Mathematics education research (MER) not only looks at the requirements of mathematics and the didactics developed in the field itself, but also at psychology on cognition, learning and teaching in general, at pedagogy on the development of pupils and students, and at other subjects, such as physics or economics for cases when mathematics is applied, or general philosophy indeed. The former weblog text said something about neuro-psychology. Today we have a look at cognitive psychology.
Stellan Ohlsson: Deep learning
Stellan Ohlsson (2011) Deep Learning: How the Mind Overrides Experience may be relevant for mathematics education. One teaching method is to get students to think about a problem until the penny drops. For this, Ohlsson discusses a bit more than the distinction between old and new experience:
“(…) the human mind also possesses the ability to override experience and adapt to changing circumstances. People do more than adapt; they instigate change and create novelty.” (cover text)
“If prior experience is a seriously fallible guide, learning cannot consist solely or even primarily of accumulating experiences, finding regularities therein and projecting those regularities onto the future. To successfully deal with thoroughgoing change, human beings need the ability to override the imperatives of experience and consider actions other than those suggested by the projection of that experience onto the situation at hand. Given the turbulent character of reality, the evolutionary strategy of relying primarily on learned rather than innate behaviors drove the human species to evolve cognitive mechanisms that override prior experience. This is the main theme of this book, so it deserves a label and an explicit statement:
The Deep Learning Hypothesis
In the course of shifting the basis for action from innate structures to acquired knowledge and skills, human beings evolved cognitive processes and mechanisms that enable them to suppress their experience and override its imperatives for action.” (page 21)
Definition & Reality methodology
The induction question is how one can know whether all swans are white. Even a statistical statement runs into the problem that the error is unknown. Skepticism that one cannot know anything is too simple. Economists have the question how one can make a certain general statement about the relation between taxation and unemployment.
My book DRGTPE (2000, 2005, 2011) (PDF online) (though dating from 1990, see the background papers from 1992) proposes the Definition & Reality methodology. (1) The model contains definitions that provide for certainty. Best would be logical tautologies. Lack of contrary evidence allows room for other definitions. (2) When one meets a black “swan” then it is no swan. (3) It is always possible to choose a new model. When there are so many black “swans” that it becomes interesting to do something with them, then one can define “swan2”, and proceed from there. Another example is that in one case you must prove the Pythagorean Theorem and in the other case you adopt it as a definition for the distance metric that gives you Euclidean space. The methodology allows for certainty in knowledge but of course cannot prevent surprises in empirical application or future new definitions. The methodology allows DRGTPE to present a certain analysis about a particular scheme in taxation – the tax void – that causes needless unemployment all over the OECD countries.
Karl Popper (1902-1994) was trained as a psychologist, and there met with the falsification approach by Otto Selz (1881-1943). Popper turned this into a general philosophy of science. (Perhaps Selz already thought in that direction though.) The Definition & Reality methodology is a small amendment to falsificationalism. Namely, definitions are always true. Only their relevance for a particular application is falsifiably. A criterion for a scientific theory is that it can be falsified, but for definitions the strategy is to find general applicability and reduce the risk of falsification. In below table, Pierre van Hiele presented his theory of levels of insight as a general theory of epistemology, but it is useful to highlight his original application to mathematics education, with the special property of formal proof. Because of this concept of proof, mathematics may have a higher level of insight / abstraction overall. Both mathematics and philosophy also better take mathematics education research as their natural empirical application, to avoid the risk of getting lost in abstraction.
Addendum September 7: The above assumes sensible definitions. Definitions might be logically nonsensical, see ALOE or FMNAI. When a sensible definition doesn’t apply to a particular situation, then we say that it doesn’t apply, rather than that it would be untrue or false. An example is an econometric model that consists of definitions and behavioural equations. A definition that has no relevance for the topic of discussion is not included in that particular model, but may be of use in another model.
|Mathematics||Euclidean space||Θ = 2π||?|
|Physics||Conservation of energy||Speed of light||Local gravity on Earth|
|Economics||Savings are income minus consumption||Institutional (e.g. annual tax code)||Behavioural equations|
|Mathematics education||Van Hiele levels of insight||Institutional||Student variety|
To my great satisfaction, Ohlsson (2011:234) adopts basically the same approach.
“The hypothetical process that supposedly transforms particulars into abstractions is called induction and it is often claimed to operate by extracting commonalities across multiple particulars. If the first three swans you ever see are white, the idea swans are white is likely to come to mind. However, the notion of induction is riddled with problems. How are experiences grouped for the purpose of induction? That is, how does the brain know which experiences are instances of some abstraction X, before that abstraction has been learned? How many instances are needed? Which features are to be extracted? How are abstractions with no instances in human experience such as the infinite, the future and perfect justice acquired?”
Definition of abstraction
There is an issue w.r.t. the definition of abstraction though. Compare:
- My definition of abstraction is leaving out aspects, see here on this weblog, and see FMNAI. My suggestion is that thought itself consist of abstractions. Abstraction depends upon experience since experience feeds brain and mind, but abstraction does not depend upon repeated experience.
- Ohlsson (2011:16) takes it as identical to induction, which explains the emphasis upon experience in his title, rather taken as repetition: “Memories of individual events are not very useful in themselves, but, according to the received view, they form the raw material for further learning. By extracting the commonalities across a set of related episodic memories, we can identify the underlying regularity, a process variously referred to as abstraction, generalization or induction.” For Ohlsson, thoughts do not consists of abstractions, but of representations (models): “In the case of human cognition – or the intellect, as it would have been called in the 19th century – the relevant stuff consists of representations. Cognitive functions like seeing, remembering, thinking and deciding are implemented by processes that create, utilize and revise representations.” and “Representations are structures that refer to something (other than themselves).” (page 29)
Ohlsson has abstraction ⇔ induction (commonality). For me it is dubious whether induction really exists. The two pathways are too different to use equivalence. (i) Comparing A and B, one must first abstract from A and then abstract from B, before one may decide whether those abstractions are the same, and before one can even say that A and B share a commonality. (ii) An abstract idea like a circle might cause an “inductive” statement that all future empirical circles will tend to be round, but this isn’t really what is meant by “induction” – which is defined as the “inference” from past swans to future swans.
For me, an abstraction can be a model too, and thus would fit Ohlsson’s term representation, but the fact that he chooses abstraction ⇔ induction rather than abstraction ⇔ representation causes conceptual problems. Ohlsson’s definition of abstraction seems to hinder his understanding of the difference between concrete versus abstract as used in mathematics education research (MER).
Concrete versus abstract
Indeed, Ohlsson suggests an inversion of how people arrive at insight:
“The second contribution of the constraint-based theory is the principle that practical knowledge starts out general and becomes more specific in the course of learning. There is a long-standing tradition, with roots in the beginnings of Western philosophy, of viewing learning as moving in the opposite direction, from particulars to abstractions. [ftnt 38 e.g. to Piaget] Particulars are given in perception while abstractions are human constructions, or so the ancient story goes.” (p234)
“The fundamental principle behind these and many other cognitive theories is that knowledge moves from concrete and specific to abstract and general in the course of learning.” (Ohlsson 2011:434 that states ftnt 38)
If I understand this correctly, and combine this with the earlier argument that general knowledge is based upon induction from specific memories, then we get the following diagram. Ohlsson’s theory seems inconsistent, since the specific memories must derive from specific knowledge but also presume those. Perhaps a foetus starts with a specific memory without knowledge, and then a time loop starts with cumulation over time, like the chicken-egg problem. But this doesn’t seem to be the intention.
There is this statement on page 31 that I find confusing since now abstractions [inductions ?] depend upon representations, while earlier we had them derived from various memories.
“The power of cognition is greatly increased by our ability to form abstractions. Mathematical concepts like the square root of 2 and a four-dimensional sphere are not things we stumble on during a mountain hike. They do not exist except in our representations of them. The same is true of moral concepts like justice and fairness, as well as many less moral ones like fraud and greed. Without representation, we could not think with abstractions of any kind, because there is no other way for abstract entities to be available for reflection except via our representations of them. [ftnt 18]”
Ftnt 18 on page 402: “Although abstractions have interested philosophers for a long time, there is no widely accepted theory of exactly how abstractions are represented. The most developed candidate is schema theory. (…)”
My suggestion to Ohlsson is to adopt my terminology, so that thought, abstraction and representation cover the same notion. Leave induction to the philosophers, and look at statistics for empirical methods. Then eliminate representation as a superfluous word (except for representative democracy).
That said, we still must establish the process from concrete to abstract knowledge. This might be an issue of terminology too. There are some methodological principles involved however.
Wilbrink on Ohlsson
Dutch psychologist Ben Wilbrink alerted me to Ohlsson’s book – and I thank him for that. My own recent book A child wants nice and no mean numbers (CWNN) (PDF online) contains a reference to Wilbrink’s critical discussion of arithmetic in Dutch primary schools. Holland suffers under the regime of “realistic mathematics education” (RME) that originates from the Freudenthal “Head in the Clouds Realistic Mathematics” Institute (FHCRMI) in Utrecht. This FHCRMI is influential around the world, and the world should be warned about its dismal practices and results. Here is my observation that Freudenthal’s approach is a fraud.
Referring to Ohlsson, Wilbrink suggests that the “level theory by Piaget, and then include the levels by Van Hiele and Freudenthal too” (my translation) are outdated and shown wrong. This, however, is too fast. Ohlsson indeed refers to Piaget (stated ftnt 38) but Van Hiele and Freudenthal are missing. It may well be that Ohlsson missed the important insight by Van Hiele. It may explain why Ohlsson is confused about the directions between concrete and abstract.
A key difference between Van Hiele and Freudenthal
CWNN pages 101-106 discusses the main difference between Hans Freudenthal (1905-1990) and his Ph.D. student Pierre van Hiele (1909-2010). Freudenthal’s background was abstract mathematics. Van Hiele was interested from early on in education. He started from Piaget’s stages of development but rejected those. He discovered, though we may as well say defined, levels of insight, starting from the concrete to the higher abstract. Van Hiele presented this theory in his 1957 thesis – the year of Sputnik – as a general theory of knowledge, or epistemology.
Freudenthal accepted this as a thesis, but, mistook this as the difference between pure and applied mathematics. When Freudenthal noticed that his prowess in mathematics was declining, he offered himself the choice of proceeding his life with the history of mathematics or the education of mathematics. He chose the latter. Hence, he coined the phrase realistic mathematics education (RME), and elbowed Van Hiele out of the picture. As an abstract thinking mathematician, Freudenthal created an entire new reality, not caring about the empirical mindset and findings by Van Hiele. One should really read CWNN pages 101-106 for a closer discussion of this. Van Hiele’s theory on knowledge is hugely important, and one should be aware how it got snowed under.
A recent twist in the story is that David Tall (2013) rediscovered Van Hiele’s theory, but wrongly holds (see here) that Tall himself found the general value while Van Hiele had the misconception that it only applied to geometry. In itself it is fine that Tall supports the general relevance of the theory of levels.
The core confusion by Ohlsson on concrete versus abstract
The words “concrete” and “abstract” must not be used as absolutely fixed in exact meaning. This seems to be the core confusion of Ohlsson w.r.t. this terminology.
When a child plays with wooden blocks we would call this concrete, but our definition of thought is that thinking consists of abstractions, whence the meanings of the two words become blurred. The higher abstract achievement of one level will be the concrete base for the next level. The level shift towards more insight consists of compacting earlier insights. What once was called “abstract” suddenly is called “concrete”. The statement “from concrete to abstract” indicates both the general idea and a particular level shift.
Van Hiele’s theory is essentially a logical framework. It is difficult to argue with logic:
- A novice will not be able to prove laws or the theorems in abstract mathematics, even informally, and may even lack the notion of proof. Having achieved formal proof may be called the highest level.
- A novice will not be able to identify properties and describe their relationships. This is clearly less complex than (1), but still more complex than (3). There is no way going from (3) to (1) without passing this level.
- A novice best starts with what one knows. This is not applied mathematics, as Freudenthal fraudently suggested, but concerns the development of abstractions that are available at this level. Thus, use experience, grow aware of experience, use the dimensions of text, graph, number and symbol, and develop the thoughts about these.
Van Hiele mentioned five levels, e.g. with the distinction between informal and formal deduction, but this is oriented at mathematics, and above trident seems sufficient to establish the generality of this theory of knowledge. A key insight is that words have different meanings depending upon the level of insight. There are at least three different languages spoken here.
Three minor sources of confusion are
- Ohlsson’s observation that one often goes from the general to the specific is correct. Children may be vague about the distinction between “a man” and “one man”, but as grown up lawyers they will cherish it. This phenomenon is not an argument against the theory of levels. It is an argument about becoming precise. It is incorrect to hold that “one man” is more concrete and “a man” more abstract.
- There appears to exist a cultural difference between on one side Germans who tend to require the general concept (All men are mortal) before they can understand the particular (Socrates is mortal), and the English (or Anglo-Saxons who departed from Germany) who tend to understand only the particular and to deny the general. This cultural difference is not necessarily epistemological.
- Education concerns knowledge, skill and attitude. Ohlsson puts much emphasis on skill. Major phases then are arriving at a rough understanding and effectiveness, practicing, mastering and achieving efficiency. One can easily see this in football, but for mathematics there is the interplay with the knowledge and the levels of insight. Since Ohlsson lacks the levels of insight, his phases give only part of the issue.
I have looked only at parts of Ohlsson’s book, in particular above sections that allow a bit more clarity on the relevance w.r.t. Van Hiele’s theory of levels of insight. Please understand my predicament. Perhaps I read more of Ohlsson’s book later on, but this need not be soon.
- In mathematics education research (MER) we obviously look at findings of cognitive psychology, but this field is large, and it is not the objective to become a cognitive psychologist oneself.
- When cognitive psychologists formulate theories that include mathematical abstraction, as Ohlsson does, let them please look at the general theory on knowledge by Pierre van Hiele, for this will make it more relevant for MER.
- Perhaps cognitive psychologists should blame themselves for overlooking the theory by Pierre van Hiele, but they also should blame Hans Freudenthal, and support my letter to IMU / ICMI asking to correct the issue. They may work at universities that also have departments of mathematics and sections that deal with MER, and they can ask what happened.
- When there is criticism on the theory by Van Hiele, please look first at the available material. There are summary statements on the internet, but these are not enough. David Tall looked basically at one article and misread a sentence (and his misunderstanding still was inconsistent with the article). For some references on Van Hiele look here. (There is the Van Hiele page by Ben Wilbrink, but, as said, Wilbrink doesn’t understand it yet.)