1. The Math Forum @ Drexel - mathforum.org My Students Can Notice/Wonder, Now What? Marie Hogan, West Covina, CA Suzanne Alejandre, Philadelphia, PA

2. The Math Forum @ Drexel - mathforum.org IntroductionsIntroductions  Marie Hogan  K-8 teacher multiple subjects 1987–1998  Principal of a private school 1998–2000  middle school mathematics teacher 2000–2012  Math Forum Teacher Associate 2009–present  Suzanne Alejandre  middle school mathematics and computer teacher 1973–1977, 1988–2000  Math Forum Staff 2000–present  Audience ?

3. The Math Forum @ Drexel - mathforum.org Notice/Wonder: Sound Familiar? CMC-North 2011: Get Your Students Hooked On Noticing and WonderingGet Your Students Hooked On Noticing and Wondering ComMuniCator, December 2010: Problem Solving–It Has to Begin with Noticing and Wondering

4. The Math Forum @ Drexel - mathforum.org Notice/Wonder: An Example

5. The Math Forum @ Drexel - mathforum.org Notice/Wonder: Quick Overview Eating Grapes

6. The Math Forum @ Drexel - mathforum.org What did you hear?

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8. If you get stuck, you might try to notice The quantities (known or unknown counts or measurements). Relationships between quantities. Information that is not given in the problem but that might be related or that the problem reminds you of. Key words from the problem. Your wonderings may include: I wonder what will happen if … I wonder what this word means … I wonder if this pattern will continue … What does this mean? What do they want? Does it have to be that way? Do I need to figure that out? How does this situation work? Is there another way to think of it? How will I know if this is true? What is a good way to express that? When is this true?

9. The Math Forum @ Drexel - mathforum.org Adopting Notice/Wonder as a Classroom Practice  personal white boards  textbook “Given” as notice “Strategy” as wonder  a bridge to algebra

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13. MP1 Make sense of problems and persevere in solving them. Connecting to CCSSM

14. The Math Forum @ Drexel - mathforum.org MP3 Construct viable arguments and critique the reasoning of others.  What does this mean in your classroom?  What do you do to develop this practice?  What’s one challenge you have?

15. The Math Forum @ Drexel - mathforum.org Notice/Wonder initially helps students to “make sense of problems and persevere in solving.” But that’s just the beginning.

16. The Math Forum @ Drexel - mathforum.org Change the Representation Changing the representation can mean use of a different form of representation (e.g. using a line drawing for a word problem) or it can mean trying different ways of presenting the information in the same form (e.g. rewriting all of the numbers as fractions with a numerator of 1). Considering multiple representations and choosing representations that fit the problem well are important problem-solving skills.

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18. Sample 1 I represented the problem visually with a fraction bar: I started with a bar that represented the total number of pets. I divided the bar into halves and made one half the fraction of cats. I divided the remaining half into two fourths and made one fourth the fraction of dogs. I divided the remaining fourth into two eighths and made one of them the fraction of horses. I divided the remaining eighth into two sixteenths and made one of them the fraction of birds. I divided the remaining sixteenth into two thirty-seconds and made one of them the fraction of birds. I was left with 1/32. There were no more pets left other than the gerbils. So the gerbils represent 1/32 of the total.

19. The Math Forum @ Drexel - mathforum.org Sample 2 One way fractions are represented is by groups of things in a set. I don't know how many pieces of the set are gerbils, and I don't know how big the whole set is, and I don’t know how many pets each piece represents. Birds are the smallest part of the set so far, with 1/32 of the set. So it makes sense to me to start out with 32 pieces in my set. I drew 32 circles, and then for each animal, I filled in the number of circles represented by that animal with counters.

20. The Math Forum @ Drexel - mathforum.org Sample 3 I am going to represent the problem using an equation. I know: total number of pets = sum of all types of pets in contest. If I call the total number of pets P then I can represent the number of each type of pet as its fraction times P. This means:

21. The Math Forum @ Drexel - mathforum.org Get Unstuck When we don’t see how to solve a problem, we try to forget about solving the problem for now. All we want is to get our brain in gear. We ask ourselves these three questions: 1.What is going on in this problem? 2.What can I try? 3.What does this remind me of?

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23. Sample 1 I tried to use pictures to show how many people got on the train. I didn’t know how many people got on at Monkey House, so I drew a box. Then for each person I knew I made a dot. I made each stop a different color. At each stop there are three more people than at the stop before, so you have to draw what you had from the stop before and then draw three more dots. I don’t know what number goes in the ? boxes though, so I can’t add them up. Maybe guess and check will help.

24. The Math Forum @ Drexel - mathforum.org Sample 2 At first I thought I had solved the problem really easily. First someone got on at the Monkey House. Then 3 more people got on at the Alligator Pond than at the Monkey House, so that means 4 people got on at the Alligator Pond. At each stop three more people got on than got on at the stop before, so I made a table to show how many people got on. Now I’m worried I made a mistake, because 20 people are supposed to get on at Big Cats, but I only have 16. Is Big Cats supposed to be part of the pattern? Student 2 asks Student 3, “Can I talk it through with you?”

25. The Math Forum @ Drexel - mathforum.org Dialogue Student 3: Sure. How did you know how many people got on at each stop. Student 2: Well first it said the first passenger got on at Monkey House. Student 3: I thought it said “passengers.” Student 2: Oh, you’re right! It must be more than one person got on there. Student 3: How will you find out how many got on at Monkey House. Student 2: Well, I think 20 people should have gotten on at Big Cats, but only 16 people did. Maybe if I made 4 more people get on at Monkey House (5 altogether), there would be 20 by the time we got to Big Cats.

26. The Math Forum @ Drexel - mathforum.org Make a Table Making a table and looking for patterns can help students:  Organize their calculations  Find patterns  Discover and generalize important relationships

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29. Use Logical Reasoning Problem solvers can use logical reasoning to:  Work from what must be true to figure out what might be true and what can’t be true.  Take what might be true and figure out whether it is true.  Identify assumptions and figure out what to trust and what to doubt.  Consider unlikely or unusual ideas that might lead to new solutions.  Break a problem into sub-problems or cases: “I can try to show that ____, and if it works, I will have solved the problem. If it doesn’t, then I will know more about what the answer must be.”

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32. Next Steps  making sense of problems  persevering  construct viable arguments  critique the reasoning of others

33. The Math Forum @ Drexel - mathforum.org Online Resources

34. The Math Forum @ Drexel - mathforum.org http://mathforum.org/workshops/cmc/2012/south/

35. Opportunity