Elaborating the “Don’t Need Boundaries” element of the Pirie–Kieren Theory

This paper seeks to contribute to the theoretical literature on mathematical understanding by elaborating an essential, yet under-researched and sometimes queried, element of the Pirie-Kieren theory, the Don’t Need Boundaries (DNBs). In the Pirie and Kieren (1989) model, the DNBs are depicted as three bold rings that are positioned between the layers of Image Making and Image Having, Property Noticing and Formalising, Observing and Structuring. The ‘Don’t Need’ label suggests that an individual working outside a DNB can work independently of the specific inner layer of understanding that produced the knowing in an outer layer<br/>Our discussion is based on close examination of evidence drawn from a qualitative, microgenetic research study, with the participation of nine 9-year-old children. Children participated in eight problem solving sessions that involved more than one trial with novel for them, partitive quotient tasks. We discuss two types of identified DNB crossings: bi-directional and uni-directional DNBs crossings and associated variations in children’s problem solving behaviours at these crossings.<br/>For the given tasks, after having crossed the first DNB, children were more likely to return to inner layers of the model than after having crossed the second DNB. We thus propose that the first two DNBs are qualitatively different in that the first DNB crossing appears to be less ‘secure’ than the second. We suggest a visual amendment to the model that accounts for this observation. We propose that folding back and spending time beyond the first DNB are essential processes through which the first DNB crossing can become more secure and the first DNB less porous. This paper adds further detail in delineating the DNBs as significant points of potentiality for crossing to a layer of operation that does not require reference to basic concepts, rather than as points that necessarily involve secure crossings to an outer layer of abstraction. <br/>