### How We Teach Arithmetic

One way a five-year old could be taught addition is by using flash cards. It is true that for some, this may sound as pure memorization. What is obviously needed is not only to remember but also to acquire the ability to relate and manipulate which then leads to a deeper understanding of arithmetic. A child of course can develop this understanding if the child has first memorized addition facts. This is not different from knowing sight words which immensely helps young minds to read. Having the facts frees up some space in a child's working memory which a child can then use to find patterns and relationships, and begin to appreciate the world of mathematics. Since a kindergarten or first grade teacher already knows what lies ahead, memorization drills can be tailored such that students can likewise anticipate what is coming. Instead of a randomly or iteratively (which may aid in memory) arranged set of flash cards, addition facts can be presented with the objective of preparing students for higher math. Such approach is nicely illustrated in a poster presented by McNeil and coworkers at the University of Notre Dame. The following is a figure taken from the poster:

 Above copied from McNeil et al.
Flash cards or activities can be easily modified to achieve so much more than just practicing addition facts. In another poster by Mcneil and coworkers, the following pair of activities nicely highlights what can be achieved by introducing minute changes on elementary flash cards. The following activity, for example, aims only for the minimum (helping a child learn to count and add):

A child learns to count dots and the numbers associated with each count. A child also learns to combine the dots from each box and therefore performs the addition operation. The following, on the other hand, with just simple modifications, can achieve so much more:

 Above copied from McNeil et al.

The activity demands exactly the same thing from a student: Count the dots and provide the number that is associated with a particular count. However, by simply rearranging what is known, the child is now being introduced to subtraction. In addition, it may look trivial but the mere replacement of the "=" sign by the phrase "is the same amount as", is actually a big deal in the mind of a young child.

The modifications are small yet the gains appear substantial in the assessment results. For instance, by simply grouping addition facts by equivalence, the following is obtained:

 Above copied from McNeil et al.
By rearranging addition facts so that the children sometimes solve for numbers on either side of the equation also yields significant improvements:

 Above copied from McNeil et al.

It does make a difference when teachers tailor their lessons and students' activities with specific objectives. These still look like drills but these help students gain a better understanding of the lesson. And it seems to require not much effort. To read more on this topic, McNeil and coworkers have published their work in the Journal of Educational Psychology:

The above results are indeed convincing. What is troubling, however, is the fact that the mean age of the children participating in this study is over 8 years old (second grade). To see that the children are able to get less than 50 percent correct on the assessments is truly distressing. Children in this study are from a disadvantaged background, showing that a lot of work still needs to be done. Interventions shown above yet simple and cheap are making a difference.