"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

Sunday, March 8, 2015

Working Memory and Word Problem Solving

When a child finds math challenging, it is useful to identify the underlying reason. Recent trends of classrooms shying away from drills in elementary grades and focusing on understanding arithmetic problems assume that a child already has the capacity to exercise a higher level of thinking. A child's brain, however, is still very busy forming and refining connections at this point. For children with disabilities, the development of various functions of the brain seems to lag behind. A thoughtful consideration of how a child's brain develops and works may be helpful in figuring ways to help a child who is struggling in mathematics.

One aspect of the brain that can limit a child's performance in math is working memory (WM) capacity. The following test (an example of a span-board task) from Cambridge Brain Sciences measures an individual's WM capacity by how well a sequence of flashing boxes is remembered:


There is likewise a digit span task in which an individual is asked to repeat a given sequence of digits:


In both tests, the limit or capacity of one's working memory (the maximum number of flashing boxes (span-board) or number of digits (digit span) correctly recalled) is measured. Children with attention deficit disorder (ADD), for instance, often exhibit low WM capacity.

A low WM capacity can seriously impair a child's learning especially when this condition is not addressed first. Math word problems are introduced in the early years of elementary schooling in an effort to make children see math in a much more tangible manner. Solving word problems can be challenging since these problems not only require arithmetic but also a translation of text to mathematical equations. Thus, strategies have been introduced to help children address word problems. For instance, there is cognitive strategy training. Here is an example from My VideoMath.

Above captured from My VideoMath

Strategies often begin with "reading the problem carefully". What is given and what is asked are the beginning and end, respectively, of a word problem. Understanding what the problem is asking can be aided verbally and visually. A verbal strategy could be as simple as underlining the important words and numbers in the world problem as shown below:

Above captured from My VideoMath

The above likewise shows a visual strategy. Near the bottom right hand corner is a light blue ellipse with two smaller ellipses inside. The large ellipse represents the amount of money Ellen had altogether ($1.35) and the small ellipse ($0.45) represents the money her friend gave. The unknown m is the amount of money Ellen had originally. The figure therefore represents the equation (m + 0.45 = 1.35) and m can now be determined arithmetically.

These strategies can indeed help a child who is struggling in solving math word problems, but not always. In a paper published in the Journal of Educational Psychology, "Does Cognitive Strategy Training on Word Problems Compensate forWorking Memory Capacity in Children With Math Difficulties?", it is found that children with low working memory capacity do not benefit from these strategies. Drawing diagrams and following instructions are difficult for children with low working memory capacity since these tasks are cognitively demanding. Making these children go through these strategies is therefore a losing battle. What clearly needs to be addressed first is a child's working memory and there are indications from research that there are interventions that can help increase a child's working memory capacity. Klingberg and coworkers have reported that training could improve working memory (WM). And it is as simple as mental drills. The specific intervention used is described as follows:
The treatment consisted of performing WM tasks implemented in a computer program developed for this study (RoboMemo, Cogmed Cognitive Medical Systems AB, Stockholm, Sweden). The program was provided on a CD and used by the child on a personal computer either at home or in school. The program included visuospatial WM tasks (remembering the position of objects in a 4 by 4 grid as well as verbal tasks (remembering phonemes, letters, or digits) (see Olesen et al., 2004) for further description of the visuospatial tasks). Responses were made by clicking on displays with the computer mouse. The children performed 90 WM trials on each day of training. Total time depended on the level and time between trials. Medium total training time (excluding breaks) was about 40 minutes. The difficulty level was automatically adjusted, on a trial by-trial basis, to match the WM span of the child on each task.
The teachers I had forty years ago probably did not know much about working memory capacity but they did drill me to stretch my working memory, and it probably helped.





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