hard math problem

Solving Hard Math Problems: Strategies and Techniques

What makes a hard math problem? This is not an easy question to answer and the answer is often different depending on who is responding. In the mathematics business, hard problems are those that strike resistance; the ones where, in order to solve them, it is necessary to mount what can feel like a massive assault. In contrast, easy problems are those where such resistance is not encountered, the problems lie open like the proverbial oyster waiting for the hungry mathematician. Peaceful Solutions is a compilation of carefully worded and challenging problems that, when taken from their cameos of solitude, respond with a surprising gentleness.

Before we begin discussing strategies for solving hard math problems, one of the most important things to remember is that hard problems are often solved through creative and original solutions. As a result, math problems that can baffle many mathematicians also provide an excellent training ground for one’s “problem-solving muscles”.

Many children are fascinated by mathematics from an early age. However, sometimes this fascination fades away as they are faced with the mind-numbing routine, drill, and memorization that many adults undertake. In contrast, developing mathematical problem-solving skills and abilities can be seen as a highly desirable goal. Developing and using problem-solving techniques will enable students to understand the skills needed to be successful in his or her work and see how they can benefit from putting what they have learned into action.

If you don’t have much experience with the operation, you will probably begin by trying to find one or two rational values. Once you have the first few terms in the sequence, you might notice some pattern (it looks like the previous term plus “1”, or it looks like 2 times the previous term, etc.). At this point, you can probably jump right ahead and conjecture the correct formula. Then you could prove this formula is correct for all positive N either with induction or act more boldly and look for an a priori argument. This is a typical solution strategy in the contest problems that involve induction over N such as identity proofs.

2.1.1. Iterating the function formula Imagine you have been given a problem that involves an iterated arithmetical operation – for example, one where you are asked to compute or find a formula for the sequence:

With the theory and the existence proofs out of the way, we can now move on to more complex and specific problem-solving techniques. Many times, the concepts given in the previous sections are more than sufficient for a gifted student to solve beautiful, fair, and really difficult problems. At other times, one must use big gun specific techniques from particular areas of mathematics.

Tackling Equations Containing Higher Powers: The first step in solving an equation of this nature is simply to try the easiest route. In this case, I would suggest this technique: If you have an equation that contains an unknown variable raised to a power of 2 (meaning a variable times itself or squared), first check to see if the equation can be simplified by solving for the variable as a square root formula. In an alternative situation, upon occasion, you will encounter a number of problems where the goal is to solve for the higher root or roots of an equation than the square root. This is accomplished by using the same technique employed to solve for the square root.

In advanced math classes and on a few conventional exams or competitive math contests, you may encounter problems containing either equations or formulas that are too ambitious to consider solving by conventional methods. Still, if you have confidence and a positive, persistent attitude, almost every time, with the aid of the techniques, you can dodge and solve this type of math problem.

Student involvement and inquiry are key words when one tries to create a classroom where students are responsible for their own learning – a student-centered classroom, as opposed to the teacher-centered classroom. To move from a teacher-centered classroom to one that is more student-centered, students must be actively engaged in their learning, for instance through connecting, conjecturing, and discussing hidden strategies in mathematics with their peers. The modeling process is an important tool in the design of student-centered learning environments. Unwittingly, teachers may constrain students’ mathematical thinking by implicitly expecting them to act as passive listeners and recipients of mathematical knowledge. Often students are accustomed to respecting the teacher’s authority and to produce just one unique solution to a particular problem. Such traditional views about the nature and the processes of mathematical learning are difficult to alter, and so modeling plays an integral role in not just teaching the content but also in the process of developing the learner.

The goal of teaching mathematics is to ensure that students develop authentic problem-solving skills, that they are engaged in learning, and that they become critical thinkers. Critical thinking, as it relates to problem solving, is regarded as the ability to think both rationally and creatively about a problem and a means to implement an active search for new knowledge. Research tells us that students do not develop conceptual understanding of mathematical ideas and processes simply by hearing someone tell them the concepts or the procedures. If concepts and procedures are to make sense and both are to be logically connected, students must grapple with and learn these mathematical ideas and procedures by doing mathematics.

There are some other approaches to solving problems that are very domain-specific; for instance, general techniques that apply to solving Diophantine equations, factorizing polynomials or evaluating some combinatorial sum come to mind. In general, the broadest strategy one can use to attack hard math problems is to be versatile. This involves approaching many different problems with many different tools and staying open to ideas inspired by any possible direction the problem might take. Versatility gives the problem solver a wide range of plans to choose from in the first place, before picking the best one to latch onto and run with.

In this article, we have considered a lot of approaches to solving hard math problems. We have wrapped up our discussion with ten tips for solving math problems, and the most important one is to approach solving problems regularly and with a definite plan in mind. This is because even though we might have a great wealth of strategy and technique at our disposal, experience and the perspective it affords are the best tools a problem solver can have. We have also discussed an approach that allows one to solve many problems whose solutions were not immediately obvious. It involves comparing the growth rates of different functions and determining how quickly desired or unwanted behavior occurs. For example, if one wants to show that a function grows without bound, one can try to find a related function that grows without bound but is easier to work with. Although this approach is very general, many students have never seen a worked example, and they do not know how to get started when they see an unfamiliar problem or even one that is only superficially unlike a problem they have seen before.

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