Talk aloud to solve mathematical problems quickly

[BenVitale]

I didn't know that students who think aloud while solving math problem solve faster and have more possibility of finding right solution.

Can’t solve a mathematical problem? Just talk aloud and see your troubles disappear, suggests a new study


Your thoughts, please.

 

[Rasalhague]

My thoughts are:

Three students is a very small sample, but maybe it's more of a demonstration of investigative technique (recording people to see what's revealed about the problem-solving strategies they use).

I get the impression (although I'm not sure) that they passively looked at what each student was doing, rather than the psychologists choosing which student would speak aloud. I wonder what would happen if the talkers-aloud weren't self selecting.

My feelings are:

Grrrr, I hate it when journalists get hold of a story and can't be bothered to give such obviously useful details as which issue, what title, or who the authors of the study were. One vaguely written article can be duplicated all over the net, making it hard to sift through all those pages to see if any have further details. The English version that's doing the rounds in December 2009 seems to be a translation of a Spanish article which dates back at least to June 2009 [ http://www.plataformasinc.es/index.php/esl/Noticias/Hablar-en-voz-alta-ayuda-a-resolver-mas-rapido-los-problemas-matematicos ]. Where it originated, I don't know.

The April edition of the journal mentioned is dedicated to mathematical education [ http://www.investigacion-psicopedagogica.org/revista/new/index.php?n=17 ], but I haven't been able to find the paper here yet. Maybe someone with more patience can root it out... On the plus side, if we can find it, it should be possible to read it online for free.

 

[snipez90]

I agree with Rasalhague's assessment. The sample size and the apparent selection bias seem to be glaring problems. I was also disappointed that the article includes so few details about the actual study. Also, mathematics at the pre-graduate level seems to retain a competitive aspect, ranging from math competitions to undergraduate exams, so talking aloud isn't particularly practical in many situations. On the other hand, mathematics seems more collaborative at the graduate level and beyond, but the point isn't necessarily to be able to solve problems as quickly as possible.

 

[Gerenuk]

I think every word in your mind blocks math thinking completely. In a few exceptions it might be useful to memorize an equation with words. But in general maths is only visual thinking and those geeks who can do crazy maths in their heads report that they "see the numbers". Whereas the more students say that they talk to themselves while doing maths, the worse they usually did.

 

[HallsofIvy]

Hey, at least if the guy next to me talked aloud while solving problems on a test, all my troubles would disappear!

 

[Rasalhague]

Hey, at least if the guy next to me talked aloud while solving problems on a test, all my troubles would disappear!

Heh, heh... until you find out he was talking nonsense.

But in general maths is only visual thinking

Roger Penrose has some interesting stuff in The Emperor's New Mind, in the final chapter, in a section called "Non-verbalirty of thought". He quotes Einstein ("the words or the language, as they are spoken, do not seem to play any role in my mechanism of thought"), Galton ("I do not think as easily in words as otherwise") and Hadamard ("every word disappears the moment I begin to think it over), and says of himself that "almost all my mathematical thinking is done visually and in terms of non-verbal concepts, altough the thoughts are often accompanied by inane and almost useless verbal commentary, such as 'that thing goes with that thing and that thing goes with that thing'. (I might use words sometimes for simple logical inferences.)"

But it's not clear from the article whether the unnamed psychologists recorded whether the verbalisation was of the kind Penrose describes or something fuller and more coherent.

I've noticed that I can do algebra while listening to spoken language, without too much distraction, but I'd find it hard to read a book at the same time and take in both the spoken and written word.

 

[BenVitale]

Heh, heh... until you find out he was talking nonsense.

Right! And you would get a bad grade

He quotes Einstein ("the words or the language, as they are spoken, do not seem to play any role in my mechanism of thought"), .

So that supports the view that math relies on visuo-spatial rather linguistic processes

The building blocks for mathematical thoughts are certain signs or images, more or less clear

They become clear as we combine them ... pretty much as you have stated:

thing goes with that thing and that thing goes with that thing'.

I got the feeling it is both: visual and language

Consider arithmetic, word problems

Skills to solve or create problems in arithmetic, or with word problems require in a language-specific format.

 

[Rasalhague]

I got the feeling it is both: visual and language

In Penrose's example, his "inane" linguistic commentary, "that thing goes there", seems like a kind of underlining or highlighting of the visual image, directing his attention to certain aspects of it, but not embodying the complexity of the ideas he's having at that moment.

A very different mode of thought is described by Brent Silby in Revealing the Langauge of Thought.

http://www.scribd.com/doc/3033742/Revealing-the-Language-of-Thought-by-Brent-Silby

He writes from a philosophical viewpoint, arguing for the identity of natural language and thought. For example, he imagines an interior monologue about realising that he's forgotten to bring coffee: "...oh no, there's no coffee. What will I do? Where will I get some? Damn! What a hassle, I'll have to get some. But I put some in my bag last night..." (3.3.1). But Penrose's description sounds much more like my own experience than Silby's. I don't laboriously spell out all my thoughts in an internal monologue. When I subvocalise, I tend to leave gaps. If anyone could hear the words in my head, even if they could see the pictures too, they'd probably not make much sense without knowing the context of what I was simultaneously thinking about, what connections I was making between ideas.

Silby: "If someone asks us what we are thinking, our expression of the thought is the same as our inner experience of that thought" (4.3.4).

Galton: "It often happens that after being hard at work, and having arrived at results that are perfectly clear and satisfactory to myself, when I try to express them in language, I feel that I must begin putting myself on quite another intellectual plane. I have to translate my thoughts into a language that does not run very evenly with them. I therefore waste a vaste deal of time seeking words or phrases, and am conscious when required to speak of a sudden, of being often very obscure through mere verbal maladroitness, and not through want of clearness of perception."

Penrose: "I had noticed, on occasion, that if I had been concentrating hard for a while on mathematics and someone would engage me suddenly in conversation, then I would be unable to speak for several seconds."

In my experience, sensory imagery, including especially visuals and fragmentary subvocalisation seem like a notepad for my thoughts, or an anchor, an aid to memory, and a tool for directing and structuring them. But the visuals and bits of natural language usually don't fully embody the thought, because I could conceive of seeing the same mental picture and subvocalising the same words while thinking something different (having a different idea about them, or noticing different conections between things). And like Galton, I'm familiar with the experience of knowing clearly what I wanted to say, but being unable to find the right words.

Consider arithmetic, word problems

Even there, unless the problem was immediately obvious, I suppose we might convert it into some other format, arithmetic, algebraic, geometric to solve it, then translate back into words. But when the problem involves definitions or concepts that are new to me, I do like to repeat them to myself and try to find the clearest and simplest way of verbalising them to help bed them in.

This made me chuckle. Penrose: "That is not to say that I do not sometimes think in words, it is just that I find words almost useless for mathematical thinking. Other kinds of thinking, perhaps such as philosophizing, seem to be much better suited to verbal expression. Perhaps this is why so many philosophers seem to be of the opinion that language is essential for intelligent or conscious thought."

 

[wisvuze]

Talking out loud helps me a lot- but I have a serious case of ADHD

 

[BenVitale]

Thanks for your insights, references and links.

I need to read Penrose and Francis Galton. I wish I had more time to read!

So, the basic question is, "How much of human thought is dependent on language?"

And, why do we need to talk loud to solve math problems quickly?

Neuroscience can helps us with the questions we have posted. Neuroscience is not my area of study, math is: I'm just a math student curious about these questions.

On a different subject, let's consider autistic savants.

They are able to quickly learn new languages, and remember scenes from years earlier in cinematic detail. They see numbers in dynamic form : Numbers assume complex, multi dimensional shapes in my head that I manipulate to form the solution to sums, or compare when determining whether they are prime or not. Numbers and words have form, color, texture and so on. They come alive to them.

On the other hand, basic arithmetic and "number sense" appear to be part of the shared evolutionary past of many primates.

Read the article : Number sense has a biological basis.

Monkeys are confused with zero!

Babies have a truly abstract sense of numerical concepts even before they learn to speak
Baby Got Math

How do we map numbers, I mean, how do we represent and order numbers?

If you close your eyes and imagine the numbers 1 through 9 on a line, what does the image that appears in your head look like?

Our Innate Sense of Numbers is Logarithmic, Not Linear

suggesting that children's early mental number line is logarithmic, then it transforms to linear with age and experience.

 

[Rasalhague]

More on Elizabeth Brannon's work and related ideas here:

http://discovermagazine.com/2009/nov/17-the-brain-humanity.s-other-basic-instinct-math

Maybe the idea of an innate logarithmic number line has something to do with why Adrian Veidt's argument in the Watchmen film that he killed millions "to save billions" falls flat. One feels very different from a hundred, and might make the moral parts of us wonder, but we don't have much of an instinctive sense of millions being different from billions; they're just BIG NUMBERS.

This makes me think of the role of logarithmic scales in human perception:

http://en.wikipedia.org/wiki/Weber–Fechner_law

Your description of autistic savant attitudes to numbers reminds me of what Daniel Tammet has said about his own thought processes. I associate numbers with colours, and give them personalities a little bit, but nothing that elaborate.

 

[wisvuze]

https://www.amazon.com/dp/0465037712/?tag=pfamazon01-20

is a good read for the psychology behind mathematical thinking.

 

[Noxide]

I have ocd. Whenever I solve any kind of math problem, simple or difficult, I write down just about every theorem I know that could be applied to the problem. From that it kind of just solves itself. Sometimes if I get a really difficult problem I kind of... uh... take it with me (in my mind) and go for a walk and solve it while walking or jogging etc.

Also, I'm on the autism spectrum. I learn things in two different ways. I have a pictorial understanding of concepts that only makes sense to me (eg the fibonnaci number sequence- I understand that as quickly flipping pages on a slanted book kind of like this but not the gun picture- if you look at the paper on the bottom right of the book- kind of forming a pyramid) or if I'm lazy I just memorize the 'normal' way so that I can explain it to other people or write it down on a test. I usually just memorize theorems or definitions and sometimes parts of conversations and spit them out later. I know tons of equations from memory because I've seen them from flipping through a book, but I don't know what most of them mean because I've just memorized them. I guess that's what I do with questions sometimes too. I just memorize them and work on them in my head. It's much faster that way but sometimes I get all messed up because I usually end up with an answer rather than all the work. It takes me a long time to do simple questions though, because I write down all the theorems when I'm doing simple questions. But it takes about the same amount of time to do a difficult question.

 

[BenVitale]

https://www.amazon.com/dp/0465037712/?tag=pfamazon01-20

is a good read for the psychology behind mathematical thinking.

It is a difficult read and would require some time to digest it.

Have you read it? If so, could you post some comments, the highlights?

----------------------
The authors reject the Platonic view of mathematics. They say all we know and can ever know is human mathematics -- that mathematics independent of human thought cannot be answered.

I don't understand how they can reject Plato's ideal.

Philosophers like to ask mathematicians the following question: Is mathematics discovered or invented?

I had this question when I took philosophy classes. It seems to me that the people who are concerned with this question are the philosophers and philosophy professors. Mathematicians and math students do math.

I like to answer that math is both discovered and invented. And it is interesting to study more closely in the history of mathematics the part which belongs to each.

 

[Noxide]

Is english discovered or invented (question mark)

English is a language.
Math is a language.

Math is invented.

 

[BenVitale]

...

... Math is invented.

What about the incommensurables? Weren't they discovered?

 

[Noxide]

What about them?

You're talking about numbers and not math.
Math is a language that deals with numbers.

 

[BenVitale]

I'm talking about math in general which can be viewed as a language and/or as a game -- a game with rules which we can change and create new rules -- and of the unreasonable effectiveness of math in sciences. I'm also talking about mathematical objects and mathematical truths.

 

[Noxide]

I'm talking about math in general which can be viewed as a language and/or as a game -- a game with rules which we can change and create new rules -- and of the unreasonable effectiveness of math in sciences. I'm also talking about mathematical objects and mathematical truths.

Yet you cite a specific, and poor, counterexample? So you point something out specifically, and once said counterexample is thwarted you assert that you're actually proposing an implied generality?

lol.

 

[Rasalhague]

I sometimes gesture with my hands, e.g. to remind myself which numbers go with which in matrix multiplication, or taking a determinant. When working out problems in special relativity, I sometimes hold one hand horizontally flat with the palm down to represent a spacelike hyperplane and then tilt it to remind myself which in which direction such planes are tilted under a Lorentz boost.

 

[Rasalhague]

Is english discovered or invented (question mark)

English is a language.
Math is a language.

Math is invented.

The parochial details of English, the features that set it apart from other human natural languages, have evolved through culture. Some words or expressions have been deliberately invented, but most features of the language can't be traced back to any individual speaker's deliberate decision. It's theorized that there are universals in human natural language, underlying structures we come preprogrammed with, which have evolved biologically through natural selection.

How much of human natural language is necessary or inevitable in a communication system of this flexibility is hard to say till we find and understand some independently evolved system.

We also have some inborn mathematical insincts, some of which we share with related species. But the part of mathematics that's transmitted by culture seems to involve much more artifice than the cultural parts of language. On the other hand, this artifice often surprises us in a way that inventions usually don't, and can gel uncannily well with diverse aspects of the universe we live in. Here too there's the question of which aspects of our mathematical thought are arbitrary and parochial, and which are necessary and universal and liable to be found in any independently created/evolved system.

Maybe mathematics, in the sense of the notation and parochial details, is like a pidgeon English we've concocted to communicate with nature (visible and invisible); and we could just as well have invented a different notation or way of thinking about an idea. But the success and richness of this method suggests that there are indeed patterns, underlying the parochial details, that are more than arbitrary invention.

 

[Noxide]

The parochial details of English, the features that set it apart from other human natural languages, have evolved through culture. Some words or expressions have been deliberately invented, but most features of the language can't be traced back to any individual speaker's deliberate decision. It's theorized that there are universals in human natural language, underlying structures we come preprogrammed with, which have evolved biologically through natural selection.

How much of human natural language is necessary or inevitable in a communication system of this flexibility is hard to say till we find and understand some independently evolved system.

We also have some inborn mathematical insincts, some of which we share with related species. But the part of mathematics that's transmitted by culture seems to involve much more artifice than the cultural parts of language. On the other hand, this artifice often surprises us in a way that inventions usually don't, and can gel uncannily well with diverse aspects of the universe we live in. Here too there's the question of which aspects of our mathematical thought are arbitrary and parochial, and which are necessary and universal and liable to be found in any independently created/evolved system.

Maybe mathematics, in the sense of the notation and parochial details, is like a pidgeon English we've concocted to communicate with nature (visible and invisible); and we could just as well have invented a different notation or way of thinking about an idea. But the success and richness of this method suggests that there are indeed patterns, underlying the parochial details, that are more than arbitrary invention.

Again, that's not what we're talking about. We're talking about math. If OP had said mathematical CONCEPTS, CONCEPT being the key word, then I would say that we do discover concepts. To draw a parallel to language, I will use love. Love is a concept that we have discovered. We did not invent the physiological reactions associated with it. However, we use english, a language which we invented, to associate a word, love, which we invented, to the concept of love. Much like we use math, a language we invented, to communicate mathematical concepts.

It's pretty simple. We invented math. We did not necessarily invent mathematical concepts.

 

[Rasalhague]

Again, that's not what we're talking about. We're talking about math. If OP had said mathematical CONCEPTS, CONCEPT being the key word, then I would say that we do discover concepts. To draw a parallel to language, I will use love. Love is a concept that we have discovered. We did not invent the physiological reactions associated with it. However, we use english, a language which we invented, to associate a word, love, which we invented, to the concept of love. Much like we use math, a language we invented, to communicate mathematical concepts.

It's pretty simple. We invented math. We did not necessarily invent mathematical concepts.

Sounds like we think something similar after all, behind the words (I just took mathematics in a broader sense), but is it that simple? Can we always distinguish between discovery and invention? Suppose you live in an isolated settlement, perhaps in the distant past before the world was well explored: it'd be all too easy to suppose that all the people and territory you know are more-or-less representative of the whole world. A student beginning to learn about physics might get the impression that there's something inevitable or necessary about working with a Cartesian coordinate system, till they come to realize that it's just one possibility among many, and that physical laws can be expressed in a mathematical form that's frame independent. Until we meet an alien civilisation, we can't always be sure aspects of our ideas are universal and which parochial. The idea embodied in the English word love (good example!) might seem universal, but languages divide up the word in different ways. Yes, the English word covers some universal human feelings, as well as accidental cultural ideas, but it isn't inevitable that a language will have a word with the vast range of concepts and connotations of the English word love (sexual or non-sexual affection, religious devotion, preference or enjoyment of inanimate object, etc.).

http://en.wikipedia.org/wiki/Greek_words_for_love