Thursday, August 20, 2015

The Power of Reflection in Problem-Solving

This post is dedicated to educator extraordinaire, my beloved father, Ibrahim Nadji (RA!) 

In all assessment items my students are expected to write a reflection that must adhere to the following guidelines as much as possible. The more inclusive of these elements a reflection is the more highly regarded it would be.

• Have a paragraph for each segment of a given assessment item
• Provide a brief summary of the outcomes of the given assessment
• Share a personal introspection on what has been learned from the given assessment
• Have a minimum of one Wonderism (thoughtful deep question) per assessment item
• Close with a minimum of one statement connecting the concepts gleaned from the given assessment item to art or real life in general for each reflection paragraph

In addition and as some of the literature on the subject of reflections indicates, "Reflections must emphasize: (1) returning to an experience, (2) attending to feelings and re-evaluating the experience, and (3) linking  this processing to action. [Source: David Boud, Rosemary Keogh,   and David Walker, Pro-moting reflection in learning:  a model," in Boundaries  of Adult Learning, edited by Richard Edwards, Ann Hanson, and Peter Raggatt (Routledge, 1996) pp. 32-56.]

Following are samples of reflections students wrote as follow ups to their solution processes to assigned math problems (each time a synopsis of the problem is stated for context purposes followed by the reflection.) 
Note: For reflections associated with other types of assessment, please refer to my Astronomy blog post on the subject matter.

*** Start of Sample 1 ***

Synopsis:  Determine the duration of a telephone call (in minutes) when the total cost of the call is $6.29 and you are charged .41¢ the first minute, .32¢ for each minute thereafter, and a $3.00 long distance fee.

Reflection: I like practical things and this was a practical and useful problem.  Many times people do not understand how a cost has been calculated.  This problem helped break down the process.  This type of problem can also be useful in determining the cost of electricity.  Most electric companies have a base fee for the privilege of being a customer (sometimes referred to as a “meter charge”) and then the customer is charged one rate for so many kilowatt hours and then a different rate for additional kilowatt hours.

This exercise guides a person through the various charges and rates when calculating a utility so the person can be better informed of how the total charge was calculated.

*** End of Sample 1 ***

*** Start of Sample 2 ***

Synopsis:  Your classmate decides to solve the equation x(x+2) =8. (Student’s Example) It is your job to grade your classmate’s solution for mathematical accuracy using a system based on 10 being the maximum points to be earned when the problem solution is perfect.

Reflection: It was kind of weird being on the other side of things and having to decide what I would give the student for his or her work. I thought a 5 was a fair evaluation because half the answer was given. I’m probably not a good grader because I like to be nice, but I know that’s not always the point of grading obviously. I need to remember to set equations like this up in standard form also because it makes it so much easier to work out and understand!    

*** End of Sample 2 ***

*** Start of Sample 3 ***

Synopsis: Invent application problem of own for which solution depends on solving the system of equations shown:  x + y = 1000 and 0.05x + 0.06y = 55

Reflection: I am proud of myself for being able to get through this problem, as I have struggled the most with application problems! This gives me confidence to know that I can do it!

*** End of Sample 3 ***

*** Start of Sample 4 ***

Synopsis: Use the elimination (addition) method to check that your solutions to the above system of [two linear] equations are correct. Note: Synopsis stated by another student and statement in square brackets is mine.    

Reflection: By reflecting I see that the addition method and the substitution methods dont always work the same for real world problems as they do matching up in math. For example; if you had two recipes that you wanted to combine using the addition method you couldn't multiply the ingredients in one recipe by a number (using the addition method) to eliminate variables and expect the recipe to be an accurate combination of the two recipes.

*** End of Sample 4 ***

*** Start of Sample 5 ***

Synopsis: This problem asks us to prove whether or not the reciprocal of an irrational number is also an irrational number. 

Reflection: This question had me stumped at first but by defining piece by piece the “scary” words in it, it became relatively simple. In order to be considered a rational number there are certain requirements that must be met. The reciprocal of any number still contains the number so it’s impossible that a number which doesn’t meet the requirement could ever have a reciprocal that does.   

*** End of Sample 5 ***

*** Start of Sample 6 ***

Synopsis: We are asked to solve either #19 or #21 from pg 343 of our textbook. I’ve choose #19 which is: “A water tank can be filled by an inlet pipe in 8 hours. It takes twice that long for the outlet pipe to empty the tank. How long will it take to fill the tank if both pipes are open?” 

Reflection: I love the examples in the book. Thinking about this as a fraction for the hourly amounts is something that is so simple when explained, but when I first read the example I had no idea how to start. The application parts are the ones where you really get to see how the things we are learning are actually useful, something that wasn’t mentioned when I took math the first time around. It makes it more enjoyable AND more understandable because you see how it relates to the real world instead of being a bunch of rules you memorized for no reason.    
*** End of Sample 6 ***

I hope the above samples convey the varying sophistication with which students approach reflections and through them how their solution processes evolve or take the shapes they do.

Please, use the comments area to share your thoughts and your take on the idea of having students reflect on their work. Thank you