H ere w e u n p a ck N o z ic k ’s relia b ilist d e fin itio n o f k n o w le d g e an d u se it to c a st lig h t o n the e d u c a tio n a l co n tex t. T h is lea d s to a m o re n u a n ced ap p ro a ch to su p p o rtin g learning:
o n e in w h ic h c o u n terfa ctiv e s (~p) are se e n to b e sig n ific a n t. In d e c id in g w h eth er or n ot N o z ic k ’s fo u r c o n d itio n s are m e t in the particular c a s e , w e ju d g e w h e th e r or n o t th e learn er h as a c la im to k n o w le d g e p b y r e lia b ilist lig h ts.
N o z ic k ’s track in g m o d e l sets o u t the f o llo w in g fou r c o n d itio n s fo r k n o w le d g e :
W e ca n add to th is list th e n o tio n that S o sa d er iv e d fro m (3 ) and (4 ) - and that w e d is c u s s e d earlier (p .3 8 ) - that o f C artesian ‘s a fe ty ’:
5. Bsp * - □ - * p
For s to k n o w that p , then, it m ust b e th e c a s e that (1 ) p is true, ( 2 ) s b e lie v e s that p , ( 3 ) w e r e p n o t to b e true, 5 w o u ld n ot b e lie v e it, (4 ) w e r e p to b e true, 5 w o u ld b e lie v e it and (5 ) w e r e s to b e lie v e it, it w o u ld b e true. S in c e w e are n o w d e a lin g w ith an
e d u c a tio n a l se ttin g , s can b e un d erstood as ‘stu d en t’ a n dp a s ‘s o m e p ie c e o f
p r e p o sitio n a l k n o w le d g e ’ - su ch a s that ‘L arissa is north o f F a r sa la ’. T h ere is a ls o an im p lic a tio n (d u e to S o sa ) that the k n o w er d e se r v e s cred it fo r k n o w in g that p.
A c o n sid e r a tio n o f the fiv e c o n d itio n s, w h ic h to g eth er d e fin e k n o w le d g e , ra ises a n u m b er o f q u e stio n s w h en a p p lied to ed u ca tio n .
1. W h a t ran ge o f p ro p o sitio n s o u g h t to b e regard ed a s su ita b le fo r th e learn ers’
e d ific a tio n ?
2. W h a t le v e l o f b e lie f is required o f th e learner? H o w ca n it b e m a d e m a n ifest?
3. I f p is stip u la ted to b e true, w h y is it n e c e ssa r y to en tertain th e p o s s ib ility that it m ig h t b e fa ls e and h o w o u g h t this b e d on e?
4 . B y w h a t m ea n s o u g h t the learner fo rm th e b e l i e f ‘that p ' from th e fa ct that p i 5. H o w ca n th e learner en su re that h is b e lie fs are sa fe - that is, n o t e a s ily m istak en ?
W e can d e a l w ith th e first three under the h e a d in g s ‘R a n g e o f P r o p o sitio n s p \
‘B e lie v in g T h a t p ’ and ‘C ounterfactual S e n s itiv ity ’ . T h e la st t w o are c o n sid e r e d to g e th e r a s ‘A c q u irin g S a fe B e lie f s ’. T a k in g th e se in turn:
Suitable Range of Propositions for the Learner — Including ‘Threshold Concepts’
T h e q u e stio n ab ou t w h a t range o f p ro p o sitio n s th e e d u c a tio n a l p ro ject o u g h t to treat ca n b e a n sw e r e d w ith referen ce, inter a lia , to (a ) th e p e r c e iv e d n e e d s o f the learner (b ) n a tio n a l p rio rities or (c ) the su b ject d isc ip lin e s. F or th e first (a ), an o v era rch in g ed u c a tio n a l (a s o p p o se d to su b ject-cen tred or p o litic a l) r e a so n fo r in c lu d in g a p r o p o sitio n o f a particular ty p e m a y b e that it s im p ly g iv e s th e learn er p ra ctice in
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d e a lin g w ith that ty p e o f co n cep t. Or, it m a y c o n cern a to p ic that is lik e ly to p iq u e h is in terest and e x te n d his k n o w le d g e in w o r th w h ile w a y s. F o r th e se c o n d (b ) w e ca n rea d ily fin d e x a m p le s o f p o litica l attem pts to m a k e e d u ca tio n a l o u tc o m e s m atch n a tio n a l p riorities. T h is h appens at all p h a se s o f ed u c a tio n , in c lu d in g u n iv ersity . T h e eu p h e m ism o f a lig n m e n t is u sed here in stea d o f c o m p lia n c e to in d ic a te the req u irem en t o f h ig h er ed u c a tio n in stitu tion s to co n fo rm to g o v e r n m e n t e d ic ts v is - à - v is their
e d u ca tio n a l o u tc o m e s, o n pain o f funds w ith d ra w a l for n o n -c o m p lia n c e :
To drive the reform and modernisation o f the third level sector by enabling institutions to align their strategies with national priorities through the implementation o f the new funding arrangements and the impact o f major investment under the Strategic Innovation Fund.34S
It w o u ld b e n a ïv e to argue that ed u ca tio n o u g h t to b e d e c o u p le d c o m p le t e ly fro m th e e c o n o m ic a g en d a . H o w e v e r , i f n a r r o w ly -c o n c e iv e d n a tio n a l p riorities are th e s o le d eterm in an t o f th e range o f p ro p o sitio n s, p , o f w h ic h w e o u g h t to treat, s o m e oth er ca n d id a tes - fo r e x a m p le , the su b ject d isc ip lin e s, th e n e e d s o f the learner and
‘th re sh o ld c o n c e p ts ’ - m ay b e rela tiv e ly and u n v ir tu o u sly n e g le c te d . For th e last, (c ), in d iv id u a l p ro p o sitio n s ca n b e id en tified as n o d e s in a c o h e r e n t su b je c t-w e b , th e m o st im portant o f w h ic h w ill in c lu d e ‘threshold c o n c e p ts ’, a to p ic that n e e d s further elab oration .
A v a r ie ty o f p r o p o sitio n s are co n sid ered b y ed u c a tio n a l p ro g ra m m es, but th ey d o n o t a ll h a v e th e sa m e im portance. S o m e are so trivial that s u g g e s tin g a c o n n e c tio n b e tw e e n th eir a c q u isitio n and w h a t m a y b e co n sid ered th e ed u c a tio n a l telos o f en h a n c in g the lea rn er’s in te lle c tu a l flo u r ish in g w o u ld b e u n te n a b le, w h ile oth ers are c la im e d to h a v e a sp e c ia l s ig n ific a n c e . H o w e v e r , so m e im portant in sta n c e s o fp in v o lv e ‘th resh o ld c o n c e p t s ’, w h ic h are ‘ak in to . . . portal[s], o p e n in g up . . . n e w and p r e v io u sly in a c c e s s ib le w a y s o f th in k in g ab ou t so m e th in g . . . w ith o u t w h ic h th e learn er ca n n o t p r o g r e s s ’,346 and h e n c e c o u ld w e ll have a m ajor b ea rin g o n h is e p is te m ic flo u r ish in g . T w o c a v e a ts o u g h t to b e m ad e at this p o in t, th o u g h , s in c e v a lu a b le as th e n o tio n o f
‘th resh o ld c o n c e p t’ m a y b e, the coiners o f the term ( i) in ten d it to b e u se d to h elp university te a ch ers in th eir p lan n in g and (ii) are w o r k in g w ith in a rather te c h n ic ist
‘lea rn in g o u tc o m e s ’ paradigm . H o w ev e r, the id e a o f a p r e v io u s ly b lo c k e d route to tr o u b le so m e k n o w le d g e b e c o m in g n e w ly -p a s s a b le is a fa m ilia r o n e in a n c ie n t G ree k
345 Government o f Ireland (2007) ibid,, p. 189.
346 Jar Meyer & Ray Land (2003) ‘Threshold Concepts and Troublesome Knowledge: Linkages to Ways o f Thinking and Practising within the Disciplines.’ O ccasional R eport no.4. (Edinburgh: ETL Project).
thought - the aporia reveals a euporid!47 ~ which suggests that, used with caution, the concept might have more general educational import. We shall explore the notion a little further.
Meyer and Land characterise threshold concepts as transformative, irreversible, integrative, bounded and troublesome. After his epiphany, the learner sees things in a new light: one in which disparate and puzzling phenomena now appear lucidly
connected. Following his transformation, the knower is changed forever and cannot return to his old ways o f thinking: Saul becomes Paul. The idea is an attractive one, but we ought to look at the details to see how this plays out in practice.
Interestingly, many o f the examples cited by Meyer and Land relate to absence or negation rather than presence: ~ p rather than/?.348 Medical undergraduates, for instance, need to leam to see pain as ‘an ally that aids diagnosis and healing’,349 rather than as an unredeemably negative phenomenon which needs to be removed. Similarly, in physics, the absence o f order - entropy - is a problematic concept which is
nevertheless essential for accessing the key ideas o f thermodynamics. For economists, the threshold concept o f ‘opportunity cost’ deals with what Homo economicus could be doing with his time, but is in fact not doing, rather than with the costs and benefits o f what he is actually doing. More formally: ‘Opportunity cost is the evaluation placed on the most highly valued o f the rejected alternatives or opportunities’.350 For the
economics student, coming to know that evaluations o f actual and counterfactual choices (including the choice o f not choosing) can be made - and that plumping for X means eschewing Y and Z - involves crossing an intellectual threshold. In doing it he acquires the notion o f ‘opportunity cost’ and has access to economic thought o f a more abstract and generalisable nature than he had before.
343 Meyer & Land make no mention o f the Greek provenance o f their idea.
348 Meyer & Land do not talk in these terms, nor do they seem to have noticed this common feature o f many o f their examples.
349 Jan H.F. Meyer & Ray Land (2005) ‘Threshold concepts and troublesome knowledge (2) Epistem ological considerations and a conceptual framework for teaching and learning’, Higher Education, 49: pp.373-388, p.374.
350 Eatwell (1998) in Meyer & Land (2003) op. cit, p.3.
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In mathematics, Meyer and Land identify the term ‘limit’ as a threshold concept, as in the following.351
In the limit, as 0 -» 0, F(0) = (sin 0)/0 = 1
(0 [the angle between the base and the hypotenuse] in radians)
Again, we see that what is important here is not what exists (a right angled triangle with 0 having a non-zero value such that 0 < ji/2 radians) but what cannot exist (a right angled triangle in which 0 is zero). The triangle which represents the limit is not only counterfactive, it is also impossible. If 0 is zero, there is no triangle but only a single straight line. Moreover, attempting to calculate (sin 0/0) involves the mathematical felony o f dividing by zero. The student o f mathematics ought eventually to appreciate that as 0 approaches zero, the fraction (sin 0/0) approaches 1. The significant word here is ‘approaches’, for were ft,per impossibile, to arrive at zero, the triangle would disappear, making nonsense o f the idea o f measuring its sine. Being able to accept such incompossible pairs as (i) p [that if 0 were zero, then (sin 0/0) would be 1] and (ii)
~ p [that if 0 were zero, then (sin 0/0) would not be l] 352 liberates the tyro
mathematician from the everyday notion o f a limit as something that can be reached.
Furthermore, having crossed this conceptual threshold, the differential calculus - which essentially deals with rates o f change [dy/dx] in the limit, as x approaches zero - is now more genuinely available to him.
From an educational viewpoint built on epistemic virtue, what is particularly interesting in the last example is the question o f epistemic authenticity. The student who has not passed the intellectual threshold and acquired the mathematical notion ‘limit’ (which cannot itself be transgressed) may still manage to demonstrate an ersatz mathematical proficiency, one that simulates the genuine work o f his classmate, who has stepped through the conceptual portal and now has what we might term ‘echt-proficiency’. The aporectic student has perhaps learned the appropriate algorithms in the form o f what David Perkins calls ‘ritual knowledge’. He has become habituated to mathematical rituals o f ‘a routine and rather meaningless character’; he knows ‘the routine that we
351 I have written this expression slightly differently from the version in Meyer & Land, for clarity and completeness. They omitted the important condition that 0 must be expressed in radians.
352 ‘(sin 0 / 0 ) = 1’ would not be true, since it trades on meaningless expressions.
execute to get a particular result’.353 To the positivistically-inclincd assessor, both students deserve the same credit, but from a virtue perspective one is less
epistemically-praiseworthy than the other, for his correct answer comes from an uncomprehending and shallow mimesis rather than from intellectual virtue.
Another example from Meyer and Land’s paper is the threshold concept o f
‘signification’ in the field o f cultural studies. Again, ~p makes an appearance, if we glossp as ‘that this is being said’:
... techniques o f deconstruction for analyzing literary texts (with a strong emphasis on the ironic, the contradictory, the ludic) often appear counter-intuitive, looking for absences, or what is not there, in order to gain insights into how the text is currently structured by a prevailing set o f (occluded or tacit) values or priorities.354
In other words, it is not the readily-apparent conclusionp (that this is being said in the text) on which the the reader ought to reflect, but ~p (it is not the case that this is being said in the text). ‘ This' is the explicit message o f a text, but it is the unsaid, the hidden and silent assumptions that are o f greater interest.
A final instance concurs with my own experience o f teaching Newtonian mechanicsj55, and, for learners, falls into Perkins’ category o f ‘foreign’ knowledge which ‘comes from a perspective that conflicts with our own’.356 Pupils often spontaneously adopt explanations for the motion o f bodies, which are at odds with classical physics and o f a distinctly Aristotelian type: for example, that a rocket needs to bum continuously to sustain its movement in travelling to a distant planet. These misconceptions are eminently reasonable, though, for everyday experience seems to contradict the Newtonian law that ‘in the absence o f a force’, an object will either remain at rest or
‘continue to move in a straight line at constant velocity’.357 Objects do seem to need a
353 David Perkins (1999) ‘The many faces o f constructivism’, Educational Leadership, 57 (3), p.8.
Meyer & Land (2003) quote Perkins (1999) too, but they give incorrect page numbers and change Perkins’ ‘foreign knowledge’ into ‘alien knowledge’. The latter change is arguably for the better, since Perkins gives a history-teaching example o f what he means by this: ‘presentism’, in which pupils view past events through present knowledge and values’. (Perkins, ibid, p. 10). A period o f remote history may be ‘alien’ to us, but being ‘foreign’ has a different connotation,
354 Meyer & Land (2003) op. cit., p.3. [Italicized words in original].
355 This is not just anecdotal evidence, however. For further evidence o f such misconceptions, see Kevin M. Leander and David E. Brown (1999) “‘You Understand, But You D on ’t B elieve It” : Tracing the Stabilities and Instabilities o f interaction in a Physics Classroom Through a Multidimensional Framework’, Cognition and Instruction, v ol.17, n o .l, pp.93-135. p.95.
356 Perkins (1999) op. cit., p.10.
357 Isaac Newton (1687) Philosophiae Naturalis Principia M athem atica, Law I. [This is a merely a special case o f Law II [ F = ma] in which F - 0, but it usefully underlines N ew ton’s rejection o f the Aristotelian treatment o f forces and motion].
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motive force to keep them moving, although within Newtonian mechanics this is explained by the necessity o f overcoming the opposing force o f friction. Here we leave Meyer and Land’s notion o f threshold concepts and consider briefly how the faulty propositions held by the learners in this final example can be ousted.
In setting out to displace these ‘alternative frameworks’ (such as pupils’ Aristotelian ideas about motion), it is important to bear in mind that their mistaken adoption by learners is neither arbitrary nor intellectually vicious but is based quite reasonably on a priori categories and sensory experience: a tethering that makes them particularly tenacious. This tenacity is strengthened by the fact that the elegant mathematical constructions o f classical physics are not easily vulnerable to attack or engagement by pupils in school science laboratories. There is something abstract and Platonic about Newton’s work that insulates it from empirical falsification in the classroom. Only on the quantum scale, or at speeds approaching that o f light, does the Newtonian world
view break down, so any apparent classroom departures from Newton’s laws are usually (and correctly) explained away with reference to friction or experimental error.
Thus, our experiments with ‘friction-compensated’ ramps, trolleys and timers are designed not to put Newton to the empirical test but to make sure that we confirm his laws in the eyes o f our pupils.
This dependence on authority (Newton’s authoritative position in the scientific pantheon; our dual role as both authority-figure and an authority in our subject) is not likely to engage learners in responsible, intellectually-virtuous dialogue. It will, rather, encourage unresponsive acceptance of a pre-defined version o f reality. Teachers might seek to reassure their pupils that if they just take it on trust for the time being, it will all make sense eventually. Unfortunately, an unwillingness to accept the deferred
gratification required, and revulsion at the often forbidding nature o f the scientific corpus, has led many to seek alternative explanations for fundamental questions not in the scientific domain, but in pseudoscience. To avoid this epistemically-unfortunate, but understandable, abdication on the part o f pupils, we need to be honest about the conflicts that abound and concede the apparent attractiveness and plausibility o f a range o f counterfactive propositions ~p, instead o f merely dismissing them and substituting p by teacherly edict. This search for genuine knowledge rather than second-hand
opinions requires authentic dialogue. In R epublic 435a, Plato offers the image o f the two fire-sticks which cannot individually make fire but require contact to produce the
necessary spark. It is the intersubjective nature o f this construction358 o f knowledge that gives this approach its power. To flourish epistemically requires intellectually- virtuous informed dialogue and guided experience. However, the teacher would be unwise to emulate the rigorous insistence on definitions and steamrollering reductio moves frequently found in Plato’s dialogues. The root meaning o f ‘elenchus’ relates to shame, we recall, which is not usually a desirable emotion in classroom contexts.
So, a consideration o f Nozick (1) has led us to consider the range o f propositions which are proper for the education project. Some o f these - such as the threshold concepts - appear to be particularly important for the intellectual flourishing359 o f learners and may justifiably be promoted on that ground. However, the construction o f a ‘shopping list’ o f propositions,/?, (for example in an examination syllabus) might cause teachers to approach them in too direct a manner and inadvisedly brush o ff the counterfactive propositions, ~p, held by learners. These misconceptions need to be addressed to clear the way for a ‘pro-factual’ concept acquisition o fp, and dialogue is a good candidate for a way o f dealing with them virtuously.
Believing That p versus Being Able to State “Thatp ”
Asserting as a condition for knowledge th at‘s believes that p ’ is not, I suggest, to be conflated with the requirement that‘s is able to state that p In Dickens’ Hard Times, for example, Sissy Jupe’s deep, first-hand knowledge o f horses is over-ridden by Gradgrind’s demand for propositional evidence that she really does know what a horse is - a demand allegedly not met by her, but (by positivist lights) adequately
demonstrated by another pupil: ‘Bitzer,’ said Thomas Gradgrind. ‘Your definition o f a horse.’
‘Quadruped. Graminivorous. Forty teeth, namely twenty-four grinders, four eye-teeth, and twelve incisive. Sheds coat in the spring; in marshy countries, sheds hoofs, too. Hoofs hard, but requiring to be shod with iron. A ge known by marks in mouth.' Thus (and much more) Bitzer.
‘N ow girl number twenty,’ [Sissy Jupe] said Mr. Gradgrind.
‘You know what a horse is.’
358 Socrates is o f course no radical constructivist, but he believes that The Truth can best be sought by dialogue. In this respect, he is what we might term a ‘pedagogical constructivist’.
359 Flourishing is o f course an Aristotelian rather than a reliabilist notion. A reliabilist justification for including the threshold concepts in a syllabus would be that they open up access to a wider range o f true beliefs and hence ‘help maximise one’s surplus o f truth over error’, to use Sosa’s phrase. Ernest Sosa (1985) ‘Knowledge and Intellectual Virtue’, Monist, 68, no.2, p.227.
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The significance o f this vignette is that Sissy Jupe, by virtue o f her experience, is in a better position than any o f the other protagonists to know what a horse is, but her voice is silenced in favour o f the ‘facts’ o f the propositional description offered by Bitzer.360 We can locate her in Michael Polanyi’s ‘... area where the tacit predominates to the extent that articulation is virtually impossible; we may call this the ineffable
domain ’,361 Were Gradgrind a more virtuous teacher, he would act in a maieutic role here, by helping her to deliver her knowledge in a form suitable for respectful public discussion. He would enable her to make the tacit more explicit. Sissy has a good web o f beliefs about horses: representing sound cognitive contact with reality, and woven during long first-hand experience, helped by intersubjective triangulation with the interpretations o f her father and others. But Gradgrind has isolated her and treated her like a feral knower. Bitzer, however, is a horse o f a different feather, who might be considered to occupy ‘... the area in which the tacit and formal fall apart, since the speaker does not know, or quite know, what he is talking about’.362 In typical
educational contexts, having no real feel for what one is talking or writing about is not a barrier to obtaining reward, for often what counts is that ‘knowledge’ be explicitly demonstrated. The ‘diffident schoolboy’363 can state ‘thatp ' in an exam, yet not be entitled to claim it as knowledge, since he does not believe the proposition, beyond having a faint intuition that it is more likely than not that p might be the right answer to the question. Although this is not a creditworthy, qua virtuous, response, it usually will earn him credit in the form o f academic reward. I return to this problem later (p.207).
In educational contexts, then, authentic belief and espoused belief can come apart in a manner that may deny the learner knowledge on the criterion o f Nozick (2), but not on
360 We can regard her more favourably as a knower, however, when w e consider her ‘being-in-the w orld’, a la Maurice Merleau-Ponty’s L a structure du com portement (1942). Merleau-Ponty stresses
360 We can regard her more favourably as a knower, however, when w e consider her ‘being-in-the w orld’, a la Maurice Merleau-Ponty’s L a structure du com portement (1942). Merleau-Ponty stresses