Students who struggle in math often lack number sense.-Jessica Shumway in Number Sense Routines (p.8)
As students build their number sense, mathematics takes on greater meaning. Mathematics becomes more about reaching understandings than following rigid sets of rules. With strong number sense, children become more apt to attempt problems and make sense of mathematics. It is the key to understanding all math.
These words tell exactly how I feel about math and math teaching and learning. The phrase "reaching understandings" is just where our focus should be in math. All too often we go straight to "following rigid sets of rules" and this is where we lose students. The "rules" are complicated and meaningless to most students. We need to be helping students explore their own ways of solving problems rather than memorizing sets of rules.
I compare this to reading instruction. What do we really think matters the most in reading instruction? I think most of us would agree that comprehension or making meaning of the text is our ultimate goal. We have learned that focusing only on the rules (phonics, etc) in reading can and usually does lead to readers who tend to be word callers - those who read words correctly but don't understand what they are reading. We know that we need to focus on the meaning of the story being read along with using some decoding strategies. Often the most successful decoding strategies are based on using meaning - thinking about what the story is about, using background knowledge, using context clues, using picture clues, etc. We need to do a blend of meaning and rule following in order to become readers, but we always need the emphasis to be on meaning.
Having number sense and "reaching understanding" in math is similar to comprehension in reading. We may need a blend of rule following and understanding to become mathematicians, but the emphasis should always be on understanding. Always. While not a popular argument, students will be able to use calculators and computers to do those tasks that fall under the "rule following" category, but they will still need to be able to decide what makes sense in a given situation or if the "answer" that they get from their calculator or computer is reasonable. When students have number sense and deep mathematical understandings they will not only be able to decide if an answer is reasonable but they most likely won't need a tool to help them with the computation very often.
I am passionate about this and have been working to make shifts in my math teaching to ensure that students have time to develop deeper mathematical understandings. Thankfully, there are many books out there to help (written by other passionate math teachers). The picture below shows some of the books I have read that have influenced my thinking. I would highly recommend them all. Who or what is influencing your mathematical thinking right now?
Number Talks by Sherry Parrish
Math Exchanges by Kassia Omohundro Wedekind
Number Sense Routines by Jessica Shumway
Teaching Student-Centered Mathematics by John A. Van de Walle