Select three of the verbs for doing mathematics. For each, think about what it looks like when a student is "doing" it, then explain what it might look like.
I tried to choose verbs that touched on different levels of Bloom’s taxonomy. From different Internet sources, it was interesting to learn that some verbs are categorized under more than one level of Bloom’s taxonomy (e.g., construct, explain, solve). I chose these verbs for this assignment.
Construct. From the few verb lists I referenced, construct could fit under Apply or Create in Bloom’s taxonomy. On the lower level of the taxonomy, construct could mean applying understanding to a new, but similar problem. For Activity 11.3, students have estimated and measured one length/object, and they apply that understanding to something different. I am not sure how construct might be under the higher order Create category in this context. Could a very large object be selected, and the students are asked to construct a plan to measure it?
Explain. Explain could fit under Understand and Evaluate in Bloom’s taxonomy. Explain could mean a student discusses their Understanding of a concept, problem, or solution. For example, in Activity 11.15 a student could be asked to explain their answer. The extension to Evaluate could be explaining how to find the diagonal neighbors to any given number of the 100s chart.
Solve. Solve could mean Apply or Create under Bloom’s taxonomy. As with construct, to solve a problem could mean simply to apply understanding to a similar, but new problem. And I am still struggling with the Create level of the taxonomy, particularly at this 1st-2nd grade level.
At this foundational level of learning, how can a teacher extend student learning to the upper levels of Bloom’s taxonomy? In my mind the higher levels of Evaluate and Create require a fair bit of background information to apply – bring to bear - to a problem.
Great thinking and I love the idea that you tried to apply the ideas. I was thinking that with addition, 3 + B = 5, the introduction of a variable to the addition problem is a way to make it a higher level thinking problem for a first grader. They are so concrete and would not use a symbol for an unknown because they do not think of what is not there. They likely think that 5 - 3 is 2, so 3 + 2 is 5...they are taught it is a family of numbers and belongs together. But introducing an unknown or something they do not see or can't touch is a higher level thought process but something I believe should be done to start reducing the fear of representing the unknown.
I think this chapter was one of the best academic chapters I have ever read because it was so relevant to what I do every day. Unfortunately laughs could be heard for what should be being done and has not been done. The section about "What does it mean to be proficient" was screaming at me. If I ask a special education student who is a hunter "what time is it if it is Quarter past three" they often look at me and say "I do not know". When I know they look through their rifle scope and divide a circle into quarters all the time and would be able to do the same with a clock. Problem solving with representation/models and an ability to ask questions and look for patterns is not taught as much as, "I will model this algorithm and you copy it". This chapter made so much sense and will be shared and discussed with my co-teacher and math team.
I love this!!! It is hard to pick, but a few verbs that I think are especially important in teaching or "doing" mathematics include: justify, collaborate, and explain.
When a student is asked to explain their work, it allows them to get used to mathematical language and become more familiar with the processes they use to solve problems. Explaining work is important for a teacher as well. It allows teachers to track the thought/reasoning process of students. Teachers can even use student explanations as an informal assessment, to determine what areas they may need additional work in, what concepts they have mastered, and what the instructor needs to do better in order to teach them. If students can explain their thought processes, they are likely to have a solid understanding for any given mathematical concept. This may play out at any point in a math lesson; students should be able to explain every step they take to solve a problem.
This brings me to justify, a verb that is bound to happen if students are to explain their work and processes. To justify is to stick with an answer or conclusion and show why it is this way. This is an important skill for students to have, all across the board. Students need to stick up for what they think is right, and be able to back up their opinions and thoughts. This is a perfect verb for mathematics, as answers can be justified rather easily, and processes [mostly] justified by getting the right answer. Asking students why they came to a certain answer encourages them to justify their solutions.
If students are to collaborate, they will need to justify their work to their peers. Collaboration is an interesting math verb that I am not accustomed to as a result of my prior mathematical experiences, though I can see the value in it. Allowing students to collaborate might mean that they can tackle ever larger problems, and perhaps even real community problems. When students collaborate they build on the ideas of their peers, and solutions reflect the best ideas of all participants. When students talk problems out, they may be more easily and creatively solved. Students could potentially come up with solutions that not a single one of them would have come up with independently. Providing lots of time for small group work allows for collaboration among students, and it looks much like a discussion. Meghan
I think that being able to discuss and explain their thinking also ensures that students are thinking deeply about the process of figuring out the problem rather than just completing it one way or giving the expected response without reflecting on their work.
I think these are great verbs to associate with math, especially within this chapter. When I reflect on what I learned throughout my math years, I think about how much memorizing I did. I memorized based on tests and then forgot the material. When we are asking our students to justify, collaborate, and explain, in math, we are forcing them to explore and think about what they are doing and how they doing it. The verb justify really sticks out to me. When the book asked us to solve the problems in this chapter, I wanted to see those answers. However, I understand the reasoning that to know and do math you must be able to justify how you got to your answer and through that process you learn more about yourself as a mathematician, even if the answer is wrong.
Collaborate, predict and explain are the three verbs I believe best describe "doers" of mathematics. Collaboration with peers allows students to explore the problem from different perspectives than they might bring with their own initial understanding. Collaborating with a community of learners allows even the learner with the least understanding of the problem to be use the layered scaffolding of peer to peer related experiences to grasp a concept that might otherwise be inaccessible on his own. "Doers" of mathematics are predictors. Students who see mathematical problems and want to understand them usually have an idea or "prediction" of what the answer might be or how they would get to it. By predicting, the student is taking a risk, a jumping off point toward finding the answer. Explaining ones understanding of math is one of the best indicators of proficiency of math. "Doers" of math must take time to show they understand the concepts of the problem by explaining "what" and the "why" to their approach or approaches to the problem. Explaining oneself is sharing of what the problem looks like in your head, how you come about it and what you see as the best way to get to the answer and why.
I also remembered learning so many procedures and there being no focus in my math classes on what anything meant. I love it that my son in second grade immediately goes to problem-solving and "strategy" when doing math, and he is excited about explaining (I don't think he says justify yet!) his thinking. The material in this chapter is so useful to me to think about as a new math teacher - and I see how it works with my kiddo, budding mathematician than he is.
I tried to choose verbs that touched on different levels of Bloom’s taxonomy. From different Internet sources, it was interesting to learn that some verbs are categorized under more than one level of Bloom’s taxonomy (e.g., construct, explain, solve). I chose these verbs for this assignment.
ReplyDeleteConstruct. From the few verb lists I referenced, construct could fit under Apply or Create in Bloom’s taxonomy. On the lower level of the taxonomy, construct could mean applying understanding to a new, but similar problem. For Activity 11.3, students have estimated and measured one length/object, and they apply that understanding to something different. I am not sure how construct might be under the higher order Create category in this context. Could a very large object be selected, and the students are asked to construct a plan to measure it?
Explain. Explain could fit under Understand and Evaluate in Bloom’s taxonomy. Explain could mean a student discusses their Understanding of a concept, problem, or solution. For example, in Activity 11.15 a student could be asked to explain their answer. The extension to Evaluate could be explaining how to find the diagonal neighbors to any given number of the 100s chart.
Solve. Solve could mean Apply or Create under Bloom’s taxonomy. As with construct, to solve a problem could mean simply to apply understanding to a similar, but new problem. And I am still struggling with the Create level of the taxonomy, particularly at this 1st-2nd grade level.
At this foundational level of learning, how can a teacher extend student learning to the upper levels of Bloom’s taxonomy? In my mind the higher levels of Evaluate and Create require a fair bit of background information to apply – bring to bear - to a problem.
Great thinking and I love the idea that you tried to apply the ideas. I was thinking that with addition, 3 + B = 5, the introduction of a variable to the addition problem is a way to make it a higher level thinking problem for a first grader. They are so concrete and would not use a symbol for an unknown because they do not think of what is not there. They likely think that 5 - 3 is 2, so 3 + 2 is 5...they are taught it is a family of numbers and belongs together. But introducing an unknown or something they do not see or can't touch is a higher level thought process but something I believe should be done to start reducing the fear of representing the unknown.
ReplyDeleteI think this chapter was one of the best academic chapters I have ever read because it was so relevant to what I do every day. Unfortunately laughs could be heard for what should be being done and has not been done. The section about "What does it mean to be proficient" was screaming at me. If I ask a special education student who is a hunter "what time is it if it is Quarter past three" they often look at me and say "I do not know". When I know they look through their rifle scope and divide a circle into quarters all the time and would be able to do the same with a clock. Problem solving with representation/models and an ability to ask questions and look for patterns is not taught as much as, "I will model this algorithm and you copy it". This chapter made so much sense and will be shared and discussed with my co-teacher and math team.
I love this!!!
ReplyDeleteIt is hard to pick, but a few verbs that I think are especially important in teaching or "doing" mathematics include: justify, collaborate, and explain.
When a student is asked to explain their work, it allows them to get used to mathematical language and become more familiar with the processes they use to solve problems. Explaining work is important for a teacher as well. It allows teachers to track the thought/reasoning process of students. Teachers can even use student explanations as an informal assessment, to determine what areas they may need additional work in, what concepts they have mastered, and what the instructor needs to do better in order to teach them. If students can explain their thought processes, they are likely to have a solid understanding for any given mathematical concept. This may play out at any point in a math lesson; students should be able to explain every step they take to solve a problem.
This brings me to justify, a verb that is bound to happen if students are to explain their work and processes. To justify is to stick with an answer or conclusion and show why it is this way. This is an important skill for students to have, all across the board. Students need to stick up for what they think is right, and be able to back up their opinions and thoughts. This is a perfect verb for mathematics, as answers can be justified rather easily, and processes [mostly] justified by getting the right answer. Asking students why they came to a certain answer encourages them to justify their solutions.
If students are to collaborate, they will need to justify their work to their peers. Collaboration is an interesting math verb that I am not accustomed to as a result of my prior mathematical experiences, though I can see the value in it. Allowing students to collaborate might mean that they can tackle ever larger problems, and perhaps even real community problems. When students collaborate they build on the ideas of their peers, and solutions reflect the best ideas of all participants. When students talk problems out, they may be more easily and creatively solved. Students could potentially come up with solutions that not a single one of them would have come up with independently. Providing lots of time for small group work allows for collaboration among students, and it looks much like a discussion.
Meghan
I think that being able to discuss and explain their thinking also ensures that students are thinking deeply about the process of figuring out the problem rather than just completing it one way or giving the expected response without reflecting on their work.
DeleteI think these are great verbs to associate with math, especially within this chapter. When I reflect on what I learned throughout my math years, I think about how much memorizing I did. I memorized based on tests and then forgot the material. When we are asking our students to justify, collaborate, and explain, in math, we are forcing them to explore and think about what they are doing and how they doing it. The verb justify really sticks out to me. When the book asked us to solve the problems in this chapter, I wanted to see those answers. However, I understand the reasoning that to know and do math you must be able to justify how you got to your answer and through that process you learn more about yourself as a mathematician, even if the answer is wrong.
ReplyDeleteThis comment has been removed by the author.
DeleteCollaborate, predict and explain are the three verbs I believe best describe "doers" of mathematics. Collaboration with peers allows students to explore the problem from different perspectives than they might bring with their own initial understanding. Collaborating with a community of learners allows even the learner with the least understanding of the problem to be use the layered scaffolding of peer to peer related experiences to grasp a concept that might otherwise be inaccessible on his own.
ReplyDelete"Doers" of mathematics are predictors. Students who see mathematical problems and want to understand them usually have an idea or "prediction" of what the answer might be or how they would get to it. By predicting, the student is taking a risk, a jumping off point toward finding the answer.
Explaining ones understanding of math is one of the best indicators of proficiency of math. "Doers" of math must take time to show they understand the concepts of the problem by explaining "what" and the "why" to their approach or approaches to the problem. Explaining oneself is sharing of what the problem looks like in your head, how you come about it and what you see as the best way to get to the answer and why.
I also remembered learning so many procedures and there being no focus in my math classes on what anything meant. I love it that my son in second grade immediately goes to problem-solving and "strategy" when doing math, and he is excited about explaining (I don't think he says justify yet!) his thinking. The material in this chapter is so useful to me to think about as a new math teacher - and I see how it works with my kiddo, budding mathematician than he is.
ReplyDelete