One Resource a (Week)Day #13: Rich Tasks from YouCubed.org

I have loved these rich tasks from YouCubed.org. I found myself puzzling over this particular puzzle about flipping coins this afternoon. It ties nicely to modular arithmetic. I believe in the end that for any number of coins n, its minimum number of flips is a function of (n mod 3) and floor(n/3), but I'll let you try it and see for yourself! As usual, I find that starting with smaller problems really helped me to generalize and see a pattern.

I also enjoyed the much simpler but accessible paper-folding task from Mark Driscoll. What I find to be really enjoyable to read is also Jo Boaler's commentary on each task, what questions she asks, and when she allows the kids to work independently versus talking to each other.

If you haven't already signed up to be on the YouCubed.org mailing list, I highly recommend it! I find myself jumping with excitement to open their emails. Even though some of the site's content targets middle-school teachers, it is so enjoyable to hear and read of the way Jo talks about mathematics.

On the upper end of open-middle problems, I had a lot of fun today thinking about the size of the tiny sphere that can fit snugly inside the space of a tetrahedron built from 4 larger spheres. As in, if you packed 4 equally sized spheres together like shown here (this is the top-down view) and then in the middle space fit in another tiny sphere, then how big is that tiny sphere's volume relative to the other bigger spheres? PCMI had problems like this in their Geometry sets from this year, and I enjoyed playing around with this particular problem. It's probably too challenging for most of our students to do, but a good one to keep in my arsenal nonetheless.

I hope you have enjoyed the tasks from today. Ciao!

First-Pass Brainstorm for SY2014-2015

For some reason, my sleep pattern has been erratic here in NOLA. Some nights I sleep beautifully and still can go back to sleep after I wake up in the morning. Other nights I get only around 5 or 6 hours of sleep, and any noise like the AC kicking in or a bug buzzing around or the train passing by a few blocks away is enough to wake me up. 

Two nights ago, I made the mistake of thinking about what I would do on Day 1 of the next school year just as I was lying down, and that was enough to keep me wide awake most of the night. The next day, in an attempt to calm my nerves, I pulled open a spreadsheet and started jotting down some of those floating ideas and to attempt to organize them, while staying away from the areas that could get me into trouble. (As in, I really don't have full control over what I will teach in Algebra 2, since there will be 4 of us teaching Algebra 2 at our school next year. Similarly, there will actually be 5 Geometry teachers next year, and we'll all have to agree on what to teach, which seems daunting to me.) Thus, I started my brainstorm with the bigger-picture things that I can fully control -- like how I plan to build a collaborative and reflective classroom culture, and ended with the more curricular things like which enduring concepts I would want my students to walk away with. Instead of focusing on each class individually and going into all the curricular details of that class, I tried to figure out which ideas or themes seem to flow through more than one class and are therefore the most important and transferable. I looked afterwards at both Kate's and Anna's lists of essential Algebra 2 understanding questions, just to make sure that I didn't leave out anything majorly important. (Thanks, gals, for the fabulous lists!) I decided not to separate the process goals from the content goals in the end, because realistically, each type of goal will require from me the same level of concrete steps and careful planning in order to implement it with gusto.

This is a work in progress, but now that I have a basic framework, I think the next step is to find and refine some rich tasks that I can use. I would love any resources you may have for me! 

One Resource a (Week)Day #12: NASA's Sustainability Math Curriculum

NASA offers a great collection of resources for teaching real math topics embedded in real science. I was skimming through their archive of materials today when I came across these interesting lessons:

Did you know that the length of an Earth day changes over time, meaning that the ratio between Earth's revolution and its rotation around the sun is not constant? That is so cool!

Also, based on the idea of cell (waste) equilibrium, we can estimate cell sizes.

Here is another nice tie-in to geometric proportionality: Estimating rate of glacial retreat via photos.

They, in fact, offer an entire book of well-organized "Earth math" for free to help educators with teaching sustainability math. A lot of the concepts can be tied easily to middle-school math.

Yay, NASA!

Just For Fun: Cake Problems

I was thinking about the variety of cake problems that are open-ended, accessible, and that encourage mathematical thinking and reasoning.

Easy: How to cut a cylindrical cake, Salvadoran-style, into n equal pieces.

Medium: How to cut a circular cake using 2 parallel cuts, such that each of the 3 resulting pieces has the same volume. 

Medium: How to cut a square cake into n pieces, so that each piece has the same amount of cake and same amount of frosting.  

Hard: How to cut a cake that already has a hole in it into 2 equal parts (from PCMI's pizza session, originally from Car Talk Puzzler). The PCMI version of the problem contains a diagram, so I give it to you here. Note that you don't know the sizes of the rectangles nor the relative angle of rotation of the inner rectangle.

Expert: How to cut a cake while maintaining maximum freshness.

Random Puzzler: How can you cut a cylindrical cake 3 times to make 8 equally large slices? 

Yum, cake! Lots of math fun in sharing things!

One Resource a (Week)Day #11: Math and Sustainability

I started looking into resources on teaching math and sustainability together, and it is like going down a rabbit hole. I think this is going to take me the rest of the week, possibly, to cull through all the fabulous resources out there. I'll review them piecewise.

Since I moved back to Seattle, I have really been thinking more about sustainability in the curriculum. Part of it is because I live in Seattle and most of the residents of the city recycle and compost on a very regular basis. (Since we moved to Seattle, Geoff and I have learned a neat trick from our friends for composting. We put our partially filled compost bags in the freezer, and that helps to make composting do-able for us. If you have been on the verge of composting, this could really make a difference for you! We have been amazed by how much our trash is cut down, once we started composting last year.) At my school, we have an extensive environmental stewardship program wherein every single student and full-time faculty member (including most of the non-teaching support faculty) gets out and cleans the school 3 times a week, which includes scrubbing bathrooms, cleaning classrooms and hallways and offices, emptying recycling and compost bins, etc. We do it because we want to instill in the kids a sense of caring for their environment, and this sentiment extends to the greater environment around us. The school's cafeteria institutes Meatless Mondays and provides compostible to-go utensils for the occasions when the staffers need to eat during a meeting. At some point this year, we used an all-school assembly to teach the students how to properly sort trash, and another meeting was devoted to looking at what happens to our sewage waste and where it goes. We encourage the students to minimize packaging on any food that they bring to school (ie. during "parties"). Geoff and I try to walk, bike, and bus everywhere in Seattle, and on the occasions when we do need to drive, we typically use a smart car-sharing program called Car2Go that both saves us money from having to maintain a car ourselves and minimizes our carbon footprint over time. Although we're not vegetarian, we subscribe to a bi-weekly CSA basket for fresh produce to encourage us to eat sustainably (it's local, organic, and cheap... win-win-win!). As a math department, we talked at length about what happens to those textbooks we order and don't use. This coming year, most of the teachers opted away from ordering textbooks as a result. So, at least I don't feel like a hypocrite when I talk to kids about sustainability these days.

Just like teaching math together with social justice, teaching math in the context of sustainability can encourage our students to think critically about issues around them, using the lens of mathematical reasoning. That said, teaching sustainability with math is certainly a challenge, for me at least. The summer is a great time for me to delve into these resources, and I would love to hear what you already do in your classroom and what has been successful.

I found a couple of terrific resources today. This website has some model lesson plans, and I looked specifically at #8, #9, #10 which pertained to the higher-level math classes. I really liked them. I think they are thoughtful and relevant, and with little or no modification can be used in our high-school classes to align to existing content.

On a separate note, Professor Pete Kaslik has written an excellent book on math and sustainability intended for the college level. (If you scroll down on that web page, there is a link to download the book. I linked to the webpage because it includes some copyright disclaimers from the author.) I think that with some modification, you can adapt most parts of Kaslik's great content to be teachable at the high school level. (The statistics part is the only part that I think is difficult to adapt without leaving behind most of the juicy bits.) More generally, his book is a great, in-depth view of math in the real world. Each topic feeds into the next one, covering an array of math content that is solidly rooted in real-world application.

For example, Kaslik starts by investigating individual sustainability in terms of basic financial education and savings -- classic Precalculus stuff. After that, he extends the idea of exponential growth to population and limited resources, in that process investigating a new type of pattern (logistics curve). He talks about the geometry of maximizing living area while minimizing loss of energy (minimizing wall space) in designing architecture, and then takes you through the math of looking at the carrying capacity of towns, based on other living requirements. You can then compare this carrying capacity with population patterns. To investigate sustainability issues further, he introduces the idea of surveying the population, goes through the mathematics of sampling, and then ties it altogether with complex analysis of dynamic systems and how you can model the many input variables using technology (Excel programming). All in all, all of the math is authentic and motivated with real analysis of real issues that can be scaled to critically consider the national or global implications. Brilliant!

I really enjoyed today's foray into sustainability and math, and I look forward to more digging around tomorrow!

One Resource a (Week)Day #10: Sustaining Student Motivation

After the last post, I went and spent some time reading up on motivation. One resource I decided to read is Daniel Pink's Drive, because even though that sounds really familiar to me (I am a bookstore browser), I couldn't easily find any reviews of the book from educators via Google. I read (most of) it between last Friday and today. The book follows a sort of 20-80 rule; the first 20% of the book contains 80% of the central ideas, and the rest of the book kind of just goes over the concepts in finer granularity, and I found my interest steadily tapering as I got farther into the book.

I am kind of iffy about recommending this book. Like I said, because it's so front-packed, I think you can more or less borrow it from a friend and glean the applicable ideas in a day. One thing I did like about reading this book is that it breaks down the best motivation for specific tasks in quite some detail, beyond the one-liner that is most commonly relayed ("extrinsic motivation is bad for kids").

Pink actually provides a quite detailed break-down of types of tasks. Say you're a teacher and you have a complex problem that you want your students to solve creatively. In order to be successful, they need to be inquisitive, resourceful, and to consider a wide range of approaches. (Yes, yes... Ideally, all of our teaching looks like this.) In that case, you definitely don't want to introduce any extrinsic motivation. The task, if given at the correct level of intellectual challenge ("just beyond the comfort zone of the student by 1 or 2 levels"), will be its own reward. The effort and process required to solve the problem are its own reward, because that level of concentration is necessary to our well-being as people, and the kids will enjoy purely being in the "flow" of the moment by engaging in the task. Giving them an "if-then" extrinsic motivation (like mentioning the impact this will have on their grades before they start the task) will negatively impact their performance, by limiting their ability to be flexible and open to all cognitive options. In a more long-term impact, it'll also diminish their intrinsic motivation to do work without grades attached. If, for some reason, after class you decide to collect the task and to grade them (a sort of extrinsic motivation after the fact), since it hasn't impacted their experience during the active learning, it will have minimal impact on their intrinsic motivation to learn. (They will still associate that learning task/experience with being intrinsically motivated.)

On a more rote task (such as skills practice that cannot be avoided before an exam), you should provide flexibility in timing and method as much as possible, to retain the level of autonomy of the student and therefore to increase their motivation. Pink asserts that autonomy is a central human need, which in turn encourages motivation. An example of this might be having multiple review sheets, and letting kids choose which ones to work on, and for how long. When they start to get bored, they should be encouraged to switch to another task to help to break up the task. Explaining directly why this more rote task is necessary/purposeful instead of relying on a reward/punishment system ("I will count it as extra points on your upcoming quiz!") is also more beneficial to encouraging quality work, as well as not impacting the students' sense of intrinsic motivation.

As for verbal feedback, Pink asserts that students' intrinsic motivation is encouraged by direct praise of their efforts and specific (positive but well-earned) feedback on their work. If you have to give rote homework assignments as additional practice, for example, besides explaining why this is necessary, be sure to also praise their efforts (individually, as you go around the class) and to collect the homework for individual feedback.

If all else fails, Pink recommends that an "if-[you-do-this]-then-[this-will-happen]" extrinsic motivation should only be used as a last resort to encourage rote tasks that have no meaning (ie. stuffing envelopes and putting stamps on... I had trouble picturing what you should use this for in education, that could be truly that meaningless).

Very importantly, Pink believes that all people (employees and students and even some higher-order lab animals) are intrinsically motivated. We have the natural need to grow and improve. What we need to do as organizations and teachers is to find ways not to suppress that natural need, but to encourage its natural expression over time.

In a separate reading of a short handbook on motivation (available free to me through Amazon Prime), the authors Albert and Robbins had a practical idea for helping to work through fears of failure. I think this is a common problem that affects a fair number of students. Talking them in advance through the worst-case scenario can help to alleviate the panic that often comes with doing poorly on a test. For example, take the day before your first quiz or test to go over what they should do, if they do fail or struggle on the first exam. Should they come to talk to you? Should they do corrections on their own to prepare for a re-quiz? Remind them of the growth mindset then. Reviewing the assessment policy in your class before a test can help to minimize the panic / fear of failure and to establish trust early on in the class.

Setting small, achievable goals is also important for sustaining motivation over the long run (after the initial energy invested in the "newness" of the task has run out). Achieving smaller goals builds up the confidence and stamina required to tackle bigger goals. I shared a strategy from Albert and Robbins with my husband, about keeping a log of current incremental goals and also a log of the goals that you have already achieved. He says that he does this on his computer, and sometimes when he feels frustrated about work, he would still pull up the "Already Finished" goals list and look at it, to help sustain his energy and to feel good about how far along he has come. (He is really crazy about goals. His current work goals list runs about 44 pages, and his personal goals list runs another 22 pages. So, I don't know how big his archived goals file must be.)

That's it! I hope this has been as enlightening and practical to you as it has been for me. I'll be back here tomorrow for another Resource of the Day.

One Resource a (Week)Day #9: Open-Middle Problems

Hi, I live under a rock so I am sure you all already know about this, but I just discovered this fabulous collection of open-middle problems, which is my offering to you as today's resource of the day. I was disappointed, however, that there were few problems under the High School category, so I am going to look and try to locate some to add to their collection. I already wrote them about one set of polynomial problems that I had used this year, but I'm not sure whether they'll post it up since the problems (as I noted to them) are not original problems from me, and I had found them on a website that is not operating under Creative Commons.

Anyhow, I should at least write about those problems here, and I'll just link you to them!

During the last weeks of this year, I had taught two methods of polynomial division (long division and the box method) in Algebra 2 and then decided to hand out these "backwards" problems to my students in randomly assigned groups without any hints. I gave each group a big piece of butcher paper, and told them to try to work out the problems, in any order, on that sheet. I'd go around and check them off for the problems that were completely correct, with all work shown on the butcher paper. I also encouraged them to talk to their team mates if they successfully completed a problem and got it checked off. Because they were working off of the same piece of paper, the students were looking at each other's work (especially the problems that I had checked off as being correct) in order to share approaches and to help look for procedural errors. If I do this activity again next year, I'll definitely escalate the level of challenge by asking each group to try to find two different ways of solving each problem.

Since we had just learned and practiced polynomial division, I was surprised and delighted that they came up with really a variety of approaches to these "backwards" problems. Some of them used the idea of a root to set up systems of equations, which provided for a rich discussion afterwards when we compared methods across groups. They also understood that if you know the remainder already, then you can figure out what the exact (perfect) product was during the division process, even though it was something that we had really not talked about explicitly. I loved the creativity they had!

It helped me appreciate that every unit, I should be providing some substantial "backwards" assignment (similar to this one, given with no hints) in order to help them think more flexibly and to help transfer the learning. If you can go forwards and backwards, then that's how you know that you really understand it, right?

Hasta el lunes (si Dios quiere??)!