GeoGebra

           I have never heard of GeoGebra until my co-worker told me about it. GeoGebra is an interactive geometry, algebra, and calculus application. Just like GSP (Geometry SketchPad), you can construct geometric figures, and uses the software to calculate the angle measures. Besides that, you can also input an algebra equation, and then the software can graph it, and solve it for you. There are so many other built in features and functions that you may need to spend some time figuring out. What makes this so special is that the application is free. You can just install it onto your computer, and then keep it forever. Unlike most Math software such as Maple, or GSP, you have a limited access time. I really like it once I start using. As a future teacher, I would love to incorporate it into my lessons, and introduce it to my students. because it is free, and you can do so much with it. It is a good software to have because students may use it to check their answers. They can use it as a free graphing calculator as well. If I'm teaching Geometry in the future, I will definitely use it to teach geometric theorems. Students may learn how to prove or make conjectures by doing geometric constructions using the software. It is a great tool that teachers shall really consider using in their classrooms.

Technology Integration Plan

For this project, I chose a lesson plan called "Paying for Your Wheels," which is the same one that I used for my inclusion class. It was a lesson plan that I got from Illuminations designed for high school students, and focused on data fitting in the context of real life application, owning a car. This lesson assumes students already have some basic prior knowledge about data fitting and linear regression model. Therefore, the goal of this lesson is to increase students’ prior understanding of fitting a regression model to a set of data. In this lesson, students will be introduced to the costs associated with a car, study the amount of fuel used by a car at various speed, perform regression analysis on speed versus fuel economy data, use the True Cost to Own calculator available at Edmunds.com to estimate the monthly cost of owning a specific car, study the relationship between speed and gas mileage, predict a car’s best gas mileage and find the line of best fit using a graphing calculator, and interpret the correlation coefficient. The lesson is aligned with the Common Core State Standards (CCSS) and the National Educational Technology Standards (NETS).

The link to the lesson plan and the technology integration matrix that I have created for the lesson is provided at the end of this blog post. Each row of the matrix is under different subheadings: assess, analyze, evaluate, produce, and communicate. For the “assess” row, CCSS that I have chosen is “use functions fitted to data to solve problems in the context of the data.” The overall learning goal of the lesson is to use real life data to determine the monthly cost of owning a car by considering all the expenses such as fuel cost associated with it, and fitting a best fit line to the data to make prediction and derive a conclusion. In order to accomplish the goal, students will have to do research online, collect and analyze data, and derive a conclusion based on the result, which is the skills listed under the NETS-S4 and NETS-S5. Teaching strategies that I will use to facilitate students’ achievement of the standards are direct teaching (lecture), problem posing (explanation), class discussion, Q&A, note-taking, small group work, information gathering, and student collaboration. This part of the lesson will be mostly teacher-centered. I will give a short lecture on the costs associated with owning a car, and linear regression model by going over one short example on finding the line of best fit using a calculator, and interpreting the fit of the line by observing the correlation coefficient. And then, I will pose the problem to students, and provide opportunity for students to ask questions. Also, I will let students work in groups or pairs throughout the entire lesson, and go over the results with them. Students will work collaboratively to gather information and data online, and take notes on their findings. Supporting technologies that I have chosen are computers with internet access, a whiteboard or blackboard, and a TI-83 view screen overhead projector. I will use the view screen projector to show students how to calculate the line of best fit on the TI-83 calculator, and students may also use it to share their results with the class.  

For the “analyze” part of the matrix, CCSS that will be addressed is “interpret the correlation coefficient of a linear fit.” Students will have to do data fitting first, and then analyze how well the line fits the data by interpreting the meaning of the correlation coefficient. NETS-S4 will also be addressed because students will have to analyze data to derive a conclusion and make prediction. Strategies associated with this are guided class discussion, student participation, Q&A, and small-group work. This part of the lesson will be mostly student-centered, and requires student participation because students are entirely on their own in terms of gathering and analyzing data. They will work in small groups and at the end, I will start a class discussion on their findings, and ask specific questions to lead the discussion, and provide opportunity for students to raise questions. Technologies that are needed are the same except computers. Students do not need computers to do data analysis.

          For the “evaluate” row, CCSS that will be addressed is “informally assess the fit of a function by plotting.” Students will have to graph the data points on a scatterplot using a calculator, find the equation of the linear regression line, and evaluate the results given by the calculator. In order to do that, they will have to select the appropriate tool (calculator), and use the right built-in features of the tool, which meets NETS-S3. Also, this part of the lesson will be student-centered. Supporting technologies are pretty much the same.

         For the “produce” row, students will have to be able to compute the correlation coefficient using a calculator, fit a line to the data, and make conclusion based on the results, which meets CCSS.HSS-ID. C.8 and B.6a.  Technologies are the same, and this part of the lesson will consists of some teacher-centered strategies such as direct teaching, demonstration and explanation, some student-centered strategies such as student participation and collaboration, and a strategy that counts as both such as Q&A. I will demonstrate and explain to students how to use the calculator to find the equation of the line, and even teach them how to interpret the results correctly. It also requires students to ask and answer questions.

        For the “communicate” row, CCSS.HHS-ID.B.6 will be addressed. Students will have to be able to describe the relationship between two variables such as speed and gas mileage, and explain their results to the class. A supporting technology that I added to the list is a document camera. Students will be able to share their written work via the document camera while they are presenting their results to the class. Strategies associated with this part of the lesson are mostly student-centered such as student participation and collaboration, and class discussion, but I have also added another student centered strategy, a short presentation to allow students to communicate to their peers. The presentation meets NETS-S2, which is “collaborate and interact with peers.”  

Lesson Plan: http://illuminations.nctm.org/Lesson.aspx?id=2459

Technology Integration Matrix: https://docs.google.com/spreadsheet/ccc?key=0AsoaTIT87eE1dGEtOGJQMWpTc0hhcWF5ZjZadDZHb3c&usp=sharing

TI-83 View Screen Overhead Projector

When I was in high school, I was fascinated by the graphing calculator view screen overhead projector that my Math teacher used. He was able to connect a TI-83 calculator to a "special" projector, and we were all able to see his calculator's screen. Later, I actually did some research on that to get the actual name of the projector, and it is called the Viewscreen Overhead Projector. Below is an image of the projector. I love it because it is extremely useful in a Math classroom. When I teach a lesson on the graphs of trigonometric functions, instead of manually draw the graphs, which are often complicated to draw, I can just graph it on the calculator, and then project it onto the board, just so every one can see it. It will save teachers a lot of time, and teachers may also show their students how to input the function into the calculator as well since many students may don't even know how to do that. I feel that it is the most useful technology that I have encountered so far in a Math classroom. I have also seen teachers at my fieldwork site used it as well, and the lessons often went really well. Therefore, I will definitely use it in my future classroom when I am teaching students how to graph functions.

Virtual Manipulative

         In the method course that I took last semester, I came across virtual manipulative such as the Algebra Tiles, fraction bars, and base ten blocks that teachers may use to teach Math more effectively. We always claim that we want to incorporate more hands on activities in our lessons, but how do we do that? I believe that we should use manipulative effectively to support students'  learning. Manipulative provides a concrete representation of the abstract concept to assist students' understanding of the concept. For example, seeing 1/2 is equivalent to 2/4 may be difficult for many middle school students, and the fraction bars maybe a useful tool to teach equivalent fractions. The fraction bars allow students to see how 1/2 and 2/4 is visually represented by bars of the same length (1), but divided into different number of pieces. It helps students to understand that cutting the bar into 4 pieces and take 2 pieces is the same as cutting the bar into 2 pieces and take 1 piece. Manipulative is especially effective in helping students to understand abstract concepts and stimulate  students' interest in the subject. Most students like to learn with something in their hands that they can play with. Therefore, it is absolutely important to use virtual manipulative in a Math classroom to supports hands-on activities.

How should students take notes in a Math class?

               As I was reading through many of my classmates' blog posts, I realized that many of them touched upon the issue of taking notes with technology, and using eBooks or actual books. I would like to comment on the issue of taking notes with technology, either by typing everything on a computer or taking a picture and saving it in their phone. I am not against the idea of taking notes with a computer, but for Math, I believe that students shall actually write down notes with a pen or pencil. I type on my computer to take notes for many classes all the time, but I feel like it is much more effective and easier to actual write down notes with a pencil than trying to figure out how to type sigma in a word document. For math, there are many symbols and notations that you can't really "type." You have to click several tabs in Microsoft Word to insert a Greek symbol or square root, and that often takes too much time. It defeats the purpose of taking notes with a computer if students are wasting time on figuring out how to type all these math notations. Therefore, I believe that students shall not take notes with a computer in a Math classroom. The traditional note-taking is much more effective. In addition, students will be able to retain more information and remember the content better if they actually write the notes. I find that true all the time at my fieldwork site. Whenever my cooperating teacher asked students about an example that they had done before, the students always said something like "oh, I remembered I wrote that down somewhere in my notebook," and they would start flipping through their notebook and know exactly where to find it. As a result, I will strongly encourage my future students to take notes using a pen or pencil.

Socrative

              Coincidentally, I was able to observe an inclusive special education class. I really like how the teacher incorporated technology with Mathematics. The teacher gave out IPads to students, and they logged into a website, www.socrative.com. There was a chat room that the teacher had created, and they were able to do math problems and interact with others via the chat room. The teacher would post a multiple choice question in the chat room, and the students would be able to access it on their IPads and choose their answer. The website would actually tell the students whether they had chosen the right answer or not, and on the teacher’s screen, it would show the teacher what all the students had chosen. The teacher would go on to the next question after every student had answered it. The website also showed how many had chosen an answer, and how many were still working on it. I like how the website actually shows you the statistics or the spread out of the results. It allows teachers to monitor students’ progress, and which question troubles students the most. Therefore, teachers will be able to use the result to make appropriate adjustments in the lesson, and determine whether further clarification or reteach is required or not. In addition, students all are working at the same pace, as a group because the teacher would't go on to the next question until everyone had answered. I will definitely use it in my classroom if I have enough IPads or computers for every student. Students enjoy doing it, and this can be a great warm-up activity as well. 

Collaboration Canvas

      At first, I had a hard time figuring out how to collaborate math with other content area since my unit is so specific to Math. Therefore, I begun to review my classmates’ canvases, eventually choose the Ancient Egypt unit plan canvas to do the remix. The unit focused on the characteristics of Egyptian Art, and this can definitely be collaborated with my unit on polygons because shape is an important element of art. Shapes used in drawings can actually mean something. Therefore, I added a mathematical aspect to the Ancient Egypt unit by first figuring out the grade level that the original unit plan aimed for. The standards actually aimed for both 8^th grade, and 12^th grade, so I decided to align the collaborated unit with high school geometry standards. Since the two units are very different, I added three videos, and images to make the collaboration. I added the study of the Golden Ratio, and Egyptian architecture like the Great Pyramid to the original unit plan, so that students can understand the geometry behind it. I also added images of paintings and sculpture that could be used to study polygons. To connect the two units, I added two new essential questions, and changed the original essential question by adding a mathematical perspective to it. I asked students to think about the importance of polygons in the context of real life, and art. I did not remove anything from the original canvas because it collaborated with my unit pretty well. 

Remixed Canvas: 

http://www.play.annenberginnovationlab.org/play2.0/challenge.php?idChallenge=2569&mode=view#network6