Memo #3 Proportional Reasoning

Proportions & Proportional Reasoning

Questions
1. What is proportional reasoning?

Proportional reasoning is just like what it sounds, it's a student's ability to reason about problems and situations that require them to think about proportions and how quantities directly relate to one another. This can be through fractions, ratios, related rates, even tables and graphs can be a part of this process. A key part of this process is the concept of equivalence. Teaching a young child about proportional reasoning may not be very successful, because they must be cognitively mature enough to think about quantities and how they relate, or are equal to other such quantities. Students need to be able to make predictions and connections and comparisons using abstract representations, such as numbers, graphs, etc. The last key concept students may need to master is the unit rate, which is highlighted in the Catehcart article as being crucial to developing other skills when working with ratios, rates, proportions, percents and fractions.

2. What are the central concepts and connections (between representations, between
procedures and concepts, etc.) for teaching proportional reasoning?

I think that the central concepts for teaching proportional reasoning is wait time and critical thought building. Again what I believe teachers should be emphasizing is student discovery. Students are not going to develop their own understandings by listening to a teacher lecture about certain types of procedures they can use to solve these problems. The unfortunate problem that I keep running into is how exactly do you inspire student initiated discovery, without seeming superficial as well as forced and confusing. The second problem that becomes difficult arises when students come into this from different levels, some more prepared for these ideas, and some needing a little more time. Some students have even been taught procedures already, and with those students, is it even possible to back track and show them the understanding they need to use with these problems, or is it just more difficult?


3. What are recommendations for teaching this topic for understanding?

a) What should I emphasize when teaching proportional reasoning?
Teaching this topic for understanding is actually very difficult to do, especially with middle school students who are just beginning to think abstractly about mathematics, and manipulate quantities using multiplicatives, instead of addatives. For certain, procedural based learning will not benefit students as well as allowing them to honestly work through the ideas for themselves. Teachers should let students discover for themselves what it means to invert and multiply, giving them the tools they need to do so.

b) How can I teach proportional reasoning using objects, pictures, and word
problems?
Another tricky problem here is that proportional reasoning is abstract, but it also involves real world ideas. Bringing in pictures that show students how things can be grouped in order to compare quantities (ie: for every two kids there is an adult watching them, for every frog in the pond there are two fish, and so on). I have actually seen this at work through a program called "Success Maker." In "Success Maker" they see gumballs in a machine and have a plethora of questions relating to proportions, probability, ratios and fractions.

c) How can instruction address common student difficulties?
Instruction addressing common student difficulties is another hard subject, especially when we don't necessarily think the same way that our students do. So I would say the first step in addressing common student difficulties is to begin thinking like a student and start noticing things that might give them trouble. I really do believe that it's the small things, and the frequent reminders that help students, and to do that we need to get inside their head. The last way that we can address common student difficulties is by involving students in the teaching. If a student comes up to the board to solve a problem and demonstrates a misunderstanding, not only does it illuminate misunderstandings, it allows for opportunities for inprovement in a very real way.