Kaj_Sotala comments on How my math skills improved dramatically - Less Wrong

21 Post author: JonahSinick 05 March 2014 08:27PM

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Comment author: byrnema 07 March 2014 02:16:44AM *  0 points [-]

I agree that while not exactly 'volume conservation', this addresses the exact same skill.

If the child gave a good explanation for a problem, there was only a 43 percent chance of his advancing the same explanation when later confronted with the identical problem.

Would you interpret this as meaning the children had not acquired the concept, after all? It seems that if the child actually truly understands the concept that moving things around doesn't change their number, then they wouldn't be inconsistent. (Or is the study demonstrating what I found unintuitive, that children can grasp and then forget a concept?)

Comment author: Kaj_Sotala 07 March 2014 02:42:49PM *  1 point [-]

I interpreted it as indicating that there are multiple ways of thinking about the problem, some of which produce the right answer and some of which produce the wrong answer. There's an element of chance involved in which one the child happens to employ, and children who are farther along in their development are more likely but not certain to pick the correct one on any single trial.

"Acquiring a concept" is a little ambiguous of an expression - suppose there's some subsystem or module in the child's brain which has learned to apply the right logic and hits upon on the right answer each time, but that subsystem is only activated and applied to the task part of the time, and on other occasions other subsystems are applied instead. Maybe the brain has learned that this system/mode of thought is the right way to think about the issue in some situations, but it hasn't yet reliably learned to distinguish what those situations are.

Not sure how analogous this really is, but I'm reminded of the fact that IBM's Watson used a wide variety of algorithms for scoring possible answer candidates, and then used a metalearning algorithm for figuring out the algorithms whose outputs were the most predictive of the correct answer in different situations (i.e. doing model combination and adjustment). So it, too, had some algorithms which produced the right answer, but it didn't originally know which ones they were and when they should be applied.

That kind of an explanation would still be compatible with a sudden boost in math talent, if things suddenly clicked and the learner came to more reliably apply the correct ways of thinking. But I'm not entirely sure if it's necessarily a developmental thing, as opposed to just being a math-related skill that was acquired by practice. Jonah wrote:

Because there was substantial overlap in the algebraic techniques utilized in the different subjects I was studying, my exposure to them per day was higher, so that when I learned them, they stuck in my long-term memory.

And if there is a specific "recognize the situations that can be thought of in algebraic terms and where algebraic reasoning is appropriate" skill, for example, then simultaneously studying multiple different subjects employing the same algebraic techniques in different contexts sounds just like the kind of thing that would be good practice for it.

Comment author: byrnema 07 March 2014 04:01:33PM *  1 point [-]

I appreciate your responses, thanks. My perspective on understanding a concept was a bit different -- once a concept is owned, I thought, you apply it everywhere and are confused and startled when it doesn't apply. But especially in considering this example I see your point about the difficulty in understanding the concept fully and consistently applying it.

Volume conservation is something we learn through experience that is true -- it's not logically required, and there are probably some interesting materials that violate it at any level of interpretation. But there is an associated abstract concept -- that number of things might be conserved as you move them around -- that we might measure comprehension of.

There are different levels at which this concept can be understood. It can be understood that it works for discrete objects: this number of things staying the same always works for things like blocks, but not for fluids, which flow together, so the child might initially carve reality in this way. Eventually volume conservation can be applied to something abstract like unit squares of volume, which liquids do satisfy.

Now that I see that the concept isn't logically required (it's a fact about everyday reality we learn through experience) and that there are a couple stages, I'm really skeptical that there is a physical module dedicated to this concept.

So I've updated. I don't believe there are physical/neurological developments associated with particular concepts. (Abstract reasoning ability may increase over time, and may require particular neurological advancements, but these developments would not be tied with understanding particular concepts.)

Seems kind of silly now. Though there was some precedent with some motor development concepts (e.g., movements while learning to walk) being neurologically pre-programmed.

This seems an appropriate place to observe that while watching my children develop from very immature neurological systems (little voluntary control, jerky, spasmodic movements that are cute but characteristic of very young babies) to older babies that could look around and start learning to move themselves, I was amazed by how much didn't seem to be pre-programmed and I wondered how well babies could adapt to different realities (e.g., weightlessness or different physics in simulated realities). Our plasticity in that regard, if my impression is correct, seems amazing. Evolution had no reason to select for that. Unless it is also associated with later plasticity for learning new motor skills, and new mental concepts.

Comment author: Kaj_Sotala 07 March 2014 09:29:24PM *  1 point [-]

I appreciate your responses, thanks.

I appreciate hearing that you appreciate them. :)

Boaler 1993 is another interesting discussion about the rules that people might use in order to decide what kind of skill or mental strategy might apply to a situation.

It argues that, because school math problems often require a student to ignore a lot of features that would be relevant if they were actually solving a similar problem in real life, they easily end up learning that "school math" is a weird and mysterious form of mathematics in which normal rules don't apply. As a result, while they might become capable of solving "school math" problems, this prevents them from actually applying the learnt knowledge in real life. They learn that school math problems require a mental strategy of school math, and that real-life math problems require an entirely different mental strategy.

Lave [1988] has suggested that the speciļ¬c context within which a mathematical task is situated is capable of determining not only general performance but choice of mathematical procedure. Taylor [1989] illustrated this effect in a research study which compared students' responses to two questions on fractions: one asking the fraction of a cake that each child would get if it were shared equally between six, and one asking the fraction of a loaf if shared between five. One of the four students in Taylor's case study varied methods in response to the variation of the word, "cake" or "loaf". The cake was regarded as the student as a single entity which could be divided into sixths, whereas the loaf of bread was regarded as something that would always be divided into quite a lot of slices - the student therefore had to think of the bread as cut into a minimum of, say, ten slices with each person getting two-tenths of a loaf. [...]

One difficulty in creating perceptions of reality occurs when students are required to engage partly as though a task were real whilst simultaneously ignoring factors that would pertinent in the "real life version" of the task. [...] Wiliam [1990] cites a well known investigation which asks students to imagine a city with streets forming a square grid where police can see anyone within 100m of them; each policeman being able to watch 400m of street (see Figure 1.)

Students are required to work out the minimum number of police needed for different-sized grids. This task requires students to enter into a fantasy world in which all policemen see in discrete units of 100m and "for many students, the idea that someone can see 100 metres but not 110 metres is plainly absurd" [Wiliam, 1990; p30]. Students do however become trained and skillful at engaging in the make-believe of school mathematics questions at exactly the "right" level. They believe what they are told within the confines of the task and do not question its distance from reality. This probably contributes to students' dichotomous view of situations as requiring either school mathematics or their own methods. Contexts such as the above, intended to give mathematics a real life dimension, merely perpetuate the mysterious image of school mathematics.

Evidence that students often fail to engage in the "real world" aspects of mathematics problems as intended is provided by the US Third National Assessment of Educational Progress. In a question which asked the number of buses needed to carry 1128 soldiers, each bus holding 36 soldiers, the most frequent response was 31 remainder 12 [Schoenfeld, 1987; p37]. Maier [1991] explains this sort of response by suggesting that such problems have little in common with those faced in life: "they are school problems, coated with a thin veneer of 'real world' associations."

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