This is a topic many of us reflect on in regard to our students. There are times we think they have it and should be able to apply the procedural knowledge to a similar task and they can't. One place I see this is in adding numerical fractions where students have to make sure they have a common denominator to do this. They learn it but when they apply the same principal to adding algebraic fractions, they have no idea what to do.
So what is going on?. According to an article I read there are several reasons students have difficulty with transference.
1. Initial learning is necessary for transfer but it is not known about the type of learning needed to promote transference.
2. Abstract representations of knowledge helps improve transference.
3. Transference is considered to be an active, dynamic process.
4. All new learning is based on transfer of previous learning.
It now appears that transference is based on the degree to which people understand the material rather than memorizing procedures and facts. In addition, it takes a lot of time to really learn the material, more than we usually schedule in class for any one topic. It has been found that students need to know the when, where, and why to use the new knowledge. In other words, build a connection.
According to another study done to see if students could transfer the mathematical processes to other subjects, it was found students had a difficult time because it required them to translate a problem stated in words into a math problem. A second study claims the reason students are unable to transfer their learning from high school over to work because they do not have enough authentic based and project based learning.
This leads me to believe there are two types of transference:
1. Transference of a skill from a simple problem to a more complex one such as requiring a common denominator for regular fractions and algebraic fractions.
2. Transference of the skill from the classroom to other subjects or even to work.
In addition, there are two types of transference which might explain why I see the two versions. The first type is near transfer where students are able to apply their knowledge to problems and situations similar to what they learned. The other is far transfer where students apply their knowledge to a situation quite different from the context they learned it in.
It is noted that schools offer more opportunities for near transfer. Students find far transfer more difficult because they have to seriously consider the situation in order to remember the rules and concepts needed to solve the new problem.
Furthermore, there are three factors involved in transference, the person's representation of the problem, their background experiences, and their understanding of the problem. Representation refers to how the person mentally solves the problem which is related to their knowledge of the content of the problem while experiences is often based on prior knowledge and understanding is linked to representation.
This gives us some understanding on why many of our students are unable to transfer math as easily as we think they should. I know my eyes have been opened. Let me know what you think!