This IS the definition of being able to utilize mathematics for problem solving. Make observations, which means collect quantified data, analyze the data to make a mathematical model of reality, and then use the model to investigate the nature of the problem, not trial and error. Finally, design an improvement based on the mathematical model. Additional testing and data are collected to evaluate the veracity of the mathematical model. If necessary, the model is revised based on the new data and the cycle is repeated.
Problem solving using mathematics is a learned, performing art, just as is playing a violin.
Students must practice the actual performance to become proficient. The students need to be in contact with people who perform in this arena, i.e., people in business and industry. After more than 40 years in business and industry I have never been paid to solve a single quadratic equation, not one, ever. However, on a recent engineering project it was estimated that we were writing and solving more than 300 quadratic equations a day. Those solutions were used to evaluate and improve upon a problem in a chemical plant.