Course
Philosophy:
Reconstructing Mathematics Understanding
Quality and depth of our learning depends on our ability to think about our own thinking and learning, to foster our own metacognitive reflection. Are you aware of some strategies that promote metacognition?
As teachers we want to be independent lifelong learners. That involves a collection of skills that are often named as self-regulated or autonomous learning skills. What might these skills be?
This class is about learning mathematics for understanding in an environment that nurtures development of autonomous learning skills and promotes metacognition.
Questioning: If you ask me "What is ..." I might answer with "Where are you in your thinking about it?" OR say, "Let's think about it together."
Course Changes: I try to make my courses better every semester which always involves a number of changes. So, please check the things before accepting what you hear from students that took this course in earlier semesters.
During the semester, I usually do not make any changes, unless absolutely necessary, in which case I will definitely inform you about it and probably negotiate that with you in advance.
Input from my current and former students: I appreciate very much any input from my students. Please feel free to ask questions both in class and outside of the class.
What is your challenge/concern after reading this? What are possible misconceptions surfacing from this philosophy? Think about it. Talk with your colleagues about it. Email me. Let’s discuss it in class.
Wichita State University
CI 319 Mathematics Investigations
Instructor: Dr. Mara Alagic
Associate Professor
Fall 2007
Faculty Member: Dr. Mara Alagic
Office: 205 Corbin
Office
Hours:
Telephone:
(316) 978-6974
E-mail
Address: mara.alagic@wichita.edu
Department:
Curriculum and Instruction
Note: Weather Cancellations – Call 978-6633 (select
2) to obtain information on weather related class cancellations.
Course Title:
Mathematical Investigations (2 credit hours)
Catalog
Description:
This course is founded on National Council of Teachers of
Mathematics (nctm.org) principles and standards for
school mathematics. It will model an investigative problem-based approach to
mathematics focusing on process standards: problem solving, reasoning and proof,
communication, connections and multiple representations.
Students should gain an
active understanding of problem-posing and problem-solving in mathematics, as
well as a familiarity with heuristics for problem-solving.
Course will also utilize appropriate
technology-based cognitive tools.
Prerequisites: MATH 501 Mathematics for Elementary teachers
Major Topics:
Mathematical
Processes: Problem Posing and Problem Solving:
What do non-routine, ill-posed mathematics projects look like? Reasoning and Proof:
Inductive and deductive reasoning; What constitutes a mathematical proof
at different levels of understanding? Communications:
Language of Mathematics Connections:
What might integration mean in the mathematics classroom?
Multiple
Representations: Open-ended, rich problems
Cognitive science: How
people learn mathematics?
What is mathematical understanding?
What’s THE right way to teach mathematics?
Doesn’t every mathematics classroom look the same?
How can I assess and evaluate my students’ learning?
How can I effectively use technology within my mathematics program?
How can I add breadth, depth, and dimension to my students’ mathematical
learning?
Technology Expectations: CORE 2
students will be able to
Use common media storage systems such as
CD-ROM and Zip disks.
Create hyperlinks and graphics in word
processing documents.
Use electronic resources ethically.
Manipulate data in a spreadsheet program.
Use presentation software to make class presentations.
Use word processing or desktop publishing to
create instructional materials or newsletters.
Use video projectors, VCR, projection
devices, digital cameras, CD/DVD, calculators, and other common
instructional technologies.
Trouble-shoot basic computer problems.
Evaluate electronic materials for accuracy,
appropriateness, and credibility.
Use concept mapping technologies for planning
or instruction.
Develop and use WebQuests.
Apply technology and content area standards
related to technology to planning and management.
Research and apply strategies that support
digital equity for all students. (Including assistive technologies)
Use multimedia to support instruction.
Related
Assessment
KSDE
Elementary Education Standards
National
Council of Teachers of Mathematics (NCTM) standards
Conceptual
Framework Connections
(Guiding
Principles)
The student
demonstrates the ability to use effective, developmentally appropriate
instructional strategies to help all k-6 students learn and use
their mathematical skills in many different situations and applications
to solve real life problems.
Digital
file/resource
S2-P3
Teaching
Principle;
Technology
Principle
HDD
The student
knows a variety of developmentally appropriate assessment tools that
align with curriculum and instruction.
Digital
file/resource
S2-K4
Assessment
Principle
HDD
The student
uses diverse and developmentally appropriate assessments that align with
curriculum and instruction.
Digital
file/resource
S2-P4
Equity
Principle
HDD
The student
knows and understands the mathematical concepts of number sense, number
systems and their properties, computation, geometric figures and their
properties, transformational geometry, measurement, data analysis, data
representations, probability, patterns, functions, and representations
of algebraic and geometric situations/solutions.
Digital
file/resource
S2-K1
Content
standards: Number sense and operations, Algebra, Geometry, Measurement,
Data analysis and probability
CKS
T
The student
understands the five process standards (problem solving, reasoning and
proof, communication, connections and representations).
Digital
file/resource
S2-K2
Process
standards
CKS
T
Appropriate
to k-6 students' age and development, the student can use and
apply, demonstrate, and teach the concepts of number sense, number
systems and their properties, computation, geometric figures and their
properties, transformational geometry, measurement, data analysis, data
representations, probability, patterns, functions, representations of
algebraic and geometric situations/solutions.
Digital
file/resource
S2-P3
Curriculum
Principle
HDD
CKS
T The student
integrates the five process standards (problem solving, reasoning and
proof, communication, connections and representations) into math
instruction.
Digital
file/resource
S2-P4
Curriculum
Principle
HDD
Required Readings
National Council of
Teachers of Mathematics.(2000). Principles and
Standards for School Mathematics. National
Council of
Teachers of Mathematics. Reston, VA
Recommended Readings
Math 501 textbook (any
version)
National Research Council (2000).
How people learn: Brain, mind, experience, and
school. Washington, DC: National
Academy Press.
Daniels, H., & Bizar, M. (1998). Methods that matter: Six structures
for best practice classrooms. York, ME: Stenhouse.
Carpenter, T. P., Fennema,
E., Franke, M. L., Levi, L., & Empson, S. B. (1999). Children's
mathematics: Cognitively guided instruction. Portsmouth, NH:
Heinemen.
CI 319
ASSESSMENTS
(for all 3 sections; M, T, & online)
NOTE:
Late
work will NOT be accepted. Plan your personal due dates accordingly.
Assignments
Assessment tool &
points
Due dates
Points
Class
participation (required readings are scheduled within the calendar)
check list;
Ongoing
150
100 points
Beginning and
end of the semester
Reflections
check list;
weekly - due by
Sunday midnight (each Pod has to
decide how much time they will provide for a person summarizing)
225
Self Evaluations
(include your grades - spreadsheet)
check list;
September 28;
November 9
100
Problem sets
rubric;
September 21
(Volume)
400
Final Exam -
presentations
rubric;
100
Total possible (tentative)
1000
NOTE:
Late work will
NOT be accepted. Plan your personal due dates accordingly. See the calendar - readings will be
announced regularly, one week in advance. You are a member of a
Reflective Pod (online group on the Blackboard site for this class). Your
weekly entry will consist of your reflective postings on (a) the topic assigned
and (b) the readings of postings of other pod-members. Every week one person, on
a rotating basis, summarizes. Read rubric carefully to better understand
requirements.
Online Discussions
Exemplary
Competent
Emerging
Substantive Postings
Contributes more than one idea that is original to the discussion
Contributes one idea that is original to the discussion
Contributes to the discussion but offers no new ideas
Acknowledging Ideas of Others
Recognizes the contribution of another and expands on the idea with
further examples OR uses examples to explain reason for disagreement
Recognizes the contribution of another and provides some reason for
agreement/disagreement
Recognizes the contribution of another with agree/disagree statement
Supporting Ideas
More
than one idea supported with multiple examples from personal experiences
and from other resources
More
than one idea supported with an example from personal experiences or
from other resources
One
idea supported with an example from personal experience or from other
resources
Timely Contributions
At
least two substantive (competent level) postings completed on time and
with separation of at least 24 hours
At
least one substantive (competent level) posting completed on time
Posting done but not on schedule
The rubric above is constructed to
guide you in self-evaluation of your contributions to your Online Discussion
Group. I hope this will encourage creative, high quality discussions related to
the learning of mathematics. I hope to build a community of learners engaged in
joint knowledge building through discussion. In order to build such a community
it is important to include discussions about the broader context of your lives
as future teachers and life-long learners. Therefore, I encourage you to broaden
your discussions outside of the required reflective discussions.
You will be turning in your scores
with self-evaluation. I will periodically check the scores with
reference to your actual online contributions.
You may choose your own format but
it has to include enough detail for me to understand how you are progressing in
this class; at least one paragraph long report on each of the following
questions. For the full number of points, question #1 will probably require more
than one paragraph - select concepts that you find most significant, and go from
there...:
What did I learn? Be very
specific and give enough details. Think about this as being
a test on
what you have learned so far. Or, if you do not like tests, consider
this a journal entry about the
mathematics content knowledge
and
mathematics-specific pedagogical content knowledge
(scaffolding)
that you have
acquired so far. Carefully select what you want to write about (2-3
concepts). Remember to support your statements.
What would I like to
learn/change? Be very specific. Include dispositions (both for yourself and
me).
The following two weeks I will
focus on . . . What can YOU do to enhance your learning related to this
class? Include dispositions (both for yourself and me).
What is your point-average at
this moment? How do you feel about it? (Attach a spreadsheet with your
grades; include self-evaluation for online reflective journaling).
Each problem set starts
with an open-ended, real-life related challenging problem focusing on a big
mathematical idea (grades 8-12). The problem set continues with 7
additional problems scaffolding down the main concept.
appropriate solutions
scaffolding for conceptual
understanding
concepts defined in
clear and precise language
clear list of key
concepts/vocabulary
See the Problem sets grading
rubric
(below) for further details.
Each problem set utilizes technology tools in an essential way (e.g.
multi-media, digital manipulatives, graphing calculators, spreadsheets, dynamic
geometry). More
details will be provided in class.
Problem sets: Grading
rubric
CI
319
Problem
Designation:
Excellent
Mediocre
Acceptable
Non-acceptable
Challenge problem selection and quality of its solution
The following attributes met: open-ended, real-life related,
significant mathematical idea/concept addressed; Each step of the
solution identified and justified;
metric system used.
At least 3
attributes met
At least 2
attributes met
Less than
2 attributes met
Scaffolding -
representations leading to the main concept
Rich
collection of simpler word problems leading step-by-step to the
challenge (at least 7 in addition to the challenge problem).
A collection of a couple of simpler problems leading to the challenge
Development of
representations
incomplete
None of
the attributes met
Quality of
the solutions
All solution steps and corresponding justification details included;
two-column solution format; metric system used
None of
the attributes met
Vocabulary
Precise
connections; Concepts clearly introduced after an experience provided
with a challenge problem (or other).
Connections not precisely introduced; OR Concepts introduced before
activities/experiences
Connections not precisely introduced; AND Concepts introduced before
activities/experiences
None of
the attributes met
ICT
tools/virtual manipulatives
ICT
representation appropriate for the task, part of a problem; interactivity clearly described
Not interactive (electronic work sheet)
OR not described OR problem not included
Insignificant value of the ICT
integration and one of the previous three requirements not in
None of
the attributes met
References
Detailed references (APA style)
Basic
references
Incomplete
Not
included
Metacognitive
Reflection
Justification: What is the main quality of this set? Describe your
own thinking during the design of the set. What did you
learn in terms of (a) content (b) yourself? How are
problems connected?
Justification incomplete;
Justification incomplete or unclear; Obstacles not recognized; No ideas
for resolutions
Not
included
Essential
recommendation:
Support each of the statements you make with a
very specific detail.
Due Dates:
September 21; October 20;
Criteria: Each student will demonstrate an acceptable or better rating on
each of the entries.
Presentation in class
as part of the final exam should include:
a selected challenging
problem - include most significant point of your justification! (ppt)
interactive ICT integration,
and (ppt) - Be ready to have our class try it out!
reflection on strategies that
promote metacognition (ppt) - What did you learn and how are you going
to use that in the future?
artifact that you made to support
your ppt presentation
(poster, manipulatives, game, ...) - include explanation into your ppt -
show your creativity
See the Presentation
grading rubric
(below) for further details.
Presentation:
Grading
Rubric
Academic Honesty: A standard of honesty, fairly applied to all students,
is essential to a learning environment. Students abridging a standard of
honesty must accept the consequences; penalties are assessed by appropriate
classroom instructors or other designated people. Serious cases may result in
discipline at the college or University level and may result in suspension or
dismissal. Dismissal from a college for academic dishonesty, constitutes
dismissal from the University. (WSU Student Code of Conduct)
Special Needs:
ADA: If you have a
physical, perceptual, psychiatric/emotional, medical, or learning disability
that may impact your ability to carry out assigned course work, contact the
Office of Disability Services (ODS), Grace Wilkie Annex, room 173. (Voice/TDD
978-3309). ODS will review your concerns, confirm your disability, and
determine, with you, what accommodations are necessary. All information and
documentation of your disability is confidential and will not be released by ODS
without your written permission.
National Council of Teachers of
Mathematics http://www.nctm.org/
NCTM standards
http://standards.nctm.org/
Electronic Examples:
http://standards.nctm.org/document/eexamples/
Illuminations
http://illuminations.nctm.org/
Virtual Manipulatives
http://nlvm.usu.edu/en/nav/vlibrary.html Enrich
www.nrich.maths.org
Monday and Thursday: Skype or a
Blackboard Chat: 10:00pm – 11:00pm
Tuesday: 10:00am – 11:00am
Wednesday: 10:00am – 11:00am
Skype contact name is my email:
mara.alagic@wichita.edu
I do have open door policy. If I
am in my office, which is whenever I do not teach, please come in.
Other times by appointment.
Learner
Outcomes
CTA
CTA
CKS
T
15 days x 10
For
online section, weekly summary of
readings (250-300 words) has to be dropboxed by Friday night everyeek.
Dispositions self-assessment and epistemology survey
Done
in class
(Bb - reflective pods)
15 weeks x 15
2 entries x 50
4 entries x100
October 20 (Exponential Growth)
November 12 (A geometry concept)
November 26 (A probability concept)
1 entry x 100
Dropbox and email
final presentation (ppt) by December 1
midnight
15 pts
10
pts
5 pts
OR
One idea is supported with multiple examples from personal experiences
and/or other resources
20pt
20pt
20pt
Some solution steps OR corresponding justification details missing; OR
solution format not followed OR
metric system not used
2 or more attributes missing
10pt
10pt
5pt
15pt
Unclear
possible resolutions
Emerging
Competent
Exemplary
Challenging problem
30 pts
Have a problem and a solution
Clearly stated challenging problem and a solution
Creative, attractive presentation of a clearly stated
problem and a solution
Interactivity (ICT)
30 pts
PowerPoint
Website that supports challenging problem
The class is engaged in that activity during presentation
(internet, GSP)
Artifact
35 pts
Mentioned
Poster, game, manipulative, ..
Engaging audience, well connected to the problem
Metacognitive reflection (all problem sets)
30 pts
Reflective statement - not clear metacognitive connection
One well supported metacognitive reflective statement
A couple of well supported metacognitive reflective
statements
The Shodor Education Foundation, Inc. http://www.shodor.org/curriculum/
CALENDAR *
TENTATIVE CALENDAR – FALL 2007
I teach 3 sections of CI 319; two sections are combination of face-to-face (hybrid) instruction and one section is completely online. All the work, rubrics, assignments and calendar are the same, except that online section gets class-work in a different format – both through the Blackboard and my website (http://www.education.wichita.edu/alagic/). Please email me (mara.alagic@wichita.edu) or call (36 978 6974) at any time if any of the information, calendar or syllabus is confusing. Be very specific with your questions so I can help you better.
NOTE: The only email that I will be using for you is your wichita.edu email address. Make sure that if you use other email address, your wichita.edu is merged in, or you check it regularly. In the subject line please always include what class you are in - CI319 Tuesday, CI319 Wednesday or CI319 online. If your email is urgent, please add that word "urgent" in the subject line.
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Class-work |
Reflection |
Readings |
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Week 1: 8/21&22
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Volume Problem (warm-up): Which one holds more? Fold a piece of paper horizontally into a cylinder (“hamburger” fold) so that edges perfectly touch (you can tape it). Fold another piece of paper of the exactly same size into a cylinder, but now vertically (“hotdog” fold). You have two cylinders. Which one can hold more (rice, water …)? Make a prediction, write it down and write why you think that. Than TEST your answer. Was your prediction correct? Can you provide better explanation
Getting
familiar with the syllabus: Details are available at the course webpage; when you have your own understanding, feel free to check it with your peers and/or with me. Your patience is required since requirements might appear overwhelming at the first glance. Students that took this course recommend
(1)
to have and keep personal due dates for the assignments,
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To access discussion board for your group, go to Blackboard place for this class, click on the class. When in, click on communication (on your left) and than click on Group pages. Click on your group and when you get in, click on R1. You can after that start a New Thread (top left) or, if someone already posted something, you can just hit “reply” to get your reflection in. Email (mara.alagic@wichita.edu) or call me (978 6974) if you have any questions or suggestions. NOTE: I use Blackberry for emails and I can answer short urgent questions almost immediately, most of the time (unless I am in class teaching or in the middle of a meeting). REFLECTION #1. (a) Share with your group members one positive experience about mathematics (it can be any grade and it can be outside of school). (b) Decide on a group name. The first person from the group list should email me that name (mara.alagic@wichita.edu) NOTE: Study reflections rubric carefully before writing this reflection. Reflections are posted within group discussion area on the Blackboard. |
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| Week 2: 8/28&29 |
Start thinking about volume concept and collecting (or designing) problems for your first problem set.
Volume problems
Understanding Problem Sets |
Reflection TWO: Can frustration be productive? Most of us get frustrated in a challenging situation, especially if we do not recognize a purpose of the challenge. What are some constructive learning strategies to move us from an ineffective frustration stage to a productive zone of proximal development? (NOTE: If you are not familiar with the term “zone of proximal development” please find out; get into agreement with your group about this concept, before proceeding to discuss the reflective question.) Summary due Sunday midnight which means that all reflections have to be posted at least by Saturday or before that, as negotiated by the group members.
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| Week 3: 9/4&5 |
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REFLECTION THREE
Describe what you learned from the volume experiments, |
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| Week 4: 9/11&12 |
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REFLECTION
FOUR: Problem Solving Standard When writing second posting, read all the entries and decide which one to reply to.
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Week 5: 9/18&19 PS1 due9/21 |
Each student will work on a portion of PPTs about process standards;
assignments will be posted here by the end of the next week.
Slides will be due September 26. ONLINE SESSIONS:
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