Transcript Our Common Goal Mathematics Learning AACTME October 2006

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My Students Can
Notice/Wonder, Now What?
Marie Hogan, West Covina, CA
Suzanne Alejandre, Philadelphia, PA
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Introductions
 Marie Hogan
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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 ?
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Notice/Wonder: Sound Familiar?
CMC-North 2011: Get Your Students Hooked On Noticing
and Wondering
ComMuniCator, December 2010:
Problem Solving–It Has to Begin
with Noticing and Wondering
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Notice/Wonder in Your Classroom
 Help all students engage comfortably so
that they are able to communicate their
mathematical reasoning.
 Students who would race to finish the
problem, slow down, while students
who would give up immediately have
something that hooks them in.
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Notice/Wonder: Quick Overview
Eating Grapes
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What did you hear?
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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?
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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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Are you starting
to think that
you just do NOT
have time?
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Connecting to CCSSM
MP1 Make sense of problems and
persevere in solving them.
 Do students write IDK on their papers?
 Do students only focus on answers?
 Do “expert” students say I’m done, now what?
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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?
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Have you seen what will be expected?
http://www.cccoe.k12.ca.us/edsvcs/commoncore/MathematicsGeneralItemandTaskSpecificationsGrades3-5.pdf
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http://www.cccoe.k12.ca.us/edsvcs/commoncore/MathematicsGeneralItemandTaskSpecificationsGrades3-5.pdf
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http://www.cccoe.k12.ca.us/edsvcs/commoncore/MathematicsGeneralItemandTaskSpecificationsGrades3-5.pdf
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Notice/Wonder initially helps
students to “make sense of
problems and persevere in solving.”
But that’s just the beginning.
Once students are comfortable
with the Notice/Wonder strategy,
there are many more to introduce.
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Change the Representation
 draw a picture
 act the problem out
 build a model
 represent the problem using blocks,
counters, etc.
 represent the problem using a number
line or lines
 make a graph or graphs
 organize quantities in a table
 write relationships as equations
 tell a different story
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Student Thinking 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.
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Student Thinking 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.
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Student Thinking 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:
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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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Student Thinking 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.
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Student Thinking 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?”
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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.
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 What strategies work for you?
 What are your challenges?
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Next Steps
making sense of problems
persevering
construct viable arguments
critique the reasoning of others
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Online Resources
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http://mathforum.org/workshops/cmc/2012/north/
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Opportunity