Why is subtraction hard? This question can be heard from many young children, but often even from adults. Whenever adults do mental math, they tend to have an easier time with addition, multiplication, and simple division than with subtraction. For the past three months we have been engaging young children in subtraction situations while considering this question.
Subtraction problems in first grade are often phrased as “taking away.” For example, if I have 16 cookies and 6 are given to my Aunt Mary, how many cookies do I have now? In this situation, children are often instructed to take 6 away from 16 by counting backwards. From an adult perspective, this might seem the easiest way to think of the problem, but for young children this can be very confusing. Thinking in terms of takeaway means that you have to be able to count backwards while remembering the number of times you count backwards. So, what are the benefits of presenting subtraction situations as takeaway?
Dr. Les Steffe in PSSM From a Constructivist Perspective (2003) states there are “good reasons for developing very particular learning situations involving take away subtraction.” One goal he states for presenting subtraction as “takeaway” is that children can begin to use their forward counting acts as items to count. For example, after counting forward 1, 2, 3, 4, …16 they may then use these counting acts as the material for which to count backwards, starting from 16 and continuing 15, 14, 13, 12, 11. A second goal Dr. Steffe states is to encourage the uniting of the counting acts together into a composite unit. If done, these six counting acts may become a unit the child can consider a single unit which contains the six counting acts. A third goal in presenting takeaway situations is that the child may begin double counting. Double counting as they count backwards to solve the problem might sound like “sixteen is one, fifteen is two, fourteen is three…eleven is six. There are ten left!” Double counting is an indicator that the child has interiorized their number word sequence.
These goals, along with the fostering of strategic reasoning, need to be emphasized for early numeric children when working with subtraction situations. In doing so, they may use their current understanding of number to develop a more sophisticated understanding. It is not appropriate to teach these children standard subtraction algorithm, an abstract process, in isolation. It robs them of understanding the concept of subtraction and the opportunity to make sense of number.
Join me in future blog posts as I discuss things we’ve learned from our observations of numeric students in subtraction situations. I will highlight ways that we have explored to make subtraction meaningful and foster strategic reasoning. In doing so our goal was not to train children in subtraction but rather to encourage progress in the construction of their understanding of number.