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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In yesterday's post on a percent game, I shared two great GeoGebra sketches that students found. I remixed each of them a little, so I thought I'd share them here.
dhabecker's neat rational number arranging sketch lets students place arrows to try to put fractions, decimals, and percents in order. He has a very clever way to check if the randomly generated numbers are in the right spots. It notifies students when they've got all 4 right, and I wanted to them to be able to check their answer along the way. So - thinking of Mastermind - I thought about what if it can give the number correctly placed? Since I was doing that I added a reset button and a bit of color. Next I would add a fifth number, as that makes so many more permutations possible.
On GeoGebraTube: download or applet. (Unfortunately doesn't seem to work in HTML5, because the polygons won't move.)
The other sketch I modified was David Cox's great percent estimation sketch. Almost immediately on trying it on the Smartboard, the students turned it into a game.
So I turned it into a game with turns and scoring. There's 6 rounds. I thought about forcing players to take turns going first but ultimately just decided to ask them to.
Next I would be interested in seeing it go from a percent number line to another quantity. So the game asks you to find 38%, but the number line goes from 0 to 630. The percent and the whole would change each time. Worth a go? Probably that's in my head because of David's nice double numberline percent sketch.
As always, I'd be interested in feedback on either one of these.
But I'd also be interested in what could help develop a remix culture in GeoGebra. I learned it (am learning it) mostly on my own by experimentation, from suggestions on Twitter, and googling stuff from the online help. But in the Learning Creative Learning class they put a big emphasis on remixing as a way to learn that gives a lot of support to learners. With my middle school GeoGebraists, they are struggling to do work of value all on their own.
Are you a remixer by nature? What would it take to get you trying it in GeoGebra?
Beforehand I was wondering how much would be new? I love his two TED talks and the RSA animation and overshare them with students, but you do notice recurring ideas. Appropriately. I also was wondering if he was tall - as he looks imposing in his TED videos. (He's not just not tall, it turns out.)
He's charming, funny and a natural performer. Really funny, like Ricky Gervais as an academic. Inspiring, too, and if you get a chance to see him, take it. He has new PBS special coming up, for his new book, and I will be watching. He is a self-promoter, and has a robust ego, probably appropriately.
He incorporated his traditional messages:
creativity is important
all people are inherently creative
disbelieve the big three myths about creativity
creative people are the exception
creativity is only valid or valuable in the arts
your creativity is fixed
current education crushes creativity, or at least discourages it
people who find their element, the connection between their passion and their creativity, are happier, healthier, more productive and have greater impact.
we cannot plan our career, as we have no idea what's coming
His new push was to create a culture of innovation. He seems to see stages of development of creativity:
He applied this mostly to education, talking about Finland and the Blue School (a school designed by the Blue Man Group), and a USC art history grad who went on to become an art evaluator for an auction house, traveling the world.
His upcoming book answers my disappointment with The Element. It's Finding Your Element on how to develop that connection. He also mentioned a major rewrite for the new edition of Out of Our Minds.
My struggles are that his stories, like the art school grad, Blue Men, Johnny Ivo, Nobel winning chemists, etc., confirm the myth of exceptionalism. They don't remind us of the students in our classrooms that we can't get to try. They also tend to wind up with 'and now they're famous and rich.' That is not going to happen for everyone. It can not. Why does his vision look like for me? For that student? (You know the one.) I was glad to hear the Blue Man story, too, because so many of his stories are of the individual, while much of what I know about creativity relies on collaboration.
I'm also left wondering why we can't convince our politicians and decision-makers of the value of this approach to education. Someone asked a question about that, sort of, and Sir Ken joked a response that boiled down to we don't know how.
Yet, for a couple of hours last night at least, I felt like this is possible. And his encouragement for teachers who are trying to make this difference is very valuable. I was reminded of and consoled by the many teachers I know locally and through the mathtwitterblogosphere who are making these changes happen for real.
Creative Learning - Week 1
Former student and learning pal Nick Smith found the Creative Learning course from the MIT media lab. It didn't take too long for me to shout "I'm in!" and join the par-tay. The leader of the course is Mitch Resnick, one of the leaders of Scratch development.
I never did get around to writing my reflection of the nice Python course I took from Rice on Coursera in the fall. So when I saw Dave Ferguson's reflection on week 1, it was the prompt I needed to try to be better about keeping record of this course.
overview of online component. Materials by email, idea of the weekly seminar (mostly panel discussions), Google+ for the course and small groups. (25,000 students)
Resnick: anecdote about current state of learning - fun stuff is for after school. School is for drill and information delivery. This course should be about rethinking learning.
Question traditional model, how to prepare people to be adaptive learners.
Inspiration from kindergarten. [Ironically was talking to a K teacher this weekend about how Michigan's new all day kindergarten is making it much more academic.]
Goal is to provide information and resources, but also a chance to create and make. We'll be using Scratch for that. (Brief explanation about Scratch.)
Great story about a scratch user who went from making a card, making and sharing sprites, working as a consultant making sprites for others, to a teacher making a sprite tutorial, to a collaborator on an adventure game.
Course will have weekly readings, Mon morning (10 am ET) panel conversations, design & learning activities, small group discussions. Overview of the course week by week.
A little long winded. I'd watch Resnick's TEDx talk first and you'll hear a lot more of his ideas in a lot less time. If you have more time, watch the Marshmallow Challenge TED talk.
"The goal is not to nurture the next Mozart or Einstein, but to help everyone become more creative in the ways they deal with everyday problems."
"As we develop new technologies for children, our hope is that children will continually surprise themselves (and surprise us too) as they explore the space of possibilities."
This excellent advice from 12 year olds for kids about to start a Crickets workshop. (The MIT media lab project that inspired Lego Mindstorms.)
I like the theme of the first week of my Math for High School course to be teaching for creativity.
This semester's group did a nice job with Dan Meyer's Toast video. I had them ask questions and record what they noticed. There was a neat dichotomy: the WCYDWT responses were all pretty traditional math textbook questions. The what they noticed branched far afield, wondering what would make a good soundtrack, wondering about darkness of toast and toaster design.
We followed that with a workshop (I think based on an Esther Billings and Pam Wells workshop) based on verbalizing and algebrafying number patterns.
The last workshop of the day was planned to be this, which in the past has been a pretty good activity:
Objective: TLW develop mathematical patterns from the teacher’s perspective.
Schema Activation: when (if you do) do you notice patterns in real life?
Focus: What we talked about in Class 01 was really curriculum. What are we going to teach? Here’s the NCTM’s take:
The Curriculum Principle
A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. A school mathematics curriculum is a strong determinant of what students have an opportunity to learn and what they do learn. In a coherent curriculum, mathematical ideas are linked to and build on one another so that students' understanding and knowledge deepens and their ability to apply mathematics expands. An effective mathematics curriculum focuses on important mathematics—mathematics that will prepare students for continued study and for solving problems in a variety of school, home, and work settings. A well-articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they continue their studies. (From the Principles and Standards for School Mathematics (PSSM for short), NCTM. All these principles have expanded information and explanation at the NCTM website.)
As we discussed, there are small shifts and large shifts. This activity applies to both: it’s an easy activity that allows for creativity (subtle shift) but may lead to you designing your own problems and questions for students (big shift).
Activity: we have blocks. Play with them!
1. Make a pattern of images with the blocks. A successful pattern for this task is one in which the next shape is determined, or is reasonable.
2. Describe your pattern in words. What’s happening, what’s changing? What can you say about the next step compared to the previous? What will the 10th step be like? A general step?
3. Describe your pattern mathematically. What can you say about the next step compared to the previous? What will the 10th step be like? The Nth step?
4. What connections do you see amongst 1 (the visual), 2 (the verbal), and 3 (the symbolic or mathematical)?
5. Repeat as time allows.
Reflection: choose 2
• Was this task too open-ended? Does it need more structure?
• Was this task engaging? Was it mathematically worthwhile?
• What were the strongest or most interesting connections you saw in a step 4?
But we didn't have time for that! (Not to butter them up, but they were jamming on the toast and I didn't want to cut that short.)
So what we did was:
Schema Activation: whole class discussion - what makes a pattern a pattern?
Focus: Pattern blocks, make what you consider a pattern.
Activity:
1. Make patterns.
2. Put a piece of scrap paper by your pattern with a Y and an N.
3. Gallery walk. Put a hashmark by yes or no if you consider that a pattern or not.
4. Stop by someone else's pattern, add three blocks.
Reflection: whole class discussion about what happened.Record your personal definition of pattern as it is now.
I'm kicking myself now that I didn't take more pictures. Most students made repeating linear patterns, some tessellation patterns, one person made a circular pattern, and then there was...
This was the only pattern that students did not see as a pattern. It was described as just random fitting together, and someone asked about the yellow. "They were supposed to be orange." And then the author shared how they were lines of blocks fit together. Another student shared how to her it was a skewed checkerboard. Then the class unanimously agreed it was a pattern. I thought that was really interesting and asked how they would verbalize it. Good descriptions followed. Could they capture it symbolically? No, not that I expected it. So I shared a bit about the wallpaper groups, and how mathematicians seek the power of good notation so they can symbolically manipulate. I also shared how I had thought the yellow was a pattern in a pattern.
The characteristics of patterns they thought most important was that it can be explained (there is an idea or structure) and someone who understands it can extend it or fill in a missing piece.
If one week determines a pattern, it's going to be a good semester!
Photo credit: The tremendous image up top is from Tanya Khovanova, in this post. The original question was rather brilliant: "Which one of these things does not belong?" Many thanks to Sue, who pointed out the author below.
Hello! It's good to be writing again. We actually took a bit of a vacation, then, of course, it's crazy trying to catch up coming back from a break. There's a bit of writing inertia to overcome, but it's definitely better to be writing than not. Starting my math for high school class, we watched Ken Robinson's 2006 TED talk on how Schools Kill Creativity.
Through the blessing of a larger than expected number of mathematics student teacher assistants, I get to teach or coteach the whole secondary math ed program this semester. 329 - Math for Middle School Teachers, 229 - Math for Secondary Teachers (not renamed since before 329 existed) and Ed 331 - student teacher observation and seminar. As a department we've needed to review the sequence as a curriculum for coherence, so now it's a good opportunity. The content for the first two courses is almost obvious to divide, with the exception of linear equations and functions. The pedagogy... wow. That's thorny. The field experiences are centered on classroom observation in HS (229) and and individual student assessment in MS (329). Right now, the high school is centered around the NCTM Principles, and the NCTM process standards in the middle school. We're lucky enough to have Char Beckmann in our department, so there's texts that follow that plan.
At the initiative of my colleague Dave Coffey (who is just starting blogging - you should read it) I started thinking about math ed classes as having themes of doing mathematics, learning mathematics and teaching mathematics. For me, the correlation between that framework and the processes/principles should be where the instruction happens.
How do you teach preservice secondary teachers? Organize your curriculum? Emphasize as themes?
At this point, if you haven't watched Sir Ken, now's the time.
Having watched that, the students thought about what was important to them. What is creativity (original ideas that have value); the conditions for creativity (if you're not prepared to be wrong, you'll never come up with anything original); the diversity of intelligence; the importance of teaching to encourage and support students. Later this week, I watched Charles Limb's TEDx talk, "Your Brain on Improv." He was actually able to observe the self-critiquing and monitoring areas of the brain decrease function, and the expressive parts of the brain increase function during jazz and rap improvisation. (Worth watching just to see a neuroscientist rap.) Also stumbled across, again, Jordan Matter's Dancers Among Us photo series, which is great in light of Sir Ken's Gillian Lynne story.
We then considered the "So What?" What does this matter for math teachers? They did a great job thinking about this. It makes process more important; requires multiple modes of instruction; enhanced by more real life connections; values problem solving and reasoning; shows more than one way to do a problem. It was interesting to me the subtle bias from their education - these are all things the teacher does. Still no autonomy or choice for the learners.
I'm wary now of revving preservice teachers up too much. When we see graduates in the schools, one of the most common things to happen is to have them apologize. They apologize because I've given them guilt over that they should be doing more activities or writing their own problems or using more technology. Which really means I should be apologizing. (And I do.)
So we discussed that our response to this issue of change can be big or little. Subtle shifts or big changes. One of the examples that came up as restrictive was number computation. As an example of a subtle shift, we tried going from "what's the answer?" to "how else could we do it?" (Really a better question in terms of differentiation as well as mathematics content.) So what is 72x26, and how else could you do it? Firstly, students were suprised by how some people were taught, and secondly, they (who do already know how to multiply) really got into it. Coming up with new methods, seeing connections, making sense of what other people had done. Really doing math. I picked 72x26 because it was adjacent to doubling, to x25, etc.
In black are the responses to how they were taught. The first response was the partial product method, which dre some "weird" comments. Then the lattice just freaked them right out. How else got them thinking about strategies like the red. These were closer to their mental strategies. When their methods were exhausted, I shared the green strategies. There was some surprise at how different these were.
Next, for an example of a big change, I shared the example of coming up with a whole new activity. Writing curriculum (Curriculum is one of the NCTM Principles). As an example, we looked at What Can You Do With This. In particular David Cox's and Dan Meyer's WCYDWT toast. (David's original toast post, Dan's regression spectacular) I don't think it is reasonable to expect all teachers to create curricula. On this scale. But with our networked community, we don't have to.
Students watched patiently. "I never knew how long toast took." "Does my toaster take this long?" Even before the first piece popped, "how do the settings control the toast? Time? Heat?" And then a mountain of questions once the toast popped. From toaster design, to physics, to burner arrangement, to people's toast darkness preference, to what they noticed about the times in between, to why doesn't the bottom edge toast, etc. Possible answers to those questions. How they could collect data. The image of the toast set them off anew. What they noticed and what they wanted to know. The more they figured out, the more there was they wanted to know. That's a good sign that you're doing mathematics. More pleasing, they made strong connections to the ideas we discussed following Sir Ken. They saw the potential for buy in, the significance of student proposed questions.
It was a great start to class, but it left me a little nonplussed. This is what class could always be like if we weren't shackled with the expectations of previous generations. The math that was useful at the dawn of industrialization. It's like Jacob Marley in reverse, in this teaching life we will bear the chains our forebearers forged in theirs. But it also held the promise of Buffy. In each generation of teachers, some will be called. They can lead us out of the hellmouth and empower the generations that follow. (When you start mixing Dickens and Whedon, it's time to finish.)
And I think the journey might be fun. One of the students demanded the Toast Song for our recessional music.
They were able to tap into the arguments that people have contrarily, though. Math is about number-crunching, plug and chug, is boring, about right and wrong. I asked them to discuss at there tables how to counter those arguments, and they said:
Everything can be mathematical - there are numbers everywhere. I think in numbers! Any lesson can tie in.
Numbers can be manipulated in many different and unique ways.
The basic skills are concrete, but the application is creative. Like physics: need math to describe creatively.
The reasoning is creative. People discovered mathematics.
The communication of math is creative. Explaining how things work.
Creative how mathematicians come up with the formulas. Might have just discovered it accidentally.
The different ways to represent: graphs, tables, etc.
Math is the universal language. Everyone can communicate in it.
Creativity is important to me (see these other posts) as a goal for my students, and I think opportunities for creativity add a lot to the likelihood of engagement.
It was very insightful to me how they focused on the communicative aspect of mathematics. If math is a language, it may or may not be creative, much as uses of language may or may not be creative. As we share our own genuine thinking, and the way we perceive the world (or a problem), we create opportunities for creative expression.
The class moved on to look at how they communicated their work on a problem of making quadrilaterals by folding a square (an extension of this nice 3rd grade problem filmed by Annenberg - Teaching Math, Lesson 20). And immediately focused on what the answer was and were they right. After our years of school mathematics, we have definitely been well trained.
In the second workshop, we considered the quadrilateral types, and in particular the idea of nested categories or hierarchical sorting. I show this weird little travelogue:
(I want to update it and maybe make that into an animoto, but the ppt file is corrupted, so it will be more work than I have time for right now.)
The students then made posters of some of the quadrilateral types, striving for a variety of examples, and to be creative in making the posters. Critique and discussion of the posters brought out the discussion points I was hoping for, like which properties are necessary, considering symmetry as an important characteristic, and whether trapezoids should have exactly or at least one pair of parallel sides.
Trying to make space for creativity is not going to be a one lesson effort, but hopefully a theme for the whole semester. I can't wait to see what happens.
EDIT: updated the slideshow to have more visual cues and a couple extension slides. I wanted there to be more to notice.
EDIT2: added student posters for the quadrilaterals. We worked on generating a variety of examples. Some made only the specific types, but some made a variety of types that fit the required properties. I like having both kinds of posters!
I've written about Ken Robinson a few times (One and Two). The idea of creativity in mathematics was a sub-theme for my summer calculus class. Students at the end felt that some of the open-ended assignments (projects of their choice), non-standard problems (like the mobiles) and emphasis on problem solving helped open them up to creativity in math. But they suggested more specific demonstrations of how to be creative. (Boy, is that insightful.)
TED occasionally has question and answer sessions with their speakers who really ignited something with their presentation, and Sir Ken recently did this. (Here's the article.) He addresses math specifically:
"If you want to promote creativity, you need, firstly, to stimulate kids minds with puzzles and questions which will intrigue them. Often that's best done by giving them problems, rather than just solutions. What often happens in classrooms is, kids sit there trying to learn in a drone-like way things of not much interest that have already been figured out.
The best math teachers I know, like the best English teachers, are always giving kids puzzles. They're given things to work on where math skills are required but may not be the focus of the activity. There giving them problems to solve. Or they're made to engage with age-old mathematical problems. For example, I'm thinking about the problem of latitude. How do you go about measuring the planet? I mean, somebody had to do that. How do you do it? Professional mathematicians have such a cornucopia of fascinating puzzles, questions, proposals and conundrums. A great math teacher really has endless opportunities to stimulate kids minds and get them engaged with things they'd probably never thought about before. Rather than just giving them techniques." -Ken Robinson
It's not hugely original, but it's nice to get confirmation of things we believe from outside sources. He touches on engagement several times in the Q&A, and I do believe that's the central issue in teaching, and I love to ponder what is the key for math. Going to more and more of these reading conferences, I am insanely jealous of the teachers who talk about the book that turned a student on to reading. Problems don't seem to have the same effect.
So I showed the Ken Robinson video to my class on the first day and it was quite interesting. The video started with medium engagement, but by the end everyone was tuned in. The story about the choreographer (Gillian Lynne) seemed to really connect.
The first connections were to elements of their own stories, and were interesting. Then one of the students brought up a connection with how Robinson describes us as being educated out of creativity. Her sister is interested in being a soloist, and it took ears of retraining or untraining to get her out of what she had been taught to do as a choral member. I'd never heard of that before, but it connected exactly to the point I wanted to make.
They have been educated - and successfully so, as college calc students - in school mathematics, which has almost nothing to do with mathematics as practiced by mathematicians. (Which really needs a descriptor. Real mathematics isn't going to interest anyone. Creative mathematics? Cool vs. school or cruel mathematics?) Instead of being able to repeat what someone shows you, it should be about solving problems that you don't know how to solve. In which situations mathematicians excel, because they are entirely willing to be wrong, and even glad to be wrong if they learn from it. They're willing to, as my friend Dave says (quoting his favorite Australian, Brian Cambourne) to "Give it a go!"
As long ago as 1982, my calc instructor, John Hocking, worked mightily to convince us to have no fear. To be wrong 100 times if it teaches you something. That math was exploratory, and creative. He shared his topological research with us... which was amazing and fun. He's the reason I added the math major in the first place. Often he put it as "blank pages are just waiting to be filled."
Remember about TED? I stumbled across Thomas Dolby's blog recently (I was a geek in the 80s, of course I like Dolby), and have been reading through back posts. He posted this link from TED that I thought was unreal. Ken Robinson is an expert on creativity, and funny besides. If Ricky Gervais had become an academic...
His point is about how we are born creative and educated out of it. An outside observer would think "The whole purpose of public education is to produce university professors." But he goes on to describe how we have convinced the majority of people that the things they are good at and interested in are not valued or even stigmatized.
So... what to do about it? I'm going to show this to my Calc 2 students on Monday and see what they think.
If you haven't been there, you should try TED. (Technology, Entertainment, Design.)
While math is not their greatest area of stength, they do have a few insightful talks. Try these. Also or alternatively how we learn, especially Stuart Brown on play. Although somehow they missed Nate Silver. Get more Nate at his excellent blog, which covers the intersection of statistics and politics.
My favorite is about the nature of genius and creativity, though I think she's off by a smidgen. (Needs God/Holy Spirit in there. But who doesn't?)
It's basically impossible to watch everything on there that's worth watching. So if you see something worth it, let me know!
dhabecker's neat rational number arranging sketch lets students place arrows to try to put fractions, decimals, and percents in order. He has a very clever way to check if the randomly generated numbers are in the right spots. It notifies students when they've got all 4 right, and I wanted to them to be able to check their answer along the way. So - thinking of Mastermind - I thought about what if it can give the number correctly placed? Since I was doing that I added a reset button and a bit of color. Next I would add a fifth number, as that makes so many more permutations possible.
So I turned it into a game with turns and scoring. There's 6 rounds. I thought about forcing players to take turns going first but ultimately just decided to ask them to.
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