This paper focus on the reflection of my teaching practices using students’ Math Moments. I began to invest time in the past mathematical experiences of my students to better help me understand my own teaching practices. Throughout this paper I will reflect on my own teaching practice, delve into relevant literature and will use poignant math moments to illustrate student’s thinking and beliefs about learning mathematics.
Being an educator for more than 21 years at various levels, there have been numerous occasions where I have tried to reconcile meeting the objectives of a mathematics curriculum with how to teach it. I have wondered often, that given a specific teaching objective, what teaching strategy would best work in a particular situation with the students in front of me. At times, I found myself forsaking methods of instruction for curriculum objectives. During these moments, reality dictated that the Socratic Method was most efficient despite the theoretical frameworks encouraged by the NCTM Standards. At the high school level, content in a mathematics classroom was of paramount concern, and at times was the facilitating factor in determining how the subject was taught. Sometimes, not covering the prescribed content was seen as a disservice especially to graduating students. At other times, I wanted to embed an important concept in a different way other than the Socratic Method. Although this was time consuming and devoured valuable teaching time, I felt those particular moments were necessary to my student’s understanding of mathematics. At the college level, mathematics curriculum is dependent on the program of study. Regardless of the program, however, I believe that college instructors still struggle to balance and reconcile the curriculum with engaging the adult learner. This constant meta-probing juggling act, is not new to educational professionals teaching at any level, who seek to do the best by their students.The College Mathematics Learner
I began to wonder about the psyche of the college mathematics learner. What particular moments in their math career have contributed to the feelings they might have about the learning of mathematics? What type of positive moments and what type of negative moments define who they are? How do these math moments affect their learning? Can we ever unpeel the various layers of their understanding? Bound by these persistent questions, thus began my journey to delve into the “math moments” defining the mathematics college learner. Throughout this paper, I will reflect on my own teaching practice, delve into relevant literature and will use poignant math moments to illustrate student’s thinking and beliefs about learning mathematics.
In the past five years, I have been involved with teaching mathematics to “at risk” students at the college level. These are students who lack the necessary skills in mathematics and are therefore in danger of completing their program of study. Weak raw scores on the Canadian Achievement Test, Third Edition (CAT 3) or College Placement Test scores (CPT) determine that they take a remedial level or preparatory mathematics course prior to beginning their program of study. What is lacking, is how best to teach the required mathematics skills necessary to this specialized group. They are considered specialized and unique because many of these students have encountered countless years of frustration and lack of success in their classrooms that often translate to negative mathematics experiences. These negative experiences varied for each individual. Some experienced personal failure, some had unconstructive relationships with their teacher, some disliked the subject and others experienced bullying in the classroom. (Reference will be made to student’s specific math moments later on in this paper as it relates to these experiences.)
Mindful of the unique qualities of this group, often, I found it problematic engaging these students in meaningful ways so that their past mathematical learning would not be trivialized. As adult learners, some students felt inadequate and misjudged as overall weak learners when it was specifically mathematics that gave them difficulty. Many students felt the exposed sting of one subject; mathematics, holding them back from their life ambition. Although the NCTM Standards exist as a guide, there are no set program standards in mathematics at the college level. Mathematics as a discipline falls into a skills set within each program of study and as such does not stand alone as it does at the high school level. So mathematics is taught within the confines of business, or biotechnology, or media studies, or ambulatory studies etc. I began to invest time in the past mathematical experiences of my students to better help me understand my own teaching practices.Math Moments
At the beginning of every semester, I invite my students to describe a situation in their mathematics past that was either a positive or negative memory. They are asked to describe their feelings and thoughts of that particular “math moment”. Many of their stories were enlightening and helped me understand them as individual students with individual needs. It is an exercise that opens a dialogue between my students and me, and sets up the bond that is necessary in a trusting relationship.
How we teach mathematics impacts deeply in the understanding for the student. The NCTM standards ask teachers to be mindful in providing students with:
Numerous and various interrelated experiences which allow them to solve complex problems; to read, write and discuss mathematics; to conjecture, test, and build arguments about a conjecture’s validity; to value the mathematical enterprise, the mathematical habits of mind, and the role of mathematics in human affairs; and to be encouraged to explore, guess, and even make errors so that they gain confidence in their own actions. (Batista, 1994, p. 463)
I found many of my students feeling that they rarely experienced mathematics in this way. Many felt that the curriculum was too demanding and did not allow opportunity for students to experience success. One student wrote:
In high school, everything was taught so quick that the harder I tried to pay attention and stay on top of it the worse I got. Getting confused half way through class was the worst because then that meant the rest would just be a blank.
Another student alluded to the problem of relating math to everyday life:
Grade 10 I began to think that what they were teaching wasn’t important and majority of what we were learning was not going to be used in the real world, (I) became bored in class. (I) had problems remembering steps when it came to tests and exams. (I) continuously have this problem.
The experiences of these two students were not unique, as others also echoed the same sentiments and it had an effect on how I was going to deliver the curriculum. I acknowledged that many students found it difficult to relate to mathematics in everyday life. Yet they never stopped to really think about how it actually surrounds us everyday.Discussions of Accuracy versus Estimation
An exercise in an article by Marilyn Burns nicely facilitates students thinking (Burns, 1998) about real life applications. (I engaged in this exercise early in the semester. My purpose was to get students to articulate for themselves the usefulness of math in their daily lives. I found this exercise challenging and exciting.) It entails two steps. The first step involves listing situations in students’ livesoutside workwhen they need to do arithmetic. From this, the student sees that “doing arithmetic is a regular part of our lives” (Burns, 1998, p. 6). The next step was to separate these situations into tasks involving mental work/estimation calculations, paper and pencil calculations, finally calculator calculations. When I performed this exercise with my class, the results were the same as stated in the article: many found that most people performed mental or estimation calculations on a daily basis. A dialogue emerged whereby students realized that “there are disturbing mismatches between how we were taught arithmetic in school and what the arithmetic that we need to do in our daily lives really calls for” (Burns, 1998, p. 8). This prompted a discussion about the relevance of the math taught in school. This discussion was necessary for students to see that arithmetic methods we use “depends on the numbers we’re dealing with, the context in which we encounter them, and the extent to which we need to be accurate (Burns, 1998, p. 11). Students also recognized that in real-life, we have to be able to estimate and know when estimates are appropriate (Burns, 1998, p. 11). Students also acknowledged that sometimes we actually solve problems differently inside and outside the classroom. As Heibert put it:
Outside of school, many…..seem to use their intuitions and conceptual understandings to decide what to do, what strategy to use. Inside of school, many…..try to recall and execute rules for solving problems “like this one” to find the answer. (Heibert, 1989, p. 39).
Once students acknowledge that real life situations and artificial classroom situations have different experiences, they begin to make sense of their own experience with mathematics.Real Life Experiences of Students
In the confines of the classroom, we sometimes forget the importance that real life experiences have on students’ learning. One student stated:
One memory I have of math class was in my grade 12 class, we went to Wonderland for Math Day and we had to do this scavenger hunt….It was a positive experience and it made me realize just how much math we were exposed to and how even without realizing, we (were) doing math in our heads.
Another student wrote:
An outstanding memory that I had that was very positive was in high school, studying areas of a circle. This was a topic that was giving me a lot of difficulty, but after comparing it to a day to day activity; I realized that it was easy. Mathematics can be related to day to day life and this made me somewhat more interested in math because in general math is not a subject that I am interested in at all.
So now that students recognize that math is real, what exercises in student’s work will reflect this? Once again, I borrowed from Burns. In her article Talking Turkey About Arithmetic she suggests that students recognize how to use arithmetic to solve problems in a variety of contexts (Burns, 1998, p. 13). I asked students to plan an important meal for their family. We talked about uncertainty and variability and I provided them with guided questions as well as a rubric for evaluation. I was pleasantly surprised as I found students going above and beyond the boundaries of the assignment. The quality of the papers suggested that given a real-life problem, students were able to relate and make sense of it better. (This assignment was asked of students only after I taught them computational skills of whole number and fractions.) Solving a non-routine problem like meal planning and preparation enabled students to find individual ways to demonstrate their understanding of arithmetic. So the student who “became bored in class” and “had problems remembering steps” may be better able to relate back to this task when asked to perform the same arithmetic procedures in a test or exam. The feedback I received and conversations I overheard from students about this assignment, affirmed that the experience gained was relevant and real for them.Attitudes, Beliefs, Emotions of Students
In my experience, attitudes, beliefs and emotions of the student play an integral role in the learning of mathematics. I believe that student’s attitudes, beliefs and emotions are an underestimated domain that often goes unchallenged in the classroom. This is more acute at the college level, where students already come prepackaged with certain views in mind. Having spent twelve years (or more in some cases) in formal mathematics learning, these students have very specific attitudes, beliefs and emotions embedded. In some instances, many years of negative experiences has built over the years to obstruct the learning in mathematics. In other cases, it took only one event to set the stage for future mathematics endeavors. Sometimes, I need to tease, sift out and negotiate these “affective factors” before I can begin to teach. McLeod and Ortega offer us a way to think about these set of factors:
Beliefs, attitudes, and emotions are terms that reflect the range of feelings and moods that make up our affective responses to mathematics. These terms vary from cold to hot in the level of intensity of the affect that they represent. They also vary in stability: Beliefs and attitudes are relatively stable and resistant to change, but emotional responses to mathematics may change rapidly. (McLeod & Ortega, 1993, p. 22)
One belief that I observed consistently throughout the “math moments” exercise, was one that questioned the student’s ability to do mathematics. These students often expressed a sense of powerlessness in performing mathematics. (See McLeod & Ortega, 1993, p. 29 on the myths associated with ability in mathematics in the U.S.) In three separate incidences students emphasized this point:
My outstanding memory in math class was not that great because when a topic got thrown my way and I do not understand it, it distracts me and I loose focus in the whole topic.
Another student outlines the frustration felt:
One memory that I have was failing grade twelve math. This class was a very bad experience for me because I struggled throughout the whole semester and barely passed each test or failed it. Towards the end, I tried really hard to pass the course, but I ended up failing the course and cried for a few hours. I was frustrated because I tried hard, but nothing seemed to work.
Another student shares memories of constant struggle:
I have struggled with math since elementary. So, all of my memories of math are pretty bad. Basically when it comes to math, I have this thing where “if I don’t get it, forget it”.
It is clear from the three stories above, that if students fail to experience success; their negative belief about their own ability in mathematics is hard to change. We must provide instances where students can be successful so that they can feel empowered to do mathematics. These moments must be built upon so that strong belief in negative ability can be changed little by little. In my classroom, I sometimes have students pair up to help talk each other through some math problems. This provides both individuals with an opportunity to share in “math talk” and in small ways also empowers them in their mathematical sense making journey. Sometimes students need to know that these feelings are a normal reaction to hard problems. (See McLeod, 1993 article where he outlines that students need to experience frustration in problem solving in order to change their affective response.) What is needed is for a teacher to negotiate honestly and sincerely by validating the attitudes and beliefs of students first. I firmly believe that once student’s affective response is met with honesty and sincerity, only then can the learning of mathematics occur in a meaningful way. Then both teacher and student work together in a common bond.
In many of the “math moments”, students believed that the teacher made a difference in the learning of mathematics. Some had negative memories, while others had positive ones. Below are three examples of negative moments and three examples of positive ones. Three negative responses that hold the same view are as follows:
In math if the class moves at my speed then I can understand it and I will do well, but in grade twelve, my teacher was all over the place and it didn’t really get through to me.
In the second example the student stated that:
I was never very good at math. I am horrible at multiplying and dividing. In grade five is where you learn how to multiply. I was made fun of by my grade five teacher. That’s where I really hated math.
Finally, the last example of a negative memory where the student felt that the teacher had an impact was in the following:
One time in math class everyone knew the answer to a question, but the teacher called on me to answer the question, but I couldn’t and someone shouted out the answer after a few seconds. It made me feel really stupid.
The instances of positive moments include three different scenarios. The first example is when a student stated that:
During my elementary school years, especially grade seven and eight, my experience in grade seven was not all that good. I wasn’t doing that well and I simply did not enjoy it. Coming into grade eight the last thing I wanted to do was go to that math class. However, one thing changed my view on math; my teacher. He helped me a lot. I guess just the influence of one person can change everything.
The second example a student stated that:
I had a really good experience with my grade twelve math course because the teacher was amazing. She was helpful and tried different ways explaining something until the person understood the material. I never knew I could actually enjoy math as much as I did.
In the final example of student moments involving positive teacher influence, it was stated that:
My memory regarding math is not a certain moment but a certain teacher. He was always there whenever I needed help and never got frustrated if I asked a lot of questions. He made math fun and easy to learn.
It is evident from the above “math moments” that the role of the teacher is essential in helping students understand and engage in the mathematics.Teacher’s Beliefs and Attitudes
Often, a teacher’s own beliefs and attitudes come into play. An article by Karp entitled “Elementary school teachers’ attitudes toward mathematics: the impact on students’ autonomous learning skills” (Karp, 1991) examined teachers’ attitudes toward mathematics and the impact that the attitude had on their students. The sample of teachers was purposefully selected so that the two opposite spectrums of attitude were clearly delineated. Teachers with positive attitude and those with negative attitudes regarding mathematics were specifically selected. The results of the study speak to two major themes emerging of teachers with negative attitudes regarding mathematics: teacher dependence and learned helplessness. It was found that teachers with negative attitudes were observed providing instruction which was based on rules and memorization (Karp, 1991). This type of teaching reinforces the belief that the teacher is the only source of information and learning. The implication of this is that students may be unable to transfer the learned skills or apply higher level thinking to novel situations without the help of the teacher (Karp, 1991). The other finding of teachers with negative attitudes was one of learned helplessness. By their behavior, (limiting students’ active involvement and opportunities to respond to questions, spending extended periods of time with individual students, asking questions and immediately answering them) they affected a students’ ability to have control over a learning situation (Karp, 1991).
In contrast, teachers with positive attitudes used instructional methods that encouraged student independence. Some of the behaviors exhibited by these teachers included: focusing on the “why” of algorithms working, less rule-based instruction, resources other than the teacher for self-instruction and requesting that students prove their answers (Karp, 1991). The above teacher behavior encouraged self-instruction, reflective thinking and self-correcting in their students (Karp, 1991). By doing so, students are able to learn and think independently and are better prepared to deal with real life problems. The teacher must model not only “conversations by listening to students, following students’ arguments and encouraging students’ attempts to support (or challenge) assertions of knowledge”, (Smith, 2003, p. 13) but also must provide the emotional and attitudinal support necessary to deepen their knowledge construction. Once these are met, the student will be empowered to build, adapt and construct their own relevant mathematical knowledge more positively in terms of their world. In doing so, the teacher can become a positive influence in the student’s learning of mathematics. The fact that I received more positive comments about the teacher’s role in math learning (in the “math moments”) than negative leaves me optimistic. If student’s attitudes, beliefs and feelings are to change for the better, the teacher must also convey positive attitudes, beliefs and feelings.Final Thoughts on Engaging the College Mathematics Learner
Positive classroom culture that enriches class beliefs, attitudes and behavior patterns as well as effective collaborative activities is therefore seen as key factors in building class community that moves toward engaged ownership in learning. Students can be directed to participate in small or large groups where the conversation of mathematics is encouraged. Creative tasks need to be designed in such ways that do not “require the imposition of a definite right answer upon the student” (Smith, 2003, p. 19). Too much of what we ask our students to do involves computation or following the algorithms in a directed way. Students need to understand that “mathematics is a lens with which we can investigate our world rather than an exercise in calculation" (Smith, 2003, p. 20). We need to encourage dialogue of problem solving among and between students. To this end, perhaps a journal writing activity might be asked of students to complete that shares their thoughts, ideas and discussions in group situations. This would provide a forum in which students could communicate in a different way, their knowledge production of mathematics.
Teaching and learning are complex activities. Engaging the adult learner in mathematics is often tricky as they come with previous knowledge and experience. It is hoped that in hearing a student voice in the day to day activities of the classroom, it can better inform the way we teach. Sometimes, a written reflection of student’s attitudes, beliefs and feelings is a good starting point in that direction (as in the case of writing about “math moments”). This type of dialogue engages the learner at the very beginning and is necessary in setting the positive atmosphere so that learning can occur. I do hear their voices and I try to make it permeate every facet of my teaching practice. Once we have accepted the challenge of investing in issues that confront students in their mathematical journey, only then, can we hope to involve teaching as an activity that moves beyond the classroom to produce students who are globally trained critical and autonomous thinkers. Their collective voices should sound the beacon for how we should approach the teaching and learning of mathematics.References
Batista, M. (1994). Teacher Beliefs and the Reform Movement in Mathematics Education. Phi Delta Kappan, February 1994. pp. 462-470.
Burns, M. (1998). Talking Turkey About Arithmetic. In Math: Facing an American Phobia. Math Solutions Publications. Sausalito, CA.
Hiebert, J. (1989). The Struggle to Link Written Symbols with Understandings: An Update. Arithmetic Teacher, March 1989.
Hiebert, J. & Lindquist, M. (1990). Developing Mathematical Knowledge in the Young Child. In Teaching and Learning Mathematics for the Young Child, edited by Joseph N. Payne. Reston, Va: National Council of Teachers of Mathematics, 1990, pp. 17-36.
Hiebert, J. (1990). The Role of Routine Procedures in the Development of Mathematical Competence. In Teaching & Learning Mathematics in the 1990s, edited by ThomasJ. Cooney and Christian R. Hirsh. National Council of Teachers of Mathematics, 1990, pp. 31-40.
Jackson, C.D. & Leffingwell, R.J. (1999). The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College. In The Mathematics Teacher, Vol. 92, No. 7, pp. 583-586.
Karp, K.S. (1991). Elementary School Teachers’ Attitudes Toward Mathematics: The Impact on Students’ Autonomous Learning Skills. In School Science and Mathematics, Volume 91(6). October, 1991.
McLeod, D. B. & Ortega, M. (1993). Affective Issues in mathematics Education. In Research Ideas for the Classroom: High School Mathematics. pp. 21-36.
McLeod, D. B. (1993). Affective Responses to Problem Solving. In The Mathematics Teacher, Volume 86(9), December 1993, pp. 761-763.
Nardi, E. & Stewart, S. (2002). (article in 2 parts): I Could Be the Best Mathematician in the World…If I Actually Enjoyed It. Mathematics Teaching 179, pp. 41-44.
Stewart, S. & Nardi, E. (2002). I Could Be the Best Mathematician in the World…If I Actually Enjoyed It: Part 2. Mathematics Teaching 180, pp. 4-9.
Russell, S. (2000). Developing Computational Fluency with Whole Numbers. Teaching Children Mathematics, November 2000, 154-158.
Skemp, R.R. Schematic Learning. In Chapman, L.R. (Ed.). The Process of Learning Mathematics. Pergamon Press, 1972.
Smith, W. (2003). A Constructivist Teaching Theory: Application in a Post-secondary Undergraduate Mathematics Course for Liberal Arts Students. A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy, School of Education.
Stuart, V.B. (2000). Math Curse or Math Anxiety? In Teaching Children Mathematics, Jan. 2000, pp. 330-335.
Whitin, P.E. (2007). A Mathematics Survey: A Tool for Assessing Attitudes and Dispositions. Teaching Children Mathematics, 13/8, 426-432.
Zambo, R. & Zambo, D. (2007/08). Mathematics and the Learning Cycle: How the Brain Works as it Learns Mathematics. Teaching Children Mathematics, Vol. 14, No. 5, 265-270.
Vanessa Farren has been an educator in Ontario since 1987 at the elementary, high school and college. She teaches Mathematics in the Faculty of Business at Seneca College in Toronto and can be reached at email@example.com.
Reflection of Teaching Experience
by Deena Sallomy
Teaching, like any truly human activity, emerges from one's inwardness,
for better or worse teaching holds a mirror to the soul. Parker Palmer (1998)
Preliminary Research and Preparation
In order to prepare to teach a unit on math to grade one students, I had to become familiar and comfortable with the Curriculum Guide and Program of Study. By reading the material I would need for this particular group of students, I began to slowly feel more comfortable with the task at hand. I also talked to my peers and related my fears and insecurities to them. I quickly discovered that a great deal of my inhibitions are linked to past experiences with math. When this became clear to me, I then was able to face those emotions and move beyond them. I took the next step, breaking the unit into smaller ideas - individual lessons - and diving into the research.
I began to research my unit by exploring the Program of Studies and Curriculum Guide. When I realized what my expectations were, I began to research various activities I could use - this required me to dig into journals and textbooks. Once I had an idea of the format I wanted to use for the unit, I began to get lost on the Internet--there is a great deal of useful information of math online. I quickly learned that having to teach only six lessons in this unit was not very long, and the resources were endless. I had been transformed from a panic and lost state to an overwhelmed and lost state!
The activities I ended up choosing were picked in relation to resources, time allocation, students' interests, and in relation to the existing classroom environment. I had already decided to adopt a style of teaching that these students were not very familiar with. The teacher usually introduces a unit of study and has set up activity tubs (stations) where she would rotate the groups. I was bringing in a more interactive approach - my unit was to be taught in both small and large group settings. I believed it would be interesting to compare and observe the similarities and differences between my approach and that of my partner teacher.
Before finalizing my plans for this unit, I discussed my findings, resources and ideas with my partner teacher. From the beginning, my partner teacher offered to share her lesson plans and activity booklets. I asked her if I could instead look at these after I had completed research and study of the unit on my own. When I presented her with my plans, she agreed that everything was in order. She did suggest that I might be taking on too much at once with the whole store set-up. Also, she was concerned that I was constantly involved in the daily instruction instead of having the students explore centers. Nevertheless, she encouraged me and offered her assistance throughout the unit.
Daily Journal of the Teaching Experience
The goal for this first lesson was to discuss the importance of learning about money, and how we use it on a daily basis. This lesson is in two, 30 minute parts, with a 30 minute break in the middle for gym class. The first part of the lesson is an introduction to the penny, nickel, and dime - physical features, and value. The activity that follows gym class, Coin Capers, is related to the lesson that occured prior to gym.
I thought that this would be a simple start to this money unit. Well, to my surprise my lesson plan (which I had copied out on a piece of paper, and set next to me) had been altered within the first ten minutes. The structure I had chosen to begin our discussion - sitting in a large circle - and the idea of starting a new unit of study with another teacher was a little too much for these students to handle. Some students were overtly excited and unable to concentrate on the task at hand. In comparison to a regular lesson taught by their teacher, the students would "usually" settle down once the teacher has requested for their attention, or sometimes her presence would do the trick. With me, I had asked for their attention and waited for a few minutes yet I still hadn't captured their attention. The conversations that were occurring during those few minutes were relative to the lesson. The students were discussing the fact they were going to be able to playwith money - so I guess I had some form of attention.
I felt that time needed to be taken out to deal with discipline issues, and if I didn't do it right away I would be dealing with it continuously. A strategy I used to deal with the behaviors was to stop the lesson and wait. This usually got the students to stop and listen. After the second or third incident, I simply stopped once more and discussed the fact that if we have to continue doing this we would be using up our activity time. This was very frustrating to me, because I felt that having to spend time managing the students was taking time away from the real lesson at hand.
When it came time to learn about the physical features of the coin and discuss its value, I began to feel a little anxious because I could detect a sense of frustration from certain students. This was evident to me by their immediate withdrawal from participating in the lesson, or losing focus. Some students voiced their frustrations by declaring "this is too hard". Others seemed to understand the concept and felt frustrated with their peers who were struggling with the idea that a dime is the same as ten pennies. For some this was a little too much to take on in one day, while others swallowed up all the information and were able to answer all my questions. It was apparent that some students had had very little experience in handling and working with money outside school. This was one aspect that I had not considered.
During the first part of the lesson, I looked at the clock and realized that I only had a little time left before we were off to gym class. After gym class, the students would return to work in small groups on completing an activity. Yet again, I managed to alter my lesson quickly in order to provide the students with enough information to work on the activity. In the initial plan, I had expected that we would start to count by 1s, 5s, 10s, up to 100. I realized that was not going to happen in four minutes, so I introduced the activity and demonstrated the task. I repeated the examples once we came back from gym class.
The activity was called "Coin Capers". The students worked in groups of 3 or 4 to figure out various combinations of coins - for example, 1 dime, or 10 pennies, or 2 nickels, etc., all make up a dime. Overall, the activity was a success with some groups needing guidance. Some groups refused to face the challenge and decided not to participate. When I added an element of casual competitiveness, it seemed to alter the class atmosphere. The reluctant students became more motivated, and tried to complete the task before their peers did. My teaching strategy was to declare that "this group has two answers" or "that group is just about done". This activity was also somewhat of a formative evaluation in that it allowed me to observe whether I had made any impact on the class, or if they understood the lesson.
The feedback from the teacher was positive at the end of the first lesson. She stated that I handled the group very well, and maintained their interest. I was able to recognize behaviors and deal with them adequately. In regard to the lesson, she felt that it went very well. I had challenged some students who needed to be challenged. When I approached her, I had a handful of critical observations that I had made about this lesson. I felt that I tried to speak above the students' voices during the first part of the lesson, and that I may have given them too much information to handle. I was frustrated at the beginning of the lesson, but I slowly realized that I had to remain calm and instinctually deal with student reactions to the lesson. I also had to be able to modify the lesson on the fly because of time constraints.
After going through today's lesson I felt a little better as to my ability to teach math, and to attain adequate classroom management. For the next lesson I planned to teach less and encourage the students to take an active role in the lesson itself. I hope that this will sustain their interest.
The goal of this lesson was first to review the activities from the day before, and then introduce the quarter. When this unit was originally planned, the quarter was not to be introduced until the third lesson. However, I was using a worksheet - which I had been very excited about - that dealt with the quarter, and therefore I was forced to introduce it. The decision to use the worksheet was solely my own. When I suggested the worksheet to my partner teacher during the planning process, she thought that it was a great idea. However, she did mention that some students might find it difficult. I integrated music at the beginning of this lesson, a coin rap to recite the value of each coin, and as a class we practiced it and added some actions as well.
Sitting in a large circle, we discussed the features and value of the quarter in comparison to the other coins. Just as planned, the lesson took on a more interactive component. The students were responsible for answering questions and illustrating their answers on a white board, which was placed in the center. There seemed to be a little more order and cooperation from the students today. During the second half of the class the students were given a sheet that illustrated various coin combinations, and a bag of coins to use to calculate the total value of the combination. The students added the coin values and wrote their answers in the piggy bank at the end of each question.
To my dismay, some students shed tears of frustration, while others counted the number of coins, and a few actually calculated the value of the coins. In other words, there was chaos. When I collected the sheets and marked them, I discovered that only three students out of twenty-six, had achieved a perfect score on the sheet. I had apparently put a little too much emphasis on the results of these worksheets. I think that I may have translated the scores on this worksheet into an indication of my teaching and how much the students had attained from my lessons. Achieving a "perfect score" was the goal I had set for myself. The sheet itself heightened my expectations of the students to a level they are not quite ready for yet.
This experience taught me a valuable lesson about myself and about my teaching. I had automatically transformed a lesson on paper into a personal disaster. Instead of stepping back and looking at the lesson and the level of the students, I took the outcome of the lesson as a personal failure. I felt that I had failed my partner teacher and the students. I had actually reached a point where I began to doubt my career choice. After a big bowl of ice cream and some tears, I calmed down and took a big step back and began to revamp the next day's lesson. With further research and consideration of where the lesson had gone array, I took on a new approach for lesson three. I was aware of the pace I needed to maintain, and I knew that I needed to be a little more conscious of the students, and their experiences in regards to the activities I presented to them. By reviewing the past two lessons, I was able to follow a pattern in my teaching style. I had been focused on the lesson itself at times and had overlooked the real life student experiences that were occurring before me. I also needed to relaxjust a little more.
After only two lessons into this unit I have began to see changes in myself, and in the way I approach various situations. The greatest lesson I learned today is how NOT to take a lesson so personally. This is much easier said than done. For lesson three I plan to review, repeat, and reflect!
The goal of today's lesson was to review the concepts from the previous two classes by working on a set of exercises that allowed students to practice their coin values. I did not believe that this would take the whole hour to complete. Therefore, I chose a group activity from the Money Monster book, which talks about purchasing pets with various coin combinations. I planned to have the students follow along as I read the book and demonstrated the coins that were being used to purchase a particular pet.
I had few expectations coming into the lesson today. I guess you can say that I was protecting myself and wanted to experiment with the notion of "teaching in the moment". Constrained by my extensive and well laid out plans for the previous two lessons, I had been a little tense and unaware of my surroundings and students. I was focusing on myself instead. To my surprise, today's lesson flowed smoothly. The students seemed a little more at ease and followed directions. The difference was that I focused less on myself, and more on my students, as I taught in the moment.
During review, I put five questions of varying levels on the board, and asked students to pick one they could solve. There was this one student who usually did not participate in any class activity or interact with any other students. He was diagnosed as being "selectively mute" - he had not spoken one word at school for about a year or so. I noticed him sitting at the far end of the carpet and counting with his fingers. I was reluctant to call upon him to answer a question, but when he looked up and smiled at me I took the initiative. I asked him if he would like to come up and solve the first problem, and, to my surprise, he did. As he walked up he received encouraging words from his classmates. As he wrote the answer on the board I became all choked up with tears of joy - for me that was my first heartfelt teaching moment!!! Needless to say, I was on a natural high for rest of the day.
This lesson went along pretty well, as did the activity. The students were very much engaged in the activity, and displayed their new found confidence by asking if they could help others. The activity itself was completed and handed in after about fifteen minutes, as opposed to the last activity, which took about twenty-five minutes. I had supplied the students with bags of coins, and various other manipulatives to use while working on their activities. This activity seemed to boost certain student's confidence and learning about money, which was really satisfying to see.
As a result of shedding tears over my last two lessons, I began to view the process of teaching and lesson planning in a new light. The plan in your head is mainly for yourself to have, but what happens in front of the class is for all to share. At the end of the day I began to wonder "why do I need to bother with the planning aspect when I know that when it comes down to it, whatever is supposed happen will happen?" I am not saying that we need to disregard all aspects of lesson planning. I personally have experienced the benefits of having a lesson plan. It offers you stability and at the same time flexibility. In essence, it is a blue print of ideas, goals and objectives to cover in a lesson; it also allows you to be responsive to new ideas as presented by students. Preparing a lesson beforehand allows the teacher to organize the classroom and be physically prepared, as well as psychologically prepared. Based on my personal experience, it is not effective to write yourself a script to read aloud. Knowing the "big idea" behind a lesson is the key to a successful and innovative lesson that can be student-centered.
The biggest challenge of lesson four was the preparation. The activity of the lesson was to play Money Bingo. To prepare for this, I had to put together a Bingo sheet that displayed the coins. The idea was that I would call out a certain coin value (ie. 45 cents), and the students had to cover the coins that would total that value. Once they had covered a line or the entire card than they could call out "Mingo"! Preparing the card took about 45 minutes, including several trips to the photocopying machine. This was one aspect of teaching that I never thought to account for. After all the time, the cutting, and pasting, I realized that I had no "free" square - at that point, I wasn't too concerned with that!
The first half of the lesson was spent reviewing coin values, and comparing the price of objects (i.e., which object costs more or less?). I discovered that bringing in objects to place in front of the students made the experience much more real to them. The students who usually find a million other things to look at during a lesson were actually interested and participating in the discussion today. It really made me realize how authentic our learning needs to be. Teachers need to "show" what they are teaching, and attach a reason or purpose to the lesson - especially in math.
The Money Bingo activity was a little chaotic with 26 students yelling out "Mingo". I quickly had to set regulations on the game (i.e., you can only call out once). During the game, I discovered that certain students had caught on to the concept, while others were still struggling. But, one particular student, who usually did not care about anything and did what he liked while sitting in the back sucking on his thumb, seemed to really want to understand this activity. So much so, in fact, that he started to cry. It was hard for me to see students crying because of what I was teaching them. I thought they must feel like I was torturing them. My partner teacher had to reassure me that certain students who were finding the material challenging had finally started to apply themselves instead of giving up. She thought I was making good progress! However, I wasn't sure if that made me feel any better. Overall, the students seemed to have had fun participating in the activity.
As I reflect on this lesson I have come to realize the importance of leaving the structured worksheets behind and allowing for time to "play" with math. Of course, I understand that there needs to be a balance of both elements, structure and play, within a learning environment. For some students, playing Money Bingo seemed to ease their frustrations and allow them to just have fun while practicing their coin values. A good lesson for me was that I could present the lesson and materials in another context and truly enrich the student's experiences.
The goal of lesson five was to prepare students for the final activity, which was setting up a store. We started by singing our coin rap, just as we had done at the beginning of every other lesson. The first half of this lesson was spent listening to me (unfortunately) explain their activity of the day. We discussed purchasing items with a certain amount of money, and finding out how much money we would have left (i.e., I had 10 cents, I spent 7 cents, I have 3 cents left). Simple concept, you would think. I had accounted for only fifteen minutes of explaining the activity, but due to the response I was getting from certain students I found myself repeating and repeating the ideas through various examples. In so doing, I found that some students slowly began to let down their guard and accept the task. The students were a little more open about asking questions and answering for that matter. I also found that if I had the students come up to the front and figure out the problem, it would give their peers motivation to think about the problem themselves.
The students were responsible for naming their store. After much discussion and a group vote, they named their store "The Grade One Pick-Up Store"! I almost died! What did the principal think when she saw this name on the blackboard? However, I felt that this would allow the students to have a sense of ownership in their store and their learning. We decided what to sell at our store and compiled a list of merchandise. Because of time restraints, I took responsibility for bringing in and pricing the merchandise. Usually, I would have had the students do this. I would have liked to have the students take an even larger part in planning the store itself and possibly developing advertising; yet, once again, time was an issue. That aside, I truly think that it was an excellent learning experience for us all.
Today's practice exercises were simple for some students, and challenging for others. This has been one aspect that I have constantly faced throughout the unit. How do you adapt a lesson or activity to reach everyone's needs? As a teacher, I could only work with a certain number of students at one given time. So, what happens to the rest (especially when I am the only one in the room)? Preparing a unit of study to meet a particular student's needs in the subject matter for a certain grade level, which also meets the provincial standard, was a huge challenge. I felt I had to include information that I was legally required to teach, and at the same time adapt the ideas into a format that would meet the needs of a diverse group of students. I was also working with a constrained amount of time in teaching a full unit - six lessons - which is a realistic teaching scenario. I haven't answered all of my own questions about this dilemna yet.
The last lesson of the math unit! It has been both exciting and exhausting. The sign for the store had been made, colored, and decorated with coins. The items for the store had been setup and priced, and I had filled their coin bags with a total of $2.00 worth of various coins. The goals of this lesson were to provide closure for the unit, to provide an opportunity for the students to practice their learned skills in a realistic atmosphere, and to evaluate their learning and my performance as a teacher.
The "The Grade One Pick-Up Store" activity took over the entire lesson. The objective was to allow the students to partake in a designed environment where they may use their coins in purchasing two separate items, providing correct change, and interacting with their peers. Once a student had had the chance to purchase an item, they were then responsible for taking on the shopkeeper's role. As every student purchased an item, the roles were switched. Each student was able to participate by buying items two times.
As the students settled into the activity, they began to incorporate social skills into their role-playing, such as "thank you for shopping at our store" and "have a nice day". I received positive feedback from the students and my partner teacher. When I heard students telling their classmates during gym class that they had "the best store ever", and were excited to leave gym early to come back to the store - well, it meant a great deal to me. The students seemed to be engaged and motivated to take part in this activity. They even offered one another positive feedback and waited quite patiently for their turn to participate. Who would have thought that they would choose math over gym?
Part of my assessment of the success of students, and the unit, involved observing the students interact in the store. Students who had struggled throughout the unit really enjoyed this activity and seemed to finally gain better understanding of money. Sometimes a paper and pencil test is not the best method of evaluation, because students are unable to demonstrate what they really know. The store activity gave the students an authentic way to demonstrate to me what they had learned.
One of the biggest challenges of teaching this math unit was probably the planning component. I still question the purpose of planning to some degree. The perfect plans I had designed before I started teaching seemed to include all the aspects of a "good" lesson. However, what my lessons lacked was the humanistic component - the students themselves. I learned that a lesson does not usually begin when you thought it would; daily routines do need to be accounted for, and classroom management is an important part of a lesson which no one really tells you about.
A comment I made in my daily journal questioned whether classroom management is part of a lesson or should be considered a separate element. Not even a month ago, I would have told you that management is not part of a lesson, and is mainly a disruption to the lesson. Now, I can confidently say that it is an integral part of a lesson. If there is a behavioral problem, I found that it could easily be connected to lack of interest, frustration, or boredom with the lesson or topic. Student behavior could also be related to many other facets stemming from events outside the classroom, and even health issues. If it is occurring during a lesson, it is then part of the lesson. A lesson, then, is not instruction alone. A lesson encompasses a multitude of factors that can aid the learning environment, or be a hindrance. These factors include lesson structure, subject, activity, classroom environment, time of day, day of the week, holidays, and most importantly, the students themselves.
The plan that is put together at the beginning of a unit, or a lesson, is merely a road map for the teacher - it is not a one-way road to follow. If I have learned anything about lesson planning, it is that flexibility is the essential element to a successful and healthy learning experience. If a teacher has an idea of what they want to achieve in a particular lesson, than that alone should be enough. Having a written script of the lesson eliminates the personable and responsive aspect, and focuses on the teacher's needs instead of the needs of students. The most important thing I learned was to follow my own instincts while teaching.
Parker J. Palmer (1998) speaks about teaching in the microcosm, and placing the subject in the center of the learning space. In chapter five of The Courage To Teach, Palmer brings a new perspective to teaching, which can be associated with the lesson planning component. He believes that we should foster an understanding of the subject we are teaching and "bring it to life" for our students. He favors a subject-centered approach to teaching. In order to bring a subject to life, there needs to be a strong belief in what you are teaching; "Passion for the subject propels that subject, not the teacher, into the center of the learning circle - and when a great thing is in the midst, students have direct access to the energy of learning and life" (Palmer, 1998, p. 120). Palmer's message became more meaningful to me towards the end of my unit when we started planning our store. The goal of creating a story gave the students the opportunity to truly take an active role in their learning, and it placed the subject, being partially the students and learning about money, in the center of the learning space. The last lesson was not taught from a script, but more so from the excitement and passion that both the students and I felt - we had a blast, and we all learned and strengthened our connections to each other. In my opinion, it is what happens during a lesson with the students that is the most important part of teaching. It is the time when I really began to understand my students, and relate with them.
Jardine, LaGrange, & Everest (1998) discuss having an element of integrity for the subject you teach, for the most exciting result is to pass that integrity on to your students. This subject integrity is closely linked to the subject-centered approach that Palmer (1998) explains in his book. All of these authors reflect on the benefits of an encouraging, and open-minded learning space. In regards to my teaching experiences in the past eight months, I can only hope that I have begun to tap into that integrity of the subjects I have taught. On a conscious level, I had strived to view my students with integrity through the good and bad days. I see that as being the initial step to fostering the type of classroom that Jardine, LaGrange & Everest (1998) and Palmer (1998) talk about.
I have had the opportunity to reflect on my strengths and weakness as a teacher and a learner. I think that I need to learn how to incorporate a multitude of learning styles and interests into a lesson. This had been another challenge that I was faced with during my teaching time. As a teacher, I found myself wanting to work with everyone and help them - especially when those tears started to roll down. The most interesting thing is that I also shed tears during this experience. This has made me think that it is okay to see your students and yourself struggle through a lesson or activity, that this is a natural part of learning. At the same time, as a teacher, I need to offer support and guidance for my students so that they may strive for success and not begin to foster negative feelings towards a particular subject. Ayers (1993) speaks about creating an environment for learning in which both the students and the teacher may be honored and seen as whole people. In chapter three of his book, To Teach, Ayers (1993) includes a passage from an exemplary teacher, who states that, "Teaching, if it is to be done well, must be built on vision and commitment; learning, if it is to be meaningful, depends on imagination, risk-taking, intention, and invention. Stripped of these elements, teaching is mechanical and sterile and learning is the stuff of pigeons pecking for food or mice running a maze" (Ayers, 1993, p. 65). During the past couple of weeks, I have been fortunate enough to experience this from both the teacher and learner perspective. As a teacher, I needed to have a goal and a vision in mind for my unit and the students. As a learner, I had to be willing to take risks and put my fears and identity on the line. Without imagination, I fear that my math lesson would have consisted of numbers and rote work.
On a personal level, I have been able to reflect on my experiences and learn so much more about who I am becoming as a teacher, and how much I have grown as an individual. Having the opportunity to initiate a unit of study and follow it up to the end has provided me with a sense of ownership over my learning - an important element for anyone to have. Through my teaching experience I have formed special bonds with the students and fellow peers. I have also been able to develop my professional identity within a classroom setting - which is very closely linked to my personal identity. I believe Palmer (1998) is right: "Teaching, like any truly human activity, emerges from one's inwardness, for better or worse teaching holds a mirror to the soul" (p. 2)
Through the process of planning and teaching I have gained a deeper understanding and awareness of my role in the teaching profession. There is a challenge to be found in every day I teach, a talent to be discovered, and a moment to cherish that which touches my soul and reinforces my devotion. Reflection seems to be the key to personal and professional growth: reflect on your lessons, your students, and yourself and your experience will guide you the rest of the way. All of this has been a treasured learning experience that will be looked upon years to come. It is now a part of me.
Learning is the only thing that which the mind can never be tortured by, never fear or distrust,
and never dream of regretting.Parker Palmer (1998).
Ayers, W. (1993). To Teach: The journey of a teacher. New York: Teachers College Press.
Jardine, D. W., LaGrange, A., & Everest, B. (1998). "In these shoes is the silent call of the earth": Meditations on curriculum integration, conceptual violence, and the ecologies of community and place. Canadian Journal of Education, 23 (2), 121-130.
Palmer, P. (1998). The Courage To Teach. San Francisco, California: Jossey-Bass Inc.
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