tag:blogger.com,1999:blog-2516188730140164076.post4167422961418068667..comments2023-05-24T06:22:53.845-04:00Comments on Devlin's Angle: The Power of DotsMathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger9125tag: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 Garelickhttps://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 Devlinhttps://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'Keeffehttps://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 Devlinhttps://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 Devlinhttps://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.com