I recently came across a video titled "Unlocking Your Intuition: How to Solve Hard Problems Easily" which beautifully explains a learning method that I've found to be very powerful for building intuition in order to solve challenging problems with creative problem-solving methods. This teaching method works by directly accessing, testing, and improving the student's intuition. In many ways, it is much more effective and easier than the methods that are traditionally used when teaching students, and even more so when the subjects being taught require creative and out-of-the-box thinking. I've used this method to go from knowing nothing about coding interviews to receiving software engineering job offers from FAANG companies in a few months. I've also noticed a dramatic difference in the ability of kids to absorb mathematical patterns in subjects like geometry and algebra when I switch from a traditional teaching style to this one. One way to implement this teaching style goes like this:

Intuition-based teaching method

1. Categorize the problems and the solution methodologies into easily memorizable patterns

2. Teach one or two examples of how to solve each type of problem and how to use each type of solution

3. Repeat the following steps until the student feel comfortable solving new problems:

   1. Generate a random list of problems of all types, distributed according to the likelihood of encountering each type of problem in real life (or on the test that the student is preparing for)

   2. For each problem:

      1. Have the student quickly recognize the problem type and solution method(s) which should be used to solve that problem

      2. Check if they're right.

         1. If they're wrong, show them the answer and skim through the detailed solution until they have a sense of what type of problem this is and why, and why that choice of solution method(s) makes sense for this problem

         2. If they're right, tell them to remember that they were right and store that information so that they will later recognize the same problem type and solution method(s) even quicker

Tools like https://www.worksheetworks.com/math/pre-algebra/two-step-equations-division.html or http://www.math.com/students/worksheet/algebra_sp.htm which generate problem lists are very useful for this form of teaching- you can take note of which problems the student is struggling with and quickly generate a new list of problems which features those types of problems more prominently. It's important to keep the problem types that the student is already familiar with in the list when doing this, so that their intuition doesn't become specialized to only one type of problem.

The reason this learning style works so well is because it taps directly into the student's intuition and builds it as quickly as possible, bypassing what would otherwise take days, weeks, months, or even years of highly motivated and grueling problem-solving. For a deeper understanding, let's compare this with how a concepts are traditionally taught:

Traditional teaching method

1. Teach all of the problem-solving techniques and work through an example of each one

2. Repeat the following steps for each of the techniques, until the student feel comfortable solving problems of that type:

   1. Have them solve a problem which uses that technique

   2. Check if they're right

      1. If they're wrong, try again and/or go over the solution to the problem in detail

      2. If they're right, move on to the next problem

There are a few problems with this approach:

• The student's problem-solving intuition is built up very slowly. Intuition is fast, memory-based thinking that arises as a "gut reaction" to seeing something that you've already encountered a variation of before and know how to approach. Therefore, it requires many examples of right and wrong decisions in order to become useful. When taking this traditional approach, however, the time between encountering two intuition-building moments - moments in which you realize that you made a wrong decision about something and learned to change your approach - is the time it takes to fully solve a problem in detail, compare with the solution, retry the problem, and go over the solution in detail. On the other hand, in the intuition-based learning method, the time between two intuition-building moments is simply the time it takes to read the problem, categorize it, predict a solution methodology, and check if you're right

• The student's problem-recognizing intuition is never built at all. For concepts with many types of problems and solutions (like coding challenges), this can be an even bigger problem than the previous concern. Because the student is presented with problems of the same type and are told beforehand which type they will be given, they never learn to develop the skill of recognizing problem types and solution method(s) by themselves. This is a common mistake that can also lead to students thinking that math is split up into many distinct categories of problem-solving methods instead of seeing it as a creative problem-solving puzzle.

Switching to an intuition-based teaching style is very important, not only because it reduces the time needed to become comfortable with a new topic, but also because it makes math and other challenging subjects accessible to students who are unable to focus for the long periods of time necessary to solve many problems in detail in one sitting. This can be due to a variety of reasons - ADHD, an unstable home environment, malnutrition, other learning disabilities, etc.

In practice, I use a combination of intuitive and traditional learning methods to learn new topics, where I take the intuitive teaching method outlined above and add an extra step in step 3.2 after the student checks if they were right about the problem type and the problem solving methodology: For one out of every N problems, I have the student solve the problem in detail, regardless of whether they were right or wrong. I find that N=5 or so gives me a good balance of building my intuition and learning how to solve problems in full detail. When teaching students with this method, it might be useful to figure out what value of N works best for them.

Example: Integration

A good example of a subject that involves creative problem-solving is integration. A student who is comfortable with solving integrals in general struggles with solving specific types of integrals might not need to completely transition to an intuition-based learning style, though it would definitely help them if they wanted to improve the time it takes them to solve problems. However, most students who struggle with integration are generally uncomfortable with the practice in general. They see almost all integration problem as a complicated mess of symbols and algebra that is impossible to make sense of, except for a few easy integrals that they've managed to memorize over time. This is the type of student who would benefit most from the intuitive-based teaching approach outlined earlier. For integration, it can be made more concrete:

1. Categorize the problems and the solution methodologies into easily memorizable patterns
   Problem Types: Polynomials, Rational Functions, Trigonometric Functions, etc.
   Solution Methods: Integration by Parts, u-substitutions, Trigonometric substitutions, Inverse trigonometric functions, etc.

2. Teach one or two examples of how to solve each type of problem and how to use each type of solution
   For this step, you could teach how to solve a couple of each type of integral. I would start with the simplest possible example of each solution methodology and then a few examples of possible combinations of solution methodology (Ex. A combination of a u-substitution and an inverse trigonometric substitution, or integration by parts and trigonometric substitutions)

3. Repeat the following steps until the student feel comfortable solving new problems:

   1. Generate a random list of problems of all types, distributed according to the likelihood of encountering each type of problem in real life (or on the test that the student is preparing for)
      Generate a worksheet with a list of integrals and a reference list of problem types and solution methodologies for them to refer to

   2. For each problem:

      1. Have the student quickly recognize the problem type and solution method(s) which should be used to solve that problem
         Have the students spend 10-20 seconds trying to classify each integral into a type of problem and also select one or more types of solution methodologies which would be necessary in order to solve that integral

      2. Check if they're right.
         Check if the integral actually was of the type that they selected, and if solving it would involve using the methods they chose

         1. If they're wrong, show them the answer and skim through the detailed solution until they have a sense of what type of problem this is and why, and why that choice of solution method(s) makes sense for this problem
            Have them look at the in-depth solution to that integration problem, which should outline all of the steps involved in the integration. Ask them to go over it until they feel like they can reproduce the solution themselves. You should have them actually reproduce the solution every once in a while to make sure that they aren't basing their intuition on concepts which they don't actually understand. Intuition can only point them in the right direction, but they need to also know how to move in that direction.

         2. If they're right, tell them to remember that they were right and store that information so that they will later recognize the same problem type and solution method(s) even quicker

I hate to see math being turned into a series of repetitive rules for kids to learn, which causes them to disengage from the subject and see it as a topic of rote memorization instead of creative problem-solving. I think that the most direct way we can address this problem is by designing problems that test students' understanding of the core concepts behind an idea, instead of their ability to apply the rules that they're taught. They still need to learn the rules, of course, but as soon as they're done learning the rules they should be asked to answer questions that test their understanding of the concepts in context of those rules- A deep understanding of the concepts will then naturally follow from them failing and learning to succeed at those problems.

For example, I've been teaching a fourth grader to solve problems involving area and perimeter recently- At first I gave her a few rectangles of different sizes and showed her how I calculated the area and perimeter and how it relates to "fencing" and "grass" needed in a yard. Then I gave her many rectangles and other shapes to calculate the area and perimeter of herself. But I wanted to make sure that she had a feeling for the geometry involved in these problems, instead of just a vague idea such as "area=multiplying the numbers, perimeter=adding the numbers".

So I came up with these problems to help her get a feel for what area, perimeter, and visual thinking in general really mean. Each of these problems addresses a few different geometrical concepts: