3. Facilitating Mathematics Education
with Spreadsheets: from the ground up
Dr Steve Sugden
Mathematical Sciences School
Queensland University of Technology
Editor, Spreadsheets in Education
(Open Access Electronic Journal)
http://epublications.bond.edu.au/ejsie
Mathematical Sciences School
Research Seminar Series, 26th July 2013
4. • About Steve
• Why spreadsheets? What have maths educators said about them?
• Examples of use in the classroom
• Ideas for using Excel in teaching basic finance to Maths A students (not just
black-box, but with understanding)
• Other examples, using conditional formatting
• Spreadsheets in Education (open access electronic journal)
• Conclusions and recommendations
Overview
4
5. About Steve
• Broad background in mathematics, science & computing
• At Bond > 24 years; foundation staff member; wrote PhD there
• Re-united with QUT in Feb 2013 (my 3rd life here )
• It’s nice to be at a university which has some real mathematics
• Current research interest: use of spreadsheets in education
• Leverage the tremendous potential in the spreadsheet to help students
understand principles of mathematical modeling and problem-solving,
right from grade one
• John Baker & I set up Spreadsheets in Education eJournal at
Bond for this purpose in 20025
7. Why spreadsheets?
• Ubiquity & familiar interface
• Rapid display of sequences, tables,
graphs; patterns quickly discerned
• Humans are hardwired for pattern-
recognition; why not exploit these innate
abilities?
• Quick feedback on conjectures; saves
time
• Spin-off benefits: skills learned are
transferable to other disciplines
7
8. Relationships
• Spreadsheets require the user to
create a model and to express
relationships
• supports transition from arithmetic
to algebra [1].
• Best-practice spreadsheet
modelling will strongly encourage
students to use spreadsheet
names.
• Relationships are created looking
very much like the algebraic ones in
the textbook.
[1] There are many references in the literature to
support this claim, e.g., Haspekian (2003), Wilson,
Ainley, & Bills (2005), Sutherland & Rojano (1993),
Wood & d'Souza (2001), Friedlander (1998).
9. Spreadsheets at school?
• Schools seem to like graphics calculators (GCs)
• Now that laptops, netbooks, tablets, etc. very cheap, I
suggest that we should be using Excel instead
• unlike the GC, spreadsheets are widely used in industry:
business, scientific, engineering areas etc.
• Not much innovative seems to be happening at schools in
this regard
• best-practice spreadsheet modelling for math patterns and
visualization of structures
• use of names, recurrences, colours (conditional formatting), etc.
9
11. Kirsty Wilson et al (UK)
“… we do suggest that rich
spreadsheet activity can be
invaluable in supporting
pupils’ construction of
meaning for algebra”
“Our data points to the
important role of the teacher
in guiding this evolution of
meaning” Spreadsheets, pedagogic strategies and the evolution of meaning for
variable, Wilson, Bills, Ainley. Proceedings of the 29th Annual
Conference of the International Group for the Psychology of
Mathematics Education, 10-15 July 2005, Melbourne
12. Alex Friedlander (Israel)
“Spreadsheets build an ideal bridge
between arithmetic and algebra and allow
the student free movement between the
two worlds.
Students look for patterns, construct
algebraic expressions, generalize
concepts, justify conjectures…as intrinsic
and meaningful needs rather than as
arbitrary requirements imposed by the
teacher.”
Friedlander, A. (1998). An EXCELlent bridge to algebra. Mathematics Teacher
91(50): 382—383.
12
13. Sergei Abramovich (SUNY, USA)
“Some teachers who attempt to
incorporate technology into the
curriculum limit its use to routine
computations only due to a lack of
experience with this technology.
The math education community
views this problem as a great
challenge to educational reform.
For math education majors, the
spreadsheet is the ideal vehicle for
illustration of math fundamentals …”
Abramovich, S., Brantlinger, A., and Norton, A. (1999). Exploring
Quadratic-like Sequences through a Tool Kit Approach. In G.
Goodell (Ed.), Proceedings of the 11th Annual International
Conference on Technology in Collegiate Mathematics, Addison-
Wesley.
13
14. Mariam Haspekian (France)
Doctoral thesis HASPEKIAN Mariam
ANRT Ref: 48600 INTEGRATION OF
COMPUTER TOOLS IN TEACHING
MATHEMATICS. CASE STUDY
SPREADSHEETS.
Haspekian, M. (2003). Between
arithmetic and algebra: a space for the
spreadsheet? Contribution to an
instrumental approach. Proc. of the 3th
Conf. of the European Society for
Research in Mathematics Education. Pisa,
Italy: CERME.
14
15. Lagrange & Erdogan (Turkey)
A first step to concept of variable
by using spreadsheets: an example
of teaching experiment in Turkey
Emel Ozdemir Erdogan & Pelin
TURAN, Anadolu University
15
16. • Spreadsheets as cognitive tools: A study of the impact
of spreadsheets on problem solving of math story
problems
• Published online: 23 August 2011, Springer Science +
Business Media, LLC 2011
• Findings indicated that the computer spreadsheet
environment facilitated the transition from arithmetic
to algebraic reasoning.
Lavidas, Komis, Gialamis
16
17. • Students who had proposed solutions consisting of distinct, simple
calculations, in the pen and paper conditions, solved the same
problems in the computer spreadsheet environment using algebraic
expressions.
• The formulation of algebraic expressions constitutes a critical step for
the transition to algebraic reasoning (Sutherland & Balacheff, 1999).
• Similar research also concluded that computer spreadsheets facilitate
transition from arithmetic to algebraic reasoning (Friedlander 1998;
Sutherland & Rojano 1993).
Lavidas, Komis, Gialamis
17
18. • The paper reports a study of the impact of spreadsheets on problem solving
practices of students, with a focus on transition from arithmetic to algebra.
• There is a specific focus on verification of solutions.
• Authors conclude that spreadsheets have a positive influence on solution
verification and the transition from number to algebra.
• Thus, the paper presents further evidence of what has been well-known
among a global coterie of mathematics education researchers for some time:
namely, that spreadsheets promote abstract reasoning and assist learners in
the transition from number to algebra.
Summary of my review of LKG (June 2013)
18
19. John & Ann Baker (Natural Maths, Qld)
Obvious in hindsight, but
spreadsheets save time!
Models to illustrate or investigate
theoretical concepts (“what -if”) are
often set up very quickly
Probably one of the most profound,
clear benefits of using spreadsheets
is just that of saving time.
(Baker & Sugden 2003).
19
20. Len Vacher
(University of Southern Florida)
In 38 papers connected with Spreadsheets in
the Journal of Geoscience Education (JGE)
1989 to 2003, …
“…Geoscience educators (and educators in
other partner disciplines) have gravitated to
spreadsheets.”
“The virtues of using spreadsheets as
teaching tools are forcefully presented by
[Baker & Sugden] in the kickoff issue [of
Spreadsheets in Education].”
“These virtues are resoundingly illustrated by
the collection of articles from the JGE.”
20
21. Fratesi & Vacher (2004)
“Spreadsheets allow educators to
promote more open-ended
investigations, problem-oriented
activities, and active learning by
students”.
Spreadsheets “provide insights
into the ... context without
necessitating attention to
extraneous distractions”.
“Building a spreadsheet promotes
abstract reasoning”.
21
23. Elliot Tonkes (Energy Edge consulting)
“The use of spreadsheets
throughout industry is possibly
far more pervasive that
researchers in the educational
arena suspect.”
Tonkes, E. (2011) Professional Development in Electricity
Markets with Spreadsheet Models, in Applications of
Spreadsheets in Education The Amazing Power of a Simple
Tool, 2011, 261-273, Mark Lau and Stephen Sugden (eds).
PhD (Mathematics) - University of Qld
Graduate Certificate in Economics - Griffith University
Bachelor of Mechanical Engineering (1st Class Honours) -
University of Qld
Bachelor of Science (1st Class Honours) - University of Qld
23
24. Excel bridges gaps between scientists,
engineers, accountants, managers
“In the finance and energy
industries, Excel is an
invaluable form of
communication between
professionals….”
“Key to the utility of Excel is
that spreadsheets are
ubiquitous across different
disciplines…”
24
25. Professional development in electricity markets
“The commonality amongst a
diverse range of workshop
participants is their
understanding of Excel.”
“It forms an ideal mechanism
to communicate technical and
mathematical concepts.”
25
26. Excel is a common language
of business and science
“The attendees at our courses span a
wide range of employment positions,
ranging from traders, to market
analysts, to engineers and
accountants.”
“With spreadsheet software as a
modelling platform, we are able to
construct simple models to convey
complex concepts.”
26
27. Dennis Almeida (UK)
Prime need is for a medium which will:
• facilitate consideration of conceptual
issues while requiring minimal
technical expertise
• provide insight into mathematical
context without distractions
• permit modification without the need
for major changes in design
• flag and encourage the pursuit of
connected enquiries.
The spreadsheet meets all of the above.
27
Relf, S., and Almeida, D. (1999). Exploring the birthdays problem and some of
its variants through computer simulation. International Journal of
Mathematical Education in Science and Technology 30, 81—91.27
28. • Response to the Australian National Mathematics Curriculum
• http://sydney.edu.au/iisme/downloads/MathematicsCurricResponse.pdf
“…we believe that spreadsheets, graphing and drawing tools are a critical feature
shaping Australia and the rest of the world as a knowledge society.”
“Spreadsheets are already the numeracy tool of choice for a large proportion of the
populace, from tradespersons to professionals. The curriculum should acknowledge
such media and work within them much more”.
University of Sydney Institute for Innovation in
Science & Mathematics Education (IISME)
28
30. Some background:
fundamental knowledge is lacking
• Australia 2013: mathematical knowledge of students
entering university is woeful.
• Personal anecdote: for Business Mathematics at Bond
University, index laws and logarithms is assumed
knowledge.
• Question for the mid-semester paper in 2011: “Solve the
equation 2 𝑥
+ 2 𝑥+1
= 5 for 𝑥”.
• Only four students from approximately 80 were able to answer
this question correctly!
30
31. Mortgage schedule with one addition & one multiplication – no logs or exponentials
(Workshop on this: Just Google “Sugden TTA”)
Maths A: Painless payments
31
32. Teaching basic finance
Finance skills are life skills.
Traditional approaches to teaching this
material used algebra and calculators.
Canned formulas; with little
understanding of problem structure or
with much inkling of the answer.
Overuse of calculators, including the
much-overrated graphics calculators, has
exacerbated this problem.
32
33. Year 10: Money and Financial Mathematics
“Connect the compound interest formula to repeated applications of
simple interest using appropriate digital technologies” (ACMNA229)
We use, in the following order:
1. Linear recursive schedule (+ Goal Seek)
2. Traditional algebraic formulas
3. Excel intrinsic financial functions
Australian Curriculum ACMNA229
33
34. • The full power of the spreadsheet recurrence in the service of
pedagogy has yet to be fully exploited.
• Spreadsheets lend themselves to this form of expression,
especially where there is a natural underlying sequence, e.g.
sequential periods of time
• Information-theory: recurrence relations pack a lot of
information into very simple-looking equations/relations
• Exponential relationships become linear if we employ
recurrences!
Method 1: Linear recurrence in Excel
34
35. It doesn’t get any simpler than linear!
• Linear recurrence in Excel.
• No logs or exponentials
needed.
• Naturally expressed in Excel
with only point-and-click plus
the simple operations of
arithmetic:
• “add 1% to what you have”, or
• “multiply what you have by
1.01”.
36. Example finance problem
Tom borrows $24,000 for
new car
There are 12 equal monthly
payments
Interest is 12% p.a.,
compounded monthly
Find his monthly payment
amount
36
37. Method 1: Linear recurrence in Excel
Amount of loan, $d0 = $24,000
$dm = debt after m months (0 ≤ m)
$X = monthly repayment
Linear recurrence (0 ≤ m ≤ 12):
dm = (1 + r)*dm - 1 + X
X negative to get zero residual, i.e., d12 = 0.
Use Goal Seek in Excel to make residual zero
37
38. • A side benefit of the recursive approach is that the
schedule produced can be used to answer other
questions, e.g., Peter wishes to pay out his loan
early, so what is the payout figure?
• This amount can simply be read from the schedule
with no further modelling or algebra whatsoever.
• Another benefit: students gain experience in
expressing relationships and build modelling skills.
Benefits
38
39. $P =present value (amount of loan = $24,000)
$F = future value (residual; often zero)
r = rate per period (we use 1%)
$X = monthly payment (? unknown)
n = # months (we use 12)
𝐹 = 𝑃 1 + 𝑟 𝑛 + 𝑋
1 + 𝑟 𝑛 − 1
𝑟
Method 2: Algebraic formula
39
40. Method 3: Excel intrinsic function
Generally simple enough, but there are
some traps.
Not advisable on its own, but good for
verification.
Totally opaque, black-box method.
Cannot see how it operates except by
experiment.
It is akin to pressing a few buttons on a
pocket calculator, and simply believing the
answer.
=PMT(1%, 12, -$100, 0)
40
41. Another PVA example
A widow is beneficiary of a $2,500,000
life insurance policy.
She receives $500,000 now & $100,000
every 3 months thereafter.
Interest at 4% p.a. compounded
quarterly.
• how many full payments will she receive?
• what additional sum paid with the last
full payment will exhaust her benefits?
• what sum paid 3 months after the last full
payment will exhaust her benefits?
41
42. • ACMNA012 Develop confidence with number sequences to and from 100 by ones from any starting point. Skip count
by twos, fives and tens starting from zero
• ACMNA071 Investigate and use the properties of odd and even numbers
• ACMNA101 Solve problems involving division by a one digit number, including those that result in a remainder
• ACMNA122 Identify and describe properties of prime, composite, square and triangular numbers
• ACMNA149 Investigate index notation and represent whole numbers as products of powers of prime numbers
• ACMNA237 Solve linear simultaneous equations, using algebraic and graphical techniques including using digital
technology
• ACMNA294 Find the gradient of a line segment ....using a range of strategies, including graphing software
Some other new curriculum maths topics …
42
44. Cryptography
Modular arithmetic lies at the heart
of computer security and
cryptography
It can be made accessible to
secondary students via patterns in
Excel
I used Excel + VBA to model the RSA
public key system (including digital
signatures) in my classes at Bond for
10 years
44
45. • It is easy to extend Excel’s function library by using VBA
• We look at the divisor function numDivisors(n)
• This counts #factors of n
• numDivisors
Example VBA function
1
mod 0
1
n
k
n k
45
46. Games can motivate and engage students if carefully chosen
Some fun & games
46
48. Spreadsheet as a Sudoku helper
• Goal is not to use Excel to automatically solve
completely (although it will do this for simpler puzzles)
• Rather, get Excel to do the boring bits of computing
possibles (like a spell checker)
• Give the player some chance to do more of the creative
thinking
• Excel alleviates boredom of computing sets, thus freeing
the player to explore possibilities and improve reasoning
skills
48
49. Letters & numbers
SBS game show new to
Australia, but not to
Europe
According to Wikipedia
“one of the longest-
running game shows in
the world.”
Australian personnel:
• Richard Morecroft
(host)
• David Astle (cryptic
crossword writer)
• Lily Serna (UTS maths
Masters student)
Note to speaker: Words at 2:20; Numbers at 11:40
50. • Lily at a school
• http://www.youtube.com/watch?v=SdcRv-ZhEnE
• http://www.youtube.com/watch?v=n8-mx3RSvOQ
• http://www.youtube.com/watch?v=nG4QCob_dNw
Just numbers (Delphi)
50
51. A bit off the beaten track, but …
Fractional derivative (a weird form of
calculus, and certainly university
level material)
• but shows what can be done with
Excel.
• Trig model
• Poly model
Factorization of integers
(applications to cryptography and
digital signatures)
• this model has no VBA – it is made
from just Excel formulas
51
54. Aim & motivation for SiE eJ
• Set up by John Baker and self in 2002
• Create avenue for scholarly research into the use of
spreadsheets at all levels of education
• Fully double-blind peer-reviewed papers
• Classroom materials (also peer reviewed)
• Provide directly-usable classroom resources freely
downloadable for both teachers and students
• Lesson plans, models + templates, examples
• Fully open-access
• no download fees
54
55. Spreadsheets in Education
– general papers
1. Spreadsheets and the development of skills in the
STEM disciplines, by Abramovich et al, Vol 3, #3
2. Spreadsheets in Education –The First 25 Years, by
John Baker and Steve Sugden, Vol 1, #1
55
56. SiE - Maths & Stats papers
Graphing Functions of Two Variables in Spreadsheets, Jan Benacka, Vol 4, #3
Teaching Statistics in a Spreadsheet Environment Using Simulation, Graham Barr & Leanne Scott, Vol4, #3
Basic Finance Made Accessible in Excel 2007: "The Big 5, Plus 2“, Steve Sugden & David Miller, Vol 4, #2
Exploring the Fundamental Theorem of Arithmetic in Excel 2007, Steve Sugden & David Miller, Vol 4, #2
Colour by Numbers in Excel 2007: Solving Algebraic Equations Without Algebra, Steve Sugden, Vol 4, #2
Teaching the Logistic Growth Difference Equation Using Spreadsheets, by Rispoli et al, Vol 3, #3
3D Graphics with Spreadsheets, by Jan Benacka, Vol 3, #1
Enhancing Mathematical Graphical Displays in Excel through Animation, by Deane Arganbright, Vol 2, #1
Spreadsheet Conditional Formatting: An Untapped Resource for Mathematics Education, by Sergei
Abramovich and Stephen Sugden, Vol 1, #2
56
57. Spreadsheets in Education- Physics papers
1. Travelling Wave and Fraunhofer Diffraction Simulations - Garry Robinson and
Zlatko Jovanoski (UNSW, ADFA), Vol4, #3
2. Radiometric Dating for Novice Learners: Visualizing, Modeling, and
Simulating via Animated Spreadsheets, Scott Sinex, Vol 4, #2
3. Spreadsheet Solution of Basic Axial Force Problems of Strength of Materials,
by Ernesto Sanchis et al, Vol 4, #1
4. Using a Spreadsheet to Study the Oscillatory Movement of a Mass-Spring
System, by Oliveira & Napoles, Vol 3, #3
5. Simulating Projectile Motion in the Air with Spreadsheets, by Jan Benacka,
vol 3, #2
6. Three Spreadsheet Models Of A Simple Pendulum, by Jan Benacka, vol 3, #1
7. Illustrating the Beats Phenomenon with Excel: the construction of meaning
through experimentation, by John Baker, Vol 1 #1
57
58. Excel models for the asking
You can freely download
these from the SiE ejournal
site
http://epublications.bond.edu
.au/ejsie
About 100 articles there, most
with Excel models and some
with lesson plans
58
59. Conclusions A
1. The spreadsheet is best tool for learning basic finance and also
for mathematical modelling and simulation (with R add-in ).
2. For finance & other maths apps, the vast power of recursion
can be exploited in the spreadsheet environment, very often
with just linear equations.
3. Emphasize understanding of model structure over opaque
intrinsic functions.
4. For maths education majors, the spreadsheet is the ideal
vehicle for illustration of mathematical fundamentals.
59
60. Conclusions B
5. Spreadsheets encourage building of
mathematical models.
6. Pre-service maths teachers need to be
taught Excel and how to leverage its power
it in the classroom.
7. Students with a knowledge of this tool at
university have a skill greatly valued in the
business world.
8. Abandon calculators and move to
spreadsheets!60
61. Conclusions C- get rid of those calculators!
8. It is time to take Excel seriously as
a maths education tool for the
masses.
• Excel friendly and ubiquitous
• Students like it
• Hardware now very cheap
9. Given suitably-prepared teachers,
I recommend the use of
spreadsheets for maths education
without reservation, and ideally,
in early grades.
64. Spreadsheet modelling: student comment #1
I would just like to thank you very much for the wonderful
semester. I greatly enjoyed your class. You might not realize it
but it has really added value to my qualities in Excel and
business. It was great that you recommended it to me in 092 and
I’m happy that I took the opportunity when it was offered.
As I said I continue to recommend your class to my closest of
friends who study business. I am referring students of high
quality and perspective. I hope that they attend your class and
that business spreadsheet modelling continues in operation.
64
65. Spreadsheet modelling: student comment #2
“I graduate from my bachelor degree this semester and directly do my
master degree in Accounting at Bond.
I was delighted that I took business and spreadsheet modelling in last
semester.
It enhances my Excel skills so much and help me a lot in my current
degree.
I was wondering if you would allow me to put you as a reference on
my CV?
I am applying for a graduation program at SEQ Hotels – Coolie Hotel.
They provide a Business accountancy internship and are looking for
people who have advanced Excel skills.”
65
66. Spreadsheet modelling: student comment #3
“I finally finished my studies and graduated in February. I must
admit, I am missing all the challenges that I had at Bond. I started
working for an insolvency firm, but realised very quickly that it
wasn't for me as it was very boring, although I am about to start
work as an accountant, which is more along the lines of
Management and Financial accounting.
This will be great because the job involves setting up a lot of
business spreadsheets, therefore making your subject Business and
Spreadsheet modelling well worth it. I really hope the university
realises how important a subject such as this is in becoming an
accountant.”
66
67. Spreadsheet modelling: student comment #4
“At least every second day I have something that comes
along which requires something learnt from the spreadsheet
modelling course. Thankful everyday that I did it. Out of all
the subjects I have done and with only 3 more to go, best
subject I have done and the only one I have used outside of
uni.”
“… seeing my time at Bond is almost over I do get people
come and ask what subjects I thought were best and I always
recommend Spreadsheet Modelling, in fact I can't even
remember the names of the other subjects I have done so
that says something.”
67
