Personal Journey to Mathematical Modeling
Elements of a Dynamic Process:
Life --> Math --> Life
The following narrative is evolving. I continually seek feedback on what to add, what to delete and what to clarify.
START WITH LIFE.
Establish real purpose. (LIFE)
Start with the real-world: a situation to understand, a problem to solve, a decision to make, a product or process to improve or assess, a plan of action to develop, a prediction to consider, or a curiosity to resolve/pursue.
The ideas of "real" and "relevant" are important. Fantasy contexts can seem real to some students. A real situation may be limited to a small section of a population and irrelevant to young students. Situations can range from, "what is the best strategy to win this game?" or "what price should we charge?" or "which venue should we choose?" or "what is the most efficient system for delivering packages?" to social justice issues such as, "is a subgroup over-represented in remedial math classes?" to critical issues such as, "How do we best manage scarce resources such as water or food?" or "How do we prepare for natural disasters?"
Focus the investigation. (LIFE)
Decide which real-world factors and variables are important to the situation. May require simplification and preliminary assumptions. Some factors may be set aside and revisited later as needed.
Quantify/Represent/Mathematize (DO MATH)
Which of the variables can be quantified or represented with mathematics? Create a mathematical model or models in the form of equations, graphs, tables, spreadsheets, diagrams, flowcharts and/or pictures.
Solve/Extrapolate (DO MATH)
Apply mathematics to the model with the goal of solving the problem. May require the learning of new mathematics.
RETURN TO LIFE
Is the information provided by utilizing the model sufficient to recommend a solution path? Is the model adequate? Did more variables need to be considered with others discarded? Is there a need to start over?
• Act upon the results of the investigation
• Recommend and defend a solution or pathway.
• Propose further exploration
A shorthand for the modeling process:
Life --> Math --> Life
In real life the process is not necessarily linear as illustrated above. The situation and the process can be messy. In real situations it is typical to revisit assumptions or even the initial question itself. Sometimes a model needs to be revised or improved to provide a better solution. This may require learning new mathematics (how cool is that!).
Without returning to life and resolving the initial problem:
Life --> Math
In this scenario the process uses life to motivate the learning of mathematics. (Not a terrible thing but not the purpose of mathematical modeling.) Many authentic/realistic textbook applications fit this scenario.
Another common situation similar to math modeling:
Contrived Life --> Math --> Resolution
In this scenario the "real" situation is contrived, ridiculous, or unimaginable. These vary in their level of ridiculousness and are, unfortunately, all too common . The situation may have some realistic components but may fail to be real by identifying a context that is unlikely to ever occur or by introducing a question that a person (other than a math teacher or textbook writer) is not likely to ever ask.
Some claim the use of "models" is sufficient to be identified as "mathematical modeling." It is true that models are an essential component of mathematical modeling. If the purpose of the model it to improve understanding of mathematical ideas or connections then it likely fits this scenario:
Math ---> Math
The majority of activities, investigations, exercises and problems addressed in math class or textbooks fits this scenario. These are important to helping students with conceptual understanding of mathematical ideas, relationships and skills. The assumption is that when students truly understand the mathematics they will be able to apply mathematics as needed to everyday and career-related problems. I contend that students are far more likely to be able to understand math concepts and apply those concepts and skills to real world problems when introduced and developed using the process of mathematical modeling. Try it and see!
This is my take on Mathematical Modeling. You are invited to investigate some of the resources linked on this website. Enjoy the adventure into meaningful mathematics!
People love mathematics for different reasons. Some love the beauty or structure of mathematics. Some love the reasoning and discipline. Some love patterns and routines. I love the application of math to life which is, for the most part, missing in classrooms and textbooks. I continue to advocate for more applied mathematics and mathematical modeling for all our students
In some countries, mathematical modeling allows teachers and students to investigate issues related to social justice and equity and/or the lived experiences of the students. Mathematics can be a powerful tool for positive change!
Real World and Mathematics
Some have argued that all mathematics is real world. It is difficult to challenge anyone's definition of real world since there doesn't seem to be universal agreement. For me, it is more about what mathematical experiences do I wish for all students. I want students to experience the mathematics they learn in school as a tool for understanding and solving situations they now encounter or will encounter in their everyday lives or work. This is missing in most classrooms and textbooks. Professional development opportunities for teachers almost exclusively focus on helping students understand mathematics (nothing wrong here) but rarely to help students understand life with math as a tool. I also want all students to experience and appreciate the beauty of mathematics in its structure and patterns and as a vehicle to develop reasoning. I'm seeking a balanced, more inclusive, view of mathematics that honors multiple reasons why mathematics is important to our civilization and why mathematics should be a critical component of student learning in school.
MATHEMATICAL MODELING IN THE CLASSROOM
It is my hope the resources on this website will serve to:
• Understand mathematical modeling.
• Real-life contexts for the math classroom.
• Teaching mathematics using mathematical modeling.
• Teaching mathematical modeling.
Connecting Life to Math to Life:
A Personal and Professional Journey
Currently being edited.