Further Pure
Mathematics with
Technology
(FPT)
Richard
Lissaman
richard.lissaman@mei.org.uk
Programming in Mathematics
The links between programming and mathematics have always been
strong – computing was seen as a branch of mathematics
With the advent of IT as a school subject the emphasis shifted to
students learning to be users of programs and applications.
This coincided with a decrease in both the amount of computing,
and hence programming, taught in schools and reduced the explicit
link with mathematics
Technology in the maths classroom
Although there have been a number of projects in England in the
past 20-30 years to integrate the use of technology into the teaching
and learning in mathematics it is not used effectively in many
classrooms. This issue was raised by Ofsted (2008):
Several years ago, inspection evidence showed that most pupils had
some opportunities to use ICT as a tool to solve or explore
mathematical problems. This is no longer the case … despite
technological advances, the potential of ICT to enhance the
learning of mathematics is too rarely realised.
Promoting Code
Coding and Mathematics
This talk is about the work MEI is doing in this area:
• Getting coding (and CAS) into assessment of A level
Further Mathematics
• Ideas for exploring ideas in A level Mathematics
through coding games
A word about CAS
‘Technology’ to speed up numerical processes has been common
place throughout history. For example:
•
•
•
•
•
Abacus
Slide Rule
Napier’s Bones
Calculators (First 4 function calculator built around 1623!)
Leibniz’s Step Reckoner
These technologies speed up the process of numerical calculation,
but what about algebra?
A word about CAS
CAS is an attempt to extend the use of technology beyond
the numerical and into the algebraic.
CAS can handle
expanding, factorising, simplifying
calculus
solving equations
limits
Demonstration of CAS
Text
Real world application accessible from A
level
Google Page Rank Algorithm
Much more valuable example
Google Page Rank Algorithm
Parallels between proof and code
Property
If m and n are both even or m and n are
both odd then m + n is even
Proof
If m and n are both even then there exist
integers k and l such that m = 2k and n =
2l.
Then m + n = 2(k + l).
Since k + l is an integer m + n is even.
If m and n are both odd then there exist
integers k and l such that m = 2k + 1 and
n = 2l + 1.
Then m + n = 2(k + l + 1).
Since k + l + 1 is an integer m + n is
even.
Python Code to explore the sums of
evens/odds
for i in range(0,11):
for j in range(0,11):
if i /2== int(i/2):
text1=“even”
else:
text1=“odd”
if j/2 == int(j/2):
text2=“even”
else:
text2=“odd”
if i + j == 2*(int(i/2)+int(j/2)) or
i + j == 2*(int(i/2)+int(j/2)+1):
print i, text1, j, text2, i + j, “even”
Technology in assessment
At A level students are allowed a graphical calculator in all
but one of their examinations;
However, these examinations are designed to be graphical
calculator neutral, i.e. having a graphical calculator should
offer no advantage to a student.
It is not surprising that if the technology is expected to not
offer an advantage in the examination then many teachers
do not exploit its use for teaching and learning.
Technology in Assessment
In addition to this there are (or were) no examinations
where programming environments or CAS are allowed.
As a consequence of this programming and CAS are rarely
used in the teaching and learning of mathematics in English
schools.
Background
MEI wanted to drive the debate forward by exploring the
possibility of having part of the A level involve the use of
technology, including programming and CAS, in a way that its
use would be expected in the assessment and consequently this
would drive its use in the teaching and learning.
Further Pure with Technology (FPT)
FPT is an optional A2 Further Pure unit that can be studied
as one of 12 (or 15) mathematics units.
FPT has been developed with the full support of OCR and
Texas Instruments. MEI is very grateful for this support.
FPT will inform MEI’s approaches to the use of technology in
future developments of A level.
FPT has been approved by Ofqual. The first examination
was on Monday 24th June 2013.
FPT: Use of the technology
Students are expected to have access to
software for the teaching, learning and
assessment that features:
• Graph-plotter
• Spreadsheet
• Computer Algebra System (CAS)
• Programming Language
The expectation has been that students
have used TI-Nspire software and
teaching resources have been created to
support this.
FPT: Technology in Pure Maths
Criteria for inclusion of mathematical topics:
• Technology allows you to access a large
number of results quickly
• Be able to make inferences and
deductions based on these
• Not included elsewhere in A level Maths
or Further Maths
x = t − k sin t, y = 1 − cos t
FPT: Content
• Investigations of
Curves
Investigate the curves for 0 < k < 1.
Describe the common features of these
curves and sketch a typical example.
• Functions of Complex
Variables
• Number Theory
f ( z ) = z 3 − (3 − 3i) z 2 − 6iz + 2 + i
Solve f(z) = 0.
Show that f’(z) = 0 has a repeated
root.
Create a program to find all the
positive integer solutions to
x² − 3y² = 1 with x<100, y<100.
FPT: Assessment
A timed written paper that
assumes that students have
access to the technology.
For the examination each
student will need access to a
computer with the software
installed and no communication
ability.
FPT: Engagement with schools
We have worked with 10 schools/colleges and expect
around 30-40 students have studied the unit this year.
We have worked with the teachers to support their
development and produce effective teaching and learning
resources.
The resources will be available on Integral from
September.
x = t − k sin t, y = 1 − cos t
Investigate the curves for 0 < k < 1.
Describe the common features of
these curves and sketch a typical
example.
3
2
f ( z ) = z − (3 − 3i) z − 6iz + 2 + i
Solve f(z) = 0 and plot the roots
on an Argand diagram.
Show that f’(z) = 0 has a
repeated root.
Create a program to find all the
positive integer solutions to
x² − 3y² = 1 with x<100, y<100.
Investigate r = a cosθ + b sin θ
dr
sin θ + r cosθ
dy dθ
=
Use
dx dr cosθ − r sinθ
dθ
to describe the tangent as you move round the
curve.
f (z) =
1
z
Use a spreadsheet to investigate f(z) as z
moves along the line z = 1 + a i , a ∈ »
Prime pairs are integers n – 1 and n +1
that are prime. Write a program to find all
the prime pairs less than a maximum
integer, m.
List all the prime pairs less than 200.
FPT: Further Information
•
Updates on the MEI website: www.mei.org.uk/fpt
•
Specification and specimen papers on the MEI website:
www.mei.org.uk/fpt
•
Teaching and learning resources: www.integralmaths.org
•
TI-Nspire: www.nspiringlearning.org.uk/
•
Project Euler: projecteuler.net/
richard.lissaman@mei.org.uk
Want to get your students
programming maths?
Exploring A level Mathematics by
programming games
Chosen because
• ease of start up/installation
• fully integrated, very quick to see results of code
• used by professionals
• good free version available
Motivating mathematics by
programming games
About MEI
• Registered charity committed to improving
mathematics education
• Independent UK curriculum development body
• We offer continuing professional development
courses, provide specialist tuition for students
and work with industry to enhance mathematical
skills in the workplace
• We also pioneer the development of innovative
teaching and learning resources