## Wednesday, 23 February 2011

I'm sure most middle school teachers have encountered mistakes like the one in the picture above due to students using an algorithm to replace understanding subtraction.
Many years ago I became alarmed when I saw it happening frequently in my Alg II classes on a test question that appeared in a testing program used at the school... it wasn't in my syllabus, but still... these are bright kids. I had always been aware of a few students who were dependent on their calculators for very trivial questions, but in following up on the particular subtraction issue, I became aware of how many of my students had almost no skills in what I call "mental math".

Now in my history I had been a regional director for the math council, and had talked with elementary teachers at conferences and in school visits about many areas of math outside my personal teaching experience, and on a few occasions when it was requested, I laid out a suggested topic map for units on teaching subtraction by the method commonly called adding-on using an idea I called number-roads. (not original, but I don't know from whom I claimed it)

In essence the idea was to develop facility with tens compliments and hundreds compliments as part of the pre-development and then use the "roads" to allow something similar to skip-counting, but with variable skipping, to get the differences between numbers. In my high-school classes I dispensed with "number roads" and simply encouraged the use of "linking numbers" to pave the way for counting on.

A student given 307 - 128 would be encouraged to set up the problem with the numbers spaced apart more than is typical, and then put a few easy linking numbers in between. These would depend on the student, and would change as they got better at mental math, but might include 130, 150, 200, 300. They could then make little gap-markers between each pair (2 for the gap between 128 and 130... twenty more for 130 to 150...etc) and simply add the gaps...

When I met resistance by students who felt they were fully functional with conventional algorithms, I would make them solve a couple of problems with time or distance involving bases other than ten. If they handled these, I would assume they understood and used subtraction well enough to do well with whatever approach they used.

I was amazed at how many of the students became fond of challenging themselves with doing the problems mentally after awhile. We would do two or three problems in the review at the beginning of each class and they often became quite skilled. When we had guests and they wanted to show off I would pick a problem which particularly favored counting on, something like 1004 - 397, but it really became unnecessary as they quickly progressed.
Their interest gave me a beautiful lever when we got to working with polynomials. I would show that (10x+5)^2 allowed you to mentally calculate the square of numbers ending in five... and 38x42 could quickly be viewed as (40-2)(40+2)... We would continue to the end of the year with a few moments of mental drill each day, and the kids who returned for pre-calc and calculus seemed to retain the skills for the most part.

I don't teach Alg II anymore, and still mention mental math once in awhile in pre-calc, but I've never gone back to the regular teaching of mental math skills.

I wonder if counting-on is taught in the elementary grades at all, or how subtraction is approached.
I did find a UK educational site where the term "counting on" was used to describe an approach very much like what I have taught...and they even have an interactive Excel spreadsheet to teach the method.