Math
Number sense, like commonsense, is easy to recognize, but not always easy to describe or support. In a world of computers and handheld devices, good number sense is more important today than at any time. As Keith Devlin, Stanford mathematician, wrote recently:
The most basic of today’s new mathematical skills is number sense. (The other important one is mathematical thinking.) But whereas the latter is important for those going into STEM careers, number sense is a crucial 21^{st}century life skill for everyone.  (Huffington Post, 2017)
Moreover, we now know that number sense begins in the first years of life; if not developed in school, a lack of number sense can cause students to struggle in the upper grades and in college.
What do people with good number sense do?
 compose and decompose numbers to solve calculations
 compute mentally using a variety of strategies
 make reasonable estimates
 understand quantity and magnitude
 reason with numbers
What does it mean to decompose and compose numbers to calculate? Let’s look at an example. In a fifth grade class students were solving a word problem that required the product of 25 x 26. Instead of assuming everyone would find the product in the same way, students are encouraged to find a variety of methods.
Before you read on, think of some ways you might multiply 25 x 26. Students had a variety of responses:
25 x 26 is the same as 26 x 25
Think of 26 as 20 + 6 20 x 25 = 500 6 x 25 = 150 26 x 25 = 650
Another student said he did it similarly but instead of 20, multiplied by 10 twice.
Still another suggested that beginning with 30 x 25 was easier than 26 x 25:
30 x 25 = 750, but that is four 25’s too many 4 x 25 = 100 750 100 = 650
Some other solutions included the following:
Think quarters: 4 x 25 = 100 26 ÷ 4 = 6.5 6.5 x 100 = 650
or 100 x 26 = 2600 2600 ÷ 4 (since there are 4 25’s in 100) = 650
One student suggested that 25 x 26 was too difficult, but it was the same as 13 x 50 because she doubled the 25 and halved the 26.
13 x 50 = (10 x 50) + (3 x 50 ) = 500 + 150 = 650
Finally one student said she studied a table of square numbers in the room and saw that 25 x 25 = 625 and 1 more 25 is 650.
Classrooms that nurture number sense Of course this variety and number of strategies doesn't occur naturally, but only in classrooms where students are encouraged to take risks, to look for structures like doubling and halving factors, to use known facts to figure out unknown ones, and most of all to share and examine each others’ thinking. Students work in pairs to come up with solutions, first with easier problems (think 11 x 15) then with those of increasing complexity.
Student solutions are posted, discussed, and whenever possible, generalized. (26 = 20 + 6 or knowing that 4 x 25 equals 100 enables you to multiply by 100 and divide by 4 when multiplying by 25.)
Finally, providing visual models for procedures deepens understanding. For instance an area model helps students understand the distributive property.
20 x 25 = 500 6 x 25 = 150 26 x 25 = 500 + 150 = 650
Is the answer the same as one derived with the paper pencil algorithm? Of course, but which develops number sense? Which encourages students to use properties like the commutative and distributive property? Which enables students to solve more complicated problems mentally?
Think how different it would be if students merely wrote 25 x 26 vertically and then did something like this:
26 x 25
5 x 6 = 30, carry the 3, put down 0, 5 x 2 = 10, add 3 = 13. And so on. No number sense. No place value. No idea if the answer is reasonable.
Taking the time to thoroughly examine one or two calculations in depth, similar to the above, encourages students to focus on thinking—not merely on getting the correct answer. It leads both to number sense and to underlying math concepts.
Andy Clark will be presenting at the 2017 Math in Focus Institute, July 26–28, in Lafayette Hill, PA.
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