"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

Wednesday, July 30, 2014

Anxiety, Math and Working Memory

Cognitive psychologists theorize that our minds work with something similar to the "working memory" of a computer. This is some sort of a mental scratch pad on which items are placed simultaneously. The capacity of this pad is obviously quite limited. We can probably easily remember three or four things to do on a day, but higher than that, we would probably need a schedule, unless these activities have already been part of a routine. When something is already done out of habit, it becomes automatic, requiring very little from our minds. How a children learns is no different. It is therefore important to keep in mind how much load is placed on a child's working memory when being instructed. The same goes when a child is being asked to solve a problem or answer a question. The significance of the working memory crystallizes clearly in learning math. It is one subject in which there is growing evidence that anxiety alone can dramatically lower the performance of a child or even an adult.

Beilock and Willingham recently wrote a review on math anxiety in the American Educator:

To read the review, visit the American Educator
Anxiety, without doubt, can occupy one's mind. Worrying takes precious space from our working memory. This explains why children who are nervous about math do not do well in math. With the introduction of new ways of teaching math that encourages pupils to work out problems with more elaborate strategies that highlight what really is going on (instead of just working from memorized addition and multiplication, or from simply counting with one's fingers), it becomes even more important that teachers are made aware of the limits of a child's working memory. As Beilock and Willingham note in their review, the destructive power of anxiety may even be greater for children who are gifted with higher working memory capacity. For instance, students who have been trained to solve "8+4" by employing a strategy such as decomposition (8 + 4 = 8 + 2 + 2 = 10 + 2 = 12) spend more working memory than a student who does the addition straight. The students who employ more advanced strategies are therefore more susceptible to the deleterious effects of anxiety. Children who have mastered the fundamentals of arithmetic, those who can add, subtract, multiply and divide "out of habit" demand much less from their working memory.

The strategies that have been developed to help children appreciate what really goes on in arithmetic are definitely appealing to those who already understand a lot about mathematics. Those among us who know by heart the multiples of nine can easily recognize that the sum of the digits for any of these multiples is nine. We can easily see this because we know that the multiples of nine are 9, 18, 27, 36, 45, 54, 63, 72, 81 and 90. In fact, we could even see that when the multiples are arranged in increasing order (like the one above), as the digit in the tens' place goes up by one, the digits in the ones' place goes down by one. Perhaps, we can even explain why the sum of the digits in any of these multiples is nine. Any one of these multiples can be described by (x - 1) in the tens place and (10 - x) in the ones places, where the multiple is 9x. So when we add the digits, we are in fact adding  (x - 1) to (10 - x), which of course sums up to 9. One reason why this may seem easy and insightful for us is the fact that our minds are no longer bothered by figuring out what the multiples of nine are. Arithmetic have already been permanently stored in our minds therefore neither adding nor multiplying require anything from our working memory. Having these procedures done almost mindlessly, our minds' working memory can therefore be devoted to other higher and more complicated tasks. Teaching math therefore requires that a teacher be tuned to the needs of each and every student. With a changing curriculum that emphasizes higher thinking, parents may need to ensure first that their children already have a strong background in basic mathematics before starting school. Of course, teachers may also need to work on helping their students master arithmetic first before going into a deeper discussion of how arithmetic works.






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