Transcript Math in the Middle

The Mathematical Education
of Teachers
Lessons Learned from Math in the Middle
Michelle Reeb Homp
University of Nebraska – Lincoln
Sandi Snyder
Shickley Public Schools, NE
Math in the Middle Institute
Partnership (M2)
• The information we’ll be sharing is based on our
experiences with the Math in the Middle Institute
Partnership
• Primary focus of M2 is professional development for
middle level teachers, but the courses we will address
have also been offered (with adjustments) to H.S.
teachers
• Many of the components of the M2 curriculum are also
part of the teacher preparation program at UNL
Math in the Middle
Institute Partnership
A Brief Summary
Principal Investigators
• Jim Lewis, Department of Mathematics
• Ruth Heaton, Department of Teaching, Learning &
Teacher Education
M2 Partnership Vision
• Create and sustain a University, Educational Service Unit
(ESU), Local School District partnership
• Educate and support teams of outstanding middle level
mathematics teachers who will become intellectual
leaders
• Place a special focus on rural teachers, schools, and
districts
(Funding began August 1, 2004)
M2 Major Components
• The M2 Institute, a 25-month institute that offers a coherent
program of study to deepen mathematical and pedagogical
knowledge and develop leadership skills
• Mathematics learning teams, led by M2 teachers and supported by
school administrators and university faculty, which develop
collegiality
• A research initiative for educational improvement and innovation
M2 Institute Courses
• Eight new mathematics and statistics courses
designed for middle level teachers
• Special sections of three pedagogical courses
• An integrated capstone course
M2 courses focus on these objectives:
• enhancing mathematical knowledge
• enabling teachers to transfer mathematics
they have learned into their classrooms
• leadership development
• action research
Math in the Middle Institute
We will focus on three courses in particular:
• Number Theory & Cryptology
• Using Mathematics to Understand Our World
• Concepts of Calculus
Some objectives which significantly influenced
the development of these courses
We want our students to:
• Know that effort matters
• See connections to school math
Presentation Outline
1. Effort Matters
2. Connections to School Mathematics
•
•
•
Number Theory & Cryptology
Using Math to Understand Our World
Concepts of Calculus
3. Sandi’s Comments
Effort Matters
The National Mathematics Advisory Panel
commissioned by the president in April 2006.
Draft #42 of Final Report states:
“Children’s goals and beliefs about learning are
related to their mathematics performance.
Experimental studies have demonstrated that
changing children’s beliefs from a focus on ability to
a focus on effort increases their engagement in
mathematics learning and improves mathematics
outcomes. Effort matters.”
Effort Matters
One of the five strands of mathematical
proficiency described in Adding it Up is
Productive disposition
Habitual inclination to see
mathematics as sensible, useful,
worthwhile, coupled with a belief in
diligence and one’s own efficacy.
National Research Council
2001
Effort Matters
M2 Innovations: “Habits of Mind” Problems
A person with the habits of mind of a mathematical thinker can
use their knowledge to make conjectures, to reason, and to
solve problems. Their use of mathematics is marked by great
flexibility of thinking together with the strong belief that
precise definitions are important. They use both direct and
indirect arguments and make connections between the
problem being considered and their mathematical knowledge.
When presented with a problem to solve, they will assess the
problem, collect appropriate information, find pathways to
the answer, and be able to explain that answer clearly to
others.
While an effective mathematical toolbox certainly includes
algorithms, a person with well developed habits of mind
knows both why algorithms work and under what
circumstances an algorithm will be most effective.
Effort Matters
“Habits of Mind” Problems
Mathematical habits of mind are also marked by
ease of calculation and estimation as well as
persistence in pursuing solutions to problems. A
person with well developed habits of mind has a
disposition to analyze situations as well as the selfefficacy to believe that he or she can make progress
toward a solution.
This definition was built with help from Mark Driscoll’s
book, Fostering Algebraic Thinking: A guide for teachers
grades 6-10.
Effort Matters
Benefits of using Habits of Mind problems
• They typically have a variety of solution
strategies – allowing teachers to see the benefits
of having students present different solutions
• The focus is on PROCESS – thus encouraging
persistence and promoting the value of EFFORT
• Serve as a resource for teachers; adaptations of
the Habits of Mind problems consistently find
their way into teachers’ classrooms
Effort Matters
Examples of Habits of Mind Problems
•
•
•
•
•
•
Crossing the River
Locker Problem
Handshake Problem
Pentominoes
Rice on a Checkerboard
Others involving combinatorics, patterns,
repeated applications of the Pythagorean
Theorem, geometric series, etc.
A problem to get started:
Making change
What is the fewest number of coins that it will take
to make 48 cents if you have available pennies,
nickels, dimes, and quarters? After you have
solved this problem, provide an explanation that
proves that your answer is correct.
Extension: How does the answer (and the
justification) change if you only have pennies,
dimes, and quarters available?
Note: We first encountered this problem in a conversation with
Deborah Ball.
Student
work
sample
(grade 7)
Student
work
sample
(grade 8)
Habits of Mind Problems
Sandi’s Response
• Allow students to see how others think and
organize thoughts
• Reinforce the idea that there is not just
one way to approach a problem
• Serve as a bridge between arithmetic,
algebra and geometry
• Encourage cooperative learning
Presentation Outline
1. Effort Matters
2. Connections to School Mathematics
•
•
•
Number Theory & Cryptology
Using Math to Understand Our World
Concepts of Calculus
3. Sandi’s Comments
Making Connections
Why is it so important?
“Most teachers see very little connection between
the mathematics they study as an undergraduate
and the mathematics they teach. … a consequence
is that, because individual topics are not recognized
as things that fit into a larger landscape, the
emphasis on a topic may end up being on some
low-level application instead of on the
mathematically important connections it makes”.
Al Cuoco, AMS Notices 2001
Making Connections
What should be done?
“The mathematics taught should be connected as
directly as possible to the classroom. This is
more important the more abstract and powerful
the principles are. Teachers cannot be expected
to make the links on their own.”
Roger Howe
AMS/MAA Joint Meetings 2001
Number Theory and Cryptology
Primary Goal:
Learn the concepts of Number Theory
which are necessary to understand RSA
cryptography, making connections to the
classroom whenever possible.
Number Theory and Cryptology
Secondary goals:
• Provide an example of a modern, real-world
application of mathematics
• Convince teachers they are mathematicians,
capable of understanding some complex ideas
Fear of factors: Cracking prime-number case raises online security
doubts, Lee Dembart, International Herald Tribune, March 10, 2003
“… the details of the mathematics (used in online security)
may be understood by relatively few people…”
Number Theory and Cryptology
Connections
1.Congruences and boxes
Today is Wednesday. What day of the
week will it be in 11 days, in 95 days?
11
0
8
95
14
-6
Sun
Mon
Tues
Wed
Thurs
Fri
Sat
4
5
6
0
1
2
3
Number Theory and Cryptology
Connections
2. Modular arithmetic and divisibility rules
Divisibility rules for: 3, 4, 8, 9, 11
– Write the rules as you would explain them to
your students.
– How do you write that an integer n is divisible
by 9 using congruence statements?
– How does the statement n  0 mod 9 translate
into a divisibility rule?
Number Theory and Cryptology
Connections
3. Definitions and Proofs
• Arrive at definitions of even and odd and use
them to prove:
even x odd = even
odd x odd = odd (etc.)
Demonstrates, in a very tangible way, the value
of precise definitions and their importance in
mathematical proof
Number Theory and Cryptology
Connections
4. Proof and the counter example
• Does a display of 100 green frogs prove that
all frogs are green? What about 1000 frogs?
To disprove, all that is required is
one blue frog
• Prove or disprove: If d|ab then d|a or d|b.
(Where d, a & b are integers)
Number Theory and Cryptology
Sandi’s Response
• Use of prime factorization to determine all
factors of a number
• Numbers that are relatively prime
• The infinitude of primes—they keep
going and going!
Using Math to Understand
Our World
Course Description
• Students will examine the mathematics underlying
socially-relevant questions from a variety of
academic disciplines
• Students will construct and study mathematical
models of the problems
• Sources will include original documentation
whenever possible (such as government data,
reports and research papers) to provide a sense of
the very real role mathematics plays in society, both
past and present
Using Math to Understand
Our World
Primary course Goals: broaden students’
perspective by applying mathematical concepts
to various interdisciplinary settings
Additional course goals:
(1) develop mathematical modeling and problem
solving skills
(2) improve ability to read technical reports and
research articles
(3) refine written mathematical communication skills
Using Math to Understand
Our World
A few project titles:
• Measuring Temperature & Newton’s Law of
Cooling
• Using Body Temperature to Estimate Time
Since Death
• Building Up Savings and Debt
• Childhood Growth Charts
Using Math to Understand
Our World
Project III: Containing Infectious Diseases
(a.k.a. School Pox)
Using Math to Understand Our World
Sandi’s Response
• Answers the question, “When am I ever
going to use this?”
• Extensive writing about the
mathematics
• Real life data that shows calculations
aren’t always going to come out
perfectly
M2 Innovations:
Learning & Teaching Projects
Select a challenging problem or topic that you have studied
in MSL and use it as the basis for a mathematics lesson that
you will videotape yourself teaching to your students.
How can you present this task to the students you teach?
How can you set the stage for your students to understand
the problem? How far can your students go in exploring this
problem? You want your students to discover as much as
possible on their own, but there may be a critical point where
you need to guide them over an intellectual “bump.”
Produce a report analyzing the mathematics and your
teaching experience.
Learning &
Teaching
Project
H.S. Teacher
sample
(page 1)
Learning &
Teaching
Project
H.S. Teacher
sample
(page 2)
Learning &
Teaching
Project
H.S. Student A3
work sample
Learning &
Teaching
Project
H.S. Student A1
work sample
Learning &
Teaching
Project
Gr 5 Teacher
sample
(page 1)
Learning & Teaching Project
Gr 5 Teacher sample
(page 2)
Learning & Teaching Project
Gr 5 Teacher sample
(page 3)
Presentation Outline
1. Effort Matters
2. Connections to School Mathematics
•
•
•
Number Theory & Cryptology
Using Math to Understand Our World
Concepts of Calculus
3. Sandi’s Comments
Sandi’s Comments
• The Heart of Mathematics
An invitation to effective thinking
Burger & Starbird
Key Curriculum Press, 2000
• Action Research project—a wonderful learning
experience
• Need to know where our students came from
mathematically AND where they will go in the future
• M2 has changed how I teach
Concepts of Calculus
Primary Goals:
• develop a fundamental understanding of key
mathematical ideas utilized in calculus
• develop conceptual knowledge of the
processes of differentiation and integration,
along with their applications
• help teachers gain insight into topics found in
school mathematics which are related and
foundational to the development of calculus
Concepts of Calculus
Features:
• Course topics progress from limits to the
Fundamental Theorem of Calculus
• Rigorous treatment of various integration and
differentiation techniques (such as the product
rule for middle level teachers) are excluded
• Graphing calculators are used extensively
Concepts of Calculus
Features:
• Very little lecture: course consists of a series
of problem sets students work through to
explore calculus concepts
• Connections to school mathematics topics
are made consistently throughout the course
Concepts of Calculus
Connections
The problem sets on which the course is
based connect to school mathematics;
many teachers take them directly to their
classrooms
• Sample problem set
Note: The problem sets are adaptations of worksheets first
developed by Kristin Umland and Matt Bardoe,
University of New Mexico, La META project
Concepts of Calculus
Connections
One vehicle for classroom connections:
homework assignments.
Calculus Connections
Asma Harcharras, University of Missouri
Dorina Mitrea, University of Missouri
Publisher: Prentice Hall, 2007
Concepts of Calculus
Connections (taken verbatim out of M.L. or H.S. texts)
Examples
•Fix area and minimize perimeter
using integer dimensions
•Fix perimeter and maximize area
using integer dimensions
Calculus Connections
Harcharras & Mitrea
Prentice Hall, 2007
•Estimate area of an irregular shape
using grids; find lower and upper
estimates; improve estimates with
finer grids
Concepts of calculus
“As I took the Calculus class, I was reminded over and
over of the importance of Algebra in preparing students
for upper level mathematics. One particularly important
aspect was linear graphing. I know my students need a
good understanding of slope and rate of change in order
for them to understand instantaneous rate of
change/derivatives.
Along with linear and quadratic graphing, Algebra students
need to have experiences with a variety of graphs with
many dips and turns. They also need to be able to
describe these using words like increasing and
decreasing. Polynomials is another concept of which my
Algebra students need a good grasp in order to work
with a broader range of mathematical situations. I also
saw the importance of solving equations and inverse
operations for them to also understand the idea of
integrals and derivatives.” M. Bornemeier, 2006 M2 graduate
Concepts of Calculus
Sandi’s Response
Linear and Polynomial Functions
• Interpret slope. What does it mean?
• Celsius and Fahrenheit are related by the equation
9
F(C)= C + 32 Interpret slope and identify the units for
5
the slope.
• Define, distinguish, create polynomial functions
• Given a standard 8.5 X 11 inch piece of paper,
determine a function which gives the volume of a box
(no lid) made by cutting squares from each of the
corners and folding up the sides. Let x be the length of
side of the square.