This week Julie and I took on our first day of Number Sense. On Tuesday, we arrived early in the morning to meet with Mrs. Arnold and Mrs. Carmack to discuss the process and expectations for the next few months. After conversing with the Kindergarten teachers at Longfellow, we were excited about the opportunities that would be presenting themselves while interacting with the students. Mrs. Arnold and Mrs. Carmack had recently attended a professional conference workshop, where they received a packet containing a copious amount of activities, ranging from number recognition activities to counting and base-ten activities, which can be used later in the term as students begin to develop a stronger number sense. Because we had agreed with our cooperating teachers that Julie and I would be taking the reign when it comes to planning lessons each week, we plan on using this packet as a reference when planning number sense activities for the students. We will be using the same ability grouping that we used during EDUC360 with Dr. Egan this past fall. On Tuesday, we wanted to begin with an activity that would allow for an initial assessment of all of our students. Thus, we began with a number recognition bingo activity. We were confident that the students would enjoy this fun game for their first day, but it also provided us with a lot of feedback regarding the students’ current understanding of number sense. The students in groups A-F struggled to recognize numbers beyond ten, where as the higher ability students provided us with evidence that they have mastered number recognition for numbers 1 through 20. However, many students who struggled with number recognition began to pick up on the pattern of teens. For example, students began to grasp the concept that a one and six is 16. However, this pattern became problematic for numbers such as 12, as some students would guess “two teen”. After we completed the bingo activity with each group, we had the students complete a Join – Result Unknown word problem. Julie and I decided that we would be doing one word problem a day with the students. Each week, we will add the new word problem pages to the students’ preceding pages, creating a book for each student. On Thursday, we did a number recognition activity with the students who have not yet provided evidence of mastering number recognition. These students rolled a large die, stated the number, and then colored in the corresponding symbolic number on a sheet. Again, we continued to see similar observations that we had seen on Tuesday. Many of the students were unable to recognize most numbers beyond ten, particularly numbers ten, eleven, twelve, and thirteen. The students who had demonstrated mastery of number recognition completed a base ten-frame worksheet. The students filled out ten frames, demonstrating their understanding that a given number eleven through nineteen is one group of ten plus “x” ones. The majority of the students showed a strong understanding of the value of a number through the ten-frame activity. I was astonished by the performance of one particular student, student I.E. After filling in nineteen dots on the ten-frame, I asked this student “If I were to add 1 more dot, how many would we have?” She answered 20. She was then able to tell me that twenty would be 2 groups of ten. While not all of our students have developed the profound number sense knowledge that student I.E. has, this experience sparked my excitement for the remainder of the number sense experience. Julie and I both look forward to helping these students practice their mathematical skills and develop a stronger number sense over the next two terms.
This week with Mrs. Carmack’s students we proctored this quarter’s acuity assessment on the iPads. As this is what my research has been focused on and leading up to, I was interested to see the new test as well as how the students would score. I asked the other two teacher candidates to complete learning notes for each of the students they sat with for the assessment, this way I would be able to gain a larger amount of data from a wide range of students and skill sets. As well as I was able to get others insights on the process of completing learning notes in the classroom. While working with my students personally I saw some of the same mistakes that I had seen last quarter, but one source of the errors was eliminated by the aid who usually gives the assessments told us to have the students tell us which answer to pick. By having us click the answer, the amounts of errors were somewhat diminished, but not completely gone. For example some of the wording in the questions lead to confusion for the students. Also the layout of the test lead to misunderstanding based on the students. One of the questions asked students to select the cone shape, but the answer options were letters that corresponded to the shapes in the question. Other problems that arose during the assessment was the fact that these tests are completed on the iPad. The program did not always work the right way, would kick out the student multiple times, or take a while to load, all of which could have an effect on the students performance. One aspect that I ad a multiple students comment about was the size and layout of the questions. Students would want to zoom in on the options to better see, but then would not be able to see the question and would lead to frustration or guessing. By completing this new assessment I was able to gain a better understanding of the benefits for learning notes, and the sufficient need for them during the assessment process in the lower grades. I am glad this week was the time the students needed to complete the test. I also gained confidence that my student’s hard work one on one with me has been helping their number sense:)
This week we were finally able to get back into the swing of things and have work with both classes on their correct day. My plan for this week was to take another question from the standardized assessment that the students took earlier in the year and approach it in a more hands on approach. The question we worked on this week had to do with a ten frame. That original question asked students to look at a partially filled ten frame and figure out how many more counters they need to completely fill the ten frame. From working with the student I found that many of them did not even know the name of the ten frame. When simply asking the students how many more they would need the students struggled to visualize what they were being asked to do. One thing that I found to help students in the completion of the task was to tell students that they needed to fill all the boxes not “complete the frame”. Once students had the different terminology and access to manipulatives they were more successful. One interesting thing that occurred this week was not in regards to the assessment questions I was recreating. Randy has created a new app that Has a string of beads that can be moved and counted. Another portion allows the app is a random number of the beads are hidden and the user is asked to identify the number hidden. One of the students that I worked with figured out that her figure was equal to the width of the bead and then placed her figure over the hidden bar to decide how many beads were actually hidden. Granted she wasn’t vert precise, but the thinking and logic were impressive.
Our main goal this week was to transition the students who were still using counting-all strategies to begin using counting-on. In order to accomplish this, we once again started by presenting students with addition problems and observing how they solved them. Some students began solving the problems by using their fingers, others preferred to draw dots, and some even had the sums of a few addition problems memorized. For those that were using their fingers or drawing dots, the challenge was to have them realize that they didn’t have to count both of the addends. We had a student working on the problem 5+3 and she began by putting five fingers up on her left hand and three up on her right. After she told us that the answer was eight we asked her why she had to count all of her fingers starting at one. She seemed perplexed so we then asked her how many fingers she had on her left hand. She responded that she had five, so we then offered a suggestion. “If you know that you have five fingers on your left hand do you really need to count them all starting at one?” The student responded by saying no. Then we prompted her by saying, “Okay, so do you know what number comes after five?” She told us that six comes next and then we showed her how she could just count-up three from five instead of counting all of the addends. After demonstrating a few times the student was able to use counting-on on her own.
In the kindergarten classrooms this week, Jackie and I have been continuing work with our students on counting on strategies. We are continuing practice for students who do not yet utilize the strategy of counting on for addition, and for those who have been using the strategy we are practicing the strategy with the goal of helping the students discover that it is more efficient to count on from the larger addend.
This week, we had two new sets of materials to help us work with the students. One of the materials was a container which held three dice, two of which were numbers 1-6 and the other was either an addition or subtraction sign. We also had a set of addition and subtraction flash cards which had numbers on one side and dots on the other.
We used the dice with students who did not have their addition facts for addends 1-6 memorized. The students would shake the container and then read the math problem that resulted. We had white boards for the students to write the problem out, and we then asked them how they would solve their problem. Many students said that they wanted to draw dots on the white board or use their fingers in order to represent the addition problem. Since most of the sums that resulted from the problems were less than 10, we found that many of the students would want to hold up both addends on their fingers and then count all in order to solve the problem. If they chose to draw dots, they would represent both addends and then count all in order to solve. We let the students solve the problems counting all the first time if this was the strategy they chose, but the next time they began using counting all as a strategy we would prompt them by questioning whether they needed to count the first addend or if they already knew how many they had. A good visual seemed to be when the first addend was five and the students chose to use their fingers. The students already know that holding one hand represented five without having to count each finger, so they did not need to count the first addend. We even heard students thinking out loud, saying “I already know that I have “x”…” when referring to the first addend. For these students, we focused on counting on from the first addend instead of counting on from the larger.
For students who could have performed the dice addition in their heads, we used the addition and subtraction flash cards. Since the cards had dots on the back, we started by having the first addend represented by a number card and the second represented by dots, hoping to prompt students to start at the first number and then count up the number of dots to find the sum. After students were demonstrating this, we flipped the second addends over so the number side was showing for both addends. At this time, we started placing the larger addend second. If students did not choose the larger addend to count on from, we would ask the students if the reverse of the addition problem gave them the same answer. We then modeled by counting on from the larger addend, and asked if this seemed easier or took less time. Many of the students would note other students in the group using the counting on from the larger strategy and would then use this same reasoning when solving their own problems. It will be interesting to see if the practice with the strategies in both situations will result in a change in strategy when we work with the students next week.
Jess and I have begun to utilize the activities that we found in a study conducted by Secada, Fusion, and Hall. Since we have already identified which of our students chose counting-on in order to solve addition problems, we have moved on to see if they possess the subskills that Secada, Fusion, and Hall identified in their research. The first subskill is knowing that the first addend does not need to be counted again because the problem already states its quantity. For example, if the problem was 5+6 we would see if the student would start counting at one, or if they would start at five and count up (6,7,8,9,10,11). In order to determine which of our students possess this subskill, we presented the students with several addition problems. First, we would create addition problems using numbers that had been written on note cards. Initially we didn’t give the students any manipulatives that would aid them in solving the problem and we also chose addends that had a sum larger than ten so that they couldn’t add the numbers just by placing an addend on each hand. After the students completed a couple of these problems, we started placing two-colored counters under the note cards. When the students were solving these problems we asked them to tell us how many manipulatives there were without counting. A majority of the students looks at the note card that was placed above the card and started to correspond the manipulatives with the note card above it. The goal of this was to introduce students to the fact that they don’t need to start at the beginning of the counting sequence; they can just start by counting-up from the first addend. Almost all of the students from Jess’ group were able to count-up after we accompanied the note cards with the manipulatives, but several of mine still struggled and reverted back to a counting-all strategy. Since they have not mastered this subskill, we will continue to work with those students until they can consistently demonstrate it. Something that we found interesting, was that some of the students chose to count-up from the second addend. When asked why they replied, “Cause that numbers bigger, it takes less time.” Upon hearing this, the two people in the group started to count-on as well, but from the larger of the two addends.
Our work this week has been focused on the subskills that our research linked to counting on. Jackie and I chose to take our students out together, three at a time, so that one of us could work with the students while the other videotaped and we would both get to witness what the students were doing.
In the groups we worked with we identified students who used the strategy of counting on and those who did not yet employ the use of this strategy. When we first took the students out, we presented them with an addition problem and asked them to solve it. We did not provide any manipulatives, and all of the problems had sums greater than 10 so the students could not hold both addends up on their fingers at the same time. After the students gave an answer, we asked them to share how they solved their problem and asked them what number they started with and how many more they added on. Our goal in having students solve this initial problem was to see who was already using the counting-on strategy. Since some students demonstrated use of this strategy, we continued on to test the subskills to see if the students would also demonstrate these because our research claimed that they were linked to counting on. We also continued on with the subskills for those who did not demonstrate the strategy already in order to see if work with the subskills would lead them to discovering the strategy of counting on.
The first subskill our research linked to counting on was the ability to identify the first addend does not need to be counted out because its quantity is stated in the problem. To test this, we placed a number of counters out in a line in front of the student, then placed a card above the counters telling how many were there. We then pointed at the last counter in the line and asked what count it would receive if we had counted all of the counters. All of the students who demonstrated an ability to count-on were also able to identify that the count given to the last counter was the same as the number on the card placed above them. Although students sometimes needed to be reminded that they did not need to count the counters to give the count of the last counter, all of the students we worked with were able to demonstrate this skill.
The other subskill we worked on with the students was the ability to identify that the first count when adding a second addend is the count given to the first addend plus one more. To test this, we placed a plus sign down next to the first counters and addend and then placed more counters out along with another card above them. We then pointed to the first counter in the second addend and asked what count the students would give it if we were adding. This question often took prompting in terms of asking “what is x and one more?” but this also may have just been due to initial confusion of students not being sure what was being asked of them, because many of them wanted to think of the second group of counters as a separate entity at first and give the first counter a count of 1. We observed all students eventually demonstrating this skill, as well. For these students who were already counting on and demonstrated knowledge of the subskills, the next step may be to have students discover that counting-on from the larger number is a more efficient use of time when adding. We witnessed a small handful of students who could count on already doing this when given the addition problem alone, but using the counters and practicing the subskills only encourages counting on from the first addend. For students who demonstrated the subskills but did not yet count-on, we hope to continue work on the subskills to see if this will lead to counting on in the future.
The past few weeks in the kindergarten classroom have been crazy due to altered schedules for different classrooms. The past few times that I have worked with my group of students we have worked on taking questions from their standardized assessment that a majority of the students missed to see if presented in a different ‘form’ the students could be successful and demonstrate their understanding that was not shown in the results of the test. Many of the students were able to show me that they knew their shapes and the names for them verbally, but had a hard time identifying the written words. All of my students were able to draw the shapes I asked on a white board and then have a conversation with me about what makes the shapes different from each other. Another big question on the assessment that seemed to cause students problems was a question about flowers. The question showed eight flowers and said something along the lines of ‘Joe has eight flowers, how many would he have if he planted one more flower.’ The majority of the students counted the drawn flowers on the paper and said the answer was eight. The idea of having the answer not be what was shown in front of them caused confusion. While working with students individually, I tried to recreate this question is a more hands on method. I used manipulatives that looked like colorful game pieces and in the shape of a person. I would set up a given amount of the manipulatives in front of the student and then have them count the manipulatives. Then I posed the question of ‘how many students would be in my class if one student moved away?’ the students then continued to move one ‘person’ from the line and recount. Then next question that I asked was “how many students would I have in my class if two new students moved in?” The students wanted to place two more people in the line, but without having the correct number of manipulatives, they would count the given number then touch the place where the second new student should be. Some students even explained to me that they counted the number they had and just went one number higher in their head. By having students talk and explain their thinking to me, I was able to gain a better understanding of what they were doing and what the demonstrated.
This week we had the opportunity to work with the students in Mrs. Carmack’s class. My goal for these one-on-one sessions with the students was to determine which of the students in my group demonstrated the counting-on strategy. Out of the five students that I worked with, only one of them utilized counting-on to solve an addition problem. This particular student used the strategy without being prompted by me or by being shown the strategy on the Number Line App. Also, her use of the strategy was consistent, and she employed the strategy to solve all of the addition problems I gave her. When I asked her how she learned the strategy, she told me that it was the fastest way to add and that’s why she did it that way. From all of the problems that I gave her, she always started with the first addend and counted-on starting with that number. She may not have been introduced to the idea of working with the bigger of the two addends. In future work with her, I will be sure to ask her why she always starts with the first addend, and if she thinks it would be easier to do it another way.
This week we worked with Mrs. Carmack’s class on both days this week. During my time with the students I individually pulled students out and asked them to all complete the same task. I asked each student to draw three shapes on the white boards. After they drew the shapes and we talked about the differences between each of the three shapes, I wrote the three words that corresponded to the shapes on the white board and asked each student to point to the word as i said it. I was surprised that all of the students were able to correctly draw the shapes that i requested as well as explain the differences and characteristics of those shapes to me verbally. When it came time to students selecting the words of the shapes that they just drew, only two students from the other math groups were able to be successful in this task and correctly point to each of the words. I decided to have students recreate this problem that was asked on the standardized assessment that they needed to complete for their quarterly grades. When I gave the standardized assessment I thought that some of the students did not know their shapes due to their answers to this questions, but when I started to asked what worked they were looking for on the answer key I realized they did know the answer, they just couldn’t find the correct option. With my research on Learning Notes, I realized that the student would benefit from implementing something similar when completing standardized assessments to show a more accurate picture of what the students actually understand. I am excited to see where my research continues to take me and the kindergarteners understandings!