Mathematical Modelling uses real life problems and is open to interpretation where as Problem Solving is more of a mathematical approach.

The difference between mathematical modelling and problem solving is that whilst problem solving components are apart of mathematical modelling, mathematical modelling is using maths concepts and structures to represent and analyse real world situations.

The concepts aren't explicitly mentioned and the situation is broader and requires more assumptions to solve the problem.

Ella Southall (I was on my 4th year placement week 6 and so was not in a group).

Mathematical modelling in primary classrooms helps students connect maths to real-life contexts, making learning more engaging and meaningful. English (2013) shows that working with complex data builds critical thinking and problem-solving, while Parish (2023) highlights how authentic contexts motivate students and show the usefulness of maths. Together, they suggest modelling fosters reasoning, collaboration, and creativity beyond routine procedures.

Mathematical modelling can be seen through examples, such as how much petrol is left in the tank, when writing a birthday card to know how much space to fill and when cooking and baking, specifically when making quantities different to what the recipe states. 

• Lily, Samuel, Isabel

  
  

There are many assumptions that mathematical modelling is a skill that should be introduced during high school, and that primary school students are not adept at learning (English, 2013). There has been a recent push to involve mathematical modelling into all year levels of both high school and primary school (Parish, 2023). However, mathematical modelling is a great concept to introduce into primary school because it encourages students to think more abstractly. It is extremely useful to connect the content that students are learning in other subjects to maths (English, 2013). For example, the gold-rush, the outside environment, natural disasters, or garden beds that they see in the school etc. (English, 2013). In the workshop, we discussed how students can find maths anywhere, and to encourage students to think about maths when they walk past a garden or when they go to the zoo.

Parish's (2023) findings, that mathematical modelling can be relevant in a primary school context, were evidenced in today's workshop. This can be seen through the Mountain Dew example as it was simple and relevant to primary school students.

We also engaged with 3-Act tasks where we posed mathematical questions such as “I wonder how tall the Moon Man’s head is..”. We then explored mathematics in a meaningful way and translated the solution back into the real-world example.

As discussed by English (2013), students in a primary school setting will display variation in the models they create. There was a wide variety of approaches evident in the swimming task, similarly throughout the seminar, groups were demonstrating differentiated ways of thinking to come to a conclusion. 

We did this through using real-life experiences such as the amount of toothpaste used and how many times a day we brushed our teeth to provide context to the problem (C&D).

There was different solutions found as some of us attacked the problem different from others, based on what made sense. This helped us revise our solutions as we had to identify what we wanted to know, the amount of toothpast needed or the how many days one tube would get us (E&F). 

A report has not be started yet as some people did different tasks but additionally, there is still variables to be investigated such as people travelling. 

1. Real world solution

   1.  looking into the amount of toothpaste in grams that people use when brushing their teeth.

   2.  how many times a day do people brush their teeth on average

   3.  cost of toothpaste

2. Mathematical Solution

   1. How many grams per brush x number of brushes per day x number of days 

3. Real world meaning

   1. China has minimal health and safety laws for toothpaste, hence the traveller wanting to bring their own

4. revise model

   1. unhappy with the amount of toothpaste suggested

5. revise model

   1. still unhappy with the amount of toothpaste suggested

6. accept solution

   1.  came to an agreement that it is likely that we use 1 gram of toothpaste per brush (2 grams a day) 

Now we are ready to begin our final report. 

-It is accessible to all students as the mathematics involved works off of pre-existing knowledge

-Helps support student productive disposition and motivates students to see that mathematics is meaningful and applies to everyday contexts

-It helps equip students with the power to exercise the fundamental duty as a citizen to appreciate, critique and use models and modelling that permeate our modern world

Challenges:

-Students may not have the knowledge necessary to apply the modelling process. This can also contribute to increased maths anxiety.

-The process of implementing the steps necessary to make the mathematical modelling process meaningful can take considerable amount of time and planning that teachers may not be able to do. They may also lack the knowledge and skills to implement it.

-Initial presentation of mathematical modelling structure can manifest student confusion and reduce self efficacy due to the subjective nature of the problems posed

1. Connects students to mathematics in a real-world context

2. Students engage a range of critical thinking skills to complete modelling tasks 

3. Mathematical modelling tasks can address a range of curriculum areas 

Challenges in implemented MM in primary school mathematics 

1. Pushback regarding 'lack of explicit teaching' 

2. Tasks often run over an extended period of time and can take up multiple lessons 

3. Lack of confidence from teachers about how to tach MM as a process 

- Connecting to real world problems and situations,

- Collaboration and communications,

- More involvement and understanding if it’s related to real world scenarios,

Challenges:

- Time pressure,

- Issues/lack of with student foundational skills,

- Balancing task design to students capabilities

• Students are able to think critically and creatively in class. They can think outside the box and have to make considerations to their models.

• Students are able to work as a team and collaborate to create a solution to the model. It also gives students more control over their learning and they can lead their investigation.

• Since the models reflect everyday life situations, students are able to understand more of the meaning behind their learning. Students are likely to be more engaged in the learning because they can relate to it more.

Challenges:

• Students might not know all of the content that is needed for students to complete a mathematical model. It might require knowledge of different maths, so students might struggle to engage in the model.

• Not all students are going to be working at the same level in maths. Especially in composite classes where there are 2 different year levels. Therefore it can be difficult for students to be on the same page.

• Mathematical models often cover multiple curriculum areas, to which students will likely need to know prior to engaging in the modelling. This can be extremely time consuming, and the modelling process can often span over multiple lessons. Therefore, it can consume a lot of time in a small amount of school space.

Personally, I believe I have created a deeper understanding of mathematical modelling and how it can be used in a classroom. It's more about letting the students explore the answers/ solutions rather than the teacher just talking at them. There shouldn't be a time limit on projects like these, as sometimes they can extend for multiple weeks; however, there is STILL learning going on. Similarly, I believe more teachers should understand mathematical modelling more so they feel comfortable teaching it and exploring it. Mathematical Modelling also correlates with multiple other subjects and curriculum. 

Through participating in mathematical modelling tasks, students use a range of skills to move from one stage of the cycle to the next,  some of these include: making assumptions, mathematising, analysing and justifying. Not only are these skills key for the development of students mathematical understanding and skills, but these skills can also be applied across a range of subject areas, as seen in the sea of mud activity which expanded across multiple areas of the curriculum. 

Additionally, mathematical modelling outlines to students the real-world implications of mathematics. It is important for teachers to demonstrate to students the importance of what they are learning to their everyday lives so that they are able to understand the purpose which will enable them to develop a productive disposition towards mathematics. Mathematical modelling draws on these real-world scenarios and students connect these situations to mathematical concepts to better understand the situation and produce a solution or representation. 

Another key takeaway is that modelling is not always about finding a perfect answer but about making reasonable assumptions and testing possible solutions. The process often involves refining models, adjusting variables, and interpreting results in context of the problem. This helped me realise that mathematics is an iterative process, where flexibility and critical thinking are important. This is a good reminder to students that even if you don't have all the information we can still come to appropriate conclusions.

I started this topic with apprehension because I like having a simple answer, but grew to really enjoy modelling because the process was extremely fun and intriguing. I liked that it related to real life problems and inspired a lot of discussion. I can visualise this being really productive in classrooms and inspire more love for maths in students who might not typically enjoy maths.

The review process is also very valuable because it challenges students’ thinking and encourages them look back at their learning. It also finds areas of improvement or gaps in knowledge.

• Harry Fleming

In this unit I found mathematical modelling really interesting. I liked that it wasn’t just about getting the “right” answer but more about exploring different variables and seeing how the problem could change. It made me realise how maths can actually connect to real-life things and why it matters.

I also noticed the modelling cycle isn’t just one straight line, you often go back and rethink what you’ve done, which made sense to me. The very first week of activities showed how maths can link to other subjects and also get students working together and thinking in different ways. Overall, I learnt that modelling can make maths feel more useful and help students build problem-solving skills. I think it'll be a fun avenue to explore with students.

Mathematical modelling involves using maths to understand and solve a real-life situation relating to real-world scenarios, whereas problem solving involves using maths to answer and develop solutions to a mathematical question.

It was evident through both readings that collaboration is key to developing different skills and learning that they may not have thought of if they had just completed a task individually. It was also clear that students first attempting problems individually and without direct instruction allow them to identify what they know and can connect to the real world. They can then share ideas and opinions with peers to expand learning.

B. Created a mathematical situation and some assumptions of:

• 365 days brushing your teeth 2 times a day 730 brushes

• Using the recommended amount of toothpaste 0.25g

• 1 tube of toothpaste is 175g

C. created a model of (730 * 0.25 = 182.5)

D. Solution - 2 tubes of toothpaste for a year.

E & F. When applied to the real world we felt it unrealistic to use only 2 tubes of toothpaste for an entire year. So we revised our definition of 'pea sized amount' to 0.5.

C. created a new model/table to show the relationship between toothpaste brush size (0.25 - 1g), amount that needs per year, and how many packets.

• The connection and application to the real world context. Being able to explore issues, topics, interests through maths and doing it creatively is beneficial to the productive disposition of students.

• Engagement to a range of problem solving and critical thinking

• Students from all different achievements can access mathematical modelling

Challenges:

- Timing often teachers complain about timing and issues with timing.

• Lots of planning and resources required to put in a mathematical modelling unit.

• Confusion

• lack of guidance students could get lost or take a wrong turn

Mathematical modelling makes meaning out of our current context, we can apply it to our own life and situations around us. It encourages exploring our world and ideas that students have. It involves collaboration and team work.

Overall, my understanding of mathematical modelling has been profound and I find excitement in wanting to implement my own unit on mathematical modelling.

Provide examples with your explanations, and remember to refer back to the numbers and letters in the chart below.

I was aware of all of the stages of mathematical modelling when working through the toothpaste task, repeating in my head where I was up to in the cycle when completing the question. The step that has stuck with me and become the most clear when working out, not just the toothpaste tasks but the other tasks throughout my portfolio, is step 2, where I begin to assume real examples to work them into the mathematical model.

For example, in the toothpaste task, I assumed that the individual was quite a hygienic person because they are worried about the health and safety laws of the country they will be living in for a year. So I assumed that they brush their teeth at least twice a day, meaning they will need enough toothpaste to brush their teeth twice a day for the year. I also assumed that they would want some leftovers in case, for any chance, they run out early. I assumed that the average amount of toothpaste used to brush your teeth is 0.5g.

• It benefits students by educating them on other important learning opportunities that promote their safety.

• Students can learn in a range of learning areas within mathematical modelling.

• Students are encouraged to use their own critical thinking skills rather than relying on being spoon fed by their classroom teachers.

List 3 challenges for teachers in implementing mathematical modelling
in primary school mathematics

• The tasks can often be quite long and require more time to complete.

• It can be hard for teachers to understand and learn the mathematical modelling cycle, let alone for students to learn and be confident in using it within their education.

• The tasks can be very messy, which may stress and confuse students because there are so many ways to go about solving the questions given.

The idea I have developed as a result of the mathematical modelling unit is that, through mathematical modelling, it is okay to have multiple answers as long as you have the evidence to back them up. Multiple ways and strategies can be used for a real-world messy mathematical modelling problem. It is a great way to introduce collaboration to the class. I've really enjoyed sharing my ideas with my friends and family within this unit because it is interesting to see their perspectives relating to the cycle and the questions posed. It has been so satisfying when I complete my answers, but also being able to have those 'aha' moments with the people I have collaborated with, joining two and two together in areas I have gotten stuck, or comparing our answers and working out to extend my own learning.

The consistent use and repetition of the mathematical modelling cycle has allowed me to grow in confidence. Using it within my working out, and now enjoy going back through the questions I have answered and identifying what stage of the cycle I am in.

1. Research information based on how much toothpaste the average human should use every day.

2. Research information based on how much the average human should be brushing their teeth

3. Create a mathematical model.

1. The average human uses around 0.25 grams. 0.25 grams = approx. 0.3 ml. 210 grams in my tube (the size I would bring) = 246 ml.

2. The average Human should brush their teeth twice a day.

3. Toothpaste Needed (x) = (Grams per Brush) x (Brushes per day) x (days per year)

   x = 0.25 x 2 x 365

   x = 182.5 g

Since my tube contains 210 grams, one tube of toothpaste should be enough to last me the whole year

applying to a real-world definition. I did not think it would be applicable to use only one tube of toothpaste for the entire year, so I revised my model, changing the grams used to 0.5 instead of 0.25.

Now I am ready to begin my final report

Encourages problem-solving skills

Promotes critical and creative thinking

Supports Collaboration and Communication

Challenges:

Complex real-world situations

Balancing assumptions with accuracy

Time Consuming