Cog Sci and Mathematics

After yesterday’s post I had been discussing Michael Ramscar’s papers with some friends, including Alex Hartley, and I had some interesting thoughts on the applications of his research as related to mathematics. Today’s blog post will be a follow-up to yesterday’s, where I will ask a question and

Alex asked,

“So how would this affect color-word learning in languages (like French and Spanish) that put the noun before the adjective (la pomme rouge)? I wonder if any work has been done in that domain? ”

I think that would definitely speed up color word learning, assuming there is no utterance in French/Spanish that change the FL order. As far as I can remember, there exist “C’est une pomme rouge” and “La pomme est rouge” – both of which fit our feature-label order qualifications. A quick Google search for “rouge pomme” turns up only a trendy restaurant, which I am going to assume means that the LF order for colors in French is done for hipness and is not commonly used.

One point that I do want to point out is that this (LF/FL learning order) goes for everything, not just color words. In mathematics, if you take a student who sees just a soup of numbers when he looks at the page, and you ask him, “Which fraction is ‘one-fifth’?”, he will not be able to tell you, just like he can’t pick out which is “the blue ball”. He attends to ‘fractions’ first, and, seeing many, is not sure which you mean. If you say to him, “one-fifth is the same as what fraction?”, he will get it. (Here, “fraction” is the label and “one-fifth” is the object.) It’s a matter of what he attends to first – if he attends to “fractions” he gets confused, just like if he attends to “blue”. If he attends to “one-fifth”, he gets it, since there are few things called “one-fifth” around.

OH! This brings me to a realization – this is an instance of a fundamental problem we see in mathematics education often: we teach the outcomes of mathematics as if they were the mathematics itself.* What does that mean? I mean that we learn things like the long division algorithm without understanding how they work. This is dry and boring, because it has no logic to it. Real math involves proving concepts and putting together ideas to create new ones. It is exciting and creative. If that sounds foreign to your math experience, it’s because we don’t teach it that way. What we teach are the things that mathematicians discovered centuries before us, and we now have to memorize.

It seems to me that this is related to students’ trouble learning fractions. In the example above, “fractions” is the Label, and “one-fifth” is the Feature. That is, “one-fifth” is a concrete entity that you can touch, and “fraction” is more general, an abstract category label. But we persist in telling students that they are going to learn about “fractions”, without having anything concrete for them to match it with. What is the one, singular “fraction”? Can you point me to it without giving me an example? We tell students they are going to learn about “fractions” without first knowing about “one-fifth”, “one-half”, “one-eighth”, or “one-hundredth”. That’s like telling kids they are going to learn about “furniture” before they know about “chairs”, “tables”, or “beds”! There is no way that they can learn the general category of “fractions” or “furniture” before they understand the elements that go into that category.

Perhaps this is where the Feature/Label orientation actually comes from – it is a matter of abstractness of the quality. The Feature is concrete – if we point it out first, we can understand what is being talked about. But the Label is abstract – if we point it out first, we are not sure which of the many objects with that label to apply it to.

But we as adults can figure it out. How come? It is because we are capable of processing many other sources of information – social context, verbal cues, previous experience with “blue” and “balls” – that we are able to understand what the speaker is referring to with the “blue ball” or the “fraction that equals one-fifth”. But children, with their limited experience and limited processing capacity, do not understand what you are talking about. To understand it you must have an understanding of both “blue” and “ball” alone to be able to apply them together – so it’s a question of previous experience, and working memory, that create this understanding. It’s similar to the subitization of 2, 4, and 6 that Ramscar cites, in that you have to be able to recognize both objects independently before you can put them together. What’s exciting about this is that it’s kind of a scientific proof of scaffolding knowledge!

So to bring this full circle, this may be why fractions/decimals/percents may be so difficult for students to learn – because instead of showing them the examples (Features) and letting them fit them into a category (Label), we teach them the category and expect them to derive the examples! By doing this, we tell them that Fractions, Decimals, and Percents are actually very different things in very different categories — and then we throw them for a loop again when we tell them, whoa, wait, these different things are actually all equal! Small wonder they have such trouble getting it. To make it simple, we have to flip around the way we teach this to them – we have to start with concrete quanities they can touch, then show them how to represent each of those as a fraction, a decimal, and a percent. When they see that we started from a quantity (say, what we adults would call “half” a pie), then Label that quantity as “one-half”, “point five”, and “fifty percent”, they will have a much greater understanding of what is going on!

As always, thanks for reading and please leave any responses/challenges/academic trash talk for me in the comments. Thanks!

Frank

* PS — For a beautiful illustration of how we teach the outcomes of mathematics instead of the mathematics itself, mathematician Paul Lockhart tells a wonderful story where he compares it to music education gone wrong: http://www.maa.org/devlin/LockhartsLament.pdf.

“A musician wakes from a terrible nightmare. In his dream he finds himself in a society where
music education has been made mandatory. “We are helping our students become more
competitive in an increasingly sound-filled world.” Educators, school systems, and the state are
put in charge of this vital project. Studies are commissioned, committees are formed, and
decisions are made— all without the advice or participation of a single working musician or
composer.
Since musicians are known to set down their ideas in the form of sheet music, these curious
black dots and lines must constitute the “language of music.” It is imperative that students
become fluent in this language if they are to attain any degree of musical competence; indeed, it
would be ludicrous to expect a child to sing a song or play an instrument without having a
thorough grounding in music notation and theory. Playing and listening to music, let alone
composing an original piece, are considered very advanced topics and are generally put off until
college, and more often graduate school.”

Unbelievable…right?

BTW, this goes for everything, not just color words. if you take a student who is bad at math, and sees just a soup of numbers when he looks at the page, he is not going to be able to pick out the correct numbers to use in a problem, just like he can’t pick out which color word refers to what. Meaning, if you say to him, “what fraction equals one-fifth?” he will be confused, since he sees a lot of fractions; and if you say to him, “one-fifth is the same as what fraction?”, he will get it. It’s a matter of what he attends to first – if he attends to “fractions” he gets confused, just like if he attends to “blues”. If he attends to “one-fifth”, he gets it, since there are few things called “one-fifth” around.

(this appears to be a reversal of what they say, but I am thinking it’s the same since you are telling them to attend to the less common quality first, then narrow their scope. But I could be wrong!)

Then the interesting part comes with “blue ball” – this is a unified unit. To understand it you must have an understanding of “blue” and “ball” and be able to apply them together – so it’s a question of working memory as well as understanding. It’s similar to subitization in that you have to be able to recognize both objects independently before you can put them together. It’s kind of a scientific proof of scaffolding knowledge!

The scaffolding of the “blue ball” goes the same for fractions, which are super hard for kids to learn. First you have to understand what a fraction is. Then you can learn about the different kinds of fractions. Then you can learn about the different kinds of decimals. Then percents. Perhaps this is why fractions/decimals/percents are so confusing to students – because they appear to be in different categories, and it is not made clear that they are all a subinstance of the category “number”. It appears instead that the fractions/decimals/percents are all very different things in different categories, and then we tell them that they’re actually all equal! Whoa! No wonder they don’t get it…