I just read somewhere the question, "What was wrong with the old way?" That was followed by the statement that "Clearly the old way did not work". This was, I think, based on years of following what was going on with students and math.
I have my own experience from which to draw (my children did not need my help). I learned what subtraction was in my second grade classroom by placing 10 articles on a table, removing 3 of those 10 articles, and counting the 7 articles remaining on the table. 10 take away 3 equals 7. I was able to generalize that experience to cover what I was doing every time subtraction was required by a math problem. I just saw an example of "You paid \$20.00 for a \$3.18 purchase." Then it was said that most people go through some mental exercises to find out the change. What would be wrong with \$20.00 - \$3.18 = \$16.82. My first job, at age 14, was at S. S. Kresge Co. I counted back that change with "\$3.18 (handed 2 pennies) \$3.20 (handed 1 nickel) \$3.25 (handed 3 quarters - 1 at a time) \$3.50, \$3.75, \$4.00 (handed a dollar bill) \$5.00 (handed a five dollar bill) \$10.00 (handed a ten dollar bill) \$20.00.
I have observed that there are few who can count back change that way these days. What happened? If the power fails and the cash register doesn't do the math, they seem to be lost!