How do children progress from counting to addition?
I understand that an grasp of counting is achieved before an understanding of cardinality (size of a set of objects). Thus, there does seem to be some division between procedural understanding and conceptual understanding of counting. However, how does this carry on to addition? Is addition also typically learned procedurally rather than conceptually?
Yes, addition becomes procedural knowledge. Below is an extract from the book "Smart Thinking by Art Markman", that clearly explains the process :
When you are learning to do addition, the two procedures you have for adding compete with each other. One of those procedures requires some effort. You start with the bigger number and then count up. Adding two and four means starting with four and then counting to five and six. The other procedure is effortless. You try to remember the answer. If you finish counting before you pull up an answer from memory, then the counting procedure wins. If you are confident you have pulled up the right answer from memory before you finish counting, then the habit wins. After you solve the problem (by either method), you store another memory that 2 + 4 = 6. So, each attempt at an addition problem provides memories that will make it faster for you to remember the correct answer in the future.
The difficulty with math is there are lots of similar facts. You are learning 2 + 4 = 6, but at the same time, you are also encountering problems like 2 + 7 = 9 and 2 + 5 = 7. Sometimes, when you see 2 + 4, you will also recall some of those similar problems. When you retrieve these conflicting answers, you are going to be uncertain about which answer is correct. So you will finish carrying out your counting procedure before you have an answer from memory. Once you have a lot of examples of addition problems in your memory, most of what you pull out of memory when you see 2 + 4 will be other situations in which you also saw 2 + 4. At that point, you retrieve information from memory faster than you can count, and so you have a habit.
Chapter references:
Logan, G. D. (1988). Toward an Instance Theory of Automaticity. Psychological Review 95: 492–527.
Schneider, W., and Shiffrin, R. M. (1977). Controlled and Automatic Human Information Processing: 1. Detection, Search, and Attention. Psychological Review 84 (1): 1–66.
Shiffrin, R. M., and Schneider, W. (1977). Controlled and Automatic Human Information Processing: 2. Perceptual Learning, Automatic Attending, and a General Theory. Psychological Review 84: 127–190.
The two competing systems mentioned above, are actually system 1 and system 2 that are explained in detail by Daniel Kahneman, in his book Thinking, fast and slow.
The work of Siegler in "The Perils of Averaging Data Over Strategies: An Example from Children's Addition", examines 5 different strategies associated with addition:
In the table below, which I've copied from the paper, you can see the progression of strategies from grade to grade.
There does seem to be a progression towards retrieval, decomposition and min-count, away from count-all and guessing. However, it is not a completely straight-forward progression.
