The APOS model proposed by Dubinsky is a way of using a Piaget-like concrete to abstract model for math learning at all levels, and in particular higher levels (high school and college levels). Piaget's original work was focused on how children develop, and doesn't make any distinctions for students working at higher levels, but the notion of working from concrete to abstract ways of thinking is so powerful that many people, including Dubinsky, find it a useful lens for considering all levels of learning. Dubinsky's specific ways of breaking down what it means to be concrete and abstract are so useful that I'm going to be modifying it to talk about fine distinctions about why some ways of looking at math at the early childhood level are more concrete or more abstract.
The APOS model has four parts, of which I will primarily be using two. I'll use counting and number as my example to explain what each of the four parts mean:
A....Action. An action way of looking at number is rote counting. I count to get a number, so the number 8 means the process of counting to 8.This is the first way that children learn about counting. Things will get a little fuzzy in a minute because there are two ways to get at the idea of number, and only one of them is counting. Counting is, however, the first way children get to the idea of a number as big as 8.
P....Process. A process way of looking at number is a bit more abstract than an action way of looking at a number. If I have a process understanding of the number 8, then I know 8 is a number, and I could figure out how big it was by counting to 8. I don't have to actually do the counting or lay out 8 things to think about the number 8--if you tell me you have 8 kittens I know that I could count to figure out exactly how many 8 is, and so I can talk and think about the 8 kittens without doing the counting.
I'm going to kind of smoosh these two categories together and refer to them as a process understanding of number. That means the the basic understanding of number is that it is the result of an process (counting).
O....Object. An object way of looking at number is when you can think about 8 without thinking about how you would count to get eight. You can compare 8 and 12 and k now which is bigger without counting either of them; you can add or subtract (or at least imagine adding or subtracting) numbers without imagining a counting process that would lead you to the answers. If I have an object understanding of 8 then I know that 8 tells how many in a group without any counting process being attached to that knowledge. With a process understanding of 8 kittens I would imagine myself counting the kittens; with an object understanding of 8 kittens, I am imagining the number of kittens without any thought that I would be likely to count them.
I'm going to mostly focus on the distinction between process and object understandings of math ideas. Being able to move past thinking of a number as a result of doing something (counting) to something that just exists (an amount of something) is a really key step.There's a big jump in the sorts of problems you can solve and the solution strategies you can start using when you move from a process to an object understanding.
S....Schema. A schema way of looking at number is to fit numbers into systems and find connections between the numbers. So, if I have a schema understanding of numbers up to 10, when presented with the number 8, I would have at my mental fingertips that 8 is two fours, and 8 is 5+3, and that 8 is two less than ten.
Schema are good, schema are important (schema is the plural as well as the singular, I believe), but I'm not going to use the word schema very much because in math education we mostly already have words that describe this.
In Dubinsky's theory of math learning, students pass through these steps in learning about each new mathematical object, so there are action, process, object and schema ways of looking at number, addition, subtraction, multiplication, division, fractions and almost everything you might learn about in math. Students generally have to pass through each of the earlier stages before they can reach the final stages. His theory of math learning is going to be important to us in this class because it will give us a vocabulary to discuss some of the fine distinctions in children's understandings at different stages.
Key vocabulary:
Process: A process is an action that you perform. A process understanding of a math idea means that you understand how to perform the process that results in that thing. For example, the process that results in a (whole) number is counting.
Object: An object is something that you think about as a single encapsulated thing. An object understanding of a math idea means that you understand the thing without having to do or to imagine an associated process. An object understanding of a number means that the number takes a single chunk of your mental space, and doesn't require you to consider an associated action.