NEURO MATHEMATICS EDUCATION AND TECHNOLOGY

Alexander Vaninsky

Presentation at the Joint Mathematics Meeting (AMS, MAA, SIAM), Boston, January 4-7, 2012

ABSTRACT

Teaching and learning mathematics may be viewed as an interactive process of creation of specific domains in the human brain. The domains later act as mathematics knowledge centers. Learning mathematics may be regarded as a development of connections among them and other centers. Topology and the dynamical system of mathematics—related brain structuring are promising areas of research. We refer to this approach as Neuro Mathematics Education (NME) and focus on the role of technology. The NME approach allows for a new insight into mathematical abilities and paves the way for development of original teaching tools, strategies and techniques. In particular, it stresses the principal importance of eliminating mathematics anxiety—the main barrier to success in mathematics. Among the new tools for teaching mathematics are the active development of mathematical intuition, which is a skill of finding solutions to problems without following formal rules, using hypnopedia and hypnosis, and instruction delivery in the multifaceted interactive environment, to name a few. The goal of the NME is the creation of a positive mental environment for the perception and storage of mathematical information: concepts, notions, rules, techniques, etc. We present evidence that using technology contributes to the implementation of the NME in practice and has a positive impact on perception of mathematics and its applications.

PERCEPTION OF MATH

Mathematicians tend to underestimate the mental challenges related to learning mathematics.

Evidence:

“…In spite of trying a myriad of popular methods (modified Socratic, self- paced instruction, mastery learning, etc.), what I produced …was ineffective teaching. I was a good lecturer, enthusiastic about teaching, serious in my attempt to do it well, and I cared about my students. They liked me and my courses, but from everything I could see, they were not learning much more than students of other teachers, and that was woefully inadequate” (Dubinsky & Moses, 2011).

EXAMPLES

An opinion of a person who has grasped a concept:

“What is so hard about equivalence classes of ordered pairs of integers?” (Wu, 2011, re: teaching fractions).

An opinion of a person who has not:

“…As a young student of functional analysis, I had considerable difficulty with the idea of the duality of a locally convex space. I was fine with the notion of a linear functional that acted on elements …to produce numbers… But the idea of applying actions to these transformations, …equipping <them> with arithmetic and

…topologies, was really tough for me” (Dubinsky & Moses, 2011).

Pedagogically, there is no big difference between the two situations. The problem is in the difference in the individual perception by different people.

WHAT SHOULD BE DONE?

Generally speaking, we, the Instructors, should clearly understand the mental activities taking place in students’ brains and manage them appropriately to make our students successful mathematics learners. One of the ways to achieve this goal is to use a neuroscience approach to teaching mathematics. It was proposed, among other publications, in Laughbaum (2011) that we follow in this presentation. Our objective is to suggest some new ways of instruction delivery stemming from the neuroscience approach and to stress the role of technology. We refer to this approach as Neuro Mathematics Education (NME).

NEUROSCIENCE APPROACH TO TEACHING MATHEMATICS

The neuroscience approach to teaching considers the transfer of knowledge as having a direct impact on the human brain and the forming domains that act as knowledge centers. Using technology in a mathematics classroom helps the implementation of the teaching tools suggested by neuroscience. Technology contributes to the optimization of the process of accumulation of information in working memory and sets up the appropriate pace of its transfer from working to long-term memory. Technology helps to increase the intensity of knowledge transfer from instructor to student, and to avoid a congestion of the student’s working memory, leading to mathematics anxiety on one side, and the instructor feeling overloaded, on the other side.

GENERAL COMMENTS

As mentioned in Laughbaum (2011), the brain always tries to associate an additional portion of information to that already stored and related to the same or associated area of knowledge. The success of a learning process depends on:

• The state of development of the long-term memory;

• The capacity of working memory;

• The conductivity of the channels connecting working and long-term memory;

• The conductivity of the channels through which new information is delivered;

• Noise in the channels.

The multi-channel delivery of new information using verbal, visual, and spatial means simultaneously is preferable because total conductivity of a set of channels is greater than that of any particular one. This phenomenon is well-known in conventional pedagogy; neuroscience approach allows for its further elaboration.

REASONS OF UNSUCCESSFUL LEARNING

Neuroscience approach reveals some cases when any method of conventional teaching has been practically ineffectual. First is a case of a restricted capacity of working memory. Such students cannot comprehend concepts or notions presented in a classroom, and experience mathematics anxiety, according to Klinberg (2009). Insuciency of the working memory may be compensated by shorter lessons or limitations imposed on the amount of new material per lesson.

The second reason is a weak long-term memory. Students may be successful during a classroom period but unable to memorize procedures or strategies for a long period of time.

Teachers assigned teaching mathematics classes comprising students suffering this hidden incapacity will face poor performance and low achievement rates.

The third reason is general memory-related problems. Such situations should be recognized in a timely manner. Only specially designed measures including special education and using medications aimed at memory improvement can help.

The fourth reason is the high level of noise in the information channels. In such cases information cannot reach the brain in full because it is mixed in the channels with a lot of subject-unrelated noise. Most of its content is lost along the way.

PRACTICES TO AVOID

The main way to successful teaching mathematics is to avoid mathematics anxiety that blocks further perception. For example, any quiz suggested in the beginning of the class should aim not to upset students. Any assessment should demonstrate students’ progress, though possibly, very small. In our experiment, all quizzes were intentionally made very easy and were counted towards extra credit.

It is also important to avoid comparing publicly successful students with those who are still not on track.

It is essential to stress any progress in the mathematical study and give some small extra credit for any improvements.

HOW TECHNOLOGY CAN HELP?

What follows is an example of implementation of some elements of the neuroscience approach for teaching mathematics in a community college. It is based on using technology to arrange the educational process in a recursive way referred to as MARTA: the Multilevel Alternating Recursive Teaching – Assessment, Vaninsky (2010).

It comprises:

• Classroom or interactive video contact as a basic and most crucial element of teaching and learning mathematics.

• Online exercises for enrichment and self-paced practice.

• Alternating teaching and assignment at all levels of proficiency.

NEW TEACHING TOOLS

In the technology-based environment, the following new teaching tools have become available for all instructors:

• Video lecturing with personal contact with students (using Acrobat Connect or equivalent);

• Prerecorded video classes on CDs or streamlined;

• Supervised video—practice;

• Online independent practice.

Technology also provides access to mathematical poems, songs, games, and plays, and makes fully innovative teaching tools available: i.e., meditation, hypnopedia (sleep-learning), hypnosis, etc. Some examples of application of these tools are described in the literature but it should be stressed that such techniques require much more additional study before being recommended for practical use. The neuroscience approach emphasizes their practical importance.

CONCLUSIONS

The study presents a neuroscience approach to teaching and learning mathematics (NME) with a stress on using technology. Learning mathematics is regarded as having a direct impact on students’ brains, their development of mathematical knowledge domains, and establishing connections among them and other domains. New teaching tools are available in the framework of the NME, such as mathematics meditation, hypnopedia, hypnosis, etc. Technology allows for the optimization of the pace of information delivery to working memory and its transfer to long-term memory using a nested multilevel process oriented to learners of different types and styles. Using technology decreases the probability of mathematics anxiety and enables the instructor to avoid overload.

SELECTED REFERENCES

Dubinsky, E. & Moses, R. (2011). Philosophy, math research, math ed research, K–16 education, and the civil rights movement: A Synthesis. Notices AMS, 58(3), 401-409.

Klinberg, T. (2009) The overflowing brain: Information overload and the limits of working memory. New York: Oxford University Press.

Laughbaum, E. (2011). Capitalizing on basic brain processes in developmental algebra – Part One. MathAMATYC Educator, 2(2):4-7.

Vaninsky, A. (2011). Multilevel alternating teaching and assessment in web- enhanced environment. Proceedings of The 23rd Annual International Conference on Technology in Collegiate Mathematics (ICTCM), March 17 - 20, Denver, CO.

Wu, H. (2011). The mis-education of mathematics teachers. Notices AMS, 58(3), 372-384.