Transcript Slide 1

Welcome!
Thanks for making this your final workshop of
NCTM 2009!
Please take each handout, get some paper ready,
and join each other at the tables. (I won’t be
offended if you leave early, but please don’t
sit on your own!) If you brought a graphing
calculator, grab it.
Ladies and gentlemen, start your counting…
many fun problems await! Openers are on #1.
Counting: It’s Not Just for
Breakfast Any More
Sendhil Revuluri
Chicago Public Schools
NCTM 2009 Annual Meeting, Session 801
Saturday, April 25, 2009
Disclaimer
While I am happily employed by Chicago Public
Schools, and they do pay me to think about math
and its instruction, any opinions I express in this
session are mine and are not necessarily shared
by my employer or anyone else (though they
should be). Anyone who says otherwise is
itching for a fight.
[title of show]
Why not just for
breakfast?
(Well, it is noon
already…)
Some assumptions, notes, requests
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I trust you to find the “standard” stuff in textbooks
You may have gone to some counting sessions
I’m stealing (but not just from counting sessions)
I tried to be brief to give your brain a chance to think
I may not start at the basics; I rely on your questions
Even without questions, I’ve still got way too much
A little about you
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How many of you teach mainly high school?
How many teach mainly middle school?
Who else is here?
Another problem: If those who said “middle school”
already shook hands with each other, and those who
said “high school” already shook hands with each
other, how many handshakes would be left?
My goals for the rest of this session
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Everyone has a chance to do some math
together, have fun, and be clever
I press you to think about what your students
need to know, what you teach, how, and why
I want to force some of my dearly-held, but
entirely-unsupported opinions upon you
First opener problem: Squares
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How were your guesses?
What approaches did you use?
What other questions could you ask?
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Which ones are pretty easy?
Which ones are pretty hard?
A clever way to use the graphing calculator to
save you some algebraic work
Second opener problem: Candies
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What did you notice?
What approaches did you use?
Does it matter if you can tell the kids apart?
(What if you can tell the candies apart?)
Two different methods
Don’t just count…
How
many
ways to
pick two?
Don’t just count, count again
How
many?
How
many
ways to
pick two?
Proof without words
Proof without words
Why bother counting twice?
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How many ways are there to pick three people
out of a group of six?
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We routinely teach how to count once
Can we make it more general by counting again?
Suppose there are five people.
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How many ways are there to pick 0, 1, 2, 3, 4, 5?
What happens when Blaise walks in?
Combinatorial identities
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The basic idea: if you can generally count
something in two ways, you know those ways
must be equivalent – you have an identity
A corollary: if you can count two things and
then put them in one-to-one correspondence,
you know they must be equivalent too
Does this sound easier to do in a concrete
context than with fractions or symbols?
Can candies lead us to an identity?
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Casework (“distinguished element”) method:
Give oldest kid 0 candies, have n left for the
other k – 1; give her 1, have n – 1 left; …; or
give her n candies, so have 0 left
“Spacer” method: n + k – 1 choose k – 1 ways
These are two fully general ways to count the
same arrangements: must lead to an identity!
The hockey-stick identity
Sum of terms
from casework
method
Spacer
method
What counting is usually included?
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What topics are included and when?
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Middle school: Multiplication principle
Algebra 1: Combinations and permutations
Algebra 2: Binomial expansion & probability
What kinds of problems are used to explore?
What do students learn?
What could we be doing more effectively?
Third opener problem: Coloring
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How many ways are there?
Does it matter that they’re equilateral?
So, does symmetry make it easier or harder?
What other questions could you ask?
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Which ones are pretty easy?
Which ones are pretty hard?
Here, the calculator can help you conjecture
This is fun, but I’m a busy (wo)man.
Why is combinatorics, and
discrete math more generally,
important for our students to
know, for us to teach, or to
spend time in our classes?
Anyone know what this is?
The transistor changed the world
We live in an increasingly
digital world. If algebra
and calculus are the
languages of classical
chemistry and physics,
then discrete math is the
language of computing.
Anyone recognize him?
It’s the Terrible Trivium
from The Phantom
Toolbooth.
Discrete math problems
are often more
conceptual, not
requiring so much
“machinery” and
avoiding the tedium.
It can be complex, but concrete…
… and
more real,
and more
fun!
But don’t just take it from me
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Per David Patrick, author of The Art of Problem Solving:
Introduction to Counting & Probability, discrete math:
… is essential to college-level mathematics and beyond.
… is the mathematics of computing.
… is very much “real world” mathematics.
… shows up on most middle and high school math contests
… teaches mathematical reasoning and proof techniques.
… is fun.
See article at http://www.artofproblemsolving.com/Resources/.
Okay, let’s do some more math
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Let’s count some geometric objects in the
circle marked with points (Handout #2).
Do the numbers look familiar?
Do you notice anything else?
Pascal’s Δ may be a source of infinite patterns
… and they’re not just beautiful, they can
form deep connections among key ideas.
Just one pattern: even & odd
You can keep going: mod 3, mod 4
What are we noticing here?
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What’s the numerical pattern?
Where is the underlying pattern coming from?
How could you…
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Justify?
Generalize?
Extend?
Apply?
A recursive relationship
How do we want students to feel?
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Believing that math has entry points for them, and
that they can learn it through effortful practice
Believing that math can be beautiful, should and
does make sense, rather than teacher as the authority
Using justification not just to ensure correctness, but
also to see why, and wanting to keep finding more
Motivating symbolic or algebraic representations
and the other tools we offer them
Two quotes and an example
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“The mind is not a vessel to be filled, but a
fire to be ignited.” – Plutarch
“If you want to build a ship, don't drum up
people together to collect wood and don't
assign them tasks and work, but rather teach
them to long for the endless immensity of the
sea.” – Antoine de Saint-Exupery
Why does a baby point? Vygotsky’s theory
How Mathematicians Think
“If we wish to talk about mathematics in a way
that includes acts of creativity and
understanding, then we must be prepared to
adopt a different point of view from the one in
most books about mathematics and science.
When mathematics is viewed as content, it is
lifeless and static…”
– William Byers
Imagine Math Day at Harvey Mudd
“[We need] opportunities to remind [our]selves
why teaching, learning, and creating math can be
useful, rewarding and fulfilling. [We] need to be
aware of the powerful role that math can play in
the lives of their students… because [math can]
be an effective vehicle for teaching students
valuable ‘habits of mind.’”
– Yong and Orrison, MAA Focus, 2008
Another counting investigation
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Dimension 0: Count point
Dimension 1: Count points, segment
Dimension 2: Count points, segments, region
Dimension 3: Keep on countin’
But why stop here?
Dimension 4: “Whoa.”
A peek into dimensions 3 and 4
Whoa indeed. Let’s see that again!
What are we noticing here?
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What’s the numerical pattern?
Where is the underlying pattern coming from?
How could you…
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Justify?
Generalize?
Extend?
Apply?
A recursive relationship, motivated by context
How does this open up our classes?
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Low threshold, high (or no) ceiling
More students can succeed at math if there are
more ways to be successful (Cohen, Silver)
Connects to multiple solution methods
Naturally problem-centered, student-centered
Connected to multiple habits of mind
Problem-solving
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Pólya’s process (How to Solve It):
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Understand, plan, solve, check
Looking for patterns and connections
Developing heuristics
“work backwards”, “try a simpler case”, etc.
Developing flexible thinkers
Justification emerges naturally
Developing problem-solving skills
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A few principles, many connected techniques
Students learn that experience solving really
contributes to their skill (growth mindset)
Helps orderly, algebraic thinking, and can
address and motivate algebraic fluency too
Develops inductive thinking (conjecturing) as
well as deductive thinking (proof), and these
problems often connect them really well
What cognitive habits do we seek?
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Questioning
Forming conjectures
Trying a simpler problem
Seeing similarities among related problems
Finding connections
Generalizing
Sense reduces sensitivity
Sensitivity
The more sense students make of
a concept… even the more sense
which you say it will make… the
less sensitive their understanding
is to surface variables.
Corollary 1: Retention
Corollary 2: Rigor
Corollary 3: Choices
Combinatorial habits of mind
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Flexibility of perspective; multiple
representations and multiple methods
Complementary thinking and symmetry
Satisfying constraints
Look for patterns as an aid to conjecturing
Look for patterns as an aid to proof
Favor constructive proofs or even
“programming” (algorithms) over proof
So is this applied math?
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It’s math…
applied to math!
Counting, and especially generalizing,
connecting, and justifying counting, can:
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motivate use of the math they’ve learned before
expose, afford discussion, and build key habits
show the math begins after you find the answer
What are some basic questions?
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Can we do it? (existence)
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How many ways? (counting)
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How does it work?
(properties and structure)
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Can we make it better?
(satisfactory or optimal arrangements)
What are some basic principles?
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Multiplication principle
Addition principle
Permutations
Over-counting
Combinations
Complementary counting
Probability
Various other topics & problems
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Pascal’s Triangle coloring and identities
Counting parts of regular polytopes
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Σ from 1 to n & other
proofs without words
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Finding and counting factors
The Set® Game
More examples (geometric & otherwise)
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There’s more “standard” stuff too
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We discussed counting the same quantity in
two different ways to find an identity
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Pascal’s: Method of the distinguished element
Binomial probability
Pigeonhole principle
Counting as correspondence
Inclusion-exclusion principle
Beyond the basics (just a list)
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Generalized permutations & combinations
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multisets
multinomial coefficients
partitions
Recurrence relations
Generating functions
Group theory in combinatorics
Graph theory
Advanced topics (if you like)
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Recurrence relations and closed form
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Example: The Fibonacci sequence
The idea of the “ansatz”
How this is like solving differential equations
Generating functions
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Combinatorial problems come in sequences
Translate the counting into function land
We can then apply many tools of analysis
Did you learn anything?
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What’s one idea you’ve gained
or one connection you’ve made?
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What’s one thing you’re going to try?
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What’s one thing you’ll tell someone about?
Thank you!
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Please do email me with feedback, questions,
comments, ideas, and more problems and
resources!
I’m happy to send you the slides & handouts
[email protected]