For a long time before I wrote this article I really disliked the whole idea of Skinner's Behaviourism. This was a strong emotional feeling.

I saw behaviourism as drill and practice imposed by an authority figure, a teacher. I saw myself as a rebellious and creative type who didn't need any external authority to impose their drill and practice on me! I also saw Skinner's absolute refusal to speculate on what happened inside the brain as a huge copout, as some sort of proof of the sterility of his whole approach.

As a methodology behaviourism seemed to symbolise the main thing that was wrong with School and Education. That it was BORING.

I have a vague memory of really liking something I read about Chomsky's "refutation" of Skinner, even though today I can't remember any of the details of what Chomsky said.

At any rate, I much preferred any approach that employed the rhetoric of creativity. Early on I was drawn to 'The Act of Creation' by Arthur Koestler and later to the writings and Constructionist philosophy of Seymour Papert. These thinkers seemed to take creativity seriously not like boring old Skinner.

This history forms an emotional backdrop to this article. When I realised that I had drifted into combining logo programming and behaviourist methods successfully in my classroom then it was a real shock, for a while I was in a state of disbelief.

So I had to write about it and theorise it. I'm still theorising it. For me this article was a difficult self reflection, an accomodation, where my view of the world suddenly crashed in the face of reality.

I now believe that Skinner was misrepresented, that the baby was thrown out with the bathwater, that I didn't understand behaviourism and for a variety of reasons I'm now critical of my former hero Chomsky. I don't go into all of this below but would like to see a broad discussion on these issues.

This article covers a lot of ground - behaviourism, constructionism, learning maths, how to use computers in school, School with a capital 'S' (the institution of school and it's ingrained ways) and what works for the disadvantaged.

One word of advice. The footnotes are important, you need to read them to understand the article.

(for instance, in the teaching of Quadratics)



During 1996 and 1997 I wrote my own Quadratics drill and practice software in Logo to assist my teaching of the Quadratics topic to Year 10 and 11 Pure Maths classes.

The software was very successful in helping the students learn Quadratics (see companion article for evaluation of the software -- 'Quadratics Software Evaluation')

Paradoxically, I became uneasy about the success of the software, as I came to realise that I was using Behaviourist methods successfully. My uneasiness came from the fact that as a Logo enthusiast I am committed to a Constructionist educationally philosophy which is way down the other end of the spectrum of teaching methodologies from where Behaviourism lies. At one point I desperately thought to myself, "I have become Skinner, is there any way out?"

My uneasiness led to further study and reflection of the nature of behaviourism, constructionism and school -- this is the resultant synthesis of my dilemma.

What is Behaviourism and what is it good for ?

Behaviourism is the idea that rewards strengthen certain behaviours. That is correct as far as it goes. But behaviourism has never explained how brains learn new ideas (Minsky, 1987, p.75). As a creative method or explanation it is clueless. Footnote(1) Minsky's critique of Behaviourism

So behaviourist methods, like a computer drill and practice program, may work well for a prepackaged curricula, which is the norm in senior maths courses, provided the boredom can be kept at bay. I'll use the teaching of Quadratics at Year 10 / 11 level as an example of what I mean.

Does School have a mind of its own?

Seymour Papert (1993) talks about how School assimilates the computer to do things according to how School has traditionally done them, as though School is an independent organism with it own set rules, procedures and homeostasis. How does School manage to achieve this, using this case as an example?

  1. By putting Quadratics into the Curriculum. Who ever questions that?
  2. By buying maths textbooks with lots of Quadratics in them. Invariably these textbooks break down the complex topic of quadratics into small parts and then relentlessly drill the students in practising those parts until "understanding" is reached.
  3. By telling students they have to do Pure Maths in Year 11 to obtain certain desired for academic and career pathways.
  4. By creating a pre-Pure Maths extended class in Year 10 for the top group to prepare them for the "very important" Year 11 Pure Maths class.

This raises a big question which is hardly ever asked: Is learning Quadratics in this way, real maths, anyway? Well, clearly Quadratics is in the Curriculum because it is pregnant with maths skills. There are number skills of substitution and calculation (BEDMAS), there is graphing using the Cartesian co-ordinates, there is looking for the change in patterns as the 'a', 'b' and 'c' values vary. There is derivation of formula, like Axis of symmetry = -b / 2a. Then we have square roots, unreal numbers, the full quadratic formula ... there is even Halley's comet, parabolic reflectors and chucking a ball up in the air, not to forget "problem solving" ...... what a list. Clearly, no respectable maths teacher or School would take Quadratics out of the Curriculum !!

Alienation and social sorting:-

Not one student asked me, "Why do we have to do Quadratics?" or "How do they relate to real life?" questions that I would have found very difficult to answer. However, many students did say (and some more than once), "the work in this class is too hard, I want to go back to my other (not extended) class". This put a lot of pressure on me as the teacher. I was trying to set and maintain a higher standard of work to prepare students for Year 11 Pure Maths. But if I pushed to hard I would have students coming to me and asking to be moved out to an "easier" class. The losers in this process were the advanced section of the class who in effect were being held back by the tail. All of these problems were substantially overcome shortly after I introduced my quadratics software.

One of the social functions of Schooling is to condition the clients for their role and social niche in later life. Maths with its traditional emphasis on sacred knowledge (like Quadratics) and marks is particularly well suited for this. I can see these forces at work in the student responses in the previous paragraph. There was a passive acceptance of the right of School to put the Quadratics hurdle in place. The advanced element of the class believed they could jump this hurdle and were comfortable with that. The less skilled and motivated members of the class had strong doubts about their ability to jump the hurdle and tried to organise a soft option. Even though many students at Disadvantaged schools may reject School ("school sux") with varying degrees of hostility it does not seem there is significant group consciously rejecting the right of School to make fundamental judgements about their future social niche in life.

My Quadratics software resolved some of these problems for students by making it easier or perhaps more interesting to jump the hurdle. But in the process it begs the question of whether real maths ought to look like this in the first place.

What is real maths and deep learning? The Constructionist model.

I believe that real maths involves self directed exploration, creating ones own Projects, play and problem finding as well as problem solving. The problem with the current maths learning environment in secondary schools is that it is very strong on teaching maths skills but very weak in creating learning environments where students will come to enjoy maths and become self motivated in learning it.

One way to look at it is to outline some fundamentally different outlooks about learning at the opposite end of the Behaviourist spectrum. Papert (1993) calls this approach constructionist and here are some of the insights he offers us:-

  1. Play is OK. We should take a hint from children who learn more by play before they get to school than they learn at school. Play is the enemy of the government model of power and measurement. Play laughs at power and defies measurement. Play is a favourite of bored or rebellious students. But what is the Educational utility of play? Difficult question but a real problem for School at the moment is that its environment makes it virtually impossible to even seriously pose that question. Footnote(2) The utility of play
  2. The emotional precedes the cognitive. Many kids say they dislike school and dislike many of their teachers. How can real learning occur for these kids in these classes? Perhaps personal appropriation, making something your own, is the single most crucial point of successful learning. Deep learning will not occur unless there is that feeling of intimate engagement or falling in love with the subject. Real knowledge is personal knowledge. Any learning regime that neglects the motivational aspect will end in disappointment.
  3. Relationships. Our knowledge is like our relationships with other people, full of subtle nuances and never ending contradictions. We learn new things by becoming good at making connections and discerning healthy relationships. This is a much healthier way of looking at knowledge than the hierarchical checklists suggested by centralised curriculum like Statements and Profiles.
  4. Trust your intuition. Frankly, logic is over-rated. Logic doesn't give us insights in the first place. It just lets us formalise and rationalise our thoughts after the event. Footnote(3) Deficiencies of Logic
  5. Take risks! This goes beyond the passive truisms, 'that we all make mistakes' and that 'mistakes are a natural part of learning.' Everyone agrees with these truisms in a disconnected theoretical sense separated from real life situations where we have to admit that we were wrong. How many teachers actually say to their students, 'I was wrong' or 'I don't know'.
  6. 'Take risks' encompasses more of the spirit that it is good to make lots of mistakes and make them quickly as part of getting on with learning. The faster we fail the better it is because then we will get onto something worthwhile quicker. Risk taking is an active virtue. Can you imagine an Education System adopting a policy of fast failure? How would that fit with a model of Standards and Accountability which engenders low risk activity?
  7. Take your time. I am now convinced that learning how to learn techniques that advocate heuristics such as concept mapping, Gowin's knowledge Vee (Novak and Gowin, 1984) or mind mapping mainly work because the process of using the heuristic means the learner spends more time with the problem. A common reaction of students to failure, boredom or disempowerment is to use an endless variety of techniques to spend less time on the task. It follows that anything that bores or disempowers students is bad for their learning. At School (particularly Secondary) we teach kids that specific subjects should be studied for fixed time intervals at certain times of the day and then suddenly stopped, when the bell rings.
  8. A good discussion promotes learning. Good plants won't grow in poor soil. A rich soil for intellectual growth can only come about through active discussion, negotiation, argument and exploration.

Papert calls learning environments built around these principles, constructionist. When student use of Quadratics software is evaluated against these sort of criteria it rates poorly as an exercise in real maths or deep learning.

Prior to writing the Quadratics software I had used Logo extensively to help create environments more in keeping with the constructionist model. Footnote(4) Constructionist uses of the computer in maths

Student needs and Teacher deeds:-

The software seems to meet the needs of many students in a Disadvantaged school who want to do well in a preparatory Pure Maths class at Year 10 level. Hurdles were jumped by many who without the software would have failed to jump them.

Although my own teaching mode is constructionist by preference I find that in Disadvantaged schools a fair bit of repetitive drill and skill is required anyway, more so than what is required in a middle class school. Otherwise students simply forget basic concepts. At any rate a balance between constructionist exploration and drill & skill is always required.

Resolving the dilemma: the Environment Rules !

To restate the dilemma -- I don't like behaviourist approaches but I worked hard to make one work and it worked well!

The learning environment often determines what teaching methods will be successful or unsuccessful. Often the teacher has little control over this, the learning environment being dictated by the set curricula, which in Senior Maths is tight. Constructionist approaches require time and freedom to explore. So, behaviourist approaches are more likely to succeed when operating in a Senior Maths curriculum environment. So, Papert is correct when he points out that School is an independent organism with a mind of its own. The School Environment is in control and the teachers and students are just unwitting participants in this competitive drama whose principle purpose seems to be social sorting.

The Quadratics software was good, yes, for the given environment. It would be dangerous to promote it as a general model to be pursued in all maths classes as much as possible. Nevertheless, the software did the job that needed to be done and that needs to be affirmed clearly as something very positive.

From a Constructionist perpective a very important part of the environment is the teacher. The students have to do the construction but unless the physical, cognitive and emotional environment is setup right then it won't work out. Constructionist teaching requires higher skill levels even though to an observer the teacher is far less obtrusive.

From a Behaviourist perspective the purpose of drill and practice software is to replace the teacher (at least for that part of the instruction). Footnote(5) Computer Aided Instruction and Disadvantaged Schools If I were to evaluate my Quadratics software from a Behaviourist perspective my conclusion would be to produce a commercial version of it and market it to other schools so they, too, could enjoy its benefits. This conclusion, however, would overlook some important considerations:-

Opposites interpenetrate:

Philosophically, the Behaviourist approach and Constructionist approach are oppositional, deservedly so. But opposites are not just opposites that passively negate each other. Opposites interact and interpenetrate with each other. We can also look at behaviourism and constructionism as two legitimate learning modes at opposite ends of a continuum. The quadratics software can be located somewhere along that continuum, closer to the behaviourist end than the constructionist end, but in the hands of a constructionist teacher it can be viewed as moving a little bit towards constructionism. For instance, the quadratics software does enables students to explore quadratics in a more dynamic fashion than traditional lecture, textbook mode. Although the time line does not permit students to develop their own software this event is not ruled out in principle and students can ask the teacher to modify the software in a direction they think will help.


Minsky, Marvin. The Society of Mind. Picador 1987

Novak, Joseph and Gowin, Bob (1984). Learning how to Learn. Cambridge: Cambridge University Press.

Papert, Seymour. The Children's Machine: Rethinking School in the Age of the Computer, 1993, Basic Books

Solomon, Cynthia. Computer Environments for Children: A Reflection on Theories of Learning and Education, 1986, The MIT Press, Cambridge, Massachusetts


A note about the Disadvantaged school setting:

Paralowie is a disadvantaged school ( 549 school card holders out of approx. 1100 students -- 1996 figures ) and although my new composite class was "extended" (representing the top 1/3 in ability at this year level) I didn't think the class was progressing particularly well in the 20 weeks I had taught them for up to the end of Term 3 before I started using the Quadratics software. I have already mentioned the poor skills of a substantial number of students when substituting negative numbers. Eg. substitute -2 into -2x^2 + 3x. Others resented the fact that they had been performing in the top half of their previous class but now were performing in bottom half of their new class. I had several requests from students to return to their previous class because the new work was "too hard". Poor attendance was a problem with about 3 students being away on a good day and up to 8 or 10 being absent on a bad day. Homework effort was poor from many because they had managed to get through with little homework in Years 8 and 9 and at any rate it is not cool to do homework. Moreover, in disadvantaged schools I find that it takes 2 to 3 terms for students to adapt and accept a new teacher and there is a continual behavioural testing out period during this time before things settle down.

Footnote(1) Minsky's critique of Behaviourism. On page 75 of Society of Mind, Minsky says:-

"Harvard psychologist B.F. Skinner ... recognised that higher animals did indeed exhibit new forms of behaviour, which he called 'operants.' Skinner's experiments confirmed that when a certain operant is followed by a reward, it is likely to reappear more frequently on later occasions. ... this kind of learning has much larger effects if the animal cannot predict when it will be rewarded. ....Skinner's discoveries had a wide influence in psychology and education, but never led to explaining how brains produce new operants ..... Those twin ideas - reward/success and punish/failure - do not explain enough about how people learn to produce the new ideas that enable them to solve difficult problems that could not otherwise be solved ..."

My comment -- Behaviourism (and much of School is Behaviourist) is a very limited approach to learning. School as it currently exists is largely condemned by Minsky's observations.
go back1

Footnote(2) The utility of play.
The most interesting account I have seen about this is Kevin McGee's PhD thesis (1992) 'Play and the Genesis of Middle Manager Agents' (available from Massachusetts Institute of Technology, send email to Florence Williams at daria@media.mit.edu for details -- ask for their catalogue too). The prerequisites of this analysis are Piaget's assimilation / accomodation dialectic and Minsky's idea that our mind is made up of innumerable agents with a need for middle manager agents to "manage" lower level agents. McGee says, "... individuals create symbols when the world requires them to accommodate; since the existing representations are not adequate, new ones must be constructed. However, these representations are given their meaning through assimilation - or play." (37)
go back2

Footnote(3) Deficiencies of Logic
Papert (1993) argues that Relationship is more central to how our minds develop than Logic. See Ch. 8 'Computerists' of 'The Children's Machine' for the full argument.

Minsky (1987) defines logical thinking as follows:-

"The popular but unsound theory that much of human reasoning proceeds in accord with clear cut rules that lead to foolproof conclusions. In my view, we employ logical reasoning only in special forms of adult thought, which are used mainly to summarise what has already been discovered. Most of our ordinary mental work -- that is, our commonsense reasoning -- is based more on 'thinking by analogy' -- that is, applying to our present circumstances our representations of seemingly previous experiences." (329)

Also Ch. 18 Reasoning, 185-94
go back3

Footnote(4) Constructionist uses of the computer in maths:-
I have used LogoWriter successfully in Constructionist manner at Year 8 and 9 level. Here is an example for teaching the Fractions topic. Year 8 students were asked to design their own computer screens to teach Year 3/4 students fractions. They had to think about fractions, what Year 3/4's would find accessible and challenging and then design their own screens (using words, pictures and symbols) using the Logo programming language. They had to record their experiences in a Journal, identifying problems they encountered and who they helped / who helped them. This was a rich Project which simultaneously covered fractions, computer programming, design skills, expressive writing about mathematics and cross age tutoring skills. Within the framework of a given task students had considerably freedom as to how they expressed themselves. See Kerr (1995) 'Educational Software: Designed by Kids for Kids' unpublished -- available on request -- for a full account.

The approach adopted in this learning sequence was inspired from Idit Harel's PhD thesis titled: Software Design for Learning: Children's Construction of Meaning for Fractions in Logo Programming (MIT, June 1988). A copy of the thesis can be obtained by enclosing US$20 and writing to:

Epistemology and Learning
MIT Media Lab
20 Ames Street
Cambridge, MA 02139
Idit Harel's thesis was subsequently published as a book called Children Designers (1991), published by Norwood: Ablex.

The analysis was further developed in:-
Harel, I. & Papert, S. (1990) Software Design as a Learning Environment. Interactive Learning Environment, 1, 1-32
and in:
Kafai, Yasmin B., Minds in Play: Computer Game Design as a Context for Children's Learning (1993).
This thesis is available from the same 'Epistemology and Learning' address given above for the Idit Harel thesis.
go back4

Footnote(5) Computer Aided Instruction and Disadvantaged Schools: The leading advocate of Computer Aided Instruction (CAI) in the USA has been Patrick Suppes. I was helped by Papert's non dogmatic appreciation of what Suppes was trying to achieve, as expressed in 'The Children's Machine':-

"The concept of CAI, for which Suppe's original work was the seminal model, has been criticised as using the computer as an expensive set of flash cards. Nothing could be further from Suppe's intention than any idea of mere repetitive rote. His theoretical approach had persuaded him that a correct theory of learning would allow the computer to generate, in a way that no set of flash cards could imitate, an optimal sequence of presentations based on the past history of the individual learner. At the same time the children's responses would provide significant data for the further development of the theory of learning. This was serious high science." (164)

Papert goes onto explore his reasons for rejecting Suppes approach which is an argument that Relationship is more central to how our minds develop rather than Logic. See Ch. 8 'Computerists' of 'The Children's Machine' for the full argument.

I then turned to Solomon who has documented Suppes work in greater detail and discovered something that was very interesting. Computer based drill and practice programs (developed to a fine art by Suppes) do work and in particular they work best for disadvantaged students and schools! These programs do not work as well for middle class students! (Solomon, pp. 22 & 27).

I interpret the finding by Suppes, as reported by Solomon, that CAI drill and practice assists the Disadvantaged but not the middle class students in this way:-

  1. Middle class kids would be more likely to do their homework (put in the time at home to generate a significant number of parabolas so that the patterns would start to make sense) and so would not need the quick fix provided by a quadratics software program, so much.
  2. Middle class kids question the system of School but are more likely to stay and perform within it. Disadvantaged kids are more likely to question the system, reject it and drop out of it, either physically or mentally.
    go back5
Bill Kerr, email: billkerr at gmail.com
Woodville High School,
South Australia

Creative Commons License
This work is licensed under a Creative Commons License.