Embed Size (px)
Citation preview
Q: How would you mark this answer?
1
A: Like this!
2
Facilitating Mathematics Education with Spreadsheets: from the ground up
Dr Steve SugdenMathematical Sciences SchoolQueensland 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
• 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
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 2002
5
WHY SPREADSHEETS?They’re not the sharpest tool in the shed, so …
6
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
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).
8
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
SPREADSHEETSWhat have STEM educators said about them?
Science
Technology
Engineering
Mathematics
10
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
11
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
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
Mariam Haspekian (France)Doctoral thesis HASPEKIAN MariamANRT 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
Lagrange & Erdogan (Turkey)
A first step to concept of variable by using spreadsheets: an example of teaching experiment in Turkey
Emel Ozdemir Erdogan & PelinTURAN, Anadolu University
15
• 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
• 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
• 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
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
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
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
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 Qld23
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
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
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
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
• 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
MATHS EXAMPLES
29
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 2𝑥 + 2𝑥+1 = 5 for 𝑥”.
• Only four students from approximately 80 were able to answer this question correctly!
30
Mortgage schedule with one addition & one multiplication – no logs or exponentials(Workshop on this: Just Google “Sugden TTA”)
Maths A: Painless payments
31
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
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 formulas3. Excel intrinsic financial functions
Australian Curriculum ACMNA229
33
• 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
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”.
35
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
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
• 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
$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 + 𝑋1 + 𝑟 𝑛 − 1
𝑟
Method 2: Algebraic formula
39
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
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
• 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
Conditional formattingDiscovering and exploiting patterns
43
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 RSApublic key system (including digital signatures) in my classes at Bond for 10 years
44
• 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
1mod 0
1n
kn k
45
Games can motivate and engage students if carefully chosen
Some fun & games
46
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
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
49
• 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
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
SPREADSHEETS IN EDUCATIONFully open-access electronic journal
52
Spreadsheets in EducationAn open-access eJournal http://epublications.bond.edu.au/ejsie/
53
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
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
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
Spreadsheets in Education- Physics papers1. Travelling Wave and Fraunhofer Diffraction Simulations - Garry Robinson and
Zlatko Jovanoski (UNSW, ADFA), Vol4, #32. Radiometric Dating for Novice Learners: Visualizing, Modeling, and
Simulating via Animated Spreadsheets, Scott Sinex, Vol 4, #23. Spreadsheet Solution of Basic Axial Force Problems of Strength of Materials,
by Ernesto Sanchis et al, Vol 4, #14. Using a Spreadsheet to Study the Oscillatory Movement of a Mass-Spring
System, by Oliveira & Napoles, Vol 3, #35. Simulating Projectile Motion in the Air with Spreadsheets, by Jan Benacka,
vol 3, #26. Three Spreadsheet Models Of A Simple Pendulum, by Jan Benacka, vol 3, #17. Illustrating the Beats Phenomenon with Excel: the construction of meaning
through experimentation, by John Baker, Vol 1 #1
57
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
Conclusions A1. 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
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
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.
61
Thank you Questions?
62
DISCARDS
63
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
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
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
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
