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300-Level Elementary Math Education (Writing-Intensive)
ANALYZING MATHEMATICAL CONCEPTS AND PROBLEM-SOLVING
PROCESSES
My idea in teaching and learning has a
lot to do with reflection. As a teacher I think back on the day and
ask "why does that work?," "why didn't it work?,"
"how could I have done that differently?" It's a matter of
looking back on what's happened. Part of the larger process I want my
students to be involved in is looking back on what they've done and
making some kind of analysis, seeing what kinds of questions they
have. -- Professor Joseph Zilliox
The journals helped me learn more about
myself. Math was fun. I learned that children see math differently
from me -- I need to think about relating to children. My expository
writing has improved. I can express myself more easily on paper than
before. --Student
| COURSE GOALS
The focus of the course is to
encourage students to question the past and present pedagogy in
mathematics education. Students reflect on their own processes for
problem-solving and relate their experiences to the learning
processes of their future students. Through a workshop approach,
students discuss mathematical concepts and operations appropriate
for the elementary school; students collaborate on class
activities to develop teaching strategies and problem-solving
methods. |
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| WRITING
ACTIVITIES
1. PERSONAL RESPONSE IN THE
LEARNING LOG
Students' write their reflections
on course content and methods and respond to ideas emerging from
class interactions and assigned reading. Students may also
include responses to particular issues raised by the instructor.
Although one log is required each week, students are encouraged
to write more often. Logs are collected occasionally, responded
to by the instructor, then returned to students. Logs are not
graded but failure to keep logs will negatively affect a
student's grade. Students write their logs on loose leaf paper
or on the computer and collect them in a binder or folder.
In the following example, a
student asks a real-world question in the learning log: How do
we [teachers] do it all? |
Logs helped
because the writing forced me to reflect on what I was doing in
the class and helped me get an idea of the progression of my
thoughts about the teaching of math.--Student |
. . . . There is a lot of
preparation that I need to do before I even think of doing the
heavy-duty geometry. I have a big question for you. How do we
do it all? I mean, how do we add in geometry while teaching
addition, subtraction, division, and multiplication? In my
third grade OP class, most of the kids barely get to
multiplication by the end of the year. I really want to put
more geometry in the classroom, but as far as
"sequencing" is concerned, I find it hard. Any
suggestions?
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The instructor responds directly
on the learning log entry with these suggestions:
Throw out some of the computation.
How much practice do we need? Connect the geometry to the
operations. Ex: Build rectangles with 36 small squares. What
is the length and width of these rectangles?
2x18...........3x12..........4x9...........6x6
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As
much as I don't want to write them at times, logs have proved to
be a great resource. Logs catalog events along the way; teachers
can look back at daily log entries and know where to modify
class teaching and techniques. Children can dialogue privately
with the instructor and take notes for future
reference.--Student |
| PURPOSE: The
learning logs provide students with opportunities to explore
ideas, clarify thinking, pose questions, express concerns and
interests, and assess the course, the instructor, and
themselves. The logs allow the instructor access to information
about the student unavailable in class discussions or in group
activities. The instructor writes comments or poses questions in
the logs. Students write frequently about how they are thinking
about mathematics and about the teaching of mathematics. Logs
may also be used as notes for the mid-term and final exams. |
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2. PROBLEM-SOLVING TASKS |
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Every two weeks the instructor
distributes a problem-solving task dealing with mathematical
ideas related to the content of secondary and elementary school
mathematics (e.g. solving a magic square, finding and applying
the rules for divisibility). Students may work individually, in
pairs, or in groups of three. The problem-solving report,
required to be written on a word processor, must include the
solution(s) and details of the solving process; personal
reflections on what the student(s) experienced in attempting to
solve the problem; descriptions of areas where the student(s)
got stuck or felt frustrated; descriptions of particular
strategies used; a sentence or two where the problem-solving
task fits into the elementary mathematics curriculum;
suggestions for making the problem easier; suggestions for
making the problem more challenging (click
here to go to an example of a problem-solving write-up).
Students are encouraged to submit illustrations or scratch work
of solutions to the problem. The instructor evaluates the
reports based on the completeness and attention to details
suggested in the description of the assignment rather than on
the correct solution to the problem. |
The
problem-solving activities were refreshing; I had forgotten how
frustrating it can be to solve a seemingly simply math problem.
It also made me more aware of the steps a student goes through
as he/she solves a problem.--Student. |
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PURPOSE: Students identify
and develop a set of strategies for solving problems, develop
skills for posing and editing problems, and reflect on their
individual approach and style in problem solving. Developing
metacognitive skills rather than finding the correct answer is
the emphasis of problem-solving. |
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3. READING COMMENTARIES |
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| Students are required to
submit three reading commentaries on journal articles focusing
on elementary mathematics content or methods. Each student
selects two articles, and the instructor assigns the third
article. Commentaries are comprised of two sections: the first
part is a one-page summary of the article and the second is a
one-page reaction which should include comments, criticism, and
questions of the article. The instructor suggests sources of
articles, such as the Arithmetic Teacher, the Journal
for Research in Mathematics Education, For the Learning
of Mathematics, or students may find other appropriate
journals. Students are invited to the instructor's office where
they may select copies of journals from a large selection or
borrow journals from the library.
The third commentary is assigned
at the end of the second quarter. Since course participants
include professional teachers as well as preservice teachers,
the instructor provides textbooks relevant to the student's
interest in teaching level. The first part of the commentary
consists of an outline of the content: What is covered in the
grade level? How much is new information? How much is repeated?
The second part of the commentary is a reaction to the textbook
in terms of topics explored in the curriculum course: What
should be covered in the grade level? How are materials used?
What is the role and function of small group work if it is
encouraged in the text? What is the importance of context,
concept development, soft algorithms, and Do-Say-Write (the
practice of acting with materials, talking out loud about the
actions, and recording the action)? Through class discussions of
their findings, students review important mathematics content
and methods topics. |
All aspects --- reading, logs,
problem-solving, and group work --- complemented and reinforced
important concepts of mathematics as well as of human
interaction -- two facets that teachers must combine in order to
keep learning meaningful. -—Student
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PURPOSE: The first two
commentaries provide students with a context for analysis and
discussion of the teaching of mathematics or theoretical
concepts. In the third commentary students interact with the
text, examining the curricular content, methods, and its
underlying principles. |
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RELATED ACTIVITIES |
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1. COLLABORATIVE LEARNING GROUPS |
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In addition to problem-solving
and reading commentary assignments which may be written
collaboratively, most of the class activities are planned for
groups of three to four students. The instructor provides
mathematical tasks that can be approached with the use of
manipulatives -- unifix cubes, straws, geometric figures --
which help students deal with abstractions. Every member is
responsible for contributing some insight, question, or
solution.
PURPOSE: The class is
designed as a workshop so that students will experience many of
the same activities they will be providing in their own
classrooms. Assuming the role of teacher as student also helps
them understand the thinking processes necessary for developing
mathematical concepts. Students are expected to experience what
can be learned through cooperation. They can discover that
collaborative learning promotes active learning and that the
individual contributions of group members help to teach one
another in unique ways other than the teacher-directed model. |
I feel that
the group work with different people contributed to my overall
learning in mathematics. If I can learn more through this
approach rather worksheets, then so should my
students.--Student |
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2. COMPUTER EXPERIENCE
The instructor introduces
students to the computer lab located in Wist Hall where class
will meet occasionally for large group instruction. Students
learn how to use "Superpaint" to enhance class
presentations or to create student activity sheets which include
one page of illustrations and one page of text. These activity
sheets may be collaborated on with another student; the work
must be titled, complete, organized, and neat. Other programs
involving the use of a spread sheet are used for investigating
mathematics content. The instructor also requires students to
submit all written assignments (except weekly logs) using any
word processing program.
PURPOSE: Students are
expected to develop some competency and comfort using a word
processing program as well as other computer applications. The
computer is another writing tool students need to use more
frequently to aid their writing and thinking.
3. BIWEEKLY QUIZZES
Students write a brief response
with diagrams to a hypothetical situation encountered in a
classroom. Quizzes may be open-book and/or open-notes.
4. MID-TERM and FINAL EXAMS
The mid-term is an open-book,
open-notes examination during the regular class period. The
final is a take-home examination; students are encouraged to
work collaboratively in groups of two or three on the exam.
PURPOSE: The
goal of both examinations is to foster careful thinking about
the process of mathematical learning, concepts, and operations.
Collaborative learning, which has been the instructional mode
throughout the semester, is reinforced once more in the final
exam format. |
I am now more
aware of the importance of providing mathematical experiences
for kids and am no longer in favor of only computation drills. .
. I also believe that the emphasis should be on the thinking
instead of an isolated answer because kids need to feel
confident that they have a chance to explain their reasoning and
thought processes without being downgraded simply because their
answers don't match a teacher's answer key . . . --Student
The group work was valuable
because it gave us a chance to learn from one another. It was
interesting to see how each person had a completely different
approach to math.--Student
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Professor Zilliox comments on his class (excerpts from an
interview):
I see the role of writing in my class in
many ways. Students must express their ideas, their opinions, and
the writing gives me access to things that I don't have access to
otherwise in class. Also I see a connection between the writing
and reflection in class, and it took me along time to buy into
that. I used to hate to write and still hate to write formally. In
terms of writing for myself I find it easier to write my thinking
down on the computer. Very often when I start to see my thinking
in print, I see that's not what I meant or sometimes I think so
fast as I'm getting it down on paper, that when I'm re-reading it
I can keep sections or trash the file. It's one way of organizing,
and I see writing as having that potential. I haven't decided what
comes first with writing, and I'm having to work that through,
what's comfortable, and what comes naturally. I do have sympathy
for students who struggle with writing because they don't know
what to write and because I was in that position.
My idea in teaching and learning has a lot
to do with reflection. . . Because of the WI status we have
certain expectations to get more writing into the course. I wasn't
sure of the value of these things, but I had tried some things and
I liked the responses. . . I'm not sure what it has meant for the
students, although at the end of the semester I find so much that
I keep as an evaluation for myself and the course. Some students
respond very positively to the logs; others say it's a waste of
time. Right now I have four classes using logs and one of the
criticisms was "I really like your feedback." Students
felt writing the logs was hard work, but there was a
confidentiality that I had through the logs that I don't have in
class. . . They can say things to me directly through the log that
they can't say face to face. . . I'm disappointed that I don't
have enough time to keep up with the logs, but of course, it's my
own management problem. This semester it's been working! Some use
the logs as notes for the mid-term. Most of the students who do
use the logs as notes are also the students who say they don't
know what to write or say. Making a summary is easier to do than
thinking about what it is they've done in class. I'd like to try
doing a log together. . . I'd like to try log writing at the end
of class; it might get them to think about the text.
One of the other things I like to use
writing for is the problem-solving paper. The emphasis in the
books of mathematics, the university, and education courses is
problem-solving, not just practice. Many of these students have
never been problem-solvers themselves, and I try to teach that or
give them activities. . . The directions for the problem-solving
papers are in the syllabus. . . One of the reasons for having
students write in paragraph form is to have them reflect on what
they're doing and why this is something even worth doing. We were
doing something with magic squares; there was a lot arithmetic
computation, and some only realized what they were doing only
after they started thinking about it. One of the things I want
students to do in mathematics is to have them do this kind
practice -- a different sense of practice for students. This
raises issues we talk about in class -- why are we in mathematics?
so what if the answer isn't correct? These issues come up
especially in the problem-solving paper. Sometimes when students
start writing about what they're doing, they begin analyzing their
process and discover where the problem is. Writing can be a way to
get students to say "It wasn't until I was writing this up
that I could explain it to you" and decide "Maybe I
could have solved the problem this way . . ." Writing
actually helps them solve the problems.
Students don't like doing these
problem-solving papers because they're dealing with abstractions,
and mathematics seems to aim for the one right answer. The goal is
to get the right answer that the teacher already knows, and I
don't know the answers to all the problems. There are bunches of
mathematics problems still out there which haven't been solved for
thousands of years. Some of them we don't know are solvable or we
think they are, but no one really knows. But that's part of mathematics;
it's not just one answer. Students believe there's some rule and
if you know the rule and can apply the rule you'll get the right
answer, and that's the end of it! However, in problem-solving
there is no single rule or simple process. We can talk about what
works or what doesn't work; we come to reasonable conclusions.
This is one of the things I want to get across to my students, and
I see it happening in the journal. I'm trying to get into my
students' minds and getting them to question their own ways of
teaching in general, teaching mathematics, and what mathematics
is. |
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