tag:blogger.com,1999:blog-37842769849604212332017-02-21T06:28:14.888-08:00My Math Education Blog"There is no one way"Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger31913MathEdBloghttps://feedburner.google.comtag:blogger.com,1999:blog-3784276984960421233.post-2041556949306985742017-02-13T13:02:00.000-08:002017-02-13T13:02:16.615-08:00Calculation <br /><div class="page" title="Page 77"> <div class="layoutArea"> <div class="column"> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Many students have weak arithmetic skills. Many teachers blame this on calculator use, but it is just as likely that the real reason lies elsewhere. For one thing, the teaching of arithmetic traditionally does not involve developing any understanding, so the learning is shallow and fragile. For another, students correctly feel that mindless arithmetic is no longer a useful skill in the age of technology, so it may not be so much calculator use, but the very existence of of the calculator which saps motivation in this arena. Finally, there was a time when many high school and college teachers didn’t need to interact with students whose arithmetic skills were weak, because those students were prevented from taking college-preparatory math classes. This is no longer true, and the population of college-intending students has grown enormously, so those teachers erroneously conclude that arithmetic skills are getting worse. </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">In any case, in my view, there is no reason to ban calculator use in the classroom. Such a rule will be perceived as punitive and only build resentment and a negative disposition. On the other hand, students have no problem at all accepting a ban on calculators during specific activities, (such as a mental math session, or a quiz on famous trig values) because in that context, the ban is readily explained and eminently sensible. </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">--------------------------------------------------------------------- </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">If you wanted to know the result of <b>multiplying 5463 by 78912</b>, it is very likely that you would reach for a calculator. No one in the world outside of schools would do this problem with pencil and paper. Children know this. They know that adults at work do arithmetic by machine, whether in a fast food joint, or a bank, or a lab, or an engineering firm – anywhere at all. It is obvious to students that doing multidigit arithmetic by hand is not a useful skill. No one will hire you to do long division. </span><br /><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Still, being able to predict that <b>the result of that multiplication is not too far from 400,000,000 </b></span><span style="font-family: 'CMTI12'; font-size: 12.000000pt;">is </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">a worthwhile skill, if only to confirm that the answer given by a calculator is in the right range. I got this result by multiplying 5000 by 80,000. I could gain a little more accuracy by following this up with an estimate for 55 </span><span style="font-family: 'CMSY10'; font-size: 12.000000pt;">× </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">79. Well, 55 </span><span style="font-family: 'CMSY10'; font-size: 12.000000pt;">× </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">80 is (5 </span><span style="font-family: 'CMSY10'; font-size: 12.000000pt;">× </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">80) + (50 </span><span style="font-family: 'CMSY10'; font-size: 12.000000pt;">× </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">80), or 4400. Subtract 55, and get 4345. So <b>the result of the original calculation should be close to 434,500,000.</b> </span><br /><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">How did I do? <b>The actual answer is 431,096,256</b>. In millions, my error was a number less than 4, divided by a number greater than 400. In other words, I was within 1%. Not too bad! (My calculator says the error was .7895555%.) </span><br /><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Few students would be able to work through this like I did. That is not surprising: this sort of computation is not taught much. If students spent less time with paper and pencil multi-digit arithmetic, that would free up some time to work on mental calculation and estimation. Working on this depends on, and helps develop number sense. For my first estimate, I had to understand rounding, and multiplication by powers of ten. In the second phase I used the distributive law twice. Paper-pencil computation, on the other hand, can be done with little or even no understanding. And almost no students gain deeper understanding by doing it. </span><br /><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">---------------------------------------------------------------------<br /> </span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">But, you say, by practicing multi-digit multiplication on paper, they can gain accuracy and speed! Yes, that is true of many students. But even the best will not be as accurate or as fast as the free app on their phone. And in any case, what</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> good would it do? Speed and accuracy in computation can no longer be a priority in math education. We should spend what limited time we have with students developing their understanding, not wasting it on useless skills. </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Mental calculation and estimation need a place of honor throughout the K-12 curriculum As far as I’m concerned, that is the main consequence of the availability of calculators. This can take place in <b>number talks</b> appropriate to what</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> is being taught at each level. <b>Estimates should precede every calculation</b>. Possible strategies should be discussed before and after the introduction of standard or non-</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">standard algorithms. </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">And this need not stop after middle school, quite the opposite: number sense and operation sense continue to be a priority all the way to graduation and beyond.</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> What is your estimate for the square root of 18? why? A mental math component can be added to topics like algebra: solve 3</span><span style="font-family: 'CMMI12'; font-size: 12.000000pt;">x </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">+ 1 = 13 without paper, pencil, or technology. Or trigonometry: you should know or be able to quickly retrieve cos(60</span><span style="font-family: 'CMSY8'; font-size: 8.000000pt; vertical-align: 4.000000pt;">◦</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">). What </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">would you guess for cos(55</span><span style="font-family: 'CMSY8'; font-size: 8.000000pt; vertical-align: 4.000000pt;">◦</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">)? cos(65</span><span style="font-family: 'CMSY8'; font-size: 8.000000pt; vertical-align: 4.000000pt;">◦</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">)? </span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> </span> </div></div></div><div class="page" title="Page 78"><div class="layoutArea"><div class="column"><span style="font-family: 'CMR12'; font-size: 12.000000pt;">---------------------------------------------------------------------<br /> </span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">One thing that the traditional paper-pencil approach and the mental math approach have in common is that they both require knowing basic addition and multiplication facts. So yes, students should learn their addition and multiplication tables. Of course, those facts are best learned in the context of understanding. </span><br /><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">But what about students for whom remembering 6 </span><span style="font-family: 'CMSY10'; font-size: 12.000000pt;">× </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">7 is an unreachable goal? No matter what we try, they can’t seem to hang on to those facts. Should we shut the door in their face, and say that if you don’t know your multiplication table you will not be allowed to go on with secondary school math? That is the way it used to be. I don’t think such a policy makes sense any more. If a student needs to use the calculator ”as a crutch”, I let them. I wouldn’t deny a crutch to someone with a broken leg. </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">---------------------------------------------------------------------</span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> </span><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">None of this is to deny that there is some interesting math to learn when working, within reason, on three-digit addition, or two-digit multiplication. But that is my point: learning some interesting math should be the goal. Speed and accuracy will ensue for some students, but it should not be prioritized.</span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br /></span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"></span> </div></div></div><div class="page" title="Page 79"><div class="layoutArea"> </div></div>--HenriHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com0tag:blogger.com,1999:blog-3784276984960421233.post-43770889901971046232017-02-09T10:40:00.000-08:002017-02-09T10:40:41.845-08:00Geometry Boot Camp!<section class="frame center"> <div class="text as-block">I will offer two workshops this summer (2017), at the Head-Royce School in Oakland, CA.<br /> Sign up for either or both!<br /><dl class="center"><dd>June 26-27: <b>Hands-On Geometry</b> (grades 6-10)</dd><dd>June 28-30: <b>Transformational Geometry</b> (grades 8-11)</dd></dl></div><small>If the times or locations don't work for you, I can offer a workshop for your school or district. <a href="mailto:henri0@mathedpage%2E%6F%72%67">Contact me</a> directly.</small></section><section class="frame center"><small> </small> <div class=" as-block">I recommend attending with a colleague, as it makes it easier<br /> to implement these ideas when you get back to school in the fall.</div></section> <div class="frame white important center"><big> </big></div><div class="frame white important center"><big>Support materials are on the <a href="http://summer.mathedpage.org/">workshop participants' Web site</a>. </big></div><big> <article class="content"> <section id="hands-on"> <div class="center clear"> <h2><a href="https://draft.blogger.com/null" id="geometry">Hands-On Geometry</a></h2><em>with Henri Picciotto</em> Monday-Tuesday, June 26-27<br /> 9:00 a.m. to 3:30 p.m.<br /> at the Head-Royce School in Oakland, CA<br /><br /> <img alt="CircleTrigGB_lrg" height="180" src="http://www.mathedpage.org/summer/CircleTrigGB_lrg.png" width="140" /><br /><br /> </div>In this two-day workshop for teachers in <b>grades 6-10</b>, I will present kinesthetic and manipulative activities. This hands-on curriculum is intended to complement related work in paper-pencil environments: it serves to preview, review or extend key concepts in geometry. The activities can be used to enrich and enliven the high school geometry course, or to lay the groundwork for it.<br /> <ul><li>Topics include angles, triangles, quadrilaterals, area, the Pythagorean theorem, congruence, similarity, "soccer angles", and tiling.</li><li>Tools include manipulatives (such as pattern blocks and geoboards) and puzzles (such as tangrams and pentominoes.)</li><li>Technology will be used to illustrate concepts.</li></ul>These lessons were developed in somewhat heterogeneous classes, and reach a wide range of students. They provide support for the less visual by complementing the drawing and studying of figures, and enrichment for the more talented by offering deep and challenging problems.<br /><br /> <div class="center"><i>This workshop does not overlap with <a href="http://www.mathedpage.org/summer/index.html#transformational">Transformational Geometry</a>.</i></div><div class="center"><br /></div></section> <div class="right up-to-top dark"> <a class="info" href="http://www.mathedpage.org/summer/index.html#top"><small><em></em></small></a></div><hr class="push-down" /> <section id="transformational"> <div class="center clear"> <h2><a href="https://draft.blogger.com/null" id="transformations">Transformational Geometry</a></h2><em>with Henri Picciotto</em> Wednesday-Friday, June 28-30<br /> 9:00 a.m. to 3:30 p.m.<br /> at the Head-Royce School in Oakland, CA<br /><br /> <img alt="pattern-block-sym" height="120" src="http://www.mathedpage.org/summer/pattern-block-sym.png" width="117" /><br /><br /> </div>The Common Core State Standards call for a complete rethinking of geometry in <b>grades 8-11</b>. Instead of basing everything on congruence and similarity postulates, as is traditional, the idea is to build on a foundation of geometric transformations: translation, rotation, reflection, and dilation.<br /><br /> In order to teach this effectively, it is important to have a solid understanding of the underlying math, as well as ideas for rich activities for students. I have been teaching transformational geometry for twenty years, and have a lot to share. This three-day workshop will cover:<br /> <ul><li>The implications for the teaching of proof.</li><li>Composition of transformations -- the four isometries and their fundamental properties.</li><li>Symmetry in depth -- around a point, along a strip, in the plane. Connections to art and design.</li><li>Computing geometric transformations with the help of complex numbers at first, then matrices -- this is the mathematics that underlies all computer graphics.</li><li>Transforming graphs: all parabolas are similar</li><li>Intelligent use of technology to support all this, including a highly motivating unit on geometric construction.</li></ul><div class="center"><i>This workshop does not overlap with <a href="http://www.mathedpage.org/summer/index.html#hands-on">Hands-On Geometry</a>. </i></div><div class="center"><br /></div><div class="center"><i>--------------------------------------------------------------------- </i></div></section> </article> <div class="frame center" id="details"> <h3><span class="big">The Details</span></h3><div class="center"><small>Check <a href="http://www.mathedpage.org/summer" target="_blank">this site</a> in late February for Continuing Education Units info.</small></div><blockquote class="tr_bq"><b>More info about me</b><em>:</em> <a href="http://www.mathedpage.org/resume/index.html">Henri Picciotto</a></blockquote><blockquote class="as-block text"><b>Registration and logistics:</b> <a href="http://www.headroyce.org/community/summerinstitute">Head-Royce School</a> </blockquote><blockquote class="as-block text"> <b>Questions?</b> Send me <a href="mailto:henri0@mathedpage%2E%6F%72%67">e-mail</a><br /> </blockquote><blockquote class="tr_bq"><b>Scholarships:</b> <i>an anonymous donor will pay 80% of the tuition</i><i> for the first seven public school teachers to register.</i><i> </i></blockquote><div class="center"><small>If the times or locations don't work for you, I can offer a workshop for your school or district. </small></div><div class="center"><small><a href="mailto:henri0@mathedpage%2E%6F%72%67">Contact me</a> directly.</small></div></div></big><br />--HenriHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com0tag:blogger.com,1999:blog-3784276984960421233.post-56879892296800303202017-02-01T19:39:00.001-08:002017-02-01T19:39:18.519-08:00Comparing two approaches <br /><div class="page" title="Page 35"> <div class="layoutArea"> <div class="column"> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Much can be said in defense of practice exercises, but when all is said and done, very few students develop deep understanding from routine practice. For example, compare these two approaches to the area of a trapezoid. </span><br /> </div></div><div class="layoutArea"> <div class="column"> <h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Approach 1</span></h3><h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;"></span></h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;">The teacher says: ”The area of a trapezoid is given by the formula </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">h(</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><span style="font-family: 'CMR12'; font-size: 12.000000pt;"></span><span style="font-family: 'CMMI12'; font-size: 12.000000pt;">b</span><span style="font-family: 'CMR8'; font-size: 8.000000pt; vertical-align: -2.000000pt;">1</span>+</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><span style="font-family: 'CMMI12'; font-size: 12.000000pt;">b</span><span style="font-family: 'CMR8'; font-size: 8.000000pt; vertical-align: -2.000000pt;">2</span>)/2, where h is </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">the height, </span><span style="font-family: 'CMMI12'; font-size: 12.000000pt;">b</span><span style="font-family: 'CMR8'; font-size: 8.000000pt; vertical-align: -2.000000pt;">1 </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">and </span><span style="font-family: 'CMMI12'; font-size: 12.000000pt;">b</span><span style="font-family: 'CMR8'; font-size: 8.000000pt; vertical-align: -2.000000pt;">2 </span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">are the bases. Here is a worksheet where you can practice this.” </span> </div></div><div class="layoutArea"><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br />The worksheet includes 20 examples, each with different numbers for the bases and the height. The students practice in silence. Many students like this, because they know exactly what to do. Other students don’t like it, because they find it boring. All of them know they will soon have to calculate some trapezoid areas on a quiz, and they hope that this practice will help them remember the formula. Some will make an effort to memorize the formula in preparation for the quiz. Most will have forgotten the formula a week, a month, or a year later. This is because they will not use the formula again, unless they take calculus many years hence. In any case, whether they remember it or not, doing the exercises does not help them understand the formula.</span></div></div><div class="page" title="Page 36"><h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Approach 2 </span></h3><div class="layoutArea"><div class="column"> </div></div><div class="layoutArea"> <div class="column"> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Students are given a sheet of paper with a few copies of a certain trapezoid on it, with all the measurements indicated, including the bases, the legs, and the height. They are not given a formula. They are asked to find the trapezoid’s area. They are allowed to use scissors to cut out the trapezoids, and if they want to, cut one of them into smaller pieces that can be rearranged. Rearranging would allow them to use formulas they already know, such as the one for the area of a rectangle, a parallelogram, or a triangle. Students will almost certainly come up with different strategies. </span><br /> <span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br /></span><span style="font-family: 'CMR12'; font-size: 12.000000pt;">(In fact, see how many strategies you can find.)</span><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br /></span> <span style="font-family: 'CMR12'; font-size: 12.000000pt;">Students who find one quickly can be encouraged to look for more. Some students may not like the activity, because they are not told exactly what to do. The teacher can offer hints to them, or encourage them to get help from neighbors.</span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Once some strategies have been found, the teacher can lead a discussion where students demonstrate their approaches. All strategies will reveal that the lengths of the legs do not contribute to the final answer. In fact all strategies will yield the same answer for the area. A general formula can be the final punch line: applying any of the strategies to a generic trapezoid always yields the same formula. </span><br /><br /><span style="font-family: 'CMR12'; font-size: 12.000000pt;">(Scissors are not absolutely necessary. For example, the activity can be carried out on paper, without any cutting. Whether that is preferable will depend on the specifics of a given class.) </span><br /><h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Conclusion</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br /></span></h3><span style="font-family: 'CMR12'; font-size: 12.000000pt;">Even if the first approach includes a brilliant teacher explanation of the formula, I claim that the second approach is preferable. Many students who cannot remember the formula at some point in the future will be able to use one of the strategies that came up in the course of the exploration, either to find a particular trapezoid’s area, or to reconstruct the formula. This approach also carries the message that formulas can make sense, that there are many ways to solve a given problem, and that not everything needs to be memorized. A perhaps unexpected bonus is that the different solutions to this essentially geometric problem yield different interpretations of the formula, and some apparently different but actually equivalent formulas. Discussing this can help improve symbol sense.</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"> Finally, if the teacher has an excellent explanation of the formula that was not found by the students, nothing prevents him or her from sharing it. <i>Starting with the hands-on activity does not prevent the teacher from offering an explanation</i>, but it does mean that more students will understand the explanation.</span><span style="font-family: 'CMR12'; font-size: 12.000000pt;"><br /></span></div></div></div><div class="page" title="Page 37"><div class="layoutArea"><div class="column"> <br /> </div></div></div>--HenriHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com5