Category Archives: Philosophy

Still Working Away In Our Silos (Thank Goodness)

If a thing is worth doing, it is worth doing badly.

–G. K. Chesterton, What’s Wrong With The World (1910)

Why are teachers beavering away in their individual silos, each one of us spending hours reinventing each pedagogic wheel, crafting schemes of work and resources for the new GCSEs?

Wouldn’t life be so much easier and better if we simply shared…?

To which I say: NO!

To be honest, my favourite part of the job is designing, crafting and re-designing resources and teaching approaches. They’re not perfect, of course. I’m reminded of a line from the opening credits of South Park: “All celebrity voices are impersonated . . . poorly.” As Chesterton remarked, if a thing is worth doing, it is worth doing badly.

But the point is, my approaches and resources are a lot less imperfect than they used to be. I flatter myself that, over the years, some of them have become . . . quite good. I believe Michael Stipe once said that in the entire history of the world there were only ever five rock and roll songs; and that REM could play two of them quite well. There’s a parallel in that most teachers have a lesson or two (or three) that they — and they alone — can teach brilliantly.

I often think that, given the right context, most students prefer shabby, bespoke individualism rather than shiny mass-produced perfection.

As teachers, I think we sometimes overestimate the impact that we have on our students. There is no royal road to learning, and neither can all our craft and pedagogic arts construct a conveyor belt either.

As educators, the most we can hope to do is clear a few stones out of the way of our charges as they set out on the rocky path to learning.

In the end, the journey is theirs. Let us wish them well as we watch from our silos . . .

The difficulty of obtaining knowledge is universally confessed [ . . .] to reposite in the intellectual treasury the numberless facts, experiments, apophthegms and positions, which must stand single in the memory, and of which none has any perceptible connexion with the rest, is a task which, though undertaken with ardour and pursued with diligence, must at last be left unfinished by the frailty of our nature.

Samuel Johnson, The Idler, 12 January 1760


Filed under Education, Humour, Philosophy

Learning Is For The Birds

​Well versed in the expanses
that stretch from earth to stars,
we get lost in the space
from earth up to our skull.

Wislawa Szymborska, To My Friends

What do we mean by learning? To tell the truth, even as a teacher of twenty-five years experience, I am not sure. 

Professor Robert Coe has suggested that learning happens when people have to think hard. In a similar vein, Daniel Willingham contends that knowledge is the residue of thought. Siegfried Engelmann proposes that learning is the capacity to generalise to new examples from previous examples. I have also heard learning defined as a change in the long term memory.

One thing is certain, learning involves some sort of change in the learner’s brain. But what is acknowledged less often is that it doesn’t just happen in human brains.

Contrary to standard social science assumptions, learning is not some pinnacle of evolution attained only recently by humans. All but the simplest animals learn . . . [And some animals execute] complicated sequences of arithmetic, logic, and data storage and retrieval.
— Steven Pinker, How The Mind Works (1997), p.184

An example recounted by Pinker is that of some species of migratory birds that fly thousands of miles at night and use the constellations to find North. Humans do this too when we find the Pole Star.

But with birds it’s surely just instinct, right?

Wrong. This knowledge cannot be genetically “hardwired” into birds as it would soon become obsolete. Currently, a star known as Polaris just happens to be (nearly) directly above the Earth’s North Pole, so that as the Earth rotates on its axis, this star appears to stand still in the sky while the other stars travel on slow circular paths. But it was not always thus.

The Earth’s axis wobbles slowly over a period of twenty six thousand years. This effect is called the precession of the equinoxes. The North Star will change over time, and oftentimes there won’t be star bright enough to see with the naked eye at the North Celestial Pole for there to be “North Star” — just as currently there is no “South Star”.But there will be one in the future, at least temporarily, as the South Celestial Pole describes its slow precessional dance.

Over evolutionary time, a genetically hardwired instinct that pointed birds towards any current North Star or South Star would soon lead them astray in a mere few thousand years or so.

So what do the birds do?

[T]he birds have responded by evolving a special algorithm for learning where the celestial pole is in the night sky. It all happens while they are still in the nest and cannot fly. The nestlings gaze up at the night sky for hours, watching the slow rotation of the constellations. They find the point around which the stars appear to move, and record its position with respect to several nearby constellations. [p.186]

And so there we have it: the ability to learn confers an evolutionary advantage, amongst many others.


Filed under Philosophy, Science

The Curse of Zombie-Ofsted

In his wonderful book, The Mismeasure of Man, Stephen Jay Gould writes of the fallacy

of ranking, or our propensity for ordering complex variation as a gradual ascending scale. Metaphors of progress and gradualism have been among the most pervasive in Western thought . . . ranking requires a criterion for assigning all individuals to their proper status in the single series. And what better criterion than an objective number? . . . one number for each individual . . . to rank people in a single series of worthiness, invariably to find that oppressed and disadvantaged groups — races, classes, or sexes — are innately inferior and deserve their status. In short, this book is about the Mismeasure of Man.

Humankind seems to have an inveterate propensity for sorting the sheep from the goats. There seems to be nothing we enjoy more than placing people, races, genders, things and classes in their allocated place on some putative “Great Chain of Being.”

The Great Chain of Being is a hierarchical worldview developed in mediaeval and Renaissance times but originating from Plato and the neoplatonists. In this view, everyone and everything has its place. An eagle is superior to the “worm eating” robin; the lion is superior to the domestic dog or cat; but those furry familiars have warrant to lord it over the wolf and rabbit because of their greater utility to Man.

In other words, according to this view, Man is the paragon of animals, but is himself subject to the authority of angels and Heaven. All shall be well if each being in the Great Chain knows its place and does its allotted duty.

I believe that the Great Chain of Being is an enduring but largely unconscious idea: we notice its presence like a fish notices the presence of water — that is to say, not at all. Our continuing propensity for ranking is a comfortable habit of thought that, regrettably, all of us slip into as easily as a favourite pair of slippers.

The other fallacy identified by Gould in The Mismeasure of Man is that of

reification, or our tendency to convert abstract concepts into entities (from the Latin res, or thing). We recognize the importance of mentality in our lives and wish to characterize it, in part so that we can make the divisions and distinctions among people that our cultural and political systems dictate.

And so it continues. For example, Regional Schools Commissioner Dominic Herrington recently wrote to a school to ask for evidence that at least 80 per cent of teaching at the school “is rated to be good or better”, including in English and maths ( 6/11/15) — to my mind, demonstrating the fallacies of both ranking and reification simultaneously.

For goodness sake, not even Ofsted does that anymore!

However, the practice is, I suspect, still common in a large number of schools as part of their appraisal systems i.e. if you don’t get a “1” or a “2” in any one of your lesson observations then you “fail”.

The depressing truth is that even when Ofsted change their collective mind about an issue in response to evidence and reasonable argument (Yay! Go edu-bloggers!), their previous ideas and systems continue onward with almost undiminished energy, seemingly with a life and mind (or non-mind) of their own: Zombie-Ofsted, if you will.

To be fair to Ofsted, they have attempted to lay these walkers to rest by publishing clear and unequivocal guidance about their expectations about such nonsense as “minimal teacher talk” or “every lesson must include group work” and so on, but even such a well meaning stake-through-the-heart has made seemingly little headway against the strong winds of the Great Chain of Being.

Zombie-Ofsted marches, or lurches, ever onward.


Zombie-Ofsted marches -- or lurches -- ever onward

Like so much else in the crazy world of education these days, it makes the mind boggle. Or curdle. Or both.


Filed under Classroom Observation, Education, Ofsted, Philosophy

Engelmann and Direct Instruction (Part 3)

I’m going to begin this post by pondering a deep philosophical conundrum (hopefully, you will find some method in my rambling madness as you read on): I want to discuss the meaning of meaning.

Ludwig Wittgenstein begins the Philosophical Investigations (1953), perhaps one of the greatest works of 20th Century philosophy, by quoting Saint Augustine:

When they (my elders) named some object, and accordingly moved towards something, I saw this and I grasped that the thing was called by the sound they uttered when they meant to point it out. Their intention was shewn by their bodily movements . . . I gradually learnt to understand what objects they signified; and after I had trained my mouth to form these signs, I used them to express my own desires.
Confessions (397 CE), I.8

Wittgenstein uses it to illustrate a simple model of language where words are defined ostensively i.e. by pointing. The method is, arguably, highly effective when we wish to define nouns or proper names. However, Wittgenstein contends, there are problems even here.

If I hold up (say) a pencil and point to it and say pencil out loud, what inference would an observer draw from my action and utterance?


They might well infer that the object I was holding up was called a pencil. But is this the only inference that a reasonable observer could legitimately draw?

The answer is a most definite no! The word pencil could, as far as the observer could tell from this single instance, mean any one of the following: object made of wood; writing implement; stick sharpened at one end; piece of wood with a central core made of another material; piece of wood painted silver; object that uses graphite to make marks, thin cylindrical object, object with a circular or hexagonal cross-section . . . and many more.

The important point is that one is not enough. It will take many repeated instances of pointing at a range of different pencil-objects (and perhaps not-pencil-objects too) before we and the observer can be reasonably secure that she has correctly inferred the correct definition of pencil.

If defining even a simple noun is fraught with philosophical difficulties, what hope is there for communicating more complicated concepts?

Siegfried Engelmann suggests that philosopher John Stuart Mill provided a blueprint for instruction when he framed formal rules of inductive inference in A System of Logic (1843). Mill developed these rules to aid scientific investigation, but Engelmann argues strongly for their utility in the field of education and instruction. In particular, they show “how examples could be selected and arranged to form an example set that generates only one inference, the one the teacher intends to teach.” [Could John Stuart Mill Have Saved Our Schools? (2011) Kindle edition, location 216, emphasis added].

Engelmann identifies five principles from Mill that he believes are invaluable to the educator. These, he suggests, will tell the educator:

how to arrange examples so that they rule out inappropriate inferences, how to show the acceptable range of variation in examples, and how to induce understanding of patterns and the possible effects of one pattern on another. [loc 223, emphasis added]

Engelmann considers Mill’s Method of Agreement first. (We will look at the other four principles in later posts.)

Mill states his Method of Agreement as follows:

If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause (or effect) of the given phenomenon.
A System of Logic. p.263

Engelmann suggests that with a slight change in language, this can serve as a guiding technical principle that will allow the teacher to compile a set of examples that will unambiguously communicate the required concept to the learner, while minimising the risk that the learner will — Engelmann’s bête noire! — draw an incorrect inference from the example set.

Stated in more causal terms, the teacher will identify some things with the same label or submit them to the same operation. If the examples in the teaching set share only one feature, that single feature can be the only cause of why the teacher treats instances in the same way. [Loc 233]

As an example of an incorrect application of this principle, Engelmann gives the following example set commonly presented when introducing fractions: 1/2, 1/3, and 1/4.

Engelmann argues that while they are all indeed fractions, they share more than one feature and hence violate the Method of Agreement. The incorrect inferences that a student could draw from this set would be: 1) all fractions represent numbers smaller than one; 2) numerators and denominators are always single digits; and 3) all fractions have a numerator of 1.

A better example set (argues Engelmann) would be: 5/3, 1/4, 2/50, 3/5, 10/2, 1/5, 48/2 and 7/2 — although he notes that there are thousands more possible sets that are consistent with the Method of Agreement.

Engelmann comments:

Yet many educators believe that the set limited to 1/2, 1/3, and 1/4 is well conceived. Some states ranging from North Dakota to Virginia even mandate that these fractions should be taught first, even though the set is capable of inducing serious confusion. Possibly the most serious problem that students have in learning higher math is that they don’t understand that some fractions equal one or are more than one. This problem could have been avoided with early instruction that introduced a broad range of fractions. [Loc 261]

For my part, I find Engelmann’s ideas fascinating. He seems to be building a coherent philosophy of education from what I consider to be properly basic, foundational principles, rather than some of the “castles in the air” that I have encountered elsewhere.

I will continue my exploration of Engelmann’s ideas in subsequent posts. You can find Parts 1 and 2 of this series here and here.

The series continues with Part 4 here.


Filed under Direct Instruction, Education, Philosophy, Siegfried Engelmann

In Defence Of ‘Inadequate Philosophy’

[B]ecause all my moral and intellectual being is penetrated by an invincible conviction that whatever falls under the dominion of our senses must be in nature and, however exceptional, cannot differ in its essence from all the other effects of the visible and tangible world of which we are a self-conscious part.
— Joseph Conrad, Author’s Note to The Shadow-Line

Anthony Radice writes a provocative blog as The Traditional Teacher: whilst I often agree with much of what he says, sadly our foundational philosophies could not be further apart.

[P]revalent theories are having a disastrous impact on the world of education. Influenced by these theories, there are many nowadays who think that materialism can be justified by statements such as ‘Evidence suggests that ‘conscience’ and ‘consciousness’ and other mental processes are products of human brain activity’.

I wrote the quoted words in the comments of the Traditional Teacher’s previous blog post [21/6/15], and I stand by them still. I would describe myself as a methodological naturalist rather than as a materialist. The label “materialist” calls to mind the seventeenth century view that there is only “atoms and the void”. This is indeed a mechanistic philosophy perhaps best described as ontological naturalism: in other words, all that exists is atoms and the void. If we know the initial states of all the particles then it would seem that we then can predict the future state of the universe at any time. This does indeed suggest that the past, present and future are pre-determined.

However, it soon became clear that such a view could not be justified. Perhaps a two-body Newtonian system can be deterministic in the sense that its past, present and future can be calculated provided enough information about its state at one instant is known. However, the lack of an exact solution to the famous Three Body Problem shows that even mechanistic ontological naturalism does not automatically entail determinism.

Since methodological naturalism does not involve a commitment to an ontology but rather to a methodology (perhaps best exemplified by the empirical sciences, but not limited to them), it does not entail a commitment to any form of determinism either.

I believe the foregoing shows that both “flavours” of naturalism do not automatically lead to determinism. Mr Radice, however, is not impressed:

Indeed, we have reached the stage where many do not hold others responsible for their actions, at least in theory. Their materialistic determinism leads them to ‘explain’ actions in psychological or social or (insert favourite flavour of determinism) terms. But this doesn’t explain anything, because it leaves out the person. It removes humanity because it removes conscience and freedom. All humanity is excused because humanity, it turns out, does not exist.

Sadly, I do not follow his reasoning. If materialism does not entail determinism (as I think I have shown above), then it does not rule out conscience or freedom or humanity. In fact, methodological naturalism leads me to conclude that there is substantial evidential warrant for supposing that they do exist. And this in spite of the fact, as Mr Radice points out, that they “are not material objects subject to laboratory experimentation”. True, but irrelevant — so are many of the entities and concepts dealt with by modern science: virtual photons for example. I believe philosopher Robert T. Pennock puts it well:

Many people continue to think of the scientific world view as being exclusively materialist and deterministic, but if science discovers forces and fields and indeterministic causal processes, then these too are to be accepted as part of the naturalistic worldview . . . An important feature of science is that its conclusions are defeasible on the basis of new evidence, so whatever tentative substantive claims the methodological naturalist makes are always open to revision or abandonment on the basis of new, countervailing evidence.
Tower of Babel, pp.90-91

Mr Radice seems to believe that since an individual neuron cannot be conscious, this means that a collection of neurons (a brain, for example) cannot be conscious simply because of the action of neurons:

But this sort of statement doesn’t explain what something is, only how it is manifested in the material realm. It mistakes symptoms for the cause. Understanding is always about finding the cause. What causes the brain activity? A human person with freedom and a conscience.

In his philosophy, neural activity is a product of consciousness rather than vice versa. This is a classic case of the Fallacy of Composition: since A is part of B, and A has property X, therefore B has property X. For example, since a single water molecule is not wet, this means that a collection of water molecules cannot be wet, therefore water is not wet. We only experience the property of wetness when water molecules combine on a large scale. Wetness is an emergent property.

Likewise, consciousness is also an emergent property. As Bo Bennett puts it:

[I]t is difficult to imagine a collection of molecules resulting in something like consciousness, because we are focusing on the properties of the parts (molecules) and not the whole system, which incorporates emergence, motion, the use of energy, temperature (vibration), order, and other relational properties.
Logically Fallacious, p.112

Essentially, Mr Radice argues that consciousness is a form of magic with no connection with the empirical universe. Such a viewpoint cannot explain why chemicals such as alcohol and other drugs affect human consciousness, or why brain injuries are demonstrated to cause permanent changes in people’s character.

And one final point:

The Nazis may have been defeated, but their idea that human beings are no more than ‘blood and dirt’ is alive and well, and very fashionable indeed. 

Nazi philosophy is not famous for its internal coherence, but the idea that empirical materialism was a major part of their worldview is not borne out by the evidence.

The party as such represents the point of view of a positive Christianity without binding itself to any one particular confession. It fights against the Jewish materialist spirit within and without . . . The leaders of the party undertake to promote the execution of the foregoing points at all costs, if necessary at the sacrifice of their own lives.
The Nazi Party Programme 1920, Article 24


Filed under Philosophy, Physics, Science, Society

Engelmann and Direct Instruction (Part 2)

In Could John Stuart Mill Have Saved Our Schools?, Siegfried Engelmann and Douglas Carnine discuss the philosophical foundations of their acclaimed Direct Instruction programme (see Part 1). They write of their serendipitous rediscovery of Mill’s work and that they

came across Mill’s work and were shocked to discover that they had independently identified all the major patterns that Mill had articulated. Theory of Instruction [1991] even had parallel principles to the methods in [Mill’s] A System of Logic [published in 1843].

— location 543 Kindle edition

What Engelmann and Carnine are attempting to do is no less than develop a scientifically reliable model of education. In their view, learners learn by constructing inferences based on the evidence or examples presented by the teacher. In other words, learners use the rules of reason and logic (consciously or unconsciously) to develop general principles from specific examples by inductive reasoning.

To me, this is a fascinating idea. Have Engelmann and Carnine hit upon the elusive essence of what learning is? Is learning genuinely a matter of constructing inferences from evidence by formal or informal logical rules?

My view is that it certainly seems a plausible idea. In the light of my own experience and thinking it has a “ring of truth”, and I suspect that I am going to find this a profoundly influential idea for the rest of my career.

Many authors and thinkers have argued that human beings construct “mental maps” or conceptual models constructed by inductive reasoning from often limited information. Anthropologist Louis Liebenberg describes an example involving the !Xõ people of the central Kalahari Desert:

While tracking down a solitary wildebeest spoor [tracks] of the previous evening !Xõ trackers pointed out evidence of trampling which indicated that the animal had slept at that spot. They explained consequently that the spoor leaving the sleeping place had been made early that morning and was therefore relatively fresh. The spoor then followed a straight course, indicating that the animal was on its way to a specific destination. After a while, one tracker started to investigate several sets of footprints in a particular area. He pointed out that these footprints all belonged to the same animal, but were made during the previous days. He explained that the particular area was the feeding ground of that particular wildebeest. Since it was, by that time, about mid-day, it could be expected that the wildebeest may be resting in the shade in the near vicinity.

— quoted by Steven Pinker in How The Mind Works p. 193

The trackers were using miniscule traces of evidence and their knowledge of the environment to make inferences about the behaviour of (currently) unseen entities. In other words, they were using inductive reasoning to put together a tentative model of what their quarry was doing or attempting to do. (And I use ‘tentative’ in the sense that the model will be adapted and corrected in the light of further evidence.)

As do we all! I would suggest that all humans use similar techniques of inference, or ‘mental modules’ in Steven Pinker’s memorable phrasing, even with vastly different subject matter. Stephen Hawking and Leonard Mlodinow even go so far as to suggest that:

we shall adopt an approach that we call model-dependent realism. It is based on the idea that our brains interpret the input from our sensory organs by making a model of the world. When such a model is successful at explaining events, we tend to attribute to it, and to the elements and concepts that constitute it, the quality of reality.

The Grand Design p.9

And where does this leave us? If Engelmann and Carnine are correct (and I believe they are} then education becomes a matter of logic. They argue that a vital criterion in designing what they call “sound instructional sequences” is that sets of examples should “generate only the intended inferences”. They note

that logical flaws in instruction could be identified analytically, through a careful examination of the teaching. If we know the specific set of examples and the inference that the learners are supposed to derive from the instruction, we can determine if serious false inferences are implied by the program.

— location 1514

And I, for one, find that a highly engaging and strangely comforting thought.

(You can read Part 3 here)


Filed under Direct Instruction, Education, Philosophy, Siegfried Engelmann

Engelmann and Direct Instruction (Part 1)

We are art’s mercenaries,
firing our thought’s arrows
at the mystery of things
— R. S. Thomas, Paving

Engelmann comes highly recommended:

In his book Visible Learning: A Synthesis of Over 800 Meta-Analyses Relating to Achievement, the researcher John Hattie evaluates the success of a range of different teaching approaches. As the subtitle suggests, he synthesised the results of hundreds of different analyses of achievement and measured the effect of different factors . . . A specific Direct Instruction programme was developed by the American educator, Siegfried Engelmann, in the 1960s. It proved incredibly successful but also incredibly controversial because it contradicted so much of what theorists like Dewey and Freire advocated. Hattie specifically endorsed Engelmann’s programme.

— Daisy Christodoulo, Seven Myths About Education, location 751 Kindle edition

Later on in the book, Hattie confronts the dominance of empirically unsuccessful constructivist ideas in teacher training. He explains the effectiveness of Direct Instruction, a structured and unapologetically teacher-led method of instruction originated in 1960s America. Despite being shunned by the American education establishment, Hattie’s analysis shows that Direct Instruction has one of the largest effect sizes (0.59) for any teaching programme.

— Robert Peal, Progressively Worse, location 2689 Kindle edition

I was intrigued and wanted to find out more, so I recently read Siegfried Engelmann’s and Douglas Carnine’s book Could John Stuart Mill Have Saved Our Schools? which can be thought of as an introduction to the philosophical underpinning of Direct Instruction.

I claim no particular expertise in this field beyond that of a working teacher with a couple of decades of experience. I suppose that it is also appropriate at this point to disclose that my practice generally tends towards the traditional-didactic rather than the progressive end of the spectrum, so I am perhaps predisposed to be sympathetic to Engelmann’s ideas. Nevertheless, this blog will attempt to engage critically with his ideas and arguments.

Engelmann and Carnine open by saying that (unfortunately, in their opinion) education has historically been excluded from the domain of science. They suggest that the five principles of induction put forward by philosopher John Stuart Mill in his A System Of Logic (1843) would form a suitable basis for a scientific systematisation of effective educational practice. The efficacy of these principles when applied to education was not recognised at the time, not even by Mill himself, until Engelmann and Carnine rediscovered them in the 1970s.

I was unfamiliar with this aspect of Mill’s work, and it was a delight to be introduced to it. I was particularly struck by this bombshell from Mill:

In another of its senses, to reason is simply to infer any assertion, from assertions already admitted: and in this sense induction is as much entitled to be called reasoning as the demonstrations of geometry
— J. S. Mill, A System of Logic, location 175 Kindle edition

Philosophers have long debated the “problem of induction”. It is generally recognised that deductive reasoning (e.g. Socrates is a man; all men are mortal; therefore Socrates is mortal) is more dependable that inductive reasoning (e.g. every swan I have seen to date has been white; therefore every swan I will see in the future will be white).

However, it is a under-acknowledged truth that in our day-to-day lives (and in science generally) we rely primarily on induction and inference and, for the most part, they serve us well. What Mill is attempting to do is address the philosophical “second class status” accorded to inductive truths by formalising a set of rules that allow us to generate valid inductive inferences.

Engelmann and Carnine argue that these rules are of fundamental importance to the teacher as they allow her to construct a system of instruction that allows students to generate valid inferences and minimise false inferences:

In summary, the fabric of well designed instruction consists of details that promote specific inferences and rule out inappropriate inferences. Effective instruction is not born of grand ideas or scenarios that appeal to development or love of learning. It is constructed from the logic and tactics of science.
— S. Engelmann and D. Carnine, Could John Stuart Mill Have Saved Our Schools? location 1944 Kindle edition

One example they present is that of a constructivist approach to the teaching of prime numbers by getting students to lay out numbers of beans in rows and columns: students are invited to notice that some numbers (e.g. 7) cannot be laid out in rows of more than one bean which have equal numbers of beans. Englemann and Carnine argue that this activity does not accord with Mill’s principles because it will encourage students to generate a number of false inferences:

The false inference is that prime numbers are odd numbers. Imagine the consternation of the student who later discovers that 9 and 15 are odd, but they generate multiple rows. In contrast, 2 is even, but it is prime. A related false inference is that there is some form of predictable pattern for the occurrence of prime numbers, rather than the fact that some numbers are primes and others aren’t. Unless students had received previous instruction on what primes are, the bean counting has a potential of inducing false inferences; however, if students first learn the properties of prime numbers, the bean counting is a pointless activity. It simply provides validation that prime numbers are different from numbers that are multiples.
— location 1779 Kindle edition

I discussed this criticism with a Maths colleague who disagreed that the constructivist approach would necessarily generate false inferences — but more on that in a later post.

In summary, I am fascinated by the potential of Englemann’s and Carnine’s approach and intend to post more as I mull over its details and implications. Lord help me, but I could not help but be stirred by what could be interpreted as a call to arms:

[Our system] could certainly be improved by a concerted effort to do so. What it needs is a comprehensive critique by serious logicians and philosophers. It needs attention to its details so they can be refined or replaced to be more in accord with logic or empirical evidence.
— location 2591 Kindle edition

And perhaps more importantly, by working teachers too.

(Part Two here)


Filed under Direct Instruction, Education, Philosophy, Siegfried Engelmann, Society