We have become very close with these students over the past two terms, thus it was extremely hard for us to say goodbye to these students on our final day. We are extremely grateful for this opportunity and feel we have grown immensely as teachers. The additional practice in planning, differentiating, and delivering a variety of lessons has really prepared us for student teaching and our future as elementary teachers. We are thankful for the constant support and guidance from Ms. Carmack, Mrs. Arnold and Mrs. Fields as well as our professors Dr. Egan and Dr. Hengst. With their support, we were able to get the most out of our experience and positively enhance student learning.

We are thankful that we decided to participate in the Number Sense Program and would definitely recommend any future perspective teachers to challenge themselves to do the same.

-Julie and Lisa

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This week our “level one” students interacted with ipad apps they had used several times early in the program. On Tuesday students explored the *Line Em Up* app, which addresses number recognition and ordering. Number recognition and ordering are both skills these students have been focusing on all year. When students first explored this app they struggled greatly because they could not recognize numbers 1-10, thus ordering was almost impossible. Many of the students resorted to counting up the boxes from one to help them both identify and order unknown numbers. This week, however, students responded much differently. They were able to recognize and order most numbers 1-20. I did notice that some students are still struggling to recognize numbers such as 12, 18, and 19. On Thursday students switched their focus to number recognition and one-to-one counting as they explored the *Count Sort* app. One-to-one counting was a skill these students struggled with early in the year, however, this week there were no obvious signs of struggle. All students were able to count the number of dots on the screen with no hesitation or error. Students’ oral sequence has also improved greatly. Because they have a better understanding and take their time students no longer forget or skip over the number “15”. During both lessons the students interacted differently with the apps than they had early in the program. They are now more confident in their abilities and much more comfortable using the ipad apps. After reflecting on our lessons and our students’ past performances, it has become increasingly clear just how much are students have grown and developed over the past several months.

Our “level two” students continued to explore base ten. Though we have worked on this concept for several weeks, we did not feel our students had an adequate understanding. Thus we did not feel comfortable moving on to a new concept or skill. This week, however, our students seem to have developed a deeper understanding. On Tuesday, students represented numbers using base ten blocks, where as on Thursday they explore the *Base Ten* app. Because both lessons and teaching tools are so closely aligned, students did not seem to demonstrate a deeper understanding using one tool over the other. Almost all students can represent any given number, however, they struggle counting out the number they had just showed. The students can easily show the number “52,” for example, because he or she knows by looking at the number that they need five longs and two units, but when students are asked to count it out many are unable to switch from counting by tens to ones. Thus, even though the students had represented the number “52” they counted to “70.” Each time students showed a number I challenged him or her to count it out for me. There were some students could represent all numbers and correctly count it out. I challenged these students by showing a number with the base ten blocks and asking them what number I had created. This reversal, though more challenging, proved that some of the students have mastered this concept.

Our “level three” students continued to work on problem solving strategies. This week students used their homemade Rekenreks and the *Ten Bead Math* app to further explore subtraction. On Tuesday students used their Rekenreks to solve random subtraction problems. I gave little instruction hoping the students would learn how to use this tool through their own exploration. Without suggesting it, students move all the beads to one side on the pipe cleaner and starting dragging one bead at a time to solve. Students solved by moving the beads back and forth one at a time. This activity seemed too easy for them so I challenged the students to solve subtraction problems with larger numbers. On Thursday students did a similar activity using the *Ten Bead Math* app, again this lesson proved to be to easy for these students. They were able to solve all subtraction problems with little to no hesitation. Something different did occur in the lesson, however. When students were given an equation such as “14-10” they dragged ten pieces at once and then counted out four more and then moved the set of ten back to conclude that 14-10 equals four. Students were able to recognize ten beads as a group of ten that could be moved one unit. Thus, students were able to solve these equations more quickly because they were no longer moving pieces one by one.

On both Tuesday and Thursday Lisa and I ended our lessons early to meet with students one on one. We choose two students whom have demonstrated the highest level of understanding to explore Randy and Mike’s newest app, Sum It Up. This app challenges students to find three addends that equal the targeted number. Lisa and I were presently surprised that both of our students were successful using this app. Though they struggled at first, it did not take long for them to discover the purpose of this app and ultimately how to “win.” Their biggest struggled seemed to be moving the pieces around in a way that lined up the numbers in the way they wanted. Both students expressed what numbers they needed to put in a row, but struggled physically doing it. This app gave the students the opportunity to think in a new way and develop a deeper understanding for numbers. Both students seemed to really enjoy this app and it is one Lisa and I hope to expose more of the students too.

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On both Tuesday and Thursday, our “level one” students explored familiar apps such as *Line ‘em Up *and *Count Sort. *Though the students demonstrated a deeper understanding of how the apps work (i.e. moving the pieces, instructions, etc) the students still showed similar struggles in recognizing and sequencing numbers. Both students continue to struggle recognizing numbers such as 11, 12, 16, 18 and 19. However, the students’ familiarity with the apps has allowed for them to be more confident in using the apps. As a result, the students were not only able to complete problems more quickly, but more accurately as well. For example, in previous lessons Julie and I noted that the students struggled with one-to-one counting when counting all the chips on the screen. In this week’s lesson, however, we noticed that the students were able to accurately count all this chips on the screen. While the students still have some work to do in terms of recognizing numbers, it is apparent to Julie and I that the students in this group have grown immensely since the beginning of the year.

Our “level two” students continued to explore the concept of base ten through both the base ten blocks and through an iPad app. All the students within these groups demonstrated a different level of understanding for this concept. Many of the students are still struggling to recognize that one long is equivalent to ten ones. Thus, a common misunderstanding many students seem to hold is the value of the blocks. One particular student within this group consistently counts out the number of blocks that is equal to the sum of the numbers in a two-digit number. For example, if asked the represent the number “28” this student will drag out all combination of 10 base ten pieces (2+8=10). Some students, on the other hand, have demonstrated their understanding for base ten. For example, a different student in this group was asked to identify what number the presented blocks were representing. When I put “16” (one long and six units) in front of this student, she wrote “1” for the long, counted six units, and then wrote “6” and told me “16”. While most students in this group are demonstrating a different understanding for the concept of base ten, it is clear that over the past weeks these students have developed a deeper understanding for base ten.

The students within “level three” continued to practice problem-solving strategies. In previous lessons, we had the students use our homemade Rekenrek and *Ten Bead Math *app to explore addition. This week, we had the students use these tools to practice subtraction. Our focus student within this group demonstrated a deep understanding not only for subtraction, but also for using these tools. The app required the students to drag beads into a shaded area based on the minuend, and then take out a given number based on the subtrahend. The focus student in this group picked up on “groups” of beads, demonstrating his ability to subsitize objects. For example, this student was able to recognize that after dragging all of the red beads into the shaded area, he had ten beads. He was then able to recognize that he needed “x” number more to represent the minuend. In another example, this student dragged two rows of 6 beads (5 of one color and one more) as was able to immediately say that it was 12 beads. Many other students in this group showed their understanding for subtraction, as I had to challenge the students with larger numbers.

This week our level one students focused on rote counting and counting on from a given point. On Tuesday we pulled our students into the library to ensure we had enough room for our movement activity and that we were not a disruption to the surrounding classrooms. We explained to the students that we would be practicing our counting in a new, more exciting manner. We planned different movements for each set of five numbers. For example, as students counted numbers one through five they clapped their hands, as they counted six through ten they stomped their feet and so on. The students kept counting until they skipped a number or miss counted. We would then explain their error and then start over again from one. We hoped this activity would better engage the students and help them focus in on their oral sequence. Throughout the activity Lisa and I were able to collect data and recognize trends in their counting. Our focus student who almost always skips “15” in his oral sequence did so again in the beginning of the lesson. After restarting several times, he slowed down his counting and was sure to include “15.” If he had skipped it while counting, he recognized his mistake and would say “wait, no, let me start over,” and would include it in his next attempt. Thus, we do feel this activity helped him develop rote-counting skills. We also noticed that all of our students struggle stating the number that starts the next set of numbers. For example, the students can successfully count to “39” rather than then stating “40” the students continue with “80” and count up from there. We tried to give them clues and orally hint at what the following number was throughout the lesson. By the end of the lesson they were able to move from 39 to 40, but still struggled with the following transition. On Thursday our students were given a “Caterpillar Counting” worksheet. This worksheet had several caterpillar bodies with one random number on each. Students were asked to first identify the number given and count forward from that point filling in the numbers as he or she went. I worked one on one with our focus student. I was very impressed with his performance and level of focus throughout the activity. He got almost all sets correct with little struggle and never skipped “15” in his oral sequence! He took his time during this activity, which I believe eliminated his number of errors. Though this student had made great progress and is beginning to pick up on the patterns, we know he still needs additional practice. We hope that, like many of his peers, he is able to count to or almost to 100 by end of the year.

Our level two students focused on base ten this week. I worked with our focus student both on Tuesday and Thursday. Last week this student explored base ten only through the use of manipulatives. This week, however, he explored this concept using the digital base ten blocks on the *Math Tools* app. On Tuesday he was able to represent numbers 1-39, which was an improvement compared to the week prior. When I pushed him to represent larger numbers he was easily able to represent them. For example, I would write the number “84” and he would automatically reach for 8 longs and 4 units. However, when I asked him to count as he pointed them out he would count “ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, (pulling the longs) ninety… (pulling the ones). This student struggled shifting from counting my tens to counting by ones when the objects switched. On Tuesday this student again worked with the tools on this app. He was able to represent and count what he had represented much better throughout this lesson. Though he still counted by tens when he was counting the units a few times, I do believe he developed a stronger understanding as a result of this lesson. Lisa and I allowed this student to have more exposure to the materials based on idea we had learned from our research. After reflecting on the past two weeks, I firmly believe that this student was more successful because he was given the opportunity to explore the materials and develop his own understanding before we had asked him to demonstrate it. The advantages of this strategy were made even clearer to me as I worked with the other level two students who were not given this opportunity. These students struggled greatly during both lessons. Some were unable to tell me that one long equaled 10 and that one unit equaled one. Others had already developed this understanding, but were unable to apply it. Some students only used units to represent numbers in the teens, while others choose blocks at random to count to the given number. Though our focus student has made great progress, Lisa and I know our other level two students need additional exposure and practice with base ten.

This week our level three students switched their focus from addition to subtraction. On Tuesday students were given connecting cubes and Skip Um cards. They were presented with random subtraction problems and were to use their connecting cubes to solve. The students solved by pulling away cubes from the original pile they had counted out. I did not need to explain this process to the students I worked with. Rather, I gave them the cubes and the problems and waited to see how they would respond and interact with the manipulative. They did just as I had hoped and just as I expected. Both students thought this activity was extremely easy so I challenged them with larger numbers. One student lost focus and started adding several times. Once I turned his focus to the subtraction sign he was back on track and just as successful. On Thursday these students did a very similar activity. Using the ipad app *Add Sub* (on the subtraction setting) students solved subtraction problems by dragging down pieces and counting the squares that remained above. The students were again successful throughout this activity and again expressed that it was too easy. Lisa and I plan on challenging these student more next week. We also plan on again working on subtraction, but with different manipulatives and ipad apps. We are interested to see if students are just as successful using different teaching tools.

The remaining time left in both lessons was spent with the students we see every other week. These students worked on problem solving strategies as they each completed several word problems in the Daily Word Problem Book. Lisa and I have also made excellent progress towards our research paper and are hoping to complete our final draft this week!

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We wanted to help the students in group one practice their rote counting skills, as this is something all students in the group continue to struggle with. On Tuesday, we took the students to the library (a more secluded area) and taught them a counting song. This song associated various large motor skills for a set of numbers. For example, when counting 1, 2, 3, 4, 5 the students clapped their hands and when counting 6, 7, 8, 9, 10 the students stomped their feet. As noted in previous lessons, our focus student in this group continually skips 15 when counting. In this lesson, however, we noticed that when this student is corrected, he is aware of the fact that he skipped over 15. All of the students in this group were able to count to 29! However, the students have no understanding for the correct pattern for counting by 10’s. Many of the students would count 28, 29, 50, 51 or 28, 29, 80, 81, and even 28, 29, 99, 100. On Thursday, we asked the students to demonstrate their ability to count using a “caterpillar counter”. Students were to fill out of a variety of caterpillars in which they were asked to count or count on from a higher number. During this, the students tended to recount from 1 rather than counting on. The students also struggled to write many of the numbers, and thus became distracting when counting. So we simply had the students state the numbers and wrote them for them. We hope that in future lessons, we can help students work towards an understanding of all numbers 1-100.

Last week we had the students in group two explore the concept of Base Ten through using the tangible base ten blocks. This week, however, we had the students continue to practice the same skill on the base ten setting of the iPad app *Math Tools. *Though the students demonstrated different understandings from one another, both students I work with appeared to demonstrate a stronger grasp for the concept of Base Ten as the lessons went on. I did not see any differences in student understanding when they used the manipulatives compared to when they used the iPad app. An important realization that Julie and I did come to, however, had to do with the time students were allotted to explore the tool. In past lessons, Julie and I tended to focus on a skill only for a week and then move on to another similar, yet different concept. Yet when exploring base ten, we allotted two weeks for the students to explore this concept. We noticed a drastic difference in student understanding by the end of the lesson, and we now feel that students have developed a strong understanding for base ten. Thus, exposure and exploration time is something that Julie and I will keep in mind as we finish the remainder of our Number Sense experience and even into our own classrooms someday.

Our students in group three did the same activity they did last week. Rather than practicing addition, however, we challenged the students with subtraction problems. On Tuesday the students were given counting chips to solve a subtraction problem. While working with our focus student in this group, I noticed that he was confused at first as to how to use the chips to solve the problem. For example, if the problem was 6-2, the student grabbed a pile of 6 chips and a pile of 2 chips. The student was unsure however of what to do next, and how to take away the chips. After modeling a problem, however, this student was able to accurately solve all the problems given to him through the use of the chips. On Thursday, we had the student practice the same skills, this time using the subtraction setting on the Add Sub iPad app. Though I did not notice any difference in the student’s understanding or performance when using the app versus manipulative, the student appeared to be more comfortable and familiar with the app, as he had previously used a very similar tool.

Although Julie and I are sad that the end of our Number Sense experience is nearing, we are amazed at how far our students have come and how much they have grown and developed over the past 20 weeks. As we finish up our research paper, we did an analysis of assessment data since the beginning of the year until now. It is amazing and reassuring to be able to tangibly see the progress that our students have made, and we are confident that these students will only continue to grow and enhance their early numeracy skills.

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Our first group of students continued to work on skills such as number recognition and number sequencing. On Tuesday of this week, we provided the students with a set of flashcards. The flashcards, containing numbers 11-20, were spread out in front of the students. They were then asked to locate a specific number that we stated. Once they were able to identify the symbolic number that was said, the students were then asked to sequence the number in a number line. Our focus student in this group had a difficult time recognizing many of the teen numbers (13, 15, 18, 19). This student had demonstrated that he has not yet grasped the concept for many of the numbers. For example, he picked up the number “13” and read it as “twenty-three”. As noted in prior observations, this student still continues to struggle with orally counting as well. When asked to count, this student consistently skips over the number 15. I asked this student to read the numbers on the card as I pointed to them in order, but he again skipped over the number 15 when counting; this student said 16 even when pointing to the flashcard containing the symbolic number 15. On Thursday, we had the students explore and practice number recognition and number sequencing through the Number ID app. The students showed similar struggles when using this app. Though the app did not ask students to sequence the numbers in order, many of the students still had a difficult time identifying the numbers. Julie and I encouraged students to utilize the number line at the top of the screen to assist them in identifying numbers. Our focus student, however, was unable to utilize this tool because he could not correctly count to twenty. Therefore, this student in particular showed signs of frustration and began randomly guessing numbers when using the app.

Our “second level” of students continues to work on skills such as representing numbers. We attempted to use a manipulative that we had not used yet with the students – base ten blocks. Julie and I have recently come to realize through our research paper and experience in the Number Sense Program that students need an adequate amount of time to explore a manipulative or iPad App before being expected to accurately utilize the tool to demonstrate their understanding. Therefore, we allowed used the base ten blocks with this group of students on both Tuesday and Thursday of this week. During both days of this lesson, I worked with a student who we have been working with regularly and also a new student who we are currently trying to place in a group based on his current skills and abilities. These two students demonstrated an extremely different understanding for base ten blocks. To begin, we drew a number on the white board and ask the students to represent the number using the base ten blocks. (I began with an easy number such as 5, so the students simply had to drag 5 units out). I then wrote the number ten and asked both students to represent this number with the blocks. Both students dragged out ten of the small unit blocks. I then asked the students to place their ten cubes in a line and set a long next to the set of the ten blocks. I showed students how when using base ten blocks, we can use the longs to represent a group of ten. The new student grasped the concept right away for how to represent numbers in the teens, and even beyond. At first, this student wasn’t sure of how to make numbers beyond twenty. However, I prompted him to think about how he could use both units and longs to represent these numbers; soon enough this student grasped the concept and was able to represent numbers all the way up to 100! The other student in the group, however, struggled with base ten blocks on both Tuesday and Thursday. She struggled to grasp the concept that one long is equivalent to ten units. This student consistently dragged out single units, rather than using one long to represent numbers that was greater than ten. This student struggled to recognize many of the numbers being written on the whiteboard. Thus, this student may have had a difficult time grasping the concept for how to represent a number if she is still having difficulties recognizing what the number looks like. Next week, Julie and I plan to allow the students to explore the same concept. This time, however, the students will be using base ten blocks on an iPad App. I am interested to see if there will be any changes or similarities in the students’ ability to represent numbers using base ten next week when the students are asked to use technology.

The students in our “level three” group continued to work on problem solving skills, specifically addition. On Tuesday, the students were presented with an addition equation using their flashcards and a pile of connecting cubes. The students were asked to use the cubes to assist them in solving the addition equation. Though the cubes were slightly distracting for the students at first as they wanted to play, both of the students were successful in solving the equations presented to them. On Thursday, the students were asked to complete a similar task. This time, however, the students were asked to use the Add Sub app on the ipad. The students were again presented with an addition problem and asked to utilize the color changing chips on the screen to help them solve the problems. Again, both students were able to successfully solve all problems presented to them. I did notice an interesting difference in problem solving strategies between the two students I was working with, however. Our focus student of the group demonstrated that he has not yet developed the ability to subtilize a group of objects, as he counted each chip in both addends of the problem. The other student I was working with, however, demonstrated his ability to subtilize by counting on from the first addend, rather than counting the first “x” number of chips. For example, if the problem was “6+4” the first student would solve the problem by counting “1 2 3 4 5 6 7 8 9 10,” where as the second student would solve the problem by saying “6, 7, 8, 9, 10”. Regardless of the problem solving strategy the students used, however, Julie and I both feel that these students are ready to move onto more advanced concepts and skills, such as subtraction.

As Julie and I continue to collect evidence of our focus students’ understanding and growth, we are excited to finish the last portion of our research paper. We have re-administered the ESGI Assessment to our focus students and now have three sets of data – a set of data from the fall, one from the middle of the year, and a set of data from the end of the year. Julie and I are both excited to analyze and compare the data and determine exactly how much our students have grown over the year!

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This week our “level one” students continued to work on recognizing and ordering numbers. On Tuesday the students used their flashcards to practice recognizing and sequencing numbers 11-20. The numbers were presented in a random order and together we worded to put them in numerical order. I noticed an obvious struggle for the students during this lesson. Two of the students are still unable to recognize number such as 15,18,19, 20. We continued to see similar trends in their behavior such as identifying 15 as 55, confusing 18 for 19 and vice versa, and naming 20 “twenty-zero.” The other student in this group could successful recognize these numbers; however, she was unable to sequence them. I modified the lesson and had her compare two numbers, rather than fill in missing spaces and she was still unsuccessful. I gave her the numbers 12 and 18 and she told me that 12 was more. On Thursday the students explored this concept through the use of the app *Number ID*. The students were again asked to identify a number 11-20. Because this app does not ask students to sequence we asked them questions such as, “is this number bigger or smaller than 15.” Students struggled to recognize the same numbers as they had in the previous lesson. To help students identify numbers they did not know, I encouraged them to use the app’s number line. I told them to count up starting at eleven and count until they got to the number they were working to identify. This strategy was not useful for either student because they were unable to accurately count to twenty. Every time the students counted they skipped 15 in their oral sequence. Thus, the students were always one number off. After reflecting on this lesson, Lisa and I feel we need to spend more time recognizing, comparing, and counting numbers beyond 10 with this group.

Our “level two” focus group experimented with base ten blocks this week. Lisa and I usually set up our lessons in a way that would allow the students to explore a concept or skill through the use of manipulatives in one lesson and the use of ipad apps in the second. Through writing our research paper, however, we have come to understand that students need multiple exposures to the same teaching tools before they are expected to demonstrate an understanding with them. I worked one-on-one with our focus student in both lessons. Because the students have already explored base ten blocks in their classrooms, he was able to confidently tell me that one unit equaled one and that one long equaled ten. I asked him to use the base ten blocks to represent the numbers I wrote on the white board. He could easily represent numbers 1-19 without much thought or hesitation at all. However, he grew very confused when I asked him to represent numbers greater than 19. When asked to show the number 24, for example, he would first grab two longs, pause, and then grab four more longs. He did not understand that he needed to use the units to represent the ones place as he had done with the teen numbers. On Thursday we continued practicing representing numbers 20-29 with base ten blocks. He developed a greater understanding and was successful early in the lesson. When I challenged him to represent numbers in the thirties, however, he again became very confused. Thus, I do not believe he has developed an adequate understanding for base ten at this point. Lisa and I are interested to see how this student will respond next week when he is again asked to explore base ten, but through the use of the ipad app *Math Tools*.

This week our “level three” students continued to work on addition strategies. On Tuesday they were given addition equations and connecting cubes. They were asked to first represent the two qualities and then count the total number of cubes to solve. On Thursday the students did a similar activity using the ipad app Add Sub. The students were again given an addition equation and squares, which were counted to solve. The students I worked with were very successful during both lesson. They used the cubes/squares for each problem, rather than their fingers when the sum was less than ten like they had done in previous lessons. They all said this activity was “easy peasy” so I challenged them with greater sums. The students were able to some these problems just as easily. Thus Lisa and I feel confident we can begin moving on to additional concepts, such as subtraction.

After working with our focus students Lisa and I met with all students we see every other week. We used this time to catch them up in the Daily Word Problem workbooks. Lisa and I both worked with students one-on-one until all missed problems were solved. This week Lisa and I also plan on finishing the last portion of our research paper. Much of this portion has to do with our focus students and the growth they have made throughout the Number Sense program. Lisa and I are very excited to analyze their scores and development over the course of the year.

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This week our “level one” group continued to focus on number recognition and counting. On Tuesday, students were given a worksheet with a number followed by blank boxes. Students were asked to identify the number in the box and count forward as they filled the boxes in with its corresponding symbolic numbers. On Thursday, students were given a similar worksheet and were again asked to identify the starting number and fill in the remaining empty boxes. In this lesson, however, students were asked to count backward, rather than forward. We found that the students were able to easily count forward orally, but struggled to write the symbolic number in the boxes. They were unsure of how they were to be written and often asked us before first attempting on their own. I also noticed that several of the students would orally stated the number “6” for example, and instead of proceeding “6,7,8,9,10,11” the students would start from one and whisper to him or herself “1,2,3,4,5,6!” (speaking louder) “7,8,9,10,11.” Thursday proved to be a bit more difficult for the students. Though they were able to successfully recognize almost all of numbers presented they struggled to state even the one number that comes before it. Even when we counted along as we point to the hundreds chart, the students did not seem to understand they were counting backwards. One student in particular was able to complete the worksheet on her own after she was given many examples. However, I noticed that this student was counting forward from one each time to figure out the number that came before the number she identified last. For example when given the number 8 the student counted to 7 starting from one, then counting to 6, again starting from one and so on. Though this student developed this strategy on her own and developed somewhat of an understanding, we hope to challenge her in future lessons to count backwards without first counting forward. We plan on continuing to focus on both number recognition and counting in the following lessons.

“Level two” continued to work on representing numbers in multiple ways. On Tuesday, students explored a new app called “Make Another B.” This is actually an app Lisa and I help Randy and Mike created based off an idea we developed after an assessment we administered to our students. Students were told to represent a number using both sets of colors. For example, when students were given the number “6” students dragged 3 green shapes and 3 pink shapes and stated “3 green and 3 pink is 6”. They were then asked to represent the same number in a different way. (e.g. 4 green and 2 pink) The biggest struggle some of the students faced was coming up with a new way to represent the same number. Many students would show 6 as 4 green and 2 pink, but when asked to show another way to represent the number many created a chain of 4 pink and 2 green. Students were reversing the colors, rather than using new numbers. Though, they were also reminded not to make a pattern, many students resorted to this strategy as well. I also noticed that a few of the students struggled with this activity because they struggled with one to one counting. They could not accurately represent the number called with the movable pieces on the app because they kept miscounting them. With much guidance, students were able to eventually come up with two ways to represent the same number. On Thursday, students were involved in a similar activity but used connecting cubes to represent the numbers. They had already explored this teaching tool before, thus Lisa and I felt comfortable explaining how they were to be used and the purpose of the lesson before they began constructing their own understanding. During this lesson, none of the students attempted to represent the numbers by creating a pattern, instead they applied what they had learned from the lesson prior. Students were able to represent numbers in multiple ways more independently then they had on Tuesday. Though we did see similar trends on Thursday (the reversal of colors and miscounting) we do feel they demonstrated a greater understanding for the content.

This week “level three” continued to work on solving problem strategies. For the past several weeks these students have been solving equations using a variety of tools. They have used dominos, the *Domino Add* app, a home made Rekenrek, *Ten Bead Math* app, and several others. This week, however, we challenged these students to solve equations using a number line. On Tuesday, students explored the *Number Line Math* app. The students were presented with an equation and drew “humps” on the screen to solve. They students did great with this activity and demonstrated an adequate understanding for the content. I knew they had developed an understanding because they were able to solve these problems independently. On Thursday, the students again solved equations on a number line, however, in this lesson students were provided with a laminated number line and were asked to draw in the “humps” to show what they were adding and how they got their answer. The students turned to the number line for assistance in solving the problems. There was one student in particular who used his fingers to add, rather than the tools we had provided him with. This student had done this is previous lessons, however, this time something different happened. When the sum was greater than ten and the student had run out of figures to count he would usually tap them on his chin trying to keep track of them or ask to use our fingers; however, in this lesson rather than asking for help he picked up his marker and used the his number line to solve! I was very impressed that this student knew what to do when his personal strategy for solving was not the best. The students responded well to both teaching tools, however, Lisa and I are interested to see how students would responded if given these tools a second time.

Overall I thought this week was a great success. I was happy to again work with the students and see such growth in them over the past few weeks. Lisa and I really do feel we are helping these students develop a greater understanding and are excited to see where the remaining weeks will take us!

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“Level one” or our first group continues to work on number recognition. However, Julie and I have decided that instead of doing repetitive number recognition activities such as flash cards, we want to challenge these students more and more each week. Through moving onto other concepts and skills, our hope is that they will continue practicing number recognition while still developing higher level thinking skills. Thus, on Tuesday, we provided each student in this group with a worksheet that would help them practice their counting on skills. The worksheet presented the students with a number and asked them to write in the remaining numerals in the number sequence. For example, if the students were presented with the symbolic number 5, the students would have to write in 6, 7, 8, 9, 10, etc. until all the boxes in the row were filled. The particular student that I worked with during this activity demonstrated excellent counting on abilities. When reflecting on activities done previously in the year, I can remember students having to begin counting at number one in order to “count on”. During this activity, however, the student was able to identify what number comes after (and even before!) the presented number immediately. On Thursday, we had the students complete a similar activity. This time, however, the students were asked to “count down” or “count backwards” from a given number. We provided the example of counting down like “3, 2, 1, 0, blast off!” Once the students had grasped the concept of how to count down, they did surprisingly well! I worked with our focus student in this group, and he was able to do the first few problems independently. Once presented with numbers in the teens, however, this student became very frustrated and confused. His frustration appeared to stem from the fact that he was unable to recognize the number “15” which he was asked to count down from. One “fault” of this activity, however, was that students had to write all the numerals. To save instructional time and minimize student frustration, I began having the students count as I recorded their answers in the boxes for them.

The students in our second group continue to work on representing numbers and number value. On Tuesday of this week, we had the students explore the “Make Another B” iPad App (Julie and I actually helped Randy and Mike develop this app in previous weeks by suggesting the content and layout of the app!) This app focuses on representing numbers in different ways. The students were given a number and asked to represent the number in two ways using colored chips. For example, if given the number “6”, students could represent it as “3+3”, “4+2”, “5+1” or even “6+0”. This is a concept that our students had demonstrated no understanding for when previously tested, hence our idea for the app! While many of the students tended to represent the number by creating two patterns (for example four orange and two green and then four green and two orange), I felt that this app decreased their tendency to do so. Because the area for them to drag the chips was an open space, rather than a formatted space, the chips were not necessarily placed in a row. This app seemed to greatly benefit the students understanding and I think it will prove to again be useful in upcoming lessons that focus on the same skill. On Thursday of this week, we again had the students represent a number in two ways. This time, however, we had the students complete the activity using connecting cubes. I immediately noticed the difference in time between using the app and using the connecting cubes; it took students a significant longer time to connect the cubes than drag the chips. Because the students took longer to handle and connect the blocks than it did to drag chips on the screen, it greatly limited the number of problems the students were able to do in the given time when using the manipulatives. Other than that, however, I did not observe any differences in student understanding or performance when they used the iPad App versus when they used the manipulatives.

The students in the third group continued to work on and develop their problem solving skills and abilities. On Tuesday, the students practiced their addition skills with the use of “Number Line Math” iPad app. The students were given an “x+y=?” addition problem and asked to solve the equation using the number line by first locating “x” and then making “y” humps on the line. On Thursday, the students practiced the same skill through the same activity. This time, however, the students were given a laminated number line and a dry erase marker. The students were asked to again solve “x+y=?” addition problems using the tangible materials. In both lessons, a handful of students demonstrated some confusion about how to use the number line to solve an equation. Because the students only had limited time to explore using both the app and the manipulative, I felt that the students were not adequately exposed to these materials. Though the students have seen number lines previously, they needed assistance and guidance in understanding how to utilize these materials to solve an addition problem. Up until this point, we have focused on solving addition problems through the use of manipulatives such as counting chips, or the shapes on the screen while using an app. When presented with a different problem solving strategy, however, the students appeared confused. I am interested to see if the students would react differently, or appear to be more comfortable or confident, if we again utilized the number line to solve addition problems at later time.

Julie and I have seen tremendous growth in all of our students over the past term and a half. We too, have made tremendous strides as teachers. We are excited and eager to see where the remaining weeks take our students and us as we conclude our research and work with our Kindergarten students in the Number Sense Program.

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This week we continued to focus on number recognition, one-to-one counting, and oral counting with our first group. On Tuesday, the students in this group used counting chips and flashcards to demonstrate their current understanding of number sense. The students were given a pile of chips and asked to count them. The students were then presented with three flashcards, and asked to select the flashcard that showed the corresponding number. While working with these students, it became clear that they continued to struggle with identifying numbers such as 6, 9, 11, 12, and 13. Both of the students did, however, continue to demonstrate a basic understanding of the teen patterns, as they have in previous lessons. We noticed that during this lesson a student consistently counted “13, 14, 16, 17”, forgetting 15 each time he orally counted. To try and help this student recognize that he was forgetting 15, I pointed to a number on the 100’s chart on the wall and had him orally state the number. However, when I pointed to the number 15, he again went from 14 to 16. Julie and I have made note of his struggle to remember the 15 when counting, and it is something we plan to pay attention to in future lessons. On Thursday, the students practiced the same skill in a similar manner, but this time the students used the Count Sort App with the counting setting. The students were shown multiple dots on the screen and asked to count how many in all were on the screen. The students then were asked to select the corresponding number as listed on the screen. Again in this lesson, the students demonstrated similar struggles in identifying numbers 6, 9, 11, 12, and 13, as well as higher teen numbers. Thus, we did not notice a significant difference in the students’ performance when using the manipulative versus using the iPad Apps.

The next group of students we worked with focused on place value, specifically activities dealing with ten-frames. On Tuesday, the students completed a Ten Frame Booklet – an activity that we had done previously in the year with a group who has now moved onto higher-level skills. The students were given a book of ten frames, each page containing blank ten frames and a number in the teens. After identifying the number, the students were asked to make marks/dots in the ten frame boxes to represent the number shown on the page. Some of the students struggled to independently complete this activity, while others were able to correctly complete the booklet with little to no guidance. Some of the students who struggled were unable to identify the number, and others were unable to make the correct number of marks in the ten frames that corresponded with the symbolic number on that page. One student demonstrated a particular example of misunderstanding of place value. The sentence on the pages read “16 is ten ones and ___ more”. The students were to identify 16, make 16 marks, and write 6 in the blank. When asked “how many more”, however, this student responded with 4, meaning that four more boxes needed to be filled in on the second frame. On Thursday, the students practiced the same skill again by using the iPad App Count Sort with the ten-frame setting. The students were shown ten-frames that were partially filled in. The students were asked to determine how many dots were in the ten-frames and select the corresponding number on the sides. I felt that this App proved to be easier for some students to use; the students seemed to have an easier time in “reading” the ten-frame, rather than constructing the ten-frame on their own. Furthermore, by providing the students with the pre-filled ten-frame, it allowed them to work through more problems. However, many of the options listed for numbers beyond ten were number reversals. This proved to be very easy for the students, as they were able to easily distinguish between 51 and 15 before even counting the dots. Julie and I learned shortly after, however, that there is the option to present the students with more than two number choices, which would eliminate the reversal concern. Thus, in the future, Julie and I will be sure to explore all settings and options before using the iPad Apps with the students.

The final group that we worked with this week focused on addition skills. On Tuesday, the students worked with the Ten Bead Math iPad App. The students used this app as an aide to solve the given addition problems. On the screen, the students were shown a “row” containing five white beads and five red beads. The students could drag white beads to the center to represent one of the addends, and the red beads could represent the other. On Thursday, the students were again given a sheet of addition problems. This time, however, the students were given beads on a pipe cleaner to use as an aide. On both Tuesday and Thursday, the students did not seem to enjoy using the beads to help with the addition problems. Rather, many of the students preferred to use their fingers to do simple addition problems. Furthermore, many of the students had difficulties moving the beads both on the iPad screen and on the pipe cleaner, and thus became very frustrated. Overall, we did not note a big difference in the students’ abilities to solve the addition problems when using the pipe cleaners versus using the iPad apps. Rather, the students preferred to use a familiar method of counting, such as using their fingers.

Julie and I are extremely grateful for the experience of working with the Kindergarten students so far! We have seen tremendous progress in our students thus far, and we are excited to see how much more they will grow over these next ten weeks! We plan to continue doing research over our spring break, and brainstorm useful and creative lessons that we will be able to use when we return!

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