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New technologies, new representations, new opportunities: iPad Maths
Touch-based technologies like the iPad offer exciting new possibilities for the teaching and learning of mathematics. Key focus areas of this discussion will be data and algebraic thinking, and we will explore together some of the powerful and exciting new opportunities that are offered by the new tools now reaching our classrooms.
We begin with an imaginary stroll around a local high school, where iPads are being used in a variety of ways in mathematics classrooms. We will peek in the door of a few classes and see what is going on...
In our first room, the teacher is using a document, called Area Formulas, that came bundled with her TI-Nspire App for iPad to build an understanding of areas of different figures. Using an app called Air Server, she is projecting her screen wirelessly through her computer to the data projector at the front of the room. The Air Server app, which runs on Mac or PC and is available for under $15, allows any AirPlay-enabled device (iPad, iPhone, Mac computer) to send their screen to the teacher's computer.
This teacher takes advantage of the portability of the device to engage students in the activity. As she walks around the room, she hands her iPad to different students, instructing them to grab and drag the shapes around on the screen as she asks the class to make conjectures about what they observe.
In the next classroom, students are studying the properties of quadratic functions and their graphs. The students begin by using a photo of the Sydney Harbour Bridge to discuss how close to a parabola that the arch appears. Using the TI-Nspire App for iPad, the students import the picture of the Harbour Bridge into their graph page and begin exploring how transformations of the graph affect the form of the algebraic equation. As they drag the graph with their fingertips, they take note of the structure of the equation and discuss how the vertical and horizontal shifts in the graph are noted in the equation. They drag the axes on the page and change its scale by "pinching" to see how these changes affect the equation of the function. The teacher uses this exploration as an opportunity to discuss the vertex form of the quadratic function.
The students will shortly be sent out to seek out other parabolic shapes in their school. Their task is not just to grab and drag a function graph to match their chosen shape, but to attempt to set the window up so that the function is as simple as possible!
Another nice variation on this task is to drag points to best fit the curve, and then see how well a quadratic regression matches.
Down the hall, a senior class uses their TI-Nspire CAS App for iPad to study volumes by revolution. The students are able to use a standard graph page alongside a 3D graphing window to study how different functions behave when rotated about the x-axis. They can easily drag the 3D figure to study it from different perspectives; enabling them to better visualise and understand these relationships.
The next room we visit is empty! The class is out following a Maths Trail that their teacher has set up in the local neighbourhood. The students are using a class set of iPads so that each of them can take photos of different tasks, and use the measurement and angle tools of the TI-Nspire App to answer the various questions.
Our last class are preparing for their "function races". Each student is assigned a function (they are starting with linear functions) and, when nominated, will come to the front and "race" against several other students, following the rule of their function. The class report will involve them "calling the race" and deciding if the participants raced correctly!
The students prepare in small groups, and have access to a document which simulates the function race. Students can enter up to four functions, and for each function, a "virtual runner" takes their position. When the race is started, they compete according to the slope of their function and the y-intercept determining their starting point. Grabbing and dragging the graph to change the slope has the immediate effect of changing the speed of the associated "actor".
You might notice in the screen shot displayed, that Air Server supports multiple displays, limited only by the available bandwidth of the network. Here we see the displays of a Mac computer, an iPad and an iPad mini all displaying live on the teacher’s computer screen. The teacher can easily move, resize and invite new presenters at any time.
Background
The first Apple iPad was launched in the United States on April 3, 2010. In three short years, this device has transformed not only the way many people do everyday things, but the ways in which we think about technology. People of all ages are making deliberate and effective use of this device to do everything from catching up on the news to buying groceries online to talking to family members who are halfway across the world. It is not hard to understand what the fuss is about!
Classroom teachers, in staggering numbers, have adopted this tool to share information and engage students in interactive, media-rich lessons in an environment that has become second-nature to their students. We may yet be a long way from one-to-one computing for our students in the majority of classrooms, but it seems we are much closer to achieving that goal for their teachers. Overwhelmingly, up to this point, the iPad has been the teachers’ tablet of choice. This is not to say that there is not and will not be plenty of healthy competition for Apple, but up to this point, they have been outstandingly successful in reaching schools. Android's day is yet to come.
The iPad is an amazing device, and (despite early critics) is ideally suited for teaching and learning. But how well is it suited for the teaching and learning of mathematics? What benefits (if any) does it offer over laptops, and even desktop computers already present in schools?
If we were to try to list some of the primary advantages gained from doing classroom mathematics on the iPad, then the following should rate a mention:
Interacting with the mathematics in a touch environment
Mobility for teacher and students (including ease of wireless presentation)
Interacting with the real world in new ways (camera, video, sensors...)
Ease of sharing and collaborating, both online and offline
Of course, if the classroom iPad has access to the Internet, then a whole additional world of value becomes available. Still, the value of the device rests - to a very large extent - on the software that is chosen for the device. This paper takes as its focus the TI-Nspire App for iPad ($31.99), which was chosen because it offers the most effective learning tool currently available for the high school mathematics classroom.
Choosing the Right Software
When the first personal computers made their debut in the early eighties, it was commonplace for enthusiasts to purchase floppy disks, frequently holding dozens of small programs, written by amateurs. Since many of the first programmers came from a mathematical background, we were quickly swamped with a multitude of simple tools for everything from calculating distance and gradient between two coordinate points to step-by-step completing the square practice. Drill and practice was king and, while there was plenty of enthusiasm, looking back we now realise that there was not a lot of quality.
We have seen this same effect with the tablet boom. Typing the term "math" into the Apple App Store produces 7,654 search results - and that is the Australian store. The US store is likely to offer many, many more.
Once again, teachers trying to select software for their classroom face quantity over quality. As in the old days, teachers trying to piece together various Apps to cover their curriculum find a bewildering choice with hugely varying quality, and a confusing array of different user interfaces to be learned.
These are the factors which quickly single out the TI-Nspire App for iPad as offering unique benefits on this platform. As with the TI-Nspire software and handhelds, teachers and students are provided with a complete mathematical toolkit that is well-supported for every aspect of high school mathematics. From algebra to statistics, geometry to graphing, there is simply no other tool available that comes close to providing such a rich and well-designed environment for high school mathematics. The Lua scripting language further extends the functionality so that, if a tool or feature is not native, it may be added through Lua!
And while this extensive array of mathematical tools is also available for Mac, PC and handheld, it is on the iPad that much of the mathematics really comes to life.
Touching the Mathematics
I have been a fan of algebra tiles for many years. I know of no more effective way for students to build firm concrete foundations for their thinking about and manipulation of algebraic expressions. Virtual manipulatives offer some important advantages with regard to student understanding of the idea of variables and the direct link between the manipulative and the symbolic form.
I have used many versions of these tiles over the years, but until I experienced grabbing and dragging the shapes on the iPad, there was a tactile level of the experience that was denied me (and my students). It is hard to quantify and explain the sense of personal ownership and tactile involvement that comes through a touch environment. I know of no research as yet that explores this concept, but I believe that it is one of those aspects of learning that needs to be experienced personally in order to be appreciated.
The TI-Nspire Apps for iPad extend that tactile learning to every domain of mathematics - including numbers, graphs, geometry and statistics - and do it within an environment that takes full advantage of the power of learning through touch. The same documents that run on the iPad will also run on computer and even the TI-Nspire handheld, but nowhere else do they "sing" as they do here.
Mathematics on the Go in a Collaborative Classroom
The mobility of the iPad is another key element in the success story. For teachers, it is critical that they can share what they are doing on any device, and the AirPlay technology standard with the Apple iOS makes it easy to send one or more screens to a projector. Certainly, this can be achieved using a hardware cable, but this limits the mobility of the device.
There are currently two options to "mirror" (or send) an iPad screen to a projector using the built-in AirPlay feature. Apple TV ($AUD109) is a separate device that connects any iOS device through the local wireless system to a projector. A less expensive software solution, through either Air Server or Reflector allows the teacher to display multiple screens simultaneously. Still, the usability of these options depends on access to a wireless network; and school networks often block third-party devices or software.
Mirroring an iPad means that teachers move around the room with their iPads, allowing them to engage students directly in the learning. When a class has multiple iPads, students can readily share their own screens at the same time, offering students a powerful learning opportunity.
The iPad is a collaborative device, so teachers must consider how they will share work with students. The TI-Nspire App for iPad links directly to Dropbox, making it easy for teachers and students to exchange documents. Documents can also be emailed from within the App or downloaded from a website and opened immediately in the App.
Touching the Real World
Another key advantage of the iPad over other platforms lies in the ease with which students can take photos and even videos, which may become part of their mathematics learning experience. Coupled with the mobility of the device, teachers can send their students to capture everything from angles to golden rectangles - and explore those relationships using the TI-Nspire Apps for iPad. How might you decide whose face is the "most Golden"?
Now think about the opportunities that abound from other free resources, such as Google Maps and Google Earth. Students can take screen shots of the maps and use these images within the TI-Nspire App for iPad to measure angles and learn about bearings. GPS information is readily accessible online, as are a range of other data sources, and students can even use the built-in accelerometer in the iPad to collect data to explore velocity. We are limited only by our imaginations!
Resources
As demonstrated here, in addition to the extensive capabilities of the TI-Nspire platform, a wealth of freely available resources also exists online, many in the form of iPad-ready documents that span the curriculum. Two sites in particular are worth a visit.
education.ti.com: From the front page, just follow the link to Activities > Math Nspired (and check out Science Nspired as well - some great resources there!)
My own site is also a treasure trove of free resources. Click on the image of the Nspire family at the front page, and then scroll down to find Lua Scripting HQ and the Dynamic Documents Sampler for Maths and Science.
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