Barry Kissane
Murdoch University
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 2009
2
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 2009
3
Typhoon Lupit
• The Internet
provided a
powerful picture
of the
approaching
storm
• This satellite
image was on
Monday morning
•
Click image to
access website
Typhoon images and information from http://www.tropicalstormrisk.com/
MATHTED 2009
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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 2009 5
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 2009
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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 2009
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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 2009
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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 2009
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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 2009 10
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 2009 11
… 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 2009 12
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 2009 13
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 2009 14
The rule of three
Symbolic representation
Graphical representation
X refers to number of years after 1980, Y1 is population (millions)
Numerical representation
Solving equation to find
population of 80 million
Barry Kissane
MATHTED 2009 15
Exploration with calculators
Importantly, graphics calculators provide
opportunities for experimentation, not just
computation, in many areas, including:
 Graphs
 Equations
 Statistics
 Probability
 Calculus
 Geometry
MATHTED 2009 16
Examples of calculator use
 There are many examples of graphics calculator
use discussed in papers on my website:
 http://wwwstaff.murdoch.edu.au/~kissane/epublications.htm
 There are also examples of use of scientific
calculators on the web:
 Go to http://edu.casio.com/dl/ 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 2009 17
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 2009 18
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 2009 19
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 2009 20
Tinkerplots
 Tinkerplots is a commercial program that




supports data analysis of various kinds
Especially important for the analysis of studentcollected 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
MATHTED 2009 21
Backpacks
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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 20th
century
 Data are from the wonderful
GapMinder website
 Data on personal wealth
(GDP per capita) are also
included
MATHTED 2009 23
Philippines population since 1900
Barry Kissane
MATHTED 2009 24
Philippines GDP per capita
The data tell a
powerful story of
the 20th 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 2009 25
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
 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
MATHTED 2009 26
Least squares line
MATHTED 2009 27
Accessing Tinkerplots and Fathom
 Further information about these two software




packages is available on the Key Curriculum
website
Evaluation copies of the software and many
examples are available.
Tinkerplots
Fathom
From the Philippines, these are available from
 Springer Hong Kong, springer@springer.com.hk
 Educational Resources Provider, http://www.123erp.com
 Learning Interactive, http://www.li.com.sg
Barry Kissane
MATHTED 2009 28
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.
MATHTED 2009 29
GeoGebra
 This is an excellent free package that handles
many aspects of geometry and algebra, and now
(with a spreadsheet) some statistics as well
 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
MATHTED 2009 30
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 2009 31
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 2009 32
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 2009 33
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 2009 34
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 2009 35
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 2009 36
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 2009 37
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 2009 38
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 2009 39
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 2009 40
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
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
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
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
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
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 2009 46
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 2009 47
http://wwwstaff.murdoch.edu.au/~kissane
MATHTED 2009 48