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