kitchen table math, the sequel: counting-all vs counting-on

Saturday, August 27, 2011

counting-all vs counting-on

When first learning to solve simple arithmetic problems (e.g., 5 + 3), children typically rely on their knowledge of counting and the associated procedures (Siegler & Shrager, 1984). These procedures are sometimes executed with the aid of fingers (finger counting) and sometimes without them (verbal counting). The two most commonly used counting procedures are termed min (or counting-on) and sum (or counting-all; Fuson, 1982; Groen & Parkman, 1972). The min procedure involves stating the larger (max) addend and then counting a number of times equal to the value of the smaller (min) addend, such as counting 5, 6, 7, 8 to solve 5 + 3. The sum procedure involves counting both addends starting from 1. Occasionally, children state the value of the smaller addend and then count the larger addend (the max procedure). The development of procedural competencies reflects a gradual shift from frequent use of the sum and max procedures to frequent use of min counting.
Numerical and Arithmetical Cognition: A Longitudinal Study of Process and Concept Deficits in Children with Learning Disability
David C. Geary, Carmen O. Hamson, and Mary K. Hoard
Journal of Experimental Child Psychology 77, 236–263 (2000)
Why does academic language have to be so opaque?

Thank God for Google.

Here's what I think they mean.

When children first begin to add numbers, they rely on sum or counting all, which a Michigan State website defines thusly: "if given the problem 5 + 2, the student may count 5 on one hand and 2 on the other hand, and then count all fingers to get 7."

Then, as children progress, they switch to min, or counting on. With counting on, the child realizes he doesn't have to count the first number; he can start with the first number and start counting from there (hence counting on). So, if the child is adding 5 + 2, he starts with the number 5 and counts two more: "5, 6, 7."

Brian Butterworth describes it this way:
Counting on from first. Some children come to realise that it is not necessary to count the first addend. [When adding 3 + 5] they can start with three, and then count on another five to get the solution. Using finger counting, the child will no longer count out the first set, but start with the word ‘Three’, and then use a hand to count on the second addend: ‘Four, five, six, seven, eight’.
The development of arithmetical abilities
Brian Butterworth

Journal of Child Psychology and Psychiatry 46:1 (2005), pp 3–18
According to Butterworth, counting on from larger is stage 3:
Counting on from larger. It is more efficient, and less prone to error, when the smaller of the two addends is counted. The child now selects the larger number to start with: ‘Five’, and then carries on ‘Six, seven, eight’.
As to the timing, Butterworth writes: 
There is a marked shift to Stage 3 in the first six months of school (around 5–6 years in the US, where this study was conducted (Carpenter & Moser, 1982). Stage 3 shows a grasp of the fact that taking the addends in either order will give the same result. This may follow from an understanding of the effects of joining two sets, that is, taking the union of two disjoint sets.
This answers my question about Dr. Wilson's observation that children with dyscalculia use inefficient strategies for simple addition.


OrangeMath said...
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OrangeMath said...

Great post. My main evidence for dyscalculia is misplacement of fractions on a number line. Adding dyscalculia starts earlier and may have different causes; which would be interesting to know.

Catherine Johnson said...

That's interesting.

I'll see if I can glean what Dehaene has to say about fractions via a quick skim.

Anna Wilson has an interesting Powerpoint that includes a number line study she and Dehaene did, but I don't understand their findings just from the powerpoint slide.

title: Dyscalculia: Why do numbers make no sense to some people? Dr. Anna J. Wilson

Catherine Johnson said...

hmmm...skimming the book, it doesn't look like he has much to say about fractions per se

Catherine Johnson said...

That seems like a major missing topic to me.

Lsquared said...

Could I get a reference for the Brian Butterworth quotes?

Lsquared said...

Disregard my request please--the reference is there as always, I just missed it.