Solutions to math problems are not explained.

A few years ago, the mathematician Steven Strogatz predicted that it wouldn’t be too much longer before computer-assisted solutions to math problems will be beyond human comprehension. Well, we’re pretty much there. In this case, it’s an answer produced by a computer that was hammering away at the Erdős discrepancy problem.

Solutions to math problems are explained step-by-step.

 Warning: this essay contains solutions to math problems.

Related Pictures solutions to math problems amor s personal website

Communication activities become more elaborate in third andfourth grade, as students become more comfortable with symbolic andwritten representations of ideas. Students should communicate witheach other about mathematics on a daily basis, exploring problemsituations and justifying their solutions. Different types of writingassignments may be used: keeping journals, explaining solutions tomath problems, explaining mathematical ideas, and writing about thereasoning involved in solving a problem. Students continue to usemanipulatives to explore new ideas and learn to relate differentrepresentations of an idea to each other. For example, after usingbase ten blocks to solve x , students might providea pictorial representation of these blocks (at left below) followed bya written explanation of what they did to get x . Linking the use of concrete manipulatives to the pictorialand symbolic representations is critical to understanding themathematical procedures.

found a solution to the math problem using the elephant! Hilarious

Writing is an integral part of conveying both mathematical concepts and solutions. Often K-12 educators are not provided with an adequate means of teaching writing in the math curriculum without delving into the English curriculum. By teaching the students to construct heuristics and algorithms (HAL) to derive solutions to mathematical problems, then the student is learning an effective means of problem solving to the extent that they must list the givens, concepts learned, and learn to think sequentially. The construction of algorithms and heuristics stands on its own merits in the area of problem solving and critical thinking. By teaching students not only how to construct algorithms and heuristics, but also a few transitional words and phrases then the HAL can then be used by the student to write an English translation of any given problem and its solution. Obviously, HAL is a form of pseudo code.

Fun Math Puzzles : Solutions to Your Math Problems
Typically, students are asked to provide written explanations for solutions to math problems

Have students verbalize decisions and solutions to math problems

Paul Erdős was famous for offering money for solutions to math problems. My understanding is that those prizes are still being administered by Ron Graham, even though Erdős passed away several years ago.

Paul Erdős was famous for offering money for solutions to math problems

These are detailed solutions to the math problems

For arguments sake, we shall say we need a heuristic anytime we solve a problem that is non-numeric. The reasoning is that such problems usually require human intervention, and all human actions/behaviors/thoughts are infinite by nature. Take the example of bending a finger. One is tempted to say this involves only the pulling of the finger toward the palm. In reality, to bend a finger the brain first generates the idea to bend the finger. Once the idea establishes itself in the brain, then signals are sent to all of the muscles involved to either contract or relax at a specific moment in time. But, the brain signal alone contains a series of steps where neural transmitters are secreted and the neurons are turned either on or off. The all-or-nothing quality of neurons themselves involves even more molecular process. Eventually, we find ourselves at the atomic level, then the quantum level, until, before we know it, we are playing with the very essence of infinity—space-time. But, we still haven’t defined how the initial thought of bending the finger was generated—to do such would guarantee us the Nobel Prize. Anyway, somewhere along the line we have to make a decision of when we can say that we have provided enough information to solve the problem. Hence, we make a generalization and, by definition, we have created a heuristic. An algorithm is any step-by-step solution. Since math education is concerned with deriving rigorously exact and accurate solutions, then the solutions to mathematical problems are, by their very nature, finite. Thus, we can solve any math problem with a finite number of steps and in doing so we establish a need for the algorithm.

Computation of solutions to Math Problems such as NP complete problems

Warning: this essay contains solutions to math problems


* on how to write solutions to math problems
* CMU (peer tutoring and academic counseling resources)
* and more illustrating calculus concepts
* at CMU