Imagine your mind is made of Christmas trees

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Imagine that your mind is a bunch of Christmas trees. Note that I wrote, ‘imagine’. I am not claiming your mind is literally a bunch of Christmas trees, which may seem obvious but is nonetheless important, as we shall see later.

Anyway, go with it for now.

Every time you learn something new, you hang it on one of those Christmas trees. Most of the time this is not a problem. One of the trees might be something like, ‘People desire power’, and the thing you hang on it is a specific instance from history of someone seeking power. You are adding specific details to a broad idea you already hold.

This is how most learning proceeds and that is that.

However, some ideas end up on the wrong tree. You may have the right tree available but still hang it on the wrong one, or you may not even possess the right tree.

Possibly the most well-researched examples of this kind involve physics (although the idea extends to any conceptual change). For instance, our intuitive views of motion tend to be wrong. We have a tree that represents, ‘Things move because they feel a force,’ and then we hang observations onto this; observations like the motion of a football. Physics teachers instead want their students to think of moving footballs in terms of Newton’s first and second laws of motion.

Empirical data suggests that we can go to quite heroic efforts to hang things on the wrong tree. We will rationalise new observations. Even attempts to induce ‘cognitive conflict’ by demonstrating that a particular observation is inconsistent with a particular tree tend to be twisted into compatibility. If I point out that nothing is pushing a football through the air, you might decide there is some residual force still left over from the kicking foot.

And yet we do, rarely, manage to change our understandings of concepts and so there must be some mechanism by which this is possible.

Stellan Ohlsson has developed a theory to explain what is going on and I’ve already been using it in this post. In his view, we collect specific bits of knowledge, observations, beliefs and so on under different theories – my Christmas trees. The theories do not have to agree with each other and the specific bits of knowledge are not logical consequences of the theory. Instead, when we encounter something new, we look for the best theory to hang it under. These need to be locally coherent, but nothing more. We simply lack a system for going through each item and checking its logical consistency.

Ohlsson suggests that each of these bits of knowledge are subsumed under different theories. If you want to move them then you need to resubsume them under a different theory.

This is hard but not impossible. Once you have both theories available in your mind, you can start to try to hang each piece of knowledge under each theory. A process of competition begins between the two theories until the most useful one wins through. This is how conceptual change occurs: Resubsumption Theory.

Ohlsson is coy about suggesting implications for teaching, noting that other factors could be far more important in conceptual change than any implications of Resubsumption. However, he does note a few things that should give us pause for thought.

The idea of developing new ideas in supposedly ‘relevant’ contexts may be misconceived because it may lead to students hanging the new knowledge under the theories they already associate with those contexts. Instead, it may be better to first make the new theory available to the students in an abstract form such as a computer generated microworld. Instead of familiar footballs and rockets, we could develop Newton’s laws in a virtual world. Once the new theory is available in the minds of students, the process of competition and resubsumption can begin.

This is an interesting idea.

Like Cognitive Load Theory, Resubsumption derives from information processing theories that model the mind as a computer. This has led to some criticism on the basis that minds are not computers.

I do not believe that minds are computers any more than I believe that they are made of Christmas trees. But I think it’s a pretty good theory to subsume some of our findings under, for now.

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11 thoughts on “Imagine your mind is made of Christmas trees”

1. Interesting! I think there’s a good parallel here in linguistics, where syntactical structures are such that, with limited data, one can jump to incorrect conclusions about the structure of one language based on analogies from one’s own mother tongue, or another familiar language. Many linguistics problems (including a few that I’ve designed) involve little traps where the solver is subtly invited to hang data off the wrong tree.

• In fact, that one I sent you on Twitter a while back was a classic example of that.

2. Tom Burkard says:

I doubt that making new theories “available to students in an abstract form” is likely to be of much use to teachers or students. I’m sure Willingham is right in claiming that “When encountering new material, the human mind appears to be biased towards learning the surface features of problems, not toward grasping the deep structure that is necessary to achieve flexible knowledge.”

Part of the problem with the Christmas tree analogy is that our schema are complex and to a degree interconnected, and that reading a book may throw huge amounts of new learning at them. There’s no telling whether the new knowledge will attach at any number of different points, but if it doesn’t make many connections, we will quickly become confused or bored and stop reading. For instance, our initial definitions of a word are usually simple and specific–little more than a synonym for a word or phrase we already know. The more times we encounter the word in different contexts, the more likely it is that we will develop a richer understanding of it–this is because it now connects with other knowledge. Occasionally, we will get it wrong, but quite obviously we usually get it right. Otherwise there’d be no point in reading or even going to school.

I’ve recently written the first draft of a paper relating to the debate as to whether teaching procedural knowledge in maths should precede or follow conceptual knowledge. Although there is a general consensus that both develop in tandem, most maths educators insist that ‘number sense’ must precede instruction in procedural knowledge–quite the opposite to usual practice in the Far East. On June 4, Barry Garelick’s and Tara Houle’s comments on this blog lent powerful support to the Oriental approach.

• Michael Pye says:

How did comments on a blog lend powerful support? When someone says something you like it is not powerful support. Links to well designed research or rigerous evidence do that. Also please elaborate on what you call number sense (I am guessing arithmetic) and how procedural knowledge can be taught while it is still weak.

• Tom Burkard says:

Well, it’s a waste of time commenting if you;re so dismissive of comments on a blog–your very comment is a waste of time in this case. You’ve just cited your own opinion and failed to cite a single piece of research. And if you’ve spent as much time on google scholar as I have, you’d understand how frequently well-designed and rigorous studies come up with quite different conclusions.
However, you clear know nothing about number sense. Don’t feel too bad about this–this is a term which can mean anything. Never mind that maths educators insist that it’s a prerequisite for learning even the simplest arithmetic. Berch (2005) has trawled a lot of studies and has found the following defitions:
1. A faculty permitting the recognition that something has changed in a
small collection when, without direct knowledge, an object has been re-
moved or added to the collection (Dantzig, 1954).
2. Elementary abilities or intuitions about numbers and arithmetic.
3. Ability to approximate or estimate.
4. Ability to make numerical magnitude comparisons.
5. Ability to decompose numbers naturally.
6. Ability to develop useful strategies to solve complex problems.
7. Ability to use the relationships among arithmetic operations to under-
stand the base-10 number system.
8. Ability to use numbers and quantitative methods to communicate,
process, and interpret information.
9. Awareness of various levels of accuracy and sensitivity for the reason-
ableness of calculations.
10. A desire to make sense of numerical situations by looking for links be-
tween new information and previously acquired knowledge.
11. Possessing knowledge of the effects of operations on numbers.
12. Possessing fluency and flexibility with numbers.
13. Can understand number meanings.
14. Can understand multiple relationships among numbers.
15. Can recognize benchmark numbers and number patterns.
16. Can recognize gross numerical errors.
17. Can understand and use equivalent forms and representations of num-
bers as well as equivalent expressions.
18. Can understand numbers as referents to measure things in the real
world.
19. Can move seamlessly between the real world of quantities and the math-
ematical world of numbers and numerical expressions.
20. Can invent procedures for conducting numerical operations.
21. Can represent the same number in multiple ways depending on the con-
text and purpose of the representation.
22. Can think or talk in a sensible way about the general properties of a nu-
merical problem or expression—without doing any precise computation.
23. Engenders an expectation that numbers are useful and that mathematics
has a certain regularity.
24. A non-algorithmic feel for numbers.
25. A well-organized conceptual network that enables a person to relate
number and operation.
26. A conceptual structure that relies on many links among mathematical re-
lationships, mathematical principles, and mathematical procedures.
27. A mental number line on which analog representations of numerical
quantities can be manipulated.
28. A nonverbal, evolutionarily ancient, innate capacity to process approxi-
mate numerosities.
29. A skill or kind of knowledge about numbers rather than an intrinsic
process.
30. A process that develops and matures with experience and knowledge.

3. An example, mentioned in Ericsson & Pool ‘Peak’: Deslauriers, Schelew & Wieman (2011). Improved Learning in a Large-Enrollment Physics Class. Science. goo.gl/Ps7qQc

4. This sounds very similar to “physics modelling instruction”. Each unit/model starts with an experiment that is carefully chosen so that teachers know students will get good results. These are the start of model building (christmas trees), which then continues with a number of activities. The questions in these activities, and the way students interact with each other, are carefully chosen so that students get a number of chances to reevaluate their old mental models (christmas trees) with the new model they developed at the start of the unit.

• Chester Draws says:

Would it not be easier to show them a good model right from the start, and then use experiments to show that the new model is better than their previous understanding?

I don’t like putting incomplete ideas into students and then correcting them as we go along, as I have seen too many students not correct them (if a student is sick and misses an important correction, what then?).

This reinforces a concept missing from the original Christmas tree model of learning, which is that we ascribe ideas right from the start with a weighting that depends on the authority of the proposer. It is why it is important that teachers are seen to know their subject inside out right from the start. Because an idea from a perceived authority is always given more weight.

The progressive “we’ll discover this as a class” as we go along risks the class suspecting that the teacher knows little more than they do. If an experiment doesn’t work well, then all weight given to the “discovered” idea is very low. Instead of being close enough to confirm an explicit teaching scheme, it risks establishing doubt in a discovery system.

And the idea that a teacher can teach knowing little more about the subject than the students is fatal to weighting. Why would anyone trust the ideas of a person who knows little more than they themselves do?

If the perceived authority of the teacher is high enough then people will much more willingly abandon old ideas. No amount of showing a previously held idea is incorrect will work if the proposer is not trusted.

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