Each Wednesday afternoon, as we have for the past 5 years, teachers and I come together for a professional development staff meeting. Meeting topics are broad and varied. We have learned about brain development, brainstormed ways to increase content area literacy, discussed and determined how to encourage wonder and curiosity, created schedules to video tape ourselves teaching in order to hone our craft, and eaten a wide range of snacks. And that is just since August!
Last Wednesday we turned our attention to math and hunkered down to increase our understanding of number sense. Some studies indicate that a child’s initial understandings of numbers have long term consequences for their success in school. For example, a 2011 study determined that, “preschool children’s knowledge of mathematics predicts their later school success into elementary and even high school” (Clements and Sarama, 2011). In other words, early number sense matters, and it matters for a long time.
So what is number sense? At TCS, we accept the widely held definition of number sense as “intuition about numbers that has developed gradually as a result of exploration, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989).
Number sense is not a unilateral relationship, instead it is a web of interconnected ideas. Over time, this web grows larger and larger. If children just learn basic facts in an isolated context, they will develop problems when numbers become larger and more complex, because they cannot connect these basic facts to other facts or number relationships. In other words, memorization alone is not enough. Our knowledge has to mean something.
So, can number sense be taught? The answer is yes, but in very specific ways. Every child can develop number sense, and some, in fact, have it intuitively. For the children that don’t come by it naturally, our job is to systematically support that learning on an individual basis. The catch is that we can’t just stand in front of the room and tell them, “8 is less than 9” and expect them to understand it without context. This is why the concept of “skill and drill” doesn’t work long term, and certainly it doesn’t work at all if we move to it without doing anything else. Brownell and Chazal (1935) found that if we move to an emphasis on speed too soon it just encourages children to become faster at their informal approaches (i.e. counting on their fingers). In other words, they are just practicing their inefficient and occasionally inaccurate strategies.
The key is that they must learn it by doing. They learn by playing games, and watching the person that rolled the 9 on the dice move one more space than the person that rolled an 8. They learn it by pulling over a chair to the dinner table when a guest comes over. They learn it by counting out 9 grapes at snack time, giving one to a friend, and then counting again. Our job is to provide experiences for them to explore and then intentionally guide them into seeing relationships around numbers. This is guided discovery at its finest.
One of the relationships that is key to developing number sense is subitizing.Subitizing is instantly seeing “how many” and it derives from the Latin for “suddenly”. Research suggests that even infants subitize on a basic level. For example, close your eyes and think of “seven”. What did you see? When we did this in staff meeting we had a wide array of responses. I saw the word “seven”. Anne Wil saw a group of 5 and 2. Kelly saw 2, 2, 2, and 1. Noah saw the numeral 7. This picture in your head of a number is key to subitizing. We need to help children who, like me, don’t automatically create a picture of sets in their head of numbers. People who are able to subitize “just know” that 7 is actually
This skill is so important, that students that cannot subitize really struggle to learn arithmetic processes. Interestingly, traditional textbooks like the ones used in non progressive schools often present information in ways that discourage subitizing!
At TCS we are deeply committed to creating time and space to explore this skill. At home, there are many things you can do to encourage subitizing.
1. Have your child create a quick image arrangement of manipulatives (pennies, cheerios, cotton balls…)
2. Arrange manipulatives into a visual image and have children quickly tell you “how many”. Challenge students to create an image that is “one more” or “two less”.
3. Encourage children to estimate with arrangements that are too large to subitize exactly.
4. For older children, create a geometric pattern with manipulatives, and have them use subitizing. For example, a square has 4 equal sides. This square has 2 dots on each side and 4 more in the corners so 12 dots total.
5. Clap a rhythm and have children tell you how many claps you made.
6. Download an app such as Pattern Sets from itunes