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Colorado Mathematics Teacher
COLORADO MATHEMATICS TEACHER - WINTER 2015
The Official Publication of Colorado Council of
Teachers of Mathematics
THE COLORADO
Sandie Gilliam
FROM THE EDITOR’S DESK
2
TEACHER
PRESIDENT’S MESSAGE
4
MATHEMATICS
Catherine Martin
*Getting Ready for PARCC
**Math Practice Standard 7 (MP7): Look for and make sense of structure
CONTENTS & FEATURES
GETTING READY FOR PARCC
*Cognitive Complexity in math - Joanie Funderburk
pg 5
*PARCC resources to help with test prep
- Jennifer Overley
pg 11
*PARCC 101
- Nicole Wimsatt
pg 13
CONFERENCE AND PROFESSIONAL
DEVELOPMENT
CCTM 2014 Annual Conference - Joanie Funderburk
pg 14
MATHEMATICAL PRACTICE STANDARD 7
**Creating number sense by incorporating
math structure - Jenni L. Harding-DeKam pg 15
EQUITY
Supporting the success of diverse,
low-income learners in a Connected
Mathematics Program (CPM) class
- Richard Kitchen
pg 22
IN THE FIELD
What can the medical field teach us about
education? - Mary Pittman pg 27
CDE CORNER
Updates
- Mary Pittman pg 28
THE CCTM BOOK CLUB
**Principles to Actions: Ensuring
mathematical success for all
- Cassie Gannett & Christy Pruitt
pg 30
HIGH SCHOOL ACTIVITY
**Sums of squares of diagonals and sides of
regular polygons
AWARDS
CCTM teaching and leadership awards: Will
you be next?
- Alfinio Flores pg 17
- Rachael Risley pg 34
The CMT presents a variety of viewpoints. Unless otherwise noted, these views should not be interpreted as official positions of CCTM or CMT.
Winter 2015 CCTM 1
Colorado Mathematics Teacher
From the Editor’s Desk
Sandie Gilliam, Editor
I
n the CMT Fall Issue, “Having Clarifying Conversations about the Practice Standards” gave a glimpse into
the CCTM Board discussion about MP 2. Our discussion
continues, but about Math Practice 7, instead. A pre-reading of Message 38 from Smarter Than We Think by Cathy
Seeley (pp. 322–330) focused our thoughts.
1. As you read the transcript below, look for the
ideas that are being presented: zooming in/out,
connections to works of art, patterns, focus on
big ideas continually used with more complexity
as students progress, questioning and teacher
facilitation of the mathematical discourse,
connections, and structure versus rules that expire.
2.
What new ideas are being presented that give you
pause as to looking at MP 7: look for and make use of
structure—in a different light?
For me as a high school math teacher, there seem
to be many things I do unconsciously—both as
a mathematician and as a teacher. Zooming in/
out is one of those things. Now that I see the
importance of this in the big picture of the CCSS
Math Practice Standards, I can consciously be aware
of opportunities that enable me, as a teacher, to
continually highlight this for my students, and stress
the importance of this piece of structure.
3. Do you have similar discussions at your school
site? What importance do your team, department,
school, and/or district place on such discussions?
What other sources could you use for pre-reading?
How could regular discussions of one Mathematical
Practice Standard impact teaching, learning, and
PARCC assessments?
At my former school, we rarely had any time for
big picture math discussions. Department meetings,
held once-a-month, consisted of announcements and
what administrators needed us to get accomplished,
such as scheduling; creating placement tests; writing
common chapter tests and end-of-course finals; textbook
selection; and budget requests. Since some of us strongly
believed in stand and deliver and others in a more
problem-based classroom, discussion on the big ideas or
possible changes in math instruction with respect to the
NCTM Principles and Standards for School Mathematics
2 CCTM Winter 2015
(PSSM) were not forthcoming. Opportunities for K–12
math articulation were also rare; thus a conscious look
at the ongoing development and expansion of one idea,
such as multiplication, didn’t happen.
To what extent are my experiences different than
yours? Did they happen because the teachers themselves
were uncomfortable constructing viable arguments and
critiquing the reasoning of others? Or did they happen
because math is seen as a hard science with rules and
procedures, and the practices (to some) seemed more
light and frivolous?
We all believed in good mathematics content for
students, but differed on what good practices of
mathematics were. The PSSM Process Standards of
communication, reasoning, problem solving and proof,
connections, and representation were not discussed,
as a department or district-wide. I think back and only
wish…to what degree might things have been different
for student understanding of mathematics, if only we
had decided to research, read, and have those hard
conversations?
Transcript of Board Discussion on Math
Practice Standard 7:
What ideas do you get from the article (Message 38)
that help you to understand Math Practice 7: Look for and
make sense of structure?
Zooming in and out
I like that zooming-in and zooming-out again,
as before in the case of MP 2. You are looking
at, in this case, a large math equation and then
just zoom in and focus on what you know about
a small piece to see if you can understand that,
and then that may help you understand the whole
thing better.
t
t That part struck me as well, and I think that’s
probably one of the harder things as a teacher to
remember–that zooming-out piece. We focus on
the objective for today, which is to get this little
piece. So sometimes I think that we forget about
how this relates to the bigger picture, and then
students sometimes see math as just a series
of steps or little snippets because of all of the
zooming-in pieces.
Colorado Mathematics Teacher
Connections to works of art
I liken this to going to the Da Vinci exhibit
that was downtown. There were these beautiful
paintings, and then there were these sketches;
the simplicity happens behind the works of art
and how they’re structured. So when Seeley talks
about the x-ray of a building, that really made
me think about it and the idea that we want all
students to be mathematicians. This isn’t a special
thing that some people can know, but if we want
all students to be able to create the mathematics,
we have to unlock the code for them—to teach
them the foundations that are behind it.
t
Focus on patterns, focus on big ideas
To tie on to this, I really feel this article gave us
maybe some step-by-step ways for teaching to
do that—starting with patterns. Just get your
students to recognize a pattern, and then help
them have the language to generalize that pattern
and the math that’s behind it. Then move into
using the big mathematical ideas of whatever
you’re working on at your course or grade level.
I underlined this part about how some big ideas
appear over and over, or span multiple grade
levels, but evolve into more sophisticated forms.
That’s so great for helping students see structure,
because they see that math is this underlying
thing that gets developed over and over again
with more complexity. So I appreciated those two
specific things that I could help my teachers to
do, to help develop Math Practice 7—focus on
patterns, focus on big ideas.
hit me that there’s a real connection to an article that’s
in the August 2014 Teaching Children Mathematics – “13
Rules that Expire” – and so you think about: if we really
want kids to understand structure, then we need to help
them think about the use of structure, rather than a rule
that applies in this one case.
t
Classroom Discussion
I liked the section Discussion and Purposeful
Questioning; it gave specific types of questions
that a teacher could ask. Because unless you are
aware of how the topics do progress from one
grade level to the next, one can get so focused
on just your little piece, and maybe you aren’t
comfortable with the connections and how the
structure builds. The questions are good things
for teachers—not only to pose to students, but to
reflect on them, themselves.
t
t It’s not enough to have a problem-centered
CALL FOR ARTICLES: The CMT seeks articles and
activities on issues of interest to K–12 mathematics
educators in Colorado.
The Spring 2015 issue will focus on Mathematical
Practice Standard 8: Look for and express regularity in
repeated reasoning. The CMT Editorial Panel is looking
for activities and articles for grades K–12 that involve
using repeated reasoning in solving mathematics
problems. What sorts of problems allow students to
pursue mathematical ideas in this way? How do teachers
know when students are proficient in this standard?
What strategies do you use to help students develop
greater repeated reasoning skills? Articles related to this
focus will be given priority, but we welcome other articles
on mathematics education. Send your articles to sandie.
[email protected].
(Deadline for submission is February 15.)
classroom, even if the problems are wonderful.
You have to have that discourse with and between
the students. Facilitation of a discussion, however,
might be overwhelming for some.
Structure versus rules
The part that really jumped out at me was about
students expansion of the use of numbers from whole
numbers to fractions and decimals and then questioning
about what rules still hold and in what ways do numbers
behave the same and differently? It just all of the sudden
Winter 2015 CCTM 3
Colorado Mathematics Teacher
President’s Message
Catherine Martin, CCTM President
C
ontinuing our focus on the Standards for
Mathematical Practice, this message focuses on Math
Practice 7: Look for and make use of structure. A good
analogy for thinking about structure in mathematics
is to compare it to the structure of buildings. When
engineers construct a building, they pay attention to
both the parts of the structure as well as to how these
parts fit together to create the structure as a whole.
Thus, in mathematics, we need to support our students
in paying attention to both the parts and structure and
the interplay between the two.
The structure of mathematics can be viewed through
the big ideas of mathematics and its patterns. Big ideas,
sometimes referred to as enduring understandings, are
themes that often span multiple grade levels and become
increasingly more sophisticated over time. Focusing on
patterns helps students to develop an understanding of
mathematics properties and supports them in seeing
how mathematics is predictable and makes sense.
Multiplication is an example of a big idea in
mathematics that spans multiple grades in K–12
mathematics. In the Common Core Standards, students
in second grade begin working with equal groups to
build a foundation for multiplication that supports their
work in third grade as they solve problems involving
multiplication and apply the properties of operations as
they multiply. Their sophistication with multiplication
continues to grow as they multiply multi-digit whole
numbers, fractions, decimals, and integers. Their
understanding of multiplication is further extended to
multiplication of polynomials and complex numbers.
When approaching multiplication through the lens of
Math Practice 7, we would want to support students in
seeing patterns in multiplication and using properties
to ensure that their understanding of multiplication
became an enduring understanding as they progress
from grade to grade. For example, when students are
developing fluency with their multiplication facts, we
would want them to notice the pattern of commutativity.
As they employ partial products as a strategy to multiply
multi-digit numbers, they do so by making use of the
distributive property. This property can be extended to
algebra and used by students to multiply polynomials
(rather than using FOIL which has limitations) and
complex numbers.
4 CCTM Winter 2015
What do students do
to look for and make
sense of structure
in the mathematics
classroom? Students
search for and identify
patterns that help
them to understand
the structure inherent
in tasks. They connect
skills and strategies
previously learned to
solve new problems
and tasks. They might
break down complex
tasks into simpler ones that are more
manageable to solve. They develop the ability to view
complicated quantities from both the perspective of a
single quantity and a composition of quantities.
To support students in looking for and making sense
of structure in classrooms, teachers would encourage
students to explore and explain patterns as a way to
understand the structure of mathematics. To accomplish
this, teachers would support productive discourse in
their classrooms and pose open-ended questions that
help students to identify structure and help students
to make connections to skills and strategies already
learned. Such questions might include:
t What observations do you make about...?
t What do you notice when...?
t Where have we seen this idea before?
t What pattern do you see? How do you know it is a
pattern?
t How is this problem similar to other problems
we’ve solved?
Our support for students in developing their expertise
in Math Practice 7 will ensure they see mathematics as
a discipline built on structure and patterns. This view
of mathematics supports them in understanding that
mathematics makes sense and that they are capable of
being mathematical sense makers.
Colorado Mathematics Teacher
GETTING READY FOR PARCC:
*Cognitive Complexity in math
Joanie Funderburk, CCTM President-Elect
I
n many aspects of our work as math teachers, we consider the difficulty level of the tasks we provide to our students. Although Bloom’s Taxonomy and Webb’s Depth of Knowledge indicators have long served to help describe
how “hard” a task is, PARCC has created and is using a different document to consider difficulty levels. Their Proposed
Sources of Cognitive Complexity in PARCC Items and Tasks: Mathematics (available online here: http://standardstoolkit.
dpsk12.org/files/Math_Cognitive_Complexity.pdf) considers multiple sources of mathematical complexity in identifying or adapting complexity levels of math tasks, including: (1) Mathematical Content, (2) Mathematical Practices, (3)
Stimulus Material, (4) Response Mode, and (5) Processing Demand. This broader scope of complexity more effectively
aligns with the shifts of the Common Core State Standards for Mathematics (CCSS-M), and is evident as a factor in the
Performance Level Descriptors (PLDs) for math. Additionally, it provides a framework for teachers to consider complexity within their own classrooms.
Sources of Cognitive Complexity
These five categories provide a broader lens through which to consider mathematical complexity. Instead of the
generalized words and descriptions used in other similar documents, the math-specific language and examples
in this document provide clarity for math teachers. The five sources combine and interact with each other in
complicated ways, but considering each in isolation might allow us to consider what makes math more or less
difficult for our students, and may provide new lenses of consideration as we create supports or extensions for them.
Following a brief summary of each of the five sources of cognitive complexity below are corresponding sample
test items (from the PARCC Practice Test for Grade 4, found here: http://practice.parcc.testnav.com/# ) and possible
complexity ranking for each. Rationale for each ranking is provided, using specific language from the document,
although individual interpretation of the tasks and complexity descriptors may result in slight variations in these
rankings.
Mathematical Content might be considered a default source of complexity. Working with different types of
numbers may make a single math problem more difficult. At the elementary level, this may mean the introduction
of fractions (for grades 3-5) instead of whole numbers. Middle school students might consider negative rational
numbers more difficult, and at the high school level, complex and irrational numbers create more difficulty.
Similarly, word problems whose underlying structure is algebraic are more difficult than those whose underlying
structure is arithmetic. Classifying the Mathematical Content of a task as Low, Moderate, or High must be done in
consideration of grade level.
Complexity Source: Mathematical Content
Possible Rank: Low
Grade 4
Enter your answer in the box.
522÷9 =
Rationale
The CCSS-M formalize the Mathematical Practices, and support teachers in considering students’ ways of
interacting with math in addition to the content standards. Although complexity is inherent in some of the Math
Practices (“make sense of problems and persevere in solving them” might be considered high complexity while
“attend to precision” might be considered low), the prompting, integration, modeling, and required explanation of
the task will also contribute to it’s complexity in regards to the math practices.
Winter 2015 CCTM 5
Colorado Mathematics Teacher
Complexity Source: Mathematical Practices
Possible Rank: Moderate
Grade 4
Enter your answer in the box.
Rationale
Stimulus Material allows for consideration of the number of different mathematical items, such as tables, graphs,
figures, etc., a student must consider in a task, as well as the role of technology tools in the item. A task with just a
single piece of stimulus material could be low complexity in this category, while one involving two or more stimulus
items, or one with transformative technology tools (tools the student must use in order to respond to the item)
rather than incremental technology tools (tools incidental to the response) are more complex.
Complexity Source: Stimulus Material
Possible Rank: High
Grade 4
Part A
Sport
Soccer
Football
Hockey
Basketball
Rationale
Fractions of all students
3/10
2/10
1/10
4/10
Part B
used.
The way in which students are required to respond, the Response Mode, also adds to an item’s complexity. In
general, choosing a single response from a list of possible responses is less complex than generating a response,
although selected response items can be of higher complexity in regards to other complexity sources.
6 CCTM Winter 2015
Colorado Mathematics Teacher
Complexity Source: Response Mode
Possible Rank: Low
Grade 4
Month
January
February
March
Number of Computers
6,521
2,374
2,498
Part A
Enter your answer in the box.
Part B
Enter your answer in the box.
Rationale
-
Processing Demands refer to the language influence on a task. This may include the actual vocabulary used
(ambiguous words, those with meanings outside of mathematics, idiomatic words or phrases), the length of the
item stem, instructions, and response choices, and the sentence structure used in the item. As is the case with other
complexity sources, processing demands of similar items may vary based on the grade level of the student, and also
whether or not English is the student’s native language. Because of the multiple influencing factors, processing
complexity is best judged holistically.
Complexity Source: Processing Demand
Grade 4
Possible Rank: Moderate
Rationale
Rating the Complexity of an Item
Because these five sources interact to create complexity in math tasks, the PARCC publication includes a
recommended weight for considering each source in labeling the complexity of an item.
Winter 2015 CCTM 7
Colorado Mathematics Teacher
Proposed Index
Rational for the Weight
Proposed
Weight in the
Overall index
Although these weights
are recommended, there
are of course other ways of
considering the interplay
between the complexity
sources. The proposed weights
shown will be used by PARCC in
creating and reviewing items,
assembling the operational test,
and to create consistency of
complexity across test forms.
-
---
Cognitive Complexity and
the Performance Levels
Descriptors
One of the primary uses
of the cognitive complexity
document was in creating the
Performance Level Descriptors
(PLDs). The teams of educators who created these documents had to consider the mathematical skills and knowledge
required of students who are on-track for college and career readiness, or at “Level 4: Strong Command” of their
grade or course content. Additionally, descriptions were written to correspond to students who were beyond the
expectation for their grade-level or course (“Level 5: Distinguished Command”), or who were approaching the
expectation for their grade-level or course (“Level 3: Moderate Command” or “Level 2: Partial Command”). The
cognitive complexity document provided language and considerations for writing these descriptors in ways that align
to the standards and the varied sources of complexity for the content of each grade and course.
Note.
Example: Overall Complexity
Item: Hallway perimeter with fractions
Grade 4
Part A
Part B
Content
Moderate: This item requires students to reason about the mathematics and the context of the problem. The
required operation in not explicit
Moderate: Processing considers stimulus material, response mode, and processing demands together. This
item is moderate in stimulus material (students consider text plus a diagram), low in response mode (students
provide a single numerical answer), and moderate in processing demand (simple to grade appropriate language,
with use of prepositional phrases).
Overall
Moderate
Looking at the place-value descriptors for grade 4, for instance, one sees the mathematical content complexity
increase from students working with three-digit whole numbers (Level 2) to four-digit whole numbers (Level 3) to
any multi-digit whole number (Level 4). A shift in the complexity of math practices is evident when students “round
8 CCTM Winter 2015
Colorado Mathematics Teacher
Winter 2015 CCTM 9
Colorado Mathematics Teacher
to any place” in Level 3, compared to “applying conceptual understanding of place value, rather than by applying
multi-digit algorithms” in Level 4. And processing, stimulus, and response demands can be seen to increase between
Level 4: “performs computations;” and Level 5: “chooses appropriate context.”
An analysis of other pages of the PLDs reveals the influence of this document. Together, the complexity
document and the PLDs provide a model for increasing and decreasing cognitive complexity as we plan, develop, and
implement rigorous learning experiences for students.
Cognitive Complexity in the Classroom
As we implement the CCSS-M in our classrooms, we must consider the opportunities we provide our students to
exhibit high levels of mathematical understanding. The increase in rigor over our previous state standards requires
an equal emphasis on conceptual understanding, procedural skill and fluency, and applications of mathematical
knowledge. For many math teachers, this is a daunting task, particularly for students who may not typically perform
at a level similar to their peers. Both the PLDs and the Cognitive Complexity sources can be utilized to consider the
complexity of the mathematical tasks and other teaching resources we use.
For instance, a teacher may recognize that a student with unfinished learning from a previous grade may have
difficulty with a typical grade-level task. In the fourth grade example below, a teacher may recognize that knowing
how to calculate the area of a rectangle, and multiplication of two two-digit numbers are required knowledge for the
task.
Source: https://www.illustrativemathematics.org/illustrations/876
Considering the complexity of the task through the lens of the five sources, a teacher may consider decreasing the
mathematical complexity of the task by using smaller numbers, or one one-digit and one two-digit number instead.
The processing demand could be made lower by providing a labeled diagram of each garden rather than a verbal
description of the gardens’ dimensions.
It is important to use
caution when lowering
complexity, however.
Although these and
other scaffolds can
provide entry points for
struggling students, it
is important that they
also have access to problems at expected complexity levels. Working with grade level content is non-negotiable for all
students, and using scaffolds that allow access but are removed as learning progresses ensures that students progress
in their learning, rather than fall behind their peers.
Some students will reach grade-level expectations more quickly than their peers. In these cases, the five sources
could also be used increase the complexity of the task. For instance, in the garden example above, the mathematical
complexity and response mode could be increased by providing less information and requiring a more complex
response.
Summary
The concept of cognitive complexity is not new. However, the ways we think about and work with math of varying
levels of difficulty is evolving. The documents and resources described here support teachers in helping their
students access the higher levels of complexity required by the new standards and next-generation assessments.
They also help to create truly meaningful learning experiences that engage students in high levels of relevant
mathematics and prepare them for future success.
Grounded in the belief that all students can achieve at high levels, teachers working collaboratively can utilize our
collective knowledge to create effective learning experiences that result in successful students.
10 CCTM Winter 2015
Colorado Mathematics Teacher
GETTING READY FOR PARCC:
*Resources to help with test prep
Jennifer Overley, District Elementary Math Coordinator, Cherry Creek Schools
I
t’s hard to ignore the daily “Common Core” aligned
test-prep products and advertisements that come across
my email these days. I usually do not consider purchasing
these materials, as I opt for consistent, strong instruction
and practice all year; but it had me thinking…how should
preparing for a new assessment look, given all of the shifts
to the teaching and learning of mathematics?
Traditional preparation was a process of reviewing
procedural steps to solve bare number tasks and poorly
written word problems that hardly seemed worth
solving. The PARCC test promises to
assess students’ deeper knowledge of
mathematics, and most students will
likely take it entirely on the computer.
As we have learned from the PARCC
Frameworks, Performance Level
Descriptors, Assessment Blueprints,
and Evidence Statement Tables,
instruction will need to be consistent
and develop levels of sophistication
over time. Today’s assessment
tasks will require students to write
arguments/justifications, critique
other’s reasoning, and show precision
in mathematical statements; so
traditional “test prep” will likely not
prepare our students for such a test.
There’s good news though—there are
plenty of resources to help out!
PARCC has released task prototypes and new sample
items to support educators in preparing students for its
complexity. The End-Of-Year (EOY) practice tests have
already been released, and the Performance-Based (PBA)
practice assessments are scheduled for release this fall.
There’s probably no denying the fact that students will
need quality time practicing with the equation editor
tool, but we found that no more than a few minutes of
practice on successive days properly prepared students to
use this tool successfully. Furthermore, educators don’t
have to wait for the release of the PBAs. Sites like Nrich
Project, Dana Center, Inside Mathematics, Shell Center,
Illuminations, Figure this!, Illustrative Mathematics,
and more can help to balance the math classroom
resources and prepare students all year long for both
Performance-Based and End-Of-Year expectations.
Teachers can also get mathematics instructional
gems by following the experts via social media sites
like Twitter, Google+, and Facebook. Links to articles,
blogs, videos, and many other valuable resources are
often shared like mathmistakes.org, 101 questions, and
a blog by Dan Meyer that have been invaluable to many
teachers. Students can even engage in many of the sites
and critique peer work. Given the demands of future
tasks, teachers must be well-equipped to look at student
work, quickly identify assumptions behind the work, and
determine what actions to take in response to the work.
This is not easy, but PLCs or team planning can benefit
from dedicating time to analyze student responses to
common mathematical inquiry.
At a recent conference of the Association of
Mathematics Teachers of New Jersey, much discussion
surrounded the ideas associated with what a good
common core question looks like. Linda Gojak, former
president of the National Council of Teachers of
Mathematics, shared several of her favorite problems,
including one set of questions she called “always,
sometimes or never,” that foster a probe into the concept
of mathematical proof. The question asks students if
Winter 2015 CCTM 11
Colorado Mathematics Teacher
multiples of five have a five in the one’s place always,
sometimes or never, and then asks them to justify their
answer. David Wees, who designs Common Core aligned
math questions says, “You have to choose the right level
of ambiguity, enough language so that students know
what to do without making it obvious what they need to
do.” Is it sometimes, always, or never that multiplication
makes things bigger? Does a hexagon have lines of
symmetry–sometimes, always or never? Students
develop the mathematical practice of constructing viable
arguments and critiquing the reasoning of others. They make
connections between different topics in mathematics and
develop natural generalizations in their thinking.
This year with the new assessments, we end the
transitional phase of TCAP and move on to the future
of equitable mathematics in this country. Survey data
gathered from more than 20,000 public school teachers
from all 50 states on their views about teaching in an era
of change shows that the further along teachers are in
implementation, the more likely they are to be optimistic
towards the impact of the standards on their student’s
skills.
Having a plan to implement rich tasks into a welldeveloped math curriculum map (which includes
traditional and non-traditional type problems, skill and
fluency)—year long—becomes the math test-prep of the
21st century.
References:
12 CCTM Winter 2015
Scholastic, & Bill & Melinda Gates Foundation (2014).
Primary sources, update: Teachers’ views on Common
Core State Standards - America’s teachers on teaching in
an era of change. Retrieved from: http://www.scholastic.
com/primarysources/teachers-on-the-common-core.htm
Felton, E. (2014, October 29). What Makes a Good
Common Core Math Question? The Hechinger Report.
Retrieved from: http://hechingerreport.org/content/
makes-good-common-core-math-question_17841/
Colorado Mathematics Teacher
GETTING READY FOR PARCC:
*PARCC 101
Nicole Wimsatt, Region Representative
components are OPTIONAL and are designed to provide
data for informing instruction, interventions, and
professional development throughout the school year.
These optional components are not yet available.
What you need to know NOW
Two REQUIRED Summative Assessment
Components:
C
olorado is a member of the Partnership for
Assessment of Readiness for College and Careers
(PARCC), which includes over a dozen states. The
members of this consortium have worked collaboratively
to develop assessments measuring what students should
know at each grade level in order to be prepared for
college and career. This set of assessments is aligned
to the Common Core State Standards and includes
both mathematics and English language arts/literacy
components. PARCC has identified the following six goals
to guide their work: (1) Create high-quality assessments,
(2) Build a pathway to college and career readiness for
all students, (3) Support educators in the classroom, (4)
Develop 21st century, technology-based assessments,
(5) Advance accountability at all levels, and (6) Build an
assessment that is sustainable and affordable
PARCC Components
There are five components to the PARCC assessment
system that will be computer-based and incorporate
innovative questioning strategies. Two summative
components are REQUIRED and are designed to
determine if students are on track to be college and
career ready, measure the full range of standards, and
provide data for growth and accountability. These will be
administered to students this spring. Three non-summative
– Performance-Based Assessment (PBA)
administered after approximately 75% of the
school year. The ELA/literacy PBA will focus
on writing effectively when analyzing text
(three testing sessions). The mathematics PBA
will focus on applying skills, concepts, and
understandings to solve multi-step problems
requiring abstract reasoning, precision,
perseverance, and strategic use of tools (two
testing sessions). It will include both shortand extended-response questions focused
on conceptual knowledge and skills, and the
mathematical practices of reasoning and
modeling.
– End-of-Year Assessment (EOY) administered
after approximately 90% of the school year.
The ELA/literacy EOY will focus on reading
comprehension (two testing sessions). The math
EOY will be comprised of innovative, machinescorable items (two testing sessions). It will be
comprised primarily of short-answer questions
focused on conceptual knowledge, skills, and
understandings.
To further prepare for the PARCC assessment, be
watching for the CCTM Regional Workshops offered
in your area and check out the following links for
additional information and resources:
Model Content Frameworks:
http://www.parcconline.org/parcc-model-contentframeworks
Winter 2015 CCTM 13
Colorado Mathematics Teacher
These documents provide information about major,
supporting, and additional content for each level. They
delineate exactly what content is included in each of the
courses.
Performance Level Descriptors:
http://www.parcconline.org/plds
These documents provide a “rubric” for content
included on the PARCC test. The content is divided
into four or five sections (Claim A - major content;
Claim B - supporting and additional content; Claim C constructing viable arguments using precise language;
Claim D - modeling with mathematics; and Claim E fluency). Within each claim there are descriptions about
the degree to which content should be taught (a more
precise version of the DOKs that were used in the old
standards).
Evidence Tables:
http://www.parcconline.org/mathematics-testdocuments
The evidence tables are the exact words that the item
writers were given to create test items. The codes on
the Performance Level Descriptors correspond to these
documents, and the wording is from the CCSS, but
clarifications are included such as whether the question
should have a context. They are divided into End of Year
(EOY) and Performance Based Assessment (PBA). Types of Tasks:
http://www.parcconline.org/samples/math (Choose
grade-level on the left of the page.)
This is a description of the three types of tasks that are
included on PARCC assessments: t Type I tasks are on the EOY and PBA tests, and all
are machine scored. t Type II tasks correspond to Claim C (described
t
on the Performance Level Descriptors), could be
either machine or hand scored, and are found only
on the PBA test.
Type III tasks correspond to Claim D (also
described on the Performance Level Descriptors),
could be either machine or hand scored, and are
only on the PBA test.
Practice Tests:
http://parcc.pearson.com/practice-tests/
Currently there are practice tests available for the EOY
test. (The PBA is due by the start of January.)
Thank you Mary Pittman, our Colorado Mathematics
Content Specialist, for helping to define each of these
excellent resources!
CONFERENCE AND PROFESSIONAL DEVELOPMENT
CCTM 2014 Annual Conference
Joanie Funderburk, Conference Chair
T
he CCTM 2014 Annual Conference Committee is
pleased to have provided another fantastic conference to Colorado’s math educators. Nearly 600 teachers
attended this year’s conference, and enjoyed powerful
pre-sessions, a fascinating keynote presentation, and
varied conference sessions on September 25–26.
Diane Briars, President of NCTM, led approximately
100 school and district leaders through a greater
understanding of what Common Core-aligned math
instruction might look like, including connections to
NCTM’s publication Principles to Actions. The teacher
pre-session had nearly 90 attendees who explored the
three aspects of Rigor: Conceptual Understanding,
Procedural Skill and Fluency, and Application. Diane
Briars provided connections and supports in this
session as well.
On Friday, nearly 600 teachers attended over 100
14 CCTM Winter 2015
breakout sessions presented by Colorado educators. Jo
Boaler, from Stanford University, shared an interesting
and informative keynote talk about teaching math with
a growth mindset. New this year was an on-site lunch,
where attendees cashed in their $10 lunch vouchers,
included in registration costs. This new lunch structure
allowed teachers to enjoy a break in the schedule to
collaborate and network with other attendees, as well as
visit the exhibit booths. Many participated in a “Bingo
Card” raffle, where over 30 fabulous door prizes were
distributed.
Conference feedback indicated appreciation for the
continued low cost of CCTM, the ease of parking and
lunch at the Denver Mart, and the high quality of
sessions and speakers. We hope you’ll join us again next
year on September 24–25, 2015.
Colorado Mathematics Teacher
MATHEMATICAL PRACTICE STANDARD 7
**Creating Number Sense By Incorporating Math Structure
Jenni L. Harding-DeKam, University of Northern Colorado
T
he seventh Standard for Mathematical Practice,
look for and make use of structure, provides students
with a foundation of number sense. This math knowledge
goes beyond the procedural understanding of solving
problems to understanding mathematical relationships,
thus creating student problem-solvers with mathematical
mindsets (Boaler, 2009) who are able to discern the
patterns and structures between and among numbers.
This use of mathematical structure creates flexibility and
fluidity with numbers allowing mental computations and
estimation to become commonplace in the classroom.
Lattice Method:
Defining Number Sense
Number sense contributes to: (a) an understanding of
numbers, (b) an intuitive feeling for numbers and their
various uses and interpretations, (c) an appreciation
for various levels of accuracy within numerical
situations, (d) the ability to detect mathematical errors,
(e) a common sense approach to using numbers with
connections to math properties, (f) sense-making
being emphasized in all aspects of math learning and
instruction, (g) classroom climate being conducive to
numerical sense-making, and (h) math being viewed as
the shared learning of intellectual practice (Reyes, 1991;
Sowder & Schappelle, 2002).
Teacher’s Role in Number Sense
Teachers can encourage mathematical practices in
students by sense-making and looking for structure. This
may be accomplished in the following ways:
1. Ask process questions of students getting at the
“how” and “why” behind the numbers and patterns.
Examples to have students investigate: How do
decimals, percentages, fraction, and ratios relate to
each other? How are these relationships the same? How are these relationships different? Why is it
important to understand these relationships?
3. Model for the students to ask, “Does this make
sense?” Reinforce mathematics relationships
between math constructs for students to gain the
vast picture of mathematics. Ideas to use: number
lines and charts (beyond whole numbers into
decimal, fraction, prime, irrational numbers, etc.)
and manipulatives (both hands-on and technology)
like the algebra balance scale.
2. Use quality over quantity toward classroom
completion of math problems by evaluating the same
problem from multiple perspectives and strategies,
while making mathematical connections explicit. An
example of three multiplication strategies using the
same numbers:
Winter 2015 CCTM 15
Colorado Mathematics Teacher
4. Demonstrate how to look at a problem holistically
before confronting the details. What is the point of
this problem? Does it have real world connections or
implications? What is the mystery of mathematics
behind this problem?
5. Encourage and help students express problems in
several ways. Then guide students toward choosing
the most effective way of solving this type of problem
in the future. Most students use the “guess and
check” method of long division, taking time to figure
out what multiplied to what fits with the problem.
Using the expanded notation method makes the
numbers easier to solve, creating an efficient way to
solve this problem.
Long Division
6. Move from concrete mathematical understanding
(connecting meaning with manipulatives, diagrams,
drawings, graphs, etc.) to abstract understanding
(numbers themselves do not have meaning on their
own without understanding) and then back to the
concrete understanding.
8. Create a classroom environment where students
are able to take mathematical risks. Are mistakes
a normal part of learning in your classroom? Do
students generally have a positive attitude during
math time persevering to figure out problems? Do
you allow time for homework to be examined to see
what your students are doing right and what they are
missing?
Questions to Ask During Instruction
Teachers can assess student’s knowledge of
mathematics structure and patterns by asking (orally or
written) the following questions:
t Why does this make sense mathematically?
t Explain to me in your own words about…
t Show me your math understanding by using this
manipulative.
t How do you know your answer makes sense?
t Draw a picture/chart/graph that demonstrates
your understanding of…
t How could you explain this math concept to your
little sister?
Conclusion
Incorporating structures and patterns into
mathematics learning may be enhanced through number
sense. This purposeful mathematics instruction provides
students opportunities for foundational underpinnings
of Mathematical Practice Standard 7. Teachers can
help students make sense of the numeric systems by
thoughtful lesson planning of a teacher’s role in number
sense and assessment questions to ask students, in order
to evaluate their mathematics comprehension.
References:
Boaler, J. (2009). What’s math got to do with it? How parents and
teachers can help children learn to love their least favorite subject.
NY: Penguin Group.
_____
4 8764
2,000 + 175 + 15 + 1 = 2,191
4 8,000 + 700 + 60 + 4
- 8,000 - 700 - 60 - 4
0
0
0
0
7. Create space for student conversation for oral
dialogue about mathematical thinking and
justification of answers. How to conduct this oral
dialogue will have to be explicitly modeled and
practiced with students in order to have productive
conversations, hear what the other person is
saying,and to disagree with the mathematical ideas—
not the person presenting them.
16 CCTM Winter 2015
Reyes, R. (1991). The ten commandments for teaching: A teacher’s
view. West Haven, CT: National Education Association.
Sowder, J. & Schappalle, B. (2002). Lessons learned from
research. Reston, VA: National Council of Teachers of
Mathematics.
Colorado Mathematics Teacher
HIGH SCHOOL ACTIVITY
**Sums of squares of diagonals and sides of
regular polygons
Alfinio Flores, Professor of Mathematics Education, University of Delaware
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Colorado Mathematics Teacher
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Colorado Mathematics Teacher
Winter 2015 CCTM 19
Colorado Mathematics Teacher
20 CCTM Winter 2015
Colorado Mathematics Teacher
Winter 2015 CCTM 21
Colorado Mathematics Teacher
EQUITY:
Supporting the success of diverse, low-income
learners in a Connected Mathematics Program (CMP) class
Richard Kitchen, University of Denver
F
or the duration of the 2007–2008 school year,
I was fortunate to teach a 6th grade class using
the Connected Mathematics Program 2 (CMP) in
Albuquerque, New Mexico. I had 17 students in the
class, 13 girls and 4 boys. At the time, all of the students
would have qualified for Free Reduced Lunch. The class
composition was also highly diverse; 14 of the students
were of Mexican descent and 12 were English language
learners (ELLs). Two of the students are twin sisters
and African-American. For the duration of the year, I
maintained a journal of my experiences. In this article, I
will discuss the importance of students, particularly lowincome students and racial/ethnic minority students1,
having the time and support needed to learn challenging
mathematical ideas and provide a few insights about
my students’ emerging mathematical identities. First,
though, I provide a brief review of research literature
concerning the mathematics education of students of
color and students living in poverty to give some context
to this discussion.
Mathematics Education for Low-Income
Students and Students of Color
The achievement gap in mathematics has become a
taken-for-granted aspect of the educational landscape
in the U.S. (Gutiérrez, 2008). However, instead of
accepting the achievement gap as innate and immutable
to change, I prefer to view it as an “opportunity gap”
(Flores, 2008). From this perspective, the access students
have to opportunities to learn mathematics is at the root
of differences in student achievement in mathematics.
Student access to a challenging standards-based
mathematics education2 is influenced by race, ethnicity,
socioeconomic status (SES), and English language
1. I use “low-income students” as synonymous with students who
are classified as living in poverty in the United States (U.S. Census,
2010). “Students of color” is used as synonymous with “ethnic and
racial minority students” and “culturally and linguistically diverse
students.”
Standards-based reforms in mathematics refer to mathematics curriculum and instruction that promote the development of
student reasoning through problem solving and discourse (see for
example, NCTM, 1989; 2000; NSF, 1996).
2.
22 CCTM Winter 2015
proficiency (Gutiérrez, 2008; Kitchen, DePree, CeledónPattichis, & Brinkerhoff, 2007; Martin, 2013). For
instance, schools that enroll large numbers of AfricanAmerican students often have disproportionally high
numbers of remedial classes in mathematics in which
instruction is focused on rote-learning and strategies
that are intended to help students be successful on
standardized tests (Davis & Martin, 2008; Lattimore,
2005). Hogrebe and Tate (2012) found that algebra
performance is influenced by where students live; the
SES of local communities is significantly related to
students’ performance in algebra (i.e., the higher the SES
of the community, the higher the algebra performance).
These are just a few examples of how standardsbased mathematics curriculum and instruction have
historically not been priorities in U.S. schools that
predominantly serve low-income students and students
of color (Kitchen, 2003; Martin, 2013).
In schools that serve large numbers of immigrant
Latino/a students who speak with an accent, use
English words incorrectly, or speak in Spanish as a
means to express themselves, educators, peers and
community members may assume students lack the
capacity to perform well in mathematics (Moll & Ruiz,
2002; Téllez, Moschkovich, & Civil, 2011). “Deficit
perspectives” such as these attribute lower levels of
academic achievement to specific ethnic/racial groups
based upon characteristics such as lack of fluency in
English, life experiences that do not parallel those of the
dominant society, or low family income (Gutiérrez, 2008;
Spielman, & Mistele, 2013). However, instead of looking
at students and their communities through a deficit lens,
students can be viewed as having funds of knowledge
such as knowing one language and learning another,
having experiences that are richly grounded in their
culture, and having extensive mathematics experiences
in their daily lives (Moll & Ruiz, 2002). If educators
build on the attributes students possess and treat them
as mathematically competent, there is greater potential
for increased academic success and an enhanced
mathematical identity (Kitchen, Burr, & Castellón, 2010;
Turner, Celedón-Pattichis, & Marshall, 2008).
Colorado Mathematics Teacher
A student’s “mathematical identity” (Martin, 2000)
is how a student thinks about her/himself in relation
to mathematics. The notion of mathematical identity
typically concerns students’ persistence and interest in
mathematics and their motivation to engage in learning
mathematics (Cobb, Gresalfi, & Hodge, 2009). Recent
research in mathematics education has expanded the
notion of mathematical identity to include the study of
the relationship between learning and the larger learning
environment of the classroom (Boaler, 2002; Cobb &
Hodge, 2007; Martin, 2000; Nasir & Hand, 2008). For
instance, Cobb, Gresalfi, and Hodge (2009) found that
teachers who regularly support students’ attempts
to articulate their solutions to tasks help students
develop positive identities of themselves as learners of
mathematics.
were continually thinking about deep mathematical ideas
and developing understanding of them. One-on-one
tutoring with struggling learners and tutoring with small
groups supported students in learning the mathematics
under consideration, while also allowing time to address
gaps in their mathematical backgrounds (particularly
important now as Colorado transitions to demanding
mathematics standards such as the Common Core).
Frankly, the majority of my students needed the extra
time provided during tutoring to have a chance to be
successful with CMP.
In my experiences teaching 6th grade CMP, I learned
about the importance of these students having
both the time and support to learn challenging
mathematical concepts. Moreover, as they began to
experience mathematical success (defined below), their
mathematical identities were enhanced.
Students need time and support to learn
mathematics
At the school where I taught the 6th grade CMP class,
students regularly worked on mathematics for more
than an hour and up to two and a half hours daily. The
school intentionally scheduled longer class periods,
approximately an hour and 20 minutes in length,
providing the time needed for the extended explorations
that are part of CMP. In addition, hour-long tutoring
sessions were provided four days/week at the conclusion
of the school day and were considered part of the
school day; student attendance at these sessions was
mandatory. Volunteers came in to assist teachers during
the tutoring sessions. During the year I taught, students
had thrice-weekly tutoring sessions dedicated just to
supporting them to learn mathematics.
The CMP is a mathematically rich program, and time is
needed to engage students in the extended investigations
that are part of the curriculum. I found that the extended
class periods and tutorial support were essential,
especially if students were going to learn mathematics
with understanding at the high levels demanded by CMP.
Alternatively framed, my students, many of whom came
into sixth-grade behind in mathematics (as defined by
the school district), needed much more time than the
standard 50-minute instructional periods found in many
schools (or more like a 35-40 minute class after taking
care of logistical things) to be successful in an advanced
mathematics program like CMP. Moreover, the additional
three hours of support provided students during the
tutorial sessions, proved vital to assist students as they
As a CMP teacher and trainer of teachers, I have come
to think of the program as a rich feast that I want my
students to enjoy daily. For this to occur, they need time
to savior the whole meal, instead of having to quickly
gulp down their food without connecting the palatability
of the main course with the tastiness of prior and side
dishes. This is particularly important for students who
may have had limited access to opportunities to engage
in a challenging standards-based mathematics education
(Kitchen, et al., 2007). The tutorial support provided at
the end of the instructional day also proved invaluable,
allowing students to continue to engage in mathematical
investigations initiated earlier in the day that play such
an important role in CMP, while also making sense of
the ideas discovered during these explorations. Rather
than viewing students as deficit because they are behind
in mathematics (Kitchen, et al., 2007), students need to
be provided supplemental academic support so that they
can both get up to grade level and move forward to learn
challenging mathematical content comprised in CMP.
Students’ emerging mathematical identities
A particularly important revelation for me as a 6th
grade CMP teacher concerned the fragility of many of my
students’ “math egos.” It should not be underestimated
how tentative middle school students may feel about
Winter 2015 CCTM 23
Colorado Mathematics Teacher
themselves and their mathematical abilities. While
it may not be readily apparent to some how insecure
middle schoolers may be about their academic abilities,
it became more apparent to me as I observed students
engage in a variety of inappropriate behaviors that were
intended to cover up these insecurities. It is important
to note that many of them who exhibited among the
most fragile math egos were my “struggling learners.”
They had not experienced much success in mathematics.
Importantly, many of these students are students of
color.
I can imagine many students with fragile math
egos giving up quickly in CMP classrooms if they do
not have regular access to teachers and tutors who
can help them make sense of advanced mathematical
ideas. What can CMP teachers do to help students
develop a positive mathematical identity? What I
learned is that students, particularly those with fragile
math egos, need to continually experience success
in mathematics. Success for some may be as simple
as finding the product of two single-digit numbers.
This is especially the case with learners who have not
experienced much joy during mathematics class. To
support and celebrate student success, I looked for
occasions to emphasize my 6th graders’ brilliance (Martin
& Leonard, 2013). For example, during one unit, I
made a point to continually call on Britney, one of the
African-American twins, because she was doing a great
job of talking the class through how to multiply two
2-digit numbers. She became our class expert on 2-digit
multiplication; something that she took pride in as the
year progressed. During another lesson, I continually
called on Janie, an ELL who was among my weakest
students in mathematics, but who was doing a nice
job of filling in the blanks in the multiplying decimal
activities (e.g., 2.4 X 10 = 24.0, therefore .24 X 10 =
__). My goal in both cases was to purposely highlight
the accomplishments of Britney and Janie, to position
them as mathematically capable (Kitchen, Burr, &
Castellón, 2010; Turner, Celedón-Pattichis, & Marshall,
2008), and to support their sense of themselves as able
learners of mathematics. It is also important to note
that both students are female students of color. In my
effort to support the development of positive student
mathematical identity, I believe it is vital to intentionally
notice and validate the mathematical thinking and
accomplishments of historically marginalized student
groups in mathematics (Martin, 2000).
I also learned with my CMP class that reinforcing
student mathematical learning through practice helped
support emerging positive mathematics identities.
As students were learning important mathematical
concepts, they needed time to reflect upon what they
had learned through exploration and to solidify what
24 CCTM Winter 2015
they had learned through practice. For example, as
described earlier, we engaged in excellent investigations
that helped students make sense of how to find decimal
products. After participating in some nice explorations
such as 24 X 10 = 240, find 2.4 X 10, students needed
time to find the products of other decimal numbers to
solidify understanding and continue to gain confidence
in their new-found knowledge. This proved particularly
important for students with fragile math egos.
Final Remarks
Supporting students to have rich opportunities to
engage in and have success with, and potentially even
relish learning the mathematics contained in the CMP,
is demanding. A tragic educational legacy in the U.S. is
that many students, particularly low-income students of
color, have had limited access to opportunities to engage
in a challenging standards-based mathematics education
(Kitchen, et al., 2007; Martin, 2013). In my work to
engage my 6th graders, all of whom lived in poverty
and all but one is a student of color, I learned about
the importance of students having time and support
to learn challenging mathematical content. Providing
time for students to learn mathematical content proved
invaluable, as did the opportunities that were provided
to students through frequent supplemental tutoring.
My students also needed to just practice some of the
complex ideas they had devoted significant effort to
make sense of, and success through practice helped
support their developing a positive mathematics identity.
I also came to value my students experiencing
on-going success as a means to develop a positive
mathematics identity. Interestingly, experts on Response
to Intervention (RtI), a national program that provides
early interventions for students who experience learning
difficulties (Fuchs, Vaughn, & Fuchs, 2008), recognize
that mathematical success leads to more success for
struggling students (Formative Assessment Working
Meeting, 2014). I believe that we need to focus more in
the mathematics education community on providing
opportunities for students, particularly students who
have not had many uplifting experiences in mathematics,
to have more mathematical successes. This is especially
important if a teacher is using a challenging, standardsbased mathematics program in a classroom with
students who may not possess a positive mathematics
identity.
To be clear, I am not advocating here for the sort of
low-level mathematics education that has historically
been found in schools that primarily serve low-income,
students of color in which the memorization of math
facts, algorithms, vocabulary, and procedures are the
focal point of instruction, rather than teaching students
through the use of complex, challenging problems (Davis
Colorado Mathematics Teacher
& Martin, 2008; Kitchen, et al., 2007; Lattimore, 2005).
My point is simply that practice can play a role when
a standards-based mathematics program is in use to
support students’ burgeoning mathematical identities.
Specifically, I found practice helped my students,
particularly those who had previously experienced little
success in mathematics, gain confidence in their abilities
and begin to believe that they could in fact do and learn
some mathematics.
To conclude, I want to speak to the evidence that
students were learning challenging mathematics,
experiencing success, and developing positive
mathematical identities. The majority of the students
in my class continued on at the same school and had
the same CMP teacher in both grades 7 and 8. I worked
closely with this teacher, and she demanded much
of her students. She placed a premium on engaging
students in mathematical discourse3 as a means to foster
mathematical reasoning. As part of a research project
that I led, my team and I collected hours of videos of
students from my class verbally sharing insightful, wellarticulated, as well as innovative solutions, to problems
while they were in both 6th and 7th grades. But perhaps
the strongest evidence we have that students were
learning mathematics and experiencing success while
in middle school are the students themselves. Janie,
who received extensive support not only mathematics,
but also in language arts, went on to graduate from a
challenging science- and technology-focused charter
high school and started college this year. As I described
earlier, Janie was an ELL in 6th grade who was among
my weakest students. What I did not mention was that
she had been classified as a Special Education student
in elementary school and had been recommended to
repeat both 4th and 5th grades, though did not. In high
school, Janie not only did well in her classes, but also
was a leader at her school. I have stayed in touch with
Janie. Recently, she shared with me that the confidence
she completely lacked, but eventually gained in middle
school, has proved invaluable in changing her perspective
and life trajectory.
3. In mathematical discourse, the teacher seeks to foster and
continually engage in dialogue with her students (Cazden, 2001;
Herbel-Eisenmann & Cirillo, 2009). As students engage in mathematical discourse through participation in learning communities,
they build on their prior experiences and knowledge to achieve more
advanced understandings of challenging mathematical concepts
(Lave & Wenger, 1991; Franke & Kazemi, 2001).
References
Boaler, J. (2002). The development of disciplinary
relationships: Knowledge, practice, and identity
in mathematics classrooms. For the Learning of
Mathematics, 22(1), 42–47.
Cazden, C. (2001). Classroom discourse: The language
of teaching and learning. Portsmouth, NH:
Heinemann. Cazden, C. (2001). Classroom
discourse: The language of teaching and learning.
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Cobb, P., Gresalfi, M., & Hodge, L.L. (2009). An
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Cobb, P., & Hodge, L. (2007). Culture, identity, and
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Davis, J., & Martin, D. B. (2008). Racism, assessment,
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Winter 2015 CCTM 25
Colorado Mathematics Teacher
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and mathematics education 1995. Arlington, VA:
Author.
Spielman, L. J., & Mistele, J. (Eds.). (2013). Mathematics
teacher education in the public interest: Equity and
26 CCTM Winter 2015
social justice. Charlotte, NC: Information Age
Publishing.
Téllez, K., Moschkovich, J. N., & Civil, M. (Eds.). (2011).
Latinos and mathematics education: Research
on learning and teaching in classrooms and
communities. Charlotte, NC: Information Age
Publishing.
Turner, E., Celedón-Pattichis, S., Marshall, M., (2008).
Cultural and linguistic resources to promote
problem solving and mathematical discourse
among Hispanic kindergarten students. In R.
S. Kitchen, & E. Silver (Editors), Promoting
high participation and success in mathematics
by Hispanic students: Examining opportunities
and probing promising practices [A Research
Monograph of TODOS: Mathematics for All],
1, 19–40. Washington, DC: National Education
Association.
U.S. Census. (2010). How the Census Bureau measures
poverty. Retrieved from http://www.census.gov/
hhes/www/poverty/about/overview/measure.
html
Colorado Mathematics Teacher
IN THE FIELD:
What can the medical field teach us about education?
Mary Pittman, Mathematics Content Specialist, Colorado Department of Education
A
few years ago I was traveling on a plane and sitting next to me was a young doctor. We struck up a
conversation, and when he found out I was in education,
he expressed interest in discussing the similarities between
medicine and education.
He asked, “Do you have the same problem in education
we have in medicine?”
I laughed and said, “We have lots of struggles in
education. Could you be more specific?”
The doctor explained to me, “In medicine it is often
difficult and expensive to test what we want to know
about a patient, so we test for something that is merely
related or associated to it. For instance, heart disease is
a leading cause of death in the United States. Doctors
generally use a cholesterol test to determine a patient’s
risk for heart disease. The problem is that cholesterol is
not the same as heart disease, but because we test for it
we now treat it.”
I was floored. This doctor had put into words for me
a struggle I was having at that very moment. I was the
director of mathematics in Boulder, and I was struggling
with the tests available for mathematics. What I cared
about was students “doing” mathematics, and the
tests available assessed things associated with “doing”
mathematics. They could for instance assess basic fact
fluency or skills fluency. My worry was that my fellow
educators and I were just like doctors—being led astray
by the tests we were using.
Rather than using the test as an indicator to
investigate more and look for root causes, we sought out
interventions and treatments focused on raising test
scores focused on basic skills. Unfortunately, raising a
student’s score on a test of basic skills does not ensure
students will be successful mathematicians. Basic skills,
like cholesterol, represent a piece of the puzzle—but
not the whole puzzle. Students also need conceptual
understanding and the ability to apply and transfer their
skills and understandings to real life modeling problems.
Our new standards set out a vision or promise of
mathematics to our students. This promise is the reason
we became teachers. It requires us to continue the hard
but rewarding work we do each day. There is no quick fix
for heart disease; there may be a pill for cholesterol, but
truly attacking the root causes of heart disease requires a
change in diet, exercise and stress levels. Similarly, there
is no perfect test or perfect set of materials that will “fix”
our students; rather our entire culture needs a new view
of mathematics. And that change in culture starts with
each of us educators in small important ways every day.
What view of mathematics are we conveying to
students? Do students view mathematics as an ability
that some people have and others do not, or do they
believe that hard work leads to success in mathematics?
Do they see mathematics as skills that need to be
memorized and practiced, or do they see that as only
one part of the larger concept of “doing” mathematics?
Do they see assessments as indicators of their larger
understanding of mathematics, or do they see each
problem as something that needs to be learned
separately?
We will always have assessments that are merely
indicators of something we really care about, just like
cholesterol will continue to be a proxy measure for heart
disease. A principal of a turnaround school once said
to me, “If my only goal is for students to do well on a
test, then I have aimed much too low. My goal is for
students to act, think, and work like mathematicians.”
No one test will tell me all that I need to know about my
students, but each one does provide an indicator for me
about a student’s larger mathematical health.
The Mathematics Standards compel us to make
mathematics relevant to students by moving
beyond mere answer-getting—to doing the work
of mathematicians. The standards emphasize the
development of students’ abilities to use mathematics
to represent their lived experiences, and to simplify and
explain complex phenomena. I would love to say that
someday no one will have heart disease in the same way
that I hope some day every student is provided with a
rich and healthy diet of mathematics. Since I am not
a medical doctor, I have no idea if the elimination of
heart disease is
possible, but as
an educator I
continue to strive
for a healthy diet
of mathematics
for every single
student.
Winter 2015 CCTM 27
Colorado Mathematics Teacher
CDE CORNER:
UPDATES
Mary Pittman, Mathematics Content Specialist, Colorado Department of Education
New Family and Community Guides to the Colorado Academic Standards
Stay tuned for even more units this spring, including
several focused on personal financial literacy.
n partnership with the Colorado PTA, family and
community guides to the Colorado Academic Standards for grades K–5 (in English and Spanish) have
been created to help families and communities across
Colorado better understand the goals and outcomes of
these standards. The guides describe the “big picture”
purpose of the standards, as road maps to help ensure
that all Colorado students graduate ready for life, college, and careers. They also provide overviews of the
learning expectations for each of the 10 content areas
and offer examples of educational experiences that students may engage in, and that families could support,
during the school year.
The Presidential Awards for Excellence in Mathematics
and Science Teaching (PAEMST)
I
(The Standards and Instructional Support office is
currently working with Colorado educators and the
Colorado PTA to create similar documents for grades
6–12. Look for those guides in the early part of 2015.)
New Secondary Units
for Colorado’s District
Sample Curriculum
Available
PAEMST is the highest honor bestowed by the United
States government specifically for K–12 mathematics
and science (including computer science) teaching.
The award recognizes those teachers who develop and
implement a high-quality instructional program that is
informed by content knowledge and enhances student
learning.
Presidential awardees receive a certificate signed
by the President of the United States, a trip for two
to Washington, D.C. to attend a series of recognition
events and professional development opportunities,
and a $10,000 award from the National Science
Foundation. The National Science Foundation
administers PAEMST on behalf of The White House
Office of Science and Technology Policy.
The state of Colorado has four elementary
mathematics finalists for the 2013–2014 school year:
t Melanie Dolifka – 2
nd
This summer, new
units were created for
Colorado’s District
Sample Curriculum
(in each of the 10
content areas). The
new mathematics unit,
Survey Says..., is part
of both the Algebra 2
and Mathematics III
courses, and focuses
on students inferring
conclusions about
a population based on samples. Students explore the
concept of the normal distribution and use simulations
to test the validity of statistical conclusions and
hypothesis. This unit is also a great connection to the
Comprehensive Health Standards because students
analyze data from the Healthy Kids Colorado Survey.
28 CCTM Winter 2015
grade teacher at Falcon
Elementary School in
Falcon School District
49. Melanie received her
Bachelor of Science in
education from Baylor
University and a Master
of Arts in education from
Southwestern Baptist
Theological Seminary. She
has taught for almost 20
years, including three years as an elementary
math coach, and provides math professional
development classes for both her school and
district. One of her former students recalls
Melanie’s classroom in the following way: “I
didn’t notice I was learning until later. I wish
all my teachers taught this way.” And Melanie’s
principal described being in her classroom as
both “inspiring and motivating.”
Colorado Mathematics Teacher
t Carolyn Jordan – 4
grade
teacher at Normandy
Elementary School in
Jefferson County School
District. Carolyn received
her Bachelor of Science
in elementary education
and communication from
the State University of
New York and a Master
of Arts in administration,
supervision, and
curriculum development from the University of
Colorado at Denver. She has taught for over 20
years, from first through sixth grade. Carolyn’s
colleagues described her by saying that she uses
current research and professional resources to
increase and apply content knowledge to promote
higher-level thinking for students as they interact
with rigorous and challenging content. Her
classroom is welcoming, she respects students’
diverse needs, and she is culturally sensitive and
supportive.
th
t Sarah Smith – 3
grade
teacher at University
Park Elementary in
Denver Public Schools.
Sarah received her
Bachelor of Arts from
the University of Iowa,
her Master of Arts from
Regis, and a Certificate
of Mathematics Teacher
Leadership from the
University of Northern
Colorado. Sarah began teaching in 2000 in my
home state of Iowa. In 2008 she became a teacher
leader for math and science; she has also served
on the school improvement team, collaborative
school committee, and principals institute. One of
Sarah’s students described her to his mom, “Mom,
other teachers want us to be perfect; Mrs. Smith
wants us to be the best we can be.” This parent
went on to say how her son’s comments “speak to
how effective Mrs. Smith is at differentiating and
encouraging each child to develop academically.
Mrs. Smith has a way of making every child feel
capable and special.”
t Tamara Walter – 1st/2nd
grade combined classroom
teacher at Carl Sandburg
Elementary in Littleton
Public Schools. Tamara,
a national board certified
teacher, received her
Bachelor of Arts in
mathematics from
California State University
and her Master of Arts
in leadership with an
emphasis in mathematics from the University of
Colorado at Denver. Tamara has been teaching
for over 10 years. She has served on numerous
district committees for mathematics and is
seen as a go-to person in her district. One of
our own CCTM board members, Ann Summers
(whose daughter was in Tamara’s second grade
class), wrote a letter of recommendation that
talked about how much she appreciated how
Tamara understood her daughter and could talk
specifically about her child’s progress towards
understanding each concept.
rd
We wish the four finalists from 2014 luck as they move
forward in the process.
Anyone—principals, teachers, parents, students, or
members of the general public—may nominate a teacher
by completing the nomination form available on the
PAEMST website. If you know more than one teacher
deserving of this award, you may submit more than one
nomination. Teachers may also apply directly at http://
www.paemst.org. The nomination deadline is April 1,
2015 with an application deadline of May 1, 2015, for
secondary school teachers (grades 7–12). Elementary
school teachers (grades K–6) are eligible to apply during
the 2015–2016 program year.
If you have any questions or comments, please feel free
to email me at: [email protected]
Winter 2015 CCTM 29
Colorado Mathematics Teacher
THE CCTM BOOK CLUB
**Principles to Actions:
Ensuring mathematical success for all
Reviewed by Cassie Gannett & Christy Pruitt, CCTM Region Representatives
t Use and connect mathematical representations.
t Facilitate meaningful mathematical discourse.
t Pose purposeful questions.
t Build procedural fluency from conceptual
understanding.
t Support productive struggle in learning
mathematics.
t Elicit and use evidence of student thinking.
A
re you a K-12 teacher looking for a book that would
give you a blueprint for creating powerful lessons and
engaging students in high-level mathematical reasoning?
Are you an educational leader looking for a framework
that supports effective school mathematics programs?
Principles to Actions: Ensuring mathematical success
for all represents a significant step in articulating a
unified vision of what is needed to realize the potential
in educating all students, supports the Common Core
State Standards for Mathematical Practice, and provides
a framework, or roadmap, for implementing researchbased best practices for mathematics education.
This empowering book will grab the attention of any
educational stakeholder, no matter what role they fill—
principals, coaches, special education teachers, teachers
of English language learners, parents, and policy
makers—by describing the following Mathematical
Teaching Practices and Essential Elements for learning:
t Establish mathematics goals to focus learning.
t Implement tasks that promote reasoning and
problem solving.
30 CCTM Winter 2015
The unveiling of these Mathematical Teaching
Practices (the first section in this book) represents
“a core set of highleverage practices and
essential teaching
skills” for supporting
the Standards for
Mathematical Practice.
Reading this book
creates a sense of
urgency to improve
mathematical teaching
in order to shift “unproductive practices” to “productive
practices” in a way that will ensure the success of deep
mathematical learning for all students.
As a preview to the detailed articulation of each
teaching practice, a reflection on the beliefs that serve as
obstacles is provided along with strategies that can help
overcome these obstacles. These beliefs should not be
categorized as bad or good but as unproductive versus
productive. A sample of the comparison is provided
below in comparing beliefs about teaching and learning
mathematics.
For each of the eight practices, the authors present
a rich discussion in order to characterize and bring
meaning to each practice. In addition, they include an
illustration of the practice coming to life (at various
levels from kindergarten to twelfth grade) in an actual
classroom environment. Finally, sample student and
teacher actions are provided that exemplify what
teachers and students should be doing to promote that
principle into action within the classroom.
Colorado Mathematics Teacher
Beliefs about teaching and learning mathematics
Unproductive beliefs
An example of the delineation of a teaching practice
contained in the book is presented below. The teaching
practice, “implement tasks that promote reasoning
and problem solving,” begins by citing the research
behind using tasks in order to improve student learning
and understanding. Smith and Stein’s (1998) research
Task A: Smartphone Plans
Productive beliefs
classifying the level of demand of tasks is provided along
with examples of a comparison (as seen in the figure
below) in which Task A is considered a high-level task
compared to Task B, which requires low-level cognitive
demand.
Task B: Solving systems of equations
-4x –
4x + 8y = -24
x–
5x + y = 9
–
The illustration of this mathematics teaching practice
uses the high-level task above (Task A) and brings
to life the actions in two different classrooms. The
discussion of the actions by two different teachers in
implementing this task, emphasizes the importance of
the roles of both the teacher and students in learning
mathematics. A guide for teachers and administrators
on implementing and evaluating the effectiveness of
this practice is provided in the table below that then
helps focus the elements that are necessary for students
and teachers to bring this practice to its full potential.
Winter 2015 CCTM 31
Colorado Mathematics Teacher
Implement tasks that promote reasoning and problem solving
Teacher and student actions
What are teachers doing?
Every practice is outlined in a similar manner that will
serve as a road map for teachers and administrators in
promoting the teaching practices in this book.
The second section of Principals to Actions is Essential
Elements. “Consistent implementation of effective
What are students doing?
teaching and learning of mathematics, as described in
the eight Mathematical Teaching Practices, are possible
only when school mathematics programs have in place–
t a commitment to access and equity;
t a powerful curriculum;
t appropriate tools and technology;
t meaningful and aligned assessment; and
t a culture of professionalism.” (p. 59)
To better understand each of these elements, an
overview is provided along with obstacles of productive
and unproductive beliefs and how to overcome those
obstacles. An illustration is also provided to show the
potential of that element at the district, school, or
department level.
“Taking Action,” the final section in Principles to
Actions, encourages everyone invested in the welfare
and success of a school and its students to become a
force for mathematics education—a force that “develops
mathematical understanding and self-confidence in all
students.”
32 CCTM Winter 2015
Colorado Mathematics Teacher
This section provides a bulleted framework for
implementing the Principles to Actions. Frameworks are
outlined for the following:
t leaders and policymakers in all district and states
or providences;
t principals, coaches, specialists, and other school
leaders; and
t teachers.
The role that each of these three groups play within
the educational community is delineated, in hopes of
implementing the practices and elements in a systematic
manner.
Although this has been a brief glimpse of everything
this book has to offer, we hope that you feel inclined
to read it for yourself, and that you find this book as
inspirational as we have. We imagine ways of using these
ideas to inform all those whom touch our students.
This may be done by influencing instruction, coaching
teachers, and empowering administrators to help every
student reach his or her fullest potential. Consider it as a
book study with your colleagues.
Winter 2015 CCTM 33
Colorado Mathematics Teacher
AWARDS
CCTM teaching and leadership awards: Will you be next?
Rachael Risley, CCTM Awards Chair
T
he Colorado Council of Teachers of Mathematics (CCTM) teaching and leadership awards provide
educators with the opportunity to celebrate their own
accomplishments and those of their colleagues. Each year,
CCTM honors up to one elementary (K–6) and up to one
secondary (6–12) teacher from each CCTM region in
Colorado. We also honor an individual who contributes
to mathematics education, but does not have full-time
classroom responsibilities (i.e., mathematics coaches,
coordinators, administrators, or higher education personnel who are dedicated to improving mathematics learning
in school districts). Amie Storlie
Charlee Passig Archuleta
Courtney Waring
34 CCTM Winter 2015
At the 2014 CCTM Annual Meeting, teacher awardees
recognized were: Amie Storlie, Harrison High; Charlee
Passig Archuleta, Rudy Elementary; Courtney Waring,
Asbury Elementary; Jami Nelson, Options Pathways
Alternative School; and Jennifer Jackson, Liberty Middle
School. The mathematics leadership awardee was Mindi
Simons, Harrison High. (Access the CCTM website to
find more about each of the awardees.) Congratulations
all!
Jami Nelson
Jennifer Jackson
Mindy Simons
Colorado Mathematics Teacher
Will you or a colleague be next?
1. Why nominate?
Most of us work in a school or a district where
we know someone who stands out in her or his
mathematics teaching. This teacher has students
who are living the standards of mathematical
practice daily in the classroom and is committed
to the continuous improvement of their practice,
by reflecting on student learning. If you are this
person, you are also encouraged to ask a colleague or
administrator to nominate you. (Nomination forms
are available at http://www.cctmath.org.)
2. Why apply when you are nominated?
Each nominee is responsible for filling out an
application that includes:
a letter about his/her teaching,
resume, and
letters of recommendation.
t
t
t
Many of us are so entrenched in our daily work,
that we haven’t given ourselves the time to reflect
on our successes—not only to remind ourselves of
our outstanding work, but also to inspire and teach
other mathematics teachers who are interested in
continuous improvement and learning. Additionally,
the Colorado State Model Evaluation System for
Teachers has placed reflection and leadership in
the foreground of the evaluation process. Quality
Standard IV: Teachers reflect on their practice,
emphasizes this value, and Quality Standard V:
Teachers demonstrate leadership, calls for the
collaboration and sharing of teacher knowledge with
the larger community. The CCTM awards application
process can provide evidence to support meeting and
exceeding the standards in these areas.
3. Process and Details?
March 15 – Nomination deadline.
April 30 – Application deadline.
July and August – Applications reviewed and
recommendations made to the CCTM board.
September 1 – Awardees contacted.
September 24 – Awards reception at CCTM
Annual Conference; awardees receive a plaque,
one-year CCTM membership, complimentary
conference registration for that year, and
$200.
t
t
t
t
t
CCTM looks forward to the nominations and applications, and to sharing your accomplishments with the
larger community this fall. If you have any questions,
please contact me at [email protected].
Winter 2015 CCTM 35