Reflection and Analysis
Though in my initial lesson plan instruction was to take place during “centers” time with a table group of 4-5 students, logistically it worked better to do my lesson in a “whole class” format with 15 students (about half of the class). This was due to circumstances of the half-day schedule, but my actual lesson followed almost the exact same format as the plan nonetheless. Students sat at their own seats with mini-whiteboards, markers, and erasers, and they did the opening questions diligently, many of them following the norms (finger raising, listening, sharing their steps, repeating other students’ contributions in their own words, etc.) they had learned during my two previous number talks. As stated in my lesson plan, I drew most of these norms from the book Intentional Talk by Kazemi and Hintz. We then moved into the body of the lesson where I first explained and modeled the place value chart/ expanded form subtraction. Students helped me with the smaller pieces within that process, and once they indicated to me the felt comfortable, they had some independent work time to practice the new strategy. Once they had shared that problem and we spent some time discussing why regrouping works, we closed with a strategy share out for a last problem.
Students used the whiteboards both to copy my modeled problems and to do independent work. This tool served as a way for students to write their own mathematical thinking and processes without having the overwhelming piles of scratch paper that develop because these students still have trouble organizing a scratch paper in a way that is coherent. The tasks they had both independently and with the whole group built upon previous concepts of standard algorithm multi-digit subtraction, place value, expanded form, and regrouping. As is made clear in Math Matters, adults do not always use the standard algorithm but rather a variety of strategies based upon a general mathematical understanding (2006, p.47-8). In this way, I hoped to create another access point for students to think about what they were doing with the quantities when “making a 10 a 9” and “taking 3 from 4.” While some students clearly grasped the concept (see student board in artifacts), others preferred to listen and continue with standard algorithm subtraction (noted from verbal feedback from the observers). While students were asked to practice the “Ms. Wraith way of subtracting,” a number of times during the lesson, students were allowed to use and share whatever strategy made sense for them, particularly in the opening and closing. This choice was made both to allow students to feel successful at something if they were overwhelmed by my strategy and to demonstrate that multiple strategies still yield the same answer when applied correctly.
While I definitely incorporated talk moves outlined in Intentional Talk in my structuring of the discussion, particularly with open strategy sharing, I wish I had incorporated more “Why? Let’s Justify” and “Compare and Connect.” Evidence of the use of talk moves can be seen in the Edthena videos: “Do you think you understand what [student] said?” and “Do you agree? … why? Show me,” for example. I think I strayed away at moments from more using comparing and connecting because of focusing on getting students to understand my representation of numbers. The class might have benefitted from more opportunities to talk to each other, rather than though me. A moment that could have been more powerful than it was was when we compared 300+40+1 to 300+30+11. Ignoring my typo (clear both in the board image and the video #2), this could have been a moment for a deeper discussion of number composition. While we did practice showing that different number sentences can still equal the same quantity, I wish I had stuck to my original plan of letting students share multiple ways of adding to get the same quantity. I ran out of time, but I would have wanted students to focus on their own representations of numbers more than mine. A way I could have allowed the students to manipulate their representations of quantity would have been to use base 10 blocks as I had planned as a back up, but with the whole class format, I was worried about running out of time. In hindsight, I am not sure this was the best choice to omit it.
In the future, I would love to include this lesson in my own classroom with students learning multi-digit subtraction. There were students (see my assessment checklist) who were not able to, in one lesson, grasp my new strategy independently, but there were proportionally more who could do most or all of it independently. There were even students who surprised me in how quickly they picked up parts of it given their performance on formal math tests and homework. I place most of the reasoning for this on the way in which this strategy built upon previous skills they had already developed and the support of the discourse moves not typical to most of their math lessons, lowering the barrier to participate. In doing this lesson again, I think I would want students to feel empowered to use the board to show their classmates their work, but because this is not a norm in their general classroom, I did not want to break it during my lesson. Whether students could do the skill the following week independently or whether they could explain the concept to someone thoroughly I would be interested to know. This is perhaps a guide for me in the future in terms of structuring assessments, and it demonstrates a limitation of a single, isolated lesson.
The questions that linger for me are about timing and content selection. I would choose to do this lesson again perhaps in a longer format with base 10 blocks, but I would continue to worry about time pressure. I worry about how long each lesson takes (luckily, my timing allowed me to cover all the content I wanted to cover save work with the base 10 blocks), and I worry about teaching strategies that differ from their text books and PSSA test prep expectations. I worry that my desire to hurry cuts off useful discourse and stops me from probing further into thinking. This worry became evident for me in watching my Edthena video part 2 when I never got a student to talk deeply about why borrowing works and why I can regroup numbers into different parts without changing the number’s value. This might have just taken more time. Using base 10 blocks, I think we could have gotten closer to conclusions about this, and in the future I think that’s what I would do. It strikes me that while I can understand and endorse the talk moves outlined in Intentional Talk, it is very difficult to break old habits of how I was taught. My biggest take away from teaching my lesson is to slow down and delve deeply into one or two important concepts instead of trying to hit multiple content areas all in one lesson.
Kazemi, E., & Hintz, A. (2014). Intentional Talk How to Structure and Lead Productive Mathematical Discussions. Portland: Stenhouse.
Students used the whiteboards both to copy my modeled problems and to do independent work. This tool served as a way for students to write their own mathematical thinking and processes without having the overwhelming piles of scratch paper that develop because these students still have trouble organizing a scratch paper in a way that is coherent. The tasks they had both independently and with the whole group built upon previous concepts of standard algorithm multi-digit subtraction, place value, expanded form, and regrouping. As is made clear in Math Matters, adults do not always use the standard algorithm but rather a variety of strategies based upon a general mathematical understanding (2006, p.47-8). In this way, I hoped to create another access point for students to think about what they were doing with the quantities when “making a 10 a 9” and “taking 3 from 4.” While some students clearly grasped the concept (see student board in artifacts), others preferred to listen and continue with standard algorithm subtraction (noted from verbal feedback from the observers). While students were asked to practice the “Ms. Wraith way of subtracting,” a number of times during the lesson, students were allowed to use and share whatever strategy made sense for them, particularly in the opening and closing. This choice was made both to allow students to feel successful at something if they were overwhelmed by my strategy and to demonstrate that multiple strategies still yield the same answer when applied correctly.
While I definitely incorporated talk moves outlined in Intentional Talk in my structuring of the discussion, particularly with open strategy sharing, I wish I had incorporated more “Why? Let’s Justify” and “Compare and Connect.” Evidence of the use of talk moves can be seen in the Edthena videos: “Do you think you understand what [student] said?” and “Do you agree? … why? Show me,” for example. I think I strayed away at moments from more using comparing and connecting because of focusing on getting students to understand my representation of numbers. The class might have benefitted from more opportunities to talk to each other, rather than though me. A moment that could have been more powerful than it was was when we compared 300+40+1 to 300+30+11. Ignoring my typo (clear both in the board image and the video #2), this could have been a moment for a deeper discussion of number composition. While we did practice showing that different number sentences can still equal the same quantity, I wish I had stuck to my original plan of letting students share multiple ways of adding to get the same quantity. I ran out of time, but I would have wanted students to focus on their own representations of numbers more than mine. A way I could have allowed the students to manipulate their representations of quantity would have been to use base 10 blocks as I had planned as a back up, but with the whole class format, I was worried about running out of time. In hindsight, I am not sure this was the best choice to omit it.
In the future, I would love to include this lesson in my own classroom with students learning multi-digit subtraction. There were students (see my assessment checklist) who were not able to, in one lesson, grasp my new strategy independently, but there were proportionally more who could do most or all of it independently. There were even students who surprised me in how quickly they picked up parts of it given their performance on formal math tests and homework. I place most of the reasoning for this on the way in which this strategy built upon previous skills they had already developed and the support of the discourse moves not typical to most of their math lessons, lowering the barrier to participate. In doing this lesson again, I think I would want students to feel empowered to use the board to show their classmates their work, but because this is not a norm in their general classroom, I did not want to break it during my lesson. Whether students could do the skill the following week independently or whether they could explain the concept to someone thoroughly I would be interested to know. This is perhaps a guide for me in the future in terms of structuring assessments, and it demonstrates a limitation of a single, isolated lesson.
The questions that linger for me are about timing and content selection. I would choose to do this lesson again perhaps in a longer format with base 10 blocks, but I would continue to worry about time pressure. I worry about how long each lesson takes (luckily, my timing allowed me to cover all the content I wanted to cover save work with the base 10 blocks), and I worry about teaching strategies that differ from their text books and PSSA test prep expectations. I worry that my desire to hurry cuts off useful discourse and stops me from probing further into thinking. This worry became evident for me in watching my Edthena video part 2 when I never got a student to talk deeply about why borrowing works and why I can regroup numbers into different parts without changing the number’s value. This might have just taken more time. Using base 10 blocks, I think we could have gotten closer to conclusions about this, and in the future I think that’s what I would do. It strikes me that while I can understand and endorse the talk moves outlined in Intentional Talk, it is very difficult to break old habits of how I was taught. My biggest take away from teaching my lesson is to slow down and delve deeply into one or two important concepts instead of trying to hit multiple content areas all in one lesson.
Kazemi, E., & Hintz, A. (2014). Intentional Talk How to Structure and Lead Productive Mathematical Discussions. Portland: Stenhouse.