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.

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 MoreSendhil Revuluri

Chicago Public Schools

NCTM 2009 Annual Meeting, Session 801

Saturday, April 25, 2009

DisclaimerWhile 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 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 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 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 How were your guesses? What approaches did you use? What other questions could you ask?

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 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 ways to pick two?

How many?

Proof without words

Proof without words

Why bother counting twice? How many ways are there to pick three

people out of a group of six? We routinely teach how to count once Can we make it more general by counting again?

Suppose there are five people. How many ways are there to pick 0, 1, 2, 3, 4, 5? What happens when Blaise walks in?

Combinatorial identities 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? 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

methodSpacer method

What counting is usually included? What topics are included and when?

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 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?

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 worldWe 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 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 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? What’s the numerical pattern? Where is the underlying pattern coming from? How could you…

Justify? Generalize? Extend? Apply?

A recursive relationship

How do we want students to feel? 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 “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 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? What’s the numerical pattern? Where is the underlying pattern coming from? How could you…

Justify? Generalize? Extend? Apply?

A recursive relationship, motivated by context

How does this open up our classes? 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 Pólya’s process (How to Solve It):

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 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? Questioning Forming conjectures Trying a simpler problem Seeing similarities among related problems Finding connections Generalizing

Sense reduces sensitivityThe 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

Sensitivity

Combinatorial habits of mind 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? It’s math… applied to math! Counting, and especially generalizing,

connecting, and justifying counting, can: 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? Can we do it? (existence)

How many ways? (counting)

How does it work?(properties and structure)

Can we make it better?(satisfactory or optimal arrangements)

What are some basic principles? Multiplication principle Addition principle Permutations Over-counting Combinations Complementary counting Probability

Various other topics & problems Pascal’s Triangle coloring and identities Counting parts of regular polytopes

Σ from 1 to n & other proofs without words

Finding and counting factors The Set® Game More examples (geometric & otherwise)

There’s more “standard” stuff too We discussed counting the same quantity in

two different ways to find an identity Pascal’s: Method of the distinguished element

Binomial probability Pigeonhole principle Counting as correspondence Inclusion-exclusion principle

Beyond the basics (just a list) Generalized permutations & combinations

multisets multinomial coefficients partitions

Recurrence relations Generating functions Group theory in combinatorics Graph theory

Advanced topics (if you like) Recurrence relations and closed form

Example: The Fibonacci sequence The idea of the “ansatz” How this is like solving differential equations

Generating functions Combinatorial problems come in sequences Translate the counting into function land We can then apply many tools of analysis

Did you learn anything? What’s one idea you’ve gained

or one connection you’ve made?

What’s one thing you’re going to try?

What’s one thing you’ll tell someone about?

Thank you! 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]