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.
After what felt like forever, we finally got back in the kindergarten classroom! Over the students’ Spring Break we have been working on writing literature reviews over the research we’ve done to try to answer the questions we are investigating while working with the kids. Jackie and I have chosen to research the same topic, which is the development of counting-on as a strategy for addition. Our research is focusing on the general background numeracy skills that are necessary before counting-on can be addressed, the specific subskills that are directly related to counting-on, and different methods for teaching students to count on.
This week, I focused on identifying students who were already demonstrating the strategy of counting on when given an addition problem. This week we only pulled students out of the room in one class, because the other class had a substitute teacher. So, after one session this week, I have identified three groups of students based on the skills they were demonstrating. One group is comprised of the students who utilized counting on. Only one student demonstrated counting on in the first addition problem he was presented. This student had been using the Number Line app, and was first presented the problem 8+7. After solving the problem I asked him to show me what he did first, and he told me he started at 8, then 9 is 1, 10 is 2, etc. I then asked him to show me the next addition problem using his fingers instead of the number line. To demonstrate a problem where the first addend was 10, the student held 10 fingers up at once, stated that he had 10, then held up the second addend and counted on from 10 one finger at a time. This leads me to conclude that he understands that he has “10,” but does not need to count out 10 in order to determine this. In coming weeks, I plan to see if students who fall into this group demonstrate all the subskills that our research has stated are directly related to counting on.
The second group of students I identified were those who sporadically utilized the strategy of counting on and needed modeling of the strategy before doing so. The Number Line app has an option which underlines the first addend for the students and then shows an arrow counting-on the amount of the second addend. After this modeling, a handful of students began to count on. One item of confusion that arose was when one student had the problem 6+7 and started at 6 but then only counted up to 7 instead of counting 7 more. One other student demonstrated counting on in some situations, and knew that 2+6 was 8 because “6 and 2 more was 8.” With this group in the coming weeks, I plan on working together on the different subskills related to counting on and seeing if the students then utilize counting on without seeing it modeled first.
The final group of students that I identified were those who I felt needed continued work on numeracy skills before counting-on could be addressed. If students could not identify and set up an addition problem without prompting on which two numbers to add together and how to count both parts in order to find the whole, I did not model counting-on. Counting-all is a strategy that should be in place before counting-on can be introduced or conceptualized.
This week was our first week back from our own Spring break, and also the last week the students would be in school before their Spring break began. Coming off end of the quarter testing and having gone weeks without seeing us, the students were all excited to be taken out of the classroom and eager to tell us all about what they plan on doing over break. On Tuesday in Mrs. Peterson’s room, I used Dominos to work on addition and subtraction with my students. For students who had demonstrated that they could formulate and solve addition problems without scaffolding, I started with subtraction, and for students who still occasionally needed scaffolding while performing addition, we worked solely on addition. Across the board, the students all expressed excitement when they saw the Domino box. This interest really helped keep the students on task and many wanted to keep practicing addition and subtraction until all the Dominos in the box had been drawn. When the students drew the Domino, they either had to add or subtract the dots on both ends. All of the students working on addition chose to count each dot one-by-one in order to determine the total. Since my research focus is on the acquisition of the counting on strategy for addition, I gave the students who were working on subtraction one Domino to add. I had the students identify the number of dots on each end before adding them together, and in this particular instance one of the two students in the group demonstrated the counting on strategy.
In Mrs. Carmack’s class on Thursday, I worked on the addition and subtraction story problem app with the students. All of the students were able to add or subtract based on what the problems asked for, some requiring scaffolding and others determining if they needed to add or subtract without assistance. With students who were repeatedly solving the problems without consistent scaffolding, I had them start identifying whether the problem was an addition or a subtraction problem before they began solving. It helped to discuss key words such as “take away” or demonstrate the action that was being described in the problem using our fingers in order to determine the type of problem. In future weeks, I am excited to start work with counting on and looking at the strategy from a new perspective based on the readings I have done.
That’s all for now though!