Why can't 2 + 2 just equal 4?

A teacher at our school recently received a note with the question posed above.  The parents were frustrated with the new Common Core standards and longed for the good ol’ days when drill-and-kill with worksheets and flash cards was all that was required for mathematical proficiency.  While drilling mathematical facts still has its place, the Common Core standards incorporate math theory along with math practice to create math fluency.  Students are no longer expected simply to know what the answer is, but rather how numbers work together to get to that answer. 

My daughter is in first grade and was struggling one night with the instructions given on how to add numbers.  I noticed her frustration and asked, “Hannah, what do you see when you look at that number?” 

Hannah responded, “It’s a 20.” 

“Hannah,” I said, “you see a 20, but I see four fives, two tens, ten twos, an eighteen plus two more, and so on.  They are trying to show you how the numbers can be taken apart and put together in different ways to get to the answer.” 

Hannah’s eyes brightened as the purpose of the exercise registered in her little mind, and she said, “Daddy, that is awesome!” 

It is awesome.  And it is different from how I learned to add, subtract, multiply, and divide.  A teacher at our school began one of her math lessons by pulling up a map of our community and having the children find all of the possible ways to get to Walmart. Some of the routes were direct, some were indirect, but they all eventually got to the same destination. She then explained that they were going to look for different routes to get to a mathematical answer.  Some would be more direct than others, but she wanted her students to understand how numbers work; not just merely how to find the sum or the difference in a problem.

Research has shown the benefit of understanding number theory along with mathematical practice as well.  A study was done a few years ago by the TIMSS (Trends in International Mathematics and Science Study) comparing the performance of students from Hong Kong to students in the U.S. on an international math assessment.   The students from the U.S. had covered roughly two-thirds more mathematical practices than their Hong Kong counterparts, but the students from Hong Kong scored significantly higher on the assessment.  Upon further study, it became clear that while the U.S. students had covered much more material, the students from Hong Kong had gone much deeper in the practices they had covered.  They had a deeper knowledge of mathematical theory and the way that numbers work.  They were able to use this understanding to successfully solve a variety of problems to which they may not have previously been exposed.

So, 2+2 does still equal four.  But it does more than that as well.  When our students understand the concepts of number theory, they will be better equipped to tackle any mathematical problem that comes their way.