Day 1: Sense-Making and
Noticing and Wondering
Annie Fetter and Max Ray
The Math Forum
Twitter: @MFAnnie, @maxmathforum
BCPS, July 2015
http://mathforum.org/workshops/bcps/
Sense Making and
the Standards for
Mathematical Practice
There are 125 sheep and
5 dogs in a flock.
How old is the shepherd?
Robert Kaplinsky (@robertkaplinsky) gave this to
32 eighth grade students. What percentage do
you think realized it was impossible to answer?
(http://robertkaplinsky.com/how-old-is-the-shepherd/
https://www.youtube.com/watch?v=kibaFBgaPx4)
How Old is the Shepherd?
75% gave numerical answers.
100% of his sixth graders gave numerical answers.
In the original research paper, “…three out of four
school children will produce a numerical answer to
this problem.”
“The Steps Trump Thinking”
[Michelle's son] was struggling to “remember” 28/4.
When [she] asked him, “How do you think about
28/4?″ He replied, “Mom, you aren’t supposed to
think about it, you are just supposed to do it!!”
Teacher Knows Best
3 5 8
+ =
4 8 12
...
C.U.B.E.S.
• 
• 
• 
• 
• 
Circle the numbers.
Underline the important words.
Box the question.
Eliminate unnecessary information.
Solve.
Sample Grade 3 State Test Problem 3
Hot dog buns come in packages of 8. Michael buys 6 packages
of hot dog buns. How many hot dog buns does Michael have in
all?
A.  14
B.  36
C.  48
D.  56
43%
8%
40%
5%
“Cracking the Math Code”
ADDITION
SUBTRACTION
MULTIPLICATION
DIVISION
add
altogether
and
both
how many
how much
in all
increased by
plus
sum
together
total
are not
change
decreased by
difference
fewer
have left
how many did not
have
how many more
less than
remain
subtract
take away
taller/shorter
by (dimension)
double
each group
multiplied by
of
product of
times
triple
as much
cut up
divided by
each group has
half (or other
fractions)
how many in each
parts
quotient of
Separated
Share something
equally
split
(document from the web site of a large eastern metropolitan school district)
Student Perceptions of Math
and Sense Making
1.  You aren’t supposed to sense-make when
doing math.
2.  You are supposed to use rules and algorithms
and accept whatever answer results.
3.  You are supposed to do what your teacher
said, even when it doesn’t seem like a good
idea.
CCSS Mathematical Practice 1
Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a
problem and looking for entry points to its solution.
They analyze givens, constraints, relationships, and goals.
They make conjectures about the form and meaning of the solution and plan a
solution pathway rather than simply jumping into a solution attempt.
They consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution.
They monitor and evaluate their progress and change course if necessary.
Jekylls and Hydes in Grade 6
Characteristics of Strong Readers
Mathematicians
•  They are motivated to read.tackle problems
•  They are able to read words accurately and
recite facts
automatically.
•  They comprehend what they read.
•  They are
( able to read with expression.
)
•  They use a variety of strategies to tackle words problems
they don’t recognize.
•  They use active problem solving strategies to
search for information, to determine meaning, to
make sense of words, to make connections.
Encouraging Sense-Making in Math
Q: How do we cultivate a classroom focused on sense
making rather than answer-getting?
A: Get rid of the question. Literally.
Sample Grade 3 State Test Problem
The corner deli sells roses in bunches of 6. If Dylan buys 3
bunches of roses, how many roses does he have?
A.  6
B.  9
C.  18
D.  24
18%
46%
31%
4%
Combined scores of the 160 third graders in a group of
four low-performing schools I used to support.
Sample Test Problem, Revised
The corner deli sells roses in bunches of 6. Dylan bought 3
bunches. Draw a picture of the story.
Sample Grade 3 State Test Problem 3
Hot dog buns come in packages of 8. Michael buys 6 packages
of hot dog buns. How many hot dog buns does Michael have in
all?
A.  14
B.  36
C.  48
D.  56
43%
8%
40%
5%
Sense Making by Noticing and Wondering
I Notice
I Wonder
Sense Making by Noticing and Wondering
Sense Making by Noticing and Wondering
Word Problems Are So Tricky Easier!
Solve
2
x
+ 1 = 37
“Ok, let’s try this a different way. I think of a number.
I square it, then I add 2. My result is 27.
What number was I thinking of?”
(http://mathybeagle.com/2014/10/09/procedure-in-the-drivers-seat/)
Word Problems Are So Tricky Easier!
•  Reviewing for the Geometry Keynote
•  Working with 9th graders this past Thursday
Brainstorm: Pain Points
1.  What is a typical math block like in your grade?
1.  Length?
2.  Routine?
3.  Time spent on each part of routine?
2.  What is a typical math unit like in your grade?
1.  Length?
2.  How is content introduced?
3.  How content practiced?
4.  How is content assessed?
3.  What are the parts of a typical day or typical unit that feel
like they take more time than they should, or kids just
don’t “get it” in the time they have?
Student Talking = Sense Making =
Efficiency
Teacher Listening = Formative
Assessment = Efficiency
Look at your pain points. How could building in more time
for student talk and teacher listening speed up your class?
Hints: Can you minimize re-teaching? Differentiate with less
work? Move small-group work faster? Stop answering the same
question over and over? Make kids do more explaining?
How Common Core Changes Coverage
Noticing and Wondering
Story Time!
What did you hear?
What Do You Notice?
https://todaysmeet.com/WeNotice
What Do You Wonder?
https://todaysmeet.com/WeWonder
What did you see?
What did you see?
I notice…!
I wonder…!
Growing Worms: Any Questions? From http://mathforum.org/pps/!
Stationary Gallery Walk
Round 1: What do you wonder? Write a
question for another student.
Round 2: What has to do with math? Star any
that have to do with math.
Round 3: Circle statements about quantities.
Underline statements about relationships.
Getting Better at Noticing &Wondering
“We are proud of how we
______________________________________
___________________.”
“We could do an even better job next time by
______________________________________
___.”
Use what you noticed and wondered to
•  Find the number of triangles in a 100-day
worm.
•  Write a rule, equation, or instructions that
describe a worm of any age.
Mingle Instructions:
•  Stand up and move around.
• 
• 
• 
• 
• 
Find someone and introduce yourself.
Ask them one question from the list.
Listen to their answer.
Move on to find another person.
No back and forth, just ask one question and
listen to the answer.
•  Think if you might add something to our
Eavesdropping Doc?
•  When I raise my hand, finish your
conversation and raise your hand.
Mingle Questions:
•  Why notice and wonder?
•  How might you implement this with the goal
to get all students thinking and sharing?
•  How might you value students’ ideas and
build up their math confidence?
•  How could it make instruction more
effective?
•  What concerns do you have about
implementing this?
Looking Ahead:
1 thing you are still wondering
2 things you are excited to hear students talk
about tomorrow
3 things you want to see modeled tomorrow