Why there is little cause to be happy with the new GCSE grade 5

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The OECD’s Programme for International Student Assessment (Pisa) has now taken the bold step of analysing measures of “happiness,” “well-being” and “anxiety” for individual countries (see New Pisa happiness table, TES 19.04.2017 https://www.tes.com/news/school-news/breaking-news/new-pisa-happiness-table-see-where-uk-pupils-rank).

The claim is made that “life satisfaction,” for example, can be measured to two-decimal place accuracy.  This begs the question, “Can complex constructs such as happiness or anxiety really be represented as a number like 7.26?”  For two giants of 20th century thought – the philosopher Ludwig Wittgenstein and the father of quantum physics, Niels Bohr – the answer to this question is an unequivocal “no.”

 

Surely common sense itself dictates that constructs such as happiness, anxiety and well-being cannot be captured in a single number?  In his book Three Seductive Ideas, the Harvard psychologist Jerome Kagan draws on the writings of Bohr and Wittgenstein to argue that measures of constructs such as happiness cannot be represented as numbers.  He writes: “The first premise is that the unit of analysis … must be a person in a context, rather than an isolated characteristic of that person.”  Wittgenstein and Bohr (independently) arrived at the conclusion that what is measured cannot be separated from the measurement context.  It follows that when an individual’s happiness is being measured, a description of the questions on the Pisa questionnaire must appear in the measurement statement because these questions help define what the measurer means by the word happiness.

Kagan rejects the practice of reporting the measurement of complex psychological constructs using numbers: “The contrasting view, held by Whitehead [co-author of the Principia Mathematica] and Wittgenstein, insists that every description should refer to … the circumstances of the observation.”  The reason for including a description of the measuring instrument isn’t difficult to see.  Kagan points out that “Most investigators who study “anxiety” or “fear” use answers on a standard questionnaire or responses to an interview to decide which of their subjects are anxious or fearful.  A smaller number of scientists ask close friends or relatives of each subject to evaluate how anxious the person is.  A still smaller group measures the heart rate, blood pressure, galvanic skin response, or salivary level of subjects.  Unfortunately, these three sources of information rarely agree.”

 

Given that a change in the measuring tool means a change in the reported measurement, one must include a description of the measuring instrument in order to “communicate unambiguously,” as Bohr expressed it.  One can never simply write “happiness = 4.29” (as in Pisa tables) because there is no such thing as an instrument-independent measure of happiness.  We have no idea what happiness is as a thing-in-itself.  Kagan notes the implications for psychologists of the measurement principles set out by Niels Bohr: “Modern physicists appreciate that light can behave as a wave or a particle depending on the method of measurement.  But some contemporary psychologists write as if that maxim did not apply to consciousness, intelligence, or fear.”  According to Bohr, when one reports psychological measurements, the requirement to describe the measurement situation means that ordinary language must replace numbers.  Werner Heisenberg summarised his mentor’s teachings: “If we want to say anything at all about nature – and what else does science try to do – we must pass from mathematical to everyday language.”

 

(To simplify matters somewhat, while numbers function perfectly well when observing the motion of a tennis ball or a star, the psychologist cannot observe directly the pupil’s happiness.  Bohr argued that there was “a deep-going analogy” between measurement in quantum physics and measurement in psychology because both were concerned with measuring constructs which transcend the limits of ordinary experience.  According to Bohr, because the physicist, like the psychologist (in respect of attempts to measure happiness), cannot directly experience electrons and photons, “physics concerns what we can say about nature,” and numbers must therefore give way to ordinary language.)

 

The arguments advanced above apply, without modification, to Pisa’s core activity of measuring pupil ability.  A simple thought experiment (first reported in the TES of 26.07.2013) makes this clear.  Suppose that a pupil is awarded a perfect score in a GCSE mathematics examination.  It seems sensible to conjecture that if Einstein were alive, he too would secure a perfect score on this mathematics paper.  Given the title on the front page of the examination paper, one has the clear sense that the examination measures ability in mathematics.  Is one therefore justified in saying that Einstein and the pupil have the same mathematical ability?

 

This paradoxical outcome results from the erroneous treatment of mathematical ability as something entirely divorced from the questions which make up the examination paper.  It is clear that the pupil’s mathematical achievements are dwarfed by Einstein’s; to ascribe equal ability to Einstein and the pupil is to communicate ambiguously.  To avoid the paradox one simply has to detail the measurement circumstances in any report of attainment and say: “Einstein and the pupil have the same mathematical ability relative to this particular GCSE mathematics paper.”  By including a description of the measuring instrument one is, in effect, making clear the restrictive meaning which attaches to the word “mathematics” as it is being used here; school mathematics omits whole areas of the discipline familiar to Einstein such as non-Euclidean geometry, tensor analysis, vector field theory, Newtonian mechanics, and so on.

 

As with the measurement of happiness, when one factors in a description of the measuring instrument, the paradox dissolves away.  Alas for Pisa, the move from numbers to language also dissolves away that organisation’s much-lauded rank orders.  Little wonder that Wittgenstein described the reasoning which underpins the statistical model (Item Response Theory) at the heart of the Pisa rankings as “a disease of thought.”

 

This brings us to the very serious implications for the new GCSE grade 5, of the arguments set out above.  The fact that a switch from numbers to language invalidates entirely the practice of ordering countries according to the efficacy of their education systems has profound implications for the validity of claims made concerning the new GCSE grade 5.  Given the assertion that grade 5 reflects the academic standards of high performing international jurisdictions as identified by their Pisa ranks, what possible justification can be offered for assigning a privileged role to the GCSE grade 5 in school performance tables?

 

To date, Pisa rankings have not impacted directly on the life chances of particular children in this country.  This would change if individual pupils failing to reach the grade 5 standard were construed as having fallen short of international standards (whatever that means).  If one accepts the reasoning of Wittgenstein and Bohr, grade 5 can represent nothing more than a standard somewhere between grade 4 and grade 6.  Any attempt to accord it special status, thereby giving it a central role in the EBacc and/or performance tables, risks exposing the new GCSE grading scale to ridicule.

Dr Hugh Morrison, The Queen’s University of Belfast (retired)

 

 

 

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Newsletter suspected of squandering Transfer Test exclusive for political motive

Below is the text of an article submitted by The Parental Alliance for Choice in Education to the Belfast Newsletter on Monday 20th February. It is worthwhile noting that a senior Newsletter journalist spent two and a half hours with the author trying to come up with reasons not to publish a story that the following Monday would be dominating the BBC Northern Ireland newscasts and a two page spread in the Belfast Telegraph.

Peter Weir, DUP Education Minister was always intended to be the beneficiary of a claim that AQE & PPTC were making progress towards a single transfer test.Weir’s appointee Professor Peter Tymms had suggested in his report, made available to the BBC, but not the public, that three 11+ plus transfer tests be taken on one day.

Think about the stressful implications of this suggestion before you cast a vote in the Assembly Election today

THe text of the original unpublished letter.

Professor Peter Tymms of Durham University and his team were engaged by the DUP Education Minister, Peter Weir, to explore the possibilities for a single transfer test to replace the current AQE/GL hybrid.  The AQE test is used by “state” grammar schools, while the GL test is used – in the main – to determine admission to Catholic grammars.  The Tymms report has serious implications for parents intending to send their children to a state grammar.

If the Tymms report is implemented, the current AQE arrangement of three tests taken on different days (with the best two used to determine the published AQE score) will be replaced by three tests taken on a single Saturday.  Multiple choice grids will replace the AQE practice of the child simply writing his or her answer on the test paper.  It would also appear that children seeking a place at a state grammar school will no longer have access to past papers.

In 2006, the Times Higher Education reported that Professor Tymms’ research had attracted criticism from Prime Minister, Secretary of State for Education, and the head of the Office for Standards in Education.  The Times Higher Education highlighted David Blunkett’s counsel that no one “with the slightest common sense” could possibly take seriously research by Peter Tymms.

Professor Tymms had a leadership role in the design of CCEA’s ill-fated InCAS tests.  To quote the Belfast Telegraph of 05.01.12: “The Department of Education has confirmed that the InCAS contract, which expires on January 19, has not been renewed.  InCAS was administered at a total cost of £3 million by the University of Durham’s Centre for Evaluating and Monitoring (CEM).”

In his report for Peter Weir, Professor Tymms and colleagues seem to draw on modern psychometrics ( the field in psychology and education that is devoted to testing) to somehow justify the preferred model of three tests on the same day.  Joel Michell has devoted much of his professional life to an in-depth analysis, presented in books and peer-reviewed articles, of psychometrics.  He comes to the conclusion that psychometrics is “a pathological science.”

One simple illustration of the validity of Michell’s claim is that psychometrics treats the central concept of “ability” as a state.  Surely ability is a capacity rather than a state?  If someone at the Department of Education or CCEA had been alert to this curious interpretation of ability in the standard measurement model, maybe £3 million wouldn’t have been lost to the public purse.

Getting down to practicalities, Peter Weir must, without delay, tell voters whether he accepts or rejects the model proposed in the report compiled by Professor Tymms and his colleagues.  After all, these academics were paid from the public purse; the report should be public property.

The AQE test has functioned without error since 2009.  It has strikingly high approval ratings from parents, and, most importantly, 40 to 50% of pupils on Free School Meals who take the AQE test, secure scores which will admit them to grammar school.

Those thousands of parents who wish to send their child to a state grammar must know, without further delay, the DUP’s unequivocal position on this important matter.

 

Stephen Elliott

 

 

Peter Tymms misunderstands the nature of measurement in psychology and education

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Why Peter Tymms’ grasp of the fundamentals of measurement in psychology/education disqualifies him from any role in determining the future of transfer testing in Northern Ireland.

Dr Hugh Morrison

The case that Professor Tymms misunderstands the nature of measurement in psychology and education

Professor Peter Tymms is a long-time proponent of the central role that latent variables play in modern psychometrics.  The Item Response model advanced by Georg Rasch has an important place in his research.  I will argue in what follows that those who advance Item Response Theory approaches in general, and Rasch modelling in particular, have failed to understand the true complexity of the central predicate “ability.”  Wittgenstein stressed that ability is something potential, a capacity rather than a state.  Individuals are carriers of potentiality and not states.  Psychology is concerned interactions and not the intrinsic properties of the relata involved; relations have definite properties while the relata themselves are indefinite.  Peter Tymms has failed to appreciate the indeterminacy of the mental.

Measurement in psychology/education is never a process of “checking up” on what is already in the mind/brain of the individual.  Rather, unlike measurement in Newtonian physics, the act of measuring transforms a potentiality to a definite state.  Measurement in psychology and education should not be concerned with what ability is, but must settle for what can be said about ability.  Michell is right to claim that psychometrics is “pathological science,” and that measurement in psychology is “at best speculation and, at worst, a pretence at science.”  Trendler’s (2011) claim that measurement theorists should abandon all attempts to repair psychometrics is surely justified.  All proponents of Item Response Theory, including Professor Tymms, subscribe to the “reflective model” in which variation in the latent variables is viewed as prior to variation in the manifest variables.  Alas, the reverse is true.

Borsboom, Mellenbergh & van Heerden (2003, p. 217), writing in one of psychology’s most respected journals, highlight the incoherence of this entire approach to measurement: “It will be felt that there are certain tensions in this article.  We have not tried to cover these up, because we think they are indicative of some fundamental problems in psychological measurement and require clear articulation. … And although the boxes, circles, and arrows in the graphical representation of the model suggest that the model is dynamic and applies to the individual, on closer scrutiny no such dynamics are to be found.  Indeed, this has been pinpointed as one of the major problems of mathematical psychology by Luce (1997): Our theories are formulated in a within-subjects sense, but the models we apply are often based solely on between-subjects comparisons.”

Item Response Theory omits entirely the human practices (reading, arithmetic, and so on) into which the child is enculturated by teachers and parents.  This is the all-important “environment” in which the child participates, an environment which Item Response Theory is powerless to represent.  Item Response Theorists posit “abilities” hidden in the mind/brain which are the source of the child’s test responses.  However, the Nobel laureate Herbert Simon dismissed such reasoning: “Human rational behaviour is shaped by a scissors whose blades are the structure of task environments and the computational capabilities of the actor.”  The scissor metaphor is a reference to Alfred Marshall’s puzzlement over which scissor blade actually cuts a piece of cloth – the top blade or the bottom.  The lesson for psychometrics is that omitting the environment of academic practices in which the child participates will produce nonsense.

One can find the source of Item Response Theory’s difficulty in Niels Bohr’s 1949 paper entitled Discussion with Einstein on Epistemological Problems in Atomic Physics.  Few scientists have made a greater contribution to the study of measurement than the Nobel laureate and founding father of quantum theory, Niels Bohr.  Given Bohr’s preoccupation what the scientist can say about aspects of reality which are not visible (electrons, photons, and so on), one can understand his constant references to measurement in psychology; “ability” cannot be seen directly, rather, like the microentities that manifest as tracks in particle accelerators, ability manifests in the individual’s responses to test items.  Assessment is concerned with “measuring” something which the measurer cannot experience directly, namely, the ability of the examinee.

Quantum theory and psychology have not shown the same willingness to acknowledge the limitations of measurement in their respective disciplines.  While physics has made no attempt to disguise its “measurement problem” (it is acknowledged in every undergraduate textbook), Michell (1997) has accused psychometricians of simply closing down all debate on measurement and suffering from a “methodological thought disorder.”  Michell’s concerns about the reluctance of psychometricians to engage in debate about the fundamentals of measurement, when set alongside the clear acknowledgement of a measurement problem in physics, bring to mind the words of the French moralist Joseph Joubert: “It is better to debate a question without settling it than to settle it without debating it.”

Item Response Theory relies on a simple inner/outer picture for its models to function.  The inner (a realm of timeless, unobserved latent variables, or abilities) is treated as independent of the outer (here examinees write or speak responses at moments in time).  This is often referred to as a “reservoir” model in which timeless (hidden) abilities are treated as the source of the individual’s (public) responses given at specific moments in time.

As early as 1929 Bohr rejected this simplistic thinking in strikingly general terms: “Strictly speaking, the conscious analysis of any concept stands in a relation of exclusion to its immediate application.  The necessity of taking recourse to a complementary … mode of description is perhaps most familiar to us from psychological problems.”  Now what did Bohr mean by these words?  Consider, for example, the concept “quadratic.”  It is tempting to adopt a reservoir approach and trace a pupil’s ability to apply that concept in accord with established mathematical practice to his or her having the formula in mind.  The guidance offered by the formula in mind (Bohr’s reference to “conscious analysis”) accounts for the successful “application,” for example, to the solution of specific items on an algebra test.

However, this temptingly simplistic model in which the formula is in the unobserved mental realm and written or spoken applications of the concept “quadratic” take place in the observed public realm, contains a fundamental flaw; the two realms cannot be meaningfully linked up.  The “inner” formula (in one realm) gets its guidance properties from human practices (in the other realm).  A formula as a thing-in-itself cannot guide; one has to be trained in the established practice of using the formula before it has guidance properties.  In school mathematics examinations around the world, pupils are routinely issued with a page of formulae relevant to the examination.  Alas, it is the experience of mathematics teachers everywhere that simply having access to the formula as a thing-in-itself offers little or no guidance to the inadequately trained pupil.  The formula located in one realm cannot connect with the applications in the other.

Wittgenstein teaches that no formula, rule, principle, etc. in itself can ever determine a course of action.  The timeless mathematical formula in isolation cannot generate all the complexities of a practice (something which evolves in time); rather, as Michael Oakeshott puts it, a formula is a mere “abridgement” of the practice – the practice is primary, with the formula, rule, precept etc. deriving its “life” from the practice.

Returning to Bohr’s writing, it is instructive to explain his use of the word “complementarity” in respect of psychology and to interpret the meaning of the words: “stands in a relation of exclusion.”  Complementarity is the most important concept Bohr bequeathed to physics.  It involves a combination of two mutually exclusive facets.  In order to see its relevance to the validity of IRT modelling, let’s return to the two distinct realms.

We think of the answers to a quadratic equation (of course, a typical school-level quadratic equation has two distinct answers) as being right or wrong.  In the realm of application this is indeed the case; when the examinee is measured, his or her response is pronounced right or wrong dependent upon its relation to established mathematical practice.  However, in the unobserved realm, populated by rules, formulae and precepts (as things-in-themselves), any answer to a quadratic equation is simultaneously right and wrong!

A formula as a thing-in-itself cannot separate what accords with it from what conflicts with it, because there will always exist an interpretation of the formula for which a particular answer is correct, and another interpretation for which the same answer can be shown to conflict with the formula.  Divorced from human practices, the distinction between right and wrong collapses.  (This is a direct consequence of Wittgenstein celebrated “private language” argument.)  This explains Bohr’s reference to a “relation of exclusion.”  In simplistic terms, the unobserved realm, in which answers are compared with the formula for solving quadratics, responses are right-and-wrong, while in the observed realm, where answers are compared with the established practice, responses are right-or-wrong.

On this reading, ability has two mutually exclusive facets which cannot meaningfully be separated.  The distinguished Wittgenstein scholar, Peter Hacker (1997, p. 250), captures this situation as follows: “grasping an explanation of meaning and knowing how to use the word explained are not two independent abilities but two facets of one and the same ability.”  Ability, construed according to Bohr’s complementarity, is indefinite when unobserved and definite when observed.  Moreover, this definite measure is not an intrinsic property of the examinee, but a property of the examinee’s interaction with the measuring tool.  According to complementarity, the “inner” and the outer are not two separate localities which somehow connect.  As Herbert A. Simon realized, one cannot dispense with either of Hacker’s facets and hope to construe ability correctly; both are vital to the predicate “ability.”  Whitaker’s (1996, p. 184) definition of complementarity captures this situation: “mutual exclusiveness and joint completion.”

Measurement of ability is not a matter of passively checking up on what already exists – a central tenet of Item Response Theory.  Bohr teaches that the measurer effects a radical change from indefinite to definite.  Pace Item Response Theory, measurers, in effect, participate in what is measured.  No item response model can accommodate the “jump” from indefinite to definite occasioned by the measurement process.  All IRT models mistakenly treat unmeasured ability as identical to measured ability.  What scientific evidence could possibly be adduced in support of that claim?  No Item Response model can represent ability’s two facets because all such models report ability as a single real number, construed as an intrinsic property of the measured individual.

Finally, in order to highlight the incoherence of the type of measurement model advocated by Peter Tymms, it is instructive to consider a thought experiment in which a primary school child responds to the addition problem: “68 + 57 = ?”  In the appendix below the erroneous thinking of the psychometrician is adopted in that the child’s ability is considered to be a mental state which is the source of his or her response.  It is demonstrated that all of the facts about the child (his or her complete history of responses to addition problems and complete information about the contents of his or her mind) are in keeping with the answer “68 + 57 = 125.”  Unfortunately, all of the facts are also in keeping with the answer “68 + 57 = p” where p is ANY number; someone with complete information has to conclude that the child is right and wrong at the same time.

Appendix

Setting the Scene

Consider the simplest of measurement situations encountered in psychology and education.  How can one establish if a student in the early years of formal schooling understands how to use the “+” sign?  The student whose grasp of the “+” sign is being measured has been taught to add, but has not yet encountered the problem “68 + 57 = ?” This problem has been selected at random but the argument generalises to any rule-governed activity (Kripke, 1982).  The measurement situation is broken into two phases: the phase immediately before the student answers and the phase during which the answer is spoken or written.

This offers two perspectives on the student’s understanding of the “+” sign.  The idea that understanding how to use the “+” sign is an “inner” mental state, activity or process has enormous appeal.  The temptation to reason that the student’s first-person perspective on his or her understanding is superior to the measurer’s third-person perspective is almost irresistible, since the measurer has to settle for mere behaviour.  It is difficult to escape the impression that the student has privileged access to his or her grasp of the meaning of “+,” because to mean is surely to have something in mind? (Putnam, 1988).  On the other hand, the measurer must settle for the mere manifestations of that understanding.  The student seems to have first-person direct access to the thing-in-itself, namely, his or her understanding of “+” while the third-person perspective is associated with indirect access.  The third-person perspective involves observation of the student exercising his or her understanding rather than the understanding itself; it would seem that the student alone can “observe” understanding because it is a mental process.  This enticing simple Cartesian picture of the “inner world” is clearly compelling.

This idealised measurement situation will be used to argue that, before the student answers the problem “68 + 57 = ?” (immediately prior to measurement) the totality of facts about the student are in keeping with the student intending to give the right answer and with the intention to give one of an indefinite number of wrong answers.  It is established that the state “understands the ‘+’ sign” and the state “doesn’t understand the ‘+’ sign” both can be simultaneously ascribed to the unmeasured student.  In short, it is meaningless to assign a definite grasp of the “+” sign to an unmeasured individual.

In quantum theory, unmeasured quantum entities are characterised by superpositions which are portrayed as being “here” and “there” simultaneously.  For unmeasured quantum entities, the notion of a definite location is unintelligible.  However, when the quantum entity is measured it assumes a definite position and is characterized as either here or there.  Consider the measurement of an individual’s ability to respond to the addition problem: “68 + 57 = ?”  The case will be made that when the psychologist focuses on the unmeasured ability of the individual, all the facts about that individual can be shown to be in keeping with the individual being both “right” and “wrong” (with respect to the addition problem) at the same time.  It will be demonstrated that, immediately prior to the act of measurement, an individual’s understanding of the “+” sign is entirely indeterminate, with the categories “right” and “wrong” being applicable simultaneously.

It will be argued that someone with complete information about the student’s past achievements in addition, together with complete information about his or her mental states, would find it impossible to use this information to predict the student’s answer to any simple addition problem in an infinity of cases.  It will be demonstrated below that before a measurement is made – for example, before a student says or writes the answer to the question “68 + 57 = ?” – at that moment, all of the known facts about the student are in keeping with the correct answer “125” and an incorrect answer – “5,” – for example.  A rule as a thing-in-itself can never determine an action.  The student’s mathematical ability with respect to the question “68 + 57 = ?” is completely indeterminate prior to the statement of the answer.

At the moment prior to answering the question “68 + 57 = ?” all of the facts about the student accord with a correct response and an infinity of incorrect responses.  Before the act of measurement, if one restricts oneself to the totality of facts about the individual (inner and outer), then the notion of accord or conflict with a mathematical rule breaks down entirely.  Clearly, in conditions where responses can be deemed simultaneously right and wrong, the very notion of correctness has become unintelligible.  While the student is characterised as both right and wrong with respect to the question

“68 + 57 = ?” immediately before he or she responds, at the instant of responding the student is deemed correct if the answer is 125, and incorrect if the student answers 5.  Immediately before answering the student is right and wrong.  The moment the answer is articulated the student is right or wrong.  In short, measurement isn’t a matter of checking up an existing attribute (as in Newtonian physics); measurement effects radical change.

Having set the scene for what is to come, the case will now be made that it is meaningless to ascribe a definite ability to an unmeasured individual; the ascription of a definite ability is only meaningful in a measurement context.  The idea that psychological measurement owes more to quantum measurement principles than to Newtonian mechanics depends on this case being made.  This is achieved by calling on Wittgenstein’s later philosophy and the remainder of this section is given over entirely to this single task.  Wittgenstein’s writings on first-person ascription of ability are essential to developing a measurement model with the individual at its core.

No Rule or Formula can Determine its own Continuation

An idea that Anscombe (1985) traces back to Leibniz (1646-1716) is instructive for preparing the reader for Wittgenstein’s insights into the role rule-following plays in thinking about psychological measurement.  Leibniz noticed that no formula or rule can fix its own continuation: any number can be regarded as the correct continuation of a rule on some interpretation.  He pointed out that an indefinite number of rules are consistent with any finite segment of a series.  Anscombe illustrates Leibniz’s thinking using the extension of a simple series such as ‘2, 4, 6, 8, …’

[A]lthough an intelligence tester may suppose that there is only one possible continuation to the sequence 2, 4, 6, 8, … , mathematical and philosophical sophisticates know that an indefinite number of rules (even rules stated in terms of mathematical functions as conventional as polynomials) are compatible with any such finite initial segment.  So, if the tester urges me to respond, after, 2, 4, 6, 8, … , with the unique appropriate next number, the proper response is that no such unique number exists. … The intelligence tester has arbitrarily fixed on one answer as the correct one. (Anscombe, 1985, pp. 342-343)

Consider the series completion problem Anscombe (1985) proposes.  The student is presented with the first four terms of an infinite series: 2, 4, 6, 8 …  He or she is then required to “go on in the same way” by the teacher.  An infinite number of formulations will generate the four numbers 2, 4, 6 and 8 but differ on the fifth term (and all terms thereafter).  For example, the formula:

Un = 2n – (1/24)(n – 1)(n – 2)(n – 3)(n – 4)

generates: 2, 4, 6, 8, 9, …

while

Un = 2n + 45(n – 1)(n – 2)(n – 3)(n – 4)

generates  2, 4, 6, 8, 1090, …

and finally

Un = 2n – 3(n – 1)(n – 2)(n – 3)(n – 4)

generates  2, 4, 6, 8, -62, … .

 

In summary, an indefinite number of different continuations can be shown to accord with any finite segment of an arithmetical series.  By careful selection, any number can be offered for the fifth term of the series.  One’s immediate reaction to the final series given above is that, in writing -62, the student has made a mistake.  In writing 2, 4, 6, 8 the student is following the correct rule but it appears that in writing -62 he or she has erroneously switched to a new rule.  But it is also possible that the student acted consistently throughout, always applying the same formula, namely,

Un = 2n – 3(n – 1)(n – 2)(n – 3)(n – 4)

to generate all five terms.  This (albeit highly unusual) student could rightly claim to be “going on in the same way” when he or she wrote down -62 as the fifth term.

The student’s claim that he or she was simply continuing the rule exhibited by the first four terms is completely defensible since there are an infinite number of rules which begin ‘2, 4, 6, 8’ but diverge on the next term and all terms thereafter.  It can be claimed that the student did continue in the same way but the student’s way was at odds with the teacher’s intention when the teacher instructed the student to “go on in the same way.”  Unfortunately, “Finite behaviour cannot constrain its interpretation to within uniqueness” (Wright, 2001, p. 98), so what makes the student’s continuation wrong and the teacher’s right?

Wittgenstein’s writings on rule-following do not, for a moment, imagine that real children in real classrooms extend this series of four even numbers as “2, 4, 6, 8, 1090, …” or “2, 4, 6, 8, -62,” for example.

It is a conspicuous feature of these case-histories that the misunderstandings are often widely improbable, and we may wonder why this is so.  Evidently, the reason cannot be that Wittgenstein believed that such extreme misunderstandings are at all likely or that a teacher would need to guard against them in real life.  So what is the explanation of his preoccupation with improbable misunderstandings? Wittgenstein’s point is not that such misunderstandings are probable, but only that they are possible.  They are possible because, if the lesson only proceeds by examples, there will always be many different specifications of the meaning of the word that are satisfied by any finite sequence of examples, and so the student can always pick a specification that was not intended by the teacher.  However, if the lesson has been well designed with carefully chosen examples, there will only be one natural way of interpreting them – or perhaps there will be minor variations, to be excluded by further examples.  If, on the other hand, the teacher tries to close the gap by offering a definition of a problematic word, the words used in the definition will present the same problem again. (Pears, 2006, p. 18)

“The idea here is that instructions for following a rule underdetermine the correct way to follow the rule … if we consider instructions and explanations as involving the provision of a finite number of examples then there are indefinitely many compatible functions or ways of going on from those examples” (Panjvani, 2008, p. 307).  Schroeder (2006, p. 189) concludes: “So, any rule, even the most explicit one, can be misunderstood; and in endless ways too: whichever way the student continues the series, his writing can always be regarded as in accordance with the rule – on a suitable interpretation.”

This problem extends beyond mathematics to all rule following.  Bloor (1997, p. 10) stresses that: “This does not just apply to number sequences.  Teaching someone the word ‘red’ is, in a sense, teaching them the rule for using the word.  This too involves moving from a finite number of examples to an open-ended, indefinitely large range of future applications”.  The problems associated with infinite rules also apply to rules with a finite number of applications.  Kripke (1982, p. 7) comments: “Following Wittgenstein, I will develop the problem initially with respect to a mathematical example, though the relevant sceptical paradox applies to all meaningful uses of language.”  Finally, McGinn (1997, p. 77) notes: “Nor is this problem restricted to the mathematical case.  For any word in my language, we can come up with alternative interpretations of what I mean by it that are compatible with both my past usage and any explicit instruction that I might have given myself.”

Every teacher is aware that students do succeed in moving from a finite set of examples to apply the rule in new cases and so it is tempting to conclude that successful understanding somehow results in the student having in mind a rule which accords with the examples but somehow transcends them.  Furthermore, it would seem that the student’s ability to apply the rule beyond the finite example set is explained by positing that the student, when confronted with novel applications, is guided by the rule that he or she has “in mind.”  Bloor (1997, p. 11) summarises this seductively simple model of how rules are grasped:

It is tempting to suppose that when a teacher is using examples to convey the meaning of a word, the teacher has something ‘in mind,’ and the finite number of examples are just a fragmentary substitute for what is really meant.  If only the student could look directly into the mind of the teacher then how simple life would be: they too would have access to the state of understanding that is the source of the teacher’s ability to follow the rule.  Of course, they can’t look into the teacher’s mind but, the argument goes, they will only have established an understanding when they can reach beyond all the examples. (Bloor, 1997, p. 12)

 

‘Understanding’ is a Vague Concept (Wittgenstein, 1983, VI, §13)

Consider the case of simple addition of two natural numbers.  Almost everyone who has been to school for a few years is confident that they have understood the meaning of the sign “+.”  They feel sure they understand how to use the rule for the “+” sign.  This confidence exists despite the fact that no individual has carried out the infinitely long list of computations associated with the “+” sign.  Given that individuals live for only a finite number of years and there are an infinite number of natural numbers x and y for which x + y can be computed, it follows that for any individual it will always be possible to identify values of x and y for which they have not yet computed x + y.

The student in the early-years of primary school, and just coming to terms with addition, may have computed x + y for all values of x and y less than 57, for example, so that the computation “68 + 57”, say, is not part of the student’s short computational history.  The calculation “68 + 57” would be a novel computation for this student.  The point to be emphasised is that for every individual it will always be possible to identify an addition problem that this individual has not previously encountered.

Kripke (1982) considers the case of a student who has not yet performed the computation “68 + 57.”  The student has mastered additions with smaller arguments such as “23 + 34,” “19 + 27” and “50 + 51” but hasn’t carried out the calculation “68 + 57.”  Kripke (1982) conjures up a “bizarre sceptic” who suggests to the student that, based on the examples the teacher used when instructing him or her to add and all the addition questions the student has completed to date, he or she should now give the answer “5” in response to the question “68 + 57 = ?”  Kripke (1982, p. 8) has his bizarre sceptic pose a simple question:

This sceptic questions my certainty about my answer … Perhaps, he suggests, as I used the term ‘plus’ in the past, the answer I intended for ’68 + 57’ should have been ‘5’!  Of course the sceptic’s suggestion is obviously insane.  My initial response to such a suggestion might be that the challenger should go back to school and learn to add.

The sceptic’s case is based on the fact that the student (in keeping with the rest of humanity) has only ever carried out a finite number of computations.  Needless to say, seeing off the claim that, given the student’s computation history, he or she should answer “5” to the question “68 + 57 = ?” the student will claim that the function he or she associated with “+” sign in all past computations requires that the answer “125” be given.  But Kripke (1982, pp. 8-9) points out that the finite set of addition problems completed in the past can be variously interpreted:

But who is to say what function this was?  In the past I gave myself only a finite number of examples instantiating this function.  All, we have supposed, involved numbers smaller than 57.  So perhaps in the past I used ‘plus’ and ‘+’ to denote a function I will call ‘quus’ and symbolise by ‘Å.’  It is defined by:

x Å y = x + y   if x, y < 57

x Å y = 5                     otherwise

Who is to say that this is not the function I previously meant by ‘+’?

The individual’s difficulty in seeing off the sceptic is that all of his or her past computations are for values of x and y less than 57 and for these values ‘+’ and ‘Å’ yield identical values.  It is only for values of x and y greater than or equal to 57 that differences occur.  It is only for numbers greater than or equal to 57 that addition and quaddition (someone using the quus function is said to be engaged in quaddition) give different results.

Kripke (1982) points out that the facts of interest to the sceptic are to be found in two distinct realms: the “outer” realm of past computations, and the “inner” realm of the mind.  If all the facts from the student’s history of past computations are consistent with the function ‘plus’ and with the function ‘quus,’ then maybe an examination of the student’s mental history (facts about the contents of the student’s mind) will decide whether he or she should answer “125” or “5” in order to be consistent with his or her past history of computation.  Kripke (1982) permits the individual responding to the sceptic’s challenge to have unlimited and infallible access to past computations (outer) and past mental states and processes (inner). Kripke (1982) frequently makes reference to what an omnipotent, omniscient, all-seeing God – who has access to every aspect of the individual’s computational history and to his or her thought processes – would see if He were to look into the student’s mind.

The evidence is not to be confined to that available to an external observer, who can observe my overt behaviour but not my internal mental state.  It would be interesting if nothing in my external behaviour could show whether I meant plus or quus, but something about my inner state could.  But the problem here is more radical. … whatever ‘looking into my mind’ may be, the sceptic asserts that even if God were to do it, he still could not determine that I meant addition by ‘plus.’ (Kripke, 1982, p. 14)

Kripke (1982) argues that all of the facts about the student – all the (outer) facts about the student’s computational history and all the (inner) facts about the student’s mental states and processes – are consistent both with that individual understanding the orthodox addition function and the contrived “quus” function by “+” sign.

The sceptic does not argue that our own limitations of access to the facts prevent us from knowing something hidden.  He claims that an omniscient being, with access to all available facts, still would not find any fact that differentiates between the plus and the quus hypotheses. (Kripke, 1982, p. 39)

The temptation, of course, is to accept that while any finite series of computations can be made to accord with both ‘plus’ and ‘quus’ functions, an examination of the individual’s mind would turn up a fact or facts that would discriminate between the functions.  It is instructive to illustrate how the sceptic refutes such arguments.

When individuals learn to add, the following notion has considerable appeal: if they grasp the meaning of the “+” sign then this understanding of the addition rule fixes unique responses to addition problems they will encounter in the future.  The notion of understanding seems to function in “contractual” terms (McDowell, 1998, p. 221).  Given the student’s past grasp of the “+” sign and history of successful problem-solving in respect of that symbol, he or she seems contracted to reply in accord with this understanding.  In this model, to understand is to have something in mind which is the source of subsequent behaviour in that it contracts the individual who understands addition to reply “125” to the question “68 + 57 = ?”

But what makes the individual’s understanding with respect to the ‘+’ sign, a grasp of adding rather than quadding?  Consider an individual who attaches the aberrant interpretation (quus) to his or her past computations.  If the student’s understanding of the “+” sign contracts him or her to use it according to the quaddition rule then the individual should reply “5” when asked to compute “68 + 57.”  In this case, to reply “125” is to fail to go on in the same way.  But what fact about the student’s understanding could be produced to convince the sceptic that he or she is contracted to follow the quaddition rule rather than the addition rule?  What fact about a student’s past grasp of the ‘+’ sign makes his or her present response of “5,” for example, right or wrong.  The finite number of additions the individual has completed to date – all involving arguments less than 57 – are consistent with understanding the ‘+’ sign in terms of the addition function and the quaddition function.

For the sceptic can point out that I have only ever given myself a finite number of examples manifesting this function, which all involved numbers less than 57, and that this finite number of examples is compatible with my meaning any one of an infinite number of functions by ‘+.’  … If the sceptic is right, then there is no fact about my past intention, or about my past performance, that establishes, or constitutes, my meaning one function rather than another by ‘+.’ (McGinn, 1997, p. 76)

Having a Formula Before one’s Mind

The common-sense view that understanding is a process or activity which happens in the mind has enormous appeal.  The notion that to understand is to have something in mind and that this understanding somehow fixes future behaviour in respect of how that understanding is exercised seems beyond challenge.  At the same time, it also seems obvious that the future behaviour referred to is somehow inferior to understanding as a thing-in-itself; understanding is the real thing, whereas behaviour is merely a particular manifestation of that understanding.  Understanding is construed as “inner” while behaviour is construed as outer.  Few dispute the thesis that while the individual somehow has direct access to his or her understanding, the person measuring that understanding, for example, has to settle for indirect access in the form of the individual’s behaviour.

First-person access to understanding seems superior to mere third-person access.  Furthermore, the inner and outer seem to be entirely independent realms; after all, someone who has understood addition has the clear sense that he or she will provide the correct answer to the question “2 + 2 = ?” in advance of writing or saying their answer.  This person doesn’t have to wait until they’ve responded in order to confirm their understanding to themselves.  Understanding (viewed as a property of the inner), in this instance, at least, seems quite independent of the subsequent behaviour in which that understanding is exercised.

And this seems obvious for, to be sure, other people must rely on my behaviour, on what I do and say, in order to discern what I am feeling or thinking.  So, it seems that they know how things are with me indirectly.  What they directly perceive is merely outward behaviour.  But I have direct access to what is inner, to my own mind.  I am conscious of how things are with me.  The faculty whereby I have such direct access to mental states, events and processes is introspection – and it is because I can introspect that I can say how things are with me without observing what I do and say. (Bennett & Hacker, 2003, pp. 84-85)

Kripke (1982) considers the case of having a formula in mind which can be introspected.  Could this provide the elusive fact that distinguishes the individual’s understanding of “+” as a grasp of the addition function rather than a grasp of the quaddition function?  Could this be the “inner fact” that determines “125” as the answer the student should give to “68 + 57 = ?” in order to keep faith with the student’s understanding of “+”?  This is an attractive option because one can conceive of a formula as something finite but which nevertheless has the capability of generating an infinity of responses.  If God were to look into the individual’s mind and spotted, say, the quus function defined above, He could predict with certainty that the individual is contracted to reply “5” to the question “68 + 57 = ?” in order to keep faith with his or her understanding.  One can also appreciate the popular appeal of this approach for one does feel that in solving quadratic equations, for example, one “calls to mind” the quadratic formula.  One can almost “see” the formula in one’s mind’s eye when solving quadratic equations.

The case against the notion of a formula as the sought-after fact which distinguishes adding from quadding will now be set out.  Alas, having a formula in mind will not satisfy the sceptic because a mental representation of the formula in itself cannot determine the individual’s response to any given problem.  A student who has not been instructed in the solution of quadratic equations but who has merely memorised the formula for solving quadratic equations will not be able to use it to solve algebraic problems involving quadratic equations.  Merely having the formula in mind is not enough to determine use.  The student must be trained in the use of the formula; simply having access to the formula in itself – whether in mind or on paper – doesn’t fix the response the student makes when presented with a problem requiring the use of the quadratic formula.

It is the practice in school mathematics examinations around the world to issue a booklet of formulae to examinees.  It is the near universal experience of mathematics teachers that students with inadequate training in the solution of quadratic equations may derive little value from having access to the relevant formula booklet.  This is because the formulae in themselves don’t fix behaviour.  If the formula in isolation fixed the student’s response, then every student issued with a formula booklet in a mathematics test would answer the quadratic question correctly.  To underline the limitations of having formulae in mind, Wittgenstein explores the circumstances under which sign-posts offer guidance.  Wittgenstein (1953, §85) states that in the Cartesian picture “A rule stands there like a sign-post.”  In this statement he is asking the reader to reflect on the property of a wooden sign-post which enables it to serve as a guide to behaviour.

Considered in itself a sign-post is just a board or something similar, perhaps bearing an inscription, on a post.  Something so described does not, as such, sort behaviour into correct and incorrect – behaviour that counts as following the sign-post and behaviour that does not” (McDowell, 1992, p. 41).

One is forced to conclude that, despite having all the facts (inner and outer) about the student, these alone can’t determine in advance that his or her response will be “125” or “5.”  Causality has broken down.  A defining principle of Newtonian mechanics is that if one has complete information about any system one can always predict what will happen next with certainty.  Newtonian determinism fails in respect of elementary rule-following: it seems that there are matters which influence the student’s response which are beyond the totality of inner and outer facts.

A quantum physicist would feel entirely at home with Kripke’s (1982) way of expressing his interpretation of Wittgenstein’s philosophy: “If even God, who can see all the facts about the past (and into your mind), could not know that you meant addition then that doesn’t illustrate limitations on God’s knowledge.  It shows that there is in this case no fact for him to know” (Ahmed, 2007, p. 102).

Also, since one can look back upon the individual’s present understanding of “+” from a vantage point in the future, it follows that there is no (inner or outer) fact about what he or she currently understands by the “+” sign.  It follows that before this student is measured (before he or she responds to the question “68 + 57 = ?”) the totality of facts (inner and outer) are in keeping with the conclusion that the student’s state is a superposition of right and wrong.  The student is in an indefinite state with respect to his or her grasp of “+” because all the facts from the two relevant provinces (the inner and outer) are in keeping with the answers “125” and “5.”

Before continuing, it is important to realise that Wittgenstein does not deny that when solving mathematical problems one often has the sense that the relevant formula is “before one’s mind.”  Rather, he’s pointing out that this formula before one’s mind can’t be the source of one’s ability to solve the problems – it’s merely a by-product of one’s instruction in addition.  It will become clear in the paragraphs below that an introspected formula cannot fix how one solves problems requiring the use of the formula.

Wittgenstein is not here denying that there are characteristic experiential accompaniments to meaning and understanding – for images and the like do sometimes come before our minds when we utter or understand words – but he is denying that such experiential phenomena could constitute understanding.  Experiences are at most a symptom or sign of understanding; they are not the understanding itself.  The mistake of the traditional empiricist conception of meaning was thus to take as constitutive what is in reality only symptomatic. (McGinn, 1984, p. 4)

It should be stressed, once again, that this argument generalises to all rule-following: “Of course, these problems apply throughout language and are not confined to mathematical examples, though it is with mathematical examples that they can be most smoothly brought out” (Kripke, 1982, p. 19).  Ahmed (2007, p. 103) points out that “there is nothing special about ‘plus’ – if scepticism about ‘plus’ is irrefutable then so is scepticism about any word in any language.”

Having a Formula (and its Interpretation) Before one’s Mind

Kripke (1982) now searches for some mechanism in the student’s mind which will obviate the need to conclude that the unmeasured student is in an indefinite state.  He reasons that the formula in mind is just like a sign post – it cannot in itself fix the student’s response.   It is neither sufficient nor necessary to the student’s exercise of understanding.  As a consequence, it might be argued that having a formula in mind is of little value unless one is equipped to interpret it.  McDowell (1992, p. 41) makes the case using Wittgenstein’s sign-post metaphor:

What does sort behaviour into what counts as following the sign-post and what does not is not an inscribed board affixed to a post, considered in itself, but such an object under a certain interpretation – such an object interpreted as a sign-post pointing the way to a certain destination. (McDowell, 1992, p. 41)

Kripke (1982) is trying to explain how students follow the rule for the use of the ‘+’ sign given appropriate teaching and a finite number of illustrations of that rule.  The idea that they are guided by a mental image of a rule proves unworkable.  Could it be that one needs further evidence, namely, evidence that the student can interpret the rule correctly?  It follows that there must be evidence in mind that the student has correctly interpreted the rule.  However, this is of scant assistance, for the rule is capable of multiple interpretations.  It seems, therefore, that one needs to have a rule in mind for selecting the correct interpretation.  In order to explain the student’s ability to follow the rule for the use of the “+” sign, one must invoke a further rule – but this time the rule is in mind – for selecting the correct interpretation.  One has now fallen into an infinite regress.

An argument centred on the right interpretation will not work for the student must then have access to the rule for selecting this correct interpretation and one has a circular argument, because it is rule-following one is seeking to explain in the first place.  It instructive to remind the reader of the central idea here.  Facts are being sought about an individual which would determine in advance what response he or she should make to a novel problem which requires the use of a rule.  It has been decided that no outer facts fix what the individual does next because any response can be shown to accord with the rule exemplified by a finite set of illustrative examples offered by way of instruction.  The search then switched to facts about the individual’s mind.

If God looked into the individual’s mind and spotted a representation of the “quus” rule identified earlier then, at first sight, it seems He could predict with certainty that if asked the question “68 + 57 = ?” the individual must answer “5.”  The introduction of a finite entity in the mind (a mental representation of the formula) explaining the individual’s potentially infinite capacity for applying the rule is appealing until one notes that it is possible to have a formula in mind and yet not know how to apply it.  It is clear that the formula in itself is neither sufficient nor necessary.  It is “normatively inert” (McDowell, 1992, p. 42) because it cannot be used in isolation to pronounce the individual’s future response to the novel problem to be “125” or “5,” or any other number.  As McDowell puts it, the formula just “stands there” in need of interpretation.

McDowell (1992, p. 42) points out that attempting to locate the sought-after facts in the individual’s mind is fraught with problems because this is “a region of reality populated by items that, considered in themselves, just ‘stand there’.’’  Mental representations have to be interpreted.  McDowell argues that whatever attaching the correct interpretation to the formula might consist in, it is nevertheless an element of a region of reality (the mind) populated by items that just stand there like sign posts.  It follows that the interpretation itself has to be interpreted, and so on, in an infinite regress.

Wright (2001, pp. 162-163) makes such a powerful and pithy argument demonstrating the futility of interpretations that it is quoted here in full:

Suppose I undergo some process of explanation – for instance, a substantial initial segment of some arithmetical series is written out for me – and as a result I come to have the right rule ‘in mind.’  How, when it comes to the crunch – at the nth place which lies beyond the demonstrated initial segment, and which I have previously never thought about – does having the rule ‘in mind’ help?  Well, with such an example one tends to think of having the rule ‘in mind’ on the model of imagining a formula, or something of that sort.  And so it is natural to respond by conceding that, strictly merely having the rule in mind is no help.  For I can have a formula in mind without knowing what it means.  So – the response continues – it is necessary in addition to interpret the rule. … An interpretation is of help to me, therefore, in my predicament at the nth place only if it is correct. … So how do I tell which interpretation is correct?  Does that, for instance, call for a further rule – a rule for determining the correct interpretation of the original – and if so, why does it not raise the same difficulty again, thereby generating a regress?

Selecting the Simplest Rule

It may strike the reader that a criterion based on simplicity may distinguish understanding the “+” sign in terms of the addition rather the quaddition function.  The quus function, with its differing approach for numbers less than 57, and those greater than or equal to 57, seems a particularly unwieldy function when compared with the simple plus function.  Its mathematical symbolism would also be alien to any student in the primary phase of education.  Could it be that the student simply selects a unique interpretation (the “correct” interpretation) from the infinity on offer by simply choosing the interpretation with the simplest associated function?

Chaitin (2007) has extended Gödel’s incompleteness theorem (1931) and Turing’s halting problem (1950) to develop Algorithmic Information Theory.  He demonstrates that the search for the simplest rule (or most “elegant” rule in Chaitin’s parlance) which generates a sequence of numbers is equivalent to the search for the shortest computer program which can generate the sequence.  Unfortunately, Chaitin (2007, pp. 120-121) confirms that for any finite sequence of numbers, the identification of the simplest interpretation also presents intractable problems.

Let’s say I have a particular calculation, a particular output, that I’m interested in, and that I have this nice, small computer program that calculates it, and think that it’s the smallest possible program, the most concise one that produces this output.  Maybe a few friends of mine and I were trying to do it, and this was the best program that we came up with; nobody did any better.  But how can you be sure?  Well, the answer is that you can’t be sure.  It turns out you can never be sure!  You can never be sure that a computer program is what I like to call elegant, namely that it’s the most concise one that produces the output that it does.  Never, ever!

This paper details only some of Kripke’s (1982) attempts to escape the conclusion that if one restricts oneself to the totality of facts about the individual – outer facts about past practice and inner facts about mental contents (the two sets of facts treated as separately analysable) – one cannot predict the response the individual will make to the simple addition question: “68 + 57 = ?”  Since all the facts are in keeping with an infinity of answers, one correct and the rest incorrect, Kripke (1982, p. 17) is compelled to conclude that, in providing a response to this question, the rule-follower must be characterised as making “an unjustified stab in the dark.”  The student has no criterion for preferring 125 over 5; all the facts are in keeping with the correct answer and any incorrect answer.  The student is in an indeterminate state with respect to an understanding of the “+” sign.

So it seems that from a first-person perspective, individuals who have been taught to add using a finite number of examples, offer the first answer that comes into their heads when required to extend the addition rule to unseen computations.  They have no criterion which guides their selection of 125 as the correct answer to the problem “68 + 57 = ?”  There must be an error in the reasoning that produces such a counter-intuitive conclusion.  According to the logic presented above a student who, having been taught to add via a series of examples and then instructed to “go on in the same way,” subsequently encounters the addition problem “68 + 57 = ?” and responds by writing “5,” can protest that he or she did go on in the same way; the student just didn’t go on in the same as the teacher who issued the instruction.  There exists an interpretation which brings the answer “5” into accord with the teacher’s examples.  Indeed, this is true of any answer the student offers.  These two answers (125 and 5) and an infinity of other answers, are all in keeping with the totality of facts about the student.

Latent Variables Modelling Misrepresents Ability

The student’s interaction with the sceptic shows that the student has no criterion which can be used to differentiate a correct from an incorrect response.  The very concepts of right and wrong don’t seem to apply here.  This invites the obvious question: In respect of the problem “68 + 57 = ?” what makes “125” the correct answer and “5” the wrong answer?  What makes the teacher right in thinking the student should answer “125” and the student wrong in answering “5”?  This issue is resolved by bringing in the human practice of mathematics.  When we enter the picture, psychological measurement is bound to lose some of its Newtonian objectivity, an objectivity that quantum theory teaches is unattainable.

Heisenberg (1958, pp. 55-56) stresses that the cost of the participant’s inclusion is reduced objectivity in scientific measurement: The “reference to ourselves” means that “our description is not completely objective.”  Objectivity in the Newtonian sense is no longer the hallmark of science because it fails to account for the participative element of measurement.  Beyond classical physics, measurement models with no place for human practices are of questionable scientific validity: “When we speak of the picture of nature in the exact science of our age, we do not mean a picture of nature so much as a picture of our relationships with nature.  The old division of the world into objective processes in space and time and the mind in which these processes are mirrored … is no longer a suitable starting point for our understanding of modern science” (Heisenberg, 1962, pp. 28-29).

Wittgenstein’s writings make clear that, divorced from human practices, the descriptors “right” and “wrong” lose their meaning, even in disciplines like mathematics and logic.  Quantum theoretical “weak objectivity” (d’Espagnat, 1983) has replaced the strong objectivity of Newtonian mechanics because Newtonian objectivity misrepresents the psychometrician’s task.

[A]lmost all of us, after sufficient training, respond with roughly the same procedures to concrete addition problems.  We respond unhesitatingly to such problems as ’68 + 57,’ regarding our procedure as the only comprehensible one (see Wittgenstein, 1953, §§219, 231, 238), and we agree in the unhesitating responses we make.  On Wittgenstein’s conception, such agreement is essential for our game of ascribing rules and concepts to each other (see Wittgenstein, 1953, §240). (Kripke, 1982, p. 96)

Wittgenstein (1975, p. 58) writes: “The only criterion for his multiplying 113 by 44 in a way analogous to the examples is his doing it in the way in which all of us, who have been trained in a certain way, would do it.”  It follows that third-person ascriptions of the ability to add are based on the criteria afforded by the practice of mathematics, a practice into which the teacher has been enculturated.  Wittgenstein notes that “Indefinitely many other ways of acting are possible: but we do not call them ‘following the rule’” (Malcolm, 1986, p. 155).

Criteria hover somewhere between deductive and inductive grounds (Grayling, 1977) and their nature can be traced back to the introduction of the participating psychologist.  For example, there is an intrinsic “vagueness,” to borrow Wittgenstein’s term, in the accepted number of particular additions one ought to compute correctly before having the ability to add ascribed to one.  There is no fixed number of even numbers a student should write down before being regarded as someone who “understands” or “has mastered” or “has grasped the meaning of” the even numbers.  This vagueness is a constitutive property of psychological predicates; it isn’t a shortcoming.

Suppose we are teaching a student how to construct different series of numbers according to particular formation rules.  When will we say that he has mastered a particular series, say, the series of natural numbers?  Clearly, he must be able to produce this series correctly: ‘that is, as we do it’ (Wittgenstein, 1953, §145).  Wittgenstein points here to a certain vagueness in our criteria for judging that he has mastered the system, in respect of how often he must get it right and how far he must develop it.  This vagueness is something that Wittgenstein sees as a distinctive characteristic of our psychological language game, one that distinguishes it from the language-game in which we describe mechanical systems. (McGinn, 1997, p. 89)

Gilbert Ryle (1949, p. 164) writes: “To settle whether a boy can do long division, we do not require him to try out his hand on a million, a thousand, or even a hundred different problems in long division.  We should not be quite satisfied after one success, but we should not remain dissatisfied after twenty, provided that they were judiciously variegated and that he had not done them before.  A good teacher, who also watched his procedure in reaching them, would be satisfied much sooner, and he would be satisfied sooner still if he got the boy to describe and justify the constituent operations that he performed.”

Hence, third-person ascriptions of ability are based on criteria while first-person ascriptions are not; in the first-person case there are no criteria for attaching the correct interpretation to the mental image of a rule.  In short, first-person and third-person ascriptions of ability are mutually exclusive; the former do not require criteria for an ascription of ability (the individual acts for no reason) but the latter do.  First-person ascriptions of ability are associated with being right and wrong, while third-person ascriptions are associated with being right or wrong.

Returning to the counterintuitive conclusion drawn by Kripke (1982), must it be accepted that individuals respond to novel addition problems by offering capricious answers?  This conclusion needn’t be drawn because there’s an error in Kripke’s premise, namely, that his analysis treats inner facts (associated with first-person ascriptions of ability) as entirely independent of outer facts (associated with third-person ascriptions of ability).  The idea that the inner stands in a deterministic relation to the outer has been challenged earlier.  Wittgenstein considered first-person and third person ascriptions as forming an indivisible whole; they cannot be meaningfully separated (Malcolm, 1971, pp. 87-91).

In summary when the measurement process is divided into the situation immediately before the individual responds to the question “68 + 57 = ?” (the individual’s ascription of ability to himself or herself) and the situation immediately afterwards (the ascription of ability to the individual by the measurer), the relation is not one of Newtonian determinism between two independent situations.  Rather, it’s one of quantum complementarity where complementarity is the more general concept which replaced Newtonian causality.  The difficulties identified by Kripke (1982) in respect of causality at the level of the individual can be seen in a new light by eschewing causality for complementarity.

Polkinghorne (1996, p. 70) defines complementarity as a “combination of apparent opposites” and Whitaker (1996, p. 184) describes it as “mutual exclusion and joint completion.”  In psychological predicates first-person ascriptions are made without criteria while third-person ascriptions require criteria.  This is the mutual exclusiveness facet of complementarity in respect of psychological predicates.  But these two very different ascriptions cannot be separated on pains of accepting Kripke’s conclusion that rule-following in mathematics is capricious.  This is the joint completion facet.

While first-person/third-person ascriptions of psychological predicates appear to stand in a complementary relationship, this asymmetry is entirely absent in Newtonian physics.  Suter (1989, pp. 152-153) writes: “This asymmetry in the use of psychological and mental predicates – between the first-person present-tense and second- and third-person present-tense – we may take as one of the special features of the mental.  Physical predicates display no such asymmetry.”

References

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Anscombe, G.E.M. (1985).  Wittgenstein on rules and private language.  Ethics, 95, 342-352.

Barrett, P. (2008).  The consequence of sustaining a pathology: Scientific stagnation – a commentary on the target article “Is psychometrics a pathological science” by Joel Michell.  Measurement, 6, 78-83.

Battig, W.F. (1978).  Parsimony or psychology.  Presidential Address, Rocky Mountain Psychological Association, Denvir, CO.

Bennett, M.R., & Hacker, P.M.S. (2003).  Philosophical foundations of neuroscience.  Oxford: Blackwell Publishing.

Blinkhorn, S. (1997).  Past imperfect, future conditional: Fifty years of test theory.  British Journal of Mathematical and Statistical Psychology, 50(2), 175-186.

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The AQE and GL Assessment test results for 2016:17 Advice for Parents

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Good morning and congratulations on your patience over a long winter. This morning you will receive the results of your child’s transfer test. All of the effort, costs, studying, revision and application cumulate in the mark revealed this morning and all children are to be congratulated regardless of the result.

Today is a day of mixed emotions for parents; the elation and relief blended with perhaps some sense of pride and appreciation that your child is soon to be fleetingly venturing out in the journey towards adulthood. Be sure to enjoy the day.

Of course with parents a fresh set of anxieties replace the old and thoughts immediately turn to trying to figure out if the mark or grade (grades remove information) will secure a place in the grammar school of choice. Children will naturally be inquisitive and parents keen to answer with accuracy but it will be months before admission decisions are known. Schools will try to be helpful and reassuring but can guarantee nothing absolutely. Some will engage in an intense effort to market and promote their schools even at the cost of misinformation.

Peter Weir 1

Political parties are in general officially opposed to academic selection (but privately their representatives choose to use transfer tests for their children) Many will not admit to this   lest they lose a vote; those supporting compromise will talk of a single test (combining AQE and GL, not just one exam) but this is a problem they are unwilling to accept they are incapable of reconciling. Education is soon likely to be an issue on your doorstep during the current election campaign. In no other aspect of business would a government be allowed to interfere in the operation of private business. Bill Gates had a very clear message to those who would attempt to steal, duplicate or pirate his Microsoft products. The Department of Education seem to have no such reservations when it comes to meddling in transfer tests.

Former DUP First Minister Peter Robinson made much of his determination to deliver a single test. He left office defeated in this aim by the resolve of parents and a dedicated group of principled individuals who will not allow political expediency to destroy parental choice for an education suitable for their individual children.
Ballymena Guardian Common Test Oct14

When Arlene Foster became First Minister and the DUP chose the education ministry for the first time it became clear that the DUP were insistent on delivering on the single test goal to satisfy their partners in the Executive. This attitude is difficult to explain since PACE published two letters in the Ballymena Guardian in 2014 outlining very profound concerns over the use of two different tests for the same purpose. No political party or church has had a single word of response.  Peter Weir was recently reminded of the warnings but has failed to adopt a leadership position by recommending the superior instrument; the AQE test.

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The BBCNI news this morning via Robbie Meredith, Education correspondent tells listeners (parents of future tests takers) that sources inform him that

“talks are taking place between the two testing organisations to find a common exam”

The BBC are misinformed since a simple matter of fact checking exposes the inconsistency. One test is developed by AQE the other by GL Assessment. GL Assessment have not been involved in any talks with AQE involving a single test. The PPTC who deliver the test in mainly Catholic grammar schools have no ownership of GL Assessment products.

The Irish News (opposed to academic selection) were at least able to get close to a truth that the Education Minister, Peter Weir refuses to accept. Weir announced on November 17, 2016

” a team of educational professionals would seek to simplify the current transfer test process”

Mr Weir should read the Irish News more carefully.

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Parents with children transferring to post-primary in 2017-18 should insist that politicians stop interfering in the matter of transfer testing since the Department of Education abandoned their responsibilities nine years ago.

Is Growth Mindset the new Brain Gym?

The Parental Alliance for Choice in Education blogged about the flaw in Carol Dweck and Jo Boaler’s research on December 13th, 2016 http://wp.me/pateI-K5. Dweck and Boaler were forwarded the critique and invited to respond. No replies have been received.
Carol Dweck is obviously feeling the heat as evidenced here http://mindsetscholarsnetwork.org/growth-mindset-firm-foundation-still-building-house/
It seems to have escaped the attention of the author of Scenes from the Battleground.

Scenes From The Battleground

Earlier this month, an article on the Guardian website told us the following:

Schools and teachers across the world have embraced Carol Dweck’s theory of growth mindset in the hope of helping students to fulfil their potential. Popular strategies include tweaking the way teachers give feedback, encouraging self-reflection through questioning and, crucially, praising processes instead of natural ability.

But many educators feel they could be doing more. A recent survey found that 98% of teachers believe that if their students have a growth mindset it will lead to improved student learning, but only 20% of them believe they are good at fostering a growth mindset and 85% want more training and practical strategies.

This seems to suggest the idea of Growth Mindset is well-established within schools. Is it a fad that’s as disreputable as Brain Gym?  Probably not, but I couldn’t resist putting that in the title after…

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E.D. Hirsch Jr. is powerless to challenge direct instruction

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Why E.D. Hirsch’s particular brand of “science” is powerless to challenge direct instruction

Dr Hugh Morrison (The Queen’s University of Belfast [retired])

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The cover page of the Times Education Supplement (TES) quotes the distinguished American educationalist E D Hirsch’s claim that “there is no scientific basis for direct instruction.”  Given the high regard in which Hirsch is held by educational traditionalists, there will be widespread dismay that one of their own is invoking science to attack a traditional pedagogical technique that can see off any progressivist model when it comes to raising the educational standards of poor children.  (Readers who google the words “direct instruction” will see why this classroom approach is so important to traditionalists.)  Hirsch clearly appreciates that the reasoning set out in his new book may not be received with universal acclaim: “To offend everybody is one of the few prerogatives left to old age.”  The good news for proponents of direct instruction everywhere is that the “science” Hirsch appeals to makes no sense.

 

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The basis of Hirsch’s TES “scientific” attack is the field of cognitive science.  To convince his readers that his book represents “consensus science” he has invited two prominent cognitive scientists, Steven Pinker and Daniel Willingham, to “blurb” the book.  Now cognitive scientists hold that psychological attributes like thought, understanding, memory, meaning, and so on, are internal processes associated with the human brain/mind.  According to cognitive scientists the mind/brain is a self-contained realm where computations are performed on mental “representations.”  However, one of the towering figures of 20th century thought, Ludwig Wittgenstein, regarded this type of thinking as deeply misconceived.  He wrote that: “The confusion and barrenness of psychology is not to be explained by calling it a “young science”: its state is not comparable with that of physics, for instance, in its beginnings. … For in psychology there are experimental methods and conceptual confusion.”  In their 2003 book Philosophical Foundations of Neuroscience Max Bennet and Peter Hacker use Wittgensteinian reasoning to attack cognitive neuroscience’s central claims.  In the remainder of this essay the “consensus science” which informs Hirsch’s claims is undermined using the writings of Bennett and Hacker.  The reader needs neither a background in cognitive psychology nor a grounding in philosophy to appreciate immediately the validity of Wittgenstein’s “conceptual confusion” claim; a healthy dose of common sense will reveal immediately the error at the heart of cognitive science.

 

While it is clear that thinking would be impossible without a properly functioning brain, the claim that brains can think or that thinking takes place in the brain ought to be supported with scientific evidence.  No such evidence exists.  To mistakenly attribute properties to the brain which are, in fact, properties of the human being is to fall prey to what Bennett and Hacker refer to as the “mereological fallacy.” (Mereology is concerned with part/whole relations and the fallacy goes all the way back to Aristotle.)

 

“Psychological predicates are predicates that apply essentially to the whole living animal, not to its parts.  It is not the eye (let alone the brain) that sees, but we see with our eyes (and we do not see with our brains, although without a brain functioning normally in respect of the visual system, we would not see).  So, too, it is not the ear that hears, but the animal whose ear it is.  The organs of an animal are part of the animal, and psychological predicates are ascribable to the whole animal, not its constituent parts” (pp. 72-73).

 

Cognitive scientists often refer to brains “thinking,” “knowing,” “believing,” “deciding,” “seeing an image of a cube,” “reasoning,” “learning” and so on.

 

“We know what it is for human beings to experience things, to see things, to know or believe things, to make decisions … But do we know what it is for a brain to see … for a brain to have experiences, to know or believe something?  Do we have any conception of what it would be like for a brain to make a decision? … These are all attributes of human beings.  Is it a new discovery that brains also engage in such human activities?” (p. 70)

neurosurgery

In the words of Wittgenstein (1953, §281): “Only of a human being and what resembles (behaves like) a living human being can one say: it has sensations; it sees, is blind; hears, is deaf; is conscious or unconscious”.  If the human brain can learn, “This would be astonishing, and we should want to hear more.  We should want to know what the evidence for this remarkable discovery was” (Bennett & Hacker, 2003, p. 71).  It is important to appreciate the depth of the error committed here.  When the claim that the brain can think is called into question, this doesn’t render valid the assertion that brains, in fact, cannot think.

 

“It is our contention that this application of psychological predicates to the brain makes no sense.  It is not that as a matter of fact brains do not think, hypothesise and decide, see and hear, ask and answer questions; rather, it makes no sense to ascribe such predicates or their negations to the brain.  The brain neither sees, nor is it blind – just as sticks and stones are not awake, but they are not asleep either” (p. 72).

 

 

One gets the clear impression from the cognitive science literature that understanding or remembering are inner processes.  Wittgenstein, while accepting that without a properly functioning brain one couldn’t learn, nevertheless teaches that understanding is something attributed to the whole person, and not the brain.  When a teacher asks a pupil what she thinks, the pupil expresses her thoughts in language.  Were it not for the pupil’s language skills, the teacher couldn’t ascribe thoughts to her.  Since brains aren’t language-using creatures, how can it make sense to ascribe thoughts to a brain?

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While cognitive scientists may protest that the brain’s ability to make connections while it (the brain) is learning, is visible from the PET or fMRI images of the brain, scientific writing should always show restraint:

 

“But this does not show that the brain is thinking, reflecting or ruminating; it shows that such-and-such parts of a person’s cortex are active when the person is thinking, reflecting or ruminating.  (What one sees on the scan is not the brain thinking – there is no such thing as a brain thinking – nor the person thinking – one can see that whenever one looks at someone sunk in thought, but not by looking at a PET scan – but the computer-generated image of the excitement of cells in his brain that occurs when he is thinking.)” (pp. 83-84).

 

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In Neuroscience & Philosophy: Brain, Mind and Language (2007, p. 143), Bennett and Hacker write:

 

“But if one wants to see thinking going on, one should look at the Le Penseur (or the surgeon operating or the chess player playing or the debater debating), not his brain.  All his brain can show is which parts of the brain are metabolizing more oxygen than others when the patient in the scanner is thinking.”

 

aristotle

 

In order to see off Hirsch’s ill-founded claims, advocates of direct instruction can appeal to no less a thinker than Aristotle.  Around 350BC he wrote:

“to say that the soul (psyche) is angry is as if one were to say that the soul weaves or builds.  For it is surely better not to say that the soul pities, learns or thinks, but that a man does these with his soul.”

 

 

 

 

 

 

 

 

 

 

The flaw in Dweck & Boaler’s Mindset research

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The flaw at the heart of Dweck and Boaler’s research, and the real source of psychology’s “reproducibility” problem

 Dr Hugh Morrison (The Queen’s University of Belfast [retired])

Email: drhmorrison@gmail.com

The Times Higher Education (28.04.2016-04.05.2016) reported that Professor Carol Dweck’s “ideas on education have swept through schools.  … In a TED talk that has so far garnered more than four million views online, she shares inspiring tales of pupils in tough, inner-city areas who have zoomed ahead after being trained to believe that their talents are not fixed.”  The key word in this quotation is the word “believe”; Professor Dweck is particularly concerned with an individual’s beliefs or “mindsets.”  In the fixed mindset, “Success is about proving you’re smart or talented.  Validating yourself.”  In the growth mindset, on the other hand, success is about “stretching yourself to learn something new.  Developing yourself.”

It is vital to Dweck’s theory – a theory currently attracting millions of pounds of education funding in the UK – that beliefs are entities in the mind/brain of the individual.  Dweck urges teachers to spur their students to success by concentrating on their “mindsets.”  On page 16 of her Mindset book she writes: “You have a choice.  Mindsets are just beliefs.  They’re powerful beliefs, but they’re just something in your mind, and you can change your mind.”  Beliefs/mindsets are properties of the individual.  Her writing is replete with references to children “putting themselves” into one mindset or the other, or being “placed into” the growth mindset by a teacher or by psychologists working on Dweck’s “Brainology” programme.

At the outset, there are two issues which engender disquiet about Dweck’s theory.  First, her writings ignore the extensive literature demonstrating that beliefs are neither mental states, brain states, nor dispositions (see, for example, Peter Hacker’s 2013 book, The intellectual powers: a study of human nature).  Second, and much more importantly, Dweck treats “intelligence” and “ability” as things-in-themselves, to borrow a Kantian conceit.  All of Dweck’s writing assumes that the individual’s intelligence exists independent of the efforts of others to learn about it.  To borrow a phrase from Crispin Wright (2001), intelligence is an “investigation-independent” entity for Dweck.  At the very core of Dweck’s research is the child’s belief that his or her intelligence is either something fixed, or something capable of improvement.  Jo Boaler’s research – which applies Dweck’s thinking to the mathematics classroom – similarly views mathematical ability as a thing-in-itself.

To prepare the reader for what follows, consider the following simple thought experiment which suggests that it is meaningless to refer to mathematical ability as a thing-in-itself, existing independent of the teacher’s efforts to measure it.  In the UK children who find mathematics a struggle take the “Foundation GCSE” mathematics examination at age 16.  The test items on the Foundation mathematics paper aren’t considered mathematically demanding.  Now suppose that Richard, a pupil who has completed the Foundation mathematics curriculum, produces a perfect score in the examination.  It seems sensible to conjecture that if Einstein were alive, he too would produce a perfect score on this mathematics paper.  Since we think of the examination paper as measuring mathematical ability, are we therefore justified in saying that Einstein and Richard have the same mathematical ability?

Paradox results when one erroneously treats mathematical ability as something investigation-independent, entirely divorced from the circumstances of its ascription.  It is clear that Richard’s mathematical achievements are dwarfed by Einstein’s.  To avoid the paradox one would have to mention the measurement circumstances and say: “Richard and Einstein have the same mathematical ability relative to the Foundation mathematics paper.”  When one factors in the measuring instrument, the paradox dissolves away.  This thought experiment illustrates how the reasoning which underpins the research of Dweck and Boaler can be undermined since neither researcher makes mention of the measurement context.

Mathematical ability is not a property of the person.  Rather it is a joint property of the person and the relevant measuring tool.  In short, mathematical ability should be thought of as a property of an interaction.  It is a relational attribute rather than an intrinsic property of the person being measured.  Definitive statements about mathematical ability – such as “my mathematical ability is fixed” – are difficult to justify.  It is only by specifying the measuring instrument in the Foundation mathematics thought experiment that one “communicates unambiguously,” to borrow the quantum physicist Niels Bohr’s words.

It would be unfair to judge Dweck’s ideas by focusing exclusively on her book Mindset, which is clearly intended for the “popular psychology” bookshelves.  Dweck’s 2000 book on “self-theories” is published by Psychology Press.  It focuses on beliefs about intelligence and is directed at the psychological community.  Dweck (2000, p. xi) sets out her programme as follows: “In this book I spell out how people’s beliefs about themselves (their self-theories) can create different psychological worlds, leading them to think, feel and act differently in identical situations.”  “Entity theorists” are individuals who believe that their intelligence is fixed, while “incremental theorists” believe that their intelligence is capable of improvement.  The former are focused on grades while learning preoccupies the latter.  (The parallels with the fixed and growth mindsets are obvious.)

The items which make up the questionnaires Dweck uses to measure children’s self-theories give the clear impression that she takes two things for granted: (i) that intelligence is an intrinsic property of the child, and (ii) that intelligence exists in “amounts.”  For example, the first two items in the theory of intelligence scale require the child to consider the statements: “You have a certain amount of intelligence, and you really can’t do much to change it” and “Your intelligence is something about you that you can’t change very much” (p. 177).

The remainder of this article will centre on statements like “I believe my intelligence/ability is fixed” and “I believe my intelligence/ability can be changed.”  Such statements sit at the core of Dweck’s theory, but I will argue that they are, in fact, meaningless because neither intelligence nor mathematical ability are things-in-themselves.

The serious conceptual errors in Dweck and Boaler’s research have their origins in the psychologist’s insistence on thinking of the concepts they study as “strongly objective” in the same sense as the concepts of Newtonian physics are strongly objective (Gould, 1996).  The attraction of psychology’s Newtonian worldview is obvious in that classical physics holds out the promise of certainty and objectivity.  Psychology’s Newtonian “physics envy” is puzzling given that physicists themselves turned their backs on a Newtonian picture of reality, adopting a quantum theoretical one approximately a century ago.

A clear indication that psychology – in common with quantum theory – is “weakly objective” can be seen in psychology’s century-long quest to define what intelligence is and what memory is.  The explanation for the futility of this quest?  The question “what is intelligence?” and the question “what is memory?” are not meaningful in a weakly objective framework.  Outside the strongly objective framework, intelligence and memory cannot be regarded as things-in-themselves about which one can speak meaningfully while disregarding the measurement context.

A recent initiative to reproduce the findings of 100 important papers in the psychological literature succeeded in only 39% of cases.  Unless psychology responds definitively, the so-called “reproducibility” project will continue to threaten cherished psychological principle after cherished psychological principle, undermining the very discipline itself.  The source of psychology’s current reproducibility ills is its claim to strong objectivity, at a time when science’s most powerful theory – tested to destruction in the laboratory – embraces weak objectivity.

Werner Heisenberg (1958), summarising “the revolution in modern science,” writes: “Since the measuring device has been constructed by the observer … we have to remember that what we observe is not nature in itself but nature exposed to our method of questioning.”  His great mentor, Niels Bohr, drew parallels between quantum theory and psychology for all of his professional life, prompting Harvard physicist Abner Shimony to conjecture that: “quantum concepts can be applied to psychology, but not with as much geometrical structure as in quantum physics.”  Richardson’s (1999, p. 40) writings (in respect of psychological measurement) echo Heisenberg’s: “[W]e find that the IQ testing movement is not merely describing properties of people: rather, the IQ test has largely created them.”

No scientist of note has ever supported psychology in its claim to strong objectivity.  As far back as 1955, the great American physicist Robert Oppenheimer pleaded with psychologists to forsake their adherence to strong objectivity: “It seems to me that the worst of all possible misunderstandings would be that psychology be influenced to model itself after a physics which is not there anymore, which has been quite outdated.”  Gigerenzer (1987, p. 11) explains the context of Oppenheimer’s plea: “quantum theory was indeed discussed in the 1940s and ‘50s within psychology.  However, it was unequivocally rejected as a new ideal of science.”  Psychology shut its ears and (incredibly) continued to teach that the measurement of living beings can be modelled on Newtonian measurement principles developed to model inanimate matter! As physicist Henry Stapp (1993, p. 219) put it: “while psychology has been moving towards the mechanical concepts of nineteenth-century physics, physics itself has moved in just the opposite direction.”

If all this seems fanciful, consider two of psychology’s most studied mental predicates: memory and intelligence.  I choose memory and intelligence because both have been studied intensively for over a century and I do not wish to be accused of making the case for weak objectivity in psychology by selecting a poorly researched area of the discipline.  The psychologists quoted in what follows are all leading researchers in memory and intelligence.

First, let’s consider memory.  Jenkins (1979, p. 431) hinted that psychology should eschew its quest to understand memory as a thing-in-itself: “The memory phenomena that we see depend on what kinds of subject we study, what kinds of acquisition conditions we provide, what kinds of material we choose to work with, and what kinds of criteria measures we obtain.”

Roediger (2008, p. 247) expresses disappointment that after decades of study, the search for an answer to the question “what is memory?” must be abandoned.  In his 2008 review article Relativity of remembering: why the laws of memory vanished, he writes: “The most fundamental principle of learning and memory, perhaps its only sort of general law, is that making any generalisation about memory one must add that ‘it depends.’”  Roediger (2008, p. 247) – anticipating the Reproducibity Project – suggests that whatever claim a psychologist makes about memory, a sceptic can always say; “Very nice work, but your finding depends on many other conditions.  Change those and your effect will go away.”  Measurement of memory seems unavoidably dependent upon the measurement context.  One can detect the same concerns in Tulving’s (2007) paper entitled: Are there 256 different kinds of memory?

Psychology’s reproducibility problem is cast in a very different light when memory is treated as a joint property of the individual and the measurement context; change the measurement context and the meaning of what is measured must change.  Battig (1978) suggests that this dependence on measurement context generalises to all psychological attributes.  Battig’s reasoning goes beyond the conservative claims of the Reproducibility Project and would suggest that psychology faces years of damaging criticism if it continues to treat the attributes it studies as strongly objective.  What physics has long since come to accept as a fundamental truth, Roediger III (2008, p. 228) bemoans as a matter of regret: “the great truth of the first 120 years of empirical study of human memory is captured in the phrase ‘it depends.’”

Quantum theory has taught physicists to accept that the question “what is an electron?” makes no sense in a weakly objective framework.  Because the physicist participates in what he or she “sees,” it must be acknowledged that physics is not concerned with standing back from nature and objectively reporting what one “sees.”  Rather, physics is limited to studying humankind’s interaction with nature.  Roediger’s “it depends” response is embraced in quantum theory: in experimental arrangement A, the electron manifests as a particle; in experimental arrangement B it manifests as something very different – a wave.  How are the statements “the electron is a particle” and “the electron is a wave” to be reconciled?  The obvious paradox is avoided by demanding (as Niels Bohr did) that physicists “communicate unambiguously” by always making reference to the measuring instrument.

The statements “the electron is a particle relative to experimental arrangement A” and “the electron is a wave relative to arrangement B” banish the paradox entirely. “Particle” and “wave” are not intrinsic properties of the electron (leading to an obvious contradiction); instead, they are properties of the electron’s interaction with the measuring tool.  According to Bohr one is communicating ambiguously – for Bohr, unambiguous communication is the hallmark of science – if one speaks of the electron as a thing-in-itself, independent of how it is observed.  All of this has relevance for Dweck’s self-theory notion because all reference to beliefs that “my intelligence is fixed/malleable” are ruled out as meaningless in a weakly objective framework, because intelligence is being wrongly interpreted as a thing-in-itself.

One can get the distinct impression from the psychological literature that the typical psychologist sees herself as a passive, objective observer of what she studies.  There is little evidence that psychologists see themselves as participants in what they “see.”  This mistaken adherence to strong objectivity also afflicts the thinking of the Reproducibility Project’s researchers who see themselves as mere observers of a re-run of psychology’s signature experiments.  The very notion of “reproducibility” sits awkwardly with the teachings of Bohr: “In the study of atomic phenomena, however, we are presented with a situation where the repetition of an experiment with the same arrangements may lead to different recordings” (Bohr, 1958-1962, p. 18). Jammer (1999, p. 234) writes that “Bohr did not regard the world as an objective reality with a given structure … conceptually separable from us as observers. … Thus, there must be limits to the depth of understanding that we can hope to gain of the world, because of our joint role as spectators and actors in the drama of existence.”  Misner, Thorne & Wheeler (1973, p. 12) counsel: “’Participator’ is the incontrovertible new concept given by quantum mechanics; it strikes down the term ‘observer’ of classical theory, the man who stands safely behind the thick glass wall and watches what goes on without taking part.  It can’t be done, quantum mechanics says.  Even with the lowly electron one must participate before one can give any meaning whatsoever to its positon and velocity.”

When attention turns to the construct “intelligence,” the case against psychology’s claim to strong objectivity deepens.  Jensen (1998, p. 46) acknowledges that after a century of attempts, a widely endorsed answer to the question “what is intelligence?” has eluded psychologists: “No other term in psychology has proved harder to define than ‘intelligence.’  Not that psychologists haven’t tried.  Though they have been attempting to define ‘intelligence’ for at least a century, even experts in this field still cannot agree on a definition.  In fact, there are as many different definitions of ‘intelligence’ as there are experts.”  One high profile search for a definition of intelligence was documented in 1986 by Sternberg and Detterman.  In the concluding paragraph of the book, Detterman sums up the experts’ judgements: “For those who expected to read this volume – entitled “What is intelligence?” – and obtain the definitive definition, I apologise.”

It was argued above that a commitment to communicate unambiguously demands that the measurement context be clearly specified if statements such as “Richard and Einstein have the same mathematical ability” are to be justified.  A statement of the mathematics tested in the Foundation examination clarifies that the words “mathematical ability” refer to a very restricted subset of the entire domain of mathematics.  In respect of intelligence Gladwell (2007, p. 95) draws conclusions from the Flynn-effect that strike at the heart of Dweck’s research: “For instance, Flynn shows what happens when we recognize that I.Q. is not a freestanding number but a value attached to a specific time and a specific test. … The notion that anyone “has” an I.Q. of a certain number, then, is meaningless unless you know which WISC he took, and when he took it, since there’s a substantial difference between getting a 130 on the WISC(IV) and getting a 130 on the much easier WISC.”

I.Q. is a relational attribute; it is the property of an interaction (between test-taker and test) and not an intrinsic property of the test-taker.  This makes it impossible to define what intelligence is as a thing-in-itself.  For example, I suspect that Dweck is using the word “intelligence” in the restricted sense of intelligence tests.  But Howard Gardner (1983), a highly respected Harvard psychologist, extended intelligence beyond the language and logical-mathematical realms to spatial intelligence, musical intelligence, the use of the body to solve problems or make things, and interpersonal/intrapersonal intelligences.  Indeed, why stop at Gardner’s seven so-called “multiple intelligences”?  Statements such as “I believe my intelligence is fixed” are devoid of any clear meaning without a precise specification of the measurement context.

The distinguished American physicist David Mermin developed Niels Bohr’s counsel that psychology can learn from quantum theory that there are questions which cannot be answered in a weakly objective framework.  It is instructive to quote Mermin’s words in full: “What does it mean for a property to be real?  When you study an object how can you be sure you are learning something about the object itself, and not merely discovering some irrelevant feature of the instruments you used in your study?  This is a question that has plagued generations of psychologists.  When you measure IQ are you learning something about an inherent quality of a person called “intelligence,” or are you merely acquiring information about how the person responds to something you have fancifully called an IQ Test?  Until the advent of the quantum theory in 1925 physicists were above such concerns.  But since then, with the discovery that experiments at the atomic level necessarily disturb the object of investigation, precisely such reservations have been built into the foundations of physics.”

From the physicist’s perspective: “[I]f we set out to measure the momentum, say, of an electron, what we are actually measuring is the ability of an electron to answer questions about momentum.  The electron may, indeed, not have any such property as momentum, in the way we think of it in the everyday world… .  We get experimental results – ‘answers’ – which we interpret as measures of momentum.  But they are only telling us about the ability of electrons to respond to momentum tests, not their real momentum, just as the results of IQ measurements only tell us about the ability of people to respond to IQ tests, not their real intelligence” (Gribbin, 1995, p. 148).

Weak objectivity views measurement as context-dependent.  When the position of an electron is measured, the physicist is not merely checking up on an investigation-independent property inherent in the election.  What one is really measuring is the interaction between the electron and the measuring instrument.  The measurement outcome is a joint property of the electron and the measurement instrument.  Niels Bohr considered questions which treated the electron as a thing-in-itself (such as “what is an electron?”) as meaningless in a weakly objective framework.  As anyone who has watched popular science programmes will recognise, in quantum theory an electron manifests as a wave in one measurement context, and as a particle in another.  It is therefore meaningless to ask what an electron is as a thing-in-itself, without reference to the measurement context.

An electron is a wave relative to one measurement context, and a particle relative to another.  According to his close colleague, Aage Petersen, Bohr summarised the switch from strong to weak objectivity as follows: “It is wrong to think that the task of physics is to find out how nature is.  Physics concerns what we can say about nature.”  He uses the word “say” because in a weakly objective framework, in order to communicate ambiguously one must provide a description of the measurement apparatus.  Changes in the measurement context therefore have consequences for the very meaning of what is measured.

Bohr labelled this tendency of quantum entities to manifest as wave or particle, depending on the measurement context, as “complementarity.”  He regarded complementarity as the central concept in quantum theory.  Physicists treat these two characteristic manifestations as opposites given that a particle is confined to a tiny region of space, while a wave spreads throughout space.  A strikingly similar concept arising in the study of mind appears in the writings of one of the greats of modern philosophy, Ludwig Wittgenstein.  In the secondary literature derived from Wittgenstein’s later philosophy, it has become known as “first-person/third-person asymmetry.”  This asymmetry applies to intentional predicates in general and to intelligence and mathematical ability in particular.  Incidentally, this analogue of complementarity in respect of mind avoids all of the difficulties associated with both Cartesian dualism and behaviourism.

I will quote in full from Colin McGinn’s book The Character of Mind in order to illustrate that an analogue of complementarity informs fundamental thinking about mind.  McGinn (1996, pp. 6-7) writes: “Mental concepts are unique in that they are ascribed in two, seemingly very different, sorts of circumstances: we apply them to ourselves on the strength of ‘inner’ awareness of our mental states, as when a person judges of himself that he has a headache; and we apply them to others on the strength of their ‘outer’ manifestations in behaviour and/or speech.  These two [opposite] ways of ascribing mental concepts are referred to as first-person and third-person ascriptions. … It would be fine if we could somehow, as theorists, prescind from both perspectives and just contemplate how mental phenomena are, so to say, in themselves; but this is precisely what seems conceptually unfeasible … we seem to need the idea of a single mental reality somehow neutral between the first- and third-person perspectives; the problem is that there does not appear to be any such idea.”

Physics has taught us that a statement such as “the electron’s velocity is constant” is utterly meaningless in a weakly objective framework such as the quantum framework.  Velocity is not an intrinsic property of the electron. Quantum theory rules out all reference to the velocity of an electron without a clear description of the particular measurement context.  One gets nonsense when one omits the context of ascription.  Similarly, investigation-independent statements such as “I believe that my intelligence is fixed” and “I believe my mathematical ability can grow” make little or no sense.  Both of these statements wrongly present mind as a carrier of definite states.  Mind is a carrier of potentiality (just like the microentity) and not a carrier of definite states.  Intelligence and mathematical ability are not inner states which are somehow the source of behaviour.  In respect of the first statement above, Mermin’s reasoning (see above) rejects the notion that intelligence can ever be an intrinsic property of the individual.

But what of mathematical ability as Boaler construes it?  Just as in the case of intelligence, mathematical ability (as a thing-in-itself) is a potentiality rather than a state.  A pupil who has grasped the concept “even number” has the potential to non-collusively modify his or her behaviour so that it is in accord with accepted mathematical practice in respect of the even numbers. Mathematical ability is therefore a joint property of the individual and the fiduciary (to borrow Polanyi’s term) framework within which he or she has been educated.  As with intelligence, mathematical ability is not an intrinsic property of the person.

In conclusion, it is instructive to demonstrate the gulf between Dweck and Boaler’s research and the writings of one of the greatest physicists of all time.  On page 96 of the first volume of his essays Bohr (1934) considers the connection between “the conscious analysis of any concept” and “its immediate application.”  For example, how is the ability to apply the concept “even number” in accord with established mathematical practice connected to the possession of an introspectable mental “object,” namely, the formula which generates the even numbers: Un = 2n?  How does having the formula in mind (a concept capable of “conscious analysis”, in Bohr’s terms) connect with one’s ability to say or write out the even numbers in accord with established mathematical practice (“immediate application”, in Bohr’s terms)?

The following connection immediately suggests itself: the mental image of the formula is the source of the individual’s ability to write out the even numbers.  Two entirely separate realms are suggested here: on the one hand, the individual’s understanding of the even numbers (to have the formula in mind is to understand the even numbers), and, on the other hand, the application of that understanding in the writing out of the even numbers in accord with established practice.  In this picture the inner world of understanding is divorced entirely from the public realm of application.  From this viewpoint it is tempting to think of the formula-in-mind as representing mathematical understanding as a thing-in-itself.  For Bohr, this strongly objective picture of the connection between “inner” and outer must be invalid.

Bohr considered this Cartesian picture (with its Newtonian, self-standing mental “objects”) as entirely wrong.  To understand Bohr’s reasoning one must turn to the later philosophy of Wittgenstein and to his writings on rule-following in particular.  The error in the strongly objective Cartesian picture, in which mathematical ability in respect of the even numbers is entirely divorced from subsequent application, is that the inner formula cannot be the source of correct application because it has no guidance properties whatever.  The formula as a thing-in-itself simply cannot determine subsequent application.  One can only derive guidance from a mathematical formula by being trained in the practice of using that formula.  However, mathematical practice is a feature of the entirely separate public realm where the formula is to be applied.  Because the formula-in-mind doesn’t have its applications written into it, it cannot guide.

It is the experience of mathematics teachers throughout the world that the appearance of the quadratic formula in the formula sheet made available to pupils taking public examinations is no guarantee that all pupils taking the examination will be able to successfully apply that formula.  Wittgenstein demonstrates that when understanding as a thing-in-itself is separated from application, any attempt to explain how these two realms are connected leads to a destructive “regress of interpretations.”  When one defines “understanding of the even numbers” as having a mental object (the formula Un = 2n) in mind, divorced entirely from the mathematical practice which gives the formula its life, one descends into confusion (see Oakeshott, 1975).  Even experienced mathematicians often fail to recognise the role played by their long apprenticeship in the discipline.  Malcolm (1965, p. 102) questions the experienced mathematician’s mistaken intuition that formulae in themselves determine their applications: “You would like to think that your understanding of the formula determines in advance the steps to be taken, that when you understand or meant the formula in a certain way ‘your mind as it were flew ahead and took all the steps before you physically arrived at this one or that one’ (Wittgenstein, 1953, §188).”

Wittgenstein and Bohr resolve the regress of interpretations problem by treating the two realms as conceptually inseparable.  This allows the difficulties of both Cartesian dualism and behaviourism to be avoided.  Kenny (2004, p. 49) explains the pivotal role played by Wittgenstein’s notion of criteria: “According to him the connection between mental states and physical [application] is neither one of logical reduction (as in behaviourist theory) nor one of causal connection (as in Cartesian theory).  According to him the physical expression of the mental process is a criterion for that process; that is to say, it is part of the concept of a mental process of a particular kind that it should have a characteristic manifestation.  The criteria by which we attribute states of mind and mental acts, Wittgenstein showed, are bodily states and activities.”

Mind is expressed in behaviour; the teacher cannot help but “see” the child’s understanding of the concept “even number” in the ease with which she applies the rule for generating the even numbers.  This is how the connection is made.  In an appropriate school context, the ability to unhesitatingly write the next 100 even numbers, starting with 2088, for example, serves as a criterion which justifies the teacher in saying that the child understands the concept “even number.”  The Cartesian causal connection between mind, on the one hand, and behaviour, on the other, is replaced by a picture in which mind and behaviour are treated as an indivisible whole.

Malcolm (1965, pp. 101-102) summarises Wittgenstein’s (and Bohr’s) rejection of the notion that mathematical ability is a thing-in-itself, entirely divorced from application: “But the question of whether one understands the rule cannot be divorced from the question of whether one will go on in that one particular way that we call ‘right.’  The correct use is a criterion of understanding. … You would like to think that your understanding of the formula determines in advance the steps to be taken, that when you understood or meant the formula in a certain way “your mind as it were flew ahead and took all the steps before you physically arrived at this one or that one” (§188).  But how you meant it is not independent of how in fact you use it. … How he meant the formula determines his subsequent use of it, only in the sense that the latter is a criterion of how he meant it.”

References

Battig, W.F. (1978).  Parsimony or psychology?  Presidential address, Rocky Mountain Psychological Association, Denvir, CO.

Bohr, N. (1934).  The philosophical writings of Niels Bohr.  Woodbridge: Ox Bow Press.

Bohr, N. (1958-1962).  Essays 1958-1962 o atomic physics and human knowledge.  Woodbridge: Ox Bow Press.

Dweck, C. S. (2000).  Self-theories: their role in motivation, personality, and development.  Philadelphia, PA: Psychology Press.

Dweck, C. S. (2006).  Mindset: The new psychology of success.  New York: Random House.

Gardner, H. (1983).  Frames of mind.  New York: Basic Books.

Gigerenzer, G. (1987).  Probabilistic thinking and the fight against subjectivity.  In L. Kruger, G. Gigerenzer, & M.S. Morgan (Eds.), The probabilistic revolution – Volume 2: Ideas in the sciences (pp. 11-33).  Cambridge, MA: The Massachusetts Institute of Technology Press.

Gladwell, M. (Dec. 17, 2007).  None of the above.  New Yorker magazine.

Gould, S.J. (1996).  The mismeasure of man.  London: Penguin Books.

Gribbin, J. (1995).  Schrödinger’s kittens and the search for reality.  London: Weidenfeld & Nicolson.

Hacker P.M.S. (1997).  Insight and illusion: Themes in the philosophy of Wittgenstein.  Bristol: Thoemmes Press.

Hacker, P.M.S. (2013).  The intellectual powers: A study of human nature.  Oxford: Wiley Blackwell.

Heisenberg, W. (1958).  Physics and philosophy: the revolution in modern science.  New York: Prometheus Books.

Jammer, M. (1999).  Einstein and religion.  Princeton, NJ: Princeton University Press.

Jenkins, J.J. (1979).  Four points to remember: a tetrahedral model of memory experiments.  In L.S. Cremak, & F.I.M. Craik (Eds.), Levels of processing in human memory (pp. 429-446).  Hillsdale, NJ: Lawrence Erlbaum.

Jensen, A.R. (1998).  The g factor: the science of mental ability.  Westport, CT: Praeger.

Kenny, A. (2004).  The unknown God.  London: Continuum.

Malcolm, N. (1963).  Knowledge and certainty.  Englewood Cliffs, NJ: Prentice-Hall.

McGinn, C. (1996).  The character of mind.  Oxford: Oxford University Press.

Mermin, N.D. (1993).  Lecture given at the British Association Annual Science Festival.  London: British Association for the Advancement of Science.

Misner, C.W., Thorne, K.S., & Wheeler, J.A. (1973).  Gravitation.  San Francisco: Freeman.

Oakeshott, M. (1975).  On human conduct.  Oxford: Clarendon Press.

Oppenheimer, R. (1955, September 4).  Analogy in science.  Paper presented at the 63rd Annual Meeting of the American Psychology Association, San Francisco, CA.

Richardson, K. (1999).  The making of intelligence.  London: Weidenfeld & Nicolson.

Roediger III, H.L. (2008).  Relativity of remembering: Why the laws of memory vanished.  Annual Review of Psychology, 59, 225-254.

Stapp, H.P. (1993).  Mind, matter, and quantum mechanics.  Berlin: Springer-Verlag.

Sternberg, R.J., & Detterman, D.G. (Eds.). (1986).  What is intelligence?  Contemporary viewpoints on its nature and definitions.  Norwood, NJ: Ablex Publishing Corporation.

Tulving, E. (2007).  Are there 256 different kinds of memory?  In J.S. Nairne (Ed.), The foundations of remembering: Essays in honour of Henry L. Roediger III (pp. 39-52).  New York: Psychological Press.

Wittgenstein, L. (1953).  Philosophical investigations.  Oxford: Blackwell.

Wright, C. (2001).  Rails to infinity: Essays on themes from Wittgenstein’s Philosophical Investigations.  Cambridge, MA: Harvard University Press.

Why OECD Pisa cannot be rescued

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PISA cannot be rescued by switching IRT model because all IRT modelling is flawed.

Dr Hugh Morrison (The Queen’s University of Belfast [retired])drhmorrison@gmail.com

On page 33 of the Times Educational Supplement of Friday 25th November 2016, Andreas Schleicher, who oversees PISA, appears to accept my analysis of the shortcomings of the Rasch model which plays a central role in PISA’s league table.  The Rasch model is a “one parameter” Item Response Theory (IRT) model, and Schleicher argues that PISA’s conceptual difficulties can be resolved by abandoning the Rasch model for a two or three parameter model.  However, my criticisms apply to all IRT models, irrespective of the number of parameters.  In this essay I will set out the reasoning behind this claim.

 

One can find the source of IRT’s difficulty in Niels Bohr’s 1949 paper entitled Discussion with Einstein on Epistemological Problems in Atomic Physics.  Few scientists have made a greater contribution to the study of measurement than the Nobel Laureate and founding father of quantum theory, Niels Bohr.  Given Bohr’s preoccupation what the scientist can say about aspects of reality that are not visible (electrons, photons, and so on), one can understand his constant references to measurement in psychology.  “Ability” cannot be seen directly; rather, like the microentities that manifest as tracks in particle accelerators, ability manifests in the examinee’s responses to test items.  IRT is concerned with “measuring” something which the measurer cannot experience directly, namely, the ability of the examinee.

 

IRT relies on a simple inner/outer picture for its models to function.  In IRT the inner (a realm of timeless, unobserved latent variables, or abilities) is treated as independent of the outer (here examinees write or speak responses at moments in time).  This is often referred to as a “reservoir” model in which timeless abilities are treated as the source of the responses given at specific moments in time.

 

As early as 1929 Bohr rejected this simplistic thinking in strikingly general terms: “Strictly speaking, the conscious analysis of any concept stands in a relation of exclusion to its immediate application.  The necessity of taking recourse to a complementary … mode of description is perhaps most familiar to us from psychological problems.”  Now what did Bohr mean by these words?  Consider, for example, the concept “quadratic.”  It is tempting to adopt a reservoir approach and trace a pupil’s ability to apply that concept in accord with established mathematical practice to his or her having the formula in mind.  The guidance offered by the formula in mind (Bohr’s reference to “conscious analysis”) accounts for the successful “application,” for example, to the solution of specific items on an algebra test.

 

However, this temptingly simplistic model in which the formula is in the unobserved mental realm and written or spoken applications of the concept “quadratic” take place in the observed realm, contains a fundamental flaw; the two realms cannot be meaningfully connect.  The “inner” formula (in one realm) gets its guidance properties from human practices (in the other realm).  A formula as a thing-in-itself cannot guide; one has to be trained in the established practice of using the formula before it has guidance properties.  In school mathematics examinations around the world, pupils are routinely issued with a page of formulae relevant to the examination.  Alas, it is the experience of mathematics teachers everywhere that simply having access to the formula as a thing-in-itself offers little or no guidance to the inadequately trained pupil.  The formula located in one realm cannot connect with the applications in the other.

 

Wittgenstein teaches that no formula, rule, principle, etc. in itself can ever determine a course of action.  The timeless mathematical formula in isolation cannot generate all the complexities of a practice (something which evolves in time); rather, as Michael Oakeshott puts it, a formula is a mere “abridgement” of the practice – the practice is primary, with the formula, rule, precept etc. deriving its “life” from the practice.

 

Returning to Bohr’s writing, it is instructive to explain his use of the word “complementarity” in respect of psychology and to explain the meaning of the words: “stands in a relation of exclusion.”  Complementarity was the most important concept Bohr bequeathed to physics.  It involves a combination of two mutually exclusive facets.  In order to see its relevance to the validity of IRT modelling, let’s return to the two distinct realms.

 

We think of the answers to a quadratic equation as being right or wrong (a typical school-level quadratic equation has two distinct answers).  In the realm of application this is indeed the case.  When the examinee is measured, his or her response is pronounced right or wrong dependent upon its relation to established mathematical practice.  However, in the unobserved realm, populated by rules, formulae and precepts (as things-in-themselves), any answer to a quadratic equation is simultaneously right and wrong!

 

A formula as a thing-in-itself cannot separate what accords with it from what conflicts with it, because there will always exist an interpretation of the formula for which a particular answer is correct, and another interpretation for which the same answer can be shown to conflict with the formula.  Divorced from human practices, the distinction between right and wrong collapses.  (This is a direct consequence of Wittgenstein celebrated “private language” argument.)  This explains Bohr’s reference to a “relation of exclusion.”  In simplistic terms, the unobserved realm, in which answers are compared with the formula for solving quadratics, responses are right-and-wrong, while in the observed realm, where answers are compared with the established practice, responses are right-or-wrong.

 

On this reading, ability has two mutually exclusive facets which cannot meaningfully be separated.  The distinguished Wittgenstein scholar, Peter Hacker, captures this situation as follows: “grasping an explanation of meaning and knowing how to use the word explained are not two independent abilities but two facets of one and the same ability.”  Ability, construed according to Bohr’s complementarity, is indefinite when unobserved and definite when observed.  Moreover, this definite measure is not an intrinsic property of the examinee, but a property of the examinee’s interaction with the measuring tool.

 

Measurement of ability is not a matter of passively checking up on what already exists – a central tenet of IRT.  Bohr teaches that the measurer effects a radical change from indefinite to definite.  Pace IRT, measurers, in effect, participate in what is measured.  No item response model can accommodate the “jump” from indefinite to definite occasioned by the measurement process.  All IRT models mistakenly treat unmeasured ability as identical to measured ability.  What scientific evidence could possibly be adduced in support of that claim?  No IRT model can represent ability’s two facets because all IRT models report ability as a single real number, construed as an intrinsic property of the measured individual.

 

 

 

The problem of Social Mobility explained

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how-not-to-be-a-hypocriteWhile the mainstream media offer endless analysis and political party talking heads pontificate on the issue of grammar schools and social mobility, the explanation for any reduction of social mobility is made clear by actions of these Members of Parliament

It is not the grammar schools  which are responsible for restricting social mobility but those influential people who had the benefit of receiving private, independent schooling or attended a grammar school and then denying to others something that improved their own social mobility.

All those illustrated in this post would benefit from reading a copy of Adam Swift’s book, How not to be a hypocrite.how-not-to-be-a-hypocrite

 

 

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Stroud MP Neil Carmichael Conservative chairman of the Education Select Committee told Radio Four’s Westminster Hour:

We have serious issues about social mobility, in particular white working-class young people, and I don’t think that having more grammar schools is going to help them

 

Neil Carmichael boarded at St Peter’s, an independent school in York that dates back to AD627 and includes among its alumni Guy Fawkes, cricketer Jonny Bairstow and actor Greg Wise. Today to send your son to board would cost £27,375 a year.

Neil Carmichael’s Wikipedia page makes reference to him being a hypocrite.

 

 

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John Pugh, Liberal Democrat education spokesperson condemns grammars. Pugh attended Prescot and Maidstone grammars, and taught in the independent sector at Merchant Taylors’ Boys’ School

 

 

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Jeremy Corbyn MP, leader of the Labour Party was educated at Castle House Independent Preparatory School and Adams Grammar School.

In 1999, the MP split from his first wife over a conflict over their son’s education. His wife, Claudia Bracchita, explained:

We had to make the right decision in the interests of our child. We would have been less than human if we had done anything else.

 

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Sir Michael Wilshaw attended Clapham College Grammar School.

The Ofsted Chief recently made a plea to Theresa May, the Prime Minister,  to stop grammar schools He told Nick Ferrari on LBC, Leading Britain’s Conversation,

We need more than the top 10 or 20% of youngsters to do well in our economy and in our society.

Sir Michael Wilshaw has conveniently ignored the Northern Ireland education system which is entirely selective.  Northern Ireland leads the UK in performance in GCSE and A-level examinations and has only one private post-primary school.

Update 13th September via Guido Fawkes

toynbee

Polly Toynbee today attacks Theresa May’s grammar school plans, arguing that segregation by social class is “irrational” and claiming grammars add to “splits and divisions” in society. She has some front. Polly herself failed her 11-plus and attended the independent Badminton School. Earning £110,000-a-year at theGuardian meant she was able to send two of her children to private school as well. Today Toynbee writes that “inequality is monstrously unfair… it means birth is almost always social destiny”. Some children are evidently more equal than others.Is there a bigger hypocrite in the grammar schools debate?