Creating the curriculum plan was the most challenging project for me this semester in math methods. It required the most amount of time, effort, and thinking. Although challenging, I can truly say this has been the project that I have taken the most away from. Organizing the different domain areas to be covered in each quarter was not very difficult for my group and I, nor were writing the objectives for each quarter. The challenging part was creating unique, interactive modeling and exploration tasks for the students to participate in within each grade. Additionally, creating the end of the semester exploration activities was a difficult task for us. However, I believe that my group members and I did an excellent job of coming up with unique activities for each grade level. This curriculum plan turned out so well, I would be more than comfortable using it in my own K-2 mathematics classroom one day! I feel that I learned a lot throughout the process of creating the curriculum plan. I realized how important it is for material to build off each other in each of the grade levels. It is to the benefit of both teachers and students to create curriculum plans for series of grade levels. This ensures that there is little overlap, and that teachers are maximizing on using students' prior knowledge. Although curriculum plans take a great amount of planning and time, they really can be beneficial to implement in a school.
After viewing the curriculum plans for both 3rd-5th and 6th-8th grades, it was interesting to see the flow of content and domains between the bands of grade levels. K-5th grades are very similar in domains, with the content obviously differing by grade level. However, once students reach 6th grade, the domains are changed quite a bit and become much more difficult and higher-level. Additionally, the measurement and data domain is completely eliminated in the upper grades (6-8). My assumption is that measurement and data is simply combined with geometry in these grades levels, and does not necessarily need its own domain area. I did see some overlap within the grade levels and bands, but I do not believe that this overlap is unnecessary. Actually, I think that some overlap (especially of difficult concepts) can be very beneficial to students. It just means that students are covering similar material in multiple grade levels, which can serve as a very good review. I also thought it was interesting that while most of the content domains changed every few grade levels, one domain was present throughout all grade levels. This domain is geometry. Geometry is heavily emphasized in each of the K-8 standards. It is clear that this is a highly important domain, and that students should be well experienced in geometry by the time they reach the upper grade levels.
Tuesday, June 30, 2015
Friday, June 26, 2015
Standards and Classroom Changes to Deepen Math Learning Reflection
Education in the K-8 classrooms across the country has changed drastically since my days in school. Even though that was not long ago, reform in education has required teachers to make huge changes in their teaching and assessing across all subject areas, including mathematics. I feel fortunate, however, that I am pursuing my college education at this point in time. The changes have been made, and now it is a matter of implementing those changes. I feel that I am at an advantage, because my education courses and experiences at Bradley are preparing me for these changes. Many current teachers do not have this practice, and were thrown into the changes and expected to make things work.
One of the biggest changes made to education was the creation and adoption (for most states, Illinois included) of the Common Core Standards for Learning. At this point in time, standards have been created for language arts and mathematics. In mathematics, these standards are often referred to as the Standards of Mathematical Practice, or SMP's. We have worked extensively in breaking these standards down this semester, in order to make sense of them ourselves. The standards themselves highlight what students should be able to do to show that they are proficient in mathematics at each grade level. There are 8 SMP's, and each are important to implement in all activities involving mathematical instruction. The Common Core Standards also provides specific standards highlighting what students should be able to do in various content areas of mathematics at each grade level. These standards should be utilized and aligned with lesson plans that teacher implement in the classroom.
Another change that has been made in mathematics education is the utilization of the NCTM Process Standards. These 5 process standards are composed of different areas of mathematics that students should be participating in during mathematics instruction. These 5 process standards include problem solving, reasoning and proof, communication, connection, and representations. All of these aspects should be included in any mathematics work that students are participating in. This is something that I will have to greatly familarize myself with and plan to implement in all of my mathematics lessons and activities.
Student engagement and active learning are two other major changes that have been added to mathematics instruction in K-8 classrooms. The days of students sitting at their desks listening to the teacher lecture and then completing a worksheet about the material are long gone. There is a major call for teachers to keep their students engaged throughout mathematics lessons and participating in active learning. This active learning does not include sitting at a desk and filling out worksheets or memorizing formulas. This active learning is problem-based, and involves more creative and engaging lessons and activities.
Finally, one of the biggest changes in mathematics instruction in K-8 classrooms has been the transition from content-based mathematics to taking meaning from mathematics. In short, students must now find the correct solutions but also explain and justify their solutions with reasoning. Simply finding the right answer is no longer enough. Students must learn how to justify their answers with appropriate reasoning and representations. This is a complicated task, and takes many students years to practice and carry out successfully.
It is clear that mathematics education has changed drastically in the last few years. There is a lot more planning, preparation, and reflection on the teacher's part. However, I feel that my past and continued experiences in college are preparing me for these changes.
One of the biggest changes made to education was the creation and adoption (for most states, Illinois included) of the Common Core Standards for Learning. At this point in time, standards have been created for language arts and mathematics. In mathematics, these standards are often referred to as the Standards of Mathematical Practice, or SMP's. We have worked extensively in breaking these standards down this semester, in order to make sense of them ourselves. The standards themselves highlight what students should be able to do to show that they are proficient in mathematics at each grade level. There are 8 SMP's, and each are important to implement in all activities involving mathematical instruction. The Common Core Standards also provides specific standards highlighting what students should be able to do in various content areas of mathematics at each grade level. These standards should be utilized and aligned with lesson plans that teacher implement in the classroom.
Another change that has been made in mathematics education is the utilization of the NCTM Process Standards. These 5 process standards are composed of different areas of mathematics that students should be participating in during mathematics instruction. These 5 process standards include problem solving, reasoning and proof, communication, connection, and representations. All of these aspects should be included in any mathematics work that students are participating in. This is something that I will have to greatly familarize myself with and plan to implement in all of my mathematics lessons and activities.
Student engagement and active learning are two other major changes that have been added to mathematics instruction in K-8 classrooms. The days of students sitting at their desks listening to the teacher lecture and then completing a worksheet about the material are long gone. There is a major call for teachers to keep their students engaged throughout mathematics lessons and participating in active learning. This active learning does not include sitting at a desk and filling out worksheets or memorizing formulas. This active learning is problem-based, and involves more creative and engaging lessons and activities.
Finally, one of the biggest changes in mathematics instruction in K-8 classrooms has been the transition from content-based mathematics to taking meaning from mathematics. In short, students must now find the correct solutions but also explain and justify their solutions with reasoning. Simply finding the right answer is no longer enough. Students must learn how to justify their answers with appropriate reasoning and representations. This is a complicated task, and takes many students years to practice and carry out successfully.
It is clear that mathematics education has changed drastically in the last few years. There is a lot more planning, preparation, and reflection on the teacher's part. However, I feel that my past and continued experiences in college are preparing me for these changes.
Manipulative Reflection
How do you know if students deepen their understanding while using manipulatives?
I believe that a teacher can see if a student is deepening their understanding through the use of manipulatives through observation and questioning. The teacher can watch students to see if they are correctly utilizing the manipulatives they are working with. If this is not happening, the teacher can intervene to help the student and suggest alternative methods. The teacher can also see if students are deepening their understanding while using manipulatives by posing questions on the students. The teacher could ask students to represent something specific, explain their representation, and justify their representation. This could help the teacher gain insight on whether or not the student is really taking away meaning from the utilization of manipulatives.
How do you know if the students can transfer their understanding from manipulatives to other situations?
One way to see if students have the ability to transfer their understanding of an idea from manipulatives to other situations would be to ask the students to think about the manipulatives in their minds, or draw them out on paper. This would eliminate the actual use of the manipulatives, but the students could still utilize ideas they learned from the use of them. An example of this would be giving students the option of drawing base 10 pieces while solving a math problem. Although physical manipulatives would not be used, drawings could still help the student to solve the problem.
How can you assess that understanding or growth?
A teacher could assess the understanding or growth a student takes from the utilization of manipulatives by asking them to verbally explain their use of manipulatives, and what their manipulatives stand for. In addition, the teacher could ask students to write their explanation of their use of manipulatives and write also what the manipulatives stand for. These are two options for assessing student understanding of manipulatives. Another option would be for the teacher to provide manipulations on the SmartBoard or overhead projector, and ask students to explain the manipulations and their meaning to a partner, or ask students to share their ideas with the entire class. This could gauge whether or not students have true understanding of their use of manipulatives.
When students work in groups, how do you hold each youngster accountable for learning?
First of all, I think that any group work in the classroom should be done in very small groups of students, maybe 2 to 3 children in each group. When group sizes become too large, there are almost always students left out of discussions or students who feel they can "get away with" not participating. This is not acceptable, and really takes away from the overall point of group work in the first place. Having small groups should make all members of the group members participate and share their ideas. If a teacher notices a group member slacking or not participating, the teacher should pull that student aside and explain that their participation is not optional, and that their group members need their help and input during group work. I am not a huge believer in punishments, but I do believe that privileges should be taken away from students who do not participate in group work. All of these modifications will help students to be accountable for their learning, and promote participation and effort during group work instances.
When students work in groups, how do you assess each youngster's depth of understanding?
Assessment of students working in groups can be a difficult task, but I think there are two things that a teacher can use to help with this. The first item that can be used are checklists. Checklists can be used for individual students to monitor their progress and understanding throughout an activity or project that has students working in groups. While students are working, the teacher can observe and monitor student progress, behavior, participation, and understanding. The teacher can monitor understanding by asking students questions and trying to make sense of their responses. This may gauge whether or not the student is understanding the topic at hand. Conversation is a very effective tool for assessment, and a teacher can make great use of it during group work assessment. Another thing teachers can do to assess students during group work would be, if applicable, to have every student complete a handout or product. This ensures that all of the students in a group are participating and completing the task, and the teacher has something to look at for each and every student.
How are you improving students' problem solving skills with the manipulatives?
Manipulatives allow students to be hands on and visualize the mathematics they are carrying out. Many students need visualizations in order to understand mathematics, so manipulatives can be very useful. In addition, when using manipulatives, students can find errors and then fix those errors more easily. This can help lessen student frustration in problem solving. Manipulatives also provide a more concrete way of problem solving for students.
I believe that a teacher can see if a student is deepening their understanding through the use of manipulatives through observation and questioning. The teacher can watch students to see if they are correctly utilizing the manipulatives they are working with. If this is not happening, the teacher can intervene to help the student and suggest alternative methods. The teacher can also see if students are deepening their understanding while using manipulatives by posing questions on the students. The teacher could ask students to represent something specific, explain their representation, and justify their representation. This could help the teacher gain insight on whether or not the student is really taking away meaning from the utilization of manipulatives.
How do you know if the students can transfer their understanding from manipulatives to other situations?
One way to see if students have the ability to transfer their understanding of an idea from manipulatives to other situations would be to ask the students to think about the manipulatives in their minds, or draw them out on paper. This would eliminate the actual use of the manipulatives, but the students could still utilize ideas they learned from the use of them. An example of this would be giving students the option of drawing base 10 pieces while solving a math problem. Although physical manipulatives would not be used, drawings could still help the student to solve the problem.
How can you assess that understanding or growth?
A teacher could assess the understanding or growth a student takes from the utilization of manipulatives by asking them to verbally explain their use of manipulatives, and what their manipulatives stand for. In addition, the teacher could ask students to write their explanation of their use of manipulatives and write also what the manipulatives stand for. These are two options for assessing student understanding of manipulatives. Another option would be for the teacher to provide manipulations on the SmartBoard or overhead projector, and ask students to explain the manipulations and their meaning to a partner, or ask students to share their ideas with the entire class. This could gauge whether or not students have true understanding of their use of manipulatives.
When students work in groups, how do you hold each youngster accountable for learning?
First of all, I think that any group work in the classroom should be done in very small groups of students, maybe 2 to 3 children in each group. When group sizes become too large, there are almost always students left out of discussions or students who feel they can "get away with" not participating. This is not acceptable, and really takes away from the overall point of group work in the first place. Having small groups should make all members of the group members participate and share their ideas. If a teacher notices a group member slacking or not participating, the teacher should pull that student aside and explain that their participation is not optional, and that their group members need their help and input during group work. I am not a huge believer in punishments, but I do believe that privileges should be taken away from students who do not participate in group work. All of these modifications will help students to be accountable for their learning, and promote participation and effort during group work instances.
When students work in groups, how do you assess each youngster's depth of understanding?
Assessment of students working in groups can be a difficult task, but I think there are two things that a teacher can use to help with this. The first item that can be used are checklists. Checklists can be used for individual students to monitor their progress and understanding throughout an activity or project that has students working in groups. While students are working, the teacher can observe and monitor student progress, behavior, participation, and understanding. The teacher can monitor understanding by asking students questions and trying to make sense of their responses. This may gauge whether or not the student is understanding the topic at hand. Conversation is a very effective tool for assessment, and a teacher can make great use of it during group work assessment. Another thing teachers can do to assess students during group work would be, if applicable, to have every student complete a handout or product. This ensures that all of the students in a group are participating and completing the task, and the teacher has something to look at for each and every student.
How are you improving students' problem solving skills with the manipulatives?
Manipulatives allow students to be hands on and visualize the mathematics they are carrying out. Many students need visualizations in order to understand mathematics, so manipulatives can be very useful. In addition, when using manipulatives, students can find errors and then fix those errors more easily. This can help lessen student frustration in problem solving. Manipulatives also provide a more concrete way of problem solving for students.
Thursday, June 25, 2015
Technology in the Math Class Reflection
A great amount of technology has been utilized during this summer semester of math methods. At the beginning of each class period, we were asked to sign in on the SmartBoard, using something other than the pen. This allowed students to work with this piece of technology and familiarize themselves with features. Additionally, each student was assigned to show the rest of the class a SmartBoard feature at some point during the semester. This was a really useful activity, because I learned a lot of new things about the SmartBoard and some of its features. This will help me in my novice and student teaching placements, as well as in my own classroom once I graduate. Throughout the semester, one of our assignments was to watch two videos online that showed a math lesson taking place in a real classroom. Although these videos were long, I really took a lot away from both and wrote deep reflections on each. These videos were very interesting to watch and then reflect on. This was an excellent use of technology, and I enjoyed watching these videos much more than if we were asked to read an article about them. Two other forms of technology used this semester were Prezi and Jing recording software. I have used both of these in the past, but never simultaneously. We were to create a Prezi highlighting some of the Common Core Standards for Mathematical Practice, and then narrate that Prezi using Jing recording software. Although this took some practice and repeated tries (for me anyway!), this was a useful activity. These are two pieces of technology that I would definitely like to implement in my classroom one day.
We also utilized the website Blogger.com to post blogs continuously throughout the semester about our work. I have blogged before for other classes, and have always found it enjoyable. It is a great way to reflect on work. I know many practicing teachers who utilize blogs to keep their students' parents and family members up to date on class work. I think this is something I will definitely do in my classroom. Another assignment I completed this semester was searching for various math applets and apps for my assigned grade level, K-2. This was another meaningful experience, because I discovered cool applets and apps to use with my future students. We did use calculators a bit this semester, mostly for calculation work during activities. I feel very comfortable at this point in my education with the use of calculators, due to my past math classes in both high school and my early years of college. The final piece of technology utilized this semester was the creation of videos discussing our curriculum project. I have created videos before for assignments, but none as long as this video. This was an interesting and challenging experience for my group members and I. A great amount of technology was used this semester, and I feel that it will really benefit me with my future in teaching.
We also utilized the website Blogger.com to post blogs continuously throughout the semester about our work. I have blogged before for other classes, and have always found it enjoyable. It is a great way to reflect on work. I know many practicing teachers who utilize blogs to keep their students' parents and family members up to date on class work. I think this is something I will definitely do in my classroom. Another assignment I completed this semester was searching for various math applets and apps for my assigned grade level, K-2. This was another meaningful experience, because I discovered cool applets and apps to use with my future students. We did use calculators a bit this semester, mostly for calculation work during activities. I feel very comfortable at this point in my education with the use of calculators, due to my past math classes in both high school and my early years of college. The final piece of technology utilized this semester was the creation of videos discussing our curriculum project. I have created videos before for assignments, but none as long as this video. This was an interesting and challenging experience for my group members and I. A great amount of technology was used this semester, and I feel that it will really benefit me with my future in teaching.
Wednesday, June 24, 2015
Assessments in Math Reflection
Assessment in mathematics has been a rather large topic of discussion during math methods this summer. In any subject area, assessment is the key to gauging whether or not students have learned the concepts and objectives taught within a period of time. A variety of assessments have been discussed, including traditional and authentic assessments. While both have their perks and downfalls, many schools are pushing their teachers to use more authentic assessments with their students. Formative and summative assessments were also discussed this semester. Formative assessments are ongoing and constantly occurring during a lesson and/or unit. This might involve the teaching asking questions to students, having classroom discussions, or simply observing students during activities. Summative assessment occurs at the end of a lesson or unit, and is used as a way to look at students' overall learning or their take away of concepts. In addition to classroom discussions on assessment, I read several articles regarding assessment in the mathematics classroom. One article that really struck me talked about the use of conversation as an assessment tool. I think too many times, schools and teachers want all assessment to be formalized so that a written product is produced. However, this article talked about how the element of conversation can be a very effective tool to use for assessment. A teacher can learn a lot about his or her students through conversation, including what they know, what they are learning, and what they do not know yet. This kind of conversation can help the teacher to quickly change the path of a lesson, or even plan ahead for future lessons. It can also help a teacher to modify instruction for certain students. I am a big believer in conversation as a method of assessment because it is relatively informal, easy to use, and can reap major benefits.
Throughout all of the assignments during this semester, I did some assessing of my own on my personal work. While completing each assignment, I regularly utilized formative assessment by rereading and checking my work. Additionally, I checked specific rubrics throughout my work to verify that I was on track with the assignment. I utilized summative assessment when I completed any assignment before turning it in by rereading my work and consulting the rubric once again. I also had the opportunity to assess some of my peers' work during this semester. I critiqued problem situations that were created by my classmates. In these critiques, I was to comment on strengths and weaknesses, and provide suggestions for improvement. This was a new experience for me, because I do not have much practice with critiquing/assessing my peers' work. I also worked with assessment when going through the More Errors document. This was a collection of student samples of mathematics problems that had mistakes within. I was to find the mistake in each sample and then try to apply that mistake to other, similar problems. This was also a new experience for me, and I had some difficulty with finding the mistakes in many of the problems. Finally, I worked with assessments by reading through the feedback my instructor gave me on all of my assignments. I did this by reading through the given rubrics and reading comments that were listed. This helped me to get an idea of what my instructor thought about my final work, and also allowed me to reflect on the assignments.
Throughout all of the assignments during this semester, I did some assessing of my own on my personal work. While completing each assignment, I regularly utilized formative assessment by rereading and checking my work. Additionally, I checked specific rubrics throughout my work to verify that I was on track with the assignment. I utilized summative assessment when I completed any assignment before turning it in by rereading my work and consulting the rubric once again. I also had the opportunity to assess some of my peers' work during this semester. I critiqued problem situations that were created by my classmates. In these critiques, I was to comment on strengths and weaknesses, and provide suggestions for improvement. This was a new experience for me, because I do not have much practice with critiquing/assessing my peers' work. I also worked with assessment when going through the More Errors document. This was a collection of student samples of mathematics problems that had mistakes within. I was to find the mistake in each sample and then try to apply that mistake to other, similar problems. This was also a new experience for me, and I had some difficulty with finding the mistakes in many of the problems. Finally, I worked with assessments by reading through the feedback my instructor gave me on all of my assignments. I did this by reading through the given rubrics and reading comments that were listed. This helped me to get an idea of what my instructor thought about my final work, and also allowed me to reflect on the assignments.
Error Analysis Reflection
The work I participated in while reviewing the More Errors document was quite interesting to dive into. A series of various student samples of mathematical problems were posed, many of which involved errors in solving and/or incorrect answers. I was to go through each problem and determine how the sample student was going about solving the problem, and if mistakes were made, where those mistakes were found. Then, I had to try and solve a few additional, similar problems using the sample student's method (which was usually incorrect). This seemed like a simple task at first, but as I explored deeper into the problems, I realized this was more difficult than I first thought. Some of the mistakes I was able to easily pick up on, while others caused me quite a challenge. I think most of the difficulty came from trying to read and understand someone else's work. It is easy for me to reread and understand my own work and my process for finding a solution, because I can internatlize my thoughts. However, when it comes to trying to read and understand someone else's methods and processes for finding a solution, it can be much more difficult. I had this problem throughout the entire More Errors document. Some of the problems really caught me off guard, because I could not figure out the methods/strategies the sample student was using to find their written solution. In additon to having difficulty in finding the mistakes the sample students used, it became evident to me that a large multitude of mistakes can be used when solving mathematics problems. I saw mistakes in the problems that I read which would never even cross my mind to do while solving a mathematics problems. I think this really opened up my mind to the fact that just because an idea, method, or strategy in mathematics makes sense to me does NOT mean that the same applies to someone else. Reading and understanding my student's mathematical work is something that I am going to do every single day. After completing the More Errors assignment, I now realize that I need a lot of practice and experience in doing this.
Friday, June 19, 2015
Article Discussions on Assessment
Getting Started with Open-Ended Assessment
Found in Teaching Children Mathematics, April 2005
This article discussed the high points and low points of utilizing open-ended assessment in the classroom. Open-ended problems are those where multiple strategies can be used to solve, and multiple solutions may be correct or accurate. These types of problems allow students to better demonstrate their actual understanding of a topic in mathematics. Instead of just finding an answer and writing it down, students must additionally explain their thinking and reasoning using detail. Open-ended problems do not rely on content-based mathematics. Instead, they require students to not only find a correct solution but also explain their process and reasoning for finding that solution. It is important for open-ended problems to utilize meaningful mathematics, allow for multiple strategies/solutions, and have the perfect medium of too little and not too much information provided. It was noted by a featured teacher in this article that creating and implementing open-ended problems can be very time-consuming and difficult. However, they can be very beneficial to students and their overall learning in mathematics. They require students to do a lot of work in justifying their answers, which may be a new concept for students. This featured teacher also suggested to not grade these problems at first, which will allow students to be more comfortable completing them. Additionally, she found it beneficial to show student samples on an overhead (with names removed) and talk about the pros/cons of each sample response to a problem. She explained that after using open-ended problems for an extensive period of time, she saw gains in student confidence and willingness to share their answers to peers. She also noted that as the teacher, she felt this provided her with a more concrete method of assessment, which helped her explain student achievement to parents.
This was an interesting article to read. I have heard of open-ended problems before, but have never really given them a lot of thought. It is obvious that they can be quite difficult to create and implement, but the benefits do seem to be great. I believe that using open-ended problems helps students to really look into their thinking and make sense of their strategies and processes for finding answers to problems. Open-ended problem solving takes a lot of effort on both the student's and teacher's part, but it is a very useful tool that can help students to take more meaning from mathematics.
Discussion Questions:
1. What are some ways that you could potentially implement open-ended problems in your future math lessons?
2. Could you implement open-ended problems into other curricular areas?
A Smorgasbord of Assessment Options
From Teaching Children Mathematics - April 2010
As teachers, we are faced with the task of creating authentic assessments to gauge our students' learning. Creating these types of assessment can be very challenging. To add to the challenge, there are an array of assessment options to choose from in which we can utilize in our classrooms. This article talked about one type of assessment, called student-centered assessment. This format benefits the students because they are participating in meaningful mathematics, but it also benefits the teacher because it helps to gauge student understanding and their level of learning. Mathematics is a process, and students must heavily rely on building upon prior knowledge as they work through concepts. In this article, a class of students who were working with understanding geometric shapes and solids was discussed. In this class, students were at a variety of levels of understanding in regards to geometric shapes. Regardless of their individual level, students used both knowledge and reasoning to come to various conclusions about geometric shapes they were working with. A "mastery" indication of these concepts occurred when students were readily able to communicate their thinking and reasoning to their peers. It was noted in this article that in order for student-centered assessment to be effective, modes of both formative and summative assessment must be utilized. It was also suggested that they best way of creating student-centered assessment is to first identify the standard, then select a target achievement level, and finally create assessment options for students to demonstrate their understanding of the concept.
Student-centered assessment seems to be a very valuable tool for teachers to use in the classroom. It not only get students focused and motivated on a topic, but it has them thinking more deeply and communicating their thoughts and reasoning to their peers. I think this is a very important skill for students to be able to master. Once they communicate their thoughts and rationale, it is a clear indication that they truly understand a subject themselves. It is definitely evident that creating these authentic, student-centered assessment is very time consuming and not exactly "easy". It takes a lot of planning and thought on the teacher's part, and it takes a greater amount of time to implement in the classroom. However, the benefits outweigh the downfalls, and student-centered assessment is definitely something I would like to use in my classroom to help engage my students.
Discussion Questions:
1. Can you think of any specific student-centered assessments that could be utilized in a mathematics lesson?
2. Are there any other benefits you have seen through observations/field experience that display student-centered assessment?
Understanding Student to Open-Ended Tasks
From Mathematics Teaching in the Middle School - April 2000
This was another article in regards to the use of open-ended tasks in mathematics classrooms. Open-ended tasks allow for student choice in the strategy to use for solving, and also allows (usually) for multiple correct responses. This allows students flexibility in solving, and using strategies that they are comfortable with. While open-ended tasks sound to be beneficial to students, they often are hard for students to grasp and understand. Some students have little experience or practice in actually communicating their reasoning, which open-ended tasks require. This article described some sample responses from 6th graders working with an irregular area task. Several responses were shown, some being incorrect and others being correct. The teacher of these students seemed to be pretty hard on giving feedback for all of the responses. In addition, students she thought understood the concept of area extremely well performed poorly on the task. Alternatively, students that she thought were struggling easily grasped the problem and explained their reasoning adequately. It was noted that the students who made great use of words, pictures, and diagrams to aid in explaining their thinking performed better on the task. The teacher had been working with this group of students over the course of a year, and felt that her feedback was allowing her students to generate more detailed and supportive responses when completing tasks like the one posed in the article.
I thought it was interesting how the teacher's viewpoint of student understanding was drastically changed after giving her students this task. The students that she was not concerned about turned out to be the ones who did not comprehend this task or provide adequate reasoning for their responses. Alternatively, the students she was concerned about really pulled through on this task to provide a correct answer and use supportive reasoning. I think that this was a clear indication that the teacher really needed a way to look further into her student's thinking and mathematical processes. I like that it was also noted within the article that any teacher beginning to provide students open-ended tasks like this one need to expect reasonable results. Students are not going to adequately explain their thinking on the first try, and will need a lot of practice in order to respond to these types of tasks correctly.
Discussion Questions:
1. What do you presume were some of the benefits that students experienced when working with open-ended tasks like the one described in the article?
2. Do you think this teacher was too harsh on her student's responses?
Assessing Students' Understanding through Conversations
From Teaching Children Mathematics - December 2007/January 2008
Conversation as an assessment tool is often an underrated element in the classroom. However, conversation is used as a method of assessment much more frequently that any other form of assessment. Why is this? I believe that it is because conversation is informal, for the most part. It is easy to participate in, and it allows for a lot of information to be exchanged between teachers and students. Conversations over mathematics can truly gauge what students do and do not understand about the topic at hand. This article talked about three different situations in which conversation was critical to understanding student knowledge. One 3rd grade teacher used informal conversation to gauge the true understanding of one of her students working with place value. Another 5th grade teacher used conversation to understand the mistakes made by a group of students on a geometry quiz. Another teacher used conversation during a lesson on measurement to realize that students were struggling with the basic concepts of measurement. All of these examples showed how critical conversation was to the lesson, because the teacher was much better able to gauge student understanding. This article also talked about promoting conversation in the classroom required that students feel safe in the environment. Students should feel comfortable sharing their thoughts, and be respective of others doing the same.
Conversation is such a valuable assessment tool, in my opinion. It is something I plan to use very regularly in my classroom. The reason that I like conversation so much is because it takes little planning, but it can still yield so much information. I believe that conversations as a class can also yield student participation, which is crucial in a classroom. This article discussed the idea of participation, and also the utilization of wait time in order to obtain responses from all students. Conversations about mathematics can not only gauge understanding, but can help students to find errors in their problem solving or clear up any misconceptions they may have. Conversation as an assessment tool is very useful to any classroom teacher.
Discussion Questions:
1. Can you think of any times during observations/field experiences where you have seen conversation being a great assessment tool?
2. How will you work conversation as assessment into your future classroom and lessons?
An Experiment in Using Portfolios in the Middle School
From Mathematics Teaching in the Middle School - March 2008
This was a very interesting article to read about the implementation and use of portfolios in a middle school mathematics classroom. The teacher featured in this article wanted to display her student's knowledge in a different way than traditional quizzes and tests. Therefore, she turned to the use of student-made portfolios. Over the course of several weeks, the teacher had the students collect pieces of their work. Students were to have sections of their portfolios discussing their mathematical attitude, problem solving, mathematical growth, mathematical writing, and mathematical connections. Students worked to select specific pieces of work for each of these areas. The portfolios additionally required a lot of student reflection on their choices, and rationale for picking these items. Once completed, the portfolios were sent home for parents to view and answer some questions about. The teacher assessed these portfolios by organization and the overall learning and understanding of mathematics that the student received from creating the portfolio. She used a checklist to do this. As far as benefits of utilizing the portfolios, the teacher explained that she began writing her lessons so that they were able to be included in the portfolios. She did this by creating assignments that involved more writing, real-world connections, and problem solving. She also noted that by reviewing the portfolios, she was better able to see what students had learned and where they experienced the most growth in their learning.
I really like the idea of using portfolios as a form of alternative assessment. I have made numerous portfolios myself for college courses, and they really are more meaningful than taking a final exam. They allow a student to examine their progress and reflect upon their work in various forms. I think that the students in this featured class seemed to take a lot away from creating their portfolios. In addition, the teacher required the students to reflect on why they choice certain pieces. The teacher also discussed how she had students write in learning logs after completing each assignment. Students were to write what they did and did not understand. This helped students when creating their portfolios to select pieces of their work that really represented their knowledge and understanding. Portfolios are a great form of assessment, and I would really like to use them in my future classroom.
Discussion Questions:
1. Where have you seen portfolios being used in the classroom?
2. Why would having students create portfolios be beneficial to use during parent-teacher conferences?
References:
Bacon, K.A. (2010). A smorgasbord of assessment options. Teaching Children Mathematics. 16(8).
Leatham, K.R., Lawrence, K., Mewborn, D.S. (2005). Getting started with open-ended assessment. Teaching Children Mathematics. 11(8).
Maxwell, V.L., Lassak, M.B. (2008). An experiment in using portfolios in the middle school. Mathematics Teaching in the Middle School. 13(7).
Moskal, B.M. (2000). Understanding student to open-ended tasks. Teaching Mathematics in the Middle School. 5(8).
Vanderhye, C.M., Zmijewski Demers, C.M. (2007/2008). Assessing students' understanding through conversations. Teaching Children Mathematics. 14(5).
Subscribe to:
Posts (Atom)