As a mathematical modelling teacher, you will have greater success when you:
Understand the modelling principles and process.
Apply the modelling process to a problem yourself before presenting it to your students — so that you are equipped to anticipate contingencies.
Encourage students to develop their report while they progress through the stages of the modelling process.
The principles of design for successful mathematical modelling are:
Nature of the Problem
Successful mathematical modelling problems: A. Are open-ended, the extent to which may depend on students’ previous experience with modelling.
B. Provide both intra and extra-mathematical information.
Questions to ask yourself:
Are there a number of possible solutions to this problem?
Are there a number of pathways to finding a solution?
What scaffolding questions/info might I need to provide?
Is the problem too specific?
Do my students have enough experience with mathematical modelling to engage with this problem?
Is there enough non-maths information provided?
Relevance and Motivation
Mathematical modelling problems should have a genuine link with the real world of the students.
Selection should take into consideration factors such as students' age, year level, personal circumstances, etc.
Consider contextualizing the problem for specific student groups.
Questions to ask yourself:
Will this problem spark interest in my students?
Is it age appropriate?
Does it link to the real world of students?
The problem does not have to be specifically focussed on content for teenagers… However, problems that focus on such issues as driving, budgeting, the environment, school, etc encourage engagement. You probably know your students well… What will pique their interest? What other subjects/units are students currently taking? Might meaningful links be made? Is there a recent phenomena reported on social media that you could use as a starting point to build your problem?
Accessibility
For a problem to be successful in mathematical modelling, it must be possible to identify and specify mathematically tractable questions from the general problem statement.
Questions to ask yourself:
Will my students be able to work out the pertinent mathematics questions?
Do they have the right tools and knowledge to determine an approach?
Feasibility of Approach
The formulation of a solution process has to be feasible, involving:
The use of mathematics available to students.
The making of necessary assumptions.
The assembly of necessary data.
Before being presented to a class, teachers must work through the problem to ensure that it can feasibly be solved by their students.
Questions to ask yourself:
Will my students be able to find a solution to the problem?
What assumptions might they need to make?
Have they access to the right tools to help them solve the problem? What resources could be provided to supplement their learning? Do they have access to the internet? Do they have the ability to perform accurate searches? Do they have access to shared online notetaking? Whiteboards? Large sheets of paper and markers?
Have I provided enough information that they can work out how to move forward with the problem?
Have they worked in groups in mathematics classes before?
Feasibility of Outcome
Students should be able to interpret and solve the basic problem.
How they respond (e.g., arithmetical versus generalised solutions) depends on the year level and characteristics of your student group.
Teachers must work through the problem before class to ensure that there are possible solutions.
Questions to ask yourself:
How might my students go about solving the problem?
Can they check the feasibility of their solution?
Didactical Flexibility
Having worked through the problem yourself, consider how it will be implemented — determine how the problem may be structured into sequential questions.
Identify ways the problem can be broken down into smaller steps.
Determine what sequential questions can be asked or structured prompts/assistance be offered to students.
The role of the teacher is to facilitate problem solving, not to be the “expert” with all the answers. Students should be encouraged to explore varying approaches. Students should also be guided back through earlier steps in the modelling process if their solution is not adequate.
Questions to ask yourself:
Where do I think my students might get stuck?
If they get stuck, what prompts can I provide for them?
What challenges can I provide if they have taken an approach that is too simplistic?
B. Preparation and Pre-Engagement
Understanding of the modelling process and its application including preparation of support materials (learn/illustrate what the modelling process is)
Preparation
Successful mathematical modelling problems: A. Are open-ended, the extent to which may depend on students’ previous experience with modelling. - Less experienced students may need additional scaffolding questions or information. - More experienced students should be expected to engage with less defined problems.
B. Provide both intra and extra-mathematical information.
The questions you should ask are:
Are there a number of possible solutions to this problem?
Are there a number of pathways to finding a solution?
What scaffolding questions/info might I need to provide?
Is the problem too specific?
Do my students have enough experience with mathematical modelling to engage with this problem?
Is there enough non-mathematical information provided?
Pre-Engagement (Novice Modellers)
Students need to be initially familiarised with the modelling process. This can be supported via materials including:
A copy of the of the modelling process for students to map their way around during implementation.
An example of a simple modelling problem matched to the phases of the cycle:
Problem statement.
Formulation (building the model — mathematical questions, assumptions, choice of variables, parameter values etc).
Solution.
Interpretation.
Evaluation.
3. A copy of report structure — Students should have a clear idea what their report should look like at the end.
Pre-Engagement (Experienced Modellers)
Reviewing pre-engagement as required.
The length of the discussion is dependent on students’ prior experience with the modelling process.
Each student should be provided with a copy of the modelling task, and a representation of the modelling process that is a depiction of the logical process that will guide their efforts.
The Classroom Environment
Mathematical modelling works best when students are provided with the opportunity to engage with the challenge in small groups. This provides opportunities for lively discussion.
The classroom environment should encourage the sharing of all ideas, solutions, approaches — particularly the unconventional.
Consideration might be given to allocating roles within the group — who will be recording the group's efforts and who will be reporting back to the class? Where possible, students should be given access to technology — to record, research, clarify, and validate their work — and be encouraged to use it. Is it possible to use a shared drive for notetaking and report preparation?
C. Delivery: Stages of the Lesson
Review of the Modelling Process
Review the process of mathematical modelling, as required. Points that may be considered by teachers and students:
It is necessary to leave the realm of pure mathematics to build a model, e.g. by procuring extra-mathematical information and data.
Several different models may be reasonable. There is rarely a unique, or a best, answer to a modelling problem.
Modelling is not a five-minutes-to-get-an answer activity.
Simplifications are likely to be needed.
Assumptions may be necessary to reduce the complexity of the "real-world" domain or to make the mathematics tractable.
Assumptions can be made at any point in the cycle.
Students should be encouraged to ask clarifying questions.
Initial Problem Presentation
1. Teacher provides brief general description of the task scenario – the "messy real-life problem" [2-3 minutes].
2. Students are organised into small groups and given time to read the task and discuss how to approach the problem (e.g., What is the mathematical question?) [5 minutes].
3. Teacher calls the class back together to discuss their initial understanding of the task and provide possible mathematical questions. Each group contributes via a representative.
4. In groups, students consider assumptions and variables relevant to the mathematical question as well as other observations, such as trends in data, etc. [5 minutes].
5. Teacher once more calls the class back together. Each group reports back to whole class by group representative. Teacher synthesises/prioritises students’ initial assumptions and variables sufficient to begin modelling process.
Points that may be considered by teachers and students:
Teachers use facilitating questions that emerge from students’ engagement in the task rather than clarify problem contexts or ask questions up front. Responses should align with the question, “What should a modeller be asking himself/herself at this point in the modelling process?” (metacognitive connection).
There should be a focus on student decision making — with no prior indications of what specific mathematical question should be identified, or what mathematical content will be useful in addressing the problem.
Students should be encouraged to pose explorative questions as to the nature of the problem.
Body of Lesson: Modelling in Progress
How teachers support mathematical modelling:
Bring to consciousness those things that are implicit… actions are then deliberate.
Support students' progress through the modelling cycle.
Activate teacher meta-meta cognition:
(a) How will students interpret what I (the teacher) am doing/saying at this point? (b) What should students be asking themselves at this stage in the modelling cycle?
Anticipate where students might have problems, e.g., interpreting the problem, generalizing the solution.
Employ "measured responsiveness" — rather than providing specific advice about the problem, prompt students to think about where they are in the modelling cycle— structure mathematical questions that promote a viable solution pathway.
Encourage the use of digital or other tools as appropriate.
Support student development of a modelling report.
Throughout the lesson, students should:
Document progress against a visual representation of the modelling process. [Problem statement → Formulate → Mathematical solutions → Interpreting outcomes → Evaluation.]
Consider forms of collaboration:
- Should individual members of the group work separately and then come together at several points along the way to discuss? - Should group members work together from the beginning with continual negotiation and confirmation of consensus? - Determine how they might engage those external to the group [Teacher, students in other groups]. - Identify groups working on a similar problem/issue and extend collaborations.
Throughout the lesson, teachers should:
Check that students are documenting progress against the modelling process (both in the doing and in the recording).
Focus students’ attention on phases of the modelling cycle (there should be no specific direction towards a solution).
Support student decision making — encouraging multiple solution pathways.
Take account of student capability and cater for diversity.
Respond to students’ questions or requests for assistance with open questions:
- What are you doing?/trying to do? - Where are you in the modelling process? - How have you checked your answer (both mathematically and in terms of context)? - Can your solution be generalized?
Conclusion: Presentation of Findings and Teacher Summary
One representative/spokesperson from each student group shares what they have found with justification. Findings should be reported in a succinct fashion which may take any one of a variety of forms (e.g., 3-4-minute video). The focus should be on what was learnt about the modelling process. Teachers/students ask questions for clarification or to test arguments.
Points to be considered by teachers and students:
Students in the audience should provide questions, elaborations, clarifications (e.g., each student to write down one question).
Comments could also be directed towards criteria related to making judgements about the quality of findings (e.g., see Modelling Process).
Teacher clarification questions can include:
How does that work with your model? (e.g., where teacher has identified an error).
Will your solution work for other situations? (e.g., teacher encouraging students to generalise).
What did you do to evaluate the model? (e.g., teacher encouraging students to validate and verify a proposed solution).
D. Reporting
Guidelines for Report Writing
The report writing for mathematical modelling tasks should:
Provide a specific target for students to aim at.
Seamlessly connect learning and assessment — students build reporting skills in their developmental tasks that are directly needed in assessment tasks.
Apply to any modelling problem, so that successive use assists in the cumulative goal over time of producing students who are skilled modellers.
Provide a set of handles for providing feedback to students in terms of their modelling and reporting actions.
Operate from the first model in the junior school and be reinforced with every use.
What the Report Should Include
Missteps should not be erased, but reported on. The report should be a means for students to communicate their findings. It should be succinct, coherent and systematic. The report must make use of appropriate mathematical language.
The student report should:
Describe the real-world problem being addressed.
Precisely specify the resulting mathematical question/s.
List all assumptions wherever they are made.
Indicate how numerical values used in calculations were decided on.
Show and justify all mathematical working, including through the use of graphs, tables, etc.
Interpret the meaning of mathematical results in terms of the real-world problem.
Evaluate the result:
- Does the answer make sense mathematically? - Does it help to answer the real problem? - Are there qualifications you want to make about your solution? - Can the result be validated in the real-world context? - Are there recommendations that arise from the work? What further work is needed?
Finally, students should identify how they used the mathematical modelling cycle to inform their problem solving. Where was the cycle revisited and adaptations made? Were there any missteps along the way? These should not be erased for form an important part of their report.
Forms of Reporting
The Mathematical Modelling Report could take on any number of forms.
For beginning modellers, the report might simply be a record of their classwork as it took place. An example of a booklet that could be used can be downloaded HERE. This type of booklet helps novice modellers to shift their thinking away from immediately attempting to find a mathematical solution to considering the steps in the modelling cycle. As most mathematical modelling problems offer a range of approaches and possible solutions, focusing students from the outset on metacognitive processes provides them with a solid base for future mathematical modelling problems.
If the mathematical problem solving is not part of their summative assessment, students' reports might simply take the shape of "works in progress" — butcher's paper, whiteboard workings, etc. To preserve these works in progress, we recommend screen shots be taken at the end of each class.
As mathematical modelling is usually undertaken in small groups, teachers should consider how to assess individual work/contributions to the report. Finally, student reports for either formative or summative assessment for mathematical modelling could take any number of genres — full written submissions, three-dimensional models, blogs, video presentations... The options are only limited by the imagination!
A ONE-PAGE OVERVIEW OF THE DESIGN AND IMPLEMENTATION FRAMEWORK CAN BE DOWNLOADED HERE.
So, how can we best enable the mathematical modelling? Our research has focussed on this question. Find out more here.