A Theory of Physically Distributed Learning

A Theory of Physically Distributed Learning: How External Environments and Internal States Interact in Mathematics Learning

Martin, T. (in press). A theory of physically distributed learning: How external environments and internal states interact in mathematics learning. Child Development Perspectives.

Description:
Physically Distributed Learning (PDL)- Understanding that children learn new concepts in concrete contexts and transfer these concepts to abstract situations. This paper presents an alternative theory of PDL.

About Physically Distributed Learning (PDL):

• NCTM and pre-service teacher handbooks support using manipulative's to teach elementary mathematics
• Hands-on materials help children learn, but how and when?
• DCog (I like that better than "distributive cognition") is a useful theoretical perspective as it considers interactions with the environmental affect thinking and problem solving.
• Co-evolution:
• over time, children's initial ideas and actions change as they evolve
  
• maybe transferring understanding as prior knowledge improves as new concepts are introduced
  
• new actions and ideas emerge when old ones fail
  
• In emergence, children take actions that suggest new ideas or develop ideas that suggest new actions
• Children’s actions are more variable and accurate when they use dynamic
  materials than when they use static materials (Ainsworth & VanLabeke, 2004; Martin &
  Schwartz, 2005).

• Action-interpretation sequences
• Through solving multiple problems, children discover useful combinations of actions and ideas.
• Children then repeat these combinations, decreasing the variability in their actions and interpretations.
  
• Next, children coordinate their actions and interpretations into action-interpretation sequences
• General Structures
  
  • The use of number relationships, rules of the number system, and number facts to solve problems suggests that children are developing more general structures.
  • e.g. "number relationships such as “one more” and “one less”

Data Presented (Prior research):

• This paper presents results of studies of elementary school children solving number operation problems with manipulatives.
  
• These studies investigate three critical elements of the PDL hypothesis.
  
• First, action is beneficial but insufficient without interpretations.
  
• Second, actions and interpretations coevolve.
  

• Third, children can transition to solving problems mentally, suggesting that PDL helps them develop more general structures for number understanding that support transfer.
  

• 10 year olds solved operator problems without feedback as part of a longer interview on fractions (Martin &Schwartz, 2005).
  
• Each child solved two problems with manipulatives and two problems
  using a pencil to draw on illustrations of the same pieces.
• The same children solved the problems correctly 76% of the time with manipulatives, but only 16% of the time with the pictures (regardless of the order of the tasks).

• Two studies demonstrate that neither actions nor interpretations is sufficient for problem solving alone.
• The first, 10yearolds solved problems they understood well(multiplication problems) and problems they struggled with (fraction addition with different denominators) with different manipulatives.
• The second study, 10yearolds solved operator problems without feedback using manipulatives and pictures.
• When the children moved pieces, they answered correctly more often than when the pieces were already partitioned into the correct groups.
  
• This result demonstrates that manipulation helps children develop interpretations.
  

Martin's Research:

• Coevolution as a Mechanism for Change (p.9)
• Children completed three interviews with feedback and a pretest and posttest without feedback. We examined the match between children’s actions and interpretations by computing an agreement score for the interviews (the absolute value of the difference between the number of problems children solved successfully and the number of problems they used partitioning).
• Patterns in the Development of Action-Interpretation Sequences (p.10)
• To develop a more fine-grained analysis of coevolution, we examined students’ actions and interpretations on each problem attempt. Then we described and examined how these action-interpretation pairs changed over time. We found three patterns of howstudents developed coordinated action-interpretation sequences.
• Developing Generalizable Mathematical Ideas (p.11)
• Through PDL, children develop general mathematical ideas independent of the physical context. In the division study, children used more mental strategies over time.

Martin's Conclusions:

• Physically distributed learning can help explain how the concrete plays a role in learning abstract mathematical concepts.
  
• Through a process of coevolution, actions and interpretations develop each other, and eventually, children develop stable ideas and
  solve problems using mental strategies.
• The research summarized here describes studies with simple arithmetic concepts and similar tasks.
  
• It remains to be seen if PDL is more broadly applicable.
  
• One way to begin to address this question is to consider how PDL may apply to the data from the various and interesting perspectives presented in this issue.
• Another important factor to consider is the structure of the learning environment(Brown, McNeil & Glenberg, this issue).
  
• PDL predicts children’s learning will suffer if the environment is over structured such that it does too much mental work for children.
• Highly structured environments, such as cockpits (Hutchins, 1995a), are excellent for achieving well known tasks quickly, but may stifle learning opportunities.
  
• Virtual or computer based manipulatives offer excellent opportunities for investigating levels of structure appropriate for different tasks as they are easily reprogrammed to instantiate varying levels of structure (Sarama & Clements, this issue; Sarama & Clements, 2004).
• A reasonable prediction from PDL is that coevolution will be most likely to occur when interaction is allowed within an environment that structures that interaction productively.