tag:blogger.com,1999:blog-7691251033406320222.post4997241767158919492..comments2024-03-26T04:19:38.862-07:00Comments on kitchen table math, the sequel: Eat, Poop, and PlayCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger111125tag:blogger.com,1999:blog-7691251033406320222.post-1923085042929634182010-04-27T14:41:11.473-07:002010-04-27T14:41:11.473-07:00I fully agree with you regarding "reform math...I fully agree with you regarding "reform math". <br /><br />To clarify our discussion, conceptual understanding can come in two broad strokes: a working knowledge of the concepts and a formal description of those concepts. What students should be developing in arithmetic is a solid working knowledge of the concepts. This does not mean that they have to make explicit definitions of place value or the distributive property, but they should be able to use both in calculations and problem solving. A solid conceptual understanding in arithmetic can be characterized by the ability to use those fundamental arithmetic concepts and skills in solving novel, unfamiliar problems. <br /><br />Regarding Keith Devlin's article: he makes an interesting point "one of the things that high school mathematics education should definitely produce is the ability to learn and be able to apply rule-based symbolic processes without understanding them." He is, of course, referring to the mathematics learned in high school which by definition is abstract and not concrete.<br /><br />But arithmetic is by definition concrete. It deals with real numbers and how to solve problems with those numbers. It is not necessary to deal with arithmetic in the abstract as is the case with more advanced mathematics. But is is necessary in arithmetic to lay the conceptual foundation for being able to think abstractly. That is done by developing a working knowledge of the foundational concepts in arithmetic (place value, equivalency, etc...)<br /><br />I am, if nothing else, extremely practical. If memorization of procedures was the quickest road to mastery, I would have whole-heartedly embrace that approach. But after reading Liping Ma and using Singapore to teach students math, it became exceptionally clear that coupling conceptual development with procedural fluency was *substantially* faster than memorization and led to an almost seemless transition to abstract math (Algebra). Singapore is not procedural fluency coupled with a few concepts. At its core is the idea that both concepts and the procedures are necessary for developing arithmetic mastery. <br /><br />I realize that the press by the fuzzy math guys has soured many on the idea of conceptual development. And if your only exposure to teaching "conceptual math" came from TERC, I could see why you would be horrified. But we shouldn't let the Fuzzys take away one of the more insightful ideas that has come out of international comparisions: <br />concepts + procedures + problem solving => fast and efficient arithmetic masteryErin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-1099713276913564422010-04-27T07:10:46.217-07:002010-04-27T07:10:46.217-07:00The bar models that Erin talks about provide a mod...The bar models that Erin talks about provide a mode to solve problems. It does shed some light on the conceptual underpinning, but by and large functions to let the student solve the problem fairly efficiently. I would not deem it as laborious, in the same manner as requiring students to list three ways to add two three-digit numbers, or draw a rectangle model to illustrate what 4/5 X 2/3 is, which some of the reform texts do. <br /><br />The problem solving method does allow students to see more of what is going on in part/whole relationships. It is not a be-all end-all in and of itself. Singapore works well because of a number of things. Problem solving via bar modeling is one of them, but it is also the sequencing of topics and the building upon mastered material. Students still are required to solve many numerical problems as well as multi-step word problems.<br /><br />The idea of procedural fluency leading to understanding is done in tandem with concepts, such as Singapore does. But a more complete understanding will probably come later--particularly when they enter algebra when they have more symbolic tools with which to analyze and set up problems.<br /><br />The example I gave of differential equations was meant to illustrate how you get to the understanding later. In college, there is a shorter time period that allows one to go back and understand the theory because there are more schema than what kids in grade school have. Thus, the conceptual understanding that kids have of why algorithms work, even those who come from the Singapore program, is going to be somewhat limited. But the foundation is there upon which to build, which I think is what you are saying and I agree with that.<br /><br />Keith Devlin has an interesting article about procedural fluency and understanding <a href="http://www.maa.org/devlin/devlin_03_06.html" rel="nofollow">here.</a>Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-62515870040239438712010-04-26T10:44:23.820-07:002010-04-26T10:44:23.820-07:00"My use of the term was strictly in terms of ..."My use of the term was strictly in terms of educational practice, in that academic language acquisition is not well-characterized (unlike arithmetic which is well-characterized), as much about what and how students acquire proficiency in the content areas is not well known."<br /><br />I really don't comprehend this sentence.<br /><br />Now if you're talking about how much we understand about language acquisition versus arithmetic acquisition, the truth is, we don't actually know a lot about either. In fact our theories of learning anything are embarrassingly vague.<br /><br />Nevertheless, there are interesting phenomena to observe...<br /><br />Ever heard of the Piraha? Well some anthropologists studied a group sort of like them. There was an interesting article about it. The key finding of the researchers is that the "natural state" of people is to have a logarithmic sense of the number line, e.g. the *absolute* distance between 10 and 11 is smaller than the distance between 2 and 3. (There was a rather clever experimental design to sound out these intuitions.) That's because in "the state of nature" it was much more useful to compare relative sizes rather than to do sort of any absolute counting (is that bear bigger than me? is this fruit tree better to pluck from than this one? do I have enough? etc.) and arithmetic developed only because it was a wa of making sure one wasn't getting ripped off in trade. <br /><br />e.g. formal math skills developed out of the need for commerce.<br /><br />kindergarteners start out with a very logarithmic sense of the number line (yes, despite their not knowing what logarithms are), but it steadily "flattens" to a straight line as they proceed to second and third grade. <br /><br />this is not observed with tribes in Brazil that don't have math classes. They don't have distinct words for anything bigger than five. <br /><br />still despite this our knowledge of how we learn arithmetic is still very ill-characterised. Certainly we haven't characterised any sort of neurobiological circuit that would act as an adder or a subtractor, as we have done with computers. Heck, we have characterised the neural circuits tht give rise to central pattern generators that create sinusoidal waveforms for constant swimmers in dogsharks because of mutual inhibition.<br /><br />but we haven't characterised arithmetic acquisition that well -- neurobiologically.<br /><br />linguistically I think we're actually a little better off because we have historical linguistics data and much more case studies and interesting (very telling) phenomena like creolisation.le radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-73050359603632161362010-04-26T10:33:07.918-07:002010-04-26T10:33:07.918-07:00sorry. I must frankly be pained at the sheer. lack...sorry. I must frankly be pained at the sheer. lack. of. science. Linguistics isn't into wishy-washy English lit textual criticism. Linguistics is the science of language: language processing, acquisition, from the canonical forms of syllable structure you can write on paper to FOXP2 genes you can sequence. Thank goodness.<br /><br />"So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains"<br /><br />A few self-evident statements.<br /><br />1. Spoken language is THE language i.e. written language is dependent on spoken language.<br /><br />2. Academic language uses grammatical structures already present in a language. It just uses them with a lot more frequency. That is to say, the variance of grammatical forms, etc. is actually reduced.<br /><br />3. It's hard to get a divergence in actual grammar unless there is a divergence in spoken languages. This can happen among people specialising in different trades that over some period of time become isolated from each other, e.g. "argots". Kind of hard to do it with written literature.<br /><br />4. For this matter, literature tends to be unity-promoting, not divergence-promoting. Classical Chinese separated from the vernacular dialects because people actually spoke it in daily life, i.e. used it to recite poems and communicate across disparate regions lacking a common language, i.e. it was the equivalent of Latin.<br /><br />I really can't see that happening with academic English.le radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-60062699271453955992010-04-26T08:42:53.199-07:002010-04-26T08:42:53.199-07:00Perhaps you have a technical definition to "w...Perhaps you have a technical definition to "well-characterized" that I am not aware of. <br /><br />My use of the term was strictly in terms of educational practice, in that academic language acquisition is not well-characterized (unlike arithmetic which is well-characterized), as much about what and how students acquire proficiency in the content areas is not well known. <br /><br />Your comments regarding memorization first (for both programming and language) and understanding second are commonly categorized as a "whole-to-part" strategy of learning.Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-39930559820310827532010-04-26T04:50:44.517-07:002010-04-26T04:50:44.517-07:00'"...once she moves into "genre patt...'"...once she moves into "genre patterns," we seem to have entered a house build on sand." <br />So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains or that those differences are inconsequential in student learning? '<br /><br />First, your conclusion about what I would say is a non-sequitor. I was writing about how recommendations for teaching need to be based on empirical results, not armchair theory.<br /><br />Second, there are obviously differences in vocabulary and background knowledge from field to field. This is hardly a revolutionary point. But background knowledge isn't language, and, linguistically speaking, vocabulary is a superficial aspect of language. Furthermore, I don't now anyone who would disagree that people need to learn the assumed background knowledge and vocabulary of a given field before becoming an effective reader of its texts.<br /><br />Third, while academic writing involves different statistical *rates* of different grammatical structures, it doesn't involve grammatical structures that aren't found in non-academic speech.<br /><br />Fourth, "textual structures" isn't a well-defined linguistic term.<br /><br />Fifth, different academic fields don't involve different grammatical structures. E.g., you don't find different sorts of elipses in history texts than you find in economics text.<br /><br />Sixth, the only case where grammar rules need to be taught explcitly are with nonnative Standard English speakers, and children with language delays (that's what GrammarTrainer does).<br /><br />"What I find bizarre in your comments of whole vs part learning is that your GrammarTrainer is so well developed in capturing the essential parts that are needed to understand language."<br /><br />First, GrammarTrainer is informed by psycholinguistic findings about language delayed children, and about language acquisition.<br /><br />Second, I haven't made any comments "of whole vs. part learning."<br /><br />'"My original referencing of Schleppegrell was in response to your (above) comment that all linguists would disagree with the assertion that "unlike language, arithmetic is well characterized: I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement."'<br /><br />None of what you've quoted from Schleppengrell shows a linguist disagreeing with the assertion that language is well-characterized.<br /><br />"I would assume that linguistic discourse is similar to the hard sciences and that real consensus is often difficult to find, thus making statements of "all linguists" or "all scientists" rarely justified."<br /><br />There is consensus on basic, well-established notions. There is also consensus among (true) biologists about evolution, and among (true) astronomers about the (round) shape of the earth.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-70951063078189165512010-04-25T23:50:04.196-07:002010-04-25T23:50:04.196-07:00"...once she moves into "genre patterns,..."...once she moves into "genre patterns," we seem to have entered a house build on sand." <br /><br />So would you say, then, that there are no differences in language (vocabulary, syntax, grammar, textual structures, ellipsis, background knowledge, etc....) between genres/subject matter domains or that those differences are inconsequential in student learning? Her point is, of course, that those subtle differences between academic discourse can be quite confusing to students who do not have a home life that is filled with academic language. Hardly a radical position, inconsequential position.<br /><br />What I find bizarre in your comments of whole vs part learning is that your GrammarTrainer is so well developed in capturing the essential parts that are needed to understand language. Not only does it partition out the needed subskills in learning so that kids can access the substructure of language instead of just the superfical aspects, it also sequences the learning to carefully build the whole (language). So why are you such a strong advocate of "whole language" when your program necessitates delineating the parts while building the whole?<br /><br />In a previous comment you stated that "For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages." <br /><br />My original referencing of Schleppegrell was in response to your (above) comment that all linguists would disagree with the assertion that "unlike language, arithmetic is well characterized: I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement."<br /><br />I would assume that linguistic discourse is similar to the hard sciences and that real consensus is often difficult to find, thus making statements of "all linguists" or "all scientists" rarely justified.<br /><br />"I don't know how you'd implement a program to teach people how to read/write academically. I think it'd be rather awful." <br /><br />Based upon what evidence?<br /><br />"I think a 12-year-old could write academically"<br /><br />Based upon what evidence?<br /><br />"...rhetorical techniques of Miley Cyrus and her appeal"<br /><br />And this is now considered "academic"? No wonder our schools are in such dire need of reform.Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-10145702874996007542010-04-25T21:55:20.744-07:002010-04-25T21:55:20.744-07:00I think the whole issue of too much theorising and...I think the whole issue of too much theorising and not enough experiment may apply to Ed School "paradigms" as well<br /><br />besides I don't know how you'd implement a program to teach people how to read/write academically. I think it'd be rather awful. Neither people who are already good nor people who are "bad" would tolerate it very well. I really think it's a vibe you pick up ... you try to sound precise, neutral, etc. with language reflecting as little personal judgment, prejudice, etc. as possible and it just flows out that way.<br /><br />And half of the other issue is just jargon specific to the field being specialised in.<br /><br />I think a 12-year-old could write academically. Now instead of say, physical chemistry or chromosome rearrangements, get her to try neutrally analyse the rhetorical techniques of Miley Cyrus and her appeal, and we lose most of the jargon issue.le radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-63436352916946612342010-04-25T21:26:36.100-07:002010-04-25T21:26:36.100-07:00As a linguist who is also interested in education,...As a linguist who is also interested in education, I am indeed always thinking about connections between learning and language. Here, the most interesting and evidence-backed theories are in psycholinguistics. Once people start making proposals about how children learn, and recommendations about how teachers should teach, empirical research is key. I've taken a "look inside" the first of the Scheppegrell books you list via its Amazon page (the second book doesn't allow me to look inside), and, on the topic of "academic registers," it doesn't look promising on the empirical front. Instead of references to experiments, we have references to what other people "suggest" and "argue":<br /><br />"Veel (1997) suggests that academic registers construe "distinctive and favored ways of viewing the world; ways which we recognize as 'scientific,' logical,' and 'rational'" (p. 161) Similarly Lemke (1987) argues that what we call "thinking logically" is for the most part simply using language... according to genre patterns ... to teach"<br /><br />The empirical research she cites seems to pertain more to well-known facts about language development in general (e.g. increasing sentence complexity); once she moves into "genre patterns," we seem to have entered a house build on sand.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-81814861933964638822010-04-25T19:46:42.429-07:002010-04-25T19:46:42.429-07:00She has a PhD in linguistics and is currently a pr...She has a PhD in linguistics and is currently a professor at University of Michigan specializing in academic language development. <br /><br />Her work:<br />"The Language of Schooling: A Functional Linguistics Perspective"<br />"Reading in Secondary Content Areas"<br /><br />You certainly don't have to pay attention to her work if you choose not to. But as a linguist that is also interested in education, you might be interested in the application of linguistic theory towards learning in the content areas.Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-87906200060624444502010-04-25T17:50:17.097-07:002010-04-25T17:50:17.097-07:00Who is Schleppegrell and why should we care what s...Who is Schleppegrell and why should we care what she "asserts"? All sorts of people (with and without PhDs) have asserted (and even published) all sorts of strange ideas about language and "texts" and "contexts"; unless it's more than armchair theorizing (i.e., in this case, backed up by empirical studies of how children acquire different "subject specific registers", complete with a definition of key terms, including this last one) I see absolutely no reason to pay attention to her.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-74918472791845999622010-04-25T12:03:54.473-07:002010-04-25T12:03:54.473-07:00they don't have enough out-of-classroom practi...they don't have enough out-of-classroom practice. <br /><br />really it's the same with French or another foreign language<br /><br />investing a little effort into writing a poem or short story of your own accord out of your own interest makes you discover idioms, discuss usage with people, etc. that you wouldn't know otherwise<br /><br />it helps to know vocabulary and to read Wikipedia, Google Scholar etc. on your own timele radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-77208165648283416052010-04-25T11:45:48.429-07:002010-04-25T11:45:48.429-07:00Schleppegrell's assertion is not that there is...Schleppegrell's assertion is not that there is "one academic register" but multiple subject specific registers with subtle but important language differences necessary for effective communication in the content areas. I would imagine that this would be of interest to people who are teaching children to communicate in the content areas and particularly those teachers who are working with children who have not experienced those academic language registers outside the school setting. As she states: "A more nuanced understanding of the role of language in schooling recognizes that students' difficulties may be related to inexpereience with the linguistic demands of the tasks of schooling and unfamiliarity with ways of structuring discourse that are expected in school."<br /><br />"I think it's a register that can be acquired naturally, i.e. by writing correspondence."<br /><br />If this were true for all students then why is it that so many students fail to become proficient at academic reading and writing?Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-78544505401488599792010-04-25T09:46:00.992-07:002010-04-25T09:46:00.992-07:00I would turn off from a "how to write/read ac...I would turn off from a "how to write/read academically" seminar, so much. I expect most of my peers would too.<br /><br />I think it's a register that can be acquired naturally, i.e. by writing correspondence.le radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-55443555344689660602010-04-25T09:38:30.726-07:002010-04-25T09:38:30.726-07:00"Many times, mastery of the procedure allows ..."Many times, mastery of the procedure allows students to understand the underlying concept."<br /><br />Of course, some students can intuit the concepts from learning procedures. But is this the most efficient manner in which to reach mastery of arithmetic for a large cohort of children? <br /><br />Certainly, the Singapore math program, Liping Ma, etc. would argue against that postiion. <br /><br />The entire point about the bar graphs in Singapore math is to make visual the conceptual elements of the problem. Drawing out bar graphs takes time, it is slow and laborious at first, but it allows the students to conceptualize the problem before solving. And yet that visualization/conceptualization allows a greater efficiency, understanding and transferability in solving complex problems.<br /><br />"I recall in some math courses, like differential equations, reading the theoretical explanation for a certain type of equation, but not following very well. I then started working the problems, imitating the procedures outlined in some of the worked examples. After getting to a certain level of proficiency, I was then able to re-read the explanation and understand what was going on."<br /><br />You are recalling what a fairly proficient student can do after developing a substantial base of knowledge. Arithmetic is taught to fairly young children who are not highly self-aware (nor should they be asked to be) and so the structure of the program needs to be quite different.<br /><br />"If, however, you believe students must experience the laborious version in order to understand the "shortcut", I disagree."<br /><br />Laborious does not mean slow. Even learning to memorize the procedures take time. My experience is that the laborious, conceptual route takes significantly *less* time to master the algorithms than just introducing the algorithms by themselves, memorizing the steps and practicing enough for fluency. I don't think that students should be wasting a lot of time with algorithms and finding the most efficient route to mastery should be the goal of any arithmetic program. Learn why algorithms they work, how they work, become proficient with their use and move on to the more important elements of arithmetic: solving complex multi-step problems.<br /><br />"I don't know what she/you mean by "academic linguistic elements". Do you know what I mean by "abstract linguistic structure"? Unless we know we're talking about the same thing, it's silly to keep talking about it."<br /><br />Agreed and given that I did not write that monograph, perhaps I should defer that more explicit definition to you. <br /><br />I think on the larger scale, what we are discussing falls under the the debate of the whole-to-part vs part-to-whole approach to learning. Certainly, my experience is that, if possible, part-to-whole works more efficiently for the largest number of children. But to teach that way all the parts need to be well defined (and in arithmetic they have been extensively if not completely characterized). For many elements of learning, particularly in language, that "part" structure is poorly defined or non-existant. (Vocabulary learning is largely best learned by whole-to-part while decoding is best learned part-to-whole.)Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-545835401453748492010-04-25T08:24:02.198-07:002010-04-25T08:24:02.198-07:00'Schleppegrell asserts that these academic lin...'Schleppegrell asserts that these academic linguistic elements are not elements of style but are essential components of content discourse and thus are needed to become proficient within the subject matter domains. And additionally, because these academic language structures are not typically enountered in everyday conversation, they need to be explicitly taught. Is she mistaken?"<br /><br />If she thinks that "academic linguistic elements" are the same as "abstract linguistic structure," then she is mistaken.<br /><br />I don't know what she/you mean by "academic linguistic elements". Do you know what I mean by "abstract linguistic structure"? Unless we know we're talking about the same thing, it's silly to keep talking about it.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-55535183237921311302010-04-25T08:01:43.594-07:002010-04-25T08:01:43.594-07:00academic writing is simply, another register.
tho...academic writing is simply, another register.<br /><br />though whenever I try to pronounce academic words (especially those I learnt early in life) sometimes they come out funny, based on how I thought it should be read as a 10-year-old.le radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-602799948908278622010-04-25T05:36:27.947-07:002010-04-25T05:36:27.947-07:00Part of the reason algebra is taught is that it is...Part of the reason algebra is taught is that it is useful. However, another reason that algebra and higher math such as calculus is required for many professions is that they function as an IQ test filtering out people who are not smart enough to grasp those subjects (and also as tests to see who will put forth the effort to learn those subjects). Many medical schools expect students to have taken a year of calculus, but I'd be very surprised if the courses in medical school use calculus. Certainly doctors do not in their daily work. Medical schools are implicitly using calculus requirements as tests of IQ and effort.Bostoniannoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-938991735229624862010-04-25T04:01:36.754-07:002010-04-25T04:01:36.754-07:00Conceptual understanding and procedural fluency ar...Conceptual understanding and procedural fluency are not mutually exclusive. Wu in his paper about this ("A Bogus Dichotomy") emphasizes that they work in tandem. Many people mischaracterize traditional math as ignoring the conceptual underpinnings. The conceptual context is there, and procedure is rarely presented in isolation of its application in word problems.<br /><br />Many times, mastery of the procedure allows students to understand the underlying concept. I recall in some math courses, like differential equations, reading the theoretical explanation for a certain type of equation, but not following very well. I then started working the problems, imitating the procedures outlined in some of the worked examples. After getting to a certain level of proficiency, I was then able to re-read the explanation and understand what was going on. If this is what you are talking about then I agree. If, however, you believe students must experience the laborious version in order to understand the "shortcut", I disagree.Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-3827298082692739452010-04-24T23:04:20.342-07:002010-04-24T23:04:20.342-07:00"Linguistic structure is highly abstract, and..."Linguistic structure is highly abstract, and most non-linguists don't learn it explicitly." <br /><br />Schleppegrell asserts that these academic linguistic elements are not elements of style but are essential components of content discourse and thus are needed to become proficient within the subject matter domains. And additionally, because these academic language structures are not typically enountered in everyday conversation, they need to be explicitly taught. Is she mistaken?<br /><br />And back to arithmetic: Liping Ma asserts that procedural experience is insufficient to provide the experience necessary for students to intuit the underlying conceptual structure of arithmetic. And in fact she postulates that teachers are critical in making the conceptual connections to the underlying arithmetic structures so that students can truly master and become proficient at arithmetic. Is she mistaken?Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-59752096085475133472010-04-24T22:03:33.937-07:002010-04-24T22:03:33.937-07:00"This would seem to imply that for academic l..."This would seem to imply that for academic language acquisition, explicitly teaching the structure of academic language (as opposed to just the superficial elements) is necessary for students to fully master the content domain language ."<br /><br />I believe you're confusing underlying linguistic structure with rules of style. Linguistic structure is highly abstract, and most non-linguists don't learn it explicitly. You don't need comprehensive knowledge of a language's underlying linguistic structure in order to become fluent in that language--even in the case of a second language--or in order to master the formal/written register of the language.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-34263175587385523362010-04-24T20:02:48.467-07:002010-04-24T20:02:48.467-07:00Catherine: Also, creative experts - this includes...Catherine: <i>Also, creative experts - this includes mathematicians - often come up with their insights and hypotheses (and elements of proofs, presumably) via unconscious processes, such as sleep. (There is research showing that people solve problems much more creatively after they 'sleep on it.') </i><br /><br />My personal anecdote confirming this observation:<br /><br />I remember being stumped on a topology problem in grad school - I could get the rest of the problem set, but that particular one eluded me. I finally gave up and handed the homework set in. After returning home and taking a nap (much needed due to having stayed up into the wee hours of the morning), the solution occurred to me while I was in that not-fully-asleep-yet-not-fully-awake state. I woke up immediately and phoned the professor, explaining that I had just handed my homework in but now I knew the solution to the unfinished problem. I outlined the solution over the phone, and he let me write it up for full credit :)Niels Henrik Abelhttps://www.blogger.com/profile/00554447042962336254noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-18871809739365768632010-04-24T19:52:56.267-07:002010-04-24T19:52:56.267-07:00"For many people, including linguists, learni..."For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages."<br /><br />Several authors have suggested that academic language is quite unlike the conversational language learning that is aquired by young children. And given that academic language is rarely spoken but largely encoutered in written form, an explict teaching of that specific language is necessary. Could you comment on the quote from Schleppegrell in "The Language of Schooling"? <br /><br />"Students who encounter academic registers in contexts outside of school may be able to draw on this exposure and the implicit knowledge about language it engenders for success in school-based tasks. Students from other background, however, may need an explicit focus on the form language takes to raise their awareness of how different grammatical choices are functional for achieving particular goals in their writing and for analyzing the points of view that are naturalized in the texts they read. ... an explicit focus on the linguistic features of the language of schooling can raise students' awareness of the choices available to them for writing the texts of schooling and can help reveal the layers of meaning in the texts they read."<br /><br />This would seem to imply that for academic language acquisition, explicitly teaching the structure of academic language (as opposed to just the superficial elements) is necessary for students to fully master the content domain language .Erin Johnsonnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-58226989857440969722010-04-24T19:28:11.072-07:002010-04-24T19:28:11.072-07:00language is very mathematic-like actually...from s...language is very mathematic-like actually...from sound processing to syntaxle radical galoisienhttps://www.blogger.com/profile/14684821442296479803noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-27248343550391722502010-04-24T18:52:01.988-07:002010-04-24T18:52:01.988-07:00"But unlike language, arithmetic is well char..."But unlike language, arithmetic is well characterized" I'm a linguist (in addition to being a programmer), and can say with confidence that no linguist would agree with that statement.<br /><br />For many people, including linguists, learning a language's superficial features can precede learning its underlying structure--without any pedagogical cost. In fact, this happens all the time with our native languages. The superficial features then become the data on which the rules about underlying structure are based.Katharine Bealshttps://www.blogger.com/profile/02838879769628392605noreply@blogger.com