What role do you see assessment being played in the classroom? Does understanding how students think help you prepare for your role are a teacher? How will you use this information to shape your practice?
In this classroom, I see the role of assessment as a way of being successful in life. Ms. Krabapple says to Bart, to stop bothering other students in the class because they have a chance of being successful in class. Some students have a similar belief that tests scores are correlated with success. In this clip, Bart takes Martin's test and writes his name on it because he doesn't understand the questions and wants to look like he can do the test. With this in mind, it is important to emphasize with students that doing poorly on a test, or not having an understanding of a particular concept in math does not reflect failure. Ms. Krabbapple also does not present herself as an approachable teacher which makes it difficult for students to ask for help without feeling stupid.
In my future classroom, I want to make it clear to students that it is okay if they do not understand something, but it is very important for them to ask questions to clarify their confusion of that is the case. Students need to feel safe and respected in order to ask for help. I also think it is important to emphasize conceptual understanding in the classroom so students understand the reasoning behind their work.
Tuesday, October 30, 2012
Monday, October 22, 2012
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.
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.
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.
Sunday, October 14, 2012
Week 3 Memo #2
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.
Monday, October 8, 2012
Week 2 Memo #1
Hiebert, J. and
Carpenter, T. (1992). Learning and teaching with understanding.
In summary, Heibert and Carpenter's Learning and Teaching With Understanding addresses how students can successfully learn mathematics through internal and external connections. Internal connections refers to the understanding the occurs while learning. External connections refer to symbols, objects, and pictures the construct understanding. External mathematical representations influence internal mathematical representations. Heibert and Carpenter explore how understanding mathematics can be beneficial for learning more mathematics later on. One of the key ideas that are addressed is how thinking and talking about similarities, differences, and patterns can help students construct relationships. They also address issues related to understanding mathematics such as conceptual understanding and procedural understanding, prior knowledge and current learning, and math being taught mechanically.
The authors define conceptual understanding in several ways. Understanding occurs through thinking and talking about similarities, difference, patterns, etc. Conceptual understanding can also result as different representations for a concept are connected. If students have constructed strong connections, or networks, for a given concept, then conceptual understanding is likely and can influence the understanding of new relationships. For example, if students can see the connection between money, blocks, and decimals have to place value, they may demonstrate a conceptual understanding.
We can recognize that a student understands a concept through a variety of ways. A student understandings when he/she can connect prior knowledge to new knowledge. If they are able to represent something internally, understand patterns and relationships, and explain a concept, this may be evidence of understanding. For example, students have conceptual understanding of operation symbols such as +, -, x, and / if they understand each operation. If students know to add, subtract, multiply, or divide when given a certain symbol, they have conceptual understanding of what the symbols mean. They should be able to work out the operation on paper, explain what it means, and be able to represent it using manipulatives.
Understanding leads to many beneficial consequences. It can lead to the ability to more easily construct meaningful connections, promote retention, reduce the amount needed to be understood at a time, and enhance the transfer of knowledge to new situations. One example of a consequence of understanding is the connections between addition and multiplication. If a student is familiar with what it means to add, the student will more easily be able to grasp the multiplication means to add a certain number of times.
The growth of understanding occurs an understood concept is connected to new concepts. As connections are internalized, new connections can be constructed. For example, if the a student understands the idea of the addition of natural numbers, this can be extended integers and multiplication. An understanding of addition and multiplication can be extended to subtraction and division. These concepts can be built on by incorporating variables in algebra. As a result, we can see that understanding grows through connections to new concepts.
In summary, Heibert and Carpenter's Learning and Teaching With Understanding addresses how students can successfully learn mathematics through internal and external connections. Internal connections refers to the understanding the occurs while learning. External connections refer to symbols, objects, and pictures the construct understanding. External mathematical representations influence internal mathematical representations. Heibert and Carpenter explore how understanding mathematics can be beneficial for learning more mathematics later on. One of the key ideas that are addressed is how thinking and talking about similarities, differences, and patterns can help students construct relationships. They also address issues related to understanding mathematics such as conceptual understanding and procedural understanding, prior knowledge and current learning, and math being taught mechanically.
The authors define conceptual understanding in several ways. Understanding occurs through thinking and talking about similarities, difference, patterns, etc. Conceptual understanding can also result as different representations for a concept are connected. If students have constructed strong connections, or networks, for a given concept, then conceptual understanding is likely and can influence the understanding of new relationships. For example, if students can see the connection between money, blocks, and decimals have to place value, they may demonstrate a conceptual understanding.
We can recognize that a student understands a concept through a variety of ways. A student understandings when he/she can connect prior knowledge to new knowledge. If they are able to represent something internally, understand patterns and relationships, and explain a concept, this may be evidence of understanding. For example, students have conceptual understanding of operation symbols such as +, -, x, and / if they understand each operation. If students know to add, subtract, multiply, or divide when given a certain symbol, they have conceptual understanding of what the symbols mean. They should be able to work out the operation on paper, explain what it means, and be able to represent it using manipulatives.
Understanding leads to many beneficial consequences. It can lead to the ability to more easily construct meaningful connections, promote retention, reduce the amount needed to be understood at a time, and enhance the transfer of knowledge to new situations. One example of a consequence of understanding is the connections between addition and multiplication. If a student is familiar with what it means to add, the student will more easily be able to grasp the multiplication means to add a certain number of times.
The growth of understanding occurs an understood concept is connected to new concepts. As connections are internalized, new connections can be constructed. For example, if the a student understands the idea of the addition of natural numbers, this can be extended integers and multiplication. An understanding of addition and multiplication can be extended to subtraction and division. These concepts can be built on by incorporating variables in algebra. As a result, we can see that understanding grows through connections to new concepts.
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