Part I: Summary Questions
a)
MAIN POINTS or CLAIMS: What main points, claims,
or arguments is the author making?
There are four semantic types students and teachers encounter which are well-chunked measures, part-part-whole, associated sets, and stretchers and shrinkers. The point Lamon stresses is that teachers “need to move beyond the level of identifying…..that affect problem difficulty” instead they should move “toward the identification of components that offer more explanatory power for children’s performances in the domain.”
There are four semantic types students and teachers encounter which are well-chunked measures, part-part-whole, associated sets, and stretchers and shrinkers. The point Lamon stresses is that teachers “need to move beyond the level of identifying…..that affect problem difficulty” instead they should move “toward the identification of components that offer more explanatory power for children’s performances in the domain.”
b)
EVIDENCE: What does the author use as evidence?
Lamon does case studies on 69 males and 69 females that are just starting sixth grade. The sixth graders complete a series of tests that test each semantic type. Lamon gathers data on the test by recording conversation and showing a report of the work of students did on each semantic problem. The semantic type problems are designed to explore children’s mathematical process and reasoning which is essential to proportional reasoning.
Lamon does case studies on 69 males and 69 females that are just starting sixth grade. The sixth graders complete a series of tests that test each semantic type. Lamon gathers data on the test by recording conversation and showing a report of the work of students did on each semantic problem. The semantic type problems are designed to explore children’s mathematical process and reasoning which is essential to proportional reasoning.
Part II: Discussion Questions
1.
What
is proportional reasoning?
Proportional reasoning is the recognition of structural similarities involving ratio and proportion. Lamon states some of the students in his case study achieve correct answers without using proportional reasoning. There is a misconception about proportional reasoning. Proportional reasoning does not consist of being able to construct and algebraically solve proportions.
Proportional reasoning is the recognition of structural similarities involving ratio and proportion. Lamon states some of the students in his case study achieve correct answers without using proportional reasoning. There is a misconception about proportional reasoning. Proportional reasoning does not consist of being able to construct and algebraically solve proportions.
2.
What
are the central concepts and connections (between
representations, between procedures and concepts, etc.) for teaching proportional reasoning?
Students draw pictures, tables, charts, and groups to represent their mathematical processing of the problems. Students are drawing upon their pictures and connecting it to the problem. By verbally explaining their pictures or procedure, students are making connections to the problem and pictures. Having to verbally explain their steps on how they solved each problem means they have to work on their mathematical reasoning. A few students have difficulty explaining their thought process which does not exclude them from not conceptually understanding the problem. Out of those few students who did not offer explanation, some of them could not offer explanations to the problems because they were not able to find the correct words to explain, but they were able to show on paper how they arrived at their answer.
In the stretcher and shrinker problems, Lamon uses the term relative thinking. Relative thinking is being able to relate all the essential features in the stretcher and shrinker problems. Students must understand that changing the length, for example, will proportionally affect the width given a ratio change. If students are not able to connect a number changing a length, then they are not able to do the problems. Most students give up in the article when they come across the stretcher and shrinker problem.
There is no direct transfer between the knowledge of semantic problems. Within types, the variations in context and modes of representations affected students’ responses.
Students draw pictures, tables, charts, and groups to represent their mathematical processing of the problems. Students are drawing upon their pictures and connecting it to the problem. By verbally explaining their pictures or procedure, students are making connections to the problem and pictures. Having to verbally explain their steps on how they solved each problem means they have to work on their mathematical reasoning. A few students have difficulty explaining their thought process which does not exclude them from not conceptually understanding the problem. Out of those few students who did not offer explanation, some of them could not offer explanations to the problems because they were not able to find the correct words to explain, but they were able to show on paper how they arrived at their answer.
In the stretcher and shrinker problems, Lamon uses the term relative thinking. Relative thinking is being able to relate all the essential features in the stretcher and shrinker problems. Students must understand that changing the length, for example, will proportionally affect the width given a ratio change. If students are not able to connect a number changing a length, then they are not able to do the problems. Most students give up in the article when they come across the stretcher and shrinker problem.
There is no direct transfer between the knowledge of semantic problems. Within types, the variations in context and modes of representations affected students’ responses.
- What are recommendations for teaching this topic for understanding?
a) What
should I emphasize when teaching proportional reasoning?
Allow for flexibility and choice of methods to use when solving proportional reasoning problems. Allowing more choice for students to use does not limit the students’ mathematical processing. Accepting only one method as answers to one problem will hinder students’ learning because students will have to close off their original ideas and work with the idea I, as a teacher, offered. I believe learning does not happen this way and I strongly believe in using different methods to tackle the same problem. My reasoning behind using multiple representations to represent one problem is to reach out to different types of students. Some students may not understand how to solve a math problem when I show them using one method, so I use a second and a third method in hopes to conceptually and procedurally teach them.
Allow for flexibility and choice of methods to use when solving proportional reasoning problems. Allowing more choice for students to use does not limit the students’ mathematical processing. Accepting only one method as answers to one problem will hinder students’ learning because students will have to close off their original ideas and work with the idea I, as a teacher, offered. I believe learning does not happen this way and I strongly believe in using different methods to tackle the same problem. My reasoning behind using multiple representations to represent one problem is to reach out to different types of students. Some students may not understand how to solve a math problem when I show them using one method, so I use a second and a third method in hopes to conceptually and procedurally teach them.
b) How can I teach proportional
reasoning using objects, pictures, and word problems?
As a future teacher, I can use objects to show understanding for kinesthetic learners. To show understanding of proportional reasoning, I can use pictures for visual learners. I can point out the important facts and relations within word problems so that students can isolate what is being asked. The use of these multiple representations allows me to reach a variety of learners. It also gives me the ability to explain one problem using different methods. I am able to teach sophisticated strategies and multi-step methods using multiple representations, therefore, I am making sure students are challenged and at the same time, teaching for mathematical understanding.
As a future teacher, I can use objects to show understanding for kinesthetic learners. To show understanding of proportional reasoning, I can use pictures for visual learners. I can point out the important facts and relations within word problems so that students can isolate what is being asked. The use of these multiple representations allows me to reach a variety of learners. It also gives me the ability to explain one problem using different methods. I am able to teach sophisticated strategies and multi-step methods using multiple representations, therefore, I am making sure students are challenged and at the same time, teaching for mathematical understanding.
c) How can instruction address
common student difficulties?
Instruction helps coordinate all the essential relationships in proportional reasoning. Teaching students to look for key words makes students struggle less with procedures and give more time focused on conceptually understanding what they problem asks. Finding key words such as “miles per hour” relates the amount of miles traveled within an hour or finding key words such as increased relates to addition. Having clear instructions makes the problem easier because they struggle less with processing the question and more with working on the problem.
Instruction helps coordinate all the essential relationships in proportional reasoning. Teaching students to look for key words makes students struggle less with procedures and give more time focused on conceptually understanding what they problem asks. Finding key words such as “miles per hour” relates the amount of miles traveled within an hour or finding key words such as increased relates to addition. Having clear instructions makes the problem easier because they struggle less with processing the question and more with working on the problem.
Ben you brought up so great ideas here but be sure you are giving your students an opportunity to explore and discover and make connections between multiple representations on their own. It is one thing to show it is another thing to discover. While teaching planning and modeling are key but you need to let go of the reigns and let children demonstrate for each other.
ReplyDeleteI agree with Ben's final words concerning clear instructions. I realize that students will do nothing in a class period if the instructions weren't clear; it kind of gives them an excuse not do anything. Key word searching is something that I find most useful with word problems so I think these strategies will really help students gain a proportional understanding, especially because problems requiring proportional reasoning are word intensive.
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