Studying Learning through Activity: A basis for a Theory of Task DesignMartin A. Simon New York University

University of Maryland, Feb. 1, 2013

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

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

Problem

Many students do not develop deep understanding of mathematical concepts

Limits most students

Major issue of equityDisadvantagedLate bloomersSpecial education

IntroductionMultiple 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.

We Are All Piaget

Chess example (e.g., fork)

Learning through activity (reflective abstraction)

AbstractionAnticipationGoal directed activityReflection

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)

Learning through Activity

Analyze the learning in chess example

Odd-Even Example

odd X odd = ?5x5=2519x21=39939x37=1443

Learning that it seems to be true

Mathematical understanding is not the result of an empirical learning process

RA not ELP (cont.)

Mathematical understanding is the result of reflective abstraction Knowing logical necessity Development of an anticipation Abstraction from one’s activity

Analyze following example:

Odd-even example:

Even ≡ everyone has a dance partner

Odd ≡ everyone has a dance partner except one person

RA not ELP (cont.)

Example with objects: 5x3

OOO OOO OOO OOO OOO

RA not ELP (cont.)

Example with objects: 5x3

O O O O O

OO OO OO OO OO

Similar activity with different numbers

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

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)

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

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]

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.

Potential Contribution to Instruction

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

A Thought ExperimentNot atypical classroom scene

Competent teacher

Problem representing math to be learned

Students work in pairs – rich representations available

1or 2 pairs solve problems –most don’t

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)

A Thought Experiment (cont)

Equity issuemore-advanced students work novel problems, less-advanced students struggle to follow explanations of solutions

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)

Potential Contribution to Curriculum Development

Provide strong conceptual framework for task development and sequencing

Thank you for your attention!

Q&A