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.

Dr. Anna Wilson lists confirmed, likely, and unlikely symptoms of dyscalculia

from Dr. Wilson's website:

Symptoms established by research

The following are seen in primary school, and well established by educational researchers:

1. Delay in counting. Five to seven year-old dyscalculic children show less understanding of basic counting principles than their peers (e.g. that it doesn't matter which order objects are counted in). [1-3]

2. Delay in using counting strategies for addition. Dyscalculic children tend to keep using inefficient strategies for calculating addition facts much longer than their peers. [2, 4, 5]

3. Difficulties in memorizing arithmetic facts. Dyscalculic children have great difficulty in memorizing simple addition, subtraction and multiplication facts (eg. 5 + 4 = 9), and this difficulty persists up to at least the age of thirteen. [6-10]

These symptoms may be caused by two more fundamental difficulties, although more research is needed to be sure:

1. Lack of “number sense”. Dyscalculic children may have a fundamental difficulty in understanding quantity. [11, 12] They are slower at even very simple quantity tasks such as comparing two numbers (which is bigger, 7 or 9?), and saying how many there are for groups of 1-3 objects. The brain areas which appear to be affected in dyscalculia are areas which are specialised to represent quantity.

2. Less automatic processing of written numbers. In most of us, reading the symbol "7" immediately causes our sense of quantity to be accessed. In dyscalculic individuals this access appears to be slower and more effortful. [13-15]. Thus dyscalculic children may have difficulty in linking written or spoken numbers to the idea of quantity.

Other symptoms

If you have read other websites on dyscalculia you may have seen quite a few other symptoms listed. Many of these are not yet proved to be symptoms (although this does not mean they might not be later on). This is because they have been reported by teachers or special education workers, but haven't yet been studied in detail by researchers. Based on my knowledge of dyscalculia and cognition I have listed likely and unlikely symptoms below.

The following are likely to be symptoms of dyscalculia:

1. Difficulty imagining a mental number line

2. Particular difficulty with subtraction

3. Difficulty using finger counting (slow, inaccurate, unable to immediately recognise finger configurations)

4. Difficulty decomposing numbers (e.g. recognizing that 10 is made up of 4 and 6)

5. Difficulty understanding place value

6. Trouble learning and understanding reasoning methods and multi-step calculation procedures

7. Anxiety about or negative attitude towards maths (caused by the dyscalculia!)

All these symptoms (bar the last) are related to quantity.

The following may sometimes be ASSOCIATED with dyscalculia, but not in all cases:

1. Dyslexia, or difficulty reading

2. Attentional difficulties

3. Spatial difficulties (not good at drawing, visualisation, remembering arrangements of objects, understanding time/direction)

4. Short term memory difficulties (the literature on the relation between these and dyscalculia is very controversial)

5. Poor coordination of movement (dyspraxia)

The following are NOT likely to be symptoms of dyscalculia:

1. Reversals of numbers - this is a normal developmental stage which all children go through and is no cause for alarm in itself

2. Difficulty remembering names - no evidence to suggest that long term verbal memory has anything to do with dyscalculia

I wonder what she means by "inefficient strategies for calculating addition facts."

update: I think she means that children with dyscalculia continue to use counting-all to perform simple addition after their peers have switched to counting-on.

dyscalculia resources from Dehaene

Dehaene includes these resources in The Number Sense:

Dehaene Laboratory

Developing Number Sense

Number Race Software -- free download and complete access to sourceware

About Dyscalculia (written by Dr. Anna Wilson)

Center for Educational Neuroscience

Dyscalculia vs Lack of Math Facts

The other day someone emailed me about a new word he'd learned:

Dyscalculia

(He clarified that he didn't think I had it, he just wanted me to know that it existed.)

I have been starting to wonder why it is so difficult for me to improve my math SAT score though.....

.....Which lead me to Daniel Willingham's article about whether or not it's true that some people are "bad at math." After reading it, I believe I fall squarely in the "lack of automaticity" category, rather than "dyscalculic."

Headed to Kumon this summer with my son to brush up on math facts so that I can free up some working memory for the fall SAT. Surprisingly (miraculously?), my son is psyched; daughter is another story.

(Cross Posted on Perfect Score Project.)

What is new with the science on math disabilities?

Wednesday, December 02, 2009

Atypical numerical cognition, dyscalculia, math LD: Special issue of Cognitive Development

A special issue of the journal Cognitive Development spotlights state-of-the-art research in atypical development of numerical cognition, dyscalculia, and/or math learning disabilities.

Article titles and abstracts are available at Kevin McGrew's excellent IQ's Corner blog.



-----------
Joe Elliot on dyslexia:

"Contrary to claims of ‘miracle cures’, there is no sound, widely-accepted body of scientific work that has shown that there exists any particular teaching approach more appropriate for ‘dyslexic’ children than for other poor readers."

I am in agreement with Elliot.

I wonder if the same will be found to be true for dyscalculia and kids who struggle with math.

Dyscalculia: What Is It? International Dyscalculia Awareness Day

Dyslexia is a nickname for "Specific Learning Disability--Reading", listed as 315.0 in the Diagnostic and Statistical Manual-IV (DSM-IV). Most people understand that dyslexia is a persistent difficulty with reading, despite good instruction and at least average intelligence.

There is also a Specific Learning Disability -- Mathematics (315.1, DSM-IV), known as dyscalculia.

The Dyscalculia Forum chose today as the day to raise awareness.

According to the National Center on Learning Disabilities (NCLD):

Dyscalculia is a term referring to a wide range of life-long learning disabilities involving math. There is no single form of math disability, and difficulties vary from person to person and affect people differently in school and throughout life.

Here's an index page on math and LDs from NCLD. Here's LDOnline's index page on math disabilities. Here's an overview of math disabilities in children from SchwabLearning. Anna J. Wilson's Dyscalculia Primer and Resource Guide.

There are also a couple of good videos on Youtube: start with Dyscalculia: It Is Not Only Trouble With Math.