Personal Journey to Mathematical Modeling
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Mathematical Modeling:
Elements of a Dynamic Process: Life --> Math --> Life For additional information and resources to support teachers who wish to implement mathematical modeling in the classroom, select this link. START WITH LIFE. Life --> Math --> 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, a curiosity to resolve/pursue or data to pursue or gather to reveal a need to change. The ideas of "real" and "relevant" are important. Fantasy contexts can seem real to some students. Real situations may be 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?" or how do we assure fair representation with voting maps. 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. MATHEMATIZE Life --> Math --> Life 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, maps, 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 Life --> Math --> Life Interpret (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? Closure (LIFE) • Act upon the results of the investigation • Recommend and defend a solution or pathway. • Report • Propose further exploration Process Notes: 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. Culturally-relevant Mathematics Mathematics can be a powerful tool for positive change! In some countries and classrooms, mathematical modeling is a vehicle for teachers and students to investigate issues related to social justice and equity and/or the lived experiences of the students. This is less likely in the United States because some important issues that can be informed by mathematics are also considered sensitive by some groups. Additional Notes Math --> Life Math ideas are typically introduced to students without a real-world context or reason to learn the math. When students ask, "Why do I need to learn this?" teacher answers are usually vague or inadequate. Students are told they will need it for the next math course, to perform better on a standardized assessment or to succeed in STEM fields. Some teachers express the hope that students who understand the math concepts and skills will be able to apply the math when later their coursework, occupation or life situation makes the math relevant. Essentially, the math is needed for something yet to come. It is beautiful, yet I suspect rare, when these later connections happen as hoped. This sequence of Math --> Life could be improved greatly by including more real (not contrived or ridiculous) math problems for students to apply the math. I prefer the flipped version where Life precedes the math. Math --> Math Much of the mathematics students are expected to learn is to understand mathematics. Curricula is designed to introduce math skills (and maybe concepts) in a step-by-step logical sequence where the end result, the culminating experience or reason, is known to the presenter. Sadly, through no fault of their own, this does not seem to be true for most teachers of mathematics. The meaning is missing. The purpose of learning math is to understand math with little to no connection to the everyday lives of humans. The cycle repeats and is entrenched. S Fo |
Connecting Life to Math to Life:
A Personal and Professional Journey Currently being edited. School math came easy for me. I successfully completed Calculus in grade 12. The architecture major was impacted at my local university so I was accepted as a Math Major and planned to switch majors in a year. To be continued... T S |