For those students of ours who already know how to count-on, we wanted to figure out how they learned to do so. After they would count-on we would ask them, “How do you know that?” The students usually responded with something like, “it’s the fastest way” or “that’s the way my brother taught me.” It was great to hear students rationalize why they chose the strategy that they did, and it was also really exciting to see that some students chose to start with the larger addend instead of the first one. When we asked students why they did that they told us, “If you start with the bigger number you have to count less and you get the answer quicker.” I’m extremely thrilled that they have come to this conclusion, because our research indicated that students are most likely going to use the strategy that they have deemed the most effective.

In our few remaining weeks, we will continue to work with our students on addition, and hopefully have a majority of our students at least demonstrate counting-on when solving addition problems.

]]>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. ]]>

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. ]]>

To get the students in the spring mood, now that it is finally here, Mrs. Peterson asked each of us to take a group of students and do a jelly bean activity. Each group had a large bag of jelly beans and we asked the students to make predictions as to which color had the most, the least, and so on. It was interesting for me to see that some students had no idea what I was talking about, while others seemed to be old pros. After we opened the bag we made a graph so the students could see how the different colors related to one another. The students LOVED this activity, and it was a great way to help start to have students understand the concept of graphing!

My plan for the next few weeks is to continue working with students on asking standardized questions in different methods to see what students can demonstrate to me.

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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. ]]>