Read Chapters 3 & 4 (pgs. 27-70).
In the comment section below, share at least one "aha" moment (evidence) and "why." Read comments by other group members and respond to posts by at least two other teachers.
I thought that these two chapters were full of good ideas. I really like the strategy of having students rank their problems from 1 to 10 based on how secure they were that their answer was correct and how well they understood how to solve the problem. This would give the teacher a lot of insight regarding a students true umderstanding of the content that was being taught.
I especially liked the idea of giving a CRA assessment for concepts that a teacher is trying to teach. Often, a student can solve a problem procedurally, but doesn't understand conceptually what is happening. At other times, students understand the concept, but can't apply the procedure or algorithm. By utilizing CRA assessments, teachers would truly know where to start to help their students.
I also liked the confidence rating strategy. I think it would be neat and also rewarding to see confidence ratings increase over time as students begin to really understand the concepts and are able to show understanding. They would also be able to look back through their math journal and see their own growth throughout the lessons or the year. I also appreciated how the author broke down the four profiles based on student confidence and correctness (p. 35).
I also liked the CRA assessments. It would be really nice to know what the students are thinking. Sometimes when the students ask a question, they don't know how to ask what they don't understand. The CRA assessments would show us what the students do not understand.
Concrete-Representational-Abstract (CRA Assessments) were the big AHA for me in this section of the reading. Especially with MTSS sessions, this would be a great opportunity to explore exactly where my students are in their learning. Chapter 4 gave a very thorough look at how I can use the CRA assessments to gauge conceptual understanding. I especially like the stations option, as it gives students to work through problems at their own pace, and also the opportunity to move around. My students also need more practice and guidance in writing about their thinking, and this would be a good opportunity to develop that skill as well. I had one student tell me that he cannot write what he is thinking because he is "thinking nothing!" The resources section in the back of the book has examples of CRA Assessments for several grade levels--how wonderful!
I agree that this would be a great strategy for MTSS groups. Often we don't know where the "hole" is in a student's learning. They might understand how to do the procedure, but not really understand conceptually.
I believe all students would benefit from explaining their thinking. Most often students can verbally explain their thoughts/answers, but struggle with transferring their thoughts on paper. Many educators for years have asked kids to record their thinking with reading (response logs, etc) but not often in Math.
I liked this section also. So many of my students do not or could not explain how the got their answers. This usually means they will not be able to transfer their thinking to other plroblems.
My big "aha" moment was in chapter 3 is when it stated that students should be engaged with their ideas rather than corrected for wrong thinking. I love this because we always want to get the student to the right answer rather than figure out what the student was thinking to get that answer.
I agree! I think it more beneficial, as far as instruction ,to know how a student arrived at the answer rather than whether the answer is correct or incorrect.
Confidence scores...I found this idea to be intriguing. I tried it out on a fifth grader and she ranked her answers very low with the answers being incorrect. It will be interesting as I continue to work with her to compare her confidence scores now to later in the year. The flexible interview I feel is even more beneficial than the confidence score. So many times when we analyze student work it appears they were "way off", but when hearing them explain their thinking verbally we get a better picture as to their reasoning. Math stations/centers have been used for many years in education, but not specifically CRA stations. I see more of a benefit in arranging stations in this manner as it also forces the concept to be more focused (as mentioned on page 50 - choosing a Mathematical Focus). Another "aha" for me was the use of a camera to take a snapshot of student constructions. This would also be helpful when talking to parents.
I am curious about how some of the students will rank themselves using confidence scores and I am excited to try this soon! I think that this will most likely help me most with some of the students that are right in the middle rather than at the lowest or highest tier because those are the students that sometimes know the algorithm but not the concept. I also like that the CRA is supposed to have a specific focus.
I also loved the camera idea, but was comfounded at how the teacher in the book could print it out immediately and have the student write their thinking. Awesome idea, would love to see this in action and use it. Gave me something to think about and I am thinking about how I could reasonably use this in my classroom. I think, too, that taking that photo would do a lot to validate student thinking and I think that the students would really enjoy that instant feedback.
I like the idea of confidence scores because if they rate a wrong answer really high there is a misconception. Then if they rate a wrong answer low those students obviously didn't understand the problem.
My first "aha" moment was when the author wrote about using the three frames to create theories about student thinking and looking at actual evidence when creating a theory rather than making an inference about why a student is struggling. Right away I thought of my students that struggle with some math concepts and I wondered what frame they might fall under. My favorite line related to my first "aha" moment is about the instruction frame perspective; "While some might suggest that this could be construed as a criticism of math teaching, I encourage us to think of theories about learner struggles that come from instruction as opportunities" and that "struggles that come from instruction may be the easiest to remedy." My second "aha" moment was the CRA assessment discussed in chapter 4. The first thing I realized is that kids would probably enjoy this type of assessment. It allows them to move around as the author mentioned. I also like that the CRA can be either whole-group, student-determined centers or teacher determined centers. I like that students have to demonstrate understanding of a specific math focus using three different methods. It is really helpful that the CRA is meant to understand "student thinking" and "not student proficiency". The important thing is to understand how and why the student is solving the problems the way that he/she does. The author also provides a lot of information about how to sort and then use the data from the CRA to inform instruction.
I also almost wrote about the frames and that instead of feeling criticized if an instructional method is not working, I should be thankful because at least I can change it more easily than if it's learner-centered or content-centered.
My interesting monent was about the CRA assessments. Giving a better choice with three different types of centers. This would be so good for some of my students. Many would benefit from hearing others explain how the come up with their answers.
My AHA moment was the idea of using the CRA as a pre assessment tool as you are preparing to learn about a given mathematical skill. So often in my preassessments I see that the students simply do not have proficiency in doing a given skill, but it woudl be interesting to see how they think about the given skill.
I would be interested also to see how many of my students would provide me with solutions that fall into the "WHat the HECK?" category. I think it would take me putting this idea into action a few times before I could really say that I understood it. But with the different things that i have been trying in my classroom in terms of problem solving this model of assessment woudl seem to fall right in line with some of the CCSS ways of thinking in terms of allowing and encouraging students to struggle through their work. I would have to continue to work on not allowing myself to provide too much assistance and letting the students persevere in finding ways to work it out.
My "aha" moment was also using a CRA as a pre-test for student knowledge and skill and interpreting the results. I didn't realize how multi-faceted mathematical understanding could be when looking at a pre-test. Pre-tests I have used before have been very cut-and-dry, questions only showing an end result instead of showing what students may partially know about a skill or concept. Teaching the gaps in the CRA model and addressing misconceptions that the pre-test shows could help me plan my instructional units better. I feel that this model could give me a better understanding of what prior knowledge students have already so that I can determine a proper starting point for units. Right now, we simply give the unit test as a pre-test, and I must admit, this has not helped to guide my instruction as effectively as what the CRA assessment could do.
I have read about CRA before but have only applied it loosely. I really liked the method of using stations to assess students levels of using models. I think it would be good to use the same question at each station (concrete, representational, abstract) at first, then repeat the process with different questions (p. 54). Analyzing students' responses would be very useful in pinpointing their areas of mastery as well as areas of difficulty.
In Chapter 3 I felt like they did a good job stressing that many times the misunderstanding of math concepts for some students rests in the fundamental number sense of the student. Applying algorithms and rules doesn't address these concerns and can compound the problem as they progress in deeper math content.
Chapter 4 was great for me because I've only been thinking about CRA from a lesson standpoint, not as an assessment piece. As a result, I leave out the C (Concrete models) on my assessments unless a student is struggling and needs the hands on models. The trick is coming up with quality CRA assessments and recording data that is not simple to interpret. This chapter has good suggestions to overcome these concerns.
We should listen to students’ thinking which then will lead us to a better understanding of their understanding of the curriculum. We can clearly see what they don’t understand so they can get assistance of some level. The CRA assessments have the students use various methods of assessments to show if they understand the problem. The way a student uses the models shows the teacher how and if they understand how to do the problems.
I thought that these two chapters were full of good ideas. I really like the strategy of having students rank their problems from 1 to 10 based on how secure they were that their answer was correct and how well they understood how to solve the problem. This would give the teacher a lot of insight regarding a students true umderstanding of the content that was being taught.
ReplyDeleteI especially liked the idea of giving a CRA assessment for concepts that a teacher is trying to teach. Often, a student can solve a problem procedurally, but doesn't understand conceptually what is happening. At other times, students understand the concept, but can't apply the procedure or algorithm. By utilizing CRA assessments, teachers would truly know where to start to help their students.
I also liked the confidence rating strategy. I think it would be neat and also rewarding to see confidence ratings increase over time as students begin to really understand the concepts and are able to show understanding. They would also be able to look back through their math journal and see their own growth throughout the lessons or the year. I also appreciated how the author broke down the four profiles based on student confidence and correctness (p. 35).
DeleteI also liked the CRA assessments. It would be really nice to know what the students are thinking. Sometimes when the students ask a question, they don't know how to ask what they don't understand. The CRA assessments would show us what the students do not understand.
DeleteConcrete-Representational-Abstract (CRA Assessments) were the big AHA for me in this section of the reading. Especially with MTSS sessions, this would be a great opportunity to explore exactly where my students are in their learning. Chapter 4 gave a very thorough look at how I can use the CRA assessments to gauge conceptual understanding. I especially like the stations option, as it gives students to work through problems at their own pace, and also the opportunity to move around. My students also need more practice and guidance in writing about their thinking, and this would be a good opportunity to develop that skill as well. I had one student tell me that he cannot write what he is thinking because he is "thinking nothing!" The resources section in the back of the book has examples of CRA Assessments for several grade levels--how wonderful!
ReplyDeleteI agree that this would be a great strategy for MTSS groups. Often we don't know where the "hole" is in a student's learning. They might understand how to do the procedure, but not really understand conceptually.
DeleteI believe all students would benefit from explaining their thinking. Most often students can verbally explain their thoughts/answers, but struggle with transferring their thoughts on paper. Many educators for years have asked kids to record their thinking with reading (response logs, etc) but not often in Math.
DeleteI liked this section also. So many of my students do not or could not explain how the got their answers. This usually means they will not be able to transfer their thinking to other plroblems.
DeleteMy big "aha" moment was in chapter 3 is when it stated that students should be engaged with their ideas rather than corrected for wrong thinking. I love this because we always want to get the student to the right answer rather than figure out what the student was thinking to get that answer.
ReplyDeleteI agree! I think it more beneficial, as far as instruction ,to know how a student arrived at the answer rather than whether the answer is correct or incorrect.
DeleteI also agree. We need to know how the students are thinking if there is any chance of redirecting their thouoghts.
DeleteConfidence scores...I found this idea to be intriguing. I tried it out on a fifth grader and she ranked her answers very low with the answers being incorrect. It will be interesting as I continue to work with her to compare her confidence scores now to later in the year. The flexible interview I feel is even more beneficial than the confidence score. So many times when we analyze student work it appears they were "way off", but when hearing them explain their thinking verbally we get a better picture as to their reasoning. Math stations/centers have been used for many years in education, but not specifically CRA stations. I see more of a benefit in arranging stations in this manner as it also forces the concept to be more focused (as mentioned on page 50 - choosing a Mathematical Focus). Another "aha" for me was the use of a camera to take a snapshot of student constructions. This would also be helpful when talking to parents.
ReplyDeleteI am curious about how some of the students will rank themselves using confidence scores and I am excited to try this soon! I think that this will most likely help me most with some of the students that are right in the middle rather than at the lowest or highest tier because those are the students that sometimes know the algorithm but not the concept. I also like that the CRA is supposed to have a specific focus.
DeleteI also loved the camera idea, but was comfounded at how the teacher in the book could print it out immediately and have the student write their thinking. Awesome idea, would love to see this in action and use it. Gave me something to think about and I am thinking about how I could reasonably use this in my classroom. I think, too, that taking that photo would do a lot to validate student thinking and I think that the students would really enjoy that instant feedback.
DeleteThe idea of confidence scores was new to me, and I would like to try it out.
DeleteI like the idea of confidence scores because if they rate a wrong answer really high there is a misconception. Then if they rate a wrong answer low those students obviously didn't understand the problem.
DeleteMy first "aha" moment was when the author wrote about using the three frames to create theories about student thinking and looking at actual evidence when creating a theory rather than making an inference about why a student is struggling. Right away I thought of my students that struggle with some math concepts and I wondered what frame they might fall under. My favorite line related to my first "aha" moment is about the instruction frame perspective; "While some might suggest that this could be construed as a criticism of math teaching, I encourage us to think of theories about learner struggles that come from instruction as opportunities" and that "struggles that come from instruction may be the easiest to remedy."
ReplyDeleteMy second "aha" moment was the CRA assessment discussed in chapter 4. The first thing I realized is that kids would probably enjoy this type of assessment. It allows them to move around as the author mentioned. I also like that the CRA can be either whole-group, student-determined centers or teacher determined centers. I like that students have to demonstrate understanding of a specific math focus using three different methods. It is really helpful that the CRA is meant to understand "student thinking" and "not student proficiency". The important thing is to understand how and why the student is solving the problems the way that he/she does. The author also provides a lot of information about how to sort and then use the data from the CRA to inform instruction.
I also almost wrote about the frames and that instead of feeling criticized if an instructional method is not working, I should be thankful because at least I can change it more easily than if it's learner-centered or content-centered.
DeleteMy interesting monent was about the CRA assessments. Giving a better choice with three different types of centers. This would be so good for some of my students. Many would benefit from hearing others explain how the come up with their answers.
ReplyDeleteMy AHA moment was the idea of using the CRA as a pre assessment tool as you are preparing to learn about a given mathematical skill. So often in my preassessments I see that the students simply do not have proficiency in doing a given skill, but it woudl be interesting to see how they think about the given skill.
ReplyDeleteI would be interested also to see how many of my students would provide me with solutions that fall into the "WHat the HECK?" category. I think it would take me putting this idea into action a few times before I could really say that I understood it. But with the different things that i have been trying in my classroom in terms of problem solving this model of assessment woudl seem to fall right in line with some of the CCSS ways of thinking in terms of allowing and encouraging students to struggle through their work. I would have to continue to work on not allowing myself to provide too much assistance and letting the students persevere in finding ways to work it out.
My "aha" moment was also using a CRA as a pre-test for student knowledge and skill and interpreting the results. I didn't realize how multi-faceted mathematical understanding could be when looking at a pre-test. Pre-tests I have used before have been very cut-and-dry, questions only showing an end result instead of showing what students may partially know about a skill or concept. Teaching the gaps in the CRA model and addressing misconceptions that the pre-test shows could help me plan my instructional units better. I feel that this model could give me a better understanding of what prior knowledge students have already so that I can determine a proper starting point for units. Right now, we simply give the unit test as a pre-test, and I must admit, this has not helped to guide my instruction as effectively as what the CRA assessment could do.
ReplyDeleteI have read about CRA before but have only applied it loosely. I really liked the method of using stations to assess students levels of using models. I think it would be good to use the same question at each station (concrete, representational, abstract) at first, then repeat the process with different questions (p. 54). Analyzing students' responses would be very useful in pinpointing their areas of mastery as well as areas of difficulty.
ReplyDeleteIn Chapter 3 I felt like they did a good job stressing that many times the misunderstanding of math concepts for some students rests in the fundamental number sense of the student. Applying algorithms and rules doesn't address these concerns and can compound the problem as they progress in deeper math content.
ReplyDeleteChapter 4 was great for me because I've only been thinking about CRA from a lesson standpoint, not as an assessment piece. As a result, I leave out the C (Concrete models) on my assessments unless a student is struggling and needs the hands on models. The trick is coming up with quality CRA assessments and recording data that is not simple to interpret. This chapter has good suggestions to overcome these concerns.
We should listen to students’ thinking which then will lead us to a better understanding of their understanding of the curriculum. We can clearly see what they don’t understand so they can get assistance of some level. The CRA assessments have the students use various methods of assessments to show if they understand the problem. The way a student uses the models shows the teacher how and if they understand how to do the problems.
ReplyDelete