Learning Trajectories

From Kaput Center Wiki

Participants: Jere, Theresa, Annie, Luis & Jim

Definition of Learning Trajectories Jere has been working with:

"Researcher conjectured, empirically supported descriptions of ordered experiences a student encounters through instruction in relation to activities, tasks, tools, forms of interactions, and methods of evidence in order to move from informal ideas through successive refinements of representations something in reflection to increase [inaudible] complex concepts over time."

• Most important aspects of definition:
  • Researcher conjectured; we're not assuming something
  • It must be empirically supported
  • It must have an order and has an instructional element to it
  • Tools have a major impact on this, technology or not

• Teresa: is there something in literature about reformulating Simon's construct with now incorporating technology?
  • Learning trajectories-a way to talk about a long-term evolution process

• Annie: entirely for researchers? Or are you aligning this with state standards?
• Jere: using method, hexagon maps, to map state standards as/ in hexagons and looking at implicit trajectories that are within them.
  • Long term goal--get them sorted out, use to guide development of standards and assessments.

• Luis: revolution of representational system and meanings (ex. from symbolic viewpoint, revolutionary viewpoint in sense of students and how they treat math symbols). Iconic--> symbolic. More practical viewpoint: task at look at symbols to looking through symbols. Related to civility of enhancing production of meaning.

• Jere: missing from their definition is notions of purpose. Informal to formal is useful--broad, lots of people would agree.
  • Question of purpose--how much freedom do people have and how much are we staying that the endpoint is formal definition of math concepts.
  • Evolution in terms of Perturbation- keeps people on the same track.

• Luis: related to issue of intentionality

• Annie: what's being perturbed?

• Jere: Piagetian perspective. Perturbation or problematic-causes people to take action; action satisfies certain needs through reflection, build up scheme for an idea.
  • Perturbation: how do you create a need for an idea so that the student encounters it as a solution or could be potential solutions to a problem?
  • Constructivism: think that is a lot of what we do, sequence tasks so you can cycle on.

• Teresa: uses idea of moving from formal to informal.
  • Important to have a model of a usage of the concept and the tool in order to foresee the evolution that you want to promote in the learners in order that they can move from their informal thinking to their formal thinking.
  • But need a model of formal way of thinking, conceptualizing an idea, competent usage of concept and/ or instrument (technology).

• Jere: is this a counter model to/ alternative model to what you just described: you want a set of empirical observations of real use of language instead of a model of a competent user.

• Teresa: yes, but in order to see which direction they are moving the formal model is a reference to contrast to compare with actual use. Theoretical artifact: to have something formal and theoretical to compare your data.

• Jere: learning trajectory is combination of theoretical model and empirical data. In synthesis work, need data. Can’t synthesis without data.

• Luis: from some perspective you need theoretical framework. Data can only be meaningful—interpretive tool, need a theory. One way to measure: ways of use of representation. How people are using representation, math language. Could give you an edge on how they’re conceiving of the mathematics by itself. One way of looking at that—looking at semiotic behavior.

• Jere: one level could do it on level of success or non-success of solving tasks.
  • Example Jere is struggling with: Less and she have a different argument whether rational number gets built out of counting. Less has the reorganization hypotheses. Jere argues that splitting should be the independent construct of counting. Big argument comes down to partitioning: piagetian perspective: how do you reverse partition? Less: gets put back through iteration. Jere: wrong reversibility for partitioning. ***Result: build different learning trajectories.
    • Driving Jere to think: is there a way to determine one trajectory is better than another? Are they both learning trajectories? How do you decide what is better? Are they logical things, consistency things, explaining large amt of data things?
    • Got to be a way to decide alternative, evaluate strengths and weaknesses.

• Teresa: after so many years of research on specific math topics, possible to predict obstacles and reactions of learners to these obstacles. In some way, if we are designing a task, you can design the task in order to create the conditions in which students have to face another difficulty so that they can progress in intended direction. Possible to make predictions; in this sense, even if one task-learners can develop through a variety of paths, they will have something in common. Through data, identify patterns.

• Jere: is that strong enough? Jere started on error patterns. Trajectories are supposed to do more than that. They are supposed to be facilitating and explanatory at some level of mechanism. What does that look like? Certainly should predict errors.

• Teresa: digital technology environments. Designed for paper and pencil environments. Students are decodifying at the same level the mathematical issues that are occurring on the screen, and all the issues that come from the software. Student argument is really at technology level of the artifact. Feedback that the tool gives to the student.

• Jere: exchange of teacher feedback to software environment feedback.

• Teresa: students mixing two levels of work. 1) Math content
  • Students are processing information on the same ‘plane’; those things make difficulties that were predictable in paper and pencil environment are not predictable in new environments.

• Luis: this level of studying informal to formal is a key aspect of learning trajectories. Look at problem from semiotic viewpoint. Ways of using representational systems. When we go from paper and pencil to digital, who do the mediation of the two keep changing? Beginning, students are guided by technology. Wishful goal: reach level of students guiding the technology. At the beginning, two forms of knowledge: math knowledge, way of managing the technology itself. Sometimes a huge obstruction. This transformation is a big step.

• Jere: requires a separation of math and technology?

• Luis: no, blending more than a separation. Through blending, access level of real digital representational system.
  • Example: in paper presented in Germany (draws on board?). Students: area of triangle is fixed is constant because there is a kind of compensation when the point is being moved; problem of variation. The center of the circle itself was the problem. Everything they were thinking about just blew apart.
    • At what moment is the student feeling the need to prove? Giving arguments to prove
    • math is becoming more and more symbolic with the way you argue; argument is changing.

• Jere: surprised to hear Luis say it’s becoming more symbolic.

• Luis: working on an indexical level, classification of signs.

• Jere: if I was a student: as I move towards the center the base gets larger, altitude gets smaller→area is invariant. When they get to a certain point both changes and breaks notion.

• Annie: Icon, index and symbol

Teresa: unknowns are icons

• Jere: we’ve agreed on value of informal to formal.
• Annie: why value formal?
• Jere: willing to accept a certain authorized expertise that finds it useful and compact. Has to do with issue of structure. Generalizability in math comes from the potential to see it from different circumstances. Value Mathematics as Modeling. Formalisms capture structure.
  • Modeling: going from an indeterminate to a determinate situation through a series of transformations. Model implies you have to get somewhere, you don’t just process.
  • Teresa: you have a goal.

• Teresa: likes math modeling as a way to understand math and the formalization of math.

• Jere: can we agree on informal and formal? Yes!

• Annie: is formal a construct to ‘keep people out’. Some people believe this.

• Jere: important problem. A lot of things are easy mathematically when you show how to frame them in contextually interesting ways.

• Annie: symbols keep people out?
• Jere: no, too limiting. Math symbols are un-interpretable. Symbols are used so much (bathroom signs, etc).

Modeling discussion under intentional design vs. redesign category

• Luis: Problem with modeling (is in designing a strategy to look through the symbols. You are given a more concrete basis of interpretation of math but at the same field are provided fields, reference, and support for abstraction.
  • Model: use meaning of model to guide you through the way the math representations are working. Roughly linked to the feeling of proof.
  • Interesting to consider the evolution of proof, of argument, in math. Students don’t feel need to prove. It is important to characterize the moment when you feel the need to prove; Important for learning trajectories.

• Teresa: learning trajectories have to deal with the transition between from the particular to the general. All these transitions. Transitional processes.

• Jim: design component of learning trajectories.
  • Idealized path with very specific design intentions
  • Compare how kids traverse paths laid out or extent to which behaviors embody models at specific benchmark times

• Luis: think of it as a piecewise trajectories--important.
  • Modeling is the way at looking at symbols. To understand an important idea, this notion of learning trajectory. Informal to formal is a key aspect.
  • Another important idea: evolution of the feeling of proof
  • Important to evolution of representations

• Annie: students don’t see the need for proof other than knowing that they have to do it to show they know something.

• Jim: levels of proof—proof for that particular triangle is a proof. Class to which you’re appealing logical argument (this triangle, all triangles, isosceles, etc).
  • Struggling with why having students solve equations in algebra. Important for students to understand solving an algebraic equation is a proof.
  • Applying structure of proof to take a particular instance given this information and this structure, this outcome is fixed/ determined. Other way—these kinds of situations has this kind of structure. Unclear of flow to moving from specific to general

• Luis: domains of abstraction. Find that moment of the feeling of the need for proof. Integrity of knowledge in mathematics.

• Annie: why would we care if a triangle has 180 degrees and why would they care about proving it?

• Jim: if you value the community for which you’re immersed, take on norms of that community. Issues of values in which the individual or group appeal, similar to concept of identity, appeal to norms of that social group. Conscious need for proof?

• Jere: do you question the need for angle measurement in general?

• Teresa: what would you accept as evidence that a student is really feeling need for proof? Perhaps if the student asks themselves if this property is valid for another triangle?
  • Perhaps we have in our mind the need for a proof. Perhaps at the learners level there are other expressions of those needs.

• Luis: thinking of proof in context of problem solving. Need for finding the solution.

• Annie: desire to know if our answer is correct.

• Jim: trial and error prototype, and guess and check. If you don’t’ have a lot of knowledge, build it as robust as you can and then test it. There is some difference to saying I’ve done all the math and therefore it will work. Inputs, and solution follows. Outgrowth of internal guessing and checking. Or, if you have enough ‘little routines’ you know that separate pieces work and the question is if they all work together.

• Annie: doesn't know how to get students to value proof and to do it.

• Jim: are values learned?

• Jere: we got here by agreeing on informal to formal; talk about transition processes

• Jim: present certain topics to class; depending on order of presentation affects their knowledge.

• Jere: There are some places where sequence matters

• Teresa: talking about trajectories because we’re talking about path. Intended learning trajectories. Some teaching goals in mind.

• Jere: raises interesting question—idea of paths and building diagnostic assessments such that the assessments constitute checking to see the ideas have been robust enough.
  • Wouldn’t think even of issues of compatibility of this is curriculum, keep curriculum independent/ general.
  • Evidence=assessments rather than straight instructional guides. Relationship between curriculum and instruction is really hard.

• Annie: are these trajectories developmental and in what sense?
• Jere: certainly developmental. Not stages; they are tendencies/ accomplishments.
• Jim: attitude of a ship; personal aspect of captain but riding something specifically with something in mind.