I think this article does a nice job of encapsulating some of our own feelings. I can relate to this article as I reflect on my own career. I know how I taught when I first started and the progression I saw in my own practices. I have come to realize that the cliche that there is 'only one right answer' in math might be true (a topic for another conversation), however, there are certainly numerous ways to get there.
The Math Standards and Moving
Beyond the Worksheet
Teaching the common standards in math
http://www.edweek.org/ew/articles/2013/03/27/26crowley.h32.html
By Alison Crowley
When I started teaching
algebra 12 years ago, I was given a textbook, a day-by-day plan listing the
sections in the textbook that I was expected to teach, and a roster of
students. I attended various trainings the summer before about state
assessments, technology, and special education laws, and boom! I was off and
running.
One of the things I
remember most from those early years was a laminated poster I had that listed
all of the state standards for algebra. I was instructed to cross them off as
the year progressed so it would be very clear to myself, my students, and any
visitors exactly what was happening in my classroom.
I have to admit that, as a
math person, I loved my standards chart. It gave me a sense of accomplishment
at the end of each lesson to cross off that related standard, confident that I
was doing exactly what I was supposed to be doing. It gave me a sense of
reassurance. If I graded a set of assessments with surprisingly low scores, I
would be able to look at my chart and say to myself, "Huh, I wonder why
they missed that question about exponents on the test. I mean, I can see right
there on my chart that I covered the material. And I remember that I assigned
all of the problems in the book. My students really need to spend more time on
homework." Just like that, the responsibility had shifted from me to my
students.
"The good news is that the common-core standards
provide an open playing field that encourages teachers to move away from the
step-by-step model."
It wasn't until much later that I realized that
"teaching math" and "covering textbook sections" were not
synonymous.
Before I started
implementing, or had even heard about, the Common Core State Standards, I had
already begun shifting my instructional practices to include more hands-on
activities and group work, and less book work. Project-based learning began
trending in my math-teacher circles, and pursuing national-board certification
forced me to rethink my instructional practices. Were my students actually
learning the material for mastery, or were they just good at following
directions and memorizing steps?
Fast forward to the 2011-12
school year, when I heard Ann Shannon, a mathematics educator and consultant
then working with the Bill & Melinda Gates Foundation, describe what she
refers to as math
teachers' tendency to "GPS" students.
Think about it: If a
teacher is explaining how to solve a system of equations using the substitution
method, she might list on the board a set of steps for students to follow. Step
1: Solve one of the equations for one of the variables; Step 2: Substitute the
value or equation found in Step 1 into the other equation. If you peeked
inside her classroom on this particular day, you would likely see all the
students copying notes, and then probably completing a worksheet with problems
similar to the example. From an observer's perspective, you might think the
lesson was going well.
But do the students really have a solid understanding of
the mathematics they are using? And, more importantly, do they understand why they're using it?
Do they have a graphical understanding of what it means to solve a system of
equations? Can they explain their methodology to another student? Can they
apply it to real-world situations? Is
their knowledge transferable so that they will be able to draw upon it
when they are solving more difficult systems of equations in future math
classes?
My guess is that the answer
to most of these questions is no. What Ann Shannon would say is that in this
particular situation, the students
have been "GPS'ed" from problem to solution. Just as when I
drive in a new city using my global positioning system, I can follow the
directions and get to where I need to go. But I can't replicate the journey on
my own. I don't have a real understanding of the layout of the city. If a road
were blocked because of a parade, for example, I would be in trouble because I
have no real understanding of the city's geography.
So, how can we keep from
GPS-ing our students, so that they understand the mathematics behind a series
of steps? How can teachers help them grasp the why, instead of just the how?
The good news is that the
common-core standards provide an open playing field that encourages teachers to
move away from the step-by-step model.
Explain
why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions
of the equations f(x)=g(x); find the solutions approximately, e.g.,
using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and g(x) are linear,
polynomial, rational, absolute value, exponential, and logarithmic functions.
Remember the earlier
example about the teacher showing the students how to solve a system of
equations using a set of steps? The first sentence of the new standard, "Explain why the x-coordinates
where the graphs intersect are the solutions," really pushes the teacher to introduce
and explain a new concept in a way that goes beyond one-dimensional instruction.
What is an x-intercept, and what does it look like on a graph? How is
that related to the algebraic equation? Perhaps rather than starting a lesson
with the steps for solving the equations, the teacher might first have students
consider graphs of related equations, or better yet, a real-world example of a
system of equations and what the values of the x-intercepts mean in that
situation. This standard also challenges the teacher to present multiple types
of equations from the beginning of the lesson so that the students can apply
the concept of an x-intercept to many types of functions.
For many teachers, myself
included, this is a fairly
significant change in instructional practice. Although I have taught
lessons on solving systems of equations using real-world applications and
emphasizing graphical connections, I have not yet truly focused my instruction
on the "why"
behind the mathematics or given students opportunities to create their own
understanding.
So how do math teachers
make that shift away from GPS-ing students a reality? It won't be easy, and we
can't do it alone. We need opportunities to collaborate, plan, and reflect with
colleagues, both in our buildings and nationwide. We need quality resources and
relevant, engaging professional development. We need time to learn from
teachers who are already successfully implementing the common core, like Kansas
educator Marsha Ratzel, who recently shared insights in her essay "The Talking
Cure: Mathematical Discourse"
(Education Week Teacher online, Dec. 31, 2012), on how her students' mathematical-thinking skills
evolved when she gave the students time and space to have conversations about
math. We need administrators and parents to support us and play an
active role in helping us transform our classrooms into places where students
are truly engaged in what they are learning.
In my daily classroom instruction,
I am still sometimes guilty of GPS-ing students. But I am hopeful that as I
learn how to fully implement the common standards, I will become less and less
dependent on steps and crossing standards off a poster. After all, my students
really deserve to navigate themselves.
Alison Crowley teaches Algebra 2 and Advanced Placement
Calculus at Lafayette High School in Lexington, Ky. A national-board-certified
teacher with 12 years of experience, she is also a member of the Center for
Teaching Quality's Common Core Lab. For more stories on teachers' efforts to
adapt to the common standards, see Education
Week Teacher's new online package, "Common-Core Instructional
Opportunities."
Vol.
32, Issue 26, Pages 29,31
