tag:blogger.com,1999:blog-5303307482158922565Fri, 18 Apr 2014 17:17:47 +0000linksreviewcarnivalsalonteachingmythswcydwtinternationalmy sonpoemscienceanthologybase eightgender issuesimaginary numbersmath edmsrioctalproblem-solvingstoryMath Mama Writes...http://mathmamawrites.blogspot.com/noreply@blogger.com (Sue VanHattum)Blogger478125tag:blogger.com,1999:blog-5303307482158922565.post-2661218744927477523Fri, 18 Apr 2014 17:17:00 +00002014-04-18T10:17:47.909-07:00Linkfest for Friday, April 18<ul><li>Jen Orr's first graders <a href="http://jenorr.com/?p=251" target="_blank">tell number stories and make a video of it</a>.</li><li>David Cox's post,<a href="http://coxmath.blogspot.com/2014/04/fostering-hypothesis-wrecking-mindset.html" target="_blank"> Fostering the Hypothesis Wrecking Mindset,</a> includes a list of a dozen good problems.</li><li>Dan Finkel has been enjoying how much thinking goes on when people play his <a href="http://mathforlove.com/2014/04/horseshoes-and-hand-grenades/" target="_blank">Horseshoes game</a>, a variation on a classic number game. (Winner is the one who gets <i>closest</i> to the target number, using 4 given numbers.)</li><li>Got some interesting data in a pdf file, and want to put it into a spreadsheet? Flowing Data points to <a href="http://flowingdata.com/2014/04/08/extract-csv-data-from-pdf-files-with-tabula-2/" target="_blank">a nice program to help with that</a>.</li><li>The <a href="http://tonysmaths.blogspot.co.uk/2014/04/109th-carnival-of-mathematics.html" target="_blank">109th Carnival of Mathematics</a> points to a game the author seems to like as well as 2048. I'll try it - <a href="http://www.johnrausch.com/puzzleworld/app/lunar_lockout/lunar_lockout.htm" target="_blank">Lunar Lockout</a>.</li><li>Toomai made a video - <a href="http://toomai.wordpress.com/2014/04/07/parabolas-iii/" target="_blank">All Parabolas are the Same Shape</a>.</li><li>Mimi's <a href="http://untilnextstop.blogspot.com/2014/04/parametric-equations-project.html" target="_blank">Parametric Equations Animation Project</a></li><li>Mr. K is <a href="http://blog.mathpl.us/?p=1029" target="_blank">introducing exponent properties with geometric series</a>. I think I'm going to try to involve series in as many pre-calc topics as possible next semester.</li><li>Jonathan Claydon wrote <a href="http://infinitesums.com/commentary/2014/3/23/a-year-with-desmos" target="_blank">A Year with Desmos</a>.</li></ul><br /><br /><br /><br />http://mathmamawrites.blogspot.com/2014/04/linkfest-for-friday-april-18.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1669776177698487383Sat, 05 Apr 2014 15:38:00 +00002014-04-05T08:38:44.778-07:00Linkfest for Saturday, April 5<ul><li>This video shows <a href="http://toomai.wordpress.com/2014/03/31/parabolas-ii/" target="_blank">multiplying by using a parabola</a>. Completely impractical, but I was curious why it worked. I figured it out and then wondered if my pre-calculus students could figure it out too. I wanted a demo instead of a video, so I <a href="https://www.desmos.com/calculator/f8auuzhllk" target="_blank">built something in Desmos</a>. (Hide the equations, and click on the three dots. The middle dot will always multiply the absolute values of the other two.) It's not perfect, but it might be good enough to impress my students.</li><li>I've seen <a href="http://blog.mathedpage.org/2014/03/the-function-dance.html" target="_blank">this cute list of functions, with the person's arms illustrating the graph</a>, on a number of blogs lately. I see two that are wrong. Henri sees one wrong, and has quibbles with four of them. What do you see?</li><li>Common Core for math... I keep hearing that the math standards are pretty good. But if the tests ignore the most important standards (the process standards, which describe mathematical thinking), then they're being used badly. <a href="http://dianeravitch.net/2014/04/02/jonathan-katz-on-some-problems-of-common-core-mathematics/" target="_blank">This post by Jonathan Katz</a> goes into some detail.</li><li><a href="http://studyofchange.wordpress.com/2011/09/17/week-1-practicing-engagement/" target="_blank">Nice exercise. </a>One person looks at the board, and describes the graph drawn there. Their partner must draw it from the verbal description.</li><li>Quintic polynomials. There is no formula for the roots. <a href="http://www.johndcook.com/blog/2014/04/03/quintic-root/" target="_blank">But there is this.</a> I want to learn more!</li><li><a href="http://fawnnguyen.com/2014/04/02/prices-proportions-percents.aspx?ref=rss" target="_blank">Fawn's lesson for proportional thinking.</a></li><li><a href="http://dailypapert.com/?p=1286" target="_blank">Papert on "hard fun."</a></li><li>I like <a href="http://www.geogebratube.org/student/m1992" target="_blank">this diagonal problem</a>, but when I tried it in class my students were not persistent enough to succeed with it. David Cox's post on <a href="http://coxmath.blogspot.com/2014/04/hypothesis-wrecking-and-diagonal-problem.html" target="_blank">how he used it with his students</a> makes me want to try it again. </li><li>In whatif?, xkcd's creator, Randall Munroe, takes a silly question and analyzes it with math and physics to come up with an answer. <a href="http://what-if.xkcd.com/90/" target="_blank">In this episode</a>, he figure how how big a splash you'd get from a tree as big as all trees on earth falling into an ocean with the water of all the ocean's on earth.</li><li><a href="http://www.moebiusnoodles.com/2014/04/inspired-by-calculus-math-circle-week-3/" target="_blank">In this post from her calculus for kids series</a>, I like Maria's thoughts on how we help kids learn problem-solving.</li></ul><br /><br />http://mathmamawrites.blogspot.com/2014/04/linkfest-for-saturday-april-5.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-567048187495357112Sun, 30 Mar 2014 16:21:00 +00002014-03-30T09:21:11.552-07:00Linkfest for Sunday, March 30<ul><li>Michael Pershan asks some good questions about <a href="http://rationalexpressions.blogspot.com/2014/03/stealing-tricks-from-teaching-fractions.html" target="_blank">how to approach the teaching of complex numbers</a>.</li><li>Fawn Nguyen has another <a href="http://mathtalks.fawnnguyen.com/2014/03/29/week-14.aspx" target="_blank">Math Talk</a> with her students. (I'm wondering if students think about which sorts of visual patterns will give a quadratic function and which will give a linear function.)</li><li><a href="http://www.openmathbook.org/" target="_blank">OpenMathBook</a>, a blog <span>"to promote, discuss, and develop free and open source mathematics textbooks".</span></li><li>Ihor describes a high school class <a href="http://climeconnections.blogspot.com/2014/03/noon-day-adventure-at-panther-academy.html" target="_blank">recreating the experiment Eratosthenes conducted 2200 years ago</a>. (Some day I want to try it...)</li><li>Maria is <a href="http://www.moebiusnoodles.com/2014/03/inspired-by-calculus-math-circle-week-2/" target="_blank">conducting calculus math circles with 7 to 11 year olds</a>. I am fascinated. My understanding of calculus is so steeped in algebraic thinking, it was hard for me to imagine at first. I love what she's doing.</li></ul><br /><br /><br />http://mathmamawrites.blogspot.com/2014/03/linkfest-for-sunday-march-30.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1361869414968272550Fri, 28 Mar 2014 15:50:00 +00002014-03-28T08:50:53.585-07:00Guest Post: John Spencer Addresses "Frustrated Parent"<i>John's was the first post I saw about the silly complaint going around from "Frustrated Parent". (Now I've seen about three more. They all have good things to say.) John has graciously allowed me to share his whole post here. (But <a href="http://www.educationrethink.com/2014/03/in-defense-of-new-math.html#gpluscomments" target="_blank">the comments over at his place are an interesting mix</a>, so go on over there too.) Here's John:</i><br /><br />There are many things I hate about the Common Core standards. I hate the way teachers were pushed out of the creation and adoption phase and how we have little voice in the implementation. I hate the fact that the standards will continue to be assessed with standardized, multiple choice tests and that these scores will be used with Value Added Measures in both teacher salary and teacher evaluation. However, I think it's important that in our criticism of bad policy we are careful to avoid blasting good pedagogy.<br /><br /><br />I'm seeing many of these posts making their rounds on Facebook.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-hLgGn-k8GrQ/UzT5dgS59lI/AAAAAAAAA9I/x3tzrDfTDqw/s1600/frustrated+parent+complaint.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-hLgGn-k8GrQ/UzT5dgS59lI/AAAAAAAAA9I/x3tzrDfTDqw/s1600/frustrated+parent+complaint.jpg" /></a></div><br /><br /><br />I'm seeing statements like, "What the hell is a number line and why do kids need it?" Or, "just teach them the basics." The notion of using a manipulative, playing with numbers, breaking them them apart and comparing processes is somehow viewed as non-mathematical.<br /><br />The truth is that number lines are powerful tools for understanding integers. True, when subtraction is something simple that requires no "borrowing" it feels like a joke. However, the goal is to build up number sense. It's to help them understand math conceptually. If you flip the numbers and end with a negative number as an answer, suddenly a number line helps make the negative-positive relationship more powerful.<br /><br />This parent's snarky answer about "the process used would get you terminated" is based on a faulty assumption that a first grader needs the same approach as an engineer. And yet . . . this "new math" approach that people mock is something we use constantly in real-world, mental math. <br /><br />Consider it this way: You have fifty-three dollars and you need to give someone twenty-seven dollars. What are you going to do to figure it out? If you find yourself breaking by tens and going backward, chances are you are using a mental number line.<br /><br />Oh, you could pull out a piece of paper and do that math that way, but chances is are that as an engineer, you'd be fired . . . or at least laughed at.<br /><br />I remember someone posting an angry rant about doing multiplication by breaking it up into different pieces. "Just teach the algorithm!" the parent posted.<br /><br />I posted a response. "If the bill is 27.42 and you want to leave a twenty percent tip, what's the answer? How did you find it?"<br /><br />Some people divided by five. Others multiplied by .2. Still others moved one decimal over and doubled it. Some rounded up to thirty. In other words, there were multiple processes that worked and each of them involved understanding the properties of numbers. In other words, most people used a process mentally that they were openly mocking on Facebook.<br /><br /><br /><div style="text-align: center;">* * *</div><br />Oddly enough, many of these same people who are mocking "new math" in their posts are also lamenting the fact that Singapore is kicking our butts in math. What they fail to realize is that the places where math is working are the places where they are building number sense.<br /><br />I've seen what happens when students lack number sense. They learn a lockstep process and think that math is the same as baking a cake. They follow the recipe without understanding why they are doing what they are doing. However, when they get into something as simple as linear equations, they struggle to know what to "do first," when there are often two or three options.<br /><br />When students lack number sense and they get the wrong answer, they fail to understand why an answer is illogical. You end up with a student who misplaces a decimal number and never finds his or her mistake. Asking students to think conceptually and engage in diagnostic problem-solving isn't superfluous. It's actually part of "the basics."<br /><br />I know that the "new" math looks different, but instead of criticizing it for being hard or being complicated, try thinking about the theories behind it. There's a reason we're using manipulatives, breaking things apart, using number lines and comparing processes. This is how math works.http://mathmamawrites.blogspot.com/2014/03/guest-post-john-spencer-addresses.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1773731578650875242Fri, 28 Mar 2014 04:10:00 +00002014-03-27T21:10:19.796-07:00Linkfest for Thursday, March 27<ul><li>I got my master's degree at Eastern Michigan University. It makes me happy to <a href="http://emumath.blogspot.com/?spref=fb" target="_blank">see them on the web</a>.</li><li><a href="http://www.karlscalculus.org/calc8_1.html" target="_blank">Colorful related rates problems, and colorful descriptions of solution strategies.</a></li><li><a href="http://craftingagreenworld.com/2014/03/17/cool-math-games-diy-multiplication-touch/" target="_blank">Turning multiplication table practice into a scrabble-like game</a> sounds like a good idea to me. </li><li><a href="http://langerkogutmath.wordpress.com/2014/03/18/one-teachers-attempt-at-teaching-metacognition-in-her-math-class/" target="_blank">Teaching metacognition in math class</a>.</li><li><a href="http://fawnnguyen.com/2014/03/18/20140316.aspx" target="_blank">Fawn is teaching absolute value.</a> (And giving us some great ideas.) And James Cleveland has a great inspiration, <a href="http://rootsoftheequation.wordpress.com/2014/03/26/estimation180-and-absolute-value-graphs/#comment-442" target="_blank">using estimation180 to help teach absolute value and analysis of errors</a>.</li><li><a href="http://mathmunch.org/2014/03/24/2048-2584-and-variations-on-a-theme/" target="_blank">Math Munch collects other games like 2048</a>. </li><li>Mr.Honner once again finds a <a href="http://mrhonner.com/archives/12130" target="_blank">weird graph on a standardized test</a>. It purports to be exponential, but it's not.</li><li><a href="http://ichoosemath.com/2014/03/24/math-circle-celtic-knots/" target="_blank">Justin Lanier did a math circle on Celtic knots</a>.</li><li><a href="http://christianp.github.io/building-houses/" target="_blank">Interesting puzzle - place the houses on the number line</a>. I know there's some math here, but I haven't figured it out yet.</li><li><a href="http://nautil.us/blog/to-save-drowning-people-ask-yourself-what-would-light-do" target="_blank">Optimization, a calculus problem solved by dogs, ants, and light</a>.</li><li>Henri Picciotto has a nice <a href="http://www.mathedpage.org/constructions/pythagore/index.html" target="_blank">visual Pythagorean Theorem proof</a>.</li></ul><br /><br /><br /><br /><br />http://mathmamawrites.blogspot.com/2014/03/linkfest-for-thursday-march-27.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-3030623655108416643Mon, 17 Mar 2014 19:59:00 +00002014-03-26T16:55:21.163-07:00Linkfest for Monday, March 17<ul><li>Looking for pictures of water coming out of a hose in a parabolic arc, I stumbled across this post about <a href="http://fractad.wordpress.com/2013/02/20/water-water-everywhere-and-parabolas-to-boot/" target="_blank">using geogebra to come up with an algebraic function that describes the path of the water</a>.</li><li>I have not yet ever taught Calc III. I'm thinking about working through <a href="http://ximera.osu.edu/course/kisonecat/m2o2c2/course/" target="_blank">this linear algebra based course</a>, which pointed me to <a href="http://www.dimensions-math.org/Dim_E.htm" target="_blank">this video on higher dimensional spaces</a>.</li><li><a href="http://gabrielecirulli.github.io/2048/" target="_blank">The newest addictive game, 2048</a>.</li><li>Another cool looking site from NCTM, with <a href="http://figurethis.org/challenges/challenge_index.htm" target="_blank">a wide variety of math challenges</a>.</li><li>Cheesemonkey describes <a href="http://cheesemonkeysf.blogspot.com/2014/03/stalkers-and-dreamers.html" target="_blank">what gave her the courage to persist when learning involved lots of failures</a>.</li><li><a href="http://cmcallister.typepad.com/blog/2010/04/inscribe-semicircle-in-square-by-geometric-construction.html" target="_blank">How to inscribe a semicircle in a square, using geogebra</a>.</li><li>Gas Station Without Pumps thinks about <a href="http://gasstationwithoutpumps.wordpress.com/2014/03/11/why-few-women-in-engineering/" target="_blank">what keeps women out of engineering</a>.</li><li>Some nice <a href="http://1ucasvb.tumblr.com/post/79557434791/the-sine-and-cosine-functions-for-the-circle-as" target="_blank">sine and cosine animations</a>.</li><li><a href="http://roice3.blogspot.sg/" target="_blank">3D printing for hyperbolic honeycombs</a>.</li><li><a href="http://design a game, using only a fair coin, that you have a 1/3 chance of winning." target="_blank">How would you design a game, using only a fair coin, that you have a 1/3 chance of winning</a>? No answers at the link, only thoughts about the value of hard questions.</li><li><a href="http://mathnotations.blogspot.com/2014/03/52x4-find-value-of-53x-1-non-calculator.html" target="_blank">Nice challenge problems with exponents</a>. Can you solve it without logarithms?</li><li>More from This Old Train... on <a href="http://untilnextstop.blogspot.com/2014/03/creative-problem-solving-on.html" target="_blank">the roller coaster project</a>. And a new project called <a href="http://untilnextstop.blogspot.com/2014/03/parametric-playground.html" target="_blank">Parametric Playground</a>.</li></ul><br />I still have too many tabs open, but the rest of them are not math...http://mathmamawrites.blogspot.com/2014/03/linkfest-for-monday-march-17.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-5412914639692790545Fri, 14 Mar 2014 15:59:00 +00002014-03-14T15:09:27.860-07:00Calculus: e and pi Are Both TranscendentalYesterday I was introducing the number e, and telling the students that my preferred definition of e is "the number that makes a slope of 1 for the graph of y=a<sup>x</sup> at x=0." I also told them that this definition makes mathematicians unhappy, and wrote out the limit definition. But I like them seeing a concrete meaning for this strange number.<br /><br />As I made this introduction, I mentioned that, like pi, e is irrational (not a ratio of whole numbers) and transcendental (not the solution to an algebra equation using whole numbers). I showed them the proof that the square root of 2 is irrational. But it is the solution to x<sup>2</sup>=2, so it's an algebraic number, not a transcendental number. I told them that I have not yet completely understood the proof that pi is irrational. (Updated link: Brent Yorgey, one of my favorite bloggers, posted a good series explaining it at <a href="http://mathlesstraveled.com/2009/12/07/irrationality-of-pi/" target="_blank">The Math Less Traveled</a>. But I never managed to get all the way through it.) "So it's not really math for me to say pi and e are irrational, when I don't know it by a proof but only by believing what I've been told."<br /><br />And then I decided that since pi was in the air, and Pi Day was coming, we'd do an activity I'll be doing today with a different group. I had them all stand up around the outer edges of the room, and hold hands, arms outstretched. The extra people made a diameter along the middle of the room. While I was getting them to stretch, I pulled people out by saying "You're out." They laughed. I told them it was a good kind of out, that we were the circle makers.<br /><br />The reason I'm posting this silly, super simple activity? When we were done, and they counted off, it turned out that there were 22 around and ... (you know I was holding my breath) ... 7 across the middle! I couldn't believe it. I was glowing, further proof to my students that I am a total nerd for math. ;^)<br /><br /> In case you're wondering why getting 22 around and 7 across was so special, that makes circumference / diameter = 22/7 = 3.1428, almost perfect. And before calculators made people think decimals were cooler than fractions, 22/7 was <i>the</i> estimate for pi.<br /><br /><br /><br />(Added on 3/14: Tried it again today in a smaller space. We got 17/4. Bummer.) <br /><br />http://mathmamawrites.blogspot.com/2014/03/calculus-e-and-pi-are-both.htmlnoreply@blogger.com (Sue VanHattum)3tag:blogger.com,1999:blog-5303307482158922565.post-2703330590115259972Wed, 12 Mar 2014 01:25:00 +00002014-03-11T18:25:01.629-07:00Join the Math Circle Institue at Notre Dame, July 7 to 11<div style="text-align: center;"><b><span style="font-size: large;">Math Circle Institute</span></b></div><div style="text-align: center;"><span style="font-size: large;">July 7th – 11th, 2014 </span></div><br />The seventh annual Math Circle Institute will be held on the Campus of Notre Dame, in South Bend, Indiana, from July 7th to 11th, 2014. Both novice and experienced Math Circle leaders are welcome. Bob and Ellen Kaplan, Amanda Serenevy, and Nathan Pflueger will demonstrate the Math Circle approach, and participants will prepare and teach their own Math Circle classes with 1st to 12th grade students who attend each afternoon.<br /><br />Applications can be made by e-mail to kaplan@math.harvard.edu.<br /><br /><br /><br /><br />The institute is wonderful. If you can go, do it. You'll be so glad you did.<br /><br />[Past blog posts about my experiences there: <a href="http://mathmamawrites.blogspot.com/2010/07/math-circle-summer-institute.html" target="_blank">year three</a>, <a href="http://mathmamawrites.blogspot.com/2012/07/math-circle-institute-day-1.html" target="_blank">year five</a>, <a href="http://mathmamawrites.blogspot.com/2013/07/sixth-annual-math-circle-summer-teacher.html" target="_blank">year six</a>. (I can't believe I never posted about the first two amazing years.) And a post sharing <a href="http://mathmamawrites.blogspot.com/2012/05/math-circles-blogs-and-summer-camp-oh.html" target="_blank">Rodi Steinig's experiences at the Institute</a>, including her great list of quotes from Bob, which became the lovely list now known as <a href="http://www.moebiusnoodles.com/2013/09/becoming-invisible/" target="_blank">Becoming Invisible</a>.]<br /><br />Here's a repost of what I wrote about it a few years ago...<br /><br />Bob & Ellen Kaplan (founders of the <a href="http://themathcircle.org/" target="_blank">Boston area math circles</a>, and authors of <i>Out of the Labyrinth: Setting Mathematics Free</i> and many other intriguing math books), along with Amanda Serenevy (founder of <a href="http://riverbendmath.org/" target="_blank">Riverbend Community Math Center</a>), run the fabulous week-long <a href="http://themathcircle.org/">Math Circle Institute</a>, held on the beautiful campus of Notre Dame. This summer is their 7<sup>th</sup> annual Institute. <br /><br />For only $850, you get room and board... (The food is amazingly good for such a huge operation - not quite up to my personal organic, local, sustainable standards, but really yummy, with an incredible variety of choices.) ... and then you get to play with math <i>all day</i>. My friend, Ellen H, and I used to call it the math spa. (We went swimming every morning before breakfast.)<br /><br />The first year I was amazed that I could do math all day long and never get tired of it. There's lots of freedom to explore in whatever way works best for you.<br /><br /><br /><u>Schedule</u><br />Arrive: Sunday, July 6 (or as early as you can on Monday, July 7). Notre Dame is in South Bend, Indiana, 2 hours from Chicago.<br /><br />Five days of math circle mania:<br /><ul><li>participate in math circles in the mornings,</li><li>run one math circle for kids late one afternoon, and watch your colleagues try it out on the other afternoons,</li><li>discuss the ins and outs of math circles and other fun ways of doing math in the early afternoons,</li><li>informal math play in the evenings, </li><li>play with Amanda's collection of math toys and browse her collection of math books, whenever you want,</li><li>plot, plan, and socialize to your heart's content in between all that.</li></ul>Depart: Saturday, July 12http://mathmamawrites.blogspot.com/2014/03/join-math-circle-institue-at-notre-dame.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1571717918600518527Mon, 10 Mar 2014 04:36:00 +00002014-03-09T21:36:00.267-07:00Linkfest for Sunday, March 9I have to clear out all my open tabs!<br /><br /><ul><li>First off, John Golden has created a great <a href="http://mathhombre.blogspot.com/2014/03/carnival-of-mathematics-108.html" target="_blank">Carnival of Mathematics #108</a>. (John, how can I get any work done if I play with all these posts?!)</li><li>Maria Andersen <a href="http://busynessgirl.com/full-version-of-algeboats-is-out/?utm_source=feedly&utm_reader=feedly&utm_medium=rss&utm_campaign=full-version-of-algeboats-is-out" target="_blank">announced the release of Algeboats</a>.</li><li>I think of <a href="http://untilnextstop.blogspot.com/2014/03/rollercoaster-project-success.html" target="_blank">the roller coaster</a> <a href="http://untilnextstop.blogspot.com/2014/03/rollercoaster-sweetness.html" target="_blank">projects</a> as being about trig derivatives, but they might work well when I get around to the limits chapter, where we'll talk more about differentiability. (I do it after derivatives and before integrals.) </li><li>Glenn is thinking about <a href="http://blog.mrwaddell.net/archives/912" target="_blank">how to make some smooth connections from Algebra I thought Calc</a> (and stat). I like what he's doing.</li></ul><br /><br /><br />I really like Maria Droujkova's response to someone who asked about the Common Core. She (oh so diplomatically) mentioned what she likes. The mathematical practices:<br /><blockquote class="tr_bq">1. Make sense of problems and persevere in solving them.<br />2. Reason abstractly and quantitatively.<br />3. Construct viable arguments and critique the reasoning of others.<br />4. Model with mathematics.<br />5. Use appropriate tools strategically.<br />6. Attend to precision.<br />7. Look for and make use of structure.<br />8. Look for and express regularity in repeated reasoning.</blockquote>Although I'm not sure their list is any better than <a href="http://www.withoutgeometry.com/2011/05/teaching-problem-solving-part-3.html" target="_blank">Avery Pickford's Mathematical Habits of Mind</a>. In other words, it's a good list, but not the only good one.http://mathmamawrites.blogspot.com/2014/03/linkfest-for-sunday-march-9.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-2645211455228468564Sun, 09 Mar 2014 18:55:00 +00002014-03-13T18:01:52.864-07:00Chain Rule: Experiments, Proof, Practice<b>Part One: The Plans</b><br />Tomorrow I start our third unit in calculus with an introduction to the chain rule. This deep into the semester, I have usually lost most of my creative energy, and don't find time to think this lesson out carefully. I'm teaching great groups of students this semester, which is helping keep my enthusiasm high. So today I'm planning my chain rule lesson.<br /><br /><b>Describing Composition</b><br />First, some functions that we cannot find the derivatives for, because there's more than the variable inside the function. Like y=sin(2x). Here we will review the language of composition.<br /><br /><b>Experiment by Graphing</b><br />Then a graph of the function, to see if we can figure out what the derivative ought to be.<br /><br />I don't usually write blog posts before I teach the class. This is an interesting point in my process. Where I go next with this may depend somewhat on the students' response to the graph exercise. I know right now that I'm not sure of their response, and that their response may inspire me to move a different direction than what I'm planning right now.<br /><br /><b>Experiment Algebraically</b><br />Where I think I'll go is this. We next take a look at functions which exhibit composition, but don't really have to (ie, they can be simplified). My three examples for now are:<br /><div style="text-align: center;">y = (5x-6)<sup>2</sup></div><div style="text-align: center;">y = (2x)<sup>3</sup></div><div style="text-align: center;">and</div><div style="text-align: center;">y = √(9x)</div><div style="text-align: left;"><br /></div><div style="text-align: left;">I'm going to work through finding the derivative without chain rule, and trying to get the derivative in a form that matches the function. I think this will be another way to experimentally figure out the chain rule. </div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Prove It </b></div><div style="text-align: left;">Then we'll do the proof. I found a much nicer proof <a href="http://archives.math.utk.edu/visual.calculus/2/chain_rule.4/index.html?PHPSESSID=ad20f82376de2f8b953cfb6cca4a089b" target="_blank">here</a> (click on the discussion link) than the one in our textbook (Anton). <i>Except that one vital step seems to be missing.</i> </div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Practice It</b></div><div style="text-align: left;">And after that, we'll do lots of practice. The textbook is probably fine for that, if I can't think up enough examples on my feet.</div><div style="text-align: left;"><br />[I found some inspiring posts, but I'm not sure if or when I can use them: <a href="http://samjshah.com/worksheets-projects/#Calc" target="_blank">Sam's filing cabinet</a> led me to lots of good posts, one of which was <a href="http://exzuberant.blogspot.com/2012/07/monkey-and-mathematician-learn-calculus.html" target="_blank">this on the monkey and the mathematician</a>. I might suggest my students check <a href="http://webspace.ship.edu/msrenault/GeoGebraCalculus/derivative_intuitive_chain_rule.html" target="_blank">this linked wheels applet</a> out on their own.]</div><br /><br /><b>Part Two: How It Went</b><br />I got up to the first algebraic example on Monday. On Tuesday, we did the three algebraic examples. They weren't enthusiastic about walking through the proof, so I asked them to look it over on the handout shown below. We finished up by practicing with lots of examples.<br /><br /><b></b><br /><div nbsp="" style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;"><b> <a href="http://www.scribd.com/doc/212366348/Calculus-Chain-Rule-for-composition-of-functions" nbsp="" style="text-decoration: underline;" title="View Calculus: Chain Rule (for composition of functions) on Scribd">Calculus: Chain Rule (for composition of functions)</a></b></div><b><iframe class="scribd_iframe_embed" data-aspect-ratio="undefined" data-auto-height="false" frameborder="0" height="600" id="doc_64647" scrolling="no" src="//www.scribd.com/embeds/212366348/content?start_page=1&view_mode=scroll&show_recommendations=true" width="100%"></iframe></b>http://mathmamawrites.blogspot.com/2014/03/chain-rule-experiments-proof-practice.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1314324878615675767Wed, 05 Mar 2014 04:54:00 +00002014-03-05T09:23:38.280-08:00Linkfest for Tuesday, March 4So many good posts lately! How will I ever remember all this good stuff?!<br /><br /><ul><li>Sam, <a href="http://samjshah.com/2014/03/05/mulling-things-over/" target="_blank">on having courage, and math as a creative endeavor</a>.</li><li><a href="http://maxwelldemon.com/2014/03/03/rational-parameterisation-of-the-circle/" target="_blank">Parametric Functions, and a rational parametrization of the circle</a>.</li><li><a href="http://infinitesums.com/commentary/2014/2/26/better-polar-explorations" target="_blank">Better polar explorations</a>. (This would even be good in Calc II, I think.)The key is to make it into a puzzle.</li><li>And here's <a href="http://blog.amathknauft.com/2014/03/they-sounded-like-mathematicians.html" target="_blank">a puzzle approach to inverse functions</a>.</li><li><a href="http://teaching.proftalbert.com/mth201f13/guided-practice/guided-practice-4-1/" target="_blank">Determining distance traveled from velocity</a>, from Robert Talbert, for calc I.</li><li>Shireen has made a Where's Waldo themed worksheet to help calc students think about <a href="http://mathteachermambo.blogspot.com/2014/03/straight-line-motion-calculus.html" target="_blank">position, velocity, and acceleration</a>.</li><li>I can't watch video right now, my son is sleeping. I have to watch <a href="http://clopendebate.wordpress.com/2014/03/04/videos-of-my-classroom/" target="_blank">these classroom videos, with students debating in math class, and creating theorems</a>.</li><li>I don't always agree with Curmudgeon, but I'm sure glad I follow this blog. Here's a great post <a href="http://mathcurmudgeon.blogspot.com/2014/03/should-math-really-be-required-subject.html" target="_blank">on requiring Algebra II</a>.</li><li><a href="http://mathinyourfeet.blogspot.com/2014/03/from-my-feeds-to-yours-six-interesting.html" target="_blank">Malke</a>'s thinking about spatial reasoning. <a href="http://themathguy.blogspot.com/2014/03/spatial-reasoning.html?spref=tw" target="_blank">Here's a post</a> she linked to, with a cute question. If we fold the paper <a href="https://www.youtube.com/watch?v=sJWAhc9kMZs" target="_blank">like this</a>, and hole punch it while folded, where will the holes be when we unfold?</li><li>Maria Anderson likes to think about the future. She's made four predictions about <a href="http://busynessgirl.com/4-predictions-about-the-age-of-technology-enhanced-learning/" target="_blank">technology-enhanced learning</a>.</li><li>Have you heard of <a href="https://play.google.com/store/apps/details?id=com.hemispheregames.osmos" target="_blank">Osmos</a>? It's a physics game app. I want to buy it.</li></ul>Eleven really good ideas in one day?! It's too much!<br /><br /><br />http://mathmamawrites.blogspot.com/2014/03/linkfest-for-teusday-march-4.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-7050067446932446494Tue, 04 Mar 2014 02:56:00 +00002014-03-03T18:56:54.972-08:00Linkfest for Monday, March 3<ul><li>Maria Droujkova was interviewed in The Atlantic! Woo hoo! Go, Maria. The title? <a href="http://www.theatlantic.com/education/archive/2014/03/5-year-olds-can-learn-calculus/284124/#disqus_thread" target="_blank">Five-Year-Olds Can Learn Calculus</a>.</li><li>In the LA Times, Edward Frenkel describes how he thinks our <a href="http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-why-study-math-20140302,0,5177338.story#axzz2ullNKtDX" target="_blank">tired old math curriculum cheats kids</a>.</li><li>John Conway video, <a href="https://www.youtube.com/watch?v=E8kUJL04ELA" target="_blank">on his Game of Life</a>, and a second one, <a href="https://www.youtube.com/watch?v=R9Plq-D1gEk&feature=youtu.be" target="_blank">describing what led up to him inventing it</a>. (Thanks, <i>Aperiodical</i>!)</li><li>I want to know more of the backstory behind <a href="http://xkcd.com/1337/" target="_blank">today's xkcd</a>. Can anyone tell me more about this?</li><li><a href="http://chronicle.com/blognetwork/castingoutnines/2014/03/03/the-inverted-calculus-course-and-self-regulated-learning/" target="_blank">Robert Talbert on self-regulated learning</a>. I am thinking about sharing this with my calculus class.</li></ul>http://mathmamawrites.blogspot.com/2014/03/linkfest-for-monday-march-3.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-7061458182703820923Sun, 02 Mar 2014 18:05:00 +00002014-03-02T10:05:14.099-08:00Please Join: Help the New Math Ed stackexchange Site Become a Reality<a href="http://researchinpractice.wordpress.com/2014/03/02/new-math-learning-site-on-stackexchange-com-needs-you/" target="_blank">Ben said it better than I can</a>:<br /><blockquote class="tr_bq">... based on my experience of the <i>incredible</i> usefulness of the StackExchange sites <a href="http://http//math.stackexchange.com/" target="_blank">Math StackExchange</a> and <a href="http://http//mathoverflow.net/" target="_blank">MathOverflow</a>, I think this site could become a great resource.</blockquote><br />Another site to follow? Oh my! But if it's a good place to get our math ed questions answered, I'm all for it. <br /><br />Stackexchange will only start up a new site if 200 people have committed to it. This site is 85% of the way there. (I was number 171.) If you want to help make this site a reality, <a href="http://area51.stackexchange.com/proposals/64216/mathematics-learning-studying-and-education?referrer=5lZuT3o_IUms3OClF1O32w2" target="_blank">click here</a>.<br /><br /><br /><div style="text-align: center;"><a href="http://area51.stackexchange.com/proposals/64216/mathematics-learning-studying-and-education?referrer=5lZuT3o_IUms3OClF1O32w2"><img alt="Stack Exchange Q&A site proposal: Mathematics Learning, Studying, and Education" src="http://area51.stackexchange.com/ads/proposal/64216.png" height="250" width="220" /></a></div>http://mathmamawrites.blogspot.com/2014/03/please-join-help-new-math-ed.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-3481049057254906390Fri, 28 Feb 2014 20:44:00 +00002014-02-28T12:44:01.434-08:00Linkfest for Friday, February 28<ul><li><a href="http://www.contracostatimes.com/contra-costa-times/ci_25154444/unique-math-program-at-los-medanos-college-boosts" target="_blank">A good math acceleration program</a> from another college in same district as mine, getting students from arithmetic to the algebra needed for statistics in one semester. We are beginning to implement this program too.</li><li><a href="http://www.brainpickings.org/index.php/2012/09/04/the-ravenous-brain-daniel-bor/" target="_blank">Some research on 'chunking'.</a> Although the article doesn't make the connection, chunking helps make math easier - when the basics become automatic, your brain gets to focus on higher level tasks.</li><li><a href="http://www.arborcenterforteaching.org/publications/books/jousting-armadillos/" target="_blank">An interesting curriculum for pre-algebra and algebra</a>, at middle school and high school level.</li><li><a href="http://infinitesums.com/commentary/2014/2/26/textbook-makeover-five" target="_blank">Lesson on population growth</a> using guesses and World Bank data.</li><li><a href="http://mathinyourfeet.blogspot.com/2014/02/spatial-reasoning-ready-setgo.html" target="_blank">Malke asks how spatial reasoning helps in math</a>, and how this skill is developed. Her Math in Your Feet program is making a big difference for upper elementary students.</li><li><a href="http://www.pleasanton.k12.ca.us/avhsweb/james/calculus/End%20of%20Year/Projects%20and%20other%20Stuff%20New/Roller%20Coaster%20Project.pdf" target="_blank">A roller coaster project for calculus</a>, from <i><a href="http://untilnextstop.blogspot.com/2013/07/calculus-project-resources.html" target="_blank">I Hope This Old Train...</a></i></li><li><a href="http://onegoodthingteach.wordpress.com/2014/02/27/how-big-is-a-billion/" target="_blank">How big is a billion?</a> from Rebecka Peterson </li><li><a href="http://haggisthesheep.wordpress.com/2014/02/28/giant-4d-buckyball-sculpture/" target="_blank">Giant 4D Buckyball sculture</a></li><li><a href="http://plus.maths.org/content/bridges-konigsberg-movie" target="_blank">Bridges of Konigsberg video</a>, from the <i>Plus Math</i> blog </li><li>In beginning algebra, I like asking students to figure out whether a big or small pizza is a better deal. <a href="http://flowingdata.com/2014/02/28/why-you-should-buy-the-bigger-pizza/" target="_blank">Here's some data</a>, from the <i>Flowing Data</i> blog.</li><li><a href="http://talkingsticklearningcenter.org/more-exploring-the-circle-and-pedagogical-questions/" target="_blank">Rodi asks some great questions</a> about what value others see in her math & art math circle.</li></ul><br />http://mathmamawrites.blogspot.com/2014/02/linkfest-for-friday-february-28.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-2900845108271561153Fri, 28 Feb 2014 18:49:00 +00002014-02-28T10:49:41.268-08:00Calculus: Optimizing the GoodnessOptimization may be the best application topic taught in standard math courses.<br /><br />Why do we use math in the real world? Because we want to make something better. I don't care for the problems that are about maximizing profit, as if other considerations aren't relevant. But when we make something, we are trying to satisfy a need, and we have particular goals - maximizing how well we achieve those goals can sometimes be modeled mathematically.<br /><br />Making a box that maximizes volume with a given amount of material is a simple problem that makes sense to all of us, and that represents well the class of problems we can solve with optimization. So that's how I begin our study of optimization (probably thanks to another blogger, though I don't remember now who it might be). Yesterday I brought in origami paper and showed the class how I would fold a simple box. Their task was to make the box with the biggest volume.<br /><br />I've done this lesson before, but I added a few details this time that improved it dramatically. I want to write up what happened so I'll remember it next semester.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-M_-FUPiJJ_g/UxDWpuOL6QI/AAAAAAAAA6o/sJzWi642KC8/s1600/box+folds.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-M_-FUPiJJ_g/UxDWpuOL6QI/AAAAAAAAA6o/sJzWi642KC8/s1600/box+folds.png" height="174" width="200" /></a></div><br />This is a class with over forty students. I set them in groups of four, had one person come up to get four sheets of the origami paper, and then showed them my steps. I fold in half twice in each direction, reverse two folds so the sides can come up, and make a triangle at each corner that can then fold out over the edge to lock the corner in place.<br /><br /><br /><br /><br />That method gives a particular shape of box, so then I show them how to vary the height:<br /><ul><a href="http://4.bp.blogspot.com/-giQ4q7Nm2NM/UxDWummQUkI/AAAAAAAAA60/7x_prG-7jQk/s1600/box+folds+with+corner.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://4.bp.blogspot.com/-giQ4q7Nm2NM/UxDWummQUkI/AAAAAAAAA60/7x_prG-7jQk/s1600/box+folds+with+corner.png" height="200" width="190" /></a><li>Fold wherever you want for your height. </li><li>Fold in half so you can copy this fold to the opposite edge.</li><li>Fold one corner onto this fold, creating a 45-45-90 isosceles triangle.</li><li>This give the position for the other sides, so that they'll be the same size.</li><li>Now fold your four sides up, and make the triangular corner bits as before, folding over the edge to lock.</li><li>When the box is made, measure to determine length and width (which are equal), and also the height.</li><li>Determine volume.</li></ul>As they worked, I ran off to get the rulers I'd forgotten. I think that had unintended benefits. Before I left, I asked who had done origami before. I asked the others to look at the hands up, and turn to them for help. Then I ran off, and left them to it. I think the learned helplessness that schooling engenders in so many students was minimized by my absence.<br /><br />After I passed out the rulers to each group that didn't have their own (love those students who carry rulers around with them), I asked whether we wanted to measure using inches or centimeters. I heard lots of them say centimeters and went with that. Good thing, because that worked better with decimals, and gave us an even measurement on the paper, which was exactly 15cm by 15cm. Then I asked who had their box measured, and began to make a table on the board.<br /><br />After I wrote the height Edwin had given me, I asked how I would find volume. The student I asked didn't know (which surprised me). I asked them to visualize little cubic centimeters filling the box. I could see that wasn't enough, so I had them all visualize a big box on my desk, 3 feet by 4 feet by 2 feet high. Then I used my hands to describe a cubic box, 1 foot in each direction. How many of these fit along the back of the big box? Yes, four. And how many rows of four? Three. So how many are in this bottom layer? Twelve. And how many layers? Two. So the total is ... twenty four. And now we can see that the volume of any rectangular box must be... I think they had all memorized V = LWH before, but few of them had seen why it must be so. This visualizing helps them see where the formula comes from, and is a start for many on the journey toward being able to visualize things like volumes of rotation.<br /><br />I put the height Edwin had given me onto my diagram of a paper with the fold lines, and talked my way through finding the proper length and width. I asked Edwin to find the volume. I repeated with seven more volunteers. For each one, I'd figure the height as I did the first time, saying I didn't want to go by their measurements, in case they measured high and got an unfair advantage. That allowed me to prime them to notice that we were always subtracting twice their height from the width of the paper.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-y9ehUTz6rmw/UxDHQXPOtII/AAAAAAAAA6E/myChcMCCDFQ/s1600/box+data.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-y9ehUTz6rmw/UxDHQXPOtII/AAAAAAAAA6E/myChcMCCDFQ/s1600/box+data.png" height="244" width="320" /></a></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-8NgnWOl4h_w/UxDNMUdbDMI/AAAAAAAAA6U/G71e6IkZMSU/s1600/box+data+graph.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://1.bp.blogspot.com/-8NgnWOl4h_w/UxDNMUdbDMI/AAAAAAAAA6U/G71e6IkZMSU/s1600/box+data+graph.png" height="166" width="200" /></a></div>As I added entries to the table, I also added them to a graph of height versus volume. I noticed that it looked like it could be a parabola, though we didn't yet know its shape. I talked about the extreme points - how a height of 0 or 7.5cm would give a volume of 0 - and added those to the graph.<br /><br />I knew the lesson was going well when one of the students wondered how we could figure out the best volume. We had not yet written a function, but we were getting close - as a group! I got to talk about modeling and mathematizing. I also mentioned that experimenting with particular cases was one of my favorite ways to get more grounded in what might be going on in a particular problem.<br /><br />I got to point to my graph, with a curve running through the points (Can I do that on Desmos?), and asked what calculus would say about this problem. They were able to tell me that the highest point was where the derivative was 0. I drew the tangent onto the graph and wrote y'=0. <br /><br />So that student's question got us to introduce a variable. Height was where we had started, so that became x. Then I got to ask them to think in their groups about how to find the length and width. Many of them were ready to tell me that was 15-2x. (This is one of the harder steps to my students, who are unfortunately not used to thinking for themselves in math.)<br /><br />I had them work in groups on simplifying the equation we had written, and finding the x values that would make the derivative equal zero. I pointed out that it was a cubic, with a double root, which they should know how to graph. I sketched in the full graph past x=0 on the left and past x=7.5 on the right, and got to mention the domain of the math function (all reals) versus the domain based on this problem, which is 0 < x < 7.5. <br /><br />The quadratic we got from the derivative isn't easy to factor, so we used the quadratic formula (giving me a chance to be silly and sing it). We saw that one of the solutions, x=7.5, would give a volume of 0, which is a minimum. I talked about how we would know that the other point at x=2.5 is a max, even without a graph, since the second derivative, V" = 24x-120, is negative there.<br /><br />As I summarized what we had discovered together, I said that this problem was unusual in that it didn't seem to have two equations, where most optimizing problems will. That had been my impression in the past. But as I described where we began, wanting to maximize V = L*L*H, I saw that it <i>was</i> like most optimizing problems - a function with two variables. And our constraint, which I now labeled, was L = 15-2H.<br /><br />I moved on to a quick walk through the problem of fencing a pen attached to a barn. I said I was imagining I had a barn and a big enough yard to have a goat. I want the pen as big as possible given 200 feet of fencing. Students were able to give me all of the steps for this problem.<br /><br />I think the best thing I did in this lesson was to draw the graph of height versus volume, so they could see that the derivative of the volume function gives us the maximum.<br /><br />Teaching a big group energizes me. I feel like this is the best I've done yet teaching Calc I. Whether they are really getting it is unclear to me - the downside of such a big group.<br /><br />On Monday I'll have them work in groups on two or three problems.<br /><br /><b>Any suggestions for your favorites?</b><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />http://mathmamawrites.blogspot.com/2014/02/calculus-optimizing-goodness.htmlnoreply@blogger.com (Sue VanHattum)1tag:blogger.com,1999:blog-5303307482158922565.post-5761623251510868792Tue, 25 Feb 2014 22:41:00 +00002014-02-25T14:42:24.215-08:00One Good Thing: A Student Who Learned to ThinkThere's a blog called <a href="http://onegoodthingteach.wordpress.com/" target="_blank">One Good Thing</a>, where teachers tell about something good from their day with students, that has often provided me with sweet bits of inspiration. Maybe I'll ask to be added to their set of authors. For now, I'll share here...<br /><br />After my calculus class, as I headed through the halls back to my office, a student from last semester said hi, and we stopped to talk. He told me that I had taught him to think, and that he was acing his next math course (Finite Math), which felt like a breeze to him after pre-calc. He said he had learned to question everything. He had been struggling in my course, and suddenly it all clicked during the final, and he earned a B on it.<br /><br />He said at first he thought I wasn't such a good teacher, because he had been confused by the way I left things open. It took a while for him to understand that I was trying to get them to think about why things work.<br /><br />He said that he's doing better in all his classes, that I changed his life!<br /><br />Wow! I needed to write this down so I can read it on days that I don't remember how much I can do for students. I told him he had made my day.<br /><br /><br />http://mathmamawrites.blogspot.com/2014/02/theres-blog-called-one-good-thing-where.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-9070515086618026047Sat, 22 Feb 2014 19:01:00 +00002014-02-22T11:16:37.777-08:00Papert on Girls and Computers<div itemprop="articleBody">In <a href="http://www.nytimes.com/1985/07/02/science/education-interview-seymour-papert-on-computers.html" target="_blank">Edward Fiske's interview of Seymour Papert</a> in the New York Times, almost thirty years ago:</div><div itemprop="articleBody"><br /></div><div itemprop="articleBody">Q: Boys tend to be more interested in computers than girls. Is that something that troubles you? </div><div itemprop="articleBody"><br /></div><div itemprop="articleBody">A: It does trouble me, and it's a reflection of the general phenomenon. It's not the computer as such that's more attractive to the boys than to girls. It's the fact that the computer comes out of a male technological, technocratic, white-dominated culture. The computer as we know it was made by engineers who like to think in a very systematic, organized, top-down, highly planned way. </div><div itemprop="articleBody"><br /></div><div itemprop="articleBody">Not everybody likes to think like that, but science and mathematics instruction in our schools is powerfully biased against people with a more artist-like style of thinking. They react against a culture that has no room for intuition, no empathy, no communication about what you're doing. They react against a culture where the emphasis is on linear thinking, on individual work and on making a product that works rather than a product that you can talk about with other people. </div><div itemprop="articleBody"><br /></div><div itemprop="articleBody">The computer, though, allows you to approach technical subjects, and mathematical ones too, more like the artist who creates by a negotiation of the object you're trying to create. There's no incompatibility between that intuitive kind of thinking and being able to do mathematics in a very creative way. We're making pockets of computer culture where learning is very personalized, where you can build up from the bottom and still structure it from the top. You can make something and change it. You can let it grow the way a painting on the canvas grows in a kind of negotiation between you and the product. </div><div itemprop="articleBody"><br /></div><div itemprop="articleBody"><br /></div><div itemprop="articleBody"><b>Has our perception of computers changed enough in these thirty years to make programming more welcoming for girls? </b><br /><br /><br /><br />I found this when I searched on "Seymour Papert girls". I was looking for a passage that I think is in his book <i>Mindstorms</i>, about how showing the kids projects that involved designing rooms got the girls much more involved. Something like that. I couldn't find the passage. Can anyone help me?</div>http://mathmamawrites.blogspot.com/2014/02/papert-on-girls-and-computers.htmlnoreply@blogger.com (Sue VanHattum)2tag:blogger.com,1999:blog-5303307482158922565.post-2031319596938881396Sat, 22 Feb 2014 04:50:00 +00002014-02-21T20:50:46.161-08:00Revisiting a Lesson: Derivatives of Sine and CosineA year ago, <a href="http://mathmamawrites.blogspot.com/2013/02/derivatives-of-sine-and-cosine.html" target="_blank">I posted the handouts I had made for this lesson</a>. They provide detailed explanations of:<br /><ul><li>how we know sin(x+h) = sin x cos h + cos x sin h</li><li>the squeeze theorem </li><li>how we know (sin x) / x approaches 1 as x approaches 0</li><li>how we know (cos x - 1) / x approaches 0 as x approaches 0</li></ul><b><br /></b><b><br /></b><b>Thursday Morning Lecture</b><br />I love lecturing! <a href="http://grantwiggins.wordpress.com/2014/02/03/the-lecture/" target="_blank">I agree that it's not usually an effective way to teach.</a> I keep that in mind when I lecture, and do everything I can to draw my students in. I started out by telling them that as they go on in math it becomes more and more about proof. I warned them it would take a lot of effort to work their way through the reasoning I presented, and asked them to please tell me whenever something wasn't making sense.<br /><br />I used index cards to call on people, and asked them to provide some of the simpler steps. I asked for the whole class at once to call out the very simplest parts: "sin<sup>2</sup> x + cos<sup>2</sup> x = ... " I used different colors of marker to point out various triangles we were considering. I had the students do a few algebra steps individually at their desks before I wrote them on the board. Except for redrawing the diagram and doing those algebra steps, I told them there was no need to take notes, since I would be giving them a handout at the end.<br /><br />I asked afterwards how many were able to stay with it. I think over half of them raised their hands.<br /><br />As we worked our way through this diagram, the students were getting a much-needed (for most of them) review of trig.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-fb3a-98PIF4/UwgfbtZqx-I/AAAAAAAAA5E/DhO2WIyUyG8/s1600/trig+diagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-fb3a-98PIF4/UwgfbtZqx-I/AAAAAAAAA5E/DhO2WIyUyG8/s1600/trig+diagram.png" height="312" width="320" /></a></div><br />I need to modify this so that it doesn't look like there's a straight line formed by the terminal sides of α + β and -β. Darker lines and bigger angle and point identifications would be nice too.<br /><br />We used this to prove our trig identity, and used that to get as far as...<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-0CsVQgatJeU/UwgjWnR7IZI/AAAAAAAAA5Q/js4l56_E8E8/s1600/derivative+equation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-0CsVQgatJeU/UwgjWnR7IZI/AAAAAAAAA5Q/js4l56_E8E8/s1600/derivative+equation.png" /></a></div><br /><br /><b>Next Week </b><br />We are through two of the four pages of my handout. For homework, I told them to use their calculators to fill in a table of values, evaluating the two expressions in the limits for h = .1, .01, .001, etc. On Monday, we'll look at the squeeze theorem to help us find these two limits, which will finish off our proof. (Maybe I'll give the second handout at the beginning of class, and ask them at the end of class which worked better for them - getting the handout afterward like we did on Thursday, or getting it before.) <br /><br />This proof is a good review of trig, a good way to see the power of the squeeze theorem, and a good way to think about limits from some new perspectives. (We have not done a unit on limits. I skip that chapter, and come back to it during our last unit, before considering integration.)<br /><br /><br /><br /><b>Video?</b><br />I was so excited after my 80-minute performance, I wanted to make a video of it. I don't know if it would be useful to anyone else, but a number of my students would have liked to watch it this weekend. A lot of what I do right in my lectures would be hard to reproduce on a video - I need the audience to get me pumped up, and I need to see a confused face to realize that I should say more. But one of my students agreed to be my audience, and another offered to join her. So I might find a way to do my video lesson with just a bit of student participation. If it works out well, maybe I can do a bunch. We'll see... <br /><br />Just in case, I set up a youtube channel for Math Mama. (Thank goodness that name wasn't taken!)<br /><br /><br /><br /><b>An Alternate Proof </b><br />When I got home from work, I re-read my blog post from last year. One of the commenters had posted <a href="http://thephysicsvirtuosi.com/posts/trigonometric-derivatives.html" target="_blank">a very different proof for the derivative of sine and cosine</a>. It's very short and very visual. I love it. One of my reactions was to worry that dragging my students through the longer proof was unnecessary. But I think our work will help the students better understand the sorts of proof that use lots of algebra, and will give them a great context for the squeeze theorem. I also think this alternate proof isn't quite complete. (At some angles, adding a bit to the angle increases the cosine instead of decreasing it. What then?) But it is so cool, I have to show it to my students. I think I'll wait a day or two, so they'll have done all they want with our conventional proof first. <br /><br /><br /><b>What do you think? Would you use this alternate proof?</b><br /><br />http://mathmamawrites.blogspot.com/2014/02/revisiting-lesson-derivatives-of-sine.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-2594539333587423475Fri, 21 Feb 2014 20:48:00 +00002014-02-21T12:48:23.611-08:00Full-time Math Teaching Position at Contra Costa College<a href="https://www.4cdcareers.net/postings/1937" target="_blank">The college I work at is hiring</a>. Time got away from me, and I forgot to post this when it first came up. The deadline for applying is next Friday. (Application process is completely online.) If you're interested, check it out. You can email me if you have questions: mathanthologyeditor on gmail.http://mathmamawrites.blogspot.com/2014/02/full-time-math-teaching-position-at.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-8936535739552472978Wed, 19 Feb 2014 16:25:00 +00002014-02-19T08:25:10.850-08:00Linkfest for Wednesday, February 19<ul><li><a href="http://www.geogebratube.org/student/m83330" target="_blank">This derivative plotter</a> will come in handy in Calc I,</li><li><a href="http://nicoraplaca.com/learned-way/" target="_blank">The case for student-invented strategies</a> (I found this blog while working on MTAP), </li><li><a href="http://samjshah.com/2014/02/18/doodling-in-math/" target="_blank">Sam is doodling in math class</a> (I need to play with this! When will I have time?)</li><li>Dave Richeson has collected <a href="http://divisbyzero.com/2014/02/18/undergraduate-math-bloggers/" target="_blank">a list of math blogs by undergrads</a>,</li><li>I'm teaching parametric and polar stuff in calc II right now, and have been thinking about drawing a straight line in polar. I liked <a href="http://www.robertobigoni.eu/Matematica/Lines/lines08/lines08.html" target="_blank">this article</a> and I've already forgotten who kindly sent me <a href="https://www.desmos.com/calculator/ux8e7g3een" target="_blank">this desmos work</a>,</li><li>I'll be teaching derivatives of trig functions in Calc I over the next few days, and want to connect it to sound waves, <a href="http://www.physicsclassroom.com/mmedia/waves/gsl.cfm" target="_blank">reading up on that here</a>.</li></ul><br />http://mathmamawrites.blogspot.com/2014/02/linkfest-for-wednesday-february-19.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1759713980447214629Wed, 19 Feb 2014 03:53:00 +00002014-02-18T19:53:00.688-08:00Re-post: My Top Ten Issues in Math EducationAlexandre Borovik invited me to join the writers at The DeMorgan Forum. My first post over there, at his request, is <a href="http://education.lms.ac.uk/2014/02/top-ten-issues-in-math-education/" target="_blank">a slightly revised version</a> of <i>My Top Ten Issues in Math Education</i>, <a href="http://mathmamawrites.blogspot.com/2010/02/sues-top-ten-issues-in-math-education.html" target="_blank">originally posted in 2010</a>.<br /><br /><br />http://mathmamawrites.blogspot.com/2014/02/re-post-my-top-ten-issues-in-math.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-3537961162900842952Tue, 18 Feb 2014 00:22:00 +00002014-02-18T16:06:16.730-08:00Linkfest for Monday, February 17<ul><li>I love <a href="http://fivetriangles.blogspot.com/2014/02/140-segment-length.html" target="_blank">the problems from <i>Five Triangles</i></a>. (Frustrated that I can't figure out how to communicate with them, though, as the blog does not accept comments.)</li><li><i>Numberplay</i> appears each Monday on the NYT site. <a href="http://wordplay.blogs.nytimes.com/2014/02/17/card" target="_blank">Good puzzle today</a>.</li><li><i>Tanya Khovanova</i> has created <a href="http://blog.tanyakhovanova.com/?p=485" target="_blank">an interesting truthtellers and liars puzzle</a>. </li><li><i><a href="http://toomai.wordpress.com/2014/02/17/abbys-puzzles/" target="_blank">Math and Science with my Kids</a></i> has an interesting pentomino (and decamino) problem that he and his daughter programmed.</li><li>I got a workout thinking about <a href="http://mathmistakes.org/?p=1648#comments" target="_blank">this <i>Math Mistakes</i> problem</a>.</li><li>Bree wants to know: <a href="http://betweenthenumbers.wordpress.com/2014/02/15/a-call-for-answers/" target="_blank"><i>How did your math courses/major prepare you for teaching?</i></a></li><li><a href="http://mathhombre.blogspot.com/2014/02/fibonacci-week-spiral-curriculum.html" target="_blank">John Golden's Fibonacci Fest</a></li><li><a href="http://mathteachermambo.blogspot.com/2014/02/accumulation-functions.html" target="_blank">Calc 1: Fundamental Theorem</a> (from <i>Math Teacher Mambo</i>)</li><li><a href="http://symmetricblog.wordpress.com/2014/02/14/team-quizzes/" target="_blank">Group quiz</a> (from <i>Solvable by Radicals</i>)</li></ul><br />http://mathmamawrites.blogspot.com/2014/02/linkfest-for-monday-february-17.htmlnoreply@blogger.com (Sue VanHattum)7tag:blogger.com,1999:blog-5303307482158922565.post-3623300844634064100Sun, 16 Feb 2014 20:42:00 +00002014-02-18T16:38:53.550-08:00Math Teachers at Play #71 (with 71 links)Back in 2009, the first time I hosted a <a href="http://mathmamawrites.blogspot.com/2009/07/math-teachers-at-play-11.html" target="_blank">Math Teachers at Play Blog Carnival post</a>, we were at #11. Seems like those smaller numbers almost always had something interesting going on. For 71, it's a stretch... <br /><br /><i><a href="http://www.archimedes-lab.org/numbers/Num70_200.html" target="_blank">What's Special About This Number</a></i> has:<br /><ul></ul><ul><li>71 divides the sum of all the primes before it (i.e., 2 + 3 + 5 + 7 + 11 + ... + 67 is divisible by 71)</li><li>71 = (4! + 4.4)/.4 (representation of numbers using only <a href="http://www.cut-the-knot.org/arithmetic/funny/4_4.shtml" target="_self">four 4's</a>)</li><li>71 = 36<sup>2</sup> - 35<sup>2</sup> = 36 + 35</li><li><b>71</b> - 1 = 1 x 2 x 5 x 7 and <b>71</b> + 1 = 3 x 4 x 6 products of partitions of consecutive numbers)</li><li>71<sup>2</sup> = 7! + 1</li><li>71<sup>2</sup> = 2<sup>7</sup> + 17<sup>3 </sup>(sum of prime powers of two prime numbers)</li><li>71<sup>3</sup> = <span style="color: red;"><b>3</b></span>5<span style="color: red;"><b>7</b></span>9<span style="color: red;"><b>11</b></span> (consecutive odd numbers) </li></ul><br /><i> <a href="http://www.numbergossip.com/71" target="_blank">Number Gossip</a></i> has:<br /><ul><li>71 is the only two-digit number n such that (n<sup>n</sup>-n!)/n is prime. </li><li>71 is the 2<sup>nd</sup> Google number. The <i>n</i><sup>th</sup> <i>Google</i> number is the first n-digit prime found in the decimal expansion of e. They are named <i>Google</i> numbers because of the unusual hiring ad that <i>Google</i> put up. </li></ul><blockquote class="tr_bq"><blockquote class="tr_bq"><blockquote class="tr_bq"><blockquote class="tr_bq">2<b>, 71</b>, 271, 4523, 74713, ... </blockquote></blockquote></blockquote></blockquote><br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-AyB80Nng0po/Uv-pfHNf8wI/AAAAAAAAA3c/fWn8VJZ0foY/s1600/71+mark+gonyea.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-AyB80Nng0po/Uv-pfHNf8wI/AAAAAAAAA3c/fWn8VJZ0foY/s1600/71+mark+gonyea.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">71, Mark Gonyea</td></tr></tbody></table><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-7wU-e69u3z4/Uv-pyWlBUXI/AAAAAAAAA3k/9RAGvVpZtac/s1600/71+brent+yorgey.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-7wU-e69u3z4/Uv-pyWlBUXI/AAAAAAAAA3k/9RAGvVpZtac/s1600/71+brent+yorgey.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">71, Brent Yorgey</td></tr></tbody></table><br />And here are a few images of the number 71 itself. <a href="http://www.markgonyea.com/" target="_blank">Mark Gonyea</a> is a designer. <a href="http://mathlesstraveled.com/2012/11/05/more-factorization-diagrams/#comments" target="_blank">Brent Yorgey</a> and <a href="http://www.richardevanschwartz.com/monsters.html" target="_blank">Richard Schwartz</a> are mathematicians. Posters are available for the numbers 1 to 100 from each of these artists.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-fE4pM11u34E/Uv-p12w3A4I/AAAAAAAAA3s/u5de2j_u8rk/s1600/71+richard+schwartz.png" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-fE4pM11u34E/Uv-p12w3A4I/AAAAAAAAA3s/u5de2j_u8rk/s1600/71+richard+schwartz.png" height="199" width="200" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Richard Schwartz</td></tr></tbody></table><br /><br />Math teachers at play know that math is best learned when the student is thoroughly engaged, through their body, their imagination (story-telling), or the world of games. I've started out this month's post with those three categories. (Most of the submissions this month described hands-on, or feet-on, activities. It's as if there had been a theme agreed upon without anyone mentioning it.) Some of the following posts are from submissions, and others are posts that I wanted to share from my internet wanderings. This post has 71 links. (You might need to digest it in smaller bites.) <i>Enjoy</i>!<br /><br /><br /><br /><h2><b>Learning with Our Bodies</b></h2><b>Julie</b>, at <i>Highhill Education</i>, shares <a href="http://highhillhomeschool.blogspot.de/2014/01/combining-art-and-math-mandala.html" target="_blank">her family's mandala art</a>, and the geometry they learned while doing it. (Julie is based in Germany. In the U.S., I've found some of the best inexpensive books come from Dover - <a href="http://search.doverpublications.com/search?keywords=mandala+coloring+book" target="_blank">here are some of their mandala coloring books</a>.)<br /><br /><b>Jennifer</b> Bardsley, at <i>Teaching My Baby to Read</i>, works (plays) with her son, <a href="http://teachingmybabytoread.com/2014/01/23/rotational-symmetry-with-cookie-cutters/" target="_blank">exploring rotational symmetry using cookie cutters and flour</a>.<br /><br /><b>Ticia</b>, at <i>Adventures in Mommydom</i>, shares <a href="http://adventuresinmommydom.org/fractions-lesson/" target="_blank">her hands-on fraction lessons</a>.<br /><br /><b>Margo</b> Gentile, at <i>Margo's Math and More</i>, was inspired by all those snow days, and created some wonderful <a href="http://margosmathandmore.com/blog/article/-snow-mazing-a-few-amazing-activities-to-do-in-the-snow-" target="_blank">mazes in the snow</a> for her dog and kids to navigate. (Bummer! I don't have any snow here in California to try this in.)<br /><br /><b>Lilac, </b>at <i>Learners in Bloom</i>, wrote <a href="http://learnersinbloom.blogspot.com/2014/02/combinatorics-in-kindergarten-what-will.html" target="_blank">Combinatorics in Kindergarten</a>, her story of making clothes for the bears, so her daughters could count how many outfits Little Bear could wear. She also made a <a href="http://learnersinbloom.blogspot.com/2014/02/homemade-math-game-feed-clown.html" target="_blank">Feed the Clown</a> game to help her daughters have fun practicing basic addition facts.<br /><br /><b>Maria</b> Droujkova, at <i>Moebius Noodles</i>, describes how a student combined ideas from the <i>Moebius Noodles</i> book, <a href="http://www.moebiusnoodles.com/2014/02/mirror-book-fractal-stars/" target="_blank">making mirror books to create fractal stars</a>.<br /><br /><b>Nicora</b>, at <i>Bridging the Gap</i>, suggests that we should <a href="http://nicoraplaca.com/fractions-let-students-break-things/" target="_blank">let students break things</a> to help them learn fractions. "Give them lots and lots of experiences where they have to break up things evenly and share things fairly." Discuss, put back together, discuss some more.<br /><br /><b>Steven</b> Strogatz describes <a href="http://opinionator.blogs.nytimes.com/2012/09/10/singular-sensations/?_php=true&_type=blogs&_r=0" target="_blank">the math found in our cowlicks and fingerprints</a>.<br /><br /><br /><br /><h2><b>Storytelling</b></h2><b>Denise</b>, at <i>Let's Play Math!</i>, <a href="http://letsplaymath.net/2014/01/20/the-linear-inequality-adventures-of-ohio-jones/" target="_blank">shares the power of stories</a>: "The mere hint of fantasy adventure can change graphing equations from <i>boring</i> to <i>cool</i>." She used <b>Dan</b> Wekselgreene's <a href="http://exponentialcurve.blogspot.com/2010/04/some-funish-worksheets.html" target="_blank">inequalities lesson based on the adventures of Ohio Jones</a>. (Denise has written extensively about Fibonacci and Alexandria Jones. I think they must all be related somehow.)<br /><br />Advertisers tell stories to convince us to buy what they're selling. Often their stories are deceptive. <b>Mr. Honner</b> describes how <a href="http://mrhonner.com/archives/10246" target="_blank">Prudential's 'oldest person you know' ad subtly points in the wrong direction</a>.<br /><br />There is a story mathematicians like to tell, of how the young Carl Gauss was asked, along with his classmates, to add up the numbers from 1 to 100, perhaps to give the schoolmaster a bit of time to relax. As the story goes, Carl saw a nice trick, and wrote just the answer down, turning it in almost immediately. <b>Brian</b> Hayes was curious about the historical accuracy of this story, and researched it. <a href="http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning" target="_blank">His article</a> (in <i>American Scientist</i>) is quite intriguing.<br /><br /><b>Alexandre</b> Borovik has written a delightful story about <a href="http://www.maths.manchester.ac.uk/~avb/anthony.html#Anthony%27s_Life" target="_blank">Anthony the Ant</a>, and his discoveries of his world (a piece of paper, which gets folded into a cube).<br /><br /><br /><br /><h2><b>Games & Puzzles</b></h2><b>John</b> Golden, at <i>Math Hombre</i>, has <a href="http://mathhombre.blogspot.com/p/games.html?showComment=1391393622484" target="_blank">a games page</a> that looks marvelous!<br /><br />Did you know that some coins cost more to make than their face value?! <b>Dan</b>, at <i>Math for Love</i>, used this to make a math lesson, and one of his students came up with a great question. <a href="http://mathforlove.com/2014/02/a-dollar-that-costs-a-dollar/" target="_blank">The puzzle she posed</a> is whether or not you can come up with coins that are worth one dollar and cost one dollar to make. And another coins puzzle, from <b>Nathan</b> Kraft, at <i>Out Rockin' Constantly</i>: <a href="http://nathankraft.blogspot.com/2012/09/exploiting-my-son-for-math.html" target="_blank">Which is worth more, a pound of quarters, or a pound of dimes? </a><br /><br />I don't know if this counts as a puzzle - more like a problem-solving challenge. <b>Nat</b> Banting, at <i>Musing Mathematically</i>, asks <a href="http://musingmathematically.blogspot.com/2012/07/bike-trail-task.html" target="_blank">what pattern the wet part of a tire makes as the tire rolls along</a>.<br /><br /><a href="http://subitizing.com/exercise" target="_blank">How fast can you decide how many dots you saw?</a> (The creators of this simple game have not left their names. They write: "Subitizing is the ability to immediately recognize the quantity of a small number of objects without counting. Research has shown subitizing to be foundational to basic arithmetic and other math skills. Many children who struggle with basic math also have trouble subitizing.") <br /><br /><b>Mike</b>, at <i>Spiked Math</i>, has created <a href="http://spikedmath.com/518.html" target="_blank">a very visual puzzle</a>.<br /><br /><b>Greg</b> Ross, at <i>Futility Closet</i>, has given us <a href="http://www.futilitycloset.com/2012/09/10/lattice-work/" target="_blank">the lovely puzzle you see below</a>. You might word it differently, depending on your student's stamina and mathematical sophistication. (Instead of asking for proof, maybe just ask them to find all the midpoints first. Then ask if they can pick points so none of the midpoints will occur at an intersection. Finally, see if they can figure out why five points will always produce at least one line whose midpoint is on an intersection.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-gFn60GITrVI/UwEAAmdabWI/AAAAAAAAA4c/QYK4HdRP1eM/s1600/lattice+puzzle.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-gFn60GITrVI/UwEAAmdabWI/AAAAAAAAA4c/QYK4HdRP1eM/s1600/lattice+puzzle.png" height="264" width="640" /></a></div><br /><br /><br /><h2><b>Math Education </b></h2><b>Crystal</b> Wagner, at <i>Triumphant Learning</i>, knows that <a href="http://www.triumphantlearning.com/the-heart-of-mathematics/" target="_blank">problem solving is at the heart of mathematics</a>, and gives some guidelines and resources for keeping problem-solving at the heart of your math lessons. (I'd like to add two more resources to her list: <a href="http://www.betterworldbooks.com/the-art-of-problem-posing-id-9780898597257.aspx" target="_blank"><i>The Art of Problem Posing</i></a>, by Stephen Brown, who advocates for students to pose problems, <a href="http://letsplaymath.net/2009/04/27/kitten-poses-a-puzzle/">like this</a>. And for advanced math students, <i>The Art and Craft of Problem Solving</i>, by Paul Zeitz.) <br /><b></b><br /><b>Jenny</b>, at <i>Elementary, My Dear, Or Far From It</i>, <a href="http://emdffi.blogspot.com/2012/07/more-thoughts-from-chapter-four-of.html" target="_blank">describes the benefits of confusion</a>. She also linked to <a href="http://mcclurken.blogspot.com/2012/07/confirmation-for-uncomfortable-but-not.html" target="_blank">this piece</a> by Jeffrey McClurken, and <a href="http://blogs.kqed.org/mindshift/2012/07/what-do-emotions-have-to-do-with-learning/" target="_blank">this piece</a> by Annie Murphy Paul. (It's hard for students in the U.S. to understand how useful confusion is - as a stage in learning anything new. I want to share all their ideas with my college students. Maybe I'll write a post consolidating it all...) <br /><br /><b>Dan </b>Finkel, at <i>Math for Love,</i> gives <a href="http://mathforlove.com/2012/10/goat-river-crossing/" target="_blank">an inspiring description</a> (using his goats for comparison) between the fearful learner, who leaves others in charge, and the adventurous learner who takes charge both of their own learning and of the math problem at hand.<br /><b><br /></b><b>Mama Squirrel</b>, at <i>Dewey's Treehouse</i>, wrote <a href="http://deweystreehouse.blogspot.ca/2014/02/how-i-became-math-teacher.html" target="_blank">How I became a Math Teacher?</a> to describe her journey into math teaching and her thoughts about the matter. <br /><br /><b>Cathy</b>, at <i>Math Babe</i>, <a href="http://mathbabe.org/2014/02/11/interview-with-bill-mccallum-lead-writer-of-math-common-core/" target="_blank">interviews the lead writer of the common core math standards</a>.<br /><br /><b>Megan</b> Hayes-Golding describes why <a href="http://kalamitykat.com/2012/12/30/designing-ranking-tasks/" target="_blank">ranking tasks are especially valuable as learning tools</a>.<br /><br /><b>Alexandre</b> Borovik, at the<i> De Morgan Forum</i> <a href="http://education.lms.ac.uk/2013/09/herbert-s-wilf-can-there-be-research-in-mathematical-education/" target="_blank">points to</a> <a href="http://www.math.upenn.edu/~wilf/website/PSUTalk.pdf" target="_blank">a paper [pdf] by Herbert Wilf</a>, who argues that there is no useful math education research out there. The abstract states:<br /><br /><blockquote class="tr_bq"><span style="font-family: 'Helvetica'; font-size: 12.000000pt;">We examine a number of papers and a book, all of which have been cited, by people who are knowledgeable in the field, as being good examples of “research in mathematics education.” We find specific serious flaws, indeed fatal flaws, in all of them, so that no conclusions of any interest follow as a result of any of the “research” that is reported in these works. We have found no evidence that the research paradigm, involving test and control groups, randomized trials, etc., which is invaluable in the life sciences, is of any use whatever in studying mathematics education and we urge that it be abandoned, in favor of human-to-human discourse about how we can improve curricula and teaching. </span></blockquote>Also at the <i>De Morgan Forum </i>are the results of a study that found that <a href="http://education.lms.ac.uk/2013/08/practice-at-guesstimating-can-speed-up-math-ability/" target="_blank">practice at "guesstimating" can speed up math ability</a>. (Hmm, isn't this research on math education? Maybe Wilf has different sorts of research in mind. I'd enjoy discussing his paper with anyone interested.) And another: Can you imagine a whole post on <a href="http://education.lms.ac.uk/2013/05/3-1-2/" target="_blank">3 - 1 = 2</a>? Alexandre Borovik has translated a paper originally written in Russian by <b>Igor</b> Arnold, which gives 20 different problems that all boil down to finding 3 - 1.<br /><br /><b>Bruno</b> Reddy, at <i>Mr. Reddy's Math Blog</i>, posted <a href="http://mrreddy.com/blog/2012/09/language-revelation/" target="_blank">some interesting videos</a> from a workshop he attended. If you teach students with limited English language proficiency, you may find this valuable. <br /><br /><b>Geoff</b> Krall, at <i>Emergent Math</i>, is thinking about <a href="http://emergentmath.com/2013/10/03/a-critical-ingredient-missing-from-my-math-blogging/" target="_blank">how to make his classroom a safe place for taking risks</a>. That's <a href="http://cheesemonkeysf.blogspot.com/2014/01/writing-kids-notes.html" target="_blank">a common theme at <b>Cheesemonkey</b>'s blog</a>.<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-1D4ioWeDiIw/Uv_1jlUoqbI/AAAAAAAAA38/Rik_lmBywWk/s1600/straight+lines+making+curves.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" src="http://2.bp.blogspot.com/-1D4ioWeDiIw/Uv_1jlUoqbI/AAAAAAAAA38/Rik_lmBywWk/s1600/straight+lines+making+curves.jpg" height="152" width="200" /></a></div><h2>Visual Math</h2><b>Dan</b> Walsh, at <i>Dan's Geometrical Curiosities</i>, saw this, and just had to figure it out. The mathematical description of what's happening is called <a href="http://danielwalsh.tumblr.com/post/3270666038/the-mice-problem-more-on-curves-of-pursuit" target="_blank">curves of pursuit</a>.<br /><br />Here's your chance to make a bit of mathematical art: <a href="http://weavesilk.com/" target="_blank">at weavesilk.com</a>. Ahh...<br /><br />If you were going to try to figure out how far it is to the horizon, what sort of picture would you draw? <b>Bryan</b> Meyer, at <i>Doing Mathematics</i>, thinks we can learn a lot about students' thinking by <a href="http://www.doingmathematics.com/2/post/2012/09/creating-rich-discussion-by-honoring-individual-thinking.html" target="_blank">discussing the pictures they draw</a> to solve problems with.<br /><br />The two mathematicians described at the beginning of this post, <b>Brent</b> Yorgey and <b>Richard</b> Schwartz, have made their images to help people visualize factors, prime numbers, and composite numbers. <b>Jeffrey</b> Ventrella has the same goal with <a href="http://www.ventrella.com/numbertree/" target="_blank">his composite number tree</a> and his book, <a href="http://www.divisorplot.com/index.html" target="_blank">Divisor Drips and Square Root Waves</a> (link is to a fascinating online version). There is also this intriguing <a href="http://www.jasondavies.com/primos/" target="_blank">Prime Number Patterns applet</a>, by <b>Jason</b> Davies.<br /><br /><br /><br /><h2><b>News</b></h2>A <a href="http://plus.maths.org/content/maths-solves-frozen-mystery" target="_blank">simulation of glacier movement</a> can be run backwards to predict where things were in the past. When the remains of some hikers who were lost almost 90 years ago were recently found, the simulation was used to figure out where they most likely were when they died.<br /><br /><b>Rachel</b> Thomas, at <i>+plus magazine</i>, writes about <a href="http://plus.maths.org/content/swimming-mathematics" target="_blank">the math of bubbles</a>, which inspired the architecture of the National Aquatic Centre in Beijing, built for the Olympics.<br /><br />The Rubik's cube has over 43 quintillion (4.3x10<sup>19</sup>) positions. <a href="http://bruce.cubing.net/ham333/rubikhamiltonexplanation.html" target="_blank">It has recently been shown</a> that there is a way to move it to each different position in sequence, without ever repeating a position. (This is called a Hamiltonian circuit.) Thanks to <b>Robert</b> Talbert for pointing this out on his <a href="http://chronicle.com/blognetwork/castingoutnines/2014/02/07/weekend-reading-february-7/" target="_blank"><i>Casting out Nines</i></a> blog.<br /><br /><b>Caroline</b> Chen has written a very readable account of <a href="http://projectwordsworth.com/the-paradox-of-the-proof/" target="_blank">the strange proof of the ABC Conjecture</a>. <br /><br /><div class="column">The <a href="http://scholarship.claremont.edu/jhm/" target="_blank">Journal of Humanistic Mathematics</a> has some interesting articles in its current issue, including one on <a href="http://scholarship.claremont.edu/cgi/viewcontent.cgi?article=1104&context=jhm">Gallileo and Aristotles' Wheel [pdf]</a> describing a paradox and how mathematicians think about it.</div><div class="column"><br /></div><div class="column"><b>Joselle</b> Kehoe, at <i>Mathematics Rising</i>, writes about <a href="http://mathrising.com/?p=862" target="_blank">brain research</a> showing that the part of the brain that deals with counting deals also (and earlier in our evolution) with representing the fingers.</div><div class="column"><br /></div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; margin-left: 1em; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-5phLtseluOI/UwBnwtXvcWI/AAAAAAAAA4M/i830iySz6Nk/s1600/math+heart.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-5phLtseluOI/UwBnwtXvcWI/AAAAAAAAA4M/i830iySz6Nk/s1600/math+heart.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">(from <a href="http://pballew.blogspot.com/2014/02/on-this-day-in-math-february-14.html" target="_blank">Pat Bellew</a>)</td></tr></tbody></table><br /><br /><h2><b>Valentine's Day Math</b></h2><b>Mr. Honner</b> wishes us all a <a href="http://mrhonner.com/archives/11899" target="_blank">Happy Permutation Day</a>.<br /><br /><b>Laura</b>, at <i>Math for Grownups</i>, shared this quickie video (she calls if a gif, my son calls these vines) <a href="http://www.mathforgrownups.com/happy-valentines-day/" target="_blank">valentine</a>.<br /><br /><br /><br /><div class="column"><h2>A Few More Tidbits</h2></div><div class="column"><a href="http://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword=" target="_blank">Math Fun Facts </a> (list and <a href="http://www.math.hmc.edu/funfacts/" target="_blank">home</a>)</div><div class="column"><a href="http://spoonful.com/family-fun/the-142857-times-table" target="_blank">The 142857 Times Table</a></div><div class="column"><a href="http://prairiecreek.typepad.com/herons/problem-of-the-week-video-help.html" target="_blank">Video Helpers for Algorithms and Problem-Solving</a> (from Prairie Creek Community School)</div><div class="column"><br /></div><div class="column"><br /></div><div class="column"><br /></div><div class="column"><h2>Recursion (links to other collections of math links)</h2></div><div class="column"><b>Brie</b> Finegold, at <i>Blog on Math Blogs</i>, has ideas about <a href="http://blogs.ams.org/blogonmathblogs/2014/02/05/how-to-get-your-friend-to-like-math-a-multipronged-approach/" target="_blank">how to get your friend to like math</a>.</div><div class="column"><br /><a href="http://mathmunch.org/" target="_blank"><i><span id="goog_1782900753"></span>Math Munch<span id="goog_1782900754"></span></i></a> comes out weekly. <b>Anna</b> Weltman, <b>Justin</b> Lanier, and <b>Paul</b> Salomon say: "We write Math Munch to help more kids find something mathematical that they love." Here's a post I liked on <a href="http://mathmunch.org/2012/08/01/bridges-meander-patterns-and-water-sports/" target="_blank">art and math</a>, one of their favorite topics, I think. And <a href="http://mathmunch.org/2012/08/06/mike-naylor-math-magic-and-mazes/" target="_blank">this post is full of puzzles</a>.</div><div class="column">Don’t miss the <a href="http://www.whitegroupmaths.com/2014/02/107th-carnival-of-mathematics.html" target="_blank" title="Carnival of Mathematics #106">107<sup>th</sup> Carnival of Mathematics</a> (our sister blog carnival).</div><div class="column"><br /></div><div class="column"><br /></div><div class="column"><br /><br /></div><div class="column">That rounds up this edition of the <b><i>Math Teachers at Play</i></b> carnival. I hope you enjoyed the ride.<br />The next installment of our carnival will open sometime during the second week of March. If you would like to contribute, please use this <a href="http://letsplaymath.net/mtap-submission-form/" target="_blank">handy submission form</a>. Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival.<br /><br />Past posts and future hosts can be found on our <a href="http://letsplaymath.net/mtap/" target="_blank">blog carnival information page</a>. We need more volunteers. Classroom teachers, homeschoolers, unschoolers, or anyone who likes to play around with math (even if the only person you “teach” is yourself) — if you would like to take a turn hosting the <b><i>Math Teachers at Play</i></b> blog carnival, please speak up!</div>http://mathmamawrites.blogspot.com/2014/02/math-teachers-at-play-71-with-71-links.htmlnoreply@blogger.com (Sue VanHattum)1tag:blogger.com,1999:blog-5303307482158922565.post-8483422875567261187Sat, 25 Jan 2014 18:53:00 +00002014-01-25T11:11:12.609-08:00Potpourri: Links and a French Curve Question<span style="font-family: Verdana,sans-serif;"><span style="font-size: large;">Links </span></span><br />Here's the one I most want to share with everyone, but now I have some reservations...<br />A lovely documentary on mathematical origami, called Between the Folds, has been posted to <a href="https://www.youtube.com/watch?v=bJRBiIeFe7Q" target="_blank">youtube</a>. The DVD costs $20, which seems quite reasonable to me. This youtube posting is not from the producers, <a href="http://www.greenfusefilms.com/" target="_blank">Green Fuse Films</a>. I've contacted them, in case they want to have it removed. Watch it quickly if you'd like to. The <a href="https://www.youtube.com/watch?v=tE4lqYzS2m0" target="_blank">official trailer is also on youtube</a>, so you can get a taste, even after the pirate version is gone.<br /><br /><br /><br />Many of the links I'm saving these days are ideas I hope to share with my students:<br /><br /><b>Calculus I</b><br /><a href="http://girlsangle.wordpress.com/2014/01/25/intuiting-the-chain-rule/" target="_blank">Intuiting the Chain Rule (Girls' Angle)</a><br /><br /><b>Calculus II</b><br /><a href="http://www.intmath.com/blog/volume-of-a-pendant/7629" target="_blank">Volume of a Pendant (SquareCircleZ)</a><br /><br /><b>Linear Algebra</b><br /><a href="http://math.stackexchange.com/questions/151294/whats-so-special-about-the-4-fundamental-subspaces" target="_blank">What's so special about the 4 fundamental subspaces?</a> (math.stackexchange)<br /><br /><br /><br />Some of the links aren't for a course, but for me to try out, some day when I have time...<br /><br /><b>Geometry</b><br /><a href="http://pballew.blogspot.com/2014/01/circles-and-equilateral-triangles.html" target="_blank">Circles and Equilateral Triangles</a> (Pat's Blog)<br /><br /><br /><br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-2H0eoPNQhZc/UuP35SEW0-I/AAAAAAAAA2U/onrv6F2ag7E/s1600/french+curves.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-2H0eoPNQhZc/UuP35SEW0-I/AAAAAAAAA2U/onrv6F2ag7E/s1600/french+curves.png" height="212" width="320" /></a></div><br /><span style="font-family: Verdana,sans-serif;"><span style="font-size: large;">Question</span></span><br />A student came to me last week for help. He wants to use fabric (a very loose weave, which will stretch some) to cover a sphere. He needed to know what shape to cut. I had no idea, and first suggested he look at information on world maps. He wants nice seams and no puckering. I thought a shape that wrapped around the equator, with sort of triangular tabs up and down to the poles might work. We knew the sides of those needed to curve, and he wanted an equation. I thought he should experiment with 8 tabs in each direction at first, and those would need to have a 45 degree angle at the pole. I had no idea what kind of equation would fit this.<br /><br />A former student, who designs and sews clothing, was in the math lab, and I brought him into our conversation. He said he would use a <i>French curve</i>, which is what you see in the image above. We looked it up, and <a href="http://www.mapleprimes.com/questions/102902-Origin-Of-The-French-Curve-Or-Mathematical" target="_blank">no one seems to know the equations</a> for the curves. <br /><br />So my question is about the French curves. <b>Can anyone help me figure out their equations?</b><br /><br />The coolest thing happened as I began to write this post. I was trying to sketch my pattern idea freehand, and I'm a terrible drawer. So I turned to Geogebra, and started putting in the points and lines. When I got to the curved segments, I was in the right frame of mind. I knew I needed something between the points (0,.5) and (1,3.5), with a vertical tangent at (0,.5) and a slope of 1 (same as 45 degrees) at (1,3.5). I knew that the square root function starts out with a vertical tangent, and so I figured I'd try to modify that. Getting it to go though those two points, I was having trouble getting the slope at the top right. (I think there's a way...) So I figured I could change what root I used. And I solved the problem I had posed! (Always a rush.) The first curve is y = 3 times the cube root of x. The next one in the same direction is y = 3 times the cube root of (x-2), and the one between, that goes in the other direction, is y = 3 times the cube root of (2-x). They may not be just the right shape. My student will have to experiment at this point, I think.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-sDLaBzZqIak/UuQFno623XI/AAAAAAAAA2k/8BM4TfMUiKQ/s1600/pattern+for+fabric+on+sphere.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-sDLaBzZqIak/UuQFno623XI/AAAAAAAAA2k/8BM4TfMUiKQ/s1600/pattern+for+fabric+on+sphere.png" height="156" width="320" /></a></div>Or is there a way to figure out a better shape mathematically? The problem is that we are really just approximating, because the fabric is a flat surface, and the sphere is everywhere curved (in all directions). So I don't see how we can<br /> describe perfectly what we want. <b>Ideas?</b><br /><br />[Oops! I just tried to explain this to my son, and realized those points have two 45 degree angles in them, for a total of 90 degrees. Back to the drawing board! (More to come...)]<b> </b>http://mathmamawrites.blogspot.com/2014/01/potpourri-links-and-french-curve.htmlnoreply@blogger.com (Sue VanHattum)0tag:blogger.com,1999:blog-5303307482158922565.post-1964190327242092909Sun, 12 Jan 2014 18:09:00 +00002014-01-12T10:09:51.891-08:00Calculus: Solids of Revolution<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ldY2ZlDcAqE/UtLaVRhxImI/AAAAAAAAA1E/crG4ZB7MA-A/s1600/Solids-of-Revolution1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-ldY2ZlDcAqE/UtLaVRhxImI/AAAAAAAAA1E/crG4ZB7MA-A/s1600/Solids-of-Revolution1.jpg" height="320" width="210" /></a></div>In about a week, I'll be working with my students on solids of revolution. (At least, I hope to be. My calc II class is too small right now, and could be canceled.) Patrick Honner's <a href="http://mrhonner.com/archives/11931" target="_blank">Math Photo</a> post of three beautiful bottles exemplifying solids of revolutions inspired me. I looked up <a href="https://www.google.com/search?site=imghp&tbm=isch&source=hp&biw=1080&bih=549&q=beautiful+bottle&oq=beautiful+bottle&gs_l=img.3..0l6j0i5l2j0i24l2.1539.4965.0.5312.16.13.0.3.3.0.92.1038.13.13.0....0...1ac.1.32.img..0.16.1051.UkoziPxa2Ys" target="_blank">beautiful bottles on Google Images</a>, and there are so many. Some of them are solids of revolution and some aren't. I wonder if my students would benefit by identifying which are which.<br /><br />Maybe that could be a first step, and then drawing a curve they think would make a good-looking volume when revolved. Hmm... (Anyone know a super-easy 3D modeler we good put our curves in, and get visuals from?)<br /><br />Maybe this Friday I can get some students to come in and work with me on making models of volumes (not volumes of revolution, though), like <a href="http://bowmandickson.com/2013/04/13/volume-in-calculus-conceptualizing-before-formalizing/#comments" target="_blank">these that Bowman Dickson made</a>, or <a href="http://www.epsilon-delta.org/2013/07/made4math-volumes-in-calculus.html" target="_blank">these (both types) that Rebecka Peterson made</a>.<br /><br /><br />http://mathmamawrites.blogspot.com/2014/01/calculus-solids-of-revolution.htmlnoreply@blogger.com (Sue VanHattum)6