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Topic: [ncsm-members] Math Standards / Moving Beyond the Worksheet
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Jerry P. Becker

Posts: 16,576
Registered: 12/3/04
[ncsm-members] Math Standards / Moving Beyond the Worksheet
Posted: Mar 27, 2013 2:41 PM
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From Education Week [American Education's Newspaper of Record],
Tuesday, March 26, 2013, Volume 32, Issue 26, pages 29,31. See

The Math Standards and Moving Beyond the Worksheet

Teaching the common standards in math

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.

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?
SIDEBAR: "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."
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.

Consider the following high school algebra standard for solving
systems of equations:

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

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 - SEE
]), 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
Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
625 Wham Drive
Mail Code 4610
Carbondale, IL 62901-4610
Phone: (618) 453-4241 [O]
(618) 457-8903 [H]
Fax: (618) 453-4244

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