One feature of new-math logic involved sets, groupings of things. For example, in Set A, list all the four-legged objects in the living room.
A = chair, sofa, dog, table
In Set B, list all the animate objects in the living room:
B = dog, Aunt Jane
The intersection of Sets A and B is written as:
A ∩ B = dog
Sets are never added, subtracted, or multiplied; they are intersected, united, or complemented. The union of Sets A and B is written as:
A ∪ B = chair, sofa, dog, table, Aunt Jane
Teachers believed that by helping students understand collections of things, students could better understand the symbolism of numbers expressed as numerals.
Another important new-math feature was the attempt to explain how number systems worked. One way to make our decimal system clear is with expanded notation:
352 = (3 x 10 x 10) + (5 x 10) + (2x1)
There are 2 groups of ones, 5 groups of tens, and 3 groups of hundreds. Understanding this structure enables students to “regroup by tens.” Thus 943 + 729 becomes:
Students learned to add the “old way” also but were supposed to understand what they were manipulating.
Once the base-ten system was clear, students also were taught to manipulate and solve problems in different base systems. In the base-two system, each digit represents a group of units two times larger than the one to the right. How would you write the base-ten number 7 in base two?
New-math teachers also used number lines. Here’s how number lines were used to make a visible solution of the problem: What is 2/3 of ¾?
The student takes three of four segments, in this case three out of four segments of 7, or twenty-one out of 28, and marks them out along a line. By so doing, he “discovers” that two of these thirds, or two segments of 7, equal one-half of the full unit of twenty-eight. So 2/3 of ¾ equals ½. The segments of 3 below the number line show how 7 x 3 = 21.