None of my other students demonstrated their ability to count-on during my session with them. During the future weeks, I will be testing the students for the subskills that were identified in the Secada, Fusion, and Hall research “The Transition from Counting-All to Counting-On in Addition.”

]]>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. ]]>

In order to determine which of the students in my group were able to count-on, I worked with them one at a time to solve addition problems. First, I started by writing an addition sentence on a white board and asking the students to read me what they saw. If the student was correctly able to read the addition sentence then I asked them how they would solve the problem. Most of the students didn’t say anything so I suggested that they use the counters that I had provided, their fingers, or the number line that was drawn on the white board. Given those strategies, five out of the nine of my kindergarteners solved the addition sentence by using their fingers, one used the counters, one utilized counting-on, and two said they didn’t know how to solve the problem.

If students were able to use their fingers or the counters to solve the problem then I used the Number Line app with them. After watching how the app solves the problem, the students began to count-on by beginning to add with the first addend and then moving up the same number of spaces as the second addend. For example, if the problem were 3+5, the students would put a line under the number three on the number line and then using their fingers they would count up five spaces to find the answer. By the end of our time together, all seven of the students who used the app were able to utilize counting-on, but only one was able to count-on without the presence of a number line.

One student in particular was able to count-on without my prompting and without the use of the Number Line app. When I wrote the problem 5+4 on the white board, she immediately answered nine. When I asked her how she knew that so quickly she gave me this explanation: “Well I know that 5+5=10 and if I put one knuckle down then that makes nine.” I was very impressed with her reasoning skills and decided to give her a harder problem. Next I asked her, what is 10+4. She took a few seconds and said, “It’s fourteen cause I started with ten, then its eleven, twelve, thirteen, and fourteen.” As she was saying the number sequence following ten, she put a finger up for each number said after ten. Given her explanation I asked her, “Why did you start with ten and not four? She answered by saying, “Miss Jackie, you’re always supposed to start with the bigger number.” When I asked her why she said, “Cause that’s what my brother told me to do.

In the upcoming weeks, I will be interested to see if what the students did on the number line translates to counting on without one.

]]>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. ]]>

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! ]]>

Some interesting things that I noticed during my time with the students is that many of them are beginning to memorize addition facts and several of them are catching on to repetitious patterns. For instance, when I was working with a student she was given the problem five plus five and without even counting the squares she told me that the answer was ten. When I asked how she found her answer without counting the squares she told me, “That’s easy! I have five fingers on this hand and five on the other and I have ten fingers.” I was very impressed that the student was able to come to this conclusion. This same student was also able to tell me several other sums without counting the squares. Her reasoning for knowing the answer to six plus two equals eight is because she knows “one more than six is seven and one more than that is eight.” This student has demonstrated the sub skills that she needs in order to be able to count on, whereas several of her peers have not. An overall pattern that many students picked up on was addition sentences where one of the addends is zero and also addition sentences where one of the addends is one. A majority of students were able to tell me that zero means “nothing” so they didn’t have to count the squares because it was going to be the other addend in the problem. Students were also able to solve the addition problems where one was an addend because they were taught that when you add one it’s the same as finding the number right after the other addend.

I’m very excited to start jumping into more activities that revolve around counting up. In one of the readings that I found for my research there were several activities that can be done in order to determine if students are ready to be taught counting on. After the students’ spring break, I would like to try and simulate these activities with my group of students to see how many of them are ready to count on.

]]>Now we have a few weeks away from the students due to spring breaks, but I am excited to see what they come back understanding that they did not before break. Over Christmas there was some huge jumps! Lets hope spring brings the same! ]]>