First Comes Number Sense by Gillis Kallem
At Hamlin, we want to cultivate a rich math experience, which includes a vibrant mathematical adventure into the world of numbers and reasoning, problem solving, and creativity, as well as real-life applications. At the foundation of this important work is number sense. In its most general definition, number sense is the ability to intuitively work with numbers. It reflects an understanding of numbers, their magnitude, and relationships. As educators, we expand the meaning of number sense to include a well-organized concept of numbers that allows a person to solve mathematical problems accurately and efficiently in a variety of ways that are not bound by traditional algorithms. (Bobis 1996)
Number sense involves the understanding that numbers are flexible. For example, a number is malleable like a ball of clay. You can change its shape – make it long and skinny, or short and fat—or you can break it apart into smaller pieces and in the end roll it back to its original ball. The same is true for any number. Take 28, it can be seen as 2 groups of 14, 4 groups of 7, or as 20 + 8 or 10+18 or almost 30. Take your pick. Having this ability to see 28 or any number in its many forms allows the user to think freely and creatively when asked to solve problems involving operations with numbers.
When we are presented with a problem such as 131 – 28 in which we might be tempted to write it out in the standard algorithm, then regroup/borrow/cross out numbers and so on, we can instead think about the many forms that any number can take and find one that makes sense for solving this problem in our heads:
Think of 28 as 20 + 1 + 7.
We can first think: 131 – 20 = 111
Then, 111 – 1 = 110,
Finally, 110 – 7 = 103.
In this method, we break apart the subtrahend.
Or we can think that 28 is almost 30 by adding 2 more.
We can change the problem: 131 – 30 = 101
Then we add back 2:101 + 2 = 103.
In this method, we adjust the subtrahend to make a friendly number or a round number, and then add back what we added to the subtrahend.
Or we can play even further!
Change both numbers by adding 2 to each.
Thus, 131- 28 becomes 133 – 30 = 103.
In this method, we adjust both sides, keeping the distance the same but making the numbers easier to work with. This method is known as constant distance.
What about 28 x 5? How would we efficiently and accurately solve this mentally?
One way might be to think of 28 as 20 + 8
Then, 20 x 5 = 100 and 8 x 5 = 40
Combine, 100 + 40 = 140.
This makes use of the distributive property of multiplication.
Another way might be to think of 28 as 30.
Then, 30 x 5 = 150.
Now, subtract two groups of 5: 150 – 10 = 140.
Again, this is using a friendly number and then adjusting afterwards.
Or you could think of the problem in an entirely different way!
Turn 28 x 5 into 14 x 10 = 140.
This is a clever method called doubling and halving.
Various classroom activities and lessons in Grades K-5 at Hamlin support the development of number sense and flexible thinking. Additionally, our assessments measure the girls’ learning of robust number sense. Building computational fluency is a core skill that fluid and flexible number sense girds for long-term success in mathematics.
In my next Curriculum Connection, I will highlight the specific classroom practice of Number Talks/Number Strings as it relates to number sense.