Alright, so we are now into the start of February, and I have some new thoughts about proportions. First I have just finished reading the Billings article on proportional reasoning, and i found it fascinating. This is due to the fact that she recommends using a “no numbers first” approach to introducing proportions to middle school students. Her argument for this is that this allows instructors to better understand where their students are in terms of their initial proportional reasoning skills. Thus this is an excellent way to both administer a diagnostic assessment to gather useful data on our students, but also to use as a formative assessment(s) to gauge how well our students are grasping the concept.
Furthermore, she argues against teaching, initially, the cross multiplication strategy/algorithm used to commonly solve proportional problems. The basis of this follows from the fact that students will then simply memorize the procedure for dealing with proportions, without necessarily comprehending the logical reasoning behind why proportions work the way they do.
Also, we did an in class activity on Day 7 (1-28-15) in which I had the opportunity to “play teacher” to the class. For this activity, we were tasked with identifying, in a series of 7 problems, which set of solutions had more “blueness” too them. What I had to do was illicit student responses and get them to explain their reasoning behind which answer they believed to be right in a ‘pure’ mathematical sense. Overall, the activity was fun, but as I explained on my thoughts about that day’s activity, I felt like it is difficult to simultaneously divorce your own experience and knowledge from the problem and keep and open mind when dealing with an actual class of middle school students who do not really have a firm grasp on what proportions are. Hence my solution to clear up some confusion is to be explicit in my questions on formal and informal assessments as to whether or not I am looking for a proportion or an actual integer or something else. That being said, a good bonus question or formative assessment would be to investigate whether our students can correctly identify which contexts require proportional reasoning compared to integer or other kinds of reasoning abilities.
Thanks for taking a few minutes to read my thoughts on proportions and teaching the topic. Hope something is useful/thought provoking.