“Modified Lesson Plans”
Promoting Systemic Change in
Middle School Mathematics Teaching *
Harel Barzilai, Homer Austin, Barbara Wainwright
Dept. of Mathematics/CS,
1.
Introduction
The
Allied Delmarva Enhancement Program for Teachers (ADEPT) consists of
interconnected mathematics content courses offered for graduate credit to
in-service middle school teachers.
Based at
The eight
courses in Math ADEPT are the following:
conceptual algebra, number theory, data analysis, mathematical modeling,
geometry, mathematical reasoning, technology, and Cartesian triad (algebra,
geometry, and Cartesian plane). So, the
content of the ADEPT courses reflects the mathematical content of the
middle-school curriculum. The ADEPT curriculum also promotes collaboration
among teacher participants and stresses the development of conceptual
understandings and contextualization of fundamental mathematics. Teachers modify existing lesson
plans with the support and feedback of colleagues and ADEPT instructors. Thus,
the ADEPT program is wed to the middle-school mathematics curriculum.
In this
paper the authors discuss the use of MLPs in three ADEPT courses: number
theory, conceptual algebra, and data analysis. A detailed sketch of each of
these three ADEPT courses is provided to highlight and bring to life the
environment in which the MLP assignments take place. Additionally the focus of
each snapshot is to illustrate how the use of MLPs brings about the ARC triad –
authentic curriculum, reflection by teachers, and PUFM-enhancing content
– which is key to effecting systemic change in the
standards-based teaching of mathematics.
2. Review of Literature
Research in
mathematics education has many faces ranging from traditional positivistic
forms to action research (Wiliam, 1999).
Practitioner research, often referred to as educational action research,
is but one of these faces. This kind of
research aims to achieve systemic changes, modifications or improvements by
intervening in the existing instructional system (Wilkins, 2000). Practitioner research naturally posits a
match between the needs of the practitioners and their policy makers, a match
that is often lacking in traditional kinds of research. Thus, in practitioner research, the quality
or nature of instructional processes is examined and investigated.
There
has been considerable concern over the teaching of mathematics in the U.S.
during the last several years, and the dearth of mathematics content courses
for in-service middle-school teachers has been of particular concern (Before
it's too late, 2000; Cuoco, 2001). Perhaps part of the reason for this lack
of mathematics content courses stems from a belief that middle-school teachers
“occupy an ill-defined niche between elementary and secondary [school] that has
no established national consensus concerning appropriate preparation” (Steen,
1995).
Apparently
there are many beliefs about teacher learning that are commonly held, beliefs
that are not grounded in fact, but rather in assumption (Ball). Some of these beliefs deal with the
professional development of in-service teachers. According to Ball, the following beliefs
about professional development are held by many, but need to be tested if they
are to be accepted as fact: a. Working closely with colleagues
fosters teacher learning. b.
One effective form of professional development is lesson study. c. Lesson study is an
effective form of professional development in
In
order to examine these beliefs, the literature in mathematics education
suggests directions for investigations.
Much has been written on the Japanese lesson study groups as a model for
professional development for in-service teachers (Fernandez, et. al, 2001). The TIMSS study noted that in the
There are
recurrent themes in mathematics education literature that highlight three other
aspects of professional development and point toward a theoretical
framework. Notable among these themes
are the ideas of authentic curriculum, practitioner reflection, and PUFM
(profound understanding of fundamental mathematics).
Even
though the literature on authentic curriculum has different definitions
depending upon the context, the idea is pervasive that the curriculum should be
in keeping with reality. In order to
have an authentic curriculum, it has been argued: “teachers have to modify their traditional
strategies so that they are multi-sensory, individualized, pragmatic, and
relevant to the common practice or real-life situations. Instructional activities need to be more
performance-based, have applications that have a direct connection to the
students' experience base, and be able to be generalized to daily living
situations. When teachers are able to
recognize students' specific needs, they can develop and implement
diagnostic-prescriptive teaching, …and modify
instructional procedures as needed” (Spinelli, 2001). Also, in another sense, a
professional development program for in-service should be authentic to the
extent that it is based on the teachers' real life (i.e., classroom realities).
The modified lesson plans (MLP's) are a part of the authentic curriculum in the
Math ADEPT program of professional development.
Research
on the reflective practitioner reveals the importance of reflective thinking as
critical for directing and informing practice.
Using reflective thinking, teachers can learn to interpret and create
new knowledge from their own experiences in teaching (Stein, 2004). As teachers become reflective practitioners,
they grow and expand themselves to a greater range of possible choices for
their classroom instructional activities and student behaviors (Larrivee,
2000). They modify their old ways of
doing things, and replace them with new ways that have arisen from their own
reflections. It has been said that
self-reflection is enhanced by feedback from others, both students and
colleagues (Walkington, et.al., 2001). Also, as teachers do self-reflection, they
need to reflect not only on what the activity is, but also on what is behind
the activity (Spilkova, 2001). In order for teachers to modify lessons, it is
crucial that they become reflective practitioners, for they must decide what is
good and not good about their lessons, what needs to be changed and what should
stay the same. They must decide how the lessons should be modified, what
the changes might accomplish, and how the lessons fit together with other
lessons on this topic.
When the
literature on the knowledge base for middle school teachers is examined, it
becomes clear that there is room for expansion.
A comparison of elementary teachers in the
The
Math ADEPT program is but one answer to the dearth of mathematics courses for
in-service middle-school teachers. One
of the many features of these mathematics courses is the requirement that
teachers must complete the assignment labeled “modified lesson plans” (MLPs). The MLPs in ADEPT provide an effective mechanism
for effecting ARC. They also lend
themselves well to conducting practitioner research in that this intervention
into the classroom is aimed at improving quality and bringing about systemic
change in the teaching of mathematics. Through ADEPT’s content courses, the
lack of sufficient mathematics content background of teachers, one of the main
obstacles to mathematics teaching reform, is being alleviated.
3.
Snapshot #1: Number
Theory course
The course,
Number Theory with a Multicultural and Historical Perspective, has been
designed and taught (by
All of the course content is wed to the middle-school curriculum: conceptual understandings that the teachers are gaining at a higher level are discussed relative to the middle-school learner. Likewise, all of the course content is considered relative to its historical and multicultural significance. The reasons for these features of the course are twofold: teachers need to understand that the development of mathematics has been, and still is, a multicultural or cross-cultural adventure; also, their students, who come from many different cultural backgrounds, bring insights and understandings (from their backgrounds) that influence their learning. Teachers and students together can build bridges to learning from cultural understandings. When teachers and their students understand this aspect of mathematics, the study of mathematics becomes a human activity. In this regard, speakers from various ethnic backgrounds are invited to speak to the class. These speakers tell about their schooling in another culture, and in particular, their instructional experiences in mathematics. Teachers have commented that this aspect of the course is beneficial to them in that it gives them a different perspective on many aspects of teaching.
The
learning environment for the classroom instruction in number theory consists of
lots of group work; so much time is given for teachers to talk with each
other. Group work is used for
problem-solving activities, discussion of topics, and exploration of O. C.'s
(Openings in the Curriculum). O.C.'s
occur quite often in this class because unexpected questions quite often arise,
and the questions beg for answers. In many ways, some could argue that the
classroom is unstructured, and it would appear that way if someone visited the
class. The classes, though, have been
all organized to explore certain content.
The ways in which the teachers do that are sometimes unpredictable.
In
all ADEPT courses, teachers modify existing lesson plans that they already
teach in their classrooms. The use of
modified lesson plans in the number theory course provides an avenue to bring
about authentic curriculum, reflective thinking, and PUFM in several ways. Teachers in this course bring their lesson
plans, and in groups of three to four, present their lesson plans to each
other. The teachers then give
suggestions for modifications of the lessons based on the content of the number
theory course. The instructor goes from
group to group and also gives ideas or questions for the teachers to consider
in their lesson modifications.
The
activity of modifying lesson plans fosters authentic curriculum in that the
content of the number theory course is tied directly to the content in the
modified lesson plans. The content of
the lesson plan deals with content from the course, whether it is factoring,
least common multiple, number sense or some other topic. The lesson plan is
also tied to pedagogical aspects of the course.
The modifications that the teachers make quite often include historical
or cultural considerations that were not in the original lesson. Modifications often are made to incorporate
more of the standards from the various states (MD, VA, DE). Because the modified lesson is one that is
taught and will be taught again, there is a transfer of the content of the
university course to the middle-school classroom.
In order to modify the lesson plans, teachers must reflect not only on their specific lesson, but also on how this lesson fits in with other lessons in this same content area. They also have to decide what they like about the lesson, and what they believe needs changing. A good aspect of having this activity built into a course is that it gives teachers the chance to revise lessons in an organized and helpful way. They feel supported in this effort by other professionals. They receive creative ideas and other viewpoints from their colleagues. Teachers have repeatedly written on course evaluations that they have found the modification of lesson plans to be valuable in that they were able to talk to other teachers.
Modification
of lesson plans also allows the teachers to develop a profound understanding of
the mathematics in the lessons (PUFM).
Many have found that their original lessons were heavy on rote
manipulation or calculations and light on conceptual understandings of the
topics. So, a modification to the lesson plan often includes some hands-on
activities to shed light on the mathematical content underlying the concepts
being taught. Some might modify their
lessons to include physical models of the concepts. Others might include lists of probing
questions to bring about class discussions of the concepts. When teachers
incorporate any of these strategies into their lessons, their understanding of
fundamental mathematics is sharpened.
Example
of modified lesson:
Ms. Brown (pseudonym) has taught for approximately
ten years in a middle school in a rural area.
She modified a lesson on "factoring polynomials." Her motivation for this topic came from
seeking shortcuts for multiplying positive integers, thus making a connection
between algebra and arithmetic. Ms. Brown’s lesson includes a pre-assessment of
the backgrounds of her students before the lesson. She clearly states the objectives for her
lesson. The lesson includes group work
with manipulatives (algebra tiles) where each group has been given a list of
tasks to do with the tiles. These lists
of activities lead them to discover factors of quadratic polynomials. Ms. Brown indicates in her lesson plan the
modifications she has made to improve the lesson. She has included more work on problem solving
by asking the students to “apply and adapt a variety of appropriate strategies
to solve the problems: try using
negative factors using the 'cover-up' approach.” She has included a modification to make and
investigate conjectures by having the students “try different types of
factoring; i.e., perfect square trinomials and difference of squares.” She has also included an activity to help her
students make connections. She guides
her students “to use factoring to solve quadratic equations in word problems. For example, use word problems dealing with
the dimensions of a picture frame or a structure bordered by a walkway.”
Ms. Brown's modifications indicate that she has done constructive reflection on her lesson, increased the complexity of the work required of her students, made connections to other situations, and led her students to a more profound understanding of the mathematical content of the lesson. She has modified a lesson that could have been dull with mechanical manipulations to one that is conceptual in nature with hands-on activities for the students. Her modifications of the lesson plan indicate that she has brought the lesson into an “authentic curriculum” in that she will teach the lesson again with a more “profound understanding” on her part. This understanding has been the result of her reflections on the lesson, her willingness to make modifications, and her interactions with other teachers.
4. Snapshot 2: Conceptual Algebra
ADEPT's algebra
course, Conceptual Algebra, is designed and taught (by Barzilai) to help
teachers encourage their students to move beyond the “symbol-manipulation”
school of algebra -- and to ground algebra in meaning and understanding.
Towards that end, the course incorporates, and is fundamentally based upon,
several thematic strands. The primary, and often
overlapping strands are as follows.
Problem solving and modeling.
Teachers analyze number patterns and seek to find algebraic representations.
They also seek to model word problems by translating a given situation into
algebraic language. Teachers are not merely allowed but encouraged to use
multiple strategies and to find several solutions. This theme of multiple
paths provides teachers with experiences which they can use to support a
diversity of learning styles among their students. More than that, it also
reinforces their own understanding that there is often more than just "one
way" to solve a problem in mathematics. Teachers also experience
first-hand how the multiple representations which grow out of having a
variety of strategies, can help deepen one's understanding of a given problem,
method, or concept.
Error Analysis. Using examples
provided by the instructor, along with a standard reference work (Ashlock,
2001), teachers analyze common student mistakes, and seek out the conceptual
root-causes behind these frequent and recurring mistakes. This keeps the
teachers who are taking this course honest about keeping meaning – rather than
memorized mechanical manipulation – at the heart of algebra. After all, if the
focus is to shift away from memorization and towards understanding in the
teaching and learning of algebra, then it is with an eye toward the diagnosis
of conceptual errors that teachers must tailor their approach to common student
errors.
Connections between algebra and
other fields of mathematics are explored -- particularly connections to
arithmetic, combinatorics, and geometry. One teacher remarked on her weekly
written summary of her learnings in the class that, “After rereading my notes
from last week, I felt like I was beginning to make connections between things
I have done as challenge problems in my classes that I didn't even know were
related. For example I have done the Gauss problems and the handshake problem
and never did I see that both answers were arrived at (or could be) arrived at
using n(n+1)/2.” Another teacher wrote that “I was surprised at the number of
connections that are being made in class to problems that I have seen before
but thought were very separate types of problems”; another wrote, “Algebra
doesn't seem so unconnected to me as it once was.” The class also employs picture
proofs or visual representations which demonstrate why the distributive
property holds, why (a+b)2 does not equal a2+b2
and where the extra terms come from.
Written mathematical
exposition
assignments are used throughout the semester. Teachers summarize what I have
learned this week as well as questions or confusions I have in
complete sentences and paragraphs, along with tables, graphs, labeled diagrams,
definitions, examples, proofs, and so forth. These, along with careful, precise
write-ups of selected problem-solving exercises completed in class are written
up and handed in, with individualized handwritten comments and feedback
provided by the ADEPT instructor. Teachers select from among their mathematical
expositions as they build their Course Portfolio, which along with the Modified
Lesson Plans (MLPs) make up the concrete products teachers take back to their
schools after completing the course. Related to mathematical exposition is the
notion of correct mathematical grammar. For example, given a problem
like if
c = 12t+4, and if t=3, find c we should not write “12*3 = 36
+ 4 = 40”; teachers in conceptual algebra not only explain why not, but
find their ears become better tuned to seek out such mistakes, and report being
"more careful with my own mathematical grammar" as well as that of
their students, in their own classes.
The above are the
main thematic strands of the course. The course uses a combination of a primary
textbook (Driscoll, 1999) along with NCTM materials (Friel, 2001), and
activities and materials designed by the course instructor. In content, conceptual
algebra progresses from patterns to “algebrization” or the process of
getting from “I see the pattern!” to finding a correct general formula. The
course discusses graphical representations, functions and functional notation,
linear functions, quadratic functions, and some discussion of exponential
functions and qualitative aspects such as “increasing at a decreasing rate”.
Particular attention is given to studying relatively elementary concepts from a
more advanced, teacher's point of view. For example, teachers are asked such
questions as: what can be said about a function having the property that f(k*a)
= k*f(a) for all k and a? What can be said about a function which satisfies
f(x+a) = f(x) + ca for all a and x (and where c is fixed)? Are these two the
same classes of functions? Why or why not?
Pedagogically, the
course is very hands-on. Virtually every class starts with a “warm-up” – mathematical problem-solving which builds on
earlier material and extends it in a new direction. These are usually completed
individually, although collaboration is allowed for some of the more advanced
problems. Anonymous comments from teachers on mid-semester feedback forms indicates
teachers appreciate this feature, with statements like “I can count on my brain
being challenged every class..very enjoyable” and “In spite of my trepidation
in taking this course, I am enjoying it and it is stretching my brain” being
typical. A similar candid remark from a teacher was that “At first I was
scared, but I have come to enjoy the challenge and the conversations/lessons in
class. I am learning [smiley face]” The warm-up leads to a discussion, and
often to presentations of one, two, or more solutions at the board by the
teachers. The course includes highly interactive lecture-presentations, and
in-class groupwork for problem-solving along with the written assignments
discussed above.
Although the class
meets once per week, a web-based discussion board allows teachers enrolled to
stay in touch between class meetings for informal exchanges, and occasionally
for a required commentary on readings, which come from the textbooks, and from
sources such as articles in NCTM journals. Technology is also incorporated
through the use of java applets such as those at illuminations.nctm.org (Illuminations)
and http://argyll.epsb.ca/jreed/math9/. Manipulatives, which are the most concrete of
technologies, complement the course. The Hands On Equations system (Borenson)
is introduced, along with both its strengths and the care on the part of the
teacher that is required for fostering conceptual learning in algebra.
For the Modified
Lesson Plans assignment, the course designer/instructor of conceptual
algebra (Barzilai) uses a step-wise approach. Teachers complete a
three-part concept map of algebra customized to the algebra course they teach.
This concept map includes "inputs/prerequisites" coming before their
course, “outputs” including future applications, and a central web of
inter-related concepts and how they fit together. Feedback is provided by the
ADEPT instructor as well as by fellow teachers, and this input is used for
creating a revised version. This process really encourages teachers to
introspect and reflect about the content they teach, and about where it fits
into the wider curriculum including other classes as well as state and national
standards. Remarked one teacher, “One of the main lessons I've learned this
week is how to think about my own thinking. The problems we solved in class
made it possible to look at the way we teach. The way we understand
mathematical problems as teachers helps us guide our students when they are
faced with the same problems.”
At the outset,
teachers are provided with a rubric of the key elements that will be valued in
their MLP assignment (http://barzilai.org/courses/541/activs/rubric.lessons.html). These elements
include organization; an introduction; a professional look; new mathematical
content and mathematical correctness; innovative pedagogy; clearly highlighted
differences between the old lesson plans and the new; clear objectives for the
lessons and some plan for assessment; a listing of relevant local, state, and
national standards; and a reflective concluding narrative.
Next, one-on-one
meetings with the instructor are held in class with each teacher. The course’s
ADEPT Teaching Assistant facilitates groupwork during these consultations,
which help teachers focus on which potential topic(s) for MLP revision would be
most appropriate and more professionally relevant to their needs. The topic is
then due usually at the Week 4 or Week 5 class. The choice of topic depends, of
course, on what “algebra” means for each particular teacher: a pre-algebra
class, Algebra I, Algebra II, or a special pre-teaching class for students
needing extra help, for example. A few ADEPT participants are resource teachers
and their MLP consists of their working with another teacher at their school to
modify her or his own lesson plans.
Teachers are also
encouraged to collaborate by working in teams to give feedback and constructive
suggestions on their lesson plans. Additional consultations with the instructor
are always available, and the web-based discussion boards are sometimes used
for brainstorming as teachers share their local situations, challenges, and the
lesson plans they are considering revising. This provides yet another layer of
professional development as a culture of mutually supportive colleagues is
strengthened among the teachers, while also providing further opportunities to
introspect about their teaching practice.
By week 9, a formal
Outline is due and additional one-on-one consultations between teachers and the
instructor normally take place at this time. The Outline includes a
specification of the class, grade level, and a three-column table organized
under the headings of Topic; Old Plans; New Plans. In this way the teachers
visually organize a 1 or 2 page overview of what they believe the most
important areas of content and pedagogy are which they plan to change, add to,
or leave as is. The final MLPs are
handed in during week 13, and presentations at the board, sometimes
complemented with computer presentations, take place during weeks 14 and 15 to
round out the semester. Here again professional growth and sharing take place
as teachers are encouraged to hand out hardcopy of parts of their lessons which
their fellow teachers may find useful.
Examples of MLPs
One teacher, Ms.
Jones, focused the enhancements she made in her MLP project on operations with
polynomials. Instead of restricting herself to examples, guided practice, and
workbook, her new lesson plans use algebra tiles which were introduced in the class.
Several others teachers have incorporated Hands On Equations. Almost all the
teachers added conceptual dimensions highlighted in class for their MLP
project. Most popular among these were the use of picture proofs; the use of
color to highlight structure when going form patterns to algebraic formulas;
the use of written assignments and more open-ended explorations for their
students; and the importance of their modeling correct mathematical grammar for
– and expecting it from – students in their middle school algebra classes.
Teachers leave the
course stronger and more confident in algebra, and aware of the importance of
ongoing professional development to deepen their background. Remarked one
teacher in her course evaluation, “This class certainly met the needs of me as
an elementary-certified teacher. Although at times this class was difficult, I
understand more algebra and how it fits into my curriculum than I did [upon
enrolling]. Consequently, I certainly know more about algebra for teachers than
I thought I would ever understand. It also makes me aware of exactly how much
more I need to learn..I look forward to the remaining ADEPT courses as it can
only enhance my ability as a teacher”
Finally, the process
of experiencing new ways to learn algebra personally, combined with the MLP
assignment which links the course directly to their own middle school
classroom, leaves teachers more confident to make changes they might not have
otherwise been ready to make. An email from one teacher a half year after her
course reads:
“This summer, I took the
Concepts of Algebra class. Immediately upon returning to school I began
implementing the ideas in my classes. With trepidation I invested a lot more
time than I thought prudent in the development of algebraic thinking through
examining and verbally explaining patterns , writing, twisting the
circumstance, then revising... before the lessons on translating expressions
into symbols, and finding function rules for simple functions.
I am still
amazed at how immediately the task was accomplished. My most unmotivated
students are involved, all have been successful at some level...”
5. Snapshot #3:
Data Analysis Course
The Data Analysis course has been developed and taught (by Wainwright) with the objective to provide middle school teachers with a profound understanding of descriptive and inferential statistics. Since the call for data-driven statistics courses in the early nineties (Moore & Witmer, 1991), there has been a shift in courses from textbook problems to data analysis. This course was designed around the same perspectives on the teaching of statistics as those described by Wainwright and Austin (1991); that is to say, the course is data-driven and focuses more on interpretation than calculation. An emphasis is also placed on the various types of data and the kinds of descriptive and inferential statistics that are appropriate for these types of data. What makes this course different from an introductory-level statistics course is the level of understanding and synthesis of the material that is expected at the graduate level. Although this is not a methods course, pedagogy is woven throughout the content material.
Many middle
school teachers do not have much content knowledge in the area of
statistics. They are often teaching
lessons involving basic data analysis.
These teachers routinely follow handouts/lessons provided by the
Maryland State Curriculum or use lessons straight out of the textbook. Throughout the Data Analysis course, these
teachers learned more content and asked relevant questions pertaining to topics
they teach on a regular basis.
Individuals expressed interest in learning certain content areas with
which they were uncomfortable. During
the class sessions they also conducted hands-on experiments and simulations in
groups, many of which they could modify for their own students.
The
focus of this course is on the interpretation of statistics. The calculated statistics are often obtained
using a calculator or computer; however, how would one then use these to
describe the data or make reliable inferences about a population? These are things that both the middle-school
students and the middle school teachers need to learn. The teachers also need to learn correct
terminology, as well as correct mathematical and statistical grammar. These
middle-school teachers admitted that they did not understand the concept
of "standard deviation" or its
usage. They could construct boxplots and
have their students construct boxplots using the graphing calculators, but they
could not read and interpret the graphs!
On one assignment, it was discovered that they could not correctly read
and compare histograms. Many were
interpreting the heights of the bars, i.e., frequencies, as the actual
measurements. Using simulations, the
teachers studied probability. They also
discussed and compared theoretical and empirical probabilities. Several of the teachers stated that they had
heard of both theoretical and empirical probabilities, but they did not know
what the difference was until they conducted these explorations.
This
class met once a week for three hours.
The class met in a regular classroom for the first half of the
period. A computer lab was reserved for
the second half of the period. For most
of the weeks, the teachers went to the lab after taking a short break half way
through the period. Whether they moved
to the lab or stayed in the classroom depended upon the activities planned for
that half of the period. The computer
software MINITAB was used in the lab for some of the simulations and for
statistical analyses of data. Although
most teachers do not have access to MINITAB in their own classrooms, some
teachers were downloading trial versions of MINITAB for possible use in their
schools. Activities were also performed using the graphing calculators, since
many teachers have access to them in their own classrooms.
As
with each ADEPT course, each teacher is required to modify an existing lesson
plan. In this course, they are also
required to conduct a group project. Students are required to generate a
problem or question they want answered, design an experiment or a survey
instrument, collect data, perform the appropriate descriptive statistics, carry
out the appropriate statistical analysis, and make inferences or conclusions to
answer their question. Each group gives
a report of the project results at the end of the semester. By midterm, the students were required to
have selected the members of their group and to have a project topic or
question approved. At this time they
were also asked to inform the instructor concerning the lesson that they
planned to modify. The teachers were then
encouraged to share the lesson with classmates to get feedback. During the latter part of the semester, the
teachers were given the last 30-45 minutes of the lab time to work on the
modified lesson plan and/or the group project.
During this time the teaching assistant and the instructor circulated
around the room and offered assistance.
The
requirement of a modified lesson plan gives the teachers the opportunity to
reflect on their own teaching and work on ways to improve or modify their
lessons. These modifications would
include added content and different activities to enhance student
learning. For some teachers who have not
yet taught statistics lessons, a lesson from a textbook or website was used and
further enhanced based on content and pedagogical methods acquired in the Data
Analysis course. In these cases, the
curriculum is not as realistic as it is for those who are revising existing
lessons that they actually teach, but they do have the experience of changing a
lesson to bring about different levels of cognitive tasks for the middle-school
students. In this way these teachers,
who are not teaching statistics now, can reflect on current teaching practices
and concentrate on how they might in the future teach a statistics lesson,
paying attention to both content and pedagogy. Modifying these lessons allows
the teachers to develop a more profound understanding of the statistics content
with which they have previously struggled and misinterpreted.
Example of Modified lesson plan:
Ms. Smith chose to modify a lesson
involving probabilities associated with families having three children. In the past she has had her students simulate
the experiment. The students are
generally provided with results of 40 trials and are asked to simulate ten more
trials by flipping a coin and creating a family: if the coin falls “heads”, the
child is a girl; if the coin falls “tails”, the child is a boy. The original lesson then had “the students
calculate the experimental probability based on the forty given trials plus the
ten additional trials that they conducted.”
To modify the lesson, Ms. Smith extended it to “not
only look at the experimental probability but to also calculate the theoretical
probability.” She plans to have the
students use tree diagrams to aid in counting all the possible combinations of
a family with three children. Ms. Smith
also plans to combine the total of fifty trials from each student to have a
total of several hundred trials (depending on class size). Students will then be able to compare the
results of their fifty trials to the combination of all the trials and to the
theoretical probabilities. This will
lead to a discussion of the Law of Large Numbers.
Ms. Smith's modification of her lesson shows that she has gained a deeper understanding of the ideas of probability. Her comparison of empirical results to theoretical probabilities shows that she understands the idea that the theory of probability is only a mathematical model. The empirical results most likely will be close to, but slightly different from, the theoretical probabilities. Her extension of this concept to lead into the law of large numbers shows that Ms. Smith is building a context for a future lesson. Thus, she has laid the groundwork for having her students perform cognitive tasks that are more challenging than the ones in her previous lesson.
K-12 Professional development programs for in-service mathematics teachers present multiple challenges for both institutions of higher education and participating school systems. Mathematics content courses designed for in-service teachers, in particular, need to combine mathematical rigor with a real-world relevance and applicability to schools, without which systemic changes will not easily follow. Through the use of MLPs, the ADEPT program grounds the experiences of participating teachers in professionally relevant projects which tie the rigorous mathematical content, as well as the authentic curriculum and pedagogy of the ADEPT courses, directly to the everyday practice of revising lesson plans in which teachers must engage. The process of revisiting and revising existing lesson plans also necessitates practitioner reflection, which is a prerequisite for “profound understanding” of mathematics by teachers.
Based on informal feedback and preliminary assessment from a variety of data sources, the ADEPT program is working well in a tri-state region and serving the needs of teachers who have a variety of mathematical backgrounds and needs. The authors believe, based on this experience, that this program, and particularly the MLP aspect of ADEPT, can be adapted for use in other universities’ in-service professional development programs for mathematics teachers; nevertheless, additional field-testing and research are needed to examine more closely how replicable this model is. Future research is planned in which PUFM will be defined through the performance of K-12 students on higher-level cognitive tasks and reflective teaching practice and authentic curriculum will be likewise defined and measure through parallel assessments and data collection.
Finally, a natural extension of the use of MLPs would be the establishment of Japanese-style Lesson Plan Study groups. It would seem that a community of teacher-colleagues, all having gone through a personal revision of their lesson plans, should – if suitably supported by post-secondary institutions – be able to transition to a regular and ongoing Lesson Plan Study model.
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* Support for Math ADEPT and the six
Math ADEPT courses was provided by the National Science Foundation ESIE
division, Teacher Enhancement program grant #5-28050. Support for Maryland Math
ADEPT (MD-ADEPT) and the two MD-ADEPT courses was provided by the Maryland
Higher Education Commission (MHEC) via Eisenhower funds, grant 4-30150.
"This material is based in part upon work supported by the National
Science Foundation under Grant No. 0101907." -- "Any opinions,
findings, and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the National
Science Foundation."