A blog for sharing my math interests on the web, to post new materials for elementary, secondary and teacher ed, and vent mathematical steam when needed.
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The Mathematics and Multimedia Blog Carnival is coming to Mathhombre.
Just not yet!
Today was the originally scheduled date, but the guiding director and founder of the Mathematics and Multimedia Blog Carnival has reset it to be in the fourth week of each month. It will be here in 3 weeks, November 29th.
That means 3 more weeks to submit your own ideas or report a blogpost that you think would be good. Here's the link for submissions.
Our study of motions led to one of my favorite topics in all of mathematics: tessellations. I've posted some previous work on this blog, have an old webpage with some good tessellation resources, and found a new source of beautiful Alhambra images to share with students.
As a math topic I just love them. The visual aspect, the geometry, the connections with algebra, the historical context, the art connections with Escher... it's darn near perfect. Working with 2nd - 12th graders they are amazing for rigid motions because you use the motions to make the tiles, then to repeat the tiles in a pattern, then can see the motions in the finished tessellation. It's visual and kinesthetic. There ought to be a song.
Instead, how about some geogebra. For some reason, this time around, I got interested in kites. Which tessellate by side to side rotation and what would a mixed glide reflection/rotation tessellation look like.
The question of which kites tessellate by double rotation boils down to what happens at the joint vertices between the two (possibly) different edges. We can pick the angles so that the kites blossom (tessellate around a point) at the vertices between congruent sides. In this sketch, you set those numbers, then observe the effect on the other angles. What condition is necessary for the angles to work out? Is it sufficient?
This sketch lets you make alterations to that classic 60-120-90-90 kite tiling. The sketch will adjust and give you a chance to both design and watch the effects.
This sketch does a tiling that can be done with any kite. (Pretty good question as to why it works for any kite!) Two of the congruent sides rotate to themselves around a midpoint, and the other two fit together with a glide reflection. Escher was fond of this pattern, as it allowed him to create creatures going in opposite directions for his contrasting tilings.
I would love to hear from readers if they prefer these geogebra sketches as links or embedded applets. Could you take a second to comment? Also, I love making geogebra tessellations, so if you have any ideas for ones you'd like to see, let me know.
I wanted to gather an overview of tech resources for students, and thought it would be worth posting for everyone. Nothing completely new, really, but I'd like this to be a good overview. Please add other relevant resources I overlooked or don't know about in the comments.
New teachers: don't overload! Subtle shifts. Look for something that will help you do what you want to do, and slowly incorporate or try out new things.
File storage and sharing: have something to share with many people or everyone? There are many free internet services.
Dropbox - best for working with colleagues. Simultaneously shares files and backs them up in the cloud. Allows you to undo to previous saved versions for up to a month. Invisibly syncs files on your computer with those in the cloud.
Box.net - similar to dropbox without the seamless syncing. May be better for sharing with the public, though.
Scribd - document sharing that you can embed in a blog or other web 2.0 application. Works with Microsoft Word and PDFs perfectly.
Google docs - clearly the best for simultaneous editing and sharing from anywhere.
Presentations:
Slideshare - upload powerpoint presentations to embed or share publicly or have available to students and yourself from anywhere. Embeddable.
Prezi - cool graphic-style presentation which is like a dynamic concept map. You can embed youtube videos, present on or offline, embed the prezi and easily share them. The editor is suprisingly easy to learn.
Glogster - internet posters. Multimedia possible. Free to teachers and their students. Easy enough for quick student projects.
Animoto - turn powerpoint files into movies. A little limited at the free level. Also easy enough for students.
russelldavies @ Flickr
Math specific:
Geogebra - my favorite, bar none. Geometry, algebra and free. That's just incredible. Exports images for use in documents, exports sketches as a dynamic webpages, runs on any platform. Becuase it's opensource, there's a large library of free resources developed by other teachers. (Some of my sketches are on this blog.)
Wolfram|Alpha - the app/webpage that makes a computer algebra system available to anyone with the internet or a smartphone. Solve equations, access databases, investigate almost anything quantitative. (Here's my W|A think aloud for a question, but here's a better introduction to Wolfram|Alpha.)
Core Plus math tools - free dynamic online applets shared freely from the authors of the Core Plus Mathematics Project curriculum, for algebra, geometry, discrete math and statistics.
NRICH - a sortable, searchable problem bank of rich problems for K-16 mathematics from the national maths organization of the United Kingdom. Updated with new problems monthly. Many problems accompanied by a dynamic visualization.
Virtual Manipulative Library - good for computer demonstrations, smartboards, and students with internet access. Some are clunkier than others, but many are quite smooth.
Other Tools
Jing - free screen capture software and screen video capture software that allows you to save to your computer or share via their linkable website. The souped up version is Snag It, and the same company makes Camtasia which is a quite nice movie editing software.
`This time the K-8 preservice teachers were discussing Chapter 3 - Driven to Understand, and Chapter 4 - Dwelling in Ideas. Van Gogh was the art connection for driven to understand. He said "I want to get to the point where people say of my work: that man feels deeply." For dwelling in ideas, she meditates on Edward Hopper, since in his paintings what is not happening is as important as what is.
Reader comments:
Stereotypes of low income schools - that teachers don't want to go there. But kids need someone to believe in them, and we can be those teachers.
Chapter 4 describes a classroom so perfect that it seems unreal. And it does seem out of reach.
Have worked in classrooms with conferencing... it is hard to get that culture where students do silent reading as you meet with them one on one for conferencing. Can meet goals of conferencing with all, though, if you're flexible on the time frame.
Crafting/Composing/reflection - all relevant things to any kind of learning.
These teachers use silent reading time after vigorous activity. Can be more meaningful learning time by adding reflection to it.
Shouldn't just have reflection at the end of the day, but throughout. While it's still fresh. Kids will notice if it's not "dead air" at the end of the day.
Giving students time to think and hear themselves think can be part of the reflection.
How do we connect this book to math? The painting that they're trying to understand is like the Pythagorean Theorem. Don't have to do it a certain way, people will have their own way to make sense of something.
I do think about the book in relation to math, but the reading is also relevant to me as a preservice elementary teacher.
There are parallels to math.
The book can be redundant almost... the initial question "what does making sense mean?" gets brought up a lot. Want the book to be more straight forward.
"As important as what is happening is what is not" - what do they mean?
When an artist provides little detail we are left to sort it out ourselves. "What's missing" may be those details, so that we understand it on our own.
The theme is that there is not one way to understand. I won't be surprised to get to the end without a definition. The idea is to make it your own. Even if that's stressful or frustrating.
Discussion - what's happening is great. But the kids who are not involved in the discussion is "what's missing."
Fervent learning... there was the story of the student who went above and beyond. Is our goal to make every subject that interesting? Then the students will direct the learning instead of us. We set the stage, give the tools, and let them work.
Tight schedule or teaching to the test fights against teaching for understanding. What about kids left behind.
I'm currently observing a teacher give the ideas to students, then they move on. How can we stray away from that, if the schedule is any tighter?
I thought it was great that they see a tension between coverage and real learning. But I want to encourage them that real learning can meet the goals of the test-givers, also. Real learning will beat the standardized tests at their own game.
We related these chapters to the idea of structure and gradually releasing students to greater freedom. Started talking about how we can wean the students away from learned helplessness.
This chapter also has one of my favorite teaching moves. Seems just straight dumb, too. When a student says "I don't know," Ms. Keene responds, "If you did know, what would you say?" Oh! I thought there was no way that would ever work, but now I have seen it create a kind of freedom for the student to answer more times than I could count. Incredible.
On the whole, the students confirmed that they are finding the reading worthwhile, and are glad that we take time in class to discuss it.
Photo credits: uhuru1701 & Bert K @ Flickr. Cartoon from Rubescartoons.com
This week my K-8 students were working on motions again. Using the Geogebra activities at Motion Sketches and More Motion Sketches. For K-8, there's really not a need for glide reflections because they're usually not a part of the curriculum.
This doesn't fit with Euclid's vision of motions, which was strongly tied to congruence. Any two objects are congruent if and only if there is a motion from one on to the other. (Aka rigid motion or isometry or Euclidean transformation...) This requires four motions, not just three. But a glide reflection is just a slide and a flip, you may say, we don't need it. Well all the motions can be made from just reflections, but we still teach turns and slides.
But I've been stumped as to a good way to present these glide reflections. Students can recognize them by a process of elimination and students can make a motion that is a glide reflection. The next level of knowing a motion is to be able to specify it. Students are good at finding lines of reflection, and can specify direction and distance for a translation. It is difficult for many/most to find the center of a rotation, without being told. They can do it in a dynamic environment (cf. MotionControl, a geogebra webpage) but it is difficult for them to construct. The first guess seems to be connecting corresponding points and trying where the lines cross. (Which doesn't work.) So it's really hard to get students to know how to specify a glide reflection. Mathematicians usually describe a glide reflection with a vector and a point or position of that vector. The vector indicates the direction and distance of the slide, and the line containing the vector is the line of reflection.
This sketch is my attempt at a glide reflection sketch - the goal was to create an environment where students might be able to notice things that would lead them to construct the idea. I would love it, if you try it out, to get feedback about ways to make the sketch more supportive. Thanks!
The math game is at the end of the post - feel free to skip the rambling!
Adapting Games
This post actually started as I was thinking about my middle school bible study (online at ghbiblestudy.blogspot.com) and trying to think about a game that fits this week's study on Lazarus and the Poor Man. An hour seems to be a bit long for my students to engage in bible study, at least for now. (I haven't helped them build that capacity. Yet.) One of my favorite games occurred to me: The Great Dalmuti by Richard Garfield. (Currently in print... so buy it! Funagaingames.com is a great online game store.) It's actually an adaptation of a family of games that have been played for many years. So I was looking for the rules for playing with a regulation deck, so that my students could play at home. (It also got me thinking about making a version of the game with Monopoly money... more on that later!) Richard's version uses a pyramid deck, with one 1, two 2's, etc., that helps gameplay but is hard to simulate with a regulation deck.
Ed Yourdon @ Flickr
In looking for the rules, I stumbled across an online archive of Richard's game design articles from The Duelist, which was the print magazine dedicated to one of Richard's other games: Lost in the Shuffle: Dalmuti This article is about how games could be as big as movies. Is that what's finally happening with the gamification of the internet? (Note that Richard's current thoughts on game design, along with those of several other designers, are there to hear at gameswithgarfield.com)
I sort of think the fourth year of HS math (for non-STEM majors, at least) should be a game playing course with Go, chess, hearts, bridge or Euchre, rummy, and Magic. (Maybe Minecraft... I'm investigating.) What else would you add?
The version of Dalmuti I wound up with for my youth group is below. No specific math content, but problem solving and strategy is strongly helpful. Maybe a good setting for probability dilemmas.
One of my favorite games to adapt is Rummy. (Rules here at Pagat, good resource for card game rules.) It's a game about connections, seeing the cards in your hand in relation to each other and the cards in the discard pile, those relationships and then what your opponents have. There's deduction and induction as you assemble pieces of information from what people play, discard and pick up versus the hidden information in their hand. How much more mathematical can you get?
Units of measurement is one place where many preservice teachers feel underprepared, especially with respect to metric measurement units. As we work in my K-8 geometry/measurement class on units, we use cards with a variety of units. For them I have on some weird ones (drams, furlongs, hectares) to help them think about the process of learning these units. We sort them by quantity that they measure, pair them as equivalent amounts, think of things to measure with the unit, and arrange them by relative size of the unit. We play Concentration (aka Memory) with the equivalent pairs. But the most fun, clearly, is playing Unit Rummy!
Unit Rummy Deck: All unit cards. Players: 2 to 5
Deal: Shuffle all cards together. Deal 5 cards to each player. Flip over one card to start the discard pile. Place the stockpile and the discard pile card in the center of the table, where all players can get to them.
Object: To have the most played cards by the time someone goes out.
Course of play: After the deal, all players receive time to organize their cards. The person on the dealer’s left goes first, continuing clockwise. A player may start his turn by drawing a card from the pile or picking up discards, going as far back into the discard pile as he wants. If a player goes back to a certain card, he must immediately play that card in a meld. (see “The Melds” below). You can’t discard this card. He may play any melds he wants to. It is not required to play a meld. A player must always end their turn by discarding a card (their choice). Lay the discards down in a way so each card is visible.
The hand is over when a player has no cards.
Optional Scoring: Score one point for each card in play, and subtract one point for each card in hand. The player who goes out gets 3 bonus points.
Calling “Rummy”: If a player discards a card that may be laid off on other cards, the first player to notice it calls “Rummy” and plays the card for his benefit. A card counts a discard once a player takes his hand off it. Once the next player draws or picks up a card it is too late to rummy.
The Melds: a meld is a set of three or more matching cards. Cards match if they are equivalent measures or all use the same unit. For example 1 c = 8 fl oz = .25 quart for equivalent measures, or 1 ft, 1000 ft, .1 ft for all the same unit. On your turn, you can play a card that matches another player’s meld.
Cards
Instead of the weird units I have them sort and reason with, we play this game with a more elementary set of units. (Email me if you want the college version with the weird units.)
The K-8 Geometry class is reading To Understand this semester. This book is Ellin Oliver Keene's follow up to her mega-book with Susan Zimmerman, Mosaic of Thought. Yes they're reading a reading book. But this book is really about comprehension, much like Mosaic of Thought, and most of these preservice teachers will be in elementary school, and teaching reading. This is the second semester using this book, which had mixed results last semester, but overall positive reviews from students.
The aspect of the book that students had trouble with is that each chapter (from 3 on) starts with the story of someone in the arts advancing their own understanding. It was difficult for some students to connect with these stories if they were unfamiliar with the artist. Last semester students did make lots of connection to what they'd read from Mem Fox; Reading Magic is on my list, but I haven't gotten to it yet.
The structure we're using is groups of up to 8, where each group has a leader - responsible for keeping things moving, questioner - for prompting discussion, recorder, and a reporter who leads the group's sharing with the whole class. After the free discussion period, the group talks about what to share, then the whole class discussion starts.
Teach concepts for greater depth. Current understanding is the key to understanding in the future. Don't do too many topics.
Even in a calc class, you can move on before students have a conceptual understanding.
Quality vs quantity is a dangerous choice. Eg. pizza parties for number of books vs what was learned.
Teach in different styles with kids having choice of structure. In elementary and HS everything was more structured vs in college when the info is presented and you do with it what you will. Elementary school needs more choice and freedom.
Everything has trial and error. Repetition is everywhere, for example math workbooks that ask many of exactly the same form of question.
Testing is crappy, and we focus more on scores than understanding. The test generate so much pressure from so many sources. Home, school, etc. You never get them back, so you can't learn from it. Test too much, at least standardized testing. What if the kid doesn't have supportive conditions (sleep, food, etc.)
All children can achieve greatness. A principal who says "someone's got to flip the burgers..."; this kid just isn't good at that. Beliefs will be communicated to students. Vs. believing in all students.
"We can extinguish the notion that some kids are going to make it and some kids won't." Psych study that showed teacher beliefs have a huge impact on achievement.
After Ellin described a great day, her husband David asked: "why isn't everyday like that?" Easy to make excuses. Ultimately it's because we don't expect them to. Belief motivates.
What will you do about barriers to eliminate the real reasons for student struggle? Eg. a teacher who brought in bagels and cream cheese before a long standardized test.
Also be careful not to expect too much, so students feel defeated.
In elementary kids are excited to go to school. They will/do lose that if it's not supported.
Do teachers care more about noise or intellectual activity?
One student hated the book so far because it felt like the author posed questions without answering them. Another student said that that was creating a desire to read on and a curiosity. That maybe the whole book was about what it means to understand.
First workshop (focused on determining importance) was more beneficial than the 2nd (stopping at each page to monitor your comprehension). Better recall of Ch 1. Stopping at each page broke the flow of reading. (This was ironic because last semester it was reverse, so I had changed Chapter 3 workshop to be like Chapter 2. But I let them know they can always choose a different way to meet the objective for the workshop.)
This workshop (Ch 2) can be an example of what not to do. If it becomes a task to do the motions, then it defeats its own purpose of looking for comprehension.
Our study of motions led to one of my favorite topics in all of mathematics: tessellations. I've posted some previous work on this blog, have an old webpage with some good tessellation resources, and found a new source of beautiful Alhambra images to share with students.
The question of which kites tessellate by double rotation boils down to what happens at the joint vertices between the two (possibly) different edges. We can pick the angles so that the kites blossom (tessellate around a point) at the vertices between congruent sides. In this sketch, you set those numbers, then observe the effect on the other angles. What condition is necessary for the angles to work out? Is it sufficient?
I thought it was great that they see a tension between coverage and real learning. But I want to encourage them that real learning can meet the goals of the test-givers, also. Real learning will beat the standardized tests at their own game.
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