WAN ZANARIAH BINTI MD JAHAYA
S2108409
PQD7006: TEACHING CALCULUS WITH TECHNOLOGY
ASSIGNMENT 3:
USING TECHNOLOGY IN DEVELOPING STUDENTS’
MATHEMATICAL MODELLING ABILITY IN THE CALCULUS
CLASSROOM.
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Contents
No.
Title
Page Number
1.0 MATHEMATIC MODELLING
1
2.0 IMPORTANCE OF MATHEMATICAL MODELLING
IN THE CALCULUS CLASSROOM
2
3.0 DIFFICULTIES IN DEVELOPING MATHEMATICAL
MODELLING IN CALCULUS CLASSROOM
2
4.0 DEVELOPING TASK
4.1 TASK 1: LITTLE MISS AND MR BAKER
5
5.0 TEACHER’S GUIDE (SOLUTION AND IMPLEMENTATION
OF THE TASK)
7
6.0 HOW TECHNOLOGY COULD OVERCOME THE DIFFICULTIES 11
7.0 REFERENCES
14
2
1.0 MATHEMATICAL MODELLING
Mathematical modelling means a process of creating a mathematical representation
of a real-world scenario to make further predictions. Mathematical modelling differs in
with problem-solving in several ways. Unlike traditional problem solving, modelling
focuses on the process as the product. Hence, getting the correct answer is not
important. When students get stuck in the process of mathematical modelling, it should
be viewed as an opportunity for refining initial ideas. By allowing students to make
mistakes, it exposed the students’ misconceptions, and it provides opportunities for
others to argue or discuss those ideas. Mathematical modelling has been practised in
many countries, but it seems quite new in Malaysia. Some researches have
summarized the effective approach to teaching mathematical modelling by including
the 3 major elements which are contextual, student-centred, and active.
There are many modelling cycles that can be used in teaching. For example, the
modelling cycle introduced by Common Core State Standards where it's containing
the problem, formulate, compute, interpret, validate and report. There is another one
famous and most used in educational research are The modelling cycle by Blum and
Leib was also called the “ideal modelling process”.
It is a process of mathematical modelling that can be a guide for the teachers to
facilitate the students. There is no single modelling cycle that fit the same for every
person. Do not be confused as the modelling process is not a step to be followed
linearly. Teachers might skip steps or create new steps or maybe do a whole bunch
of things at the same time depending on the students and situation. Teachers are
encouraged to execute genuine investigation of modelling situations and use the
modelling cycle to engage them in meta-thinking.
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2.0 IMPORTANCE OF MATHEMATICAL MODELLING IN THE CALCULUS
CLASSROOM.
There are rapid changes in mathematics education where the development of
problem-solving and critical thinking among students are prioritized. According to
research(Kaur & Leong, 2020), modelling can solve any mathematically rich issue
such as engaging students in mathematical thinking, drawing upon earlier discovered
understanding and supporting the understanding of their mathematical ideas.
Mathematical modelling is effective at supporting the students to discover
mathematics, connect what they have learned in school with their everyday lives and
in developing a suitable environment for improving mathematical ability.
One of the important factors in the mathematics curriculums is making conjectures and
reasoning. Using mathematical modelling in problem-solving can benefit students’
skills in constructing a model or physical representation(Kwan Eu, 2013). For example,
modelling population growth using a model of an exponential function is suitable to
show the pattern of population continues growing larger and faster over time.
3.0 DIFFICULTIES IN DEVELOPING MATHEMATICS MODELLING IN CALCULUS
CLASSROOM
Time consuming
Other than the nature of mathematical modelling itself that requires multiple
intelligence as supported by many research, implementing math modelling in class
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can be very time consuming. In order to practice mathematics modelling in solving
math problems, students cannot focus only on getting a correct answer but need more
time to collect data, modelling it from real life to mathematics formula. As teachers are
always running out of time in catching the syllabus, they tend to kill the student’s
curiosity by simply giving solution to problems which oppose with the method of
developing students meta-thinking.
Lack of pedagogical and technological knowledge among teachers to teach
modelling in the classroom.
Students nowadays cannot be thought as what we teachers being taught previously
as future career demand students to have higher order thinking skills. There are many
reason why mathematical modelling is so hard to be implemented as it is differ from
teachers education experiences (Blum & Borromeo, 2009). Mathematics modelling
requires teachers to improve their pedagogical ideas and require knowledge that
beyond the school curriculum. They need to balance out between giving guidance to
students and allowing students to think independently. Teacher may encounter
different way of students thinking and solution might be varied among them. Teacher
must be able to create question that encourage students to think reflectively and do
further justification. Explicit explanation by teachers, proper planning, monitoring, and
evaluating one’s thinking are important in successfully developing students metathinking.
Most teachers themselves did not familiar in enhancing students’ mathematical
knowledge using. Hence, they tend to ask direct questions that accessing student’s
memorization instead of understanding (ÖZGÜR, 2020). In terms of technology,
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inhouse training need to be provided for existing teachers to teach mathematics
modelling using technology. Meanhwile, improvement in the syllabus are needed for
preservice teachers to include integrating technology in mathematics teaching
(Gulkilik, 2020)
Lack of facilities provided by school
As representation can involve as cheap as pencil and papers, but some mathematics
problems best presented using dynamic software such as GeoGebra (Khalil et al.,
2017). Since GeoGebra is a free package and specially designed for mathematics
subject for high school levels schools need to facilitate students by providing a proper
computer lab. Meanwhile stable internet connection is crucial for students finding other
resourceful material on the internet and using web-based application such as Desmos.
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4.0 DEVELOPING TASK
TASK 1 : LITTLE MISS AND MR BAKER.
Objective: Linear Programming (Maximizing sales using mathematical modelling)
Learning Outcome:
1) Understand equation and inequalities.
2) Using mathematical modelling to solve the problems.
Technology used: GeoGebra
Problem 1:
Hana and Ian intend to sell cakes and caramel pudding during schools’ carnival this
coming August. Hana has large mixer to prepare the dough and she usually takes 2
hours to make one whole cake and 3 hours to make caramel pudding. She has only
24hrs to prepare the cake and caramel pudding as it best to consume within 2 days.
Ian can only decorate the cakes and caramel pudding with some fruits and whipping
cream maximum up to 10 pieces either one of those or the combination of it in a day.
The one whole cake will be selling for RM65 and caramel pudding will be selling for
RM80. How many cakes and caramel pudding should they make in order to get the
maximum amount of money? How much money will they make then?
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Step 1 : What's the problem to be solved?
Step 2:What are the information that we have?
Step 3:Any assumption that we made in this situation ?
Step 4: What are the variables / parameter involved?
Step5:Write down your possible solutions. Model you situation into mathematics
Step 6: Use technology ie: Microsoft Excel, GeoGebra
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Step 7: What can you say about the graph and its intersection?
Step 8:Report finding. Which combination do you think that can give the maximum sales?
Step 9: Does anyone have a different way of thinking about this?
5.0 TEACHER’S GUIDE (SOLUTION AND IMPLEMENTATION OF THE TASK)
Question 1
What’s the problem to be solved?
Suggested answer: Number of cakes and caramel pudding should they make in
order to get the maximum amount of money.
Question 2
What are the informations that we have?
Suggested answer: Hana takes 2 hours to make one whole cake and 3 hours to
make caramel pudding and Ian can decorate either of those up to 10 pieces only.
Question 3
Any assumption can we made in this situation given?
Suggested answer: 2 days best consume means 1 day for preparation and 1 day for
the selling time
Question 4
What are the variables/ parameter involved?
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Suggested answer:
i-Time in hours.
ii-Number of cakes to be made by Hana and decorated by Ian.
iii-Price in Ringgit Malaysia
Question 5
Write down your possible solution to solve problems given.
Hana
Ian
Within 24 hrs cake and
pudding can be made
Cake: 24/2=12
Pudding: 24hours/3 =8
10 max
Price in Ringgit
Cakes x RM 65
Pudding RM80
10
0
10x65= 650
9
1
(9x65)+ 80 = 665
8
2
(8x65)+(2x80) = 680
7
3
6
4
TEDIOUS
5
5
Let’s use
4
6
Spreadshet/
3
7
Graphical
2
8
Calculator/
1
9
GeoGebra.
0
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Total
Quantity
Cake: 12 cakes x RM65 = RM780
Pudding:8 pudding x RM80 = RM640
So, 12 cakes not possible instead 10 cakes
only.
10 cakes x 65 = RM650
8 cakes x 65 + 2 pudding x 80 = RM680
But there are a lot of other combinations too
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working,
Question 6
Use Microsoft Excel by inserting the data and formula to autogenerate the values. You
may use any other technology as well.
Suggested answer:
Method 1: Spreadsheet
Method 2: Using linear programming by GeoGebra
What equation can simplify the situation above.
X = representing cakes
Y= representing caramel pudding
Ian can decorate; x+ y ≤ 10
Time Hana can bake ; 2x + 3y ≤ 24
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Question 7
What can you say about the graph and its intersection?
Suggested answer: The intersection points shows value of x and y that representing
number of cakes made and pudding that satisfy for both equation above.
Question 8
Which combination do you think that can give the maximum sales?
Suggested answer: To get the maximum number of cake x=6 and chocolate pudding
y=4 . Hence the maximum amount of sales will be 6 x RM65 + 4 x RM80 = RM710
Question 9
Does anyone have a different way of thinking about this? Discuss.
Simultaneous equation for inequality.
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6.0 HOW TECHNOLOGY CAN HELP TO IMPROVE MATHEMATICAL MODELLING
IN THE CALCULUS CLASSROOM.
Spreadsheet
Some people might underestimate the power of spreadsheet, but it is actually
powerful. Inserted data in its rows and columns makes our data readable and editable.
There are plenty formulas and functions to be used to manipulate and make
conclusion of our data. Its data filtering tools, and graphical tools makes thousands of
data easily be extracted and visualized. In our case, calculating one by one into our
mathematical formula can be tedious. Hence, spreadsheet help us in calculating the
data we have without having us multiply it one by one using calculator(Chaamwe &
Shumba, 2016).
GeoGebra
Education sectors play a vital role in shaping our future generation. With the
advancement of technology, it is very useful for a person to fully utilize the technology
that can support teaching and learning. GeoGebra as one of the many dynamic
geometry software able to diverse students’ mathematical thinking which later
preparing them to work in the science and technology rich environment of 21st
century(Khalil et al., 2019). By using GeoGebra students can observe the possible
values of x and y after successfully modelling the problems given in mathematical
representation. The graph easily represented the inequalities, and it doesn’t limit to 2
functions. If let say in future students dealing with more mathematical representation
or equation, it can easily be drawn using graphing tools GeoGebra.
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References
Blum, W., & Borromeo, R. (2009). Mathematical Modelling: Can It Be Taught And
Learnt? Journal of Mathematical Modelling and Application, 1(1), 45–58.
Chaamwe, N., & Shumba, L. (2016). ICT Integrated Learning: Using Spreadsheets
as Tools for e-Learning, A Case of Statistics in Microsoft Excel. International
Journal of Information and Education Technology, 6(6), 435–440.
https://doi.org/10.7763/ijiet.2016.v6.728
Gulkilik, H. (2020). Analyzing preservice secondary mathematics teachers’ prompts
in dynamic geometry environment tasks. Interactive Learning Environments.
https://doi.org/10.1080/10494820.2020.1758729
Kaur, H., & Leong, K. E. (2020). Effects of blum’s phase based instruction on the
mathematical modelling abilities among year 5 pupils. Annals of Mathematical
Modeling, 2(1), 16–28.
Khalil, M., Khalil, U., & ul Haq, Z. (2019). Geogebra as a Scaffolding Tool for
Exploring Analytic Geometry Structure and Developing Mathematical Thinking of
Diverse Achievers. International Electronic Journal of Mathematics Education,
14(2), 427–434. https://doi.org/10.29333/iejme/5746
Khalil, M., Sultana, N., & Khalili, U. (2017). Exploration of mathematical thinking and
its development through GeoGebra. Journal of Educational Research, 20(1),
83–99.
https://www.researchgate.net/profile/Muhammad_Khalil8/publication/336210058
_Exploration_of_Mathematical_Thinking_and_its_Development_through_Geoge
bra/links/5d946945299bf10cff1fefe9/Exploration-of-Mathematical-Thinking-andits-Development-through-Geogebra.
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Kwan Eu, L. (2013). Mathematical Modelling in the Malaysian Secondary Curriculum.
Learning Science and Mathematics, 8, 66–74.
ÖZGÜR, H. (2020). Improving Teachers’ Qualifications for Preparing ICT Based
Educational Materials. Malaysian Online Journal of Educational Technology,
9(1), 48–69. https://doi.org/10.17220/mojet.2021.9.1.245
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