OISE AQ assignment 4: Proportional Reasoning

[fox_sejant]

What is Proportional Reasoning?
“Proportional reasoning is a form of mathematical reasoning that involves a sense of co-variation and of multiple comparisons, and the ability to mentally store and process several pieces of information. Proportional reasoning is very much concerned with inference and prediction and involves both qualitative and quantitative methods of thought.” http://www.cehd.umn.edu/ci/rationalnumberproject/88_8.html
          Key to proportional reasoning is the ability to make inferences and predictions.  Proportional reasoning is the end stage for elementary mathematics and the beginning stage for high school mathematics.  In part this is due to proportional reasoning being one of the skills children acquire when they move from concrete operations (needing tools and diagrams and manipulatives) to formal operations (abstract thought) http://slideplayer.com/slide/4535169/

What will students learn from proportional reasoning?
Students will learn to recognize relationships between numbers and between values.  Recognizing that certain numbers and values can be replaced with units is a key step.  For example four nests with 3 eggs in each nest can be thought of as 1 “unit” as well as 3 eggs at the same time.  http://www.edu.gov.on.ca/eng/teachers/studentsuccess/ProportionReason.pdf page 5. 
Key concepts in proportional reasoning include: recognizing equivalent fractions, division, place values and percents, measurement conversions, and ratios & rates.  Anyone who has ever tried to follow a recipe for 6 people when cooking for 8 people will recognize many of the skills learned in proportional reasoning.  The same is true of shopping for many items and doing comparisons between products.  Consider for example two cereals.  The first has 22g of carbohydrates per 1/3 cup.  The second has 12g of carbohydrates per ½ cup.
Students will also learn to recognize similarities between sets of numbers and apply those similarities to other sets.  Students will have the opportunity to make predictions about these relationships and then test to see if they are true.
For example watch this video: 

Click to view

While many factors are at play here it should be obvious that in a truly proportional relationship the giant rubber band should have flown nearly 1500 meters.  Students could create their own experiments with various rubber bands to see if they can replicate the failure of the experiment in the video or if they can in fact demonstrate the suggested proportional relationship.
          Students will build upon their knowledge of fractions, multiplication and division.  They will build toward a greater understanding in algebra and geometry.

What problems will students encounter?
          Failing to properly recognize abstract units and instead focusing on the specific numbers in a problem is a common difficulty students encounter.  Similarly many students will learn how to apply multiplication skills to proportional reasoning problems without actually recognizing the proportional relationships at play. 
          In some cases students will look at a problem and see and additive relationship and mistake it for a proportional one.  Consider this example from http://www.proportionalreasoning.com/and-multiplicative-thinking.html
Two crocodiles are presented.  One is 4 meters long.  The second is 5 meters long.  When they reach adulthood the first 7 meters long.  The second is 8 meters long.  Both have grown 3 meters but that is only an additive relationship.  Recognizing the proportional relationship requires additional thinking.
          As for Paige Fox in the comic strip above?  She simply forgot a step in her thinking.  5:100, compared to 30:x results in 600.  But Paige was already doing 5 minutes of homework so the increase is only 500%.

Resources for further reading

http://www.proportionalreasoning.com/and-multiplicative-thinking.html
http://www.proportionalreasoning.com/uploads/1/1/9/7/11976360/proportional_reasoning.pdf
https://www.youtube.com/watch?v=rZnQatZgy4U
http://www.cehd.umn.edu/ci/rationalnumberproject/88_8.html
http://www.edu.gov.on.ca/eng/teachers/studentsuccess/ProportionReason.pdf