tag:blogger.com,1999:blog-5378306332287859742024-03-13T11:40:27.038-05:00Angles of ReflectionThoughts about mathematics, teaching, and teaching mathematics.Unknownnoreply@blogger.comBlogger95125tag:blogger.com,1999:blog-537830633228785974.post-9913596568399845182015-03-12T18:55:00.003-05:002015-03-12T18:55:56.242-05:00What Happens in a Math Class?<!--[if gte mso 9]><xml>
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I think there are two dramatically different perceptions
about what is supposed to happen when students are doing mathematics, be it in
class or when they are doing their homework. I think this difference colors the
way mathematics learning and teaching is perceived and causes considerable
confusion when students, teachers and parents discuss progress. One point of
view is that the goal is to get the correct answer to the problems posed,
basically that the problem is a “test” to see if the solver can get the correct
answer. The other belief is that the problems have been posed so that the
solver will learn something by working on the problem, and that working on the
problem will make the solver better at doing math. </div>
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I was observing a first grade mathematics lesson last week
and students were learning about how to use “near doubles.”<span style="mso-spacerun: yes;"> </span>Essentially this was a formal lesson that
encouraged students to do problems like 13+14 by thinking that twice 13 is 26
so 13 +14 =27, or one more than 26. Or they could reason that twice 14 was 28
and so the sum would be one less than 28. As I observed, I thought about how
wonderful it was that these students were learning about the structure of
addition, as well as a handy method for finding sums that many adults use
regularly, rather than just memorizing a rule for adding two digit numbers.
This has to empower them as they learn more and more mathematics. </div>
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I walked around and observed that virtually all students got
the correct answers to the addition problem. I also observed that many were
doing it backwards, they were writing down the sum and then the doubles fact
that would help them get that sum. <span style="mso-spacerun: yes;"> </span>As in
the example above, the student solves 13+14 = 27 first, and then the student solves
it backwards, using 13+13 = 26 (a well-known doubles fact) and then 26+1 = 27.<span style="mso-spacerun: yes;"> </span>Some had the correct sum but a doubles fact
that did not make sense to me, or was not a doubles fact but some other fact of
addition. (For example, they might have 10+10=20 as their doubles fact. While
this can be used to determine the sum and uses doubles, it shows me that the
student has not grasped the concept of near doubles)</div>
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My take away from this observation is that often students
are not focused on anything other than getting the right answer, one way or
another, while the point of the lesson was to make the computation easier as
well as more meaningful by learning about the structure of addition instead of
just memorizing a rule. In fact, using doubles makes a boring addition problem
into an interesting challenge. <span style="mso-spacerun: yes;"> </span>It then
struck me that there is a dichotomy in the world of mathematics education that
has serious consequences throughout the learning experience of an individual. I
would like to think that the learner was always focused on what the point of
the lesson was, on what the creator of the lesson wanted the user of the lesson
to learn and understand at the end of the lesson. I suspect that the learner is
often not even considering that but is focused on answer getting as opposed to
finding a really great way to get to that answer. I think many parents also
have that same belief, even without realizing it. If their child gets right
answers on tests, they are happy. And what could be wrong with that?</div>
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For one thing, the student who only has one way to approach
many problems may find it boring, and may not realize that there are techniques
needed later that benefit greatly from thinking about a concept in many
different ways. An example in later mathematics that comes to mind is the many
different ways to create a graph of a line, or solve a system of equations. The
student who has mastered only one way is severely handicapped if the situation
does not fit comfortably into the solution method they have memorized. <span style="mso-spacerun: yes;"> </span>For another, the student who learns one way to
do it may find math to be dull and uninteresting, while the student who
explores many different ways of looking at a problem will, I believe, find
mathematics as an outlet for creativity and inventive thought. I believe that
students who enjoy what they are learning, understand that they have a certain
amount of control over how they proceed, and know that there is a utility to
what they learn, will work much harder and learn more than those who just do it
on order to get the right answer and therefore<span style="mso-spacerun: yes;">
</span>a good grade. </div>
<div class="MsoNormal">
This explains something that has troubled me for some time.
I frequently hear from parents that their child is not challenged by the
curriculum that is being offered in their school. When I have looked at the
curriculum I find that it is often very rich and full of many challenges.
Perhaps the reason for this disconnect is that when the student looks at<span style="mso-spacerun: yes;"> </span>the material, the only thing the student is
thinking about is how can I get this done and get a right answer, while I am
looking at how many different things a student can learn from the variety of
approaches taken. </div>
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The problem is, that if I am right about this, it does not
give me an immediate plan of action to correct it and we educators believe that
it is our job to correct things that are not working. Perhaps that is food for
another blog, but at the least I am interested in you comments about my thoughts.</div>
John Bensonhttp://www.blogger.com/profile/03424169848462972662noreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-78952907979442777482015-02-10T07:46:00.002-06:002015-02-10T07:46:49.463-06:00Thoughts on Formative AssessmentIt has been many months since I have written something here. I am no longer in the classroom observing and noticing and so am not sure I have things to contribute I haven't already written.<br />
<br />
Last week I observed some classes and I noticed something, something I think is worth sharing.<br />
This was a second grade class in a very good school, with a good teacher. The students were capable and attentive. They were learning addition strategies. The context was fascinating to me.<br />
<br />
Each student had a page with photographs of fish. They were excellent photographs and the page was welcoming to a reader. Next to each fish was an identifying letter, from A to K, the length of the fish in inches and the weight of the fish in pounds.<br />
<br />
The teacher posed the following question:"If fish E ate fish B, how much would fish E weigh?" Students thought for a moment and then the class decided that the appropriate strategy was to add the two weights. The teacher was diligent about units, so when a student said add 4, she admonished the student to say 4 what, until the student said 4 pounds. I really liked this aspect of the activity. It emphasized the importance of units as well as reminding students they were talking about weight.<br />
<br />
They then did another problem just like it. This was followed by the following question, " If fish F ate a fish and then weighed 64 lbs. what fish did fish F eat?" I was pleased with the direction of the discussion, and especially impressed when many of the students quickly answered the question correctly.<br />
<br />
The students then were asked to do several more similar problems, on their own and in groups.When they were done, they were asked to create their own question. I then walked around the room and noticed that the students were able to answer the questions, show their work, and create a new question. I decided to interject something a little different, so I asked one table the following question: " If fish K ate fish B, how long would fish K be?" I expected laughter. They did not laugh. Some of them added the weights. I asked about units and what question I had asked, and they agreed that the answer would be the sum of the lengths of the fish. I went to another table and had the same experience. Teacher then got the attention of the class, and I asked the question to everyone. No one got the correct answer to my question. I asked them if they had ever eaten Twizzlers? Many had. After some discussion we agreed that a Twizzler was about a foot long. I asked them if they then got a foot taller. A few said yes, most said no, but no one wanted to change their answer to the fish story.<br />
<br />
I will let you draw your conclusions from this experience, but I reiterate, these were able students who were being well taught. I conclude that this would happen in most first grade classrooms. I am not sure why but I find it a bit frightening and am not sure what we can do about it.I m curious to read comments from others. Please help me to understand.<br />
<br />John Bensonhttp://www.blogger.com/profile/03424169848462972662noreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-60546739430025364932014-06-23T11:05:00.001-05:002014-06-23T11:05:10.884-05:00Testing, TestingAs my ITRW friends know, I'm in my second "semester" of a graduate program, which means that I'm buying textbooks, frantically getting readings done ... and taking tests. Last week was a doozy: I had two weeks, and a maximum of two weeks, to pass the statewide online test certifying me as a "teacher-evaluator". What did I learn from this process?<br />
<br />
<ol>
<li>For starters, as high-stakes as this test is -- every principal, assistant principal, suburban department chair, etc. has to pass it OR LOSE THEIR JOB -- we had two tries. That is, you could take any portion of the test, fail it, and after 24 hours take it again. The cynical among us would say "Of course you get two tries, otherwise there wouldn't be anyone left to turn on the lights in schools." But the reality is this: when it's really important to get person X up to a particular level of competence in skill Y, we almost always give them multiple tries. How many residents are dismissed for missing the vein the first time? And how many times do <i>we</i> give students the opportunity to retake a test at no penalty?</li>
<li>Second, we were encouraged to do the training together; in fact, virtually everyone from the network chiefs on down told us to take the tests together. Most of the hardest test consisted of watching videos and arriving at performance ratings (along the eight components of Danielson's Dimensions 2 & 3) for the class, and watching with other people--and talking with them about what we saw--was incredibly valuable. I wound up doing the videos on my own (have you tried to schedule a six-hour testing window during the day, with other people, when you have kids?) but the times I did sample videos with other people I learned much more than studying on my own. How often do we provide structures and encouragement for kids to work together on their tasks--and assessments?</li>
<li>Third, we didn't <i>have </i>to study: once you opened the training modules, you could go straight to the assessments. You could also dip into any of the lessons in any order. So when you felt ready, you could go ahead and try the test. (Remember, you get two tries.) How often do we give students this option?</li>
</ol>
<div>
But the most important thing I learned from the experience wasn't about designing assessments, or even structuring instruction. My big takeaway was this: I was <i>terrified. </i>Even with everyone I know assuring me that I'd pass, even knowing that my score beyond pass/nopass was irrelevant, even knowing that my practice tests were all well above the mark, I found the experience of waiting to take the test and then actually getting going practically unbearable. </div>
<div>
<br /></div>
<div>
Now, we don't tell high school students that they'll lose their jobs if they don't pass Friday's math test. But we do tell them that they need straight A's to get into "top colleges" (and if we don't tell them this, they certainly hear it from everyone else around them), and so an A (or B+) is, for our top students, effectively the passing grade--a much higher threshold than I had to cross. And as adolescents, they're not as good at dealing with stress as most adults (or at least as the "optimal" adult). What I'm getting at is this: I left my office convinced that most of our students--and most of our strong students--find testing incredibly stressful. And while I may have "known" that before, I'm not sure I felt it. I'm not sure what to do about this, but I know that <i>I'll</i> think about testing very differently this year.</div>
<div>
<br /></div>
<div>
So maybe that's the pedagogical payoff for attending graduate school.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-61217304396028701602014-06-16T21:56:00.000-05:002014-06-16T21:56:07.462-05:00The Way, Way BackAs we wind up this year, I've been thinking about last summer's awesome movie <i><a href="http://www.imdb.com/title/tt1727388/" target="_blank">The Way, Way Back</a>,</i> and not just because it's about a fourteen-year-old on summer vacation. The movie starts with a long scene introducing Duncan (the aforementioned fourteen-year-old) as a kid who's just kind of, well, a shlump. His mom's boyfriend thinks so, and while we realize that this adult's criticism is way too harsh, you have to see his point: Duncan is sullen, introverted, and, apparently, completely nonspecial.<br />
<br />
Over the movie--and I won't spoil the whole plot for you--Duncan finds a second home at a local water park, where he takes on a new identity as "Pop 'n Lock", the park's indispensable factotum and MVP. We see him going back and forth between "Duncan," still-awkward fourteen-year-old, and "Pop 'n Lock", master of every detail and every need in the park. Although Duncan gains a little confidence at home, his mom (and his mom's boyfriend) don't get to see any of his alter ego until ... well, I won't give it away. It's good.<br />
<br />
The point is that I think we have a lot of students like Duncan, students who are mostly unremarkable, even substantially less than remarkable, at school, but who totally shine in some part of their life outside of school. This is the kid who is super-responsible and on-the-ball at her job at Starbucks; or who's a realistically-aspiring professional dancer; or who keeps track of two younger siblings and three cousins, does the grocery shopping, and puts food on the table every night while mom is at work. These kids are stars. But what we teachers see is the late homework assignment, the "C" quiz, the half-checked-in, half-checked-out stare at school. We don't see the stars.<br />
<br />
More often than not, these kids are not middle class, and not white. One of the privileges many (certainly not all) white, middle-class kids experience is not having these kinds of burdens, or if they have them, they often know that they're as much assets as deficits, and that the student's job is to alert me early about the constraints and plan with me to work around them. They send us emails like "Hey, Mr. K, I wanted to give you the heads up that we have rehearsals every afternoon and evening for the next two weeks, so I might wind up one or two assignments behind. Can you tell me what I should focus on the most if I have to do triage?"<br />
<br />
I think other students don't know that we care, and often don't think that we necessarily should care. Many of my minority students have an overt "no excuses" attitude, which is refreshing except for when there really are extenuating circumstances. Yes, if I'd known your sister was in the hospital all last week, I would have been happy to let you take Friday's quiz on Monday. But on the positive side, I don't think they know that we want to see them at their best--even if their best isn't what we see at school.<br />
<br />
I don't have an easy solution for this problem. I think it helps to tell kids what we look for and want to know about, and if we coach them on the kinds of things that might qualify as reasonable exceptions. (A long time ago I stopped writing "late assignments will not be accepted" on my assignment sheets, and started writing instead things like "under ordinary circumstances, I won't accept this assignment late," with details that described some truly extraordinary circumstances.) And I think it helps to ask kids about what's going on in their lives outside of school. But the most helpful thing may be to remind ourselves, when we see a kid who seems like a Duncan, that more often than we might think, there's probably a setting in which that kid is really a Pop'n'Lock.Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-80932956597635498792014-06-10T23:18:00.004-05:002014-06-10T23:21:20.878-05:00Cutoffs, and the last week of classesSo now that CPS is in all-out last-week-of-classes mode, I have a few gripes. They all relate to cutoffs.<br />
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Remember cutoffs? When you got a hole in the knees of your jeans, you'd just cut them off into shorts, and they'd look like this:<br />
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After a while, though, the ends would start to fray, and they'd look more like this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQYrvuTOWBBK_pwWyF5-hmWwZ_j2R-S5GDYT0ue2WzZd27EpcxFk8xHLujocUtiGDBDznsBx8ZO-kP0E9D-_uMAVijdP4lKeXulj3WIBmUyMsbJbxec9t6dSWfHpEjTWb2IosCtBxiC4G7/s1600/images+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQYrvuTOWBBK_pwWyF5-hmWwZ_j2R-S5GDYT0ue2WzZd27EpcxFk8xHLujocUtiGDBDznsBx8ZO-kP0E9D-_uMAVijdP4lKeXulj3WIBmUyMsbJbxec9t6dSWfHpEjTWb2IosCtBxiC4G7/s1600/images+2.jpg" height="200" width="160" /></a></div>
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So you'd trim up the shorts, and the cycle would begin again. Eventually they would look like this:</div>
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<img alt="516x387_ac.jpg (516×387)" src="http://d74bwl3dcueqd.cloudfront.net/images/guide/bcfbcefbeb3a451a8d9bc2b096ffce59/516x387_ac.jpg" height="150" width="200" /></div>
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OK, maybe not exactly, but you get my point.</div>
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The problem with the end of the year is that we have this urge to stop teaching just one or two days before the end--either because we're tired, or because we feel like the kids are tired, or because grades are already either submitted or essentially impossible to move much, so why bother giving another test, and why bother teaching when you're not going to give a test? There are a hundred rationalizations, all of them bad. Because then when there are three or four days left, you say "Oh, well, in a couple of days I wouldn't be teaching anyway, so ... " and the end of the year frays up a little more. And so it goes, until suddenly you're spending an entire WEEK (or TWO weeks!) of instruction in games, "free time," movies, semi-reasonable documentaries ... anything but bona fide teaching.</div>
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What's actually wrong with this? A few things:</div>
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<ul>
<li>Not teaching because there's no test (or tests are over) just teaches the kids that the test was the only reason you were teaching in the first place.</li>
<li>We all wish we had more classroom time. Well, we do: this week, a solid 250 minutes per class.</li>
<li>Just because the curriculum for the test is over and done with doesn't mean the subject is over and done with. What kind of teacher doesn't have a favorite poem, theorem, artist, historical event, ... that just doesn't fit into the regular curriculum? (In fact, for me, a life-changing moment was when, the last week of school, my Brit Lit teacher read us <i>The Love Song of J. Alfred Prufrock</i> aloud.) There's always more to teach--it's not like we're in danger of running out of poems, problems, or paintings.</li>
<li>Finally, how is it fair that we make kids go to school and then waste their time? My kids can play video games, watch movies, read books, doodle, and mess around with their friends just fine on their own--they don't need state-compelled school attendance to do any of these things. If the <u>law</u> is that kids are compelled to be in school this week, then <u>fairness</u> and <u>respect</u> for them as human beings demand that we make it worth their while.</li>
</ul>
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It's not so hard. This week, my students are finishing up a discussion about "shape space", where we measure the distance between different shapes to talk about fractal limits; learning about Markov chains and why regular Markov chains converge; and giving short presentations about explorations they did with fractional linear transformations, queuing theory, and hyperbolic geometry. They're advanced. In other classes, I've used the time to read a cool book (like <i>Arcadia--</i>thanks, Mary, for the idea!), or do interesting problems we never saw before, or think about mathematical puzzles. Or you could just get a head start on one or two interesting ideas that the kids will see the next year.</div>
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I know I'm spitting in the wind -- Emma told me yesterday that she'd "figured it out: when teachers don't want to teach, they just go on Netflix, put in a documentary, and voila--instant lesson!" That experience makes me sad. But if we all just held the line ... or the hem ... </div>
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<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-49166917327017986672014-06-01T23:16:00.002-05:002014-06-01T23:16:38.722-05:00Gender GamesTwo issues of gender and games that have come up in the last weeks ...<div>
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1. My oldest daughter reports some frustration that in one of her classes, the teacher divides them into teams every day to play games (no, this is not PE, so be kind) and that, every day, one of the iterations is boys versus girls. This way of forming teams seems like no big deal to most people, and to me that's a pretty big deal. It bothers my daughter because most of her friends are boys, but even more so because it reinforces our society's idea that there's a huge inherent difference between boys and girls. We wouldn't separate kids by short and tall, or straight hair versus curly hair -- so why does boys versus girls seem to "make sense" to so many people?</div>
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A radio show I listened to a long time ago made the point that even in addressing classes as "boys and girls" reinforces that preoccupation with gender roles. Don't believe it? Imagine replacing "boys and girls" with "white children and black children". If describing the kids by gender doesn't make a difference--as many people seem to think it doesn't -- then why does it feel so funny to use race? Only because when we separate boys and girls in this way, we are reminding ourselves that gender differences are crucial in ways that we're less comfortable reminding ourselves that race differences are crucial. (Although if we were honest and not all "it's the end of racism", we'd have to admit that race plays a huge role in kids' experiences in school. But that's a different blog post.)</div>
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The point is this: dividing kids up by gender except for activities that require gender separation (bathroom use, for example) is just lazy; it makes some kids uncomfortable, and should make all of us wonder why this difference is the one we keep reinforcing in our schools. Don't do it!</div>
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2. It's the end of the school year and my kids are handing in portfolios; the assignment includes a short essay on "How I got here", namely, into a hyper-advanced college-level post-calculus class while still in high school. This year's crop follows a trend I first noticed four years ago. Virtually every boy says something like "The first time I really loved doing math was when I was competing against [my best friend/my classmates/my archenemy/another school] in [some math competition]." Whether it's a simple race to finish multiplication tables or a full-blown Mathcounts meet, this experience is clearly one that has turned the boys I teach on to mathematics. On the other hand, almost <i>none</i> of the girls even mentions competitions; when they do, it's almost always negative. (One girl--now a math major--once wrote "I decided I hated math when my teacher had us play this stupid game called 'Around the World'.") And my own daughter loves math but hates competition. So there's some interesting data.</div>
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My friend Cathy O'Neill, aka <a href="http://mathbabe.org/" target="_blank">mathbabe</a>, wrote on this issue a couple of years back, "<a href="http://mathbabe.org/2011/07/17/math-contests-kind-of-suck/" target="_blank">math contests kinda suck</a>." As a contest author, I have to agree with the majority of her argument: our society's reliance on math contests for identifying and encouraging mathematical talent discourages a lot of girls, who either don't like contests, or don't like doing math the way that you have to in order to be successful at most math contests. That's why we need more math circles and other places where girls can get their hands dirty doing awesome mathematics--Chicago's <a href="http://qed.wpcp.org/" target="_blank">new math research symposium</a> is one such. There's a place for math contests in the pantheon of cool, challenging math activities--but the pantheon has got to get a lot bigger than that.</div>
Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-537830633228785974.post-29891009853825205932014-03-22T17:08:00.001-05:002014-03-22T17:08:14.491-05:00Guest Blog: She likes math, but she hates math homeworkEmma, 6th grade, likes math (does math circle voluntarily on Saturdays, for example), but hates math homework. When I asked her why, she sent me the following well-thought-out response--unedited by me (okay, I deleted two commas). Math teachers, take note!<br />
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<span style="background-color: white; color: #222222;"><span style="font-family: Helvetica Neue, Arial, Helvetica, sans-serif;">As a 6th grader, here are the reasons I hate math homework, and solutions to the problems:</span></span><br />
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<span style="font-family: Helvetica Neue, Arial, Helvetica, sans-serif;">1) Quantity- Homework is supposed to be for us to practice what we've learned. So if you know the material, then it's a waste of time to do 20 similar problems when all you need to prove that you know the material is 4 or 5. And if you don't know, then you'll practice incorrectly, and it really won't help you to learn how to graph inequalities if you're repeating incorrect steps for any number of problems greater than that needed to prove that you know the material.</span></div>
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<span style="font-family: Helvetica Neue, Arial, Helvetica, sans-serif;">2) Quality- Having similar problems doesn't help anyone. Sure, if you want to memorize steps, it may be helpful, but in order to understand the math behind the equation and prove why you need to take the steps, you should have a variety of problems. Because in order to understand how to use a method, you should take the time to practice with different types of problems, or else you run across a slightly different problem and you don't know how to do it. </span></div>
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<span style="font-family: Helvetica Neue, Arial, Helvetica, sans-serif;">3) Comprehension- Having problems to solve like "x+2=5. x=?" doesn't help you learn math. That's <i>arithmetic. </i>Arithmetic is something that you can do with a calculator. <i>Math</i> is knowing <i>why</i> x=3. And just showing your work doesn't help, because again, you're just showing your calculations, which again, could be done with a calculator. What helps is asking, "Why does x=3?" Because then you have to look up from your calculator and think about what makes the problem work.</span></div>
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<span style="font-family: Helvetica Neue, Arial, Helvetica, sans-serif;">4) Grading (as a class)- When homework is being graded as a class, then you should again focus on the logic hidden behind the problem. Ask a kid to come up to the board and prove why their answer is correct (having fewer problems would also prove to be helpful here). When you go over them by just stating the answer, it doesn't help the kids who got the answers wrong. Unless they just made an arithmetic error, they still don't get why it works, still don't understand the material, and now only know that they were wrong, which doesn't help. And as teachers, your job is not to make kids pass or fail, but to get them to learn something. </span></div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-67322353453126665332014-03-16T23:13:00.004-05:002014-03-16T23:13:50.222-05:00Three Parables of TeachingA peril of teaching for a while--and thinking about teaching--is that everything you read becomes filtered through the question "What does this say about teaching and learning?" But here are three short pieces that more-or-less deliberately engage issues of teaching and learning in somewhat parabolic (if not elliptic) ways.<br />
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"<a href="http://hilo.hawaii.edu/~tbelt/Pols360-S08-Reading-ShootingAnElephant.pdf" target="_blank">Shooting an Elephant</a>," by George Orwell, captures the experience of being a new teacher (whether new to the profession, or new to a school) confronted by a discipline problem <i>and</i> a class or hallway full of students what you're going to do about it. I think it also reminds us that we're never really as prepared as we think we are, that first time, and that the chances of things ending well--for the elephant surely, but also for us--are slim indeed.<br />
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<i><a href="http://www.semcoop.com/book/9780380698714" target="_blank">Sideways Stories from Wayside School</a>, </i>by Louis Sachar, is full of wonderful and off-the-wall tales and fables, but it's the first one that has stuck with me: Joe is held back during recess by Mrs. Jewl because he can't count properly. When Joe counts a set of objects, the numbers come out in any old random order, but he always gets the number of objects right. The (very <a href="http://users.rcn.com/rathbone/lw81-88c.htm" target="_blank">Wittgensteinian</a>) irony is that whenever Mrs. Jewl tries to explain to him how to count the "right" way, he does exactly what she does, and comes up with the wrong answer. Ever since my mentor Steve Starr read me this story, I've tried to listen more and worry <i>more</i> about whether the student has what appears to be a robust way of getting the right answer than whether he or she is doing it my way.<br />
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Finally, Philip Roth's "<a href="http://www.fredonia.edu/west/pdffiles/roth.pdf" target="_blank">The Conversion of the Jews</a>" is the hilarious and deeply sad story of Ozzie, a young pre-teen Jewish boy in 1950's New Jersey who has a history of asking his rabbi the wrong--and by that I mean "the hard"--questions. The story's precipitating incident is an argument in which his rabbi asserts the existence of a <i>historical</i> Jesus but says that he couldn't have been the Son of God as the New Testament describes because, as Ozzie quotes the rabbi, "‘The only way a woman can have a baby is to have intercourse with a man." Ozzie asks "if He could make ail that in six days, and He could pick the six days he wanted right out of nowhere, why couldn’t He let a woman have a baby without having intercourse." And the rabbi refuses to give a consistent answer, leading Ozzie to burst out "You don't know anything about God!" What strikes me here is that Ozzie demands only two things of his rabbi: intellectual honesty, and a modicum of kindness. And really, for a teacher, is it reasonable to expect anything less?Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-76378150573386999362014-03-09T21:58:00.000-05:002014-03-09T21:58:03.218-05:00It's not how big your class is, it's what you do with itBig news over the last couple of weeks--besides the nascent testing rebellion going on in CPS and other districts--has been the publication of Diane Schanzenbach's paper, <a href="http://nepc.colorado.edu/publication/does-class-size-matter" target="_blank">"Does Class Size Matter?"</a> by the National Education Policy Center. Among this paper's key findings:<br />
<ul>
<li>Old studies claiming zero or negative correlations between small classes and achievement relied on faulty meta-analysis of published data.</li>
<li>The STAR experiment in Tennessee, which was a randomized trial, showed a 0.15-0.20 standard deviation gain from assignment to classes of 13-17 rather than 22-25, with higher gains in African-American and low-income subgroups.</li>
<li>Teachers of smaller classes are able to (and, in the STAR case, did) use a variety of individualizing strategies, including tracking individual achievement, differentiating instruction, and making personal connections with students. </li>
<li>Contrary to popular belief, these effects were <i>larger</i> with more experienced teachers.</li>
<li>Although the STAR study is the most comprehensive randomized study in the US, its findings are backed by other studies that managed to control for other variables in the process.</li>
</ul>
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These findings are summarized as simply "All else being equal, increasing class sizes will harm student outcomes."</div>
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But I'm still a skeptic, mostly because of the phrase "all else being equal." In particular:</div>
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<ul>
<li>The STAR gain of 0.15 sd requires reducing classes to about half of what they currently are in Chicago, which would require--roughly--doubling the number of teachers. CPS currently has 22,000 teachers, and it's hard for me to imagine that the district would be able to shazam up anywhere near 22,000 additional teachers without dredging the bottom of the applicant pool. But would these new, bottom-of-the-pool teachers actually improve outcomes?</li>
<li>The reason why you have to get down to 13-17 students per class to see the payoff is that once the denominator of an expression is large, reducing it a little doesn't increase the quotient very much. For example, if a class lasts 50 minutes and has 30 students, each student gets at most 1 minute and 40 seconds of "air time" or attention. If the same class has only 25 students, each student gains 20 seconds of "air time" or attention--which isn't very much. So in the real world of the class size reductions that are plausible in the short term, you're not going to see much payoff.</li>
<li>The main pedagogical advantages the small-class-size teachers had over the the regular-class-size teachers in the STAR survey were all things <i>all teachers should be doing anyway: </i>monitoring what individual students are doing and learning, giving students second or third opportunities to learn material they didn't get the first time, and making personal/emotional connections with their students. It's easier to do those things in smaller classes, sure, but they're hardly small-class-only techniques. In fact, studies have shown that weaker teachers placed in small classes take no more advantage of these techniques than they did in larger classes. So the STAR study suggests that teachers who are using these techniques move further than teachers who don't, which I kind of feel like we already knew.</li>
<li>In a world of finite resources, smaller classes are a <i>trade-off.</i> In the US, we trade smaller classes than teachers get in other countries for more of them: the standard load is five classes of 28-30 students (in Chicago) or 40-odd students (in much of California). So obviously it's better to have five classes of 17 students than five classes of 28 or 40 (for one thing, you have many fewer students to keep track of). But what if you traded back, having three classes of 45 or so students instead of five classes of 25? You'd get an additional two prep periods a day to plan,conference with other teachers, and analyze (grade + reflect on) assessments. That time would allow you to better pace the next day's lesson--which you could plan that day, rather than having to do a week's worth on Sunday just to be able to keep up during the week--and to learn more about teaching. And you wouldn't have to stay up until midnight to do that. The situation I'm describing is pretty close to what they have in China: two sections of 50 students, with most of the day devoted to planning and preparation.<sup>1</sup></li>
<li>Back to the issue of air time: how would getting an additional ten or twenty seconds of verbal feedback each day compare to getting <u>written</u> feedback on <u>individual work</u> every single day, which is what many Chinese classes offer?</li>
<li>Another trade-off we make is teacher quality. Just as hiring 22,000 teachers would reduce quality, it seems reasonable that if we were willing to live with much larger classes, we might be able to increase overall teacher quality. That's what Finland did when they first started turning their educational system around (although now <a href="http://parentsacrossamerica.org/what-finland-and-asia-tell-us-about-real-education-reform/" target="_blank">class sizes are back down to about 20</a>, which is the stated average in CPS elementary schools, as a concession to eliminating class tracking).</li>
</ul>
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So what's the moral? Better teaching clearly produces better outcomes. Smaller class sizes have costs. (At a district average of about $80-100K per teacher, even just an extra 10,000 teachers runs to about a billion dollars <u>annually</u>, which seems like a lot to pay for a five percentile point gain on tests.) Expanding class sizes <u>without increasing total student loads</u> might have substantial benefits to teachers and students: increasing opportunities to work together and to plan and assess better and more frequently, with more and more individualized feedback. Why are we talking about keeping "all other things equal" when they so rarely are?<br />
Because veteran teachers know: it's not just how big your class is; it's what you do with it that matters.<br />
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1. "But China and Finland are so much more homogeneous, it's easier to teach big classes of those students!" I hear you cry. Well, in the eight-county Chicago area, more than 50% of African-American students are in classes that are over 95% African-American, and over 25% of Latino students are in similarly segregated schools. (<a href="http://www.wbez.org/series/race-out-loud/greater-segregation-regions-black-latino-students-100452" target="_blank">WBEZ</a>) So while the CPS <u>system</u> is much more diverse than China's or Finland's, it's not obvious that its <u>classrooms</u> are.</div>
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<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-32946495456096901692014-02-22T21:08:00.002-06:002014-02-22T21:08:25.603-06:00Bayes's Theorem and Hiring Your Way To Great Teachers<a href="http://en.wikipedia.org/wiki/Bayes%27_theorem" target="_blank">Bayes' Theorem</a> is the most important statistics theorem that nobody knows. In essence, it says that when computing the probability of an event occurring, you must take into account the information you already know about what theoretical outcomes are actually possible. In mathematical terms: the probability of A given that B has occurred equals the probability that both occur, divided by the probability that B occurs (with or without A).<br />
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Bayes's theorem has wide applications in statistical inference, but today we're going to talk about one that's of crucial concern to anyone trying to improve teaching in a particular school: hiring teachers. Suppose that your interview protocol allows you to identify satisfactory-or-better teachers with 80% accuracy (if you have such a protocol, tell me!). And suppose that 80% of the teachers in the applicant pool are satisfactory-or-better. Then if you interview 100 teachers, here's what happens:<br />
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<ul>
<li>80 teachers are satisfactory-or-better. Of these, your interview protocol says that 80%, or 64, really are satisfactory-or-better, and 20%, or 16, it rates as unsatisfactory.</li>
<li>20 teachers are unsatisfactory. Of these, your interview protocol says that 80%, or 16, are unsatisfactory, and 20%, or 4, are inaccurately rated as satisfactory-or-better.</li>
</ul>
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What's the takeaway? Well, there are <b>68</b> teachers rated as satisfactory-or-better, but of these, 4 are actually unsatisfactory. Thus if you hire one of the 68 teachers, you have about a 94% chance of getting a satisfactory-or-better teacher.</div>
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So far, the odds sound pretty good. But those odds are highly dependent on the assumptions we made: that 80% of the candidates were satisfactory-or-better, and that your protocol helps you tell good from bad 80% of the time. In my experience, neither of these things is necessarily true:</div>
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<ul>
<li>Many of the satisfactory-or-better teachers are pretty happy where they are, and aren't looking for new jobs. In my experience, the applicant pool for Chicago Public Schools jobs is more like 40% satisfactory-or-better. (Note: I'm <b>not, not, not </b>saying that only 40% of CPS teachers are satisfactory. There are lots of satisfactory-or-better CPS teachers--but in my experience, many of them are committed to the schools where they teach. What I'm saying is that when I was interviewing applicants for jobs at my school, only 40% of the <b>applicants </b>were satisfactory-or-better.)<br /></li>
<li>Teachers can have great credentials and interview well without being great in the classroom. You can catch that with demo lessons (I've known of candidates who hit home runs in the interview only to totally whiff the demo lesson), but even that won't tell you how well they relate to students and parents over the long-term, how well they collaborate with colleagues, and how committed they really are to improving their practice. So my guess is that interview protocols are less than 80% reliable, say 70%.</li>
</ul>
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Running that same thought-experiment with our revised assumptions yields very different results:</div>
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<ul>
<li>Only 40 applicants are satisfactory-or-better, and our protocol identifies 70%, or 28, of them as such. 12 applicants are incorrectly identified as unsatisfactory.</li>
<li>60 applicants are unsatisfactory, and our protocol identifies 70%, or 42, of them as such. 18 applicants are incorrectly identified as satisfactory.</li>
</ul>
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Thus 46 applicants are identified as satisfactory-or-better, of whom only 28 really are satisfactory. So the probability that a given applicant who passes the interview process is actually satisfactory is 28/46, or about 61%. Under these conditions, then, if you hire five candidates, only three will probably work out. And you'll be stuck with two whom you sort of wish you didn't hire. </div>
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What are some conclusions we can draw?</div>
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<ul>
<li>The usefulness of hiring procedures has as much to do with the overall quality of the applicant pool as it does with the theoretical reliability of the procedure itself. If the pool has lots of unsatisfactory teachers, even a good test will end up with many of the apparently-satisfactory teachers being actually unsatisfactory.</li>
<li>If you don't have a great applicant pool, firing a teacher who isn't working out will only result in a substantial improvement 60% of the time. That's better than nothing, but a whole lot less than the "Just fire all the bad teachers" voices usually let on.</li>
<li>If lots of bad teachers are suddenly fired, the applicant pool will get worse, both because the fired teachers are now in it, and because lots of people are trying to hire the good ones (remember that 3/5 of the teachers we hire, even under the pessimistic assumption, are good teachers). And then as we've seen, hiring procedures become less effective at securing satisfactory teachers for jobs. So as a system-wide policy, "fire the bad teachers" is unlikely to produce substantial improvements for a large fraction (probably more than half) of the kids in the system.</li>
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So it's unlikely that we can hire--or fire--our way to great teachers. We need to take the teachers we already have and develop them instead.</div>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-41718047070590229882014-02-15T21:40:00.001-06:002014-02-15T21:41:24.541-06:00Teaching Mathematical "Grit": A Dialogue<span style="font-family: Times, Times New Roman, serif;">This week, the <a href="http://www.cpam.teachersdg.org/" target="_blank">CPAM </a>listserv has been bubbling with discussion about how to teach <a href="http://www.ted.com/talks/angela_lee_duckworth_the_key_to_success_grit.html" target="_blank">grit</a> in mathematics; unsurprisingly, both of us have pretty strong opinions on this subject. So here was our part of the dialogue:</span><br />
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<span style="font-family: Times, Times New Roman, serif;"><i>John:</i> <span style="background-color: white;">During class, I have them work on one challenging problem at a time. They work. I walk around and listen. They are encouraged to try it themselves first, then discuss their work with their neighbors. I do not give hints or show methods to solve the problem. They know that they have been given a problem, not an exercise. That is, I expect that it will take time to solve it, it is related to what we are working on but is not a copy of other problems they have been asked to work on. I do not ask them to do the problem, but insist that they work on it. Every fifteen seconds, or so, I walk by and observe progress. If you want them to become persistent you must provide situations where persistence is the only way. At some point during the class, we discuss solutions that various students have proposed. If it is a worthy problem, one that requires persistence, there will be several ways to approach it. When the problem has been solved, they have learned the content for tonight's homework. That is the intent of the problem, to teach the new lesson by having them figure out what the next thing is. Then I give them another problem. I never give them a worksheet nor do I give notes. We spend class time working on worthy problems. They learn the content, they learn how to solve problems, they learn "grit" and most of all, they love it.</span></span><br />
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<span style="font-family: Times, Times New Roman, serif;">It takes a while for them to get used to the concept that I will not "show them how to do the problem before I ask them to do it." but once they do, they are actually learning math.</span></div>
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<span style="font-family: Times, Times New Roman, serif;"><i>PJ.:</i> One lesson I learned about this is that it's hard to learn more than one thing at a time. So a good problem to use to teach grit is one in which the actual skills embedded in the problem aren't that difficult, but maybe they have to think through a lot of different possibilities, go down a blind alley, or something. It helps a lot if the problem has someplace fairly obvious to start (even if it's ultimately the wrong place) and if there's some obvious way to check that your answer is correct. Many math contest problems require a lot of grit but fail on these two counts, because there's basically only one thing to do, and the "problem" is seeing what that one thing is. On the other hand, problems like the camel & bananas problem (one version + solution <a href="http://mathcentral.uregina.ca/QQ/database/QQ.09.99/rich1.html" target="_blank">here</a>), or the farmer with the broken eggs (look <a href="http://mathforum.org/library/drmath/view/58849.html" target="_blank">here</a>), or the locker problem (look <a href="http://connectedmath.msu.edu/CD/Grade6/Locker/" target="_blank">here</a>) can be good places to start.</span></div>
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<span style="font-family: Times, Times New Roman, serif;">In terms of classroom practice, I'm not as hands-off as John, but I think it's really important NOT to scaffold the problem, especially in problem-specific ways. Scaffolding the problem by asking leading questions just leads the students, and teaches them that they needed your help to do the problem, which isn't what you want them to learn. Following Polya and my friend Doug, I have a series of nudges I tend to give in these situations, and I choose the problem with those nudges in mind. (That is, I'm thinking about which general strategies I want kids to apply.) There aren't many of these nudges, and I use the questions a lot, to the point that I can start asking students "What do you think I'm going to tell you to do?" A not-totally-exhaustive list, in no particular order:</span></div>
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<li><span style="font-family: Times, Times New Roman, serif;">Try a simpler case.</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Try several simpler cases, and look for a pattern.</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">What are the conditions? Can you state ______ as a mathematical sentence?</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">What's the unknown?</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Look for congruent triangles/similar triangles. [I teach a lot of geometry]</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Try chasing angles. [Same]</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Reduce the problem to finding a point. [Same]</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Try dropping one condition and satisfying the others.</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Can you satisfy even one of these conditions?</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">What's a related problem? Can you transform this problem into that one?</span></li>
<li><span style="font-family: Times, Times New Roman, serif;">Use algebra</span></li>
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<span style="font-family: Times, Times New Roman, serif;"><i>John:</i> I think it is more what we choose to emphasize when we try to get other people to understand what we are trying to do. I believe teachers tell kids too much so I emphasize that I do not tell them anything. The things you listed are certainly questions I ask my students , as well as things like, can you prove it, can you do it another way, does your answer make sense, does your answer agree with the answers others got, perhaps you should draw a bigger picture.....</span></div>
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<span style="font-family: Times, Times New Roman, serif;">[After this, another teacher wrote in about Carol Dweck's work on <a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&ved=0CD8QtwIwAg&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DMTsF2TaEaJA&ei=DzIAU_rmAcG4yQH0soGYCw&usg=AFQjCNGRDuxAIoOGL_Gv_PgD7Cy5ws7nsg&sig2=6vsWEp2Dg9koT2sXqoqQJA&bvm=bv.61535280,d.aWc" target="_blank">mindset</a>, and fourth teacher said that she regularly assigns her kids hard problems to work over a week or two; students don't ask her for help unless they're "totally stuck". I wrote back the following pair of replies:]</span></div>
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<span style="font-family: Times, Times New Roman, serif;"><i>P.J.</i>: Two brief extra thoughts:</span></div>
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<li><span style="font-family: Times, 'Times New Roman', serif;">It turns out that kids actually respond when they are taught the science behind cognitive development -- that in itself can be a way to change their mindset -- which is maybe surprising?</span></li>
<li><span style="font-family: Times, 'Times New Roman', serif;">I actually encourage my students to see me relatively early on (before they're totally stuck), because I worry that their peers will give them too much help. But I have a small enough roster that I don't have to worry about being overwhelmed.</span></li>
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Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-56549704308107779242014-02-12T13:17:00.002-06:002014-02-12T13:24:11.544-06:00δ, ε, and Mathematical Thinking<span style="font-family: Arial, Helvetica, sans-serif;">So a few of you have expressed interest (surprise, concern on my behalf...) that I'm doing formal limits with my class. To be fair, the class is <i>very</i> advanced: only one of the fifteen students has yet to take Calculus, and almost all have spent at least one summer doing math. But it's still a scary prospect. So here's a report from day 1.</span><br />
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<b>Why?</b> Why teach the formal definition? And how do you motivate it? We began this unit by doing an empirical investigation of iterates of the function <i>f</i>(<i>x</i>) = <i>rx</i>(1 - <i>x</i>) for 0 ≤ <i>x</i> ≤ 1 in the context of rabbit populations: if <i>x</i> represent this year's population density in a particular warren, <i>f</i>(<i>x</i>) represents next year's population density in that warren. Students quickly discovered that a variety of long-term behaviors are possible (convergence to a single limit, oscillations between 2 or 4 points, and apparently "random" behavior that we couldn't quite nail down), and depend mostly on <i>r</i>. So then as we want to refine our ideas and start writing proofs about limits, we realized that we needed a formal definition--otherwise, as I put it in class, "we're not doing mathematics, we're doing what those people across the hall [the science teachers] do."</span><br />
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So that's the class's official motivation. But why slog through this? A traditional answer is this: if you're going to go on in proof-based mathematics, you need to be able to write proofs using the formal definition of limit. A less-traditional answer is that we're going to need some formal properties of limits, continuous functions, and sequences in this very class, and this is our first opportunity to start working those muscles. But the least-traditional, and most important answer is this: I'm using limits to teach mathematical thinking.</span><br />
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<b>What?</b> A fundamental mathematical activity is defining, and it's very hard. A good mathematical definition describes a phenomenon precisely, excluding everything else. In lower-level classes, I tell students that, unlike in English, a mathematical definition is like a poem: every word is crucial in exactly the place that it is. When possible, we actually analyze those poems: why do we define a trapezoid as "a quadrilateral with at least one pair of parallel sides", or an isosceles trapezoid as "A quadrilateral with at least one pair of parallel sides, such that the two angles formed by one of the parallel sides with the other two sides are congruent"? We drop conditions, try to draw things that "break" the revised definition, argue about the merits of an inclusive versus exclusive definition (what theorems about parallelograms follow from our definition of trapezoid?).</span><br />
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But it's rare that students get to construct their own definition, and so that's exactly what we did yesterday. We iterated through four stabs ("It's getting pretty violent in here!" quipped one student). Each time we started by taking the fuzzy new idea and rephrasing it in mathematical language. For example, when a student proposed "The terms in the sequence get closer to the limit," we rephrased as "|<i>x<sub>n</sub></i> – <i>L</i>| decreases." But each time, one student or another would come up with an objection: "Look, we're saying that these terms are getting closer to 1733 1/3 ... but they're also getting closer to 2000, 2100, or anything bigger than that!" Then we rephrased: "What do you mean by closer?" "As small as you want." "Okay, then, so how do we say that mathematically?" "Smaller than any number." "Okay, then we're going to have to name that any number...."</span><br />
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<b>Who?</b> It helped that a few students had done the formal definition in a class the previous year, but I found an interesting way to handicap them: I told them they could only give two kinds of contributions, "genuine questions" and "counterexamples" in response to other students' proposals. That restriction didn't totally quiet them (although it did, somewhat) -- it forced them to think through what they had learned last year, and to apply the underlying ideas to the definition we were working with. At least two key ideas (and one major counterexample) were found by students who had never studied limits formally.</span><br />
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<b>How?</b> I've already said that I hamstrung students who had already studied the topic in a way that made them think mathematically without taking the work away from other students. But I made a few other crucial decisions that really made this half-hour of discussion go well:</span><br />
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<li><span style="font-family: Arial, Helvetica, sans-serif;">We started with a rich set of concrete examples--a bunch of sequences with a variety of long-term behaviors--on which we could draw as we worked on our definition. This context put the meat of the activity--asking whether our current stab included the things we wanted to include, and excluded the things we wanted to exclude--within just about every student's grasp.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;">We started with sequences. In the past, I've found that limits of sequences are much easier for students to get going on than the limit of a function at a point. Sequences are simpler: the values are discrete, they only go in one direction, and the definition only involves one set of absolute values, not two. Moreover, there are lots of relatively interesting (nonconstant) sequences whose N's can be explicitly calculated from their ε's. Conceptually, too, there's a natural quality to "we're wondering what happens as time goes on" that "we're wondering what happens when <i>x</i> gets close to, but not exactly equal to, <i>a</i>" seems to lack.</span></li>
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<span style="font-family: Arial, Helvetica, sans-serif;"><b>Well?</b> Did it work? I'm not sure yet. The kids were engaged and spent half an hour discussing definitions before we settled on what we called the "working draft" we'd use until further notice. Most of the students contributed to the discussion at least once, so that's something. And when we went through a proof, together, the kids seemed to follow. But I'll know more tomorrow, after kids try one on their own. I can tell you this, though: it was way more engaging for everyone, way more interesting, than starting class by writing the definition on the board and then slogging through a bunch of examples.</span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-47459033244747180722014-02-09T17:44:00.005-06:002014-02-09T17:44:35.379-06:00A Mathematical Adventure Underway!My apologies for the relative silence. We're back and running, somehow.<br />
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Today's entry is relatively brief but pretty exciting. For the last two summers, I've taught a Chaos "Maxi" course at <a href="http://www.hcssim.org/" target="_blank">HCSSiM</a>: three weeks, 2.5 hours every morning and 3 hours of problems most every evening, with 15 of my best teenage friends. The course was based on the Chaos half of my "KAM-Geometry" course that I teach at Walter Payton, but I sussed it up with more analysis and other theoretical math, and of course I cared a little less about students "getting" everything (so long as they weren't totally lost). What a pair of crazy adventures those courses were!<br />
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<span style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;">Our annual "Toga Day" (do mathematics while dressed like Ancient Greek mathematicians)</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;">Batman comes to help us prove the completeness of Fractal space under the house-dwarf metric</span></div>
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This spring, I'm teaching the high school version of the course -- but I've decided it won't be a high school version. Instead, we'll do as much of the sussed-up, proof-heavy, analysis stuff as we can: because it's cool, because it makes the course math (instead of "here's another cool observation, wonder why it happens?"), and because I think the kids can do it.<br />
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We do the formal definition of limit tomorrow ... I'll let you know how it goes!Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-18911201828841664392013-12-08T17:18:00.000-06:002013-12-08T17:21:01.498-06:00Want to make math interesting? Try this!<div class="separator" style="clear: both; text-align: center;">
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This week's mandatory hell-in-a-handbasket piece is the <i>New York Times'</i>s <a href="http://www.nytimes.com/2013/12/08/opinion/sunday/who-says-math-has-to-be-boring.html?_r=0" target="_blank">"Who Says Math Has to Be Boring?"</a> gives several clear, cogent recommendations for improving math education in the United States: move to curricular models more flexible than the traditional Algebra-Geometry-Algebra/Trig-Precalculus sequence; improve math teacher preparation in mathematics (now there's a chicken-and-egg problem if I ever saw one); expose kids to numbers (not just numerals) and numerical relationships earlier, before first grade if possible; give students exposure to math in the real world. But despite the title--and the early admonishment that "Finding ways to make math and science exciting for students who are in the middle of the pack could have a profound effect on their futures," it leaves off what is to me, the most important point. Here it is: to make math interesting, do interesting math, what I would call "actual math".<br />
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Now, I'm not taking sides in the perennial "pure-versus-applied" debate, although <a href="http://blog.mrmeyer.com/?p=18252" target="_blank">Dan Meyer's blog last week</a> includes the important point that the new-traditional view "kids are interested by applications, not abstractions" is simply false:<br />
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What makes an activity engaging isn't how "real-world" it purports to be. It's a combination of factors, including:<br />
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<li>Challenge level</li>
<li>Immediacy and quality of feedback</li>
<li>Stakes--low but nonzero is better, especially in the short-term</li>
<li>Visibility of progress towards ultimate mastery</li>
<li>Novelty</li>
<li>Familiarity</li>
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(Yes, I did mean both of the last. Things that are completely new are often off-putting; things that are the same every time are drudgery. So you need to provide novelty within familiar frameworks, or something like that ... but that's another blog post.)</div>
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Math that's been reduced to algorithms for students to memorize and apply with perfect precision on tests whose scores "will follow you for the rest of your life" has none of these qualities. So of course it's not interesting.</div>
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What is interesting is thinking about problems, trying to carve them into meaningful abstractions, attacking them again and again with slight gains, etc. Doing problems that seem familiar, but with a twist: this time I'm not asking for the hypotenuse given two legs, but for a leg; or for a right triangle with integer sides and a particular leg; or for the dimensions of a rectangle, given its area and the fact that its sides and diagonal are all integers .... Or this time I'm wondering: suppose I approximate the square root of 2 as 1.414. How bad an error will I get for the hypotenuse of a 10m-10m right triangle? 100m-100m? 1000m-1000m? If it takes a minute to walk 100m, how long do the legs have to be before the error is five minutes of walking? </div>
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All those are abstract problems, with varying degrees of "real-world" relevance. I'm not claiming they're great questions--I just came up with most of them right now--but they avoid most of the pitfalls above, and if done in class, in a problem-solving environment (i.e. where the kids have been exposed to the fact that math problems don't always yield on the first attempt, several times), would be a heck of a lot better than a page full of Pythagorean theorem practice problems. Some of them lead to deep questions: are there numbers that <i>can't</i> be sidelengths of a right triangle whose other sides are integers? Are there numbers that can only appear once as members of pythagorean triples?</div>
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Math is at its heart a process of abstracting and connecting ideas through logic and generalization. We worry that most kids can't do these things, and so we often do our best to avoid engaging in such practices overtly, or even requiring kids to engage in them at all. But that's exactly the wrong approach. Engaging kids in the process of actually abstracting, connecting, reasoning, generalizing, conjecturing, and applying--rather than just practicing spitting out formulas--now <i>that </i>would make math class interesting. It would make math class about <i>math</i>.</div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-3883663318280262552013-11-29T11:25:00.001-06:002013-11-29T11:26:06.699-06:00"Irresponsible Teenagers": an oxymoron?<blockquote class="tr_bq">
<span style="color: #222222; font-family: arial, sans-serif; font-size: 13.333333969116211px;"><i>Dubois then turned to me. "I told you that `juvenile delinquent' is a contradiction in terms. `Delinquent' means `failing in duty.' But duty is an adult virtue -- indeed a juvenile becomes an adult when, and only when, he acquires a knowledge of duty and embraces it as dearer than the self-love he was born with. There never was, there cannot be a `juvenile delinquent.'"</i></span><span style="color: #222222; font-family: arial, sans-serif; font-size: 13.333333969116211px;"> Robert A. Heinlein, <i>Starship Troopers</i></span></blockquote>
I hear a lot from my colleagues about making teenagers be responsible, and indeed I think that's a really important (primary?) goal of high school. But the thing we as teachers often fail to realize is that teenagers <i>aren't</i> responsible. They're not really capable of planning ahead long-term, they often make poor decisions for reasons that, more and more, we understand as weaknesses in brain development, a mismatch between the complexity and long-term consequences of what kids can do on the one hand and their brain's inability to think through complex decisions with long-term consequences on the other. (See <a href="http://pmbcii.psy.cmu.edu/dahl/Dahl_Adolescent_brain_development.pdf" target="_blank">here</a> for one of many scholarly articles on the subject.)<br />
<br />
So when we give kids a long-term project that we don't help them break down into pieces--and I think there's a huge distinction between handing it to them, all sliced into pieces, and walking through the planning process with them--or make passing or failing a single test a huge piece of their grades, or create any other single point of failure, we're really playing into the thing that we know they can't do. And then when they don't do it, we call them irresponsible.<br />
<br />
The thing is, if kids can't really be responsible (yet), they can't really be irresponsible, either. It only makes sense to talk about <i>irresponsibility</i> in the context of something that we can reasonably expect someone to be responsible for. Kids can be responsible for lots of things with short-term consequences, and they can be taught (helped) to see the connection between lots of short-term decisions and long-term consequences. But those lessons are hard ones to learn, and the process often mirrors a saying attributed to Mark Twain: "Good judgment comes from experience. Experience mostly comes from bad judgement." What that saying suggests is that we need to <i>create</i> and <i>preserve</i> opportunities for kids to fail safely and learn from those failures, rather than making those failures catastrophic. And we need to be right there so that the kid can connect the dots between what he or she did or didn't do, and the negative consequences that resulted.<br />
<br />
So often, when I hear a teacher talking about a kid's being irresponsible, I wonder two things:<br />
<br />
<ol>
<li>What did you, as the responsible adult, do to bring about this situation? More importantly, what did you, as the responsible adult, do to avert it?</li>
<li>What could we have reasonably expected this child to do in this situation? Why didn't he or she do the "responsible" thing?</li>
</ol>
<div>
#2 is shorthand for my exasperated "Of COURSE he was irresponsible -- he's a child!" But it doesn't really help kids to throw around this moralistic label -- it only makes them feel cruddy about things that happened in the past, rather than accepting the consequences and doing better in the future.</div>
<div>
<br /></div>
<div>
One last pitfall. We all know kids who are remarkably responsible--who manage to pull it together and keep it together despite an insane array of pressures, conflicts, and demands. But that phenomenon is just the end of a bell curve of development and personal characteristics. We all know--or know of--six-foot-two seventh graders, or freshmen taking Calculus. Yet we don't hold those kids up as examples against which other kids are judged. No matter how much we wish that kids were more "responsible" than they often are, blaming them for being irresponsible -- especially ordinary-kid-kind-of-irresponsible -- isn't any more reasonable (or responsible) than blaming them for their height, or for "only" being in Algebra.</div>
<div>
<br /></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-61245604141370081912013-11-19T20:19:00.001-06:002013-11-19T20:21:56.545-06:00Objectives and Experiences<blockquote class="tr_bq">
<i>And we turn him into an anecdote, to dine out on, like we're doing right now. But it was an experience. I will not turn him into an anecdote. How do we keep what happens to us? How do we fit it into life without turning it into an anecdote, with no teeth, and a punch line you'll mouth over and over, years to come: "Oh, that reminds me of the time that impostor came into our lives. Oh, tell the one about that boy." And we become these human jukeboxes, spilling out these anecdotes. But it was an experience. How do we keep the experience?</i></blockquote>
<blockquote class="tr_bq">
John Guare, <i>Six Degrees of Separation</i></blockquote>
When I was a starting teacher, it was as much as I could do to articulate what I wanted my kids to know at the end of the day, much less at the end of the week or month. A key development for teachers in transitioning from that beginner level to something like "proficient" is learning to anticipate what's needed for instruction over the next week, month, and year: for example, knowing that you have to teach a bunch of chunking in Algebra II and Precalculus so that students can do integration by substitution in Calculus.<br />
<br />
That's where standards come in: they tell you what you have to do in each year so that kids can go on and do what they need to do the next year. Whether you're using the <a href="http://www.corestandards.org/the-standards" target="_blank">Common Core standards</a>, <a href="http://www.nextgenscience.org/next-generation-science-standards" target="_blank">Next Generation Science Standards</a>, or an in-house list, this shift in perspective -- from "What am I going to do today?" to "What do kids need to learn?" is crucial if you're going to accomplish anything--and if your students are going to make any genuine progress.<br />
<br />
But I've noticed that master teachers--like my co-blogger John, or my friend Doug O'Roark--ask a different question, not necessarily first, but early in the planning process. This question is: "What experience do I want kids to have?" I find that this question more than any other has changed my perspective about planning for classes.<br />
<br />
What experience do I want kids to have? Is the goal to give them the experience of discovering something? Of exploring in a rich "sandbox" of cool math ideas, regardless of what they wind up conjecturing and proving? Of solving involved numerical problems? Of developing a set of ideas to describe a new situation? Of applying old ideas to solve a new problem? Or ... ?<br />
<br />
Attending to the quality of my students' experience rather than simply on what I want them to learn leads me to new ideas--that, even in a mostly-remedial algebra class, it's important to have fun. (The way Doug and I did this was to do magic tricks with Algebra.) And it puts common teaching pitfalls into perspective. I mean, who would answer the question "What experience do you want the kids to have today?" with "I want them to watch a powerpoint for 30 minutes"? And who would want kids to have the exact same experience, every day, for 180 days?<br />
<br />
I think that one way to understand the <a href="http://www.corestandards.org/Math/Practice" target="_blank"><i>Standards for Mathematical Practice</i> </a>section of the CCSS is very much in this vein: they describe and, to a certain extent, prescribe the kinds of experiences we want kids to have while they are learning the content in the other standards. For example, take SMP-1, "Make sense of problems and persevere in solving them." The "standard" doesn't describe a set level of perseverance that kids are supposed to attain, or even clearly define what "making sense" of a problem is. But it suggests that kids ought to be <i>experiencing</i> problems that are ill-defined, or at least initially resistant to mathematical analysis, and that these experiences should include trying more than one approach before being successful. Thinking of the SMP in this "experience" way helps me reconcile the essentially-fuzzy nature of those standards to the others, and also helps me think about how to mesh the two: the point isn't to do one kind of standard <i>and then</i> the other, but to approach one (content) standard <i>in the mode</i> of one or more of the others.<br />
<br />
And that "or more" leads my musings to a caveat. At its best, mathematical experience is rich: a great class is one in which kids are spotting patterns, making conjectures, trying things out, having fruitful errors, using various technologies (from compasses to computers), all woven together into great mathematics. It's a rich story. What the CC-SMP provide us is more like a set of anecdotes. Their list isn't exhaustive, and it isn't supposed to be exhaustive--but that's not my problem. My problem is rather that boiling down one of these terrific class days into a set of three or four practice standards is, as Guare's character Ouisa warns us, turning an experience into a set of anecdotes. When we do that, we lose--the <i>math</i> loses--something essential: by being just explorers, or cross-examiners, or number-crunchers, we stop being the rich, mathematical people we were when the class was going on--and the math stops being, well, mathematics.<br />
<br />Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-51710196012423792952013-11-10T19:51:00.000-06:002013-11-11T12:00:49.345-06:00What do we learn from our students?It's been a while--partly because of work, and partly because I just found out about the death two summers ago of one of my former students, tragically in the course of mourning the (more-recent) death of another former student.<br />
<br />
I've been hesitant to write about him, but I haven't been able to write not about him either, so here goes:<br />
<br />
Like many of my colleagues and friends, I went into teaching in large part to "make a difference" in the lives of young people. And I have, but in some cases, I have to admit that the difference isn't necessarily the difference I intended to make. This young man in particular started ninth grade as an angry, somewhat-alienated wannabe skate punk; I say "wannabe" because although he was an accomplished skater, he hadn't quite worked out the "punk" part besides just being angry a lot. Something about him spoke to me: I wanted to be "that teacher" for him, the teacher who got what he was about, who saw what amazing, exciting gifts he had to offer, who helped him mediate between his anger and desires and the sometimes-irrational (and always irrational-seeming) system in which he lived and learned, who he'd talk about fondly as "the only reason I stayed in school". But when I was done intervening, he had become an extremely angry, completely alienated fifteen-year-old on his way to dropping out of high school (which I believe he did two years later, shortly after moving to another city). His final exam in my class was covered with obscenities.<br />
<br />
I was "that teacher," all right.<br />
<br />
I've gotten better about such interventions, and I share this "wisdom" so far as I can put it into words--although I think that the experiential way in which I learned is probably the only way to learn.<br />
<br />
First and most important, I've learned that you can't make students trust you. You can act to earn their trust, by being trustworthy in your actions, and by reminding them that you're there. But all you can do is open the door. To put in terms of my favorite joke: it only takes one teacher to change the lightbulb, but it has to want to be changed.<br />
<br />
On a more pragmatic front, there's a specific mistake I've decided not to make again: while I'm willing (indeed, in some sense, happy) to bust students with whom I've forged close relationships, I won't jeopardize those relationships by asking them to turn in their friends. In fact, I've decided that that question--"who else was with you?"--is just not fair, unless it's literally a matter of life-and-death. And in that case, I'd rather convince my student of the life-and-deathness of the situation rather than simply use my personal leverage to get the answer out of him.<br />
<br />
Third (and my current or recent students might laugh at this), I've learned to use a lighter touch. I'm not naturally subtle, and I've had to realize that anything I say -- in particular anything negative -- is effectively amplified many, many times (maybe I should call this the "multiplier affect"?). Most of being "that teacher" is really about listening, and waiting, rather than talking. And when talking, it's hard to underestimate the importance of being positive, positive, positive: not untruthful, not unrealistic, but as relentlessly positive as possible given those two constraints. Remember how insecure and terrified you were as a teenager? That's what I'm talking about.<br />
<br />
As teachers, much of what we teach is propositional--facts and ideas that can be put into words easily. Much of the important stuff isn't so propositional--for example, how to approach a math problem, or ways to analyze a text--and arguably the most important stuff is the stuff we don't even think of as part of the curriculum. (Ted Sizer's excellent book <i>The Students are Watching</i> is all about this last part.) But when we think about "learning from students," I think we sometimes default to the propositional mode. Sure, I'll never forget Dan's incredulity that I--who knew way more than he did about math--didn't know that hogs have to be walked. And I learned a fact: that if you're raising hogs, you have to walk them. But I also learned something much more important: that students whose knowledge in one area is only a small subset of your own can be experts in areas about which you're totally ignorant.<br />
<br />
I think the most important part of what I've learned from my students--slowly, painfully, extremely imperfectly--has to do with how to be more like the person I wish I could be when I'm teaching them. And most of that's been learned the hard way.<br />
<br />
For the record, after a couple of years of wandering in the desert--I'm reminded of Tolkien's line that "not all who wander are lost"--Constantine apparently found his way and what he was about. Friends of his friends tell me that his last years were good ones, where his creativity and energy were valued and celebrated by the people around him. I wish I'd been a part of that, of course, but even more, I wish I'd been able to see it for myself. I'm sorry I couldn't be <i>that</i> teacher.Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-37632502958632740932013-10-06T22:47:00.002-05:002013-10-06T22:53:11.859-05:00The Homework ParadoxRobert Pondiscio's <a href="http://www.theatlantic.com/education/archive/2013/09/poor-students-need-homework/279566/" target="_blank">recent article in <i>The Atlantic</i></a> argues that while upper middle-class "gifted" kids may not need homework, for children in poor or less-well-educated households, homework can be an invaluable source of intellectual stimulation. He's responding to people who--like me--wonder why our children's reading ability grows far more in two months of summer vacation than in nine months of education (the answer: she read something like a book a day when she didn't have homework to do). These observations lead well-meaning liberals--like myself--to question the value of giving any homework at all. As Pondiscio rightly points out, it's not fair or reasonable to impose on other people's children conditions (or, in this case, lack of conditions) simply because those conditions make sense for our own children. He's right about that, and also when he carefully argues<br />
<blockquote class="tr_bq">
<span style="background-color: white; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px;">The proper debate about homework – now and always – should not be “how much” but “what kind” and “what for?” Using homework merely to cover material there was no time for in class is less helpful, for example, than </span><a href="http://www.aft.org/newspubs/periodicals/ae/summer2002/willingham.cfm" style="background-color: white; color: #00598c; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px; text-decoration: none;">“distributed practice”</a><span style="background-color: white; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px;">: reinforcing and reviewing essential skills and knowledge teachers want students to perfect or keep in long-term memory. Independent reading is also important. There are </span><a href="http://gse3.berkeley.edu/faculty/aecunningham/Readingcanmakeyousmarter!.pdf" style="background-color: white; color: #00598c; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px; text-decoration: none;">many more rare and unique words</a><span style="background-color: white; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px;"> even in relatively simple texts than in the conversation of college graduates. Reading widely and with stamina is an important way to build verbal proficiency and </span><a href="http://www.aft.org/newspubs/periodicals/ae/spring2006/willingham.cfm" style="background-color: white; color: #00598c; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px; text-decoration: none;">background knowledge</a><span style="background-color: white; font-family: Georgia, 'times new roman', times, serif; font-size: 16px; line-height: 23.1875px;">, important keys to mature reading comprehension.</span></blockquote>
But I still worry about homework as an educational prescription for the poor, for a few reasons.<br />
<br />
<ol>
<li>First, I think the same factors that disadvantage the children Pondiscio says need high-quality homework also make it less likely that they will get it. These children are less likely to have access to the high-quality teachers who assign the thoughtful and thought-provoking tasks that Pondiscio praises. (For a prime example, go back to the <i><a href="http://anglesofreflection.blogspot.com/2013/08/summer-homework-challenge.html" target="_blank">Summer Homework Remix Challenge</a>,</i> or simply look at the books full of repetitive worksheets textbook publishers use to sell their series.) And they're less likely to have the academic and intellectual support at home to complete those tasks: parents who understand what a genuine science experiment is, or how to think about a complex text. This isn't racism, or classism, but simple logic: if the problem that homework is supposed to fix is inadequate intellectual stimulation at home, why would we expect those homes to provide adequate intellectual support for challenging tasks?</li>
<li>Second, especially in high school, kids from poor backgrounds are likely to have circumstances that make it hard to get homework done. Many of my economically disadvantaged students babysit for younger siblings so that their parent(s) can work, or for nieces and nephews whose own parents' income is essential for the functioning of the wider family. Others work after school--not for money to blow on an iPad, but for essentials like winter coats or even the family rent. Even those who don't have these responsibilities often lack the most basic requirement: a calm, reasonably quiet place to work. So asking these students to do large quantities of homework isn't always reasonable.</li>
<li>Items #1 and #2 point to a perverse <a href="http://en.wikipedia.org/wiki/Matthew_effect" target="_blank">Matthew Effect</a> of the exact sort that Pondiscio wants homework to overcome: poor students are often in circumstances that make it harder to get homework done, less likely that they'll get what they're supposed to out of homework, and more likely that they'll find homework just another unreasonable demand of an already-harsh educational system. At my school, inability to get homework done well is often one of the leading causes--and symptoms--of a student's failure to make a real go of it at all. So relying on homework <u>widens</u> the gap between the educationally/economically advantaged and the disadvantaged.</li>
<li>Finally, I'll admit it: I'm a skeptic. Not about homework, but about middle-class arguments that essentially boil down to saying "The poor need this, but not my children." We've done a pretty crappy job educating other people's children differently from our own, and I guess my tendency is to err--pretty far--on the other side: if I wouldn't want a school, teacher, rule, or homework policy for my child, I'm hesitant to recommend it to anyone else.</li>
</ol>
<div>
Pondiscio raises good points, and I'm not--really, I'm not--saying that we should get rid of homework for everyone. But I am wary of saying that homework is likely to solve the problem of socio-educational inequality, especially before we have a coherent way of ensuring that the students homework is supposed to help have equal access to the kinds of high-quality assignments, and homework support structures, that we'd want our own kids to have.</div>
Unknownnoreply@blogger.com4tag:blogger.com,1999:blog-537830633228785974.post-47456227690290407832013-09-22T22:12:00.002-05:002013-09-22T22:12:52.204-05:00Department Chairperson<!--[if gte mso 9]><xml>
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<br />
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Frank May hired me in 1969 to teach at Evanston Township
High School. Frank was my supervisor for the first part of my career, observed
my classes, advised me, taught me, encouraged me and was a role model for what
mathematics teachers could accomplish. </div>
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It would be hard to find two people more different than
Frank and I. Frank never raised his voice; I never lowered my voice. I
continually tried to find different ways to bring mathematics to students;
Frank stood at the board and lectured. Frank was careful and meticulous. No one
has ever described me in those terms. Frank May was a master teacher, and I was
lucky to have him as my mentor and advisor.<span style="color: #c0504d; mso-themecolor: accent2;"></span></div>
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Frank taught me to pay attention to the details. In
particular, he taught me to write complete mathematical sentences using equal
signs, stating conclusions;. he took care to ensure students understood the
underlying reasons for procedures and notation. He taught me the importance of
using correct notation as well as language in the classroom. To this day, I
can’t bring myself to describe congruent shapes as being equal. </div>
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It often strikes me, how much I learned from someone with a
dramatically different personality and point of view. Part of it is that I
always respected his knowledge, and part of it is that I am uncontrollably
drawn to people who love mathematics.</div>
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His evaluation meetings after observing my class were more
like math lessons than like criticisms of what I had done. It was clear that he
had an abiding love for mathematics as well as a deep understanding of the
subject and how students learned it. His quiet, soft-spoken manner allowed his
beliefs and understandings to gradually sink into my brain without me actually
realizing the impact he was having on my teaching. </div>
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He had a deep understanding of how topics related to each other
and of why some things were difficult for students to understand. I often went
to him when I had a difficult topic to teach, and he always started by agreeing
with me that it was a difficult topic to teach and then uncovered two or three
insights that helped me understand why students had trouble. He never told me
what I should do, nor did he tell me what he always did. Instead, he made
careful observations about why students found the particular topic difficult
and sometimes made a suggestion or two as to what might be done to help them.
He then let me figure out how to overcome student difficulties after he shed
light on the nature of those difficulties. In short, he used exemplary teaching
techniques to teach me how to become a better teacher. He would always check
back to see if what I had done had worked better, and again he would offer an
insight or two that would help me fine tune my approach, but he never insisted
that I do it his way.</div>
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<span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span>At some
point in my career I was asked to teach B.C. Calculus. It had been many years
since I studied calculus, and I was not sure I ever really understood series. I
touched base with Frank about one thing or another almost every day. He single
handedly taught me how to teach Calculus.</div>
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School bureaucracy being what it is, Evanston eliminated the
position of Department supervisor. Many of his fellow administrators retired or
moved to other institutions. Only one went back to full time teaching, Frank. <span style="mso-spacerun: yes;"> </span>He could have retired, but he wasn’t ready to
stop teaching. He could have taken a supervisory position at a different school.
Frank went back to the classroom and taught five classes a day for another ten
or so years. I think he looked at the situation and determined what would be
best for the students, for the school, and for him, and gracefully took a step
down and finished his career doing what he did best: teaching students
mathematics. <span style="mso-spacerun: yes;"> </span>Of course he did more than
that: he continued to mentor many of us even though it was no longer in his job
description. He was still the person I went to when I needed math help, and he
was still eager to talk math. </div>
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While he was a master teacher, he never boasted about his
accomplishments, but he came close once. I complemented him on the incredibly
high scores of his BC Calculus students one year. He told me that he had had
the same group for pre-calc two years in a row and so was able to prepare them
appropriately. I often wondered how successful students would have been if they
had him for four years, or if they had someone as good as him for four years.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
The reason I am writing this now is that Frank May passed
away last week at the age of 89. There was no mention of his passing in the front
page of <i style="mso-bidi-font-style: normal;">The New York Times</i> or <i style="mso-bidi-font-style: normal;">The Chicago Tribune</i>, because he was not
a famous athlete, entertainer, or politician. All he did was<span style="mso-spacerun: yes;"> </span>positively influence thousands of students
and hundreds of teachers. Rarely a week passes when I don’t reflect on one of
Frank’s insights about how people learn. He is the primary reason I have
reverence for the Parallel Postulate, the Distributive Property of
Multiplication over Addition, and the Differential. I learned from Frank that
one can exhibit enthusiasm in a quiet, dignified manner, that it is not
necessary to jump up on tables and throw things across the room to share with
students how exciting mathematics is. Let this be one small instance of tribute
to a great man. Thank you, Frank, for all you have done for our profession, and
therefore for countless people. I am forever grateful for all that you did for
me.<span style="mso-spacerun: yes;"> </span></div>
John Bensonhttp://www.blogger.com/profile/03424169848462972662noreply@blogger.com1tag:blogger.com,1999:blog-537830633228785974.post-9290901654745139772013-09-04T21:07:00.002-05:002013-09-05T13:34:40.514-05:00My daughter's good, but don't call her smart!<a href="http://www.npr.org/blogs/health/2012/11/12/164793058/struggle-for-smarts-how-eastern-and-western-cultures-tackle-learning" target="_blank">This story on NPR</a> reminded me -- again -- of how much it bothers me when we call kids smart. In fact, calling successful kids smart is one of the worst things we can do: to (and for) them, and to (and for) other kids.<br />
<br />
Many years ago I decided to stop using the word "smart" to describe my students, on the grounds that the word "smart" is so imprecise, using it is just an excuse for sloppy thinking. There are a bunch of different ways a student can be "smart":<br />
<ul>
<li>Catches on to new ideas and techniques quickly.</li>
<li>Doesn't forget things he or she has learned, even a long time ago.</li>
<li>Anticipates consequences and implications of new ideas and issues.</li>
<li>Sees generalizations; synthesizes readily.</li>
<li>Sees new applications for already-learned facts and skills.</li>
<li>Makes few, if any, mistakes in applying already-learned knowledge and skills.</li>
<li>Generates new, unusual ideas.</li>
<li>Is intellectually playful: likes wordplay, quasi-argumentative banter, hypotheticals.</li>
</ul>
<div>
After a year or two I backed off, but I still use lists like this when I'm writing college and scholarship recommendations: it doesn't help my students if my praise is essentially meaningless.</div>
<br />
The NPR story took a different tack. It compared Western and Asian parents' comments to their children, both when their children are successful, and when their children are unsuccessful. This ground has been trodden many times -- notably by Po Bronson and Ashley Merriman, in <a href="http://www.nea.org/home/42298.htm" target="_blank">articles</a> and in their awesome book <i><a href="http://www.amazon.com/NurtureShock-New-Thinking-About-Children/dp/0446504130" target="_blank">Nurtureshock</a>.</i> I'll summarize typical comments in the chart below:<br />
<br />
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<br /></div>
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Western Parent<o:p></o:p></div>
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Asian Parent<o:p></o:p></div>
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Successful Child<o:p></o:p></div>
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“You’re so smart!”<o:p></o:p></div>
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“You worked so hard!”<o:p></o:p></div>
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Unsuccessful Child<o:p></o:p></div>
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“I wasn’t good at math either!”<o:p></o:p></div>
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“You must work harder!”<o:p></o:p></div>
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There are two major takeaways from an educational-policy perspective:</div>
<div style="text-align: left;">
<ul>
<li>Teaching kids that success is a result of being smart sets them up for failure. When these kids encounter genuine struggle, they often conclude that they are simply not talented enough to be successful.</li>
<li>Teaching kids that success is a result of working hard sets them up for further success, because attributing their success to the only factor that an individual can control (as opposed to talent, luck, and ease of task).</li>
</ul>
<div>
But I'd add a third issue--one that I've encountered with my students and my own child, and especially with talented girls who work hard. These students are <u>proud</u> of their hard work, and when someone says "You're so smart" or "You're a genius," they don't experience those comments as praise. Instead, they feel that their hard work has been <u>devalued</u>, because they've just been told, "Look, you're not successful because of anything you chose to do -- you're successful because you were born that way." (The variation on this that drives my daughter bananas is "Of course you're good at math--your dad's a math teacher"--which is why she never lets me actually teach her math.)</div>
<div>
<br /></div>
<div>
So telling successful kids they're smart is bad for at least five reasons:</div>
<div>
<ol>
<li>It tells successful kids that they shouldn't get credit for their success, because that success isn't due to anything they actually did. So it's insulting.</li>
<li>It sets successful kids up for failure, because it doesn't give them anything to fall back on when they encounter challenging tasks.</li>
<li>It tells unsuccessful kids that they can't do anything to become successful, because the successful kids are the ones who are already smart. So it's implicitly insulting to unsuccessful kids: you're not doing well because you're dumb.</li>
<li>It sets unsuccessful kids up for continued failure, because--in this worldview--there's nothing they can do to become smarter.</li>
<li>It's an inexcusably-sloppy way for a teacher to describe a student, because it doesn't say anything about <i>what</i> the student actually does.</li>
</ol>
<div>
So please do recognize your successful students' (and children's) achievements--just don't call them smart.</div>
</div>
</div>
Unknownnoreply@blogger.com4tag:blogger.com,1999:blog-537830633228785974.post-86499788900183176052013-08-27T21:33:00.002-05:002013-08-27T21:33:54.197-05:00Thoughts on CollaborationIt's the first week of school, and I put in some time rewriting my course policies to articulate why, when, and how to collaborate on problem sets. This issue is particularly crucial in my advanced Geometry class, where students solve college-level problems, not just exercises. Feel free to use some version of this in your own classes, and -- even more important -- to let me know what you think!<br />
------------------<br />
<br />
<div class="MsoBodyText">
<u><span style="font-size: 11.0pt;">Collaboration, Research,
and Hard Problems</span></u><span style="font-size: 11.0pt;"><o:p></o:p></span></div>
<div class="MsoBodyText">
<br /></div>
<div class="MsoBodyText">
<span style="font-size: 11.0pt;">Contrary to stereotype,
mathematics is best done as part of a community—not alone in a study. I hope and expect that students will
frequently discuss problems, ideas, and solutions in and out of class. Some guidelines:<o:p></o:p></span></div>
<div class="MsoBodyText">
<br /></div>
<div class="MsoBodyText" style="margin-left: .5in; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 11.0pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span style="font-size: 11.0pt;">Make sure that
when you are “working with” someone else, you are both really working and
contributing. Contributions can take
many forms—clarifying, questioning, justifying, and restating, to name a
few—not just coming up with “the idea”, but each of you should leave the
collaboration feeling good about what you, and the other person, contributed to
the session.<o:p></o:p></span></div>
<div class="MsoBodyText" style="margin-left: .5in;">
<br /></div>
<div class="MsoBodyText" style="margin-left: .5in; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 11.0pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span style="font-size: 11.0pt;">One test for how
much you did in solving a problem is whether you can reconstruct the entire
solution on your own afterwards. If you
need to look at your notes extensively, or get lost in the middle, you probably
should have collaborated more actively.<o:p></o:p></span></div>
<div class="MsoBodyText" style="margin-left: .5in;">
<br /></div>
<div class="MsoBodyText" style="margin-left: .5in; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 11.0pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span style="font-size: 11.0pt;">One time when it
is useful to be alone in your study is to check for your own
understanding. I recommend that students
<u>start</u> problem sets by themselves, to see what they can accomplish and
what ideas they can generate individually before working with other
students. I also recommend that students
<u>finish and write up</u> problem sets by themselves, to make sure that they
really understand the work they and their fellow students did together.<o:p></o:p></span></div>
<div class="MsoBodyText" style="margin-left: .5in;">
<br /></div>
<div class="MsoBodyText" style="margin-left: .5in; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 11.0pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span style="font-size: 11.0pt;">Give credit
generously: it doesn’t subtract from the points you get (and in the real world
of “karma points”, giving credit almost always adds to your own). Write “Wolfgang suggested this auxiliary
line” or “Credit to Seraphina for spotting the similar triangles.” Taking ideas from other people without
attribution is plagiarism; taking ideas <u>with</u> attribution is research.<o:p></o:p></span></div>
<div class="MsoBodyText" style="margin-left: .5in;">
<br /></div>
<br />
<div class="MsoBodyText" style="margin-left: .5in; mso-list: l0 level1 lfo1; text-indent: -.25in;">
<!--[if !supportLists]--><span style="font-family: Symbol; font-size: 11.0pt; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;">·<span style="font-family: 'Times New Roman'; font-size: 7pt;">
</span></span><!--[endif]--><span style="font-size: 11.0pt;">Finally, unless
specified in a particular project or assignment instruction, please DO NOT do
research on the web (or in books, if you still remember those). The essence of this course is learning to
reason mathematically by solving problems and thinking through solutions, not
regurgitating theorems and ideas you learned somewhere else. <o:p></o:p></span></div>
Unknownnoreply@blogger.com0tag:blogger.com,1999:blog-537830633228785974.post-30062278898757074092013-08-12T21:26:00.004-05:002013-08-13T07:39:20.571-05:00Summer Homework Remix ChallengeSo I'm just getting back from three weeks teaching at <a href="http://www.hcssim.org/" target="_blank">HCSSiM</a> (which in case you don't know about it, is an absolutely awesome summer math program for high school students), and one thing that really leapt out at me was the number of students trying to cram in several hours of summer homework on top of (literally) 45+ hours of mathematics each week. (You read that right: four hours of class each morning, quasi-optional lecture at 5pm, three hours of problem session from 7:30-10:30pm.)<br />
<br />
This situation is appalling.<br />
<br />
First, the kinds of high-performing kids who go to HCSSiM are working harder than ever during the year; they need a break from regular work. Second, the work that gets sent home is--in general--the worst sort of homework you can imagine. A sampling:<br />
<br />
<ul>
<li>Outline <i>n</i> chapters of a bio/history/government textbook, where <i>n</i> ≥ 3. My experience is that few kids, if any, are taught how to outline, so these wind up being lists of section headings. Teachers typically don't even read or give feedback on this work. I wonder, too: if a kid can get "enough" out of this type of learning experience, what do teachers think the are adding in the actual classroom?</li>
<li>Write out, by hand, 100 vocabulary words and definitions. (My colleague and friend Erica, soon to be a teacher in a Massachusetts middle school, asked "If the work is so low-level that it's impossible to tell whether one kid has copied another's assignment, why would you even assign it?"</li>
<li>Read a 400 page, mostly stream-of-consciousness novel chosen by the teacher, with little or no guidance.</li>
<li>Fill out dozens of worksheets practicing math facts, or vocabulary from a foreign langauge, or ...</li>
</ul>
<br />
Assignments like these replicate the worst parts of the school experience--repetitive, low-level work with little attention to context or purpose besides "get it done"--without any of the other experiences that can make going to school worthwhile. What's more, they put this school-sucked-dry experience <i>into the summer,</i> which kids typically enjoy.<br />
<br />
Why do kids who enjoy learning prefer summer to school? I can think of three reasons:<br />
<br />
<ol>
<li>During the summer, kids can choose what they will do and when they can do it. <br /><br />Of course, choice is more satisfying than constraint. But study after study has shown that giving kids choices about what learning activities they do increases student engagement and the activities' effectiveness. So taking the choice out of summer learning activities is a double-whammy.</li>
<li>During the summer, kids do things that they find relevant.<br /><br />In actual summer academic programs--like the ones my kids and my friends' kids do--the learning activities are focused on things kids actually want to know about. And kids don't choose to do things that they find irrelevant.</li>
<li>During the summer, kids do things that are challenging.<br /><br />Have you seen a kid spend hours practicing a skateboard move, or throwing a football, or playing a videogame? Kids don't do things they find easy--they find things that are at a "can't-quite-do-it-yet" level, and when they've mastered something, they move on. </li>
</ol>
By contrast, the summer homework assignments listed above involve no choice, make little attempt to be relevant to kids' own interests, current issues, or anything else of interest, and are very much one-size-fits-all (in fact, because no teacher is available, they're usually way too easy--which reinforces the idea that they're mostly "get-it-done" work, not an opportunity to actually learn anything).<br />
<br />
So here's my remix challenge, in three parts.<br />
<br />
I. Find an actual summer homework assignment passed out to kids in grades 6-12.<br />
II. Rework it to fit the three criteria--choice, relevance, adaptable challenge.<br />
III. Post it here, in the comments, or via email to me at pjkarafiol@gmail.com.<br />
<br />
Three guidelines:<br />
<br />
<ul>
<li>Obviously, the made-over assignment must address many of the same objectives and issues as the original.</li>
<li>Obviously, the assignment must be one that students can complete, with some reasonable degree of success, on their own. The assignment doesn't have to be easy, but (e.g.) feedback might be built-in or easy to obtain.</li>
<li>Third, the assignment shouldn't take more than 4-5 hours of work, 8 tops. Really, guys, this is the summer. If you want the kids to attend summer school, teach an actual class.</li>
</ul>
<br />
As extra credit, ask a student to suggest a summer homework "makeover" of their own.<br />
<br />
Here are two to get you going:<br />
<br />
1. For the AP Biology Chapter 1 assignment on basic physics and chemistry: read two articles from one of the following periodicals (Scientific American, New York Times Science Tuesday, Discover Magazine) written within the last year about a discovery or problem in biology. For each, identify what chemical or physical processes are described in the article, and be ready to give a short (3-5) minute presentation on the underlying chemistry or physics described.<br />
<br />
2. For the vocabulary list -- given the same list of words, find instances of 20 of these words in recent writing (last five years) on the web, in periodicals, or published books. For each, give the surrounding paragraph, explain what the word means in context, and write a sentence or two evaluating whether the author should have chosen a less-esoteric word (with a suggestion).<br />
<br />
Happy end of summer!<br />
<br />
<div>
PS: Full Disclosure -- I havea a summer assignment of my own, but it involves choice, is not onerous, and shouldn't take more than 3-4 hours to complete. Here it is: <span style="-webkit-composition-fill-color: rgba(175, 192, 227, 0.230469); -webkit-composition-frame-color: rgba(77, 128, 180, 0.230469); -webkit-tap-highlight-color: rgba(26, 26, 26, 0.292969); font-family: '.HelveticaNeueUI'; font-size: 15px; line-height: 19px; white-space: nowrap;">http://www.wpcp.org/StudentLife/SummerAssignments.aspx</span></div>
Unknownnoreply@blogger.com1tag:blogger.com,1999:blog-537830633228785974.post-28111830863267059452013-07-01T07:52:00.002-05:002013-07-01T07:52:55.302-05:00QuietI suggest adding the book Quiet by Susan Cain to your summer reading list.<br />
<br />
I read it at the suggestion of a former student. It is well written, researched, and short.<br />
<br />
It is not specifically about education but it is about assumptions our culture makes about quiet people. We all have quiet people in our classrooms. She challenged many of the assumptions that I had made about quiet people and forced me to rethink some of my long held beliefs about structuring my classes. She raises some interesting and profound ideas about things like group work, whole class instruction, brainstorming, problem solving, calling on students, and the role each student plays in the dynamic of the classroom.<br />
<br />
I also found it insightful with regard to interactions with friends, neighbors and colleagues.<br />
<br />John Bensonhttp://www.blogger.com/profile/03424169848462972662noreply@blogger.com1tag:blogger.com,1999:blog-537830633228785974.post-69260504489626010062013-06-27T01:02:00.000-05:002013-06-27T01:02:03.643-05:00The Four Most Important Words in TeachingMy co-blogger, John, often mentions that a crucial part of his practice is standing at the door, greeting students as they come in. Although this practice started as a way to keep order in the halls, for him it's persisted because it gives him an opportunity to check in with students individually. I don't have anything like that kind of discipline, but I have to agree that the 3-5 second "touch" is incredibly important, wherever and however you make it happen. So my candidate for the four most important words is "How are you doing?"<br />
<br />
As a math teacher, I don't "get" the opportunity to talk with my students about their personal lives; I have to make that opportunity. But I think that the kids who most need to talk are often the most fearful of actually opening up; the biggest secrets are just below the surface. "How are you doing?" is a low-stakes way of saying "I'm interested, and if you want to talk, we can." When a student has already opened up to me, "how are you doing?" is really a statement: "I know it's been rough, and I'm concerned." It doesn't demand an extended exchange. "Not so great thanks" -- "I'm sorry; find me at lunch if you want to talk" is almost always as long as it gets: five to ten seconds.<br />
<br />
It's easy to misjudge the amount of effort needed to care for our students' social and emotional health on the basis of that very small number of students whose drama is like a riptide, dragging in friend after friend and teacher after teacher. But most students aren't like that. And I find that especially when a student is in crisis, or just coming out of one, regularly asking "How are you doing?" makes a huge impact--probably more than an hour-long "session" would, at least with this nonskilled practitioner.<br />
<br />
This impact was demonstrated to me by an unusual coincidence during this last week of school. I said goodbye to two boys who had been going through rough times this year (one just for the summer, one who is graduating); both were practically in tears. One wrote in my yearbook that my checking in with him had made it possible for him to finish school and graduate, and he meant it. Literally--I promise you--no conversation with this student had lasted more than five minutes. Then our graduation speaker was an alum who, five years ago, had told his entire class and their parents (he was the graduation speaker at his own graduation, too) that my "How are you?"'s had been a lifeline during difficult times. (And again, at the time, I was bowled over: none of these conversations lasted more than 3 minutes, and only a few were even half that long.)<br />
<br />
When I first started teaching, I envisioned long, soulful one-on-ones with students about all the bumps and pitfalls of adolescence, of which I'd had my own share. But I've come to realize that most students don't want those conversations most of the time, and that they can't be forced. Two of my mentors, at Andover, pointed me in the right direction. Craig Thorn (beloved house counselor and English department chair) told me his secret the day I arrived: just be around, in their rooms or wherever, so that they see you and know you and talk around you. Doug Kuhlmann, who was the math department chair, said something like this: when you ask a student a personal question, you need to be aware of your own stake in the answer. What he meant, roughly, was that when you ask a student a personal question, you need to be aware of your own reasons: are you asking because talking will help the student, or for the emotional buzz of reinforcing your relationship with the student, or to validate your own self-image as "the teacher who cares." It's easy, he warned, to think of yourself as asking for the student's benefit, when really it's about you, which is problematic: as an authority figure, you're in a position to demand a response, even when you shouldn't. <br />
<br />
"How are you doing?" largely avoids this pitfall: asked in the hallway, or when you're checking in homework, or outside the lunchroom, it <i>doesn't</i> demand much, if anything, of the student. Just saying "Okay" is enough--in fact, it's standard protocol. The first couple of times, I might follow up with "Really? Doing okay?" -- to indicate that I am really asking, not just following the script. But then it's up to the student. So "How are you doing?" is empowering, not disempowering: it says "Remember, I'm here if you need me, but I'm not going to push it if you don't." And it's a "touch" I can make in front of other students, who may or may not know the backstory.<br />
<br />
With 140 students per teacher, it's easy to fret about how little we can do. But what I take away is that it only takes a little. The key is -- to circle back to John -- to keep asking, to make it a regular routine. The kids who thank me for it later remember this: I asked them every time. And that regularity--and the care behind it--mean more than you'd expect. Unknownnoreply@blogger.com2tag:blogger.com,1999:blog-537830633228785974.post-29577079764326899142013-06-11T12:39:00.000-05:002013-06-27T01:02:47.994-05:00ImprovementIn my first several years of teaching, when things did not go as expected, I thought about how to improve results. This [what] often meant adding another rule or expectation [to what], because my students were not doing what I perceived they needed to do in order to learn. After a few years, I had created a bureaucracy that was unmanageable for me as well as for my students. The worst part was that these rules and expectations had not helped improve instruction.<br />
<br />
I came to realize that my job was not to tell students what to do. My job was also not to show them how to do a problem. My job was to create interesting situations where they could think about mathematics and learn from the work and discussions that followed. My job was to assist students in anyway I could to develop their own understandings of important mathematics. I have documented these thoughts and processes many times in earlier posts. Here is a new thought for me.<br />
<br />
This same process applies to teacher improvement. Those in charge of teachers at the local, state and national level are ill served by piling on more and more rules and mandates about how teachers should teach. If they really want to improve instruction, they must follow the same model that I followed in my classroom. The supervisors need to see that their job is to do whatever they can to facilitate teacher learning (therefor student learning), as opposed to requiring teachers to do certain things in a certain way. Teachers, with proper resources, will find ways to reach students. As things stand now, those in charge are working very hard to make a teacher's job as hard as possible.<br />
<br />
Administrators need to observe teachers teaching. Administrators need to listen to what teachers have to say about the difficulties they have, and administrators need to work hard to help the teachers solve their problems. I have had to good fortune of working for a few such supervisors, and it makes more of a difference than I would have ever imagined. And the very best supervisors worked hard to help the top-level administrators understand that learning to teach well is a very difficult process, that it takes time, and that it requires support and nurturing. Learning to teach well does not require demands and punishment on the part of the Administration.<br />
<br />
It is part of the job of experienced classroom teachers to facilitate this process with newcomers and to help administrators understand what they need to do to be effective.<br />
<div class="MsoNormal">
<o:p></o:p></div>
John Bensonhttp://www.blogger.com/profile/03424169848462972662noreply@blogger.com1