The solid you obtain are not convex polyhedron. They are sometimes referred to as "stellated polyhedron", but this is not exactly a stellation. In fact, the polyhedron is augmented with regular pyramids on all its sides. What makes them fascinating to me is how they embody the duality of the platonic solids...
It was during my master degree. One day, my friend Isabelle told me about polyhedron origami: You start with a certain amount of square paper pieces, all folded in the same way. This gives you modules that you assemble into the required solid. To see how, you can refer to this video. Although it is in french, it is enough visual to see how it works.
The solid you obtain are not convex polyhedron. They are sometimes referred to as "stellated polyhedron", but this is not exactly a stellation. In fact, the polyhedron is augmented with regular pyramids on all its sides. What makes them fascinating to me is how they embody the duality of the platonic solids...
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I come recently across this nice result. Displayed that way, it looks like a poem. Too nice to be true, so simple in its beauty! Proof:
Many students having a bad grade just throw away their test and try to forget about it. But this is not how it should be! Test correction allow students to learn from their mistakes, gain a better understanding of their grades and, allowing to gain back part of the lost marks, reduces the test's stress. How does it work? When I mark a test giving students the opportunity of test correction, the only feed back they have is how much point they gain (for exemple 3/5). From that, they have to find how/where they loose their marks, understand their mistakes and/or their improper math writing. Then, they hand in their test back with a test correction that include an analysis of the mistakes made along with what they think is the correct answer. I check their new answer and see how much it would have been worth during the test. If it comes with a good analysis, they get back ½ the mark they lost. For example a good analysis of an exercise with mark 3/5 along with a well written correct answer (thus worth 5/5) gives back (5-3)/2=1 mark. However, the same exercise where the correction still miss something and is worth 4/5 along with a good analysis of the mistake discovered, gives only back (4-3)/2=½ mark. Preparation & tools Before giving the test: When I write my personal correction to a test, I try to have a particularly clear mark-scheme coming with it.This allows me to be effective and is anyway necessary for being consistent between the first and second correction. For the students: It is important that the students understand what is expected from them. They tend to think that "I did not understand" or "I wrote ... while I should have written ...". Therefore, I started to use grading rubrics for the correction. It uses a 4 levels scale: 0, 1/4, 1/2 or all of the marks back. The the highest one can only be reached in really rare occasions. Also I introduced the 1/4 level for cases where I see efforts but still not a good analysis or if only part of the mistake is analyzed. Objections and response Time consumption: One may argue that it demands double correction time. I would answer: not necessarily. The first correction is faster than a standard test correction as I have a clear mark scheme in advance plus I don't spend time writing feed back. For the second correction, hopefully the students have the correct answer so I just have to give few feed back on the analysis. Plagiarism & too high grade: I don't use class time for test correction. This makes that student can get help from tutor, friends... But the same objection can be made about grading homework. I think student can benefit of discussing their errors with others and the analysis part make that they cannot just copy the correct answer from a friend. They still have to understand what went wrong. As for the possibly too high grades, giving test corrections makes that I can be little more picky on the writing of math. Also, I never allow test correction for the final exam which then can balance slightly generous grades. What I think is important is that students still keep in mind their original grade as this is what they were able to achieve in test condition. Finally, my experience shows that the final grade they achieve usually reflects better the student actual level. Remaining thoughts I haven't find the ideal solution concerning what to do when an exercise is left (almost) blank. Right now, I see 2 different approaches:
My journey with Bean continues and today's subject is group work. I won't list here the argument for (or against) group work or even discuss the numerous amount of strategies (you can refer to Engaging Ideas for that ;) ). Instead, as group work is already something I use, I will start from my own experience. My most recent experience of Group work Forming the groups. Most of the time, the groups are formed by students sitting close to each others, sometimes more or less randomly assigned. "More or less" is because it happens that I still try to have groups of mixed competences. I am actually not comfortable with assigning groups completely chosen by me. In a way I don't feel I have the competences or the insight for making groups that would work better than groups formed randomly or by affinity. Concerning the size of the groups, it might depend of the activity. Usually, there are three to five students in the groups. For some group works, I even do not force the ones who are not comfortable to work in a group. Maybe I should, but I intend to discuss that later. How do I assign group works? I make students work in group in different ways.
Reflections
Flipped classroom, what a tempting concept! But this suppose before class reading.My first trials where not successful and couldn't be: I just asked the students to read this or that section in their book or look at this or that online resource. No guidance, no reading question/quiz. Students just came to class saying: "I tried, miss! But I didn't understood..." And I ended up lecturing. How worst can it be? The most recent ones went better, thanks to few reading questions, always including Mazur's last question: “Please tell me what you found difficult or confusing in this reading assignment. If you did not find anything difficult or confusing, tell me what you found most interesting.” Still, I just tried it on some particular topics, having difficulties to "let go" and not lecture on difficult or completely new concepts. Why? I think mainly because I do not trust students reading capacities. In chapter 9, Bean gives many reason why students have difficulties to read. Two points kept my attention:
(Experts) hold confusing passages in mental suspension, having faith that later parts to the text may clarify earlier parts. They "nutshell" passages as they proceed, often writing gist statement in the margins. They read a difficult text a second and a third time (...) They interact with the text. A reading guide?
Reflections on Bean "Engaging Ideas" chapter 8.In chapter 8, Bean gives different strategies for designing critical thinking tasks. As a conclusion (p. 160), he gives a list of possible strategies. I will copy the one I find the more relevant to me for the moment and try to give an idea of task I could use that relate to that precise strategy.
– Think of tasks that would let students link concepts in your course to their personal experiences or prior knowledge: At first, I had difficulties to see how I could possibly do that. But actually, there are a few that might work! This all starts with Writing in the Discipline's project at Dawson College. Reading our main reference book, Bean, "Engaging Ideas: The Professor's Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom", I tart to feel like I need a space to write down and organize my reflections. In fact, as I read and reflect upon the utility of writing for student learning, I stated to realize that I could also benefit from writing. Yes, but what kind of writing will be the simplest and more effective?
Yesterday, I started thinking about asking my students to write math haiku. So I started to googling for examples, ideas on how make it work etc... During my search, I came across great math educators blogs (see the links section). That's when I decided to use this form of writing. I see many advantages:
Now this blog does not aim to become a major math and education resource packed with new revolutionary ideas. But if readers can benefit from my quest, I will be more than happy to help! |