Math Identity Blog Entry 4 - Invented Math Strategies

In the problem-solving interview, my student N used a mix of counting strategies and direct modeling strategies to solve the problems. I particularly remember her strategy for the measurement division with a reminder problem that reads, “Abby has 23 little toys. If she puts 10 toys into each bag, how many bags of toys will she be able to make?” N used direct modeling and began by counting out 23 unifix cubes and connecting them together. She continued by counting out 10 cubes and breaking the chunk off. N then checked to make sure she had counted out 10 correctly as she counted each cube of the chunk one-by-one. That group represented her first bag and she counted out 10 more cubes to serve as her second bag. Eventually, she determined that Abby still had 3 toys leftover, which could not be put in a bag on their own.

    N could also have solved this problem using direct modeling with pencil and paper. She could easily have drawn 23 squares or shapes to represent toys. N would then have counted 10 “toys” and drawn a circle around the items, grouping them into a bag and then another bag.

Student N may have also been able to use a counting strategy. Able to count by tens, N could have simply stated, “10, 20,” drawing a bag on her paper at each counting interval and then counting “21, 22, 23” as she drew individual toys remaining. Yet another way to use counting by tens would have been to provide N with a multiplication chart and counters. If she placed a counter at the end of each complete row, N would know that only 2 complete rows, or 2 “bags,” were filled and 3 leftovers remained. Regardless of the particular strategy, the student would still be able to reach the same answer. N really enjoyed visually representing the problems with manipulatives and her explanations were excellent, with added detail to tell a true story in her explanation of the problem. I think it’s wonderful to see the variety of strategies students use and I look forward to learning new ones from my students over the years!

Math Identity Blog Entry 3 - Math Talk Moves

I have really enjoyed learning about Chapin’s Five Productive Talk Moves. Although I have unknowingly used these strategies while teaching in past years, I have never really thought about them in depth. Using revoicing, students really have a chance to process their own thinking while the teacher or other individual voices their steps in proper terms. Asking students to restate someone else’s reasoning and to apply their own reasoning requires that students understand a strategy, perhaps one different from their own. Finally, prompting a student for further participation and using wait time allows a student the opportunity to respond on his or her own. It is important that we respect the child’s thought process and encourage the child to share their opinions, proving that we value their skills. While I am aware of these techniques, I still find it somewhat difficult to use them in the midst of teaching under time restraints and other pressures. I hope to become more comfortable using these strategies over the semester and into future years of teaching.

I did not teach or observe math this week, as I was in a Data Analysis Meeting for the 5th grade team on Tuesday and the 4th grade team on Wednesday. Last week, I remember observing a lesson on operations and key words that provide hints to which operation should be used in word problems. I noticed my teacher is excellent at providing sufficient wait time with students. She will call on students but if they are hesitant to answer, she will tell the student to think about his answer, as she probes other students. If the student is still stuck, he may ask for the help of a friend. At times, the silence is almost uncomfortable. The students realize, however, that their teacher is expecting an answer and that she is willing to give them ample time to respond. I will certainly be looking to my teacher as a model for this “talk move.” One other strategy she demonstrates particularly well is revoicing. Since my students are ELLs, it is critcal that my teacher models appropriate math language and vocabulary (and English language too) to aid their instruction.

If I could redo the lesson, I would also have students restate their friend’s reasoning. I think, in this way, students would be attentive to all problems rather than only the ones that are designated to one student by name. Students could also gain better insights to problem-solving strategies that could potentially be easier or more efficient. I would integrate this talk move into the teaching practice with simple prompts, like, “Student ___ can you tell me in your own words how Student ___ solved the problem?” I find this method would get students more involved in the lesson and foster a positive classroom culture.

*** NO pictures this week because I was in the Data Analysis Meetings mentioned in paragraph 2

Mathematical Identity Blog Entry 2 – Stepping Into Teaching

      As soon as I entered this elementary school, I felt comfortable and safe. I was greeted by the receptionist in the office and noticed the smiling faces of children in the hallway. I was also particularly impressed by the cleanliness and organization of the school. Not one straw wrapper or dropped pencil lined the floors, and students knew the expectation of walking quietly down the hall in a straight line. The walls were neatly covered by bulletin boards, displaying anything from rules and expectations posters to “My Summer Shorts” stories and colorful student artwork. There were even custodian appreciation notes or thank you posters hung around the school, reflecting the school’s emphasis on valuing all members of their community. This school community extends even beyond students, educators, and staff to include the parents and families of the students. A family support services office is located on site to assist families with their financial, health, and safety needs. Since a majority of students are English Language Learners and families typically have little to no English knowledge, a translator is also offered for support. Parents and younger siblings can be seen in the office or outside the school chatting with each other or teachers while waiting to drop off or pick up their child. I can only hope that schools I work at in the future share this same positive relationship between all their members. 

Although the students under my teacher’s caseload have yet to receive much math instruction due to testing in various areas, I have been able to see a few lessons too. During the test administration, my teacher always reads the problems aloud for the students. With word problems, this detail is essential to the students’ success in math for the simple reason that they are also struggling in reading – While they are in the fifth grade, their reading levels fall between that of a first grader and that of a second grader. Students seem to be engaged in the material at first but several quickly gaze off and may be offered a break to regain their attention. The math lesson I observed was on identifying the four basic operations by their symbol in an algebraic expression, the word expression, or by other related words. The teacher explicitly reviewed the table (shown in picture) but made sure to also explain how learning to use certain words (like less than, more than) for particular operations can at times be confusing, since exceptions to the rule exist. I love how my CT encourages students to ask questions and requires participation from all! She builds to independent practice probes by first providing students with opportunities to discuss problems with a partner before sharing with the group. I find the students to enjoy solving problems on white boards best. This activity has a game-like feel and students sense less pressure to solve the answer correctly. As a bilingual class, students may use both Spanish and English as needed to express themselves fully. During assessment and some class work, students receive a math tri-folder as an aid, which includes math tables like the multiplication chart and hundreds chart, a fraction guide, and more. Having yet to teach math lessons of my own, I am currently observing or participating in these math conversations with the students. I look forward to learning more about their current math abilities and determining their areas of need.

As for my teacher, I know that she favors teaching math over the other subjects. While she views all areas of instruction equally important, math has always come easier for her to teach. She thinks the variety of manipulatives that are available and the ability to go between numerical equations and word problems provide students with many accessible ways to learning the material. I would have to agree with her on this because I too love math and have found it easier than teaching reading or writing. With tactile manipulatives at hand one is able to both show and tell students how a problem might be solved. In this way, you can individualize or accommodate lessons for students as needed. For example, my teacher has already created those math folders that students are able to use during general education instruction. Students are not embarrassed by these study guides and benefit from using them. Nevertheless, I have already experienced some of my own difficulties in teaching math when working with students at lower levels. I mentioned in my earlier blog that my greatest struggle was while working with a life skills student who showed little number sense. Without the ability to even recognize numbers or answer one-to-one correspondence questions, the student showed little progress, which both tested my patience with repetition and often led me feeling unsuccessful as her teacher.

    At the moment, I am feeling fairly confident as a mathematical teacher. Since I enjoy the subject, I hope to share that excitement with my students and to point out the connection of math in real-life experiences to make learning worthwhile. I am worried that for students like the one I had in my life skills placement, I will be unprepared or discouraged to finding alternative teaching strategies. I know that in those situations, I will most likely turn to other peers or teachers for advice on what has worked for them in the past. In my current placement, I am concerned with the minimal amount of time I have with each student. We are expected to get through our entire lesson, which leaves me little time to work with a single student who may be struggling more than the others. I am sure I will be watching my CT as she models appropriate ways to modify lessons for a range of learners. I also hope to collaborate with other cohort members for advice.

Week 2 - Reading Responses

1. How does taking a problem-solving approach to teaching math differ from first teaching children the skills they need to solve problems and then showing children how to use those skills to solve problems?
   Taking a problem-solving approach differs from simply teaching children the skills they need to solve problems in a few ways. Problem-solving must begin where the students are and does not rely on a set of memorized rules or procedures. There is no "correct" solution, but students still are required to justify their answers. (p. 33)

2. How do you think your experiences, feelings, and beliefs about math will impact the kind of teacher of math that you will be or the kind of teacher of math that you want to be?
    I believe that my math experiences will all positively contribute to my attitude and ability to teach math. I have always enjoyed math as a subject and love problem-solving to determine answers. I find math to be like a game where everyone can use different strategies to win. This fun math mentality will hopefully also be reflected in my teaching style.

3. Not everyone believes in the constructivist-oriented approach to teaching mathematics. Some of their reasons include the following: There is not enough time to let kids discover everything. Basic facts and ideas are better taught through quality explanations. Students should not have to "reinvent the wheel." How would you respond to these arguments?
    I disagree. While some methods may prove to work best or be quickest for a majority of students, it is important to see our student's as individuals. People have different learning styles and different interests that should be utilized to engage them into the math material. It shouldn't matter too much how a student comes up with an answer, as long as he or she is still accurate and can attempt to explain their thought process.

4. We sometimes want to jump in and help strugglng students by saying things like, "It's easy! Let me help you!" Is this good idea? What is a better way of helping a student who is having difficulty solving a problem?
    NO. It is not necessarily to always jump in and provide a student with help. Students should be given adequate time and perhaps tools to help them solve problems on their own before seeking out help from a peer and then teacher. One idea might be to provide a student with manipulatives, making it easier for him/her to solve the problem with a visual.

5. Reflecting on how tasks were defined in the Van de Walle chapters, how did the tasks presented in the Behrand article to Learning-Disabled students help in their mathematical development? Please give specific examples.
    Students were given the opportunity to come up with answers on their own prior to teaching. The teacher could then compile their means of solving the problem to discuss how one problem can be solved multiple ways. It's not a competition.

Mathematical Identity Blog Entry 1 – Math Life Story

What is math? I would imagine a dictionary-definition of the word might include the words measurement, numbers, symbols, or perhaps relationships. Growing up, however, I also associated the words “fun” and “exciting” to my mathematics experiences. I had always loved completing “Mad Minutes” in the elementary grades or playing “Around the World” using multiplication facts, primarily because I was good at these games. I could and often did win. Nevertheless, the highest point of my math experiences did not come until the seventh grade. Pre-Algebra began, and for the first time, I was manipulating variables and solving equations. I loved how many of the problems could be solved in different ways to reach the same answer and how a process of back solving could be used to check for understanding and accuracy. When I finished my class work early, my teacher would allow me to help other students in the class with their own work and I truly enjoyed sharing my talents with my peers. With this new responsibility, I felt empowered as a student, someone who was self-confident and a leader in the room.
    I excelled in math through high school and choose calculus my senior year. Calculus homework was beginning to take up more of my time than math assignments had in past years. As a senior, I did not wish to compromise my involvement in extracurricular activities to allow for more homework time, so I continued as best as I could. I realized I was beginning to struggle as I no longer made only As and asked for some help from my teacher. Conveniently, this decline came around the same time as the peak in soccer season. I was co-captain and it was my senior season. Although I had recently fractured my ankle and would be sitting on the sidelines until after the winter holidays, I choose to attend our away tournament in Dallas without hesitation. With this decision, I would be missing two days of calculus test review, but skipping out on my teammates despite my injury was never an option in my mind. It was a tournament to remember until I received my calculus test grade of a C+, the first C I had ever received. I was overwhelmed and flustered when my calculus teacher confronted me and directly asked if I held regret toward my decision to join the soccer team when I couldn’t even play. She proceeded to question my abilities as a student leader and member of the honor society and student council, which infuriated me as I felt that was irrelevant to my performance in calculus. I did eventually pick my grade back up but it was nearly impossible to view my teacher with the esteem I once had. I promised myself that I would never do the same to a student or friend. Soccer was the thing I looked forward to the most and this teacher was trying to take that away from me while putting greater stress and blame in my hands. From this experience, I also constantly remind myself to have patience with my students and to really focus on what their own needs or goals are, capitalizing on their strengths to improve their weaknesses.
    As for a turning point, I believe I experienced a turn in events my freshman year in college here at the University of Texas. Taking an elementary mathematics course, I was forced to look beyond determining an answer but assessing the means of getting to that answer. I began to understand the “why” behind mathematics and how it is that different cultures may solve math problems in a variety of ways. While I found this class to be frustrating at times, I really did enjoy the material. I look forward to studying mathematics and sharing numerous types of math with my students.
    Several other scenes stand out in my mind over the course of my mathematics experiences. First, I remember being in pre-school and beginning to count money. I was intrigued by the different coins and understanding that various sizes and colors were worth different amounts. My favorite game at the time became playing grocery store, where I was always the designated cashier. So much fun! I also particularly enjoyed working with the preschool students in my PPCD placement. These students appeared to enjoy math because I taught it in the format of games. We would laugh and play during instructional teaching.
    I consider the greatest challenge in math to be my experience with one student in particular just last semester.  This kindergartener was in a life skills placement and had very little number sense. She could not identify numbers and was not able to perform simple one-to-one correspondence problems. Although I consider myself a very patient person, there were times when I felt like giving up but knew that was not an option. I value my time spent with this student, for I realized that math or numbers does not come naturally to everyone and that I must often think creatively to facilitate for unique styles of learning.
    Why do I want to be a special education teacher? I want to be a special education teacher because I feel I have been given a talent of working with young children who require just a little extra patience. I believe children deserve quality education and are deserving of caring role models who seek to help them reach their full potential as citizens of a global community. To do so, I feel that this education should be appropriate and accommodated to their unique needs and learning style. By providing students with a number of opportunities to succeed, I will do my best to ensure a positive, learning environment for each child that I meet.