2 October 2018
Initial observations and records of development.
This semester, I will be journaling the mathematical development of G., a second-grade student in Mrs. Hammond’s classroom. I will observe his development while in the classroom on Tuesdays and make additional visits to the classroom as needed. G. will work on math five days a week, for approximately two hours a day. Math instruction will be provided using the Engage New York Curriculum, in small group and whole class settings. G. will also utilize the Moby Max program for supplemental online math instruction. Furthermore, G. will participate in a target math group consisting of students that are failing to meet grade level expectations. I will lead this group on Tuesdays, and Mrs. Hammond will lead the group on the remaining days.
G. was a late arrival to Mrs. Hammond's class this year, having changed classes after the second grade was downsized by one classroom. He is one of a handful of students that did not have Mrs. Hammond as their first-grade teacher before coming to her second-grade classroom. G. Is testing below grade level in both reading and mathematics. His most recent Star reading test scores indicate that he is reading at a 1.2, or two months into his first-grade year. His most recent mathematics test scores indicate that he is performing at a 1.4, or four months into his first-grade year. This is consistent with the work that he has turned into Mrs. Hammond during the first nine weeks of the semester.
G. is a bright, 7-year-old, wriggle worm. I observed his behavior from a distance during last Tuesday’s math lesson. It was clear that he behaves differently when he is close to Mrs. Hammond or myself during math lessons. Observing from a distance allowed me to observe what he is capable of on his own. His first problem in learning math was immediately apparent. G. cannot sit still. During the first couple minutes of the math lesson, he mopped his entire table with his head. When he is not wriggling, he is talking to any of his peers that will pay attention to him. When his attention is redirected to the board, he says that he cannot see what the teacher is doing. G. wears thick glasses, so his trouble seeing the board was not a surprise. However, when moved closer to the board, G. still does not pay attention. Instead he is looking around the room and talking to the peers that are now close to him. During independent work time, G. behaves much the same way. He is often the last one to finish his work. Most of the time this is due to his own distracted behavior. He also has difficulty focusing if the classroom is loud or if there is an abnormal amount of movement in the room.
G.’s behavior during math lessons is considerably different when he is close to a teacher. He attends to what is occurring in the lesson and asks questions about parts of the lesson that he does not understand. When working, he is able to show his strategy use and justify his thinking. However, it is clear that he is missing pieces of content knowledge that he should have acquired last year. In particular, he struggles with subtraction within 20. G. knows that he struggles with math and has stated that he wants to do better. He is motivated to learn, as long as he has support close at hand. When he is on his own, he gives up before he even tries.
24 October 2018
G. has ample opportunities to participate in math during school hours. These opportunities take on many forms, including small group intervention activities, large group mathematical discussions, working on a team to solve story problems and one on one conversations with Mrs. Hammond and myself. The small group intervention utilizes the Do The Math program written by Marilyn Burns. The large group mathematical discussions are conducted using the Engage New York curriculum, with supplements from Cognitively Guided Instruction and Intentional Talk. The story problems provided by the Engage New York curriculum are worked on as a team and discussed using tools from Cognitively Guided Instruction and Intentional Talk. The one on one conversations enable Mrs. Hammond or myself to elicit G.’s mathematical thinking and make sure that we are clearing up misconceptions as they develop. Overall, G. receives around two hours of math instruction, five days a week.
Yesterday, while working on Engage New York second grade module 3, lesson 15; I was able to listen in on G. as he worked with his team to solve four related math story problems. G. listened quietly as his team talked about solving the first three problems, providing his input only when directly asked for it. When his team reached the fourth problem they were stuck. The problem asks how many pencils they think the principle will need for four months. This was the first problem they had seen that asked them to form an opinion and defend it mathematically. As the rest of the team was looking to me to tell them how to come up with a solution, G. says “I think it’s 400.” His team looked as surprised as I was that he had spoken up. The look on his face made me think he was also surprised he had spoken out loud. I asked him his least favorite question, why? G. looked at his teammates and said it was kind of hard to explain. Once I reassured him that he had his team and myself to support his explanation, he relaxed a little. He had us look at problem 2, which stated that the principle wanted 300 pencils for three months. He concluded that if the principle needs 300 pencils for 3 months, it made sense that he would need 400 pencils for 4 months. His team agreed with his reasoning and decided that G. would be the speaker when they presented that solution. G. was one of two students that came up with an answer to problem 4. I was very proud of him and made sure to tell him so at the end of his explanation to the class.
G. has stated that he does not engage in any math outside of the classroom. Mrs. Hammond rarely sends home homework and when she does, it is usually reading work. G. has told me that he only likes math when he is at school. He explained that it is hard to do math when you do not have a teacher to tell you if you are right. G.’s parent teacher conference is coming up. At that time, his parents will be provided with ideas for ways to incorporate math into their life at home. This will include games they can play as well as ideas for using math at the grocery store and other locations.
I have also observed that G. has an easier time paying attention when he has something small to play with in his hands. Most days when the class is engaged in whole group instruction and discussion, G. wiggles in his chair to the extent that he is unable to focus on what is being said. Last week, a classmate gave everyone small pencil top erasers. During instruction, G. played with his eraser. As he did so, his eyes were on Mrs. Hammond and I could hear him quietly repeat everything she said was important. I was amazed at how much better he was paying attention to what was going on around him. Next week, I am going to find a small piece of soft fabric or some other small something that he can have in his hands during instruction. I am curious to see if the eraser was a fluke, or if this will help him to focus better. If it works, he will have access to something to fidget with during whole group instruction in all content areas.
5 November 2018
Math Discussion
Today G. and I talked about his strategies for counting change. G. was given a series of problems to solve that required him to find the total amount of money he has if he is given a set number of each coin. For example, how much money do you have if you have 1 quarter, 7 dimes, 4 nickels, and 2 pennies? At the beginning of the activity Mrs. Hammond lead the class in a discussion about how much each coin is worth, what each coin looks like, and why this information is good to know. This is the second lesson the students have participated in that involved coins, and they were a little apprehensive of attempting the problems on their own. Mrs. Hammond modeled how to solve the first problem in a couple different ways before letting the students work independently.
G. and I began our talk by talking about why he should know how to count change. He expressed the importance of being able to pay for things at the store and counting money in his piggy bank to know if he has enough for the game he has been saving for. The conversation next moved to the problems G. was to solve. As we approached the first problem he told me he found the way Mrs. Hammond counted change, which involved using expanded form, to be confusing. We read the first problem together, 4 dimes, 4 nickels, and 6 pennies. I asked G. to show me how he counts change. The following exchange occurred:
G – “this one is easy, there are no quarters, quarters are hard.”
T – “Okay, can you show me how you are going to figure it out?”
G – “First, I do the dimes, those are ten, 10, 20, 30, 40. Then I do the nickels, 45, 50, 55, 60. Then I do the pennies, 61, 62, 63, 64, 65, 66.”
T – “So how much money do you have?”
G – “I have 66.”
T – “66 what? 66 bananas, 66 apples, 66 monkeys?”
G – “66, I forgot what you call it when you just have change.”
A (student sitting next to G) – “cents”
G – “66 cents.”
T – “Can you explain why you counted up like you did?”
G – “what do you mean?”
T – “you started counting by tens, then you changed the way you were counting. Can you show me how you counted?”
G – “I counted by tens for the four dimes, then I counted by fives for the nickels, and ones for pennies”
T – “what would happen if you counted them separately?”
G counts the dimes and writes 40, then counts the nickels and writes 20, then counts the pennies and writes 6. “Ohhhh, this is what Mrs. Hammond did! I get it.”
T – “should we try it that way on the next one?”
G – “okay.”
G. tries the method that Mrs. Hammond used on the next problem, 1 quarter, 3 dimes, and 4 pennies.
G – “that’s 25, that’s 30 and that’s 4. 25 plus 30. That’s hard.”
T – “How can we solve that problem?”
G – “well 20 plus 10 is 30 plus 10 is 40, then we have the 5 from the 25 so that’s 45. We have 45 cents?”
T – “I see where you used the quarter and the dimes, but what happened to the pennies?”
G – “oh yeah we have 45, 46, 47, 48, 49. We have 49 cents.”
T – “that’s great work buddy. I like how hard you tried. What did you think about using that strategy?”
G – “I didn’t like it, it was hard for me. I like the way I did it on this one” points to the first problem.
T- “are you ready to try a couple without me, and I will check in on you in a couple minutes?”
G – “I think so, I will let you know if I need more help.”
This conversation with G. showed me where he is in his development. He is more comfortable using counting strategies and is not yet ready to move to more abstract methods. He does have an understanding of the value of each coin, and a working strategy for finding the total number of coins. Later in the lesson, when I checked in with G., he had completed his worksheet. We were discussing the work that he had done when he told me that counting money was fun. I asked him if he could count money at home or somewhere away from school. He said that he was going to go home and count the money is his piggy bank. He was also planning on asking his mom if he could count some of the change in her change jar.
6 December 2018
Teaching math the last couple weeks has been really exciting, both for G. and for myself. Last week, the students worked on a series of story problems involving addition and subtraction of numbers less than 1000. G. had a difficult time with the problems, in part because of reading difficulties and in part because of inefficient strategy use. Thus, I sat with G. as he worked on his worksheet. My goal was to provide the reading support he needed to be successful at the math.
The first problem on the worksheet stated, “39 books were on the top bookshelf. Marcy added 48 more books to the top shelf. How many books are on the top shelf now?” (New York State Education Department, 2014). G. had a difficult time reading this problem because he did not recognize Marcy as a name. When I asked him about where he was having trouble, he identified the word Marcy as problematic. He stated he wasn’t sure if it was important to the problem, so he didn’t want to start and get the problem wrong. After discussing where he was having trouble, G. and I read the problem together. After reading it, G. just sat there looking at me. I asked about his ideas for solving the problem, and he told me he was waiting for me to tell him how to do it. I asked him how he would solve the problem if no one told him how. He finally decided to draw a picture. I wondered away to check on other students for a couple minutes.
When I came back to G. he had drawn 39 little squares for the 39 books and was beginning to draw 48 more little squares. However, he kept losing track of how many he had drawn. After a few minutes of discussion on how he could keep track of the little books, he decided to write numbers in the box of every tenth book. He was excited when he finally finished drawing little boxes (which took about 6 minutes total). When I asked him how many books were on the shelf, he looked at his work and asked, “Do I have to count all of these?” Together we used 3 crayons of different colors to group the books by tens. Then we counted the groups of tens and the remaining books to get a total of 87 books.
While he was pleased with the work he had completed for the first problem, he decided he did not want to use that strategy again. When I asked him why, he explained it was a lot of work and it took a long time. The second problem on the worksheet stated, “there are 53 regular pencils and some colored pencils in the bin. There are a total of 91 pencils in the bin. How many colored pencils are in the bin?” Once again G. had troubles reading the problem. We read the problem together. He then stated, “I’m confused, are we plussing the pencils or taking away the pencils?” To help him clarify what the problem was asking we used one of the classrooms pencil cups to act out the problem. This helped to clarify what the problem was asking.
Overall, this worksheet was very difficult for G. I helped him read and work out each problem using strategies he wanted to try. I was not optimistic about G.’s ability to answer the questions without constant support. Fortunately, this week’s module enabled me work through the strategy G. was trying to use with the entire class. This is when it got exciting.
This week’s lesson utilized a ten’s and one’s chart to solve addition problems that required bundling. As I planned to teach the lesson, I made connections between what the students were doing this week and the worksheet they had completed last week. I decided the students would begin their work by completing a worksheet from this week’s module then they would review the work they had completed last week and make corrections.
The first thing I noticed when teaching this week was G. paying closer attention to the lesson than usual. After working several examples on the board, the students began to work on their worksheets in pairs. I took the opportunity to talk to G. about how closely he was paying attention. He explained that I was showing him how to do what he was trying to do on the other worksheet. I left him and his partner to work on their worksheets and was surprised when they were one of the first groups done. I was concerned that his partner had done all the work, so I asked G. to explain how he had solved a couple of the problems. I was astonished at the difference between this week and last. G. was getting it.
G. went on to correct all of the addition problems on last week’s worksheet and complete another worksheet for additional practice. He was so proud of what he could now do. I made sure to tell him how proud of him I was. When I asked about the difference between the work last week and the work this week, he explained that he was frustrated at not being able to read the story problems. Even with help reading them, he was not sure which operation the problem required. He stated that this week was easier because he knew they were all addition problems. However, when checking G.’s worksheet he had corrected, I found that his subtraction problems were also now correct.
I think a large part of G.’s problem is the variety of strategies that are introduced by the Engage New York curriculum. It seems that the class is not given adequate time to practice a strategy before being introduced to another one. This has led most of the class to using strategies they learned last year instead of the new strategies they were being introduced to. Going forward, I will make sure that new strategies are connected to other strategies that G. has used successfully and there is adequate time for everyone to practice the strategy.
References
New York State Education Department (2014). Grade 2 mathematics. Retrieved from https://www.engageny.org/resource/grade-2-mathematics