This document discusses strategies for mentoring young mathematicians, including conferrals, gallery walks, and congresses. It emphasizes that mathematics should be viewed as modeling and structuring the world, rather than just transmitting structures. Effective mentoring requires understanding mathematical development and landscapes of learning. Conferrals involve listening to students' strategies and supporting both horizontal and vertical mathematizing. Gallery walks exhibit viable arguments, while congresses are carefully crafted conversations to support progress. The role of the mentor is to understand a student's current abilities and guide them to new understandings within their zone of proximal development.

1. Conferrals, Gallery Walks, and Congresses:Conferrals, Gallery Walks, and Congresses:
Mentoring Young Mathematicians at WorkMentoring Young Mathematicians at Work
Cathy FosnotCathy Fosnot

2. Hans FreudenthalHans Freudenthal
Mathematics should be thought of as a humanMathematics should be thought of as a human
activity of “mathematizing”—not as a discipline ofactivity of “mathematizing”—not as a discipline of
structures to be transmitted, discovered, or evenstructures to be transmitted, discovered, or even
constructed—but as schematizing, structuring, andconstructed—but as schematizing, structuring, and
modeling the world mathematically.modeling the world mathematically.

3. Cognition does not start with concepts, but rather the other way
around: concepts are the results of cognitive processes… How often
haven’t I been disappointed by mathematicians interested in
education who narrowed mathematizing to its vertical component, as
well as by educators turning to mathematics instruction who restricted
it to the horizontal one.
Hans FreudenthalHans Freudenthal

4. The givens regardingThe givens regarding
young mathematicians at work…young mathematicians at work…
 Community ofCommunity of
DiscourseDiscourse
 Think TimeThink Time
 Problem SolvingProblem Solving
 InquiryInquiry
 ConferringConferring
 Gallery WalksGallery Walks
 CongressesCongresses

5. To really mentor well, we need toTo really mentor well, we need to
deeply understand mathematicaldeeply understand mathematical
developmentdevelopment
 The progressive development of strategiesThe progressive development of strategies
 The development of structuring (big ideas)The development of structuring (big ideas)
 The development of modelingThe development of modeling
 Landscapes of LearningLandscapes of Learning

6. The Landscape of LearningThe Landscape of Learning

7. The developmental landscape is the framework forThe developmental landscape is the framework for
the construction of sequences.the construction of sequences.
It’s also the framework for conferrals, galleryIt’s also the framework for conferrals, gallery
walks, and congresses.walks, and congresses.

8. Conferring…Conferring…
 Listen carefully for the developing strategies andListen carefully for the developing strategies and
big ideasbig ideas
 Note the modelingNote the modeling
 Support both vertical and horizontalSupport both vertical and horizontal
mathematizingmathematizing
 Consider how to support the developingConsider how to support the developing
mathematician (instead of thinking about fixingmathematician (instead of thinking about fixing
the mathematics)the mathematics)

9. Getting the Conferral StartedGetting the Conferral Started
 Listen firstListen first
 Converse about the strategy the child is using.Converse about the strategy the child is using.
 How did you decide to start?How did you decide to start?
 What have you tried?What have you tried?
 Have you found a strategy that seems promising?Have you found a strategy that seems promising?

10. Mentor…..Mentor…..
If a child is having difficulty starting….If a child is having difficulty starting….
This is a hard problem, isn’t it? But you knowThis is a hard problem, isn’t it? But you know
what mathematicians say about hardwhat mathematicians say about hard
problems?problems?
““That’s what makes them worth trying toThat’s what makes them worth trying to
crack! And oh what fun when we crack them!crack! And oh what fun when we crack them!
If problems are easy they aren’t worth doing.If problems are easy they aren’t worth doing.
They are trivial and boring!”They are trivial and boring!”
Persevere when solving problems

11. Use quotes from famousUse quotes from famous
mathematicians about the process ofmathematicians about the process of
doing math:doing math:
Keith Devlin once said, “Keith Devlin once said, “When I’m working on aWhen I’m working on a
problem it’s like climbing a mountain. Sometimes Iproblem it’s like climbing a mountain. Sometimes I
can’t even see where I’m going. It’s just one foot in frontcan’t even see where I’m going. It’s just one foot in front
of another. And then I reach a point where all of aof another. And then I reach a point where all of a
sudden the vistas open up and I can go down easily forsudden the vistas open up and I can go down easily for
awhile, only to eventually reach another climb.”awhile, only to eventually reach another climb.”

12. Or….Or….
Andrew Wile said, “Andrew Wile said, “I can best describe my experience ofI can best describe my experience of
doing mathematics in terms of a journey through a dark,doing mathematics in terms of a journey through a dark,
unexplored mansion. You enter the first room of theunexplored mansion. You enter the first room of the
mansion and it is completely dark. You stumble around,mansion and it is completely dark. You stumble around,
bumping into furniture, but gradually, you learn wherebumping into furniture, but gradually, you learn where
each piece of furniture is. Finally, after awhile, you findeach piece of furniture is. Finally, after awhile, you find
the light switch. You turn it on and suddenly it’sthe light switch. You turn it on and suddenly it’s
illuminated. You can see exactly where you were. So eachilluminated. You can see exactly where you were. So each
of these breakthroughs….couldn’t exist without theof these breakthroughs….couldn’t exist without the
stumbling through the dark that precedes them.”stumbling through the dark that precedes them.”

13. Model what a mathematicianModel what a mathematician
might do to get started.might do to get started.
Sometimes mathematiciansSometimes mathematicians
start by modeling thestart by modeling the
problem, like on aproblem, like on a
number line (array, rationumber line (array, ratio
table, etc.) They asktable, etc.) They ask
themselves, “Whatthemselves, “What
model might be helpfulmodel might be helpful
for this type of problem?for this type of problem?
Does this problemDoes this problem
remind me of anyremind me of any
problems I’ve doneproblems I’ve done
before?”before?”
Choose appropriate tools to
model the mathematics

14. When data has been collected…When data has been collected…
 Once mathematiciansOnce mathematicians
feel they have “madefeel they have “made
their way across thetheir way across the
room” they step backroom” they step back
and say,and say,
““Is there any structure orIs there any structure or
regularity here? Anyregularity here? Any
interesting noticings,interesting noticings,
patterns, anything thatpatterns, anything that
could be generalized?”could be generalized?”
Attend to precision; look for
structure and regularity

15. 24 lb. turkey at $1.25 @ lb.24 lb. turkey at $1.25 @ lb.
 What does it cost?What does it cost?

16. Examining a ConferralExamining a Conferral
Nate and NellieNate and Nellie

17. Construct viable arguments and critique the reasoning of othersConstruct viable arguments and critique the reasoning of others
Gallery Walks

18. Writing a viable argumentWriting a viable argument
 How will you convince your audience?How will you convince your audience?
 What will they need to know to be able to followWhat will they need to know to be able to follow
your reasoning?your reasoning?
 Is just telling what you did a convincingIs just telling what you did a convincing
argument?argument?

20. Planning a CongressPlanning a Congress
Not a “sharing out”. It is a carefullyNot a “sharing out”. It is a carefully
crafted conversation to supportcrafted conversation to support
progressive development.progressive development.

21. 24 lb. turkey at 15 minutes @ lb.24 lb. turkey at 15 minutes @ lb.
 How long to cook it?How long to cook it?

22. The Landscape of LearningThe Landscape of Learning

23. 2 questions to keep in mind2 questions to keep in mind
 Where is the child on the landscape?Where is the child on the landscape?
 What is just coming into view on the horizon?What is just coming into view on the horizon?
(Vygotsky’s zone of proximal development)(Vygotsky’s zone of proximal development)

29. A snapshot of theA snapshot of the
congress …congress …

30. Role of DiscourseRole of Discourse
 Dialogue BallDialogue Ball
 Ideas and strategies emerged in the communityIdeas and strategies emerged in the community
as children discussed their own attempts atas children discussed their own attempts at
sense-making, worked together, and tried outsense-making, worked together, and tried out
one another’s ideasone another’s ideas

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33. A Sampling of DreamBox's
Virtual
Manipulatives
Function Machine
Mathrack
Ten Frame Snap Blocks
Open Number Line

34. Continuous assessment.
Dynamic adaptation.
DreamBox Learning
Typical “Adaptive”
Computer Programs

35. ““Mathematics is not a careful march down aMathematics is not a careful march down a
well-cleared highway, but a journey…”well-cleared highway, but a journey…”
W.S. AnglinW.S. Anglin

36. Contact information:Contact information:
www.NewPerspectivesOnLearning.comwww.NewPerspectivesOnLearning.com