Week 4 Memo #3

Kramer, Post, & Currier 1993, Learning and Teaching Ratio and Proportions: Research Implications

In summary, Learning and Teaching Ratio and Proportions examines the differences between proportion reasoning problems and non proportional reasoning problems.  It further explains the mathematical relationships in proportional situations.  They explain that proportional situations have purely multiplicative relationships between 2 quantities in 2 measure spaces.  Also, if the relationship was to be graphed, the proportional situation with go through the origin whereas the non-proportional situation would not.  Furthermore, they explore assessment and student solutions when faced with proportional problems. Many students could see that if one quantity was tripled, it would follow that the other quantity would triple as well.  To truly understand proportional situations, one would need to be able to generalize past purely numerical concepts.  An example of this kind of problem is an orange juice problem in which juice A has 2 parts orange and 3 parts water, and juice B have 3 parts orange and 4 parts water, which juice has a stronger mixture?  This proved to be challenging for students because comparing unequal rates was more challenging than comparing equal rates.  A study was conducted on 7th and 8th grade students' solution strategies for proportional reasoning situations.  The most commonly used approach was the unit rate approach.  This involves finding the multiplicative relationship between measure spaces.  Another approach students used was the factor of change method.  This involved finding how many times greater something was in another measure space, which is very similar to the unit rate approach.  Student results in terms of correctness, showed that when presented with non integer relationships, students achieved lower and looked at the problem differently.  It was also found the situations the involved scaling were more difficult than buying, speed, or density problems.

1.      What is proportional reasoning?
Proportional reasoning is not merely when someone can set up "cross-multiply and divide" algorithm for solving proportions.  It means understanding why the algorithm works, and the mathematical and multiplicative relationships the occur between 2 or more measure spaces in proportional situations.  It also means being able to recognize which situations or proportional and which are not.  It means being able to solve a variety of problem types, including missing value problems, numerical comparison problems, and two types of qualitative situations.   Students with proportional reasoning should also be able to overcome unfamiliar settings and non-integer numbers such as factions and decimals.

2.      What are the central concepts and connections (between representations, between procedures and concepts, etc.) for teaching proportional reasoning?
Some of the central concepts in proportional reasoning involve the idea that all proportional situations have a multiplicative relationship between two (or more, depending on the problem) measure spaces.  Students also need to be able to see proportional reasoning represented in a variety of way so they can see varying instances of the application.  If students can see more representation, they are more likely to make the necessary connections to acquire proportional reasoning skills.  Students need to be able to see proportional situations represented in as many forms as possible, such as patterns, fractions, graphs, and various real life situations.

3. What are recommendations for teaching this topic for understanding?
a)      What should I emphasize when teaching proportional reasoning?
After seeing the first example problem in the reading, I think it is incredibly important for students to see the differences between proportional situations and nonproportional situations, so they are aware of when proportional strategies can be used.  This can be done, for example, by students working in small groups in which they need to figure out answers to 2 very similar word problems.  One of these will be proportional whereas the other will be similar, by not proportional.  Then, students can look at patterns and graph them.  This way, they can see the visual differences, and even be introduced to graphing lines.  This can also help them discover that proportional situations go through the origin on a graph whereas a nonproportional situation would not.  After this exercise, they can differentiate between the two problems and explain why proportional reasoning and multiplicative strategies may work for one situations and not for another.

b)      How can I teach proportional reasoning using objects, pictures, and word problems?
Proportional reasoning needs to be taught through the use of objects, pictures, manipulatives, real life situations, and meaningful relationships.  This means giving students the opportunity to use objects and manipulatives to model proportional relationships in word problems.  Even being able to draw the patterns through pictures and graphs can help students make connections so that the concept is of meaning to them. For example, the Mr. Short and Mr. Tall example using paper clips and buttons are a great way to use objects to find patterns and relationships.  Also, according to Kramer, Post, & Currier, the "cross-multiply and divide" algorithm needs to be postoned until students have made the multiplicative relationships meaningful for themselves.

c)      How can instruction address common student difficulties?
Instruction can address common student difficulties in a variety of ways.  For one, as a teacher, it may be possible to observe students are they are learning and notice misconceptions they are having.  Something I have seen my cooperating teacher do, is model students' incorrect work and ask the class why it is incorrect.  If students are able to correct work as well as see common misconceptions, they can explore why one way may be incorrect in comparison to another strategy.  Kramer, Post, & Currier also point out that many students get stuck or interpret a problem differently when they are faced with non integer values.   As a result, it may be important to emphasize that using different numbers does not change the thinking behind the problem.  Another common student difficulty arised in scaling in comparison with density, money, and rates.  It would be beneficial for students the see as many applications of proportional relationships as possible, including scaling and changing the size of objects, and noting the patterns the occur.