How do students learn math? Are there learning theories that are especially relevant to the teaching and learning of math?
Progress is occurring in providing answers to these sorts of questions. However, the prevailing teaching situation is summarized by the following quote:
For example, I've been in enough high school math classes over the last five years to know that there is no developmental theory of how students learn algebra. The kids who don't make it and don't respond to the kind of instruction they're receiving are simply not included in the instructional model. And teachers in the classrooms I've observed take no responsibility for the lowest-performing students. That's because the prevailing a theory of learning suggests that teaching mathematics is not a developmental problem but a problem of aptitude. Some people get it, some don't.Elmore, Richard F. (2002). The Limits of Change [Online]. Accessed 2/18/02: http://www.edletter.org/current/
This quotation captures the essence of a need for understanding of mathematics developmental theory and a need for understanding of learning theories appropriate to the teaching and learning of math. Both are missing in many math education environments.
There are many different learning theories. For many years, the behaviorist theory of B. F. Skinner was dominant. In more recent years, a number of new theories have been developed.
Some are called cognitive learning theories, because they take into consideration the conscious thinking abilities of a human being. These theories posit that human learners are much more than pigeons and rats where a stimulus/response approach can be used to condition certain behaviors.
Much of this workshop is built on constructivism. This is a learning theory that says that people build (construct) new knowledge upon their previous knowledge. In recent years, there has been considerable research that supports constructivism. It is a theory that can help guide curriculum, instruction, and assessment across all disciplines covered in our formal educational system. It is particularly applicable in mathematics education.
Humans and many other animals have a modest amount of innate ability to deal with numbers. Many different species can perceive the difference between two small numbers -- such as three offspring are present versus only one is present. But, the human innate capacity to count -- 1, 2, 3, 4, many -- is certainly limited relative to needs in our contemporary society. Thus, throughout recorded human history we find evidence of humans developing aids to the innate mathematical abilities of their brains. A baboon bone with 29 incised notches has been dated at 37,000 years old. A 20,000 year old bone has been discovered that has 11 groups of five marks incised on it. More recent example include the counting board, abacus, math tables, mechanical calculators, logarithms, electrical and electronic calculators, and electronic digital computers.
Mathematics is much more than counting and simple arithmetic. It is a cumulative science in which new results are built upon and depend on earlier results. We have a 5,000 year history of formal mathematical development. Humans have accumulated (discovered, developed) a huge amount of mathematical knowledge -- far far more than a person can learn in a lifetime, even if the person spent all of their time studying mathematics.
During these 5,000 years we have developed many aids to learning and "doing" (using, applying) mathematics. Thus, our educational system is faced by:
Howard Gardner has identified Logical/mathematical as one of the eight (or more) intelligences that people have. As with the other intelligences in Gardner's classification system, people vary considerably in the innate levels of mathematical intelligence that they are born with.
People like to argue nature versus nurture in terms of both general intelligence and intelligence within specific domains such as those that Gardner lists. We know that the brain has great plasticity, that there is a lot of brain growth after a person is born, that the brain continues to grow new neurons and new connections among neurons throughout life, that certain drugs can damage brain cells, that proper nutrition is needed for proper brain growth, and so on.
A certain amount of math knowledge and skill is innate--genetic in origin. The great majority of a person's math knowledge and skills comes from learning--learning to use parts of the brain that can learn to do math, but were not genetically designed specifically for this purpose.
Math is a cumulative, vertically structured discipline. One learns math by building on the math that one has previously learned. That, of course, sounds like Constructivism.
In brief summary, here is a constructivist approach to thinking about mathematics education.
It is interesting to note that many researchers and practitioners in ICT have come to the same conclusion about teaching and learning ICT. They recommend a constructivist approach.
Journaling, Project-based Learning, and Problem-based Learning are all standard components of a constructivist teaching/learning environment. (Note that both project-based and problem-based learning are abbreviated PBL.) One of the strands of this workshop is devoted to ICT-Assisted PBL
Situated Learning is emerging as a learning theory that is particularly relevant to teaching. Thus, this topic needs to be presented in some detail here. My current bibliography on the topic is given at:
Situated learning tends to have characteristics of Project-Based Learning and Problem-Based Learning. It also appears to tie in closely with general ideas of Problem Solving. Thus, in Problem Solving we talk about domain specificity and domain independence. The argument is that one needs a lot of domain specific knowledge to solve problems within a domain. Situated Learning tends to be within a domain (a situation). Thus one might call it a Domain Specific Learning Theory.