ADVANCED MATHEMATICS for YOUNG CHILDREN
CUNY THE COLLEGE OF STATEN ISLAND
DEPARTMENT OF EDUCATION
“I think, therefore I am...
Each problem that I solved became a rule which served afterwards to solve other problems.”
Instructor: Dr. Judit Kerekes
Office: 3S/207B
Phone: (718) 982 4112
Office Hours:
Wednesday 3:00 pm-4:00pm
Wednesday 8:00 pm-9:00pm
Thursday 3:00pm- 4:00pm
Time: Wednesday 4:40- 6:20 PM
Room: 3S/210
Course Rationale
This course will specifically focus on enabling the teacher to design and implement math lessons that will cover the needs of students with a variety of abilities, the integration of instructional technology into the curriculum, and multiple approaches to assessment of learning.
This course takes a developmental perspective on the teaching and learning of mathematics. The role of context, culture, and language are explored as they relate to the development of mathematical ideas, strategies, and emergent models in the elementary years. Students will use video of real classrooms (K-6), investigate teachers’ decision-making, and analyze children’s work. On-site fieldwork and explorations with standards-based curricula will provide participants with guided opportunities to acquire those teaching strategies that best support mathematics learning in the elementary school. In order to strengthen participants’ subject matter knowledge, this course will also engage participants in mathematical activities and investigations at their own level.
Required materials: Fosnot, C. T., and Dolk, M. (2001) Young Mathematicians at Work
Vol. 1: Constructing Early Number Sense Addition and Subtraction Portsmouth, N.H.: Heinemann Press ISBN 0-325-00353-X
Dolk, M., Fosnot, C. T., Cameron, A., and Hersch, S. (2004) Working with Number Line Mathematical Models Portsmouth, N.H.: Heinemann Press ISBN 0-325-00672-3
M. Dolk, C. T. Fosnot, A. Cameron, S. Hersch, (2004) Taking Inventory, The Role of Context Portsmouth, N.H.: Heinemann Press ISBN 0-325-00672-5
Recommended: Fosnot, C. T., and Dolk, M. (2001) Young Mathematicians at Work Vol. 2: Constructing Multiplication and Division Portsmouth, N.H.: Heinemann Press ISBN 0-325-00354-8
Dolk, M., Fosnot, C. T., Hersch, S., and Cameron, A. (2004) Fostering Children’s Development The Landscape of Learning Portsmouth, N.H.: Heinemann Press ISBN 0-325-00672-1
Mokros, Russell, and Ekonomopoulos (1995) Beyond Arithmetic: Changing Mathematics in the Elementary Classroom Palo Alto, California: Dale Seymour ISBN 0-86651-846-0
Kamii, C. (2000) Young Children Reinvent Arithmetic, Teachers College Press, Columbia University, New York, NY, ISNN 0-8077-3904-9
NCTM Standards 2000; www.nctm.org
Objectives
Develop an understanding of all children’s mathematical development regardless of cognitive ability
Study learning theories and their relevance to teaching mathematics to elementary students
Learn to develop lessons to teach mathematics concepts to children of different abilities
Requirements
Attendance and participation are crucial.
Videotaped inquiry on the development of number sense of choice for grades PK-3 will be prepared and submitted. The unit consists of several investigations sequentially ordered to support the development of the topic. This includes the development of big ideas, strategies, and models related to the topic. Written lesson plan, report, children’s’ book and analysis of children’s ideas are required. Include plan and your report after teaching. What went well in the lesson? What would you change if you were to teach this lesson again?
Journal is assembled on a weekly basis. Participants are expected to read and write a reflective response to a selected article, or to a video, and/or to work on mathematics investigation. Use Dolk, Fosnot, A. Cameron, S. Hersch, Mokros, Russell, and Ekonomopoulos, Kamii, and NCTM materials for reference. You may include activities from these materials verbatim and you may include activities from class. The journal should include a discussion of the following: the role of teachers’, questioning, and connection of the lessons to the NCTM standards.
Final Oral Presentation on your field-tested unit.
Method of Evaluation
Journals 20%
Unit with taped lesson as above 30%
Exams 30%
Attendance and Portfolio 20%
Total 100%
Evaluation:
Journal (1pg, typed): 0credit: nothing, 1cr: on time, 2cr: good, 3cr: very good, 4cr: excellent 5x4= 20cr
Exams: Quizzes Qs1-3: 1cr: on time, 2cr: 1good Answer, 3cr: 2good A, 4cr: 3good A 4x3=12cr
PowerPoint: 0credit: nothing, 1cr: on time, 2cr: good, 3cr: very good, 4cr: excellent 4cr
Oral presentation: 2’: 3 slides, and a 30” clip; 0credit: nothing, 1cr: on time, 2cr: good, 3cr: very good, 4cr: excellent
4cr
Mid-term; 10cr
30cr
Unit: 3LP, 15-20’ video, students’ work, reflection with 1pg typed transcript. 0-30 credit see rubric 30cr
Portfolio: Table of Content, Attendance, Participation,
5 journals, 4 quizzes, unit, final reflection, other,
20 cr
Due:
Journals: Electronic copy, the day before class until midnight, hard copy in class before the class start.
Quiz: Due at the beginning of the class.
PowerPoint: On flash drive, 2/25, and 5/6 Wednesday.
Unit: 5/6 Wednesday. (Y/N)
Portfolio: 5/20 Wednesday. Pick up: 6/9, or after
Conceptual Framework/ Performance-Based Learning
Course: EDE 642/7575 How Children Learn Mathematics
|
Teachers and Teacher Candidates are expect to know and do |
Evidence |
Diversity (Knowledge & Skills) |
Demonstrate an understanding of the role of culture and language as it relates to meaningful problematic situations and the didactical use of such contexts in mathematical teaching.
Characterize mathematical development in terms of “big ideas”, the progression of strategies, and emergent models. |
Written analysis of the role of context in various mathematical investigations and its effect on children’s learning (CDROM assignment)
Selects video clips which exemplify developmental shifts in understanding and discusses them n relation to mathematical teaching and learning ( CDROM assignment)
|
Content Knowledge (related to your professional standards) |
Demonstrate a deep conceptual understanding of the mathematics candidates will be teaching (PK-2nd ) as described in the NCTM Principles and Standards 2000
Show evidence of a developing ability to plan worthwhile mathematical task connected to the NCTM Principles and Standards, 2000
Show evidence of a beginning ability to create a learning environment that fosters the development of mathematical power, as described in the NCTM Principles and Standards, 2000 |
Final Exam and performance during in class mathematical investigations
Preparation of a teaching unit on a mathematical topic, to develop number and spatial sense and several sequential investigations.
Written analysis of a class, including a selection of clipped moments from digital environments where teacher is doing something to support the development of community (CDROM assignment)
|
Pedagogical Knowledge |
Shows evidence of a developing ability to foster classroom mathematical discourse and reflection
Understands the importance of questioning and wait time
Understands the importance of inquiry on mathematical development |
Written analysis of teacher behaviors, including clipped moments from digital environments of mathematical discourse and reflection (CDROM assignment)
Written analysis of videotaped lessons analyzing the role of teacher’s questioning and wait time (CDROM assignment)
Preparation of teaching unit on a mathematical topic, to develop number sense and several sequential investigations to show evidence of appreciations for inquiry
|
Leadership Learning |
Is aware if the political aspects of the NCTM Principles and Standards, 2000 and the need to help the public understand and appreciate the reform
Demonstrates a working knowledge of the use of new curricular materials
|
Writes an accompanying letter to parents for the prepared unit, explaining the purpose of the unit (submitted with unit)
Use new curricula to plan unit |
Uses of Technology |
Proficiently uses digital environments for professional learning
Evidences a developing knowledge of available software and its uses in mathematical investigations |
Completes CDROM assignment successfully
Demonstrates flexible in-class use of Geo Logo while involved in mathematical explorations
|
Disposition to: Become lifelong learner; |
Exhibits an excitement and enthusiasm for doing mathematics
Seeks out materials and reading to expand and grow as a professional
|
Exhibits an excitement and enthusiasm for doing mathematics
Seeks out materials and readings to expand and grow as a professional |
Become a reflective practitioner; |
Show the ability to reflect critically on teaching decisions, both on one’s own and others |
Written reflection of videotape lesson and field experience
|
Establish a professional identity; |
Sees oneself as part of caring community of discourse engaged in activity, knowledge construction, communication, and reflection
|
Behaviors exhibited during in-class group work and discussion |
Build a caring community; |
Show evidence of the ability to analyze classroom interactions (student / learner and learner / learner) and the teacher actions, physical environment, and power dynamics that produce them
|
Written analysis of videotaped lesson and field experience |
Build democratic schools; |
|
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COURSE OUTLINE
EDE 642/7575
01/28 Class1 Introduction
Handshake Problems
Cooperative Learning
Progression of Strategies
Progressive Schematization
Math Congress
Build a Mathematical Community
Assessment: Reading Article, Standards
Journal # 1 Write a Reflective Piece on Handshake Problems
02/04 Class 2 Pre-Test, Lesson Plan
“Mathematics” or “Mathematizing”
Inquiry that Involves Mathematizing
Quantity, Unitizing, and Division
The Landscape of Learning, Learning Trajectories
Assessment: Reading Chapter 1 and 2
Journal # 2 Write a Reflective Piece on Teaching and Learning as Development
02/11 Class 3 Quiz 1
Number Sense on the Horizon
Strategies, Big Ideas, and Models
Counting, Understanding “How many?”
Piaget’s Theory of Numbers
Designing Contexts, Using Games
NCTM, NY City and State Standards
Assessment: Reading Chapter 3 and see CD
Journal # 3 Write a Reflective Piece on Young Children Construct Number Sense
02/18 Class 4 Quiz 2
Place Value on the Horizon
Additive Numerations
Babylon (3300BC), Maya, and Roman Numerals
Multiplicative Numerations
Construct Understanding of Unitizing
Curriculum Design
Assessment: Reading chapter 4 and see CD
Bring your flash drive.
Journal # 4 Write a Reflective Piece on Unitizing is the Heart of Place Value
02/25 Class 5 Quiz 3
Developing Mathematical Models
64, Bus like Context
From the Model of Situation to the use of Number Line
Math for Disabled Students
The Role of Context
Instructional Technology
Assessment: Reading Chapter 5 and Work on CD
PowerPoint: The Role of Context
03/04 Class 6 Quiz 4
Addition and Subtraction Facts on the Horizon
Double + 1, Quintery Structure, Swapping, Making Tens
Using Manipulatives: The Rekenrek
Children as Problem Solvers
Lesson Planning: Procedure
Working with Number Line Mathematical Models
Assessment: Read Chapter 6, and Review 1-6
03/11 Class 7 MID-TERM
Mathematics as a Human Activity
03/18 Class 8 Quiz 5
History of Math, the Playgrounds of Mathematics
Muhammad ibn Musa al-Khwarizmi
Algorithms versus Number Sense
Developing Meanings for the Operations
Observe and Analyze Children’s Thinking
Assignments, Reflection and Revision
Assessment: Read Chapter 7
Work on Unit
Journal # 5 Write a Reflective Piece on Representing Children’s Strategies on Rekenrek
03/25 Class 9 Quiz 6
Computation with Mini-lessons
Pictures with Built in Constraints
The Role of the Imagination
Multiplication as Repeated Addition
Exploring Cross-Cultural Multiplications
Lesson and Unit Planning
Assessment: Work on Unit
Read Chapter 8
04/1 Class 10 Post-test
The Development of Early Geometry
Array, Area, Perimeter Investigation
3D and 2D Representations
Building Boxes and Nets
The Cube Problem
Classroom Management
Assessment: Work on Unit, Review Chapter 1-8
Portfolio assessment: Taking Inventory CD
Write a Reflective Piece on Taking Inventory CD
04/8-17 Spring Recess
04/22 Class 11 Children’s Museum
Principles of Learning-Teaching
Characteristics of the Classroom Culture
Doing Math in a Mathematical Environment
Assessment: Review Book
Assessment: Work on Unit
04/29 Class 12 NCTM Conference
Developing Big Ideas
Partitive and Quotative Division
Developing Strategies
Basic Statistics and Probabilities
Teachers as Mathematicians
Assessment: Unit with Video component due
Assessment: Work on Portfolio
05/06 Class 13 Oral Presentation One (with Video Component)
Pattern Block Activities
Measurement Inventory, Measuring the Body
Compare the Metric System with the Customary System
Children’s Museum, Montessori School
Child Friendly Learning Environment
We All Grow into Lifelong Learners
Assessment: Work on Portfolio
05/013 Class 14 Oral Presentation Two (with Video Component)
Fractions as Emerging from Fair Sharing Situations
The Function of Context in Dealing with Remainders
The Ratio Table and the Double Number Line
Equivalence to Decimals and Percents
Understanding Percents as Part/whole Relationships
Assessment: Come to The End of Your Portfolio
05/20 Class 15 Final Examinations Portfolio due
REFERENCE………………………………
Fosnot, C.T. (1989). Enquiring teachers, enquiring learners: A constructivist approach to teaching. New York Teachers College Press
Freudenthal, H. (1991). Revisiting Mathematics Education: The China Lectures. Dordrecht, the Netherlands: Kluwer.
Kamii, C. (1985). Young children reinvent arithmetic. New York: Teacher College Press.
Kamii, C. (1993). Young children continue to reinvent arithmetic: grade three. New York: Teachers College Press.
Kerekes, J. (2006). The role of simple own constructed manipulatives in improving student participation, Understanding and Mathematical Effectiveness. Mathematics in School, 6-9
Kerekes, J., and King, K. P. (2008). Submarine sandwiches investigation: Making fractions more palatable for adult learners. Perspectives: The New York Journal of Adult Learning, 7(1), 47-53.
Kerekes, J., and Fosnot C. (1998). Using Pictures with Constraints to Develop Multiplication Strategies. The Constructivist 13(2): 15-20
Ma, L. (1999). Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Erlbaum
Smith, S. S. (2009). Early Childhood Mathematics Pearson Education, Inc.
Tournaki, N., Youngseh, B., and Kerekes, J. (2008). Arithmetic rack: A new Mathematics manipulative tried with students with learning disabilities. Learning Disabilities: A Contemporary Journal, Council for Exceptional Children 6(2), 41-60