tag:blogger.com,1999:blog-37842769849604212332018-04-19T04:51:21.134-07:00My Math Education Blog"There is no one way"Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger34113MathEdBloghttps://feedburner.google.comtag:blogger.com,1999:blog-3784276984960421233.post-62435053258370456472018-04-02T10:08:00.000-07:002018-04-16T16:27:07.554-07:00April Travels, May Webinar, Summer Workshops<p>I'll be traveling a lot this month. Here's the plan, should you want to say hello.</p><div><blockquote class="tr_bq"><p><strong>New York City April 5</strong>,<strong> 4:30pm</strong>: I will present <strong>Geometric Puzzles </strong>at the Museum of Math Teachers’ Circle. Geometric puzzles are accessible to solvers of all ages, but they can also challenge even the most tenacious of solvers. Join math education author and consultant Henri Picciotto in an exploration of hands-on polyomino puzzles that involve area, perimeter, symmetry, congruence, and scaling — you’ll even participate in some collaborative pentomino research! </p><p><strong>New York City April 6</strong>, <strong>6:30pm</strong>: I will present <strong>Playing with Pentominoes </strong>at the Museum of Math Family Friday. Pentominoes are simple to create — just join five equal-sized squares together — but provide a host of classic challenges in the world of recreational mathematics. Discover them, play with them, and explore a variety of visual puzzles that span the whole range: from kindergarten to adult, from the most accessible to the most challenging, and from the meditative to the maddening.</p><p><strong>Atlanta April 11-15</strong>: I will attend the <strong>Gathering for Gardner</strong>. I won't be presenting, but I look forward to seeing old and new friends. I contributed a thematic cryptic crossword to the conference book.</p><p><strong>Washington, DC April 25 and 27 3:00pm</strong>: I will present <strong>Taxicab Geometry </strong>for the Math Teachers’ Circle sessions at the NCTM National Meeting (in the Networking Lounge). Many concepts depend on distance: the triangle inequality, the definition of a circle, the value of π, the properties of the perpendicular bisector, the geometry of the parabola, etc. In taxicab geometry, you can only move horizontally and vertically in the Cartesian plane, so distance is different from the usual "shortest path" definition. We will explore the implications of taxicab distance. There are no prerequisites, other than curiosity and a willingness to experiment on graph paper.</p><p><strong>Washington, DC April 26, 2pm </strong>and <strong>April 27, 1pm</strong>, I will be at the Didax booth (#253 in the exhibit hall, NCTM National Meeting) for a 15-20 minute introduction to the <a href="http://www.mathedpage.org/manipulatives/lab-gear.html">Lab Gear</a>. Participants will get a free sample! </p><p><strong>Washington, DC April 27 8:00am</strong>: I will present “<strong>Quadratic Equations and Functions Use Manipulatives and Technology for More Access and More Depth</strong>” at the NCTM National Meeting. (Convention Center 209 ABC) Algebra manipulatives provide an environment where students can make sense of two ways to solve quadratic equations: factoring and completing the square. Graphing technology allows students to link those approaches to quadratic functions. Using these tools and connecting these concepts makes the algebra come to life for all students. [I will give away manipulatives to the first 72 attendees.] <em>If you’re an experienced Lab Gear user, I would love it if you would assist me during that session. Get in touch!</em></p></blockquote>And from the comfort of your own home: <br /><blockquote class="tr_bq"><p><strong>Anywhere, May 17, 5:00pm Pacific Time: </strong><strong>Reaching the Full Range</strong>, a webinar. As everyone knows, students learn math at different rates. What should we do about it? I propose a two-prong strategy based on alliance with the strongest students, and support for the weakest. On the one hand, relatively easy-to-implement ways to insure constant forward motion and eternal review. On the other hand, a tool-based pedagogy (using manipulatives and technology) that supports multiple representations, and increases both access and challenge. [Save the date. I'll announce the specifics when I have them.]</p></blockquote>Whether you attend these events or not, you can find handouts and links on <a href="https://www.mathedpage.org/talks.html" target="_blank" rel="noopener">my Talks</a> page.</div><h3 class="center ">Summer Workshops</h3><dl class="as-block text"><dt>I'll be presenting two summer workshops for teachers, at Menlo School in Silicon Valley:</dt><dd><strong>No Limits!</strong> (Algebra 2, Trig, and Precalculus, with Rachel Chou, Aug 1-3), and</dd><dd><strong>Visual Algebra</strong> (grades 7-11, Aug 6-9.)</dd><dt>For more information about the workshops, visit <a href="https://www.mathedpage.org/summer/index.html">my Web site</a>.</dt></dl><dl class="as-block text"><dt><em>Info about registration and logistics</em>: <a href="https://www.menloschool.org/academics/summer-teacher-workshops.php">Menlo School</a>.</dt><dd><em>Early bird discount if you register by April 9.</em></dd></dl><p>--Henri</p>Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com0tag:blogger.com,1999:blog-3784276984960421233.post-5101844136093791452018-02-23T17:34:00.001-08:002018-02-25T06:58:05.891-08:00Vocabulary<p style="font-size: 16px;">In my last post, I offered <a href="http://blog.mathedpage.org/2018/02/sequencing.html">guidelines for sequencing</a> math curriculum. The response I got on Twitter (and in one comment to the post) was quite positive. However, one point I made triggered some disagreement:</p><blockquote><p style="font-size: 16px;"><strong>Start with definitions?</strong> No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about.</p></blockquote><p style="font-size: 16px;"><strong>Michael Pershan wrote</strong>:</p><blockquote><p style="font-size: 16px;">I also don't want to introduce vocab before kids are ready for it, and there are often times when I introduce vocab when it comes up in class. When that happens, I like to launch the next class with that vocab and give kids a chance to use it during that next lesson.</p><p style="font-size: 16px;">I don't disagree with what @hpicciotto says in this post either. I square the two by saying vocab probably shouldn't be the start of the unit, but it works nicely for me when it's the start of a lesson.</p><p style="font-size: 15px;">My heuristic is something like, I want the definition to be easy to understand, so I give the definition when it won't be hard to comprehend. And that often (not always) requires some prior instruction. But I do like introducing vocab at the start of a lesson.</p></blockquote><p style="font-size: 16px;">I don’t disagree with Michael. None of the guidelines I gave in that post should be interpreted as rules one cannot deviate from. The essence of what I was trying to get across is to avoid definitions if students cannot understand them. Michael’s system does not violate that principle. Starting a particular lesson with a definition, if students are ready for it, is not a problem at all.</p><p style="font-size: 16px;">As always when thinking about teaching, beware of dogma! Be <a href="http://blog.mathedpage.org/2016/08/eclectic.html">eclectic, </a>because n<a href="https://www.mathedpage.org/teaching/nothing.html">othing works</a>, not even what I say on this blog<a href="http://blog.mathedpage.org/2016/08/eclectic.html">!</a></p><p style="font-size: 16px;"><strong>Mike Lawler gave a specific example</strong>:</p><blockquote><p style="font-size: 16px;">Teaching via definitions: I found it useful to introduce a formal definition of division to help my younger son understand fraction division initially (this video is from 4 years ago) : </p><p style="font-size: 16px;"> https://www.youtube.com/watch?v=dC409YJ60mc</p><p style="font-size: 16px;">Then a few days later we did a more informal approach with snap cubes. Overall I thought this formal to informal approach was useful and helped him see fraction division in a few different ways:</p><p style="font-size: 16px;"> https://www.youtube.com/watch?v=o33WPlC5Blw</p></blockquote><p style="font-size: 16px;">You should definitely watch the videos. They provide a great example of starting with a definition, that worked. But let’s analyze this. The student, in this case, was indeed ready for a definition:</p><ul><li><span style="font-size: 16px;">H</span><span style="font-size: 16px;">e already knew that fractions are (or represent?) numbers. </span></li><li><span style="font-size: 16px;">He already knew what a reciprocal is (not merely “flipping” the fraction, but the number by which you multiply to get 1, and from there he got to flipping)</span></li><li><span style="font-size: 16px;">He already knew, that a division can be represented by a fraction.</span></li></ul><p><span style="font-size: 16px;">Many students who are told to “invert and multiply” know none of this, and for them defining division this way would not carry a lot of meaning. Moreover, students at this level usually have an idea of what division means, and <em>it would be important to show that multiplying by the reciprocal is consistent with the meaning they already have in mind.</em></span></p><p><span style="font-size: 16px;">So my approach might be to start with something students know (for example, “a divided by </span><span style="font-size: 16px;">b” can be said “b times what equals a?) and from there find a way to get to “multiply by the reciprocal”. (That is my approach in <a href="https://www.mathedpage.org/early-math/fractions/index.html">this document</a>.) However it is not easy to do in this case, and defining first, and then getting to a familiar meaning may well be preferable. </span><span style="font-size: 16px;"> </span><span style="font-size: 16px;">In fact, that is very much the approach I use when defining </span><a style="font-size: 16px;" href="https://www.mathedpage.org/alg-2/complex/index.html">complex number</a><span style="font-size: 16px;"> multiplication in high school. </span></p><p><span style="font-size: 16px;">To conclude: yes, a lesson can start with a definition, as long as the students know what you’re talking about, and will not instantly turn off. This does not invalidate my point. To return to the example I gave in my last post, in a bit more detail, compare these two approaches to introducing the tangent ratio.</span></p><p><strong><span style="font-size: 16px;">Standard</span></strong><span style="font-size: 16px;"><strong> approach</strong>: “Today we’re starting trigonometry. Please take notes. In a right triangle, the ratio of the side opposite the angle to the side adjacent to the angle is called the tangent. (etc.)” This approach is likely to lead to eyes glazing over, to some anxiety induced by the word “trigonometry”, and to a worry about remembering which ratio is which. The latter is allayed by the strange incantation “soh-cah-toa”, but alas that does not throw much light on the topic.</span></p><p><strong><span style="font-size: 16px;">My suggested approach:</span></strong><span style="font-size: 16px;"> “As you know, f</span><span style="font-size: 16px;">or every angle a line makes with the x-axis, there is a slope. </span><span style="font-size: 16px;">For a given slope, there is </span><span style="font-size: 16px;">an angle with the x-axis. [More than one, but no need to dwell on that right now.] We’ll use this idea, a ruler, </span><span style="font-size: 16px;">and </span><a style="font-size: 16px;" href="https://www.mathedpage.org/circle/index.html">the 10cm circle</a><span style="font-size: 16px;"> to solve some real-world problems.” This allows students to right away put the tangent ratio to use, without knowing its name</span><span style="font-size: 16px;">. (See <em>Geometry Labs, </em>chapter 11. Do Lab 11.2 after introducing the 10cm circle, but before making tables as suggested in Lab 11.1. <a href="http://www.mathedpage.org/geometry-labs/">Free download</a>.) Once they’re comfortable with the concept, you can tell them there’s a word for this, a notation, and a key on the calculator. And yes, at that point, you can do all that at the beginning of a lesson. And a</span><span style="font-size: 16px;"> few months later, you can introduce the sine and cosine in a similar way.</span></p><p><span style="font-size: 16px;">-- Henri</span></p><p><span style="font-size: 16px;">PS: How to introduce trig, complex numbers, and matrices in Algebra 2 and Precalculus are among the topics Rachel Chou and I will address in our workshop </span><strong style="font-size: 16px;">No Limits!</strong><span style="font-size: 16px;"> this summer. </span><a style="font-size: 16px;" href="http://www.mathedpage.org/summer">More info</a><span style="font-size: 16px;">.</span></p>Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com0tag:blogger.com,1999:blog-3784276984960421233.post-8167653152394848482018-02-15T16:24:00.001-08:002018-02-16T20:20:07.931-08:00Sequencing<p><span style="font-family: Georgia; font-size: 14px;">In <a href="http://blog.mathedpage.org/2018/02/mind-maps.html">my last post</a>, I argued that, as teachers and math education leaders in a school or district, we need to free ourselves from the sequencing preordained by the textbook, and instead pay attention to what actually works with our students. In this post, I will present some general guidelines for sequencing topics, and some specific suggestions. All these ideas are based on 3+ decades in the high school math classroom, with somewhat heterogeneous classes.</span></p><p><span style="font-family: Georgia; font-size: 14px;"><strong>Is the topic age-appropriate?</strong> Mysteriously, not much attention is paid to this. For example, tradition requires completing the square and the quadratic formula as topics for Algebra 1, In my experience, it is much easier to teach this in Algebra 2, to students who have a little more maturity. This in turn frees up precious time in Algebra 1 to reinforce the sort of basics that we find so frustrating when they are missing later on.</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Can the topic fit in a single</span></strong><span style="font-family: Georgia; font-size: 14px;"><strong> unit?</strong> Some topics are important, but difficult for some students. It’s a good idea to spread those out over more than one unit, and sometimes more than one course. One example of that is linear, quadratic, and exponential functions, which can be approached in different ways at different levels, from Algebra 1 to Precalculus. Another example is trigonometry, which can be distributed among Geometry, Algebra 2, and Precalculus.</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">When in the school</span></strong><span style="font-family: Georgia; font-size: 14px;"> <strong>year? </strong>Difficult and important topics should not be introduced in May! By then both students and teachers are tired, and there is little chance of success. Teach those topics as early as possible. There may also be some traditionally late topics that can be useful early on. For example, an exploration of inscribed angles has only very few prerequisites, and can be used to practice angle basics in an interesting context. I did this very early in my Geometry class. See my <em>Geometry Labs </em>(<a href="http://www.mathedpage.org/geometry-labs/">free download</a>.)</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Spiraling? </span></strong><span style="font-family: Georgia; font-size: 14px;">There is much to be said for coming back to already-seen topics later in the school year. However spiraling can be overdone, and result in an atomized curriculum consisting of chunks that are so small that students don’t get enough depth on each topic. One easy-to-implement policy, which provides the advantages of both spiraling and depth is to separate related topics. I have written about that in past blog posts (<a href="http://blog.mathedpage.org/2013/08/separate-related-topics.html">here</a> and <a href="http://blog.mathedpage.org/2013/08/extending-exposure.html">here</a>).</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Review?</span></strong><span style="font-family: Georgia; font-size: 14px;"> It is widely believed that one should start the year, and then each unit and even each lesson with review of relevant past material. Of course, I understand why that is a standard practice, but I believe it is counterproductive. Over time, it tells students “you don’t need to remember anything — I’ll make sure to remind you.” It is also profoundly boring for the students who don’t need the review, and takes away from the excitement of starting something new. It is far better to start with a solid <em>anchor </em>activity, and use homework, subsequent class work, and if need be out-of-class support structures to do the review. It is especially catastrophic to start Algebra 1 with a review of arithmetic. The students who need it won’t get it, and those who don’t will be disappointed.</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Anchor activity?</span></strong><span style="font-family: Georgia; font-size: 14px;"> If you can, start a unit with an interesting problem or activity (the anchor). It should be motivating and memorable, and it need not be easy. Examples of anchor activities are Geoboard Squares for the Pythagorean theorem (Lab 8.5 in <a href="http://www.mathedpage.org/geometry-labs/"><em>Geometry Labs</em></a>,) <a href="https://www.mathedpage.org/alg-2/exponential/index.html">Rolling Dice</a> for exponential functions, or <a href="https://www.mathedpage.org/calculator/index.html#super">Super-Scientific Notation</a> for logarithms. A good anchor brings together key content with good practices, and generates curiosity and engagement. It is something you can refer to later on, to remind students of the basics of the unit.</span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Start with definitions?</span></strong><span style="font-family: Georgia; font-size: 14px;"> No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about. For example, see the Super-Scientific Notation lesson mentioned above. Another example: the tangent ratio can be introduced with the help of slope, without having to mention trigonometry or the calculator, instead using <a href="https://www.mathedpage.org/circle/index.html">the 10-centimeter circle</a>. Once students can use the concept to solve problems, you can name it and reveal that there is a key on the calculator for it. </span></p><p><strong><span style="font-family: Georgia; font-size: 14px;">Concrete or abstract?</span></strong><span style="font-family: Georgia; font-size: 14px;"> Math is all about abstraction, but understanding is usually rooted in the concrete, so it is usually a good idea to start there. This can mean many things:</span></p><ul><li><p><span style="font-family: Georgia; font-size: 14px;"><em>Discrete first, continuous later. Numerical examples first, generalization later</em>. For example, work on the <a href="http://www.mathedpage.org/geoboard">geoboard</a> (both the standard 11 by 11 geoboard, and the circle geoboard) is strictly with specific examples based on the available pegs. But it lays the groundwork for a generalization using variables which would otherwise be impenetrable to many students.</span></p></li><li><p><span style="font-family: Georgia; font-size: 14px;"><em>Natural numbers to real numbers</em> -- almost any new idea is more accessible if you start with whole number examples — as in <img title="NewImage.png" src="https://lh3.googleusercontent.com/-hZRInNgLdHc/WoYkwSallCI/AAAAAAAAA6Y/VUc56aZ9QE4CwggvrG9WNOoKuqQlqZSWACHMYCw/NewImage.png?imgmax=1600" alt="NewImage" width="70" height="20" border="0" />.</span></p></li><li><p><em><span style="font-family: Georgia; font-size: 14px;">Kinesthetics</span></em><span style="font-family: Georgia; font-size: 14px;"> (<a href="http://www.mathedpage.org/kinesthetics/index.html">link</a>) and <em>manipulatives </em>(<a href="https://www.mathedpage.org/manipulatives">link</a>) do not accomplish miracles, but they can improve classroom discourse and provide meaningful and memorable reference points. In particular, <a href="http://www.mathedpage.org/manipulatives/lab-gear.html">algebra manipulatives</a> can provide both access and depth to an essentially abstract subject, by way of a visual / geometric interpretation.</span></p></li><li><p><span style="font-family: Georgia; font-size: 14px;"><em>Tables and graphs</em> can help provide a concrete foundation to the study of functions. This is sometimes described as modeling: you start with a concrete situation, use tables and graphs to think about it, and generalize with equations. This is the approach I use a lot in <a href="http://www.mathedpage.org/attc/"><em>Algebra: Themes, Tools, Concepts</em></a> and in my <a href="https://www.mathedpage.org/alg-2/index.html">Algebra 2</a> materials.</span></p></li></ul><p><span style="font-family: Georgia; font-size: 14px;"><strong>From easy to hard? </strong>Well, that is certainly implied in the previous segment. However, I will now challenge that assumption. (What can I say, sequencing curriculum doesn’t lend itself to simple choices.) In my view, it is a good idea to start with somewhat challenging material, then ease up, and keep alternating between hard and easy. Starting too easy can give the wrong impression, that the unit will not require work. In fact, most of the above guidelines are best implemented as a back and forth motion: for example, after introducing vocabulary and notation, one needs to re-introduce the concepts. Likewise for most of these guidelines.<br /></span></p><p><span style="font-family: Georgia; font-size: 14px;"><strong>This is all fine, but how does one deal with externally mandated sequencing?</strong> Alas, I have no experience with this, as most of my career was at a small private school, and moreover I chaired my department (with plenty of input from my colleagues.) I can only suggest discussing these ideas with colleagues and supervisors! Also, most of the suggestions in this post address sequencing within a unit, and thus may be implemented anywhere if there is any wiggle room at all in the mandated sequence.</span><span style="font-family: Georgia; font-size: 14px;"><br /></span></p><p><span style="font-family: Georgia; font-size: 14px;">-- Henri</span></p><p><span style="font-family: Georgia; font-size: 14px;">PS: I’m offering two <strong>summer workshops</strong> (one on algebra, grades 7-11, and one with Rachel Chou on Algebra 2 / Precalculus. The workshops will include many of the bits of curriculum I linked to in this post. <a href="http://www.mathedpage.org/summer">More info</a>.</span></p><p><span style="font-family: Georgia; font-size: 14px;"><strong>Notes</strong>: </span></p><ul><li><p><span style="font-family: Georgia; font-size: 14px;">Much of this post is based on one section of my article with the cheerful title </span><a style="font-family: Georgia;" href="https://www.mathedpage.org/teaching/nothing.html">Nothing Works</a><span style="font-family: Georgia; font-size: 14px;">.</span></p></li><li><p><span style="font-family: Georgia; font-size: 14px;">Some of it is from a previous post on sequencing (<a href="http://blog.mathedpage.org/2015/05/mapping-out-course.html">Mapping Out a Course</a>), where I propose a step by step process for doing just that.</span></p></li></ul>Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com2