Student-Invented Strategies: When the Right Answer Isn't Right Enough

In observing students engaged in mathematical thinking, both during the math interview and during the math lessons I teach at my school, I have been able to see students applying different strategies based on their developmental stages and past experiences in math.  Despite differences in age and grade level, many of the students with whom I have worked tend to show a clear affinity for solving problems using the direct modeling approach, an approach that Van de Walle describes in his book, Elementary and Middle School Mathematics, as “the use of manipulatives or drawings along with counting to represent directly the meaning of an operation or story problem” (p. 214).  For some of the students I teach, the direct modeling approach is the only strategy they have available when solving problems.  However, others among my students have been encouraged, throughout their mathematical development, to experiment with different counting strategies, and, as a consequence, they have come to see that math is not solely defined by a set of algorithms; they understand that solving problems is not restricted to formulas or drawings, but can actually be a process of invention.  The ability to use student-invented strategies, which is described by Van de Walle as “any strategy other than the traditional algorithm or that does not involve the use of physical materials or counting by ones” (p. 215) represents a more complex approach to problem-solving that requires a deeper level of understanding about mathematics and number sense.  

Using the Math Talk Moves during my math lessons has caused me to more closely analyze my students’ thinking as they solve problems, an analysis which draws my attention to their different levels of understanding and their familiarity with different problem-solving approaches.  By using Talk Moves that require all of the students to listen to and actively think about the approaches used by their peers, I have observed that the students look forward to the opportunity to play the role of “teacher” during the lessons, scaffolding for their peers and exposing each other to different modes mathematical thinking.  Their own explanations of their thinking (as they approach each problem) by my students who are using invented problem-solving strategies,  gives my other students the chance to see the potential that exists when traditional algorithms and direct modeling are forgone in favor of what can be more efficient problem-solving strategies.  In the future, perhaps such modeling will encourage other students to look at problems from this inventive perspective as well. 

Last week, I had the opportunity to observe two of my students using an invented strategy as I introduced them to the concept of subtracting with regrouping.  When the first student, whom I will call Zoe, was presented with the problem 23-19, her first step, after borrowing* 10 from the 20, which gave her 13 in the ones place, was to set up the addition problem, 9+_____=13. 

For Zoe, thinking about the problem through the “lens” of addition, seemed more natural, and this decision enabled her to use her knowledge of the fact that 9+1=10 plus 3 more equals 13, giving her the answer 4.  Using this “Add Ones to Make a Ten, Then Tens and Ones” strategy meant that, instead of counting on her fingers or taking the time to draw representations of each number, Zoe was able to quickly arrive at an answer in her head even though she did not have that particular math fact memorized!   

The second student, whom I will call Erin, initially solved the problem by simply using her knowledge of the math fact that 9+4=13.  When I asked her to explain another method that she could have used to get the answer, Erin responded, “Well, I know that 10+3=13.  Ten is 1 more than 9.  That means that 4 must be added to 9 in order to get 13.”  Because Erin set the problem up in a way that made the subtraction problem into an addition problem, the strategy that she finally used when explaining her thinking to me most closely resembles that of the “Use a Nice Number and Compensate” strategy listed in the table of “Invented Strategies for Addition with Two-Digit Numbers” in Van de Walle’s book.  Erin’s ability to see and then correctly implement different processes for solving individual problems suggests to me that her number sense and understanding of the addition and subtraction processes are quite developed and, perhaps, indicative of  her readiness to move on to the more complex multiplication and division procedures.

As I consider the problem, 23-19, which these two students solved, I noted that both opted to break the problem into two smaller problems based on the ones and tens places (a decision probably influenced by the way I had modeled the process of subtracting two-digit numbers earlier in the lesson).   Had they chosen to look at the problem as a whole, one way that they could have solved it using an invented strategy would have been to use the “Add Ones to Make a Ten, Then Tens and Ones” procedure.  With this procedure, they could have added 1 to the 19, giving them 20.  They then could have subtracted 20 from the 23, giving them 3, and making the answer 3+1, which is 4. 

Although both of the students implemented “counting up” strategies, a “counting down” invented strategy they could have used that would have given them the correct answer is the “Take Tens from the Tens, Then Subtract Ones” strategy.  With this strategy, they would have first taken 19 away from 20, giving them 1.  They then would have added 3 to the 1, giving them 4 as the answer. 

Reflecting upon what I was able to learn about my students’ math understanding based on their different problem solving approaches, I can really see the benefit of the Talk Moves strategies!  Not only did using Chapin’s Talk Moves make it possible for the students to hear descriptions of different strategies that might influence their own problem-solving in the future, but also, getting to hear vocalizing of the students’ thought-processes when solving the problems, made me see and appreciate the complex thinking that my students are capable of employing on a regular basis!  What once might have been a lesson with a quick check at the end to determine the correctness of students’ answer on a particular problem, became a lesson where I was able to assess, not only their correctness, but also, and more importantly, their degree of understanding and the quality of the cognitive processes they employ.  With the help of the Talk Moves and the knowledge I have gained through the readings on Cognitively Guided Instruction, I’m able to see that what really matters in the math classroom is less about the answers students end up with, and more about the cognitive processes and approaches that the students employ to get to those answers.  Encouraging students to think about math problems in ways that go beyond the commonly seen “plug and chug” methods of typical algorithms, not only creates an environment where student invented strategies are more likely to be seen, but also, as I’ve learned from my own observations, creates an environment where students feel more empowered and valued when they are engaged in the problem-solving process.

*Though Van de Walle encourages the use of the term, "trading,” rather than “borrowing,” the term borrowing is used in the students’ classroom, so I have adopted this term in my lessons as well so as not to confuse the students.

Math Talk Moves: Five Steps to Student Success!

Having interned in elementary schools for the past three semesters, I’ve seen--all too often—how very reluctant and unhappy some students are when the time of day arrives when they must leave their general education peers to receive “special help” in the resource room.

Compounding the already prevalent social stigma associated with requiring services in the resource room is the factiuyuyt that the students leaving their general education classrooms may be leaving behind much more interactive and enjoyable learning environments.  In the hands of a teacher who doesn’t recognize that his/her responsibility in a resource room setting should be to provide instruction that meets the individual student’s interests as well as his/her instructional needs, learning, for students with learning disabilities, can be a painful experience.    

This, however, is far from the case in the math lessons I’ve observed in the resource room at my placement school!  The supportive, collaborative atmosphere that the math resource teacher I observe has created makes it hard for students not to feel empowered and respected regardless of the difficulty of the tasks which they’re presented. During most of the lessons this resource room teacher implements, the small student-to-teacher ratio (usually ranging from one-on-one instruction to teaching groups of three students) allows instruction to be highly individualized. Consequently, throughout the lessons, each student is able to enjoy the opportunity to work directly with the teacher when solving some problems, to share his/her thinking trajectories, and to answer probing questions designed to elicit deeper reflection about the processes in which he/she is engaged.  It is during this time, that Chapin’s “Five Productive Talk Moves” often become the focus of instruction.

While reading about the “Talk Moves,” I could imagine their value in the classroom, but, really, it wasn’t until I saw this teacher actually using them during instructional time that I really understood just how empowering these questions can be to students.  During a particular lesson, I observed the teacher working one-on-one with a student who for the past few months has experienced a great amount of difficulty with the concept of a “fact family.”  According to the teacher, while the student is able to solve the math problems using manipulatives, the relationship between these numbers seems transitory to him, and, as such, while he night sometimes say four plus three equals seven, a few moments later he might give a completely inappropriate numerical response to seven minus four—almost as if he has lost track of the fact family.  In an effort to communicate the permanence of the relationships between numbers in a “fact family,” I observed the teacher first relating the three numbers that make up a “fact family” to the three members of the student’s own family. After creating a link between the number four and the student’s mother, the number three and the student’s father, and finally, the number seven and the student himself, the teacher explained to the student “You wouldn’t have some stranger come, join your family, and kick another member of your family out, would you?  Well, the same is true with these ‘fact families.’  Four and three and seven are always going to be in a ‘fact family’ together.”  She then followed this explanation with an example of the problem illustrated with manipulatives actually shaped like people—a nice touch, I thought!

With manipulatives in front of him and the answer in his mind, the teacher then introduced the first “Talk Move” in the lesson.  Asking the student what another formula in the “fact family” looked like, the teacher asked him to explain what he had been thinking when he initially said that “seven minus four must equal five.”  The student began to justify his incorrect answer when--suddenly understanding-- he turned to his manipulatives instead!  Counting out seven manipulatives, the student then used the “Counting Down” strategy to arrive at the correct answer, three.  The teacher then used another “Talk Move” when she re-voiced his strategy, saying something similar to “So, let me see if I get this right.  You first counted seven manipulatives, and, then, you looked at the problem and saw that you needed to take away four of the manipulatives.  But I didn’t see you take those manipulatives away!  How did you end up getting the answer?”  Re-voicing his strategy in a way that still asked him to explain the procedure he applied to eventually get to the answer made the student more aware of the strategy, perhaps increasing the likelihood that he will use the “Count Down” approach in the future.

 Finally, throughout the lesson, the teacher expertly employed the strategy of “Wait Time.”  As I watched the student process and then progress throughout the lesson, I came to see how critical this “Talk Move” is when working with a student with a learning disability.  So often in general education placements, it’s not unusual to see those students who take longer to answer questions being ignored in classroom discussions and allowed to fall behind in their coursework.  As I watched the teacher today wait, in complete (but comfortable) silence, I was able to witness her student really thinking about the problem, analyzing the possible solutions and offering a well-reasoned answer.

Reflecting upon the lesson I observed today, the apparent effortlessness of the teacher’s implementation of these “Talk Moves” almost obscured my ability to observe them!  While reading about the different questioning practices Chapin suggests, I had pictured a classroom, at times, more like an interrogation room, with the students’ thinking constantly being challenged.  Seeing the comfortable environment that the teacher was able to create and the way she actually used the “Talk Moves,” not to question student thought, but rather, to elicit reflective thinking, has made me anxious to try some of these strategies myself! 

In a classroom where Talk Moves are expertly employed, getting through lessons without understanding the content isn't an option. :)

Of all the strategies I saw this teacher demonstrate, I think the most valuable one to use when teaching students with learning disabilities might also be the most difficult to employ: “Wait Time.”  For students in resource room, as valuable as remediation of academic skills might be, instilling in them confidence in their ability to succeed with subjects they find difficult might be equally, if not more, important.  Despite all the potential benefits of the “Wait Time” strategy, I think that this strategy might be more difficult to implement effectively, at least initially, with students who require a lot of reinforcement and scaffolding.  For a student for whom experiencing silence causes worry and uncertainty, I think that teacher gestures such as nodding your head, smiling , and even using such phrases as “you’re doing a good job, keep going!” might be helpful as he/she tries to answer the question.  Over time, the frequency of these prompts could be reduced and the duration of  “Wait Time” could be increased. 

Without Wait Time, just being able to give an answer becomes a challenge!

Two other “Talk Moves” that I want to employ in my own lessons are “Re-voicing” and asking students to restate someone else’s reasoning.  As I watched the lesson today, I really liked how the teacher’s use of “Re-voicing” enabled the student to give voice to his own procedures and thought process, enabling him to critically consider the veracity of the thoughts which shaped his actions.  As Chapin suggested in her article, when teaching groups of students, asking students to restate someone else’s reasoning can dfprove to be an effective way to have other students restate the thinking processes of their peers.  For students who might be English Language Learners (ELLs), this technique, where ELL students would repeat what their native English-speaking peers stated, would provide both practice in considering other math thought processes and also valuable practice speaking English. 

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! :)

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.

My Math Life Story: A Story of Peaks and Pits

Part 1: Peak

 A brain like Fermi’s certainly wasn’t needed to estimate that, this first day of Plan II Math, in our classroom filled with forty Liberal Arts majors, the majority of hearts present beat more rapidly than usual.  When the bell rang in the dark UTC, there was no question for whom it was tolling:  freshman-semester Plan II majors about to face the inescapable--fearsome Plan II Math!  Most of us had dreaded this class; its presence on my own spring course schedule had cast an ominous shadow over my entire winter vacation.  My mind plagued me that whole break with images of math instruction befitting a medieval dungeon more than any classroom I’d heretofore experienced.  I worried that the humanity, creativity, and familiar ambiguity of words and ideas, which I’d come to value so much in my lit, philosophy, and history courses, were about to be replaced with the stern, black-and-white world of numbers, where formulas and rote memorization would rule.  Maybe these concerns were inevitable; after all, most of my past math experiences had been exactly that, all dry formulas and tedious memorization, my elementary school years spent solving thousands of timed Kumon problems, and my middle and high school years spent plugging numbers into formulas as practice for SAT and AP tests.

The big, fat, spiral notebook I opened as the instructor entered was a testament to exactly what I thought math was and what Plan II Math would be. I’d resigned myself to the onslaught of formulas that I thought would ensue, but which would never be explained, and to timed math tests that would be taken, and then, grades received, shoved into the same notebook, only serving to exacerbate the fear already so palpable in the classroom.  My consternation, however, was rapidly replaced with interest when my professor began to explain the purpose of the course and showed us the required texts.  Quickly, I realized that this class might require a reconsideration of my views regarding the study of math!  I’d expected to spend class time scrambling to write down every formula presented by the professor and every free moment after class memorizing and then applying these formulas. The professor, instead, engaged my classmates and me in a lively discussion about what math is, what numbers are, and what it means when we say we “solved” a problem, a discussion thread which was to dramatically embroider the entire semester. 

As the semester unfolded, my appreciation for mathematics continued to increase.  I’d never been in a math class where “solving a problem” first meant really thinking about it and considering all the novel ways that might exist to solve it.  Instead of just telling us to copy his particular methods for working out problems, this professor showed a genuine interest in the different ways each individual student in the class saw the math problems and how each arrived at solutions to them. Even when the solutions themselves were incorrect, the professor validated our contributions by allocating attention, not just to the answers to which we’d arrived, but also, to the thought processes we’d used.  Doing my homework each night, I began to feel a new sense of ownership for my solutions to math problems, and, when asked to explain how I arrived at my answers in class, the explanations actually came easily because the solutions were truly my own. 

Plan II Math helped make us, forty Liberal Arts majors, feel like a class of forty mathematicians!  This math class appealed to ur interests and strengths; for example, the professor connected math to the historical study of famous math equations’ origins, and he encouraged us to use the creative thinking we’d previously only employed in our humanities coursework.  A class that initially seemed so unfamiliar, frightening, and one-dimensional to me became a class I actually enjoyed, and its subject became an area in which I actually felt competent!  I learned a lot about math in my Plan II Math course, but perhaps the most important thing the course gave me was a new appreciation for what the essence of math really is and a surprising sense of accomplishment and interest in a field that previously had seemed so separate who I am.

Part 2: Pit

Thinking back to the peak and pit experiences in my “Math Life,” it’s strangely appropriate, that I entered both experiences with similar feelings of dread, and left both transformed, but in two very different ways.  During the two years after taking Plan II Math, I took biology, education, and literature classes, always aware that each class I completed and every semester that passed brought me that much closer to being forced to enroll in the requisite and infamous Plan II physics course. This was a science course by definition, but, by implementation, every bit a math course.  While Plan II Math had concerned me before I actually experienced it, this upcoming physics course was the stuff of nightmares, and I had yet to meet a senior who didn’t flinch when recounting his/her physics experiences to me and my classmates.  Still,  choosing to interpret my unfounded worry about and ultimate enjoyment of Plan II Math as a sign that perhaps my fears would prove unjustified yet again, I entered my physics class harboring some, albeit small, hope that everything would be ok.

That hope died around midnight after the first day of class.  Hunched over my physics homework, analyzing a page of math problems that remained as pristine as when originally handed to me for wont of my knowing what the heck to do with them, I finally accepted that my experience in this course would be quite different from that in Plan II Math.  The class session earlier that day had left me confused and anxious.  While the professor gesticulated wildly and rapidly transitioned from chalkboard to podium, lecturing in a manner more entertaining than informative, my pencil futilely hovered over my naked notebook page, waiting to transcribe anything that might resemble an explanation or formula.  My late night spent with physics homework sadly became a standing date; weeks and then months passed in the classroom without me actually being taught or effectively learning the course content.  Panic soon gave way to resignation for me and many of my classmates, all generally good students and diligent workers, but frustrated by trying to teach ourselves the course material, enduring the drudgery of yet another impossible weekly homework assignment, and suffering the horror of another stress-filled exam. 

By the end of the semester, I can’t say that my understanding of physics had grown significantly, but my feelings for the subject certainly had been colored by such a negative experience.  Employing what appeared to be a teaching philosophy at odds with everything my education courses have taught me makes someone a good teacher, the physics instructor laid all instructional responsibility upon his students. He, in effect, set us up for failure by designing a course that was inappropriate give the lack of necessary prior learning on the part of the students and by withholding clear explanations and instructional support throughout the course.  Rather than inspiring interest in the subject matter, the “fear of failure” and frustration that we continuously experienced during the semester translated for many students into general antipathy towards and a feeling of incompetence in physics, despite the high grades many ultimately received.  Before being given a scalpel, white coat, or degree, doctors must pledge to “do no harm.”  Thinking back to the harm done in my physics class, perhaps such a pledge might be warranted as well for others pursuing the title of “doctor,” including Ph.D.’s! 

Part 3: Turnabout

Some things are just difficult to teach.  For example, how do you explain to a preschooler the correct method for tying a shoe?  When the “bunny ear” trick fails, what keeps us from finding ourselves with a mangle of shoelaces and, ultimately, from making a trip to buy Velcro?  Sometimes it takes being asked to reconsider a concept, reevaluate a belief, or present something I think I know in new light to a new audience, to make me as a teacher realize that my own understanding of what I am teaching may require some reevaluation as well.  I know, because this happened to me last semester in my Life Skills placement, when, for the first time, I found myself initially but wholly at a loss when trying to define for a child with autism what numbers are and what it means when we say “add them.” 

The inherent difficulty in clearly defining and explaining math concepts and terms so that true comprehension and application on the part of the students is probably responsible for much of the “plug-in-and-go” approaches to math that you see in many schools today.  After all, when the formula or simple procedure for finding the correct answer is all that is given, you don’t really need to understand concepts to be successful.  Division of fractions becomes simply a matter of flipping one fraction and multiplying without regard to the underlying process.  

Last semester, I felt for the first time how scary it can be when I had no formula to guide my teaching.  As it became more and more apparent that my words were failing me, I did as I’d been taught in my special education training, and I considered what the concepts and processes I was presenting to my student might look like from his perspective.  In reflecting upon my own experience as a student in my Plan II Math class, I recalled that one way my professor had made his instruction more meaningful to me was by linking that instruction to things that interest me.  With this in mind, I implemented a math instruction approach where my words, or lack thereof, did not matter.  Using manipulatives and drawing on the child’s interest in technology and animated Pixar films, I was able to meaningfully demonstrate over time what the desired concepts and procedures looked like and actually teach him how to successfully add numbers in what was to him a meaningful and enjoyable way. 

I’m sure I’ll look back on the rudimentary instruction I gave that semester and think about all of the interventions and instructional techniques that I could have applied had I known them at the time.  Nevertheless, in considering why I found this event to be a turning point for me in my “math life,” I notice that it’s connected to my new-found awareness that effective math instruction is less about me and my understanding of math concepts and more about how well I am able to translate what I know into terms a diverse population of students can understand.  It is ironic that, in so many cases, math teachers understand perfectly everything they say when they teach a new math concept, while at the same time individual students too often understand none of it.  When this happens to me, I will remind myself not to blame my students or to resort to only formulaic approaches to teaching but, rather, to reconceptualize how I teach.  Finding myself without a formula was necessary before I could really formulate an understanding of the kind of math educator I hope to be.  When words fail me next time, I’ll take the opportunity to consider instruction from my individual student’s perspective.  I’ve learned that instruction which truly takes into consideration individual students’ needs, interests, learning style, developmental level, and degree of prior learning can be more effective and certainly more meaningful than somehow expecting students to change in order to receive inherently inappropriate instruction. 

Part 4: Influences

-Kumon, Or, The Hare Will Always Beat the Tortoise

When it came to math education, my elementary school seemed to subscribe to the same philosophy that many well-meaning mothers do when they tell their children to “pull the band-aid off quickly.”  After all, if pain must come, isn’t it better if it comes in a really fast burst rather than prolonging the agony?  While I’m not sure that there’s any real mitigation of pain with speed, I am sure that math at my school was over quickly.  Using the Kumon method, math “instruction” occupied approximately fifteen to thirty minutes of the school day during which time we sat in silence, furiously answering problem after problem with our trusty pencils before the timer rang.

Such was my introduction to the leveled world of math education, where students quickly and permanently got labeled as being “good at math” or “bad at math” in the classroom.  We all knew at what Kumon level the other students were; we marveled at the first graders who could already multiply and divide, and we felt deep sympathy for the other students who were still working on shapes and basic addition. 

Brought up in this system that reinforced speed over reasoning, it was easy to fall prey to the belief that a person’s skill at math could be largely appraised by the ticking of a clock.  Prizes were distributed to students who did particularly well in the program, correctly completing problem sets at the most advanced levels in the least amount of time.  The remediation for students who struggled was simple: more Kumon. 

There was a certain kind of payoff to being in this program.  The years of repetitive drill and practice meant that we became students who could work math problems very quickly, but our enjoyment of the subject was questionable.  Over time, however, just like aging track stars, our speed and agility faded, and we found ourselves in the unfamiliar ground of middle school and high school math, where thought and reasoning were more important to success than facility with math facts and where calculators supplanted stopwatches as necessary “tools of the trade.”

As I consider the impact upon my classmates and me of our school’s complete endorsement of the Kumon program, it was less the program itself, and more the message that the program espoused that had the longest-lasting and most damaging repercussions.  At a time when a supportive cognitively and affectively appropriate infrastructure would have helped us learn to appreciate the subject and feel confident in our skills in math, the judgmental nature of the program’s leveled framework-- and the message such levels gave us about our own abilities in mathematics-- reinforced at an early age the groupings and labeling that the education system sometimes appears to be all too willing to impose, and students seem all too quick to believe.

-AP Calculus, Or, There Was A Reason for Your Torture

If you had asked any one of my classmates what the bane of his/her seventeen-year-old existence was, no hesitation would have preceded their declaration of, “Calculus!”  Never in my thirteen years at the school had there ever been such a unifying force as our common dislike for first period’s AP Calculus class!  The fear of a pop quiz loomed over the door every day as we filed in, and the unrelenting glare of the overhead from the first day of school signaled that learning was definitely going to occur that day and every day, or else.  We progressed doggedly and systematically through the text book, and I came to see, that yes, it is possible to cover all content in a 400-page textbook when class always begins five minutes before the bell rings and ends five minutes into the next class.

Calculus ensured that even in the midst of applying to colleges and the creeping onset of the oh-so-contagious “Senioritis,” we still were kept aware that we were high-schoolers; as such, we had more important things to worry about, like the AP test in Calculus!  Without opening a practice guide or assigning practice AP problems, our teacher prepared us for the test, not by “teaching to the test,” but rather, by making sure that we understood the course content, and all of it, at that.  As spring and AP testing dates approached, I found myself questioning my readiness in certain content areas.  Had I learned sufficiently what the policies of the 16^th president of the United States were?  Did I remember the units for velocity or the formula for mass?  In the midst of scrounging about for missed information and of playing catch-up for what my other AP classes had let slide, I found myself >sigh of relief< at total peace with my understanding of Calculus. 

The week after the AP tests were over, my Calculus teacher was still hard at work preparing us for any Calculus II classes we might take in college.  In our final week of our final year in high school, it was Calculus that we were anxiously studying for a comprehensive test on the very last day of school.  The week following the AP test saw little change in my teacher’s passionate instructional practices, but it did see a huge change in our feelings towards her and her class.  Whereas before, it had been our intense hostility towards the class that united us, the feelings that we shared now were gratitude and appreciation for the teacher and the class that had really prepared us for our futures.

Part 5: Challenge

The single greatest challenge that I’ve faced in math is a challenge all too common among students, that is, successfully confronting the Correct Answer Syndrome (CAS).   Perhaps more so with math than with other areas of instruction, assessment in this field often becomes an absolute, daily occurrence, so that the need for accuracy and validation overrides the desire, or even ability, to truly learn and internalize the content being taught.  As I went through school, I spent hours each night on my math homework, solving the problems just as I had seen my teachers do, and panicking when I found myself confronted with a problem that I couldn’t solve.  In most math classes, what counts is the final product, the list of answers arrived at, but not the thinking process behind each answer.  Doing my math homework, I often remember feeling as though I were being set up for failure.  It wasn’t really until my Plan II Math course that I began to trust myself in math and to see that failing to arrive at the “correct answer” in math is really just another means for learning how to solve a problem.  Thanks in large part to my Plan II Math professor’s attitude towards –and even encouragement of— mistakes, especially when they stemmed from creative and well-reasoned thought, I was finally free to devote myself completely to the study of math and found that my critical thinking about math concepts actually improved, as did my enjoyment of the subject.  Most importantly I have overcome the challenge of CAS.

Part 6: Teaching

When I entered UT, I entered as a Plan II and Business major.  Devoting myself to my coursework, I kept telling myself that the nagging sensation that something was missing from my academic pursuits would disappear once I progressed further in my studies.  After going back home for winter break during my sophomore year, and after volunteering once more with my hometown’s Respite Care Program, in which I’d volunteered throughout high school to care for children with diverse disabilities, I finally realized that what was missing from my life in college was the fulfilling feeling that I was actually making a positive difference in peoples’ lives.  When I went back to college that spring, I dropped my Business major and added Special Education, a decision I’ve never regretted.  The opportunities to work with children with special needs and their families as they navigate through the education system has really reaffirmed for me my belief that each student has the right to learn, and that it is our responsibility as educators to ensure that this happens.  As I envision teaching mathematics to my students in the future, my greatest hope is that I will be able to instill in them confidence in their own abilities in math so that they never fear taking risks to try to solve problems.  Mathematics affords teachers and students alike the opportunity, not only to explore a field, but also, to really explore human thought and decision making through the development of critical thinking skills.  I’m dedicating myself to improving my students’ academic and social futures, and I can help ensure these by supplying a supportive and structured environment in which they can explore the field of math. My hope is that they will be able to look back at the time they spend with me as one of the peaks of their math lives!