The Language of Mathematics
Along the learning curve, some students in Singapore have learnt how to do the various operations of addition, subtraction, multiplication and division, but have no idea what they really mean. Hence, when they come across word problems, they get stuck. Not only that, but it is amazing how many times I have tried to explain concepts to the students, to find out that their understanding of Mathematics concept is seriously weak.
I showed the students 2 apples and 2 oranges. I told them that an apple and an orange had been "taken away", and asked them what I must do. Those students weak in concept actually suggested a division, telling me that they should "divide" the 2 apples and 2 oranges by 2. While the end result was the same, the fact that the concept was wrong spells trouble, and I had to throw in the curveball by asking them the result were I to have THREE apples and 2 oranges instead? The stumped look told me a lot.
I had to rephrase, by asking the students what would happen were they to have $50 and I "take away" $20? This time more of them understood, and I took the opportunity to let them know that when we want to "take away" or "remove" something, we use the Mathematical operation of subtraction. So when I "take away" an apple and an orange, we should use subtraction, not division.
Another concept that often got confused is that of multiplication and division. When three boys have a total of 15 apples, how many would each boy have if they shared them equally*? Some students actually did a multiplication operation and gave me 45 apples altogether. I have to actually show them that when we want to do "equal sharing", we are doing a division operation, by drawing pictures of apples and circling them on the board.
As I reflect on the situation, I can only say that Asian children are trained very well to do the operations quickly and accurately (unlike many of their western counterparts). When it comes to deep thinking, and associating the concepts behind the operations, Asian children appear to be somewhat weaker in general. I believe this could be the result of harried teachers, trying to complete an extremely tight syllabus, before a class of 40+ children, in time to prep and cram the students for their exams.
It can be really difficult for a teacher to really find the time to do all these, and yet have a life beyond the classroom.
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*I am very insistent on Mathematical questions having this part "shared equally" because it is NOT wrong for a student to indicate that the three boys have 10, 4 and 1 apples respectively if they share them, if the "equally" is missed out.
I showed the students 2 apples and 2 oranges. I told them that an apple and an orange had been "taken away", and asked them what I must do. Those students weak in concept actually suggested a division, telling me that they should "divide" the 2 apples and 2 oranges by 2. While the end result was the same, the fact that the concept was wrong spells trouble, and I had to throw in the curveball by asking them the result were I to have THREE apples and 2 oranges instead? The stumped look told me a lot.
I had to rephrase, by asking the students what would happen were they to have $50 and I "take away" $20? This time more of them understood, and I took the opportunity to let them know that when we want to "take away" or "remove" something, we use the Mathematical operation of subtraction. So when I "take away" an apple and an orange, we should use subtraction, not division.
Another concept that often got confused is that of multiplication and division. When three boys have a total of 15 apples, how many would each boy have if they shared them equally*? Some students actually did a multiplication operation and gave me 45 apples altogether. I have to actually show them that when we want to do "equal sharing", we are doing a division operation, by drawing pictures of apples and circling them on the board.
As I reflect on the situation, I can only say that Asian children are trained very well to do the operations quickly and accurately (unlike many of their western counterparts). When it comes to deep thinking, and associating the concepts behind the operations, Asian children appear to be somewhat weaker in general. I believe this could be the result of harried teachers, trying to complete an extremely tight syllabus, before a class of 40+ children, in time to prep and cram the students for their exams.
It can be really difficult for a teacher to really find the time to do all these, and yet have a life beyond the classroom.
------------
*I am very insistent on Mathematical questions having this part "shared equally" because it is NOT wrong for a student to indicate that the three boys have 10, 4 and 1 apples respectively if they share them, if the "equally" is missed out.
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