Designing Powerful Digital Environments for Designing Powerful Digital Environments for Professional DevelopmentProfessional Development

Cathy FosnotCathy FosnotDR-K12, 2009DR-K12, 2009

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 educationalists turning to mathematics instruction who restricted it to the horizontal one.

Hans FreudenthalHans Freudenthal

Complexity Theory and Complexity Theory and Embodied CognitionEmbodied Cognition

Learning is complex, not linearLearning is complex, not linear Ideas and strategies evolve through Ideas and strategies evolve through

discourse and interaction and appear on a discourse and interaction and appear on a landscape of learning as developmental landscape of learning as developmental landmarkslandmarks

Emotions and social discourse are central Emotions and social discourse are central to mathematical activity. What defines to mathematical activity. What defines mathematics is a set of shared mathematics is a set of shared preferences, ways of reasoning, and preferences, ways of reasoning, and truths accepted by the communitytruths accepted by the community

Importance of the “surround” (Maturana Importance of the “surround” (Maturana and Varela)and Varela)

Young Mathematicians Young Mathematicians at Workat Work

Community of Community of DiscourseDiscourse

Think TimeThink Time QuestioningQuestioning ParaphrasingParaphrasing Pair TalkPair Talk InquiryInquiry Gallery Walks Gallery Walks

and and CongressesCongresses

Math Lessons vs. Math Math Lessons vs. Math WorkshopWorkshop

Doing activities and problems vs. Doing activities and problems vs. allowing learners to go deeply into allowing learners to go deeply into the topic using sequences of crafted, the topic using sequences of crafted, related investigations over a period related investigations over a period of timeof time

Role of ContextRole of Context Learning in and from a community Learning in and from a community

of mathematical discourseof mathematical discourse

Didactics: Role of Didactics: Role of ContextContext

Open to allow many entriesOpen to allow many entries Developing understanding by Developing understanding by

staying in the context to help staying in the context to help children realize what they are doingchildren realize what they are doing

Crafting further contexts with Crafting further contexts with constraints to support developmentconstraints to support development

Role of DiscourseRole of Discourse

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

community as children discuss their community as children discuss their own attempts at sense-making, work own attempts at sense-making, work together, and try out one another’s together, and try out one another’s ideasideas

Assessment of the Facilitation of Assessment of the Facilitation of MathematizingMathematizing

Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert SeignoretCatherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert SeignoretMathematics Scale Pedagogy Scale Context Scale

[Depicts the development in the ability to see mathematics in students’ work, to identify relevant connections across solutions, and to bring students’ solutions to a higher level of mathematical sophistication]

[Depicts teacher development from teaching by telling to teaching to facilitate students’ mathematical reasoning, understanding and constructions.]

[Depicts development from a mechanical use of context merely as a locus for applying taught procedures, towards the use of (realistic: i.e. realizable, imaginable) contexts and ‘truly problematic’ situations both as starting points for mathematical constructions and as a didactic to facilitate mathematical development.]

Level 1

Level 2

Level 3

Assessment of the Facilitation of Assessment of the Facilitation of MathematizingMathematizing

Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert SeignoretCatherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret

Mathematics Scale Pedagogy Scale Context Scale[Depicts the development in the ability to see mathematics in students’ work , to identify relevant connecting across solutions, and to bring students’ solutions to a higher level of mathematical sophistication.]

[Depicts teacher development from teaching by telling to teaching to facilitate students’ mathematical reasoning, understanding and constructions.]

[Depicts development from a mechanical use of context merely as a locus for applying taught procedures, towards the use of (realistic: i.e. realizable, imaginable) contexts and ‘truly problematic’ situations both as starting points for mathematical constructions and as a didactic to facilitate mathematical development.]

Level 1: The teacher misses the “math teaching moments” during the lesson. This may happen either because the teacher is unaware of critical big ideas, strategies, and mathematical models due to his/her lack of mathematics knowledge or because s/he is too intent on obtaining his/her expected answers.

Level 1: Transmission or direct instruction modality of teaching; teaching by explanation (the teacher does all the explaining and showing); emphasis is on reinforcement and practice and on arriving at the answer and use of procedures that she/he has in mind.

Level 1: Lack of context or mechanical use of context: either the mathematics work is done entirely within the domain of symbols (no context) and children are expected to work symbolically without the help of context to realize what they are doing; or contexts - mostly limited to stereotypical word problems - are used for the application of previously learned concepts and procedures.

Level 2

Level 3

Assessment of the Facilitation of Assessment of the Facilitation of MathematizingMathematizing

Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert SeignoretCatherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret

Mathematics Scale Pedagogy Scale Context Scale[Depicts the development in the ability to see mathematics in students’ work , to identify relevant connecting across solutions, and to bring students’ solutions to a higher level of mathematical sophistication.]

[Depicts teacher development from teaching by telling to teaching to facilitate students’ mathematical reasoning, understanding and constructions.]

[Depicts development from a mechanical use of context merely as a locus for applying taught procedures, towards the use of (realistic: i.e. realizable, imaginable) contexts and ‘truly problematic’ situations both as starting points for mathematical constructions and as a didactic to facilitate mathematical development.]

Level 1:

Level 2: The teacher begins to focus on the mathematics in students’ work, exploring some “math moments” in an attempt to facilitate learner development, but focus is primarily on strategy use, rather than the development of important mathematical ideas and relations in relation to the landscape of learning.

Level 2: Signs of change (use of questioning, think time, diagrams, models, and/or manipulatives) towards facilitating students’ constructions and thinking, but more rote use, or routine use, of these pedagogical strategies rather than in relation to the development of student reasoning about mathematics content. For example, teacher may provide manipulatives routinely even when they hinder mathematical thinking, rather than thinking out which manipulative, when, and why; or use think time and paraphrasing even when questions are trivial.

Level 2: Word problem types of contexts are used as a starting point for construction, in contrast to application of previously learned knowledge as depicted in level one. But this serves merely the purpose of motivation or to elicit children’s thinking; no attention is paid to the process whereby mathematical ideas and/or strategies may emerge or originate from suggestions or constraints in rich, didactically-crafted contexts.

Level 3

Assessment of the Facilitation of Assessment of the Facilitation of MathematizingMathematizing

Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert SeignoretCatherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret Mathematics Scale Pedagogy Scale Context Scale

[Depicts the development in the ability to see mathematics in students’ work , to identify relevant connecting across solutions, and to bring students’ solutions to a higher level of mathematical sophistication.]

[Depicts teacher development from teaching by telling to teaching to facilitate students’ mathematical reasoning, understanding and constructions.]

[Depicts development from a mechanical use of context merely as a locus for applying taught procedures, towards the use of (realistic: i.e. realizable, imaginable) contexts and ‘truly problematic’ situations both as starting points for mathematical constructions and as a didactic to facilitate mathematical development.]

Level 1:

Level 2:

Level 3: Teacher recognizes and takes advantage of most or all of the “math moments,” thus, taking a proactive role in facilitating the development of students’ mathematical constructions and raising the level of mathematical sophistication, pushing towards deep understanding and generalization.

Level 3: Teaching to facilitate mathematical construction. Rich questioning and reflection used at opportune times to support the development of important mathematical ideas and generalizations.

Level 3: Use of realistic contexts and truly problematic situations with didactics embedded. For example, contexts are designed with built-in constraints to facilitate puzzlement and challenge, or with potentially realizable suggestions to support mathematical development.

The Turkey The Turkey Investigations, grade Investigations, grade

3…3…

““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