Background to the Idea of “Enablers” of Mathematical Modelling
A major focus during our project was the identification of aspects of teaching and learning that enabled students to model successfully, and in the longer term, promote the development of their modelling capability. The process of identification was informed by two ideas from research in mathematical modelling — implemented anticipation and anticipatory metacognition.
Implemented anticipation (Niss, 2010)* is a process by which students anticipate and carry out within the act of modelling: (a) actions that they see as potentially useful when responding to a problem including the mathematics that might be utilised in future subsequent steps; and (b) decision-making that brings those steps to fruition. Implemented anticipation is key to a modeller’s capacity to mathematise, execute the modelling process, and consequently complete a modelling problem successfully.
Anticipatory metacognition describes an associated idea that includes the capacity to seek and gather information and decide how to analyse the data collected. This means considering possible avenues to pursue during the modelling process by drawing on previous experience. This requires the capacity to think forward while in the act of modelling. In the case of teachers, the engagement of this capacity means asking questions such as, “Where, in the modelling process, will this group of students be likely to encounter obstacles?” And “What can/should I do to help them move forward?” Answering these questions should result in prompts that direct students to use the modelling process to resolve an impasse, rather than providing direction that steers them along a predetermined pathway to a solution.
The capacity to anticipate is key in all modelling processes including pre-mathematisation (e.g., posing questions, assumptions, simplifications), mathematisation, mathematical treatment, interpretation, and model evaluation. This capability is significant when seeking to promote the development of modelling capacities in students. Examples of these processes include:
Anticipating features that are essential in mathematising or anticipating mathematical representations and mathematical questions when developing a mathematical model.
Thinking forward about the genuine potential of a selected mathematisation, and the resulting model, to provide a solution to the problem scenario in focus.
Thinking forward to identify related problems and refinements that emerge during the modelling process. Some of these may not have been thought of at the outset of the problem.
Extending the Idea of Enablers of Mathematical Modelling
Through the project, we have extended the ideas behind Niss’s enablers by adapting them to Australian contexts. These enablers are focused on students’ capabilities and include attention to students’ beliefs and dispositions in addition to their mathematical capacities.
The most successful mathematical modelling occurs whenstudents:
believe that the inclusion of modelling activities is a valid component of mathematical coursework and assessment,
possess mathematical knowledge able to support modelling activities (e.g., possess mathematical knowledge and skills),
understand the systematic modelling process that includes successive stages from problem question to model evaluation,
are capable of using their mathematical knowledge when modelling. (This implies a core understanding of and engagement with the modelling process — Formulate, Solve, Interpret, Evaluate — so that the right questions can be asked and pursued systematically),
persevere and are confident in their mathematical capabilities (e.g., continue to persevere or take intellectual risk by exploring new directions in attempting to solve a problem).
In order for students to successfully model, the capabilities outlined above must be complemented by principles for the implementation of tasks. During the project we adopted a set of principles that provideteachers with guidance related to the development of tasks, how they should introduced into classrooms and the type of support students should be provided with when engaging with modelling problems.
The principles that relate to teachers are:
The mathematical demand of tasks does not exceed the mathematical capabilities of the student group.
Problems are introduced so as to engage the students fully with the task context, while ensuring that goal of the task is understood.
Assistance provided during modelling sessions (measured responsiveness) is geared to helping students use the modelling process to reach a solution, rather than treat a problem as an individual exercise.
Students are encouraged/required to organise and report their work using headings/sections consistent with the modelling process.
Productive forms of collaborative activity are used to enhance and hold to account the quality of on-task progress. Effective use of digital technologies. Students’ interest in a problem.
Enablers of Mathematical Modelling
The two sets of principles — those for teachers and students — were used as a starting point during classroom observations for identifying those activities that enabled students’ modelling.
During our classroom visits we observed the following enablers were employed by students during successful mathematical modelling sessions.
Students who model successfully*:
ENGAGE
Align their activity with the modelling process.
Develop a specific mathematical question (before identifying assumptions).
Prepare an explicit plan of attack for the activity.
Develop initial assumptions and variables.
Revise assumptions and variables throughout the modelling process.
Reflect on the analysis of a problem and the justification of its solution.
Use different strategies to validate a solution.
WORK COLLABORATIVELY
Negotiate functional roles within a group.
Act as a teacher to others in the group — when one member has knowledge needed for progress that others lack. The identity of the "teacher" can vary depending on the immediate need.
Negotiate consensus — based on knowledge, not status.
Bootstrap group performance — when group members have complementary expertise, which in combination enables the group to learn and perform at a higher level than individual members could have achieved alone. This can occur locally (e.g., resolving a specific issue) or globally (achieving a higher overall performance in modelling).
Seek help from others outside of their group.
USE TECHNOLOGY
Search for information that complements what they have been provided in the problem statement.
Develop different mathematical representations of a problem/scenario.
Explore potential ways forward with a problem through the use of different mathematical representations.
Verify or validate the plausibility of potential solutions.
Perform or check calculations.
* It should be noted that it is possible for successful modelling to be undertaken alone, and without the use of technology.
Teachers who implement modelling successfully in their mathematics classes:
DEVELOP/SELECT TASKS
ON THE BASIS OF:
Student interest in a topic.
A balance between challenge and accessibility.
Students valuing the learning experience (e.g., relevant to future assessment tasks).
Relevance to formal assessment requirements.
ENCOURAGE STUDENTS
TO:
Explore diverse approaches to addressing the problem which includes intellectual risk taking.
Ask questions related to the context of the problems and how this connects to the modelling process.
Work collaboratively to address the problem.
Use technology when exploring the problem and while developing a response.
Use resources available both inside and outside of the classroom.
RESPOND
TO STUDENTS’ QUESTIONS:
Exercising measured responsiveness— providing just enough information for students to progress without directing their work along a particular pathway to solution.
Providing clarification for the problem itself without indicating which mathematical content is relevant.
Directing students to their progress within the modelling cycle.
ANTICIPATE
Where scaffolding might be required to accommodate students learning needs (e.g., current knowledge and previous experiences).
Potential barriers to developing relevant responses to a problem.
That unforeseen issues will arise but which must be responded to in the moment.
*Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modelling. In R. Lesh, P.L. Galbraith, C.R.Haines & A. Hurford. (Eds.), Modeling Students’ Mathematical Modeling Competencies: ICTMA 13 (pp. 43-59). New York, NY: Springer. This reading can be found in the Links to Our Publications page.