Eastern Connecticut State University Education Department Issues and Applications in Secondary Mathematics (EDU 546) Principles and Practices of Teaching Mathematics (EDU 464) Course Outline Fall 2013

Instructor: Dr. Hari P. Koirala Class hours: Thursdays, 5:30-7:15 pm & Online Office: Webb 129 Classroom: GO 102 Office hours: M, T, W, 3-6 pm; F: 9-11 am or by an appointment Phone: 860-465–4556 (W) Email: [email protected]

Purpose of the course The overall goal of this course is to encourage you to embrace the challenge of learning to teach mathematics through inquiry into students’ understanding of mathematics and the mathematics curriculum. Although there are no recipes and formulas for teaching mathematics, this course will provide you opportunities to explore how students learn mathematics and how you can use various teaching approaches to engage students into mathematical thinking. A significant portion of this course will be spent on designing and analyzing secondary school (grades 7-12) mathematics lessons and units. This course is built around six high school mathematics content areas as outlined in the Common Core State Standards (CCSS): Number and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics and Probability. Also integrated in this course are the eight mathematical practices as outlined in the CCSS and the NCTM/NCATE standards and indicators (both 2003 and 2012) required for teacher candidates. All course goals, objectives, and themes are interconnected with the Education Unit Conceptual Framework Candidate Proficiencies (ECP), Connecticut Pre-service Teacher Competencies (PTC), 2010 Connecticut Common Core of Teaching (CCT), the CCSS and the NCTM standards for mathematics. The following table shows the elements of ECP, PTC, and CCT.

Eastern Candidate Preservice Teacher Common Core of Teaching Proficiencies (ECP) Competencies (PTC) (CCT) 1: Content Knowledge (CNK) 1: Development and Domain 1: Content and 2: Pedagogical Knowledge Characteristics of Learners Essential Skills (PDK) 2: Evidence-based/Standards- Domain 2: Classroom 3: Integration of Knowledge based Instruction Environment, Student (INT) 3: Evidence-based Classroom and Engagement, and Commitment 4: Technology as a Tool to Teach Behavior Management to Learning (TTT) 4: Assessment Domain 3: Planning for Active 5: Diversity (DIV) 5: Professional Behaviors and Learning 6: Professionalism (PRF) Responsibilities Domain 4: Instruction for Active Learning Domain 5: Assessment for Learning Domain 6: Professional Responsibilities and Teacher Leadership

The table below provides an outline of how the goals and objectives of this course align with the ECP, PTC, CCSS, and the NCTM/NCATE program standards for beginning teachers. Also, each goal/objective is associated with a student product that would be completed during the course.

1 Course Goals/Objectives/Candidate Proficiencies Alignment Course ECP, PTC, NCTM/NCATE Products Goals/Objectives and CCT Standards for Teachers and the CCSS By the end of the course students will: 1. Demonstrate in- ECP: 1.1 NCTM/NCATE Attendance, Participation, & depth understanding PTC: Standards 1, 2, 3 Dispositions (APD) of content CCT: Domain 1 CCSS all domains: Philosophy (PH) knowledge Number & Quantity, including central Algebra, Functions, Unit plan (UP) concepts, principles, Modeling, Geometry, Clinical Report (CR) skills, tools of Statistics & Probability inquiry, and structure of mathematics by using various mathematical contents such number and quantity, algebra, functions, modeling, geometry, and statistics and probability in designing mathematics lessons and units for students. 2. Use various ECP: 1.1, 2.1, 2.2, 3.1 NCTM/NCATE PH, UP, CR mathematical PTC: Standards 1, 2, 3 processes and CCT: Domain 1 CCSS all domains and practices such as mathematical practices problem solving, reasoning, communication, and modeling in designing mathematics lessons and units. 3. Recognize ECP: 1.1, 2.1, 2.2, 3.1 NCTM/NCATE UP, CR regularity in PTC: Standards 1, 2, 3 repeated reasoning CCT: Domain 1 CCSS all domains and and make use of mathematical practices mathematical structure.

2 Alignment Course ECP, PTC, NCTM/NCATE Products Goals/Objectives and CCT Standards for Teachers and the CCSS 4. Be aware of the ECP: 2.2, 2.3 NCTM/NCATE PH, UP, CR availability, use, and PTC: 1, 2 Standards 4 limitations of a CCT: Domain 2-4 CCSS: Mathematical variety of resources practices 5 (including technology) and strategies to enhance student learning of mathematics. 5. Plan, design, and ECP: 2.1-2.4 NCTM/NCATE APD, UP, CR implement PTC: 1-4 Standards 3, 4, 6, 7 curriculum lessons CCT: Domain 3 CCSS all domains and and units in mathematical practices mathematics which are consistent with the national and state standards. 6. Use various ECP: 2.4 NCTM/NCATE APD, UP, CR assessment PTC: 4 Standards 3, 5, 6, 7 strategies such as CCT: Domain 5 CCSS all domains and multiple choice mathematical practices questions, journals, and portfolios to monitor student learning and improve instruction. 7. Demonstrate their ECP: 5.1, 6.1 NCTM/NCATE APD, PH, UP ability to support PTC: 1, 5 Standards 3, 4, 5, 6, 7 the diverse needs of CCT: Domain 6 CCSS all domains and students in terms of mathematical practices exceptionalities, race, ethnicity, gender, culture, language, and socioeconomic status.

Course Grading The assignments are intended to be creative (and not merely reproductive) undertakings. Simply completing all of the requirements for an assignment will not insure a top grade.

Method of Grading A range An insightful and challenging piece of work that goes beyond the requirement of the course. [90–100%] B range All of the elements have been completed and there is evidence of critical or creative thought that goes beyond what was discussed in class. [80–89%]

3 C range All of the required elements of the assignments have been fulfilled but there is no evidence of critical or creative thought that goes beyond what was discussed in class. [70–79%] D range Some elements of the assignment are missing. [60–69%]

Disability Statement: If you are a student with a disability and believe you will need accommodations for this class, it is your responsibility to contact the Office of AccessAbility Services at (860) 465-0189. To avoid any delay in the receipt of accommodations, you should contact the Office of AccessAbility Services as soon as possible. Please note that accommodations are not retroactive, and that I cannot provide accommodations based upon disability until I have received an accommodation letter from the Office of AccessAbility Services. Your cooperation is appreciated.

Academic Services Center: Students are encouraged to use the support offered by the Academic Services Center (ASC) located on the ground floor of the Library. Advising Services and tutoring in math, writing, and other subjects, including supplementary instruction, are available. The ASC also offers assistance in study techniques, time management and understanding learning styles. Fall 2013 hours: Sun. 2-9; M.-Th. 9- 9, Fri. 9-5. (Closed Sat.) For further information call 465-4310 or check the ASC website at http://www.easternct.edu/asc/.

Academic Misconduct: Students should read and understand Eastern's Academic Misconduct Policy, which can be found in the Eastern Student Handbook or at: http://www.easternct.edu/judicialaffairs/academicmisconduct/ All violations will be handled under the procedures established in this policy.

Electronic communication: Effective August 1, 2009, Eastern email will become an official form of correspondence within Connecticut State University System (CSUS). Therefore, it is expected that communications to students sent via email will be received and read in a timely fashion. It is expected that students check their university email at least as often as their class meets, in recognition that certain communications may be time-critical. Students should not assume that email sent from outside providers will be received by their professor.

Library Research Guidelines. For library research guidelines, please go to the Education/Curriculum Research Guide in the following website:

http://easternct.libguides.com/education

Course Texts and References Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf Connecticut State Department of Education. (2010). 2010 Common Core of Teaching: Foundational Skills. Retrieved from http://www.sde.ct.gov/sde/lib/sde/pdf/educatorstandards/Board_Approved_CCT_2-3- 2010.pdf Connecticut State Department of Education. (2010). Regulation of State Board of Education, Rev. 2-3-2010, Part III, Pre-Service Teacher Competencies. Retrieved from http://www.sde.ct.gov/sde/lib/sde/pdf/cert/regulations/2015_proposed_regulations_11-10-2010.pdf Posamentier, A. S. (2003). Math wonders to inspire teachers and students. Alexandria, VA: Association for Supervision and Curriculum Development.

4 National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Reston, VA: Author. Retrieved from http://www.nctm.org/standards/default.aspx? id=58 National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Retrieved from http://standards.nctm.org/. NCTM NCATE (2012). NCTM NCATE Standards-Secondary. Retrieved from http://www.nctm.org/uploadedFiles/Math_Standards/NCTM%20NCATE%20Standards%202012%20- %20Secondary%20(2).pdf NCTM NCATE (2012). NCTM NCATE Mathematics Content for Secondary. Retrieved from http://www.nctm.org/uploadedFiles/Math_Standards/NCTM%20NCATE%20Standards %202012%20Mathematics%20Content%20-%20Secondary03.06.13.pdf

Reading List Afamasaga-Fuata’I, K. (2008). Students’ conceptual understanding and critical thinking. A case for concept maps and vee-diagrams in mathematics problem solving. Australian Mathematics Teacher, 64(2), 8-17. Bieda, K. N. (2010). The whole is the sum of its parts. Mathematics Teaching in the Middle School, 15(9), 540-546. Brinkmann, A. (2003). Mind mapping as a tool in mathematics education. Mathematics Teacher, 96(2), 96- 101. Britton, K. L., & Johannes, J. L. (2003). Portfolios and a backward approach to assessment. Mathematics Teaching in the Middle School, 9(2), 70-76. Brown, C. L., Cady, J., & Taylor, M. (2009). Problem solving and the English Language Learner, Mathematics Teaching in the Middle School, 14(9), 532-539. Chappell, M. F., & Strutchens, M. E. (2001). Creating connections: Promoting algebraic thinking with concrete models. Mathematics Teaching in the Middle School, 7(1), 20-25. Chua, B. L. (2008). Harry Potter and the coding of secrets. Mathematics Teaching in the Middle School, 14(2), 114-121. D’Ambrosio, B. S., Kastberg, S. E., & Santos, J. R. V. (2010). Learning from student approaches to algebraic proofs. Mathematics Teacher, 103(7), 489-495. Davis, B. (1997). Listening to differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education, 28(3), 355-76. Dirksen, J., Dirksen, N., & Cheng, I. (2010). ProofBlocks: A visual approach to proof. Mathematics Teacher, 103(8), 571-576. Donovan II, J. E. (2006). Using the dynamic power of Microsoft Excel to stand on the shoulders of giants. Mathematics Teacher, 99(5), 334-339. Edwards, M. T. (2004). Fostering mathematical inquiry with explorations of facial symmetry. Mathematics Teacher, 97(4), 234-41. Fukawa-Connelly, T., & Buck, S. (2010). Using portfolio assessments to assess students’ mathematical thinking. Mathematics Teacher, 103(9), 649-654. Gratzer, W., Carpenter, J. E. (2008). The histogram-area connection. Mathematics Teacher, 102(5), 336-340. Hackmann, D. G. (2004). Constructivism and block scheduling: Making the connection. Phi Delta Kappan, 85(9), 697-702.

5 Halpern, C. M., & Halpern, P. A. (2006). Using creative writing and literature in mathematics classes. Mathematics Teaching in the Middle School, 11(5), 226-230. Hart, E. W., & Martin, W. G. (2008). Standards for high school mathematics: Why, what, how? Mathematics Teacher, 102(5), 377-382. Hrabowski, F. A. (2003). Raising minority achievement in science and math. Educational Leadership, 60(4), 44-48. Ketterlin-Geller, L. R., Jungjohann, K., Chard, D. J., & Baker, S. (2007). From arithmetic to algebra. Educational Leadership, 65(3), 66-71. Kieren, T. E., Davis, B. A., & Mason, R. T. (1996). Fraction flags: Learning from children to help children learn. Mathematics Teaching in the Middle School, 2(1), 14–19. Lufkin, D. (1996). The incredible three-by-five card! Mathematics Teacher, 89(2), 96-98. McCrone, S. M. S., Dossey, J. A., Turner, R., & Lindquist, M. M. (2008). Learning about students’ mathematical literacy from PISA 2003. Mathematics Teacher, 102(1), 34-39. Mikusa, M. G., & Lewellen, H. (1999). Now here is that authority on mathematics reform, Dr. Constructivist! The Mathematics Teacher, 92(2), 158-163. Mittag, K. C., Taylor, S. E., & Fies, D. (2010). Mean, median, or mode: which one is my pencil? Mathematics Teaching in the Middle School, 15(9), 548-553. Murrey, D. L. (2008). Differentiating instruction in mathematics for the English Language Learner. Mathematics Teaching in the Middle School, 14(3), 146-153. Mvududu, N. (2005). Constructivism in the statistics classroom: From theory to practice. Teaching Statistics, 27(2), 49-54. Nelson, C., & Williams, N. (2008). A fair game: The case of rock, paper, scissors. Mathematics Teaching in the Middle School, 14(5), 311-319. Reeder, S. L. (2007). Are we golden? Investigations with the golden ratio. Mathematics Teaching in the Middle School, 13(3), 150-55. Roberts, S., & Tayeh, C. (2007). It’s the thoughts that counts: Reflecting on problem solving. Mathematics Teaching in the Middle School, 12(5), 232-237. Scher, D. (2003). Technology tips: The parallelogram theorem revisited. Mathematics Teacher, 96(2), 148-149. Shultz, H. S. (2005). Internal rate of return. Mathematics Teacher, 98(8), 531-33. Shultz, H. S., Shultz, J. W., & Brown, R. G. (2003). Unexpected answers. Mathematics Teacher, 96(5), 310-313. Suarez, D. (2007). When students choose the challenge. Educational Leadership, 65(3), 60-65. Teuscher, D., & Reys, R. E. (2010). Slope, rate of change, and steepness: Do students understand these concepts? Mathematics Teacher, 103(7), 519-524. Thompson, D. R., & Chappell, M. F. (2007). Communication and representation as elements in mathematical literacy. Reading & Writing Quarterly, 23, 179-196. Wade, W. R. (2006). A dialogue between calculator and algebra. Mathematics Teacher, 99(6), 391- 393.

6 Walker, E. N. (2007). Why aren’t more minorities taking advanced math? Educational Leadership, 65(3), 48-53. Walmsley, A. L. E, & Muniz, J. (2003). Cooperative learning and its effects in a high school geometry classroom. Mathematics Teacher, 96(2), 112-116. White-Clark, R., DiCarlo, M., & Gilchriest, N. (2008). "Guide on the Side": An instructional approach to meet mathematics standards. High School Journal, 91(4), 40-44.

7 Tentative Weekly Calendar

Session Course readings/Assignments Week 1 Course introduction August 29 CCSS and NCTM Standards and focal points, Professional journals Week 2 Current issues in mathematics education September 5 Constructivist view of learning Hackmann (2004); Mikusa, & Lewellen, (1999); White-Clark et al. (2008) Read CCSS and NCTM Standards and focal points Week 3 What do the Standards really intend? September 12 Planning lessons and units Afamasaga-Fuata’i (2008); Brinkmann (2003), Hart-Martin (2008); Thompson (2007) Complete reading CCSS and NCTM Standards and focal points Draft philosophy of mathematics education due Week 4 Assessing math learning September 19 Multiple-choice test, Performance and portfolio assessments Connecticut Academic Performance Test (CAPT) Britton & Johannes (2003); McCrone et al. (2008); Fukawa-Connelly (2010) Final philosophy of mathematics education due Week 5 Problem solving September 26 Roberts & Tayeh (2007); Suarez (2007) Text chapters 1, 2, and 3, pp. 1-98, 198-214, 229-270 Week 6 Teaching of numbers and their relationships October 3 Number systems, Number theory, Number patterns Text chapters 1 & 2, pp. 1-98 Continued Week 7 Teaching of fractions, decimals, percent, ratio, and proportion October 10 Davis (1997); Kieren et al. (1996) Draft unit plan due Week 8 Teaching of probability and statistics October 17 Gratzer (2008); Mittag et al. (2010); Mvududu (2005); Nelson (2008); Reeder (2007); Text chapter 7, pp. 215-228 Week 9 Teaching of algebra October 24 Integers, equation balance, algebra tiles, algeblocks etc. Chappell & Strutchens (2001); Chua (2008); D’Ambrosio (2010); Ketterlin-Geller et al. (2007); Shultz et al (2003); Teuscher-Reys (2010); Text chapter 4, pp. 98-122 Final unit plan due Week 10 Teaching of algebra continued October 31 Week 11 Teaching of geometry, measurement, and proof November 7 Shapes, symmetry, tessellation, proofs Bieda (2010); Dirksen et al (2010); Halpern & Halpern (2006); Lufkin (1996); Walmsley & Muniz (2003); Text chapter 5, pp. 123-197 Week 12 Student presentations November 14 Clinical report due Week 13 Student presentations November 21 Teaching with technology Donovan II (2006); Edwards (2004); Scher (2003); Shultz (2005); Wade (2006) Week 14 Thanksgiving Holiday November 28 Week 15 Helping all students learn mathematics December 5 Brown et al (2009); Hrabowski (2003); Murrey (2008); Walker (2007) Disposition Reflection due

8 Course Assignments

Note that all written assignments must be submitted through Blackboard Learn.

Attendance, Participation, Dispositions, and Online Threaded Discussion [31%] One purpose of this course is developing a community that is concerned about the teaching and learning of mathematics. Each member of the class is essential to the development of a learning community and, as such, regular attendance and participation is expected of all students in classroom and online. Each student must participate in an online threaded discussion every week (See Blackboard Learn for details). You are expected to check the course website at least two times a week to read and respond to messages. I have posted discussion topics and directions on Blackboard. The topics are related to articles and readings also posted on Blackboard. In each discussion topic, you’re expected to read every message and respond to some of them just like you would listen to everybody in a physical class and would respond to questions and comments posed by the instructor and your peers. More specifically, you are required to respond to every single topic/prompt provided by the instructor. In addition to your own original posting on a topic/prompt, you must respond to at least two discussion messages posted by class members under each topic. The postings in online discussion will weigh 22% of the course grade. Your postings will also affect your disposition grade. Provided below are some of the discussion topics. Please go to Blackboard Learn for more specific directions and timeline.

1. Please read the NCTM Standards, NCTM PreK-8 Focal Points, and the Common Core State Standards in Mathematics (CCSSM). After reading please post your comments on how these documents will guide your mathematics education philosophy and curriculum. You may focus on such questions: What are the major content areas in secondary mathematics? How important are the process standards such as mathematical problem solving and mathematical representation? Which NCTM principle do you think is most critical in teaching secondary mathematics? What are some of the relationships between the NCTM process standards and the mathematical practices delineated in the Common Core State Standards?

2. Choose one article from Week 2, Week 3, or Week 4 course readings. Read the article, summarize it, and then relate it to a research article that you find on your own. (If an article from the course readings has been already summarized and posted by two people, you need to choose a different article to summarize.) The article you have found must have been published within the last 3 years, unless the article has a historical significance in the topic chosen. At the end of your discussion posting, you must provide a full reference of the selected articles using the APA format. If the article is available online, a link to the website or a PDF file must be provided. A posting should not be more than one page long. In addition to your posting, you must respond to at least one discussion posted by a class member.

3. In this TD please share your ideas about Unit Plan and get feedback from each other. You can post any idea such as the topic and lessons you’ve chosen, an activity you have developed, or an assessment strategy you’ve thought about. Hope this TD will help you to come up with solid ideas about your unit plan.

4. Please select a problem from your text (Posamentier, 2003) and describe the importance of your problem in secondary school mathematics curriculum. Also, describe how you will use this problem with students.

9 5. Many secondary school students believe that algebra is boring and it has nothing to do with their everyday world. How would you convince the students that algebra is important for them? In your discussion, please cite an article from Week 9.

6. Based on your experience of working with students, discuss the profile of a student(s) with special needs (exceptionalities, race, ethnicity, gender, culture, language, and/or socioeconomic status). What kinds of challenges or opportunities arise because of these needs during math lessons? What specific kinds of strategies would you use to support their math learning? If two people have already discussed a particular need, then you must discuss a different kind of needs.

7. This is the final threaded discussion in this course. Over the semester, you were engaged in a variety of activities in mathematics education. Please think for a moment and reflect on what you learned in this course. What concepts and/or activities were most important for you? Why? How would you use these concepts/activities in your teaching? Note that your answers may vary from one another.

Disposition Rubric

At the end of this course, you must submit a 1-2 page reflection describing your strengths and challenges with respect to target or acceptable dispositions as explained in the rubric. Grades will be determined by carefully comparing your reflection with my notes. Although you will write your disposition reflection at the end of the course, you will have opportunities to demonstrate required dispositions throughout this course. If needed, meetings will be conducted with individual student(s) to discuss how dispositions can be improved.

Target (3) Acceptable (2) Unacceptable (0-1)

Class Attended every class, always Missed one class, almost Missed more than one participation came on time, submitted all always came on time, class, often came late, assignments by their due dates, submitted all assignments by and/or was inactive or was not distracted, and was their due dates, was not distracted in group/whole actively engaged in group and distracted, and was actively class activities. whole class activities. engaged in group and whole class activities. Professionalism Read professional and research Read professional and Did not read professional journal(s) in their discipline(s) to research journal(s) in their and research journal(s) in improve their own personal and discipline(s) and demonstrated their discipline(s) and/or professional growth, sought some passion and enthusiasm did not demonstrate membership of professional for their discipline(s) and passion and enthusiasm organization(s) to become methods of teaching. for their discipline(s) and involved in the professional methods of teaching. community of educators, and demonstrated passion and enthusiasm for their discipline(s) and methods of teaching. Respect Displayed professional and Displayed professional and Did not display ethical behavior in the class, ethical behavior in the class, professional and ethical always paid attention and and always paid attention and behavior in the class listened to peers and the listened to peers and the and/or did not pay instructor of the class with instructor of the class with attention to the ideas of respect, and often responded respect. peers and the instructor of thoughtfully and appropriately to the class. the ideas of peers and the instructor.

Philosophy of Mathematics Education [9%] 10 Write a two-page statement of your philosophy of mathematics education. Specifically write your goals of mathematics teaching and the roles of students and teachers in the learning of mathematics. You have to first submit a draft of your philosophy for the instructor's feedback. In the final version of your philosophy, you must address all questions/concerns raised by the instructor.

Philosophy Rubric

Target (3) Acceptable (2) Unacceptable (1) Logic and The philosophy statements are The philosophy statements are The philosophy statements clarity direct, straightforward, and generally clear but sometimes are unclear or ambiguous. unambiguous. The paper ambiguous. The paper The paper does not consist consists of well defined and consists of clearly developed of well defined and clearly clearly developed paragraphs paragraphs which are logically developed paragraphs. It which are consistent and connected to each other does not maintain the flow logically connected to each other maintaining the flow of the of the paper. It is not maintaining the flow of the paper. It is focused. focused. paper. It is well focused. Connections The statements are supported by The statements are supported The statements are not to classrooms meaningful examples and by examples from classroom supported by examples from illustrations from classroom and/or personal experiences. classroom and/or personal and/or personal experiences. experiences. Readings, The philosophy statements are The philosophy statements are The philosophy statements citations, and based on critical reflection of based on reflection of course are not based on reflection formatting course readings. The paper readings. The paper follows of course readings. The follows proper APA formatting APA formatting. paper does not follow consistently. proper APA formatting.

Overview and Design of a Unit [30%] This is a very important assignment that students must complete in this course. This assignment will consist of several elements. Its main purpose is to help candidates develop a unit of mathematics that could be used in their teaching. The unit will include: 1* A concept map; 2* A unifying theme and assumptions for the unit; 3* A list of the resources, including technology; 4* Statements of how the unit aligns with some of the state and national standards (CCSS and NCTM); 5* Citation and analysis of at least five sources/articles, including the CCSS and NCTM standards, related to the unit; (Note that graduate students are required to analyze and cite at least 8 sources in this assignment.) 6* Objectives of the unit; 7* Outline of at least 10 lessons; 8* Three fully developed lesson plans using Eastern’s lesson plan template (used during student teaching); 9* A tentative timeline, showing a possible sequence of unit topics and the amount of time allotted to each topic; 10* An account of how and where this unit might fit with other mathematical content areas; 11* An account of how this unit might fit with other subject areas; 12* An outline of how instructional tools and mathematics-specific technology are integrated in the unit;

11 13* A description of how the unit shows the importance of mathematics in everyday life and real-world contexts; 14* A description of how the unit will provide students with problem solving and modeling opportunity and enhance their problem solving skills; 15* Ways of assessing students’ understanding of mathematics (both formative and summative).

At least two of following mathematics topics should be covered in the unit: 16* Number and Quantity (integers, rational, real, sequences, etc.) 17* Algebra (equations, inequalities, algebraic equivalence, functions, etc.) 18* Geometry and trigonometry (Shapes/properties and proofs) 19* Statistics (Data collection, Presentation, graphs, analysis, etc.) and Probability (Games, concepts, rules, combinatorics etc.) 20* Calculus; and 21* Discrete Mathematics

The design of a unit should be based on the principle that “the whole is more than the sum of its parts.” That is to say a unit plan is more than a collection of lesson plans. You are encouraged to work in small groups of 2-3 people to bounce off ideas. However, you have to submit your own individual unit. The unit plan will be evaluated based on the attached rubric.

Unit Plan Rubric Target (3) Acceptable (2) Unacceptable (1) Themes, timelines, The unit contains a clear The unit contains a clear The unit lacks a clear assumptions, description of unified theme, description of unified theme, description of unified concept map, and the grade level, topic, a the grade level, topic, a theme, the grade level, unit objectives tentative timeline, entry-level tentative timeline, entry-level topic, a tentative timeline, NCTM/NCATE characteristics, features, characteristics, features, entry-level Standards 2003, 7.2 resources to be used, concept resources to be used, concept characteristics, features, NCTM/NCATE map, and objectives that are map, and objectives, some of resources to be used, and Standards 2012, 3a clear and adequate. which may not be clear or objectives, many of adequate. which are not clear or adequate. Quality of lesson The lesson plans include all the The lesson plans include at The lessons miss two or plans components: topics, grade least six components: topics, more components or do NCTM/NCATE level, connection to the grade level, connection to the not focus on student Standards 2003, standards, objectives, standards, objectives, mathematical 8.4 procedures, assessment procedures, assessment understanding. NCTM/NCATE strategies, and differentiation strategies, and differentiation Standards 2012, 3c and accommodation plan. The and accommodation plan. lesson plans focus on student The lesson plans focus on engagement and mathematical student mathematical understanding. understanding. Assessment The unit contains sufficient The unit contains adequate The unit does not contain strategies number of assessment strategies number of assessment adequate number of NCTM/NCATE (both formative and strategies (both formative and assessment strategies or Standards 2003, 7.5 summative) and some sample summative) and some sample no rubric or grading NCTM/NCATE quizzes, exams, projects, and quizzes, exams, projects, and criteria is provided. Standards 2012, 3f alternative assessment alternative assessment techniques. Each assessment techniques. Some The unit does not includes a rubric or grading assessments include rubric or demonstrate that the criteria. grading criteria. candidate has knowledge The unit demonstrates that the The unit demonstrates that of assessments used by candidate has a thorough the candidate has knowledge some of the leading

12 knowledge of assessments used of assessments used by some assessment organizations by some of the leading of the leading assessment (e.g. SBAC and NAEP). assessment organizations in the organizations (e.g. SBAC state and the nation (e.g. SBAC and NAEP). and NAEP). Mathematical Shows understanding of Shows understanding of Lacks understanding of content knowledge content, by providing content, by providing mathematical content. and processes appropriate examples from at appropriate examples from at Examples are not NCTM/NCATE least two CCSS content areas, least two CCSS content provided or they lack Standards 2003, 7.4 number and quantity, algebra, areas, number and quantity, comprehension. Errors NCTM/NCATE functions, modeling, geometry, algebra, functions, modeling, are made. Standards 2012, 1 and statistics and probability. geometry, and statistics and Does not demonstrate & 2 The unit is fully supported by probability. The unit is understanding of specific mathematics concepts supported by specific math mathematical practices and questions. Errors are not concepts and questions. described in the CCSS. made. Errors are rarely made. Also demonstrates full Also demonstrates understanding of mathematical understanding of practices described in the mathematical practices CCSS. described in the CCSS. Mathematical There is at least one problem in There is at least one problem The problem selected is modeling and the unit plan which provides a in the unit plan which not based on problem solving solid mathematical problem provides a mathematical mathematical modeling or NCTM/NCATE (based on modeling), at least problem (based on the description of using it Standards 2003, 8.8 two ways of solving it, and an modeling), at least one way at the secondary level is NCTM/NCATE excellent description of how the of solving it, and a unclear. Standards 2012, 2c problem can be used to teach description of how the secondary school mathematics. problem can be used to teach secondary school mathematics. Lessons Fully demonstrates how the Demonstrates how the Does not demonstrate connection lessons in the unit are lessons in the unit are how the lessons in the NCTM/NCATE interconnected and how the unit interconnected and how the unit are interconnected or Standards 2003, 7.2 is connected to other content unit is connected to other how the unit is connected NCTM/NCATE areas, everyday life, and the content areas, everyday life, to other content areas. Standards 2012, 3d real-world. and the real-world. Dealing with The unit provides a clear The unit provides a No clear description of diverse learners description of how it can be reasonably adequate how the unit can be NCTM/NCATE extended to serve high or low description of how it can be extended to serve high or Standards 2003, 8.1 ability students. Some extended to serve high or low low ability students. NCTM/NCATE activities are modified for this ability students. Some Students' special needs Standards 2012, 4c purpose. Students' special activities are modified for are not identified. Does needs are clearly identified. this purpose. Students' not use technology as a Uses technology as a tool for special needs are identified. tool for modification. modification. Uses technology as a tool for modification. Use of Thoroughly describes how Describes how instructional Does not describe how Manipulatives & instructional tools such as tools such as manipulative instructional tools Technology manipulative and physical and physical models, virtual enhance the teaching of NCTM/NCATE models, virtual manipulatives, manipulatives, and mathematics content in Standards 2003, 8.9 and mathematics- specific mathematics- specific this unit. NCTM/NCATE technology such as calculators, technology such as Standards 2012, 4e spreadsheets, and interactive calculators, spreadsheets, and software packages (e.g. interactive software packages Geogebra, Minitab) enhance enhance the teaching of the teaching of mathematics mathematics content in this content in this unit. unit. Use of Research Shows appropriate citation and Shows appropriate citation Does not show NCTM/NCATE analysis of research (including and analysis of research appropriate citation and Standards 2003, 8.6 ones from professional (including ones from analysis of research 13 NCTM/NCATE mathematics education professional mathematics related to the unit that Standards 2012, 3b organizations such as the education organizations such leads students in NCTM’s print, digital, and as the NCTM’s print, digital, mathematical learning virtual resources/collections) and virtual experiences. related to the unit that leads resources/collections) related students in rich mathematical to the unit that leads students learning experiences. in mathematical learning experiences. Organization and The unit plan is well organized The unit plan is organized. It The unit plan is not Presentation and is free of spelling and may have some minor organized or has many grammatical errors. spelling and grammatical spelling and grammatical errors. errors.

Note: Incomprehensible and missing responses will result in a score of 0.

Investigating Students’ Understanding of Mathematics (Clinical Report) [30%] This assignment is directly related to your clinical experience in a secondary classroom. While in school, you are expected to investigate students’ understanding of mathematics. You can accomplish this task by implementing the following steps:

i) Design a pre-test and post-test to assess students’ mathematical content you will be teaching in your clinical classroom. Administer the pre-test prior to teaching your lesson. ii) Teach a mathematics classroom and take careful notes related to the following questions: What was the mathematical content and what national and common core state standards did this fit into? What kinds of manipulatives or other teaching resources were used by you (the teacher) and students? What kinds of teaching/learning strategies were used? Who was more engaged; students or the teacher? How frequently did the students ask questions? Was this primarily a traditional/behaviorist or a progressive/constructivist classroom? iii) Administer the post-test after teaching your lesson. This will help you to decide whether or not your teaching made a positive impact on student learning. iv) Select two students (one at a higher mathematical level and one at a lower mathematical level) based on your pre- and post-tests and also with coordination from your clinical experience teacher and interview them to investigate their mathematical understanding. Collect student work and take interview notes. If it is hard to take notes during the interview you can tape record the interviews and transcribe them later in your convenience. v) Analyze pretest, posttest, and other student work and determine their understanding of mathematics. Do you think that these students achieved the lesson objectives? If so, what is the evidence? If not, what went wrong? Write a report. In your report you must cite at least five readings, including both the CCSS and NCTM standards. You also need to provide a reference page using the APA formatting. Note that graduate students are required to analyze and cite at least eight sources in this assignment. Your report must include the following: i) Describe the context and mathematical levels of students that you taught. Describe the lesson (content and standards). ii) Discuss the pretest, posttest, problem and interview questions that you asked the two students. iii) Analyze student work and interviews and report your findings. Discuss with evidence whether or not the lesson objectives were met. iv) Finally provide your reflection on how you would change the lesson to better suit the students’ needs. v) Attach your lesson plan, pre- and post-tests, and some work samples from the two students you selected. Your report should be no more than 10 pages in length (double-spaced), excluding the attachments. You will also need to give a 10 minute presentation to the EDU 546/464 class about your lesson. Your oral presentation in class should include the following steps:

14 a) Bring the manipulatives/resources used in the school classroom. If no resources were used, you must prepare similar manipulatives/resources to demonstrate to the class during your presentation. b) Describe the lesson (content and standards) you observed and the mathematical levels of students (1-2 minutes). c) Carry out a portion of the lesson in EDU 546/464 class, including an activity with the manipulatives/resources that you bring to the class (5-6 minutes). During your activity make sure that EDU 546/464 class is engaged. Your job is not to lecture what you did but to engage the class in a meaningful way. d) Ask a question and lead the discussion (1-2 minutes). Make sure that the question is related to the topic of your presentation.

CLINICAL REPORT and PRESENTATION RUBRIC Target (3) Acceptable (2) Unacceptable (1) Classroom and The description of student The description of student The description of student interview context background, classroom background, classroom background, classroom NCTM/NCATE context, and lesson presented context, and lesson presented context, and lesson Standards 2003, 8.1 to the students is clear and to the students is generally presented to the students is NCTM/NCATE coherent. clear. unclear. Standards 2012, 3a Lesson Plan The lesson plan used for The lesson plan used for The lesson plan used for NCTM/NCATE teaching clearly demonstrates teaching indicates how the teaching does not indicate Standards 2003, 7.6 how the teacher candidate teacher candidate planned to how the teacher candidate NCTM/NCATE planned to differentiate differentiate mathematics planned to differentiate Standards 2012, 3c mathematics content for content for diverse groups of mathematics content or how diverse groups of students students and how mathematics-specific and how mathematics- mathematics-specific instructional tools were specific instructional tools instructional tools were used used in building students’ were used in building all in building students’ conceptual understanding students’ conceptual conceptual understanding and and procedural fluency. understanding and procedural procedural fluency. fluency. Analysis of The analysis of teaching, The analysis of teaching, The analysis of teaching, teaching, student student work, and interviews student work, and interviews student work, and work and is clear, meaningful, and is clear. interviews is unclear or a interviews insightful. The analysis demonstrates component is missing. NCTM/NCATE The analysis demonstrates that the teacher candidate The analysis does not Standards 2003, 7.3 that the teacher candidate created learning environment demonstrate that the teacher NCTM/NCATE created learning environment in which students were candidate created an active Standards 2012, 4b in which students were engaged in building new learning environment. actively engaged in building knowledge from prior new knowledge from prior knowledge and experiences. knowledge and experiences. Monitoring and The teacher candidate The teacher candidate uses The teacher candidate does Adjusting effectively uses formative formative and summative not use formative and NCTM/NCATE and summative assessments assessments (pre-test/post- summative assessments or Standards 2003, 8.3 (pre-test/post-test) to measure test) to measure students’ does not make effective NCTM/NCATE students’ understanding of understanding of instructional decisions. Standards 2012, 3g mathematics, monitors mathematics, monitors students’ progress, and students’ progress, and makes instructional decisions makes instructional to help students develop decisions. conceptual understanding and procedural fluency. Comparison of The analysis is compared to The analysis is compared to The analysis is not analysis to course course readings including the course readings or the PTC, compared to course readings and PTC, CCT, CCSS and CCT, CCSS and NCTM readings or the standards. research NCTM standards in a standards. The analysis does not 15 NCTM/NCATE meaningful way (at least 5 The analysis indicates that indicate that the candidate Standards 2003, 8.5 citations for undergraduate the candidate utilized utilized resources from NCTM/NCATE candidates and 8 citations for resources from professional professional mathematics Standards 2012, 6c graduate candidates). mathematics education education organization. The analysis clearly indicates organization. that the candidate utilized resources from professional mathematics education organization

Impact on student The analysis and reflection The analysis and reflection The analysis and reflection learning clearly indicate that the provide some indication that do not indicate that the NCTM/NCATE teacher (candidate) is making the teacher (candidate) is teacher (candidate) is Standards 2003, a highly positive impact on making a positive impact on making a positive impact on 16.3 student learning. student learning. student learning. NCTM/NCATE Standards 2012, 5c Use of Appropriate grade level Appropriate grade level Appropriate grade level Manipulatives & manipulatives and manipulatives or technology manipulatives or technology Technology technology are used in the is used in the EDU 546/464 is not used in the EDU NCTM/NCATE EDU 546/464 class class presentation 546/464 class presentation. Standards 2003, 8.9 presentation. NCTM/NCATE Standards 2012, 4e Activity and The students in EDU 546/464 The students in EDU 546/464 The students in EDU engagement in class are actively engaged in class are engaged in class 546/464 class are not presentation class activities. activities. engaged in class activities.

Reflection Reflection is focused on Reflection is focused on Reflection is not focused on NCTM/NCATE lesson objectives and lesson objectives and lesson objectives, does not Standards 2003, 8.6 it clearly articulates it provides future provide future directions, or NCTM/NCATE future directions on directions on how the is not compared with the Standards 2012, 6c how the lesson should lesson should be PTC or CCT. Sometimes be changed. The changed. The these elements may be reflection is reflection is compared unclear. meaningful and with the standard compared with the documents including standard documents the preservice teacher including the competencies (PTC) preservice teacher and the Common Core competencies (PTC) of Teaching (CCT). and the Common Core of Teaching (CCT). Organization and The clinical report is well The clinical report is The clinical report is not Presentation organized and is free of organized. It may have some organized or has many spelling and grammatical minor spelling and spelling and grammatical errors. grammatical errors. errors. Note: Incomprehensible and missing responses will result in a score of 0.

16 EASTERN CONNECTICUT STATE UNIVERSITY EDUCATION UNIT CONCEPTUAL FRAMEWORK CANDIDATE PROFIECIENCIES (ECP)

Candidate Proficiencies for ECSU Candidates 1: Content Knowledge (CNK) 1.1 Candidates/Graduates demonstrate in-depth understanding of content knowledge including central concepts, principles, skills, tools of inquiry, and structure of the discipline(s) by engaging students through meaningful questions and learning experiences. 2: Pedagogical Knowledge (PDK) 2.1 Candidates/Graduates are able to formulate developmentally appropriate learning goals and objectives for students based upon knowledge of subject matter, students, the community, curriculum goals (both state and national), and theories of human development, and to plan and implement instructional activities which foster individual and collective inquiry, critical thinking, and problem solving to facilitate learning for all students in a safe and nurturing environment. 2.2 Candidates/Graduates use methods, activities, and grouping arrangements appropriate for lesson goals and objectives in an environment that is conducive to learning. 2.3 Candidates/Graduates conduct learning activities in a logical sequence and respond to the developmental needs, interests, ability, and background of students to promote their development of critical thinking, independent problem-solving, and collaborative inquiry. 2.4 Candidates/Graduates use multiple forms of assessment to evaluate student learning and modify instruction as appropriate to ensure the continuous intellectual, social, ethical, and physical development of the learner. 3: Integration of Knowledge (INT) 3.1. Candidates/Graduates demonstrate how different concepts, themes, and principles are interconnected within and across the discipline(s) and promote connections between content knowledge and pedagogical knowledge to help students learn concepts, principles, skills, tools of inquiry, and structure of the discipline(s) they teach. 3.2. Candidates/Graduates demonstrate an ability to integrate learning theories and other pedagogical knowledge in their clinical experiences and student teaching. 4: Technology as a Tool to Teach (TTT) 4.1. Candidates/Graduates integrate appropriate digital and non-digital technology throughout their courses and clinical experiences to support student learning. 5: Diversity (DIV) 5.1. Candidates/Graduates demonstrate their ability to support the diverse needs of students in terms of exceptionalities, race, ethnicity, gender, culture, and socioeconomic status. 6: Professionalism (PRF) 6.1. Candidates/Graduates collaborate with cooperating teachers, other teachers, school administrators and other school professionals, parents, families, and communities in a professional and ethical manner to help students reach their maximum potential.

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