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Studying Learning through Activity: A basis for a Theory of Task Design Martin A. Simon New York University University of Maryland, Feb. 1, 2013

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Collaborators Ron Tzur, Luis Saldanha, Evan McClintock, Arnon Avitzur, Nicora Placa, Jessica Tybursky, Tad Watanabe, Gulseren Karagoz, Ismail Zembat, Karen Heinz, Margaret Kinzel, Peg Smith, Barbra Dougherty, Zaur Berkaliev

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Omitted from this Talk Discussion about the research approach upon which this is all based Discussion of what we mean by studying learning in contrast to many other researchers. See Simon et al, 2010 Fair game for Q&A

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Problem Many students do not develop deep understanding of mathematical concepts Limits most students Major issue of equity Disadvantaged Late bloomers Special education

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Introduction Multiple aspects of mathematics education (problem solving, conceptual understanding, communicating about mathematical ideas …) Focus on the learning of mathematical concepts How does one promote new mathematical concepts? Limitations of problem solving approach (being the non- solver) Lack of theory supporting building from concrete. Approach to promoting concepts based on research on learning.

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We Are All Piaget Chess example (e.g., fork) Learning through activity (reflective abstraction) Abstraction Anticipation Goal directed activity Reflection

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The Road Less Traveled IF learning through activity is a useful description of learning alternative to problem solving approach. Possibility of designing to directly foster the process. Promoting activity (raw material) Promoting reflection (processing the raw material)

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Learning through Activity Analyze the learning in chess example

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Odd-Even Example odd X odd = ? 5x5=25 19x21=399 39x37=1443 Learning that it seems to be true Mathematical understanding is not the result of an empirical learning process

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RA not ELP (cont.) Mathematical understanding is the result of reflective abstraction Knowing logical necessity Development of an anticipation Abstraction from ones activity Analyze following example: Odd-even example: Even everyone has a dance partner Odd everyone has a dance partner except one person

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RA not ELP (cont.) Example with objects: 5x3 OOO OOO OOO OOO OOO

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RA not ELP (cont.) Example with objects: 5x3 O O O O O OO OO OO OO OO Similar activity with different numbers

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Current Work Contrast with important work on social interactive aspects of learning Design experiments (teaching experiments) promoting concepts fractions and ratio Begin with conjecture about design for learning through activity (next slide) Develop concept-specific learning trajectories Deepen understanding of learning through activity Develop design principles

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Conjecture about Design for Learning through Activity 1. Assess student understanding 2. Articulate a learning goal (articulation of understanding) 3. Specify an activity or activity sequence that students currently have available 4. Design tasks that will engage students in the intended goal-directed activity AND lead to learned anticipation – reflection on activity (not deterministic)

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Example from Research Goal: recursive partitioning (part of concept) Here is 1/3 of a unit, make 1/6 of a unit Kylie repeats part 3 times and then cuts the first third into two parts Repeats this process with 1/5 of a unit to make 1/10 of a unit

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Recursive Partitioning (continued) Given 1/3, asked to make 1/9 K: [Cuts the bar into 3 parts] One of those is one ninth. R: How do you know K: Because, um. How many times does three go into nine?... Three times. And it's one third! So. Three times three is nine [indicates that since the bar is 1/3, there would be 3 of the 3 parts, therefore ninths [continues this process on subsequent problems]

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Analysis of Example How do we explain the learning? Planning this learning. If the student gets stuck, we missed something. Contrast with problem solving approach.

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Potential Contribution to Instruction Goal: Improved ability to engineer task sequences that foster particular understandings for a diverse set of students.

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A Thought Experiment Not atypical classroom scene Competent teacher Problem representing math to be learned Students work in pairs – rich representations available 1or 2 pairs solve problems –most dont Class discussion – 1 pair presents solution With teachers help others seem to understand solution Who will more likely … ? Difference in cognitive demands of generating a solution versus understanding a solution (apply to abstraction)

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A Thought Experiment (cont) Equity issue more-advanced students work novel problems, less-advanced students struggle to follow explanations of solutions

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Vision for Instruction What if 80% could produce the new abstraction? Understand LTA design principles task sequences foster abstractions (build up requisite experience) Apply this in small groups (change in large group)

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Potential Contribution to Curriculum Development Provide strong conceptual framework for task development and sequencing

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Thank you for your attention! Q&A

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