CI 319

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.

Office Hours: Wednesday 11:00 - 11:45; Thursday 10:00 - 10:45; other times by appointment

Note: Weather Cancellations – Call 978-6633 (select 2) to obtain information on weather related class cancellations.

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.

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?

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

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
Reflections (Bb - reflective pods)	check list; 15 weeks x 15	weekly - due by Monday 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; 2 entries x 50	October 6; November 17	100
Problem sets	rubric; 4 entries x100	September 22, October 13, November 3, November 24	400
Final Exam - presentations	rubric; 1 entry x 125	Wednesday class class: November 29 Thursday class: November 30 Dropbox final presentation by Monday, November 27 midnight	125
Completed Digital Resource File (4 problem sets, Self-Evaluations, Final Exam and Final Reflection=last week online reflection) with corrections incorporated (to the best of your potential/time) is due at the time of your presentation (on a CD). NOTE: Final reflection has to be drop boxed, too. Final reflection should not be longer than one page.			100
Total possible (tentative)			1100
NOTE: Late work will NOT be accepted. Plan your personal due dates accordingly.

Online Discussions	Exemplary 15 pts	Competent 10 pts	Emerging 5 pts
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 OR One idea is supported with multiple examples from personal experiences and/or 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

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 attributes met	Less than 2 attributes met
Scaffolding through word problems - representations leading to the main concept 20pt	Rich collection of simpler word, real-life related 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 20pt	All solution steps and corresponding justification details included; two-column solution format	Some solution steps OR corresponding justification details missing; OR solution format not followed	2 out of 3 attributes missing	None of the attributes met
Vocabulary 10pt	Precise connections; Concepts clearly introduced after an experience provided with a problem (challenge problem or any 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, part of any problem in the problem set; 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 5pt	Detailed references (APA style)	Basic references	Incomplete	Not included
Metacognitive Reflection 15pt	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; 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. Due Dates: September 25, October 13, November 3, November 24

Presentation: Grading Rubric
	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, Excel)
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

August Problem Posing and Problem Solving: What do non-routine, ill-posed mathematics projects look like?
13/14
20/21
25/26
	Introductions - Volume activity Which one holds more (engaging, not open-ended challenge)? Introduce the concept/vocabulary Discuss briefly conceptual understanding of volume Course philosophy - syllabus Learning goals for this course Rubric for problem sets Rubric for online reflections	READ: Process Standards: p. 52 – 67

	Reflective pods: a. Group name b. Positive experiences about learning mathematics c. Summarize on a rotating basis (alphabetic order)	HW: Get familiar with the syllabus; Go carefully over the problem set rubric
30/31	1. Process Standards 2. Virtual Manipulatives: http://nlvm.usu.edu/en/nav/vlibrary.html http://matti.usu.edu/nlvm/nav/frames_asid_273_g_3_t_4.html REFLECTION #2: Educators have different ideas about when it is appropriate to introduce new vocabulary words. For example, some believe that in reading instruction, words should be introduced before reading a story, while others opine that new terms should be introduced after students have read the selection. This discussion is applicable to introducing new words in any academic subject, including mathematics. How does a teacher decide when to introduce a new vocabulary word? How important is it to connect new words with students’ prior knowledge and experiences?
September Multiple Representations: Open-ended, rich problems
6/7

	REFLECTION #3: Next week we will be discussing Reasoning and Proof Standards. In preparation for that, in your first part of the reflection, reflect on your existing knowledge and experience with reasoning and proof. Your second part of reflection should be written based on careful reading (textbook, p. 342-348). Instead of just summarizing what you read, design a higher-order level question to ask your classmates. Describe why you think that question will help us learn more about Reasoning and Proof Standards. During the next week’s class, I will expect every group to facilitate a class discussion guided by their questions.
13/14	1. Peer evaluation of the problem set #1 draft: HINT #1: Do not try to develop other concepts; assume that they know all other concepts HINT #2: Problems not activities; do not use "students will" HINT #3: Real-life related word problems - not computational or conversion problems HINT #4: BLACKBOARD: Do not use "#" in the file name HINT #5: BLACKBOARD: "add" puts it in your digital drop box "send" puts it in mine HINT #6:USE METRIC HINT #7: How many ideas did you get from the textbook HINT #8: You have to develop volume formula within your problem set HINT #9: HINT #10: 2. Sharing a challenge: How can we make a challenge more challenging?	NO reflection this week: Work on your problem set #1
20/21	Apple Pi: How can teachers get students to understand the importance of knowing "why" in their learning and explaining their thought process? Mathematic Mommas: How do you prevent a student from getting lost when presenting a proof and reasoning lesson? Mathletes: Do you always have to have proof to prove a proof, what is your reasoning? Quad Pod: Do you believe that it is okay in the primary grades to first teach memorization and later teach justification? C.A.R.E. How can we incorporate reasoning and proof into the curriculum in the early grades, so that it is not a new and overwhelming concept in high school? The Challengers: What do you do as a teacher when you are faced with a group of students, or just one student, who struggles to grasp the concept of reasoning and proof? Group #3: How much proof is enough for an answer to be solid? The Denominators: What can we do as teacher educators to inspire, facilitate and enable students to appreciate mathematics as it applies to the real world? Just the Two of Us: Would it be beneficial for teachers to start reasoning and proof techniques at an early age? How can we interest students in this subject and teach these techniques? The Investigators: Why is it so important to reason and prove when we don't really use it in daily life? Reflection #4: Teaching for conceptual understanding This week each group member will write an essay (not less than 300 words) on the assigned topic and post it for other members to be able to read it. It is expected that at least two resources be used and referenced. It is also expected that your second reflection for the week will consist of comments regarding your group members’ essays. No summary is needed this week. Topics are assigned in the order listed below. The order of the group members should correspond with alphabetical order by last name. Note: If you have less than five group members, not all topics will be used. First group member: Open-ended problems Second group member: Scaffolding Third group member: Metacognitive reflection Fourth group member: Questioning Fifth group member: Conceptual Understanding HINTS: How did you name your files. Mine would be ALAGIC PS3 Do you include title and your name? Do you have 7 problems + challenge? Do you have metacognitive reflection? Did you INCLUDE and DESCRIBE interactivity? Do you have references? Are your solutions detailed and justified? How is your scaffolding? Do you want to add more problems?
27/28	Problem Set #1: Exponential growth HINT #1: Mathematical concept is EXPONENTIAL GROWTH. First step is to write up a real-life related problem dealing with exponential growth. It can be very similar to what I did in class with the King's chessboard story. HINT #2: The simplest problem could be related to exponents, since we need to understand exponents in order to work with exponential growth. HINT #3: ICT (Information and Communication Technology) tool for this assignment is a spreadsheet. Make sure to integrate it in at least one of your problems and show how you are using spreadsheet interactivity for this problem set. HINT #4: EACH problem statement is followed by two-column table with solution (step by step) and justification (at each step). HINT #5: You are the mathematics learner in this class: Write as a problem poser and a problem solver - not like a teacher. HINT #6: To write exponents in word, highlight the number that is going to be your exponent (power, not base), go to Format>Font and click on Superscript HINT #7: To insert a two-column table in the Word File click on Table>Insert table OR click on icon for table on the (standard) bar and make 2x2 table. HINT #8: Do you include title (EXPONENTIAL GROWTH/CHANGE) and your name? Do you have 7 problems + challenge? Do you have metacognitive reflection? Did you INCLUDE and DESCRIBE interactivity? Do you have references? Are your solutions detailed and justified? How is your scaffolding? Do you want to add more problems?

October Reasoning and Proof: Inductive and deductive reasoning; What constitutes a mathematical proof at different levels of understanding?
4/5	NO reflection this week
11/12	Van Hiele's levels REFLECTION #6: What does geometry mean to you? Recall what you like(d) or did not like about geometry. ENTRY 1 Select any geometry concept and write a paragraph (250- 300words) about that concept: (a) Describe/define the concept; (b) Give examples of real-life applications of that concept; (c) Relate/connect that concept to other parts of mathematics. ENTRY 2 How is your concept related to the concepts that your group members wrote about; Design a problem or an example relating all the concepts mentioned in this week discussions.
18/19	Reflection #7 (DUE October 23) This week each group member will write an essay (250-300 words) on the assigned topic and post it for other members to be able to read it. It is expected that at least two resources be used and referenced. Your essay should include the following: a. Define your topic. b. What are some examples of real-life applications? Your second reflection for the week will reflect on your group members’ essays. No summary is needed this week. Topics are assigned in the order listed below. The order of the group members should correspond with alphabetical order by last name. Note: If you have less than five group members, not all topics will be used: First group member: Congruence and congruent figures; Second group member: Similarity and similar figures; Third group member: Pythagorean Theorem; Fourth group member: Geometric transformations; Fifth group member: Tessellations Differentiating Instruction: Acquisition - Storage - Expression of Knowledge Van Hiele's levels -study the power point to understand well the levels - K-2 students explore similarities and differences among two-dimensional shapes. 3-5 identify characteristics of various shapes/quadrilaterals. 6-8 examine and make generalizations about properties of particular shapes/quadrilaterals. 9-12 develop logical arguments to justify conjectures about particular polygons Rhombus problem If a student knows that the Ødiagonals of a rhomb are perpendicular she must be able to conclude that, Øif two equal circles have two points in common, the segment joining these two points is perpendicular to the segment joining centers of the circles medians --> centroid altitudes --> orthocenter perpendicular bisectors --> circumscribed circle angle bisector --> inscribed circle
25/26 ONLINE WORK	Problem Set #3: Geometry concept DUE NOVEMBER 10 HINT #1: SELECT A GEOMETRY CONCEPT HINT #2: The simplest problem could be related to shapes, since we need to understand shapes in order to work with any geometry concept. HINT #3: ICT (Information and Communication Technology) tool for this assignment is a GSP. Make sure to integrate it in at least one of your problems and show how you are using GSP interactivity for this problem set. HINT #4: EACH problem statement is followed by two-column table with solution (step by step) and justification (at each step). HINT #5: You are the mathematics learner in this class: Write as a problem poser and a problem solver - not like a teacher. HINT #6: To insert a two-column table in the Word File click on Table>Insert table OR click on icon for table on the (standard) bar and make 2x2 table. HINT #7: How did you name your files. Mine would be ALAGIC PS3 Do you include title and your name? Do you have 7 problems + challenge? Do you have metacognitive reflection? Did you INCLUDE and DESCRIBE interactivity? Do you have references? Are your solutions detailed and justified? How is your scaffolding? Do you want to add more problems? Reflection #8: Integration of art and mathematics (DUE October 30) How can integrating art with mathematics increase students’ understanding of mathematics concepts? Provide some specific examples of mathematical concepts explored through art activities. Please use and reference resources.
November Communications: Language of Mathematics

1/2 ONLINE WORK	PROBLEM SET 4: Pascal's triangle and patterns - DUE NOVEMBER 24 The following assignment (DROPBOX) is due November 2 midnight: Have a look at the following PPT, select a couple of slides with the same theme and write a paragraph (250 words) about that topic/theme. Think about it as a beginning of your last problem set. Reflection #9 (DUE November 6) 1st entry: Select one main characteristic of each process standard (in your opinion) and write a couple of sentences about it. Make sure to select different characteristics than people who posted their first entry before you. 2nd entry: For your second entry, pick a standard according to your order in the group: Problem solving, Reasoning and Proof, Representations, Connections and Communication. Look what others wrote about that standard, and based on that and your own understanding of that standard, write an essay about it (approximately 350 words).
8/9
15/16	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 PERMUTATIONS, COMBINATIONS, BINOMIAL FORMULA AND PASCAL'S TRIANGLE Tossing a coin: H or T Tossing 2, 3, ..; assume that order does matter Developing Pascal's triangle In how many different ways can you order three color: Amber, Blue and Coral How many different outfits can you have if you have 3 t-shirts, 4 pairs of jeans and 2 vests Permutations of unlike objects (a) In how many ways can you order a group of 5 students? P(5) = 5x4x3x2x1= 5! (Factorial symbol is just a symbol used to simplify our writing. So, P(5) = 5x4x3x2x1= 120 (b) In how many ways can you order a group of 5 students? *Permutations are arrangements in which order is important.* Permutations of a subset of a set (objects in a set) In how many ways can you select a group of 4 students in a class of 20 students? ₂₀P₄ = 20x19x18x17 = 20!/16! = 20!/(20-4)! _nP_k = n!/(n-k)! What is _nP_n ? _nP_n = n!/(n-n)!= n!/0! = n! (Because mathematicians agreed that 0! = 1; this is one of the strange examples that does not make much sense but it makes everything work) So, 0! = 1 1! = 1 2! = 2x1 = 2 3! = 3x2x1 = 6 … n! = nx(n-1)x(n-2)x…x3x2x1 = … Permutations involving like objects (a) You have 2 t-shirts and one pair of jeans. In how many different ways can you order these 3 objects? (Think of these objects on one shelf, next to each other.) P(3) =3! (b) If I had 3 jeans and 7 t-shirt that I wanted to lay them next to each other on a long shelf, I would want to permute 17 objects in all possible ways. Therefore, P(10) = 10! = 10x9x8x7x…x3x2x1 = 3 628 800 (c) You have 2 t-shirts of the same color and one pair of jeans. In how many ways can you order these 3 objects? 3!/2! In general, n!/k! (where k is set of objects in which order does not matter) Combinations = arrangements of objects in which the order makes no difference (a) In how many ways can you select a pair out of 3 students? Permutations Combinations AB AB BA AC AC CA BC BC CB 3x2 is all possible pairs Since order does not matter we have to divide by 2 and we can write that as (3x2)/2 = 3!/2! (b) One triple out of 5 students? 5x4x3 if the order does matter (Just for fun, I can write that as 5!/2!) BUT, order does not matter, so I have to divide by 3! (because there are 3! possible ways in which I can order three students) (5!/2!) 3! = 5!/3!2! (b) How many groups of 4 students can you have if your class has 20 students? (Notice, order does not matter; so, these are called combinations “of 20 elements in groups of 4) If order matters, we know we are talking about permutations (see problem #2) ₂₀P₄ = 20!/(20-4)! = 20!/16! = = 20x19x18x17x16x15x14x…x3x2x1/16x15x14x…x3x2x1 = 20x19x18x17x16 = 1 860 480 BUT, order does not matter, so we have to divide with 4! (which also can be written as P(4) or ₄P₄) SO, we have ₂₀C₄= ₂₀P₄ / ₄P₄ = [20!/(20-4)!]/4! = 20!/4!16! = 1 860 480/4! = 1 860 480/24 = 77520 (c) To generalize, If we have n people for a party and k tables, in how many different ways can we sit these people? _nC_k= _nP_k / _kP_k = [n!/(n-k)!]/k! = n!/k!(n-k)! (combinations of n elements in groups of k; REMEMBER - combinations mean that order does not matter) (a) Calculate ₅C₀= ₅C₁= ₅C₂= ₅C₃= ₅C₄= ₅C₅= (b) Is there a row in Pascal’s triangle that matches these numbers? (c) Remember, these numbers are coefficient of the expression (x + y)⁵. How can you write that down? Think different representations of this expression using what you know about combinations (case (a)) and what you know about Pascal’s triangle (case (b)). TRY THIS: How many (a) different gender (b) same gender, pairs can you have if your class has 12 girls and 8 boys? TRY THIS: Observe a row in Pascal's Triangle. What happens if we first subtract and than add and continue alternating like that in a horizontal direction? What if we first add and than subtract and continue alternating like that in a horizontal direction? Can you generalize one of these cases for any row? Can you prove that? (Hint: Find out what nCk means and use it in your proof). (Problem #7 is a challenging problem that does not have an obvious real-life connection for you at this point. It is a significant property of combinations that is represented with Pascal's triangle and it can be used in probability calculations. If you decide to use this problem as your challenging problems, I will not take points of for the fact that it is not real-life related). Problem set 4. Select either permutations or combinations for your concept in the problem set 4. Combinations are a little bit more complex than permutations, but it might be easier to scaffold them.	No reflection this week; self evaluation is due and it will be worth 60 points.
22/23	Have a great Thanksgiving!
29/30	Final Presentations

December Connections: What might integration mean in the mathematics classroom?

January 2006
	These pages are always under construction: I am trying to keep them up-to-date with my activities :) Questions and/or comments are welcome!	Maintained by: Mara Alagic Mathematics Education Curriculum & Instruction Department Wichita State University Wichita, Kansas 67260-0028

Learner Outcomes