CI 319 Mathematics Investigation

CI 319 Mathematics Investigations

Summer 2005

<< SYLLABUS AND OTHER RELEVANT INFORMATION>>

Course Philosophy:
Reconstructing Mathematics Understanding

Let's suppose that as elementary teachers we need to know as much of mathematical content as a solid high school student. We do not have to be able to teach high school students, but being able to read high school mathematics should be one of our goals. That way we will know what we are preparing our students for and what steps they will need to take from a kindergarten's AB pattern to understanding, for example, the difference between linear and exponential growth.

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 life long learners. That involves a collection of skills that are often named as 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.

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

Summer 2005

Faculty Member: Dr. Mara Alagic

Office: 205 Corbin Education Center

Office Hours: By appointment

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.

Learner Outcomes	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 CTA
The student uses diverse and developmentally appropriate assessments that align with curriculum and instruction.	Digital file/resource	S2-P4	Equity Principle	HDD CTA
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 CKS T

Required Readings

1. National Council of Teachers of Mathematics.(2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics. Reston, VA (also available online at http://standards.nctm.org/document/index.htm )

Recommended Readings

1. Math 501 textbook (any version)

2. National Research Council (2000). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.

3. Daniels, H. & Bizar, M. (1998). Methods that matter: Six structures for best practice classrooms. York, ME: Stenhouse.

ASSESSMENTS
Assignments	Assessment tool & points	Due dates	Points
Class participation (required readings are also scheduled within the calendar)	check list; 15 days x 10	ongoing	150 points
Reflections (Bb - reflective pods)	check list; 15 days x 10	daily – summary due by Wednesday & Sunday midnight (each Pod has to decide how much time they will provide for a person summarizing)	150 points
Self Evaluations (include your grades - spreadsheet)	check list; 2 entries x 50	June 15; June 22	100 points
Problem sets	rubric; 5 entries x100	June 10, June 14, June 18, June 21, June 24	500 points
Final Exam - presentations	rubric; 1 entry x 100	June 24	100 points
Completed Digital Resource File (5 problem sets and Final Exam) with corrections incorporated (to the best of your potential/time) is due at the time of your presentation.
Total possible (tentative)			1000 points
NOTE: Late work will NOT be accepted. Plan your personal due dates accordingly.

READINGS DISCUSSION

The following readings from PSSM are required. Additional recommended readings extend to the same topic within the PSSM textbook and further to the content standards. Be ready to discuss in details each of the readings. Instead of discussion an occasional quiz will be given over these readings.

1. June 7 -- Process Standards: p.52 – 67

2. June 9 -- Problem Solving: p. 334 - 342

3. June 13 -- Reasoning and proof: p. 342 - 348

4. June 15 -- Representation: p. 360 - 364

5. June 17 -- Connections: p. 354 -360

6. June 20 -- Communication: p. 348 - 354

REFLECTIONS

You are a member of a Reflective Pod (online group on the Blackboard site for this class). Your weekly entry will consists 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	Emerging 3 pts	Competent 5 pts	Exemplary 10 pts
Substantive Postings	Contributes to the discussion but offers no new ideas	Contributes one idea that is original to the discussion	Contributes more than one idea that is original to the discussion
Acknowledging Ideas of Others	Recognizes the contribution of another with agree/disagree statement	Recognizes the contribution of another and provides some reason for agreement/disagreement	Recognizes the contribution of another and expands on the idea with further examples OR uses examples to explain reason for disagreement
Supporting Ideas	One idea supported with an example from personal experience or from other resources	More than one idea supported with an example from personal experiences or from other resources OR One idea is supported with multiple examples from personal experiences and/or other resources	More than one idea supported with multiple examples from personal experiences and from other resources
Timely Contributions	Posting done but not on schedule	At least one substantive (competent level) posting completed on time	At least two substantive (competent level) postings completed on time and with separation of at least 24 hours

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.

SELF-EVALUATION

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...:

i. 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.

ii. What would I like to learn/change? be very specific. Include dispositions (both for yourself and me).

iii. 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).

iv. 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).

THE DIGITAL RESOURCE FILE

is a collection of student’s works in digital format, focusing on problem-based learning. Completed file has to be Technology will be used both as a presentation and integration tool. The resource file consists of the two required components: 5 Problem-sets and Final Exam Presentation.

1. 5 Problem-sets:

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 additional, at least 7 word 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 utilize 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 20pt	The following attributes met: open-ended, real-life related, significant mathematical idea/concept addressed; Each step of the solution identified and justified	At least 3 attributes met	At least 2 attribute met	Less than 2 attributes met
Scaffolding - representations leading to the main concept 20pt	Rich collection of simpler word problems leading step-by-step to the challenge (at least 7 problems)	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 20pt	All solution steps and corresponding justification details included	Some solution steps OR corresponding justification details missing	Some solution steps AND corresponding justification details missing	None of the attributes met
Vocabulary 10pt	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 10pt	ICT representation appropriate for the task, interactivity clearly described	Not interactive (electronic work sheet)	Insignificant value of the ICT integration	None of the attributes met
References 5pt	Detailed references (APA style)	Basics	Incomplete	Not included
Metacognitive Reflection 15pt	Justification: What is the main quality of this set? What did you learn in terms of (a) content (b) yourself? Transfer ideas? How are problems connected?	Justification incomplete; Unclear possible resolutions	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.

Criteria: Each student will demonstrate an acceptable or better rating on each of the entries.

2. Presentation (Final Exam)

Presentation in class as part of the final exam should include

· a selected challenging problem - teach us! (ppt)

· interactive ICT integration, and (ppt)

· reflection on strategies that promote metacognition (ppt)

· Artifact that you made to support your ppt presentation (poster, manipulatives, game, ...)

See the Presentation grading rubric (below) for further details.

Presentation: Grading Rubric
	Emerging 5 pts	Competent 15 pts	Exemplary 25 pts
Challenging problem	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)	Powerpoint	website that supports challenging problem	the class is engaged in that activity during presentation (internet, GSP)
Artifact	Mentioned	Poster, game, manipulative, ..	Engaging audience, well connected to the problem
Metacognitive reflection (all problem sets)	Reflective statement - not clear metacognitive connection	One well supported metacognitive reflective statement	A couple of well supported metacognitive reflective statements

Certification Requirements: Both the state of Kansas and national accreditation requires that university programs for the preparation of teachers and other school personnel be performance-based. In particular, this requires that students not only pass required courses/attain certain GPAs, but also receive satisfactory ratings on certain required assessments, many of those embedded within program coursework.
One or more of those required assessments occur in this course. A title/description of any assessments and associated rubrics and passing criteria follows:

· Digital resource file

Students failing to attain a satisfactory rating on a required assessment may be provided special assistance. The university is not able, however, to recommend individuals for licensure who fail to attain a satisfactory rating on required assessments, even though they may receive an acceptable course grade or exceed minimum GPAs.

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 (DS), 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 DS without your written permission.

CALENDAR

WEEK of June 6 Problem Posing & Solving: What do non-routine, ill-posed mathematics projects look like? Multiple Representations: Open-ended, rich problems
June 6	ONLINE ASSIGNMENT: (1) Study syllabus very carefully so that you feel comfortable understanding all the requirements. Pay very close attention to the problem sets rubric as it will be your major guide in developing problem sets. Familiarize yourself with due dates and times. It might be a good idea to mark those dates, as well as timely reminders, in your personal calendar. Keep in mind that late assignments are not accepted. (2) Access Blackboard, complete your first assignment ("My mathematics autobiography – both successes and challenges") and dropbox it. (If you do not know how to access Blackboard or "dropbox", email me and I will help you out.) (3) When in Blackboard, access discussion board for your group, introduce yourself to other members and make a couple of suggestions for a group name. As a group, you should decide on a group name by Wednesday, June 8. (4) READ: Process Standards: p.52 – 67 (due June 7) and Problem Solving: p. 334 - 342 (due June 9). (5) If you have any questions, please email them before May 28. I will not be able to access my email from May 28 to June. 6.
June 7	Which one holds more? Introducing problem set idea around the concept of volume. Course Philosophy: Mathematics knowledge; Autonomous learning; Metacognitive reflection Process Standards: p.52 – 67
June 8	Process standards - continuing discussion Spreadsheets - developing grade sheet; sequencing; simple formulas Problem sets work time Reflection #2: Reflect on problem solving standard in the context of designing your Volume Problem Set. Think of yourself as a learner/student and relate the process of designing to significant problem solving characteristics and strategies. (Due June 10.)
June 9	Pretest Fill and Pour http://matti.usu.edu/nlvm/nav/frames_asid_273_g_3_t_4.html Virtual manipulatives http://matti.usu.edu/nlvm/nav/vlibrary.html Problem sets: Q&A Dynamic Geometry introduction Van Hiele's levels Van Hiele's levels - table Rhombus problem: If a student knows that the diagonals of a rhomb are perpendicular, she must be able to conclude that, if two equal circles have two points in common, the segment joining these two points is perpendicular to the segment joining centers of the circles. Problem Solving: p. 334 - 342
June 10	Dynamic geometry: Triangle properties
June 13	Dynamic geometry: Triangle properties Reasoning and proof: p. 342 - 348 Reflection #3: Read about The Learning Principle (p.20) and The Technology Principle (p.24) of the School Mathematics. Reflect on these readings in the context of your own experiences of learning mathematics.
June 14	King's Chessboard Exponential versus Linear Growth - using spreadsheets - realistic answers Graphing - using spreadsheets
June 15	Dynamic geometry: Triangle properties - medians, perpendicular bisectors, angle bisectors, altitudes Reading discussion: Reasoning and Proof PS3: Exponential Growth - ideas, scaffolding, ... Representation: p. 360 - 364 Reflection #4: After reading the article http://nrich.maths.org/public/viewer.php?obj_id=2838 complete two postings. (1) Posting One: Pick and solve one problem from Stages 2-5 (see left bar in the above website; select one that other group members did not already use). Reflect on your thinking process and compare it with ideas from the article. (2) Posting Two: Reflect on postings of other group members; how similar or different is your thinking from theirs?
June 16	Represent a^2-b^2= (a+b)(a-b) visually; what does this expression mean in geometry? Mathematical expressions vs. mathematical equation Pascal's triangle - discovery approach Connecting with binomial formulae Searching for patterns; connecting with exponents of 2; Fibonaci's numbers, ... Multiplication Table and Pascal's triangle using spreadsheets
June 17	Reading discussion: Representations and Connections Dynamic geometry: (a) conversation about properties; (b) Excircles of a Triangle; Animation :) Pascal's Triangle in spreadsheet - generalizing properties Atomic learning Connections: p. 354 -360 PS#4: CHALLENGE PROBLEM What happens in a Pascal's Triangle if we first subtract and than add in a horizontal direction? Generalize for any row and prove. (Hint: Find out what _kC_n means and use it in your proof. NOTE: Geometer's Sketchpad - Student Edition - http://www.keycollege.com/catalog/titles/sketchpad.html#GSP%20Student%20Edition%20Anchor
June 20	Dynamic geometry: Excircles of a Triangle; Animation :) Communication: p. 348 - 354 Transfer of knowledge: Incircle and excircles - Which parts of construction are the same/similar 1/3+1/4 - misconceptions Further misconceptions - mathematical vs. common language - perimeter - area-space-volume; units Permutations and Combinations Reflection #5: Reflect on your geometry-learning in this class. What are the things that you want to remember and apply in your own future classroom. Think very broadly: triangle properties, angles, segments, perimeter, area, volume, circles, transformations, process standards ... How can you make that meaningful to the lives of your elementary students? DUE June 25. *For those of you that completed reflection #4, since it was more complex than other reflections, a very small award: DOUBLE your points for that reflection!*
June 21	Constructing a square in DG Pythagorean Theorem Preparing Final Presentations Problem sets #3, #4, #5 - Q&A PS #5: Your choice
June 22
June 23	Links for presentations www.primarygames.co.uk/pg4/SwimmingSid/swimmingpool.html (Jen Swingle) http://nlvm.usu.edu/en/nav/frames_asid_166_g_2_t_3.html?open=activities (Brianne Means)
June 24