Monday, July 6, 2015
Re-teach Reflection
In previous classes we have talked a lot about how it is important to adjust your teaching methods to best fit your students. This assignment was the first time I have actually taken a student work sample and applied this concept. It was difficult, but after we finished it was easy to see how you can use what your students do and do not understand to better your ability to teach them. It is important to look at students individually and as a whole so you can see where certain students may be struggling vs. where there may be holes in your teaching methods. It is just as much of a learning experience for us to teach our students as it is for our students to learn what we are trying to teach them.
Wednesday, July 1, 2015
Error Reflection
Throughout this course we have talked a lot about how making math more inquiry based we can learn more about what students know because the responses we expect leave room for more explanation. In fact, explanation is encouraged and essential in most if not all inquiry based assignments. We have talked about how we know what students are grasping by looking at their correct answers and explanations. However, after completing the NAEP assignment and the reteach assignment based on the 14 error problems I think that there is more to be learned about individual student knowledge based on what they are doing wrong. If a student grasps a concept and can maneuver their way through it that is great. When they can't, if we have them explain their reasoning we can better understand exactly where they are going wrong. We can better understand how they are processing what we are telling them. This gives us a starting point as to how we can begin to explain things in a way that each student will better understand. I think that the error analysis is more beneficial and useful to us and in the end to the students as well.
Curriculum Plan Reflection
This was by far the most stressful and difficult assignment in this course. It was a challenge to consider how each and every content area for a grade band fits into one another and how you might choose to teach each one. When you are trying to write individual lesson plans on a specific topic it can sometimes be difficult to know what to include and what to leave out. However, after completing this assignment it gives me a better idea of how lesson planning will go when I have an actual teaching position. I think that it will be much easier, not only in math but in all content areas, to lesson plan when you have at least a general idea of how the entire year will play out. Not only knowing what students need to cover on a yearly basis, but knowing what they have covered and what you are leading them up to will aid in a better understanding of what you need to cover specifically. This project gave me a glimpse of what it takes to plan for students learning long term and how important it is to communicate with teachers at all grade levels to know what they are covering as well.
Monday, June 29, 2015
Assessment Reflection
In this class we have talked about assessment a lot. What it means, different types, how to create quality assessments, and how to assess ourselves. Without assessment how would we really know what we are learning? Assessment is a very important part of the learning process. When we assess we can learn how to improve ourselves, what our students are grasping, and what we need to work harder at, as teachers, to help our students improve. In order to achieve all of these we must create quality and useful assessments. The NAEP project really helped me to see how difficult it can be to assess all students equally if you are not using a quality rubric. All students think differently and may arrive at an answer to a problem differently, but we have to have a way to score their understandings. I like rubrics, I think that you can assess just about anything with them, but they are very difficult to create.
Classroom Changes Reflection
I have to say the the CCSS-M SMP's and the NCTM Process Standards were very intimidating at first. It was a lot of information to look at and digest all at the same time. Once we, as a class, really looked at what each of the standards meant and clarified any misunderstandings it made them all much less intimidating. I think that we have had ample opportunity in this class to work with the standards and actually apply them with assignments like our lesson segments. Just like with anything else the more practice you have working with something the easier it will become and the more comfortable you will get with it. I think that I still have a little bit of practice ahead of me before I am really good at working with these standards, but I have no doubt that one day I will know these standards backwards and forwards without batting an eye to think about it.
Another big change I have seen in the math curriculum is the push for less traditional methods of teaching math. The CCSS-M and NCTM Process Standards most definitely lean in favor of this as well. Students are expected to answer more open-ended questions and be able to explain their reasoning behind things rather then simply being able to complete a process or plug numbers into a formula. They must know how and why they are doing things. With the NAEP project and reading various article throughout the semester I think that this is going to be a definite struggle for me to teach students math in this way. Creating an all encompassing rubric that works the same on all answers, which may be completely different, for the open-ended problems that are called for now also seems to be an impossible task.
Another big change I have seen in the math curriculum is the push for less traditional methods of teaching math. The CCSS-M and NCTM Process Standards most definitely lean in favor of this as well. Students are expected to answer more open-ended questions and be able to explain their reasoning behind things rather then simply being able to complete a process or plug numbers into a formula. They must know how and why they are doing things. With the NAEP project and reading various article throughout the semester I think that this is going to be a definite struggle for me to teach students math in this way. Creating an all encompassing rubric that works the same on all answers, which may be completely different, for the open-ended problems that are called for now also seems to be an impossible task.
Technology Reflection
If I didn't have my computer as a college student it would seem nearly impossible to complete anything. On top of that we always have our phones in our hands or the tv on or some sort of technology we are using for most of the day. It seems essential to our daily function now a days. Technology has been incorporated into many different subjects throughout my schooling days, but when I thought about math I always thought about using a pencil and paper. In this class not only are we learning about how to use technology to teach math with things like the smart board, apps, and applets, but we are also using technology to learn about teaching math. We have watched various videos and used this blog to reflect on our ideas. I am seeing the use of technology as a more beneficial aid to student learning, even in math, than I ever did before. I think that it can enhance the learning experience and offers a variety of ways to assist students in learning math.
Manipulative Reflection
I really liked the in class activity we did to help us brainstorm various different ways that we can use different manipulatives. As educators we often have to be creative and get kids involved without spending a lot or any money at all on resources. By having us switch stations and keep thinking about what each manipulative could be used for I think that it helped us to come up with some really useful ways to use the different manipulatives. It was about making the tools as versatile as we can.
I think that you can tell a student really grasps something when they can use the manipulatives to show you a concept and explain it to you thoroughly while they are doing so. When they use the manipulatives as a tool for explanation you know they are really benefitting the child and helping them understand a concept in a more concrete way. Because of their deeper understanding through the use of the manipulative I think that students can more easily transfer concepts into other situation or domains. I think that one of the most beneficial ways to assess students is through observation and conversation. If you can watch what a student is doing when they are trying to learn and also talk about their processes with them you gain more useful knowledge about what they know. I think that if you talk to students individually to assess them even when they are working in groups on various concepts you can more accurately assess what they know and understand. You are improving a students problem solving skills by using manipulatives because you are giving them another tool they can use to solve the problem they are working on or the concept they are trying to understand.
Tuesday, June 23, 2015
Readings for June 23rd
Getting Started With Open-Ended Assessment (broken calculator)
This article talks about different ways you can begin to implement open-ended assessment into your classroom. It also talks about the positives and negatives. It is important to create an environment conducive to sharing, make sure students understand your expectations, give examples of what you expect, be patient, keep trying, and there is an online database available with open-ended questions you can use until you become more comfortable writing your own. Your students will become more confident in themselves and their ability to learn and you will be able to better understand what they understand. These are difficult to implement because they take more time to create, administer, and grade. Responses vary greatly and this can make consistent grading difficult. If you don't ask the question correctly it may be difficult to get what you want out of the answer.
A Smorgasbord of Assessment Options
This article talks about how in order for assessment to be effective you need to know exactly what you are assessing, who it's for, and why you need it. It should guide instruction. Teachers should use a variety of assessment types and have a good idea of what students are thinking. (Van Heile model of geometric thought: visualization, analysis, informal deduction, deduction, rigor). Assessment should be integrated into each unit. Before. During. After. It should allow for students to deepen their understanding of the concepts they are working with.
Understanding Student to Open-Ended Tasks
This article describes an open-ended geometry question asked to a class of sixth graders. They had to find the area of an irregular object. I feel like the teacher had a hard time being able to describe what she actually expected from her students. She wanted everything to be super specific. Students should be able to explain themselves, but I don't think all of them had a clear idea of her expectations. Students should be able to show detail in their understanding when explaining their work, but they can't be expected to do so in the very first open-ended problem you give them.
Assessing Students' Understanding Through Conversations
This article talks about how it can be an effective assessment tool to simply have a conversation with your students. You can talk to them and have them explain things to you. You can have them talk to each other or share with the class. By simply listening to their responses and explanations you can spot areas of difficulty or areas they may struggle with. You can also see what they understand very well because they will tell you exactly what how and why they did something. It is important to establish an open environment in your classroom that allows students to feel as if they can share their ideas and speak openly and honestly without fear of ridicule.
An Experiment In Using Portfolios in Middle School
This article talks about a math teacher who began using portfolios as a type of assessment in her class. Students were required to include things like work they had corrected to show their growth. They included work that they enjoyed doing and understood really well to show their strengths. Many of the assignments were accompanied by introductions and reflections. All of these pieces help a teacher to get a much fuller understanding of what a student is grasping and where they need extra assistance. The teacher in this article used the creating of the portfolios not only as assessment for her students, but as assessment for herself as well. She adapted her teaching to benefit her students.
This article talks about different ways you can begin to implement open-ended assessment into your classroom. It also talks about the positives and negatives. It is important to create an environment conducive to sharing, make sure students understand your expectations, give examples of what you expect, be patient, keep trying, and there is an online database available with open-ended questions you can use until you become more comfortable writing your own. Your students will become more confident in themselves and their ability to learn and you will be able to better understand what they understand. These are difficult to implement because they take more time to create, administer, and grade. Responses vary greatly and this can make consistent grading difficult. If you don't ask the question correctly it may be difficult to get what you want out of the answer.
A Smorgasbord of Assessment Options
This article talks about how in order for assessment to be effective you need to know exactly what you are assessing, who it's for, and why you need it. It should guide instruction. Teachers should use a variety of assessment types and have a good idea of what students are thinking. (Van Heile model of geometric thought: visualization, analysis, informal deduction, deduction, rigor). Assessment should be integrated into each unit. Before. During. After. It should allow for students to deepen their understanding of the concepts they are working with.
Understanding Student to Open-Ended Tasks
This article describes an open-ended geometry question asked to a class of sixth graders. They had to find the area of an irregular object. I feel like the teacher had a hard time being able to describe what she actually expected from her students. She wanted everything to be super specific. Students should be able to explain themselves, but I don't think all of them had a clear idea of her expectations. Students should be able to show detail in their understanding when explaining their work, but they can't be expected to do so in the very first open-ended problem you give them.
Assessing Students' Understanding Through Conversations
This article talks about how it can be an effective assessment tool to simply have a conversation with your students. You can talk to them and have them explain things to you. You can have them talk to each other or share with the class. By simply listening to their responses and explanations you can spot areas of difficulty or areas they may struggle with. You can also see what they understand very well because they will tell you exactly what how and why they did something. It is important to establish an open environment in your classroom that allows students to feel as if they can share their ideas and speak openly and honestly without fear of ridicule.
An Experiment In Using Portfolios in Middle School
This article talks about a math teacher who began using portfolios as a type of assessment in her class. Students were required to include things like work they had corrected to show their growth. They included work that they enjoyed doing and understood really well to show their strengths. Many of the assignments were accompanied by introductions and reflections. All of these pieces help a teacher to get a much fuller understanding of what a student is grasping and where they need extra assistance. The teacher in this article used the creating of the portfolios not only as assessment for her students, but as assessment for herself as well. She adapted her teaching to benefit her students.
Thursday, June 18, 2015
Journal Articles
Taking It to the Next Level: Students Using Inductive Reasoning
Jaclyn M. Murawska & Alan Zollman
This article talks about using a geoboard activity along with an inquiry continuum to promote the use of inductive reasoning among students. On the continuum the lowest level is confirmation inquiry going higher to structured, then guided and the highest level open. For this activity in the first three levels the students are given the question that they must answer about their geoboards and in the third level they must come up with their own questions to investigate. As you progress through the activity the students receive less support in working with the question at hand. This increases their ability to reason inductively without overwhelming them.
I liked the idea of starting at the first level and working your way up the continuum throughout the lesson in order to really support your students learning and to slowly back off without giving students too much all at once. I think that the idea of using the continuum for instruction in math is beneficial to students and can be used for various different lessons and concepts. I also liked that they talked about not only teaching one standard in the lesson, but rather teaching standards that are connected or linked with each other. They go hand in hand and work well with each other. I like the general idea behind the lesson and the specific activity described in the article and would more than likely use it in my own classroom.
"I Don't Really Know How I Did That!"
R. Scott Eberle
This article explores using tessellations as a valuable learning tool in geometry. With these pattern blocks students can explore mathematical complexity, symmetry, validity, uniqueness, units, surprise, and connections. The activities described in this article seemed endless. It talked about having students create patterns with one shape and multiple shapes. It talked about challenging them to create patterns that had reflective lines of symmetry and rotational symmetry. The article also talked about how students like to make connections to real world objects that they recognize when they are working with the pattern blocks. This can be a good thing, but it can also block their creativity of they focus on that too much.
I think that these blocks are good for a variety of different ages and allow for a wide variety of activities that students can learn from. In the younger grades you can introduce these blocks and have students start looking at the number of sides and corners that shapes have. You can have them start creating patterns that continue in one direction. Then you can begin looking at angles and patterns that extend on forever in all directions as the students get older.
Jaclyn M. Murawska & Alan Zollman
This article talks about using a geoboard activity along with an inquiry continuum to promote the use of inductive reasoning among students. On the continuum the lowest level is confirmation inquiry going higher to structured, then guided and the highest level open. For this activity in the first three levels the students are given the question that they must answer about their geoboards and in the third level they must come up with their own questions to investigate. As you progress through the activity the students receive less support in working with the question at hand. This increases their ability to reason inductively without overwhelming them.
I liked the idea of starting at the first level and working your way up the continuum throughout the lesson in order to really support your students learning and to slowly back off without giving students too much all at once. I think that the idea of using the continuum for instruction in math is beneficial to students and can be used for various different lessons and concepts. I also liked that they talked about not only teaching one standard in the lesson, but rather teaching standards that are connected or linked with each other. They go hand in hand and work well with each other. I like the general idea behind the lesson and the specific activity described in the article and would more than likely use it in my own classroom.
"I Don't Really Know How I Did That!"
R. Scott Eberle
This article explores using tessellations as a valuable learning tool in geometry. With these pattern blocks students can explore mathematical complexity, symmetry, validity, uniqueness, units, surprise, and connections. The activities described in this article seemed endless. It talked about having students create patterns with one shape and multiple shapes. It talked about challenging them to create patterns that had reflective lines of symmetry and rotational symmetry. The article also talked about how students like to make connections to real world objects that they recognize when they are working with the pattern blocks. This can be a good thing, but it can also block their creativity of they focus on that too much.
I think that these blocks are good for a variety of different ages and allow for a wide variety of activities that students can learn from. In the younger grades you can introduce these blocks and have students start looking at the number of sides and corners that shapes have. You can have them start creating patterns that continue in one direction. Then you can begin looking at angles and patterns that extend on forever in all directions as the students get older.
Sunday, June 14, 2015
Number Operations: Multiplication and Division (video)
This lesson really focused on multiplication and division and using these operations in word problems. The students were asked to explain multiplication and division in their own words and using examples. I like how the teacher started off with the quote "A picture is worth a 1000 words" and she came back around to that in the end. I thought that the teacher did a really good idea of guiding the students through the lesson rather than soon feeding everything to them. She really made the students think about the problems they worked with and she allowed them to go in whichever direction they felt correct and then she redirected them.
After listening to the explanations that students gave for their work I think that they have a pretty solid grasp on multiplication and division. Most of them were spouting off expressions like nobody's business. However, when they were asked to explain them they couldn't. They had no clue what they were writing down they just knew how to do it. This goes back to knowing the difference between procedure and concept. These students had the procedure down, but they weren't fully understanding what the concept meant.
I really like this lesson. I like how the students have time to think on their own, with partners, in groups, and as a whole class. I think that all of these different options helps students to fully develop their thoughts. I like how they went over various different strategies and there was only one problem presented to the class at one time. The teacher tried to keep them focused and actively thinking to the best of there ability and I think that this lesson helped to do that.
After listening to the explanations that students gave for their work I think that they have a pretty solid grasp on multiplication and division. Most of them were spouting off expressions like nobody's business. However, when they were asked to explain them they couldn't. They had no clue what they were writing down they just knew how to do it. This goes back to knowing the difference between procedure and concept. These students had the procedure down, but they weren't fully understanding what the concept meant.
I really like this lesson. I like how the students have time to think on their own, with partners, in groups, and as a whole class. I think that all of these different options helps students to fully develop their thoughts. I like how they went over various different strategies and there was only one problem presented to the class at one time. The teacher tried to keep them focused and actively thinking to the best of there ability and I think that this lesson helped to do that.
Thursday, June 11, 2015
NAEP Reflection
Throughout my college career I have encountered, worked with, and created numerous different types of assessments. The idea behind assessment is to determine student knowledge based on a certain set of criteria. Some assessments are better able to depict student knowledge than others can even begin to be capable of. I honestly have to say that I think the rubrics we worked with when completing the NAEP project are some of the most difficult to understand and use/implement out of all the assessments I have come across. It was very difficult to decide between the different levels that you were given to grade student work. There was a lot of overlap that made it even harder. The rubrics we looked at were sometimes unclear and hard to understand exactly what it was looking for.
This project was a difficult, but also a very valuable learning experience for me. When we struggle with something I believe we take the most out of it because we have the most invested in it. The difficulty in this assignment helped me to truly see how important it is to create accurate assessments to evaluate student work, not only to do the student justice when looking at their hard work, but also for your own sanity. If an assessment is difficult to read and understand than how can we expect it to accurately depict a students work. If we can't figure out how to equally apply it to multiple works then it's not a very accurate scale either. When creating rubrics it is important to be very specific about what you expect from your students in order for them to be able to deliver the result you expect and for them to show you what they are truly capable of. Although this assignment was very trying it was very valuable as well.
Math Apps and Applets
2048 in the Apple App Store
This app helps students practice basic addition in doubling while also working with patterns and mathematical logic. You start with 2's. You swipe the screen up, down, left, or right to add the twos together. This continues in a pattern of 2, 4, 8, 16 and so on. You must think about which way you swipe the numbers in order to add them together and create bigger numbers without filling up your board. When you do the game is over. This app helps students with simple addition that gradually gets more difficult while working with patterns in doubling. Students must use their logic skills to avoid filling up their game board because they must think about how they will move their number tiles. This app is simple to use and to understand. I like that this app helps with patterns and addition while it is a game. The students are thinking without realizing it. I wouldn't use this game in my curriculum, but I would allow students to play this game during free time or recess in order to keep them thinking about math while they may not even realize it. It is good addition practice to keep their minds working.
Basic Division: http://www.adaptedmind.com/index.php
The specific applet I choose to look at from this website focused on division. It included a combination of word problems and basic algorithms into the game. In the beginning you choose the lesson you wish to work on and then you get to choose a character/monster as your game piece as you move through the levels in the lesson. The game levels map reminds me of the way it is set up in Candy Crush Saga. I think that this is a positive to keep students intrigued and wanting to continue to see which level they can reach. There are many positives and negatives to this applet. It is set up so that a parent can type in their email and it will save their child's progress in the game. This is a great feature. However, after the one-month free trial is up you have to pay for the use of the website which could be problematic if using this in a classroom. It only lets you play the first level without signing up, but I am assuming that each level gets progressively more difficult. This is good because it helps with student progression. I also like that when a student answers the question their is an button that says "explain." If a student is struggling with the problem their is a video that explains the problem to help them better understand. I could see using this applet in a classroom setting during center time. I think that it would really benefit students. The website as a whole includes various different lessons over grades 1-8 so it can be used throughout as you teach new concepts as a valuable practice tool for students. I also like that it benefits students at all different levels whether it be struggling, at grade level, or students who are excelling. Students can move through the levels at their own pace.
Base Blocks Addition: http://nlvm.usu.edu/en/nav/category_g_2_t_1.html
This applet helps students practice their addition skills by using base-10 blocks. On the right hand side it gives them a problem and the representing base-10 blocks on the left. The students can move around the blocks and combine them together to create larger units. Students can also create their own problems which can help with practice on problems they struggle with. This could also be used as a beneficial study tool as well. I had to play around with this applet a little bit to figure it out, but once you do it is simple to use and I think it would be easy for students to use once they are showed as well. I think that the visual representation that this applet provides can benefit students and help them to better understand the concept behind an addition problem rather than simply how to find the answer. Overall I really like this applet and would use it in my classroom as another means for students to learn and practice their addition skills.
This app helps students practice basic addition in doubling while also working with patterns and mathematical logic. You start with 2's. You swipe the screen up, down, left, or right to add the twos together. This continues in a pattern of 2, 4, 8, 16 and so on. You must think about which way you swipe the numbers in order to add them together and create bigger numbers without filling up your board. When you do the game is over. This app helps students with simple addition that gradually gets more difficult while working with patterns in doubling. Students must use their logic skills to avoid filling up their game board because they must think about how they will move their number tiles. This app is simple to use and to understand. I like that this app helps with patterns and addition while it is a game. The students are thinking without realizing it. I wouldn't use this game in my curriculum, but I would allow students to play this game during free time or recess in order to keep them thinking about math while they may not even realize it. It is good addition practice to keep their minds working.
Basic Division: http://www.adaptedmind.com/index.php
The specific applet I choose to look at from this website focused on division. It included a combination of word problems and basic algorithms into the game. In the beginning you choose the lesson you wish to work on and then you get to choose a character/monster as your game piece as you move through the levels in the lesson. The game levels map reminds me of the way it is set up in Candy Crush Saga. I think that this is a positive to keep students intrigued and wanting to continue to see which level they can reach. There are many positives and negatives to this applet. It is set up so that a parent can type in their email and it will save their child's progress in the game. This is a great feature. However, after the one-month free trial is up you have to pay for the use of the website which could be problematic if using this in a classroom. It only lets you play the first level without signing up, but I am assuming that each level gets progressively more difficult. This is good because it helps with student progression. I also like that when a student answers the question their is an button that says "explain." If a student is struggling with the problem their is a video that explains the problem to help them better understand. I could see using this applet in a classroom setting during center time. I think that it would really benefit students. The website as a whole includes various different lessons over grades 1-8 so it can be used throughout as you teach new concepts as a valuable practice tool for students. I also like that it benefits students at all different levels whether it be struggling, at grade level, or students who are excelling. Students can move through the levels at their own pace.
Base Blocks Addition: http://nlvm.usu.edu/en/nav/category_g_2_t_1.html
This applet helps students practice their addition skills by using base-10 blocks. On the right hand side it gives them a problem and the representing base-10 blocks on the left. The students can move around the blocks and combine them together to create larger units. Students can also create their own problems which can help with practice on problems they struggle with. This could also be used as a beneficial study tool as well. I had to play around with this applet a little bit to figure it out, but once you do it is simple to use and I think it would be easy for students to use once they are showed as well. I think that the visual representation that this applet provides can benefit students and help them to better understand the concept behind an addition problem rather than simply how to find the answer. Overall I really like this applet and would use it in my classroom as another means for students to learn and practice their addition skills.
Monday, June 8, 2015
"A Model for Understanding: Understanding in Mathematics" & "Thinking Through a Lesson: Successfully Implementing High-Level Tasks"
"A Model for Understanding: Understanding in Mathematics"
According to this article in order to be able to understand something you must be able to do some of the following: give examples, explain it in your own words, make connections, use it in different ways, see the opposite, see the consequences, and recognize it in multiple circumstances (Davis, 190). This applies in mathematics as well. However, "Understanding a concept is different from understanding a procedure" (Davis, 191). The way I comprehend what the author is trying to say is that when you complete a problem, understanding how to get to your answer is different from understanding the concept behind it. When we teach students how to solve a problem we use logical thinking to get them there. Students ask questions, complete exercises, explain, and demonstrate to show their understanding. We must help students understand a concept by using physical representation and then moving to more symbolic methods. The way I understand it, if we don't teach students the concepts behind problems at their level they will have understanding the significance of the procedure. They may be able to complete the procedure, but have no clue what it means.
Davis, E.J. (2006). A model for understanding: Understanding in mathematics. Mathematics Teaching in the Middle School, 12(4), 190-197.
"Thinking Through a Lesson: Successfully Implementing High-Level Tasks"
The TTLP method of lesson planning for higher level thinking tasks focuses heavily on organization and a lot of preparation in advance. It talks a lot about how you need to plan out exactly what you expect from your students, what you expect to get from them, and how you can help them get to their correct solution. Although their may be more than 1 path to the solution their if often 1 solution. It is important to not to heavily guide students through their thinking process. Instead have questions prepared for every situation you can think of in order to be best prepared on how to guide them. The less you leave up to chance the better prepared you will be. I think that one of the most important things to take from this article is that all students learn and think differently and it is important to embrace that and to put in the extra effort that it takes to make sure that each and every student is able to explore their thinking processes.
Smith, M.G., Bill, V., & Hughes, E.K.
Thinking through a lesson: Successfully implementing high-level tasks.
Designing and Enacting Rich Instructional Experiences (pgs. 11-18)
According to this article in order to be able to understand something you must be able to do some of the following: give examples, explain it in your own words, make connections, use it in different ways, see the opposite, see the consequences, and recognize it in multiple circumstances (Davis, 190). This applies in mathematics as well. However, "Understanding a concept is different from understanding a procedure" (Davis, 191). The way I comprehend what the author is trying to say is that when you complete a problem, understanding how to get to your answer is different from understanding the concept behind it. When we teach students how to solve a problem we use logical thinking to get them there. Students ask questions, complete exercises, explain, and demonstrate to show their understanding. We must help students understand a concept by using physical representation and then moving to more symbolic methods. The way I understand it, if we don't teach students the concepts behind problems at their level they will have understanding the significance of the procedure. They may be able to complete the procedure, but have no clue what it means.
"Thinking Through a Lesson: Successfully Implementing High-Level Tasks"
The TTLP method of lesson planning for higher level thinking tasks focuses heavily on organization and a lot of preparation in advance. It talks a lot about how you need to plan out exactly what you expect from your students, what you expect to get from them, and how you can help them get to their correct solution. Although their may be more than 1 path to the solution their if often 1 solution. It is important to not to heavily guide students through their thinking process. Instead have questions prepared for every situation you can think of in order to be best prepared on how to guide them. The less you leave up to chance the better prepared you will be. I think that one of the most important things to take from this article is that all students learn and think differently and it is important to embrace that and to put in the extra effort that it takes to make sure that each and every student is able to explore their thinking processes.
Smith, M.G., Bill, V., & Hughes, E.K.
Thinking through a lesson: Successfully implementing high-level tasks.
Designing and Enacting Rich Instructional Experiences (pgs. 11-18)
Thursday, June 4, 2015
RIch Activity Lesson (Tin Man)
I learned a ton about trial and error in this lesson. You may expect a lesson to go one way and it will end up going somewhere completely different. I think that the lesson went over very well over all. I think that the main idea behind the lesson worked well and that everyone enjoyed the activity. It took forever to cut out the foil pieces and the lesson felt a little rushed. This was a lot to complete in such a short amount of time. We slightly modified this lesson for class so the whole class made a tin man together rather than in groups. It allowed students to work with a couple different shapes without them being overwhelmed with all of them. We talked about precutting squares and splitting the lesson up into 2 or 3 days in order to help solve some of these issues.
I think that the other two groups learned a lot from trial and error today as well. We were all able to give each other perspectives that we may not have thought of on our own when we were reflecting after each lesson. I really liked all of the activities that were taught today and I can see myself using them in future classrooms if the situation fits. I haven't really thought of math as a hands on content area until this class. I feel like everything we do or talk about proves that theory of mine more and more wrong.
Thursday, May 28, 2015
CCSSM Reflection
Throughout this assignment I have learned a lot more than I honestly thought I would when looking at a bunch of standards. When looking at the two standards that I was assigned specifically I learned just how much they lean towards an inquiry basis. The same is true for the whole set of standards. When watching the other presentations and listening to the examples provided you can see the opportunity for inquiry based math lessons using the common core state math standards. Inquiry based instruction is what we are moving towards so it is positive that the standards work well with that. These standards focus on a deeper level of thinking and being able to explain the reasoning behind your solutions. They encourage students to understand mathematical concepts rather than simply how to plug numbers into a formula. Students are really challenged to think deeper with these standards. I think that these standards are a step in the right direction to not only communize standards across the state, but also to encourage an inquiry based structure in the classroom and more thoughtful students.
Wednesday, May 27, 2015
"Preserving Pelicans with Models That Make Sense" & "Count On It: Congruent Manipulative Displays"
"Preserving Pelicans with Models That Make Sense"
This article talks about model-eliciting activities, specifically an activity relating to pelicans and the concentration of their colonies. An MEA is an activity that allows students to work with models in order to help form their understanding of concepts. In the articles example students were trying to come up with the best way to estimate the number of pelican colonies in an area. Students worked in small groups to come up with the best way to estimate this number.
I think that this kind of activity is very beneficial to student learning if implemented properly. It allows students to share, discuss, and revise their ideas and to really think through a problem situation. I would use this type of activity in many different situations in my own classroom. I personally think that the MEA is best suited to topics that students can relate to the most because it can be very time consuming and involves a lot of thought to carry out.
Moore, T.J., Doerr, H.M., Glancy, A.W., & Ntow, F.D. (2015). Preserving pelicans with models that make sense. Mathematics Teaching in the Middle School, 20(6), 358-364.
"Count On It: Congruent Manipulative Displays"
This article talks about the use of manipulatives. As students we have had plenty of experience working with them to assist in our own learning, but thinking about how to effectively use a manipulative to teach your future students seems less simple. You must make sure that you are properly connecting the concept being taught with the manipulative being used. It is also important to understand that when you start using manipulatives with your students you begin with very concrete examples. The more the students learn the more abstract the manipulatives can be. this article also talks about the importance of guiding students to use manipulatives in their own independent study.
Throughout my personal experience using manipulatives I am able to understand the positives that they can have when working with them. I had never really thought about how difficult it is to incorporate them into a lesson before though. Obviously they help some people but how do you introduce them effectively. I can see my students using all kinds of manipulatives in my classroom, whether it be during a lesson or while working on their homework, as long as the manipulative fits the task at hand. I am a very hands on person and when you have to work through a difficult problem they can be very helpful.
Morin, J., & Samelson, V.M. (2015). Count on it: Congruent manipulative displays. Teaching Children Mathematics, 21(6), 362-370.
This article talks about model-eliciting activities, specifically an activity relating to pelicans and the concentration of their colonies. An MEA is an activity that allows students to work with models in order to help form their understanding of concepts. In the articles example students were trying to come up with the best way to estimate the number of pelican colonies in an area. Students worked in small groups to come up with the best way to estimate this number.
I think that this kind of activity is very beneficial to student learning if implemented properly. It allows students to share, discuss, and revise their ideas and to really think through a problem situation. I would use this type of activity in many different situations in my own classroom. I personally think that the MEA is best suited to topics that students can relate to the most because it can be very time consuming and involves a lot of thought to carry out.
Moore, T.J., Doerr, H.M., Glancy, A.W., & Ntow, F.D. (2015). Preserving pelicans with models that make sense. Mathematics Teaching in the Middle School, 20(6), 358-364.
"Count On It: Congruent Manipulative Displays"
This article talks about the use of manipulatives. As students we have had plenty of experience working with them to assist in our own learning, but thinking about how to effectively use a manipulative to teach your future students seems less simple. You must make sure that you are properly connecting the concept being taught with the manipulative being used. It is also important to understand that when you start using manipulatives with your students you begin with very concrete examples. The more the students learn the more abstract the manipulatives can be. this article also talks about the importance of guiding students to use manipulatives in their own independent study.
Throughout my personal experience using manipulatives I am able to understand the positives that they can have when working with them. I had never really thought about how difficult it is to incorporate them into a lesson before though. Obviously they help some people but how do you introduce them effectively. I can see my students using all kinds of manipulatives in my classroom, whether it be during a lesson or while working on their homework, as long as the manipulative fits the task at hand. I am a very hands on person and when you have to work through a difficult problem they can be very helpful.
Morin, J., & Samelson, V.M. (2015). Count on it: Congruent manipulative displays. Teaching Children Mathematics, 21(6), 362-370.
Monday, May 25, 2015
"Word Problem Clues" (video)
I think that this lesson was very well thought out and there was a lot of planning behind it. However, I think when implementing this lesson it went in an entirely different direction than previously planned for. In some aspects the lesson ended up being kind of all over the place. The initial intention of the lesson was to use the clues in the word problems to help you figure out the problem. I think that she did a really good job of trying to incorporate this into the lesson, but the students continued to be confused by the problem and I felt she just kept asking the same questions to try and clear up the confusion. I think that the students have a solid foundation of addition and subtraction and strategies to complete both, but they struggle with applying them and little mistakes upon completion.
As the teacher described in the beginning, many of the students, even after the teacher directed portion of the lesson, still wanted to simply add the numbers together and say that was their answer. They weren't focusing on what the problem actually said and meant. I understand why this lesson may not have gone in the intended direction. With more time for examples I think this lesson could have flowed more smoothly. Maybe the teacher could have gone through more examples or strategies together. Some students were beginning to grasp it and possible strategies, but were making mistakes along the way.
Overall, I really like the idea for this lesson. I love the opportunity for students to explain their work and really talk about how they arrived at the answer they came up with. I like the student reflection on their own work and the opportunity for students to reevaluate their work and fix possible mistakes. I think this could have been more beneficial if students had the opportunity to see and walk through correct answers after this lesson so they could compare and learn from their mistakes. This lesson idea is something that I can most definitely see myself using in my own classroom.
As the teacher described in the beginning, many of the students, even after the teacher directed portion of the lesson, still wanted to simply add the numbers together and say that was their answer. They weren't focusing on what the problem actually said and meant. I understand why this lesson may not have gone in the intended direction. With more time for examples I think this lesson could have flowed more smoothly. Maybe the teacher could have gone through more examples or strategies together. Some students were beginning to grasp it and possible strategies, but were making mistakes along the way.
Overall, I really like the idea for this lesson. I love the opportunity for students to explain their work and really talk about how they arrived at the answer they came up with. I like the student reflection on their own work and the opportunity for students to reevaluate their work and fix possible mistakes. I think this could have been more beneficial if students had the opportunity to see and walk through correct answers after this lesson so they could compare and learn from their mistakes. This lesson idea is something that I can most definitely see myself using in my own classroom.
Wednesday, May 20, 2015
Standards Articles
Construct Viable Arguments & Critique the Reasoning (3)
The String Task: Not Just for High School
This article talks about functional thinking or relationships amongst numbers. An experiment was conducted in classrooms grades 3-5 to promote algebra early on in school. The types of problems in this article that were used to teach functional thinking, lent themselves well to working with standard 3. The first problem had to do with cutting a piece of string a number of times. Students had to figure out the relationship between the number of cuts and the number of strings. "Students were encouraged to discuss their mathematical thinking and to use multiple representations to communicate their ideas with their peers… [and] to explain their thinking" (Isler, 285). Students had ample opportunity to talk with other student or the teacher. They came up with a relationship between the two numbers in partners, represented their data with charts and pictures, and explained their reasoning. This worked the same way for the Brady problem. Except this problem was square tables and how many people could sit at each.
Isler, I., Marum, T., Stephens, A., Blanton, M., Knuth, E., & Gardiner, A.M. (2014). The string task: Not just for high school. Teaching Children Mathematics, 21(5), 282-292. Retrieved from http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue5/The-String-Task-Not-Just-for-High-School/
Use Appropriate Tools Strategically (5)
Mapping the Way to Content Knowledge
This article talks about using a content map in order to better understand how to teach subtraction as a preservice teacher. This is an example of how the teacher and the student can benefit from a tool. There are various other examples of tools being used throughout the article. A student is working to solve 70 - 23 and get an answer of 53 using the traditional method. When she repeats the problem again using base-10 block and a hundred chart she gets the problem correct. The particular example of the subtraction content map is something that I have benefited from as a preservice teacher and the other tools described in this article can be beneficial for future students.
Poling, L.L., Goodson-Epsy, T., Dean, C., Lynch-Davis, K., & Quickenton, A. (2015). Mapping the way to content knowledge. Teaching Children Mathematics, 21(9), 538-547. Retrieved from http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol21/Issue9/Mapping-the-Way-to-Content-Knowledge/
The String Task: Not Just for High School
This article talks about functional thinking or relationships amongst numbers. An experiment was conducted in classrooms grades 3-5 to promote algebra early on in school. The types of problems in this article that were used to teach functional thinking, lent themselves well to working with standard 3. The first problem had to do with cutting a piece of string a number of times. Students had to figure out the relationship between the number of cuts and the number of strings. "Students were encouraged to discuss their mathematical thinking and to use multiple representations to communicate their ideas with their peers… [and] to explain their thinking" (Isler, 285). Students had ample opportunity to talk with other student or the teacher. They came up with a relationship between the two numbers in partners, represented their data with charts and pictures, and explained their reasoning. This worked the same way for the Brady problem. Except this problem was square tables and how many people could sit at each.
Isler, I., Marum, T., Stephens, A., Blanton, M., Knuth, E., & Gardiner, A.M. (2014). The string task: Not just for high school. Teaching Children Mathematics, 21(5), 282-292. Retrieved from http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue5/The-String-Task-Not-Just-for-High-School/
Use Appropriate Tools Strategically (5)
Mapping the Way to Content Knowledge
This article talks about using a content map in order to better understand how to teach subtraction as a preservice teacher. This is an example of how the teacher and the student can benefit from a tool. There are various other examples of tools being used throughout the article. A student is working to solve 70 - 23 and get an answer of 53 using the traditional method. When she repeats the problem again using base-10 block and a hundred chart she gets the problem correct. The particular example of the subtraction content map is something that I have benefited from as a preservice teacher and the other tools described in this article can be beneficial for future students.
Poling, L.L., Goodson-Epsy, T., Dean, C., Lynch-Davis, K., & Quickenton, A. (2015). Mapping the way to content knowledge. Teaching Children Mathematics, 21(9), 538-547. Retrieved from http://www.nctm.org/Publications/Teaching-Children-Mathematics/2015/Vol21/Issue9/Mapping-the-Way-to-Content-Knowledge/
CCSM 3 & 5
Construct Viable Arguments & Critique the Reasoning (3)
This standard focuses on students not only being able to solve problems, but how they get to their solution. Students must explain, support, and defend the solutions to the problems they are attempting to solve. This standard leaves plenty of room for students to justify their process to their classmates, and to explain the steps they took to reach their solution. It allows them to work with others to ensure they are reaching the correct solution, and it also shows that their may not always be one solution to a problem. Creating a classroom where respect for others and their thoughts will help to achieve this standard. Providing an open environment for discussion will be beneficial as well.
Use Appropriate Tools Strategically (5)
This standard helps to create a more hands on and active learning environment in the math classroom. Manipulatives and various other tools can be used during instruction and by students to help deepen their understanding of various concepts. It is possible that a tool can be beneficial in teaching/learning multiple concepts, but it is important to make sure that the tool is assisting in the learning process of your students and that it in fact serves a purpose. A variety of tools should be available and accessible to your students. They should understand how to use each tool, when it is appropriate to use each, and the benefits and limitations of each.
This standard focuses on students not only being able to solve problems, but how they get to their solution. Students must explain, support, and defend the solutions to the problems they are attempting to solve. This standard leaves plenty of room for students to justify their process to their classmates, and to explain the steps they took to reach their solution. It allows them to work with others to ensure they are reaching the correct solution, and it also shows that their may not always be one solution to a problem. Creating a classroom where respect for others and their thoughts will help to achieve this standard. Providing an open environment for discussion will be beneficial as well.
Use Appropriate Tools Strategically (5)
This standard helps to create a more hands on and active learning environment in the math classroom. Manipulatives and various other tools can be used during instruction and by students to help deepen their understanding of various concepts. It is possible that a tool can be beneficial in teaching/learning multiple concepts, but it is important to make sure that the tool is assisting in the learning process of your students and that it in fact serves a purpose. A variety of tools should be available and accessible to your students. They should understand how to use each tool, when it is appropriate to use each, and the benefits and limitations of each.
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