There are 2 ways to engage students in learning mathematics:
1) Show students methods and they repeat them. This approach is used in most schools, but the methods often lack meaning, and students reasonably ask: when are we going to use this? Additionally, students only ever get to use what they were shown, not select a method themselves, one of the most important mathematical acts.
2) Engage students in rich, open, visual and creative tasks. They use their intuition and thinking, and choose methods that can be useful in the task. When they need to learn new methods, teachers teach them inside the task. Students immediately see how important they are and learn them more deeply. They engage in the important acts of choosing and making connections between ideas.
The second approach is much more effective; however, teachers tell us that they do not have time to use open, rich tasks that students take in different directions. They see the lists of methods set out in curriculum standards and in text books and decide they only have time to show them briefly to students then move on. This is because our curriculum standards are over packed with outdated content that students will never need or use.
Mathematics in the 21st century requires a different approach. This web page is dedicated to hearing new ideas about teaching from professional mathematicians.