That maths is a generally maligned subject is without much doubt. When I meet a new person and they ask my job and can see the sweat begin to bead on their foreheads. As a maths teacher it can be difficult at times to deal with this. A part from the regular “I’m not a maths person” comment and I have often heard people discuss the feeling of never getting better, it was always “too hard”. This conception begins in the classroom. Students will often claim they cannot do a problem because it is too hard despite having successfully solving similar (or easier) problems in the recent past.1 The concept of “too hard”, seems to persist in spite of the evidence in front of the students eyes. One phenomenon that may, in part, explain this is Prevalence Induced Concept Change (PICC).
When instances of a concept become less prevalent, the concept may expand to include instances that it previously excluded, thereby masking the magnitude of its own decline.2
An example used in a series of studies3 was whether a dot was blue or not. Taking into account hysteria around what colour dresses are we can say that the blueness of a dot isn’t a radically subjective idea. However, as researchers removed the number of blue dots per sample (decrease instances) the reporting of seeing blue dots remained the same (increase in scope of concept). They then go on to discuss how this applies not just to dots but the ripeness of bananas or, even more staggeringly, the ethical character of actions.
My contention is that perhaps PICC happens in the classroom too. The “too hard” concept is based on instances of problems that are genuinely too hard. Given that reducing the instances of these problems that are “too hard”. However, for some students while the prevalence of instances of “too hard”-problems decreases, the concept of what constitutes “too hard” increases to now include problems that strictly speaking are now not too hard. So when students continue telling us that the problem is “too hard” perhaps they’re not displaying a lower ability to self-manage, perhaps their feeling is exactly the same as it was before.
Resolving this issue, or heading it off at the outset, appears vital. One strategy often employed is telling people what the problem is. This can be seen in metacognitive analysis and reflection. However, for PICC, there’s a problem:
When blue dots became rare, purple dots began to look blue; when threatening faces became rare, neutral faces began to appear threatening; and when unethical research proposals became rare, ambiguous research proposals began to seem unethical. This happened even when the change in the prevalence of instances was abrupt, even when participants were explicitly told that the prevalence of instances would change, and even when participants were instructed and paid to ignore these changes.
Are your Catch-22 alarms going off, too?
1 This problem is piqued when moving from implicit algebra (using substitution to solve geometry problems) to explicit algebra. As a result we spend a lot of time building these skills but in many cases without seeming to change a students perception of their own capacity.
2 Prevalence Induced Concept Change by David E. Levari, Daniel T. Gilbert, Timothy D. Wilson, Beau Sievers, David M. Amodio, Thalia Wheatley