Making Sense of Factoring (Part 1): Laying the Foundation
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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 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?
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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
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
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.
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
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
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!
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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
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
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
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 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
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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.
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
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
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
Light the Fire
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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