In my last blog entry, I described three goals suggested by Dr. Les Steffe which support introducing subtraction as take away. Yet, there is a belief among some math teachers that thinking of subtraction as take away interferes with future mathematical development. They argue that using the words “take away” should be eliminated completely from elementary mathematics, in favor of using words such as “the difference between” or “distance between.”
For children in the early numeric stage of counting, take away subtraction offers opportunities for developing mathematical sophistication. The early numeric stage of counting occurs when children construct an initial number sequence (INS). With this construction they can use previous experiences of counting to generate current counting experiences without actually counting. An example of this is when a they are given an addition situation such as “What is 8 and 5 more?” The child may say “eight” and then count on five more saying 9-10-11-12-13. In saying “eight,” the child uses previous experiences of counting from 1 to 8, eliminating their need to actually count from 1 to 8.
Introducing take away situations during this stage gives students opportunities to move forward and backward along their number sequence, allowing for increased sophistication of their understanding of number. As an example, we may present the situation 23 – 5. For the child using the INS, 23 represents the acts of counting 1 to 23, without needing to count. Subtracting five from 23 may be done through the removal of 5 counts from 23 by counting backwards. This may prove difficult. Adults can experience the same difficulty if asked to start with the letter P and work backwards five letters along the alphabet sequence. For many this requires going forward through the alphabet first, then using that recent experience to move backwards. In addition, saying the previous five letters requires a simultaneous monitoring of how many letters have been spoken while determining the next letter. Similarly, teachers may ask the child to count out 23 objects to begin the problem. The counting gives them a recent forward counting experience to use when counting backwards. The removing of counting acts can, at times, be heard in the child’s explanation.
Dan, a 3rd grade student, explained his thinking this way, “I counted backwards in my head. I counted five 1’s backwards.” He then continued, “Since there were 23 (he extends thumb), I took out one and there’s 22 (extends pointer finger), then I took out another one 21 (middle finger), and then 20, (ring finger), and then 19 ( pinky finger) and I have 18 left.”
A child in the early numeric stage has not constructed an understanding of number which would allow them to make sense of subtraction as a “difference between” or “distance between.” Providing take away experiences in subtraction increases children’s understanding of number. By increasing the understanding of number, “the difference between” or “distance between” can eventually be appropriate for them. Join me February 27 as I look deeper into Dan’s ways of thinking about subtraction.