**
**

After completing this lesson, I learned several things about counting and number sense. My first mathematical understanding came the very first day before we left for Xmania. After practicing counting in Xmanian, it now makes sense to me how children first learn how to rote count. When that day was finished, I was confident in my rote counting; however, I lacked any concept of number sense because all I could simply do was recite numbers. For example, like students learning to count 1, 2, 3, and so on, I could count A, B, C, D, XA, and so on. However, just like students in this stage of learning had no idea how many 10 was, I could not determine how many XC equaled.

On the second day, I achieved another mathematical understanding. They second day we again practiced counting in Xmanian. On this day, I was beginning to feel quite confident in my counting skills, as were my peers, so we began to challenge one another. We would ask each other, instead of reciting which number came next, to solve simple arithmetic equations. For example, we would say, “What is XA + C?” At this point, I finally understood why some students cannot count on and must count all. It took me quite sometime to learn to count on because in order to solve the above equation, for example, my thought processes would have included, “A, B, C, D, XA, AA, (A), BA, (B), CA, (C). The answer is ‘CA’.” I attribute this lack of counting on to a lack of number sense. I still did not grasp the concept of how many a certain number represented. In addition, I was still heavily relying on manipulatives (such as my fingers) to count.

On the third day, I achieved yet another mathematical understanding. After using manipulatives to solve problems, I understood how children achieve the concept of manyness because of how we used manipulatives to accomplish this concept. It now makes sense to me to recognize a pattern and begin grouping in order to achieve the concept of how much a number represents. These manipulatives, such as unifix cubes and drawings using blocks or dots and dashes, were assisting me in truly understanding each step of my thought process. I was finally able to conceptualize that “XA” represents J J J J J items.

Just when I thought all was going well, trouble hit Xmania a few days later. Up until this point, I had been confident about my mathematical understandings in counting in Xmanian. However, despite all efforts to extinguish any thoughts of the English counting system, my prior knowledge got the best of me. For the longest time, I could not understand the concept of number columns. I insisted on trying to read my columns from right to left (similar to ones, then tens, then hundreds, etc.). I was setting up my columns like …XXA, XA, A instead of A, XA, XXA, XXXA, … My peers attempted to show me my error, but I was simply not grasping the concept. My prior knowledge was blocking my understandings. Finally, when they showed me using the blocks and I was physically able to count: A, B, C, D, XA…XXA…XXXA did I grasp this concept of number columns. It now makes sense to me to have the columns read from right to left because I used manipulatives to demonstrate competency in this area. At this point, I finally achieved the true AHA moment! Now, counting on and a small concept of number sense finally made sense to me. I was eventually able to visualize the blocks moving from column to column instead of physically moving the blocks. It now makes sense that CDXA is C (J J J) items in the A column where J = A items, D (* * * *) items in the XA column where * = XA items, and X (-) items in the XXA column where - = XXA items, and A (#) items in the XXXA column where # = XXXA items. Walking on Cloud 9, I finally felt or at least thought I felt how a student would feel when s/he finally understands the true concept of counting.

As far as questions that I still have concerning counting, because of the AHA moment, I do not have any more questions. Most of my questions were generated during the experience and were then answered if not the next day by the end of the entire lesson. In the beginning, I only had a partial understanding of counting on, manyness, and number sense. However, through the use of manipulatives, I was finally able to completely understand these concepts.

**How
I Learned**

My understandings of counting changed drastically from this lesson. While I thought that I had a firm or at least somewhat firm grasp on this concept, my knowledge was tested by this activity. During the entire lesson, I was more concerned with my own process of learning Xmanian rather than how my peers were deriving their solutions. In the beginning, in order to learn rote counting, I learned by reciting the sequence of letters in Xmanian over and over again. We practiced in small groups and in large groups. Then, as the tasks became increasingly difficult, I relied heavily on manipulatives. Before I used the unifix cubes or the blocks, my fingers aided me through my thought processes. In addition, in an attempt to wean myself from all of those manipulatives, I attempted to use paper and pencil. However, it was not until the AHA moment, that I could truly say that I did not need the aides anymore.

When we finally began solving problems in Xmanian, I turned to the manipulatives to provide a physical representation of my mental thought processes. In addition, these materials helped me accomplish the concept of number sense. They provided a visual image of what XA blocks represents (J J J J J). Without these materials, I do not think that I would have reached the AHA moment. In addition, my peers really attributed to my the AHA moment as well.

Ultimately, my complete understanding changed as a result of what I did, thought, and said. In terms of what I did, I began by recitation to learn rote counting, then turned to my fingers to assist my partial understanding, and finally relied on paper, pencil, and blocks to achieve my AHA moment or complete understanding. My understandings changed as a result of what I thought because I was able to use manipulatives to visually display my thinking processes. Only at that point, since I could visually see how I was thinking, was I able to understand my thought processes and consequently the desired concepts. It was then that I was able to see my errors and attempt to correct them. Also, because of my verbalization of my frustration, confusion, and understandings, I was able to address my understandings of the Xmanian counting system. For example, when I expressed the fact that I could not understand what was going wrong with my number columns, my peers rushed to my aide and helped me achieve my complete understanding. Had I not voiced that frustration and confusion, I might never have felt that AHA feeling.

Since the majority of my learning
consisted of in-class experiences, they ultimately contributed greatly to my
learning. It was in class that I
had access to unifix cubes and other materials.
In addition, I had my peers as a reference and teaching guide.
By learning in-class, I was able to bounce ideas off of my fellow
classmates and receive instant feedback—information that I would not have
received if this assignment had been an out of class one.
This peer interaction greatly influenced my learning because without it,
I might not have said, “AHA!”

** ****Observations
of the Teaching**

This activity was structured in such a fashion as to enable us, as future teachers, to experience how younger children go through different thought processes and feelings when learning to count. We were able to physically feel the frustration, confusion, and final completion at the AHA moment as children feel when they finally grasp several concepts surrounding numbers and counting. Our lesson began by recitation then moved to rote counting, to the use of manipulatives, to solving simple equations, and finally to grasping the concepts of counting on, manyness, and number columns. (For further explanation as to how I accomplished these tasks, see the previous sections). In addition, the structure of the lesson was done in such a way that we were able to understand, reflect, and build upon our prior knowledge until we acquired complete understanding. Without realizing it, this lesson walked us step by step through our learning processes.

In my opinion, the nature of this mathematical task was for us to experience how children learn to count from a first hand perspective. As future teachers, it is important for us to understand children’s thought processes and the only way for us to accomplish this task, is to experience it for ourselves. This lesson was meaningful to our professional careers as educators. In a nutshell, that was the purpose or the nature of the mathematical task in the Xmania lesson.

In
this activity, I experienced two types of teaching—that of my professor and
that of my fellow peers. Like the
previous lesson on shapes, the role of my professor was as facilitator of
learning and not provider of all knowledge.
For example, when we vocalized our confusion and frustration, we were not
just given the answer. Instead we
were guided to look at *how* we were thinking.
Also, in order to assist in our instruction, we were given lots of
manipulatives to use. Our teacher
did not just give us the answer or show us how to solve the equations (for
example) on the chalkboard. Instead
we provided our own solutions. We
put our thinking on the blackboard and explained to our peers our thought
processes. If someone disagreed,
s/he vocalized his/her idea and we as a class discussed it.
This discussion really contributed to my learning because it made me
question how I had derived a certain solution and ultimately strengthened my
total understanding of the learned concepts in the Xmania lesson.
On the other hand, my other “teacher” was my peers.
When I did not understand a concept, they were there to assist me.
They also were facilitators of learning. For example, I was really struggling with the concept of
number columns as I stated in previous sections.
At my table, they sat with me and walked me through my thought processes
using the unifix cubes. They would
ask me questions and ultimately helped me realize that I was reading the columns
from the wrong direction. In the
end, they helped me achieve the AHA moment and provided positive reinforcement
to confirm my actions. This type of
instruction is what I hope to model in my future classroom.

From this activity, it was not only important for us to understand several mathematical concepts surrounding counting, but it was meaningful for us to understand how children learn to count. In addition, this lesson also aided us in practicing being facilitators or learning and not providers of knowledge when our fellow classmates were struggling with specific concepts. Mathematically speaking a key understanding was the concepts of rote counting, counting on, manyness, and number columns or place value. Our learning was guided through the use of proportional materials in order for us to grasp the concept of place value. As Van De Walle suggests on page 154, “(non-proportional) materials do not illuminate an understanding of (numbers) in the same manner as proportional materials.” I agree with this statement because without the use of the unifix cubes (proportional materials), I would most likely not have been able to understand that when I have XA of A items, I substitute an XA block in the next place value and remove the XA items from the A column.

Besides mathematical concepts, this
lesson required us to teach our fellow peers by practicing being facilitators of
learning and not providers of knowledge. While
it is easier to just give someone the answer, through this lesson, we learned
that our students (in this case our peers) truly grasp a concept when they
discover the solution and are not just provided with the answer.
It is more meaningful for the student to derive the solution. For me, on the other end of that spectrum, I now understand
why it is more important for a teacher to facilitate learning rather than just
provide the solution. My peer
teachers could have just given me the answer, but because they did not, I was
able to discover my own solution and my own true understanding (AHA moment) of
counting. Since I experienced that
feeling and how that feeling was achieved, I learned that that manner is the way
in which I need to teach my future students. This alternative lesson was the key understanding developed
by the original Xmania lesson. Also,
as quoted in a previous paper, Libitowitz says it best by commenting,
“"those who understand and can do mathematics will have significantly
enhanced opportunities and options for shaping their futures (page 5)."
While it was important to ultimately learn how to count in Xmanian, it was key
for us to truly understand our thought processes and how we derive solutions.

For me, going through Xmania had an enormous influence on the way I will teach mathematics to elementary students. Not only did this lesson help me realize from a first hand perspective how students learn to count, but it also demonstrated the manner in which I should and will teach my students. Xmania walked me through my mathematical thought processes and ultimate understanding of numbers, counting, and place value, while at the same time, in a more subliminal fashion, taught me that by being a facilitator of knowledge, my students can achieve the utmost comprehension of a mathematical concept. For me, these lessons that I learned from the original Xmania mathematics lesson are incredibly important to my future as an educator and to my professional and personal development.

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