Checking for answers, or solutions revisited

We live in a world where information and data is becoming more and more easily available. Before I had mentioned time and time again how for students we should always be checking for solutions and not answers. I think this genuinely extends to problems with cheating and academic integrity as well. It's so easy for students to take pictures of their homework and send it to friends to copy off of, look things up in the middle of the test by going to the bathroom, and so on. We live in an age where the boundary between collaboration and cheating has become very thin. In the future we will most likely find ways to easily deal with problems like these but as of right now we are at a crossroads. Do we have students completely separate themselves from the world when taking tests? That doesn't seem fair seeing as we are always trying to encourage collaboration and team work. But meanwhile, having students discuss problems and leave the potential for cheating available is also not the right way to do things. I think this is especially because our society puts emphasis on grades and not understanding. Students cheat because they don't want to fail and they don't realize that by cheating they are not understanding and by not understanding they are setting themselves up for even bigger problems down the road. I believe that in our classrooms we need to be thinking about these questions and find a balance between what constitutes cheating and what is collaboration.

Times's Article on the State of Our Generation

     Times recently published an article titled "The Me Me Me Generation" talking all about how the Millennial Generation is lazy, narcissistic, and unfocused due to our extreme attachment to technology and our self-serving, ego-boosting ways. While reading the article I can honestly say that I felt more than a little angry at the author of the piece. In the piece they use many blanket statements like "the Millennial Generation" and "they all" which are, first and foremost, never an effective way of debating a point but also felt like a personal attack. Stepping back from all this though, I decided to not judge the article based on anything other than it's statistics and to look at the piece from a purely scientific view, but still the piece doesn't make a whole lot of sense. For one, the piece tries to argue that our technological and always-on addictions are distracting us and making us less social and more self-centered. What the article fails to mention however is how exactly we are "addicted" to technology. I can honestly say that I have been in a situation where I am hanging out with a group of friends and everyone is on their phones or laptops. From the outside it may seem shallow and self-centered, but if you were to look at any of their screens you would see that many of them are reading news articles, looking up data, reading research papers, exploring Wikipedia, and more. Our generation may be screen locked, but it is not only an excuse to become more self-centered, but actually quite the opposite. The Millennial Generation is on a constant adventure to learn more, to be exposed to alternative viewpoints, and to try and be aware of what's going on around them. Again, from the outside it may look like we are in our own little world, but on the inside we are trying to place ourselves in our new global society.
     And yes, the Millennial Generation are not lining up to move out of our parents house, take on full time office jobs, and work strict 9 to 5 hour work weeks. But this is not because we don't want to, it's because we realize that we don't have to. Why should I be forced to drive 2 hours a day in traffic, to go to an office building where my main job is to enter data into spreadsheets? Why not instead fax me the data and I'll enter it at home? It's bad enough that we are forced to enter a work field that has few jobs to offer us only to be stuck doing something that doesn't fulfill or challenge us. We can use that time instead to spend time with our family, friends, and loved ones. This may sound narcissistic yes, but I believe that we are truly moving from an age of structured walls and cubicles to an age where the majority of workers will be working from home.
     But how does this relate to education? Too often I feel that students are wrongfully and harsh fully reprimanded for having cell phones out in the hallways and lap tops out during recess. We tell them to put their phones away and tell them, "Your friends can wait until after school." But what if the student was actually trying to figure out a question that had been nagging them for a while? If every moment can be turned into a teaching moment, then why not this as well? Instead of jumping to conclusions and seeing the slouched over student on his phone as a nuisance, instead ask him what he's doing? Does he have any questions? Asking these questions can lead to a great teaching moment.

Technology Integration Plan

The lesson plan I was working from can be found here.

My educational matrix can be found here.

            I chose to work with this lesson plan over other lesson plans I had worked with in the past because I find that trigonometry is often a topic that students can be easily lost in and that teachers often try too hard to integrate technology into. The lesson plan is very basic and starts under the assumption that students are already aware of certain topics. They are then quickly split into groups, told to complete a puzzle, discuss the completion of the puzzle as a group, and then moving on and completing two worksheets.

            And so for my technological adaptations, I chose three very simple adaptations: the addition of relevant GIFs and videos, phone applications, and an online discussion forum assignment. These technological adaptations would not only integrate themselves easily into the lesson plan, but also be transformative and make the learning targets and curriculum standards more easily achievable.

            The first part of the lesson, the trigonometry square puzzle, is trying to accomplish two goals: understanding that side ratios in triangles are properties of the angles in the triangle (CCSS.Math.Content.HSG-SRT.C.6) and using the relationship between the sine and cosine of these angles (CCSS.Math.Content.HSG-SRT.C.7) to complete the puzzle assignment. These standards can be achieved with the worksheet, but by integrating the use of relevant GIFs and videos, like this and this, we can make the very abstract concept of the relationship between sine, cosine, triangles, and the unit circle more concrete. Students will then have a deeper understanding of the properties of the sine and cosine graph (NETS-S1), an understanding of how the functions are derived (NETS-S1), and the relationship between the sine and cosine of a triangle (NETS-S1). Thus, when completing the puzzle they will have a visual breakdown to refer to when completing the assignment and a more secure grasp on the educational standards for this part of the lesson.

            The second half of the lesson also relies on worksheets, the Angle of Elevation/Declination worksheet. The goal of these worksheets is to have students use trigonometric ratios to solve right triangles (CCSS.Math.Content.HSG-SRT.C.8), prove the Law of Sines and Cosines (CCSS.Math.Content.HSG-SRT.D.10), and apply the Law of Sines and Cosines (CCSS.Math.Content.HSG-SRT.D.11) to complete the worksheets. Again, these goals are attainable with the use of a worksheet and some excellent teaching but with the addition of some simple technology we can work to increase student understanding. The first thing that we can add is a quick field trip outside and the use of some phone apps like this one. By using this phone app we can have students measure the angle between the ground and the tops of objects outside like trees, the height of certain school buildings, etc as well as the distance from where they are standing to the objects base. Using this information we can then have students use the Law of Sines and Cosines to compute the other measurements of the triangle, record this information, and share it with the class. This way, instead of filling out a worksheet on a topic that requires some abstract thinking, we can have a plethora of real world examples of ways in which the Law of Sines and Cosines can be applied that students can think concretely about and refer back to.

            Furthermore, by asking students to go home and record the data of an object they find around the house and posing a question using this data on an online discussion forum we can increase student understanding of, again, how the Law of Sines and Cosines can be applied and also how questions using the Law of Sines and Cosines are structured. Students will also be asked to answer at least one of their classmates unanswered questions so as to get more practice and to better form a classroom community. Thus by integrating the technology into this half of the lesson we have shown students easy ways to use technology to gather data (NETS-S3), apply the data they collected to answer problems (NETS-S3), and use critical thinking skills to ask, answer, and critique authentic questions (NETS-S4) while also reinforcing the core standards of the lesson and the learning target.

            As I've stated again and again, these core standards are attainable with just the use of the provided worksheets. Likewise, technology is not always going to be a transformative addition to a lesson. However, by using the technologies detailed above, in conjunction with the strategies provided, we can enhance the overall lesson and provide students with a deeper, clearer, and more confident understanding of the material.

My Research and Technology

Over the past year I have been doing math research with a professor on campus and a research partner. Over this time I have been slowly adjusting while I transition from math student to math researcher. It is one thing to be able to learn mathematical rigor and constructs, but it's another thing to just start with nothing and work your way up. My professor understands this and is doing his best to make this transition easy. He directs me to mathematical research websites like arXiv, a website run by Cornell that lets users upload research papers for others to work from and review, so I can see how research papers are structured and to orient me in my research. Likewise, I browse websites looking for answers and similar results. Websites like  The On-Line Encyclopedia of Integer Sequences make it very easy to plug in a few numbers and see every series that share numbers with your input. Websites like these make it very easy to quickly see connections and share information. But I am finding that this has problems as well. With publication being as close as a click away, false results are sometimes uploaded and referenced in multiple research papers whereas in the past, results had to be carefully peer reviewed before being published. And while this was slower and thus led to a "stunted" growth in the mathematical field, it can also be said that this led to more rigorous results.

As a future teacher, I need to consider these things. Is it more important for my students to get answers? Or is it more important for my students to get the concepts of mathematical rigor? The obvious answer is the second one but it is the sad truth that most schools orient themselves more towards the first. With answers so easily available online and with, what seems like, no answer unasked, should I only be checking for correct answers, correct work, or should I even assign written homework at all? Should I instead assign students oral exams and grade them on their thought process? This is my problem with Scantrons. Scantrons may be easier and quicker but they also invoke a binary system of right and wrong that I don't want to subject my future students to. I want them to be able to try things and to see their thought process. A blank Scantron bubble is wrong, but there could be so much thought and work behind it that is important to consider.

A Great Example of Technology

The other day in my READ 411, Language and Literacy, class, a group presented a unit on confidence in the classroom and showed how technology can be both helpful and detrimental to learning. First, one of the students used a projector to display some information on the board and use it as a makeshift "smart board" as our classroom was not equipped with one. She went through similar fractions, giving examples and displaying pictures but mostly just lecturing. After she was finished she asked us if we felt confident with the material. Most of us agreed, blindly, and shrugged our shoulders. After that was finished another student then presented a parallel lesson, where she taught us the same material but in a different way. She used lots of hands on material, like cut out fraction pieces and color by number sheets, and then had us access an online application relevant to the lesson. The whole group, using one or two laptops, then took turns playing the game and everyone had a lot of fun. By the end of the second lesson everyone said they felt comfortable with the material and everyone had a smile on their face too. This just goes to show you that even fun kid-oriented games can be fun and instructional. Just because students are older doesn't mean that fun apps don't hold some validity to their learning. Likewise, by using one-sided technologies, like the smart board, you prevent students from utilizing and interacting with the technology and distance them from the learning process.

Collaboration Canvas

When I found out that I had to make a remixed canvas I knew that I didn't want to chose the obvious choice of physics. So instead I chose music because, as much as people know that math and music are related, I feel like it is often something that is overlooked and could help students make a connection between what they are doing in the classroom and what they are doing in real life. First, though, I had to find a music lesson that I knew I could incorporate math into and make it age appropriate as well. There was no point in talking about subgroups of note sets if the subject of sets is barely even covered until linear algebra. That is when I found the topics of triads. By connecting musical triads to mathematical triads I can help students explore the Pythagorean theorem, and also give them an idea on how it relates to the real world and the music they listen to. By extending this to the topic of musical structure I can help them "hear" conditional probability. Coupling this with a video and some outside sources I hope show the relationship between math and music.

Linked here is the original canvas
Linked here is my remixed canvas

Unit Plan Canvas

In designing a lesson plan for areas, volumes, and surface areas, I knew that I had plenty of material to work with. For some reason, teachers find this topic incredibly appealing, most likely because it is a very concrete, and visual concept, and it shows that math can be practical (formulas, relationships, ratios, etc.) and fun (boxes, blocks, and building!). But, as much as teachers love teaching this concept, I find that often they focus more on the fun and rational and less on the relationships. And so for my lesson plan I really wanted to focus on the relationship between areas, volumes, and surface areas. To do this I specifically chose videos, links, photos, and applets that showed how volumes can be thought of as an expansion of a flat area through space. This, then, showed that by taking the area of a base, we can use this to get a formula for the volume of a 3D shape. And to get students thinking about this I prompted them with questions about "stacking areas" and what they can get by doing this. Likewise, I included links to applets, explanations, and a video showing how 3D shapes can be broken down into 2D shapes whose areas can be solved to get surface area. By doing this I can get students getting a deeper understanding of the relationship between areas, volumes, and surface areas and have them think in a more abstract and 3D way.
Linked here is my lesson plan for reference.

Technology & Pedagogy

            For my lesson plan analysis I chose a lesson plan titled "Fill 'Er Up" that involved teaching students about volumes of rectangular prisms for a few reasons. First and foremost I chose this particular lesson plan because I believe that the core standard it focuses on is what a lot of people say is wrong with math education nowadays. That is, that we are educating students to pass tests and not giving them the skills outlined by CCSS.Math.Practice.MP4, which states that we must teach students to be able to model everyday problems with mathematics and CCSS.Math.Practice.MP2, which focuses on the relationship between abstract mathematics and quantitative mathematics. On top of this, I also chose this particular lesson plan because its use of technology, which I felt was forced and only detracted from the goals of the teacher.

            The goal of the lesson plan was to teach students about a fundamental skill that pertains to their everyday life, while also teaching them about the ideas of volume, space, and quantity, but to do so by lecturing and having them use a program that worked in abstract quantities. By doing so the plan fails to meet the standard outlined above because the students are unable to, "interpret their mathematical results in the context of the situation and reflect on whether the results make sense" (Standards for Mathematical Practice). One way to improve on this would be to start of the lesson with a student centered Q and A session as opposed to a lecture and teacher centered Q and A session. By allowing students to ask questions that they may have about definitions of height, length, depth, and volume and allowing them to define the concepts themselves the lesson become more aligned with the core standard as well as an additional standard CCSS.Math.Practice.MP6 which focuses on meanings of units and clear definitions of concepts.

            This standard is further aligned with the plan with the removal of the technology it recommends, which seems unnecessary, and actually detracts from the lesson plan. The applet recommended for use, serves little purpose as it is something that is easily modeled with real world examples, is time consuming, as even I was confused as to how the applet worked and what it was trying to achieve, and deals exclusively with abstract units. By removing the use of the applet, and instead using a shoebox with number cubes, students can use the definitions they defined at the beginning of class to measure the length, width, and height of the shoebox, as well as the real world measurements of the number cubes, to get a better understanding of what exactly volume is and how it is measured and recorded and further align the lesson plane with all standards detailed above.

URL: https://docs.google.com/spreadsheet/ccc?key=0ArLDPnTD1B3ldG9NQWZ6UnBXdjY0UXJvREhtUDNXZVE&usp=sharing 

References:

"Standards for Mathematical Practice." Common Core State Standards Initiative: Preparing America's Students for College and Career. Common Core State Standards Initiative, 2012. Web. 20 Oct. 2013. <http://www.corestandards.org/Math/Practice>.

My Pre-Calculus Class or The Not So Smart, Smart Board

            When I was a junior in high school I had a pre-calculus class in the "new" wing of the school. In all actuality the wing was already several years old, but, as it was the newest wing, it was called the new wing. The new wing had everything you could ask for, nice clean bathrooms, the coldest water fountains, air conditioning, and, of course, the newest technologies. My pre-calculus teacher was one of my favorite teachers and she is one of the biggest reasons that I discovered that I loved math. She was strict, but she was also passionate, intelligent, and knew exactly how to explain something to you to make you understand it.

            For example, everyday she would give us mini-quizzes to help us get better at easily recognizing trigonometric functions so that when we took tests we were less focused on what sine of 30 degrees was and more focused on what the law of sine's actually was. And often, she would have us come up to the board and explain proofs to the class so that we could better understand the concepts of the theorem. But one day, our teacher was not there, and was instead at a seminar on how to use the current technologies in the classroom. And when she came back, she was all ready to start using smart boards, tablets, etc. in the classroom and, seeing as the school put an emphasis on these technologies, she used them.

            So instead of our mini-quizzes, she would spend the first 5 minutes of the class booting up the tablet software, and syncing it to the projector. And instead of having us come up to the board we instead sat at our desks and passed the tablet around, using this to spell out steps on the smart board. If someone hit a wrong button, we would have to spend the next 2 or 3 minutes trying to figure out what went wrong, and ultimately it was frustrating for everyone involved.

            You see, my teacher did not need the technologies that the school was trying to make her implement in the classroom. All she needed was a few extra quizzes, a whiteboard, and some class participation. I think as teachers it is important for us to constantly use these experiences to make sure that we are critically analyzing not only the what but the why when implementing technologies in the classroom. We need to be asking ourselves not only, "what am I going to use?", but "why am I going to use this?". If there is no answer, it may be better simply to not utilize the technology at the present moment.

Computer Algebra Systems and Wolfram Alhpa

    What once started as a simple tape machine adding machine, has transformed time and time again from simple digital calculator, to advanced graphing calculator, and now easily accessible computer algebra systems. This advancement has allowed math students the world over to find the hints, tips, and solutions they need in one convenient place. Wolfram Alpha, for example, can, in under 10 seconds, take an equation and derive, integrate, graph, transform, and expand an equation for you. A system like this has many advantages for students.

    For one, Wolfram Alpha does provide students with answers, but it also provides a step by step for students to look at. By doing  this students can check to see if the answers they got were wrong and then see exactly what part of the problem they got wrong. This provides students with a personal source they can reference and takes the onus off the teacher to meet the needs of each individual student independently, but at the same time.

    Likewise, Wolfram Alpha has a multitude of math related articles, definitions, and how to's any student may need. This way, even if students get confused because they forget what an integral sign means, they can quickly and easily look up the definition and guide on integrals that Wolfram Alpha provides. They can then look at a couple of practice problems and synthesize the material they read to truly understand the subject at hand.

    Of course, like all pieces of technology, there are some cons to always having Wolfram Alpha available. Students may use Wolfram Alpha to simply look up answers and not attempt to look into how the system got the answers or what is going on.

    On the other hand, using Wolfram Alpha, or Mathematica, Wolfram Alpha's downloadable computer algebra system, you can more easily show students how functions look, how they operate, and compare with each other simply by typing in the equation. Thus, the process of describing how 3 dimensional functions look and operate become much easier than just drawing a doodle on a paper. Students can look at, manipulate, and rotate surfaces to see what is really going on.

    Computer Algebra Systems are a fantastic way to help students learn and visualize complex formulas in 2 and 3 dimensions, but it is important to keep in mind that the program is designed to help teach, not simply to show. Students learn by understanding, not by being given the answer, and so Wolfram Alpha, or any computer Algebra System, is a great addition to any class. 

History of Technology in Mathematics

Calculators make it easier for students to think about and solve problems in multiple dimensions

Finding a Happy Medium

          Using technology too often has the same effects as eating too much cake, it can make you tired, lazy, lethargic, sick, etc. And that is why technology, like any good thing, must always be used in moderation.

          Facebook can be a fantastic way to get students to connect and easily complete group assignments. On top of that, its group feature can be a good way to start up a class page where students can easily discuss questions on assignments and post pictures of notes for classmates that may have been absent that day. Chat and online functions can also allow students to easily see who is online at any given point and have a one on one chat session on specific questions they may have. On the other hand, Facebook is also open to many other functions that may distract students and prevent them from completing work on time and may aid students who wish to cheat of each other’s homework or on tests and quizzes.

          Tumblr is a great site for quickly and easily sharing images. In an art class you can set up a Tumblr and use it to link students to artists that are relevant to the current class. Students can, then, compare and contrast styles and also have a reference at any time. Students can also make their own Tumblrs to get their work into the public space and keep track of artists they like through their likes and favorites. On the other hand, with Tumblr’s endless scroll functionality, students can waste hours and hours scrolling through picture after picture with no end.

          Twitter and a class blog work in similar ways. Using Twitter students can get reminders, via text message, about homework problems, tests and quizzes, and important class updates through twitter. While a class blog will always give them a reference in case anything was forgotten. But Twitter is also distracting and with its endless scroll functionality students can also scroll through tweet after tweet for hours on end. Also, students may no longer see the need in writing down homework in class and if there are any problems with the service than they may simply not do the homework since they do not know what was assigned.

          Youtube can be used to show instructional videos, upload demonstrations, and give step by step instructions to students who may not have paid attention in class. And through Youtube’s comment section students can post specific questions that can be addressed in another video. But this too can be easily distracting and on top of that students may use the videos as an excuse to not pay attention in class.

          These are just a few examples of websites, but the same methods can be applied to phones, cameras, projectors, etc and it always boils down to the fact that technology can help us, but it can also hurt us. We cannot be afraid of using technology, but we also should not be afraid to not use it. Sometimes an extravagant website may help, and sometimes a white board and a marker may get the same job done. It is our responsibility as teachers to determine what technology is truly necessary and to what degree we should apply it.

Technology Autobiography

                For me, the three most influential technologies of this day and age are the internet, phones, both mobile and household, and television. The reason these technologies are so influential is because, first and foremost, they all promote the spread and growth of information and knowledge. This reason alone is why I rank the internet as the most influential. The internet allows for peoples across the world to share and relate information quickly and at any hour. Because of this, it is possible for ideas to be worked on day after day without the need for breaks. Important problems can be worked on for 8 hours, their results emailed across the world, and then picked up and worked on by someone across the world. In mathematics, my field of study, new research is posted daily on forums and archives, readily available for any mathematician to look at expand upon and there are databases one can use to search for any sequences of numbers one might come across. In these ways the internet has played a fundamental part in the shaping of our world and the information we have available to us.

            In this same way, phones have perpetuated this constant growth of information. Through emailing, text messaging, and wireless browsing, your phone is a constant medium through which you can access information. If you ever need something explained to you, a colleague is only a phone call away; if you ever forgot an assignment at home, someone may be available to email it to your phone; or if you forgot how to spell a word, your phone may have a dictionary application that you can use to look it up. On top of that, with the recent addition of camera phones, front facing cameras, and an open source marketplace for applications, it is possible for anyone to do anything they put their minds too.

            Finally, television also plays an important role in the dissemination of information in our world. News companies, MTV, local broadcasting, etc. all have a means to convey information to the public in a quick and easy way. Using audio as well a visual means to convey this information puts television ahead of radio, but slightly behind the internet. Nonetheless, television and the internet often work together to bring the public information quickly and easily. It brings information of far away countries as well as local events into our homes and makes us constantly aware of the world around us.

            Watching the video "Learning to Change, Changing to Learn", was a lot like watching a video of myself and my friends testimonials about technology. The points brought up by the students were all points that I would make and have heard made in the past. Specifically the way in which homemade websites, or blogs, can help define you as a person, how the knowledge of technology helps you think about problems in a different way, and how technology has become such an integral part of our identities that without it we may not have anything to define ourselves by. In this way, technology can be a bad thing. Without a constant outlet to communicate by, we may feel lost or unfulfilled, and sometimes if we are unhappy with the attention we are getting online we may feel sad. These things however are just a side effect of our world adapting to technology. If you think about it 20 years ago the internet was barely even an idea and now it is something that we take for granted. With new technologies and new information we need time to adapt, to bring it into our society and understand it, and our own, place in the world.

            However, technology, for better or for worse, is a part of our society and so it is always important to look on how drastically it has helped our society. Parents are now able to work from home and share more time with their children, cures for diseases are worked out every day, while clusters of computer processors work out mathematics problems we could never possibly count. Sure, we may all use technology to varying degrees, with varying intensities, and with varying intents, as explained by the video, but technology has changed and shaped our world for the better and by using it, I believe it can truly help the way we think.