This summer I visited the PROMYS program for a day. It's a six week program for teachers and advanced high school students that explores number theory at a college level. The kids are there because they want to spend their summer doing math. The teachers are there because they want to revolutionize math teaching. PROMYS for teachers models how to use seemingly simple problems to encourage very sophisticated math discoveries. They are trying to develop math habits of mind in students (which overlap very well with our habits of mind) so that students see math as a process of observing, making conjectures, testing and re-thinking. Communication is key. "Failure" is important because it is only through testing a conjecture that we can make it stronger. As the professor giving the lecture said, "It's not finding the answers that's hard...it's finding the right questions."
One of the challenges of teaching math is that, in order to discover patterns, make conjectures and talk about math, students need a certain amount of facility with computation and vocabulary. It's not enough to just explore. Students need to solidify those explorations and be able to use that knowledge efficiently to push their learning ceiling higher.
To help facilitate that process, our math program has many parts. I realized that we sometimes don't talk explicitly about the purpose of each element and wanted to take the opportunity to do that:
Exploration Math - This is what we call our math themes. This is when we work with children to use numbers, algebra, data, graphing and geometry to make sense of the world. We focus on a math concept such as probability for 2-3 weeks. During that time, we use games, open ended problems, and other math experiences to build students' understanding of a mathematical concept. Once the concept is solidified, we often introduce algorithms and vocabulary so that students can use and communicate their understanding when solving more complex problems.
Foundation Math - This is an almost daily practice session when students are becoming more efficient computers. While calculators are an invaluable tool, we also want students to be able to solve problems using paper and pencil. In addition, working problems regularly helps information "stick," solidifying students' memory of a skill.
Quick Math - Almost every day, students spend 2 minutes recalling basic math facts. We have found that when students take more than 3-4 seconds to recall a math fact, they easily get lost in their computations and make errors. In addition, knowing one's facts leads to more discoveries in math. One sees patterns in numbers and relationships more easily.
This year, I am using a slightly different progression of mastery (40 facts in 2 minutes). Addition, subtraction, Multiplication in this order: 0s, 1s, 2s, 5s, 10s, 11s, 9s, squares, and then the "last 10 facts" which have not been memorized in the preceding sets: 3x4, 3x6, 3x7, 3x8, 4x6, 4x7, 4x8, 6x7, 6x8, 7x8. Students then "flip" their multiplication knowledge to master division and finally, common percent/fraction/decimal equivalencies (e.g. 1/5=20%)
Number Talks - We often write a problem on the board and have students solve it mentally then share their pathway to a solution. This encourages math communication and flexible thinking. It also shores up number sense.
Math Problem of the Week - a multi-step problem in which students are asked not just to give an answer but to explain how they arrived at their answer. The focus is on explaining the process of solving and it develops students' ability to communicate about math succinctly and clearly.
"Recreational Math" - A lot of elementary math focuses on computation but that is only a small fraction of the world of math. We take the time to introduce students to some of the biggest questions in math in a way they can understand. The 4 color map challenge is an example of this kind of math. Students love exploring these new ideas and, even though some of the math used in explaining will not be taught to them for years, it's valuable for them to be excited by these big math ideas. (Click on the link to find out just how big an idea our little math challenge was.)
Serendipitous Math - Part of my job as a teacher is to look for ways to pull math into our daily classroom life. Math conversations can pop up in meeting. Impromptu class surveys help us crunch data efficiently. A current event might give us a chance to noodle on the numbers involved (just how much snow does it take to make a giant snow shark?) Math is everywhere and it helps us understand things.
Whew. It's a lot of math and, when we do it right, it's exciting and engaging and real. I very rarely have a student ask, "When are we going to use this?" because they are using it right now.