Conferrals, Gallery Walks, and Congresses: Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work Mentoring Young Mathematicians at Work

Cathy FosnotCathy Fosnot

Hans FreudenthalHans Freudenthal

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

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

The givens regarding The givens regarding young mathematicians at young mathematicians at

work…work… Community of Community of

DiscourseDiscourse Think TimeThink Time Problem Problem

SolvingSolving InquiryInquiry

ConferringConferring Gallery WalksGallery Walks CongressesCongresses

To really mentor well, we To really mentor well, we need to deeply need to deeply understand understand mathematical mathematical developmentdevelopment The progressive development of The progressive development of

strategiesstrategies The development of structuring (big The development of structuring (big

ideas)ideas) The development of modeling The development of modeling Landscapes of LearningLandscapes of Learning

The Landscape of LearningThe Landscape of Learning

The developmental landscape is the The developmental landscape is the framework for the construction of framework for the construction of sequences.sequences.

It’s also the framework for conferrals, It’s also the framework for conferrals, gallery walks, and congresses.gallery walks, and congresses.

Conferring…Conferring…

Listen carefully for the developing Listen carefully for the developing strategies and big ideasstrategies and big ideas

Note the modeling Note the modeling Support both vertical and horizontal Support both vertical and horizontal

mathematizingmathematizing Consider how to support the Consider how to support the

developing mathematician (instead of developing mathematician (instead of thinking about fixing the thinking about fixing the mathematics)mathematics)

Getting the Conferral Getting the Conferral StartedStarted

Listen firstListen first Converse about the strategy the Converse about the strategy the

child is using. 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 Have you found a strategy that seems

promising?promising?

If child is having a hard time getting started, model what a professional mathematician If child is having a hard time getting started, model what a professional mathematician might do.might do.

Mentor…..Mentor…..

If a child is having difficulty If a child is having difficulty starting…. starting….

This is a hard problem, isn’t it? But This is a hard problem, isn’t it? But you know what mathematicians say you know what mathematicians say about hard problems?about hard problems?

“ “That’s what makes them worth That’s what makes them worth trying to crack! And oh what fun trying to crack! And oh what fun when we crack them! If problems when we crack them! If problems are easy they aren’t worth doing. are easy they aren’t worth doing. They are trivial and boring!”They are trivial and boring!”

Persevere when solving problems

Use quotes from famous Use quotes from famous mathematicians about the mathematicians about the

process of doing math:process of doing math:Keith Devlin once said, “Keith Devlin once said, “When I’m working When I’m working

on a problem it’s like climbing a on a problem it’s like climbing a mountain. Sometimes I can’t even see mountain. Sometimes I can’t even see where I’m going. It’s just one foot in where I’m going. It’s just one foot in front of another. And then I reach a point front of another. And then I reach a point where all of a sudden the vistas open up where all of a sudden the vistas open up and I can go down easily for awhile, only and I can go down easily for awhile, only to eventually reach another climb.” to eventually reach another climb.”

Or….Or….

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

Model what a Model what a mathematician might do mathematician might do

to get started.to get started.Sometimes Sometimes mathematicians start mathematicians start by modeling the by modeling the problem, like on a problem, like on a number line (array, number line (array, ratio table, etc.) They ratio table, etc.) They ask themselves, “What ask themselves, “What model might be model might be helpful for this type of helpful for this type of problem? Does this problem? Does this problem remind me of problem remind me of any problems I’ve any problems I’ve done before?”done before?”

Choose appropriate tools to model the mathematics

When data has been When data has been collected…collected…

Once mathematicians Once mathematicians feel they have “made feel they have “made their way across the their way across the room” they step back room” they step back and say,and say,

““Is there any structure Is there any structure or regularity here? or regularity here? Any interesting Any interesting noticings, patterns, noticings, patterns, anything that could anything that could be generalized?”be generalized?”

Attend to precision; look for structure and regularity

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What does it What does it cost?cost?

Examining a Examining a ConferralConferral

Nate and NellieNate and Nellie

Construct viable arguments and critique the reasoning Construct viable arguments and critique the reasoning of othersof others

Gallery Walks

Writing a viable Writing a viable argumentargument

How will you convince your How will you convince your audience?audience?

What will they need to know to be What will they need to know to be able to follow your reasoning?able to follow your reasoning?

Is just telling what you did a Is just telling what you did a convincing argument?convincing argument?

Planning a Planning a CongressCongress

Not a “sharing out”. It is a Not a “sharing out”. It is a carefully crafted conversation carefully crafted conversation

to support progressive to support progressive development.development.

24 lb. turkey at 15 24 lb. turkey at 15 minutes @ lb.minutes @ lb.

How long to How long to cook it?cook it?

The Landscape of LearningThe Landscape of Learning

2 questions to keep in 2 questions to keep in mindmind

Where is the child on the landscape?Where is the child on the landscape? What is just coming into view on the What is just coming into view on the

horizon?horizon?

(Vygotsky’s zone of proximal (Vygotsky’s zone of proximal development)development)

A snapshot of the A snapshot of the congress …congress …

Role of DiscourseRole of Discourse

Dialogue BallDialogue Ball Ideas and strategies emerged in the Ideas and strategies emerged in the

community as children discussed community as children discussed their own attempts at sense-making, their own attempts at sense-making, worked together, and tried out one worked together, and tried out one another’s ideasanother’s ideas

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Virtual Manipulatives

Function Machine

Mathrack

Ten Frame Snap Blocks

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““Mathematics is not a careful Mathematics is not a careful march down a well-cleared march down a well-cleared highway, but a journey…”highway, but a journey…”

W.S. AnglinW.S. Anglin

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