Mathematics Investigations

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.

Office Hours: Thursday 10:00 - 11:00 Room 251 Corbin; other times by appointment (134 Hubbard Hall)

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 points
Reflections (Bb - reflective pods)	check list; 15 weeks x 10	weekly - due by 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	October 1, November 5	100 points
Problem sets	rubric; 5 entries x100	September 8, September 29, October 20, November 10, December 1	500 points
Final Exam - presentations	rubric; 1 entry x 100	December 13 8:00 – 9:50	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.

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

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.

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

August Problem Posing and Problem Solving: What do non-routine, ill-posed mathematics projects look like?
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	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
	Reflective pods: a. Group name b. Positive experiences about learning mathematics & science c. Summarize on a rotating basis (alphabetic order)
	HW: Get familiar with the syllabus; Go carefully over the problem set rubric
	READ: Process Standards: p. 52 – 67
25	ONLINE ASSIGNMENT: Pose a real-life related open-ended problem about volume in your discussion group space; Explain why you think that is a valuable problem. The problem should be on your developmental level. Challenge yourself! After all group members posted a problem, decide as a group which problem (and solution) you will share next time in class. Each member will have to contribute to the presentation - decide who is doing what.
	Reflective pods: Reflection #2 My favorite mathematics concept is ... because...
	HW: READ Problem Solving: p. 334 - 342
September Multiple Representations: Open-ended, rich problems
1	READINGS DISCUSSION: Process Standards: p. 52 – 67
	READINGS DISCUSSION: Problem Solving: p. 334 - 342
	Which one holds more? Let's sort them from "the most" to "the least" according to the amount of rice that they can hold. A problem solving challenges: http://matti.usu.edu/nlvm/nav/frames_asid_273_g_3_t_4.html http://matti.usu.edu/nlvm/nav/vlibrary.html
	Reflective pods: Reflection #3 HW: READ Problem Solving: p. 334 - 342
8	READINGS DISCUSSION: Representation: p. 360 - 364 Linear vs. exponential Growth Keeping track of grades - Spreadsheets KNOWLEDGE TRANSFER: standards <<>>problem sets classroom work<<>>problem sets prior knowledge<<>>problem sets
15	READINGS DISCUSSION: Connections: p. 354 - 360 Estimation: Linear vs. exponential Growth - What does it really mean? Geometer's Sketchpad: What can I do with it? Reflection #5: Volume and Connections Reflect on Connections standard (see reading for this week) in the context of designing your Volume Problem Set. Remember that connections can be made within mathematics and with disciplines outside of mathematics. Share YOUR understanding of this standard (do not copy phrases from the book into your reflection) and provide specific examples from your problem set to illustrate your understanding. Think of yourself as a learner/student not as a teacher.
22	READINGS DISCUSSION: Communication: p. 348 - 354 Work in pairs: Design a word problem addressing exponential growth and using spreadsheets. Write up a solution and design a graph to represent your solution. Email me that and be ready to share. Geometry Preassessment - rhombus problem Geometer's Sketchpad: Geometric transformations; Geometric properties; Ratio and proportions HOMEWORK: Read geometry standard and geometry handout Go to the lab and play with Geometer sketchpad ONLINE REFLECTION: 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. NEXT: self-evaluation + reflections + problem sets (one more has to be on a geometry concept including GSP as a technology tool)
29	Van Hiele's levels Rhombus problem Geometer's Sketchpad: Geometric transformations; Geometric properties; Ratio and proportions ONLINE REFLECTION: What happens if you take a piece of paper cut it in half, then cut these two halves into halves and continue the process forever. Describe that process mathematically (mathematical representation or model). Think of different ways of doing this. Are there similar processes in the nature.
October Reasoning and Proof: Inductive and deductive reasoning; What constitutes a mathematical proof at different levels of understanding?
6	Geometer's Sketchpad: Triangle Properties - median, sum of interior angles, exterior angles property ONLINE REFLECTION: Scaffolding Geometry: Browse through geometry standard in your textbook. Focus on scaffolding ideas from grade k to grade 8. Select a concept that is appropriate for grade 8 (according to NCTM standards) and describe how you would scaffold it up to grade 8 through just a few (2-3) scaffolds. (NOTE: It is not necessary to match exactly with some grades between k and 8; just focus on the complexity of details/properties/formulas as you scaffold).
13	Geometer's Sketchpad: GSP: Triangles and their properties - incenter, orthocenter, circumcenter ICT interactive: (a) What is the difference between an interactive website and an electronic work sheet? What makes an ICT/computer-based activity interactive? Provide examples. (b) Which attributes of spreadsheets and dynamic geometry (such as GSP) make these tools interactive? Describe/list them and support your statements. Next problem set: Geometry topic of your choice Q: How can you adapt GSP activities that we did in class for K-6 grade levels. Think very broadly, at first: triangle properties, angles, segments, ...
20	Geometer's Sketchpad: Geometric transformations; Geometric properties; Ratio and proportions Reflection #10: 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. You can post your complete essay at one time (no need for 2 entries this week) and it is not necessary to have a summary. Topics are assigned in the order listed below. First group member: Open-ended problems Second group member: Scaffolding Third group member: Metacognitive reflection Fourth group member: Conceptual Understanding
27	WHY EFFECTIVE TEACHERS ANSWER QUESTIONS WITH QUESTIONS? What KIND of questions? Benefits Challenges Addressing Challenges Due date adjustment - PS4 due November 17; I will give you a challenging problem (topic - select a mathematical concept emerging from Pascal's triangle and fitting with the assigned challenge problem) Reflection #11: “intellectual fire” A student wrote in her self-evaluation that there is no “intellectual fire” in her reflective group discussions. Are you experiencing the same thing? (a) If yes, try to explain why this might be happening (without blaming your teacher or your peers). Assuming that you cannot do much about extrinsic factors, what can you do about intrinsic factors? (b) If not, provide support; how are you keeping your discussions at an appropriate/challenging intellectual level? What triggers your intellectual fire? Intrinsic or extrinsic factors?
November Communications: Language of Mathematics
3	Process standards - reflecting on reading - design a problem and take it through different process standards 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
10	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 _nC_k means and use it in your proof).
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December Connections: What might integration mean in the mathematics classroom?
1	FINALS: Presentations
8	FINALS: Presentations

September 2005
	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