### Unconventional Quizzes that Liven Up My Classroom

By Susan Zielinski, posted September 12, 2016 —Math quizzes can be engaging, affirming, and even fun! These quizzes are collaborative (group work), formative (identifies what still needs work), or full information (students have complete prior knowledge of the questions). These quizzes can boost
self-confidence, reward persistence, and encourage a growth mindset.Some issues to consider• Will the assigned groups be random, heterogeneous (strong-weak groups), homogenous (groups of same strength), or self-selected? There are pros and cons to each, so I mix it up. • I don’t tell

### Blog Post 3: Collecting Daily Grades and Feedback with Google Forms

By
Susan Zielinski, posted August 29, 2016 —
Do you spend too much time gathering,
entering, and sharing homework grades? Would you like an easy way to see which
homework problems were confusing to your students and gather their questions? My
solution? Google forms!
After a few terms of collecting daily
homework grades in class, either on a paper grade sheet or using an app, I was
frustrated. Some days I forgot; some days I ran out of time; and every day, I
begrudged wasting class time collecting
grades. Because I have students self-grade and then revise their

### Grading Homework for Accuracy or Completion? Yes!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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?

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!

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!

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

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

### It Gets Personal

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
AngelouI 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.

### You Can Quote Me on That!

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

### Finding My Mathematical Muse

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

### Feeling Math

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

### Making Time for Mathematics

Mathematics is at my core. I don’t know why I am wired this way—I just am. But I learned very qu ickly 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

### Numbers and Shapes Everywhere

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.

### The Twenty-First Century Mathematics Classroom

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

### You Matter

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

### Not Alone

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

### Light the Fire

Teaching
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