Monday, September 20, 2010

Mathematics Identity: Stepping into Teaching

The Culture of My School























Driving to my internship site each day, my thoughts often wander back to the elementary school I attended as a child, and I find myself wondering  how much better my experience as a student there would’ve been had its culture matched my belief system as well as that of the school where I now intern! It’s hard for me not to fall in love with my internship school, an elementary school that considers students’ development of a sense of responsibility to society and to the planet to be as important as their acquisition of discrete academic subject knowledge. Each day, upon being welcomed to the school by the cheerful, purring cat who lives in the Office, passing by any of the rabbits and lizards that inhabit the school’s halls, or discovering yet another butterfly garden tucked into a secluded corner of the campus, I feel very grateful to find myself in such a positive learning/teaching environment!  The presence of so many animals and the respect that is constantly modeled for the natural world helps make this elementary school truly a one-of-a-kind AISD educational community.


The door to our resource room is always open!
In keeping with my internship school’s regard for the development of student responsibility and independence, most teachers I’ve observed there appear, consciously or unconsciously, to have a Progressivist philosophy of education. Generally, this is supported by their teaching behaviors, which tend to reflect a Constructivist psychological orientation toward learning with an emphasis on experiential learning, discovery, and (often) cooperative learning supported by appropriate teacher modeling and scaffolding.  Much like Van de Walle describes, education at my placement school, inspired by Constructivist learning theory, appears to be especially effective, not only at getting students to learn concepts and acquire skills, but also at helping them to learn actively and independently and to genuinely value those concepts and skills.  Someday, I hope to incorporate many of these Constructivist principles into my own teaching; however, as a future special educator, I am also very much aware of the need to make sure, in the midst of discovery and self-directed learning, that the subject matter/TEKS- mandated education of all students, including students with disabilities, is simultaneously and thoroughly addressed.  As I come ever closer to having a classroom of my own (my own students!!!  my own classroom!!!), both an enticing and a nerve-wracking prospect, I wonder what kinds of modifications to my instructional approaches I’ll need to make in order to properly reconcile my appreciation for Constructivist learning with my pragmatic desire to help students, some of whom will have severe learning disabilities.



















The Math Routine of My School

Over the past two weeks of my internship in the resource room, most of my cooperating teacher’s instruction in math comes in the form of what she refers to as push in.”  This “push in” system involves my cooperating teacher going into a general education classroom to work directly with one or two students while the general education teacher instructs the class.  The intent of this “push in” system is to appropriately serve the student(s) with disabilities within the least restrictive environment in a way that effectively promotes inclusion.  On the days I am at the school for my internship, my cooperating teacher “pushes in” in a third-grade, general-education classroom, assisting a student with vocabulary, formulas, and note taking during his math lesson.  Given that these past two weeks have been the second and third weeks of class at my placement, the general education teacher has primarily been focusing on assessing and developing her students’ number sense.  She had them “go on a scavenger hunt” in newspapers within the classroom itself to find different types and uses of numbers, and she introduced them to the topic of expanded notation. 

The other “resource room teacher” in my cooperating teacher’s classroom focuses more on math instruction throughout the day than does my cooperating teacher, whose focus is primarily reading.  While observing and assisting this other resource room teacher during some of her math lessons, I’ve seen her work primarily with small groups of two to three students on basic addition and subtraction problems.  One particular day, I helped during her lesson by playing a math game with one of her students.  The game involved us rolling dice, then adding the numbers represented on each die and moving a marker that number of spaces on the game board.  As an example of a guided practice activity during the lesson, I think this game was effective, not only because it provided the student the opportunity to practice his addition skills, but also, because he seemed to genuinely enjoy such practice when it appeared in game-form.  Additionally, I appreciated that the game’s format afforded me the opportunity to immediately correct any addition mistakes that the student made in order to prevent him from learning the addition rules incorrectly and, at the same time, to immediately reinforce good problem-solving strategies that I saw him use, e.g., referring to a number line and employing the “counting on” technique.  As the weeks progress with my internship, I hope to spend more time working with this resource room teacher to see how she teaches other math facts and concepts and to pick up valuable insights about how to make the teaching of math both fun and beneficial for learners with special needs. 



My own math lessons will be taught in a fourth-grade, general-education classroom, where I will be given a small group of students with special needs with whom to work. My role will be to reinforce what is being taught by the classroom teacher and to provide remediation (including re-teaching) when necessary.

Anyone care for another math manipulative or game????












My Teacher’s Philosophy of Math Instruction

The symbiotic relationship that exists between my CT’s instructional philosophy and my school’s support of hands-on learning and self-discovery leads to a classroom learning environment where, to every extent possible, students share responsibility with their teacher regarding their progress toward their educational goals.  In the past, some of my observations and work in Progressivist classrooms, where Constructivist ideologies dictated most of the students’ learning experiences, led me to the belief that for students with disabilities, such self-created and guided learning is less beneficial than more structured and teacher-centered Essentialist and Behaviorist learning environments.  After all, when a teacher in a Constructivist classroom fails to take into consideration the level of background knowledge/prior learning and experience that many students, with or without disabilities, need in order to be successful and later construct knowledge for themselves, the potential for meaningful student learning can actually diminish despite the teacher’s Progressivist ideals.  In my classroom this semester, however, my teacher not only takes into consideration these necessary prerequisites to learning, but she also ensures that they are present by mixing direct and explicit instruction with student selected learning and discovery-based activities. 

On one particular occasion, after my teacher had taught a short “one-on-one” lesson to a student, she invited him to select the type of activity he wanted to do involving the math operations he’d just learned.  As the student and I played the math board game he’d chosen, I saw how integrating a student’s interests into a lesson and empowering him to do some decision making can make the lesson much more enjoyable, and possibly more valuable.  Being placed in a resource room for a portion of the day is a decision many, if not most, students had absolutely no part in making.  Consequently, allowing students the opportunity to decide how they want to practice the skills they’re taught in the resource room, in addition to making the learning itself more fun, can help increase their sense that they do have some control over their own educational life.  The student with whom I played the math board game did not ask repeatedly when it might be time to leave the resource room to rejoin his general education peers; rather, he enthusiastically joined me in playing the game several times and, therefore, was exposed to a wide array of math concepts and engaged repeatedly in meaningful and motivating “fact family” practice.  Integrating activities such as this one into the resource room learning environment seems to help make required “time in the resource room” seem less like a punishment and more an opportunity to have fun learning in small groups.  In my own future classroom, whether I’m working with students in a whole-group, small-group, or one-on-one inclusion setting, I hope that I can regularly make the time to discover and incorporate my students’ interests and decisions, especially after I have seen how valuable such a practice can be from an cognitive as well as an affective perspective.  

Emotional Assessment Poster for Daily Lessons:
We try to avoid getting to those "Level 5's!"




My Developing “Teacher Identity” and Accompanying Worries

It’s a fine line we walk as Special Education teachers, especially when our focus is on working with students in more inclusion-based placements, such as resource rooms.  As an intern this semester in such a placement, I’ve already become accustomed to the somber and sometimes even sullen attitude evinced by upper elementary students as they reluctantly make their way to the “Special Education Room.”  It doesn’t seem to help much when we go to them--they can’t seem to help seeing us, their special education teachers, as an indelible and painfully conspicuous sign of their “differentness” from their classmates while we loom behind them--albeit armed with good intentions--in their general education classroom.  No amount of my CT’s or my smiles, conviviality, humor, or even our liberal distribution of stickers seems to be able to negate the sense of shame that many of these students feel regarding their status as “resource room kids.” Our very presence in the general education classroom, I’ve already begun to sense--more frequently than I like to admit--seems to make the inclusion-based model of special education delivery, when applied in real world classrooms, more a vehicle for making children feel excluded than included!  I can already see that one of the greatest challenges I will face as a resource room teacher will be to find ways to minimize or eliminate this harmful situation.   


When I entered the Special Education field, I went in with a deeply felt desire to help improve the academic and social lives of children with disabilities.  As I look forward to the future, I believe that my desire to help children in need, extensive coursework, and knowledgeable professors have prepared me to effectively improve my students’ academic lives. Already this semester, I’ve been able to feel that peculiarly gratifying rush of pride and satisfaction that comes from helping some of my students learn math concepts and reading readiness skills.  Yet, when it comes to helping my students feel socially included and genuinely accepted in the classroom by their peers, I feel frustrated; few textbooks or admonitions from experts seem to offer workable solutions to his problem. 

Ultimately, my “resource room setting” experiences so far have resulted in my suspicion that true inclusion of students is only possible if the standard model of service delivery, where special education is linked to a specific place, person, or strategy, is somehow amended.  I’m leaning towards the belief that increased cooperation and collaboration between general education and special education teachers must become a priority, with responsibility for service delivery and remediation shared by both educators.  How, I find myself worrying, can I acquire the knowledge and skills needed to facilitate this type of cooperation?  And how, as a new teacher in a new school, can I help?  As the “new kid on the block,” do I dare attempt to help my general education colleagues cultivate the beliefs, willingness, and determination to implement the changes in the status quo which may be vital to establishing a more cooperative and genuinely inclusionary environment?  When the shift in focus becomes meeting students’ special learning needs concurrently in both the least restrictive environment and in the least obtrusive manner, it would seem that my job as a special educator would be transformed.  My job would become as much about helping general education teachers implement the instructional strategies that I’ve learned constitute effective practice as about delivering instruction myself.  Looking ahead to my teaching career, a career that seems more complex and demanding with each passing day, I’m confronted with the worry that learning how to effectively teach students with special needs, will not, by itself, be sufficient to give my students the greatest opportunity for success.   To make that happen, I will have to become both a good teacher and a highly skilled instructional leader. 

















When lessons get hard, my Kleenex is ready! :)


Thursday, September 2, 2010

Response to the Reading


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 problem?

By using a problem-solving approach, you are encouraging students to make connections between math concepts that they ordinarily might not see.  In the chapter, I believe that one of the student samples involved using addition in place of multiplication operations.  When a teacher explicitly tells a student how to think about and look at a problem, the connection between different operations might never be made, and the depth of understanding that such thought and experimentation when problem solving might result in students whose understanding of the concepts is not as well-formed.

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 hope to make my students feel confident in their ability to solve math problems, encouraging them to view their experimentation and incorrect responses as a part of the learning process, and more importantly, as qualities of good problem solving.  I also plan to hold high expectations for my students in order to supply the necessary instruction to ensure that they can meet those expectations with success.

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 agree, that when implemented in name only, the constructivist approach to math education can be a recipe for kids to not really come to any understanding of the basic concepts around which math is based.  For example, subscribing to the philosophy that when supplied with time and materials, students will automatically learn the important math concepts, is not a guarantee that such learning will actually occur.  Rather, in order to be successful when learning in a constructivist math classroom, students need scaffolding and supports.  Additionally, I'm not sure that a teacher who subscribes to a constructivist teaching philosophy really has to completely ignore quality explanations.  The difference in this philosophy lies in from whom those quality explanations come.  Having students draw their own conclusions and make their own connections perhaps is an even stronger method of exposing students to the "basic facts and ideas" of math.

4. We sometimes want to jump in and help struggling students by saying things like, "It's easy!  Let me help you!"  Is this a good idea?   What is a better way of helping a student who is having difficulty solving a problem?

Perhaps the most important reason why providing such "help" is not a good idea lies in the message that such actions communicate to the student.  Requiring a teacher to constantly "come to the rescue" when solving math problems runs the risk of imbuing the child with a sense that they are not competent or skilled in the area.  Additionally, when a teacher solves the problem, who really is doing the learning then?

5. Reflecting on how tasks were defined in the Van de Walle chapters, how did the tasks presented in the Behrand article to LD students help in their mathematical development?  Please give specific examples.

This approach helped students who did not feel comfortable in their own abilities and who did not enjoy the subject area take ownership of their learning and discover previously untapped abilities to critically analyze a problem.  By showing the students that to be at math did not mean that you had to always answer the problem correctly, freed them to truly discover their own abilities.