Conferrals, Gallery Walks, and Congresses:Conferrals, Gallery Walks, and Congresses:
Mentoring Young Mathematicians at Wor...
Hans FreudenthalHans Freudenthal
Mathematics should be thought of as a humanMathematics should be thought of as a human
ac...
Cognition does not start with concepts, but rather the other way
around: concepts are the results of cognitive processes… ...
The givens regardingThe givens regarding
young mathematicians at work…young mathematicians at work…
 Community ofCommunit...
To really mentor well, we need toTo really mentor well, we need to
deeply understand mathematicaldeeply understand mathema...
The Landscape of LearningThe Landscape of Learning
The developmental landscape is the framework forThe developmental landscape is the framework for
the construction of seque...
Conferring…Conferring…
 Listen carefully for the developing strategies andListen carefully for the developing strategies ...
Getting the Conferral StartedGetting the Conferral Started
 Listen firstListen first
 Converse about the strategy the ch...
Mentor…..Mentor…..
If a child is having difficulty starting….If a child is having difficulty starting….
This is a hard pro...
Use quotes from famousUse quotes from famous
mathematicians about the process ofmathematicians about the process of
doing ...
Or….Or….
Andrew Wile said, “Andrew Wile said, “I can best describe my experience ofI can best describe my experience of
do...
Model what a mathematicianModel what a mathematician
might do to get started.might do to get started.
Sometimes mathematic...
When data has been collected…When data has been collected…
 Once mathematiciansOnce mathematicians
feel they have “madefe...
24 lb. turkey at $1.25 @ lb.24 lb. turkey at $1.25 @ lb.
 What does it cost?What does it cost?
Examining a ConferralExamining a Conferral
Nate and NellieNate and Nellie
Construct viable arguments and critique the reasoning of othersConstruct viable arguments and critique the reasoning of ot...
Writing a viable argumentWriting a viable argument
 How will you convince your audience?How will you convince your audien...
Planning a CongressPlanning a Congress
Not a “sharing out”. It is a carefullyNot a “sharing out”. It is a carefully
crafte...
24 lb. turkey at 15 minutes @ lb.24 lb. turkey at 15 minutes @ lb.
 How long to cook it?How long to cook it?
The Landscape of LearningThe Landscape of Learning
2 questions to keep in mind2 questions to keep in mind
 Where is the child on the landscape?Where is the child on the lan...
A snapshot of theA snapshot of the
congress …congress …
Role of DiscourseRole of Discourse
 Dialogue BallDialogue Ball
 Ideas and strategies emerged in the communityIdeas and s...
... a future where every student
enjoys an individually-tailored,
world class learning experience.
Seattle...
......
…diff...
A Sampling of DreamBox's
Virtual
Manipulatives
Function Machine
Mathrack
Ten Frame Snap Blocks
Open Number Line
Continuous assessment.
Dynamic adaptation.
DreamBox Learning
Typical “Adaptive”
Computer Programs
““Mathematics is not a careful march down aMathematics is not a careful march down a
well-cleared highway, but a journey…”...
Contact information:Contact information:
www.NewPerspectivesOnLearning.comwww.NewPerspectivesOnLearning.com
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work
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Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work

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See strategies educators can use to transform classrooms into communities of mathematicians. School can be an exciting place where children engage in crafting solutions, justifications, and proofs of their own making, and focus on questioning, conferring, and assessing.

Discover how blended learning and technology innovations are inspiring students and increasing their achievement, powering seamless formative assessment, and supporting new standards, including the CCSSM

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  • Equity, individual children …Compare landscapes to learning lines…learning lines for curriculum development but landscapes are needed for teaching
  • Equity, individual children …Compare landscapes to learning lines…learning lines for curriculum development but landscapes are needed for teaching
  • The Development Story:
    The innovators behind DreamBox had a dream: a future where every student enjoys an individually-tailored world class learning experience.
    They recognized that the Web had completely transformed eCommerce (with “smart” recommendation engines that tracked your preferences and personalized responses), but not so for eLearning.
    (Click) And in fact, their previous work experience with Expedia, Microsoft and Amazon told them that the technology was there, waiting to be accessed.
    (Click) So they went out and gathered a team of math experts. They went to John Van de Walle first, who immediately recommended Cathy Fosnot. In addition, they added Skip Fennell to the team, who had done a lot of work on the Big Ideas in math (as a former president of NCTM and contributor to the Focal Points) and John Bransford, a cognitive psychologist, for his landmark work on how people learn and how students learn mathematics.
    (Click) Together, they worked to create a web-based learning resource that was adaptive to the needs of the student – truly differentiated instruction accessed through technology.
  • At the heart of DreamBox Learning are the virtual manipulatives which students use to explore the mathematics. These are a sampling of the virtual tools used to represent the abstract world of mathematics concretely.
    (NOTE: To demonstrate these virtual manipulatives you need to be connected to the internet and showing this PPT in presentation mode.Click the underlined word to go to a TUTORIAL Lesson on that manipulative. You may need to log in (in the future check the box so that DB automatically logs you in).
    I would suggest demonstrating the Mathrack since it’s the one that Cathy Fosnot made famous and might be unknown to your audience. But they’re all very good and quite unique.)
  • Behind the fun is some seriously powerful technology. It’s DreamBox’s continuous assessment and dynamic adaptation that distinguishes it from everything else out there, and represents a next generation of eLearning. Here’s what I mean.
    (click) Typical computer programs operate in a linear fashion: you work through a series of objectives and when you meet the requirements, you move on to the next one.
    While they may call themselves “adaptive”, it’s only in a very limited way: if students “pass” an objective quickly, they don’t have to do all the questions, but they still must demonstrate mastery at each level.
    (click) Not so with DreamBox. DreamBox continuously gathers data on every move a student makes -- every mouse click, every hesitation, every movement of the mouse – and based on those moves, determines the next appropriate instructional move a student needs to optimize learning. DreamBox continually assesses and adapts the learning environment – the math content, the level of difficulty, the pacing and sequence of lessons – all in order to maximize success. It is completely responsive and personalized and the path a student takes is uniquely his or her own.
    (click) In fact, there are over 350 lessons in DreamBox, representing literally millions of paths a student could take. All dependent on the moves they make.
  • Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at Work

    1. 1. Conferrals, Gallery Walks, and Congresses:Conferrals, Gallery Walks, and Congresses: Mentoring Young Mathematicians at WorkMentoring Young Mathematicians at Work Cathy FosnotCathy Fosnot
    2. 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. 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. 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. 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. 6. The Landscape of LearningThe Landscape of Learning
    7. 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. 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. 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. 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. 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. 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. 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. 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. 15. 24 lb. turkey at $1.25 @ lb.24 lb. turkey at $1.25 @ lb.  What does it cost?What does it cost?
    16. 16. Examining a ConferralExamining a Conferral Nate and NellieNate and Nellie
    17. 17. Construct viable arguments and critique the reasoning of othersConstruct viable arguments and critique the reasoning of others Gallery Walks
    18. 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?
    19. 19. 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.
    20. 20. 24 lb. turkey at 15 minutes @ lb.24 lb. turkey at 15 minutes @ lb.  How long to cook it?How long to cook it?
    21. 21. The Landscape of LearningThe Landscape of Learning
    22. 22. 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)
    23. 23. A snapshot of theA snapshot of the congress …congress …
    24. 24. 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
    25. 25. ... a future where every student enjoys an individually-tailored, world class learning experience. Seattle... ...... …differentiated instruction and …intelligent adaptive learning
    26. 26. A Sampling of DreamBox's Virtual Manipulatives Function Machine Mathrack Ten Frame Snap Blocks Open Number Line
    27. 27. Continuous assessment. Dynamic adaptation. DreamBox Learning Typical “Adaptive” Computer Programs
    28. 28. ““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
    29. 29. Contact information:Contact information: www.NewPerspectivesOnLearning.comwww.NewPerspectivesOnLearning.com

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