What is the best way (and age) to start teaching your children mathematics in a rigorous manner?
The starting age is pre-school. It could be as young as 3 or 4 years old for some kids. Many people think that young kids will learn the conceptual foundations of math through spontaneous play - songs, games, playing with blocks, etc. Research shows that doesn't happen. Kids need systematic instruction on the concepts to get it. Here's a quote from an article in Science:[People] often believe they are “doing mathematics” [with young children] when they provide puzzles, blocks, and songs. Even when they teach mathematics, that content is usually not the main focus, but is embedded in a fine-motor or reading activity. Unfortunately, evidence suggests that such an approach is ineffective, owing to a lack of explicit attention to mathematical concepts and procedures along with a lack of intentionality to engage in mathematical practices (p. 968).https://portfolio.du.edu/portfol...Other research shows that kids who don't have a solid foundation in what is called "number sense" by the time they finish kindergarten (or better yet, before they enter kindergarten) will have a much harder time in the mainstream math curriculum than kids who do. Here's another quote from the same paper:...preschool children’s knowledge of mathematics predicts their later school success into elementary and even high school.More about number sense specifically:Number Sense: What it is, why it’s important, and how it developsAs to good ways to teach it, that's not something that can really be answered in the short space of a Quora post. But if you are interested in reading more, here is some relevant material from my own work:What We Know About How Children Learn Math - And How It Can Help Us Close the Achievement GapTeaching Math with Minecraft (An Impromptu Education Experiment)Teaching Math with Minecraft #2: Understanding Addition, Multiplication, and the Commutative PropertyHow Can Less Studying Produce More Learning?