Developing Number Sense – Part 4

This is part four in a six-part series on the teaching of number sense.  In the previous issues we discussed the importance of number sense and 

outlined the components that comprise efficacy with number:
·      Number magnitude
·      Estimation
·      Mental computation
·      Mathematical properties
·      Effects of operations
We have also developed the first two of these components, number magnitude and estimation, and in this article we will explore the teaching of mental computation.

As we move toward implementation of the common core, skill in mental mathematics will be more important.  The Common Core calls for students not only to be able to compute, but to think mathematically.  Mental computation will facilitate mathematical thinking in general and aid in the problem solving process.

A math teacher once told me about tutoring a high school student from another country.  He said that he gave the student a copy of his district’s math assessment.  What surprised him was that not only did the foreign educated student finish in significantly less time that his own students, he didn’t use a pencil or paper on many of the questions.  Simply thinking about the problem yielded a logical answer from among the multiple choices.

To promote this skill in my own students, I offer them challenging problems that lend themselves to mathematical calculation.  For example, I ask my students to calculate 15x16 mentally.  Many of my 8th graders are able to do this using varied strategies.

• One student knew that 15^2 was 225.  He then added 15 more to get 240.
• Another student multiplied 10x16 to get 160 and then added half of that (5x16).  This incorporates the distributive property.
• A third student approached the problem by thinking that 1x16=16 and 2x16=32.  Therefore, 1.5x16 would be in the middle of these two products.  Simply annexing a zero to the 24 yields the correct answer.
• Factoring the problem would give (3)(5)x(2)(8).  Applying the associative property gives (5)(2)x(3)(8)=10x24.

Another example asked the students to find 45% of 250.  Successful strategies again varied greatly.

• Some students found four sets of 10% (25) and added half of that (12.5) to get 112.5
• A second strategy began with 50% and subtracted 5%.
• Finding 40% of 250 (100) and 50% (125) would allow a student to find the number halfway between these two.

A third problem asked the students to find a fraction between 5/7 and 13/18.  One student simply added the numerators and added the denominators to get 18/25.  Though I first thought this solution was coincidental, he showed that it had worked on other problems.  We tried it with more fractions, even negative ones, and found that it always works and can be proven mathematically.

I don’t want it minimize the importance of traditional mathematical procedures, but the ability to think about mathematics is as crucial as the ability to calculate with pencil.  In fact, it is more likely that the math we use in our daily lives involves mental calculation than paper and pencil.  By focusing on this skill, my students have become more proficient at mental mathematics.