Designing Powerful Digital Environments for Professional Development Cathy Fosnot DR-K12, 2009

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

Hans Freudenthal 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 Freudenthal

Complexity Theory and Embodied Cognition Learning is complex, not linear Learning is complex, not linear Ideas and strategies evolve through discourse and interaction and appear on a landscape of learning as developmental landmarks Ideas and strategies evolve through discourse and interaction and appear on a landscape of learning as developmental landmarks Emotions and social discourse are central to mathematical activity. What defines mathematics is a set of shared preferences, ways of reasoning, and truths accepted by the community Emotions and social discourse are central to mathematical activity. What defines mathematics is a set of shared preferences, ways of reasoning, and truths accepted by the community Importance of the “surround” (Maturana and Varela) Importance of the “surround” (Maturana and Varela)

Young Mathematicians at Work Community of Discourse Community of Discourse Think Time Think Time Questioning Questioning Paraphrasing Paraphrasing Pair Talk Pair Talk Inquiry Inquiry Gallery Walks and Congresses Gallery Walks and Congresses

Math Lessons vs. Math Workshop Doing activities and problems vs. allowing learners to go deeply into the topic using sequences of crafted, related investigations over a period of time Doing activities and problems vs. allowing learners to go deeply into the topic using sequences of crafted, related investigations over a period of time Role of Context Role of Context Learning in and from a community of mathematical discourse Learning in and from a community of mathematical discourse

Didactics: Role of Context Open to allow many entries Open to allow many entries Developing understanding by staying in the context to help children realize what they are doing Developing understanding by staying in the context to help children realize what they are doing Crafting further contexts with constraints to support development Crafting further contexts with constraints to support development

Role of Discourse Dialogue Ball Dialogue Ball Ideas and strategies emerge in the community as children discuss their own attempts at sense- making, work together, and try out one another’s ideas Ideas and strategies emerge in the community as children discuss their own attempts at sense- making, work together, and try out one another’s ideas

Assessment of the Facilitation of Mathematizing Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret Mathematics ScalePedagogy ScaleContext 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 Mathematizing Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret Mathematics ScalePedagogy ScaleContext 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 Mathematizing Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret Mathematics ScalePedagogy ScaleContext 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 Mathematizing Catherine Twomey Fosnot, Maarten Dolk, Betina Zolkower, Sherrin Hersch, and Herbert Seignoret Mathematics ScalePedagogy ScaleContext 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 Investigations, grade 3…

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