Part IV-Research on Student Learning
1. What specific misconceptions or difficulties might a student have about ideas in this topic?
Students often see graphs as literal pictures and do not realize that they are actually symbolic representations of situations. Similarly, students have a hard time time translating between graphical and algebraic representations, especially moving from a graph to an equation. They also read graphs point by point and ignore the global features such as increasing/decreasing, and minimum/maximum values.
2. Are there any suggestions as to what might contribute to students' misconceptions and how to address them?
In the Benchmarks for Science Literacy book, it states that little is known about how graphic skills are learned and how graph production is related to graph interpretations but it does mention that microcomputer-based laboratories are known to improve the development of students abilities to interpret graphs. These labs can help students learn that a graph is not a picture and that it is a symbolic representation of a situation.
4. How does the research draw attention to important prerequisite knowledge?
The research explains what prerequisite knowledge is needed and also explains the misconceptions that are associated with this knowledge. For example, the section of the book on graphing is the most important prerequisite for our lesson and it goes into a lot of detail about what students need to know about graphs and when they usually gain the misconceptions that they have. For instance, they say that students of all ages see graphs as literal pictures and that it is usually discovered in algebra that students do not see the global features of a graph. This information about prerequisite knowledge is helpful because it will be easier to discover where the students' misconceptions are stemming from.
5. What other new insights about the topic did you gain by reading this section?
Something interesting that was mentioned in the "Computation and Estimation" section was that calculators help build concept development as opposed to weakening students conceptual understanding which is usually what people believe about calculator use. The use of calculators has also been shown to improve test scores as well. Reading this section has allowed me to realize that calculator use is actually beneficial for the students and makes me want to try and include a calculator activity in my lesson plan.
Part III-Identify Concepts and Specific Ideas
Part II-Consider Instructional Implications
1. What suggestions are provided for effective instruction of the topic?
This section also discusses the importance of graphing in this concept and again makes it clear that students should be able to use data to crate a graph and also be able to use this graph to identify trends and better understand everyday situations. They say that the best way to get the students make these connections between graphs and the real data that they represent is to alert the students to the inconsistencies that generally arise from the creation of these graphs. For instance, it is important for students to realize that there is a certain amount of variability in the world and that there will never be a 100% accurate representation of any natural phenomenon.
2. What student learning difficulties, misconceptions, or developmental considerations are mentioned?
In the geometry section, the book mentions that students have difficulty creating and using mathematical proofs because they believe that doing a few examples is enough to prove a conjecture to be true. This is useful to know in all math because there are many instances in multiple topics in which students believe that they have found a new way to solve problems because it works on a few examples so it must work on all examples. It is important to recognize these misconceptions when they occur otherwise students will go on to later assignments with a false way of solving problems.
3. Does the reading suggest representations or everyday experiences that are effective in learning the ideas in the topic?
Yes the reading has many everyday experiences that could be used to teach this topic. For example, there is one situation shown that has the students create a sine graph that models the amount of daylight in Chicago. This activity would be an excellent way to teach students about amplitude, frequency, and phase because they will not only see the graphic representation of these but also see how they apply to a real world situation.
IIA
1. How does the general essay help you gain a K-12 "big picture view" of the topic?
The general essay shows the gradual progression of the topic by explaining what the students should learn throughout their schooling. It mentions that students begin by learning the vary basics of algebra such as learning how to use symbols and then move on to manipulating these symbols. It states that while many take algebra classes and learn to solve equations, they never really get to the steps where they apply these equations to real life and understand why the equations that they can solve are useful. This essay has backed up the fact that students should be introduced to real world applications more often while still keeping the problems at a level that is consistent with the students' knowledge.
IIB
1. How do the essays, tasks, and student work help you think about the instructional implications of the topic?
As mentioned in the last question, these tasks and student work help to show the importance of real world application problems in the lesson. These types of problems are easiest for students to understand because it helps them to apply what they are learning to something real instead of just being a random formula for them to memorize. Using real world applications allow the students to see that these formulas were made for a purpose and that they could even be applied to situations other than what the problem states and will hopefully help them to create their own equations for problems that they encounter in the future.
1. What specific misconceptions or difficulties might a student have about ideas in this topic?
Students often see graphs as literal pictures and do not realize that they are actually symbolic representations of situations. Similarly, students have a hard time time translating between graphical and algebraic representations, especially moving from a graph to an equation. They also read graphs point by point and ignore the global features such as increasing/decreasing, and minimum/maximum values.
2. Are there any suggestions as to what might contribute to students' misconceptions and how to address them?
In the Benchmarks for Science Literacy book, it states that little is known about how graphic skills are learned and how graph production is related to graph interpretations but it does mention that microcomputer-based laboratories are known to improve the development of students abilities to interpret graphs. These labs can help students learn that a graph is not a picture and that it is a symbolic representation of a situation.
4. How does the research draw attention to important prerequisite knowledge?
The research explains what prerequisite knowledge is needed and also explains the misconceptions that are associated with this knowledge. For example, the section of the book on graphing is the most important prerequisite for our lesson and it goes into a lot of detail about what students need to know about graphs and when they usually gain the misconceptions that they have. For instance, they say that students of all ages see graphs as literal pictures and that it is usually discovered in algebra that students do not see the global features of a graph. This information about prerequisite knowledge is helpful because it will be easier to discover where the students' misconceptions are stemming from.
5. What other new insights about the topic did you gain by reading this section?
Something interesting that was mentioned in the "Computation and Estimation" section was that calculators help build concept development as opposed to weakening students conceptual understanding which is usually what people believe about calculator use. The use of calculators has also been shown to improve test scores as well. Reading this section has allowed me to realize that calculator use is actually beneficial for the students and makes me want to try and include a calculator activity in my lesson plan.
Part III-Identify Concepts and Specific Ideas
Part II-Consider Instructional Implications
1. What suggestions are provided for effective instruction of the topic?
This section also discusses the importance of graphing in this concept and again makes it clear that students should be able to use data to crate a graph and also be able to use this graph to identify trends and better understand everyday situations. They say that the best way to get the students make these connections between graphs and the real data that they represent is to alert the students to the inconsistencies that generally arise from the creation of these graphs. For instance, it is important for students to realize that there is a certain amount of variability in the world and that there will never be a 100% accurate representation of any natural phenomenon.
2. What student learning difficulties, misconceptions, or developmental considerations are mentioned?
In the geometry section, the book mentions that students have difficulty creating and using mathematical proofs because they believe that doing a few examples is enough to prove a conjecture to be true. This is useful to know in all math because there are many instances in multiple topics in which students believe that they have found a new way to solve problems because it works on a few examples so it must work on all examples. It is important to recognize these misconceptions when they occur otherwise students will go on to later assignments with a false way of solving problems.
3. Does the reading suggest representations or everyday experiences that are effective in learning the ideas in the topic?
Yes the reading has many everyday experiences that could be used to teach this topic. For example, there is one situation shown that has the students create a sine graph that models the amount of daylight in Chicago. This activity would be an excellent way to teach students about amplitude, frequency, and phase because they will not only see the graphic representation of these but also see how they apply to a real world situation.
IIA
1. How does the general essay help you gain a K-12 "big picture view" of the topic?
The general essay shows the gradual progression of the topic by explaining what the students should learn throughout their schooling. It mentions that students begin by learning the vary basics of algebra such as learning how to use symbols and then move on to manipulating these symbols. It states that while many take algebra classes and learn to solve equations, they never really get to the steps where they apply these equations to real life and understand why the equations that they can solve are useful. This essay has backed up the fact that students should be introduced to real world applications more often while still keeping the problems at a level that is consistent with the students' knowledge.
IIB
1. How do the essays, tasks, and student work help you think about the instructional implications of the topic?
As mentioned in the last question, these tasks and student work help to show the importance of real world application problems in the lesson. These types of problems are easiest for students to understand because it helps them to apply what they are learning to something real instead of just being a random formula for them to memorize. Using real world applications allow the students to see that these formulas were made for a purpose and that they could even be applied to situations other than what the problem states and will hopefully help them to create their own equations for problems that they encounter in the future.