OECD Wrote:Do you encounter math difficulties in your students? .
As usual Dr. Wilson gives us much to think about. In this post I would like to focus on children, abstraction, metaphor, numbers, and learning. My suggestion is that the way that numbers, number processes, and teaching are presented to children create a confusion that obstructs the learning process.
I read the following in a students online site.
“I remember when I was your age being quite confused by the difference between cardinal numbers and ordinal numbers. Now I realize that it was probably because my teachers were confused as well!
When I was your age, the best explanation I got was something muddled like, "three" is a cardinal; "third" is an ordinal.
The reason for all the confusion is that the difference is a fairly deep subject in mathematics, but I'll try to explain it.â€
I believe that much of our thinking comes from metaphors that we acquire in life. The above indicates that these metaphors endure to the extent that they are not clarified even to educated adults. ( By the way the explanation that followed did little for me. The subject is “deep†in mathematics because our confusion puts it there. We place it in number theory rather than cognitive science where it belongs.)
The crux of the issue is that we are confused by abstractions of concepts and abstractions of processes. The confusion of concept is the meaning of “number†and the confusion of process is with those processes we use numbers for.
“Number†can mean many things. A number is a symbol. We can impose on these symbols the property of order which then leads to the concept of ordinal numbers. We can then say that the symbols need the properties that we require for counting and they then become cardinal numbers. If we then impose the properties that we require for measuring, i. e. a metric, we can then build things like the number line. All of these are mathematical abstracts of concept.
The problem is that for each concept there is associated at least one process. For the symbol there is language. For the ordinal number there is the process of ordering. For cardinal numbers there is the process of counting that leads to the processes of addition and subtraction that leads us into a huge morass ( a soft wet low-lying area of land that sinks under foot) of ideas.
As children develop in the real world they acquire many metaphors about numbers and their associated processes. I believe this is supported by the Geary reference
(
http://www.uth.tmc.edu/clinicalneuro/ins...Geary2.pdf )
and the following extracts from that report.
“Children's understanding of the principles associated with counting appears to emerge from a combination of inherent constraints and counting experience
inherent constraints five implicit principles.
1)one-to-one correspondence (one and only one word tag, e.g., one, two, is assigned to each counted object),
2)stable order (the order of the word tags must be invariant across counted sets),
3)cardinality (the value of the final word tag represents the quantity of items in the counted set),
4)abstraction (objects of any kind can be collected together and counted), and
5)order irrelevance (items within a given set can be tagged in any sequence).
In addition to these inherent constraints,
children make inductions about the basic characteristics of counting by observing standard counting behavior and associated outcomes “
(I will return to this idea of observation below)
“result in a belief that certain unessential features of counting are essential. These unessential features include
standard direction (counting must start at one of the endpoints of a set of objects) and
adjacency. The latter is the incorrect belief that items must be counted consecutively and from one contiguous item to the next that is, jumping around during the act of counting results in an incorrect count.â€
The above portrays a learning process that is confused and not entirely controlled. It is not cognitively well founded and the students become victimized by this environment and it appears that a part of this confusion comes from the education system itself. Geary states that “young children's conceptual understanding of counting is rather rigid and immature and is influenced by the observation of standard counting procedures.†I believe that the “observation†part of the statements includes a social aspect, i. e., external to the curriculum.
Geary then looks at reading disabilities (RD) and mathematics learning disabilities (MLD) and states “Many children with MLD/RD, however, did not understand Gelman and Gallistel's (1978) order irrelevance principle and believed that adjacency is an essential feature of counting. ... even in second grade, many children with MLD/RD and MLD only do not understand all counting principlesâ€.
My question to everyone is how much of this malady should be assigned to the child as MLD and how much should be assigned to social inadequacies and where is the curriculum is this morass? Is this a practical problem for brain scanning or cognitive neuropsychology?
One reference I use in these matters is the book “Where Mathematics Comes From†by George Lakoff and Rafael E. Nunez,