Learning through discovery has been used for decades. It means that a teacher can tailor lessons to give students some of the following and then allowing them to figure out some possible solutions: bits of information, manipulatives, worksheets with information missing, group collaboration, measuring, drawing, cutting, folding, asking questions, comparing numbers/information and making educated guesses. Research in the area of Mathematics and the human brain has been conducted and as a result, Jo Boaler, author of the book Mathematical Mindsets has written a tremendous amount of material on Math and the mind. She indicates that none of us are born as 'good in math' or 'bad in math'. In chapter 4 she talks about the struggles sometimes that students go through, with the end result being one of positivity :
"Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics. (Thurston, 1990)
These are the moments when students have those 'aha' moments. These moments turn into memories that they often remember. Formulas are often forgotten, but activities where students discover a solution to a problem by themselves, will long be remembered. Myself and other teachers have used the lesson below with Grades 7-9 when teaching Circumference, Diameter and the introduction of PI. Giving a formula is easily written down and forgotten. Letting students be in charge of the learning, results in them being able to recall the activities they did to remember what PI is all about.
You will need a few materials and space to place students into groups of 3-5 along with the pre-designed worksheet for students to fill in as they do the activity. At the end, the students will realize that a ratio is developing when they compare the circumference with the diameter.