HOW DO I DEVELOP AND ASSESS AN UNDERSTANDING OF MATHEMATICAL EQUALITY IN 1ST GRADERS? Carla Nordness John Muir Elementary
Background Information This fall (August 2000) I came to John Muir Elementary School in Madison Wisconsin. I was transferring form a neighboring school as I had decided to make a change from teaching special education to teaching 1st grade. I had worked hard for the last 6 months to get my certification classes completed so that I could make this change. As soon as the year started, I was asked if I wanted to be involved with The Early Algebraic Thinking Project. I was also asked if I wanted to do action research on this topic. For years, I had attended in-services, classes and read literature related to the best practices for teaching language arts. I felt fairly comfortable in my beliefs relating to the teaching of language arts. Throughout the years, I had felt less comfortable in my approach to teaching math. Math felt so disconnected, so big. It seemed as if there was not a common thread that ran true to all the different areas of math. The teachers I met at John Muir were so impressive, and the way they talked about children’s understanding of math was amazing. I was drawn to this project, but I was very apprehensive. These teachers were working with Tom Carpenter and Linda Levi, from the University of Wisconsin-Madison. It was overwhelming to me, yet an opportunity that I could not pass up.
Classroom Description My classroom is comprised of 14 1st graders, 6 girls and 8 boys. Among this group of students are 2 children who speak English as a second language. Two of the students in my class have attended Reading Recovery. One of the children in this class was retained in 1st grade.
The Question I flew out to Arizona in the beginning of October. Before I knew it, I was being asked to develop my research question for this project. At this point, I had met only once with a group of teachers to learn more about this whole project. I was panicked! I needed to determine a question for a topic that I knew little about. As my colleagues talked about the elements of algebraic thinking and what they were considering for their questions, I was overwhelmed. I really didn’t totally remember what all these terms, big ideas or concepts were. It had been a very long time since I took algebra. The one “big idea” that kept being mentioned that I “thought” I had some understanding of was, equality. This seemed like a very logical place to start. It seemed as though equality was an appropriate concept for 1st graders. I worked hard on developing a question that was measurable and that I could accomplish in this time frame. I left Phoenix feeling excited about the project that I had signed up for. The focus of my action research became: How do I facilitate and assess the understanding of mathematical equality in 1st graders?
My Journey As soon as the school year began, there was “algebra talk” among staff members. I felt as if I landed in a foreign country and had learned to speak the language on an audiotape. I enthusiastically signed up for the Algebra Seminar, Focus Group and Action Research Group. They were to begin meeting in the beginning of October. In early September, some things appeared in my mailbox with the instructions, "try these with your class before our first seminar meeting. Yikes!!!!! The first one I tried looked like this: What number can you put in the box to make this true?7+5= +6
So, I gave it to my students. My students had great difficulty with the idea of “true”. They assumed that anything I would tell them or ask them to do would be true. Many felt that they had successfully filled in the blank. Not one child made this number sentence true. When asked how they arrived at their answer all of the students reported using tally marks, fingers or counters.
The second number sentence looked like this:
48+58-
=48
My students were fairly frustrated with this one. When asked to explain what they did in order to arrive at their answer, the kids had great difficulty. Many of them stated, “I guessed.” One child stated very confidently, “ I learned to do this at my old school. We did math drills.” Her answer was 141. Another child stated, “I thought that if I started from the back…oh, I don’t know…this is too hard!” One child said, “I used my fingers.” When I asked him to show me how he used his fingers he held them up and counted 1,2,3,4,5,6,7,8,9.
The third number sentence we were to try looked like this: Is this number sentence true or fa lse and why? 17=5+12
Five of the children stated that this number sentence was true. Their rationales included: “ Because I thought it was the answer” “ Because there’s always a made up number; it’s always that number” " Because I matched the numbers to the same numbers on the board” “ Because I counted a little. I counted up to 75. I think it is 75.” “ I just think it is.”
Eight of the children stated that it was false. Their rationales included: “Because it is not true” “I just thought it was false” “Because 17=5 is not true” “Because it’s not true” “Because there is no box to write the number that is missing in “Because there is no box to write the missing number in and the plus has to be after the 17” “Because it is really higher than 12; 17+5=22” “Because 17 is higher than 12 or 5” One child was unable to respond, and made a large "?" on her page.
From this activity, I found that the children were not comfortable with the true/false format. I also found that they had great difficulty explaining their thinking or talking about and justifying their ideas about math. In early October, the plane left Madison for Phoenix. I was apprehensive about my participation in this project. I went, learned a lot, and had the opportunity of being in the company of some of the most skilled math teachers that I have ever met. I also got to attend the CGI and Beyond conference. This really helped me to gain very valuable information. I arrived back in Madison on Sunday evening and immediately started trying to brainstorm activities that I could try. We had been given a problem to try with our kids before our next seminar meeting. So on Tuesday afternoon I tried it. It looked like this: Solve and write a number sentence for this problem. The Cardinals and the Blue Jays Soccer teams have the same number of players on each team. There are 8 girls and 4 boys on the Cardinals. If there are 5 boys on the Blue Jays, how many girls are on the team?
Two children were able to correctly solve this problem. They both used concrete methods to direct model the problem. Neither of the two could write a number sentence for this problem. One of them did write 8+4=12 , and 5+7=12. When provided with the number sentence 8+4= +5, nobody was able to accurately solve this. This activity was very frustrating for the children. Many of them did not even respond to either part of the activity. The next day, I began our math lesson by writing 9=9 on the blackboard, and asked the kids to determine if this was true or not. I told them that they would need to be able to show the class how they came to their belief that it was true or false. The kids jumped right into this project. They took out math tools and spread out all over the room. They worked fast and furiously, except for the 3 children that often sobbed, complained, and paced for ten minutes every time they were presented with something that resembled math. All children were certain that they knew the correct answer. I asked the students to turn to someone tha t was sitting at their table to discuss what they found. I was amazed! One of my boys, a child that often appeared to be in his own world, rarely talked directly
to his peers, and often was turned in another direction began to passionately argue with the girl sitting next to him. She had moved to Muir from North Carolina, and had been ranked number 1 in her class, according to her parents. Her school had focused largely on rote drills and fact practice. She believed that 9=9 was false. The little boy felt strongly that it was true. He stood up from his chair and began speaking loudly and directly to this little girl, his hands were gesturing wildly. He said, “Of course it is true! 9 has to equal 9; it can’t equal 10 or 8.” She quickly and with an equal amount of passion said, “You just don’t get it! I had this last year! That has to be false, it would only be true if it said 9+9=18!” Of the 14 children in my class, only three felt that this could be true. When it came time to share, they felt uncomfortable as most of the other children disagreed. Many of the students shared making 2 groups of 9 with counters and then pushing them together. When asked why they pushed them together they said, “because you have to.” I was amazed at the discussion that the kids were having. We spent almost one hour on this activity, and the kids were all engaged on some level. They didn’t seem to have a clear understanding of this concept, but it was amazing how long they stay focused on this one question. I could tell that the y needed more work on the concept of equality from our discussion. Planning the next day’s math lesson was very easy. I needed to plan an activity that would help the students explore this concept more thoroughly. The next day we played a game. I made pairs of note cards, one with a number ( 9) and one with an expression (4+5). I passed the note cards out and asked the students to find someone that they were equal to. I also reminded them that they would need to prove that they were equal. The kids were excited as they loved games. Two girls instantly came running up to me. “ Look, we are equal,” they said. (One had 6 the other had 6+2.) I asked, “How do you know?” One of them said, “ I have a 6 and so does she!” I asked how they would prove it, and they decided to use cubes to make a tower. They then held their towers next to each other. One of the girls exclaimed, “we are not equal. She has more!”.
The kids quickly caught on to this and started matching cube towers. Within the course of this activity, kids started saying, “look, we are equal;” “we have the same amount;” “our towers are the same size.” After the game we sat down and discussed what does = mean. The kids as a group defined it as “the same amount as.” At the end of this activity the little girl who was certain that 9=9 was false went to the little boy and said, “I guess you were right!” Over the course of this year, we have played variations of this game many times. We have used pairs of expressions (7+7 and 11+3). We have used pairs of expressions with different signs (3+4 and 10-3). We have used combinations of coins. We have used pairs of expressions with multiple digits and multiple operations (3+2-1+6, 8+1+3-2). Once the students learned the format of this game, they easily found their partners and could prove that they were =. When the matches were found, the matching cards were placed in a pocket chart with = signs in the middle. The children easily placed the cards on each side of the equals sign. The y then wrote down some of the number sentences that they had made in their notebooks. It was monumental for me to note that during this activity my three students that were generally very unhappy about math activities participated without a tear or a comp laint. Again, the children were all engaged, which allowed me the opportunity to walk around and listen to what they were saying. Their discussions were fascinating. The students were discussing their findings, persuading each other, and showing each other how they arrived at their answer. The kids were helping each other to locate their partners. When one child had difficulty finding their partner as they had computed their expression incorrectly, several other children sat down and modeled how they would solve this same expression. I felt like we had started on our way at this point. I was getting a sense of what these students understood about this concept, by standing back and listening and watching. Shortly after this, the students began generating number sentences daily for the date. Every morning, during that hectic first part of the morning, the students attempt to find as many ways as they can to make a number found in the date. I wasn’t sure initially if this would help to develop an understanding of equality as it isn’t directly talking about
“is it equal?” However, I have found that this has added to their understandings of equality. As the children are working they may check in with me or with each other. The words I am using or hearing are, “is that equal to ____,” or “does that equal ______.” The growth for the students is clearly documented in their notebooks. One of my students wrote 4+4=8, and 3+5=8 on the first day. On March 29, his number sentence was 500+500+29=29+0. For me this was pretty good data that he has increased his understanding of equality. Another student wrote this on the first day: 4=4+8 3=5+8 6=2+8 She recently shared:110+11+1-10-100=12. Again, this looks like really good growth to me. It is also important to mention that for both of these students, English is their second language. A third student cried and pouted for the first 3 weeks that we started doing this task. He really struggled and needed my questioning and unifix cubes in order to make a number sentence like 4+4=8. Since then, he has really worked with patterns and concepts. For example, he has recently been exploring “the big idea” that “if you have a number and take that same number away you get 0” (my students’ wording for this conjecture). He recently shared 93-93+12 =12; 34-34+23=23; 50-50+14=14; 19=19+101101. Before that, he was experimenting with this pattern 14-1=13; 15-2=13; 16-3=13; 17-4=13, 18-5=13, 19-6=13, 20-7=13. This documents to me his understanding of equality. This activity also gives me good data as to what these students are thinking, specifically what concepts or big ideas that they are working on in their heads. It often has been the springboard for the conjectures that we have developed. It is also interesting, that an idea that one child shares soon becomes something that many try in their own work. For example: one child stated that 14 = 1+1+1+1+1+1+1+1+1+1+1+1+1+1, then said, “it is simple: fourteen ones =14. When I asked if this is a pattern that would work with other numbers, many of the kids started using cubes or fingers, and decided that it probably would. For the next couple of weeks everyone’s number sentences reflected this discussion.
My students love to play games so I use a game format whenever possible. Another game we played was “Make it Equal.” Each student had a partner. They shared one die. The first player shook the die, then they both wrote down the number that represented what this player shook. They then both wrote down a plus sign. Then the first player made a cube tower that also represented this number. This player shook again. They both wrote down the next number and then an equals sign. The first player added this number of cubes to the tower. The first player handed the cube tower and the dice to the second player and said, “make it equal!” The second player then shook one die, both players wrote down this number, and the second player made a cube tower that represented this number. Together they decided what number would make both sides equal. I was amazed - the kids again were all enga ged. They were helping each other to improve their understandings. They were modeling for each other how to figure out what to do to make both sides of the number sentence equal! They found the missing number by comparing block towers. They also easily handled that sometimes they would need to subtract in order to make the number sentence equal. They had no problem making it equal. One child stated, “this is so easy, if it is equal they both need to have the same number of cubes.” This activity again freed me up to walk around, listen in to children’s discussions, ask questions of the students, and observe what children were doing. This activity kept the children engaged for 45 minutes. When recess time came, the children needed to be encouraged to go out and play. Another game that we played was called “MindReader”. The kids again were in pairs. Each pair had a pair of dice. One child shook the two dice. Both students wrote down the number that the dice represented followed by an = sign. The first player also put that many cubes on the table. The other player then covered his/her eyes. The 1st player took a plastic cup and covered some of the cubes. The other player counted the cubes on the outside of the cup, then wrote that number down with a plus following it. He/she then tried to figure out how many were under the cup. If they were correct, then they got to be the player that covered some cubes. If not the other player continued in their role.
While the children were playing, they talked, taught, and discussed with each other. I roamed around the room, listened in to the conversations, asked questions, and observed what was happening. I was able to spend time with the students that were having difficulty. I had a chance to observe and discuss all of the students as they played. After playing this game, we discussed what had happened. The concept of equality came up over and over again. The kids discussed how it was easy to figure out how many cubes were missing or the missing numbers because they knew ho w many cubes there were altogether. All of the children felt successful and all had something to say at some point during this activity. At one point, it became clear that many of the children were thinking about what happened when you added 0 to a number. This became clear in the number sentences that they were creating each morning. Many number sentences contained +0. So, during our math time we started discussing this. I began by writing several examples of number sentences that contained +0 on the board. I then asked, “what do you notice about these number sentences.” One child stated “they all have a +0”. Another child stated that the number that was on the same side of the =s as the 0, was also on the other side. I was amazed! Slowly the conversation picked up and kids were chiming in from all corners. They eventually came to the conclusion that, “if you add 0 to a number, that number doesn’t change.” I asked if they thought it would always be true? They weren’t all absolutely sure… The next day I came armed with a box full of number sentences that contained a +0. I varied the placement of the +0 (on both sides of the equals sign, before the other number, after the other number). Each one of these number sentences had a variable or a box. I gave the students a challenge…. could they take these one at a time, glue them into their notebooks, and solve them all before it was time for recess? They took the challenge immediately. The children quickly took the strips of paper one at a time, glued them into their math notebooks and used cubes, place value blocks or fingers to figure out what the expression would be equal to. I was able to walk around, ask questions, and listen to discussions. One by one the students had “ah-ha” moments. They started taking new strips more quickly, and the math tools were no longer being used by more and more of the students.
One little boy stood up and stated in a loud, exuberant tone, “ I get it! Every time, we plus zero it is the same number, I don’t even have to use cubes!” The children went out to recess having figured out all of the strips that I had made. They felt quite proud. When they returned we discussed what we had found. Our first conjecture was written. The next day one of my students, showed up with a folded piece of paper in his pocket. He proudly took this paper out and unfolded it for me to see. This piece of paper contained a whole page of number sentences that he had done at home all of which contained the +0 big idea. This concept showed up in all of the students’ number sentences for the next few weeks. Over the next weeks and months, we continued to do a variety of activities that were designed to build an understanding of equality. I was often surprised and amazed at what I learned by listening, watching and talking with my students. The activities included games, true/false number sentences and number sentences that contained a variable, box or missing number. The children amazed me over and over again. One day, I had planned a lesson that included a game called “Switcheroo”. One of my peers was working with me this day, and I wanted to do something cool. As I was explaining the rules to the game (this game hopefully would help the children to build an understanding of a+b=b+a, a discussion amongst the students began. Before I knew it, we were involved in a discussion that lasted for 45 minutes and the game never got played. The children got very involved in discussing this concept. We started giving examples to help the students more concisely define their understanding of this concept. Before I knew it, the students had math tools out all over the floor and were working with each other talking, arguing, and explaining their thoughts relating to this idea. Children kept coming up to me and, “asking is this true?” I would respond, “I don’t know, what do you think?” Most of the students would go on to figure out what they thought. One little boy had been working near me for quite some time. He turned to me and said, “ from what I can tell, you don’t know much about anything!” I was amazed by the words and thoughts that these first graders came up with. They were asked, does 16+13+13=16? Comments included, “no the =sign is like a gate that separates the two sides. It would have to say 16+13=13+16.” One child commented,
“of course that is not true because 16+13+13 would have to be a much bigger number than 16.” 13 is equal to 13 and then you have to have two 16’s.” Other students needed to use concrete means to solve this, however, they often did not finish the direct modeling. They would make the first set, then notice that the second set would not be equal just by looking at the numbers involved. It was exciting! Again, I got to watch, listen, question, and encourage independent thinking. On another occasion, we were completing our “Math Warm-up,” which includes counting by 10’s. After practicing this rote skill. I always ask questions like: what would 10+10 +10 be? What would 5 tens be? During this usual drill, one child raised his hand and said, “I just noticed something.” I asked him to continue. He said, “10+10+10 is the same as 10 three times; that is the same as 3x10 and 3x10 equals 30.” He wrote this on the board. He then said, “all you do is write the 3 then the 0.” I asked if he thought this would work for all numbers. He wasn’t sure, so I started writing examples on the board: 5x10=, 8x10=. The kids were engaged. Answers were being shouted out from all parts of the room. Then a student suggested we try 46x10, then 27x10. The kids easily solved these. Then one little girl said “what about 10x100?” The room became silent for a second. Then someone said, “we could turn it around and try 100x10.” I asked, “do you all agree that we could do this?” Immediately a debate began. Students were interrupting to get their opinion heard. We were pretty much split down the middle on this one. One child asked if they could use math tools to figure it out. I, of course, said, “sure.” Before I knew it, all the children were working within small groups trying to figure this out. I again walked around, listened, asked questions, and observed what the students were doing. Then one student ran up to me and said, “I need some help.” We all went over to see what he had done. On his table he had 100 ten sticks. He said, “I know that 100x10 equals 1000, so I decided to figure out if 10x100 was the same.” He started counting the 10’s sticks, and they started falling all over as his hands w small. Then one student ere suggested, “why don’t you trade 10 tens sticks for a hundreds flat?” Everyone had now gathered and thought it was a great idea. Quickly the students helped this boy trade the 10’s sticks for hundreds flats. They finished this process and the little boy said, “they are
equal!” They blew my socks off! I would never have entered into this discussion with first graders. They initiated this discussion and solved their questions by themselves.
We also did many CGI story problems that dealt with this concept. Examples of these included: • Ted and Trevell have the same amount of money. They want to buy a present for their friend, Lucas. When they put their money together they have $28. How much money does each person have? • Mrs. Nordness and Mrs. Wiesner went hiking in the desert. They saw the same number of cacti. Mrs. Nordness saw 17 saguaros and 25 prickly pear. Mrs. Wiesner saw 25 saguaros and some prickly pear. How many prickly pear did Mrs. Wiesner see? • Danny and Terrell got the same amount of Valentine heart candies. Danny got 13 pink and 5 white hearts. Terrell had 10 pink 5 white and some purple candies. How many purple candies did Terrell get? • Lucas had 4 blue marbles and 3 red marbles. Danny had 3 blue marbles and 4 red marbles. Who had the most marbles?
We also worked with true and false number sentences.
What I found was that we were beginning to engage in richer individual, small group and large group discussions. The children were disappointed if they did not get a chance to share their strategy or how they solved the problem.
Conclusion/Discussion I could go on describing the things that I planned or that we did forever, as this was a very exciting year for me regarding the teaching of math. I learned a ton about teaching math to 1st graders. As I reviewed my data and began writing this paper, what I learned became very apparent. I learned that the activities that I planned were not anywhere near as important as the manner in which I began to teach during this project. I started listening to the students and their thoughts about math. Their discussions,
thoughts, methods, and work samples became my data or my assessment. From this information, I began planning what could come next. I had to be flexible and be willing to scrap what I had planned and let the children’s questions, thoughts, ideas take over the lesson if needed. I found that the topics or big ideas that the children were ready for, were often not something that I would have ever brought to 1st graders if I hadn’t been really paying attention to what they already knew, were beginning to understand, and what they were talking about. I found that this type of inquiry-based instruction was far more fun and rewarding for both my students and myself than any worksheet or “really fun activity” I could have planned. I also found that this type of inquiry-based instruction really “leveled the playing field,” so to speak. All of the children felt that their ideas were valuable. This type of instruction met the needs of a variety of types of learners. Through questioning and observing I could help students improve their understanding of a “big idea” in math, yet they had the ownership of learning it. It was a way for me to really individualize my math instruction, something that I had been trying to do for years. This was truly one of the most powerful experiences I have had in education. I started seeing this spill over into other areas. For example: the students determined that whenever a ‘w’ and an ‘r’ are next to each other in a word, the ‘w’ is silent. They discovered this by finding all of the "wr" words that they could and listening to each other read these words. They were able to list the characteristics of mammals by researching and studying different mammals. One day they had a passionate discussion about whether animals are just reptiles or just birds, or can an animal be a bird/reptile or a mammal/bird. I have found my role has changed drastically since beginning this project. I am often a facilitator. I am no t the person who imparts knowledge. I guide the children to their understandings. Another very powerful observation for me was the change in attitude that many of my students have demonstrated. At the beginning of the year, math time felt hard for all of us. There was a pass/fail attitude associated with math. As the year has progressed, I have had many of the students state, “ I love math.” They are seeing math as a challenge that they can handle versus a task that they may not do correctly. They also are listening to each other more. However, we continue to work on ways to improve this listening and sharing. They are passionate with each other about what they are thinking. They have
begun to use questioning techniques with each other, versus telling each other the answer. They have begun to question each other and me about the ideas that are shared. They are not viewing me as the person that has all the answers. I am amazed that one simple number sentence can be a math lesson for a day or more. My students are sharing math thoughts that they have had at other times of the day. Math is no longer something that we do from 1:00-2:00 every afternoon. That has been very exciting for me to see. When I developed my research question, I assumed that there would be a magic moment when I could say that my students knew or understood equality. What a misconception that was! During this year, I have found that the concept of equality, one that I had never really spent much time thinking about or teaching before, touches so many of the concepts in math. I don’t know for sure if I will ever be able to say that a student has met the objective of demonstrating an understanding of equality. There seems to be developmental levels of this understanding. As I worked with my students on developing other math concepts or understandings, I found that we were frequently revisiting the concept of equality. For example: when we started a unit on counting money, one of my students said, “ a nickel is equal to 5 cents.” This statement really helped the students to start thinking about different coins and their values. They began counting money in the same ways as they counted objects. They were able to easily address the concept of equal amounts. In my past experiences, getting children to understand that a quarter is equal in value to 2 dimes and a nickel, or 25 pennies, or 5 nickels, was difficult. Here, these kids took off with it. When solving many of the CGI problems that I wrote, hoping to get at other concepts (like place value, a+b=b+a), the children often would explain their strategies and relate it to equality. Many of my students also began building an understanding of multiplication. As they were explaining how they knew that 3x10=30, the words “equals” or “equal to” kept coming up. When doing some work with measurement, one child stated, “ look, our lizards are equal.” When I asked what he meant, he said, “they are the same length.” This may not fit under the algebraic definition of equality, but it gave me some good information about my children’s understandings of measurement and the purpose for measuring. The
words “equal to” clearly had become a part of the children’s understanding. They were relating it to or generalizing it to other situations. As for the research question, yes my students’ understanding of equality did increase due to the activities I provided them. In March, I gave the students the CGI problem that I gave in the beginning of the project. The problem was: The Cardinals and the Blue Jays Soccer teams have the same number of players on each team. There are 8 girls and 4 boys on the Cardinals. If there are 5 boys on the Blue Jays, how many girls are on the team? At the beginning of the year, two children were able to correctly solve this proble m. They both used concrete methods to direct model the problem. Neither of the two could write a number sentence for this problem. One of them did write 8+4=12 , and 5+7=12. When provided with the number sentence 8+4= +5, nobody was able to accurately solve this. This activity was very frustrating for the children. Many of them did not even respond to either part of the activity. In March,12 of the 14 students were able to correctly solve this problem. • • 5 of the students were able to write the number sentence correctly. 4 students wrote 8+4=12 5+7=12 The rest of the students ran out of time to get to this.
8 of the students were able to correctly solve the number sentence 8+4=5+ • • 4 of these students recognized that it was the same number sentence as the story problem on the front of the page. 2 of the students responded with the number 17
When asked, is 12=7+5 true or false at the beginning of the year, 5 students stated it was true, 8 stated it was false. The students responded in this way: 11- two students 12- five students 21- two students (both said that their answer was 12)
13- two students 28- one student no response- 2 students In March,11 students responded that it was true. One child’s rationale was, “because in the other problem it was 8+4=7+5. Once I figured it out, I knew that it was 12. So I knew that 7+5=12.” Another student wrote, “true, because 7+5=12. I used my fingers." The students were given the number sentence 58+48- =58
At the beginning of the year, the students responded this way: 9- two students 14- one student 18- one student 19- one student 45- one student 48- two students 49-one student 56-one student 141- two students no response-two students In March,12 of 13 students correctly solved this number sentence. When asked to explain one child wrote,” they are the same.” Another student stated that 48-48=0, and 58+0=58. One child wrote, “it is the same on both sides.” One girl wrote this to
explain her answer, “If I get 48 cookies and I get rid of 48 cookies, that leaves me with 0, and 58+0 equals 58.” Another students explained it this way, “if you took away the 48, it would leave 58.” Several students said, “I just knew it had to be 48.” This data showed me that the students had increased their understanding of equality. The children were in different places with their understanding yet they had all made progress. In summary, I feel very fortunate to have been involved in this project. I have really learned a lot about my students and about teaching. The action research process really helped me to focus on one area and really look at the results. It was a very powerful experience for me. I look forward to continuing to work in this area. There are many questions that I now have. These questions include: How do I best facilitate the
discussions between kids? Are there developmental levels in the understanding of equality? How do I best assess and communicate a child’s understanding of equality? I guess my journey has just begun.