Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Talk aloud to solve mathematical problems quickly

  1. Dec 19, 2009 #1
    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.
     
  2. jcsd
  3. Dec 20, 2009 #2
    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 [Broken] ]. 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.
     
    Last edited by a moderator: May 4, 2017
  4. Dec 20, 2009 #3
    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.
     
  5. Dec 20, 2009 #4
    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.
     
  6. Dec 20, 2009 #5

    HallsofIvy

    User Avatar
    Science Advisor

    Hey, at least if the guy next to me talked aloud while solving problems on a test, all my troubles would disappear!
     
  7. Dec 20, 2009 #6
    Heh, heh... until you find out he was talking nonsense.

    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.
     
    Last edited: Dec 20, 2009
  8. Dec 20, 2009 #7
    Right! And you would get a bad grade

    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:

    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.
     
  9. Dec 20, 2009 #8
    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 Language 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.

    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."
     
  10. Dec 20, 2009 #9
    Talking out loud helps me a lot- but I have a serious case of ADHD
     
  11. Dec 20, 2009 #10
    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.
     
  12. Dec 20, 2009 #11
    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.
     
  13. Dec 20, 2009 #12
    Last edited by a moderator: Apr 24, 2017
  14. Dec 21, 2009 #13
    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.
     
    Last edited by a moderator: Sep 25, 2014
  15. Dec 21, 2009 #14
    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.
     
    Last edited by a moderator: Apr 24, 2017
  16. Dec 21, 2009 #15
    Is english discovered or invented (question mark)

    English is a language.
    Math is a language.

    Math is invented.
     
  17. Dec 21, 2009 #16
    What about the incommensurables? Weren't they discovered?
     
  18. Dec 21, 2009 #17
    What about them?

    You're talking about numbers and not math.
    Math is a language that deals with numbers.
     
  19. Dec 21, 2009 #18
    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.
     
  20. Dec 21, 2009 #19
    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.
     
  21. Dec 21, 2009 #20
    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.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook