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Barry Kissane Murdoch University

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Overview Analysis of the significance of technology in mathematics education in the Philippines Complementing the paper in the Proceedings Looking forward … I touch the future, I teach. Examples of suitable technologies Calculators Graphics calculators Computer software Internet for students and teachers Technology directions and standards Barry Kissane MATHTED

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Mathematics and technology Real world applications of mathematics frequently use technology these days So it seems important that our curricula in schools also reflect technology Where this is possible The recent deadly storms through the Philippines provide a good, albeit tragic, example Let’s look at the present example … MATHTED

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Typhoon Lupit The Internet provided a powerful picture of the approaching storm This satellite image was on Monday morning Click image to access website MATHTED Typhoon images and information from

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Forecasting typhoon behaviour A great deal of mathematics is used to predict the likely path: The increasing circles indicate increasing uncertainty of predictions The typhoon classification scheme The measurement of the earth The satellite positioning systems … a lot of technology was used to make this picture MATHTED

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Probabilities and errors In the real world, probabilities in models are rarely derived from counting finite events Observe also: The probability scale Unlikely events can happen Events with very high likelihood may not happen All predictions of these kinds involve errors Only the past is certain MATHTED

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How to respond? Mathematics provides good information about the range of possibilities and their likelihoods Using statistical analysis Using the normal distribution Using previous experience at predictions But humans must interpret all that information, here shown four days ago MATHTED

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Mathematics, technology and typhoons All of this information is accessible to anyone using the Internet, not only ‘experts’ In the next few years, it will be routinely available to many Filipino students and their families Does the school mathematics curriculum provide them with the necessary skills to interpret it and to use it? Is the mathematics curriculum informed by and related to technology? MATHTED

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Why is technology important? As the typhoon example suggests, we inhabit a world that is awash with technology Travelling here on a budget flight, many passengers had their own technology with them So of course, students ought to be inducted into that world in schools But technology can also make a distinctive contribution to teaching and learning mathematics By providing opportunities for learning and teaching that are not otherwise available Let’s consider some examples … MATHTED

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Technologies for school maths Calculators Scientific calculator Graphics calculator Computer software Generic software (eg Microsoft Excel) Mathematical software (eg Tinkerplots, Fathom, GeoGebra, Cabri 3D) Java and Flash applets The Internet For students For teachers MATHTED

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Why calculators? They are easily portable, and so can be easily shared in a classroom or a school, used in different classes, and even used at home. They are less expensive than computers, especially when all the software needs are taken into account (as calculators contain their own software). As they are potentially more accessible on a wide scale to students than are other forms of technology, curricula can be designed on the assumption that students can access technology. … MATHTED

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… educational arguments They can be used in formal examinations, which are of considerable importance in many educational settings. This advantage is mostly a consequence of the preceding reasons.. Most of them have been designed, and modified, for the express purpose of school mathematics education, and so are sensitive to the needs and interests of students and teachers. MATHTED

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Computation is facilitated Calculators are helpful for doing computations When the numbers are beyond mental arithmetic When many computations are needed When mathematical ideas are not yet firmly learned MATHTED

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Computing Philippines population According to the NCSB, the 1980 population of the Philippines was 48 million. The growth rate was about 2.3% per annum over the next 20 years or so. On a graphics calculator, this information can be used symbolically, numerically and graphically The rule of three Numerical solution of equations is possible MATHTED

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The rule of three Barry Kissane MATHTED Symbolic representation Graphical representation Numerical representation Solving equation to find population of 80 million X refers to number of years after 1980, Y1 is population (millions)

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Exploration with calculators Importantly, graphics calculators provide opportunities for experimentation, not just computation, in many areas, including: Graphs Equations Statistics Probability Calculus Geometry MATHTED

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Examples of calculator use There are many examples of graphics calculator use discussed in papers on my website: There are also examples of use of scientific calculators on the web: Go to and press ENTER Accept the agreement Click on Guest button (or register) Click on ES Standard Calculators Curriculum Paper Activity and then Next Download activities of your interest These provide many opportunities to see how exploration with calculators can affect learning. Barry Kissane MATHTED

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Technology and the curriculum When students have access to technology, the content and balance of the curriculum may be affected Here are some examples: Negative numbers in primary school Complex numbers in secondary school Exact arithmetic (Numerical) optimisation Integration techniques Such changes are only reasonable when access to the technology is universal MATHTED

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Computer software Computer facilities and access differ and the differences are important: Laboratory (individual) Classroom (shared) Home (individual) Software is of many kinds, including: Bundled (with the computer) Free educational software Commercial software Applets (run from a browser) MATHTED

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Spreadsheets Spreadsheets have versatile uses in mathematics, although they were not designed for educational purposes They may be important because they are available, typically as part of a software bundle Helpful for some forms of data analysis, modelling and everyday computation Some have used them powerfully; eg the Centre for Technology and Teacher Education at the University of Virginia has several downloadable examples. Here is one for simulation: ➪ ➪ MATHTED

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Tinkerplots Tinkerplots is a commercial program that supports data analysis of various kinds Tinkerplots Especially important for the analysis of student- collected data, a key part of mathematical modelling Sorting data is the primal act of analysis Less emphasis on sophisticated quantitative analyses and more emphasis on the story of the data Let’s look briefly at an example of Backpacks Data were provided with the software 21 MATHTED 2009

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Backpacks 22 MATHTED 2009

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Philippines population data Tinkerplots can be used to analyse data obtained from elsewhere To illustrate, we will look at the population of the Philippines over the 20 th century Data are from the wonderful GapMinder website Data on personal wealth (GDP per capita) are also included MATHTED

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Philippines population since 1900 Barry Kissane MATHTED

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Philippines GDP per capita The data tell a powerful story of the 20 th century The effects of WWII are clear As are the turbulent times of the 1980s … Students can use technology to explore real data Barry Kissane MATHTED

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Fathom Tinkerplots is described as suitable for grades 4- 8, but arguably extends beyond both extremities Fathom (an earlier commercial statistics package from the same developer) is also designed for educational purposes, and has a slightly older focus, including regression and inferential statistics Fathom Let’s look again at some population data to briefly see some differences The data show the Philippines population after World War II Notice how well the ‘least squares’ concept is demonstrated The data are close to linear, although clearly curvilinear 26 MATHTED 2009

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Least squares line 27 MATHTED 2009

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Accessing Tinkerplots and Fathom Further information about these two software packages is available on the Key Curriculum website website Evaluation copies of the software and many examples are available. Tinkerplots Fathom From the Philippines, these are available from Springer Hong Kong, Educational Resources Provider, Learning Interactive, Barry Kissane MATHTED

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Student data Data analysis is most powerful when students are addressing problems that matter to them They can design and undertake their own data collection Then use Tinkerplots or Fathom to store and analyse their own data Use of the software does not require high-level ICT skills Incidentally they will learn important lessons of data editing, dealing with missing cases, entry errors, etc. 29 MATHTED 2009

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GeoGebra This is an excellent free package that handles many aspects of geometry and algebra, and now (with a spreadsheet) some statistics as well This Can be downloaded or operated online Online version requires Internet access and is usually a little more recent Significant resources are being developed (eg with a Wiki) There is not time now to do more than mention it as a general purpose tool worthy of closer exploration later 30 MATHTED 2009

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Cabri 3D Three-dimensional geometry is harder to depict on a screen or in a book Consequently has been neglected in much of school mathematics? Cabri 3D provides opportunities not previously possible Click on the pictures MATHTED

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Java and Flash applets Another species of computer software comprises small applets that are operated by a browser Java and Flash are popular environments for writing these While these are typically available online, some can be downloaded and used locally, so that an online connection is not needed while the applets are being used. Some software (eg Geogebra and Cabri 3D) can export applets, so that programming expertise is not needed to construct them Let’s look at some examples MATHTED

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NCTM’s Illuminations Project This large web-based project includes a number of Java applets that can be saved for later use The pictures show two examples (click on them) Applets can be searched MATHTED

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Applets from GeoGebra wiki Many contributions to the GeoGebra wiki include downloadable applets, which have been exported from GeoGebra These address many aspects of mathematics, not only geometry A good example concerns integration with a Riemann sum Only a Java-enabled browser is needed to operate these applet Click on the image for an example MATHTED

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Applets from a textbook Increasingly, textbook publishers are providing applets with their products or as separate objects online. This is a good way to integrate technology in with other instruction. Here is an example simulating coin tosses from Baldi & Moore’s text, MATHTED

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Applets from Nrich Nrich is a large website in the UK, offering enrichment and extension materials of many kinds to students Some of the materials are interactivities, generally in the form of Flash applets A wonderful example is Trigon Click the circle to start MATHTED

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Nrich teacher packages The (wonderful!) Nrich site in the UK offers several teacher packages, comprising a collection of related materials, including Flash applets. The Probability package contains some applets for simulation that might be well used on an interactive whiteboard. Some of these use the DIME probability charts, which show a succession of events. Click the picture MATHTED

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Mathematics Online This large Austrian web-based project includes many Java applets Concerned mostly with senior secondary school mathematics Good pedagogical design Materials can be downloaded and used for learning and teaching The Gallery contains many thoughtful examples Click on the picture to explore the Gallery MATHTED

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National Library of Virtual Manipulatives This large collection of Java applets from Utah State University is searchable by content and level Click on the image Applets offer advice to students, teachers and parents The whole set can be purchased on CD-ROM to use when Internet access is limited MATHTED

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The Internet The Internet has changed forever the meaning and use of technology in affluent western countries like mine Internet access is rapidly rising in the Philippines, as it is elsewhere Home, community and school It offers distinct advantages for both learning and teaching mathematics Depending, of course, on access issues Let’s examine what these are MATHTED

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What Internet tasks for students? Active engagement in mathematics Reading material Browsing for information Communicating Practising Using like a traditional textbook A little text and illustrations Lots of examples, questions, problems

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The teacher’s role Locating and storing sites (eg on Intranet) School website construction Using with whole class With a data projector Including IWB use Referring students to particular URLs to focus constructive student use For projects, homework, interest, … Constructing or assigning webquests

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Categories of student use There is now an astonishing and increasing array of free and excellent Internet materials: Interactive opportunity Reading interesting materials Reference Communication Problem solving Webquests Good examples in each of these categories are given on my website (Internet Maths) Click on the image

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What tasks for teachers? As well as helping students to learn from the Internet, there are a range of ways mathematics teachers might use the Internet: Planning lessons Accessing official advice and help Connecting with other professionals Getting help from companies Designing and using school websites General educational reference These differ from student tasks, as the work of teachers differs from the work of students

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Categories of teacher use The Internet offers significant help of other kinds to teachers A tentative list: Lesson preparation Official advice and support Professional engagement Commercial support Local school web sites Other general reference Examples in these various categories are stored and described in a little detail on my website (teaching maths) Click on the image

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Technology directions and standards Some contributions to the discussion … Purposes of using technology Improve student experience Facilitate student experimentation Support good teaching Focus on activities that cannot be done well easily without the aid of technology Issues of student access Teacher issues Professional development and support Integration into curriculum Examination practices Research to understand and improve practice Barry Kissane MATHTED

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Closing remarks NCTM Principle: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning. Technology does not teach: teachers do. Students don’t learn directly from technology: rather, they might learn from what they do with the technology. Barry Kissane MATHTED

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