# Ideal response to – how does mathematical modeling help students solve real world problems?

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Mathematical modeling helps students solve real world problems by providing a framework for analyzing complex situations, making predictions, and testing solutions, which can lead to more informed decision-making and problem-solving skills.

## For those who are interested in more details

Mathematical modeling is an essential aspect of modern problem-solving that helps students apply mathematical concepts to real-world situations. By using mathematical models, students can analyze data, make predictions, and test solutions to complex problems. This process is beneficial because it fosters critical thinking skills and encourages creativity, which are crucial for success in any field.

According to the famous mathematician John von Neumann, “The use of mathematical models should be viewed as a means whereby the scientist can carry out a very large number of trial-and-error experiments with great economy.” Mathematical modeling allows students to test different scenarios without having to actually implement potential solutions in the real world, thus saving valuable time and resources.

Interesting facts related to the topic of mathematical modeling and its benefits for students include:

• The use of mathematical modeling is not limited to scientific fields but is also utilized in fields such as economics, engineering, and social sciences.
• A study conducted by the National Council of Teachers of Mathematics found that students who use mathematical modeling in the classroom outperformed their peers in problem-solving abilities.
• Mathematical modeling is used in various industries, including healthcare, finance, and technology, to make informed decisions and reduce risk.
• The development of mathematical modeling dates back to the 17th century with the invention of calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz.
• Mathematical modeling can be represented in various forms such as equations, graphs, and tables.
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One way to illustrate the benefits of mathematical modeling is by using a table to compare traditional problem-solving methods with the use of mathematical modeling. See the example below:

Traditional Problem-Solving Mathematical Modeling
Relies on trial-and-error Uses data analysis and predictions
Can be time-consuming Allows for efficient testing of multiple scenarios
May not consider all variables Takes into account a multitude of factors
Solutions may be limited Provides creative solutions
Does not always result in informed decisions Leads to informed decision-making

In conclusion, mathematical modeling is a valuable tool that helps students develop problem-solving and critical thinking skills. By using this approach, students can understand complex situations more easily, make accurate predictions and test solutions, and make informed decisions. As the famous mathematician Leonard Euler said, “Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity.” Mathematical modeling contributes to the development of both intuition and ingenuity, making it an important aspect of learning.

The video delves into the problem of traffic and how mathematical models can provide insights into it. The presenter explains the process of modeling, assumptions made to simplify the model, and the concept of equilibrium, density, and flux. He demonstrates how a traffic model he created can predict real-world phenomena, such as the optimal density to maximize flux. The video also discusses the relationship between velocity, density, length of cars, and driving habits in traffic modeling. Furthermore, the presenter explains a model he programmed into Matlab that simulates traffic flow and measures the perturbation variable from equilibrium. The video concludes by discussing the strengths and weaknesses of the model and how it can be improved by increasing the specificity and generalizability of assumptions.

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## Surely you will be interested

Why is mathematical modelling important?
Response to this: A consequence of `mathematising’ is that mathematical functions are allowed to naturally emerge. For this reason, mathematical modelling is an excellent way of showing students the relevance of mathematics in, and emergence of mathematics from, the real world. Creating a mathematical model may be difficult but is an important conduit to learning:

Furthermore, How do mathematicians solve modeling problems?
Response: To solve modeling problems, mathematicians make assumptions, choose a mathematical approach, get a solution, assess the solution for usefulness and accuracy, and then rework and adjust the model as needed until it provides an accurate and predictive enough understanding of the situation.

How can a mathematical model be used to explain reality?
Thevelocityat which the coin reaches the ground can also be calculated with ease. This is a prime example of a mathematical model that can be used to explain what we observe in reality. Mathematics can be used to solve very complex problems – far more complicated than that of a coin falling under gravity.

Just so, How can technology help students develop mathematical models?
In reply to that: Technology, including graphing calculators and software such as Excel, can be a useful tool for students to explore these mathematical models ( da Silva Soares, 2015) without the need to restrict examples to artificial data that is easy to manage computationally.

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