What Are Heuristics (in Math)?

[SF49erfan]

Some math advice/tips guides say that having a knowledge of heuristics can assist with mathematical problem-solving.

I'm trying to get better at the subject over the summer before my first year of uni coming in the fall. I don't think, however, that I know what is meant when people use the term "heuristics." I'm wondering if someone wouldn't mind with a definition of this term and perhaps an example?

Greatly appreciate your help! TVM.

 

[HallsofIvy]

You are asking for the definition of a word. Is there a reason why you could not look it up yourself?

Checking Wikipedia gives:
Heuristic (Template:(IPAc-en); Greek: "Εὑρίσκω", "find" or "discover") refers to experience-based techniques for problem solving, learning, and discovery that give a solution which is not guaranteed to be optimal. Where the exhaustive search is impractical, heuristic methods are used to speed up the process of finding a satisfactory solution via mental shortcuts to ease the cognitive load of making a decision. Examples of this method include using a rule of thumb, an educated guess, an intuitive judgment, stereotyping, or common sense.

In more precise terms, heuristics are strategies using readily accessible, though loosely applicable, information to control problem solving in human beings and machines.

 

[Stephen Tashi]

and perhaps an example?

For example, figures with a given perimeter tend to have the most symmetrical shape that is allowed by other constraints. The plane figure with largest are for a given perimeter is a circle. The 4 sided polygon with largest area for a given perimeter is a square.

Another example, applying trigonometry to geometry problems tends to involve triangles, especially right triangles. So if you are tyring to figure out the length of a line segment in a diagram you should look for triangles. If you don' see any you should extend lines in the figure to make triangles and draw lines perpendicular to other lines to attempt to form right triangles.

 

[Ackbach]

Check out How to Solve It, by George Polya, and How to Solve It: Modern Heuristics, by Michalewicz and Fogel. Both are great on problem-solving and heuristics.

 

[CellCoree]

Heuristics in math refer to problem-solving strategies or techniques that are used to find solutions to mathematical problems. These strategies are often based on experience and intuition and are used to simplify complex problems into more manageable steps. Heuristics can help individuals approach problems in a systematic and efficient manner, leading to successful problem-solving outcomes.

Some common heuristics in math include:

1. Trial and error: This involves trying different approaches or methods until a solution is found.

2. Working backwards: Starting with the desired outcome and working backwards to determine the steps needed to reach it.

3. Making a model or diagram: Visual aids can often help in understanding and solving a problem.

4. Using patterns and relationships: Identifying patterns and relationships between numbers or variables can lead to a solution.

5. Applying previous knowledge: Drawing on past experiences and knowledge to solve a new problem.

It is important to note that heuristics may not always lead to the correct solution, but they can be useful in guiding problem-solving and developing critical thinking skills. As you continue to practice and learn math, you will become more familiar with different heuristics and how to apply them effectively. Keep practicing and seeking help when needed, and you will continue to improve your mathematical skills. Best of luck in your studies!