Sometimes the teaching of thinking is contrasted with the teaching of conventional subject matter; thinking ability

and domain-specific knowledge are viewed as opposing educational goals. This is unfortunate. Thinking ability is
not a substitute for knowledge; nor is knowledge a substitute for thinking ability. Both are essential. Knowledge
and thinking ability are two sides of the same coin. They are the yin and yang of intellectual competence and
rational behaviour. Raymond Nickerson, David Perkins and Edward Smith, The Teaching of Thinking'

The DIKR people seem to be in favour of thinking - yet they show surprisingly little interest in understanding what it
is and how it grows. Daisy Christodoulou's index shows that her only interest in thinking (as we shall see in a
moment) is in proving that you can't expect children to think because they are 'novices' and not 'experts'. Tom
Bennett's only contribution to thinking about thinking is his strenuous efforts to rubbish Edward de Bono's idea of
the "Six Thinking Hats" (a useful little classroom technique that has, I freely admit, sometimes been overhyped and
fetishised). In neither of David Didau's large tomes does thinking appear in the index at all. And the Michaela book
doesn't even trouble us with an index.

It is unwise to make any sweeping statements about something if you don't know what it is. So here's what I think
thinking is. We are thinking when certain trains of verbal or symbolic thought, or sensory images, happen in
consciousness and seem to lead to conclusions or insights that were latent in what we already knew, or in what new
information we are receiving, but not yet explicit. Sometimes these thoughts appear as the result of deliberate
chains of deductions; and sometimes they just 'occur to us' or 'pop into our heads. When we are thinking about
something, we are making connections between facts and ideas. We are turning new information or ideas into our
own words, or linking them to our own experiences, interests or existing frameworks (sometimes called 'schema'):
we are trying them on for size, we might say. We are looking for synergies or contradictions with what we already
know or believe. We are exploring in our minds the possible new uses to which existing information can be put. In
other words, thinking is what we do to broaden and deepen our understanding of the validity and utility of the
'knowledge claims' that come our way. Registering things without actively engaging with them in this way is called
remembering.

Now obviously there are better and worse ways of thinking. We can think in ways that are hasty, leaping to
conclusions or failing to explore the different nuances or ramifications of an idea. We may admit premature defeat if
our first line of thought doesn't seem to get anywhere. We may sneak in some of our own interpretations and skew
our thinking to suit our own ends, without giving a new idea a chance to speak for itself (what is called in
psychology my-side bias). Thinking can be sophistical, full of rhetorical tricks designed only to win an argument and
not to 'get at the truth'. Or thinking may be hazy, failing to make important distinctions. (For example, on Earth,
you can often get away with treating mass and weight as if they were the same thing, but out in space, you can't.)
Thinking can be stereotyped, relying only on familiar tramlines of associations and opinions and missing what might
be new or 'thought-provoking'? By contrast, thinking can be clear, precise, cogent, imaginative, fair-minded and
complex. And it is perfectly possible to imagine two people who know roughly the same amount about a subject
but who differ wildly in their ability or inclination to think well about it. One might be pugnacious and dogmatic;
the other questioning and convivial.

Take, as a case in point, the current widespread interest in education in the idea of critical thinking. Broadly, this
means being inclined to probe what you are told and to ask incisive questions to establish the validity of facts and
arguments. For many audiences of teachers I have spoken to, in this age of fake news and internet fraud, this ranks
pretty high as a thinking disposition of real-life value. (If I had a teenage son or daughter, I would certainly want
them to have their critical antennae twitching while they were chatting and exploring online.) Is factual knowledge
necessary to support this fact-checking? Of course it is. Both knowledge and the disposition to question it are
necessary, but neither of them on their own is sufficient to be a critical thinker. All the knowledge in the world won't
make up for being credulous, and all the analytical skill in the world won't make up for ignorance. Do people differ
in their tendency to deploy this kind of critical thinking when it is needed? Certainly, some are much more gullible
than others are.

Is this something that we as teachers might be able to help our students to cultivate? I see no reason at all why not
- if we have the will. They certainly do in Finland, a country which comes close to the top of international league
tables on 'resistance to fake news. In recent reforms to the Finnish national curriculum, according to an article in
The Guardian by journalist Jon Henley, real-world critical thinking has become a core, cross-subject component.
Henley reports that, "In maths lessons... pupils learn how easy it is to lie with statistics. In art, they see how an
image's meaning can be manipulated. In history, they analyse notable propaganda campaigns." Even in primary
schools, apparently, children are taught the difference between misinformation (simple mistakes), disinformation
(deliberate lies and hoaxes) and mal-information or 'gossip' (which may be factually correct but which is intended to
manipulate). One Helsinki headteacher said, "Even quite young children can grasp this. They love being
detectives." He went on:

The goal is active, responsible citizens and voters. Thinking critic-ally, fact-checking, interpreting and evaluating all
the information you receive, wherever it appears, is crucial ... Kids today don't look for news, they stumble across it
on WhatsApp, YouTube, Instagram, Snapchat. Or, more precisely, an algorithm selects it, just for them. They must
be able to approach it critically. Not cynically - we don't want them to think everyone lies - but critically.*

He wants his students to be on the alert for dubious claims, always ready to ask probing questions such as: who
produced this information, and why? Where was it published? What does it really say? Who is it aimed at? What is it
based on? Is it verifiable elsewhere? And what is the point? Asked (by a British journalist) why fake news mattered,
one 17-year-old pointedly replied, "Because [otherwise] you end up with false numbers on the side of a bus, and
voters who believe them." And remember that, despite such changes, Finland continues to do well on the
conventional PISA tests, compared to the other OECD countries surveyed. Contrary to DIKR dogma, it seems that,
in Finland at least, there is no conflict between developing a healthy scepticism, mastering necessary knowledge
and doing well on the tests." As Raymond Nickerson and his colleagues conclude in their comprehensive review of
teaching thinking,

Sometimes the teaching of thinking is contrasted with the teaching of conventional subject matter; thinking ability
and domain-specific knowledge are viewed as opposing educational goals. This is unfortu-nate. Thinking ability is
not a substitute for knowledge; nor is knowledge a substitute for thinking ability. Both are essential. Knowledge
and thinking ability are two sides of the same coin. They are the yin and yang of intellectual competence and
rational behaviour?

Thinking tools
There are lots of tools that people can be directly taught and which help them think better - not only in the context
of specific subjects but more broadly. Psychologists are very familiar with such tools: they call them mindware.®
There are useful maxims, caveats and rules of thumb. "Measure twice and cut once" doesn't only apply to
carpentry; it is a general reminder that being thoughtful and precise saves trouble in the long run. "You can't
deduce an ought from an is one saying that I regularly find a useful reminder. "A correlation doesn't imply a causal
relationship" is another that all psychology undergraduates get stamped into their brains in first-year lab classes.
Learning about Venn diagrams in the sixth form at school has served me in good stead. Mind maps sometimes help
me organise my thoughts on a subject.

One practical technique that has gained wide recognition in the business world is the pre-mortem. As Gary Klein,
the American psychologist who originally devised the method, explains,

A pre-mortem is the hypothetical opposite of a post-mortem. A postmortem in a medical setting allows health
professionals and the family to learn what caused a patient's death ... A pre-mortem... comes at the beginning of a
project rather than the end, so that the project can be improved rather than autopsied.

In a business context, a typical pre-mortem begins after a team has been briefed, say, on a proposed project. The
leader, projecting into the future, tells everyone that the project has failed spectacularly. Participants in the meeting
immediately have some time to try to think up, and record, every possible reason why the project failed, and then
group members take it in turn to share items from their list. The whole team then spends the rest of the meeting
redesigning the project to accident-proof it against all these possible sources of failure.
I can't see any reason why we should not introduce school students to the idea of the pre-mortem - in the context
of sixth form projects, say - and see if they might prove useful. The activity is just one amongst many potentially
useful techniques for hacking one's own mind and guarding against common pitfalls in thinking.

Experts vs. novices

A recurring theme in DiKRish thinking is reducing complex issues to binary oppositions. One example is the claim
that 'novices' and 'experts' are completely different, and that this tells us a lot about what children can and can't do
and therefore how we should treat them in school.
Children are novices and don't know much. Experts are older and have lots of knowledge. So, Christodoulou
asserts, "the way experts approach and solve problems is fundamentally different from the way novices approach
problems". And from this she infers, "Not only is it unrealistic to expect primary-age pupils to think like experts. It is
unrealistic to expect school leavers to think like experts. It is not a legitimate aim for primary or secondary
schooling."" And from this she goes on to conclude that it is a waste of time to get students thinking about what
they are learning, because they are not experts yet.

Didau in What If tells us, "we need to remember that children are not experts and that the only way they are likely
to become experts is by learning facts about the subjects we teach.""' He warns against trying to help your
students to move towards deeper understanding and more flexible expertise by

encouraging them to think about content in deeper, more abstract terms, so that they are better able to generalise
what they learn to new contexts. Regrettably this doesn't work... Any attempt to short-cut the process is only likely
to lead to mimicry and inflexibility.

He goes on to inform us rather gnomically, "mimicry is a necessary waiting room in the chaos of liminal space", but
regrettably he doesn't refer us to any research that clarifies or buttresses these rather surprising assertions.
But hold on a minute. How do experts get to be experts? Daniel Willingham says, "We can't be experts until we
put in our hours." Put
slightly more accurately, we should say, "We can't be experts unless we put in our hours." But what is the novice
doing during those hundreds or thousands of hours? Are they just passively (but attentively) sitting and absorbing
masses of information: just mugging stuff up for a test? Not at all. If you look at the research on how apprentices
become masters of their craft, it is through a long journey of skill development: time spent trying to do things
better than they currently can and using a lot of trial and error as they go. To learn from your errors requires
understanding where you went wrong and trying to improve. The long slow road to expertise is through practising,
conjecturing, experimenting and correcting. It also involves watching what more able exponents do and listening to
how they think out loud. Modelling and imitating are powerful drivers of the development of expertise. In fact
research has shown that early apprentices learn most not from the experts themselves but from somewhat more
senior apprentices, whose practice and thinking shows the beginner what the next steps along the road to
proficiency look and sound like.

Some explicit, declarative knowledge is useful in skill development, but it is knowledge that can be immediately
applied, or which helps you see how you could develop your expertise, not knowledge per se. You don't get good
at taking penalty kicks, playing the violin, talking in public - or thinking critically and logically - just by having your
head filled with lots of information.

Cognitive apprenticeship
How does the idea of learner as apprentice apply in the context of the school curriculum? The idea that school can
profitably be seen as a cognitive apprenticeship has been around for a while. One version of it has been developed
by eminent American cognitive scientist (and one of the founders of the field of the learning sciences) Allan Collins
in collaboration with John Seely Brown (ex-chief scientist at the Xerox Corporation).' It involves teachers seeing
themselves as coaches of developing expertise in their subjects, rather than just as didactic conveyors of concepts
and information.

The coaches' job is to model, design activities and provide feedback and to introduce only as much theory as
directly supports the learners' here-and-now skill development. Collins and Seely Brown argue that the
apprenticeship model could usefully be applied to the teaching of cognitive or scholarly expertise - to the
development of intellectual skill - just as much as to practical or physical prowess.

A cognitive coaching approach to mathematics, for example, may involve the teacher in not only explaining a
mathematical procedure but also making explicit the rules of thumb and learning strategies that expert
mathematicians may use when approaching a tricky problem and model-ling the use of such processes. Collins and
Seely Brown refer to the work of a renowned American maths educator called Alan Schoenfeld, who would
regularly invite his classes to throw a tricky maths problem at him out of the blue and then think aloud as he tried to
solve it. He would then give his students challenging problems to work on and coach them in the use of the
strategies which he had been modelling. He would start by offering a suitable amount of support or 'scaffolding' to
help them get going, but would then progressively and deliberately reduce (or 'fade') the support so that students
got used to activating the thinking strategies for themselves and not waiting for the teacher to prompt them. '

If all the teachers in a school were using congruent but complementary teaching approaches, suited to their own
subjects, we could see school itself as providing a more general kind of epistemic apprenticeship in which the
learning and thinking skills of each subject were gradually adding up to more than the sum of their parts. We could
see school as an all-round mental training in which cognitively more adept people (teachers) design activities
(learning hard stuff) and model ways of engaging with it (thinking out loud) so that beginning thinkers (students)
become progressively better at thinking and learning. From this perspective, the content of the curriculum has a
dual role. It is - or ought to be - patently valuable in its own right (as I argued in the previous chapter), and it also
serves as an appropriately graded series of exercises for developing ever more powerful ways of thinking and
learning (students' epistemic mentality).

We could go even further and follow the influential work of cognitive anthropologists Jean Lave and Etienne
Wenger. They suggest that the epi-stemic apprenticeship shapes not only students' ways of thinking about their
range of subjects but also their sense of themselves as increasingly adept (or inept) thinkers and learners more
generally: their epistemic iden-tities. They pick up ways of thinking and talking about learning; ideas about how
long learning takes and what it involves; tacit understandings of how knowledge is made and who has a right to
make it and critique it; what kinds of learning and knowing are worthwhile and what are considered trivial or
inferior; and I, critically, where they themselves stand and what epistemic status they can rightfully claim for
themselves. Some children might emerge from school seeing themselves as failures at learning, or as people who
have no voice in the knowledge-making and critiquing business. Others might take away a sense of their own
entitlement to argue, debate and have opinions of their own. (This brings us back to my image of the learning river,
and the importance of teachers being aware of the effects they are having on the development of these identities.)

To hark back to the discussion of the values and purposes of education (chapter 2) for a minute, we could argue
that this kind of epistemic apprenticeship is not only possible but also highly desirable, especially at the moment.
At exactly the time when societies are facing the hugely complex, tightly interwoven problems of climate change,
population increase, mass migration, growing inequities, territorial disputes and novel pandemics, leaders are
emerging all around the planet who are populist, nationalistic and divisive. Their rhetoric plays on people's fears
and prejudices. If our creaking democracies are to stand any chance of rising to these challenges in an intelligent
and nuanced way, there must be citizens in their millions who can resist and debunk these dangerous ploys, and
who in their turn are capable enough and brave enough to face these challenges head on and think deeply about
them. As Jonathan Rowson has recently written, education should be in the business of "developing the capacity to
hold the emotional tension of mental complexity"' - and that means crafting a curriculum that not only informs
young people explicitly and accurately about this nest of wicked problems but also relentlessly stretches and
strengthens their capacity to meet such complexity with minds that are calm, balanced, open, subtle and acute.
Filling their heads with the unquestioned verities of yesteryear is not, on its own, going to achieve that.
Generic skills
One of the loudest campaigns of the DIKR lobby is against the attempt to teach what they call 'generic skills':
things like the abilities to communicate clearly, to solve problems and to think critically and creatively. Yes, says
E.D.
Hirsch, "we need a citizenry that is able to communicate, solve problems, master computers, think critically,
innovate, and adapt." But the attempt to teach these skills head-on is doomed. "The emphasis on hands-on
skill-projects will shift class time away from subject matter and therefore impair the skills students need to be
productive citizens and participants in the workforce." And why 'therefore"? Because

the conception that 21st century skills are all-purpose muscles that, once developed, can be applied to new and
unforeseen domains of experience [is] an error that is fundamental and fatal... On the con-trary, reading
comprehension, communication, critical thinking and the rest are inherently constituted by specific knowledge.

In other words, knowledge and the abilities to communicate, manipulate and critique that knowledge are so closely
bound together that they can't be separated - just as scrambled egg can't be unscrambled - and so it is impossible
to peel off the skill part from the specific knowledge part and use it elsewhere. The claim is a strong one: not just
that much teaching does not in fact lead to transfer but that it is, in many instances, not possible to teach in a way
that leads to the progressive disembedding of epistemic expertise from one specific combination of content +
context + purpose, so that useful ways of thinking can be carried over to a different set of conditions. Clearly we
need to examine the evidence before we can accept such pessimism.
There is no doubt that transfer is a real problem in education, especially if we assume, as Perkins and Salomon do,
that the point of school is not just to know things but also to be able to put those things to good use in contexts
other than the classroom or the examination hall. The problem is that transfer happens much less often than we
would have hoped. Things learned in school frequently fail to come to mind in out-of-school situations where they
would have been useful. As Alan Schoenfeld has said, this is an "almost universal phenomenon: [the fact] that
students who are capable of performing symbolic operations in a classroom context... fail to connect [those]
procedures with the 'real-world' objects represented by the symbols constitutes a dramatic failure of instruction".

Even within school, small changes in the form of a problem - for example from the classroom worksheet to the
examination paper - can disrupt transfer. Many students who have learned to calculate the time it takes for a rock
dropped off a 100 m tower to fall to earth have been stymied by a problem that asks them to find the time it takes
for a rock that has been dropped down a 100 m well shaft to land." And conversely, 'real-world' knowledge also
fails to transfer into school-type situations. Ten-year-old Brazilian street children selling sweets at traffic lights in
Recife can calculate change in a jiffy but are completely stumped by the same calculations when they are written
down on paper.

Can this problem of the frequent lack of transfer be overcome? The traditional solution - built into the very fabric of
the education system - was to make school problems as abstract and remote from daily life as possible.
Traditionalists thought that by doing so the knowledge and skills used to solve these disembedded problems will
be registered in the brain untethered by particular contexts and purposes. They should float around freely in
cognitive space, like koi in a pond, ready to be hooked by any new situation to which they might be relevant. This
was the argument for prioritising subjects that had little obvious connection to life as lived by the majority of
students - subjects like algebra or Latin. School was designed on the assumption that learning could be
context-free, unfettered by personal feelings or connotations. Learning Latin, it was argued, could provide a
generic 'training of the mind' in a way that woodwork or 'current affairs' never could.

Unfortunately this assumption is clearly false. It doesn't work in either theory or practice. School can never be a
place of 'no context', for the obvious reason that it is a very specific - and rather odd - context indeed.
School learning is not 'off-the-job' at all. There are very clear jobs at which learning is aimed, like getting good
marks, avoiding punishment and mastering approved methods of displaying knowledge (such as knocking out
small essays to show that you have remembered and understood what you have been taught).
The problem, as we saw in the previous chapter, is that the brain just doesn't work the way the Trads believe. It is
not like a fishpond. It is more like a tangled organic undergrowth of associations and connections - and if new
knowledge is not anchored meaningfully to existing structures, it is hard to learn, easily forgotten and unlikely to be
retrieved when needed. That's why Trad teachers advocate the need for continual retrieving, retesting and
reminding: it is the only way you can possibly get any of this disembedded knowledge to stick. As Angeline Lillard
says, there are "many studies showing that embedding learning in a meaningful context is associated with better
learning, more interest, and greater embracing of challenges. No wonder transfer from conventional education is
disappointing.

Teaching for transfer

So are there styles of teaching that do help with the transfer problem: ones that capitalise more effectively on the
brain's natural proclivities? Jo Boaler, an ex-colleague of mine at King's College London and now professor of
education at Stanford University, has conducted several detailed, mixed-method investigations into the effects of
different ways of teaching maths. Her first study compared two schools on the south coast of England which had
very similar demographics but different approaches to teaching. At Amber Hill, teachers adopted a traditional style,
lecturing and explaining mathematical procedures to the whole class, showing them some worked examples, and
then setting them to work their way, on their own, through a series of similar problems. Students were disciplined,
hardworking, exam-focused and keen to do well. If they got stuck on a problem, they would ask the teacher and
maybe refer back to the worked example.

In contrast, Phoenix Park took a more progressive approach, prompting students with open-ended mathematical
questions for them to explore on their own or in small groups over the course of several lessons. The questions
were things like, "What's the biggest sheep-pen you could make with 36 same-sized fencing panels?" or "Find
some objects that have an area of 216 square cms." Students got stuck all the time, and worked hard to figure
things out for themselves. But they were not left to flounder com-pletely. If they needed to know a mathematical
method they hadn't met before, their teachers would explain it to them in the context of the problem they were
working on. For example, in the problem of the 36 fences, some of the students wondered if the biggest shape
might be a 36-sided polygon.

To find its area they had to learn about trigonometrical ratios, so their teacher taught them the trigonometry they
needed. Teachers continued with this problem-based approach up until just a few months before the students were
to take their high-stakes GCSE examinations at age 16. In these exams, Boaler found that the Phoenix Park
students did slightly worse than the Amber Hill students on questions that required routine calculation but
significantly better on conceptual problems that needed more thought. They were less thrown by problems being
posed in an unfamiliar way and more resourceful in applying their knowledge.

However, the biggest differences came when Boaler asked the two groups of students about transfer. She says:

I asked the Amber Hill students if they used mathematics outside school. They all said they did - many of them had
part-time jobs at that stage - but when I asked the students if they made use of methods they learned in school,
they all said they did not.

They felt that school maths belonged in a world of its own. For example Richard in Year 10 said, "When I'm out of
school, the maths from here [school] has nothing to do with it, to tell you the truth ... Most of the things we've
learned in school we would never use anywhere." Whether school maths was potentially useful or not, the Amber
Hill students did not perceive it as such. The Phoenix Park students, on the other hand, told quite a different story.
Over three-quarters of them said they made good use of the maths they had learned in school and didn't see a
difference between what they had learned and the real-world applications which they were now using (in their
part-time jobs). Gavin in Year 10 said, "When I'm out of school now, I can connect back to what I done in class so I
know what I'm doing." And John adds, "It just comes naturally, once you've learned it you don't forget."

Unusually, Boaler went to the trouble of following up the two cohorts of students eight years later, and asked them
again about the relevance of their school maths to their current working lives. Of those who replied, Boaler
selected ten from each school to create matching groups that she then interviewed. Most strikingly, none of the ten
from Amber Hill said that the mathematics they had learned at school was useful in their jobs. Typically, Trevor said,
"Stuff like pi and trigonometry - that's never been useful to me since." By contrast, all ten from Phoenix Park
reported that their school maths was useful to them now. For example, Andrew said, "There's a lot of things I can
relate back to maths in school... I suppose maths is about problem-solving to me... it's about being logical." And
Simon added,

It's not about the nits and grits of... what you're actually taught in school. It's about the way you manage it ... I think
if you struggle with something, you find ways around it, don't you? That's what I took with me.

Clearly the Phoenix Park method of teaching had left students with a lifelong confidence and capacity for thinking
and problem-solving. There was beneficial transfer from one method of teaching but not from the other. Of course,
as with all education research, there is a trade-off in Boaler's work between real-life relevance, in all its complex
messiness, and artificially slimmed-down methodological rigour. Though Amber Hill and Phoenix Park were well
matched, they are just case studies of two particular schools, and there could be other explanations of the
differences than just their different pedagogies. So since moving to Stanford, Boaler and her colleagues have
carried out a similar study on three high schools in California. Two were suburban, similar in achievement at the
start of the study, but the third, Railside, lagged behind. From a poor and diverse inner city neighbourhood,
Railside was, according to the locals, on 'the wrong side of the tracks' - and so close to the tracks, in fact, that
lessons frequently had to be stopped while the trains went by. Despite these disadvantages, after two years of the
study, students at Railside had overtaken their more fortunate peers, showing significantly higher levels of
achievement and more positive attitudes towards maths. By their senior year, 41% of Railside students were taking
advanced classes in calculus compared to 27% in the other two schools, and many more Railsiders were planning to
pursue maths in college. In addition, after two years of the study, Railside had wiped out the achievement gap
between White, Black and Hispanic/Latino students, differences that persisted in the other two schools.

How come? Many features of the teaching at Railside may have contributed to their students' different attitudes.
Lessons were 90 minutes long. Students worked in heterogeneous groups on challenging problems. Teachers
would tell them what they need to know, but didn't rescue them from difficulty - they had to think for themselves
and justify their thinking.

Ana, a Year 3 student, said, when talking about her teacher,

She won't... tell you how to do it ... There's a lot of times when she's just like - "well, think about it" - and then
she'll walk off and that kills
me! That really kills me! But it's cool. I mean it's like, it's alright, you know: I'll solve it myself.

At Railside, it appears that students were not just learning 'how to do maths'; they were learning to think, to
collaborate, to listen to and respect each other's contributions, and to persist in the face of difficulty. When all the
students in the study were asked to estimate how long they would grapple with a maths problem before giving up,
the Railsiders said, on average, just under 20 minutes. The students from the other schools said less than 10.

There was also a lot of what you might call metacognitive coaching at Railside. Teachers would notice, comment on
and, if necessary, expand on examples of good thinking that students were using. For example, Arturo was stuck
on a problem, and his teacher, Guillermo, asked him to articulate exactly what it was that he was confused about.
As Arturo searched for a way of asking his question more precisely, he realised for himself what it was he needed to
do and carried on with working out the problem. He concluded that the answer was "550 pennies" wait, that's not
very much"', but then stopped himself saying, "no, At this point Guillermo intervened. Talking to Arturo in front of
the whole class, he said:

Two things just happened. Number one is, you stopped to ask yourself a good question and then suddenly you
had ideas. And number two, you checked out the answer and you realised it wasn't reasonable and that is excellent
because a lot of people would have just left it there.?
Expanding on Arturo's thought processes, Guillermo is teaching the class as a whole to recognise and reinforce
their own productive learning practices and thereby strengthen their ability to be more independent and
resourceful in their approach to learning. Is it helpful to try to categorise this as either progressive or traditional
teaching? Not at all. It has elements of both, problem-based learning and rigorous mathematics, for example, but it
also transcends both. It is a kind of teaching that develops know-ledge, expertise and epistemic character - all
three layers of the learning river - at once, and the result is deeper learning and broader and longer-lasting transfer.

Summary
We will talk more about the distinction between skills and dispositions in the next chapter, but I need to say
something about it here in order to wrap up this chapter. In a nutshell, a disposition is a skill that one is inclined or
disposed to use (in certain circumstances). So dispositions are matters of degree: I may become more or less
disposed to accept 'news' at face value, for example, or to stick with things that I am finding difficult. And I may
refine (broaden or narrow) the range of circumstances that trigger a disposition.
As children grow up, they shift the range of people who they find admirable and worth imitating or learning from,
for instance.

So yes, it is true that there are probably no completely 'generic' skills.

And yes, it is true that 'thinking skills' cannot be learned quickly and easily through direct instruction. But it is also
true that such dispositions can be cultivated over time, becoming broader or more targeted, stronger and more
robust, and more sophisticated, if the culture is conducive. The DIKR people are right only within a very narrow
conception of education: one that only recognises the learning that is happening on the surface of the river. With a
slightly more nuanced and accurate model of the classroom, they are not right at all. Classrooms necessarily
provide an apprenticeship in ways of thinking, as well as a diet of information to be learned. The question is: are
those ways of thinking transferable to, and valuable for, out-of-school contexts and purposes, or are they designed
principally to enable smooth management of large institutions and to produce creditable test scores?