Week 4; Memo #3

Summary: The article is about defining the meaning of proportional reasoning and proportional problems and making relationships in proportional situations.

1.      What is proportional reasoning?

“Proportional reasoning involves an understanding of the mathematical relationships embedded in proportional situations” (Page 395). Proportional reasoning also involves being able to differentiating the difference between proportional and nonproportional problems by having to recognize the properties of proportional problems and non proportional problems. In addition to finding the properties of proportional problems, students obtain proportional reasoning by being able to solve a variety of types of proportional reasoning problems. This includes knowing what method to apply, identify what is being asked, being able to use a numerical comparison, and quantifying units.

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

Students must conceptually understand proportional reasoning problems before solving the problem procedurally. Teachers must show students a way to better identify what is being asked either by showing examples or drawing pictures. On page 393 of Learning and Teaching Ratio and Proportion: Research Implications, the article shows that teachers should represent the word problems pictorially and through the use of examples by looking at the problem from a different point of view. The example given relates proportion to strength of the orange juice. The teacher is building a relationship between ratio/fractions to the concentration of orange juice. Students can relate to this problem because they most likely encountered a situation similar to this problem. Adding more orange juice to water will make the solution more concentrated. What this problem also addresses is the fact that teachers are making proportional reasoning connections from accessing students’ prior knowledge. Relating the new knowledge to old knowledge makes learning proportional reasoning more meaningful.

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

a)      What should I emphasize when teaching proportional reasoning?

On page 396, teachers need to emphasize postponing teaching algorithms and focus more on internal understanding. By giving students the algorithms to solve various problems, I, as a teacher, am teaching my students to become masters at using the algorithm, but not masters at applying the algorithm. The first step to solve proportional reasoning problems is knowing what algorithm to apply. This is done by teaching for conceptual understanding before teaching for procedural understanding.

b)      How can I teach proportional reasoning using objects, pictures, and word problems?

“Teachers need to step outside the textbook and provide hands-on experiences with ratio and proportional situations” (page 396). These hands-on experiences include activities, drawings, objects, and word problems. To teach proportional reasoning via external representations, I can show how to represent numerical proportional reasoning expressions using objects. Showing is still not enough because I have to use words to explain the connection between proportional reasoning and the external representations by using objects to clarify what is being asked, pictures to explain problems, and word problems to connect to real life situations so that students are able to draw more concepts out of the topic.

c)      How can instruction address common student difficulties?

Providing clear instructions lets students know what they need to do so that they can succeed. Being clear about the instruction makes the curriculum more organized for the teacher in the way that I know what I have to do each day to get students to understand proportional reasoning.

I can also provide instructions on misconceptions and common mistakes that students make when doing proportional reasoning. This is to show students what not to do so that they do not repeat the mistakes others have made. In addition to these instructions, I will have students reflect upon their methods and correct mistakes. This is so that they can learn from their mistakes and learn how to reason through the processes that lead up to the answer.