tag:blogger.com,1999:blog-87226409132344617732024-03-26T00:57:48.122+11:00exzuberantExplorations in teaching and learning mathematics at high schoolenzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.comBlogger128125tag:blogger.com,1999:blog-8722640913234461773.post-84919487542974673872019-10-20T10:04:00.000+11:002019-10-20T11:16:14.489+11:005 reasons why Japan is the perfect term break holiday for teachersI'm addicted - I just can't stop travelling to Japan! Here's why Japan makes for the perfect term break holiday for teachers - especially if you live in Australia.<br>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqQQNfrEydtcHDEmg0TtS4XNuRfH0reCgCe4BrXSUXU-qyb0n3byqCCGhWW6zmjH4nNvarh42gq7CqQKcxhOO2jtXjTdoNm5VL1Ufap5wMr1vsRXPzWn7d69t0MHgAESA3Zvf_ZlztmZ4z/s1600/20170930_163615.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="900" data-original-width="1600" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqQQNfrEydtcHDEmg0TtS4XNuRfH0reCgCe4BrXSUXU-qyb0n3byqCCGhWW6zmjH4nNvarh42gq7CqQKcxhOO2jtXjTdoNm5VL1Ufap5wMr1vsRXPzWn7d69t0MHgAESA3Zvf_ZlztmZ4z/s400/20170930_163615.jpg" width="400"></a></div>
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1. Your 10 day break feels like a month of holidays</h3>
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Japan is <i>so </i>different from your normal routine, every hour of your break has the potential to be novel and different, if you want it to be. After a day or two in Japan, you will feel like you've been away for weeks. One of the challenges of being a teacher is you are so exhausted by the end of term, you really need to be refreshed and re-energised, but those two weeks can go very fast. A trip to Japan is so engaging, you really do feel like you have been away for a very long time. You'll be reliving your Japan holiday for at least a month or two into the next term ... just in time to plan your next holiday to Japan!</div>
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<tr><td class="tr-caption" style="text-align: center;">The famous Inari Temple in Kyoto. The classic Tokyo-Kyoto-Osaka route is the perfect first visit. It will be very busy depending when you go, but a great introduction. Once you see how things work in Japan, you'll be able to visit regions which surprisingly have almost no foreign tourists.</td></tr>
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Another terrific bonus for Australian teachers is the timezone difference to Japan - in effect there is virtually no timezone difference. Which means no loss of time for those precious 10 days because of jet lag! Indeed, once you are experienced, you could leave school on Friday afternoon and head to the airport, and return back home the Sunday morning two weeks later, just before the next term starts.<br>
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And as a final incentive, if you need one, a holiday in Japan means you can start or end each day with a super relaxing hot bath at an onsen - either in your hotel, or at public onsen nearby. It's like you get a spa holiday thrown in as a bonus. Never miss a chance for an onsen on a day trip if there is one featured as part of the visit. A little bit embarrassing the first time, but once you get over it - absolute bliss for the tired teacher.</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbMgZ5MoJ2eDg84qNgPGZOwwjdqBfz6HBA-YOw2R0Jq42m_jcEBpJP9cNcSavhJF-mjyzCJx1NedoDafnAN9bsaWmsxDJy9CW_B-S98m6V6DXDY1WOdcoLbV0yOLz938AlSXUxG5uOH0Ht/s1600/IMG_20190421_073653.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="424" data-original-width="1600" height="105" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbMgZ5MoJ2eDg84qNgPGZOwwjdqBfz6HBA-YOw2R0Jq42m_jcEBpJP9cNcSavhJF-mjyzCJx1NedoDafnAN9bsaWmsxDJy9CW_B-S98m6V6DXDY1WOdcoLbV0yOLz938AlSXUxG5uOH0Ht/s400/IMG_20190421_073653.jpg" width="400"></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The onsen in a small local hotel far from the regular tourist path. Have a wash, then soak in bubbling hot water while looking outside at a beautiful Japanese garden. Onsens are the perfect antidote for Term 3, Week 9 & 10.</td></tr>
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2. Travel in Japan is really easy - once you get used to it</h3>
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As a teacher who has just survived through to the end of Week 10, you really really need a relaxing break, not a stressful one. The first time you go to Japan you'll be in culture shock for about two days. Then once you calm down, you will realise just how tourist friendly Japan is. You begin to notice all the announcements on public transport are in four languages (Japanese, English, Mandarin and Korean) not one, you realise that with some help from Google Maps, <a href="http://www.hyperdia.com/">HyperDia </a>,<a href="http://japan-guide.com/"> japan-guide.com</a> and even (heaven forbid) Trip Advisor, you can pretty much work out how go anywhere or do anything. Work out how to buy and use a transport card (so easy!) and the train and bus networks open up to you hassle free. There are official tourist information booths and help everywhere. People are so helpful and friendly (if a little shy sometimes) - you are never alone. Oh, and did I mention that<i> everything works </i>? If the train timetable says the train is there at 9:04, it will be. As you get a little more confident, you can move off the classic tourist routes and the country opens up for exploration. And it's surprisingly easy to hire a car in more remote regions, with good discounts if you book wisely. With a little gentleness and patience, Japan is really easy to get around with only English language skills.</div>
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<tr><td class="tr-caption" style="text-align: center;">Walking the <b>Kumano Kodo </b>in Wakayama Prefecture. Options from 1 to 6 days following a 2,500 year old pilgrimage trail. Stay overnight in small Japanese homestyle inns, get your bags transported overnight, enjoy a spectacular walk. Incredibly cheap if you arrange it yourself using the <a href="http://www.tb-kumano.jp/en/kumano-kodo/">Tanabe City Kumano Kodo</a> website.</td></tr>
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3. There is something for every season, for every interest</h3>
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Whatever your interest - cities, culture, food, the outdoors - Japan has it. For starters, there are at least seven completely different regions of Japan, and don't forget the islands way down South in the Pacific. Once you step outside the Toyko-Kyoto-Osaka triangle, you'll discover so many different types of holidays in Japan. You could easily do two, or even three, trips to each region and have a completely different experience. And unlike Australia, Japan has <i>seasons. </i>So your April visit to Japan will be completely different to your September/October visit (but watch out for typhoons - go somewhere with less of them at that time.) As a teacher with predetermined times for the term breaks, sadly you won't be able to time your visits perfectly for the sakura (cherry blossoms in spring) or the autumn leaves, but with some careful planning can optimise what you do see (hint: choose several lattitudes and/or elevations for your trip - at least one of them will be the right time and place).<br>
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<tr><td class="tr-caption" style="text-align: center;">Autumn leaves in Northern Hokkaido (mid October).</td></tr>
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4. You (probably) have an in-house Japan guide at school</h3>
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If you're a high school teacher, and given that Japanese is the mostly widely taught foreign language in Australian schools, you likely have your very own in-house Japan travel advisor at school : the Japanese teacher. When planning your first trip to Japan, you might be a little anxious - so go talk to your Japanese teacher. My lovely Japanese teacher at school helped me with itinerary planning, and then while I was in Japan, kept on giving me advice and travel tips via Facebook. And it turned out our Music teacher also was a big fan of travelling to Japan. So there was a whole network at school to help plan trips to Japan and share experiences. </div>
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<tr><td class="tr-caption" style="text-align: center;">Egg dessert display at a Toyko supermarket. <br>
The food basements of Japanese supermarkets are a tourist attraction in their own right.</td></tr>
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5. Added value for teachers : taking the next step</h3>
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There is a good chance that after your first visit to Japan, you might be tempted to consider learning some Japanese. And again, because you're a teacher, you're in luck. If your school teaches Japanese, there's a colleague who will most likely be super eager to help you out. When I got back after my first trip, our Japanese teacher gave me permission to sit in with her Year 7 class (with luck the timetable mostly worked out) to learn Hiragana (the first Japanese script to learn). So many added benefits : you get more out of your time in Japan, and your students get to see you learn. I find now I often bring my Japanese learning into the maths classroom - we often talk about "maths kanji" now.</div>
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Build up your relationship with Japan and the Japanese teaching and learning at your school and you may be surprised where it leads. I was extremely fortunate to be invited to help supervise a student exchange visit to Japan last year - giving me the chance to spend time in a Japanese school with Japanese teachers.</div>
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So ... as I contemplate my <i>sixth</i> trip to Japan, I can only urge you, if you haven't got the bug yet, go talk to the Japanese teacher now!</div>
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</div></div><a href="http://exzuberant.blogspot.com/2019/10/5-reasons-why-japan-is-perfect-term.html#more">Read more »</a>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-79083669921442620782019-05-26T00:32:00.002+10:002019-05-26T00:37:58.470+10:003 reasons why it's unwise to draw Bloom's Taxonomy as a pyramidIn my <a href="https://exzuberant.blogspot.com/2019/05/how-do-we-visualise-blooms-taxonomy.html">last post</a> I looked the different ways Bloom's Taxonomy is presented in diagrams, and by far the most common is the famous <b>Pyramid of Bloom's Taxomony</b>.<b> </b>Here are 3 reasons why using the pyramid form can lead us to confused thinking about teaching and learning.<br />
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<b>Problem #1. A misunderstanding of the role of knowledge leads down the path of technological dependency. </b><br />
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Pyramid thinking has led to this type of "21st Century Learning" discourse:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhROB_g6wUYP180FJ5aTSh7-kDZsVZtZon2eJpE7i1lJ1kzGTS6tXXFUQuM6uR1ULQeM8PW6LK_OYi-X_IrklJiXzZOf0eVW1eFS9FvERkfpqZ6nvCMW1b3lXV5hXGllD4GQe5-7e8kWNJ-/s1600/bloom-new-newA.png" imageanchor="1"><img border="0" data-original-height="401" data-original-width="456" height="281" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhROB_g6wUYP180FJ5aTSh7-kDZsVZtZon2eJpE7i1lJ1kzGTS6tXXFUQuM6uR1ULQeM8PW6LK_OYi-X_IrklJiXzZOf0eVW1eFS9FvERkfpqZ6nvCMW1b3lXV5hXGllD4GQe5-7e8kWNJ-/s320/bloom-new-newA.png" width="320" /></a></div>
We remove the human element from "remembering", and indeed as we increasingly see in student work, there is an over reliance on Google and Wikipedia for analysis and evaluation. I don't think we are far off from the next phase where Google Assistant, Siri and other tools do most of this thinking for us,<br />
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<tr><td class="tr-caption" style="text-align: center;">...</td></tr>
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and finally it we just throw in the towel:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1Ak66vnKhwvpJcnkh6gsS2yRtVL1o0oQLSyzJ1gNp1CA0tqiOcomE86_Ou_wLVsp5qwD8T6EqoFU6OLa2b_Qn2S5jBdeO8FownwpeXGvzGHGHjaJzNRFIpsGz9n77JdTWQesEbXFs0S64/s1600/bloom-new-newC.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="405" data-original-width="456" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1Ak66vnKhwvpJcnkh6gsS2yRtVL1o0oQLSyzJ1gNp1CA0tqiOcomE86_Ou_wLVsp5qwD8T6EqoFU6OLa2b_Qn2S5jBdeO8FownwpeXGvzGHGHjaJzNRFIpsGz9n77JdTWQesEbXFs0S64/s320/bloom-new-newC.png" width="320" /></a></div>
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Yes, this may be an exaggeration, but people are now seriously considering that sophisticated software, built on a large knowledge depository (hmm, funny that) will soon be as creative, if not more so, than humans. Anyone who thinks by focusing on creativity we will "future proof" students is in for a rude shock as the AI software goes to the next level.<br />
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<b>Problem #2 : Thinking there is just one pyramid</b><br />
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Pyramid thinking can quickly become monolithic - the concept that there is one, all embracing pyramid which integrates all knowledge and all cognitive processes.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgIRvlJXn9MGBitioNffQjJ4Vo8XwqaU6j-0eqClnOZjY7eBt3Qbf_4Xyc3kfkC_3mkH50EFG2eJInROCe8Fecwbx0MOCPDEzAUFYJBmpq8Bn0_VHUchJMME-TIvd5ffAihe1hYtLPQcl-/s1600/bloom-new-newD.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="630" data-original-width="604" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgIRvlJXn9MGBitioNffQjJ4Vo8XwqaU6j-0eqClnOZjY7eBt3Qbf_4Xyc3kfkC_3mkH50EFG2eJInROCe8Fecwbx0MOCPDEzAUFYJBmpq8Bn0_VHUchJMME-TIvd5ffAihe1hYtLPQcl-/s320/bloom-new-newD.png" width="306" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The ONE TRUE "integrated" pyramid of cognition</td></tr>
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In this view, there is one large pool of knowledge to "remember" and learning how to "understand", "evaluate", "create" is a transferable skill that can be applied to all the knowledge in the "one pyramid". All the cognitive science papers I have read suggest this is just not true. Developing creative skills in music isn't going make me a creative essay writer or a creative mathematician. Both knowledge and cognitive skills are domain specific.<br />
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<b>Problem #3 : The pyramid completely misrepresents the revised Bloom's Taxonomy</b><br />
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The 2001 revision of Bloom's Taxonomy by Anderson & Krathwohl involved so much more than changing the words from nouns to verbs. The 2001 work corrected a major flaw in Bloom's original taxonomy : the conflation of knowledge (bottom level of the pyramid) and cognitive processes (the rest of the pyramid).<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHtmqpDABPNd2EpZXTiU9gZ_nGgrbWgwbTcdWBT2C0NHfdHmN7aorXdZuQ6g1z1DGxb6XJeosvW5p7sTnbrW1tp6N8T_na0P4EJEo3Yc54PnMr7OO0fsZt4wi8ZrzSHBDBz3f4r_Lw7fp5/s1600/blooms+real.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="251" data-original-width="702" height="113" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHtmqpDABPNd2EpZXTiU9gZ_nGgrbWgwbTcdWBT2C0NHfdHmN7aorXdZuQ6g1z1DGxb6XJeosvW5p7sTnbrW1tp6N8T_na0P4EJEo3Yc54PnMr7OO0fsZt4wi8ZrzSHBDBz3f4r_Lw7fp5/s320/blooms+real.jpeg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Two explicitly delineated dimensions<br />
in the 2001 revised Bloom's Taxonomy.<br />
Nothing like a pyramid!</td></tr>
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The pyramid representation only provides a one-dimensional view of the new taxonomy, <i>completely omitting the knowledge dimension. </i>Without the knowledge dimension explicitly in view, we are at risk of making category errors focusing too much on the development of skills at the expense of building knowledge.<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQJSEAM3n1GAMmscOwHjq8HB0Z8njHvIC1BaDtmKZmKcoa54LYEld_fhT5RslweCrbEWWBuvRUTcDz6bnQ53dmyRhjC4Dyl3LQxjTxwd-Dyi6rJChMIqAeJ8a3l_fkH8leezTRvydxIvGg/s1600/broken-pyramid.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="346" data-original-width="540" height="205" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQJSEAM3n1GAMmscOwHjq8HB0Z8njHvIC1BaDtmKZmKcoa54LYEld_fhT5RslweCrbEWWBuvRUTcDz6bnQ53dmyRhjC4Dyl3LQxjTxwd-Dyi6rJChMIqAeJ8a3l_fkH8leezTRvydxIvGg/s320/broken-pyramid.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Pyramids are just so 26th Century BC<br />
it's time for a new diagram.</td></tr>
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In terms of graphic design and correctness, this diagram is far better:<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhYBqROT7s6ND4kg5ZV0MYWqXIA3DXF2jq4z1W-ZBH-pt4DpLHX-6hNFITRiKQxjd7cvgDrI7qR2ctM386SmhAr-vu-Wpy3BsgHz5W9ukiIYVujoQLPqZofptnb9mXFgxva_5p-y3VILN0/s1600/bloom+purple+circle+chart.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="355" data-original-width="351" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhYBqROT7s6ND4kg5ZV0MYWqXIA3DXF2jq4z1W-ZBH-pt4DpLHX-6hNFITRiKQxjd7cvgDrI7qR2ctM386SmhAr-vu-Wpy3BsgHz5W9ukiIYVujoQLPqZofptnb9mXFgxva_5p-y3VILN0/s320/bloom+purple+circle+chart.jpg" width="316" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://teachingcommons.lakeheadu.ca/blooms-taxonomy-21st-century-learners" style="color: #888888; text-decoration-line: none;">https://teachingcommons.lakeheadu.ca/blooms-taxonomy-21st-century-learners</a></td></tr>
</tbody></table>
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<i>In my next and final post on this topic, I'll consider how Bloom's (revised) Taxonomy and indeed Bloom himself, is worthy of being considered a key player in the development of the Knowledge Based Curriculum.</i>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2tag:blogger.com,1999:blog-8722640913234461773.post-84114815607349087872019-05-18T15:12:00.003+10:002019-05-26T00:41:05.261+10:00How do we visualise Bloom's Taxonomy?<div class="separator" style="clear: both; text-align: left;">
I've been thinking a lot about Bloom's Taxonomy lately and just how influential it is to teachers' thinking. The more deeply I look into Bloom's Taxonomy, the more surprising things I find. However before we go there, here's a question for my teacher friends : If I say "Bloom's Taxonomy", what image do you form in your mind?</div>
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Now compare your mental image to the great Google mind....</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjC14U0LuTWiCAB24qhAHAKVtejFpWzg1W1RHKcXG1_w_-D1YGhuTIDr_4HXuyNGyOu9YlPbiMVF9KUpJ-s30j60VyeNAWPMZ-MKxveWwDhx6ONid26vCTHkNjJsu9OqWYSMQ3Jbxm1TBy/s1600/google-images-bloom.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="661" data-original-width="1196" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjC14U0LuTWiCAB24qhAHAKVtejFpWzg1W1RHKcXG1_w_-D1YGhuTIDr_4HXuyNGyOu9YlPbiMVF9KUpJ-s30j60VyeNAWPMZ-MKxveWwDhx6ONid26vCTHkNjJsu9OqWYSMQ3Jbxm1TBy/s400/google-images-bloom.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Result of a Google Images search for "Bloom's Taxonomy"</td></tr>
</tbody></table>
<div style="text-align: left;">
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<div style="text-align: left;">
Based on this search, I'm predicting y<b>our mental image is a pyramid.</b> With labels like "Facts", "Knowledge" or "Remember" at the bottom rung, and perhaps "Synthesise" or "Creativity" at the top. And there's an implied, or maybe even explicit, upwardly pointing arrow. And that's how I remember Bloom's Taxonomy. </div>
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I couldn't help myself and did a frequency analysis of the first 60 images thrown up by Google. Just under <b>50% were pyramid versions,</b> either the classic version:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicLq6M1odEjGsDbk-9AKKmV14jd_Ya2AeMKnQXyFwvOaIyqEZZ4T-eSwkDJl29qQDMbfRRGrcik96F1IqZoSifq0GJTsOhZ4hS4TkXJwwqCkADIBziXqaRmCcC9Yw6tllvg0PltJmsmJft/s1600/blooms_old.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="502" data-original-width="587" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEicLq6M1odEjGsDbk-9AKKmV14jd_Ya2AeMKnQXyFwvOaIyqEZZ4T-eSwkDJl29qQDMbfRRGrcik96F1IqZoSifq0GJTsOhZ4hS4TkXJwwqCkADIBziXqaRmCcC9Yw6tllvg0PltJmsmJft/s320/blooms_old.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Source: Learn NC, <a href="http://www.learnnc.org/lp/pages/4719" style="background-color: #f7f7f7; color: #026770; font-family: "Century Gothic", AppleGothic, Arial, sans-serif; font-size: 11px;">“Bloom’s Taxonomy,”</a><span style="background-color: #f7f7f7; color: #666666; font-family: "century gothic" , "applegothic" , "arial" , sans-serif; font-size: 11px;"> used under </span><a href="https://creativecommons.org/licenses/by-nc-sa/2.5/" style="background-color: #f7f7f7; color: #026770; font-family: "Century Gothic", AppleGothic, Arial, sans-serif; font-size: 11px;">a Creative Commons license</a><span style="background-color: #f7f7f7; color: #666666; font-family: "century gothic" , "applegothic" , "arial" , sans-serif; font-size: 11px;">.</span></td></tr>
</tbody></table>
<div style="text-align: left;">
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or the updated 2001 Anderson & Krathwohl version:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijZup25hrBMymA73oMB6HkwlCiS-Jf9bnW3PDInCvuhqacZUcFTAF0iBxMMVW1GU8izeyMMzecuMU35toc0-RYSCcFQTqRD5rHlnMelFZrzWCGCMMHyue8pM0vrZtncJai0xh1NH5pyNBs/s1600/blooms_new.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="502" data-original-width="585" height="274" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijZup25hrBMymA73oMB6HkwlCiS-Jf9bnW3PDInCvuhqacZUcFTAF0iBxMMVW1GU8izeyMMzecuMU35toc0-RYSCcFQTqRD5rHlnMelFZrzWCGCMMHyue8pM0vrZtncJai0xh1NH5pyNBs/s320/blooms_new.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Source: Learn NC, <a href="http://www.learnnc.org/lp/pages/4719" style="background-color: #f7f7f7; color: #026770; font-family: "Century Gothic", AppleGothic, Arial, sans-serif; font-size: 11px;">“Bloom’s Taxonomy,”</a><span style="background-color: #f7f7f7; color: #666666; font-family: "century gothic" , "applegothic" , "arial" , sans-serif; font-size: 11px;"> used under </span><a href="https://creativecommons.org/licenses/by-nc-sa/2.5/" style="background-color: #f7f7f7; color: #026770; font-family: "Century Gothic", AppleGothic, Arial, sans-serif; font-size: 11px;">a Creative Commons license</a><span style="background-color: #f7f7f7; color: #666666; font-family: "century gothic" , "applegothic" , "arial" , sans-serif; font-size: 11px;">.</span></td></tr>
</tbody></table>
<div style="text-align: left;">
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<div style="text-align: left;">
Around <b>33% show </b>the taxonomy in a grid with clear hierarchy:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkzo3PkU0r1ixDNA6PK4zA5IiezDYjXx4-h8BG6R5hpVy7bodhIqjGz4aamxV9K43tvMgRyTrGfPrmP0cqsoYFHKrzYCiiKjWwvMdbQOoyg5prMlkaqnTLWC_48A9Txy7vW2yeAhlbIzkI/s1600/blooms-grid.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="595" data-original-width="875" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkzo3PkU0r1ixDNA6PK4zA5IiezDYjXx4-h8BG6R5hpVy7bodhIqjGz4aamxV9K43tvMgRyTrGfPrmP0cqsoYFHKrzYCiiKjWwvMdbQOoyg5prMlkaqnTLWC_48A9Txy7vW2yeAhlbIzkI/s320/blooms-grid.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://mon.uvic.cat/clil/teaching-support/fonaments-teorics-aicle/thinking-skills/">https://mon.uvic.cat/clil/teaching-support/fonaments-teorics-aicle/thinking-skills/</a></td></tr>
</tbody></table>
<div style="text-align: left;">
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh11RX5C5geEle_wEH0jWHAwqfA8nBU7Z8pacS9_YgUt-6Qcxtq4WIzKn-lA46k3_AOMTXMNJL0oLD8lYJd-ED_NDZge-MKirUQe2ng3IZX3jTc5mqySzp7ps-lYfXuaPX_2vHhqQbJpOUE/s1600/blooms-taxonomy-verbs.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="750" data-original-width="1000" height="240" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh11RX5C5geEle_wEH0jWHAwqfA8nBU7Z8pacS9_YgUt-6Qcxtq4WIzKn-lA46k3_AOMTXMNJL0oLD8lYJd-ED_NDZge-MKirUQe2ng3IZX3jTc5mqySzp7ps-lYfXuaPX_2vHhqQbJpOUE/s320/blooms-taxonomy-verbs.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://www.teachthought.com/critical-thinking/249-blooms-taxonomy-verbs-for-critical-thinking/">https://www.teachthought.com/critical-thinking/249-blooms-taxonomy-verbs-for-critical-thinking/</a></td></tr>
</tbody></table>
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Many of these diagrams come with arrows and labels to reinforce the visual message of hierarchy:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvT6HmPOBEACRcDWxxfh6vW1meG9EFqH6KwOrgfewWPxxdHeden7NMGICVrFqHBkIY3qur6L-eb8ABBsnaKDDB2E8l8iGv3UUhQG5gyACvN27L5F84ayPjVVLTCzrG8_sT1a7l90_3tShD/s1600/bloom_s_taxonomy.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="315" data-original-width="500" height="201" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvT6HmPOBEACRcDWxxfh6vW1meG9EFqH6KwOrgfewWPxxdHeden7NMGICVrFqHBkIY3qur6L-eb8ABBsnaKDDB2E8l8iGv3UUhQG5gyACvN27L5F84ayPjVVLTCzrG8_sT1a7l90_3tShD/s320/bloom_s_taxonomy.png" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixqctfD9lFrmU1dA-SHlcg7-Z0r62O9nhVYu54GdfiEGSVtAf_lEu7Cd6s1egJdIFffADLXIKjFLI-hBrTJDxerHEqj4IQlpTawMVhJGYmOnH_J3vyvFulKQeHN9D69oBOCJ0HyaGX2MK3/s1600/blooms_revised_taxomony1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="447" data-original-width="373" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixqctfD9lFrmU1dA-SHlcg7-Z0r62O9nhVYu54GdfiEGSVtAf_lEu7Cd6s1egJdIFffADLXIKjFLI-hBrTJDxerHEqj4IQlpTawMVhJGYmOnH_J3vyvFulKQeHN9D69oBOCJ0HyaGX2MK3/s320/blooms_revised_taxomony1.jpg" width="267" /></a></div>
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Less than 10% of the images present the taxonomy as components without a particular order - or more interestingly, as an integrated view:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzYGFvBi1GBjvqAos32YzlPf2Z9iDvgCilxuT5S9Lmhng5LvNb__WrxTkoRNfwj3k8s_lHb8Iwlp5JOY97hKHW96idtCcU7iH9a9Yff_Lo2qsmZ-XE_Gtl5SvivSJIo8JoOZup7Rb6gpDU/s1600/blooms-rose.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="455" data-original-width="500" height="291" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzYGFvBi1GBjvqAos32YzlPf2Z9iDvgCilxuT5S9Lmhng5LvNb__WrxTkoRNfwj3k8s_lHb8Iwlp5JOY97hKHW96idtCcU7iH9a9Yff_Lo2qsmZ-XE_Gtl5SvivSJIo8JoOZup7Rb6gpDU/s320/blooms-rose.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://commons.wikimedia.org/wiki/File:Blooms_rose.svg">https://commons.wikimedia.org/wiki/File:Blooms_rose.svg</a></td></tr>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9cu_zFXN6IFTNQP5O89QCd68dxkTA0fiTgrtytolC4CnAtJ-jSwl_gQZ2-lOhI-hnoL7zSFq91hyT2e_rJaNessnbkUJpLO_nrwUe9sV92CvXALAHJ6_CN6W2cw7h1mcRdcBRVhqKmoOo/s1600/blooms-wheel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1511" data-original-width="1600" height="302" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9cu_zFXN6IFTNQP5O89QCd68dxkTA0fiTgrtytolC4CnAtJ-jSwl_gQZ2-lOhI-hnoL7zSFq91hyT2e_rJaNessnbkUJpLO_nrwUe9sV92CvXALAHJ6_CN6W2cw7h1mcRdcBRVhqKmoOo/s320/blooms-wheel.jpg" width="320" /></a></div>
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Much to my surprise, the more deeply I read into the history of Bloom's Taxonomy, the critique of the taxonomy, it's 2001 reformulation and the more nuanced commentary by supporters and critics of the taxonomy, the more I realise <b>my mental image of Bloom's Taxonomy is just plain wrong</b> - and I think Bloom would say that too.</div>
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Scrolling further down the Google Images search result, <i>much further down</i>, image #76 reveals something very different which hints at the important (and much neglected) aspect of the revised 2001 taxonomy:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaBBVLE98TD_6Qv0uWr1MEKY1rCtRsTZ6KoaOXBSkLbUuUo9hyphenhyphen6RWFpdacL5sVvuNdSUDXXlGmLv5tBJErEuS-uv9iYpaeyhpUAdgkSd-vypnl5yEA82H6AoXDHOFFW5iM6d1DT_iDaTJ3/s1600/blooms+real.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="251" data-original-width="702" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaBBVLE98TD_6Qv0uWr1MEKY1rCtRsTZ6KoaOXBSkLbUuUo9hyphenhyphen6RWFpdacL5sVvuNdSUDXXlGmLv5tBJErEuS-uv9iYpaeyhpUAdgkSd-vypnl5yEA82H6AoXDHOFFW5iM6d1DT_iDaTJ3/s640/blooms+real.jpeg" width="640" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">SOURCE: Anderson, L. W., & Krathwohl, D. R. (2001). A Taxonomy for Learning, Teaching, and Assessing: A Revision of Bloom's Taxonomy of Educational Objectives. New York, N.Y.: Pearson.</td></tr>
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<div>
No more pyramid, no more hierarchy (mostly). Knowledge has been pulled out of the list and <i>turned into a separate dimension</i>. The other parts of the taxonomy have moved into a cognitive process dimension.<br />
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<div>
I find this summary from Julie Stern very helpful in understanding the change:</div>
<blockquote class="tr_bq">
<div style="background-color: white; box-sizing: border-box; margin-bottom: 10px;">
<span style="color: #404040; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , sans-serif;">Few educators, including those who criticize the taxonomy, have considered the other major change to Bloom’s Taxonomy: the knowledge dimension. Anderson and Krathwohl (2001) have taken “knowledge” out of the cognitive domain and added it as a separate dimension, recognizing four distinct types: factual, conceptual, procedural and metacognitive.</span><span style="color: #404040; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , sans-serif;"> ... that instead of six ways to think about one type of knowledge, there are now six ways to think about four distinct types of knowledge.</span><span style="color: #404040; font-family: "open sans" , "helvetica neue" , "helvetica" , "arial" , sans-serif;"> </span></div>
</blockquote>
<blockquote class="tr_bq">
<div style="background-color: white; box-sizing: border-box; margin-bottom: 10px;">
<a href="https://sharemylesson.com/blog/what-no-one-tells-you-about-blooms-taxonomy">https://sharemylesson.com/blog/what-no-one-tells-you-about-blooms-taxonomy</a></div>
</blockquote>
</div>
<div>
<br />
Here's a very nice attempt to show both dimensions from the 2001 model in one diagram without too much hierarchy<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbia0CZPazBx0rf-m4EfaDX5Sow2krvwMqCQqZHpURuU4BIWK-gFPp4HzmkE1O1JAZC_mx2ZKNj5GEW-QAOVRrdDQE5q5YoOldHISoJUITBAao72iPrEf9DU-GMDlW7sxgvRLOwMQ9MFdK/s1600/bloom+purple+circle+chart.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="355" data-original-width="351" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbia0CZPazBx0rf-m4EfaDX5Sow2krvwMqCQqZHpURuU4BIWK-gFPp4HzmkE1O1JAZC_mx2ZKNj5GEW-QAOVRrdDQE5q5YoOldHISoJUITBAao72iPrEf9DU-GMDlW7sxgvRLOwMQ9MFdK/s320/bloom+purple+circle+chart.jpg" width="316" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://teachingcommons.lakeheadu.ca/blooms-taxonomy-21st-century-learners">https://teachingcommons.lakeheadu.ca/blooms-taxonomy-21st-century-learners</a></td></tr>
</tbody></table>
<br />
The most serious problem with the pyramid view of the first iteration of Bloom's Taxonomy is that knowledge is right at the bottom and seen as something we just build upon, a "low order" thing (the lowest in fact). Versions of the 2001 pyramid omit the knowledge dimension, focusing solely on the cognitive processes, leaving most teachers with the impression nothing changed except the words, with knowledge now just called "remembering (facts)".<br />
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<div>
In the next blog post, I'll be looking at some of the issues resulting from using an oversimplified view of Bloom's Taxonomy. For now it's enough to point out many of us, myself included, have been guilty of lazy thinking. Maybe it's not our fault - it's the brain's wonderful design to simplify ideas so we can cope with them efficiently.</div>
<div>
<br />
To finish off, here are two surprising 21st Century Learning versions of the taxonomy which really had me scratching my head. Observe that well motivitated and exciting as they are, both of the them <i>have completely lost the knowledge dimension.</i></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilkvW4-HXCgqAmOg9nlUAyIRYtUzwUw3sQBI1AIV4HlFzr6p67NWST12Q93_jAwdqahSXH1ycI3wm4sTSlGzg-qDOk5RGu8uqFioHEByReu_CaPGgVMDgXOGwguYSm8vrDHzEwGBXImLW0/s1600/blooms-digital.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="768" data-original-width="768" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilkvW4-HXCgqAmOg9nlUAyIRYtUzwUw3sQBI1AIV4HlFzr6p67NWST12Q93_jAwdqahSXH1ycI3wm4sTSlGzg-qDOk5RGu8uqFioHEByReu_CaPGgVMDgXOGwguYSm8vrDHzEwGBXImLW0/s320/blooms-digital.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Ron Carranza's "Bloom's Digital Taxonomy". <br />
I'm very pleased to see that by writing a blog post I'm at the top of the grid...<br />
In the 21st Century we're not even "remembering", we're "bookmarking".</td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0P7cPtY1ZJqyhfw-_5RdtiGXhyphenhyphenC3S0w681jxHh_w2o0s28gplksXdx2rmjJnAcS_LitfQEpykWamzw6V-R-nK_TP3WpKapfsyqsLg1jNDq-uRi0Vq-qPISgPq3MMYKyP7DC1yawZggKQZ/s1600/blooms-flipped.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="524" data-original-width="478" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0P7cPtY1ZJqyhfw-_5RdtiGXhyphenhyphenC3S0w681jxHh_w2o0s28gplksXdx2rmjJnAcS_LitfQEpykWamzw6V-R-nK_TP3WpKapfsyqsLg1jNDq-uRi0Vq-qPISgPq3MMYKyP7DC1yawZggKQZ/s320/blooms-flipped.png" width="291" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The "Flipped Learning Bloom's Diamond"<br />
<a href="http://www.maggiehosmcgrane.com/2015/07/flipped-learning-and-blooms-taxonomy.html">http://www.maggiehosmcgrane.com/2015/07/flipped-learning-and-blooms-taxonomy.html</a></td></tr>
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<a href="http://exzuberant.blogspot.com/2019/05/3-reasons-why-its-unwise-to-draw-blooms.html">Part 2: 3 reasons why it's unwise to draw Bloom's Taxonomy as a pyramid</a> </div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com4tag:blogger.com,1999:blog-8722640913234461773.post-6565777798263923872019-05-12T21:02:00.001+10:002019-05-12T21:17:55.392+10:00Three ideas for defusing the pedagogy wars<div style="text-align: justify;">
There's been a battle for teachers' (and parents') hearts and minds over the last few decades that, now appearing on twitter, can on some days seem particularly virulent. No, I'm not talking about the reading wars, but a broader dispute between what is often called "traditional" versus "progressive" approaches to teaching and learning. Or to use a catch phrase often used, pejoratively I'm afraid, as a choice between the teacher being "the sage on the stage" versus "the guide on the side". </div>
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In the Australian context, it feels this debate is pretty much a one-sided affair, with almost unquestioning acceptance that we need more "future focused, student centred" learning, but for those following the broader international scene, it's remarkable to witness the resurgence and reformation of traditional approaches into "knowledge based, explicit teaching" informed by findings from cognitive psychology. For the mathematics teachers among us, it's been fascinating to see some high profile teachers in Britain completely change their view on how to teach mathematics - switching from very innovative "student centred discovery learning" approaches to embracing their inner "teacher as the subject expert".</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg26qf048VgzkOOmUD9CHmMYii1Ivsx3tdo1HhEJooFeCUdf876e_LPFHl6fwoMdCqMAej7YRIUwqNbNdyawCluIjhGMT0sevm8w5u2pgGbc2A8urh9JKOlj7C09KxoHcu7iMXI2OLYEEdA/s1600/barton-book.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1129" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg26qf048VgzkOOmUD9CHmMYii1Ivsx3tdo1HhEJooFeCUdf876e_LPFHl6fwoMdCqMAej7YRIUwqNbNdyawCluIjhGMT0sevm8w5u2pgGbc2A8urh9JKOlj7C09KxoHcu7iMXI2OLYEEdA/s320/barton-book.jpg" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Craig Barton, a highly respected UK mathematics teacher, writes in 2018 about <br />
his almost 180 degree change of view on how to teach mathematics. </td></tr>
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But meanwhile the slanging matches continue - it can be quite ugly some days to read the ad hominem attacks and to see the emotive grenades being tossed over the trenches. What is a teacher to do? Especially if they have formed a view that isn't the currently dominant view? How can we move forward?</div>
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<b>#1: Respect and recognition for our colleagues ("niceness")</b></div>
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Right from the outset, I think we need to set a much much better example to the outside world as to how educated people can have a proper and respectful debate. It really disturbs me to see teachers write messages on twitter that exhibit behaviour we would not accept in our students. It's essential we recognise that even if we 100% disagree with a fellow educator, even if we think they are naive/partisan/ignorant/bigoted, that we recognise they are motivated by the very best intentions: the well being and care of young people. I may not concede that motivation for some others in the education debate (especially people with products to sell), but it's axiomatic to me that I respect and recognise that anyone who signs up to be a teacher and stays with it really has the best motivation. Let's remember that and start every discussion with the right tone. My mum called it "being nice".</div>
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<b>#2: Remember context, context, context</b></div>
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Just as the real estate agent reminds us it's all about "location, location, location", as teachers it's essential we keep "context, context, context" at front of mind. Each time we are about to say that something "for sure is correct", or "definitely doesn't work", we need to remind ourselves just how much context matters. What works for a Year 8 mathematics teacher with a specific class for a specific year for a specific group of students may well not work for a Year 4 primary teacher or a Year 12 music teacher in a different year, at a different school. It's so easy to get caught up in your own certainty, your lived experience as reality, and forget context. Am I a "relativist"? No I'm not, it is my view now that there <i>are </i>some universal things we can confidently say about teaching and learning. But on a broader scale, education truly is a "wicked problem" - there aren't many simple answers that work everywhere. We need a lot more caveats in our heated debates to explain our context and to recognise different contexts.</div>
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<b>#3: Can we change the words? Mode A and Mode B</b></div>
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Tom Sherrington, in his wonderful book "The Learning Rainforest" takes an interesting approach to defuse the angst: just stop using the words "traditional" and "progressive". Instead use the generic terms "Mode A" and "Mode B". </div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNk1HmkA2g1YVrtlg5g4NgebswXSXEvtVFwa0wGh5kPsfzLStO4wRNMht9lrmURnJyJWXK_Gp-ADmdxTz7Qec3BosCHFbad4ilFRPLEtYr4I6msIYTRdpnTdlgiMn25eJLJt1uG7JoyaVF/s1600/sherringon-book.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="346" data-original-width="244" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiNk1HmkA2g1YVrtlg5g4NgebswXSXEvtVFwa0wGh5kPsfzLStO4wRNMht9lrmURnJyJWXK_Gp-ADmdxTz7Qec3BosCHFbad4ilFRPLEtYr4I6msIYTRdpnTdlgiMn25eJLJt1uG7JoyaVF/s320/sherringon-book.jpg" width="225" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">A language reset : "Mode A" and "Mode B".<br />
Tom Sherrington's book is my #1 reading pick for 2019.</td></tr>
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This is a surprisingly powerful technique - it reduces the emotion and short circuits the automatic, non-thinking responses. It allows Tom to make statements like "for my subject, for my group of students, I think 80% Mode A and 20% Mode B is a good mix". And then we can calmly look at the ideas, strengths and challenges inherent in each different Mode A and Mode B teaching technique, without getting bogged down in polemics.</div>
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<b>So does that mean there shouldn't be a debate?</b></div>
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Certainly not! There's lots to debate about when it comes to teaching and learning and it's important, especially if we believe, as I do, that well educated young people are an essential component to addressing challenges such as climate change, socioeconomic inequality and role of technology in our future. But we need to deescalate. It's not an arms race - we need diversity in our education systems and every school will have a blend of Mode A and Mode B. Different students will respond differently to different teachers, to different teaching and learning strategies. Diversity is the strength of our system.</div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2tag:blogger.com,1999:blog-8722640913234461773.post-65448280130924329162019-03-03T12:40:00.001+11:002019-03-03T12:55:06.433+11:00The ultimate formative feedback? Tales from the whiteboard classroom.<div style="text-align: justify;">
The idea is extremely (and deceptively) simple: Cover your classroom with whiteboards, then think about how to get students to spend most of their time standing at the whiteboards, rather than sitting at a desk listening to you talk from the front.</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidcLGdmJynJE0Dqibu9t3uKn8HxpRq0GyhIw5KvdLnVKlPH7dFldcBsIjZxGc_qDXT9KrsQU4mGGAJ5u4y3fbRIGeLG7m_PTmRHcfhyphenhyphenhZEGCSIaNrAeyMPbUfP1KSyGtq3fRIIhpUJLKZK/s1600/Whole-view-CROPPED-2017-Dec.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="559" data-original-width="1600" height="171" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidcLGdmJynJE0Dqibu9t3uKn8HxpRq0GyhIw5KvdLnVKlPH7dFldcBsIjZxGc_qDXT9KrsQU4mGGAJ5u4y3fbRIGeLG7m_PTmRHcfhyphenhyphenhZEGCSIaNrAeyMPbUfP1KSyGtq3fRIIhpUJLKZK/s400/Whole-view-CROPPED-2017-Dec.jpg" width="500" /></a></td></tr>
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A "fish eye" view of my whiteboard classroom - Dec 2017. </div>
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There are 7 whiteboards (not all are shown here) and the room comfortably fits 16 students, and 24 at a pinch. In mid 2018 I removed half the desks to make more space.</div>
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This was the challenge posed to me by Tricia Forester (University of Wollongong) at the 2017 MANSW Conference, referencing the "<a href="http://peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-Feb-14-20151.pdf" target="_blank">Building Thinking Classrooms</a>" work of Peter Liljedahl. Well actually, Tricia was more clever than that - she posed interesting maths problems to groups of teachers, providing a "vertical non-permanent surface" space to try it out. It didn't take long to see the power. When I got back to school, my principal gave me permission to try it out for real, transforming a smallish classroom no-one really liked into a fully fledged whiteboard room with 7 large boards.</div>
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It's now been nearly one and half years that I've been working in a whiteboard room for my senior classes (unfortunately I can't fit 30 junior class students in the room) and there's no looking back! There are many things to share about the experience and why it supports such powerful learning, but for now I'll confine myself to a few big ideas.</div>
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The power of the whiteboard room is the way it provides <b>continuous formative feedback - to both the teacher and the student. </b>In fact, for the teacher, it's overwhelming - you will be <i>bombarded </i>by feedback. You instantly see what students can and cannot do, you see the approaches they take and you hear many simultaneous conversations discussing the problem. For the whole lesson. It's almost too much feedback. In the whiteboard room, we have a "live stream" answer to the <i>"How do I know what my students learned?"</i> question. I don't have to give them a diagnostic test every day - I can see it every minute. This continuous formative feedback allows me to dynamically alter the lesson according to needs of the students. It's a bit of a high-wire act some days, and often full of unexpected surprises, but it makes for exciting and rewarding lessons that students really enjoy and demonstrably produces learning. I'll write some more soon about the strategies I've developed to dynamically respond to so much continuous feedback.</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN5Ach6A6Tvl3lTmUeikdSopeb8KjIgu5G1EOQI53SLCClgCAS3E071OCrYcvv3dnvxVFhNySw-XJ9p_WVFoTb5G7dt57rRV2a5WiY7mOdKIBhmCS8wF19XhJtHIi0msPXExrgXsXLY5OI/s1600/IMG_20190218_105524.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1200" data-original-width="1600" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgN5Ach6A6Tvl3lTmUeikdSopeb8KjIgu5G1EOQI53SLCClgCAS3E071OCrYcvv3dnvxVFhNySw-XJ9p_WVFoTb5G7dt57rRV2a5WiY7mOdKIBhmCS8wF19XhJtHIi0msPXExrgXsXLY5OI/s400/IMG_20190218_105524.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: justify;">Students working in "interleaved" mode on a problem. The students worked out on their own that harder integration questions are easier done in "parallel" by two students - but they still end up correcting each others work! Different colour markers let me quickly see how (or if) students are working together.</td></tr>
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The <b>student engagement</b> factors at play in the whiteboard room, especially when combined with the <a href="http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf" target="_blank">Visibly Random Groups</a> technique, are truly astounding. I have seen students who previously had difficulty making friends, difficulty communicating, who were disengaged, lacked confidence, transformed by the whiteboard room experience. The combination of the social interaction, the risk free "non-permanent" writing surfaces and physical movement in the whiteboard room seems to work magic on students. One day a learning support teacher came into the room and asked a notoriously lazy student why he liked the room so much, he replied "Because I can't sleep in the back of the classroom any more!". Everyone, without exception, is engaged in the whiteboard room - there's no option to not be involved *.</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhATWICjCcJ872vSqaZSsiYV3kMEPBa9mXJzwXVy9_e1Q8xk0Ci1-DR7YB7dG3KLpF7X1IHz04JkdKazXlu6sj_PCBNORi9ITqLPJHz6YCyCq1dhZG8euO5D79e0RtoF5bSfu9qscNSYn21/s1600/20180619_082323.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="911" data-original-width="1600" height="227" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhATWICjCcJ872vSqaZSsiYV3kMEPBa9mXJzwXVy9_e1Q8xk0Ci1-DR7YB7dG3KLpF7X1IHz04JkdKazXlu6sj_PCBNORi9ITqLPJHz6YCyCq1dhZG8euO5D79e0RtoF5bSfu9qscNSYn21/s400/20180619_082323.jpg" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: justify;">We also use the whiteboard room for our before school "Year 8 Algebra Workshop" - a weekly session for students who need a little more support with algebra. The whiteboards allow them to work together while developing fluency. The social interaction and support created by the whiteboard room is an essential component of this program which seeks to build confidence and a growth mindset attitude.</td></tr>
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Is this "student-led" or "teacher-led" teaching? I think it's a blended approach, and a good example of why we need to get beyond using labels. However I do think I need to be clear: while students are doing most of the work, the lessons for my senior classes are very strongly guided by me. The student feedback steers the lesson according to what they need or where they are ready to go, but I'm still the "guide", it's just I don't "teach from the front". Several times during the lesson I will regroup the class in front of a student whiteboard I selected because it has a "teachable moment" and then "microteach" to summarise or clarify, and then teach a small amount of new content as required so students can continue with the next set of problems. Other times, a carefully selected sequence of problems means I don't need to do that - the learning follows naturally, and I can "spot teach" to the groups who need more scaffolding. My lesson designs have changed significantly. Because the students are doing so much work, the whiteboard room forces me to seriously consider the <i>"What will my students be doing?"</i> question. I spend much more time on thinking about examples and problems than on what I will say. </div>
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It's scary to change how you teach, and even after all this time, I sometimes worry I'm doing something too different from the other classes at my school. This year, the whiteboard room is less available to me (because I'm encouraging other teachers to use it!), and due to timetabling, while I still have the whiteboard room for my Year 12 class, I can only use the room with my Year 11 class for four out of every thirteen periods - so it's been interesting to see what happens when you mix things up. Sometimes old habits resurface, and I'm tempted not to use the whiteboard room when it's available - because it <i>is </i>easier to stand at the front, or you just haven't had the time to plan a whiteboard room lesson sequence. But much to my (pleasant) surprise, my Year 11 students complain and <i>insist </i>we go to the whiteboard room, so we just do it. And you know, when we finish the period, it's so clear they were right - it's a better lesson, even if I hadn't planned it as a whiteboard room lesson.</div>
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<b>Some starting resources and ideas:</b></div>
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Laura Wheeler's <a href="https://twitter.com/wheeler_laura" target="_blank">@wheeler_laura</a> blog post <a href="https://mslwheeler.wordpress.com/2017/03/15/building-thinkingclassrooms/" target="_blank">Building thinking classrooms</a> is an excellent starting point. Here's a <a href="https://mslwheeler.wordpress.com/tag/thinking-classroom/">handy link to all her posts</a> with the "thinking classrooms" tag.</div>
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Can't set up whiteboards in your room? Try out <a href="http://www.magicwhiteboard.com.au/" target="_blank">Magic Whiteboard</a> to transform any wall into a vertical non-permanent surface.<br />
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Peter Liljedahl's <a href="https://twitter.com/pgliljedahl" target="_blank">@pgliljedahl</a> papers "<a href="http://peterliljedahl.com/wp-content/uploads/Building-Thinking-Classrooms-Feb-14-20151.pdf" target="_blank">Building Thinking Classrooms (pdf)</a>" and <a href="http://www.peterliljedahl.com/wp-content/uploads/Visibly-Random-Groups.pdf" target="_blank">Visibly Random Groups (pdf</a>) are highly readable and motivating! Liljedhal's "Thinking Classrooms" framework is much more than just vertical non-permanent surfaces (VNPS) and visibly random groups (VRG). For me this is start of a journey.<br />
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* If a student really wants "to be alone", I'll give them a break and let them work solo for a while, hoping they will change their mind later, or next period. They always do. Also at the end of a really hot day, we might decide to sit on chairs for a change - sadly the whiteboard room doesn't have air conditioning.</div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-56155246351736182272019-03-03T07:51:00.003+11:002019-03-03T08:07:24.753+11:00Some lessons work best when you do (almost) nothing<div style="text-align: justify;">
<i>Returning to my blog after many years absence, I found this Feb 2017 post in my "drafts" folder, waiting for posting. I think it's still worth posting now in March 2019 (!).</i><br />
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It's a new school year, for me the start of the seventh year teaching. And this week I think I taught my best lesson ever. The surprising thing is that I barely taught anything. In the words of one of my mentors, a teacher with 35 years experience, "you know it's a great lesson when you do almost nothing. You sit back*, close your eyes, and you hear the student conversation taking place - they are talking about mathematics - and you'll hear how they think - and then you know what you need to teach, specific to each student."</div>
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For this lesson, I handed out "A/B quizzes" to the students which looked liked this:</div>
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They are designed to test skills in way that encourages students to help each other. In <a href="https://drive.google.com/file/d/0ByVkChxwrC4DVkVLQldjSWh1Sjg/view" target="_blank">the A/B quiz lesson design</a>, different students get different papers (see the letter in the bottom right hand corner of the paper). They can help each other if they like, but since the questions are slightly different, they have to actually teach each other rather than just share answers. Half way through the period, the students form into groups with those who did the same paper and compare their work, teaching each other anything they need to. Then they repeat the process for a second round, using a different version of the "A/B" quiz.</div>
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It was amazing to watch this group of students work together. I saw them struggle through the harder integrals with negative fractional powers, debating with each other what the correct answers were. Every single student was involved, no-one was left out. I mistakenly had some indefinite integrals on the quiz, which I hadn't taught yet, but the students who had worked ahead taught the others how to do them. And they loved the lesson, "Integration is so much fun, Sir!" I think they packed a week of learning into forty minutes. I only taught for the last ten minutes, reinforcing some of their findings and explaining a few finer points of the setting out and reasoning required to provide the highest quality solutions. It felt like one of the best lessons I had ever taught - but strangely I had barely done any teaching.</div>
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OK - I'm exaggerating when I said I did nothing. The lesson happened as a result of many months working with this group of students, building trust, confidence and openness. It does take a fair bit of training to get students working effectively in groups for this process - to ensure they actually help each other and look carefully at each other's work, rather than just "looking for an answer". I did design a very specific learning sequence for the topic, selecting items in the diagnostic quiz designed to elicit discussion and to expose any difficulty students had executing the required skills. And it does take many years of experience teaching a topic such as Integration to anticipate what difficulties students will have, how to diagnose them and how to provide the necessary support.</div>
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And that's the joy of the Art and Science of Teaching - sometimes the very best lessons have very little 'visible' teaching. But if you look closely, you'll see a lot of visible learning - and it will take many years of experience to feel like you got the best out of a particular lesson plan.</div>
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Would this work for every lesson? Absolutely not! It's my view, after trying many different approaches, that when teaching mathematics, four out of every five lessons should follow more traditional, explicit instruction supported with ongoing and regular formative assessment. And for the fourth or fifth lesson - try lots of different things - the more student-centred the better. "A/B" quizzes are a lesson design that works really well - the gift that keeps on giving, year after year.<br />
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<i>* Update for 2019: I've found an even better way to run a lesson like this! Coming soon...</i></div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-851814917964619012015-08-02T08:03:00.001+10:002015-08-02T08:38:39.089+10:00"Why are we learning algebra?"<div style="text-align: justify;">
It had been several weeks since my Year 7 class had the discussion of why we were learning algebra, so I was taken off guard when the perennial question came up again: "Mr Zuber, w<i>hy </i>are we learning algebra (again)?" </div>
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I have a whole range of answers I like to offer to this favourite question but this time something unexpected came out of my mind. "Have you seen those amazing new pictures of Pluto that came in this week from the Horizon spacecraft?" I was pleased to see many students in the class start to get excited - they certainly were inspired by those photos. </div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJ6BkdlC2CqPDLfkoC3v8vktFhM4xSaNbIj6Vl4tiiWqpfVQ3OxQh_OFOOCbDlxnN-TswYSCnjeDs25iAFosbgIh36F86b7PuJ_oRZny2Ro1XCOA0uXnFL2Wx4E1-kElnwe-hAPJacJ-A/s1600/Algebra-Why-Image0.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="276" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsJ6BkdlC2CqPDLfkoC3v8vktFhM4xSaNbIj6Vl4tiiWqpfVQ3OxQh_OFOOCbDlxnN-TswYSCnjeDs25iAFosbgIh36F86b7PuJ_oRZny2Ro1XCOA0uXnFL2Wx4E1-kElnwe-hAPJacJ-A/s400/Algebra-Why-Image0.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><a href="https://www.nasa.gov/image-feature/global-mosaic-of-pluto-in-true-color" target="_blank">Global mosaic of Pluto in true color (NASA)</a> July 2015</td></tr>
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"Well", I said, "that was algebra. <b>Algebra brought us those pictures.</b> Very complicated algebra, and physics and engineering worked out by smart people helped get that spacecraft just at the right place, at the right time above Pluto, millions of miles away from Earth, to get that photo and send it back to us. That's why we're doing algebra."<br />
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I think that was the best answer I gave in class all week - and the students seemed to like it. Thank you NASA!<br />
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Here are four reasons for learning algebra that I like to offer students when I start the introductory algebra topic. </div>
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Firstly we have some utilitarian reasons:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR0CJfb_1zmdYdJP7p57AUJ8WQ14qJVDma7UVI9TIqSI7ITbSmspHwAYU8BECZVOwVf2Gf9IsL3Pl0Uhen7o10z7jgGYfdvncj4o2DpBpnj6r-_24yXbgmW2fBwrrssD4AHIE_xd49QnNB/s1600/Algebra-Why-Image1.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiR0CJfb_1zmdYdJP7p57AUJ8WQ14qJVDma7UVI9TIqSI7ITbSmspHwAYU8BECZVOwVf2Gf9IsL3Pl0Uhen7o10z7jgGYfdvncj4o2DpBpnj6r-_24yXbgmW2fBwrrssD4AHIE_xd49QnNB/s320/Algebra-Why-Image1.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Algebra is a tool to help solve problems. <br />
We use it to find values of something we don't know.</td></tr>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwGxhQXp15hswJEXk2SIjBRFRBLgg592njow2w-2ag0Bdrw2uXJrYVeWtXgGrUlo2Y4VgpocpdijEfjEN6vedhFBI7rm4Ev4EVfHbBaA-dMsLgfmxxQDx8XbuP0fWeJQrwECUg2yib6io3/s1600/Algebra-Why-Image2.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwGxhQXp15hswJEXk2SIjBRFRBLgg592njow2w-2ag0Bdrw2uXJrYVeWtXgGrUlo2Y4VgpocpdijEfjEN6vedhFBI7rm4Ev4EVfHbBaA-dMsLgfmxxQDx8XbuP0fWeJQrwECUg2yib6io3/s320/Algebra-Why-Image2.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Algebra allows us to record information about relationships between numbers in a formula. <br />
We can then put values into those formulas to find related numbers. This could be the area of a triangle, or the dosage of medicine to give a child based on their weight.</td></tr>
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At a deeper level, algebra has an important place in our discovery of the world:</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGyujPx9dtKJPooWiddxyJEurhrtNoR_DjNohd2oWPMogxSEbwrK7h0wogKkQH9SHIdvAE302o2EyBp3mljXlOe9I6e5WcPyQ7KDmYt6LB1lKpa1L0ncepxKO8QL4cnqAzKJA13XitACBs/s1600/Algebra-Why-Image3.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGyujPx9dtKJPooWiddxyJEurhrtNoR_DjNohd2oWPMogxSEbwrK7h0wogKkQH9SHIdvAE302o2EyBp3mljXlOe9I6e5WcPyQ7KDmYt6LB1lKpa1L0ncepxKO8QL4cnqAzKJA13XitACBs/s320/Algebra-Why-Image3.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Algebra allows us to describe how the world works.<br />
Students like this image. The picture in the centre is a matter-antimatter collision and the formula is Heisenberg's Uncertainty Principle.</td></tr>
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and in supporting our exploration and representation of mathematical ideas.</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjIaqQ_hyMHuh3_LCbIxWKrJBSFC3VwBzWLFmACzWX0sbP5wY5gJPjZ5ao6-xD2KK0uVKmSp80lVbVVbKM3UoeFEa1KRLTIEbilbJNI5tpAU8mekuyCqyjYfAYtxxQHvG9q32Qk8YF_EY6/s1600/Algebra-Why-Image4.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="265" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjIaqQ_hyMHuh3_LCbIxWKrJBSFC3VwBzWLFmACzWX0sbP5wY5gJPjZ5ao6-xD2KK0uVKmSp80lVbVVbKM3UoeFEa1KRLTIEbilbJNI5tpAU8mekuyCqyjYfAYtxxQHvG9q32Qk8YF_EY6/s320/Algebra-Why-Image4.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Algebra allows us to represent and explore mathematical ideas and mathematical objects.<br />
At least some students in your class will have seen the Mandelbrot Set and know how complicated it is - they will be very surprised how 'simple' the algebra looks.</td></tr>
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Putting all these ideas together, I like to summarise with the one big idea: <b>algebra is a language.</b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcyVi7iKPpLK-6UieTQH7eptDBlm3VehPKebivP9nLZr3RO8DLa0Pka2OnJxOvVhWu1kasn0YkFCGkOIJh7Js1OZ07QvgazkLrCEMFPfTP2G3sYSYqwPrWj4uVDpDaV4XlhtFaP3gc8azg/s1600/Algebra-Why-Image5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="273" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcyVi7iKPpLK-6UieTQH7eptDBlm3VehPKebivP9nLZr3RO8DLa0Pka2OnJxOvVhWu1kasn0YkFCGkOIJh7Js1OZ07QvgazkLrCEMFPfTP2G3sYSYqwPrWj4uVDpDaV4XlhtFaP3gc8azg/s320/Algebra-Why-Image5.png" width="320" /></a></div>
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So - for those people who say "but I will never use the quadratic formula in my future work", I would respond: "Wouldn't you like to learn this amazing language? It will open up so many career possibilities to you (a utilitarian argument) and it's also a fascinating and rich language that will let you access a whole new level of knowledge and ideas (a sheer pleasure argument)"</div>
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What's even more amazing about this language is that it's an international language. I can speak algebra with a Russian or a Chinese mathematician. Somewhere out there in space, a class of Year 7 students with green skin and three eyes is also learning algebra. Can you think of another subject you're learning at school which is also being taught in Alpha Centuri? (OK - science... but let's pretend that's the same as maths :-)</div>
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Postscript: Should I have mentioned that algebra helps us develop reasoning skills? Possibly.... but I'm not sure most students buy the "it's good for your thinking" argument. So I take the "what algebra will offer you" line, and make sure I give emphasis to its role in abstract thinking as well as in 'practical' applications.</div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2tag:blogger.com,1999:blog-8722640913234461773.post-14528968891311062472015-07-09T13:41:00.003+10:002015-07-09T13:51:06.725+10:00Wormholes and Tesseracts in the Classroom - Part 2<br />
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In <a href="http://exzuberant.blogspot.com.au/2015/07/wormholes-tesseracts-and-power-of-love.html" target="_blank">Part A</a> I looked at some inspiring ideas about teaching in the movie Interstellar. Here are some ideas for using some Interstellar content in the classroom.</div>
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<b>Exploring Dimensions:</b> whether you're doing lessons on 2D and 3D solids, or just having a discussion why we say "x-squared" and "x-cubed" but "x-to-the-fourth", it's time to bring out The Tesseract. In the Interstellar version, it's an object that has spatial and time dimensions: as the main character Cooper moves through space, he's moving into different time "rooms".</div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/BRaAGRAA4_Q/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/BRaAGRAA4_Q?feature=player_embedded" width="320"></iframe></div>
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"Time is represented here as a physical dimension"<br />
Warning: SPOILER for the film!</div>
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My Year 7 students liked this, and were very fast on their internet devices to find more traditional mathematical representations of the tesseract (the hypercube) which made for a good discussion.</div>
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<b>Permutations and Combinations</b>: I haven't worked it out yet, but there's definitely a perms and combs activity to do with the CASE and TARS robots! Might link in well with a Quadrilaterals exploration too.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBdSHKd2oak1pLE8JN0SWKQ6azFnFtoE2d_zLRpgQ5gCa5lEUxIuYkVGFyS3NmjeLtMXT7f_sKuochMvbxN9hWijrBd4rc7jXkhsUQkv8sV5oDl1op6tTjj6ehCheQFLHk6u1MB2KSLMDn/s1600/Image-TARS.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBdSHKd2oak1pLE8JN0SWKQ6azFnFtoE2d_zLRpgQ5gCa5lEUxIuYkVGFyS3NmjeLtMXT7f_sKuochMvbxN9hWijrBd4rc7jXkhsUQkv8sV5oDl1op6tTjj6ehCheQFLHk6u1MB2KSLMDn/s320/Image-TARS.png" width="265" /></a></div>
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Watch CASE at work on Miller's Planet: </div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/rDlL94hzx84/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/rDlL94hzx84?feature=player_embedded" width="320"></iframe></div>
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Some good resources:</div>
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<a href="http://www.wired.com/2014/11/interstellar-droids/">http://www.wired.com/2014/11/interstellar-droids/</a></div>
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<a href="http://www.businessinsider.com.au/interstellar-how-tars-was-built-2015-3">http://www.businessinsider.com.au/interstellar-how-tars-was-built-2015-3</a></div>
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<b>An exciting way to introduce circular motion:</b></div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/a3lcGnMhvsA/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/a3lcGnMhvsA?feature=player_embedded" width="320"></iframe></div>
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A nice adjunct to the more classic and sedate circular motion sequences in my other all time favourite movie, "2001: A Space Odyssey".</div>
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And for something different, you may like to point your music teacher friends at this mini documentary about the making of the soundtrack:</div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/L_8t2VlwK4w/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/L_8t2VlwK4w?feature=player_embedded" width="320"></iframe></div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-31424549719173409792015-07-09T13:03:00.001+10:002015-07-09T13:51:26.095+10:00Wormholes and Tesseracts in the Classroom - Part 1<div style="text-align: justify;">
"You have to go see the film Interstellar", I told my Year 12 class recently, "Maths saves the human race!" They corrected me immediately: "No Sir - that's not true, LOVE saved the human race". Yeah - we have a lot of science fiction movie buffs in our school...</div>
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But quibbling aside, what a wonderful resource Christopher Nolan has given with this film. Here are a few highlights that inspired both my teaching and enriched the conversations in class.</div>
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<b>Interstellar: Inspiration for Teachers</b><br />
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Need to describe the projection of 4 dimensions into 3 dimensions in under 30 seconds? This explanation of why a wormhole should be a sphere is astoundingly concise (start at 03:00)</div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Rhr1FPbkMmM/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Rhr1FPbkMmM?feature=player_embedded t=3m00s" width="320"></iframe></div>
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Wow! Imagine if we could teach complex content this easily.<br />
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Probably the scariest conversation a maths/science teacher could ever hear takes place early in the film - a wake up call to all us! In a four minute sequence, Nolan describes the futility of assigning a single number to measure student success, text book wars and the battle we have on our hands to defend the scientific world view:</div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/iw5wYoJNowg/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/iw5wYoJNowg?feature=player_embedded" width="320"></iframe></div>
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So why do I teach? In an astounding sequence, Cooper <strike>mumbles</strike> muses:<br />
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<i>"We used to look up in the sky and wonder at our place in the stars. Now we just look down and worry about our place in the dirt." </i></blockquote>
(sequence starts at 3:00)<br />
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Y-WpPxhF-As/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Y-WpPxhF-As?feature=player_embedded" width="320"></iframe></div>
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We need more star gazing. If I can get even one student a year to decide to look at the stars, it's all been worth it.<br />
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<a href="http://exzuberant.blogspot.com.au/2015/07/wormholes-tesseracts-and-power-of-love_9.html" target="_blank"><b>Part 2: </b>Curriculum links : ways to use Interstellar in your classroom.</a><br />
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<i>(The videos are probably not going to be around for long ... watch them while you can!).</i>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-88591512579084839882015-04-26T09:05:00.006+10:002015-04-26T09:11:36.209+10:00Just change one word<div style="text-align: justify;">
Ever found yourself describing a student or a class to another colleague as "low ability"? It's a shortcut we use more frequently than we may realise, even if said in the most caring, well-intentioned way. Early in my teacher education at Sydney University, I was very fortunate to be given a very simple and powerful idea: change one word. Replace the word "ability" with "achievement". </div>
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The result may surprise you. Here's my favorite example: next time you hear yourself saying: "Let's save that (interesting, challenging) activity for the high ability class", change it to: "Let's save that (interesting, challenging) activity for the high <i>achieving</i> class". Wow! Would you really want to do that? Changing one word isn't about being politically correct - it's about altering our mindset from a fixed mindset to a growth mindset. When done with awareness, changing one word can make a real impact in your classroom and your school.</div>
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It's approaching five years now since I graduated from Sydney University, and this simple idea continues to pay dividends in my teaching. So it seemed only fitting to make it the topic for my presentation at our alumni conference, SUSMAC 2015. </div>
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<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Jd1ems1kaWQ/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Jd1ems1kaWQ?feature=player_embedded" width="320"></iframe></div>
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<a href="https://drive.google.com/open?id=0ByVkChxwrC4DNjZ4YjNkYWE3TG8&authuser=0" style="text-align: start;" target="_blank"><span style="font-size: x-small;">Presentation slides on Google Drive</span></a></div>
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With thanks to Judy Anderson and Maria Quigley for organising the conference, and to Eddy Woo for his work to make the conference available on the internet.</div>
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For a full set of videos and notes from SUSMAC 2105, featuring a keynote from Andrew Martin, and over 20 short presentations from teachers and preservice teachers, see:</div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://misterwootube.com/teachers/susmac2015/" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWVDM0QwadH5-HQcBfm7vmlW9iDc3gqmX2hnC-5I5EdU205XN3Z9Cbn52zv89UxLVf-Cou4V0Wrk8OBjoRr9vZwDL9LLcsG0iNh0IVSyRlwv2MnKBBVW4IiSXjw2kbUF0kwLjXjgpy8e18/s1600/SUSMAC2015.jpg" height="213" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><span style="font-size: x-small; text-align: justify;"><a href="http://misterwootube.com/teachers/susmac2015/" style="text-align: justify;" target="_blank">SUSMAC 2015</a></span></td></tr>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-51261813975926600292014-11-11T19:03:00.002+11:002015-04-26T09:19:49.095+10:00The Pi Collection : 50 maths enrichment books for your school libary<div style="text-align: justify;">
A little project I've been working on for the last year with my school librarians: we call it "<a href="https://drive.google.com/file/d/0ByVkChxwrC4Ddkk3Yi1GbFpfZnc/view?usp=sharing" target="_blank">The Pi Collection". </a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisRT_JCKmRfqaBAsyMSWhwv_1Qer1yOi0T-lPhpwxpZoMKx2J3_0gT02SzFfbisTU-QwAMG_qmn7mz8LrKahQjE4UTffeN_dt4OAlRE1XIc4gzOqJOMoh489gDdrDY_lvUmj6dUgajzWGC/s1600/Pi-Collection-Purple-Sticker.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisRT_JCKmRfqaBAsyMSWhwv_1Qer1yOi0T-lPhpwxpZoMKx2J3_0gT02SzFfbisTU-QwAMG_qmn7mz8LrKahQjE4UTffeN_dt4OAlRE1XIc4gzOqJOMoh489gDdrDY_lvUmj6dUgajzWGC/s1600/Pi-Collection-Purple-Sticker.png" height="50" width="320" /></a></div>
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We've built a carefully selected collection of around 50 books with maths themes - fiction and non-fiction, in an effort to entice our students to engage with mathematics beyond their regular classroom work.<br />
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Officially our idea is to encourage students to read more widely and deepen connections between maths and other subjects, but in truth, we just love having these great books available to share with our students. So, when, for example, we're exploring extra dimensions, we can ask "What if someone built a house in four dimensions?" - and point at the classic Robert Heinlein short story "And he built a crooked house" which just happens to be the library waiting for you to read. Or when we're talking about equations, and wondering if we could use them to describe <i>everything </i>- point students at the wonderful Isaac Asimov "Foundation" series - every thirteen-year old nerdy boy's dream of running the universe through maths. Or perhaps someone thinks girls don't math? Well have we got several books in the Pi Collection to show you otherwise!</div>
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Along the journey of building the collection, we've discovered books about people who think differently ("Born on a Blue Day"), a terrific manga-style book questioning the inner truth of logic in mathematics ("Logicomix") and short stories about what could happen if you <i>were </i>allowed to divide by zero ("Stories of your life"). And who could forget that the answer to the meaning of life is 42 ("Hitchhikers Guide to the Galaxy") - a book which also has great fun exploring probability - remember the "Improbability Drive"? Doing permutations and combinations? Well that's just begging for a reference to Arthur C. Clarke's short story "The Nine Billion Names of God".</div>
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Nothing however quite had the impact of "The Cold Equations" - this flew off the shelf as soon as we posted it up:<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuM_fd4zz3bKiiiDekWnQq8lT_ba26UTqLiqgSW3iUjn2UeZlP4PT4UF6RGlrDyYss1aNbRdHNeY4Xk98TWGGidleoBqTMYtdIrW92v3eURqGXeMdqSyq_haKag9l4Lw_LAHsPzhIMnwFa/s1600/The+Cold+Equations.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuM_fd4zz3bKiiiDekWnQq8lT_ba26UTqLiqgSW3iUjn2UeZlP4PT4UF6RGlrDyYss1aNbRdHNeY4Xk98TWGGidleoBqTMYtdIrW92v3eURqGXeMdqSyq_haKag9l4Lw_LAHsPzhIMnwFa/s1600/The+Cold+Equations.jpg" height="320" width="209" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Maths tells our hero he needs to eject his stowaway into space. <br />
His heart tells him otherwise. What decision will he make? </td></tr>
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I was a little worried about the unsubtle appeal to baser instincts - but hey - anything to get the students reading! And it turns out our English faculty teaches Science Fiction in Year 7 and The Detective Novel in Year 8 - so we made sure to select books in these genres that also had a maths element. As we expand the collection, we're finding more connections to Geography, History, Science and Art.<br />
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We've also included maths extension and enrichment books for curious students who want to go beyond the official high school curriculum. Collections of classic puzzle books, short articles on maths topics as well as some more challenging books. We even snuck in a few that might encourage some students to consider a teaching career (thank your Mr Lockhart!).</div>
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The <a href="https://drive.google.com/file/d/0ByVkChxwrC4Ddkk3Yi1GbFpfZnc/view?usp=sharing" target="_blank">full list </a>is available here. Do you know of any books we should add?<br />
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Looking for more maths themed books? <span style="text-align: justify;">An invaluable resource is </span><a href="http://kasmana.people.cofc.edu/MATHFICT/" style="text-align: justify;" target="_blank">Alex Kasman's collation of titles</a><span style="text-align: justify;">. Some care is required though, because not every book here is suitable for high school students.</span><br />
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<span style="text-align: justify;"><b>Updated</b>: <a href="https://drive.google.com/open?id=0ByVkChxwrC4DSXRBMUxDVXlleDg&authuser=0" target="_blank">2015 Semester 1 Pi Collection</a></span>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2tag:blogger.com,1999:blog-8722640913234461773.post-55635009139375776342014-10-23T14:36:00.001+11:002014-10-23T14:36:25.664+11:00AB Quizzes in the mathematics classroomNotes and resources for a presentation I gave to the North Sydney Region Maths Association this week:
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<a href="https://drive.google.com/file/d/0ByVkChxwrC4DQkl3bndTUlR2OUU/view?usp=sharing" imageanchor="1" style="clear: left; display: inline !important; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinu-HOkT_qmK7Egk6FIwChY5z6WBSaMt-q7896SVnKjekF7mQzzKkYReqLX0xjMxFrMBKLZFRJpVTOH4tTrAVEDkraTXeurPynGbDRPa4FKGHBreRgfWWr9q4Bs-YSK1MkAdhyphenhyphen_GiKgXt3/s1600/AB-Quizzes.png" height="196" width="320" /></a><br />
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<a href="https://drive.google.com/file/d/0ByVkChxwrC4DdjhwaWJBT3pYYkU/view?usp=sharing" target="_blank">PowerPoint template for an AB quiz</a><br />
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<a href="https://drive.google.com/file/d/0ByVkChxwrC4DVkVLQldjSWh1Sjg/view?usp=sharing" target="_blank">Author's copy of article in MANSW Reflections - 2013 Conference Issue</a><br />
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Thanks for the invite and feedback - some great ideas from the group to experiment further with the AB Quiz concept.enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-60771444170127923652014-05-10T09:13:00.001+10:002015-04-26T09:16:46.446+10:00Bringing "childishness" into the mathematics classroom<div style="text-align: justify;">
Why would you show a Sesame Street video to a senior Year 12 mathematics classroom? Is there any value is doing paper folding exercises with such a class? Here's my thoughts on the value of bringing "childishness" into every mathematics classroom - no matter the age group or the current achievement level, presented at a recent conference.</div>
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<b>Games played by children and mathematicians</b><br />
April 23, 2014<br />
<a href="https://www.youtube.com/playlist?list=PL5KkMZvBpo5C78NrnhuGONyywm950j4-1" target="_blank">University of Sydney, Mathematics Education Alumni 2014 Conference.</a> <br />
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/d9OoLXIIrhs?feature=player_embedded' frameborder='0'></iframe><br />
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Or just preview the presentation:<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://drive.google.com/file/d/0ByVkChxwrC4DQ1V3cGJVUEN1UXc/" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4hFHcrGbJJ5vl3VMz8Ii87HcMkAQ7X8iHmiiP8KjWl1C3r4oCyxk0bcDYf_ApSDxu4BYO3z6DJRxjI4pfwYfqq-0PlvamLiJiWhKWJrls9sPtOhzKNMlLe7fCnhl11gD11q2BWR8kIauO/s1600/Zuber-USyd-2014-games-children-play-THUMB.png" height="194" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Click on the image for the Google Drive document</td></tr>
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<a href="https://drive.google.com/file/d/0ByVkChxwrC4DQ1V3cGJVUEN1UXc/">https://drive.google.com/file/d/0ByVkChxwrC4DQ1V3cGJVUEN1UXc/</a></div>
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<br />enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2tag:blogger.com,1999:blog-8722640913234461773.post-80902741959754002052014-03-09T11:47:00.001+11:002014-03-09T11:47:35.222+11:00ChangesIt's been a very long time between posts ...
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<table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik0yasu6fPBuyAYKmbULGwSlE3LbLNhb_z-6gAXM2A7L0Bsr0AEOrSblxU6s2pAwov56f7JPeJ8NqN91xpZ0a35LCXih4G6OuTvx7Pa_J2f78RuLCa3YJWiLxWbC_umQFkwdMvX3WPe9SI/s1600/Chrysalis.jpg" imageanchor="1" style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik0yasu6fPBuyAYKmbULGwSlE3LbLNhb_z-6gAXM2A7L0Bsr0AEOrSblxU6s2pAwov56f7JPeJ8NqN91xpZ0a35LCXih4G6OuTvx7Pa_J2f78RuLCa3YJWiLxWbC_umQFkwdMvX3WPe9SI/s1600/Chrysalis.jpg" height="320" width="249" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">CC-BY-NC <a href="http://www.flickr.com/photos/ultimorollo/382464726/" target="_blank"><span style="background-color: #fefefe; font-family: Arial, Helvetica, sans-serif; line-height: 17px; text-align: left;">Gabriela Ruel</span><span style="background-color: #fefefe; font-family: Arial, Helvetica, sans-serif; line-height: 17px; text-align: left;">lan</span></a></td></tr>
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A very big change has happened in my teaching career - I have become the head of a mathematics department. I have been extremely fortunate to move to a school with a terrific faculty, a very supportive executive team and the most amazing students you could ever hope to teach. </div>
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However even in the most ideal environment, the time pressures on a new head teacher are unbelievable and there is a very steep learning curve.
As such, I'm putting this blog into a dormancy period. </div>
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Thank you for your readership in the last few years and hopefully I will be blogging again later in the year.
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-6022646512281126802013-10-07T13:10:00.002+11:002013-10-07T13:19:06.452+11:00Fruit vectors: checking for understanding in the mechanics classroom<i>Here's a small idea I had while teaching mechanics that turned out to have very surprising and fruitful results in my classroom.</i><br />
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So our class has been working on mechanics for a few weeks now, we think we know the basics from a physical and a mathematical view. It's time to look at something harder now: circular motion. But just before we do that, let's check we really understand. <br />
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Can you draw vectors, any sort you like, and tell a story about what is happening here?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeNjfs0OW9e9SMNFiIfBshrS1p6SnwPlIHQBQAaPfFF88aAQ2tThbnxOK01so5o96HXFFehjJocwKPFIYy3y5xHKGfpZlbrj5stjmNxukoncXMfuc_eN7rSVLBvC6YLkwzfy7dWCSiK-Ev/s1600/Vector-games3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgeNjfs0OW9e9SMNFiIfBshrS1p6SnwPlIHQBQAaPfFF88aAQ2tThbnxOK01so5o96HXFFehjJocwKPFIYy3y5xHKGfpZlbrj5stjmNxukoncXMfuc_eN7rSVLBvC6YLkwzfy7dWCSiK-Ev/s320/Vector-games3.png" width="320" /></a></div>
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.. and this one ....</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxepqfDwa8s1qS88ksmG11I_IdqdB1SWj0eVnQb8jckuhfaUuwGI5ogynywrIeKyYOqNieXKAyVu92VSj5QoqPNHhyJ02iaRbobm8FIgvjD5MNbNabMCYFDsRBymWfDfGDUR3JtJP4szhL/s1600/Vector-games4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="143" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxepqfDwa8s1qS88ksmG11I_IdqdB1SWj0eVnQb8jckuhfaUuwGI5ogynywrIeKyYOqNieXKAyVu92VSj5QoqPNHhyJ02iaRbobm8FIgvjD5MNbNabMCYFDsRBymWfDfGDUR3JtJP4szhL/s320/Vector-games4.png" width="320" /></a></div>
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How about this one?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGRYRhq0gRybM6BRJtSUqfyldzN6dMB8QHJyQ7MYv8m6UTt4ZrMfIUsnGXJ3PW_Wbl7XwA3_mibwGFqF92mTUkpTgYaeVCU1p0dsTXAXQG8LLchjx7k3G2w8S_MYJal1dIvM9BwkTKAuYo/s1600/Vector-games1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="161" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGRYRhq0gRybM6BRJtSUqfyldzN6dMB8QHJyQ7MYv8m6UTt4ZrMfIUsnGXJ3PW_Wbl7XwA3_mibwGFqF92mTUkpTgYaeVCU1p0dsTXAXQG8LLchjx7k3G2w8S_MYJal1dIvM9BwkTKAuYo/s320/Vector-games1.png" width="320" /></a></div>
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It was surprising just how much discussion resulted from just these three diagrams, by asking students to draw velocity vectors, acceleration vectors and force vectors - and then tell a narrative in mathematical and in physical terms. Many misconceptions dealt with!</div>
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And now for the reward, setting things up for circular motion. What are the acceleration vectors (and hence the force vectors) for this picture?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXx_RrSv6XjLYuYhGgczTDMwD_3wHmodn2vW6k908JqNjHq7xEHE5XhKW5G8cqNY9d_uJyj-QexlA-zbh0rYKGOfntVXeVBv_THV8djEVrqYxe04Pg35aYhW17D5suqA45_bCRkXSoiClg/s1600/Vector-games2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXx_RrSv6XjLYuYhGgczTDMwD_3wHmodn2vW6k908JqNjHq7xEHE5XhKW5G8cqNY9d_uJyj-QexlA-zbh0rYKGOfntVXeVBv_THV8djEVrqYxe04Pg35aYhW17D5suqA45_bCRkXSoiClg/s320/Vector-games2.png" width="320" /></a></div>
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By the time you have finished working out the vectors it's very clear what is required to move in a circle, even at a constant speed. What I love about this activity is that the students have developed an intuitive mathematical sense for what the result should be even before we start doing the detailed analysis to get the acceleration and force equations.</div>
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<b>Resource</b>: Here's a version of the diagrams I gave to students to scribble on:</div>
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<a href="http://www.scribd.com/doc/173986427/Thinking-About-Motion" style="text-decoration: underline;" title="View Thinking About Motion on Scribd">Thinking About Motion</a> by <a href="http://www.scribd.com/enzuber" style="text-decoration: underline;" title="View enzuber's profile on Scribd">enzuber</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="0.69224865694551" data-auto-height="false" frameborder="0" height="600" id="doc_88302" scrolling="no" src="//www.scribd.com/embeds/173986427/content?start_page=1&view_mode=scroll&access_key=key-1c6cuk7jeqpnee020wrc&show_recommendations=true" width="400"></iframe>
</div>
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or get this from Google drive: <a href="https://docs.google.com/file/d/0ByVkChxwrC4DWGF5OHJ2bzhsZFE/edit?usp=sharing" style="text-align: justify;" target="_blank">Thinking about motion</a><span style="text-align: justify;"> (free download)</span>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-40613028274161325722013-09-15T16:04:00.003+10:002014-01-26T07:09:55.229+11:00MANSW 2013 Presentation<div style="text-align: justify;">
<i>A quick post for those wanting to see a copy of my presentation this morning at MANSW. Thanks to all who attended and gave such enthusiastic support - all the more so given it was 9AM on Sunday morning after a very late night conference dinner!</i></div>
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<b>Three tools: The ABQuiz, the Tracking Sheet, the Feedback Form</b><br />
September 15, 2013<br />
MANSW 2013 Conference, Terrigal.<br />
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Google docs - free download:<br />
<a href="https://docs.google.com/file/d/0ByVkChxwrC4DTUJfWnpEU21Hb00/edit?usp=sharing">https://docs.google.com/file/d/0ByVkChxwrC4DTUJfWnpEU21Hb00/edit?usp=sharing</a><br />
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Links to the Feedback Form tools:<br />
<a href="https://docs.google.com/file/d/0ByVkChxwrC4DNlZBZVFUX21SRWs/edit?usp=sharing" target="_blank">Feedback Form template</a> (Word doc)<br />
<a href="https://docs.google.com/file/d/0ByVkChxwrC4DWTJHMVhDVFN2OFk/edit?usp=sharing" target="_blank">Feedback Form analysis</a> (Excel spreadsheet)<br />
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Read it now in Scribd:<br />
<div style="-x-system-font: none; display: block; font-family: Helvetica,Arial,Sans-serif; font-size-adjust: none; font-size: 14px; font-stretch: normal; font-style: normal; font-variant: normal; font-weight: normal; line-height: normal; margin: 12px auto 6px auto;">
<a href="http://www.scribd.com/doc/168279471/MANSW-2013-Presentation-Three-Tools" style="text-decoration: underline;" title="View MANSW 2013 Presentation: Three Tools on Scribd">MANSW 2013 Presentation: Three Tools</a> by <a href="http://www.scribd.com/enzuber" style="text-decoration: underline;" title="View enzuber's profile on Scribd">enzuber</a></div>
<iframe class="scribd_iframe_embed" data-aspect-ratio="1.33234859675037" data-auto-height="false" frameborder="0" height="800" id="doc_10457" scrolling="no" src="//www.scribd.com/embeds/168279471/content?start_page=1&view_mode=scroll&access_key=key-1hvnp4d07mw9rwuuskm4&show_recommendations=false" width="600"></iframe>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-353097523200530012013-07-12T18:18:00.003+10:002013-07-20T09:41:50.138+10:00Video helpers in the mechanics classroom<div style="text-align: justify;">
<i>This is the second post in a sequence about teaching the NSW (Australia) HSC Mathematics Extension 2 Mechanics topic. The <a href="http://exzuberant.blogspot.com.au/2013/07/freefalling-into-extension-2-mechanics.html">first post</a> looked at some initial challenges teaching mechanics and ways to use Felix Baumgartner's historic freefall jump in 2012.</i></div>
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How wonderful it is to be teaching in the age of the internet - being able to draw on the work of so many talented and inspirational teachers - and better yet, bring their insights and passion directly into your classroom to share with your students! Like hundreds of thousands of other people, I've been following the work of Derek Muller and his incredible <a href="http://www.youtube.com/user/1veritasium">Veritasium YouTube channel </a>for some time, however it's only now that I'm planning lessons for a sequence on mechanics that I get to draw on his work for my mathematics classroom. As I designed my lesson sequence, I was stunned just how well the Veritasium videos fitted into my lesson design. Here are a few ways I think it's going to be a winner to have Derek in my classroom this term.</div>
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<b>Misconceptions about falling objects</b></div>
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It's all too easy for students to agree with the statement that every object falls with acceleration <i>g</i>, but do they <i>really </i>believe it? The truth is they don't - not even some students who have studied physics at university. This engaging and challenging video will do the trick:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/aRhkQTQxm4w?feature=player_embedded' frameborder='0'></iframe></div>
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<b>Introducing force concepts with an interesting problem : dropping a slinky</b></div>
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Choice: draw some boring static diagrams - or watch Derek's intriguing video about dropping a slinky? No brainer! What I love about this sequence is the way it's designed for deeper teaching and learning: it's not just a passive "sit and watch" session - instead we are presented with an intriguing problem and challenged to decide on a response. I'm certain my students are going to respond enthusiastically - and provide me the perfect hook to introduce free body diagrams as a way to better understand the situation.</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/wGIZKETKKdw?feature=player_embedded' frameborder='0'></iframe></div>
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Then in the next video we get to watch what happens - and it is surprising!</div>
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And then extend the idea in several ways: What if we attached something to the slinky? What if we used a SUPER MASSIVE slinky?</div>
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<object class="BLOGGER-youtube-video" classid="clsid:D27CDB6E-AE6D-11cf-96B8-444553540000" codebase="http://download.macromedia.com/pub/shockwave/cabs/flash/swflash.cab#version=6,0,40,0" data-thumbnail-src="http://img.youtube.com/vi/JsytnJ_pSf8/0.jpg" height="266" width="320"><param name="movie" value="http://youtube.googleapis.com/v/JsytnJ_pSf8&source=uds" /><param name="bgcolor" value="#FFFFFF" /><param name="allowFullScreen" value="true" /><embed width="320" height="266" src="http://youtube.googleapis.com/v/JsytnJ_pSf8&source=uds" type="application/x-shockwave-flash" allowfullscreen="true"></embed></object></div>
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As a side benefit, this video communicates positive messages to students about studying science at University. You'll get to do interesting work, and work with people like Rod Cross.<br />
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After the slinky videos, we'll take a look at this excellent discussion of reaction forces - which is also going to support understanding of how to work out free body diagrams:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/aJc4DEkSq4I?feature=player_embedded' frameborder='0'></iframe></div>
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The next video isn't Veritasium, but so powerful I have to share it. This high definition footage from a camera on the Space Shuttle booster rockets, tracking the rise and fall of the boosters is going to make for a exciting exploration of terminal velocity ( 2,900 mph at timestamp 5:15 down to 220 mph at 6:45)</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/2aCOyOvOw5c?feature=player_embedded' frameborder='0'></iframe></div>
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Again - I want to inspire as well as educate. Maths and science is so much more than school work - it's an exciting and rewarding pursuit - with great career options.<br />
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So many videos to choose from. The challenge is to choose videos that serve both the needs of the teacher and the student - the video has to do much more than just entertain in order to justify taking time away from "the regular program". It has to serve the learning goals and promote specific outcomes - as well as being engaging and memorable - a gift that keeps on giving in the classroom.</div>
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<b>What makes the Veritasium videos so good in the classroom? </b>It's not just the sheer enthusiasm and fun of the presentations, or the fact they are short and sweet and fit nicely into a lesson segment, it's the fact Derek has grounded them in quality pedagogy. Presenting information in videos as a statement of facts turns out to have very little benefit - most definitely for science content, and quite possibly of limited value for mathematical understanding. If you're a fan of using video for teaching, definitely check out Derek's research work - you might be a little surprised at what he found:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/eVtCO84MDj8?feature=player_embedded' frameborder='0'></iframe></div>
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Finally, back on topic, if you dare, this contradictory Veritasium video "Three Incorrect Laws of Motion" would be a wonderful basis for class discussion and perfectly demonstrates what makes a richer educational video - provided there is good support in the classroom (you better really understand the <i>correct </i>laws!).<br />
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So thanks Derek for your amazing work and generosity - and welcome into my classroom!</div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-53664182136671267482013-07-03T10:44:00.000+10:002013-07-12T18:20:22.487+10:00Freefalling into Extension 2 Mechanics<div style="text-align: justify;">
<i>This post refers to the NSW (Australia) Mathematics Extension 2 course - the highest level mathematics taught in our high school system, but should hopefully have relevance for anyone teaching introductory mechanics.</i></div>
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My class is just beginning to explore the harder mechanics content in our senior mathematics course, exploring the equations of resisted motion, so it was with great delight I brought the exciting work of Felix Baumgartner into my classroom:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/FHtvDA0W34I?feature=player_embedded' frameborder='0'></iframe></div>
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Here are some reflections on introducing this topic:<br />
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<b>Did you lay a strong foundation right at the start of learning calculus? </b>Coming from a physics/engineering background, I taught all the calculus concepts right from the start as a study of rates of change, using this as a <i>motivation </i>for the mathematics. My classes started with motion using a motion detector even before we looked at the concept of the derivative. When we studied the properties of the derivative, the meaning of increasing, decreasing and stationary points, it was all done in the context of motion (lots of roller coasters!). When we studied the second derivative, we asked the question "Why?" - and looked at how much of the physical world works on the second derivative rather than the first. So by the time we moved on to specific topics of applying calculus to the physical world, my students already had wide exposure to the link between the mathematics and the physics.</div>
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<b>Did you explain why we have equations of motion that connect acceleration to displacement?</b> One of the hardest parts of the earlier work in our course on motion is understanding and working with these equations:</div>
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$$a = v \frac{dv}{dx}, a = \frac{1}{2} \frac{ d(v^2)}{dx}$$</div>
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By showing that acceleration is caused by <i>forces</i>, and that in turn forces are often dependent on <i>position </i>rather than time gives the motivation for these more complex equations. It's an excellent opportunity to tap into students' current science knowledge on gravitational and electrical fields - and again reinforces Newton's Laws of Motion (this time the second law). For students with a little more physics knowledge, it's very interesting to link the second form of this equation to the equation for kinetic energy $$KE = \frac{1}{2}mv^2$$ using some integration.</div>
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<b>You can't emphasise Newton's First Law often enough</b>. It's amazing how many studies show that while university physics students can describe and use Newton's First Law of Motion, deep down they remain firmly wedded to the Aristotelian world view. I make it a point to emphasis the First Law of Motion each and every time I start a problem - and I keep an eye out for anyone hesitating or wavering. If necessary, I repeat my stories about ice-skaters and about the Voyager spacecraft continuing on their journey even though they have run out of fuel.</div>
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<b>More diagrams, more diagrams! </b>It is sad to see how few diagrams are presented in most mathematics textbooks when presenting the theory of dynamics. While there are diagrams in most worked examples, they aren't explicit in the construction of the diagram, leaving it for students (and teachers) to try to interpret why the diagram was done that way. I spent most of the introductory lessons on this topic just drawing pictures.</div>
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<b>"That's why it's called RESISTANCE sir!" </b>My favourite quote from one of my students. We were drawing diagrams and I was trying to find a clear way to show that the resisting force always opposes the current velocity direction. I was saying the word "opposes" a few times when one of my student yelled this out - we won't forget that in a hurry!</div>
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<b>"Bait-and-switch" constants. </b> Your students are probably used to the little bait-and-switch games we play with constants of integration. The same games are played with the constants used for resistance forces:</div>
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$$R = kv, R = mkv$$</div>
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Sadly many of our standard text books just switch on the fly between the two forms without explanation, adding in the mass whenever it's needed. It's very confusing for students (and teachers) when this is done so arbitrarily. And then it hit me: this is a totally legitimate game - we're just using a <i>different </i>constant to make our life easier. It sure would be nice to write:</div>
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$$R = k_1v, R = mk_2v$$</div>
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I shouldn't complain since I happily went along with the game when integrating! This is yet another small but cumulative thing that makes this topic challenging. I think it's important to be explicit about this little game.<br />
<i>Update: As explained in Robin's comment below, this trick is a bit too clever - verging on not being legitimate - because it gives the false impression that the resistive force depends on the mass - it doesn't. The trick only works for a specific case of the mass in this problem.</i></div>
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<b>Students have difficulty seeing that the physics has nothing to do with the coordinate system</b>. And it's not their fault - our teaching and our text books rarely show how arbitrary coordinate systems are - we just happily keep changing them to our convenience, potentially confusing our students. I think it's critical to draw lots of vector diagrams without any coordinate systems, and then make a clear and obvious choice with the class that we can choose <i>any </i>coordinate system that works for us. We had a deep-learning moment in my class last week when I unwittingly applied a different orientation of the axes than was in our text book - a great opportunity to highlight this issue.</div>
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<b>Terminal velocity is a really fun concept.</b> Students are absolutely fascinated with it - the physical understanding is interesting, and the mathematical development is revelatory. We had some great discussion on different terminal velocities for different situations and these led directly into a more rigorous discussion of Felix's jump.</div>
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<b>Don't think that girls aren't interested in watching extreme sport events.</b> My class of fifteen girls was were absolutely riveted watching Felix make his jump. They insisted on watching the full length 10 minute video, totally transfixed for the duration. I recommend reading the <a href="http://en.wikipedia.org/wiki/Red_Bull_Stratos">Wikipedia page on Felix's "Red Bull Stratos" jump</a> with your students prior to watching the video - it provides an excellent opportunity to discuss the language of motion, examine the different stages of the jump and provides meaningful context for this thrilling event.</div>
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Hold your breath and enjoy the whole jump:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/YIj9XGLTGH8?feature=player_embedded' frameborder='0'></iframe></div>
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<i>The <a href="http://exzuberant.blogspot.com.au/2013/07/video-helpers-in-mechanics-classroom.html">next post in this series l</a>ooks at the wonderful Veritasium You Tube resources available for teaching mechanics.</i></div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-52794822292891660802013-06-10T07:58:00.001+10:002013-06-10T07:58:54.062+10:00Still alive ...<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhK2t_8bFd0UAXEsHC8cOMrVEP6ea2wOMpf-xxsUzcB4BRJK0uHMJ1y_sTUOT6UBo7xddiqpKDXVBcfgyEsnew99lvfygUE_G651X1q0ZpniAffUv4IHw3S9cjwO6CJK3YXep3HQ9-7zne8/s1600/pulse.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="120" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhK2t_8bFd0UAXEsHC8cOMrVEP6ea2wOMpf-xxsUzcB4BRJK0uHMJ1y_sTUOT6UBo7xddiqpKDXVBcfgyEsnew99lvfygUE_G651X1q0ZpniAffUv4IHw3S9cjwO6CJK3YXep3HQ9-7zne8/s200/pulse.jpg" width="120" /></a></div>
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It's been along time between posts - just a short note to regular readers to say I'm still breathing. It's been a very hectic year so far - teaching Year 12 Extension 2 mathematics for the first time (18 hours a week of very intense mathematics just with one class), and this term taking on the role of relieving head teacher. So not much time for reading blogs let alone writing them. Hopefully I will get back to blogging next term - lots of posts I would like to write based on recent experiences. </div>
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Some forthcoming topics I think readers may find interesting:</div>
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<li>Helping students (and parents) when faced with a poor result in a Year 9 half-yearly exam,</li>
<li>The joy of the binomial gnomes - and the mystery of Yang Hoi's triangle,</li>
<li>Teaching ideas for geometric series and inverse functions.</li>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-15616351684928819322013-03-17T10:54:00.001+11:002013-03-17T11:00:59.423+11:00How to get your math class SCREAMING<div style="text-align: justify;">
I know students are not <i>supposed </i>to be using mobile phones in class for private communications*, but I couldn't help but smile when one of my students showed me a text message she had just received from a friend in the class next door : "What are you guys doing in there? We can hear you screaming!" </div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigGgaqvdDPo_fxB1kHal_XdFx901iOWQDGjdF5GW001Y3tZypTwp7xNHBXdNiqVdtgBa8rdPR4wh89AK2SboGyqWFzsCLOGhnl4Qhtzvn2DmGlvSP3M8nZh6lSiuBeNP8N2nrn2cSCrj0b/s1600/One-Direction-maths.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="272" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigGgaqvdDPo_fxB1kHal_XdFx901iOWQDGjdF5GW001Y3tZypTwp7xNHBXdNiqVdtgBa8rdPR4wh89AK2SboGyqWFzsCLOGhnl4Qhtzvn2DmGlvSP3M8nZh6lSiuBeNP8N2nrn2cSCrj0b/s400/One-Direction-maths.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">If you don't know who these five boys are,<br />
then you're definitely not teaching at a girls school.</td></tr>
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So a little context first (context is everything!): This class doesn't really like maths that much - they tolerate it - I try my best to make it relevant and pleasant, trying to raise their confidence and skill levels. We've been studying a fairly dry topic for the last few weeks - they've done reasonably well in the topic test but need more practice. Looking for something engaging to make the second half of a long double period interesting, I turned to <a href="http://quizboxes.com/">Stu Hasic's Quiz Boxes.</a></div>
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://quizboxes.com/" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="251" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinU7JqcAmxDD2XRHYtlvxeQ-X_aNGUVnLFCT7Geh0Xyp04giXb7jg2rUy9GhwOY88CtJgrjBXskucDY4uCzFOQLXkJyJq251ZsHjuU2SiPD1LXQgiX-iApD0KvyNzrfxmfMKg9Lhob7r_t/s320/quizboxes1.jpg" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Download Stu's Quiz Boxes at <a href="http://quizboxes.com/">http://quizboxes.com/</a></td></tr>
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Quiz Boxes offers a Jeopardy! style game with questions of increasing complexity organised into categories, with a high stakes question at the end. Students love this game - and with careful planning and implementation (you will need to design the questions) it makes for a terrific fun period with high levels of engagement and gets students doing a lot more maths revision than they might have otherwise intended :-). There are many ways you can use Quiz Boxes so I would like to share an approach I have found that works well for classes of all levels of maths achievement. </div>
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<b>Quiz Design</b></div>
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<li style="text-align: justify;"><b>Choose categories that students are interested in. </b>Current hot topics are "One Direction", "Justin Bieber", "Beyonce", "You Tube Hits" and "In the Movies". Find whatever your class is interested in. Once they play the game, they will suggest topics to you. Since I don't know that much about One Direction, I go to Wikipedia and collect the factual information I need. Find some obscure information for the harder questions. Your students will be amazed you know something so detailed about One Direction - and infuriated most of them don't know it. I like to use student interests for half the categories, and use more explicit math topic categories for the rest.</li>
<li style="text-align: justify;"><b>Work maths into the "non-maths" categories. </b>For example, my third question on One Direction was "What percentage of One Direction are boys?". OK - it is a simple question - but it reinforces the idea that 100% means "all". One question I found generated interesting responses was "How many records has Beyonce sold?" - which gave a good opportunity to explore estimation. Another One Direction question: What is the name of the band member who is last in alphabetic order?" Again - it's easy, but it gets some mathematical thinking happening.</li>
<li style="text-align: justify;"><b>Make the maths category questions easy at the start </b>You want students to engage with the maths categories. I always start with easy questions - if you make them too hard, students will turn off - it's not a game any more. I save the harder questions for the 800 and 1000 point questions. I make the end-game question a more challenging - but doable - math question on the current topic.</li>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAfoYB_S7KOFwKR1rjz1szc6pms-EUSW7re6eJkY07A1n1pUWD4zokYlPVSfm8E1TDgt5myopsDbDM4bJ_K85ASmANAuJybmZdpk_WIrY36n_RMXc1ADq5qKtwgh3cyQW1aDoNPoxAFVkb/s1600/Quiz-Image-Small.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="248" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAfoYB_S7KOFwKR1rjz1szc6pms-EUSW7re6eJkY07A1n1pUWD4zokYlPVSfm8E1TDgt5myopsDbDM4bJ_K85ASmANAuJybmZdpk_WIrY36n_RMXc1ADq5qKtwgh3cyQW1aDoNPoxAFVkb/s320/Quiz-Image-Small.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Quiz editor in Stu's Quiz Boxes.<br />
I find I can reuse the quizzes across many grade levels,<br />
making this an efficient use of lesson preparation time.</td></tr>
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<b>Playing the game</b></div>
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This game is so much fun, and the students get so excited, it's <i>essential </i>to have a management strategy.</div>
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<ul>
<li><b>Every group gets a chance to answer the question. </b>This is perhaps the biggest change I make to playing the game: I don't have a "first-answer-wins" approach. In a classroom of 30 students, it's impossible to work out who gave the first answer and the noise levels are impossible if you go this way. Instead every group has a mini whiteboard to write their answer (you could use just a sheet of paper). Once I see a group has a quality answer (doesn't have to be correct - just interesting), I yell "2 minutes" and give all the other groups time to complete. When I call "time up", we look at all the answers and <i>every </i>group that has a correct answer gets the points.</li>
<li><b>Encourage group checking of answers. </b>As the questions get harder and are worth more points, I ask each group to ensure everyone agrees on the answer before presenting it. This gives the group a chance to teach the content to each other. It's wonderful to see students try to convince each other their answer to a maths question is correct.</li>
<li><b>Noise level management. </b>This is hard because it's so exciting. Never have you seen a class so interested in knowing what 8% of $200 is! As the noise level rises you'll have to calm the class down.</li>
<li><b>Prizes.</b> I confess to motivating with a very small chocolate prize. I give one to every student at the end and don't buy into "but we won...." discussions - as far as I'm concerned everyone is a winner if they participated :-) Waving the packet at the start of the game gets their attention - but it's amazing how quickly the students forget about the chocolate and become obsessed with winning game points.</li>
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Special thanks to Stu Hasic who so kindly donated Quiz Boxes to the education community. I highly recommend you try Quiz Boxes with your classes. And over time you will develop a bank of quizzes which you can share with other teachers in your faculty - or maybe even at Stu's website.<br />
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<b>Practicalities</b></div>
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Here's what you need:</div>
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<ul>
<li>A data projector (or an Interactive White Board)</li>
<li>A copy of <a href="http://quizboxes.com/">Quiz Boxes - free download from Stu's web site</a></li>
<li>A pre-prepared quiz. It can take a good hour to design a quiz, but you will find you can reuse quizzes across many year levels and they stay current for several years. You might like to challenge your class to design quiz questions for a category - although this will take some time and planning.</li>
<li>Students arranged in groups - maximum six groups for Quiz Boxes.</li>
<li>Mini-whiteboards OR a pad of paper per group.</li>
<li>Solid walls between you and the classroom next door. Close your windows :-)</li>
</ul>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com7tag:blogger.com,1999:blog-8722640913234461773.post-60528979170468325372013-02-03T11:15:00.000+11:002013-02-03T11:27:19.592+11:00Getting the most out of graphing software<div style="text-align: justify;">
"GeoGebra is your friend!" - my students must have heard me say it a hundred times. If a student asks me about a homework question, they know my immediate response : "Did you check what it looked like in GeoGebra?". If they haven't, then I will usually ask them to sit with me while we explore it together using the software.</div>
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Some teachers worry using mathematics software will weaken student's skills, but here's a mantra I recite in class which I believe not only develops mathematical skills but also stimulates deeper learning:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQVglkuYnkLAYS9JBmA14oFmJNEt8P0OKDegBijJw7Mnegql2llhIGIBm4lQPNL6rsFFwE_Xc6mm8JYi29-YGtNqPTRQJninlh5y2p2gW_fnBqz7XFItLWRMi8IJCisDRI8LlBcWXTiEBC/s1600/GeoGebra-is-your-friend-v2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQVglkuYnkLAYS9JBmA14oFmJNEt8P0OKDegBijJw7Mnegql2llhIGIBm4lQPNL6rsFFwE_Xc6mm8JYi29-YGtNqPTRQJninlh5y2p2gW_fnBqz7XFItLWRMi8IJCisDRI8LlBcWXTiEBC/s1600/GeoGebra-is-your-friend-v2.png" /></a></div>
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I believe the essential ingredient in using graphing software to answer questions is to stop and think before using the software and then <b>predict </b>what you expect the software to display. If you are fortunate, you'll find the software <i>doesn't </i>match your prediction. I say fortunate because you have discovered a misconception, an error - or in some cases, managed to confuse the software. Prediction and the subsequent reveal of an incorrect prediction is a powerful learning tool. With a <a href="http://exzuberant.blogspot.com.au/2012/03/whos-afraid-of-error-monster.html">positive attitude to the error monster</a> this revelation will stimulate questions and further exploration.</div>
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Another key learning idea I advocate is to take a few extra minutes once you have your answer to <b>extend the problem with some "what if?" questions:</b> "What if I changed that positive <span style="font-family: Times, Times New Roman, serif;"><i>x</i></span> to a negative <i><span style="font-family: Times, Times New Roman, serif;">x</span></i>? What if that was to the power 3, not power 2? What if that parameter was 4 not 5? Can I reflect that curve?" Here the power of the software comes to the fore: we can ask many questions and rapidly get answers - something not possible in reasonable time without the software. Of course students won't have the time to do this for every question, but even just doing this once in a study session is rewarding.</div>
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One more powerful pedagogical factor is at work when students use a graphing tool to help with their homework: they are forced to <b>translate their problem </b>into a representation suitable for the tool. For example, an algebraic equation has to be split into two (or more) graphs and intersections found. This serves to build and reinforce understanding of the links between the different forms of mathematical representation. Often a student needs break down the problem into steps, introducing parameters and intermediate results or constructions, providing 'hooks' they can use to explore how the problem changes as parameters are changed. </div>
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A topic I recently taught was based totally on drawing graphs by hand - and students have to be able to do this in an exam situation, without software. For a course like this, I think the graphing software is an even more valuable learning tool. Why check your answers in the back of the book when you can do this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMZAvIvO1ums2mv5cPMqWCsaeSk-CjbwnsMSPkh2_aDutZRochFh2sGi7CQPC1yHGLewmTPzQmZDt0Jrkaz93NhddebGHvo4p-KfkBjoHcAXVawvHN1f5FuyMsCv2URVOI8qlGD-M_47wf/s1600/GeoGebra-is-your-friend-pencil-v2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMZAvIvO1ums2mv5cPMqWCsaeSk-CjbwnsMSPkh2_aDutZRochFh2sGi7CQPC1yHGLewmTPzQmZDt0Jrkaz93NhddebGHvo4p-KfkBjoHcAXVawvHN1f5FuyMsCv2URVOI8qlGD-M_47wf/s1600/GeoGebra-is-your-friend-pencil-v2.png" /></a></div>
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This approach means students are still learning to work by hand - and maximising the benefits of having software during the learning of the topic - without becoming dependent on it - a bad thing at exam time!</div>
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So to my way of thinking, there's no question dynamic geometry software is a powerful learning tool: when coupled with a mindset that thinks and predicts <i>prior </i>to using the software, and then extends a problem through questioning and exploration with the software - it's like having a personal tutor. GeoGebra is indeed your friend!</div>
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<b>Practicalities:</b> There's lots of good quality dynamic geometry and algebra software available to students: I'm a big <a href="http://www.geogebra.org/cms/">GeoGebra</a> fan, and I also like the<a href="https://www.desmos.com/calculator"> Desmos tool</a>. I'm beginning to really appreciate <a href="http://www.autograph-maths.com/">AutoGraph</a> - but sadly the cost factor rules it out for most of my students. For intensive algebraic work, I point my students at WolframAlpha - especially the <a href="http://products.wolframalpha.com/ipad/">WolframAlpha iPad app</a> which is great value.</div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-86280820973572898542012-12-16T14:28:00.001+11:002012-12-16T14:38:04.724+11:00Images of Integration<div style="text-align: justify;">
Slice and dice: that's how I think about the calculus topic of Integration - take something complicated, slice into increasingly fine slices, then put it all back together. In my quest to encourage my students to see this theme in the wonderful world around them, here is a selection of images I used this term to help show the idea, generously made available by people around the world through a <a href="http://www.flickr.com/commons">Creative Commons License on Flickr</a>. If you're taking great photographs - think about sharing them under Creative Commons - a wonderful resource for teachers to help inspire students.<br />
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<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/72213316@N00/5593887213/" style="margin-left: auto; margin-right: auto;" title="Graceful flowing curves on the Bay St. Louis bridge by Alaskan Dude, on Flickr"><img alt="Graceful flowing curves on the Bay St. Louis bridge" height="375" src="http://farm6.staticflickr.com/5067/5593887213_f29b20c257.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Bay St, St Louis Bridge by Alaskan Dude on Flickr</td></tr>
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<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/manuelmeyer/5427217603/" style="margin-left: auto; margin-right: auto;" title="La Ágora by el.manu, on Flickr"><img alt="La Ágora" height="333" src="http://farm6.staticflickr.com/5012/5427217603_0b3d888f3f.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">La Agora, be el.manu on Flickr</td></tr>
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<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/manuelmeyer/5427216531/" style="margin-left: auto; margin-right: auto;" title="L'Hemisfèric by el.manu, on Flickr"><img alt="L'Hemisfèric" height="333" src="http://farm6.staticflickr.com/5013/5427216531_762b9c9dd4.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">L'Hemisferic by el.manu on Flickr</td></tr>
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<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/timtom/2079713671/" style="margin-left: auto; margin-right: auto;" title="Tower by timtom.ch, on Flickr"><img alt="Tower" height="333" src="http://farm3.staticflickr.com/2329/2079713671_8a166dae22.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Tower by timtom, on Flickr</td></tr>
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<tr><td><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/symo0/4852286705/" style="margin-left: auto; margin-right: auto;" title="Untitled by Symo0, on Flickr"><img alt="Untitled" height="375" src="http://farm5.staticflickr.com/4073/4852286705_e47485b8d0.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Untitled, by SymoO, on Flickr</td></tr>
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The idea of looking for visual representations inspired one of my students to take a photo of the magnificent Neuroscience Research Australia under construction across the road from our school - which just screams at me "Area under the curve!" every time I walk past it.
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm1rbTykQx7fnvcqdOx1_XKtFALoHP73FUDVqljZIu1wkFxAf5bgMeId1yKrrE0Q9Rwkgh8xYIZRGXnKedkX9WXRsEhM_gaHnXC2WblStio4X5cy7CzB1p0wxBurU4BM_oWObWUpJXVaEv/s1600/NRA-Joy-Yu.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm1rbTykQx7fnvcqdOx1_XKtFALoHP73FUDVqljZIu1wkFxAf5bgMeId1yKrrE0Q9Rwkgh8xYIZRGXnKedkX9WXRsEhM_gaHnXC2WblStio4X5cy7CzB1p0wxBurU4BM_oWObWUpJXVaEv/s400/NRA-Joy-Yu.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Neuroscience Research Australia building 2012 - under construction. <br />
Photo by J Yu - used with permission.</td></tr>
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<i>This is part 3 of a series of posts on teaching Integration.</i><br />
<a href="http://exzuberant.blogspot.com.au/2012/12/exploring-inequality-entry-point-to.html"><i>Part 2: Exploring Inequality - an entry point to calculus</i></a>enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com0tag:blogger.com,1999:blog-8722640913234461773.post-78157972638039777412012-12-15T14:09:00.001+11:002012-12-16T14:34:09.793+11:00Exploring inequality : an entry point to calculus<div style="text-align: justify;">
"Have you ever noticed .... ", I said to my senior maths class, as I walked in bearing a huge and very obvious glass bowl containing about 40 packets of Smarties, ".. how some people seem to have so much more than other people?"<br />
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<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/ghalog/6271929376/" style="margin-left: auto; margin-right: auto;" title="Taking it Back, Occupy Oakland (19 of 20) by glennshootspeople, on Flickr"><img alt="Taking it Back, Occupy Oakland (19 of 20)" height="332" src="http://farm7.staticflickr.com/6223/6271929376_2f5cb67c60.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">"Taking it Back - Occupy Oakland" by Glenn Halog<br />
<a href="http://www.flickr.com/photos/ghalog/6271929376/in/photostream/">http://www.flickr.com/photos/ghalog/6271929376/in/photostream/</a> CC-BY-NC-2.0</td></tr>
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I then proceeded to "share" out the Smarties: first I gave 20 of the 40 packets to one student - making a huge pile on her desk. Her eyes popped out - while the other students looked with disbelief and some concern for their own anticipated share. I gave a wicked grin and 10 packets to the student next to her. To the rest of the class I handed out 2 or 1 packets - apart from a few students at the end of line who received nothing. Oh the looks they gave me!<br />
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And so we started a lesson exploring the question of how we could measure income distribution - a hook (although the class didn't know it yet) - to introduce our next calculus topic: integration. Here are some notes on my first attempts at a lesson design using an idea from economics as a motivation why we might want to find the area between two curves. But first a big thank-you to mathematics teacher <b>Alastair Lupton</b> who showed me how to bring the Gini Coefficient into the classroom and encouraged me to try it out in my classroom.<br />
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So here's the sequence I tried this year.</div>
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<b>Step 1: Build interest in the problem. </b>With strict instructions not to eat or worse yet - share - their Smarties, we looked at a short OECD video about the rising inequality in income distribution:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/ZaoGscbtPWU?feature=player_embedded' frameborder='0'></iframe></div>
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Depending on the time available, you might want to explore some other video material, perhaps some recent news footage of the Occupy movement protests, or look at some studies of global income distribution.<br />
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<b>Step 2: Thinking how to organise the data: </b>I lined up the students, holding their very unequal distribution of Smarties. We ordered the line by 'income' and partitioned into 5 groups - helping the students see the organisation of the data into quintiles. We returned to our desks and looked at some local and international data on income distribution, also organised into quintiles. Here is some recent Australian data:</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEPsDvLRUuWHCG-Vdqkf0cGsliYzwNBPNVv5qCkfXmBel1ZcEttu9MW5hRzNVq-6bW_L3USu5LGK4CSo05bavBwZe6xe1AFKKJpSWqxR1ta1mWd-qgrZWMuZ-KjKEOf0MhCd5sG4Gql1Ya/s1600/ABS-65030-TableA1.3.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="341" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEPsDvLRUuWHCG-Vdqkf0cGsliYzwNBPNVv5qCkfXmBel1ZcEttu9MW5hRzNVq-6bW_L3USu5LGK4CSo05bavBwZe6xe1AFKKJpSWqxR1ta1mWd-qgrZWMuZ-KjKEOf0MhCd5sG4Gql1Ya/s400/ABS-65030-TableA1.3.png" width="400" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Click on the image for a larger view.<br />
Source: <a href="http://www.abs.gov.au/ausstats/abs@.nsf/mf/6503.0">Australia Bureau of Statistics 6503.0Household Expenditure Survey and <br />Survey of Income and Housing User Guide 2009-10</a></td></tr>
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<b>Step 3: Ask the question: "How could we measure inequality?" </b>This isn't easy or obvious. Give the class some time to explore ideas. Then it's time to look at how economists do it...<br />
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<b>Step 4: Develop the idea of graphing cummulative quintiles. </b>After trying some different ways to plot our quintiles, I showed the students how the economists do it: reorganising the data into <i>cummulative </i>quintiles. This allows us to make normalised curves which work for all situations, regardless of the size of the total income pool. We drew our first Lorenz Curves:<br />
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMrYgAC-WNl84AnSatgdls3oVcYYT6VmZwPbgIM5-6_CxfGELayTeeor6nfmm9UwSN_k0RVOnH_wMQekEoQ7pfgQOkuEhzsXhCC4ndOld57FI2zObuiqNJvuyx6WojMuQiO5igXegVJPc_/s1600/500px-Economics_Gini_coefficient2.svg.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMrYgAC-WNl84AnSatgdls3oVcYYT6VmZwPbgIM5-6_CxfGELayTeeor6nfmm9UwSN_k0RVOnH_wMQekEoQ7pfgQOkuEhzsXhCC4ndOld57FI2zObuiqNJvuyx6WojMuQiO5igXegVJPc_/s320/500px-Economics_Gini_coefficient2.svg.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">The Lorenz Curve is used to calculate the Gini Coefficient. The area A is the difference from total equality.<br />
The larger the area A as a proportion of the total area A+B, the greater the inequality.<br />
Source: <a href="http://en.wikipedia.org/wiki/Lorenz_curve">Wikipedia Lorenz Curve</a> Image by Reidpath,</td></tr>
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To help explore the idea, we discussed what the Lorenz Curve would look like if one person had all the Smarties, and if all the Smarties were shared equally. We also considered if the curve would ever go above the "Line of Equality" (it won't!). We selected different data sets (see references below) and plotted them. Here is the 1993 World Bank data for Nigeria plotted in GeoGebra, with a polynomial fitted to the curve:</div>
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<tr><td style="text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUZD5WjLIDjbHqvnqGHlY2dxik1eNLRR8eqTCfrdczdXz0_7PMcvnSeEhg_wsTv42w58dLbdZIqg-QQRIoD270ysoezgGcewNGhDJy7cibZK0qtRQHFyeRL178OKE0F-N28eLULA_vT6Vr/s1600/Lorenz-curve-Nigeria.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="248" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUZD5WjLIDjbHqvnqGHlY2dxik1eNLRR8eqTCfrdczdXz0_7PMcvnSeEhg_wsTv42w58dLbdZIqg-QQRIoD270ysoezgGcewNGhDJy7cibZK0qtRQHFyeRL178OKE0F-N28eLULA_vT6Vr/s320/Lorenz-curve-Nigeria.png" width="320" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">By modelling the curve with a polynomial, we can use integration <br />
to calculate the area under the curve and hence the area between the curves.<br />
Data is entered into the GeoGebra Spreadsheet window, then plotted and <br />
a function calculated to fit the data using FitPoly[].</td></tr>
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<b>Step 5: Ask the question again: how could we measure the inequality?</b> After looking at a few different data sets, students will quickly come to the conclusion that measuring the area between the line of equality and the Lorenz Curve will give us a nice single number. And now you have them hooked: here's a very interesting and practical reason we might want to be able to calculate the area between two curves.</div>
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<b>Step 6: Declare a communist revolution. </b> I then ordered a redistribution of the Smarties so everyone was equal. This was actually quite funny because several of my diet conscious students insisted they did not want any Smarties. Tongue-in-cheek I told them this was not an option - it was a revolution and everyone had to be equal whether they wanted it or not! A nice opportunity to open up the discussion to different views about income distribution. I gave my students a selection of recent articles from The Economist which seemed to provide a good balanced discussion on the topic.</div>
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<b>Step 7: Begin the mathematical discussion on ways to calculate the area between the two curves.</b> Your students will have many useful ideas! Try them out with the tools available. And now you're ready to start a calculus based exploration: What is the area under a curve? </div>
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<b>Where could you go with this lesson idea?</b></div>
<ul>
<li style="text-align: justify;">Get students to make up a small poster using their data and stick them up on the wall. Then as you move through the Integration topic, you can refer to them in the context of each new idea.</li>
<li style="text-align: justify;">Once students know how to integrate, get them to model their curves as a polynomial - I like to use the GeoGebra <span style="font-family: Courier New, Courier, monospace;">FitPoly[]</span>function - and then do calculate the integral, comparing their result to given Gini Coefficient for the data set.</li>
<li style="text-align: justify;">The student data makes for a great application of the <b>Trapezoidal Rul</b>e : they can calculate the area without knowing the equation of the curve. A good example of why you might want to use the numerical approaches to calculating integrals.</li>
<li style="text-align: justify;">Challenge activity: calculate the area under the curve using <b>Simpson's Rule</b>. If you only have the standard Simpson's Rule, <i>you can't do it because there are an even number of data points</i>! But there is more than one Simpson's Rule - challenge your students use the internet to find one that will work for this data. [Hint: Simpson's 3/8 rule will work].</li>
<li style="text-align: justify;">Apply the concept of the Lorenz Curve to another field of study. An interesting application is to social networks - some people contribute significantly more than others, while others 'lurk' in silence. I use edmodo with my class and there is a high degree of inequality in the number of postings per student - counting postings per students could make for an interesting Lorenz Curve.</li>
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<b>Thinking beyond the mathematics:</b></div>
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<li style="text-align: justify;">Talk to the economics teachers at your school. I discovered mine do teach the Gini Coefficient, but they don't go into how it is calculated. I think it could be a very powerful lesson to develop a sequence of combined economics/calculus lessons with an economics teacher at your school. The more I explored the subject, the more interesting I found it. Options to consider include: the effects of taxation policy on the Lorenz Curve; the differences in the Gini Coefficient between different types of economies; differences within one country over a time series; challenges to the validity of the measure; economic and social arguments on the topic of income distribution. All highly suitable for deeper mathematical and social science exploration.</li>
<li style="text-align: justify;">Take some time out to look at the <a href="http://www.gapminder.org/">Gap Minder website</a> which options to view the data through the Gini Coefficient.</li>
</ul>
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<b>Resources</b><br />
<ul>
<li>The Wipedia pages on the <a href="http://en.wikipedia.org/wiki/Lorenz_curve">Lorenz Curve </a>and the <a href="http://en.wikipedia.org/wiki/Gini_coefficient">Gini Coefficient </a>are a good starting place, with good entry points to more nuanced discussion on the use of the Gini Coefficient.</li>
<li>Some excellent Australian data and good explanatory notes in the ABS publication <a href="http://www.abs.gov.au/ausstats/abs@.nsf/mf/6503.0" style="text-align: center;">6503.0 Household Expenditure Survey and Survey of Income and Housing User Guide 2009-10</a>. See Table A1 in Appendix 1, and a high quality discussion on the Gini Coefficient in Appendix 3.</li>
<li>International data from <a href="http://www.rrojasdatabank.info/wdr01/tab5.pdf">World Bank: World Development Report 2000/2001: Attacking Poverty - Table 5 Distribution of Income or Consumption </a></li>
<li>The<a href="http://www.economist.com/printedition/2012-10-13"> October 23, 2012 issue of The Economist</a> contains some excellent articles on the challenges of income inequality seen from a pro-Free Market view. I found these particularly interesting given one could hardly call The Economist left wing!</li>
</ul>
<b>Some teaching reflections:</b><br />
<ul>
<li>The students really loved the lesson - they were engaged and it was interesting.</li>
<li>I planned carefully for my 'inequitable Smarties distribution'. Our class was well established and we knew each other well enough that my students would know I was up to something and trust me when I played this game. I also made sure the students who didn't receive Smarties were the most resilient, confident students.</li>
<li>I did however make the mistake of trying to do this opening lesson in a single 50 minute period - it wasn't enough time and I rushed it, making it less student centred than I had hoped. This lesson needs a double period to do it justice. </li>
<li>Is it worth taking the time out from a busy course to do this activity? I think so. Once I realised I could leverage this work into my teaching of the Trapezoidal Rule, Simpson's Rule, the area between two curves and also do some polynomial modelling, I saw it was a lesson that just "keeps on giving".</li>
<li>Coming from a physics background, it was wonderful to find an interesting and practical application of calculus to a completely different field. Many of my students are planning a career in business and are interesting in economics - here was something to show them the calculus applied to money as much as to speeding particles!</li>
</ul>
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<i>This is part 2 of a sequence of posts on teaching integration. </i><br />
<i><a href="http://exzuberant.blogspot.com.au/2012/12/slicing-and-dicing.html">Part 1: Slicing and Dicing.</a> <a href="http://exzuberant.blogspot.com.au/2012/12/images-of-integration.html">Part 3: Integration in the world around us</a></i></div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-36589359525204949992012-12-09T10:35:00.002+11:002012-12-15T14:17:12.374+11:00Slicing and Dicing<div style="text-align: justify;">
To my way of thinking, the topic of Integration is all about 'slicing and dicing' - thinking about what happens when you take an object and chop it into increasingly thinner slices, then put those slices all back together again. Here's a fascinating and gruesome hook I used in my senior mathematics class this year to consolidate* the theme of "slicing and dicing": What would happen if you sliced up a human being?<br />
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<span style="color: #e06666;"><b>Warning:</b> </span>This content is only suitable for a senior class, and you should warn students there are medical images coming up. There won't be any blood, but it might affect sensitive students and the dissection of human bodies may not be culturally appropriate in your classroom.</div>
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First we start in reverse, using a scene from one of my favourite science fiction films <a href="http://www.imdb.com/title/tt0119116/">"The Fifth Element"</a></div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/t-3AMq8XGaA?feature=player_embedded' frameborder='0'></iframe></div>
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Then let your students know the images of the human body used aren't computer generated, but actually come from <a href="http://en.wikipedia.org/wiki/Visible_Human_Project">The Visible Human Project</a>. Cue in this video clip:</div>
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<iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/dPPjUtiAGYs?feature=player_embedded' frameborder='0'></iframe></div>
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My students were grossed out and fascinated - and then asked to see it several more times! It took them a while to come to terms with the fact the images weren't generated using a medical scanning device, but by actually slicing up a body. Lots of questions followed!</div>
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Depending on time and if you think this is a good idea or not, there are some websites where students can use an online Java application to dynamically explore the data by selecting their own slices in any orientation and see the resulting image created by reassembling the original slices to your specification. </div>
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Here are two websites I found worth exploring:</div>
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<tr><td style="text-align: center;"><a href="http://visiblehuman.epfl.ch/index.php" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinutQYZy_kobK5i4k7OALSI4vU6oA9HXmucdU4UZLHqDKCgy-xBOkrrlUvlpws5tckKmibk283usxiH0VO42oXjPk_pFUBVS364IyRm-Fva0g1PI1aBT5Z6Jjj2_HOJkwXN0GjcCE_1h_Z/s320/Visible-Human-Website1.png" width="315" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;"><div class="MsoNormal" style="margin-bottom: 0.0001pt;">
<span style="color: blue; mso-ascii-font-family: Calibri; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-hansi-font-family: Calibri;"><a href="http://visiblehuman.epfl.ch/index.php">http://visiblehuman.epfl.ch/index.php</a></span></div>
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<tr><td style="text-align: center;"><a href="http://visiblehuman.epfl.ch/index.php" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="236" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEzoQuopUa6a2QONj7G8K9gFO61xCv2G9Ux38RT4AGP7UNANy4XdSVNpzHR6jNVRbo5dfl0odxf_FhMnhavK2EfvtlQ16LJ70kLo2mUTkvoR3Y38DTfCJtUAYEpl1KsszXQZ6pa8p4WZLF/s320/Visible-Human-Website2.png" width="320" /></a></td></tr>
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<a href="http://visiblehuman.epfl.ch/index.php"><span style="color: blue; mso-ascii-font-family: Calibri; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-hansi-font-family: Calibri;">http://visiblehuman.epfl.ch/index.php</span></a><u><span style="color: blue; mso-ascii-font-family: Calibri; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman"; mso-hansi-font-family: Calibri;"><o:p></o:p></span></u></div>
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Where to next? Many options for discussion about: </div>
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<ul>
<li>the mathematics and computation required to reassemble the data so that different views can be constructed.</li>
<li>the ethics of using bodies from condemned prisoners for science.</li>
<li>the value of the data from The Visible Human Project - there were scientific as well as ethical criticisms of the project.</li>
<li>Recent advances in 3D printing technology to "print" biological components using layers of living cells. A long term goal is to print transplant organs using cells from the donor. A quality video from ABC Catalyst program at <a href="http://www.abc.net.au/catalyst/stories/3618385.htm">http://www.abc.net.au/catalyst/stories/3618385.htm</a> (starting at 00:03:00).</li>
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One of my students later told me the data from The Visible Human Project is also used in a (rather violent) Japanese manga film <a href="http://en.wikipedia.org/wiki/Gantz_(live_action_films)">Gantz</a>.</div>
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<i>(*) I used this lesson idea in the middle of the topic sequence. For my first Integration lesson, I went down a different path - but that's for the next post! </i><a href="http://exzuberant.blogspot.com.au/2012/12/exploring-inequality-entry-point-to.html">Part 2: Exploring Inequality</a></div>
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enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com1tag:blogger.com,1999:blog-8722640913234461773.post-67105370310161772932012-12-02T08:30:00.001+11:002012-12-02T08:30:54.774+11:00Still going ...<div style="text-align: justify;">
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<tr><td style="text-align: center;"><a href="http://www.flickr.com/photos/cdm/84202849/" style="margin-left: auto; margin-right: auto;" title="yellow by darkmatter, on Flickr"><img alt="yellow" height="357" src="http://farm1.staticflickr.com/42/84202849_dbaab9ec15.jpg" width="500" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">Photo: "Yellow" by darkmatter CC-BY-NC-ND <br />
<a href="http://www.flickr.com/photos/cdm/84202849/">http://www.flickr.com/photos/cdm/84202849/</a></td></tr>
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It's been a very long and tough final school term. I'm still running the "<a href="http://exzuberant.blogspot.com.au/2011/07/becoming-teacher-its-marathon-not.html">marathon</a>" - albeit limping on some days. Ran headlong into some very steep hills (teaching Mathematics Extension 2 for the first time, in addition to teaching Mathematics Extension 1 for the first time... madness!). Combine this with the normal teaching load, writing over a hundred school reports and accumulated sleep deprivation - not good. Running too fast, too hard - feels like I've done a year's work in a term. In recovery mode now - still hundreds of end-of-year papers to mark but only a few weeks to go!</div>
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Like all marathons though, the experience is amazing - the views incredible. Lots of teaching ideas share in this blog once my energy levels are restored.</div>
enzuberhttp://www.blogger.com/profile/15258424253649444449noreply@blogger.com2