MATH-ish: Finding Creativity, Diversity, and Meaning in Mathematics by Jo Boaler

High school math hasn’t changed much since the 1800s. There is more math than high school years (algebra, geometry, algebra 2, pre calc, and calculus). This results in tracking in middle school based on flimsy evidence such as a single exam. Jo recommends adding courses like data science and statistics and saving calculus for college since 2/3rds of the 16% of students who take it in high school retake it in college.
Students who take data science are better at making sense of real-world data than those taking algebra 2. All math courses can use an infusion of real-world data. Students should know the difference between correlation and causation. To protect themselves from misinformation they should ask questions like who produced the data, is all the data being displayed, are the axes exaggerated, and what relationships are evident?

Deliberate practice lets you develop mental representations. Such models aid in learning and thinking. Mathematicians use the visual parts of their brains for all math. Other academics don’t do this. Real trailblazers have better developed visual brains. Students perform better when they are given lots of visual material.
Students should use number lines early and often. Fingers can help develop mental models. Fractions are often taught in non conceptual ways. Thinking about approximate (ish) answers can help. Teachers often resort to algorithms and avoid working on conceptual understanding. This is backwards. Making good visuals is not difficult. Encourage students to try. Instead of going over homework, start class with fun and interactive questions.

Mathematics is a set of concepts, big ideas and connections. It is important to teach these before bringing in methods or rules, which tend to hide the concepts. Your brain can compress concepts, it can’t compress methods and rules. This compression can lead to joy. High achievers approach problems with the number sense associated with concepts.
It is important to focus on the big picture prior to the small methods. Students should see math as a set of connected ideas. They should be encouraged to look for connections. Unfortunately, math is presented to teachers as disconnected standards. Students should create and share visual sketchnotes as they learn. There are examples here.