"Bear in mind that the wonderful things you learn in your schools are the work of many generations, produced by enthusiastic effort and infinite labor in every country of the world. All this is put into your hands as your inheritance in order that you may receive it, honor it, add to it, and one day faithfully hand it to your children. Thus do we mortals achieve immortality in the permanent things which we create in common." - Albert Einstein

Thursday, October 20, 2016

How a Child Learns Math: Conceptual versus Procedural

My mother-in-law taught my daughter while she was in kindergarten how to add numbers. When adding for example 8 and 4, the five-year old child was instructed to put her hand on her chest and say "8", then she should proceed counting from there, "9, 10, 11, 12" (four more, to add 4 to 8). That was how she was able to arrive at the correct answer, "12". My daughter was actually proud when she was able to do it correctly on her own with any addition question thrown at her. At first glance, one might think that my daughter was simply learning a procedure, but the mere assignment of "8" as a starting point represented a very important concept in mathematics, the cardinal principle:  The idea that the last number reached when counting the items in a set represents the entire set. My daughter did not need to count from 1 to 12 to determine what 8 + 12 was. She could simply start with "8".

My mother-in-law with my daughter counting how many eggs she has found in an Easter egg hunt.
Figuring out what skills are important for a young child to learn in order to succeed in mathematics is an important question in education research. A paper scheduled to be published in the Journal of Educational Psychology addresses this:

Above copied from
The Importance of Additive Reasoning in Children’s Mathematical Achievement: A Longitudinal Study.
Ching, Boby Ho-Hong; Nunes, Terezinha
Journal of Educational Psychology, Oct 13 , 2016, No Pagination Specified. http://dx.doi.org/10.1037/edu0000154
As the above abstract suggests, conceptual knowledge of counting, which includes the cardinal principle, is among the important skills that correlate with achievement in early mathematics. However, when it comes to solving word problems, one must go further, additive reasoning and working memory are the good predictors. Additive reasoning is basically the ability to figure out what calculations need to be carried out. The following is an example from the British Journal of Educational Psychology:

Above copied from
Nunes, T., Bryant, P., Barros, R. & Sylva, K. (2012). The relative importance of two different mathematical abilities to mathematical achievement. British Journal of Educational Psychology, 82(1), 136–156.
In both problems, the final distance between the boy and the girl is being asked. In the first case, both boy and girl walk from a house in the same direction, the boy walks 6 km and the girl walks 2 km. In the second case, the boy and girl are walking in opposite directions from the house. The boy walks 3 km and the girl walks 5 km. To answer the first case, one must subtract 2 from 6, so the final distance between the boy and the girl is 4 km. In the second case, one must add 3 to 5, such that at the end, the boy is 8 km from the girl. Being able to decide what calculation needs to be carried out (in this case, either addition or subtraction) is key to solving this mathematical problem. And it obviously goes much farther than just knowing one's arithmetic.

Procedural knowledge does play a factor in math achievement in the early years but clearly, it is the conceptual knowledge that is much more important. Boby Ho-Hong Ching and Terezinha Nunes, in their recent paper, especially underscore the lesser role played by procedural knowledge since their study involves Chinese students. Some people sometimes ascribe the higher math achievement of young Chinese children to the language they have. An example is an article published by the Philippine Daily Inquirer:

For this reason, it is important to do scientific studies so that we become better informed with regard to how children learn. Otherwise, we can fall easily for arguments that sound good but really have no evidence. The case of my daughter and her grandmother may be different, however. My mother-in-law, after all, had been a teacher all her life. From her vast experience, she obviously knows what works.

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