This is interesting as he uses every day things, and shows us mathematics in them. Then shows the marvelous and incredibly complicated and beautful shapes that can come out of simple simple formulas--"Bottomless wonders spring from simple rules...repeated without end." This is something I would probably not formally use in my class, but would have it as something extra that could be shown to students who might be curious.

• Myer, D., (March 2010). Dan Meyer: Math class needs a makeover. http://www.ted.com/talks/lang/en/dan_meyer_math_curriculum_makeover.html (retrieved 9 April 2012).
  
  This is a Ted Talk where a high school teacher is talking about how math classes here in the US really need  a "makeover." He discusses ways to teach students math such that "The math serves the conversation, the conversation doesn't serve the math." He also mentions a quote from Einstein, where Einstein says that the formulation of the problem may be more important than the solution. Yet despite thus we focus more on solution here. This text I would use as something to help myself and other teachers keep in mind how to teach effectively.

Pictures: 

• Demotivated Pictures (2 April 2011). "Division by zero: It just happened" http://www.demotivationalinc.com/photo/view/1030 (retrieved April 2012). Picture.
  
  This I would love to have a poster of in my classroom. This is just a funny reminder that trying to divide by zero is invalid. 

• Harris, S., (2012). Science cartoons plus - S. Harris math cartoons http://www.sciencecartoonsplus.com/gallery/math/index.php (retrieved April 2012).
  
  There are several very funny cartoons on this site. These could be fun to occasionally put at the bottom of assignments or agendas. When we give students things to laugh at, the math will be more enjoyable.

• Imgur, (2012). "Sin(b)/tan(b)=" http://imgur.com/gallery/30umX (retrieved April 2012). Picture.
  
  This is a really funny picture that would also be fun to hang in my classroom. It would be nice to help students learn about how tangent is sine over cosine. This poster is funny, and they will undoubtedly want to know why it works, or it won't be funny at all. When they understand, they won't forget.

• Imgur, (2012). "Beautiful dance moves" http://imgur.com/gallery/tEfBW (retrieved April 2012). Picture.
  
  This is a fun way to help students learn and remember what different graphs look like. I'd love to keep a poster of this hanging in my room, also.

Songs:

• Eddington, E., Gillespie, B., Muir, A., Olsen, A., (2011). That's how you know - Pythagorized. Roosevelt, UT: Union High School, Mr. Busenbark's class. Song.
  
  This is a funny video done by my little sister and her friends for their high school math class. This is a play off the song "That's How You Know" from the movie Enchanted. In this song they sing about Pythagoras and the Pythagorean Theorem. I would love to use this in my class as a fun tool to teach about the Pythagorean Theorem. It might be fun to have this be a demonstration, and have my students do a similar assignment to this, where they write a song about some mathematician and his contribution to mathematics. There is one minor error in this song that might also provide a start for a discussion of what the error was, and what it should have said, to make it correct.

• Fine, S. (producer) (2006). Twin prime conjecture. http://www.pbs.org/wgbh/nova/physics/twin-prime.html. PBS. Song.
  
  This is a song about the Twin Prime Conjecture. The clip is only about 3 minutes long, but I found it quite entertaining. The conjecture is that pairs of primes will appear into infinity. (Meaning that an infinite amount of twin primes (two primes right next to each other, separated by one even number inbetween) exist.) I could see myself using this to get a discussion started about the twin prime conjecture (or when wrapping up a discussion/discovery lesson, involving primes).
• WSHSmath (13 March 2012). "All I do is solve (WSHS math rap song)"  http://www.youtube.com/watch?v=1qHTmxlaZWQ&feature=relmfu  (retreived April 2012).
  
  This is a funny, yet instructional video created by high school teachers, and includes many students. This is a fun way to show that math is cool. The songs are also very catchy and could be used to help students learn and remember how to solve systems of equations.

• WSHSmath (13 May 2011). "Do the quad solve (WSHS math rap song)"  http://www.youtube.com/watch?v=jGJrH49Z2ZA (retreived April 2012).
  
  This is a funny, yet instructional video created by high school teachers, and includes many students. This is a fun way to show that math is cool. The songs are also very catchy and could be used to help students learn and remember how to solve quadratics.

• WSHSmath (31 January 2011). "Gettin' Triggy Wit It (WSHS math rap song)"   http://www.youtube.com/watch?v=t2uPYYLH4Zo&feature=relmfu  (retreived April 2012).
  
  This is a funny, yet instructional video created by high school teachers, and includes many students. This is a fun way to show that math is cool. The songs are also very catchy and could be used to help students learn and remember things in trig.
• WSHSmath (31 October 2011). "Super base (WSHS math rap song)"   http://www.youtube.com/watch?v=QIZTruxt2rQ  (retreived April 2012).
  
  This is a funny, yet instructional video created by high school teachers, and includes many students. This is a fun way to show that math is cool. The songs are also very catchy and could be used to help students learn and remember how to deal with exponents on numbers and variables, and when they can be added or multiplied or subtracted.

• WSHSmath (8 November 2010). "Teach me how to factor (WSHS math rap song)"  http://www.youtube.com/watch?v=OFSrINhfNsQ&feature=relmfu  (retreived April 2012).
  
  This is a funny, yet instructional video created by high school teachers, and includes many students. This is a fun way to show that math is cool. The songs are also very catchy and could be used to help students learn and remember how to factor.

Email... google... texting... writing papers... facebook... research...these are things I do on a regular basis and they all involve digital literacies.It is really surprising to take a step back an see how much time I spend on my computer or phone. Even just today, a day that I really haven't spend that much time on my computer, I have been on her thus far probably about two hours. This morning I got online to print off some notes for a test today...and while I was at it I checked Canvas for a message from one of my students, and also distracted myself looking at other emails. Then later today I got online to actually work on this blogpost...but got sidetracked into researching how to manage hair in places that are humid...and that lead to researching the best ways to take care of and style curly hair. Getting on here yet again still later today, I got distracted with facebook, looking up my best friend from my K-3 school years. Now I am writing this blogpost.
Then a suprising amount of time is spent texting people...even though I don't really consider myself to be a super big texter I have still sent 17 text messages just today.
I never would have thought, growing up, that I would ever spend this much time using my phone or my computer. My laptop was out of commission for a little while not too long ago, and boy did I learn how much I really rely on my computer!! Luckily there is a USU computer lab that I can get to within about five minutes of leaving my apartment.
In an age where students are growing up using, at least to some extent, phones and computers and other technological things almost as soon as they can talk, being familiar with technology and incorporating it into my classroom will be to use something right up their alley. They grow up playing computer games, so why not show them some computer games that involve math? They use computers to look things up and learn about them, so why not also use a computer to help them learn math? There are so many online tools that I am becoming aware of and learning how to use that will be so beneficial in terms of teaching. Some of these are GeoGebra, Geometer's Sketchpad, Wolfram Alpha, heck--I, myself, have even made a website dedicated to interesting problems in probabiliity! In all honesty, I don't know that I'll be able to incorporate as much technology into my teaching as other teachers might, just because I am less comfortable with it, but the more I am learning, the more I become comfortable with, and the broader the spectrum of technologies to use in my teaching.