tag:blogger.com,1999:blog-7691251033406320222.post2535287272506804062..comments2024-03-08T00:21:56.482-08:00Comments on kitchen table math, the sequel: How to Solve ItCatherine Johnsonhttp://www.blogger.com/profile/03347093496361370174noreply@blogger.comBlogger16125tag:blogger.com,1999:blog-7691251033406320222.post-23976294753898828972009-07-20T19:22:39.884-07:002009-07-20T19:22:39.884-07:00Catherine, the notion of a related accessible prob...Catherine, the notion of a related accessible problem is really hard in practice. That's why Polya's book is worth reading, not just the summary. He walks through trying to show a student just *what* related problems the student *does* know. <br /><br />But as Barry pointed, knowing what you know takes years. The whole idea of knowing which way to modify a problem to make it look like a problem one already knows how to solve is subtle--not all ways are equally good. At the beginning, watching your teacher do it is mystifying. Their intuition leads them in certain ways based on the sheer volume of problems they've already managed to see as interrelated. Building up that level of flexible knowledge requires massive practice. No wonder that you couldn't get your kids to see what a related problem was--you probably could not describe why you knew which related problem was the right one, too.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-88314348327091307542009-07-20T17:22:18.741-07:002009-07-20T17:22:18.741-07:00Liping Ma is famous for asking: list problems wher...Liping Ma is famous for asking: list problems where division by a fraction make sense. Few American elmentary teachers can name any. Chinese ones can name many.orangemathhttps://www.blogger.com/profile/05099727076265177042noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-34695653089793997302009-07-20T17:14:14.757-07:002009-07-20T17:14:14.757-07:00Another way to 'conquer' fraction division...Another way to 'conquer' fraction division is to demonstrate a problem using lowest common denominators. It's an unconventional use of LCD but effective in getting the light bulbs turned on at times.<br /><br />Here's an example, say 2/3 divided by 1/4. Well this is 8/12 divided by 3/12. At this point I usually just say that the denominators are just like units for the numerators, so this problem is really just 8 whatevers divided by 3 whatevers or 2 2/3.<br /><br />It works pretty well because by the time they tackle fraction division they are fluent with the whole LCD concept and are used to it being treated like 'units' for the numerators.<br /><br />I don't teach it this way of course but it's a useful alternative for kids who are convinced that the standard method is just magic and try to call BS on you.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-75107713429382344072009-07-20T16:28:05.861-07:002009-07-20T16:28:05.861-07:00great example
I'm sure all of us were just de...great example<br /><br />I'm sure all of us were just desperately trying to teach our kids this "truc" as a last-minute survival strategy.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-79527093947275602092009-07-20T14:48:39.220-07:002009-07-20T14:48:39.220-07:00Could you imagine a more accessible related proble...<i>Could you imagine a more accessible related problem?</i><br /><br />It takes many years to get to that point on your own. But you can get there in baby steps. When teaching division by fractions, a problem might be "How many 3/4 inch pieces of ribbon can be cut from a piece of ribbon that's 9 inches long?" <br /><br />When confronted with the blank stares such a problem engenders, then I would switch and say, "How many 3 inch pieces can you cut from a 9 inch piece?" They would quickly figure it out. "How did you do it?" I would ask. "Divide 9 by 3".<br /><br />"OK, now let's go back to the first problem. How would you do it?"<br /><br />Hesitant, unsure answer: "Divide by 3/4?"<br /><br />"Yes, absolutely". <br /><br />The student has to build up a repertoire of problem solving techniques which then can be drawn upon, as well as being shown as I did above, how to look for similar problems. It isn't by any means easy for students to do this.Barry Garelickhttps://www.blogger.com/profile/01281266848110087415noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-68498496769121020252009-07-20T14:20:25.606-07:002009-07-20T14:20:25.606-07:00wow - Paul!
I love that!wow - Paul!<br /><br />I love that!Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-49744201881539269122009-07-20T14:19:13.275-07:002009-07-20T14:19:13.275-07:00Could you imagine a more accessible related proble...<i>Could you imagine a more accessible related problem?</i><br /><br />At some point, way back when, I figured this one out on my own. It's MAJOR for anyone who does **not** have a serious education in math.<br /><br />Interestingly, it's quite hard to teach this approach to your own child (or so I found).<br /><br />At least two of my friends said the same thing.<br /><br />Some of the issue, of course, is that kids don't want their moms re-teaching them math.<br /><br />But there was something else, too...<br /><br />I'm guessing we all tried to get our kids to use this approach when they were trying to do a problem that was far too difficult for them.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-54374725231893131772009-07-20T13:22:19.990-07:002009-07-20T13:22:19.990-07:00The 'repairs' that I've experienced ar...<i>The 'repairs' that I've experienced are always in the form of isolated fads, like throwing M&Ms at your hunger when what you really need is a Snickers.</i><br /><br />right<br /><br />I'm (currently) with Richard DuFour on this: 'it's the culture, stupid.'<br /><br />You can bring in every good reform there is, but if you don't have a 'student-focused' culture none of them is going to work.Catherine Johnsonhttps://www.blogger.com/profile/03347093496361370174noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-53046591350377667072009-07-19T13:48:35.538-07:002009-07-19T13:48:35.538-07:00Perhaps the biggest problem educational theorists ...Perhaps the biggest problem educational theorists DON'T KNOW THEY HAVE is that Polya and people that try to use this work (including me), have never been able to find what content goes with what strategies - both are needed, but... In other words, there is no general theory of heuristics.<br /><br />The obvious fact that in all of the hubbub surrounding strategies and content, "heuristics" is never (good luck finding it) mentioned merely indicates how pathetic ed researchers are. It's not the billions ($) lost, but the unwillingness to confront the limitations of teaching heuristics.<br /><br />Humility goes with using heuristics in instruction. It's the best we can do, but the multiple-choice world is a dead endorangemathhttps://www.blogger.com/profile/05099727076265177042noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-40145944779195786802009-07-19T03:34:39.981-07:002009-07-19T03:34:39.981-07:00It occurs to me that this is exactly the process o...It occurs to me that this is exactly the process one has to use to solve any problem, not just math.<br /><br />It's also precisely what does not happen when folks seek to solve the problems of public education. I wrote about this a year ago in <a href="http://arp148.com/blog/?cat=4" rel="nofollow">The Maze</a> and it still seems relevant today. The 'repairs' that I've experienced are always in the form of isolated fads, like throwing M&Ms at your hunger when what you really need is a Snickers.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-15627427594777552342009-07-19T02:56:01.924-07:002009-07-19T02:56:01.924-07:00I use an organizer that approaches this technique....I use an organizer that approaches this technique. I think it's adapted from an ELA template called four square. Here's what I have the kids do.<br /><br />Take a piece of paper and fold into quarters. Open it up and draw a rectangle in the center. Now you've got four quadrants and a rectangle which is not exactly 'four square' but in the interests of marketing I guess four square sounds sexier.<br /><br />Read the problem twice. Then in the rectangle restate the question in the form 'find blah blah blah in units of xxxxxx'.<br /><br />Read the problem again and in the upper left quadrant identify and define all the symbols (variables and constants) you'll use in your solution.<br /><br />Read the problem again and use the lower left quadrant for a diagram/picture.<br /><br />Read the problem again and use the upper right quadrant to define your strategy. This can be words or preferably a set of equations to solve.<br /><br />The lower right quadrant is where you show your arithmetic.<br /><br />Finally, the back of the paper is used to justify your answer.<br /><br />It works well for entry level problem solving of the kind you would encounter through maybe grade six. If kids master this it should instill some good habits for more complicated things.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-52317272845560203502009-07-19T01:16:10.990-07:002009-07-19T01:16:10.990-07:00From Bertrand Russell or was it Alfred North White...From Bertrand Russell or was it Alfred North Whitehead...."the goal is to have no problems only exercises".<br /><br />By expanding long term memory one can turn potential problems into exercises. <br /><br />It seems that the goal of many pushing the discovery/inquiry approach is to have no exercises but only problems. <br /><br />I am still in shock from Dr. Ruth Parker's powerpoint given at NCTM national that the Standard Algorithm always harms conceptual understanding, which seems to be a recipe for lots of problems.<br /><br />The person most everyone would like to hire is the one that has so much extensive background knowledge that most everything is an exercise.dan dempseyhttps://www.blogger.com/profile/15536720661510933983noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-35885598671893687332009-07-18T10:12:48.034-07:002009-07-18T10:12:48.034-07:00I always discuss with teachers the distinction bet...I always discuss with teachers the distinction between Polya's "Look Back" as opposed to what is more commonly used: "Check your work." <br /><br />Here's how it looks at a 3rd grade level:<br />Does my answer satisfy the question? <br />Does it make sense? <br />Is there another way to solve the problem? A simpler way?<br />Have I shown my work? <br />(and in my classroom: Can Mrs. Turner read it?)Anonymoushttps://www.blogger.com/profile/01239123267984420065noreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-18329565239098222692009-07-18T06:47:17.148-07:002009-07-18T06:47:17.148-07:00This relates directly to the points Barry Garelick...This relates directly to the points Barry Garelick made when citing the Kirschner, Sweller, Clark study.<br /><br />Polya seems to be giving explicit directions on how to decide what information in long term memory is relevant to the problem so it can be retrieved. He is giving criteria to help determine what to pull into working memory.<br /><br />K-S-C point out that "the aim of all instruction is to alter long-term memory".<br /><br />It takes problem practice in math and decoding and blending practice in reading to alter long term memory.<br /><br />Discovery or inquiry learning by a novice rarely alters long term memory.<br /><br />Spiral learning is not sufficient to alter long term memory for most kids.<br /><br />Basically instructional techniques like inquiry learning in math and the spiral approach ignore empirical evidence of what works and the cognitive architecture of the human brain.<br /><br />It seems that too many educators in this country are pushing reading and math instruction techniques that are nothing more than superstition.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-66325356365115607642009-07-17T20:37:05.377-07:002009-07-17T20:37:05.377-07:00Please please read the book, too. Give it to every...Please please read the book, too. Give it to everyone you can! (I've lent the book and never had it returned more times than I can count!) It's so terrific. It has these dialogues between student and teacher that really help show how and where students get stuck and how the heuristics help in real problems. <br /><br />It sounds simple, but of course, real students don't know how to adapt from the tiny space of their mastery to bigger and different problems. They don't know *which* part of a problem to collapse, *which* part to ignore for now, *which* part is analogous to a problem they already know. "Solve a simpler problem"--but which way is the way to simplify the problem? All of that is confusing and takes practice practice practice, but here at least, they can start to actually practice. <br /><br />Personally, my recitation sections in stat mech, and to some degree, quantum, were built around this model. And especially in those subjects, as there just aren't all that many problems that can be solved in a small amount of time, mastery of this model meant mastery of the test material, too. With practice, doing this in recitation meant it started to gel how to recognize the basic problems to master and how to tell what related problems were (microcanonical ensemble! grand canonical! isothermic system! Harmonic Oscillator! phonons!)<br /><br />This also relates back to the modeling instructions stuff of Hestenes. The reason why modeling instruction can help is because it so strongly helps define "this is the basic problem to solve; this is a related problem". Learning how to see similarities and trust your mastery of the simple problem is 80% of the way there.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-7691251033406320222.post-81925933390307132682009-07-17T19:23:42.783-07:002009-07-17T19:23:42.783-07:00I'm going to steal this, Allison. I like to s...I'm going to steal this, Allison. I like to start my class with a talk on problem solving, and I like this script. That's exactly where I see a difference between what I do and discovery learning -- I want them to be able to solve new problems, but expect them to do that by recognizing how the new problem relates to problems they have done already, not by flailing around and getting lucky! As a colleague of mine likes to say, you can't figure out chemistry during the test.ChemProfnoreply@blogger.com