tag:blogger.com,1999:blog-2516188730140164076.comments2014-09-15T18:38:06.428-04:00Devlin's AngleMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger122125tag:blogger.com,1999:blog-2516188730140164076.post-31619612191107391712014-09-14T02:38:23.205-04:002014-09-14T02:38:23.205-04:00This is a great read!
I grew up at a time when cal...This is a great read!<br />I grew up at a time when calculators are not allowed for basic computations and students are forced to memorize the multiplication table. I finished my studies with great math grades but all because I was able to memorize algorithm and formulas. I never understood fully what the Pythagorean Theorem is for one. I am so used in memorization that once I forgot a formula, then I am stumped and cannot move on.<br />Hopefully with the onset of the Common Core State Standards students would be encouraged and trained to use their critical thinking and mathematical thinking. That they would be able to deeply explore, justify and prove why one thing is true.Annie Beloniohttp://www.blogger.com/profile/17032847667945591045noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-12295109496960661672014-08-17T12:37:42.182-04:002014-08-17T12:37:42.182-04:00This is great. I remember being really annoyed wh...This is great. I remember being really annoyed when a calculus class in university showed me that every series had an infinite number of correct next entries (or something very like that - it *was* back in the early 14th century, after all). I was still smarting from generating so many wrong answers in grade-school exercises designed to elicit one, and only one, answer. I thought then and think now that kids would be able to examine the different solutions they came up with and learn to see the different degrees of elegance in them - to learn why some answers were better than others. I hope this new approach is trying to teach kids to think.Isabel Gibsonhttp://www.traditionaliconoclast.comnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-37430418244974949062014-08-02T18:57:56.776-04:002014-08-02T18:57:56.776-04:00Very nice read!
This reminds me very much of what...Very nice read!<br /><br />This reminds me very much of what I discussed with my friends in my last year in high school here in the Netherlands.<br /><br />I'm afraid that the educational system in the Netherlands is still very much focused on repetitive problems (that indeed only have 1 correct answer!) and not at all on the ideas and concepts behind the problems (what am I actually doing when I calculate this primitive?).<br /><br />It's frustrating to realize that: yes, I'm able to do the calculation, but if you were to ask me what I'm actually doing I'd be almost clueless. On the one hand I have all this knowledge on how certain systems work and how number relate to each other, but the isomorphism between the numbers and the real world is missing. <br /><br />Rickhttp://www.blogger.com/profile/09941329825490289800noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-88212790228355073282014-08-01T09:53:56.320-04:002014-08-01T09:53:56.320-04:00Well done. As well as "mathematics", I ...Well done. As well as "mathematics", I also found I could substitute "mountain biking" for "rock climbing". All three share very similar attitudes.<br /><br />The first time I finish a hard move, I almost always feel "lucky", then rethink what I did, what made it work, rest, then try again until I master it. And when all the moves are good, there's still the matter of piecing them all together. And when that's done, I still have this desire to climb the whole route as gracefully and elegantly as possible. And that can always be improved.Patrickhttp://ptruchon.pagekite.menoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-54377030030974842952014-07-31T09:26:09.097-04:002014-07-31T09:26:09.097-04:00Use of geometry –geometric lines/arcs- overlaying ...Use of geometry –geometric lines/arcs- overlaying your pictures combined with colours to distinguish LH vs RH would strengthen your story to the layman.<br />Well done.<br />B<br />P.S. Would your method work during the darkened hours of a 24 hour solo MTB race?Mr. Armstronghttp://www.blogger.com/profile/02899421112010848381noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-44715423078262748462014-07-07T12:37:35.107-04:002014-07-07T12:37:35.107-04:00Mary O'Keefe states: "I certainly don'...Mary O'Keefe states: "I certainly don't advocate going back to the old ways in which the mother was educated."<br /><br />Can you elaborate on the ways you believe the mother was educated and why you think they were bad? I can provide you examples of textbooks from the 30's through the 60's which show that not only was math NOT taught by rote, but provided the contextual and conceptual explanations that are continually mischaracterized as missing.Barry Garelickhttp://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-5001400231925450782014-07-07T09:41:37.003-04:002014-07-07T09:41:37.003-04:00Not sure what Audrey Tan finds as ironic – dot dia...Not sure what Audrey Tan finds as ironic – dot diagrams of different kinds provide ways to help gain understanding, by focusing attention on the possible structures and the patterns. The Times illustration showed a dot diagram that focuses on one of several important instantiations of multiplication. I think Ms Tan is too focused on the dots themselves, and missing the depth of structure that a dot diagram can lead to. We do, after all, view Newton’s (mythical) observation of the falling apple as having scientific significance – he was indeed, if not “experimenting with space”, then very definitely “observing and reflecting on space”. The depth lies not in the activity but is what is going on inside the person’s head. Of course, mindless drawing of dots has no benefit, and a worksheet that demands students use dot diagrams as the solution method would surely not be productive. But that would be a problem with the teaching, not the CCSS.<br /><br />I agree with everything Mary O’Keefe writes, and never suggested anything at odds with what she says. What the CCSS set out to do, and what I was advocating, is shift the focus from learning specific procedures (now done for us by machines) to acquiring the ability to think like a mathematician. And the fact is, mathematicians spend a lot of time reflecting on concepts and problems until they achieve sufficient understanding (not infrequently using dots diagrams to aid the cognitive process). That kind of thinking is the marketable mathematical ability for the 21st Century.<br />Keith Devlinhttp://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-20667799201553219942014-07-04T10:02:45.762-04:002014-07-04T10:02:45.762-04:00I certainly don't advocate going back to the o...I certainly don't advocate going back to the old ways in which the mother was educated. There is, however, a huge difference between a research mathematician analyzing dot patterns because s/he has decided that it looks like a promising way to approach an unsolved problem and a child being *assigned* to draw massive numbers of dots to solve problems s/he may feel she already knows how to do without being given any clearly explained motivation for WHY she should be doing this exercise. There are many other approaches to do this same kind of learning in ways that might be more engaging as well as easier for young children who may have fine motor skills issues or visual alignment difficulties (eg crossed eyes) which could make it hard to draw and count dots. Arranging pennies or Lego squares or other manipulatives into arrays could serve the same purpose. The problem is not the pedagogy behind the dots. The problem is mass-produced mindless worksheets which do not motivate the kind of mindful problemsolvers we need.Mary O'Keeffehttp://www.blogger.com/profile/14662977706706048151noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-32057860632291773302014-07-04T04:24:33.701-04:002014-07-04T04:24:33.701-04:00I find it ironic that someone who, some years ago,...I find it ironic that someone who, some years ago, insisted that multiplication is not repeated addition, is now supportive of children drawing so many dots to "study" multiplication.<br /><br />Personally, I have no problem with drawing dots to support a child who is learning the concept of multiplication (even if it does have a whiff of repeated addition about it). However, when drawing dots becomes a prescribed <i>method</i>, then we have a serious problem. <br /><br />To describe the image chosen by The NY Times as a "sensible and deep use of dot diagrams" and "all about creative thinking" is akin to describing a baby "experimenting with space" when he/she throws a toy across the room. I think the only sense in which this example is deep is the way the author continues to dig himself!Audrey Tanhttp://www.mathmo.co.nznoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-25534173126221644432014-07-03T22:00:15.336-04:002014-07-03T22:00:15.336-04:00R Craigen confuses learning mathematics with doing...R Craigen confuses learning mathematics with doing mathematics. Methods that are efficient for doing mathematics are not necessarily well suited for learning mathematics. <br /><br />In particular, computational efficiency is definitely not a good metric for learning mathematics. In terms of doing mathematics, the most efficient way to multiply in today’s world is use a calculator, so if efficiency were the metric, we should just train kids to use a calculator well. (Which we should do, but wisely we do more.)<br /><br />To prepare students for life in the 21st Century, learning researchers have developed methods that are optimized not for efficiency but for learning. And dot diagrams play a useful role in that learning. <br /><br />True, as I noted in my article, it is possible to overdo it and generate a lot of dotty busywork. But the illustration of a dot diagram the Times used showed an excellent use of a dot diagram to develop an understanding of one of the main instantiations of multiplication -- a particularly important example, since multiplication is a notoriously complex operation that many adults do not understand well, let alone school kids.<br /><br />The article by Christopher Danielson that I referenced explains how dot arrays can help kids come to understand (some important aspects of) multiplication.<br /><br />R Craigen may be right, and some of my colleagues may have thought me foolish for spending months staring at my morass diagrams, but I was not trying to impress those colleagues, I was trying to understand something that I found complex and difficult to grasp. Those diagram helped me do that. In fact, I was not alone. Some of those colleagues also spent a lot of time staring at similar diagrams, and likewise seemingly making no progress for a long time. (I guess those would be the ones who did not think me foolish!) <br /><br />Foolish or not, many of us did eventually mange to achieve the required understanding, and we made progress, so I would argue that the method is not without merit. In fact, I find it had to imagine a better one. The brain seems to need that external stimulus, impoverished and skeletal though it may appear to someone who has never had such an experience.<br /><br />The commentator’s final comment seems to indicate the degree to which he has missed my point. Neither I, nor any child in a school classroom, is working with dot diagrams to *carry out* calculations. The goal is understanding. <br /><br />Once someone understands, say, multiplication, she or he has various choices of efficient ways to do it. The classical Hindu-Arabic algorithms are one way. But there is a much faster and more accurate way: use a calculator.<br /><br />In my school days, calculators were not available, so it was important for us to master the classical algorithms. In contrast, in the 21st century we are preparing today’s students to go out into, unlimited computation power is as freely available as running water and electricity, so it makes more sense to ensure our students can make good use of that utility. <br />Keith Devlinhttp://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-20398563852445095542014-07-03T16:11:00.239-04:002014-07-03T16:11:00.239-04:00In my work as a professional mathematician I too q...In my work as a professional mathematician I too quite often use dot diagrams of various sorts. Some look exactly like the ones Devlin displays here; we call them Ferrers diagrams and they have a somewhat different purpose. In advanced math many imaginative notations and representations are used as excellent, precise tools to aid thinking.<br /><br />Devlin's invocation of this fact in reply to the parent of a 9-year-old bored to tears with doing arithmetic in the most boring possible way reminds me of the canonical story about the lost balloonist who shouts to someone on the ground, "Sir, can you tell me where I am?" <br /><br />The bystander shouts back, "You're in a balloon!" He replies, "I perceive that you are a mathematician, sir!" When queried, "How can you tell?" he says, "Because what you told me is absolutely true ... and completely useless".<br /><br />It's quite true that advanced math may, at times, use complicated-looking diagrams involving dots. And this is quite useless as an explanation of why little boys and girls need to be forced to engage in boring, wasteful and insightless tedium to perform tasks for which there are enlightening, easy-to-learn and efficient procedures near at hand.<br /><br />If Dr. Devlin was found to be staring at his dot diagrams for days when a well-known elementary and efficient process would obviously have arrived at the same insight in a few lines and at very little expenditure time and effort, either he or his colleagues would consider him to have been foolish for doing so, or perhaps ready for retirement. <br /><br />Let us teach 9-year-olds the best and most suitable mathematics for their level of mathematical development and not lead them down garden paths to inefficient thinking and mindless tedium in service of an abstract and unsubstantiated belief that "deep understanding" will magically arise if they are limited to performing arithmetic essentially as our cave-dwelling ancestors did (substituting pages of dots for piles of rocks).R. Craigenhttp://wisemath.orgnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-18518832099868814202014-07-03T09:11:42.909-04:002014-07-03T09:11:42.909-04:00Jon Awbrey's comment hits the nail on the head...Jon Awbrey's comment hits the nail on the head. <br /><br />Ze'ev Wurman has clearly missed the entire point I was making and appears to have a very misguided (though not unique) view of doing mathematics, probably a result of poor education.Keith Devlinhttp://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-89591929461676111072014-07-03T08:20:55.925-04:002014-07-03T08:20:55.925-04:00The real problems of CCSS have more to do with the...The real problems of CCSS have more to do with the commercial exploitation of public school tax dollars than anything else. There is a wealth of information and informed commentary on this score at Diane Ravitch's blog:<br /><br />http://dianeravitch.net/Jon Awbreyhttp://inquiryintoinquiry.com/noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-30272202652596350792014-07-02T17:24:45.335-04:002014-07-02T17:24:45.335-04:00I am eagerly waiting for your defense of the use o...I am eagerly waiting for your defense of the use of Roman number system instead of the decimal one in elementary grades. After all, we do use the letter X a lot in higher level mathematics, so surely it must be good for young children to also use it in their arithmetic.Ze'ev Wurmannoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-83666178548278534602014-05-28T16:33:54.438-04:002014-05-28T16:33:54.438-04:00Thanks Keith I find all your lectures really usefu...Thanks Keith I find all your lectures really useful. You showed some Disney animations of derivatives and integrals in some fo your lectures. Are those animations available anywhere? I'd like to show them in my classes.George Lilleyhttp://www.blogger.com/profile/03981535542840463843noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-85404680036121388812014-03-04T13:24:26.837-05:002014-03-04T13:24:26.837-05:00pp, You have misread my post. I did not say the am...pp, You have misread my post. I did not say the amygdala will carry out mathematical processes. Rather my argument is that the neo-cortex does the mathematical processing, and then leaves the amydgala to "do its stuff". In this case, that would amount to making the key connections between the various mathematical processes that have been not just worked out, but rehearsed. While this has to be just speculation, what I propose is certainly feasible, and consistent with everything we have observed. And, as I tried to argue, no different from carrying out complex physical tasks, such as mountain biking up a steep, technical trail.Keith Devlinhttp://www.blogger.com/profile/17423495316890452375noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-86964990694971940752014-03-04T13:24:09.271-05:002014-03-04T13:24:09.271-05:00pp, You have misread my post. I did not say the am...pp, You have misread my post. I did not say the amygdala will carry out mathematical processes. Rather my argument is that the neo-cortex does the mathematical processing, and then leaves the amydgala to "do its stuff". In this case, that would amount to making the key connections between the various mathematical processes that have been not just worked out, but rehearsed. While this has to be just speculation, what I propose is certainly feasible, and consistent with everything we have observed. And, as I tried to argue, no different from carrying out complex physical tasks, such as mountain biking up a steep, technical trail.Keith Devlinhttp://www.blogger.com/profile/17423495316890452375noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-48518722830145333022014-03-03T07:54:10.811-05:002014-03-03T07:54:10.811-05:00Have you read Bounce by Matthew Syed? He deals bea...Have you read Bounce by Matthew Syed? He deals beautifully with the sports learning you describe. It seems to me you are asking a lot of the amygdala to expect it to carry out mathematical processes. Is this not more likely a left / right brain division where the processes are still parallel but of a higher order whilst still providing a eureka moment, as there is no evolutionary need to solve a mathematical problem at speed. I have heard of scientists at RAF Westcott cycling up and down the runway whilst thinking, seeking the eureka moment. -Just a thought.pphttp://www.blogger.com/profile/07173389276293429052noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-37955075879034388022014-03-01T07:28:29.041-05:002014-03-01T07:28:29.041-05:00I haven't re-read Arthur Koestler for a long w...I haven't re-read Arthur Koestler for a long while, but this all reminds me of his notion of sudden "bisociation" of two otherwise unrelated cognitions as the key to creativity and discovery in science, humor, and the arts."Shecky Riemann"http://www.blogger.com/profile/07065658607024191185noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-17720585195192838012013-12-28T20:45:58.171-05:002013-12-28T20:45:58.171-05:00Cathy, Indeed my primary focus is the US education...Cathy, Indeed my primary focus is the US educational system.<br />Keith Devlinhttp://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-88980360539977663872013-12-26T17:40:40.943-05:002013-12-26T17:40:40.943-05:00You write: "The trouble with writing about, o...You write: "The trouble with writing about, or quoting, Liping Ma, is that everyone interprets her words through their own frame, influenced by their own experiences and beliefs.”<br /><br />One way of reducing the variability of interpretations might be to compare definitions for certain words. In the case of this post, I think your meaning of “curriculum” might refer more to selection of topics or subject (e.g., arithmetic) and not include much emphasis on how that content is construed or organized (e.g., construing arithmetic as a collection of algorithms without rationales). That meaning of “curriculum” seems plausible for the US, given traditions of local control and standards (and accountability) that focus on performance. It seems less accurate as a description of its meaning in countries where curriculum objectives place more emphasis on sequencing and topics taught, and policy-makers give more information about intended meanings of topics via textbooks and teachers’ guides, and where teachers are afforded opportunities to share meanings via practices like lesson study and demonstration lessons.Cathy Kesselhttp://mathedck.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-35761942725549243982013-12-23T14:24:14.298-05:002013-12-23T14:24:14.298-05:00Rebecca, Thanks for writing. I am vaguely aware of...Rebecca, Thanks for writing. I am vaguely aware of what is happening in the UK from reports in The Guardian, but surely, unless the UK has imposed a very totalitarian regime of local control, good teachers will prevail. No? <br /><br />Which is not to say I don't understand your frustration and anger at having a national education policy controlled by someone totally ignorant of education.Keith Devlinhttp://www.blogger.com/profile/16899343259650938644noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-61955348623441668952013-12-20T15:54:38.787-05:002013-12-20T15:54:38.787-05:00Dear Keith,
I really must take issue with this co...Dear Keith,<br /><br />I really must take issue with this comment:<br /><br />"BAD CURRICULUM + GOOD OR WELL-TRAINED TEACHERS = GOOD EDUCATION"<br /><br />It pays no respect whatsoever to the horrors of what is now going on in England. Key politicians have developed a primary mathematics curriculum based on the naïve idea that the more abstract content you give children and the younger you give it to them the better they will be at maths. Instead of paying any attention to those with relevant experience and their own expert advisers they have chosen to completely ignore such people and instead to viciously attack them in the press as being 'enemies of progress'. <br /><br />The primary curriculum which we are to teach from January is not remotely age-appropriate and is not like any other national curriculum. It is statutory and teachers are required by law to teach it by year group. This is overseen by an exceptionally brutal regulator. <br /><br />To simply say that if you're a good teacher it won't bother you is very out of touch. There's a full report here:<br />http://authenticmaths.co.uk/report-primary-schools-new-national-curriculum/<br /><br />BTW my classes had all those structures LiPing Ma found in China (like 3 conceptual methods of solving division of mixed numbers rather than procedural methods). More here if you're interested: http://mathseducationandallthat.blogspot.co.uk/2011/05/how-do-chinese-do-it-introduction.html <br /><br />RebeccaRebecca Hansonhttp://www.blogger.com/profile/15973235163335279852noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-87069382342190289542013-02-12T05:16:43.211-05:002013-02-12T05:16:43.211-05:00Teaching science is frequently about correcting mi...Teaching science is frequently about correcting misconceptions since students have a lifetime of experience with things like forces and growing plants.<br /><br />The challenge with mathematics instruction, in my experience, is that students frequently do not have any misconceptions to correct. Frequently, students are a "blank slate" since they do not typically consider questions like "Is 1 = 0.999...?" or "What is the average value of a continuous function over an interval?" There is nothing in their daily lives to compel them to confront such ideas. At first blush one might be tempted to think that mathematics is easier to teach since students have fewer misconceptions to overcome; however, if you have ever tried to teach mathematics you know this is not the case.<br /><br />Abstract mathematical thinking is sufficiently different from the empirical thinking of science (broad brush strokes) that the effective instructional techniques of one do not necessarily inform the other.<br /><br />So to me it seems we are left with two takeaways...<br /><br />(i) Who, if anyone, is investigating the essential elements of an effective mathematics video? I think vihart.com is on the right track. With my students, I have seen her videos create mathematical curiosity, which is an important first step in mathematics teaching.<br /><br />(ii) The key to learning is cognitive effort on the part of the student under the guidance of an invested teacher/professor. If the technology employed does not force the student to mentally engage then very little learning will be achieved. It is for this reason that pencil and paper may just be the best educational technology ever conceived.<br /><br />My two pence, for what it's worth.Robert Kennedyhttp://www.blogger.com/profile/00279301671110421767noreply@blogger.comtag:blogger.com,1999:blog-2516188730140164076.post-89111926981967231972013-02-06T23:43:22.124-05:002013-02-06T23:43:22.124-05:00I agree with the point you raise in the Edge artic...I agree with the point you raise in the Edge article, which I would generalize as, "Our technologies can have the effect of painting us into a corner." I think you are saying that the overuse of the technology you cite, machine presentation, is an authentic tragic flaw in the ancient Greek sense.<br /><br />Maybe there will come a science fiction novel where some bar napkin notes trump the Power Point presentation. We can only hope!<br /><br />On a similar note: I wrote software for many years, mostly in the era where you had to wait quite a while to get your results back from a central computer. To use your time well, you needed to read your program carefully to cut down on the number of times you ran it. You really needed to know your code, and there were some shops where everybody knew everybody else's code as well. These days, with a fast computer on every desk, programmers seem much more inclined to crank out their first guess and trial-and-error it into submission. The QA people love having to deal with the resulting fallout and the dismissiveness of the programmers towards them.Paulhttp://www.blogger.com/profile/04473326737299320884noreply@blogger.com