Proportion Reasoning from Susan J. Lamon's Representing and Presenting

Natasa's comments on the article by Sue Lamon were telling enough to get me read this carefully. She talked about the research using 6 classrooms where 1 class was taught fractions traditionally and the other 5 were all taught/learned fractions through some other method than parts of a whole. The 5 non-traditional classes all performed better on an assessment of their proportional reasoning than the control class. While I have been able to reason out problems from my own grasp, I was challenged by how unfluent I was in describing the other methods. I'm going to publish a few of problems they pose as an example of this thinking.

1.) Does the shaded area show 1 (3/8 pie), 3 (1/8 pies), or 1 1/2 (1/4 pies)? Does it matter?
[the shaded region was 3 sectors of a circle divided into 8 equal sectors]
2.) You have 16 candies. You divide them into 4 groups, select one group, and make it three times its size. What single operation would have accomplished the same result?
4.) If it takes 9 people 1 1/2 hours to do a job, how long will it take 6 people to do it?
5.) Without using common denominators, name three fractions between 7/9 and 7/8
6.) Yesterday Alicia jogged 2 lapps around the track in 5 minutes, and today she jogged 3 laps around the track in 8 minutes. On her faster day, assuming that she could maintain her pace, how long would it have taken her to run 5 laps?
7.) Here are the dimensions of some photos: (a) 9 cm x 10 cm, (b) 10 cm x 12 cm, (c) 6 cm x 8 cm, (d) 5 cm x 6.5 cm, and (e) 8 cm x 9.6 cm. Which one might be an enlargement of which other one?

Think of the nuances these represent?
My grade 8 classes this year have really exposed the gaps in this understanding:

For example: One case contains 8 cans. How many cans are in 3 cases? How many cans are in 2 3/4 cases?

Most could do the first question correctly. A common answer for the second question expectedly was 19 but some had answers of 40??? Clearly, I hadn't created learning in this area. The next period I had them again draw out the cases showing the cans in the cases. This simple improvement with a concrete example made it possible for the students to do much harder examples successfully.

Teaching Proportional Reasoning

Thank you to all who have expressed an interest in discussing lesson design on line.

Here is my proposal:

Based on Deanna Burney's "In Craft Knowledge: The Road to Transforming Schools":

"Learning those kinds of skills is not a solitary endeavor; rather, it needs to be a highly social one. It depends on continual discussion and demonstration. People learn by watching one another, seeing various ways of solving a single problem, sharing their different "takes" on a concept or struggle, and developing a common language with which to talk about their goals, their work, and their ways of monitoring their progress or diagnosing their difficulties. When teachers publicly display what they are thinking, they learn from one another, but they also learn through articulating their ideas, justifying their views, and making valid arguments."

We start with Walter's idea of proportional reasoning and flow it into fractions as an extension as per Karen’s suggestion. This then is universal in buy in as it is an enduring concept that is foundational to mathematics at every level. Elementary teachers will have as much to say as anyone as they are in on the ground floor of developing the ideas in students and secondary/post secondary teachers can ensure that our language is appropriate and transferable through the years and into other branches of mathematics. Mathematicians can strengthen our grip on any aspects we seem to internalized incorrectly ourselves due to our successful memorization and rule building through our schooling.

Anyone is welcome to participate because it is the articulation and justification of ideas that lead to our professional development while learning from one another and also lead to improvements in students’ learning. Those that want to ‘listen’ in may, and those who are more active learners can be as active as their time permits. Please respond whenever something strikes a strong cord or gives an ‘aha’ moment. Already we have had China, Ontario, Vancouver Island, Cranbrook, the Lower Mainland and elsewhere involved.

At PIMS on Saturday we had a fascinating conversation around “perception based perspective (ie. empirical based learning processes) versus reflective abstraction. These big words were made clear through the example: What is the sum of two odd numbers? Having students add a bunch of odd numbers to discover the ‘rule’ two odds add to an even would be the empirical. Having students think about a chess club (or any two person game group). With an even number of players, they could all be playing at one time. An odd number of players means that one person is waiting to play. But if two chess clubs with an odd number of players were to join (add) then the two waiting could play each other immediately. This would be the example of reflective abstraction. At what point can students ‘see’ an odd number as one more than an even? In any case I see this level of discussion on proportional reasoning as our goal. Katherine B. taught some of us about the Pirie-Kieran Learning theory (1994 I believe). The P-K theory describes the growth of student understanding through a series of ever expanding circles from primitive knowing, to image making, to image having, to property noticing, to formalizing, to structuring, to observing, to inventizing [and humourizing! - my addition].

Where do we start with proportional reasoning? I’m initially thinking it would be in the real world with sharing food with a friend, with three other friends, etc. before representing the food with an image like a rectangular chocolate bar. The key is for students to understand “Parts of a whole”. I’m aware that Singapore math would swear by the ‘bar’ approach and wonder why we are so taken with circles for teaching fractions. I’ve had good effect in expanding an idea from Jump Math of utilizing nested equivalent bars like the diagram below. I can use the biggest bar as “1” to show the parts of this whole but the proportional reasoning is exposed by providing different starting numbers in various parts of the diagram. The example here could show 16 in the biggest bar to start the puzzling out of division or I could supply the bottom little one as “1” and then the concepts of repeated addition and multiplying would be exposed. Providing one of the “4’s” would cover both ideas.

Would this type of problem with a simpler diagram be an example of proportional reasoning in younger grades? At what level can students do this thinking? We don’t want rote memorization to be our foundation but a conceptual depth to seat any memorization/learning that takes place.

How long is the longest bar?



or



A strong understanding of –– three 3’s is the same as “9” is necessary before proportional reasoning can be seated (and remembered, Magnus.)

Comments? Other ideas around starting points? Any points made here that trigger a positive or negative response? I’m no expert on early numeracy but am learning from the gaps in understanding that emerge with the increased complexity of secondary and through professional associations like with yourselves here.