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The Power of Problem Solving

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Blog Post #3 in the series "Finding Inspiration and Joy in the Words of Others" 

 

The mathematics I do remember is the mathematics in which I understand how and why it works.—Sarah (2001)

These words are pinned to the bulletin board in my office. The sentence was written several years ago by a preservice teacher in a reflection about her mathematical understanding and serves as a reminder of the contribution of how and why to one’s mathematical knowledge. Often, how and why are not always embraced as relevant understandings by those who want to get to an answer quickly or who simply want to use procedures and step-by-step processes.

Earlier this summer, I taught a graduate course on mathematical problem solving to a class of young teachers. We tackled a variety of problems from several branches of mathematics—contest problems, recreational mathematics problems, and open-ended problems. We expanded our repertoire of problem-solving tactics and strategies, and we developed perseverance in our efforts to find satisfying solutions. Through authentic engagement in mathematical problem solving, these teachers encountered and recognized rich connections within the mathematics they know and to the subjects they teach. Each teacher developed a plan for providing students with more problem-solving opportunities.

NCTM has long espoused the power of problem solving. Its Agenda for Action (1980) states, “True problem-solving power requires a wide repertoire of knowledge, not only of particular skills and concepts but also of the relationships among them and the fundamental principles that unify them.” Clearly, the relationships among skills and concepts, unifying principles, and mathematical processes are where the how and why of mathematical understanding reside.

In the foreword to Mathematical Mosaic: Patterns and Problem Solving, Ravi Vakil states, “Most ‘big ideas’ and recurring themes in mathematics come up in surprisingly simple problems or puzzles that are accessible with relatively little background” (p. 10). Each of us has favorite problems that we like to use in class—problems that connect to the real world, problems that generate a mathematical topic, problems that contain rich connections, and problems that yield surprising or unexpected results. Problem solving invites our students to encounter the big ideas of mathematics and to uncover the how and why of mathematical concepts.

With all its mathematical potential, problem solving can do more than unite mathematical concepts and processes. It has the power to create and nurture a community of learners, sharing and celebrating the journey toward deeper understanding.

References 

National Council of Teachers of Mathematics (NCTM). 1980. Agenda for Action: Recommendations for School Mathematics of the 1980s. http://www.nctm.org/standards/content.aspx?id=17278

Vakil, Ravi. 1996. Mathematical Mosaic: Patterns and Problem Solving. Ontario: Brendan Kelly Publishing.
 

 


Tom EvitsTom Evitts, TAEvit@ship.edu, is a mathematics teacher educator at Shippensburg University of Pennsylvania and is the current president of the Pennsylvania Association of Mathematics Teacher Educators (PAMTE). He is a frequent presenter at NCTM annual and regional meetings and enjoys helping others find, make, and strengthen mathematical connections.

It Gets Personal

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Blog Post #2 in the series "Finding Inspiration and Joy in the Words of Others" 

I’ve learned that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.—Maya Angelou

I received the news of Dan’s death on Monday, and my thoughts have returned to him often this week. Dan was my student in a college mathematics class last year—a student who often appeared at my office door asking for a little extra help with his math assignments. In June, a tragic accident claimed his life and the bright future that lay ahead of him.

I refer to Dan as my student. We all use the possessive my when we talk about the young people whom we teach. They are ours from the moment we meet them on the first day of class, and that relationship does not end when the school year concludes. Years after they graduate, we still call them my students. Those same students, if they ever have cause to talk about us, refer to us as my teachers.

I will be the first to admit that appreciation and respect are not always attached to this possessive relationship between teachers and students. We have all experienced the many dimensions of classroom relationships, including frustration, exasperation, hard work, shared laughter, disappointment, anger, joy, pride, inspiration, and sorrow. In spite of the emotional, logistical, and curricular challenges, the relationships that we cultivate—student to student, teacher to student, and student to mathematics—form the vital connections for classroom learning.

As high school and college teachers, we have the opportunity to be a contributing part of our students’ journey into young adulthood. We also realize that we are not always going to know exactly what our individual contribution will be. Students come and go, names and faces get mixed up, and memories fade. In 2012, I received an e-mail from a woman who identified herself as a student in my high school algebra class—in 1980! As she recently worked with a young family member doing math homework, she was reminded of my encouragement and compassion; she wanted to say thanks for being the teacher I was and for my persistent but unsuccessful efforts to have her seek after-school assistance. Somewhat stunned, I sat quietly as I read her message. In 1980, I was still a novice teacher, developing my classroom presence and practices. I was humbled that she remembered me and took the time to write, but, more so, I was struck by her naming several lasting characteristics that were certainly in their early stages of development at the time.

In 1982, Neil Postman wrote, “Children are the living messages we send to a time we will not see.” Whether you are at the beginning, managing through the middle, or approaching the end of your career as a mathematics teacher, ask yourself about the messages that you are currently writing to the future. Are your passion for mathematics and your love for learning included in your message to your students? Do your enthusiasm and your support for your students serve as permanent markers for your message? You most likely will never know how far your message will travel, but you must write it through your students—day after day after day.


Tom EvitsTom Evitts, TAEvit@ship.edu, is a mathematics teacher educator at Shippensburg University of Pennsylvania and is the current president of the Pennsylvania Association of Mathematics Teacher Educators (PAMTE). He is a frequent presenter at NCTM annual and regional meetings and enjoys helping others find, make, and strengthen mathematical connections.

You Can Quote Me on That!

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Blog Post #1 in the series "Finding Inspiration and Joy in the Words of Others"

The recent death of American author and poet Maya Angelou (1928–2014) reminds us all about the power of words. As she has said, “Words mean more than what is set down on paper” (http://www.brainyquote.com). Words can inspire, provoke, exhilarate, arouse curiosity, evoke a smile or a laugh, bring tears, and convey one’s innermost thoughts and dreams.

For many years, one feature of my high school mathematics classroom was a daily quotation in an upper corner of my whiteboard for all to see. A new one appeared each morning without fanfare and remained visible throughout the school day. The quotes came from a variety of print sources (this practice predated the availability of the Internet as a source!) Often students would contribute quotations and quotation books to my growing collection.

I rarely called attention to the quotation; it was simply there as a thought for the day. When students would ask, “What does that mean?” or comment, “I don’t get [or like] that one,” a brief conversation might ensue. Each year, I observed that several students would diligently write each quote in their notebooks. I was happy that this small, subtle attribute of my classroom may have been stimulating student thinking, but its impact on at least one student’s lifelong learning was apparent at a graduation ceremony in which I recognized one of my classroom board quotations at the beginning of a student speaker’s address to the class. The statement was Eleanor Roosevelt’s: “No one can make you feel inferior without your consent.”

I continue to use quotations in my college classroom, particularly in the methods classes for preservice teachers. A wonderful source of mathematics-related quotes is Theoni Pappas’s The Music of Reason. Two of my favorites are these:

Wherever there is number, there is beauty.—Proclus (410–485)

The true spirit of delight . . . is to be found in mathematics as surely as in poetry.—Bertrand Russell (1872–1970)

We occasionally need to be reminded that the seemingly little things we do as part of our classroom practice have the potential to have a lasting effect on our students. A quotation of the day offers an invitation to all students to find their own meaning and value in others’ words.

Reference 

Pappas, Theoni. 1995. The Music of Reason: Experience the Beauty of Mathematics through Quotations. San Carlos, CA: Wide World Publishing.


Tom EvitsTom Evitts, TAEvit@ship.edu, is a mathematics teacher educator at Shippensburg University of Pennsylvania and is the current president of the Pennsylvania Association of Mathematics Teacher Educators (PAMTE). He is a frequent presenter at NCTM annual and regional meetings and enjoys helping others find, make, and strengthen mathematical connections.

Finding My Mathematical Muse

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When I was in fourth grade, I ordered a copy of Martin Gardner’s Perplexing Puzzles and Tantalizing Teasers through my school’s book order program. I remember reading the book many times—memorizing the puzzles and their solutions and sharing them with my friends and family (who were probably much less enthusiastic about my discovery than I was). Two years later, I received my first copy of what was a new periodical, Games Magazine. Since then, I have been hooked.

These publications tapped into an interest that had already begun for me. I created word searches and mazes starting in third grade. I had teachers who cultivated my interests—including letting me create more puzzles for my classmates or designing a game as part of a school project. But then my world was opened up to other puzzle types. And as much as I enjoyed crossword puzzles, acrostics, and other language-dependent puzzles, it was the occasional logic puzzle that really caught my interest. And as Sudoku (originally appearing as Number Place) and other language-independent logic puzzles slowly made their way into Games and Games World of Puzzles, I was struck by how these puzzles spoke to me.

I loved the mathematical structure that lay beneath the surface of these puzzles. I was intrigued by their uniqueness and the creativity behind their creation. When Gardner’s books introduced me to the field of recreational mathematics, I discovered that I had a language for talking about why mathematics was my favorite subject in school.

But as a teacher, I learned that not all students have the same enthusiasm for puzzles that I have. When I shared puzzles with my middle school students, reactions were mixed. I was reminded that students are truly individuals—that each person has different interests and triggers that get him or her excited about learning. And I learned that part of my job as a teacher is to help students find what it is that ignites that spark.

So I thought it would be fitting to end with a puzzle that I have created for this blog entry. It is a Sudoku puzzle that uses the letters in the phrase “MODERN FIT.” When the puzzle is completed, two words appear in the shaded diagonals, each of which completes the phrase “A teacher is a ___________.”

 

 Sodoku Puzzle: A Teacher is a _________ 


Jeffrey WankoJeffrey J. Wanko teaches mathematics methods courses at Miami University in Oxford, Ohio. He is interested in the development of students’ logical reasoning skills using puzzles.

 

Feeling Math

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I was recently reminded how important it is to have a Feel Good File. I started mine when I was teaching middle school twenty-five years ago, and I still have one for items that I receive from my university students (although it is largely a digital folder now).

My Feel Good File contains handwritten notes, photographs, e-mails, and drawings from my students, their parents, my colleagues, and people whom I have never met. It is a clearinghouse for items to pick me up when I need it most. You never know when it’s going to come in handy—all teachers have those days when they need a pick-me-up. And although I could imagine a Feel Good File packed with Father’s Day cards from my own children, I choose to keep it for items that focus on my teaching or the impact that I have had on a student. When I have a day when my teaching is less than stellar, I can open this folder and be reminded that my teaching really does make a difference.

I also wondered recently about making a Feel Math File—a place where I could put reminders that mathematics is awesome. Too often, we are bombarded with the perception that mathematics is strictly about calculations or procedures, and we forget about the beauty and wonder that may have gotten us excited about mathematics in the first place. We should have a reminder of what it means to feel math, not what it means to do math.

A month ago, I received a book in the mail—Love and Math: The Heart of Hidden Reality (2013), by the mathematician Edward Frenkel. There was no indication of who had sent the book or its purpose. A week later, I received a note from a former student, asking if I had received the book. He explained that he had heard about this book and that it reminded him of me. He is a first-year teacher, working hard to reach and teach his students. He had taken a history of mathematics course from me in which I spent a lot of time talking about the awesomeness of mathematics—and Frenkel’s book tries to rekindle that same spirit for its readers.

On the book jacket, Frenkel wonders:

What if you had to take an art class in which you were only taught how to paint a fence? What if you were never shown the paintings of van Gogh and Picasso, weren’t even told they existed? Alas, this is how math is taught, and so for most of us it becomes the intellectual equivalent of watching paint dry.

Although I know that Frenkel’s perception of how mathematics is taught is monochromatic, informed only by a traditional pedagogy, we all know that his perception is still the reality for some students. But his idea that we need to return to what makes mathematics awesome and wonderful is critical here. His premise is that we have engendered in our students the idea that mathematics is about rote memorization, not about its beauty and power. To me, this means that we need to refocus our energies on the need to feel math, rather than do math.

So another note, another gift from a former student—the type of thing that has helped stuff my Feel Good File for twenty-five years—has yielded something new for me: a Feel Math File.

What is something that you would put in your Feel Math File? What is something that you can turn to as a reminder of what inspired you to love mathematics? In my fourth and final blog entry, I will share with you one of my earliest mathematical muses—puzzles.

Reference 

Frenkel, Edward. 2013. Love and Math: The Heart of Hidden Reality. New York: Basic Books, Perseus Books Group.


Jeffrey WankoJeffrey J. Wanko teaches mathematics methods courses at Miami University in Oxford, Ohio. He is interested in the development of students’ logical reasoning skills using puzzles.

 

Making Time for Mathematics

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Mathematics is at my core. I don’t know why I am wired this way—I just am. But I learned very quickly that not everyone has the same appreciation for mathematics that I do. I absolutely have no problem with that. But I will not shy away from professing my love for something that defines and shapes me while also keeping my role as a mathematics educator at the forefront.

We have seen the posters, the T-shirts, the bumper stickers, and the jewelry that proclaim an individual’s idolatry of mathematics. I think these are great. They help substantiate one’s place in the world and can even help unite people with common interests. But, as a teacher, I like to use them to start conversations about mathematics and even to provide teachable moments.

For example, the clock shown here (marketed as the Geek Clock) hangs in my office:  

Geek Clock 

I often catch people looking at this clock when they come in to talk about a nonmathematical issue. Their expressions always give them away—as mathematics enthusiasts, as mathematics appreciators, or even as mathematics loathers. My goal is never one of conversion but of conversation. When I sense that the moment is right, I will ask, “Which of those expressions make sense to you?” I will often admit that there are several that I always have to look up myself. This clock is a great conversation starter, and the discussion often focuses on the person’s mathematics experiences as a student. We can then discuss all sorts of mathematics (from cube roots to infinite decimals) or even why some people have anxiety about mathematics.

Many other mathematical clocks are out there, including some that can instigate a conversation about mathematical errors or lack of precision. A very popular clock that I have seen in a number of places (including colleagues’ offices and classrooms) is this one:

Math Clock with Errors 

I appreciate that the mathematics on this clock is more accessible than that on the Geek Clock, but I am bothered by several things that appear. First, this clock has more than just expressions on it; it contains several equations as well. I could assume that I am supposed to solve equations like –8 = 2 – x for x and use that value, but that is an assumption on my part. More bothersome, though, is the equation 52 – x + x2 = 10. This quadratic equation has two real-number solutions: 7 and –6. When the hour hand is pointing at this equation, could it really be negative six o’clock (other than in the world of mod 13)?

But perhaps the most egregious error can be seen in the expression at 9 o’clock: 3(π–.14). The implication is that pi is exactly equal to 3.14, an inaccuracy that mathematics teachers often hear (and may unwittingly perpetuate). When I see this clock, I can’t help but seize the teachable moment and ask, “Do you see anything wrong with this clock?” I brace myself for the response, “Yes, it has math on it,” and steer the conversation back to the mathematics that is presented on the clock—mindful that my goal is not to shame but to educate.

We must continue to make time for mathematics in our own lives as well as those around us. I love the exactitude that much of mathematics is built on, but I am also mindful that not everyone sees things the same way that I do.

I still can’t help being not only a math geek but also a mathematics teacher.

 


Jeffrey WankoJeffrey J. Wanko teaches mathematics methods courses at Miami University in Oxford, Ohio. He is interested in the development of students’ logical reasoning skills using puzzles.

 

Numbers and Shapes Everywhere

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I look for—and find—interesting mathematical (numeric and geometric) properties and patterns everywhere.

This blog entry was written on April 14, 2014. That’s 4.14.14 (when written using a common U.S. notation), which is a numeric palindrome. That makes me wonder how many other palindromic dates I have already lived through. A quick investigation shows that this is my 76th palindromic date—and that we are in the midst of a run of nine palindromic dates (April 11 through April 19, 2014) over a two-week period.

My children and I also talk about how long it is until the next upside-down time. For example, at 12:21 and 8:08, the time appears on our digital clocks in such a way that if you stand or your head—or even easier, turn the clock upside down—the digits look the same as they did originally (12:21 and 8:08, ignoring the colon when necessary). This gets my kids and me thinking about number representations and doing mental math for fun. It has also produced a special time of day for us, which we call, “When the clock says Bob.” (Talk to me again at 8:08 for more details.)

I also look at geometric patterns and think about them all the time. This morning, I saw a set of six squares arranged like this: 

Six_Squares 

Mentally, I tried to imagine how this hexomino design might tessellate. Pretty soon, I had an image in my mind of how these pieces would fit together. 

Tessellated_hexomino 

 

I have come to understand that this is just how I see the world—in numbers and shapes. I also understand that this is not how everyone sees the world, but I think that introducing my students and my own children to my world is not a bad thing. It lets them know that it is OK to view the world through a mathematical lens and that, in doing so, we can practice the skill and the art of looking for patterns and connecting ideas.

So the next time you see a license plate and mentally factor the number that appears there, or the next time you push the buttons on your cell phone to call a friend and notice that the pattern makes a rectangle or trapezoid shape—know that you are not alone.

You are in good company!


Jeffrey WankoJeffrey J. Wanko teaches mathematics methods courses at Miami University in Oxford, Ohio. He is interested in the development of students’ logical reasoning skills using puzzles.

 

The Twenty-First-Century Mathematics Classroom

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Students sitting quietly in rows, raising hands to answer questions, and dutifully taking notes. Is this a description of the perfect classroom? Perhaps in a classic movie or in 1950. Today? Not so much. The world has shifted from manufacturing to one that integrates technologies and cultures in a social setting. How has the mathematics classroom changed?

Through coaching, I have seen a teacher in Minnesota use grouping strategies and sentence frames to focus student conversation and interaction around solving tasks and justifying reasoning. Students learn not just to look at the answer but also to begin conversations with “I agree with you because …” or “I disagree with you because …” as they make sense of the task at hand. A teacher in Oregon guides students to reference informational text and classmates as resources before requesting her support. The teacher and students are collectively building a community of learners who can challenge one another to make sense of problems. A teacher in Illinois encourages students to wonder about mathematics and use inquiry to learn.

Recently, I watched a geometry teacher draw an xy-coordinate plane on the carpet with chalk and depict a three-dimensional graph by standing as the z-axis to clarify the concept for students. Another teacher in that same department showed an interactive video of fireworks on a SMART Board™® to model quadratic equations and had students develop the models. A third teacher used calculators to see how students were answering questions and connecting the multiple representations of functions.

These mathematics teachers are everywhere, helping students reason and make sense of problems while building time for them to productively struggle toward that understanding. Mathematics teachers are working to bring students into the process of learning and use formative assessment to help the students themselves articulate what they understand and are still working to learn. The classroom is transformed into a lab, and students develop the habits of mind to connect the concepts they have learned to real-life contexts and reason logically.

Such understanding doesn’t happen in quiet rows. It happens in the structured interactions facilitated and directed by you.

 
SchuhlSarahSarah Schuhl has worked as a secondary mathematics teacher and instructional coach for twenty years, is an author, former MT Editorial Panel Chair, and consultant.  She enjoys working with teachers to find instructional and assessment practices that result in student learning 

 

 

 

You Matter

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In every school, educators with diverse backgrounds and a wide array of expertise collectively work to teach students fundamental skills and prepare them to lead independent, productive lives. Every teacher—from the language arts classroom to the drama stage to the woodworking shop and back to social studies, science, and Spanish—plays an important role in cultivating intelligent, well-rounded thinkers and citizens.

But I’ll let you in on a secret, mathematics teacher: Your work is as vital to your students’ future success as the air they breathe.

Mathematical ability has emerged as the single most critical skill that schools must develop in students to open doors to future opportunities. Whether the students you work with are headed to college or a career, their ability to choose a path for themselves and pursue their dreams is rooted in the depth of their understanding of how mathematics works and quantifies the world around them.

Today, more than at any other time in human history, we live in a world of technology that constantly reinvents itself, a world of scientific inquiry and discovery. Mathematics is the bedrock on which science and technology advance.

Students need computational skill and numeric fluency, but, even more, they need to own mathematical understanding in a deep and personal way that allows them to pursue their interests in science, technology, engineering, medicine, design, or any other field they might select.

Our world may indeed be flat, but, to the students you serve, it is also an infinite plane that joins us all together. The work you do to help students see the mathematics all around them makes it possible for them be successful today and into tomorrow.

The responsibility that you own as a mathematics teacher is a massive and altogether worthwhile one. Every subject and every teacher are important to each student’s overall growth, just as all the members of the band must work together to create a song. But there is only one rock star in the group: It’s you.


SchuhlSarahSarah Schuhl has worked as a secondary mathematics teacher and instructional coach for twenty years, is an author, former MT Editorial Panel Chair, and consultant.  She enjoys working with teachers to find instructional and assessment practices that result in student learning 

Not Alone

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Sometimes, it seems, mathematics teachers live on an island, separated from teachers of other subject areas. When other teachers use or reference mathematics, they generally do so with the expectation that students have already learned the content in our classrooms. Where are the help and support from colleagues? Why must mathematics teachers bear the responsibility for helping all students learn mathematics when all teachers are supporting students in learning the reading and writing standards in a schoolwide literacy framework?

This is a question I hear often, a belief I once held, and a mindset in need of changing. Perhaps it is time to look at the issue through a new lens. How are the other subject area teachers working to support students’ learning in mathematics classrooms?

The November 2013 issue of Educational Leadership focused on teaching students to read and access informational text. As I read the journal, several statements struck me as similar to teaching students to read high-level tasks and mathematical texts.

Ehrenworth mentions teachers in subject areas needing to find text that is “accessible, engaging and complex”; if sufficiently complex, the informational text allows students to integrate ideas and feel success in meeting the challenge, gaining “new insights and epiphanies” through solving the task (Ehrenworth 2013, p. 18). Don’t we mathematics teachers do the same thing when finding worthwhile high cognitive tasks to use in class? What are students learning in other areas that will help us?

Frey and Fisher (2013) establish several considerations for reading informational text, three of which include a connection to mathematics learning: establish purpose, use close reading, and use collaborative conversations (pp. 35–37). Why should students engage in a task? What is the purpose of the work, and why does an answer need to be found? Can students generate their own questions? How can they interact with one another to challenge one another to solve problems using collective strategies? In close reading, teachers provide short passages and model posing questions while reading the text to monitor one’s own thinking. Students learn to annotate text and answer text-dependent questions that require critical thinking and to reread passages as needed.

Sound familiar? Perhaps the literacy framework used in cross-curricular areas can support students in learning mathematics. One challenge is for us to recognize the connections and tap into the strategies that students are already learning. Another challenge is to allow students time to struggle productively when solving problems and to develop curiosity to ask more questions and explore the true beauty of mathematics.

A teacher in Oregon recently shared with me the gains made by students in her Algebra 1 and Calculus AP classes when she started to focus on reading informational text as part of a schoolwide literacy program. Perhaps we all are working to deepen students’ critical thinking skills, and perhaps the island is really a community.

References 

Frey, Nancy, and Douglas Fisher. 2013. “Points of Entry.” Educational Leadership 71 (3): 35 – 38.
Ehrenworth, Mary. 2013. “Unlocking the Secrets of Complex Text.” Educational Leadership 71 (3): 16–21.


SchuhlSarahSarah Schuhl has worked as a secondary mathematics teacher and instructional coach for twenty years, is an author, former MT Editorial Panel Chair, and consultant.  She enjoys working with teachers to find instructional and assessment practices that result in student learning 

Light the Fire

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MTclassroomTeaching is exhausting work, and on the wrong day it can quickly become exasperating. Classes are crowded, supplies are short, and the expectations of administrators and parents alike are soaring. What is a well-trained and well-intentioned mathematics teacher to do?

            The answer is in the eyes of the student.

You know the one—quiet, eyes on the floor, sitting in the back row and avoiding every opportunity to join the class discussion or volunteer an answer. But look closer and see the opportunity before you. That student, the one whom you struggle to reach, is both the antidote for your fatigue and the reason you teach every day. That student, in the face of all the challenges of the job that confront you, is your fountain of youth and your gold strike wrapped inside a backpack.

Put aside your justifiable frustration with what has been handed to you at work and see the student who needs you most. Reach that student on his or her terms, at the point of ability he or she presents you, no matter how high or low. Teach that student right there something new.

Light the flame hidden inside that student with something you know or something you made. Stoke that fire until there is a blaze of new knowledge and skill roaring where yesterday there was nothing.

And watch the chains of work come undone, replaced by the satisfaction of a job well done.

Overcoming the challenges of the classroom is not easy. Reaching that reluctant or discouraged student will require all the knowledge, skill, experience, creativity, and perseverance you can muster and sustain. Perhaps all at once.

Every day that you enter the classroom you take on an arduous task as complex as surgery, as combustible as rocket science. You are the teacher, the expert, the person who can show that student the magic in mathematics and help him or her advance toward dreams that seem out of reach.

Are you exhausted? Are you exasperated? Get up and teach anyway.

That student is counting on you.


SchuhlSarahSarah Schuhl has worked as a secondary mathematics teacher and instructional coach for twenty years, is an author, former MT Editorial Panel Chair, and consultant.  She enjoys working with teachers to find instructional and assessment practices that result in student learning 

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