MTH 212 Paper

Here's something I just wrote on "Proportional Reasoning," for Elementary Mathematics 212.

Proportional Reasoning
The world of ratios is a very different world than that of mere common fractions. I cannot even begin to imagine teaching something as complex as proportionality to a group of fifth graders, after having seemingly just taught them fractions. It almost seems to me that the knowledge of one could almost uproot the knowledge and understanding of the other. Yet, somehow I learned both, even though I have no remembrance of the process. I was educated in Saxon mathematics, which is apparently considered out dated and poor for teaching concrete number sense and deeper understanding of mathematics, by nowadays standards.
I really enjoyed the reviews for the problems in “Proportional Reasoning,” by Jane Lincoln Miller and James T. Fey. I think the most important part of Elementary Mathematics is learning about how kids understand and see a certain problem. If we could actually have kids in the class occasionally to learn from, it would probably be extremely helpful. In the review of Task 2, the children’s responses are very enlightening. One response given is, “Yes, because each drink you buy is 40c cheaper, so you are saving money.” I like this response because it is so simple and correct. The child clearly understands the comparison of the two drinks, but fails to comprehend the quantity inherit in the values. This is a common mistake by children, whereas more is usually better, meaning the Gatorade would probably be chosen if the question was not phrased “most economical choice.” This of course brings me to my next discussion, which is the poor word choice, “most economical choice,” as you can see one kid didn’t have any idea what that even meant. As I’m sure several students here at PCC Cascade would struggle with that in a math class, without any background whatsoever in economics.
In Task 1, the problem deals with very concrete ratios. We are not working with any total values. Yet, we have to deal with ratios or direct comparison, which represent the relationship between two numbers. It is difficult to recognize a greater relationship than simple addition, the idea of something being directly proportionate is a much higher complex understanding. Children and adults alike tend to think that if an amount is increased by another amount, the other amount given must be increased by that same amount. It is often they do not see the direct proportional rate that needs to be applied to both given amounts. A given amount that has been doubled, must apply that same doubling to the other given amount. Doubling the number 4, takes you to 16, an increase of 12. Yet if we are given a second amount of 3, and simply add 12, as opposed to doubling 3, we get 15, which is very different from 9. The relationship between 4:3 and 16:9 is directly proportionate. The relationship between 4:3 and 16:15 is just not in any way. Proportionately represents a greater understanding of fractions, percents, multiplication and the beginning of exponents. Task 3 is a great problem because enlarging a picture is a concrete real world activity expressing a direct proportionality between the two sides of the photo. I would think it would be the most concrete hands-on activity for a classroom to develop into their curricula.