· Mental Computation
· Mathematical Properties
· Effects of Operations
Before we explore each of these components, we need to understand the strategies by which number sense is developed. It is helpful to compare the teaching of number sense to the coaching of physical skills. In order to be in good physical condition, we must have an ongoing training regimen. We can’t assume we are “in shape” simply because we once ran a mile. Similarly, research shows that number sense is developed only through frequent instruction over a period of years. We cannot cover number sense as a separate unit of instruction as we can with other areas of mathematics. It cannot be learned from a chapter in a textbook nor assessed with standardized testing. Yet we know when a student has it and when he does not. Thus it is our job as teachers to provide students with these ongoing opportunities to develop skills with numbers.
These practice opportunities need not be lengthy but they must be frequent. Thus our daily warm-ups are an ideal time to foster the development of number sense. In these sessions I focus on letting students use creative strategies, explore multiple strategies, search for patterns, play with numbers and operations, and use language to process their learning.
As our first example, let’s take a look at number magnitude. A sense of number magnitude is crucial for navigating the mathematics of our daily adult lives. For example, we think about number magnitude in planning costs, fuel, and time for a trip or for determining if a car purchase will result in an affordable payment.
On the first day of school, I ask my eighth grade students to consider this problem: are there more inches in a mile or seconds in a year? During this thinking time they are not to use pencil or paper or discuss their ideas with anyone else. After 60 seconds, I ask them to stand on one side of the classroom if they think there are more inches in a mile and the other side if they think there are more seconds in a year. Because I allowed them to think without interruption, their strategies are varied. If I had allowed immediate response, they would have stopped thinking the moment a sharp math student responded.
I then ask students to explain why they chose one side over the other. Some just offer a gut-level guess, but others suggest strategies. For example students note that to find seconds in a year they would have to multiply 60 seconds per minute by 60 minutes per hour by 24 hours per day by 365 days per year. For the other question, they would multiply 12 inches per foot by the number of feet in a mile. Most of them don’t know how many feet are in a mile. However even without that knowledge we see that the two problems look like this:
12x(feet per mile)
The 24 hours in a day can be written as 12x2. By rearranging the other factors we get these two problems:
12x(feet per mile)
The question then becomes whether the number of feet in a mile is less than or more than 2x60x60x365.
Sometimes we calculate the actual answer to evaluate our thinking but other times we do not.
The next day I ask them, “How old will you be when you have lived a million seconds?” Again they must think silently about this without pencil, paper, or calculator. Then I ask them to line up in order based on their estimates. In the initial days of number sense instruction, their guesses cover a wide range, but as the year progresses, they become more refined in their strategies and the range narrows.
The students are usually surprised to find that they passed one million seconds on their 11th day of life. This helps answer the question from the previous day. The next day I ask them when they will be a billion seconds old. This doesn’t occur until they are nearly 32 years old. It would take over 30,000 years to log a trillion seconds. This gives them a sense for the differences in magnitude among a million, a billion, and a trillion.
Each day these tasks take five to ten minutes at the beginning of class. As they practice these skills on a daily basis they begin to be more sophisticated in their thinking and more discriminating about the strategies they apply. The also become more comfortable with the idea of approximation and estimation, which we will explore in our next issue.
Brad Fulton is a veteran math teacher and nationally recognized educational consultant and keynote speaker. In 2005, he was selected as the C.L.M.S. middle school educator of the year.