Becoming a Teacher of Mathematics to Elementary Students

Abstract: 

This is an autoethnography of a pre-service teacher during the first semester of Professional Development School, documenting the journey as a learner and teacher of mathematics for this pre-service teacher. Due to the qualitative nature of the research, there is no hypothesis for this study. Narrative data from sixteen weeks of communication about teaching and learning mathematics in the community served as data, which were analyzed for points of change from fear of mathematics to readiness to teach it to young children. Pre and post depictions of perception of math learning and teaching can be represented through art, using a running narrative as an indicator of change. The findings show that due to research and experience on the Response to Intervention Model, assignments from a University of North Texas math methods course, and experience with tutoring elementary students in the pre-service field, reconciliation between math and the pre-service teacher occurs.

Table of Contents: 

    Introduction

    As a future teacher, I would like to say that my experience with every content subject has been a beautiful and uplifting one, but, truth be told, that was just not the case. From elementary school to high school and from high school to college, math and I have been engaged in an ongoing battle with one another. In fact, math was winning until I completed my first semester in Professional Development School at the University of North Texas in fall 2011. A new and glorious change occurred within me that semester that changed my entire outlook on math and on my self perception as a teacher of mathematics to young children.

    What follows is an autoethnographic study of my transformation. “Autoethnography is a qualitative research method that uses data about self and its context to gain an understanding of the connectivity between self and others within the same context” (Ngunjiri, 2010, par. 2). With this in mind, I used an analytical approach to examine where the change occurred in the sixteen-week semester and to identify the evidence that supported the change. According to Sarah Wall, “initial engagement with a research topic occurs with the discovery of an intense interest, a passionate concern that is not only personally meaningful but has broader social implications” (2006, p.4). Professional Development School allowed me to immerse myself in math. I was able to focus and concentrate completely on math and learning how to effectively teach it. Every week, I wrote a narrative response to a prompt from my professor regarding the experiences that I had as pre-service teacher in the classroom, field, and/or mathematics class. I used these narratives, artwork, and tables of documentation as my data for this autoethnography. I have reviewed the data and found the source of my transformation. This is my lived experience, a journey into a familiar land, where I gained a new perspective.

    Professional Development School

    I was terrified going into my senior year of college and my very first semester of Professional Development School. So many rumors were circulating about the amount of work I was in for, including a tremendous amount of math. I was praying and praying it was not true. On the very first Tuesday of my Math Methods course my prayers were squashed and thrown into the trash when my professor announced, “You will be tutoring elementary students in math during your second rotation of PDS 1” and “You will create a math game that actual students will play.” Instantly, I was in a state of panic. “How in the world am I, someone who is absolutely terrible at math, going to pull this off?” By the end of the semester, I would manage both of these expectations and come out a changed person.

    Artwork

    The very first assignment of the semester in my math methods course required me to find a piece of artwork that resembled me as a math learner. I chose a piece calledMurnau-Garden II by Wassily Kandinsky (1910) (Figure 1).This piece of art has an abstract, floral feeling to it where all the colors collide and blend together. Although a beautiful and colorful piece, it left me confused and made me feel somewhat lost. I related this to my math learning because the way the colors ran together reminded me of my inability to remember terms or formulas. The painting was blurry in its abstractness and related to my forgetfulness when it came to math. It was a subject I hated and could never conquer, so remembering or even trying to remember anything about it was never important to me before. With all these aspects of the painting, I strongly believed that it represented me as a math learner. I could handle the earliest years of math, including the basics of addition, subtraction, multiplication, and division but I felt lost with anything beyond that. This first connection between my math learning and a piece of artwork helped me realize that I did not want to look at math as a confusing unknown anymore. If I was going to be an effective teacher, I needed to gain passion for all subject areas. Little did I know that this small response was the first step on a transformational journey.

    Fiesta Math Nights

    The first stop on my journey took me out of the university classroom and into the field. In three DISD elementary, Title I schools, I met hundreds of predominately Hispanic children and parents in what was known as Fiesta Math Night. These were nights full of math games that reflected cultures from Mexico, Central America, and South America based on trade relationships and economics. My pre-service teacher colleagues and I had to create mathematics games that tied the economics and trade relationships of a Latin American country to the United States. I had to formulate my own game and make it culturally relevant to the country I chose. To describe my game simply, it dealt with identifying place value of culturally relevant imports and exports from El Salvador. The students spun a spinner two times and landed on either an import or export. They would draw two cards, read the facts on the cards, complete the mathematics/economics tasks required, and then win a prize. It seemed easy to me but in reality it was not.

    September 24, 2011, the first “Fiesta Math Night” was the debut of my math game, “Spin It.” This night proved to be discouraging and a down right “beating.” Walking into the school that night, I already had predetermined that my game was going to fail. That attitude made it easy for me to get flustered and anxious as students of all ages tried to play my game. When parents and teachers came over to the table where my game was located, I felt ashamed for them to see it. I went home nearly in tears because none of the students seemed to understand my game plus many of my game pieces failed to work properly including my spinner, which was one of the most important pieces. Throughout my reflection that week, I wrote over and over again about the need to fix my game to make it more enjoyable and playable by the children. There were two game nights left and I was determined to revise my game and make it work for every student next time. Those two game nights ended up being incredible confidence boosters, something I desperately needed.

    Tutoring

    In the fifth week of the semester, I was given a second field assignment in my math methods course which required me to tutor a group of elementary students in mathematics for about eight weeks. These tutoring sessions were based on the Response to Intervention Model (RTI). RTI helps identify students with learning disabilities by using “a problem-solving process that uses curriculum based measures to identify students whose level and rate of learning are below those of their peers” (Stickney, 2005, p. 1). Students who use this model receive evidence-based instruction in the general education classroom that is modified to meet their needs. During the RTI process, Stickney (2005) stated:

    If the student’s rate and level of learning increase, the student would not be considered for special education. If the student’s rate and level do not improve, the student would be considered for special education services or for a special education evaluation. (p. 1)

    The outcome of my tutoring sessions would not completely determine if students needed to be referred to special education or not, but it would be a great source of information for the students’ teachers in the future if they just so happened to need it to provide evidence of the students’ learning. These sessions helped me experience firsthand examples of the RTI process. Two of the eight week tutoring sessions would be testing weeks (pre-tutoring and post-tutoring), demonstrating where the students were before intervention began and after intervention. My tutoring class consisted of students from three separate third grade classrooms. There were a total of six students, three boys and three girls. I was also given my own room that had dry erase boards, desks, paper, pencils, and many manipulatives to use with my students.

    Diagnostics

    September 30, 2011, was the start of the diagnostic testing week to determine the areas my students were struggling with the most. The scores ranged from 10 percent to 70 percent accuracy on a 100 point scale. None of the students made a perfect score and each student appeared to be on a different level. Overall, they appeared to struggle with the concept of place values which affected their performances in addition and subtraction. This made me nervous because I was not sure how I would meet all of their needs. After studying the overall results, I found that all students struggled in similar areas. From there, I looked at the specific Texas Essential Knowledge and Skills (TEKS) and identified the skills I needed to focus on the most as the tutoring sessions continued. In the university math methods course, we learned to structure the diagnostics and to read them to determine what to focus on during tutoring. The first week that I actually began tutoring my students was quite eventful. Like the first “Fiesta Math Night” I went in with a not-so-positive attitude. I could not wrap my mind around the fact that I—a person who was absolutely horrible at math and who could not even remember most of what I learned as a student myself—was going to tutor elementary students. So, can you guess how it went? Awful. Rushed, stressed, nauseous, and frustrated were feelings I took away from that first week of tutoring. I was angry and upset because, for the life of me, I could not figure out what I was doing wrong. Changing only a few things, I pressed on to the next week. Things went better, but I still did not feel like I was being an effective teacher.

    Manipulatives

    When I went into my methods course the following week I was miserable and unsure what I would teach my students in the weeks to come. Looking through my math notes, I realized “Hey! What we’re doing in methods is exactly what my students need help with! Use it!” Dr. Tunks, my professor for math, was constantly shoving manipulatives into our hands for just about everything we did. It was fabulous! Never had I understood math as well as I did in her class. We took math concepts apart piece by piece and studied them by relating them to real world things and using physical objects to bring the problems to life. From that day on, I knew what I had to do in tutoring. Relating the tutoring content to the real world and using manipulatives in every lesson would prove to be an effective strategy for the progression of my students. My reflective notes on my teaching and the students’ learning on October 21, 2011, showed that I had figured out that I needed to give the students things to manipulate, interact with them more, be more supportive of their learning, and begin to feel more confident about helping students learn mathematics.

    Teaching Transformation

    October 23, 2011, was the second “Fiesta Math Night,” which came shortly after my tutoring sessions began. After weeks and weeks of tearing apart my game and revising it, I truly believed I was ready for round two, because I had made so many worthwhile changes to my game. For instance, the spinner that would not cooperate at the first math night now was fully functional. Also, I changed the wording on the draw cards where every student could understand it or at least have it explained to them more easily. Providing more choices seemed like a logical idea. Instead of just solving place value the students could now compare or add numbers. With all these changes how could the game not work? With that thought, I marched into the school thinking, “Tonight will be a success!” And it was. The students seemed to understand the game much better. They loved having a choice. Several students even came up to me and said it was their favorite game of the whole night, which provided a huge boost to my confidence level. Parents even complimented me on my game and said it was great practice for all levels. The night ran a lot more smoothly than the first. I was not stressed out or overwhelmed; in fact, I was enjoying every minute of the night. Hearing feedback from the parents and students warmed my heart and made me feel like I was heading in the right direction with this math game. I was excited to see what the last math night in the following month would have in store for me.

    One of the strongest math transformations I experienced occurred on the very last day of tutoring, November 11, 2011. This was a day for the students to review everything we had learned by playing games related to the content we went over during each session. It was also a way to help them prepare for the post diagnostic test that would be administered the following week. During this week I learned a lot about the progress of my students. Having them play games that went all the way back to what we learned in the first few weeks worried me a great deal. But my students surprised and shocked me with how well they did on all the games. I even altered some of the games as we played them, asking the students to expand on the answers to questions that met different TEKS than the original game did. This was to see if they could identify the relationship between the concepts. That was something I would not nor could have accomplished in the beginning. It was a big indicator of my growth as a teacher. Witnessing my students dig out prior learning and knowledge to help them solve the problems and situations in the games was incredible. They referenced past tutoring sessions as they were playing the games and made connections. Students who, in the beginning, did not try to answer questions at all were now raising their hands and participating. The success of this day for my students spilled over into the following week when they took their post diagnostic exam. Examining the results, I was nearly brought me to tears. Every student had progressed. A chart from November 18, 2011, shows the progress of my students during the entire tutoring time frame, including the pre-diagnostic test, the six tutoring sessions, and the post-diagnostic (Table 1).

    Conclusion

    The effects of this tutoring project went far beyond just the help the students got and their improvement that was apparent in the results of the diagnostic testing. It was an unforgettable learning experience for me and my teaching career. I learned that using real world ideas and items is one of the most effective ways to teach math concepts. It was not only helpful for my students but for me as well to discover this principle. Middleton (1995) suggests, “When children are motivated intrinsically to perform an academic activity, they spend more time engaged in the activity, learn better, and enjoy the activity more than when they are motivated extrinsically,” (p. 1). Research has also shown that when students can relate to the topic they are learning, they will perform better. Students need to know why they are learning about a topic and how it relates to them. In relation to real world items, using manipulatives is also important. Toni Battle (2007) sums it up best when she says, “manipulatives are the way to our future and the way to new knowledge. No matter where we turn or what we do, we as a society are using some form of manipulative in our lives,” (p. 4). Children need these extra tools to help them master mathematics. For example, in my tutoring sessions I used base ten blocks and place value charts to help my students understand place value. This helped them see the physical value of the numbers they saw written on the white board in front of the classroom and on their worksheets. Base ten blocks along with counters aided the students in physically constructing addition and subtraction problems. It is important that manipulatives be used to help increase students’ academic achievement and my growth though this experience in mathematics helped me see this.

    Teaching should always be about the success of the students and should never just be about the teacher. When I first started tutoring, I was not putting the students first. I was just worried about making it through the lesson. Once I realized what I was doing and stopped, the learning began. It is important to take note of what works and does not work when working with students so their needs can be better met. The weekly reflections forced me to do this, and as I analyze those reflections while writing this autoethnography, I was again reminded of the importance of reflection for improved teaching. Another important lesson I learned was realizing teachers need to familiarize themselves with the math concepts they are teaching and to create a good relationship with those concepts no matter if they like the subject or not. According to research, “the mathematical knowledge of most adults is weak. We are simply failing to reach reasonable standards of mathematical proficiency with most of our students, and those students become the next generation of adults, some of them teachers” (Ball, Hill, & Bass, , 2005, p.14). When teachers understand the concepts it is more likely that the students will understand them, too. By working to understand the specific areas of math, ironically, I also developed a more positive attitude toward it as well. My math methods course was there for me to use as a resource, which I loved. The comments from my teacher every week helped me realize I was thinking in a different way. This opportunity was truly a blessing in disguise and a great learning experience.

    Round three of “Fiesta Math Night” on November 17, 2011, was perfect. My confidence was up and I was excited to be in the presence of students and parents, eager to play math games. Every student seemed to enjoy my game and played it more than once. In fact, I had several who played it at least four times in a row. I challenged them by adding more to what they originally had to do in the game. They loved it and I loved it. The confidence I acquired from previous game nights allowed me to go and grab students who were standing around the cafeteria and encourage them to participate. I allowed myself to engage in conversations with the students as they played the games. Teachers approached me and talked about all the games and how great they all looked. This made me feel appreciated and good about all of my hard work.

    When I wrote my reflection for Dr. Tunks, I was excited to explain the change that occurred within me after experiencing all three math nights. Her response (J. Tunks, personal communication, November 2011) was:

    It is so exciting to read that your confidence level was so elevated, even to the point of collaring children to play your game. It is wonderful to see how you felt you had progressed across time and the differences your adjustments made each time you presented. This is marvelous indeed. It is the early energy that sees you through offering new ideas. Don’t let your quietness keep you from taking a stand and offering ideas. You have a taste of success, at your own hands, so run with it.

    Her words brought me to the realization that I was no longer as afraid of math as I had been at the beginning of the semester. With the tutoring of a few students and participating in game nights, I had overcome the fear and proved to myself that I could learn and teach math.

    By the end of the first semester in Professional Development School, math and I had engaged in a truce. We both waved the white flag of surrender and began a friendship. I started out the semester being terrified of the idea of learning math again and having to teach it. After experiencing the sixteen-week semester, participating in “Fiesta Math Night” and leading tutoring sessions, I developed a new love for math. If you had asked me a year ago if I would ever like math I would have said, “Absolutely not! And for that matter, it’s the worse subject known to man and woman.”

    The painting, Square Sierpenski Subdivision Variation #1 2006 by Michael A. Coleman represents my current feelings about math (Figure 2). In the bottom left corner is a blank square that represents how my learning of math started; empty and blank, feeling like I knew nothing. As the semester went on more shapes and colors were added in, making a beautiful collection of silhouettes that join together. Dr. Tunks summed up my transformation best on her reply to my comments:

    It was as though I was glimpsing into the soul freed from pain and suffering and renewed by the love of seeing children learn. It is marvelous to read that the darkness you encountered at the beginning has been brightened by so much complexity of light and embedded shapes. (J. Tunks, personal communication, December 10, 2012)

    Math will no longer bring me to tears again.

    The theorist James Zull, in an interview from 2006 about his book “The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning,” says that we go through four different stages of learning (Zull, 2006). “We have a concrete experience, we develop reflective observation and connections, we generate abstract hypotheses, we then do active testing of those hypotheses, and therefore have a new concrete experience, and a new learning cycle ensues” (October 12, 2006). We get information, create an understanding of that information, develop new ideas from these meanings, and act on those ideas. That being said, Zull proposes that the pillars of learning are gathering, analyzing, creating, and acting. Being able to learn like this requires us to step out of our comfort zones and put forth much effort.

    “Fiesta Math Night” and the mathematics tutoring were definitely two activities that required me to step outside of my comfort zone and work hard to improve my performance as an educator. It was a necessity in order to create a better relationship with math. I believe for one to truly learn, an individual needs to be self motivated and driven. Being self-motivated and willing to try to improve was one of the biggest assists in my transformation.

    Zull also states, “To feel in control, to feel that one is making progress, is necessary for this learning cycle to self-perpetuate” ( 2006). He goes on to say that when we initially learn about something and develop fear against that learning, it takes our brains a while before it will see it as anything less than frightening. Once we have a positive, real-life experience the negative experience is overshadowed by that positive, real-life experience. This certainly holds true for me and my math relationship.

    My real life experiences involved math, children, parents, my colleagues, mentor teachers, and my math methods instructor. The negativity I felt toward math was surrounded by challenging, engaging, and motivating experiences that forced me to reconcile with the dreaded fear of my nemesis, math. I gained new knowledge about how mathematics works in my methods course that created an understanding through practice and reflective thought each week. I was in control throughout my experience, although I was bit shaky initially, but by the end of the sixteen weeks, I had transformed. As W.S. Anglin put it, “Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” (Moreno, n.d.) I was a lost explorer in the never-ending rough and jagged math journey. Now that I have seen that I can effectively teach math to young students and understand it myself, I know I can be an effective teacher of math. I am thankful for the Professional Development School, my professor, and my students. This experience provided me with a second chance to love math.

    References

    • Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching. American Federation of Teachers, Fall, 14-46.
    • Battle, T. S. (2007). Infusing math manipulatives: The key to an increase in academic achievement in the mathematics classroom. Final Research Proposal. Online Submission Eric.
    • Coleman, Michael A. (2006). Square Sierpenski Subdivision Variation #1. Retrieved from http://miquel.com/crystals.html
    • Fernandez, A. (2006, October 12). The art of changing the brain: Interview by J.
    • Zull. Retrieved from http: www.sharpbrains.com.
    • Moreno, M. (n.d.). Favorite Math Quotes: W. S. Anglin. Furman University. Retrieved 12 June. 2012, from http://gateways2learning.com/Quotes.htm
    • Kandinsky, Wassily. (1910). Murnau-Garden II. Retrieved from http://www.art.com/products/p12491404-sa-i1678006/wassily-kandinsky-murnau-gardenii1910.htm?sorig=cat&sorigid=0&dimvals=0&ui=470d0feca6904c828304da4b4f35d784&search
    • string=murnau+wassily#
    • Middleton, J. A. (1995). A study of intrinsic motivation in the mathematics classroom: A personal constructs Approach. Journal for Research in Mathematics Education, 26(3), 254 – 279.
    • Ngunjiri, F.W. (2010). Living autoethnography: Connecting life and research. Journal of Research Practice, 6(1), E1.
    • Stickney, D. & Doerries, D. (2005). Response to Intervention Retrieved from http://education.wm.edu
    • Wall, Sarah. (2006). An autoethnography on learning about autoethnography. International Journal of Qualitative Methods 5(2).

    Table 1: Tutoring Data

    Tutoring Summary Report Fall 2011

    Student

    Pre-diag score

    Post-diag score

    Overall
    gain

    Concept(s)
    tutored

    Concept(s)
    tested gain

    Concept(s)
    tested loss

    Concept(s)
    tested same

    1

    10

    30

    20

    pv

    add

    sub

    X

    X

     

     

     

    X

     

    2

    70

    90

    20

    pv

    add

    sub

    X

     

    X

       

    3

    65

    80

    25

    pv

    add

    sub

    X

    X

    X

       

    4

    45

    85

    40

    pv

    add

    sub

    X

    X

    X

       

    5

    60

    100

    40

    pv

    add

    sub

    X

    X

    X

       

    6

    70

    75

    5

    pv

    add

    sub

     

     

    X

    X

     

     

     

    pv = place value
    add = addition
    sub = subtraction

    Figure 1: Murnau-Garden II

    (Kandinsky, Wassily, 1910)

    Figure 2: Square Sierpenski Subdivision Variation #1

    (Coleman, 2006)