Lamon (1993) Ratio and Proportion: Connecting Content and Children's Thinking
Susan Lamon's article aims to understand the level of understanding students have in terms of proportional reasoning. A study was conducted among 69 male and 69 female students. These students were given 1 hour to work through and explain their mental processes for eight problems that fit into Symantec problem types of well-chunked measures, part-part-whole, associated sets, and stretchers an shrinkers. They were then interviewed and their solution processes were analyzed following the following mathematical dimensions: use of relative or absolute thinking, types of representation, quantity structure, and sophistication of strategy. Student answers are analyzed to see how they processed each symantec problem type, their understandings and misconceptions. They found that students had difficulty with quantities such as speed, price, stretching and shrinking in ratios and proporations because they could not be represented physically. They also found that students can have a strong sense of the size and nature of quantities without actually being aware of how numbers are related to one another proportionately.
1.
What
is proportional reasoning?
Proportional thinking is defined to be the ability to construct and algebraically solve proportions. In other words, it is "the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree" (Lowery 1974). It involves the strategies that are used to tackle a number of problems in mathematics and plays an important role in students' mathematical development. Proportional reasoning is used in fractions, word problems, rates, functions, etc.
Proportional thinking is defined to be the ability to construct and algebraically solve proportions. In other words, it is "the comparative relation of one thing to another in part or whole and is expressed in terms of magnitude, quantity, or degree" (Lowery 1974). It involves the strategies that are used to tackle a number of problems in mathematics and plays an important role in students' mathematical development. Proportional reasoning is used in fractions, word problems, rates, functions, etc.
2.
What
are the central concepts and connections (between
representations, between procedures and concepts, etc.) for teaching proportional reasoning?
Central concepts for teaching proportional reasoning include unitizing and relative thinking. Unitizing is defined as being able to interpret a situation by the size of a unit, or being able to reinterpret the situation by changing the size of the unit. It is important to see a ratio as a type of unit. Another key concept that is emphasized by Lamon is the idea of relative thinking. This means being able to see a ratio as a new separate quantity. Relative thinking is a sign that a student is beginning to connect the relationships between addition and multiplication in ratios and proportions.
3. What are recommendations for teaching this topic for understanding?
In terms of teaching ratios and proportions, there are a variety of suggested strategies to teach this topic for understanding. It might be beneficial to pose questions that tie together absolute and relative thinking. Another suggested strategy may be to encourage the process of unitizing by offering problems with multiple solutions and the use of pictures. Giving students a chance to explore multiplicative situations can also help students understand the relationship in the traditional symbolism of proportions (e.g., a/b = c/x) Students should also be given the opportunity to use pictures and manipulatives when working with associated-set problems.
Personally, if I am teaching proportional reasoning, I would emphasize the relationship between multiplication and addition in ratios and encourage students to explore the relationship between the two. It may help students to work with graphical representations or charts to see patterns of how pieces of proportional information relate to one another. I would have students work on a proportional reasoning word problems in which they will need to work out the solution using each method including the use of manipulatives, creating a chart, seeing patterns and even creating an algebriac formula that will work. If students can even understand the idea of equivalent fractions, it provides a strong basis for this topic.
Instruction can address student difficulties in a number of ways. Depending on how the teacher assesses the class, the teacher can model student misconceptions and explain why it is incorrect. Student can even be given worksheets with incorrect answers and common misconceptions they can correct in small groups. I find that if students are seeing incorrect examples and fixing them, they are less likely to have the same misconceptions.
Central concepts for teaching proportional reasoning include unitizing and relative thinking. Unitizing is defined as being able to interpret a situation by the size of a unit, or being able to reinterpret the situation by changing the size of the unit. It is important to see a ratio as a type of unit. Another key concept that is emphasized by Lamon is the idea of relative thinking. This means being able to see a ratio as a new separate quantity. Relative thinking is a sign that a student is beginning to connect the relationships between addition and multiplication in ratios and proportions.
3. What are recommendations for teaching this topic for understanding?
In terms of teaching ratios and proportions, there are a variety of suggested strategies to teach this topic for understanding. It might be beneficial to pose questions that tie together absolute and relative thinking. Another suggested strategy may be to encourage the process of unitizing by offering problems with multiple solutions and the use of pictures. Giving students a chance to explore multiplicative situations can also help students understand the relationship in the traditional symbolism of proportions (e.g., a/b = c/x) Students should also be given the opportunity to use pictures and manipulatives when working with associated-set problems.
Personally, if I am teaching proportional reasoning, I would emphasize the relationship between multiplication and addition in ratios and encourage students to explore the relationship between the two. It may help students to work with graphical representations or charts to see patterns of how pieces of proportional information relate to one another. I would have students work on a proportional reasoning word problems in which they will need to work out the solution using each method including the use of manipulatives, creating a chart, seeing patterns and even creating an algebriac formula that will work. If students can even understand the idea of equivalent fractions, it provides a strong basis for this topic.
Instruction can address student difficulties in a number of ways. Depending on how the teacher assesses the class, the teacher can model student misconceptions and explain why it is incorrect. Student can even be given worksheets with incorrect answers and common misconceptions they can correct in small groups. I find that if students are seeing incorrect examples and fixing them, they are less likely to have the same misconceptions.
Hello Cera,
ReplyDeleteYour post this week included some great suggestions for encouraging proportional reasoning in the classroom. Giving students multiple opportunities and representations will only strengthen their capacity to think proportionally. I like the idea of framing questions in a real life context and having students think about the application to real life examples. If we can encourage students to make connections outside of the classroom they will only strengthen their ability and develop their understanding with greater ease and comfort. Speaking a second language is much more meaningful when it is in a context embedded. I think this is also true with mathematics.