Introduction
For the past few years, I have literally stood on my head to teach my fourth grade bilingual students that one is unable to subtract a larger number from a smaller number in the positive number system. Many of my students, when given a problem such as the following, will simply arrange the numbers to fit a pattern and arithmetic operation with which they are comfortable: Siggy has 54 crayons. His brother Ziggy used some of the crayons for a school art project. Now Siggy has 29 crayons. How many crayons did Ziggy use for his art project? This is a Separate problem where the result is known and the change is unknown, frequently referred to as a Missing Addend problem.
The students are unsure of what they are trying to find out. It seems they consider 29 and 54 as interchangeable in terms of minuend and subtrahend. Therefore, some children will make this problem: 29-54=35.
They are having at least two difficulties here. First of all, they do not know how to decompose tens, so they subtract the four ones from the nine ones and get five, and then, realizing perhaps that they can't subtract five tens from two tens, they simply flip the numbers around, subtract two from five and call it a day. It was this baffling (to me) maneuver that inspired me to stand on my head.
Secondly, the children don't understand the idea of missing addends, or change unknown. They do not realize that they are working with a problem where the result is known; hence the decision to make one number the minuend and another number the subtrahend, in a sort of "whatever works" spirit. On the positive side, they do understand that the problem is best solved by subtraction, although some students will make the problem 54+29= 83, because carrying or composing a ten is an easier operation than decomposing a ten.
For all of the above reasons, I chose this very basic and fundamental mathematical operation of subtraction to be the focus of this curriculum unit. I want to explore the "cousin" relationship between addition and subtraction because it is the foundation for understanding inverse relationships in arithmetic, and because it is a metaphor my students well understand.
Many of my students are Mexican immigrants and their schooling has been uneven and choppy with gaps of time when families are moving around and the children do not attend school. Therefore, there are also gaps in their mathematical knowledge. It is my desire to fill in those gaps, and subtraction is a good place to start. With a solid background in understanding the process of subtraction, students will be able to move forward into the more complicated operations of multiplication and division. They will attain this understanding by thoroughly exploring the subtraction process and by beginning this unit with an in-depth study of the decimal, i.e. base-ten system.
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