Development of Addition and Subtraction Understanding

Children that are not taught algorithms become better at mathematics. Those that are taught algorithms rarely use strategies such as: Making nice numbers (368 + 204 = 368 + 200 + 4 = 568 + 4=572). Keeping the whole (71 - 36 = {subtract 1 from both to get} 70 - 35 = 35) (342 - 37 = {add 3 from both to get} 345 - 40). Or other strategies that would be more efficient depending on the value of the numbers to be added or subtracted.

Many critiques of this type of mathematics have used emotional propaganda techniques by saying that the new mathematics is soft or is dumbing down mathematics instruction. However, the mindless use of algorithms is the real dumbing down.

With a good teacher students can learn a variety of strategies as well as an algorithm. For example, if students are given the following problem: "I went to the store with \$32.00 and spent \$17.00, how much do I have left?" Most will draw tallies and cross out to arrive at the answer. Later, as children are given the opportunity to invent algorithms they usually split the numbers into place value and develop not only usable efficient algorithms but an understanding to go with them.

The following problem (7 + 52 + 186) was given to different groups of second graders and 45% of the students solved the problem without using an algorithm, 26% used part of an algorithm, and 12% used an algorithm. The following problem (504 - 306) was given to different groups of students, some that were taught addition and subtraction with an algorithm and some that were taught without an algorithm. 74% of the second and 80% of the third graders taught without algorithms got the right answer compared to 42% of the second and 35% of the third graders taught with an algorithm.

How do you get students to develop efficient strategies? Students who were taught relationships in which automaticity was the goal significantly outperformed the traditionally taught students in being able to produce correct answers to basic addition facts within three seconds. 76% to 55%. (Research from Fosnot…)