Thanks to Beyond Traditional Math for asking, "How do you help your students discover concepts on their own?"
Some students may learn from listening and watching but all students learn from doing. I think the key is to get students actively engaged in doing math. In my mind doing math means a lot of things. It can mean inquiry based lessons, math games, authentic math experiences, real world problem solving.
One thing I have implemented in full force this year is using math games to teach and practice basic facts. I'm sure you use games too, but I'm not talking about here and there or once in a while but every single day.
Dr. Constance Kamii gives the research behind using math games on a daily basis in her book Young Children Reinvent Arithmetic. She recommends 40 minutes per day. When I started thinking about her research findings which showed that children will invent their own strategies, I ended up going back about 10 years to when my oldest was about 7 years old. We were trying to teach her some games that we thought she could play with us (Hearts, Yahtzee, Cribbage) and needed a fourth player. We recruited our 4 year old. By the time my youngest hit kindergarten and first grade she was not only a pro at playing these games but had become efficient and proficient at her basic addition and subtraction. She was definitely ahead of many of her peers. To give a little more evidence to further back the research by Dr. Kamii, I must confess that I have never purchased workbooks, math activity books, flashcards, computer games, get your child ready for K products, etc. Good children's literature, math games, puzzles and trees and trees worth of paper. That was it. If research backs it and it worked with my daughter, then why not in the classroom?
To take this argument even a step further, the NCTM has an article by Dr. Kamii on their website. To summarize her article, it is either superfluous or premature to teach children to solve 3 + 4 by teaching them to count out 3 and then count out 4 and finally count them all up. The same goes for teaching children to count on. Either they are have constructed their logic to sufficiently understand this already or they simply are not ready for it yet.
So, what does this mean for the classroom teacher? We need to find ways to help students discover math concepts. Engaging, active, authentic, motivating, inquiry based learning.
What else does "doing math" mean to you?
Kamii, Constance. Young Children Reinvent Arithmetic. 2nd ed. New York:Teachers College Press, 2000.
Link to Dr. Kamii’s article, “Teachers Need More Knowledge of HowChildren Learn Mathematics.”