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!
This week I had the opportunity to work with the students in Mrs. Carmack’s room and gather data related to counting on. The schedule this week was adjusted a bit, and I was not able to work with all of the students in my group, but of the students I was able to work with I saw the students fitting into groups similar to the ones I had seen in Mrs. Peterson’s class.
One of the goals of finding out if students can count on, is seeing if different situations seem to encourage counting on. In Mrs. Carmack’s class this week one student, E, demonstrated the strategy of counting on in multiple situations where the first addend was 10. For all of the problems, the sum was greater than 10 to discourage the use of counting all on fingers, but when the addend was anything other than 10, E did not use counting on.
One other student I observed, T, attempted to utilize counting on after it had been modeled on the Number Line app. When she did so, however, she would place her finger on the first addend on the number line, but then would count up the number of the first addend instead of the second. I wonder if reinforcing the fact that we start at the first addend because we already know that we have that many without needing to count, will help T to understand the strategy of counting on.
All of these findings will be interesting to take a closer look at now that I have started to identify students who are counting on.
We are finally back into the swing of things after what felts like months without working with the kindergarten students. This week was my first chance to start focusing on my research topic and see what i can find out. My topic is focused on examining multiple forms of assessments and trying to conclude a sound method for assessing kindergarteners rather than standardized tests. On tuesday we did not have the chance to work with many students one on one due to Mrs. Carmack being absent. I was able to talk to the new student in the class during math when I saw something interesting on her paper. The students were working on counting cartoon bugs and then graphing how many there were of each. I saw that in the new student’s tally section she had drawn four vertical tallies, then the fifth was horizontal across the four. I asked her why she did her tallies that way, and she told me ‘because there are five and its easy to see’. I was IMPRESSED! Connecting this to my research, on the standardized assessments, there would not be an opportunity to ask why the student completed the question in such a manner, but only make judgements. On Thursday I was able to work one on one with the students from Mrs. Peterson’s class. To start to gain information linked to my research topic i asked the students to complete multiple different types of questions that overlapped content. Then I specially took a question similar to one of the Acuity questions to recreate and observe what was happening. For this question I asked students to draw me three shapes on a dry erase board, a square, triangle, and circle. One student was unable to draw a square, so I asked her to draw a rectangle for her third shape. After each student drew the three given shapes, I erased the board, wrote the words of the three shapes they drew, and then asked them to pick the word as I called them out. There was only a few (3-4) that were able to get one right. Many of the students told me they couldn’t read the words, or went in the order they were written. I was not surprised to see this happen, but I was glad to see that the students did know their shapes just not their names. This evidence was different than what would have be given from the standardized assessment. I am excited to what the next weeks bring in both my research findings and the student’s number sense understandings:)
Our work with the kindergarteners has resumed and I’m extremely excited to start diving into our research topics! Jess and I have chosen to focus on counting-on, and more specifically we would like to identify what subskills are needed in order to count-on and what strategies teachers can utilize to reinforce the addition strategy. The first step to begin our investigation was to identify which of the students in our groups were able to count-on. Once we establish this, we can begin to teach the students who were unable to count-on using the strategies that we found during our research, and we can ask the students who know how to count-on how they learned it.