welcome! thanks for coming to this interactive workshop! please take a handout, join each other, and...
TRANSCRIPT
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 http://www.artofproblemsolving.com/Resources/.
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?
5th grade ILS: Multiplication principle 8th 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 email 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)