[Opening Page] [Table of Contents] [Project Overview] [Professional Development] [Number Sense Activities] [Evaluation] [Sources] [MSMME info] [Contact Author]
Providing
Professional Development
Logistics: The formal meetings supporting this project involved all 4 elementary math teachers (and myself) and took place during the school day. The elementary teachers have a planning time at the end of the day on Fridays while the students take part in activity periods (physical education, music, art, and library). The administration allowed my At-Risk teacher aide (a fully certified teacher, although not in math) to substitute for me during the last 40 minutes of our Geometry block on the days that the math teachers met. Other meetings with individual teachers took place on an “as needed” basis—before school, after school, during planning periods, etc.
This capstone project is a
small part of a much larger set of goals for the district’s math
department. We are in the process of
rewriting our curriculum with the hopes of providing a useful tool for new and
returning teachers, thus strengthening our students’ learning experiences, and
continuing to meet state evaluation requirements. Paramount to implementing a good curriculum is an awareness and
usage of good teaching methods. Along
with providing sound content information, NCTM’s 2000 Principles and Standards
point toward good instructional strategies that were discussed in greater
detail in NCTM’s 1991 Professional
Standards for Teaching Mathematics.
I hope our math teaching staff has opportunity to study these documents
and continues to refine mathematics knowledge and pedagogy long after the
details of this capstone project are completed.
Introducing The Standards
In keeping with this project’s goal of
improving mathematics instruction, the aim of my first formal meeting with the
elementary math teachers was to determine their familiarity with the content of
either the National Council of Teachers of Mathematics’ (NCTM) Principles
and Standards for School Mathematics or the state level Show Me
Standards. From the very beginning,
I also wanted the teachers to think about the methods they use as they teach
and the expectations they have for students entering their classrooms. I used a simple survey to collect this
information. To allow for teacher
reflection in the midst of demanding schedules, I asked that it be completed
within two weeks of our first meeting.
In summary, the survey indicated that all of the teachers were nominally aware of the Show Me Standards because they are the basis of our state assessment program. None of the teachers knew about NCTM or its Principals and Standards. Currently, our elementary teachers have no specialized training in mathematics beyond what is required for an elementary education degree. The results reaffirmed in my mind the need for continuing professional development and a coherent, well-articulated curriculum. I believe both of these items are necessary to improve our mathematics instruction and consequently, our students’ mathematical performance.
As the result of personal reading, presentations at NCTM professional meetings, and several classes in my graduate program, I have come to believe that number sense is the cornerstone of math success at most, if not all, levels. It involves far more than adequate paper and pencil computation skills. The phrase “number sense” has entered math teacher jargon and is seldom defined. While difficult to define succinctly, page 32 of Principles and Standards describes it as
the
ability to decompose numbers naturally, use particular numbers like 100 or ½ as
referents, use the relationships among arithmetic operations to solve problems,
understand the base-ten number system, estimate, make sense of numbers, and
recognize the relative and absolute magnitude of numbers.
NCTM further indicates that the Number and Operations Standard should receive its greatest, though not sole, emphasis early in a student’s schooling. (For a visual image, scroll down to figure 3.1 on page 30 of Principles & Standards.) Despite NCTM’s call for this emphasis, most of this district’s students exhibit a weakness in number sense and theory that creates an obstacle to math success when they reach junior high and high school. These particular student limitations are what first drew my attention to a need for change in our mathematics program.
Focusing on “Number and Operations” and Instructional Strategies
With my focus set on number sense, the intent
of our second meeting was to begin a study of the Number and Operations
standard and to begin discussing methods of teaching those skills and
concepts—paying particular attention to student-centered, active-learning
instructional methods. However, I first
found I had to put out a “fire” started by a survey question that concerned
discrete math. The Show Me Standards
were greatly influenced by NCTM’s 1989 Curriculum
and Evaluation Standards for School Mathematics, so they include a
separate strand for Discrete Mathematics.
None of the elementary teachers had any idea of what “discrete math” is,
and I refused to tell them until they finished their surveys! After a brief standards-based discussion
(scroll down to page 31 in Principles &
Standards, “Where is Discrete Mathematics”), we were ready to begin
some specific dialogue about Number and Operations**. At this meeting I also introduced the elementary teachers to
NCTM’s periodicals, Teaching Children Mathematics, Mathematics
Teaching in the Middle School, and Mathematics Teacher. I made my copies available to the teachers
as well as several books I have acquired over the years and encouraged them to
use the materials as resources for classroom activities and reflection. All four teachers took at least one book or
periodical, and two reported at a later date that they found useful ideas and
interesting “food for thought”.
**If you teach under the Show Me Standards, please note that they separate NCTM’s “Number and Operations” into 2 separate standards: Math 1 is Number Sense and Math 5 is Mathematical Systems and Number Theory. You can view these state Math Frameworks at http://www.dese.state.mo.us/divimprove/curriculum/frameworks/index.html.
The specific discussions of further meetings centered around three questions I asked the teachers to consider and respond in writing:
1—What do I expect students to know (in regards to number sense and theory) when they enter my classroom?
2—What do I expect students to know when they leave my classroom?
3—What learner activities do I use to instigate the growth I expect?
I asked for the entrance and exit expectations in order to help us identify possible gaps or overlaps between grade levels. Most of the teachers simply teach from the book or workbook that is in place at their grade level. The same text series (a 1992 edition of a very traditional series) is used in grades K-6. It provides little direction for activity-based learning or teaching “outside” the text. I asked for learner activities in order to have fodder for discussion of teacher-centered versus active student-centered instruction. The Teaching and Learning Principles of Principles and Standards (pages 16-21) emphasize the importance of choosing worthwhile mathematical tasks that engage students intellectually, nurturing a classroom environment where mathematical thinking is the norm, and expecting conceptual understanding on an equal basis with factual knowledge and procedural proficiency. In spite of this professional standard, most of the students entering junior high in this district appear to have had little experience with “doing” math in any context other than a worksheet or text page of problems. Their actions early in junior high indicate they believe school math is something you do on paper where the only important part is the answer. They seem to believe the teacher’s job as a mathematician is to give them a set of steps to follow in order to arrive at that answer in the shortest amount of time with the least amount of effort. These observations also fueled my belief in the need for change in our mathematics program.
Following our group discussions of the Standards, my focus on professional development narrowed to working with only one teacher. It is my hope that as we begin the next school year, we will continue professional development that will broaden all our teachers’ working knowledge of a variety of instructional strategies. Strengthening and diversifying teaching methods will provide our students with more occasions to learn mathematics in meaningful settings. It will also spark staff experiences that should help us meet the state requirement of including aligned learner activities and instructional practices in our curriculum document.
One-on-One Professional Discourse
Besides instigating discussions to improve
our teachers’ knowledge of the math they teach, this capstone also sought to
discover and test several activities that when used appropriately in the
classroom would help students cultivate number sense. (It is in no means an attempt to outline a complete “number
sense” curriculum.) In collaboration
with one teacher (I’ll call her “Betty Johnson”), my goal was to find number
sense activities that go beyond the traditional teacher-centered lesson, then
to assist in implementing them through modeling or team teaching. This shift to working with only one teacher
was at the advice of my graduate committee and the request of my school
administration. The teacher involved
had several years experience in the classroom but had never taught mathematics
at any level prior to the 2001-2002 school year.
The district administration requested that I spend at least one class period a week with the students in a specific grade level in order to assist Mrs. Johnson and to provide additional instruction toward our state math goals. Early in the school year, at her request, we separated Betty’s two-grade classroom during math time. While I wanted to work with her, Betty was tremendously concerned about the younger students’ need for individualized attention. I agreed with that assessment, so I worked with the older grade in a separate room while Betty worked with the younger grade in the regular classroom. It was during this time that I tried many of the number sense activities described in this website as well as other types of activities. To keep up with professional development, Betty and I discussed activities for both grade levels with regard to content, pedagogy, and success both before and after the classes met. Later in the school year both classes remained in the regular classroom during math time, but we worked with separate groups as before. Finally, during the latter part of fourth quarter, we combined the classes and did activities with both age groups together.
As a result of these activities and discussions Betty began to see how the rote paper and pencil methods by which we learned arithmetic do very little to support an understanding of number. She was reluctant at first to trust other algorithms for teaching computation, but later came to understand how they foster number sense. Number sense reflects how we think about numbers, and thinking develops in a child long before writing does. (Read Burns’ Math: Facing an American Phobia, pp. 8-12, for an excellent discussion of mental versus paper and pencil math.) Betty and I agree that we don’t want to forsake paper and pencil methods, but we do want our instructional strategies to be varied, to support student understanding, and to go beyond student memorization. These methods must begin when the student begins school, earlier if possible. We found and discussed several articles in NCTM periodicals that support these ideas. Out of those discussions I developed a packet of summer reading for all the elementary math teachers. We hope to use these articles for discussion starters as the elementary teachers continue to meet this coming school year (see the summer reading list on the Sources page).
This district’s students need better opportunities to develop number sense in the school setting. Many students do not have the same experiences at home that have helped develop number sense in past generations. Likewise, our mathematics teachers need opportunities to discuss what number sense entails and to collaborate on methods that efficiently meet students’ number sense and number theory needs. Number sense is only the beginning. Following our group discussions of number sense, we started a similar strategy for the Algebra standard (Patterns & Relationships in the Show Me Standards). Our next task will be the Reasoning and Proof standard. Efficient instruction and learning can only happen when in addition to sound content knowledge teachers have a clear understanding of what came before, what will come after, and what is expected overall. Ongoing professional development and a unified curriculum should provide that information not only for current teachers, but also for teachers new to the district or to the teaching profession.