Basic Math in Monkeys and College Students (Cantlon and Brannan)
Human and multiple non-human species have demonstrated a variable capacity for intuitive number sense (approximate quantitative representation, counting, and simple addition). Mathematics has a significant link with verbal and visual symbolic representations (ie, the arabic number symbols) that have been used by humans to extend mathematical analysis far beyond the most basic intuitive quantitative analysis. Humans and a much more limited number of non-human species demonstrate any capacity for "symbolic sense" for even basic mathematics - the (variable) ability to associate verbal or visual symbols with the species' (variable) number sense. Major limiting factors on the results and comparison of results among previous studies have included small sample sizes and methodological differences (particularly when compared to human experiments). Although results should be taken with requisite skepticism, some of the differences found are still clearly in the domain of basic mathematical operations; multiple species can track quantity in a +1 addition experiment, but few non-human species seem to exhibit an intuitive concept of set addition. Furthermore, humans, although more capable than many other species, are clearly limited in their intuitive capacity for set addition. In the study above, adult human (n = 14) demonstrations of intuitive set addition was similar to that of adult female rhesus macaque monkeys (n = 2).
The study used the same method to compare the species, but sample size continues to be an issue. Results, though, do seem plausible: that humans performed a bit more accurately and quickly, and could more accurately add larger sets. The limitations found in human capacity also seem plausible, showing a declining accuracy as set size increased and also as set size and similarity in set quantity increased. If my assumption is correct that symbolic sense in mathematics is constructed upon numerical sense, results would support the inference that learning the rules of symbolic manipulation for mathematics doesn't necessarily mean that there is a depth of understanding in the meaning of results; in other words, it's likely that there is a significant difference between the ability to do symbolic math and the ability to understand the meaning of both the process and the result.
There are profound educational implications of the relationship between intuitive mathematics and symbolic mathematics. For one example, I've encountered this many times in teaching about the history of life on Earth and the concept of biological evolution. Current estimates put the age of our planet at about 4.5 billion years, and evidence of life has been found in fossils nearly 4 billion years old. It's easy to look over what I've just done because it's so commonplace: I represented the astronomically incredible age of our planet with 10 squiggly marks on the screen, and the age of life on our planet with only 8 squiggly marks! To really understand evolution, we need to understand huge quantities of time. I have heard a variety of analogies used to help students understand large quantities. However, many of these analogies are rooted in symbolic math, and, unsurprisingly, still involve thinking about huge quantities: for example, a mole, about 6.023 x 10^23, is huge.
To increase the intuitive understanding for the enormity of "billions" of years, I have students do an activity that specifically builds symbolic math upon a simple but powerful physical experience. The student prepares with a blank piece of paper and a writing implement, and for five minutes draws lines - "hash marks" - on a page. (Of course, students can take a break if their hand cramps a bit or if they are uncomfortable at all - I try to cheer them on a bit like a sport coach...) After five minutes I have them count up their marks, and then help them to think about each of them as representing one full Earth year. We then work through a calculation of how much time it would take them - with no breaks for eating, sleeping, etc - to make 4 billion marks. The usual result is usually around 40 years, so I ask them to think about what they'll be like when forty years from now -- it's enough to get the gears moving in a much more powerful way than just saying and/or writing 4 billion.
I think there are some interesting philosophical implications of these studies, too, though I'm admittedly superficial and short in my ability to deal with them. But I think they come down to the remaining "open" question of whether or not mathematical logic is created or discovered - ie: is math a property of the universe or a property of the human mind? Given that the human mind is based in the human brain, and the human brain shares at least some degree of common ancestry with the brains of all species tested (and certainly a great deal with the monkey), perhaps the question on the "universality" of mathematical logic is refined to whether it is a property of the universe or of the vertebrate mind.
Exploring relationships among neuroscience, cognitive psychology, and learning. Sprinkled with education policy, reform, and leadership, STEM education research, and technology.
Sunday, March 30, 2008
Wednesday, March 26, 2008
From TED - "My Stroke of Insight" by Jill Bolte Taylor
A colleague sent me this video, and in so doing, introduced me to the TED theme "How the Mind Works". I'd been loosely familiar with TED already, but I'm thrilled to see that they're sharing so many ideas from so many incredible people on a subject that I find so interesting and so critical to the future development of education. I have provided a link to the TED theme in my links section, and look forward to accessing more of the resources therein.
I hope that sharing this talk by Jill Bolte Taylor, entitled "My Stroke of Insight", will help others to learn about the large-scale functional differentiation of the human brain, and will promote interest in learning about some of the smaller-scale functional differences that I suspect will become more and more important in facilitating student learning. While I do tend to be a bit skeptical about some of the generalizations included in Dr. Taylor's presentation, I think they help to build the foundation for inspiration and critical thinking. In particular this presentation demonstrates the educational power of personal experience and sharing it.
Visit the TED page for the transcript of this talk and much more.
I hope that sharing this talk by Jill Bolte Taylor, entitled "My Stroke of Insight", will help others to learn about the large-scale functional differentiation of the human brain, and will promote interest in learning about some of the smaller-scale functional differences that I suspect will become more and more important in facilitating student learning. While I do tend to be a bit skeptical about some of the generalizations included in Dr. Taylor's presentation, I think they help to build the foundation for inspiration and critical thinking. In particular this presentation demonstrates the educational power of personal experience and sharing it.
Visit the TED page for the transcript of this talk and much more.
Labels:
brain
Sunday, March 23, 2008
Neuron behavior - randomness, Godel's theorum, and cognitive modeling
Stochastic Differential Equation Model for Cerebellar Granule Cell Activity (Saarinen et al)
Along with my efforts to read and to think about current research in neuroscience and cognitive psychology, I'm still reading books and novels as is my norm. I just finished "Next" by Michael Crichton - an easy read, but provocative in its dealing with themes related to the implications of modern biological technologies. After I finished it, I decided to pull out "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. I first read it in my first semester of graduate school, and found it to be a very inspiring and challenging text on consciousness. I'm still only in the first few chapters of my re-read, but I'm thinking a lot about one of the major points of the book, related to the work of Godel, showing that formal systems can't be both complete and consistent (IANAM - I'm trusting D.H. and the Wikipedia page on Godel's Incompleteness Theorems - they seem to match in their interpretations of Godel's work).
So, again - I Am Not A Mathematician - but it would seem that the stress on interpreting the implications of Godel's aforementioned work is that it applies to formal systems that attempt to provide a perfect list of starting information and processing rules that are not contradictory of themselves (hence, complete and consistent). There are a variety of systems - mathematics perhaps as the most pure - that we think of as consistent and complete in our superficial, everyday analysis, but are not ... for example, as given by Hofstadter, language and the "liar's paradox" ("This statement is false."). These paradox are given a new name by Hofstadter: "Strange Loops"; I hope to write more on the liar's paradox and my thoughts on its relationship to epistemological development - for now, though, I want to stay focused on the implications of Godel's proof to attempts to model cognition.
In the article linked at the beginning of this post, the authors propose that stochastic - random - modeling of neural cell activity provides a more accurate representation of the in vitro behavior of said cells. The classical model for neural activity - the Hodgkin-Huxley formalism - is deterministic in the sense that it posits that voltage-dependent ion channels (responsible for the 'ON' / 'OFF' status of the neuron) behave predictably. It's been known for some time that they are not perfectly predictable, but models for neural networks have been based on these easier-to-deal-with deterministic formalisms for some time. The authors of this paper point out that improvements can be had with regard to the modeling, but also with regard to computational speed, when a particular method for representing the random behavior of these voltage-dependent ion channels is built into the larger-scale models. Good stuff for those limiting their research only on networks of neurons, but how can this be relevant for others who want to transcend the chasm between brain and mind?
I think the link is provided, with due credit to Hofstadter, by Godel and now Saarinen (et al). Or perhaps with credit to Godel and now Saarinen (et al) by way of Hofstadter? As I proceed with actually writing about some of the research I'm familiar with, and continue to read and write about more current work, I believe we'll find that the pursuit to formalize conceptual change continues to use deterministic underpinnings, in which the state of one element of the model (E1), with consideration to its relationship with another element of the model (E2), determines the state of the other element. Deterministic modeling makes sense, given that there is a significant degree of predictability in the behavior of neurons, and even (though to a lesser degree, I'd think) in the behavior of the mind. However, I suggest that the stochastic modeling found beneficial by Saarinen et al, will also be beneficial to the efforts to construct both static and dynamic models of cognition - the models that will ultimately help us to understand the process of learning (constructing and retaining knowledge). [Those familiar with my old blog on the concept of evolution may have seen this before - I'll be bringing some of those ideas and posts over to this blog in the near future.]
Finally, I do think that all of this can relate back to the classroom and to teaching (which I am beginning to tend towards defining as 'the process of manipulating environment and available resources to promote student learning'). Many times in the classroom, teachers are confronted with random and seemingly-meaningless student questions; a stochastic model for cognition, based on the stochastic behavior of the functional units of the brain, could incorporate the random activation of prior knowledge that is (from the expert perspective) unrelated to the concept or skill at hand. As these stochastic cognitive models (and their implications) are developed and studied, I suspect we'll begin to see meaning in the seemingly-meaningless, as perhaps these "random" student ideas and/or questions are critical to the brain-mind's learning process. Perhaps the brain-mind has, in fact, evolved in a way that allows it to transcend the limitations that Godel proved inherent to formal systems. In other words, and appropriately paradoxically so, it might be true that our very capacity for logical thought rests on the fundamental informality of the workings of the brain.
Along with my efforts to read and to think about current research in neuroscience and cognitive psychology, I'm still reading books and novels as is my norm. I just finished "Next" by Michael Crichton - an easy read, but provocative in its dealing with themes related to the implications of modern biological technologies. After I finished it, I decided to pull out "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. I first read it in my first semester of graduate school, and found it to be a very inspiring and challenging text on consciousness. I'm still only in the first few chapters of my re-read, but I'm thinking a lot about one of the major points of the book, related to the work of Godel, showing that formal systems can't be both complete and consistent (IANAM - I'm trusting D.H. and the Wikipedia page on Godel's Incompleteness Theorems - they seem to match in their interpretations of Godel's work).
So, again - I Am Not A Mathematician - but it would seem that the stress on interpreting the implications of Godel's aforementioned work is that it applies to formal systems that attempt to provide a perfect list of starting information and processing rules that are not contradictory of themselves (hence, complete and consistent). There are a variety of systems - mathematics perhaps as the most pure - that we think of as consistent and complete in our superficial, everyday analysis, but are not ... for example, as given by Hofstadter, language and the "liar's paradox" ("This statement is false."). These paradox are given a new name by Hofstadter: "Strange Loops"; I hope to write more on the liar's paradox and my thoughts on its relationship to epistemological development - for now, though, I want to stay focused on the implications of Godel's proof to attempts to model cognition.
In the article linked at the beginning of this post, the authors propose that stochastic - random - modeling of neural cell activity provides a more accurate representation of the in vitro behavior of said cells. The classical model for neural activity - the Hodgkin-Huxley formalism - is deterministic in the sense that it posits that voltage-dependent ion channels (responsible for the 'ON' / 'OFF' status of the neuron) behave predictably. It's been known for some time that they are not perfectly predictable, but models for neural networks have been based on these easier-to-deal-with deterministic formalisms for some time. The authors of this paper point out that improvements can be had with regard to the modeling, but also with regard to computational speed, when a particular method for representing the random behavior of these voltage-dependent ion channels is built into the larger-scale models. Good stuff for those limiting their research only on networks of neurons, but how can this be relevant for others who want to transcend the chasm between brain and mind?
I think the link is provided, with due credit to Hofstadter, by Godel and now Saarinen (et al). Or perhaps with credit to Godel and now Saarinen (et al) by way of Hofstadter? As I proceed with actually writing about some of the research I'm familiar with, and continue to read and write about more current work, I believe we'll find that the pursuit to formalize conceptual change continues to use deterministic underpinnings, in which the state of one element of the model (E1), with consideration to its relationship with another element of the model (E2), determines the state of the other element. Deterministic modeling makes sense, given that there is a significant degree of predictability in the behavior of neurons, and even (though to a lesser degree, I'd think) in the behavior of the mind. However, I suggest that the stochastic modeling found beneficial by Saarinen et al, will also be beneficial to the efforts to construct both static and dynamic models of cognition - the models that will ultimately help us to understand the process of learning (constructing and retaining knowledge). [Those familiar with my old blog on the concept of evolution may have seen this before - I'll be bringing some of those ideas and posts over to this blog in the near future.]
Finally, I do think that all of this can relate back to the classroom and to teaching (which I am beginning to tend towards defining as 'the process of manipulating environment and available resources to promote student learning'). Many times in the classroom, teachers are confronted with random and seemingly-meaningless student questions; a stochastic model for cognition, based on the stochastic behavior of the functional units of the brain, could incorporate the random activation of prior knowledge that is (from the expert perspective) unrelated to the concept or skill at hand. As these stochastic cognitive models (and their implications) are developed and studied, I suspect we'll begin to see meaning in the seemingly-meaningless, as perhaps these "random" student ideas and/or questions are critical to the brain-mind's learning process. Perhaps the brain-mind has, in fact, evolved in a way that allows it to transcend the limitations that Godel proved inherent to formal systems. In other words, and appropriately paradoxically so, it might be true that our very capacity for logical thought rests on the fundamental informality of the workings of the brain.
Monday, March 17, 2008
Locating the light bulb: neural activity correlates to insight
Neural Activity When People Solve Verbal Problems with Insight (Jung-Beeman et al, 2004)
The "Aha!" moment is, perhaps, one of the most tangible and exciting educational experiences for teachers and students alike; the students finally "get it" - whatever concept or skill "it" might be. Using fMRI and scalp EEG, Beeman et al describe that the experience of insight in solving a problem (followed by an "Aha" moment experience) correlates with a particular type of activity in a specific region of the brain called the right anterior superior temporal gyrus. This front and top region of the right hemisphere's temporal lobe is active in the early stages of problem solving, but also experiences a burst of activity approximately 0.3s prior to the "Aha" moment.
The location of this region in the right hemisphere means that there are some interesting things that we can do to promote insight, such as presenting helpful information and/or potential solutions to the left part of the visual field. Much as our right limbs are controlled by the left hemisphere of our brain, the left part of our visual field is processed by the brain's right hemisphere. The study also points out that individuals vary in their response to solving problems with insight, and even found one individual's brain responded more strongly to non-insight problem solving. So, we shouldn't be surprised, then, that different students need different stimuli and time to achieve insight, nor that some students may not achieve the "Aha" experience in our classrooms. Finally, it's important to keep in mind that the problems studied were word problems, and that the area of the brain found to be active preceding the conscious "Aha" experience is very close to an area of the brain found to be strongly associated with language skills; it's possible that "Aha" experiences are caused by different regions of the brain when the problem solving takes on different modalities.
The "Aha!" moment is, perhaps, one of the most tangible and exciting educational experiences for teachers and students alike; the students finally "get it" - whatever concept or skill "it" might be. Using fMRI and scalp EEG, Beeman et al describe that the experience of insight in solving a problem (followed by an "Aha" moment experience) correlates with a particular type of activity in a specific region of the brain called the right anterior superior temporal gyrus. This front and top region of the right hemisphere's temporal lobe is active in the early stages of problem solving, but also experiences a burst of activity approximately 0.3s prior to the "Aha" moment.
The location of this region in the right hemisphere means that there are some interesting things that we can do to promote insight, such as presenting helpful information and/or potential solutions to the left part of the visual field. Much as our right limbs are controlled by the left hemisphere of our brain, the left part of our visual field is processed by the brain's right hemisphere. The study also points out that individuals vary in their response to solving problems with insight, and even found one individual's brain responded more strongly to non-insight problem solving. So, we shouldn't be surprised, then, that different students need different stimuli and time to achieve insight, nor that some students may not achieve the "Aha" experience in our classrooms. Finally, it's important to keep in mind that the problems studied were word problems, and that the area of the brain found to be active preceding the conscious "Aha" experience is very close to an area of the brain found to be strongly associated with language skills; it's possible that "Aha" experiences are caused by different regions of the brain when the problem solving takes on different modalities.
Saturday, March 15, 2008
Research resources
As I'm devoting some limited free time to developing this blog, I'm finding a number of good resources for current research involving the overlap of cognitive neuroscience and education research. I have a number of articles I'm parsing through from the Public Library of Science (PLoS) journals, including PLoS Biology, PLoS Computational Biology, and PLOS one. I've also discovered the International Mind, Brain, and Education Society, a fairly recently-formed group that organizes the journal Mind, Brain, and Education. I'm thinking about joining the Society - seems right up my alley - but I'm going to have to keep looking for access to their journal as it doesn't seem available through databases I'm currently using to access subscription-based materials.
This brings up, naturally, the question of open access. I believe that open access to research - especially when in any part publicly funded - is critical to current and future development, especially that which deals with issues related to education. So, while I'm not going to stop reading through subscription-based journals as I have the opportunity, and reporting on interesting findings cogent to this blog when possible, it is likely that I will continue to focus my reading and reporting efforts using sources that will be openly available to as wide an audience as possible.
This brings up, naturally, the question of open access. I believe that open access to research - especially when in any part publicly funded - is critical to current and future development, especially that which deals with issues related to education. So, while I'm not going to stop reading through subscription-based journals as I have the opportunity, and reporting on interesting findings cogent to this blog when possible, it is likely that I will continue to focus my reading and reporting efforts using sources that will be openly available to as wide an audience as possible.
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professional development
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