The process of problem posing: development of a descriptive phase model of problem posing

TL;DR

The paper develops a descriptive phase model for problem-posing processes based on structured situations, using task-based interviews with pre-service mathematics teachers. It identifies five content-related episode types and derives a cyclic model with high interrater agreement.

Key findings

• Five content-related episode types were identified: situation analysis, variation, generation, problem-solving, and evaluation.
• The derived descriptive phase model has a cyclic structure; there is no predetermined order, and transitions between any episode types were observed empirically, though a “natural order” often appeared: situation analysis → variation/generation → problem-solving → evaluation.
• Interrater agreement for the episode coding was Cohen’s κ = 0.81, described as almost perfect; separate category kappas were κ SA = 0.87, κ V = 0.83, κ G = 0.72, κ PS = 0.87, κ PS/E = 0.73, κ E = 0.49, and κ O = 0.97.
• The largest coding differences were for variation vs. generation and for distinguishing problem-solving, problem-solving/evaluation, and evaluation.

Methods

• Task-based interviews were conducted with pre-service primary and secondary mathematics teachers working in pairs on two structured problem-posing situations: (A) Nim game and (B) Number pyramid. The abstract states 36 task-based interviews; the data collection section reports 64 PSTs and 32 analyzed processes (15 for situation A, 17 for situation B).
• Processes ranged from 9 to 25 min, averaged 14.5 min, and totaled 7 h 46 min of video material.
• Analysis adapted Schoenfeld’s verbal protocol analysis with event-based sampling: interviews were segmented into episodes and coded through inductive-deductive category development. Interrater agreement used the EasyDIAg algorithm with a 60% overlap criterion; a second coder analyzed about 2 h 23 min (30.7% of the material, 10 of 32 processes).

Limitations

• The paper states the model offers one possible perspective on problem-posing processes; depending on the problem-posing situation, sample, or study design, slightly different or additional episode types may occur.
• The study considers two specific structured situations with a non-routine initial problem and uses PSTs as the sample, though the authors report successful applications to other situations, contents, and sample groups in bachelor’s and master’s theses.
• Evident from the text: the abstract’s “36 task-based interviews” is inconsistent with the data collection section’s report of 64 PSTs in pairs and 32 analyzed processes; the evaluation category also had the lowest reported kappa (κ E = 0.49).