Welcome! Thanks for coming to this interactive workshop! Please take a handout, join each other, and introduce yourself to someone new. Then get your paper out and your pencils warmed up. Ladies and gentlemen, start your counting… many fun problems await! (I may pause you at some point and ask you to read all of the opener problems.)

Count on It! Use Combinatorial Problems to Build Your Students’ Engagement and Understanding Sendhil Revuluri, Senior Instructional Specialist CPS Department of H. S. Teaching & Learning CMSI Annual Conference, May 2, 2009

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

Opener problems  What did you notice about these problems?  How are they similar to problems you use?  How do they reinforce core ideas you teach?  How are they different from other problems?  Could they help you make students think?  Could they inspire students to ask questions?

Opener problems 1, 2, and 3  How did you approach these problems?  Were these problems related? How?  What are some key principles they illustrate? Strategic counting Being organized The multiplication principle  What are other problems you could make?

Opener problems 4 and 5  How did you approach these questions?  Which questions were easier? Why?  What other questions could I have asked?  What relationships do you notice?  What are some key principles illustrated? Finding a simpler case Symmetry Recursion  What are other problems you could make?

Opener problem 6  Did you want to do c the same way as a & b?  Did the order I asked these questions matter?  Do the numbers you got look familiar?  Could you make a picture of this situation?  What are other problems you could make?  Could your students ask questions like this?

Triangular numbers

What does #6 have to do with #5?  Let’s take a little change of perspective  What if the grid were infinite?  How do we continue counting forever?  This is called Pascal’s Triangle  Do you notice any common numbers?  Do you see any other patterns?

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

Another pattern: even & odd

You can keep going: mod 3, mod 4

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 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

That’s nice, 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?

Why counting is important  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

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

What counting is usually included?  What topics are included and when? 5 th grade ILS: Multiplication principle 8 th grade ILS: Combinations and permutations By PSAE: Binomial expansion & probability  What kinds of problems are usually used?  Do we have to wait so long?  What else can counting help students learn?

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 [our] students… because [math can] be an effective vehicle for teaching students valuable ‘habits of mind.’” – Yong and Orrison, MAA Focus, 2008

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

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

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 with feedback, questions, ideas, comments, and more problems and resources!  I’m happy to send you these slides and our handout (and more, from a longer version)