Author: J.P. Das
E mail: j.p.das@ualberta.ca
All of us who have tried our hand in helping our children, and grand-children with math have experienced how frustrating it is not to be able to use procedures out of the context that was provided by the teacher.
‘No, grandpa, that’s not the way my teacher told me how to do this’
‘Tell me if in this problem, I should use subtraction or division, then I can do it’
Examples like the above illustrate the narrowness of math learning procedures without understanding.
From the teachers’ point of view, “One of the complaints of mathematics teachers is that students often do not know which mathematical technique to use in a new situation...On the other hand, some critiques say that Math should not be taught by posing questions to answer — ‘Class, tell me what is 25 divided by 5?’.
Mathematical concepts are based on foundations of actual experience before teaching math is over burdened with teaching numerical operations. The numbers are dead unless these come alive through relating numbers to a child’s present behavior.
Nunes and Bryant discuss the development of math concepts from preschool through schooling, stressing the importance of building upon children’s knowledge before they receive formal instruction on measurement. Then they can use it appropriately
“If mathematical problem-solving is always used in the classroom as a way of practicing a procedure just taught by the teacher, the social definition of mathematics becomes the use of school-taught routines” (Nunes & Bryant, 1996, p. 247).
Let us begin by presenting some theoretical assumptions: Learning math is concerned with two basic concepts, magnitude or size, and value. Math also demands two processes; one is the procedure, which is a step by step thinking out of an arithmetic problem (divide 26 by 3) and the other is conceptualizing or comprehending the problem. We will explain each one in the following section. Besides the basics of magnitude and value, and the two processes that help math, working memory (WM) is a major cognitive process that is essential for math/arithmetic. Those who are weak in making out the difference between size and value or are weak in WM need to be assisted. They need a programme, Booster Math.
About Size or Magnitude, and Value
Magnitude is size, bulk, and quantity: We express magnitude comparatively by terms such as big, medium, little, large, and small. Each of these words indicates magnitude: A big object is larger than a small object. Even an infant has a sense of magnitude, knowing that a big object can hide a small one behind it. If this is not a modular concept present at birth, there is at least a blueprint that exists to start with that is fed by experience.
Even some animals notice size.
A white rat is in a maze with two arms, like a T junction. He has grown up in the lab and knows how to run through the different arms of a maze. He is hungry. As he reaches the junction, he can clearly see that at the end of one arm there is a huge bulk of rat-food. On the other arm there is a smaller bulk, but it is quite enough for him to eat. Where does he go? Runs to the big bulk of food. He is given just two minutes to eat before he is picked up by the examiner—usually, he eats enough in 2 minutes time. Next time he is placed in the same T-maze, the big bulk of food is at the other arm. He notices this and runs through the arm that has the larger bulk.
The infant held on mother’s lap looks at a heap of toys on the left side of his visual field, and a visibly smaller heap of same toys on the right side. The examiner has a camera set up to monitor to track his gaze; where does his gaze gets fixed? and for how long?
In a few trials, we find that the infant looks at the larger heap. If at all the infant looks at the smaller heap in some trials, the gaze is for a significantly shorter duration.
So, the sense of size may be inborn, or preference for the bigger size appears very early.
Value?
The size of the font of a small number, say 3, may be shown in a large font printed on a card, whereas number 9 is printed in a much smaller font. When a child is in Grade2, is asked to choose the card with a big number, he/she picks 9 notwithstanding the font size. Number 9 is always bigger than number 3—bigger in terms of value, whereas a kindergarten kid who hasn’t yet learnt the value of numbers may point to the bigger font.
Value is a concept that needs to be learnt.
We will see a group of slides where the idea is clearly presented--items are shown in a gradually smaller to bigger heaps. Even a kindergarten kid who has not learnt as yet of the value of numbers is asked to arrange 4 piles of shapes from the smallest to the largest pile, usually may not make mistake. The kid is on a boundary limit separating size from value.
Click HERE to access the slide show.