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.
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