1. Mathematical modelling
A matMathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.
A mathematical model is an abstract model that uses mathematical language to describe the behaviour of a system.
Mathematical models are used particularly in the natural sciences and engineering disciplines (such as physics, biology, and electrical engineering) but also in the social sciences (such as economics, sociology and political science); physicists, engineers, computer scientists, and economists use mathematical models most extensively.
Eykhoff (1974) defined a mathematical model as 'a representation of the essential aspects of an existing system (or a system to be constructed) which presents knowledge of that system in usable form'.
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.
Mathematical models can take many forms, including but not limited to dynamical systems, statistical models, differential equations, or game theoretic models.

2. Mathematical modelling-what is it ?
a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.
In simple words – to model or represent a real world situation in terms of mathematics and its language
Traditionally, mathematical modelling in the elementary grades has focused on arithmetic word (story) problems that are represented with concrete materials and then modelled by more abstract operational rules. Solving these word problems entails a mapping between the structure of the problem situation and. the structure of a symbolic mathematical expression.
Oftentimes, solving these word problems is not a modelling activity for children, rather, it is one that relies on syntactic cues such as key words or phrases in the problem (e.g., “times,” “less,” “fewer”). Furthermore, there is usually only one way of interpreting the problems and hence, children engage in limited mathematical thinking. While not denying the importance of these types of problems, they do not address adequately the mathematical knowledge, processes, representational fluency, and social skills that our children need for the 21st century (English, 2002b).
Traditionally, mathematical modelling in the elementary grades has focused on arithmetic word (story) problems that are represented with concrete materials and then modelled by more abstract operational rules. Solving these word problems entails a mapping between the structure of the problem situation and. the structure of a symbolic mathematical expression.
Oftentimes, solving these word problems is not a modelling activity for children, rather, it is one that relies on syntactic cues such as key words or phrases in the problem (e.g., “times,” “less,” “fewer”). Furthermore, there is usually only one way of interpreting the problems and hence, children engage in limited mathematical thinking. While not denying the importance of these types of problems, they do not address adequately the mathematical knowledge, processes, representational fluency, and social skills that our children need for the 21st century (English, 2002b).

3. Mathematical Modelling for children today
To foster the mathematical modelling abilities children require for today’s world, we need to design activities that display
Authentic problem situations;
Opportunities for model exploration and application;
Multiple interpretations and approaches;
Opportunities for social development;
Multifaceted end products; and
Opportunities for optimal mathematical development.
Mathematical modelling is frequently viewed as the construction of a link or bridge between mathematics as a way of making sense of our physical and social world, and mathematics as a set of abstract, formal structures (Greer, 1997; Mukhopadhyay & Greer, 2001).
To foster the mathematical modelling abilities children require for today’s world, we need to design activities that display the above features:

4. Opportunities for Model Exploration and Application
Begin with an activity that elicits the development of significant mathematical constructs.
followed by a model-exploration activity or activities, where children can consolidate and refine the conceptual systems they have developed as well as construct powerful representation systems for these systems
The next activity in the sequence is a model-application task where children deal with a new problem that would have been too difficult without their prior development of a conceptual tool.
A program of modelling experiences for children is most effective if it comprises sequences of related activates that enable models to be constructed, explored, and applied. The activities should be structurally related, with discussions and explorations that focus on these structural similarities (Lesh, Cramer, Doerr, Post, & Zawojewski, 2002).
Example: “What factors are important to you in buying a pair of sneakers?“
In small groups, children generate a list of factors and then determine which factors are most important. This inevitably results in different group rankings of the factors. The teacher then poses the problem of how to create a single set of factors that represents the view of the whole class.
This activity is followed by a model-exploration activity or activities, where children can consolidate and refine the conceptual systems they have developed as well as construct powerful representation systems for these systems. In essence, at the end of a model-exploration activity, children should have produced a powerful conceptual tool or model that they can apply to other related problems.
The next activity in the sequence is a model-application task, where children deal with a new problem that would have been too difficult without their prior development of a conceptual tool.
This new activity requires some adaptation to the tool and involves the children in problem posing as well as problem solving, and information gathering as well as information processing (Lesh et al., 2002).

5. A program of modelling experiences for children is most effective if it comprises sequences of related activates that enable models to be constructed, explored, and applied. The activities should be structurally related, with discussions and explorations that focus on these structural similarities (Lesh, Cramer, Doerr, Post, & Zawojewski, 2002).
Example: “What factors are important to you in buying a pair of sneakers?“
In small groups, children generate a list of factors and then determine which factors are most important. This inevitably results in different group rankings of the factors. The teacher then poses the problem of how to create a single set of factors that represents the view of the whole class.
This activity is followed by a model-exploration activity or activities, where children can consolidate and refine the conceptual systems they have developed as well as construct powerful representation systems for these systems. In essence, at the end of a model-exploration activity, children should have produced a powerful conceptual tool or model that they can apply to other related problems.
The next activity in the sequence is a model-application task, where children deal with a new problem that would have been too difficult without their prior development of a conceptual tool.
This new activity requires some adaptation to the tool and involves the children in problem posing as well as problem solving, and information gathering as well as information processing (Lesh et al., 2002).

6. Multiple Approaches
Modelling activities involve multiple, simultaneous interpretations.
With modelling activities contemplate which of several approaches could be taken in reaching the goal
must also interpret the goal itself and the accompanying information.
Each of these components might be incomplete, ambiguous, or undefined; there might be too much data, or there might be visual representations that are difficult to interpret.

7. Opportunities for Social Development
modelling activities play an important role in children's social, as well as mathematical, developments
children have a shared responsibility to ensure their models meet the desired criteria and that what they produce is informative and user-friendly
Numerous questions, conjectures, arguments, revisions, and resolutions arise as children develop and assess their models, and communicate their models to a wider audience.
Although their discussions during model construction can be off task at times, children nevertheless develop powerful skills of argumentation in which they challenge one another’s assumptions, ask for justification of ideas, and present counter-arguments.
Although their discussions during model construction can be off task at times, children nevertheless develop powerful skills of argumentation in which they challenge one another’s assumptions, ask for justification of ideas, and present counter-arguments.

8. Approaches to teaching-generalised applications approach
concentrates upon a particular application. Usually the teacher has taught the model and students are manipulating it under controlled conditions. This approach is common in secondary classrooms
(i) gathering, interpreting and analysing mathematical information and
(ii) applying mathematics to model situations.

9. Approaches to teaching- structured modelling approach
uses real life situations and the full modelling process
In this approach the teacher exerts considerable control over the mathematical model

10. Approaches to teaching-open modelling approach
Allows students to study a problem at the level of mathematics that they are comfortable using.
All stages of the modelling process are completed.
Students are asked to work with a problem with limited teacher assistance, because the teacher does not control the mathematics chosen by the students

11. The modelling process
Image source :

12. Authentic Problem Situations
situations that are:
motivating, interesting,
relevant to children’s world,
where there is a genuine need for particular mathematical processes
For a number of years, mathematics curriculum documents and mathematics educators have been emphasizing the importance of couching children's problem experiences in situations that are motivating, interesting, and relevant to their world, and where there is a genuine need for particular mathematical processes (Boaler, 2002; Kolodner, 1997). Such authentic contexts provide sense-making and experientially real situations for children, rather than simply serve as cover stories for proceduralized and frequently irrelevant tasks.
While the benefits of such experientially real contexts have been well documented (most notably in the Realistic Mathematics Education research, emanating from the Freudenthal Institute; Freudenthal, 1983; Gravemeijer, 1994), there have also been some concerns expressed (Boaler, 2002; Silver, Smith, & Nelson, 1995).
A main issue cited is that children are frequently required to both engage with the problem contexts as though they were real and to ignore factors that would pertain to real-life versions of the task (Boaler, 2002).
When children situate their reasoning within their own authentic contexts, there are of course several “correct” answers. It has thus been recommended that we need to not only provide real-world contexts, but also “real-world solutions” (Silver et al, 1995, p. 41). Mathematical modelling activities take both aspects into account.

13. For a number of years, mathematics curriculum documents and mathematics educators have been emphasizing the importance of couching children's problem experiences in situations that are motivating, interesting, and relevant to their world, and where there is a genuine need for particular mathematical processes (Boaler, 2002; Kolodner, 1997). Such authentic contexts provide sense-making and experientially real situations for children, rather than simply serve as cover stories for proceduralized and frequently irrelevant tasks.
While the benefits of such experientially real contexts have been well documented (most notably in the Realistic Mathematics Education research, emanating from the Freudenthal Institute; Freudenthal, 1983; Gravemeijer, 1994), there have also been some concerns expressed (Boaler, 2002; Silver, Smith, & Nelson, 1995).
A main issue cited is that children are frequently required to both engage with the problem contexts as though they were real and to ignore factors that would pertain to real-life versions of the task (Boaler, 2002).
When children situate their reasoning within their own authentic contexts, there are of course several “correct” answers. It has thus been recommended that we need to not only provide real-world contexts, but also “real-world solutions” (Silver et al, 1995, p. 41). Mathematical modelling activities take both aspects into account.

14. An Example
How much space this cardboard box holds ?

15. Modelling the situation
We know how to find the volume of a cuboid
V = l x w x h
We can apply
this method to the present situation by finding the l (length), w(width) and h(height) h w l

16. Modelling the cardboard volume-accuracy
The answer may not be accurate –Why ?
A mathematical model may not be exactly same as the real thing/ object/situation
Think the context !
However, the answer is generally good enough to be useful.
Did we take into account Card board’s thickness?, what about the measurement errors ?

17. How can we improve the model ?
Assuming the cardboard is “t” units thick . The Mathematical model may be improved by changing inside height, inside length and inside width by h-2t, l-2t and w-2t
Hence inside Volume of the box (more accurately) be given by
V = (l-2t ) x (w-2t) x (h-2t)
Now we have a new model ( which still may be accurate) but is better.

18. Using the model
When we have a model we can use it, play with it, see the changes the different variables cause etc.

19. Example
A company uses 200x300x400 mm size boxes, and the cardboard is 5mm thick.
Someone suggests using 4mm cardboard ... how much better is that?

20. Solution
With the current volume h = 200, w = 300 and l = 400 and t = 5 so
Current Volume = 
(200-2×5) × (300-2×5) × (400-2×5) = 21,489,000 mm3
The new value of t = 4 , hence
New Volume = 
(200-2×4) × (300-2×4) × (400-2×4) = 21,977,088 mm3
The change in volume is
(21,977, ,489,000)/21,489,000 ≈ 2% more volume. Thus the model is useful as it let us know the change caused by taking 1 mm less thickness cardboards for the boxes
Discuss with the students , how do we know what is h, and what are l and w in 200x300x400 ( or does it matter if student differ in taking h, w and l duifereltly)

21. Confusing situations for children
On our street there are twice as many dogs as cats.
How do we write this as an equation?
To set up a mathematical model we also need to think clearly about the facts!

22. Think clearly
Let D = number of Dogs and
C = number of Cats on our Street
According to the given situation
C= 2D
Or Should it be
D = 2C ???

23. Another Example – Most economical
Problem:
Your company is going to make its own boxes!
It has been decided the box should hold 0.02m3 (0.02 cubic meters which is equal to 20 liters) of nuts and bolts.
The box should have a square base, and double thickness top and bottom.
Cardboard costs $0.30 per square meter.
It is up to you to decide the most economical size.

24. Step1- Draw a visual mark the sides with symbols
Thoughts !
The base is square, so we will just use "w" for both lengths h w
mark the sides with symbols w

25. Step 2 – making the formulas
Ignoring the thickness – The Volume is given by
V = w x w x h = w2 x h
Area of four sides = 4 x w x h
Area of top and bottom = w2
Total area of the card board
A = 4wh + w2
Cost = x area of the cardboard

26. Step 3 – what else is given
Given is that V = 0.02 m3
Which means = w2 x h
Cost = 0.3 x Area = 0.3 (4 wh + w2 )
= 0.3 (4 w x 0.02/ w w2 )
= 0.3 ( 0.08/ w + 4 w2 )
Step 4 – Making single formula

27. Step5- Plot the graph (technology)
What to plot? Well, the formula only makes sense for widths greater than zero, and I also found that for widths above 0.5 the cost just gets bigger and bigger.
when the width is about 0.22 m (x-value),
the minimum cost is about $0.17 per box (y-value).
Just by eye, I see the cost reaches a minimum at about (0.22, 0.17). In other words:
when the width is about 0.22 m (x-value),
the minimum cost is about $0.17 per box (y-value).
In fact, looking at the graph, the width could be anywhere between 0.20 and 0.24 without affecting the minimum cost very much.

28. Connecting all together to the context
Step (Last)
Connecting all together to the context
Using this mathematical model you can now recommend:
Width = 0.22 m
Height = 0.02/w2 = 0.02/0.222 = m
Cost = $0.30 × (0.08/w+ 4w2) = $0.30 × (0.08/ ×0.222) = $0.167 about 16.7 cents per box.

29. improvements to this model:
Step (to improve)
 improvements to this model:
Include cost of glue/staples and assembly
Include wastage when cutting box shape from cardboard.
Is this box a good shape for packing, handling and storing?
Any other ideas you may have!

30. Conclusions Mathematical Modelling
take children beyond the traditional classroom experiences
extends their thinking and abilities
We need to implement worthwhile modelling experiences
modelling activities for children should build on children’s existing understandings
should engage them in thought-provoking, multifaceted problems that involve small group participation
activities should be set within authentic contexts
engage in important mathematical processes such as describing, analysing, coordinating, explaining, constructing, and reasoning critically
It is imperative that we take children beyond the traditional classroom experiences, where problem solving rarely extends their thinking or mathematical abilities. We need to implement worthwhile modelling experiences in the elementary and middle school years if we are to make mathematical modelling a way of life for our students. Younger learners, irrespective of their class achievement levels, can successfully complete modelling problems . Mathematical modelling activities for children should build on their existing understandings and should engage them in thought-provoking, multifaceted problems that involve small group participation. Such activities should be set within authentic contexts that allow for multiple interpretations and approaches. As children work these activities, they engage in important mathematical processes such as describing, analyzing, coordinating, explaining, constructing, and reasoning critically as they mathematize objects, relations, patterns, or rules.

31. In Conclusion
It is imperative that we take children beyond the traditional classroom experiences, where problem solving rarely extends their thinking or mathematical abilities. We need to implement worthwhile modelling experiences in the elementary and middle school years if we are to make mathematical modelling a way of life for our students. Younger learners, irrespective of their class achievement levels, can successfully complete modelling problems .
Mathematical modelling activities for children should build on their existing understandings and should engage them in thought-provoking, multifaceted problems that involve small group participation. Such activities should be set within authentic contexts that allow for multiple interpretations and approaches. As children work these activities, they engage in important mathematical processes such as describing, analyzing, coordinating, explaining, constructing, and reasoning critically as they mathematize objects, relations, patterns, or rules.

32. References
L. D. English (2003) Mathematical Modelling With Young Learners Mathematical modelling : a way of life ICTMAII Chichester: Horwood Publishing available at
Mathematical Models at
MATHEMATICAL MODELLING and the General Mathematics Syllabus Available at

33. Activities- Catch the correlation
Students will need:
Scissors
Copy of the graphs sheet, printed on transparent sheet (on the next slide).
Sheet of white paper
Pen and notebook
What to do
Cut out each graph from the sheet.
Choose two graphs, and lay one on top of the other, with a plain white sheet of paper behind. Make sure the months line up. Do the curves on the graph match?
Move the two graphs so the curves match the best you can make them. Do the months line up now?
Choose two different graphs and repeat the matching process. Try several different pairs.
Can you describe the relationships between each of the graphs? Can you think of a good reason why they don’t match perfectly?
Activity source: Maths +Stats By (10 March 2015)

34. Activities - Catch the correlation

35. Activities Catch the correlation
What’s happening?
None of these graphs line up perfectly, but they are all linked. What we have found is a correlation. The graphs in this activity are all seasonal, rising and falling in a yearly cycle.   We can work out some of the reasons the graphs don’t line up perfectly. Temperature lags a bit behind day length, which means the hottest days are a bit after the longest days. This is because it takes a while for sunlight to heat the Earth. Several long days in a row will make the ground get hotter and hotter.   London’s weather is similar to Canberra, but the seasons are flipped. This is because they are in different hemispheres on Earth. When the days are long in the southern hemisphere, they are short in the northern hemisphere.