The Mathematical Design and Implementation Framework#

1. Design Principles#

As a teacher, you'll have greater success if

Understand the modelling princples and process
Apply it to a problem before presenting to students
Encourage students to develop their report as they go

Design principles are

1.1 Nature of the problem#

Open-ended and involve both intra- and extra- mathematical information. Vary open-endedness dependent on students’ previous experience with modelling. Add more scaffolding if required.

1.2 Relevance and motivation#

Genuine link with students' real-world. Heavily dependent on students and context. Further adaptation may be required.

1.3 Accessibility#

Mathematically tractable questions are evidence from a general problem statement. Accessible to student math content and perspective.

1.4 Feasibility of approach#

The solution process is available to students given (a) their mathematical K&S, (b) the making of necessary assumptions, and © the assembly of necessary data. Teachers must work through the problem.

1.5 Feasibility of outcome#

Will students actually be able to solve and evaluate their solution to the problem?

1.6 Didactical flexibility (a form of generativity?)#

Able to structure the problem in different ways (sequential questions). What other didactical scaffolds might be required?

2. Preparation and pre-engagement#

Ensure students are understand are prepared to engage in the modelling process

2.1 Preparation#

Ensure problem is sufficiently open-ended and accessible to the students. Providing scaffolding where required for less familiar students. Provide both intra and extra-mathematical information.

Considering, possible solution pathways, scaffolding questions, sufficient non-mathematical content, description of the problem solving method, examples etc.

Diagram representing The mathematical modelling cycle

2.2 Classroom environment#

Small group best social structure
encourage discussion, share of all ideas, solutions, approaches - especially the unconventional
group role allocation may be considered
Where possible provide access to technology to record, research, clarify and validate work

3. Delivery: stages of the lesson#

3.1 Review of the modelling process#

Review the process and consider making points such as

will likely require non-mathematical information - can't rely just on math
may be different models, rarely a single best model
not a quick, get an answer activity
will likely require simplifications and assumptions to make tractable
assumptions can be made at any point
encourage question asking

3.2 Initial problem presentation#

brief teacher provided description
students in group read and discuss the problem and likely approach
whole class discussion to check understanding, approach and perhaps provide pointers
groups consider assumptions, variables as well as other observations
whole class discussion with teacher synthesising and prioritising
focus on student decision making

3.3 Body of the lesson: modelling in progress#

Teacher will

monitor and support student process
anticipate problems
prompt and scaffold, rather than answer
encourage tools
support student engagement in report preparation

Students should

Document process
consider forms of collaborations

3.4 Conclusion: presentation of findings and teacher summary#

Findings presented in succinct way. Focus on what was learnt about the modelling process. Questions posed for clarification and to test arguments.

4. Reporting#

4.1 Guidelines for reporting writing#

provide a specific target
connect with assessment
apply to any modelling problem
provide space for providing feedback

4.2 What the report should include#

Report missteps. Seen as a means to communicate findings. Succinct, coherent, systematic.

describe the problem
precisely specify the resulting mathematical question/s
list assumptions
indicate how numerical values were decided on
show all mathematical working
interpret the meaning of mathematical results in terms of the real world problem
evaluate the answer
- Does the answer make sense mathematically
- Does it help to answer the real problem?
- Are there qualifications
- Can the result be validated in the real-world?
- Are there recommendations that arise? Further work required?
Students identify how the modelling cycle informed their problem solving.

4.3 Forms of reporting#

Different forms.

beginning modellers might use this type of booklet
if not a summative task, butcher's paper, whiteboard etc.
if group work, how to assess individual work
genres other than reports - blogs, video presentations etc.

References#

Geiger, V., Stillman, G., Brown, J., Galbriath, P., & Niss, M. (2018). Using mathematics to solve real world problems: The role of enablers. Mathematics Education Research Journal, 30(1), 7--19. https://doi.org/10.1007/s13394-017-0217-3

Geiger, V., Galbraith, P., Niss, M., & Delzoppo, C. (2022). Developing a task design and implementation framework for fostering mathematical modelling competencies. Educational Studies in Mathematics, 109(2), 313--336. https://doi.org/10.1007/s10649-021-10039-y

Niss, M. (2010). Modeling a Crucial Aspect of Students' Mathematical Modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling Students' Mathematical Modeling Competencies: ICTMA 13 (pp. 43--59). Springer US. https://doi.org/10.1007/978-1-4419-0561-1_4