[Opening Page] [Table of Contents] [Project Overview] [Professional Development] [Number Sense Activities] [Evaluation] [Sources] [MSMME info] [Contact Author]
Evaluation
Much of what this capstone project hoped to begin could
not be assessed immediately. Changes in
teaching practices are often slow in developing and usually qualitative, rather
than quantitative, in nature. Curriculum
development and teacher training in reality needs to be an unending process—the
moment a teaching staff believes they are “done” is exactly the moment they
must begin again or risk growing stagnant.
Changes in society, new developments in mathematics, and advances in
learning theory should cause changes in our math classrooms. I hope to continue helping develop
meaningful activities that use a variety of appropriate teaching methods. I hope to be a part of building a
mathematics curriculum in which every teacher understands how each topic fits
into our math goals from K-12. I have
no doubt that our students will benefit from the clarity of purpose ongoing
training and a cohesive curriculum will provide.
With that said…
An essential part of any
project is short-term as well as long-term evaluation. I found assessing professional development
in the short-term a difficult task to complete. Teachers appraise student performance day in and day out—but if
you mention “assessment” with teachers on the receiving end, sheer panic often
ensues! Professional development
initiated from within the ranks of a district is also prone to personalities
and politics. I can, however, provide
anecdotal evidence of some improvement in mathematics knowledge and instruction.
Evaluating Professional Development
During the introduction to
the Standards, the teachers were willing to learn about the content of
the different strands. They were, in
fact, eager to learn about discrete mathematics since they discovered it
was in the Show Me Standards and had no previous knowledge of it. Prior to the introductory survey, the
teachers had done little thinking about what topics were considered
“mathematically important”, at least as defined by anyone besides themselves or
the chapters they used from their classroom textbooks. Their discussions reflected that they knew
they would be better able to prepare the students for state assessment if they
understood what the state considered mathematically important. They acknowledged that our set-aside meeting
times forced a consideration of the Standards that previously had been
put off for “more pressing” tasks. Our
discussions on the Number and Operations strand centered around questions of
identification—“Is this task part of Number and Operations?” or “When I teach
this, where does it fit in the Show Me Standards?”. The reflective writing I asked them to do
after our discussions suggested an understanding of what the Number and
Operations standard entails.
All the teachers indicated
a readiness to engage in professional reading, but only half appeared to follow
through and report on their findings.
One problem often cited by large-school and small-school teachers alike
is the lack of time to do anything 
besides prepare tomorrow’s (or next hour’s)
lessons. Teachers fill many
responsibilities that have little or no relationship to their teaching duties;
that problem is often exacerbated in the small school where the same number of
jobs is filled by fewer people.
However, I believe the time spent reading good professional documents is
justified by the ideas found that make the teaching process more efficient or
more productive. I have found the NCTM
periodicals contain a large number of helpful articles as do several Math Solutions
publications. The Internet contains a
wealth of perspectives and ideas, although finding really good sources can take
much time. Unfortunately, many
materials on math and teaching math are poorly written, and many
well-intentioned teachers have fallen asleep in their chairs trying to wade
through to pertinent information!
When it came to discussing
instructional strategies and learner activities, I met with far less
receptiveness. The generally accepted
definition among our teachers of a “student-centered” activity was one where
the child was doing something instead of the teacher. On the surface that may seem a reasonable (albeit generic)
definition, the majority of our teachers translate that definition to mean
worksheets and textbook assignments (since the students, not the teacher, do
them). Thus, they are satisfied that
their instructional practices include a balance of both teacher- and
student-centered activities. It is a
delicate thing for a fellow teacher to suggest changes in another teacher’s
classroom. It was my intention to model
lessons that used active-learning strategies.
While I enjoy learning by following and modifying the example of others,
not every teacher appreciates that method of acquiring knowledge.
Another obstacle we
discussed to experiential learning is the need for additional materials. Our elementary school has very few
manipulatives due to a prior administrative philosophy. Our teachers do make a few hands-on
resources, but making them requires time and personal expense. Facilitating their use can be difficult when
the teacher is not comfortable with an active-learning setting. The physical structure of our elementary
school also makes student-centered activities difficult because sound travels
with very little impediment. These
obstructions may take creativity to overcome, but I believe they can and must
be overcome. Inconveniences and
violated comfort levels must not stop us from providing each student with
opportunities to experience mathematics rather than observe it. One teacher of the four endorsed the same
emphases on a student-centered classroom as I.
She and I worked together to develop active learning opportunities that
supported number sense concepts. Her
growth took place not so much in instructional methods as in the knowledge of
sound mathematics. She has become a
mainstay in communicating and demonstrating diversified teaching strategies.
Evaluating Number Sense Activities
It was easier to evaluate the
activities that foster number sense. I
cannot provide quantitative evidence as to how much they improved our students’
number sense, but I can attest to individual growth and new potential for
growth in many students in grades 3-12.
As we developed and tested these activities, it was my goal to find a
way to gauge their success and possible impact without bogging down already
time-challenged teachers. I wanted a
written tool that encouraged reflection and would provide documentation for
revising and using the activities in other classes. A rubric is one tool used to communicate expectations as well as
gather information in a particular performance situation. I believe this kind of assessment tool helps
keep a teacher (or student) focused on what is important. In the links below, you will find the rubric
and responses for the number sense activities described on this website.
How
many laps make a mile?—grade 4
Going
Up, Coming Down dice game—grade 4
Don’t
Sweat Long Division—grade 5
Let’s
Get Close dice game—grade 7
Counting
Stars & Circles—grade 7
Cutting
uncut pans of brownies—grade 7
Warm Up
Puzzles—grade 8 (and up)
Unexpected results
I expected the results of
improved instruction, particularly in number sense, to be gradual. I hoped to see more mathematically powerful
students—2 and 3 years from now—after we had had a chance to polish our
projects and get a solid program in place.
I was surprised to find evidence of improved number sense on year-end
exams this year! The tenth grade
students were expected to perform a decimal calculation in the context of a
scale drawing
. They
needed to multiply 3.2 by 1.5.
Calculator use was permitted and encouraged. Many of the students showed for their work 3.2 + 1.6 = 4.8. At first I was puzzled—until I realized they
were doing the work mentally, using the distributive property to do the
computation. What really startled me
was this—I had done very little in the way of number sense projects with the
tenth grade. What I had done was
use the same “talk out loud” method of mental computation that I was
emphasizing in grade 7 & 8. The
tenth grade algebra or geometry class would be faced with a computation. They would reach for their calculators; I
would strive to complete the problem using mental math before they could get
the same answer. If I was successful, I
usually did a gloating little dance of glee—to the cheers of the students! If I was wrong, they were glad to back up
and find what I did wrong. The
result: students who a year ago would
have blanched at the thought of doing a decimal calculation without a
calculator did a decimal calculation mentally even when calculators were
available and their use encouraged.
Anecdotal, yes—but, wow!
This small evidence of
success—and similar ones—is edifying. I
truly believe that number sense is the gateway to more mathematically powerful
citizens. I hope that my fellow
teachers and I—educated in rote skills that require less understanding—will be
alert enough to recognize the valid “new” ways our students think about numbers
as a result of strengthened reasoning skills.
I hope we will be mathematically confident enough to keep from snuffing
out new ideas and settling for ways that are more familiar to us, particularly
if those ways are less sound in terms of promoting number sense. I hope we will be daring enough to try
teaching practices that may take us out of our comfort zones if they are for
the improvement of our students’ learning experiences.
What did I learn?
The goal of this project
was to improve mathematics instruction with a particular instructional strategy
(student-centered learning) and a particular curricular topic (number sense) in
mind. However, there are things that
can be said in regards to what I learned about professional development in
general. I would encourage any teacher
who sees the need for change in his own classroom to study the Principles
and Standards, to read what other professionals say about the topic under scrutiny,
and to try new methodologies. Real
teaching means continually seeking a better way, even if it requires stepping
out of a comfort zone. In terms of
affecting change on a district-wide level, I feel professional development will
be more meaningful and productive if the need for change is perceived by more
than one teacher. This is difficult in
a small school setting where the hard work of elementary teachers culminates
beyond their sight in the hands of one high school teacher. Teachers, like most adults, resist change if
they see no need for change. However,
if it is possible for teachers to collaborate so that all those involved are
aware of the assets and shortcomings to the curriculum and their instructional
practices, real professional growth may become a unified extension of the
collaborative effort. Tools that may be
helpful in pointing to these positive as well as negative attributes include
feedback from state or national assessments.
However, teachers must have confidence that these tools are truly
measuring what they purport to assess.
Frequent local assessment built on clearly understood and agreed upon
standards provides critical information in a timelier manner. I believe the key to success in professional
development is multi-fold: good
communication, free from any animosity and political agenda; well-defined
standards to provide teachers a foundation for instructional planning;
frequent, trustworthy assessment by which teachers can gauge student and
professional progress; and teacher ownership of change.
Where to from here?
While the requirements for
this Masters’ degree have been fulfilled, this “project” is far from
completed. The ideas and goals that began
with this capstone are still 
a work in progress.
Teaching staffs change; materials change; ideas change; state education
department requirements change. It is
my goal to study each of the NCTM strands with our teaching faculty over the
next few years. To provide continuity
and to keep us from reinventing the wheel each year, we are in the process of
developing a curriculum document that I hope will actually be utilized as well
as useful. The Show Me Curriculum
Administrators Association (www.smcaa.org)
makes available a software tool (called the Electronic Alignment Tool) that
will assist us in aligning our curriculum to the Show Me Standards. In addition it allows for the easy insertion
of instructional practices, learner activities, and assessment methods on the
same screen. I hope we can build a
“catalog” of student-focused activities from which old and new teachers can
choose while planning instruction.
Beginning in August 2002, I
will begin a 3-year training program in performance-based assessment designed
to enhance standards-based, performance-oriented instruction. The state Department of Elementary and
Secondary Education sponsors the training and will continue to do so as long as
funding is available. Part of my
responsibilities will be to return to my district and provide parallel training
for all the teachers in our district.
The need to improve and expand our methods of assessment has been a
concern of mine for several years.
While I was not able to address the issue fully in my graduate program
at Montana State, I hope this state-initiated program will provide me with the
knowledge and training skills to affect positive change in our district.
Not to be taken for
granted, the primary focus of all we do is our students. They are our children, our community, and
our leaders now and in the future. The
real winners in any plan to improve our school should be the students. I truly believe that with enhanced teacher
knowledge and teaching methods, students can encounter mathematics in
comfortable and confident settings.
Positive experiences in mathematics from kindergarten through grade 12
will strengthen our students’ abilities to reason and make decisions. Good reasoning skills and informed decision-making
can provide people greater opportunities to change their lives and our
community for the good. Our community
is a wonderful place to live. My hope
is to keep it that way—perhaps even make it better.
