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.
e=mc2andallthat 
Reblogged this on The Echo Chamber.
At this very moment I am using Engelmann’s materials to teach my seven year old fractions. They are wickedly good. I used them with my elder children too and they are just so unphased/confident. At the moment my son spends a small part of each lesson identifying which fractions are more or less than one. After four weeks of if it he is just so confident. Every new idea is do carefully taught. After teaching through the course already I can identify all the goundations
Thanks! I must confess that I have never taught using D.I. materials. My explorations to date have been based on Engelmann’s book “Could John Stuart Mill Have Saved Our Schools?” His ideas seem plausible to me and I am pleased that they seem to live up to their billing, from your experience at least. Can I ask where you got hold of the Maths course you used?
I had to buy it on Amazon. It was a pain as it is designed for schools and so lots of different bits. It is called ‘Connecting Math Concepts’.
Thanks!
It’s encouraging to see Engelmann being picked up and noticed now and then these days. The idea of a science of learning, based on logic, is definitely a long way from where we are at the moment. I think we have two or three generations of teachers to work through before logic is given primacy over ego. Heather is right: using clearly designed etching materials has such an impact on students’ confidence. Which raises serious questions about why students lose, or fail to gain, confidence in our schools.
I am still in the process of discovering Engelmann. I have been looking primarily at the philosophical underpinnings so far, and haven’t looked at any DI teaching materials yet. It’s a deficit that I want to address soon. Whatever their quality, the research I have done on DI will affect my own practice for the better, I think. It’s a depressing thought that it might take two or three generations for the teaching profession to accept a better mousetrap…
Good blog thought provoking. I don’t know Engelmann that well so I’m taking at face value what you have written as far as I understand it
Here’s my view. Wittgenstein had two distinct phases. In the first he was part (ish) of the logical positivist crowd.
Later he rather did for them along with Popper and a number of others. In my view Wittgenstein is not suggesting that a symbol (pencil) has more than one meaning he is suggesting that it can only mean what it does in its context of use.
For example we can fix the meaning of a “belief in Jesus” and it can have the same meaning but at the same time it will mean different things to different groups of Christians. It in itself it has a notional meaning but it’s real meaning is derived from its relationship with other things within that context. Some groups will see it as more ephemeral than others. Indeed Muslims could see at as something else again even though it has the same formal meaning that meaning is also dependent on many other related meanings.
Take Saussure’s Dog for example. I can point at a dog and it becomes a dog.
I can point at a girl but she only becomes a Dog dependent on very complex social understandings not least rank Misogyny. No amount of pointing and association would do because a girl isn’t a dog. Her behaviour is not dog like. The abstraction of the word is such that understanding the concept would be difficult.
However most are intrinsically capable of understanding the complexity of the usage of the word dog in this horribly mysoginistic way..
I accept that Math works in the context that you use it but part of learning is to try and de-construct complexity not create a re-contextualised curriculum that has not other meaning other than in the context of education.
It’s my view that often pedagogic knowledge is so de-contextualised that it becomes meaningless. Its not that students forget; it’s just gibberish outside the classroom.
Try for example asking a 16 year old about what they learned in Biology at schools; say Mitosis. What you get back is meaningless gibberish probably meaningful within the context of classroom and curriculum but otherwise just gibberish. You can almost hear the teacher prompting and unpicking the gibberish.
Without meaning it cannot be used in discourse without usage it goes or becomes ever more confused.
So I am sceptical about sacrificing complexity to facilitate ease of understanding. Classroom concepts unless, of course, they are very specific types of pedagogic knowledge that have a relatively fixable meaning in life are dependent upon complex relationships. Part of education is to be able to de-construct those relationships to see what lies beneath.
Hope that makes sense.
Thanks for a very thoughtful reply. Philosophical Investigations is very much the later Wittgenstein. As I understand the point he was making (section 28 of the PI, as I recall), he was highlighting the inadequacy of traditional, simplistic models of language and led to him developing the concept of language games. The reason I mentioned it is because I thought it provided an interesting example of how inappropriate inferences can be drawn even in apparently straightforward, simple situations like naming an object. As I read Engelmann, he is keen to expose students to complexity so that correct inferences can be drawn rather than presenting oversimplifications. I entirely agree that deconstructing complex ideas is a hugely important part of education. It seems to me that Engelmann might give us and students the tools to do it more reliably and effectively.
I’m interested in seeing how you develop this idea because it is, as you say interesting.
I think your use of the Wittgenstein is correct however you then seem to argue for the kind of logical approach that Wittgenstein was to some extent arguing for, and then turned against in PI. I think that is what is confusing me.
I agree that it is easy to make inappropriate inferences from the complex business of symbolic interaction. What you seem to be suggesting here is that the way to avoid this is to make learning less conceptually conflictual, which I see as a kind of dumbing down of knowledge.
Perhaps I can learn more about Engelmann from reading your blogs. Good stuff.
A very perceptive comment. As I understand Wittgenstein in the PI, he would agree that language has a logical structure, but was troubled by the recognition that all parts of language do not share a common logical structure, even if these parts have their own coherent internal logical structure — hence his coining of the term “language games”. What I find interesting about Engelmann’s appropriation of Mill’s rules of logical induction for education is that it suggests that communication can be made more reliable with less scope for misunderstanding. In other words, Engelmann is offering a set of reliable rules for the “language game” of education. I don’t think this will eliminate conceptual conflicts where they exist, but it may well minimise spurious misunderstandings. At least in theory (!)