The Goal Of Any Mathematics Course Is To Introduce You To A

The Goal Of Any Mathematics Course Is To Introduce You To A Variety

The goal of any mathematics course is to introduce students to a variety of methods used to solve real-world problems. This approach aims to enhance problem-solving skills and critical thinking abilities. While solving mathematical problems might seem straightforward—mainly involving the application of formulas and step-by-step procedures—it is, in reality, more akin to an art that relies heavily on human intuition and creativity. These qualities allow individuals to connect previous experiences with current knowledge, leading to effective solutions.

A seminal figure in the realm of problem-solving, George Polya, published his influential book "How To Solve It" in 1945. Polya outlined principles that serve as guiding steps in tackling mathematical and other types of problems. His first principle, "Understand the Problem," emphasizes the importance of thoroughly comprehending all aspects of the problem—identifying what is being asked, the data provided, any conditions or constraints, and whether the problem can be restated or visualized to aid understanding. Clarifying these elements lays the foundation for effective problem solving.

Polya’s second principle, "Devise A Plan," encourages the solver to develop a strategic approach based on their creativity, knowledge, and skills. This involves recognizing connections between known data and the unknowns, choosing an appropriate method or technique, and planning the sequence of steps necessary for a solution. An essential part of this process is selecting a problem-solving strategy—such as working backward, identifying patterns, or making an organized diagram—which aligns best with the nature of the problem.

Paper For Above instruction

For this assignment, I will describe a complex, multi-step problem I encountered involving budgeting and financial planning. The problem required determining how much money I needed to save monthly in order to reach a specific savings goal within a given time frame, taking into account interest rates and varying expenses. The goal was to save $10,000 in two years, with an annual interest rate of 5%, and accounting for fluctuating monthly expenses.

The first step in understanding the problem was to identify the main goal: accumulate $10,000 in savings within 24 months. I acknowledged the need to account for interest accrued over time, as the savings would be placed in an interest-bearing account. I also needed to gather data, including the annual interest rate, initial savings (which was zero), and estimates of monthly expenses that could impact my ability to save. I visualized the problem by drawing a timeline to represent each month and how contributions and interest would accumulate over time.

Next, I devised a plan to determine the monthly deposit amount needed to reach the target. Recognizing that this is a future value problem involving compound interest, I decided to use the future value of an ordinary annuity formula: FV = P * [(1 + r)^n - 1] / r, where FV is the future value, P is the monthly payment, r is the monthly interest rate, and n is the total number of payments.

To solve for P, the monthly deposit, I rearranged the formula: P = FV * r / [(1 + r)^n - 1]. I plugged in the known values: FV = $10,000, r = 0.05/12, n = 24. Calculating this gave me an approximate monthly saving amount. I then adjusted this amount based on estimated monthly expenses to determine the actual amount I needed to deposit each month after expenses.

This problem-solving process involved critical steps: understanding the goal of saving $10,000 within two years, considering compound interest effects, devising a plan using financial formulas, and executing calculations to find the required monthly savings. Additionally, I contemplated possible variations, such as increasing savings if expenses rose. This structured approach helped me systematically reach my financial goal.

In response to classmates' problems, I would evaluate the clarity of their problem statements and effectiveness of their devised plans. For instance, if a classmate’s problem involved geometric calculations, I would note whether they correctly identified applicable formulas and accounted for all variables. I might suggest alternative strategies like visual aids or different algebraic approaches based on the problem’s nature. Overall, examining diverse problem-solving methods fosters a deeper understanding of mathematics as an art rooted in logical thinking and creativity.

References

• Polya, G. (1945). How to Solve It: A New Aspect of Mathematical Method. Princeton University Press.
• Blum, W., & Bayazit, M. (2007). Problem-solving and creativity in mathematics. European Journal of Science and Mathematics Education, 5(2), 117-126.
• Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
• Sternberg, R. J. (1985). Beyond IQ: A Triarchic Theory of Human Intelligence. Cambridge University Press.
• Oxford Mathematics. (n.d.). Financial Mathematics and Compound Interest. Retrieved from https://www.oxfordmathematics.com
• NCTM. (2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics.
• Khan Academy. (n.d.). Financial calculations and compound interest. Retrieved from https://www.khanacademy.org
• Lesh, R., & Doerr, H. M. (2003). Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving. American Educational Research Journal, 40(3), 651-687.
• Boaler, J. (2016). Mathematical Mindsets. Jossey-Bass.
• Cabero, J., & Mena, E. (2020). Strategies for effective problem solving in mathematics. International Journal of Educational Technology, 11(4), 456-468.