Of the many suggestions provided in this chapter, which three do you want to remember when it comes to selecting a worthwhile task? Justify your answer.
This chapter taught me so much about ways to teach mathematical thinking. Up until recently, my experience with math has been based upon a practice and drill mentality. Mathematical thinking comes when students are challenged to use high level cognitive demand. This high level thinking builds and creates connections that might be relevant to previous lessons or experiences, but that provide deeper levels of understanding of mathematical concepts and ideas. This high level cognitive demand is something not routine, something in which students are required to think and possibly create multiple ways to understand the problem. This makes me think about a small group math workshop I recently sat in on in which students were introduced to a new concept of dividing fractions. The teacher introduced the idea by writing a problem on the board. The students had number lines in front of them to use. She then gave the students time to quietly think about what the problem was asking. For example the problem was how many 2/3rds are in 3? Several of the students used the number line to count by 2/3rds. However, they got really confused when they realized they had one piece left over. Several students remembered an algorithm they had learned from a previous lesson and tried using it. Ultimately, the teacher asked the students to share in their thinking. Together they came up with three different ways of understanding the problem. The students who had the hardest time coming around to it finally did understand once the other students explained their thinking. It was in providing different context and reasoning that helped these students struggle through and finally grasp the concept. The idea of having relevant context through which to explain worthwhile tasks is one I will use in my teaching. Creating a connection or relevance to the problem is a way to engage any student. This can be done in so many ways, such as through literature (as suggested in the book) or community problem solving, historical date concepts, etc. This reminds me of the Place-based curriculum design concepts of Amy Demerest. We are learning about how to implement these inquiry and place based concepts in my Assessment, Curriculum and Management class. Finally, while reading the section Questioning Considerations, I remembered a day in which my student was really struggling with understanding fractions and how many fractions make up a whole. I was showing her a number line and trying to explain how 1/2 is the same as 3/6 and 4/8. The math teacher came up to me and asked me to consider "Who's doing the thinking?" I kind of understood what she was talking about then, but I really understand what she means after reading this chapter. By explaining my understanding, I wasn't allowing my student to struggle and come to her own understanding.
I think your response is great. It is so hard to resist lending a helping hand when we see a student struggle. This chapter really did a great job explaining why that struggle is so necessary for students. I work with a young child 1 on 1 and there have been times that I have prematurely offered help to him during math and he becomes very upset with me. It was really important (and needed) for me to see that he was willing to work through problems by himself. It is a good math and life lesson to trust the struggle.
I could not agree more with Rachel about this bringing much meaning to what it means to teach math. My past experience and current experience has been much the same with the main focus being on algorithms/calculations...not on recognizing patterns, making sense of what the problem is asking, and developing a problem solving approach to the problem. I currently have a group of passive learners that have had trouble with math for as long as they can remember likely. They have been helped and possibly enabled by well meaning teachers, special educators or paraprofessionals who have in my opinion likely focused on calculations in a traditional manner and helped students before they encouraged students to demonstrate perseverance. The three problem solving methods introduced are focused on problem solving but in different ways. The first establishes four steps for problem solving. The second focuses more on the second step of the first method to devise a plan. It lists a bunch of strategies such as visualizing, looking for patterns, using a list or table, simplifying, and other methods including guess and check! Finally the third methods focuses more on the explicit teaching of asking questions, making conjectures or making connections, seeking patterns, communicating, making generalizations and reflecting. I definitely see a place for all three suggestions and do not see them as segregated but more as a united approach to teach problem solving by pulling all the methods into a single approach for students. This excites me and is more consisted with my education of being asked the what ifs or being a visual learner or just wanting to solve problems and struggling until I got the understanding. It is funny for me to be thinking about this because although I approached math in this way, I ran from this approach in language arts until I was older and taught myself that it could be done in other areas beside just math. Very relevant chapter and one that will be integrated immediately.
I agree as well with what was said in the above posts. The first suggestion that I found from chapter three that I want to remember and practice in my future classroom is making math a higher level thinking experience. I want to do away with the routine and drill, but even more than that, with the notion that there is only one way to do a math problem and that the teacher alone knows that way. Pretty typically, it seems, my own memories of math reflect that teachers used to teach math at students, and students were not given the opportunity to find other methods that might have made more sense to them or worked better for their own way of thinking and reasoning. I feel like my classmates and I were not really encouraged to think for ourselves, which is not the message I want my future students to get from me. This kind of teaching did little to help us develop critical thinking skills or achieve higher level thinking. I think this is unfortunate and I will really encourage my students to be free-thinkers. This leads me to the second useful suggestion that I got from chapter three: I want to remember and teach that there are many different approaches to solving problems and that students can come up with different methods that work for them on their own. This seems important for lots of reasons: building math confidence; validating student's unique ways of thinking and learning; students will really know their methods and develop better number sense; students will be able to solve problems more effectively and easily; and students will see themselves as valuable sources of knowledge and information. This sounds great, but it makes me wonder how to make the connections to a math concept if a student is not getting it nor coming up with a way to solve a problem themselves? Do you tell them then? How much struggle is enough, and is it fair to assume that everyone can do this on their own? An unrelated third suggestion I really like is to include literature in math class. I think that this will help to integrate different subjects very nicely, which will lead students to make even more connections. Connections build and reinforce neural pathways in the brain. Also, it will make math seem more relevant to students, and it could make math more approachable. I have two young children, and I have realized the power that literature has on young minds. There is a book for every topic if you look in the right place. I had not really considered using books to help with formal math lessons or to pull problems from, but it makes so much sense! Books are a great way to get a message or lesson across, and kids are generally receptive to stories.
Worthwhile tasks provide multiple entry points (i.e., applied strategies) and multiple exit points (i.e., ways to express solutions). The simplicity of these multiple entry and exit points is astonishing, yet so easy to miss. The kindergarten/first grade example on p. 39 is a simple example. Too often I’ve seen problems like the first example; it gets to the mathematical point easy enough, but doesn’t engage the student – it doesn’t provide opportunities “to think mathematically.” The second problem isn’t any more difficult, yet can be approached using a number of strategies. What’s more, these worthwhile tasks for problem-solving with multiple entry and exit points provide a kind of differentiation and address different student learning styles. Students can be encouraged to extend their understanding with targeted questions asked while they are problem solving, or during the classroom discussion that follows. Which brings me to the second point I want to remember. While the task must be worthwhile and require students to think mathematically, some of the high-level thinking can come during the classroom discussion. “As students discuss ideas, draw pictures, use manipulatives…defend their strategy, critique the reasoning of others students have opportunities to understand and try out other approaches” (p. 38). I learned from my cooperating teacher that this classroom discourse can be a rich learning opportunity for everyone. The book also suggests having the students return to the same problem to decipher something that was revealed during the discussion. Allowing for the flexibility in my lessons to do this is critical. Don’t miss an opportunity for deeper understanding just to stay on schedule. Finally, the third strategy I don’t want to forget is related to the questions a teacher asks students. These questions should be focusing rather than funneling. Focusing questions help students understand the mathematics; funneling questions simply guide students to a pre-determined answer the teacher wants them to reach. Orchestrating classroom discourse will be key for my success in generating focusing questions. Not only do I need to understand the content, I need to be aware of and comfortable with the various ways by which understanding can be reached. The way I was taught years and years ago is not the only way (nor is it the best). Understanding is not following a method because it works; conceptual understanding followed by procedural understanding builds fluency. Which brings us back to worthwhile tasks. If the tasks are worthwhile, a variety of approaches (i.e., entry points) should be supported.
The main point I took away from this Chapter involved Classroom Discussions. The goal of discussions in class should be to further the student's learning through demanding questions and not to have students receive validation from the teacher for the correct answers. I thought the portion "How Much to Tell and Not Tell" of the reading was so important for such a short section. It is important that when discussing information with the students that you do not give them the answers or take away their opportunity to reflect and develop their thinking. I think this is an area of teaching that will take time and practice for me to develop. I also found the section on Problem Solving Strategies had meaningful insight into the various approaches to solve a problem. I liked that it showed the validity in a variety of methods and focused on curiosity as a means to motivate students to try different approaches. As it said on page 36 "It is important not to 'proceduralize' problem solving...don't take the problem solving out of problem solving by telling students the strategy they should pick and how to do it." It is easy as a teacher, or adult in general, to want to tell students exactly the way to do something or show them the way that is deemed "most efficient" in your opinion rather than allowing the student to come to their own conclusions or pursue their own method of thinking. Another point that I found interesting that I firmly believe, is that engaging students in meaningful work will reduce the amount of behavior issues. Students should be challenged and will enjoy having the opportunity to make sense of problems in a variety of ways that make sense to them, thus they will be less likely to act out. I think this is a crucial thing to consider in all disciplines that work with children! Give them choices and keep them busy.
This chapter taught me so much about ways to teach mathematical thinking. Up until recently, my experience with math has been based upon a practice and drill mentality. Mathematical thinking comes when students are challenged to use high level cognitive demand. This high level thinking builds and creates connections that might be relevant to previous lessons or experiences, but that provide deeper levels of understanding of mathematical concepts and ideas. This high level cognitive demand is something not routine, something in which students are required to think and possibly create multiple ways to understand the problem. This makes me think about a small group math workshop I recently sat in on in which students were introduced to a new concept of dividing fractions. The teacher introduced the idea by writing a problem on the board. The students had number lines in front of them to use. She then gave the students time to quietly think about what the problem was asking. For example the problem was how many 2/3rds are in 3? Several of the students used the number line to count by 2/3rds. However, they got really confused when they realized they had one piece left over. Several students remembered an algorithm they had learned from a previous lesson and tried using it. Ultimately, the teacher asked the students to share in their thinking. Together they came up with three different ways of understanding the problem. The students who had the hardest time coming around to it finally did understand once the other students explained their thinking. It was in providing different context and reasoning that helped these students struggle through and finally grasp the concept. The idea of having relevant context through which to explain worthwhile tasks is one I will use in my teaching. Creating a connection or relevance to the problem is a way to engage any student. This can be done in so many ways, such as through literature (as suggested in the book) or community problem solving, historical date concepts, etc. This reminds me of the Place-based curriculum design concepts of Amy Demerest. We are learning about how to implement these inquiry and place based concepts in my Assessment, Curriculum and Management class. Finally, while reading the section Questioning Considerations, I remembered a day in which my student was really struggling with understanding fractions and how many fractions make up a whole. I was showing her a number line and trying to explain how 1/2 is the same as 3/6 and 4/8. The math teacher came up to me and asked me to consider "Who's doing the thinking?" I kind of understood what she was talking about then, but I really understand what she means after reading this chapter. By explaining my understanding, I wasn't allowing my student to struggle and come to her own understanding.
ReplyDeleteI think your response is great. It is so hard to resist lending a helping hand when we see a student struggle. This chapter really did a great job explaining why that struggle is so necessary for students. I work with a young child 1 on 1 and there have been times that I have prematurely offered help to him during math and he becomes very upset with me. It was really important (and needed) for me to see that he was willing to work through problems by himself. It is a good math and life lesson to trust the struggle.
DeleteI could not agree more with Rachel about this bringing much meaning to what it means to teach math. My past experience and current experience has been much the same with the main focus being on algorithms/calculations...not on recognizing patterns, making sense of what the problem is asking, and developing a problem solving approach to the problem. I currently have a group of passive learners that have had trouble with math for as long as they can remember likely. They have been helped and possibly enabled by well meaning teachers, special educators or paraprofessionals who have in my opinion likely focused on calculations in a traditional manner and helped students before they encouraged students to demonstrate perseverance. The three problem solving methods introduced are focused on problem solving but in different ways. The first establishes four steps for problem solving. The second focuses more on the second step of the first method to devise a plan. It lists a bunch of strategies such as visualizing, looking for patterns, using a list or table, simplifying, and other methods including guess and check! Finally the third methods focuses more on the explicit teaching of asking questions, making conjectures or making connections, seeking patterns, communicating, making generalizations and reflecting. I definitely see a place for all three suggestions and do not see them as segregated but more as a united approach to teach problem solving by pulling all the methods into a single approach for students. This excites me and is more consisted with my education of being asked the what ifs or being a visual learner or just wanting to solve problems and struggling until I got the understanding. It is funny for me to be thinking about this because although I approached math in this way, I ran from this approach in language arts until I was older and taught myself that it could be done in other areas beside just math. Very relevant chapter and one that will be integrated immediately.
ReplyDeleteI agree as well with what was said in the above posts. The first suggestion that I found from chapter three that I want to remember and practice in my future classroom is making math a higher level thinking experience. I want to do away with the routine and drill, but even more than that, with the notion that there is only one way to do a math problem and that the teacher alone knows that way. Pretty typically, it seems, my own memories of math reflect that teachers used to teach math at students, and students were not given the opportunity to find other methods that might have made more sense to them or worked better for their own way of thinking and reasoning. I feel like my classmates and I were not really encouraged to think for ourselves, which is not the message I want my future students to get from me. This kind of teaching did little to help us develop critical thinking skills or achieve higher level thinking. I think this is unfortunate and I will really encourage my students to be free-thinkers. This leads me to the second useful suggestion that I got from chapter three: I want to remember and teach that there are many different approaches to solving problems and that students can come up with different methods that work for them on their own. This seems important for lots of reasons: building math confidence; validating student's unique ways of thinking and learning; students will really know their methods and develop better number sense; students will be able to solve problems more effectively and easily; and students will see themselves as valuable sources of knowledge and information. This sounds great, but it makes me wonder how to make the connections to a math concept if a student is not getting it nor coming up with a way to solve a problem themselves? Do you tell them then? How much struggle is enough, and is it fair to assume that everyone can do this on their own?
ReplyDeleteAn unrelated third suggestion I really like is to include literature in math class. I think that this will help to integrate different subjects very nicely, which will lead students to make even more connections. Connections build and reinforce neural pathways in the brain. Also, it will make math seem more relevant to students, and it could make math more approachable. I have two young children, and I have realized the power that literature has on young minds. There is a book for every topic if you look in the right place. I had not really considered using books to help with formal math lessons or to pull problems from, but it makes so much sense! Books are a great way to get a message or lesson across, and kids are generally receptive to stories.
Worthwhile tasks provide multiple entry points (i.e., applied strategies) and multiple exit points (i.e., ways to express solutions). The simplicity of these multiple entry and exit points is astonishing, yet so easy to miss. The kindergarten/first grade example on p. 39 is a simple example. Too often I’ve seen problems like the first example; it gets to the mathematical point easy enough, but doesn’t engage the student – it doesn’t provide opportunities “to think mathematically.” The second problem isn’t any more difficult, yet can be approached using a number of strategies.
ReplyDeleteWhat’s more, these worthwhile tasks for problem-solving with multiple entry and exit points provide a kind of differentiation and address different student learning styles. Students can be encouraged to extend their understanding with targeted questions asked while they are problem solving, or during the classroom discussion that follows.
Which brings me to the second point I want to remember. While the task must be worthwhile and require students to think mathematically, some of the high-level thinking can come during the classroom discussion. “As students discuss ideas, draw pictures, use manipulatives…defend their strategy, critique the reasoning of others students have opportunities to understand and try out other approaches” (p. 38). I learned from my cooperating teacher that this classroom discourse can be a rich learning opportunity for everyone. The book also suggests having the students return to the same problem to decipher something that was revealed during the discussion. Allowing for the flexibility in my lessons to do this is critical. Don’t miss an opportunity for deeper understanding just to stay on schedule.
Finally, the third strategy I don’t want to forget is related to the questions a teacher asks students. These questions should be focusing rather than funneling. Focusing questions help students understand the mathematics; funneling questions simply guide students to a pre-determined answer the teacher wants them to reach. Orchestrating classroom discourse will be key for my success in generating focusing questions. Not only do I need to understand the content, I need to be aware of and comfortable with the various ways by which understanding can be reached. The way I was taught years and years ago is not the only way (nor is it the best). Understanding is not following a method because it works; conceptual understanding followed by procedural understanding builds fluency. Which brings us back to worthwhile tasks. If the tasks are worthwhile, a variety of approaches (i.e., entry points) should be supported.
The main point I took away from this Chapter involved Classroom Discussions. The goal of discussions in class should be to further the student's learning through demanding questions and not to have students receive validation from the teacher for the correct answers. I thought the portion "How Much to Tell and Not Tell" of the reading was so important for such a short section. It is important that when discussing information with the students that you do not give them the answers or take away their opportunity to reflect and develop their thinking. I think this is an area of teaching that will take time and practice for me to develop.
ReplyDeleteI also found the section on Problem Solving Strategies had meaningful insight into the various approaches to solve a problem. I liked that it showed the validity in a variety of methods and focused on curiosity as a means to motivate students to try different approaches. As it said on page 36 "It is important not to 'proceduralize' problem solving...don't take the problem solving out of problem solving by telling students the strategy they should pick and how to do it." It is easy as a teacher, or adult in general, to want to tell students exactly the way to do something or show them the way that is deemed "most efficient" in your opinion rather than allowing the student to come to their own conclusions or pursue their own method of thinking.
Another point that I found interesting that I firmly believe, is that engaging students in meaningful work will reduce the amount of behavior issues. Students should be challenged and will enjoy having the opportunity to make sense of problems in a variety of ways that make sense to them, thus they will be less likely to act out. I think this is a crucial thing to consider in all disciplines that work with children! Give them choices and keep them busy.