Education ∪ Math ∪ Technology

# Day: December 5, 2009 (page 1 of 1)

I recently had students do a project where they apply the distance formula to finding the shortest path for a traveling salesperson to travel between 8 cities.  The basic idea is, the students use the longitude and latitude coordinates as substitutes for the x and y coordinates of the city, then they can use the distance formula to find a pseudo-distance between the cities.  Of course, on any kind of largish scale, this makes no sense, but on a small enough geographic scale the error in the distances is small, and I made sure the kids were aware of this deliberate error.  This project was intended to be a chance for the students to get lots of practice using the distance formula.

You can view a sample of the students work here which I have permission from the student to share.  This was rather an exceptional sample for many reasons.  The first reason is that the student went to extraordinary measures to format his document and arrange his work in a clear logical sequence.  Another reason I really enjoyed looking at this student’s project was because he had obviously done research about the traveling salesperson problem, including appropriate mathematical terminology such as the ‘Hamiltonian path’.  He talks about the limitations of the approach we took, the errors he found in the project, and suggests an improved method than the brute force solution.  He uses technological tools to his advantage, learning a bit of JavaScript to find the coordinates of the cities, and using Excel to greatly reduce his calculation time for the distances between the cities.

If all of our students work was so neatly arranged and so carefully done, I think very soon we’d soon have much different jobs.  Instead of ‘instructing our students’ we would be learning from them as equal partners.  This what I strive for in my teaching.

I’m working on my personal learning theory again, as a reflective activity in my Masters degree.  I created a very short summary of my personal learning theory before, and am now updating it to include vocabulary and ideas from the semester long course I just finished about learning theories.  I hope most teaching colleges offer this kind of course as part of their teacher training, it has been incredibly valuable to me.

Here is what I have so far:

Personal Learning Theory

I personally think people learn through an unconscious process called experiential learning.  They hypothesize about how the world should work, collect data, compare the data they have collected to see if it fits in their theory, and then revise their theory if they feel enough evidence has been found.  In this theory, as described by Kolb (1984), people construct an understanding of the world around them using what they know as a basis.

Each piece of knowledge people gain has to be fit into their personal hypothesis.  At first, people will "bend" their hypothesis to make facts fit which seem inconsistent, but eventually if enough contradictory data is collected, people are forced to revise their ideas.  This is part of the reason why students have so much difficulty learning topics for which they do not have any background; they are constantly required to create and revisit their hypothesis, and to build theories about the information they are receiving "from scratch".  "Ideas are not fixed and immutable elements of thought but are formed and re-formed through experience." (Kolb, 1984)

It is crucial during this process that the learner feels comfortable to make mistakes.  Although it is possible that an individual learner will have a theory which fits all the facts as they are collected, it is much more likely that conflicts exist between their theory and the data.  As the Lewinian experiential model suggests, observations of what one has learned or not learned are a critical aspect of the learning process (Smith 2001).

As drawn from the work of Vygotsky, situated learning suggests that "experience in the activities of the practice" (Kolb, 2005) are integral to the learning process.  Without learners being embedded within a community of practice, their ability to make connections, draw conclusions, and verify hypothesis will be greatly hampered.

References:

Kolb, D.A. (1984). Experiential Learning: Experience as The Source of Learning and Development, Case Western Reserve University, retrieved from http://www.learningfromexperience.com/research-library/ on December 2nd, 2009

Kolb, D.A., Boyatzis, K.E., Mainemelis, C. (2000). Experiential Learning Theory: Previous Research and New Directions, Case Western Reserve University, retrieved from http://www.learningfromexperience.com/research-library/ on December 2nd, 2009

Kolb, A.Y, Kolb, D.A, (2005) Learning Styles and Learning Spaces: Enhancing Experiential Learning in Higher Education, Academy of Management Learning & Education, 2005, Vol. 4, No. 2, 193–212.

John-Steiner, V., Mahn, H. (1996). Sociocultural Approaches to Learning and Development: A Vygotskian Framework, Educational Psychologist, 31(3/4), 191-206, retrieved on December 2nd, 2009

Smith, M. K. (2001) ‘Kurt Lewin, groups, experiential learning and action research’, the encyclopedia of informal education, retrieved from http://www.infed.org/thinkers/et-lewin.htm on December 4th, 2009