tag:blogger.com,1999:blog-37842769849604212332018-10-16T01:21:37.698-07:00My Math Education Blog"There is no one way"Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.comBlogger34413MathEdBloghttps://feedburner.google.comtag:blogger.com,1999:blog-3784276984960421233.post-72118988363718489832018-10-15T14:41:00.000-07:002018-10-15T14:41:32.272-07:00Spiraling Out of Control?In most math curricula, students work on a single topic at a time. (When I taught elementary school, decades ago, I noticed that if we’re working on subtraction, it must be November! But the same applies at all grade levels.) The idea is that is that by really focusing on the topic, you are helping students really learn it, before you move on to the next unit. Unfortunately, that is not how retention happens. It is much more effective, when learning a new concept, to see it again a few weeks later, and again some time after that. Thus the concept of spiraling. Years ago, the Saxon books distributed homework on any one topic across the year, typically with one or two exercises per topic on any given day. Some more recent curricula do facilitate that sort of homework spiraling by including review homework in addition to homework on the current topic after each lesson. The <a href="http://www.mathedpage.org/attc/">algebra textbook</a> I coauthored in the 1990’s is spiraled throughout: not just in the homework, but in the makeup of each chapter and many lessons. This idea was so important to us, that there is an image of a spiral at the start of each chapter! (If you have the book, check that out! Or just look at it <a href="https://www.mathedpage.org/attc/attc-color.html">online</a>.)<br /> <br /> In this post, I want to argue that while I agree with the fundamental underlying idea of a spiraled curriculum, there is such a thing as overdoing the spiral. I will end with specific recommendations for better spiraling.<br /><h3>Impact on Learning </h3>Too much spiraling can lead to atomized, shallow learning. If there is too much jumping around between topics in a given week, or in a given homework assignment, it is difficult to get into any of the topics in depth. Extreme spiraling makes more sense in a shallow curriculum that prioritizes remembering micro-techniques. In a program that prioritizes <a href="https://blog.mathedpage.org/2018/09/understanding.html">understanding</a>, you need to dedicate a substantial amount of time to the most important topics. This means approaching them in multiple representations, using various learning tools, and applying them in different contexts. This cannot be done if one is constantly switching among multiple topics.<br /> <br /> In particular, in homework or class work, it is often useful to assign nonrandom sets of exercises, which are related, and build on each other. For example, “Find the distance from (<i>p, q</i>) to (0, 0) where <i>p</i> and <i>q</i> are whole numbers between 0 and 10.” (This assignment is taken from my <i><a href="http://www.mathedpage.org/geometry-labs/">Geometry Labs</a>.</i>) At first sight, this is unreasonable: there are 121 such points. But as students work on this and enter their answers on a grid, they start seeing that symmetry cuts that number way down. In fact, the distances for points that lie on the same line through the origin can easily be obtained as they are all multiples of the same number. (For example, on the 45° line, they’re all multiples of the square root of two.) Nonrandom sets of problems can deepen understanding, but they are not possible in an overly spiraled homework system.<br /><h3>Impact on Teaching</h3>The main problem with hyper-spiraling is the above-described impact on learning. But do not underestimate its impact on the teacher. For example, some spiraling advocates suggest homework schemes such as “half the exercises on today’s material, one quarter on last week, one quarter on basics.” Frankly, it is not fair to make such demands on already-overworked teachers. Complicated schemes along these lines take too much time and energy to implement, and must be re-invented every time one makes a change in textbook or sequencing. Those sorts of systems are likely to be abandoned after a while, except by teachers who do not value sleep.<br /><div style="text-align: center;"></div><div style="text-align: center;"><img alt="NewImage" border="0" height="154" src="https://lh3.googleusercontent.com/-7obrpWJi4cE/W8T1QJA45bI/AAAAAAAAA_I/sheLBpfgah8zT1HwBDKI9h5rkMxfNEX8gCHMYCw/s0/NewImage.png" title="NewImage.png" width="199" /></div><div style="text-align: left;"></div><div style="text-align: left;">Another problem for teachers is that it makes using a hyper-spiraled curriculum difficult to use, because it is difficult to find where a given concept or technique is taught. (In the case of <i>Algebra: Themes, Tools, Concepts</i> we tried to compensate for that by offering an <a href="https://www.mathedpage.org/attc/tg/topics-tools-index.pdf">Index of Selected Topics and Tools</a>. We also included notes in the margin of the Teachers’ Edition: “What this Lesson is About”. But even with all that, a hyper-spiraled approach makes extreme and unrealistic demands on teachers’ planning time. In fact, some hyper-spiraled curricula lack even those organizational features. Without them, a teacher needs to spend the whole summer working through the curriculum in order to be ready to teach it. This can be fun if the curriculum is well designed (e.g. the Exeter curriculum), but no one should feel guilty if they’re not up to that level of workaholism. </div><h3 style="text-align: left;">Spiraling Made Easy and Effective</h3>So, you ask, what do I suggest? In the decades following the publication of my overly-spiraled book, I developed an approach to spiraling that:<br /><ul><li>is unit-based, and allows for going in depth into each topic</li><li>is easy to implement and does not make unrealistic demands on the teacher</li><li>is transparent and does not hide what lessons are about (most of the time)</li></ul>I have written a fair amount about this, under the heading <a href="https://blog.mathedpage.org/2013/08/extending-exposure.html"><i>extending exposure</i></a>. The ingredients of this teacher-friendly approach are:<br /><ul><li><a href="https://blog.mathedpage.org/2013/06/lagging-homework.html">Lagging homework</a> and <a href="https://blog.mathedpage.org/2015/12/once-again-heterogeneous-classes.html">assessments</a></li><li><a href="https://blog.mathedpage.org/2013/08/separate-related-topics.html">Separating related topics</a></li><li><a href="https://blog.mathedpage.org/2018/10/more-on-extending-exposure.html">Teaching two units</a> at any one time (just two!)</li></ul>Implementing these policies does not require more prep time, or more classroom time, and it creates a non-artificial, organic way to implement “<a href="https://blog.mathedpage.org/2018/08/catchphrases.html">constant forward motion, eternal review</a>”. It helps all students with the benefits of spiraling, but without the possible disadvantages. You really should try it!<br /> <br /> -- HenriHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com0tag:blogger.com,1999:blog-3784276984960421233.post-63778616270323780892018-10-07T11:10:00.000-07:002018-10-07T11:20:31.928-07:00More on Extending ExposureI have written several posts in which I argued that <i>extending student exposure</i> to mathematical concepts is one key to reaching the whole range of students. This is based on the simple observation that students learn math at different rates, and that <a href="https://blog.mathedpage.org/2013/08/extending-exposure.html">extending exposure</a> by making simple changes to our routines can benefit all students: those who pick up new ideas quickly, and those who need more time. If schools heed NCTM’s recommendation to eliminate tracking, making these simple changes becomes even more important.<br /><br />On this blog, the most popular post on extending exposure is <a href="https://blog.mathedpage.org/2013/06/lagging-homework.html">Lagging Homework</a>, and it links to other posts where I describe additional strategies (<a href="https://blog.mathedpage.org/2013/08/separate-related-topics.html">separating related topics</a>, lagging assessments, and more.) If you haven’t read those posts, you should. The problem is, it does involve a bit of clicking around. Thus, I decided to combine all that information in a single longer article on my Web site.<br /><br />However, before I do that, I need to share a few more related tidbits which hadn’t made it into those posts. Writing about those here will help when I’m ready to put it all together.<br /><h4>Homework as Preparation</h4>A number of people, over the years, have told me that they don’t agree with my lagging homework system because they like to assign homework that prepares the students for the next day’s lesson. That, my friends, is not a disagreement! I love that idea.<br /><br />Lagging homework is not a rigid system that requires homework to be assigned exactly one week (or day, or month) after the corresponding class work. My main point is that on most days, you should not assign homework based on the day’s lesson. A week’s delay, more or less, provides many advantages which I described in my original post, the main one being extended exposure to each topic. This in no way precludes homework that sets up the next day’s lesson, as long as it is not (usually) based on the day’s lesson. I described the characteristics of such preparatory problems in <a href="https://blog.mathedpage.org/2014/12/asilomar-report.html">this post</a>, based on Scott Farrand’s approach to in-class warm-ups.<br /><br />In general, these activities (whether assigned as homework, or as warm-ups) are essentially long-lagged work, based on ideas that were introduced the previous semester, or the previous year, or whenever. Such long lags can also be used for review (better than taking precious class time for that). On the other hand, if preparing for the next day’s lesson requires completing homework about today’s lesson, and this needs to happen frequently, then I strongly discourage that as the collateral damage on some of your students is substantial. (Unfortunately, the students who suffer from this policy will get the blame: they didn’t work hard enough, they’re not abstract thinkers, they don’t belong in this class, and so on.)<br /><h4>Two Units</h4>I had the privilege of teaching in <a href="https://blog.mathedpage.org/search?q=long+period">long periods</a> for my whole career. One way I used what is sometimes known as a block schedule was to focus on two units at any one time. For example, here is the outline of semester 2 in a “Math 2" class I taught before my retirement:<br /><div style="text-align: center;"><img alt="NewImage" border="0" height="79" src="https://lh3.googleusercontent.com/-5y8ud5zdS6w/W7eUhsUDBHI/AAAAAAAAA-c/QCpucK-f0l4iKGxVyQ4a51cpjaP_7V3rQCHMYCw/s0/NewImage.png" title="NewImage.png" width="596" /><br />(My point is not to recommend this exact sequence, which depends on many <br />department-specific assumptions, but to use it as an example of what is possible.)</div><br />Here are some of the advantages of that approach.<br /><ul><li>Any one day or week is more varied, which is helpful in keeping students interested and alert. Note in particular that we tried to match topics that are as unlike as possible.</li><li>It takes roughly twice as many days to complete a unit. This is good for students who need that extra time.</li><li>This takes nothing from students who pick up ideas quickly. In fact, they appreciate the variety.</li><li>It makes it easier to balance challenging and accessible work: if you hit a difficult patch in one topic, you can ease up on the other one. More generally, you gain a lot of flexibility in your lesson planning.</li><li>If your work on one topic hits a snag, you can emphasize the other topic while figuring out what to do.</li><li>Perhaps most importantly, it carries a message to the students: you still need to know this when we’re working on something else.</li></ul>In the long period, it is possible to hit both units in every class period, for example by introducing new ideas on one topic during the longer part of the period, and applying already-introduced ideas on the other topic in the remaining time. (Homework is typically on one or the other topic, not both.) Can this approach, or a version of it, be used in traditional 50-minute classes? I don’t know. I am guessing that the answer is yes, but I have not tried it. It might involve, for example, focusing on each topic on alternate days, while the homework is on the other topic .<br /><br />But, you ask, is this not confusing to students? Don’t they prefer focusing on one single topic? If they do, that is only because that is what they’re used to. In teaching, the biggest obstacles to making changes are the cultural ones: the expectations of students, colleagues, parents. administrators, and of course one’s own deep-seated habits, . The only way to tackle these obstacles is departmental collaboration, and a step-by-step approach: don’t make all the changes at once! At my school, working on two units, separating related units, lagging homework, all those were department-wide policies. Once they’re used to it, students do not question any of them.<br /><h4>Coming Soon</h4>My next post will hopefully be the last in this series, and it may surprise you: <b>Beware of Over-Spiraling!</b> Once that is written, I’ll be ready to put all the pieces together in one article on teaching heterogeneous classes (in other words, any classes!)<br /><br />-- HenriHenri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com3tag:blogger.com,1999:blog-3784276984960421233.post-62587740567795601382018-09-23T20:05:00.001-07:002018-09-25T14:25:04.659-07:00Understanding "understanding"<p>It is not uncommon to read articles about math education in the mainstream press, arguing that students must master basic skills <em>before</em> they can develop conceptual understanding. And moreover, that the road to such mastery is teacher explanation followed by repetitive drill. These essays frequently argue that it’s like learning to play the piano: you must practice scales before playing real music! When I mentioned this to a friend who is a piano teacher, he considered it to be an insult to his profession. He said that obviously these people are not piano teachers! Teaching piano is about music! Yes, students do need exercises, but if that’s all you had them do, you’d drive them away from music. The biggest motivator is the recital, when they play real music, not scales! </p><p>My friend is right: the authors of these op eds are not piano teachers, but they’re often not math educators either! Still, it is important to address their ideas, because they reflect a broad cultural consensus among many parents, administrators, students, and teachers. Some proponents of the “skills first” approach equate teaching for understanding to what they call “fuzzy math”, a flaky anything-goes sort of teaching, with no specific learning goals, no accountability, just feel-good teachers who allow students to wallow in their ignorance.</p><p>It behooves those of us who disagree with this caricature to clarify what we mean by understanding. That is the main purpose of this post.</p><p>To get a straw man out of the way, my position is that understanding cannot be divorced from the acquisition of skills. Without understanding, it is hard to develop an interest in the skills, or to retain them; without skills, understanding is out of reach. Good teaching requires a skillful weaving of those two strands. It is, in fact, like learning to play the piano! Skills are important, but it’s all about the music.</p><p>But on to the main point: what is understanding? This is a difficult question, and the true fact that experienced math teachers can "recognize it when they see it" is not a sufficient answer. Here is an attempt at spelling it out. <strong>A student who understands a concept can:</strong></p><ul><li><strong>Explain it.</strong> For example, can they give a reason why 2(x+3) = 2x+6? Responding “it’s the distributive rule” is evidence that the student knows the name of the rule, but a better explanation might include numerical examples, or a figure using the area model, or a manipulative or visual representation. Therefore, we should routinely ask students to explain answers, verbally or in writing, even though many don't enjoy doing that. It is a way for us to gauge their understanding, and thus improve our teaching, and more importantly, it is a way for them to go deeper and guarantee the ideas stick.</li><li><strong>Reverse processes associated with it.</strong> For example a student does not fully understand the distributive law if they cannot factor anything. More examples: can they create an equation whose multi-step solution is 4? Can they figure out an equation when given its graph? And so on. Reversibility is a both a test of understanding, and a way to improve understanding. </li><li><strong>Flexibly use alternative approaches. </strong>For example, for equation solving, in addition to the usual "do the same thing to both sides" for solving linear equations, students should be able to use the <a href="http://www.mathedpage.org/attc/cover-up.pdf">cover-up method</a>, trial and error, graphs, tables, and technology. If they have this flexibility, they can decide on the best approach to solve a given equation, and moreover, they will have a better understanding of what equation solving actually is.</li><li><strong>Navigate between multiple representations of it</strong>. Famously, functions can be represented symbolically, or in tables, or in graphs. Making the connections between these three is a hallmark of understanding. I have found that a fourth representation (<a href="http://www.mathedpage.org/func-diag">function diagrams</a>) can also help deepen understanding, and be used to assess it. Multiple representations on the one hand offer different entry points that emphasize different aspects of functions, but making the connections between the representations is part and parcel of a deeper understanding. </li><li><strong>Transfer it to different contexts</strong>. For example, ideas about equivalent fractions are relevant in many contexts, such as similar figures and direct variation. Or, the Pythagorean theorem can be used to find the distance between two points, given their coordinates. If a student can only handle a concept in the form it was originally presented in class or in the textbook, then surely no one would claim they fully understand it. </li><li><strong>Know when it does not apply</strong>. When faced with an unfamiliar problem, students will tend to reach for familiar concepts, such as linear functions and proportional relationships. Sometimes, this makes sense, of course, but students need to be able to recognize situations where a given concept does <em>not</em> apply. </li></ul><p>Clearly aiming for all this is a high bar, and it is tempting to just have students memorize some facts and techniques, and then test them to see if they remember those a few weeks later. (This is often how the “skills first” approach plays out.) But what good would that do? It would just add to the vast numbers who got A’s and B’s in secondary school math, went to college, and now tell us “they’re not math people”. Yes, teaching for understanding is ambitious, and it must be our goal for all the students. </p><p>Alas, there are obstacles. For one thing, understanding cannot be easily conferred by explanations. (A naive traditionalist once suggested that it was easy to teach about variables: patiently explain to the students that variables behave just like numbers! Would that it were that easy.) Moreover, understanding is not always valued by students, parents, and administrators, many of whom believe that everything would be so much more straightforward if we could just have the students memorize facts and algorithms. Finally, understanding is difficult to assess. (Actually, the list above is one way to improve assessments: each item on the list suggests possible avenues for authentic assessment. To reduce complaints that it’s "not fair" to assess students that way, such assessments can be ungraded. As the current jargon would have it: consider them formative assessments.) Being able to reproduce a memorized set of steps is a good test of memory and obedience. To test understanding, non-rote assessments are the most revealing.</p><p>The above list is also a tool in <a href="https://www.mathedpage.org/teaching/assessment/index.html#forward">forward design</a>. When planning a unit, ask yourself how you can incorporate reverse questions, alternate approaches, multiple representations, varied contexts, and so on. A <a href="http://www.mathedpage.org/tools/tools.html">tool-rich pedagogy</a> is helpful, as different manipulative, technological, and paper-pencil tools provide a way to do this and avoid boring repetition. Of course, the implication of such planning is that it takes more time to teach any given concept. Because of the enormous pressure of coverage at all costs, it is generally necessary to take less time on less important topics, and approach the most important topics in as many ways as possible. </p><p>In any case, I hope you find this post helpful in your teaching, and also in your conversations with colleagues, administrators, parents, and students.</p><p>Good luck as you teach for understanding!</p><p>-- Henri</p><p>[This post includes part of my <a href="https://www.mathedpage.org/teaching/nothing.html">Nothing Works</a> article, updated and expanded.]</p>Henri Picciottohttps://plus.google.com/107858350012538689018noreply@blogger.com10