- Exploring Representations of Addition and Subtraction – Concepts, Algorithms, and Mental
Math (Integers, Fractions/Rational Numbers)
- Exploring Algebraic Reasoning through Arithmetic, Geometry, and Data Management using
manipulatives and graphing calculators
- Making Sense of Student’s Differentiated Responses to Solving Problems within Inclusive
Settings
- Collection of Data for Teacher-Based Teacher Inquiry (solving the lesson problem and
analysing the design of the lesson)
Understanding and implementing Ministry of Education curriculum expectations and Ministry of
Education and district school board policies and guidelines related to the adolescent
Understanding how to use, accommodate and modify expectations, strategies and
assessment practices based on the developmental or special needs of the adolescent
Understanding Diversity of Knowing
and Learning Mathematics –
Mathematics for All Students
ABQ Intermediate Mathematics Fall 2009
SESSION 8 – Nov 4, 2009
Preparation for Wednesday Nov 4, 2009
Treats – Donovan and Brenda
Reminder - Gathering math topic articles for teacher inquiry
Read and Record for Nov 4 (Point form / chart or table)
See sample next page
a. Describe 2 characteristics of each theory: behaviourism,
constructivism, and complexity theory.
b. Infer how these theories can be used to analyze a
mathematics teaching/learning experience.
Behaviourism and Constructivism:
- Funderstandings. Behaviourism, Constructivism (Piaget, Vygotsky)
- Clements, D. & Battista, M. (1990). Constructivist learning and teaching.
Arithmetic Teacher, 38(1), 34-35
Complexity Theory
- Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An
International Journal of Complexity and Education, 2, pp. 85-88.
- Davis, B. (2003). Understanding learning systems: Mathematics education and
complexity science. Journal for Research in Mathematics Education (34)2, pp. 137-167
Assignments
Oct 28 - Technology
Webquest
Nov 4 - Math Task 2
Nov 7 - Annotated
Bibliography
Nov 18 - Learning
Theories paper
Teacher Inquiry - Annotated Bibliography
•
•
•
6 articles (one person), 8
articles (two people or more)
1st paragraph - describe key
math idea
2nd paragraph explains how
you are using the key idea to
to design your lesson
Due Sat Nov 7, 2009
Analytic journals
TI plan
Math Task 1
Math Task 2
Learning Theories paper
Annotated Bibliography
TI Oral/Visual
•
•
•
•
One bansho plan for one of the 2 lessons (only 2 lessons, not 3)
Two lesson – 2 different lesson problems
Display bansho of original student work on the whiteboard (if in groups, put in
grade level order to see development) – take digital pictures of blackboard and
student work for ppt, but bring originals (put before, during, after on chart paper)
Organization of the solutions is mathematical (to see how the idea you are
teaching develops) NOTE difference (NOT across grades – that was only for math task
1 and 2 to get you see development)
•
•
Focus for 35 minute OV Presentation - Rationale (what did you want to find out);
description the 3 part lesson; have us do the problem and discuss solutions;
analyze student solutions through your whiteboard bansho (math task type –
lesson problem), description of math that student learned (practise solutions –
evidence of learning from lesson); conclusions
Paper is due either Dec 2 or (last class) or Dec 5 (delivered to MLK’s 45B
Benlamond Ave #3 Toronto ON M4E 1Y8) – SUMMATIVE ASSESSMENT
Teacher Inquiry Topics
RED Oct 21
GREEN Oct 28
BLUE Nov 4
1. Sarjeet – Gr7 – Addition and Subtraction of fractions
2. Christina – Gr7 – Area of trapezoid
3. Joe – Gr7 – Patterning and using a table to represent a sequence
4. Donovan – Gr7 – Dividing Fractions
5. Brian – Gr8 – Geometry ??
6. Maria – Gr9Applied – Collecting and organizing data using charts,
tables, and graphs
7. Elina and Marijana – Gr9 Basic – Area and perimeter
8. Spencer – Gr9 – Area composite shapes
9. Yudhbir – Gr7 – Area of composite shapes
10. Jim – Gr9 Applied – Adding and Subtracting Integers
11. Brenda – Gr 8 – Using Algebraic Expressions to describe pattern
12. Michelle – Gr 8 – Multiplying and Dividing fractions
Topic Discussions - Schedule & Readings
Oct 14 - Adolescent Learning, (BLUE)
Oct 28 - Maslow’s Hierarchy of Needs
and Communities of Practice,
Behaviourism and Constructivism,
Vygotsky and Piaget (Green)
Nov 4 - Complexity theory (Yellow)
Nov 7 – Comparisons of learning
theories
Topic Discussion Process
Adolescent Learning
•
Jensen, E. (1998). How Julie’s Brain Learns. Educational
Leadership, 56(3), pp. 1-4.
•
Knowles, T., and Brown, D. (2000). What every middle school
teacher should know. Portsmouth, NH: Heinemann.
•
Reinhart. S. (2000). Never say anything a kid can say.
Mathematics Teaching in the Middle School. Pp 478-483.
•
Stahl, R. (1994). Using think-time and wait-time skilfully in the
Classroom. ERIC Clearinghouse of Social Studies/Social
science Education, Bloomington, IN.
Behaviourism,
Communities of Practice - Funderstandings, Wenger
Maslow’s Hierarchy of Needs – Funderstandings
Vygotsky and Piaget
Constructivism
•
Clements, D. & Battista, M. (1990). Constructivist learning
and teaching. Arithmetic Teacher, 38(1), 34-35
•30 minutes (usually start here)
•1 facilitator per 3 or 4 colleagues (one from each
of the other groups)
•Preparation (by all) - do the readings, viewing
of webcast, website search
Complexity Theory
•Facilitator - develops some thought provoking
•
Davis, B. (2005). Emergent Insights Into Mathematical
questions or a task to stimulate discussion of the
Intelligence from Cognitive Science. Delta-K, 42(2), pp. 1019.
topic, making reference to preparation
•
Davis, B. (2005). Teacher as “consciousness of the
•Colleagues - participate in task to be prepared
collective’. Complicity: An International Journal of Complexity
to share learning with group members in record
and Education, 2, pp. 85-88.
•
Davis, B. (2003). Understanding learning systems:
learning in journal
Mathematics education and complexity science. Journal for
Research in Mathematics Education (34)2, pp. 137-167.
Complexity Theory – discussion 1
Bus Problem
There are 36 children on school bus.
There are 8 more boys than girls.
How many boys? How many girls?
a) Solve this problem in 2 different ways.
b) Show your work. Use a number line, square grid, picture,
graphic representation, table of values, algebraic
expression
c) Explain your solutions. 1st numeric; 2nd algebraic
Compare your solutions.
How are they similar? How are they different?
Math Task 2 - Bus Problem
1 Design an Before (activation) task for your TI grade level (Before
2.
3.
4.
5.
problem) - activate students’ knowledge and experience related
to the task and show 2 different responses
Develop curriculum expectations knowledge package –overall,
and specific for grades 6 to 10
4 solutions (grade 7, 8, 9, and 10) to the problem (precise and
clear in your mathematical communication)
Bansho plan (labels at the bottom, categories of solutions,
mathematical annotations, and mathematical relationships
between solutions) with your anticipated solutions to the problem
Design an After (Practice) problem for students (grade level of
TI) to practise their learning and provide 2 different responses
Sample Bansho Plan
11”
Knowledge BEFORE
Activation
Package
Gr 7 to 10
8-1/2”
-codes and
description
-lesson
learning
goals in rect
highlighted
-Task or
Problem
-2 solutions
Relevant to
TI grade
Math
DURING
Vocabulary -Lesson (bus)
Problem
list
-What
8-1/2”
information
will WE use
to solve
the problem?
List info
AFTER Consolidation
-Math annotations on and around
the solutions (words, mathematical
details to make explicit the mathematics
in the solutions
-Mathematical relationship between
the solutions
4 different solutions exemplifying
mathematics from specific grades
Gr7
Gr8
Gr9
Gr10
labels for each solution that
capture the mathematical approach
AFTER
Highlights/
Summary
-3 or so
key ideas
from the
Discussion
For TI grade
AFTER
Practice
-Problem
-2 solutions
- focused on
TI grade
Lesson Analysis Using Learning Theories
Intro statement identifying the focus of the
paper
2. Description of the lesson flow in your MAIN
lesson
3. Lesson Analysis (At least 4 examples)
1.
Aspect of lesson that does align with a learning
theory principle (summary statement)
b) Detail of the lesson aspects explained in relation to
learning theory principles (APA referenced)
c) Aspect of lesson that does not align with a learning
theory principle (summary statement)
d) Detail lesson aspects explained not aligned to
learning theory in relation learning theory principles
(APA referenced)
a)
Suggestions for Improving Lesson - using
learning theory principle (APA referenced)
5. Conclusion
4.
• Adolescent
Learning theory
• Behaviourism
• Communities of
Practice
• Complexity
Theory
• Constructivism
• Maslow’s
Hierarchy of
Needs
What Can We Learn From TIMSS?
Problem-Solving Lesson Design
BEFORE
• Activating prior knowledge; discussing previous days’
methods to solve a current day problem
DURING
• Presenting and understanding the lesson problem
• Students working individually or in groups to solve a
problem
• Students discussing solution methods
AFTER
• Teacher coordinating discussion of the methods (accuracy,
efficiency, generalizability)
• teacher highlighting and summarizing key points
(Stigler & Hiebert, 1999)
• Individual student practice
Criteria for a Problem Solving Lesson
•
•
•
•
•
•
•
•
Content Elaboration- developed concepts through teacher and student
discussion
Nature of Math Content - rationale and reasoning used to derive
understanding
Who does the work
Kind of mathematical work by students - equal time practising procedures
and inventing new methods
Content Coherence - mathematical relationships within lesson
Making Connections - weaving together ideas and activities in the
relationships between the learning goal and the lesson task made explicit by
teachers
Nature of Mathematics Learning - seeing new relationships between math
ideas
Nature of Learning first struggling to solve math problems
then participating in discussions about how to solve them hearing pros and cons,
constructing connections between methods and problems
− so they use their time to explore, invent, make mistakes, reflect, and receive needed
information just in time−
Teacher
Inquiry Lesson
Analysis
Using
Problem
Solving
Bring your ONE
TI Lesson Plan
- 9 copies for
June 3 class to
get analysis
feedback
Pool Border Problem
- What Should the revised lesson look like?
Lesson Description
- What the students do to learn
<what the teacher does to teach>
-Include math details
-Framed within a 3-part problem solving
lesson
Preparation for Saturday Nov 7, 2009
Treats – Joe and Maria
Due – Annotated bibliography for teacher inquiry
Read all learning theories papers and bring along
Behaviourism and Constructivism:
- Funderstandings. Behaviourism, Constructivism (Piaget, Vygotsky)
- Clements, D. & Battista, M. (1990). Constructivist learning and teaching.
Arithmetic Teacher, 38(1), 34-35
Complexity Theory
- Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An
International Journal of Complexity and Education, 2, pp. 85-88.
- Davis, B. (2003). Understanding learning systems: Mathematics education and
complexity science. Journal for Research in Mathematics Education (34)2, pp. 137-167
Assignments
Oct 28 - Technology
Webquest
Nov 4 - Math Task 2
Nov 7 - Annotated
Bibliography
Nov 18 - Learning
Theories paper