I have been teaching transformational geometry using Geometer’s Sketchpad (GSP) since 1994. I have used graphing calculators, Fathom and GSP to enhance how I teach in my classroom by using technology to substitute and augment what students had done before with graph paper, pencil, ruler and compass. Now that I am planning for my final COETAIL program project, I would like to transform how I teach math by using technology in ways that modify and redefine what I do in the classroom as described by the SAMR Tech Integration Model. My initial project ideas are for units that expand on the use of tech tools students already use in my classroom by using the Web 2.0 tools to share what they have learned.

I have used Fathom since teaching A.P. Statistics in Oregon in 2001. Students use the software to discover patterns in data using dynamic links between tables, graphs and statistical tests. My I.B. Math SL students have also used Fathom to complete their internal assessment type 2 tasks in mathematical modeling by using sliders to transform functions they think will fit the data. By using Fathom, I have redefined how students learn statistics concepts in my classroom. Now I would like to have them use Fathom in a statistics unit at a transformational level.

With the addition of Web 2.0 tools to my statistics unit I can transform my statistic lessons by having students use Fathom to explore the living conditions of Romanians and share their graphs and statistics they create online. I have recently been approved to use the data from the Integrated European Census Microdata website. This year the IB Math SL internal assessment changed from a 10 day type 1 or 2 portfolio task involving mathematical modeling or pattern investigation to an exploration of a mathematical topic of their choosing. Students will use Fathom and micro data to do a practice exploration project to describe a region of Romania in statistics and graphs that will be posted on a webpage of a Google Sites website. The American International School of Bucharest recently became a Google apps school under the domain name of go.aisb.ro. The students’ Google Sites page will include info graphics created by using Fathom, Gap Minder and one of the other info graphic creation tools.

Next semester due to scheduling conflicts I will not be teaching the middle school robotics course and the seniors in I.B. Math Studies will be reviewing for their I.B. exam. I will be teaching new material in only the previously mentioned Math SL course and my MYP Math Level 4 course so another Course 5 project option would be to do a tech transformation makeover of my geometry unit in MYP Math 4. MYP math students are graded on four criteria of knowledge and understanding, investigations, communication and reflection. My assessment of their reflections is the weakest link in how I grade my MYP math students. Currently I assign a project each semester that they write a reflection paragraph on and that determines their reflection grade for the semester.

I would rather give my students multiple opportunities to be assessed on reflection but I do not have the time to grade more multi page projects. To transform how I teach and assess reflection I would have students keep their own problem solving blog. The students would be assigned a problem from Dan Meyer’s Three-Act Math Tasks or a Three-Act Math Task I have created myself. They would would then be given 2 weeks to post an explanation of their math task solution using Java Sketchpad or GeoGebra to creative interactive elements on their blog posts. Students would then be assigned a new Three Act Math Task to work on for the next 2 weeks during which they will also be required to comment on two of the solutions posted on their classmates’ blogs. When 2 weeks have passed students will post their solutions to the second Three Act Math Task and write a paragraph reflecting on their solution to their first Three Act Math Task by responding to some of the comments posted by their peers.

I will need to learn new skills for both of these Course 5 project ideas. I have not used Fathom with micro data before and I am not sure how the graphs and tables produced using Fathom could be used with info graphics creation tools. I have created websites before using iWeb and Freeway Express but I have never used Google Sites. I have used Geometer’s Sketchpad for 18 years (an eternity in tech time!) but I have never used it to create interactive blog posts. Lastly I have never created a Three Act Math Task like the ones created by Dan Meyer.

Implementing either the Info graphic Romania Student Website or the Three Act Math Task Reflection Blogs will raise tech integration in my lessons to the transformational level. Both of these projects would be impossible to implement without the use of technology. Students creating the info graphic web pages will not only learn statistical concepts but will also learn about the living conditions of Romanians who were surveyed by the EU census. Students creating Three Act Math Task problem solving blog posts will be incorporating comments from their peers and possibly from students at other schools into their reflection writing. Removing the technology component from either Course 5 project idea will make the project impossible to implement which is the essential quality of a Transformational Mathematics unit.

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I manage my classes using 1 to 1 laptops through a mixture of movement and motivation. I have had more luck using low or no tech techniques to make sure students are on task whether they are using paper and pencil or a laptop. Two years ago when I first bought my iPad and was testing out how I could use it in my classroom, I came across an app for LanSchool. I imagined using the app to make sure students stayed on task and to have them answer questions using their laptops to check for understanding during the lesson but eventually I went back to my low tech management techniques.

In my 4 years of teaching students using 1 to 1 laptops, I have come to accept the short time delay it takes students to open their laptops, turn them on and connect to the network. During this time I have them use paper and pencil to do a warmup problem and ask questions about the homework. By the time the last homework question is answered most students are ready to download activity files from Moodle and get started with their 1 to 1 laptop activity.

After testing the LanSchool app for a semester I realized the software was having a negative impact on learning in my classroom. Students had to log on to both the network and LanSchool and the school’s network could not keep up with the bandwidth use by LanSchool. The time delay between the start of class and students using their laptops to explore mathematics was now longer and I was still not seeing all of the class laptop screens on my iPad. I also realized that a student response system which consisted of multiple choice answers was not giving me enough information on what problem solving steps individual students were getting stuck on. In the end I decided it was faster for me to walk around the classroom and look at students’ steps in their notebooks and their laptop screens at the same time to ensure they were on task and understanding new material.

My management on the move is sometimes made more difficult due to other technology I use. SmartBoard’s were in most of the classrooms when I was first hired even before AISB had a 1 to 1 laptop program. With a SmartBoard I can show students how to use their TI-84 or TI-nSpire calculators by using emulators that display the calculator key presses on the SmartBoard as I use the calculator. The SmartBoard also made it easier for students and myself to demonstrate problem solving steps and ideas using virtual algebra tiles, Sketchpad and Fathom. Recently the department has also purchased licenses for FluidMath software that will give myself and students the ability to write mathematical notation on the SmartBoard and convert the handwritten material into graphs, sliders and function tables. The use of the SmartBoard has made my management on the move method more difficult as I now have to run between the SmartBoard at the front of the class and back to students at their desks as I teach.

Using a SmartBoard and 1 to 1 laptops both present management challenges but the benefits outweigh the challenges and I hope to find other tech solutions that will eliminate the SmartBoard’s effect of tethering me to the front of the classroom. Last year the school purchased a Fujitsu Q550 slate for me to test as a teacher laptop. I also bought a Vianect Air TV USB adapter so I could manage on the move and still control what my students saw on the SmartBoard by using a wireless video connection between the slate and projector. The atom processor on the slate could not keep up with the processing needs of the SmartBoard, Sketchpad and AirTV software running simultaneously so I reluctantly gave up on controlling the SmartBoard remotely. This year the rest of the math department teachers received a ThinkPad X Convertible Tablet which has an Intel Core i5 processor that should keep up with the demands of running a wireless video connection. One of the department’s teachers has borrowed my AirTV adapter and we shall see…..

In the end managing a classroom is easier when students are motivated. As I mentioned in a previous post about flipped classrooms, student access to 1 to 1 laptops has changed how I teach. My students use their laptops to discover math concepts and construct their own understanding of mathematics using Sketchpad, Fathom and their calculators. In my robotics classroom the students use their laptops to both program their robots and learn new programming techniques using Carnegie Mellon’s RobotC curriculum. I believe that using 1 to 1 laptops as tools to help students understand course material results in more motivated students that are easier to manage while I move around the classroom. And if I am wrong at least all my movement in class might help me keep up with my Colombian wife’s salsa moves on the dance floor!

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This year I noticed that my alma mater, U. C. Berkeley, had joined the MOOC movement by joining edX to offer online courses in computer science. The news reminded me of my first physics course at “Bezerkeley” with over 100 students in the same lecture hall. On the first day of the course the professor announced we were at the wrong university as at another UC campus we could have attended a smaller introductory physics class and receive more feedback than he could provide us. (Currently U.C. Berkeley has 258 courses with a class size of more than 100 undergraduate students.)

I did not take the professor’s advice and a year later I was in smaller classes having survived the freshman “weeder” classes. I survived those massive offline classes by seeking out the help of the massively overworked teacher assistants. Unlike the professors, the teacher assistants had time to give me knowledgeable feedback on how to solve problems in the class assignments. I also studied with class mates who are still some of my best friends to this day but when I was really stuck it was time to get help from a teacher’s assistant.

I am excited that high school students can now choose to take courses from U.C. Berkeley, M.I.T., Harvard and yes I hate to admit it…. Stanford but they will have to be more resourceful than I was as a college freshman to benefit from MOOC courses. By enrolling in a MOOC course my students could potentially learn C++ and Computer Graphics, Introductory Statistics, Introductory Physics, or the theory behind the Control of Mobile Robots. To make this potential learning opportunity a reality students will have to find fellow online leaners that can provide the knowledgeable feedback that the U.C. Berkeley teaching assistants provided to me as a freshman student.

My experience in massive offline classes also resulted in a benefit that no MOOC program offers, a UC Berkeley B.A. degree certifying my knowledge of Astronomy and Physics to the rest of the world. Currently MOOC courses assess students’ knowledge through a mix of online tests and peer grading. As a New York Times article points out, students who learn from participating in a MOOC course will have trouble certifying their knowledge until MOOCE or the “massively open online course evaluation” problem is solved.

My students are currently part of a massively minimized educational experience. As in most international schools, AISB limits the class size to 21 students which is much smaller than the class sizes of 40 that I taught as an Oregon public high school teacher. I may not be able to teach my students Artificial Intelligence programming or Analytic Combinatorics but I do know their strengths and weaknesses when it comes to problem solving and they can get my help when they are stuck on a problem during lunch, break, after school or by email. I can also certify my students’ knowledge of IB Mathematics every November when I am asked to make a prediction of how well they will score on their IB external exam in May for their university applications. It is statistically scary that for the 41 students of last year’s graduating class the teachers at AISB predicted the students’ IB external exam scores with an accuracy of less than ± 1 point from a maximum possible IB point score of 45.

MOOC courses have the potential to open up a university education to anyone who has an internet connection but first the evaluation and course assistants issues must be worked out. Even with our virtual high school courses the teacher who supervises students in the program has problems evaluating their knowledge in subject areas he is unfamiliar with. The challenge for MOOC courses is to incorporate the benefits of small off line courses to make bigger better.

P.S. Here is a link to Dan Meyer’s comments on the MOOC site Udacity

]]>Last year I first heard about the flipped classroom from our school’s chemistry teacher who was using Moodle to post lecture videos based on powerpoint slides for his students to watch outside of class. Later that year I tried using a flipped lesson during a snow day in which I used an Echo pen to record a lecture on how to draw a cumulative frequency graph for my IB Math Studies class. My experience with flipping my Math Studies lesson were far from transformational. In this lowest level IB math class students who are difficult to keep on task when they are in the classroom ignored the assignment to watch the video at home leaving me with only 4 students out of 12 prepared to apply the skill to solve problems in class.

At the end of last year the math department teachers agreed to make learning how to use technology tools to teach discovery math lessons a departmental initiative and part of their personal teaching initiatives (PTI). As the head of the math department one of my responsibilities is to help teachers with their PTI’s and two teachers chose to implement a flipped classroom approach. Both teachers are using the new tablets they received this year to create videos and post problem sets for students to work on outside of class. Encouraging students to be responsible for their online assignments has been a challenge for these teachers also and what to do when only some of the class has seen the online introductory material has been a particular stumbling block to applying the flipped classroom model.

Wether you call it flipped or reverse instruction this new fad on how to use technology to teach makes some basic assumptions about the non flipped classroom which I believe are false in the classrooms of effective teachers. The basic rationale given for reversing instruction is that teachers use lecture to introduce concepts in class and that class time would be better used doing explorations and activities that apply what has been introduced online via the lecture method. This is the argument Sal Khan makes for using his website of math tutorial videos to introduce students to new concepts. For both the SAMR or Technology Integration Matrix model of integrating technology this is a very low level use of technology to replace a teaching strategy that I try to use as rarely as possible. I believe, as Dan Meyer does, that students should engage in an introductory activity before watching a live or recorded lecture. As Dan Meyer states;

I’m still unsure why we’d waste 25 minutes on direct instruction anyway? That’s 25 minute of

teacher talkingwith no interaction from students, no questioning, no formative assessment. Whether that goes home on a video or it stays in the classroom, it seems like a misallocation of a teacher’s resources.

The power in using technology to teach mathematics does not lie in changing the order of when to use limited teaching techniques but instead to use technology to replace those limited teaching strategies all together. When I introduce a new concept in my math classrooms I start with a discovery activity not a lecture. Students use Geometer’s Sketchpad, Fathom, Cabri 3D, and AutoGraph software both in my classroom and at home to make conjectures about math concepts and later we go through the problem solving steps that are based on the concepts. It does not matter if I do the discovery learning activity in class or assign it as homework as the goal is to minimize my use of lecture. Dan Meyer also advocates changing online instruction to emphasize how technology can be used to explore math concepts in a way books and paper never allowed.

So when it comes to my classroom it’s neither flipped, reversed, inside-out, backwards, upside-down, or mirrored. I use technology not to make geometric transformations of the order in which I use teaching strategies but to transform the actual teaching strategies I use. It’s not the direction that matters it is how you get there.

]]>Since the start of course 5, three weeks ago, I have had a lot to think about in terms of using technology in math classes. During the first days of the course I was attending the Computer Based Mathematics Summit in London. Over 2 days, discussion panels covered many of the issues raised on how to truly integrate the use of tech tools into the math curriculum as described by the SAMR model and Tech Integration Matrix approaches to transforming learning using technology.

Much of the discussion at the summit revolved around how tech tools has made the teaching of mathematical algorithms such as factoring, solving equations and finding the derivative of a function either less important or unnecessary. Students with the latest iPhone can now use Siri to answer basic calculus questions and the Khan Academy has videos on the step by step process for any math skill I learned as a high school student. Dan Meyer recently wrote in his blog how textbook companies are missing the opportunities to transform how students use e-books to learn math. With his taco cart problem he points out how technology can help students with the process of abstraction that they need to use to turn a real world problem into a math problem. Using technology as an indispensable tool to help students learn problem solving skills that cannot be programmed into an iPhone should be the goal of any computer based math course.

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GapMinder Screen Fathom Screen

One of the nice things about teaching statistics is how easy it is to create interdisciplinary lessons involving mathematics and social/environmental issues. Mathematical modeling is part of the syllabus for all levels of math in the International Baccalaureate Diploma Program. I have just finished teaching transformations of exponential graphs using the TI nSpire calculators in my Math SL class so I decided to follow it up with a lesson on mathematical modeling of population growth.

In previous years I have used a Fathom statistical software activity from the “Exploring Algebra 1 with Fathom” activity book to teach the lesson but this time I decided to introduce the topic of population growth with the GapMinder software. Instead of starting with static scatter plots, I used GapMinder’s animation feature to show students how the population grew in India and China from 1600 to 2011. As the population trails for India and China grew closer and closer I asked in what year will the population gap between the countries be closed? This led to a discussion of how an exponential function could be used to predict India’s population after 2011 and into the Fathom lesson on how to fit the function.

In a later lesson, students will fit an exponential function to China’s population growth data and use the two mathematical models to predict when India’s population will match China’s. We will then compare the answer they get with what GapMinder shows after running the animation past 2011.

The GapMinder software makes it easy to make dynamic scatter plots with a wide variety of built in data sets. The data from a GapMinder chart can be downloaded as an Excel file but the file does not import correctly into Fathom. As the author of the blog Roughly Normal comments I would also appreciate it if a mathematical Santa Claus could close the gap in the capabilities of Fathom and GapMinder by leaving me a statistical software package that does it all under my Christmas tree this year!

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3 minutes of video only took 7 hours to create! I knew I wanted to create a movie in the style of Dan Meyer’s 3 acts videos on Vimeo. I also knew I had to teach 3D geometry in an upcoming Math Studies class so I decided to do my digital storytelling on the topic of volume.

As in Dan’s videos I wanted students to use information in the video to make predictions. Using math I wanted to students to answer questions like; What fraction of each container will be filled with melted ice cream? How long will it take for the ice cream balls to completely melt? Will Mr. Hester make a mess?

I edited the movie in iMovie but to get the video clips I used the Camera app on my iPad along with a new iPad app called iTimeLapse to get the time lapse video of the ice cream melting. I then created photos from frames in the movie using QuickTime and added measurements and graphics to the still shots using Graphic Converter. I used CompFight to find a few Creative Commons photos to add and a new resource called Audio Farm to find a Creative Commons soundtrack for the credits screen. Hope you enjoy watching it!

*Click here to view the embedded video.*

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So with paper and pencil in hand I soaked in the tub and wrote down all the presentation ideas I had in sloppy hand writing. I realized that now I would be able to submit my presentation assignment much earlier than I planned. I had bought Gar Reynolds iBook on Presentation Zen and had read a bit and watched some of his videos describing his design philosophy but I had only one day to come up with what I wanted to say for the Kick-Off. So like Archimedes I jumped out the bath (but first I got clothed!) grabbed my MacBook Air and cranked out the presentation using photos from *FIRST* and last year’s Eastern European *FIRST *Tech Challenge Tournament. I even put in two short video clips of the World Championships and the AutoVortex team winning the BootStrap Award which appear as photos in the Google Presentation below. (I guess videos get lost in the Keynote to PowerPoint to Google Presentation translation.)

**Serendipity**** is wonderful ….** The first lesson I need to teach next week is the Law of Sines in my Math Studies class and I also need a second blog post for my current COETAIL course about using creative commons images for teaching students. One of the new capabilities of Geometer’s Sketchpad Version 5 lets users import photos into the software and to create geometric constructions on top of them. Images can also be modified using transformations. Texas Instruments thought the ability to do math on images was such a good idea that they added it to the new operating system for the TI nSpire calculator.

I have also been wanting to try and use Dan Meyer’s Levels of Abstraction idea to encourage students to be what Dan calls “patient problem solvers” The lesson will start with a warmup activity where I show the tree on the left and ask what measurements would you need to find the height of the tree without climbing it? (They have already solved other IB right triangle application problems in previous classes.) I increase the level of abstraction by adding a coordinate grid in Geometer’s Sketchpad. Can we still use right triangle trig for the left tree on the slanted hill?

After students decide that the left tree’s height can be measured with a right triangle, I have them go up another level of abstraction by having students construct a right triangle on top of the image. *See left image below.* (Good questions here are; What are the ¨top¨and ¨base¨of the tree? What angles could be measured? What triangle lengths could be measured without climbing the tree? If Sketchpad measures the base of the triangle constructed over the photo in centimeters, what else do you need to know about the photo to get the tree’s real world height in meters?)

After using the tangent ratio to find the height of the tree in the photos on the left, I show them the tree in the right photo at the top of this post. Can we use right triangle trig? I increase the level of abstraction by adding a coordinate grid in Geometer’s Sketchpad and ask how can we tell that right triangle trig will not help in finding the height of the tree? Students then increase the level of abstraction by constructing an acute triangle on top of the image. For this tree what angles and distances could we measure without climbing it? Does right triangle trig give us the correct answer? (Students can use Sketchpad’s measurement ability to check it themselves.)

Students will then complete a Geometer’s Sketchpad Lesson Link activity to discover the formula for the Law of Sines. As a lesson wrap-up students are asked to use the Law of Sines formula that they have discovered to find the height of the tree in the photos on the right of this blog post.

*Note: Part of this post was written with the help of a Mac OS blog editor called MarsEdit. If you have Geometer’s Sketchpad you can download the GSP file here.*

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**As the confused math student once said the only constant is change. **It is the start of a new school year and the start of a new COETAIL course. *FIRST* has released a new game entitled RING IT UP! for this year’s *FIRST* Tech Challenge robotics competition which is featured on my new Eastern Europe *FIRST* Tech Challenge website.

Decisions by Apple have determined the design of all of the gadgets I use from my iPhone to my iPad to my MacBook Air but now one of Apple’s decisions has prompted my own design changes. For years I hosted my websites using Apple’s Mobile Me service but as of June 29th the service has been shutdown. I decided to use MacMate to host my new tournament site.

Many of the design choices I made for my old tournament website were driven by the capabilities of Apple’s website design tool iWeb but after changing the website host I decided to also change the web design software I used. Freeway Express is a WYSWYG editor like iWeb but it gives users more control over the design elements of a webpage in a similar way as InDesign gives publishers control over the design of the printed page. From the day I first created webpages I have never been drawn to the nuances of HTML coding and by using Freeway Express I can focus instead on the overall web site design.

I started creating the new website before the start of Course 3 but I think I have already made changes incorporating some of the advice in this week’s article “Understanding Visual Hierarchy in Web Design” I have reduced the amount of text in small font and increased the size of important visual elements. I am hoping that both coaches and student team members will find the new site easier to navigate. Eventually, I hope to look at the design of my AISB Math website so that my math students will also benefit from what I learn about design but currently I am dealing with what has not changed since last year, not enough time!

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