MICA Then & Now
‘Coding’ and Computational Modeling in Mathematics at Brock University
The 22-Year ‘MICA’ Story: Innovation, Evolution, Contribution
For over 20 years, math majors and future math teachers at Brock have been learning to use programming for pure or applied mathematics inquiries in the context of 3 project-based courses known as MICA I-II-III.
2001 marked the beginning of an exciting new direction for mathematics at Brock University: The Department of Mathematics and Statistics launched a sequence of courses known as ‘MICA’ – Mathematics Integrated with Computers and Applications – that sought to encourage mathematical creativity, develop mathematics concepts hand-in-hand with computers, and ultimately equip students with the skills needed to solve modern-day problems in research, industry, business, and education . Such courses were at the forefront of embracing the power of computation in mathematics, described ten years later by the European Mathematical Society: “Together with theory and experimentation, a third pillar of scientific inquiry of complex systems has emerged in the form of a combination of modeling, simulation, optimization, and visualization” (, p. 2).
The first postcard advertisement for the MICA courses, printed in 2001.
The core of the MICA courses is project work, counting for 70–80% of students’ grades. Indeed, it’s primarily through the completion of inquiry projects (4–5 in each course) that students learn to design, program, and use interactive computer environments – so-called ‘exploratory objects’ (EOs) – to investigate mathematical concepts, conjectures, theorems, or real-world situations . For example, a MICA I project has invited students to develop EOs for examining, numerically and graphically, the behaviour of discrete dynamical systems involving parameters; and another project, implemented in MICA II, has introduced students to building EOs for simulating and analysing battles between two opposing armies, as modelled by discrete equations .
EOs for investigating discrete dynamical systems (left) and battle simulations (right), constructed as part of assigned MICA projects.
In the final project of each MICA course, students have the chance to take ownership of their inquiries, working individually or in pairs to investigate a topic of their choice ( provides some examples). For some students, this means constructing an EO to engage in pure or applied mathematics research: Adam, for instance, used his EO to formulate a conjecture concerning the bounded area of the function defining the Mandelbrot set, and Matthew and Kylie programmed a simulator to answer the question: “Is it better to walk or run in the rain?” Other students, often future mathematics teachers, decide instead to create a ‘learning object’ (LO): an interactive program that provides a guided learning experience of a school mathematics concept , such as the one created by Ramona and Cassie on perimeter. Such ‘passion projects’ can serve as important milestones on a student’s journey through the MICA courses, enabling them to demonstrate their accumulation of skills as creative problem solvers in an increasingly computational world .
EOs for investigating the Mandelbrot Set (left) and whether it is better to walk or run in the rain (right), constructed as part of original projects in MICA II.
Selected parts of a LO for supporting the learning of perimeter, constructed as part of an original project in MICA I.
MICA courses have evolved over the years, including the addition of a MICA III course for future teachers which, in part, addresses the recent integration of ‘coding’ in the Ontario Gr. 1–9 math curriculum.
Even after 22 years, certain elements of the MICA courses have remained the same. To support the core – students’ project work – the courses involve two hours of weekly lectures and two hours of weekly sessions in computer labs, which are typically capped at 35 students. Lectures primarily introduce mathematical content that serves as a background and motivation for the programming-based mathematical tasks completed in labs . And tasks are purposefully designed to build students’ confidence and fluency by becoming progressively more challenging: While in MICA I students focus on developing programming skills in accessible mathematical contexts, MICA II and III challenge students to code and use simulation to explore more advanced mathematics and more complex systems . The following table illustrates such a progression by listing examples of topics students may encounter in their MICA course projects. Topics in MICA I have been relatively stable since 2001, whereas topics in MICA II and III have changed with time and according to 9 different instructors’ interests (see  for lab and project guidelines for MICA I and examples of project guidelines for MICA II and MICA III for future math teachers).
|I||1||Conjecture about prime numbers or hailstone sequences|
|2||RSA encryption method|
|3||Discrete dynamical systems (cubic with two parameters)|
|4||Original, end-of-term project, on a student-selected topic|
|II||5||Buffon needle problem and Monte Carlo integration|
|6||Markov chains applied to income demographics and chronic illnesses|
|7||Dynamical system of the logistic function and bifurcation diagram|
|8||Simulation of battles (Lanchester equations)|
|9||Original, end-of-term project, on a student-selected topic|
|III||For math and science majors||For future math teachers|
|10||Exponential growth with harvesting model and application to world population||Calendar problem in Scratch|
|11||Power laws in spatial models applied to patterns on the shell of certain snail species||Simulations about Bertrand’s Paradox|
|12||Determining dynamical exponents in random and ballistic deposition models||Prey-predator biological model (Lotka-Volterra)|
|13||Percolation and wetting models||Randomness of DNA sequences|
|14||Original, end-of-term project, on a student-selected topic||Original, end-of-term project, on a student-selected topic|
Examples of topics encountered in the sequence of three MICA courses.
All of this said, there have been some significant changes to the MICA courses since their inception . For example, in 2018, MICA III was split into two courses to serve the varying needs and interests of two student populations: math and science majors, and future math teachers. To better support the teachers, MICA III was revised to include new objectives and related activities. In addition to completing inquiry projects to further their own experience learning to use programming to investigate mathematics, the future teachers also engage in complementary reading and reflection to deepen their understanding of their experience, including the affordances of computing for mathematics learning. After the Ontario Ministry of Education made ‘coding’ an official part of the Gr. 1–9 Mathematics curricula in 2020 and 2021 , another objective was added: Building an awareness of the place of programming in the curricula and relevant teaching approaches. Key to addressing this last objective is a new kind of original end-of-term ‘passion project’ in which the future teachers collaborate with teachers in local schools to prepare and implement coding-based activities in mathematics classrooms .
The above evolutions have arisen alongside increasing support for the integration of coding and computational modelling in all levels of mathematics education. Consider, for instance, PISA’s 2021 international mathematics assessment framework , which has taken the following position: By the time students are 15 years old, they “should possess and be able to demonstrate computational thinking skills as they apply to mathematics as part of their problem-solving practice,” (p. 5) which means being able to engage in “defining and elaborating mathematical knowledge that can be expressed by programming” (p. 12). From being celebrated as a pioneering program of its kind in Canada back in 2001, the MICA courses are now part of a university math ed community aiming to inspire students who may have been learning to use programming for mathematical investigation since the age of 6. It will be interesting to see the kinds of evolutions this may stimulate in the future.
MICA implementation has been providing a rich context for research ofdifferent aspects of learning to use programming as a meaningful tool for mathematics learning and its teaching.
Not long after the MICA courses were launched in 2001, some faculty members involved in their design and implementation began to report on the innovative approach in educational publications and events: e.g., a presentation on MICA was a natural fit at the 2006 ICMI Study Conference seeking to revisit the terrain of digital technologies and mathematics teaching and learning . Reflecting more deeply on the project work students encounter in MICA led to the development of a first theoretical model for thinking about students’ learning activity .
Development process model of student engagement in programming for a pure or applied mathematics inquiry project (, p. 1025). See  for a video presenting a dynamic illustration of the model in the context of two student projects.
And so the stage was set for a longer-term research project aiming to use the MICA context to better understand how students learn to use programming as a meaningful tool for university mathematics, and how instructors and universities can support that learning.
Over the past 5 years, the SSHRC-funded research project (2017–23) has involved 10 research collaborators and 13 research assistants in the exploration of various topics, which can be grouped under the following 4 foci:
- Students’ learning: e.g.,
- appropriation of programming as the development and evolution of a complex web of instrumented-action schemes, with individual and social aspects 
- constructionist experiences through 14 programming-based projects 
- ways of thinking, and facing and overcoming challenges in inquiry projects 
- empowerment, identity, and mindsets 
- contributions of an institutional approach, including the development of programming-based praxeologies 
- predicative and operational knowledge in a project-based approach 
- Instructors’ teaching: e.g.,
- roles and demands for supporting a constructionist learning environment 
- decision-making and actions for supporting students’ development of schemes 
- assessment of students’ computational thinking for mathematics 
- effective features for supporting students’ learning in a project-based approach 
- Task and task design: e.g.,
- Teacher education: e.g.,
As the SSHRC project is coming to an end, the team is excited to think about the kinds of topics, collaborations, and contexts that can continue to move the research forward!
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Written and edited in April 2023 by Dr. Laura Broley, in collaboration with (alph. order) Dr. Chantal Buteau, Dorothy Levay, Neil Marshall, Dr. Eric Muller, and Jessica Sardella.