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Three Acts to Fermi

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So we’ve talked a bit about the “aha!”moment. This week, I want to discuss one of my biggest struggles—turning a really fun, cool, interesting idea into a full-fledged math lesson. I love going to math conferences, and I love hearing about intriguing problems. However, it’s one thing to see a cool problem; it’s quite another to turn the problem into a teachable lesson.

One great place to see how rich tasks are turned into full-on lessons is the website of one of my favorite bloggers and speakers, Dan Meyer. His new collection, Math in 3 Acts, views mathematics teaching from a storytelling perspective. Meyer explains (The Three Acts Of A Mathematical Story) that act 1 is the motivation for the problem. He typically uses a picture or a video to intrigue students and then asks students to write their own questions. Act 2: Students attempt to answer the question. And act 3: A visual payoff for the hard work put in during act 2, providing the answer to act 1. Take a look at the Water Tank task, for example. It is very simple, very visual, and I guarantee that students will be on the edge of their seats during act 3!

Water Tank Task Video 

But I am not as tech savvy as Mr. Meyer, so although I use his lessons liberally, I don’t use the same process for the lessons that I create. Instead, I take all the cool problems that I have collected and try to fit them into my curriculum. I solve each problem myself and determine the main mathematics content. I try to introduce the problem without giving away too much information. And I think about scaffolding—What product do I want from my students, and what support do they need?

One of my favorite types of tasks is Fermi questions—questions of magnitude that are impossible to answer directly, forcing an estimate. Some examples of Fermi questions: How many piano tuners are there in New York City? and How long would it take to walk Earth’s circumference? I’m going to focus on the question, How many golf balls would fill the Washington Monument?

Photo of Washington MonumentWhat is my goal for the activity? I want students to calculate the volume of the Washington Monument (pyramid/prism) and of a golf ball (sphere) and to compare these volumes (http://www.corestandards.org/Math/Content/HSG/GMD/A/3/). I also want students to realize that there are empty spaces between the golf balls, so simple division will not work. I will prepare a worksheet that sets up the problem, gives students space to record important information and to show their work, and asks for an explanation of the answer. I will be ready with empty prisms and pyramids, golf balls, and the measurements that they will need.

Your turn! Who are your favorite math teacher bloggers? (My students love Vi Hart’s videos.) Have you used Math in 3 Acts in your teaching, or do you call it something else? And, of course, how many golf balls would fill the Washington Monument?

 


Katie HendricksonKatie Hendrickson, katieahendrickson@gmail.com, is a middle school mathematics teacher at Athens Middle School, in Athens, Ohio. She is currently serving as an Albert Einstein Distinguished Educator Fellow in Washington, D.C. and co-edits “Cartoon Corner” for NCTM’s journal Mathematics Teaching in the Middle School. Her dream is that one day fear of mathematics will cease to exist and everyone will have a love and appreciation for the beauty of mathematics. 

Eureka! (The “Aha!” Moment)

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As an undergraduate in college, I discovered that I enjoyed solving puzzles. I had become addicted to that “aha!” moment, when everything that had been frustrating me suddenly became crystal clear and clicked into place. Although this moment may be accurately described as “I’ve found it,” the phrase is markedly blander than its ancient Greek translation, “Eureka!” famously coined by Archimedes, whose “aha!” moment was so rousing that he forgot his clothes. Whether you call it the “aha!” moment or the “Eureka!” moment, it can be thrilling to experience. 

Archimedes' Eureka Moment 

Used with permission of Flickr
(https://www.flickr.com/photos/fingertrouble/6777959098)

I became a math teacher so that I could foster more of these “aha!” moments. Over the past several years, however, I have felt inundated with new curriculum, new standards, and new tests—and have increasingly felt that there is no time for the “aha!” Fortunately, the Common Core State Standards also incorporate mathematical habits of mind and problem-solving skills with the strong emphasis on the Standards for Mathematical Practice.

Fawn Nguyen, blogger and math teacher, has a brilliant method for tackling some of this practice (in particular, communication, making sense of patterns, and reasoning abstractly) with a collection of 145 patterns on her website, www.VisualPatterns.org (the answer key is available on her blog, www.fawnnguyen.com). These “Pattern Talks” have provided a venue for me to engage students in rich thought every day.

Here’s what I do. As students enter the room, a pattern is projected on the board along with a list of questions:

Pattern from VisualPatterns.org 

1. Draw the next step.

2. Draw step 43.

3. How many blocks (or toothpicks or other objects) will be in step 43? In step n? 

4. Write an equation for the pattern.

Students work individually for three minutes and then talk with a partner for two minutes. Then I start calling on students randomly to answer the questions. I ask three students to share their answers to question 1, and I write all three answers on the board. I ask often, “Who has something different?” and encourage students to speak up and explain their thinking. I ask students to connect their ideas and their patterns to their peers’ ideas and equations. I allow exactly five minutes for students to share their work, their thinking, and their answers, and when the time is up, we move on to the day’s lesson.

When I start class this way, the students begin class engaged, talking to one another and thinking critically about one another’s work. Because I keep a strict ten-minute time limit, I have become increasingly comfortable with leaving problems unsolved. Sometimes we come back to the pattern another day with a fresh start, and sometimes I leave the problem on the board and invite students to work out a solution over the weekend. Sometimes I do neither. But the one thing I have stopped doing is telling the students the answer. If the students don’t agree on the answer after ten minutes, I don’t give it away. It’s their “aha!” moment to discover, not mine to steal.

So my question to you: How many patterns can you find in the problem above? How might you use this for your class, whether geometry, algebra, calculus, or remedial? And how do you work in these mathematical practices—and “aha!” moments—daily?


Katie HendricksonKatie Hendrickson, katieahendrickson@gmail.com, is a middle school mathematics teacher at Athens Middle School, in Athens, Ohio. She is currently serving as an Albert Einstein Distinguished Educator Fellow in Washington, D.C. and co-edits “Cartoon Corner” for NCTM’s journal Mathematics Teaching in the Middle School. Her dream is that one day fear of mathematics will cease to exist and everyone will have a love and appreciation for the beauty of mathematics. 

What Constitutes Compelling?

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A month into school, and I’m exhausted! I kept my stay-up-late summer schedule but get up early for school. Luckily, math classes have been pretty fun so far, so I’m still smiling.

I had a great class a few days ago. Before the students got into the classroom, I rearranged the desks into islands and then, as they came through the door, asked them all to sit in a new spot. It took only a minute, but the combination of a new configuration with new people seemed to create a tantalizing air of excitement. Perfect!

The sheets I had worked on the evening before (up late, again) asked straightforward but abstract questions. One sheet, for instance, asked students to “explore” the limits of sin(x)/x and sin(1/x) as x approaches zero. I asked for compelling evidence that the limit does or does not exist. I like that word compelling. It’s great for starting discussion, inviting different perspectives, and weighing evidence, all Common Core ideals for mathematical practice.

Students found the visual evidence pretty compelling. They showed me successive calculator-screen graphs and also suggested looking numerically as x → 0 in a table.

I see the question of what constitutes compelling as an emergent property of discussion. Collecting evidence is pretty straightforward, making access for students pretty easy. There is no “perfect” answer, but there are certainly better and worse arguments, and students are pretty quick to sort through them.

My favorite part of a class like this is simply listening. Monitoring several conversations at once is relatively easy, and I can move from island to island and compliment a neat argument or help a group that’s struggling with consensus.

A few students asked me to look at the differences in behavior between the graphs of y = sin(x)/x and y = sin(1/x). They wanted to say that their graph of the latter (see fig. 1) looked as if it were tending toward zero, but their numerical analysis would not support that statement.

Graph 1
fig. 1 

We switched to a different graphing technology—desmos.com—on my Chromebookä (see fig. 2), and the difference became clear: There is no sense of convergence, and, in fact, the limit does not exist.

Graph 2 - from desmos.com

fig. 2
 
 

I heard rich but very different discussions around the question from another sheet, “Is y = 2.4x – 1.6 tangent to y = x3 – 2x + 1.95?” It’s hard to use a graphing calculator to tease out the behavior, but students came up with ingenious solutions. One group set up the equality 2.4x – 1.6 = x3 – 2x + 1.95 but could not solve the cubic. They turned to Wolfram Alpha (www.wolframalpha.com), using an app on their phones. They used the results to argue convincingly that a point of tangency should result in a root with multiplicity two. Their equation clearly showed three solutions (see fig. 3); therefore, the line could not represent a tangent.

Graph 3 - from WolframAlpha
fig. 3
 

I had not expected this line of attack, but the students made a great argument. They were just as correct as groups that showed that the derivative of the cubic was similar to but clearly different from the slope of the line at each point of intersection.

I really try to keep changing the dynamics of my classes. Students will come up with surprising connections and interesting arguments if I just make sure that I’m ready to listen. They’re still arguing about evidence in class many days later, and I’m still smiling and enjoying school!

Best wishes for the rest of the school year. 


Greg StephensGreg Stephens, stephensg@hohschools.org, is a high school mathematics teacher, department chair, and instructional leader for the Hastings on Hudson School District in New York. He just rotated off the Mathematics Teacher Editorial Panel but is keeping busy in a doctoral program at Fordham University in New York City. At the moment, his thesis topic is the impact of digital literacy on the high school mathematics classroom, but the hardest thing of all is picking just one topic to focus on!

Technology Has Transformed My Teaching

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How cool were those first graphing calculators from Texas Instruments? I loved the immediate connection between equation and graph. I also spent my own money on MathType to spiff up my worksheets. Then I spent more of my own money for a personal copy of The Geometer’s Sketchpad so I could get up to speed using the school license we had just bought.

I spent the summer reading research papers and working on curriculum projects, all on a Dell Chromebook 11 that I borrowed from the school’s tech department. What a joy to leave the laptop behind! I traded a couple of pounds for a daylong battery. I’m expecting to have twenty of these in the various geometry sections this year, but it will be a few more weeks before they’re unboxed, up and running.

I had thought that the next great revolution in technology for the high school mathematics classroom was going to be a suite of killer math apps for kids’ phones. Instead, I’m finding that a few sophisticated—and free—programs are changing the way I teach. My two favorites are Desmos and GeoGebra, both intended to make mathematics visual.

Desmos is a graphing program whose ease of use lets students play around with mathematical ideas from algebra to calculus. Both the graphing and equation screens display together, so it’s easy to see how changes in one influence the other—we pull it up in class and build functions on the spot to explore how something works. (You need an Internet connection for this one.)

GeoGebra is a dynamic geometry program that also knows algebra and functions. It runs in a variety of operating systems or streams as an app, meaning that you and your students can use it on and off the network. This is a sophisticated program that lets kids start testing ideas right away—as they learn more, they can do more with it.

Sometimes the exactly right thing to do is to give students a handful of printed problems that they dig into with a pencil and solve. I have textbooks at hand and hard drives full of handouts. As do we all. What’s harder is to give students the tools to explore something that they are just beginning to understand. Students explore construction faster and deeper in GeoGebra than they do with a compass and ruler. They latch onto the power of coefficients with Desmos more effectively than with the handheld graphing calculator. These programs are empowering, and they run on stuff kids already have—phones, tablets, and curiosity. I love teaching, but technology is giving students the power to learn along with me, instead of simply from me.

I’m excited again, and the school year has only just begun!


Greg StephensGreg Stephens, stephensg@hohschools.org, is a high school mathematics teacher, department chair, and instructional leader for the Hastings on Hudson School District in New York. He just rotated off the Mathematics Teacher Editorial Panel but is keeping busy in a doctoral program at Fordham University in New York City. At the moment, his thesis topic is the impact of digital literacy on the high school mathematics classroom, but the hardest thing of all is picking just one topic to focus on!

Are We Seeing Our Kids Thinking Yet?

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I joke with my students that I forget everything over the summer: not just the stuff I teach but even how to think about the stuff! We laugh at both aspects of it—that we all forget stuff during summer break but also the irony of thinking about thinking itself.

I remember the emphasis on reflection in George Pólya’s slim book How to Solve It (1945) back when I was in graduate school (and dinosaurs still roamed the earth). More recently, thinking and reflection are wound through the Common Core’s Standards for Mathematical Practice. This isn’t always what we’re used to in our math classes, so I focus on specific activities in class aimed directly at helping kids make their thinking visible.

My favorite activity for making thinking visible is using didactic triangles to help students draw out connections between ideas. Choose three ideas and put one at each vertex of a triangle. Use the sides to ask kids to list ideas and relationships that connect only those two vertices. You can always put a few ideas common to all three in the middle.  

Triangle with each point labeled: Right Triangle, Pythagorean Theorem, Distance FormulaFor the triangle at left, kids are quick to point out that the Pythagorean theorem goes with right triangles, but they don’t always see that the distance formula simply captures the lengths of the sides of right triangles.

It’s amazing how differently kids see connections and how quickly this exercise can make their thinking visible. Some observations are superficial (“the bottom two are formulas”); others are deep (“the distance formula says that any distance between two points is the hypotenuse of a right triangle”).  

Triangle labeled Mathematics with each point labeled: Algebraic, Textual, Visual. Didactic triangles are discussed in research papers, but they tend to highlight teacher, student, and concept relationships. See, for example, Reider Mosvold’s blog. Cool research, but not for class! Didactic triangles reimagined as a graphic thinking organizer was an idea I took away from an NCTM conference in Baltimore. Leigh Haltiwanger and Amber M. Simpson presented on writing as support for mathematical thinking, and I took notes as fast as I could! (My own focus is on high school mathematics, but you can find a middle school version of their presentation in our sister journal Mathematics Teaching in the Middle School [“Beyond the Write Answer: Mathematical Connections,” MTMS April 2013, vol. 18, no. 8, pp. 492–98].)

Now my students expect these triangles as warm-ups for a class activity and look for them on exams, as a way of wrapping up. I know the technique is successful when I can overhear the evolution of a concept as students discuss the connections they think they see.

We need powerful ways to get kids thinking and to make this thinking visible, but not just students. In our department meeting last spring, we had a lively discussion about the appropriate connections to make with the didactic triple “natural, rational, integer.” Try it with your own colleagues. After all, why should students have all the fun?

Regardless, best wishes for the new school year, and please send along your own ways to get the thinking going.


Greg StephensGreg Stephens, stephensg@hohschools.org, is a high school mathematics teacher, department chair, and instructional leader for the Hastings on Hudson School District in New York. He just rotated off the Mathematics Teacher Editorial Panel but is keeping busy in a doctoral program at Fordham University in New York City. At the moment, his thesis topic is the impact of digital literacy on the high school mathematics classroom, but the hardest thing of all is picking just one topic to focus on!

Nerves . . . and a Plan!

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I love teaching, but I always get nervous as the new school year approaches. After almost twenty years, you’d think those last-week-of-summer-vacation jitters would be gone, but they aren’t. I love meeting my new groups of crazy teenagers, and yet I still agonize over how to start those first few classes. I worry about everything—that the summer has eroded away half of what I knew in June and that I’ll never be able to juggle the lesson planning, standards, exams, and activities in ways that keep the classroom on fire. (At least it’s not like those first few years, when I had to study twice as much as any of my students every day for the first few months of every year!)

What I love most about starting each year is studying up on new ways of doing things so that they’re fresh in my mind when the kids arrive. Last year, I worked on building Web pages for my classes, and the year before I dedicated myself to learning some computer programs that I’ve used in class ever since. I’ve got to plan something new every August, or the jitters will burn out my brain.

This year, I’m committed to increasing the number of ways I get kids working together in class. This is always hard to do in our math classes; if we’re not careful, kids get into the habit of working alone and then either arriving at an answer or waiting patiently for the correct result to appear. I want a classroom where people talk through solutions, argue with one another (or even with me!), and brainstorm ways to solve stuff that’s different from anything they’ve worked on before. Doing this right is easier with a few activities I can adapt, and that’s my August plan for jumping into this new school year—now only a few short days away!

I often borrow discussion strategies from my English Department friends on the English Language Arts side, but right now I’m using a document from Expeditionary Learning. It’s an appendix of protocols and resources (eponymously named “Appendix: Protocols and Resources”), free from New York State’s Education Department resource pages, EngageNY. Here’s a link: www.engageny.org/sites/default/files/resource/attachments/appendix_protocols_and_resources.pdf This site provides a ton of ways to get students thinking and discussing—exactly what I like in class!

The two activities I’m leaning toward right now are “Rank-Talk-Write” and “World Cafe.” Remember, these were designed with ELA in mind, but I’m co-opting them for math class. In “R-T-W,” kids examine a text (insert: fully-worked-out solution) and first summarize and then rank the work in different sections. We compare our summaries and ranking to start discussions about what’s important in our mathematics work. In “World Cafe,” small groups work on a problem together, with a representative taking notes on their discussion and work. The groups change, with each representative reporting on the work of the previous group. I like the similar structures in both: Kids interpret and then share out.

My goal—get kids talking about mathematics, and they’ll practically be teaching themselves. As long as I’m armed with a plan, I’m set. But still awfully nervous!

 


 

Greg StephensGreg Stephens, stephensg@hohschools.org, is a high school mathematics teacher, department chair, and instructional leader for the Hastings on Hudson School District in New York. He just rotated off the Mathematics Teacher Editorial Panel but is keeping busy in a doctoral program at Fordham University in New York City. At the moment, his thesis topic is the impact of digital literacy on the high school mathematics classroom, but the hardest thing of all is picking just one topic to focus on!

Back to the Future!

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Blog Post #4 in the series "Finding Inspiration and Joy in the Words of Others" 

We’re going to be able to ask our computers to monitor things for us, and when certain conditions happen, are triggered, the computers will take certain actions and inform us after the fact.—Steven Jobs (http://www.brainyquote.com/quotes)

For the past few weeks, I have been wearing a fitness bracelet. I am still getting used to its presence on my wrist, and my current skill set is limited to reviewing my record of daily activity—specifically, number of steps taken and calories burned—and my sleep patterns. These data are collected from the bracelet through a smartphone app, and I will soon learn how to enter daily nutrition data—including caffeine intake!—all in an effort to develop healthier life habits.

This gadget and its informative capabilities have caused me to daydream about a bracelet that our students might wear, continuously collecting data about their mathematical learning and habits. Before reading further, ask yourself what this futuristic monitor might collect and organize for your students.

My daydreams envisioned a device that would—

  • reveal how much time was spent on a problem or assignment;
  • report levels of engagement and persistence—incorporating contributions from others, the use of various problem-solving strategies, and intervals of subconscious processing;
  • record connection-making moments when links to the real world or to other mathematical ideas are activated; and
  • assess strength of content knowledge for a particular mathematical topic or area.

You get the idea. How would our classrooms change if students were able to monitor and act on data such as these?

My daydreaming continued, and I created a second bracelet for mathematics teachers to wear. Our bracelets would help us answer these questions:

  • To what extent do I provide equal opportunities for every student to learn mathematics?
  • How well (and how completely) do I activate students’ prior knowledge before teaching my lesson?
  • What are the different ways in which I assess my students’ understanding throughout my lesson?
  • How many minutes do I spend talking? How many minutes do I spend listening?
  • How many different representations do I use in my lessons?
  • How frequently do I make connections within mathematics and to the real world?
  • How prepared am I to teach my lesson?

Would you wear such a bracelet? What would you do with the data and the trends that are provided to you?

In spite of Steven Jobs’s prediction, my fitness bracelet cannot take the actions that the data and trend lines might suggest. (Will a future version be able to zap my wrist when I reach for a second cookie?) At present, the data-driven decisions are mine to make, and, as a result, I have the power to develop and maintain the life habits that I desire. Wearing their classroom bracelets, students and teachers would wield the same decision-making powers about their mathematical and pedagogical habits.

Until these products of a summer daydream become a reality, students and teachers have two tools that can be used to produce desirable results in student achievement and teacher effectiveness. Metacognitive strategies invite students to monitor and become engaged with their mathematical thinking and learning. For teachers, postteaching reflection—supported, when possible, by data collected during the lesson—can provide the means to answer the questions posed here. As is the case with my fitness bracelet, time and commitment are required to monitor and develop lasting habits.

The power and the future are already in your hands! 


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.

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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