Education ∪ Math ∪ Technology

# Tag: learning(page 2 of 2)

This year I have really tried to step up the process of bringing the real world into my mathematics class.  A major focus has been on using technology appropriately as a tool to help solve real life problems.

Here are some examples:

Distance formula:  Finding an optimal (or near optimal) solution to the Traveling Salesman problem for a small number of cities.

Basically here the students were given the assignment of choosing 6 or 7 cities fairly near each other on a Google map and finding the x and y coordinates of each city, then using the distance formula to determine the distances between the cities.  Once they had this information, they were to try and figure out a shortest path, or at least something very close to the shortest path, and then justify their solution.

Linear graphs & Piecewise functions:  Compare 4 or 5 difference cell phone plans.

Students should take a few cell phone plans and compare the plans, including the cost for text messages (which may include similar graphs), the cost for extras, start up costs, etc…  I found the students end up needing to create piecewise functions in order to represent a cell phone plan which has a fixed rate until the minutes are used up at which point the customer has to pay extra for each minute.

Shape and Space: Design a new school building.

Here I showed the students the new lot our school is in the process of purchasing and our project is to design a building for that spot, and calculate how much their building design will cost (within the nearest \$1000).  It involves finding area, volumes, perimeters, scales, perspective, etc… We are using Google Sketchup for the designs but I am now trying to work out how to import the students designs into a virtual world (like OpenSim) so we can have each student group lead walk-arounds of their building.

Polynomials:  Determine how many operations multiplying a 100 digit number times a 100 digit number takes.

Students are learning about computational complexity theory by analyzing the number of steps it takes to multiply numbers together.  They record each step in the operation and increase the size of the numbers of each time and re-record their results.  They then compare the different number of steps in each operation and try to come up with a formula, so that they can answer the 100 digit times 100 digit question.  Our object: Figure out why our TI calculators can’t do this operation.  It turns out that the formula itself is a polynomial, and their substitutions to check their various formulas count as a lot of practice substituting into polynomials, which was a perfect fit for our curriculum.

Quadratic functions:  Create an lower powered air cannon and use it to fire potatoes a few meters.

Here the students are attempting to use quadratic math to try and analyze their cannon, then the objective is to try and hit a target with a single shot later.  The cannons should be very low powered for obvious safety reasons, capable of firing a potato (or Tennis ball) a few metres at most.  There is also a slight tie-in to Social Studies where my students will be studying cannons in their unit on medieval warfare.

Bearings and Angles: Set up an orienteering course in your field or local park.

Students attempt to navigate a course through a park and pick up clues at each station, which they use to figure out a problem.  Students have to be able to recognize the scale on the graph, navigate using bearings, and measure angles accurately.  Also lots of fun, we did this in Regents park for a couple of years in a row.

Integration: Calculate the area (or volume in a 3d integration class) of an actual 2d or 3d model.

Basically you have the students pick an object which they then find the functions (by placing the object electronically in a coordinate system) which represent the edge of the object, then place the object in a coordinate system and calculate area of the object using integration.

Percentages: Find out how much your perfect set of "gear" (clothing) costs when it is on sale and has tax added.

Students take a catalog and calculate how much it will cost for them to buy their perfect set of clothing.  They can buy as many items as they want (with their imaginary money) but have to keep track of both the individual costs and the total cost of their clothing.  You can also throw some curve balls at them, like if they buy more than a certain amount, they get  discount, etc…

If you have any other examples of real life math being used in a project based learning context, please let me know.  I’m always interested in other ideas, especially for the more challenging areas of mathematics.  I’ll add more ideas here as I remember them.

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

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

I personally think people learn through an unconscious process very much like the scientific method.  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 way, 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".

It is crucial during this process that the learner feels comfortable to make mistakes.  Instead of feeling pressure to have exactly the right answer each time, learners must be willing to work through the entire process of learning.  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.

In the classroom, this is when we normally say that a student has "made a mistake", which is unfortunate language.  Rather than criticizing students who have a cognitive discord occurring, we should encourage more reflection of the learning process, and provide opportunities to establish a new theory which fits the given facts and can be worked into the learner’s personal theory of how the world works.