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  • MT Blog: Joy and Inspiration in the Mathematics Classroom

  • Grading Homework for Accuracy or Completion? Yes!

    at 8/10/2016 11:19:00 AM
     
    By Susan Zielinski, posted August 15, 2016 —I was inspired when I read D. Bruce Jackson’s “homework sandwich” article in MT (Jackson 2014). He wrote, “Given two slices of bread—a problem and the answer—students fill in the fixings: their own mathematics reasoning.” This system is a brilliant solution to the common dilemma of how to grade homework: for completion or accuracy. Neither method alone gives students incentive to revisit problems they missed on the first attempt, which is exactly what they need to do. The sandwich addresses this beautifully, in the end grading for both

    The Queen’s Reward: Cannonballs and Quadratics

    at 7/28/2016 2:20:28 PM
     
    By Susan Zielinski, posted August 1, 2016 —To tackle this deceptively simple problem, students need to solve a system of equations, use the quadratic formula, the equation for objects in free-fall, and the distance formula. It’s a meaty enrichment lesson suitable for the middle of algebra 2 and beyond. The complete lesson write-up, including extensions and solutions, can be found at http://tinyurl.com/queensreward. This problem can be solved in one class period or can be enhanced by including the extension questions and digital production. The ProblemStudents must write and solve a

    10,000 Kicks: Practice in the Mathematics Classroom

    at 7/15/2016 12:10:51 PM
     
    By Tim Hickey July 18, 2016 — “The best way for a student to get out of difficulty is to go through it.” —Aristotle“I fear not the man who has practiced 10,000 kicks once, but I fear the man who has practiced one kick 10,000 times.” —Bruce Lee“Mastering a skill requires a fair amount of focused practice. . . . It’s not until students have practiced upwards of about 24 times that they reach 80-percent competency.” —Robert Marzano, Debra Pickering, and Jane Pollack in Classroom Instruction That Works (2001, p. 69)“There is a consistent progression to the lessons of . . . champion

    Technology for Learning, not for Technology’s Sake: Toolbelt Theory and the SAMR Model

    at 6/30/2016 2:15:37 PM
     
    By Tim Hickey, posted July 5, 2016 — Toolbelt Theory and SAMR, two of the foundational ideas for my district’s Educational Technology Plan, will guide the integration of technology into our schools over the course of the next few years. I have worked on developing both concepts in a tangible way in my classroom recently.Toolbelt Theory asserts that educators have a responsibility to help students develop both a “lifespan toolbelt” as well as the ability to choose tools in that toolbelt. Thus, in addition to understanding both the nature of a given task as well as the

    Donuts, Submarines, Airplanes, and Trumpets

    at 6/17/2016 2:19:02 PM
     
    By Tim Hickey, posted June 20, 2016 — The study of volumes in calculus (or any math class, for that matter) is ripe with potential for students to find joy and inspiration in a math setting through building and hands-on learning. Here are just a couple of the ways I have created fun opportunities for students to apply calculus during our study of volumes over the years. The Donut problem: I set out a few dozen donuts, napkins, plastic knives, calculators, and graph paper. Students work in groups and are given a slip of paper that reads simply, “Estimate the volume

    The Gini Index: An Economic Application of the Area Bound by Two Curves

    at 6/1/2016 11:51:22 AM
     
    By Tim Hickey, posted June 6, 2016 — Students in a calculus class are not expecting to engage in a discussion on the causes of the 1929 stock market crash. So I like to begin my class on applying the area bound by two curves by facilitating that very discussion. Students typically can describe many factors (a credit to my history-teaching colleagues!), such as overspeculation in the stock market, an overreliance on credit, a languishing agricultural economy, stagnant wages, or inflated tariffs. We then read a passage from the excellent “nutshell” book by John J. Newman and

    Trigonometry Miniature Golf

    at 5/19/2016 4:39:27 PM
     
    By Tim Hickey, posted May 23, 2016 —Cutting wood with a band saw is just plain fun. I have not found a trigonometry student yet who disagrees. This is one of the reasons that I have built the Trigonometry Miniature Golf project into my Trigonometry curriculum. The concept is simple, and I have seen a wide variety of versions of this  idea over the years in different geometry and trigonometry classrooms, such as the “Back Page: Virtual Miniature Golf” article by Edwards and Quinlan article (MT September 2015, Vol. 109, Issue 2). The current version that I use fits well into my unit on

    The Evolution of the Coffee Cup Problem

    at 5/3/2016 1:10:39 PM
     
    By Tim Hickey, posted May 9, 2016 — Find the radius of a right circular cylinder with a volume of 100 cubic milliliters and a minimum surface area. Bored yet? I’ll bet many of my students were not terribly inspired by this problem during my first year of teaching calculus. For a math purist, the problem is interesting enough. It is an optimization problem with the surface area as a primary equation and volume as a secondary equation. But for teenagers trying to figure out why they are learning applications of derivatives, the problem is lacking. So, the second year I taught calculus,

    MATH AND EMPATHY

    at 4/22/2016 12:16:12 PM
     
    By Kasi Allen, posted April 25, 2016 – We live in times of polarized politics, road rage, and extreme economic inequality—when seemingly every disagreement can quickly become colored with emotion. Here, math provides a powerful opportunity, especially for teenagers—to justify their ideas on the basis of indisputable facts, to see a problem from a different point of view, to make sense of someone else’s thinking, to come to the same solution via alternative paths, to create shared understanding. The ultimate beauty of a “math fight” is that there need not be only one winner.

    Passion for Math

    at 4/4/2016 5:03:43 PM
     
    By Kasi Allen, posted April 11, 2016 — When creativity and risk taking become the norm in a math classroom, students show passion for their mathematical ideas because they have a new sense of ownership. Under the right circumstances, they might even be willing to disagree with a classmate for the sake of their own thinking. A few years ago, when I was conducting a research project in a middle school math class, I noticed that the students sometimes argued in their small groups. When I asked the teacher about this, he smiled and told me that the ultimate sign of a

    Taking Risks—for Learning’s Sake

    at 3/25/2016 4:09:26 PM
     
    By Kasi Allen, posted March 28, 2016 — Creativity requires risk taking—whether it’s having the courage to try new technology or posing a problem with multiple solutions or using mathematics to explore a social justice issue. We know that students benefit from strong daily classroom routines. However, too much of a good thing can lead to monotony. And when students can predict exactly how each math lesson will unfold, boredom and frustration follow, leading to negative emotions and limiting access to working memory—not exactly a recipe for mathematical success. Decades

    Prioritizing Creativity

    at 3/11/2016 3:06:26 PM
     
    By Kasi Allen, posted March 14, 2016 — Nearly thirty years ago, in my first algebra class, a frustrated ninth grader got my attention when she challenged me with this statement: “You know, I’m a really creative person, and I get to be creative in every class—every class, except MATH.” Her words cut deep. In my heart of hearts, I knew she right. Mathematics, as she was experiencing it, rarely gave her a chance to let the creative juices flow. Rather than generating her own ideas, she felt forced to reproduce the ideas of others. As her teacher, I could see how my

    Making Sense of Factoring (Part 3): Building on Prior Knowledge and Connecting Representations

    at 2/26/2016 11:40:55 AM
     
    By Barbara A. Swartz, posted February 29, 2016 — This three-part series started with using the graphical representation of “multiplying” two lines to create a new function and parabolic graph as a way to lay the foundation for factoring quadratic equations. Now let’s look at using algebra tiles for helping students “see” how multiplying linear factors creates a quadratic function and how we can use this as another representation to build on for getting students ready for understanding factoring. Let’s return to the two lines that we were multiplying in my first post in

    Making Sense of Factoring (Part 2): Using Context to Create Purpose and Meaning

    at 2/5/2016 4:14:05 PM
     
    By Barbara A. Swartz, posted February 16, 2016 — For many students of algebra, factoring quadratic equations can seem like a completely arbitrary thing to do. Why would anyone want to know the zeros of a quadratic function? Teachers can set up introductory factoring lessons to provide the rationale through projectile motion. Everybody wants to know, Will we hit the target? Projectile Motion, the PhET Interactive Simulation by the University of Colorado-Boulder is a great way to pique students’ interest in quadratic functions and their solutions. In this simulation,

    Making Sense of Factoring (Part 1): Laying the Foundation

    at 1/28/2016 1:42:47 PM
     
    By Barbara A. Swartz, posted February 1, 2016 — Learning algebra poses unique challenges to students: It requires them to reason abstractly, learn a “new” language of mathematical symbols and vocabulary, and understand mathematical structures such as equations, functions, and equality (Rakes et al. 2010). In my experience, factoring has been one of these particularly difficult topics. When I was a beginning teacher, following the textbook’s lead only seemed to further confuse and even frustrate my students; to their credit, they weren’t satisfied with simply memorizing

    Focus on Learning, Not Grades

    at 1/14/2016 1:57:41 PM
     
    By Barbara A. Swartz, posted January 19, 2016 — Last month on the Mathematics Teacher blog, Jerry Brodkey wrote about how and why we should start deemphasizing grades in our mathematics classes. I was excited by his post because I too have been trying out this approach and gradually converting some other teachers to the idea. My solution was to stop assigning grades (on selected assignments)! Jo Boaler advocates changing our practices from “assessment of learning” to “assessment for learning” in chapter 5 of What’s Math Got to Do With It? (2008) and mentions a study

    Creating a Kinder Classroom (Part 4): Learning from Mistakes

    at 12/31/2015 3:04:17 PM
     
    By Jerry Brodkey, posted January 4, 2016 — Since I began teaching (nearly forty years ago), I have seen unmistakable changes in my students. More students have IEPs and 504 plans, many documenting student anxiety and stress. There is more pressure to get good grades, more pressure to be perfect. Even many strong students in my AP Calculus classes are fearful. What has happened, and what can be done? In 1983 my school offered just one or two sections of AB Calculus; we had no BC Calculus. Last year my school offered eight sections of courses following

    Creating a Kinder Classroom (Part 3): Creating Trust

    at 12/16/2015 2:28:28 PM
     
    By Jerry Brodkey, posted December 21, 2015 — One of my students’ first assignments at the beginning of the school year is to write a “math biography.” What path have they taken to get to this math class? What works for them in a math classroom and what doesn’t? I get many comments about the stress, anxiety, and even fear that many students feel about learning math. Many who have been successful in math classes believe that they are weak in math, and many are worried about the upcoming year. I remember those feelings from when I was in high school many years ago. I have to

    Creating a Kinder Classroom (Part 2): Deemphasizing Grades

    at 12/3/2015 12:47:54 PM
     
    By Jerry Brodkey, posted December 7, 2015 – Math can be a source of fear and anxiety. I worry about my students and my own children. How will they navigate these treacherous waters? In my classes, I often see tired, overwhelmed young people. Students are juggling academics, sports, jobs, family responsibilities, college applications, and more. Some students’ families are undergoing severe economic problems. Many students are worried about grades, worried about not being perfect. I believe that just one teacher in a single classroom can do some things to

    Creating a Kinder Classroom (Part 1): Basic Philosophy

    at 11/18/2015 11:33:38 AM
     
    By Jerry Brodkey, posted November 21, 2015 — For many years, I taught AP Calculus to some of our school’s strongest students as well as Geometry and Algebra to eleventh- and twelfth-grade students who had struggled in math. I am not a big believer in memorization, so I had posters of math concepts all over the walls as reminders and references. These posters would go up and come down as needed. One poster, the most important poster, was never removed from its central location over the front board that I always used. This poster was titled “Basic Philosophy.” It

    Everyone Has a Personal Green’s Theorem

    at 10/21/2015 2:55:26 PM
     
    By Dan Teague, posted November 9, 2015 — It was early September 1963. At John Hanson Junior High, I was part of a new program in which a small group of eighth graders were taking Algebra 1. Mr. Green was my teacher, explaining the difference between a number and the numeral representing the number and why x = 3 wasn’t the solution to the equation 2x = 6; rather, it should be {x ∈ ℜ| x = 3}. (New Math—those were the days). As far as we knew, we were the first kids in the history of the world allowed to take Algebra 1 in eighth grade. We thought we were hot stuff.Then Mr. Green, in

    Demonstrating Competence by Making Mistakes

    at 10/21/2015 2:46:06 PM
     
    By Dan Teague, posted October 26, 2015 — Common advice for new teachers is to be sure to do all the homework problems before you assign them. This is good advice. Much of what is possible in our classrooms comes from our reputation among students (and their parents). When students trust you, you have leverage and leeway in trying new things. A solid reputation allows for creativity in your teaching, which is often rewarded with creativity in student work. Everyone wins. Building a reputation takes some time. The first requirement from parents and students

    The Complexity of Simplicity

    at 10/7/2015 12:47:37 PM
     
    By Dan Teague, posted October 12, 2015 — I distinctly remember the first time I thought about what I was teaching. I had often thought about how I was teaching, but I had never really thought about the content. The content was always whatever was in the text I was assigned. In the summer of 1984, in the middle of a talk by Henry Pollak, then head of the mathematics division of Bell Labs, that changed.   Henry said something like, “To be a high school teacher of mathematics, you must learn to say, with a straight face, that 1/a + 2/b + 3/c, an expression involving five

    Modeling and the Mathematical Toad; Or, Use Your Own Mind

    at 9/25/2015 12:40:15 PM
     
    By Dan Teague, posted September 28, 2015 — I was at my desk on the first day of school when a student walked in and said, “I’m not in your class, but my father asked me to say hello.” After a short conversation (her father had been my student twenty-some years earlier), as she turned to go, she said, “Oh, he said to tell you that mathematical modeling changed his life.”Her father did not say that I changed his life. There was nothing special about me; it was a course and the experiences it offered that were magical. We all know teachers with magnetic personalities who attract

    The Residue of Mathematics

    at 9/10/2015 8:43:47 AM
     
    By Dan Teague, posted September 14, 2015 — A few weeks ago, in preparation for the new school year, I took some time for my annual mid-August ritual. Each year it’s the same. Once I know my new teaching schedule, I think about my goals for each course and what I would like the residual for each course to be. The residual, of course, is what is left over, what sticks around, after the course has been completed. The residue is the knowledge, skills, and beliefs the students take with them, not just into the next course but throughout their lives. It includes the lasting

    Geometry and Proof

    at 8/25/2015 4:51:12 PM
     
    By Andrew Freda, posted August 31, 2015 — A mathematician is an animal which turns coffee into theorems.—attributed to Paul Erdõs What does it mean to prove something? This is a question that I ask my Geometry students often and in different contexts. Early in the year, we work through Euclid’s Proposition 1 from Book 1 of The Elements (see, Geometry and Euclid). As rigorous as that exercise seemed at the time, students are stunned to discover that Euclid falls short of modern standards for a mathematical proof. Specifically, he uses the intersection of two circles, but he

    Geometry and Algebra

    at 8/13/2015 9:48:49 AM
     
    By Andrew Freda, posted August 17, 2015 — As long as algebra and geometry have been separated, their progress have been slow and their uses limited, but when these two sciences have been united, they have lent each mutual forces, and have marched together towards perfection. —Joseph Louis Lagrange (1736–1813) I find that many students, parents, and even colleagues see Geometry as a “year off” from math or certainly a year where algebraic skills will rust and fade. I urge all teachers to fight these myths! My vision of the complete math student is one who is strong whether

    Geometry and Chemistry

    at 7/30/2015 3:30:50 PM
     
    By Andrew Freda, posted August 3, 2015 – A chemist who understands why a diamond has certain properties, or why nylon or hemoglobin have other properties, because of the different ways their atoms are arranged, may ask questions that a geologist would not think of formulating, unless he had been similarly trained in this way of thinking about the world. —Linus Pauling (“The Place of Chemistry in the Integration of the Sciences,” Main Currents in Modern Thought [1950])One of my favorite “Geometry and . . .” units that I do with my students involves chemistry. I find that students come

    Geometry and Euclid

    at 7/14/2015 3:29:44 PM
     
    By Andrew Freda, posted July 20 – You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight.—Abraham Lincoln (Henry Ketcham, The Life of Abraham Lincoln [1901])Should we make time for Euclid in our Geometry classrooms? Yes! When I teach Geometry, the first nontextbook unit I use is always “Geometry and Euclid” (and I encourage everyone to visit a wonderful website, which has all of Euclid’s

    Ask, Don’t Tell (Part 4): The Equation of a Circle

    at 7/1/2015 10:30:46 AM
     
    By Jennifer Wilson, posted July 6, 2015 – I used to tell my students how to write the equation of a circle, given its center and radius. Then I would give them the center and radius of a circle and ask for an equation. Now I provide my students an opportunity to figure it out by practicing The Common Core’s Standard for Mathematical Practice 8: Look for and express regularity in repeated reasoning. Jill Gough and I have worked this year on leveled learning progressions for giving students a path to using the Standards for Mathematical Practice when they don’t know where to start.

    Ask, Don’t Tell (Part 3): Special Right Triangles

    at 6/17/2015 4:40:36 PM
     
    By Jennifer Wilson, posted June 22, 2015 – I used to tell my students the relationships between the legs and hypotenuses of special right triangles. Now I provide them the opportunity to figure out those relationships themselves.We started our lesson practicing the Common Core’s Standard for Mathematical Practice 7: Look for and make use of structure. (See my post on SMP 7.) What do you know about a 45-45-90o triangle? What can you figure out about a 45-45-90o triangle? • The triangle is right.• The triangle is isosceles.• The triangle is half of a square when I draw

    Ask, Don’t Tell (Part 2): Pythagorean Relationships

    at 6/2/2015 1:28:53 PM
     
    By Jennifer Wilson, Posted June 8, 2015 – A few weeks ago, I overheard one student telling another, “Will you help me figure this out? Don’t just tell me how to do it.”How many of the students in our care are thinking the same thing? How often do we tell them how to do mathematics? How often do we provide them with “Ask, Don’t Tell” opportunities to learn mathematics?I used to tell my students how to determine whether a triangle is acute, right, or obtuse, given its three side lengths. Now I provide them an opportunity to determine the relationship between the squares of the side

    Ask, Don’t Tell (Part 1): Special Segments in Triangles

    at 5/21/2015 12:44:23 PM
     
    By Jennifer Wilson, Posted May 25, 2015 – My daughter, Kate, decided to make hot chocolate. She found a 1/3 measuring cup and asked, “Where is the 2/3 measuring cup?” Without thinking, I almost said, “You can just use that measuring cup twice,” but I caught myself. Instead I asked, “Could you use the 1/3 measuring cup to get 2/3?” She thought for just a few seconds and said, “Use this one twice!” I had almost short-changed Kate’s opportunity to make her thinking visible by telling her what to do. Changing “you can” into “could you” made all the difference.How many times have I

    Teach Like a BoS

    at 5/6/2015 3:25:03 PM
     
    By Matt Enlow, Posted May 11, 2015–How do we go about becoming better teachers? There is no shortage of books that will gladly tell us (three in particular whose titles begin Teach Like . . .), and there is absolutely nothing wrong with reading these books. But none of us should blindly follow someone else’s script for How to Be a Good Teacher. We should be writing our own.We can do so by looking at everything we do with a critical eye. Why do we teach particular subjects or units the way we do? What do we hope our students come away with by the end of the school year? By the time

    Freeing My Students to Take On a Challenge

    at 4/23/2015 2:54:08 PM
     
    In my last post, I shared that it was only through personal experience that I truly understood the important role that confidence plays in developing one’s problem-solving abilities. Understanding it is one thing; actually helping our students build their own confidence is quite another.The most common symptom of low math confidence is giving up too soon when presented with an unfamiliar-looking problem. “This problem looks hard. I don’t even really understand what it’s asking. I couldn’t possibly get the right answer, so why should I even try? I will only get further confirmation of

    A Lesson for the Teacher

    at 4/8/2015 4:47:57 PM
     
    Recently, a question and answer from a math test made the Internet rounds. The question read, “Come up with an equation that is true when x = 7. (Be creative; you can make the equation as simple or as complex as you want.)” The student’s answer was simply “x = 7”; the comment from the teacher, in bright red marker, was “Really?” Someone somewhere in Internetland commented that this lesson had turned out to be a lesson for the teacher. I laughed out loud when I saw the item because I have been that teacher many times: writing an assessment, trying to think outside the box,

    Mathematics, Imagination, and Freedom

    at 3/24/2015 2:17:34 PM
     
    Fifteen years ago I left a computer programming job to enter the teaching profession. My primary reason for doing so was that I loved math and wanted everyone to derive as much pleasure from it as I did. Math was a subject that everyone loved to hate, and I decided that I needed to do my part to try to fix that. At the time, I thought that my enthusiasm alone would win my students over—once they saw how passionate I was about my subject, they would naturally become curious and want desperately to see the beauty I saw in mathematics.I hope you’re smiling at my naïveté right now,

    Modular Origami

    at 3/11/2015 8:01:57 AM
     
    I run an after-school math club for fourth and fifth graders to share ideas and puzzles with a strong mathematics content without necessarily appearing so. It is not a competition-oriented group. Instead, I offer differentiated challenges, encouraging exploration and, I hope, some joy and inspiration. Looking for another activity for the club, I found the NCTM Illuminations Pinwheel activity. Here’s an easy-to-follow YouTube™

    Quadratic Surprise

    at 2/24/2015 2:01:47 PM
     
    The start of my teaching career coincided with the mass introduction into math classrooms of handheld graphing calculators. I have learned and explored so much with these technologies that I cannot imagine teaching without these deeply inspiring tools. I first encountered Computer Algebra Systems (CAS) in 1999 when one of my AP Calculus students showed up with a newly-released TI-89. Since then, CAS have inspired, supported, and revolutionized my students’ thinking and my teaching even further. The following problem beautifully combines the powers of these technologies.The standard

    Great Problems Keep on Giving

    at 2/20/2015 12:04:59 PM
     
    From my last post, you know that I’m a big fan of problems that can be solved in multiple ways, especially for students of many ages. Here’s a surprisingly pretty geometry problem that I found on Twitter under #mathchat (https://twitter.com/hashtag/mathchat)—a phenomenal source of math conversations and professional support.A square of side length 20 has two vertices on a circle and one side tangent to the circle. What is the circle’s diameter? (https://casmusings.files.wordpress.com/2014/12/circle1.jpg?w=500 )What I particularly like about this problem is that it is accessible to

    How Do I Solve This? Let Me Count the Ways.

    at 2/15/2015 1:49:06 PM
     
    I encourage students and teachers to explore multiple ways to think about and solve problems. I believe that teachers should not necessarily hold back from questions that professional training suggests “belong” in another course or require skills beyond what we think might be required. Being able to struggle with a problem that is just beyond our reach gives us opportunities for joy, inspiration, creativity, exploration, and mathematical insights. By sharing “stretch” problems with young people, sometimes I learn (or relearn) strategies that my mind might not have seen because my

    Honoring Student Voice: Student Contributions

    at 1/23/2015 11:30:32 AM
     
    In an effort to put my money where my mouth is, for my final post on student voice I asked students to contribute to the blog. The first two pieces are from recent class experiences, and the last two are general reflections on mathematics. The courage and insights of my students inspire me. Today and every day.Reflection by Lucy Hoffman: x4 – x2 – 12 = 0When Mrs. Erickson wrote this equation on the board and asked us how to solve it, the first thing I thought of was this:a = x2    b = –x     c = –12This was the logical answer for me, but apparently no one else thought of it

    Honoring Student Voice: Questions

    at 1/23/2015 11:25:23 AM
     
    I enjoy student questions. They can be insightful, intriguing, and stimulating. Questions can reveal a misconception or illuminate a connection among ideas. But let’s be honest: Although student questions are often energizing, they can also be enervating. They can suck the wind right out of your sails.Raise your hand if you have heard any of the following: “When am I ever going to use this?” “Will this be on the test?” “Are we doing anything important in class today?” “I was out yesterday. Did I miss anything?” “How long will the test be?” “Do we have to do this?” “Can I work with a

    Honoring Student Voice: Friday Afternoon

    at 1/23/2015 11:23:32 AM
     
    It is Friday afternoon. The last bell has rung. Students are rushing from the building. Teachers are trying to find the energy just to pack up their bags. I am standing in my room, exhausted. I should erase the board, straighten the desks, take time to reflect on the week. What went well? What could I have done better? Most important, what do my students need from me next week? I can barely think about more immediate questions. Do I have all the papers that need grading? Should I carry the laptop home, or will I just bring it back on Monday morning untouched, telegraphing guilt every

    Honoring Student Voice: the Green Dress

    at 1/23/2015 11:14:09 AM
     
    I bought a dress when I started teaching. Sea foam green, high collar, shoulder pads, flaps in the front. And the defining characteristic: a gigantic silver belt buckle. A very fashionable dress—in 1991. I got compliments every time I wore it.Time passes. The dress goes to the back of the closet. Ten years later, I bring it out again and wear it to school, ready to receive all those compliments. The possibility that the dress is no longer fashionable does not enter my mind. I am teaching a class of juniors. The school is small; the students all know one another and have been

    Technology Has Transformed My Teaching

    at 9/22/2014 3:04:25 PM
     
    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 TheGeometer’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 Chromebook11 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

    Are We Seeing Our Kids Thinking Yet?

    at 9/16/2014 3:29:46 PM
     
    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

    Nerves... And a Plan!

    at 9/16/2014 3:23:37 PM
     
    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

    Back to the Future!

    at 9/16/2014 3:21:59 PM
     
    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.

    The Power of Problem Solving

    at 9/16/2014 3:20:50 PM
     
    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