Their world our worldccedilbridgingthe divide

ADRIANOLDKNOW

College Lane Chichester West Sussex PO19 6PE UK

[Submitted September 2009 accepted October 2009]

Students worldwide are gaining access to powerful computing devices and services Theyare learning from each other how to find and share information how to carry out usefultasks such as editing images and video where to find the best entertainment etcClassrooms especially in the UK are changing to make better access to computer tech-nology for teachers to use in presenting information to students The term lsquodigital dividersquowas used to distinguish between students who did and did not have access to informationand communication technology (ICT) Now there is an increasing digital divide betweenthe ways students use ICT for themselves and that which they experience at schoolThis article considers how that divide may be narrowed by involving students morecentrally in their own learning It is illustrated by material from recent school-based pro-jects including examples of ICT-based activities used to help make mathematics moreexciting relevant and challenging to young learnersccediland also to develop itscross-curricular use in science technology engineering and mathematics

1 IntroductionThis is a very exciting time in the development of the educational use of information and communi-

cation technology (ICT) because of recent breakthroughs in technology which are making mobile

computing devices ever smaller powerful robust affordable and practicable (Oldknow 2008a) Much

of this is the result of new technologies developed within the Asian region as foreseen in the lsquoShift

happensrsquo presentation (Microsoft 2008)

In the UK we have already seen considerable developments in the educational use of ICT to support

classroom teaching of mathematics with nearly all teachers having access to laptops data projectors

and the Internet and most also having the use of interactive white boards (IWBmdashFig 1A) and virtual

learning environment platforms (VLE) Most students in secondary education (grades 11ndash19) will not

experience much if any hands-on use of computer technology during normal mathematics lessonsmdash

even though many schools have bought site licenses for mathematics software Some schools use

graphical calculators and some are adopting TI-Nspire handhelds Increasing numbers of schools are

now adopting laptops (Fig 1B) net-books (Fig 2) etc as when the price falls the functionality

stabilizes (eg Windows XP as a common platform)

By contrast increasing numbers of students have personal ownership of a mobile phone many of

which can also be used as web-browsers and digital cameras as well as access to games consoles and

Email aoldknowyahoocouk

Teaching Mathematics and Its Applications (2009) 28 180^195doi101093teamathrp027

The Author 2009 Published by Oxford University Press on behalf of The Institute of Mathematics and its ApplicationsAll rights reserved For permissions please email journalspermissionsoxfordjournalsorg

home computers The development of new generations of processors such as the Intel Atom will

ensure a convergence between the functionality of personal computing devices The article considers

some specific examples in which this might be used to educational advantage

2 Mathematics from holiday snapshotsThe first example from (Oldknow 2008b) is based on a photograph taken in Prague after the

ICTMT-8 conference See Oldknow (2003) for further ideas

The photograph in Figure 3 depicts a street lamp with an interesting geometric form together with a

human figure to provide an idea of scale Can you estimate the volume and mass of the concrete post

Photographs can now be imported into a wide variety of mathematical software for further analysis

(including Excel and Autograph)mdashbut in this case I have chosen to use geometry software

Using Cabri II Plus we can easily convert measurements taken from the image into their actual

scale as in Fig 4A These enable us to build for example physical scale models of objects using

FIG 2 An Intel Classmate PC (This figure appears in colour in the online version of Teaching Mathematicsand its Application)

FIG 1 (A and B) A well-equipped London classroom (This figure appears in colour in the online version ofTeaching Mathematics and its Application)

A OLDKNOW 181

FIG 3 A piece of cubist art in Prague (This figure appears in colour in the online version of TeachingMathematics and its Application)

FIG 4 (A and B) Modelling the geometry in 2- and 3-dimensions (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE182

materials such as card In this case though we can use Cabri 3D as a vehicle both to build the model

and from which to take measurements

The basic building block is a frustum of a hexagonal pyramid as in Fig 4B Ten of these form the

main lamp post and we can perform successive reflections of the basic polyhedron in the parallel faces

to build the object By scaling the measurements we can easily come up with a good approximation to

the actual volume and by finding out the density of concrete we can also make a good estimate of its

mass Therefore it is a good idea to take a digital camera with you on your trips and to keep an eye out

for interesting objects An example of this can be seen (Teachers TV 2009) where students used the

apparatus at a childrenrsquos playground as the basis for a geometry project

3 Mathematics from video clipsIn some cases a still image will allow us to capture a great deal of data from a dynamical systemmdashsuch as

of the water-spout from the Merlion fountain in Singapore harbour in Fig 5 Knowing an actual distance

such as the spout above the water we can fit a quadratic function and use it to calculate information such

as the velocity with which water leaves the spout In other circumstances such as with a ball being

thrown there is not enough information in a still but we can get a lot more from a short video clipmdash

especially with free tools to edit and convert video formats and to digitize and analyze them

Digital video cameras have also become more widely availablemdashas well as being cheaper smaller

and more user-friendly The new range of Casio Exilim digital cameras will capture video in high

quality at 210 fps and in lower resolution at 420 and 1000 fps However for most classroom use the

lowest resolution (eg 320 240) short lsquoavirsquo video clip taken in the video mode of a digital camera or

mobile phone is all that is required A childrenrsquos playground with its dynamic apparatus like slides

swings and roundabouts is one good source of materialmdashespecially if the person being filmed can also

be equipped with some data-logging equipment eg for acceleration Another is the flight of a ball in a

wide variety of sports and games The important thing to bear in mind is that the camera needs to be

FIG 5 Modelling the speed of a water spout (This figure appears in colour in the online version of TeachingMathematics and its Application)

A OLDKNOW 183

held still preferably using a tripod and should not be used to track or zoom on the moving object

under question Ideally too there should be an object of known dimensions within the field of view to

help in calibration and the object being tracked should be far enough away to reduce effects of

perspective and parallax

A different approach to the use of video is illustrated by the work of the UK Teachersrsquo TV service

which broadcasts digital free programs which can be viewed online and also downloaded by UK

teachers who have registered on the site This has rapidly become an unexpectedly effective CPD

resource and there are many excellent programs to support the effective teaching of mathematics

Some of these are particularly relevant to the subject of this current paper Thus for instance the

program (Teachers TV 2007) demonstrates the use of data-capture from video clips of free throws at

basket ball as a source of data for modelling using quadratic functions In the example shown the

teacher uses the US Open source physics software called Vidshell (Davis 2000) to calibrate and

annotate a video clip The resulting still screen image is captured and pasted into geometry software

where the students choose appropriate origin and scales before modelling the trajectory of the ball with

quadratic functions The more recent Tracker 2 software (Brown 2009mdashsee Fig 6) is another US

Open Source Physics (hence free) productmdashthis time in Java Other software such as CMA Coach

and Vernierrsquos Logger Pro 3 (see Fig 7) also supports video analysis alongside other data capture

Some students are beginning to use the free Jing software to produce their own instructional videos on

how to use software toolsmdashboth for their peers and for their teachers

FIG 6 Modelling a basketball throw using Tracker video analysis (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE184

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

home computers The development of new generations of processors such as the Intel Atom will

ensure a convergence between the functionality of personal computing devices The article considers

some specific examples in which this might be used to educational advantage

2 Mathematics from holiday snapshotsThe first example from (Oldknow 2008b) is based on a photograph taken in Prague after the

ICTMT-8 conference See Oldknow (2003) for further ideas

The photograph in Figure 3 depicts a street lamp with an interesting geometric form together with a

human figure to provide an idea of scale Can you estimate the volume and mass of the concrete post

Photographs can now be imported into a wide variety of mathematical software for further analysis

(including Excel and Autograph)mdashbut in this case I have chosen to use geometry software

Using Cabri II Plus we can easily convert measurements taken from the image into their actual

scale as in Fig 4A These enable us to build for example physical scale models of objects using

FIG 2 An Intel Classmate PC (This figure appears in colour in the online version of Teaching Mathematicsand its Application)

FIG 1 (A and B) A well-equipped London classroom (This figure appears in colour in the online version ofTeaching Mathematics and its Application)

A OLDKNOW 181

FIG 3 A piece of cubist art in Prague (This figure appears in colour in the online version of TeachingMathematics and its Application)

FIG 4 (A and B) Modelling the geometry in 2- and 3-dimensions (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE182

materials such as card In this case though we can use Cabri 3D as a vehicle both to build the model

and from which to take measurements

The basic building block is a frustum of a hexagonal pyramid as in Fig 4B Ten of these form the

main lamp post and we can perform successive reflections of the basic polyhedron in the parallel faces

to build the object By scaling the measurements we can easily come up with a good approximation to

the actual volume and by finding out the density of concrete we can also make a good estimate of its

mass Therefore it is a good idea to take a digital camera with you on your trips and to keep an eye out

for interesting objects An example of this can be seen (Teachers TV 2009) where students used the

apparatus at a childrenrsquos playground as the basis for a geometry project

3 Mathematics from video clipsIn some cases a still image will allow us to capture a great deal of data from a dynamical systemmdashsuch as

of the water-spout from the Merlion fountain in Singapore harbour in Fig 5 Knowing an actual distance

such as the spout above the water we can fit a quadratic function and use it to calculate information such

as the velocity with which water leaves the spout In other circumstances such as with a ball being

thrown there is not enough information in a still but we can get a lot more from a short video clipmdash

especially with free tools to edit and convert video formats and to digitize and analyze them

Digital video cameras have also become more widely availablemdashas well as being cheaper smaller

and more user-friendly The new range of Casio Exilim digital cameras will capture video in high

quality at 210 fps and in lower resolution at 420 and 1000 fps However for most classroom use the

lowest resolution (eg 320 240) short lsquoavirsquo video clip taken in the video mode of a digital camera or

mobile phone is all that is required A childrenrsquos playground with its dynamic apparatus like slides

swings and roundabouts is one good source of materialmdashespecially if the person being filmed can also

be equipped with some data-logging equipment eg for acceleration Another is the flight of a ball in a

wide variety of sports and games The important thing to bear in mind is that the camera needs to be

FIG 5 Modelling the speed of a water spout (This figure appears in colour in the online version of TeachingMathematics and its Application)

A OLDKNOW 183

held still preferably using a tripod and should not be used to track or zoom on the moving object

under question Ideally too there should be an object of known dimensions within the field of view to

help in calibration and the object being tracked should be far enough away to reduce effects of

perspective and parallax

A different approach to the use of video is illustrated by the work of the UK Teachersrsquo TV service

which broadcasts digital free programs which can be viewed online and also downloaded by UK

teachers who have registered on the site This has rapidly become an unexpectedly effective CPD

resource and there are many excellent programs to support the effective teaching of mathematics

Some of these are particularly relevant to the subject of this current paper Thus for instance the

program (Teachers TV 2007) demonstrates the use of data-capture from video clips of free throws at

basket ball as a source of data for modelling using quadratic functions In the example shown the

teacher uses the US Open source physics software called Vidshell (Davis 2000) to calibrate and

annotate a video clip The resulting still screen image is captured and pasted into geometry software

where the students choose appropriate origin and scales before modelling the trajectory of the ball with

quadratic functions The more recent Tracker 2 software (Brown 2009mdashsee Fig 6) is another US

Open Source Physics (hence free) productmdashthis time in Java Other software such as CMA Coach

and Vernierrsquos Logger Pro 3 (see Fig 7) also supports video analysis alongside other data capture

Some students are beginning to use the free Jing software to produce their own instructional videos on

how to use software toolsmdashboth for their peers and for their teachers

FIG 6 Modelling a basketball throw using Tracker video analysis (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE184

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

FIG 3 A piece of cubist art in Prague (This figure appears in colour in the online version of TeachingMathematics and its Application)

FIG 4 (A and B) Modelling the geometry in 2- and 3-dimensions (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE182

materials such as card In this case though we can use Cabri 3D as a vehicle both to build the model

and from which to take measurements

The basic building block is a frustum of a hexagonal pyramid as in Fig 4B Ten of these form the

main lamp post and we can perform successive reflections of the basic polyhedron in the parallel faces

to build the object By scaling the measurements we can easily come up with a good approximation to

the actual volume and by finding out the density of concrete we can also make a good estimate of its

mass Therefore it is a good idea to take a digital camera with you on your trips and to keep an eye out

for interesting objects An example of this can be seen (Teachers TV 2009) where students used the

apparatus at a childrenrsquos playground as the basis for a geometry project

3 Mathematics from video clipsIn some cases a still image will allow us to capture a great deal of data from a dynamical systemmdashsuch as

of the water-spout from the Merlion fountain in Singapore harbour in Fig 5 Knowing an actual distance

such as the spout above the water we can fit a quadratic function and use it to calculate information such

as the velocity with which water leaves the spout In other circumstances such as with a ball being

thrown there is not enough information in a still but we can get a lot more from a short video clipmdash

especially with free tools to edit and convert video formats and to digitize and analyze them

Digital video cameras have also become more widely availablemdashas well as being cheaper smaller

and more user-friendly The new range of Casio Exilim digital cameras will capture video in high

quality at 210 fps and in lower resolution at 420 and 1000 fps However for most classroom use the

lowest resolution (eg 320 240) short lsquoavirsquo video clip taken in the video mode of a digital camera or

mobile phone is all that is required A childrenrsquos playground with its dynamic apparatus like slides

swings and roundabouts is one good source of materialmdashespecially if the person being filmed can also

be equipped with some data-logging equipment eg for acceleration Another is the flight of a ball in a

wide variety of sports and games The important thing to bear in mind is that the camera needs to be

FIG 5 Modelling the speed of a water spout (This figure appears in colour in the online version of TeachingMathematics and its Application)

A OLDKNOW 183

held still preferably using a tripod and should not be used to track or zoom on the moving object

under question Ideally too there should be an object of known dimensions within the field of view to

help in calibration and the object being tracked should be far enough away to reduce effects of

perspective and parallax

A different approach to the use of video is illustrated by the work of the UK Teachersrsquo TV service

which broadcasts digital free programs which can be viewed online and also downloaded by UK

teachers who have registered on the site This has rapidly become an unexpectedly effective CPD

resource and there are many excellent programs to support the effective teaching of mathematics

Some of these are particularly relevant to the subject of this current paper Thus for instance the

program (Teachers TV 2007) demonstrates the use of data-capture from video clips of free throws at

basket ball as a source of data for modelling using quadratic functions In the example shown the

teacher uses the US Open source physics software called Vidshell (Davis 2000) to calibrate and

annotate a video clip The resulting still screen image is captured and pasted into geometry software

where the students choose appropriate origin and scales before modelling the trajectory of the ball with

quadratic functions The more recent Tracker 2 software (Brown 2009mdashsee Fig 6) is another US

Open Source Physics (hence free) productmdashthis time in Java Other software such as CMA Coach

and Vernierrsquos Logger Pro 3 (see Fig 7) also supports video analysis alongside other data capture

Some students are beginning to use the free Jing software to produce their own instructional videos on

how to use software toolsmdashboth for their peers and for their teachers

FIG 6 Modelling a basketball throw using Tracker video analysis (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE184

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

materials such as card In this case though we can use Cabri 3D as a vehicle both to build the model

and from which to take measurements

The basic building block is a frustum of a hexagonal pyramid as in Fig 4B Ten of these form the

main lamp post and we can perform successive reflections of the basic polyhedron in the parallel faces

to build the object By scaling the measurements we can easily come up with a good approximation to

the actual volume and by finding out the density of concrete we can also make a good estimate of its

mass Therefore it is a good idea to take a digital camera with you on your trips and to keep an eye out

for interesting objects An example of this can be seen (Teachers TV 2009) where students used the

apparatus at a childrenrsquos playground as the basis for a geometry project

3 Mathematics from video clipsIn some cases a still image will allow us to capture a great deal of data from a dynamical systemmdashsuch as

of the water-spout from the Merlion fountain in Singapore harbour in Fig 5 Knowing an actual distance

such as the spout above the water we can fit a quadratic function and use it to calculate information such

as the velocity with which water leaves the spout In other circumstances such as with a ball being

thrown there is not enough information in a still but we can get a lot more from a short video clipmdash

especially with free tools to edit and convert video formats and to digitize and analyze them

Digital video cameras have also become more widely availablemdashas well as being cheaper smaller

and more user-friendly The new range of Casio Exilim digital cameras will capture video in high

quality at 210 fps and in lower resolution at 420 and 1000 fps However for most classroom use the

lowest resolution (eg 320 240) short lsquoavirsquo video clip taken in the video mode of a digital camera or

mobile phone is all that is required A childrenrsquos playground with its dynamic apparatus like slides

swings and roundabouts is one good source of materialmdashespecially if the person being filmed can also

be equipped with some data-logging equipment eg for acceleration Another is the flight of a ball in a

wide variety of sports and games The important thing to bear in mind is that the camera needs to be

FIG 5 Modelling the speed of a water spout (This figure appears in colour in the online version of TeachingMathematics and its Application)

A OLDKNOW 183

held still preferably using a tripod and should not be used to track or zoom on the moving object

under question Ideally too there should be an object of known dimensions within the field of view to

help in calibration and the object being tracked should be far enough away to reduce effects of

perspective and parallax

A different approach to the use of video is illustrated by the work of the UK Teachersrsquo TV service

which broadcasts digital free programs which can be viewed online and also downloaded by UK

teachers who have registered on the site This has rapidly become an unexpectedly effective CPD

resource and there are many excellent programs to support the effective teaching of mathematics

Some of these are particularly relevant to the subject of this current paper Thus for instance the

program (Teachers TV 2007) demonstrates the use of data-capture from video clips of free throws at

basket ball as a source of data for modelling using quadratic functions In the example shown the

teacher uses the US Open source physics software called Vidshell (Davis 2000) to calibrate and

annotate a video clip The resulting still screen image is captured and pasted into geometry software

where the students choose appropriate origin and scales before modelling the trajectory of the ball with

quadratic functions The more recent Tracker 2 software (Brown 2009mdashsee Fig 6) is another US

Open Source Physics (hence free) productmdashthis time in Java Other software such as CMA Coach

and Vernierrsquos Logger Pro 3 (see Fig 7) also supports video analysis alongside other data capture

Some students are beginning to use the free Jing software to produce their own instructional videos on

how to use software toolsmdashboth for their peers and for their teachers

FIG 6 Modelling a basketball throw using Tracker video analysis (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE184

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

held still preferably using a tripod and should not be used to track or zoom on the moving object

under question Ideally too there should be an object of known dimensions within the field of view to

help in calibration and the object being tracked should be far enough away to reduce effects of

perspective and parallax

A different approach to the use of video is illustrated by the work of the UK Teachersrsquo TV service

which broadcasts digital free programs which can be viewed online and also downloaded by UK

teachers who have registered on the site This has rapidly become an unexpectedly effective CPD

resource and there are many excellent programs to support the effective teaching of mathematics

Some of these are particularly relevant to the subject of this current paper Thus for instance the

program (Teachers TV 2007) demonstrates the use of data-capture from video clips of free throws at

basket ball as a source of data for modelling using quadratic functions In the example shown the

teacher uses the US Open source physics software called Vidshell (Davis 2000) to calibrate and

annotate a video clip The resulting still screen image is captured and pasted into geometry software

where the students choose appropriate origin and scales before modelling the trajectory of the ball with

quadratic functions The more recent Tracker 2 software (Brown 2009mdashsee Fig 6) is another US

Open Source Physics (hence free) productmdashthis time in Java Other software such as CMA Coach

and Vernierrsquos Logger Pro 3 (see Fig 7) also supports video analysis alongside other data capture

Some students are beginning to use the free Jing software to produce their own instructional videos on

how to use software toolsmdashboth for their peers and for their teachers

FIG 6 Modelling a basketball throw using Tracker video analysis (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

BRIDGING THE DIVIDE184

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

4 Mathematical modelling and visualization in 3DThe UK secondary school curriculum contains a subject called lsquoDesign and Technologyrsquo (DampT) which

involves aspects such as resistant media (wood metal plastic) systems amp control (electronics mech-

anisms hydraulics) food technology and textilesmdashwith an emphasis on design testing and making

A major recent government initiative has been to introduce CADCAM in schoolsrsquo DampT departments

httpwwwcadinschoolsorg In mathematics students get very little 3D workmdashmaybe a few ques-

tions where they have to identify and solve a right-angled triangle within a 3D diagram Yet we live in

a 3D world which many people have to control or simulate as part of their daily lives The availability

of powerful 3D modelling tools particularly Cabri 3D which won a coveted BETT award in 2007

now enables mathematics teachers to develop approaches to using ICT to encourage pupilsrsquo powers of

visualization and modelling in 3D as recommended in the Royal Societyrsquos Geometry report (Royal

Society 2001) The Mathematical Association and the Association of Teachers of Mathematics sup-

ported by the British Educational and Communications Technology Agency (Becta) have been work-

ing with a group of teachers to produce case studies of the use of ICT to support the teaching of lsquohard

to teachrsquo aspects of the secondary curriculum These case studies are featured on a Teachersrsquo TV

program (Teachers TV 2008amdashsee Figs 8A and 8B) For example a particularly powerful feature in

the Cabri 3D software to aid visualization is the ability to fold back the sides of a 3D solid shape to

show its net and to reveal its interior

FIG 7 Finding the speed of a cricket ball using Vernierrsquos Logger Pro 3 software (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 185

Oldknow and Tetlow reported on other aspects of the use of 3D modelling tools for mathematics in

their contribution to the 2007 ICTMT-8 conference (Oldknow amp Tetlow 2008) Again we can relate

this to the technological world the students inhabit by building simulations and models with 3D

geometry softwaremdashshowing for example central orbits and spherical triangles The route to the

conference from my hotel passes by the Temple Neuf The spire Fig 9A on the main tower looks on

first sight like a cone but it is probably a pyramid on an octagonal base as are the pair of smaller ones

below The tower at the end is more unusual appearing to have faces which are rhombusesmdashFig 9B

We might conjecture that if we continued the four slanting ridges downward they would strike the

plane through the bases of the four vertical triangles in a square and so that the tower is actually

formed by slicing a square-based pyramid with four vertical planes It is something which Cabri 3D

makes very easy to investigate as in Fig 9C

FIG 8 (A and B) Excerpts from Teachersrsquo TV lsquoHard to teach Mathematicsrsquo (This figure appears in colour inthe online version of Teaching Mathematics and its Application)

FIG 9 (AndashC) Modelling a tower in the Temple Neuf Metz (This figure appears in colour in the online versionof Teaching Mathematics and its Application)

BRIDGING THE DIVIDE186

5 Modelling motion with real dataWe have seen one example of capturing data from a moving object using video analysis Another

significant development in the educational use of ICT has been in the area of data-logging and

sensorsmdashwhich are now an integral part of the English science curriculum There are many experi-

ments which do not require specialized apparatus and which can be performed safely inside a math-

ematics classroom or outdoors One of the simplest and most powerful applications of data-logging

for mathematics is capturing time and distance data for a moving object using a lsquomotion detectorrsquo such

as the TI CBR2mdashsee Figs 10AndashC This device costs around pound75 and collects data on the distance to

the nearest object It is ideal for producing displacement-time graphs as shown in Fig 11 You can see

this in use for whole class teaching on Teachersrsquo TV together with a review (Teachers TV 2006a b)

In this example we have captured data on a student walking away from the CBR at a fairly state pace

then stopping briefly before coming back a bit faster We do not know which way she was facing but

of course those present in the room have captured a lsquovideorsquo in their memory against which they can

compare the distance versus time graph

Ideas for cross-curricular data-logging and modelling at Key Stage 3 (11ndash14) can be found in

(Oldknow amp Taylor 1998) Oldknow and Taylor reported on a number of data-logging activities to

FIG 10 (AndashC) TI-SmartView in the classroom TI-84 Plus and TI-Nspire for outside (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 11 Capturing data for a distance time graph

A OLDKNOW 187

support cross-curricular work (11ndash19) in the ICTMT-8 conference (Oldknow amp Taylor 2008) The

CBR can also be used to track the rebounds of a bouncing ball as in Figs 12AndashC In this case we are

looking at a distancendashtime graph of a ball which is (more-or-less) bouncing on the spot and not a

trajectory in space like the basket-ball throw

We can fit a quadratic function to any one of the complete bouncesmdasheither by

trial-and-improvement as shown or by using a built-in lsquobest-fitrsquo (or lsquoregressionrsquo) functionmdashwhich

is available in the latest version of Excel and in many good mathematical software packages such as

TI-Nspire which was used for the images above Another interesting (but harder) problem is to find a

function to model the successive maximum points on the graph At first sight they look like an

exponential decaymdashbut surprisingly they are well-fitted by a quadratic function Clearly the lsquobest-fitrsquo

graph y = f3(x) which has a minimum at y = 017 cannot be a realistic modelmdashas it would suggest that

all the bounces would be at least 017 m high But as each bounce takes a shorter time there is the

possibility that an infinite number of bounces could take place in a finite time The function f4(x) is a

quadratic function which passes through the point where t = 0 and the point P on the t-axis By sliding

P we can make the function pass as close as possible to the data pointsmdashand see that all bounces will

have died out in about 75 s

We have seen two different approaches to data-capturemdashone using a sensor (in this case a motion

detector) and the other using a video clip Now there is software which brings them togethermdashand

which also supports wireless connection with the data-logger to a computer or hand-held device One

way to do this is to set the logger to capture and store data automatically and to download it to the

device after the experiment Another is to send the data in real-time wirelessly to the device using

eg a Bluetooth connection

Vernierrsquos Logger Pro 3 software supports video capture alongside other forms of experimental

data-logging This means for example that you can choose whether to let the software control

real-time video capture from say an attached web-cam while simultaneously logging data such as

force and acceleration from a springmass experiment or whether to video it independently and

subsequently to synchronize the resulting video clip with the displayed data Vernier also produces

a Wireless Dynamics Sensor System (WDSS) which can be connected using Bluetooth so that data can

be logged wirelessly in real time This has an altimeter 3 accelerometers and a force sensor The

picture in Figure 13 is a still from a video clip of an elephantine lsquobungy jumperrsquo connected to a strong

spring via the Vernier WDSS The wireless kit is detected by Logger Pro and used to capture and

display data such as acceleration and force The video clip is imported and adjusted so that a given

frame corresponds to a particular reading such as when displacement is at a minimum or maximum

The corresponding video graphical and numerical information are shown in the following Logger Pro

FIG 12 (AndashC) Bouncing ball data analyzed with TI-Nspire

BRIDGING THE DIVIDE188

screenshot Needless to say Logger Pro also contains tools to fit your own functions to graphs and to

compute best fits

Of course the modern world abounds in sensorsmdashjust think about some you can find in cars the

engine and other systems are continuously monitored such as brakes lights tyres etc the interior

climate can be controlled your position can be located by satellite and your parking assisted by

information about distances to neighbouring objects Modern game controllers use a variety of tech-

niques to let you simulate playing golf tennis 10-pin bowling etcmdashyour movements can be captured

wirelessly using accelerometers or by stereo video cameras Data whether captured from sensors or

video can be used to create scatter-grams onto which we can graph test functions to model the data

We can also use modelling and CAS tools to predict results from ideal models (usually in the form of

differential equations) either solved numerically (eg by Runge-Kutta techniques) or where possible

analytically using formal calculus Texas Instrumentsrsquo new TI-Nspire hand-helds and software work

directly with USB probes such as the CBR to capture data from experiments such as the oscillating

elephant The data-handling and analysis software greatly facilitates explorations of modelsmdashsuch as

with the phasendashplane diagram illustrated in Fig 14 AB

During the conference I took a drive with an unusual passenger in the back seat an Intel Classmate

PC attached to a Vernier wireless dynamic system and a USB GPS probemdashfor satellite tracking

Making sense of the plethora of resulting data for force accelerations latitude longitude altitude

time velocity and distance travelled could be very taxing and confusing However we can export data

such as for latitude longitude and altitude into mapping software such as Google Maps and trace our

route as in Fig 15 The colours used for the route are generated from the altitude data so we can see

from the Google Earth display in Fig 16 that most of the journey was at low altitude following the

River Moselle but that we climbed out of the valley at the end

The current version of Google Earth allows you to connect your own GPS receiver such as the

Garmin ExTrex used by hikers and cyclists and to show your route effectively in 3D

FIG 13 (A and B) A spring-mass system wirelessly modelled with Logger Pro (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 189

6 Putting the ideas togetherccedilis it rocket scienceThe UK as with many other countries is concerned to attract young people to study subjects perceived

to be vital to our future economic well-being These subjectsmdashscience technology engineering and

mathematicsmdashare together known as STEM

We looked for ideas of activities which would provide opportunities to integrate both the STEM

subjects and some of the varied mathematical approaches using ICT illustrated here Rockets made

a popular theme 2007 was the first year of a new national competition for UK schools run by the

UK Rocket Association (UKRA) httpwwwukayrocorgukmdashthe UK Aerospace Youth Rocket

Challenge

A team of Year 8 pupils (12ndash13) from one school entered for the challengemdashto design build and fly

a rocket which would reach a maximum altitude of 260 m and deliver an unbroken egg to earth in 45 s

The following year the school entered two teams one from Year 8 and one from Year 9 Both qualified

FIG 15 (A and B) Tracking with WDSS and GPS with Logger Pro and Google Maps (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 14 (A and B) TI-Nspire and CBR used for modelling

BRIDGING THE DIVIDE190

5 Modelling motion with real dataWe have seen one example of capturing data from a moving object using video analysis Another

significant development in the educational use of ICT has been in the area of data-logging and

sensorsmdashwhich are now an integral part of the English science curriculum There are many experi-

ments which do not require specialized apparatus and which can be performed safely inside a math-

ematics classroom or outdoors One of the simplest and most powerful applications of data-logging

for mathematics is capturing time and distance data for a moving object using a lsquomotion detectorrsquo such

as the TI CBR2mdashsee Figs 10AndashC This device costs around pound75 and collects data on the distance to

the nearest object It is ideal for producing displacement-time graphs as shown in Fig 11 You can see

this in use for whole class teaching on Teachersrsquo TV together with a review (Teachers TV 2006a b)

In this example we have captured data on a student walking away from the CBR at a fairly state pace

then stopping briefly before coming back a bit faster We do not know which way she was facing but

of course those present in the room have captured a lsquovideorsquo in their memory against which they can

compare the distance versus time graph

Ideas for cross-curricular data-logging and modelling at Key Stage 3 (11ndash14) can be found in

(Oldknow amp Taylor 1998) Oldknow and Taylor reported on a number of data-logging activities to

FIG 10 (AndashC) TI-SmartView in the classroom TI-84 Plus and TI-Nspire for outside (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 11 Capturing data for a distance time graph

A OLDKNOW 187

support cross-curricular work (11ndash19) in the ICTMT-8 conference (Oldknow amp Taylor 2008) The

CBR can also be used to track the rebounds of a bouncing ball as in Figs 12AndashC In this case we are

looking at a distancendashtime graph of a ball which is (more-or-less) bouncing on the spot and not a

trajectory in space like the basket-ball throw

We can fit a quadratic function to any one of the complete bouncesmdasheither by

trial-and-improvement as shown or by using a built-in lsquobest-fitrsquo (or lsquoregressionrsquo) functionmdashwhich

is available in the latest version of Excel and in many good mathematical software packages such as

TI-Nspire which was used for the images above Another interesting (but harder) problem is to find a

function to model the successive maximum points on the graph At first sight they look like an

exponential decaymdashbut surprisingly they are well-fitted by a quadratic function Clearly the lsquobest-fitrsquo

graph y = f3(x) which has a minimum at y = 017 cannot be a realistic modelmdashas it would suggest that

all the bounces would be at least 017 m high But as each bounce takes a shorter time there is the

possibility that an infinite number of bounces could take place in a finite time The function f4(x) is a

quadratic function which passes through the point where t = 0 and the point P on the t-axis By sliding

P we can make the function pass as close as possible to the data pointsmdashand see that all bounces will

have died out in about 75 s

We have seen two different approaches to data-capturemdashone using a sensor (in this case a motion

detector) and the other using a video clip Now there is software which brings them togethermdashand

which also supports wireless connection with the data-logger to a computer or hand-held device One

way to do this is to set the logger to capture and store data automatically and to download it to the

device after the experiment Another is to send the data in real-time wirelessly to the device using

eg a Bluetooth connection

Vernierrsquos Logger Pro 3 software supports video capture alongside other forms of experimental

data-logging This means for example that you can choose whether to let the software control

real-time video capture from say an attached web-cam while simultaneously logging data such as

force and acceleration from a springmass experiment or whether to video it independently and

subsequently to synchronize the resulting video clip with the displayed data Vernier also produces

a Wireless Dynamics Sensor System (WDSS) which can be connected using Bluetooth so that data can

be logged wirelessly in real time This has an altimeter 3 accelerometers and a force sensor The

picture in Figure 13 is a still from a video clip of an elephantine lsquobungy jumperrsquo connected to a strong

spring via the Vernier WDSS The wireless kit is detected by Logger Pro and used to capture and

display data such as acceleration and force The video clip is imported and adjusted so that a given

frame corresponds to a particular reading such as when displacement is at a minimum or maximum

The corresponding video graphical and numerical information are shown in the following Logger Pro

FIG 12 (AndashC) Bouncing ball data analyzed with TI-Nspire

BRIDGING THE DIVIDE188

screenshot Needless to say Logger Pro also contains tools to fit your own functions to graphs and to

compute best fits

Of course the modern world abounds in sensorsmdashjust think about some you can find in cars the

engine and other systems are continuously monitored such as brakes lights tyres etc the interior

climate can be controlled your position can be located by satellite and your parking assisted by

information about distances to neighbouring objects Modern game controllers use a variety of tech-

niques to let you simulate playing golf tennis 10-pin bowling etcmdashyour movements can be captured

wirelessly using accelerometers or by stereo video cameras Data whether captured from sensors or

video can be used to create scatter-grams onto which we can graph test functions to model the data

We can also use modelling and CAS tools to predict results from ideal models (usually in the form of

differential equations) either solved numerically (eg by Runge-Kutta techniques) or where possible

analytically using formal calculus Texas Instrumentsrsquo new TI-Nspire hand-helds and software work

directly with USB probes such as the CBR to capture data from experiments such as the oscillating

elephant The data-handling and analysis software greatly facilitates explorations of modelsmdashsuch as

with the phasendashplane diagram illustrated in Fig 14 AB

During the conference I took a drive with an unusual passenger in the back seat an Intel Classmate

PC attached to a Vernier wireless dynamic system and a USB GPS probemdashfor satellite tracking

Making sense of the plethora of resulting data for force accelerations latitude longitude altitude

time velocity and distance travelled could be very taxing and confusing However we can export data

such as for latitude longitude and altitude into mapping software such as Google Maps and trace our

route as in Fig 15 The colours used for the route are generated from the altitude data so we can see

from the Google Earth display in Fig 16 that most of the journey was at low altitude following the

River Moselle but that we climbed out of the valley at the end

The current version of Google Earth allows you to connect your own GPS receiver such as the

Garmin ExTrex used by hikers and cyclists and to show your route effectively in 3D

FIG 13 (A and B) A spring-mass system wirelessly modelled with Logger Pro (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 189

6 Putting the ideas togetherccedilis it rocket scienceThe UK as with many other countries is concerned to attract young people to study subjects perceived

to be vital to our future economic well-being These subjectsmdashscience technology engineering and

mathematicsmdashare together known as STEM

We looked for ideas of activities which would provide opportunities to integrate both the STEM

subjects and some of the varied mathematical approaches using ICT illustrated here Rockets made

a popular theme 2007 was the first year of a new national competition for UK schools run by the

UK Rocket Association (UKRA) httpwwwukayrocorgukmdashthe UK Aerospace Youth Rocket

Challenge

A team of Year 8 pupils (12ndash13) from one school entered for the challengemdashto design build and fly

a rocket which would reach a maximum altitude of 260 m and deliver an unbroken egg to earth in 45 s

The following year the school entered two teams one from Year 8 and one from Year 9 Both qualified

FIG 15 (A and B) Tracking with WDSS and GPS with Logger Pro and Google Maps (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 14 (A and B) TI-Nspire and CBR used for modelling

BRIDGING THE DIVIDE190

support cross-curricular work (11ndash19) in the ICTMT-8 conference (Oldknow amp Taylor 2008) The

CBR can also be used to track the rebounds of a bouncing ball as in Figs 12AndashC In this case we are

looking at a distancendashtime graph of a ball which is (more-or-less) bouncing on the spot and not a

trajectory in space like the basket-ball throw

We can fit a quadratic function to any one of the complete bouncesmdasheither by

trial-and-improvement as shown or by using a built-in lsquobest-fitrsquo (or lsquoregressionrsquo) functionmdashwhich

is available in the latest version of Excel and in many good mathematical software packages such as

TI-Nspire which was used for the images above Another interesting (but harder) problem is to find a

function to model the successive maximum points on the graph At first sight they look like an

exponential decaymdashbut surprisingly they are well-fitted by a quadratic function Clearly the lsquobest-fitrsquo

graph y = f3(x) which has a minimum at y = 017 cannot be a realistic modelmdashas it would suggest that

all the bounces would be at least 017 m high But as each bounce takes a shorter time there is the

possibility that an infinite number of bounces could take place in a finite time The function f4(x) is a

quadratic function which passes through the point where t = 0 and the point P on the t-axis By sliding

P we can make the function pass as close as possible to the data pointsmdashand see that all bounces will

have died out in about 75 s

We have seen two different approaches to data-capturemdashone using a sensor (in this case a motion

detector) and the other using a video clip Now there is software which brings them togethermdashand

which also supports wireless connection with the data-logger to a computer or hand-held device One

way to do this is to set the logger to capture and store data automatically and to download it to the

device after the experiment Another is to send the data in real-time wirelessly to the device using

eg a Bluetooth connection

Vernierrsquos Logger Pro 3 software supports video capture alongside other forms of experimental

data-logging This means for example that you can choose whether to let the software control

real-time video capture from say an attached web-cam while simultaneously logging data such as

force and acceleration from a springmass experiment or whether to video it independently and

subsequently to synchronize the resulting video clip with the displayed data Vernier also produces

a Wireless Dynamics Sensor System (WDSS) which can be connected using Bluetooth so that data can

be logged wirelessly in real time This has an altimeter 3 accelerometers and a force sensor The

picture in Figure 13 is a still from a video clip of an elephantine lsquobungy jumperrsquo connected to a strong

spring via the Vernier WDSS The wireless kit is detected by Logger Pro and used to capture and

display data such as acceleration and force The video clip is imported and adjusted so that a given

frame corresponds to a particular reading such as when displacement is at a minimum or maximum

The corresponding video graphical and numerical information are shown in the following Logger Pro

FIG 12 (AndashC) Bouncing ball data analyzed with TI-Nspire

BRIDGING THE DIVIDE188

screenshot Needless to say Logger Pro also contains tools to fit your own functions to graphs and to

compute best fits

Of course the modern world abounds in sensorsmdashjust think about some you can find in cars the

engine and other systems are continuously monitored such as brakes lights tyres etc the interior

climate can be controlled your position can be located by satellite and your parking assisted by

information about distances to neighbouring objects Modern game controllers use a variety of tech-

niques to let you simulate playing golf tennis 10-pin bowling etcmdashyour movements can be captured

wirelessly using accelerometers or by stereo video cameras Data whether captured from sensors or

video can be used to create scatter-grams onto which we can graph test functions to model the data

We can also use modelling and CAS tools to predict results from ideal models (usually in the form of

differential equations) either solved numerically (eg by Runge-Kutta techniques) or where possible

analytically using formal calculus Texas Instrumentsrsquo new TI-Nspire hand-helds and software work

directly with USB probes such as the CBR to capture data from experiments such as the oscillating

elephant The data-handling and analysis software greatly facilitates explorations of modelsmdashsuch as

with the phasendashplane diagram illustrated in Fig 14 AB

During the conference I took a drive with an unusual passenger in the back seat an Intel Classmate

PC attached to a Vernier wireless dynamic system and a USB GPS probemdashfor satellite tracking

Making sense of the plethora of resulting data for force accelerations latitude longitude altitude

time velocity and distance travelled could be very taxing and confusing However we can export data

such as for latitude longitude and altitude into mapping software such as Google Maps and trace our

route as in Fig 15 The colours used for the route are generated from the altitude data so we can see

from the Google Earth display in Fig 16 that most of the journey was at low altitude following the

River Moselle but that we climbed out of the valley at the end

The current version of Google Earth allows you to connect your own GPS receiver such as the

Garmin ExTrex used by hikers and cyclists and to show your route effectively in 3D

FIG 13 (A and B) A spring-mass system wirelessly modelled with Logger Pro (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 189

6 Putting the ideas togetherccedilis it rocket scienceThe UK as with many other countries is concerned to attract young people to study subjects perceived

to be vital to our future economic well-being These subjectsmdashscience technology engineering and

mathematicsmdashare together known as STEM

We looked for ideas of activities which would provide opportunities to integrate both the STEM

subjects and some of the varied mathematical approaches using ICT illustrated here Rockets made

a popular theme 2007 was the first year of a new national competition for UK schools run by the

UK Rocket Association (UKRA) httpwwwukayrocorgukmdashthe UK Aerospace Youth Rocket

Challenge

A team of Year 8 pupils (12ndash13) from one school entered for the challengemdashto design build and fly

a rocket which would reach a maximum altitude of 260 m and deliver an unbroken egg to earth in 45 s

The following year the school entered two teams one from Year 8 and one from Year 9 Both qualified

FIG 15 (A and B) Tracking with WDSS and GPS with Logger Pro and Google Maps (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 14 (A and B) TI-Nspire and CBR used for modelling

BRIDGING THE DIVIDE190

screenshot Needless to say Logger Pro also contains tools to fit your own functions to graphs and to

compute best fits

Of course the modern world abounds in sensorsmdashjust think about some you can find in cars the

engine and other systems are continuously monitored such as brakes lights tyres etc the interior

climate can be controlled your position can be located by satellite and your parking assisted by

information about distances to neighbouring objects Modern game controllers use a variety of tech-

niques to let you simulate playing golf tennis 10-pin bowling etcmdashyour movements can be captured

wirelessly using accelerometers or by stereo video cameras Data whether captured from sensors or

video can be used to create scatter-grams onto which we can graph test functions to model the data

We can also use modelling and CAS tools to predict results from ideal models (usually in the form of

differential equations) either solved numerically (eg by Runge-Kutta techniques) or where possible

analytically using formal calculus Texas Instrumentsrsquo new TI-Nspire hand-helds and software work

directly with USB probes such as the CBR to capture data from experiments such as the oscillating

elephant The data-handling and analysis software greatly facilitates explorations of modelsmdashsuch as

with the phasendashplane diagram illustrated in Fig 14 AB

During the conference I took a drive with an unusual passenger in the back seat an Intel Classmate

PC attached to a Vernier wireless dynamic system and a USB GPS probemdashfor satellite tracking

Making sense of the plethora of resulting data for force accelerations latitude longitude altitude

time velocity and distance travelled could be very taxing and confusing However we can export data

such as for latitude longitude and altitude into mapping software such as Google Maps and trace our

route as in Fig 15 The colours used for the route are generated from the altitude data so we can see

from the Google Earth display in Fig 16 that most of the journey was at low altitude following the

River Moselle but that we climbed out of the valley at the end

The current version of Google Earth allows you to connect your own GPS receiver such as the

Garmin ExTrex used by hikers and cyclists and to show your route effectively in 3D

FIG 13 (A and B) A spring-mass system wirelessly modelled with Logger Pro (This figure appears in colourin the online version of Teaching Mathematics and its Application)

A OLDKNOW 189

6 Putting the ideas togetherccedilis it rocket scienceThe UK as with many other countries is concerned to attract young people to study subjects perceived

to be vital to our future economic well-being These subjectsmdashscience technology engineering and

mathematicsmdashare together known as STEM

We looked for ideas of activities which would provide opportunities to integrate both the STEM

subjects and some of the varied mathematical approaches using ICT illustrated here Rockets made

a popular theme 2007 was the first year of a new national competition for UK schools run by the

UK Rocket Association (UKRA) httpwwwukayrocorgukmdashthe UK Aerospace Youth Rocket

Challenge

A team of Year 8 pupils (12ndash13) from one school entered for the challengemdashto design build and fly

a rocket which would reach a maximum altitude of 260 m and deliver an unbroken egg to earth in 45 s

The following year the school entered two teams one from Year 8 and one from Year 9 Both qualified

FIG 15 (A and B) Tracking with WDSS and GPS with Logger Pro and Google Maps (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 14 (A and B) TI-Nspire and CBR used for modelling

BRIDGING THE DIVIDE190

6 Putting the ideas togetherccedilis it rocket scienceThe UK as with many other countries is concerned to attract young people to study subjects perceived

to be vital to our future economic well-being These subjectsmdashscience technology engineering and

mathematicsmdashare together known as STEM

We looked for ideas of activities which would provide opportunities to integrate both the STEM

subjects and some of the varied mathematical approaches using ICT illustrated here Rockets made

a popular theme 2007 was the first year of a new national competition for UK schools run by the

UK Rocket Association (UKRA) httpwwwukayrocorgukmdashthe UK Aerospace Youth Rocket

Challenge

A team of Year 8 pupils (12ndash13) from one school entered for the challengemdashto design build and fly

a rocket which would reach a maximum altitude of 260 m and deliver an unbroken egg to earth in 45 s

The following year the school entered two teams one from Year 8 and one from Year 9 Both qualified

FIG 15 (A and B) Tracking with WDSS and GPS with Logger Pro and Google Maps (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 14 (A and B) TI-Nspire and CBR used for modelling

BRIDGING THE DIVIDE190

for the finals which took place in such foul weather that they were unable to make a final set of

launches and instead had to make team presentations to a panel of UK rocket scientists So while

STEM is at the heart of this competition the students also have to have excellent ICT and commu-

nication skills You can see the schoolrsquos video at httpwwwcroftonschoolcoukvideosrocket

rocket-projecthtml and the teamsrsquo presentations at httpwwwcroftonschoolcoukvideosrocket

rocket-presentationshtmlmdashfrom where the images in Fig 17 were taken Teachersrsquo TV also a pro-

gramme of some of the schoolrsquos cross-curricular approaches (Teachers TV 2008b) While the rocket

competition makes an excellent cross-curricular project it is rather expensive in terms of materials

equipment and time It also requires considerable care from the safety viewpointmdashas illustrated by the

images in Fig 18 So we needed to find a simpler experimental vehicle Also the mathematics

of powered flight proved too difficult to model at an elementary levelmdashso we needed to combine

projectile motion with ballistic rocketsmdashhence STOMP

The best solution we have found so far is a simple cheap air-powered plastic rocket called STOMP

Students jump on the plastic box to launch the rocket They make estimates of the launch angle the

distance travelled horizontally and the time in the air They also have a simple data-collection kit

consisting of a protractor a tape measure and a stop-watch They record their results in the field and

can analyze them using hand-held technology such as TI-84 Plus or TI-Nspire

On return to the classroom they can enter their data into software models eg in Excel dynamic

geometry or TI-Nspire software These aspects are illustrated in the six images in Fig 19 Using

expressions for average velocity and acceleration they derive Galileorsquos formulae and use them with

recorded values of launch angle distance travelled and time taken to compute values for g the launch

FIG 16 Tracking a journey with a GPS receiver and Google Earth (This figure appears in colour in the onlineversion of Teaching Mathematics and its Application)

A OLDKNOW 191

velocity and the maximum height and hence plot the trajectory From the trajectory they can even find

approximations for the distance travelled The analysis of the STOMP air rocket data depended on

using a simplified model from Newtonian mechanics in which spin wind and air resistance are

ignored

An excellent vehicle for testing the validity or otherwise of such a model is to use a rocket

equipped with a nose cone containing sensors such as height pressure and acceleration The

Sciencescope company has produced just such a productmdashwhich comes together with a more

high-powered rocket powered by compressed air httpwwwsciencescopecoukrocketloggerhtm

The data are downloaded following a flight and can be exported to Excel for analysis and display

The data can also be saved in a format which produces a simulated video of the launch in Google

Earth These aspects are illustrated in the three images in Fig 20 By comparing the recorded trajec-

tory against the GalileoNewton model students can see to what extent factors like air-resistance

can safely be ignored They can also explore possible models for air resistance as powers of velocity

Suddenly itrsquos really caught on

A real challenge then is to find more such rich activities linking mathematics to realistic and

stimulating applications which involve students to the fullmdashand ICT clearly has an important central

role Possible criteria for such rich activities are summarized in the acronym Al Fresco

Accessible Lively Fun Reliable Easily set up Safe Cheap Open-ended

FIG 18 (AndashC) Fireworks a mortar and a field gunmdashall launched using gunpowder (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 17 A rocket launch carrying two eggs (This figure appears in colour in the online version of TeachingMathematics and its Application)

BRIDGING THE DIVIDE192

7 ConclusionThe paper has given examples of ICT-based activities developed recently in a number of UK second-

ary schools They use readily available software and other ICT tools and have proved inspiring for

both students and teachers However the teachers have been prepared to take risksmdashand not every

activity attempted has been met with complete success One important dimension added by the

teachers has been to relate the activities to the world familiar to their students While most of them

realize that their mobile phones have digital cameras few realize that these phones would not work

were it not for the rocket science which put the communication satellites in orbitmdashor that it is the

FIG 20 (AndashC) The Sciencescope rocket gathers real data for analysis and display (This figure appears incolour in the online version of Teaching Mathematics and its Application)

FIG 19 STOMP (AndashF) simpler cheaper and safermdasha lot of fun even in the rain (This figure appears incolour in the online version of Teaching Mathematics and its Application)

A OLDKNOW 193

geometric properties of the parabola that enable them to watch satellite TV ICT cannot displace the

mathematics teacher but I hope to have exemplified how it can make him or her more effective in

winning the hearts and minds of the next generation of technologists

We have the technology Much of it is already in the hands of the learnermdashmore soon will They

have the skill to put it to any use As educators we need to find ways to unlock those skills so that they

increase studentsrsquo learning about the world of mathematics science and technology their curiosity

about how the objects around them work and the excitement challenge and pleasure they get from

relating the first with the second

REFERENCES

BROWN D (2009) Tracker ndash Video Analysis and Modeling Tool Open Source Physics Cabrillo College Aptos

CA USA Available at httpwwwcabrilloedudbrowntracker [accessed 13 November 2009]

DAVIS DV (2000) Webphysics ndash Vidshell New Hampshire Community Technology College Brunswick NH

USA Available at httpwwwwebphysicsccsnheduvidshellvidshellhtml [accessed 13 November 2009]

Microsoft (2008) Shift Happens Available at httpblogsmsdncomukschoolsarchive20080218shift-

happens-and-freezing-frogsaspx [accessed 13 November 2009]

OLDKNOW A (2008a) Itrsquos 2008 - So What You Got to Offer Then ndash Using ICT to Put Learners in Touch with

Mathematics National Centre of Excellence in the Teaching of Mathematics London UK Available at

httpwwwncetmorgukfiles251777Becta+article+AOpdf [accessed 13 November 2009]

OLDKNOW A (2008b) Cubism and Cabri Mathematics Teaching 206 January 2008 the Association of

Teachers of Mathematics (ATM) Derby UK Available at httpwwwadrianoldknoworgukATM-

MT206pdf [accessed 13 November 2009]

OLDKNOW A (2003a) Geometric and Algebraic Modelling with Dynamic Geometry Software Micromath

V19N2 ATM Derby UK Available at httpwwwatmorgukmtmicromathmm192oldknowpdf

[accessed 13 November 2009]

OLDKNOW A (2003b) Mathematics from Still and Video Images Micromath V19N2 ATM Derby UK

Available at httpwwwatmorgukmtmicromathmm192oldknowapdf [accessed 13 November 2009]

OLDKNOW A amp TAYLOR R (2008) Mathematics Science and Technology Teachers Working Collaboratively

with ICT eJMT V2N1 Radford University VA USA Available at httpsphpradfordedu~ejmt

deliverAbstractphppaperID=eJMT_v2n1n3 [accessed 13 November 2009]

OLDKNOW A amp TAYLOR R (1998) Data-capture and Modelling in Maths and Science NCET Coventry UK

Available at httpwwwadrianoldknoworgukdatacapturepdf [accessed 13 November 2009]

OLDKNOW A amp TETLOW L (2008) Using Dynamic Geometry Software to Encourage 3D Visualisation and

Modelling eJMT V2N1 Radford University VA USA Available at httpsphpradfordedu~ejmt

deliverAbstractphppaperID=eJMT_v2n1np4 [accessed 13 November 2009]

ROYAL SOCIETY (2001) Teaching and Learning Geometry 11ndash19 The Royal Society London UK Available at

httproyalsocietyorgdocumentaspid=1420 [accessed 13 November 2009]

TEACHERSrsquo TV (2006a) New Mathematics Technology in the Classroom Teachersrsquo TV London Available at

httpwwwteacherstvvideo154 [accessed 13 November 2009]

TEACHERSrsquo TV (2006b) Resource Review ndash Secondary Mathematics 2 Teachersrsquo TV London Available at

httpwwwteacherstvvideo4872 [accessed 13 November 2009]

TEACHERSrsquo TV (2007) Hard to Teach ndash Secondary Mathematics ndash Quadratic Function Teachersrsquo TV London

Available at httpwwwteacherstvvideo19119 [accessed 13 November 2009]

TEACHERSrsquo TV (2008a) Hard to Teach ndash Secondary Mathematics using ICT Teachersrsquo TV London Available

at httpwwwteacherstvvideo29853 [accessed 13 November 2009]

BRIDGING THE DIVIDE194

geometric properties of the parabola that enable them to watch satellite TV ICT cannot displace the

mathematics teacher but I hope to have exemplified how it can make him or her more effective in

winning the hearts and minds of the next generation of technologists

We have the technology Much of it is already in the hands of the learnermdashmore soon will They

have the skill to put it to any use As educators we need to find ways to unlock those skills so that they

increase studentsrsquo learning about the world of mathematics science and technology their curiosity

about how the objects around them work and the excitement challenge and pleasure they get from

relating the first with the second

REFERENCES

BROWN D (2009) Tracker ndash Video Analysis and Modeling Tool Open Source Physics Cabrillo College Aptos

CA USA Available at httpwwwcabrilloedudbrowntracker [accessed 13 November 2009]

DAVIS DV (2000) Webphysics ndash Vidshell New Hampshire Community Technology College Brunswick NH

USA Available at httpwwwwebphysicsccsnheduvidshellvidshellhtml [accessed 13 November 2009]

Microsoft (2008) Shift Happens Available at httpblogsmsdncomukschoolsarchive20080218shift-

happens-and-freezing-frogsaspx [accessed 13 November 2009]

OLDKNOW A (2008a) Itrsquos 2008 - So What You Got to Offer Then ndash Using ICT to Put Learners in Touch with

Mathematics National Centre of Excellence in the Teaching of Mathematics London UK Available at

httpwwwncetmorgukfiles251777Becta+article+AOpdf [accessed 13 November 2009]

OLDKNOW A (2008b) Cubism and Cabri Mathematics Teaching 206 January 2008 the Association of

Teachers of Mathematics (ATM) Derby UK Available at httpwwwadrianoldknoworgukATM-

MT206pdf [accessed 13 November 2009]

OLDKNOW A (2003a) Geometric and Algebraic Modelling with Dynamic Geometry Software Micromath

V19N2 ATM Derby UK Available at httpwwwatmorgukmtmicromathmm192oldknowpdf

[accessed 13 November 2009]

OLDKNOW A (2003b) Mathematics from Still and Video Images Micromath V19N2 ATM Derby UK

Available at httpwwwatmorgukmtmicromathmm192oldknowapdf [accessed 13 November 2009]

OLDKNOW A amp TAYLOR R (2008) Mathematics Science and Technology Teachers Working Collaboratively

with ICT eJMT V2N1 Radford University VA USA Available at httpsphpradfordedu~ejmt

deliverAbstractphppaperID=eJMT_v2n1n3 [accessed 13 November 2009]

OLDKNOW A amp TAYLOR R (1998) Data-capture and Modelling in Maths and Science NCET Coventry UK

Available at httpwwwadrianoldknoworgukdatacapturepdf [accessed 13 November 2009]

OLDKNOW A amp TETLOW L (2008) Using Dynamic Geometry Software to Encourage 3D Visualisation and

Modelling eJMT V2N1 Radford University VA USA Available at httpsphpradfordedu~ejmt

deliverAbstractphppaperID=eJMT_v2n1np4 [accessed 13 November 2009]

ROYAL SOCIETY (2001) Teaching and Learning Geometry 11ndash19 The Royal Society London UK Available at

httproyalsocietyorgdocumentaspid=1420 [accessed 13 November 2009]

TEACHERSrsquo TV (2006a) New Mathematics Technology in the Classroom Teachersrsquo TV London Available at

httpwwwteacherstvvideo154 [accessed 13 November 2009]

TEACHERSrsquo TV (2006b) Resource Review ndash Secondary Mathematics 2 Teachersrsquo TV London Available at

httpwwwteacherstvvideo4872 [accessed 13 November 2009]

TEACHERSrsquo TV (2007) Hard to Teach ndash Secondary Mathematics ndash Quadratic Function Teachersrsquo TV London

Available at httpwwwteacherstvvideo19119 [accessed 13 November 2009]

TEACHERSrsquo TV (2008a) Hard to Teach ndash Secondary Mathematics using ICT Teachersrsquo TV London Available

at httpwwwteacherstvvideo29853 [accessed 13 November 2009]

BRIDGING THE DIVIDE194

TEACHERSrsquo TV (2008b) Cross-curricular ndash Bath Bombs and Rockets Teachersrsquo TV London Available at http

wwwteacherstvvideo28859 [accessed 13 November 2009]

TEACHERSrsquo TV (2009) Mathematics for All ndash Geometry from the Playground Teachersrsquo TV London Available

at httpwwwteacherstvvideo37909 [accessed 13 November 2009]

Adrian Oldknow is Emeritus Professor at the University of Chichester UK He has retired from teaching

but continues to work with teachers and schools on embedding technology effectively to enhance learning

and teaching in mathematics and in integrated approaches to STEM He was co-editor of Teaching

Mathematics and its Applications

A OLDKNOW 195