Monday, June 30, 2014

One Resource a (Week)Day #6: What's Math Got to Do with it?

I was going to review a Calculus text that I flipped through today, but then I got distracted reading the introduction to a book What's Math Got to Do with it? that I had received as a Xeroxed hardcopy a while ago. I was so moved by the introduction and even without the byline I had guessed it was Jo Boaler's work. I decided to abandon the plan to review the Calculus text, and for the next few days I'll be spending my free time reading the digital version of this book.

See you on the flip side!

Dipping My Toes into Sewing

I took an amazing weekend sewing class! It was 8 hours this weekend, 4 on Saturday and 4 on Sunday. We learned how to thread and to use a sewing machine and how to make a bobbin (the spool of thread that runs on the opposite side of your garment; you make your own bobbin whenever you change the color of the thread, so that both sides of the seam can match on color...). We also learned some basic sewing terms and did some simple projects. I picked out the fabric beforehand myself, since this studio doesn't sell any fabric on site. Also, very interestingly, based on our individual choices of material, we each had slightly different challenges in the sewing process. It helped me to appreciate/imagine the complexity of problem-solving when sewing on your own at home.

Project #1: We made a cinch bag! The bag is two-layered, meaning the inside has the same pattern as your outside. (ie. I used four sheets of the striped material total.) This project seems complex (and it does have a lot of pieces), but it was really not bad to assemble! We also got to use a decorative seam along the cinch, which was good fun.

Project #2: We also made an infinity scarf! It was an easy project, but a good learning experience in choosing appropriate fabric. I chose a fabric that was a bit too heavy, which has an unfortunate impact on the appearance of the final scarf. Next time, I'll choose a lighter, softer fabric!

Project #3: They call this a "circle skirt", but actually it's a "conical skirt" when in adult size, because you end up cutting out a half circle and then sewing the radii together to form a seam on the side of the skirt, thereby forming a cone. I helped the group figure out how to use math to find the inner radius that would help us match to the desired waist measurements, since the teacher was showing them trial/error and they were getting a bit confused. Yay math!!

It was WAY cool! I really enjoyed the individual touch on these projects. Since we all brought very different fabric, the projects all looked vastly different in the end, to suit our individual styles. So fun!!!

My next sewing project will be one that I attempt individually, by reading pattern instructions. I am going to go to the studio for drop-in help to make sure that I am on track while completing this, and so that I can delay buying a machine. (The studio also has all the cutting, measuring, etc. supplies that you may need.) I have chosen this Miette skirt pattern, and I am excited because although the design seems pretty simple to make (no buttons or zippers), here you can see the variety of beautiful skirts that have come from this! Amazing!

This is officially my new hobby. I am completely obsessed. There seems to be a large sewing community in the Seattle area, including meetups and wine socials where people come together and help each other with projects, which is just super exciting. The hubby says that if I can keep up with the learning and drop-in studio sessions through Christmas, then we should invest in some sewing supplies for me to tinker at home. Wait and see my progress!

Friday, June 27, 2014

Easing Students' Transition to Constructivist Learning

I plan to sit down at some point to write about my end-of-year Calculus and Precalc projects, and the Habits of Mind/Math Practices portfolio project that we plan to do for next year. But, for now, a question that has been circling in my mind...

I reflected upon my journey in transitioning to the new school this year and upon the difficulties that I had. I realized that a big part of this is that I bring with me a constructivist approach to learning, which the kids were not at all used to. It helped me to reflect on the fact that learning to learn by yourself is really not easy. Some of the kids ramped up quickly on the new learning mode; some struggled for a while (a good half-year) and eventually acclimated. I won't say that they all fell in love with it by the end, but a significant portion did seem to grow to enjoy it (this was most evident in my Calculus class). From the rest, I had endured major pushback, including a few going to the principal and saying that they felt unsupported in my Precalculus class, that I wasn't teaching effectively, etc. It caused issues in my Precalculus class's social dynamics, and by the time that had happened, it was challenging to dig ourselves out of the hole. Although things definitely got better by the end of the year, I never felt that we built a good learning community, only that I managed to reach them with my varying degrees of personal relationship to each of them. By the end of the year, I was actually SHOCKED when some of those kids came in during the last week of school and said that they would really like to have me again for Calculus, because I had felt largely unsuccessful with them all year.

In the end, my Head of School told me (at the end of the year, during our one-on-one meeting) that he wants me to stay strong in what I do, because teaching via constructivism is more difficult and more worthwhile than teaching by direct instruction all the time, and that it helps kids to build the perseverance, resourcefulness, and communication skills that we talk about teaching besides the content. It also helped that my Head of School's son and his son's buddies were in two of my classes, and that privately they had told each other when he was nearby that they really liked my classes.

Anyhow, this experience has certainly helped me to appreciate how difficult it is for some kids to ramp up on the constructivist approach of learning. In the past, I probably had similar students who felt similarly challenged by the transition, but were simply less vocal about it. Although our school culture definitely breeds kids who are especially vocally reactive, I do see it as an opportunity to challenge myself to ease that transition for the kids next year who will be in the same situation.

Things that I think may be helpful:
* Being explicit about the learning model that we're using and that frustration is considered healthy in this learning process. (State this at the start of the year, but also at the start of tasks that are particularly challenging.)
* Being explicit about how and when to ask for help at the start of the task, and what type of work I would have expected them to already have completed (tables, defining variables, diagrams, asking each other, etc.) before asking for help.
* Being exaggerated in my compliments/celebration of how well they're working in groups and what great strategies I am seeing.
* Doing more "I notice... I wonder..." shareouts in groups and as a class, to lead in to times when I do need to stop the whole class to go over something. Even if they don't get through the whole process, what they are noticing should be celebrated.

Do you have other tips for helping ALL kids transition over from a lecture-based classroom to a constructivist classroom?  I need to do better than just letting them acclimate at their own speeds. I find, by the way, that the unhappiness in the transition is not limited to the students with weaker math skills. One of my least happy/most vocal students is a student who thrives under direct instruction but struggles with ambiguities.

Thursday, June 26, 2014

One Resource a (Week)Day #5: Blackbox Challenge

I took a break from reading math stuff today to do something hands-on.

I wrote an entry yesterday about an idea that I had read about tying math to simple machines, but I decided that if I were to use this in my class, I would simply offer it first as a challenge for the kids to build a box that magically scales its input to output. I would show them the one that I made, as a blackbox, (they wouldn't get to see what's inside), and then give them a week or two outside of class to meet in groups to brainstorm ideas and to do some research.
Bonus points for any group who brings me a working design (built/assembled, ideally with recycled material) by the end of the two weeks. And then we can discuss ratios involved as a class and relevance to other simple machines. Sorry, my video is not great, but I had fun building this out of stuff laying around at home! (Paper clips, cardboard, masking tape, glue gun, and string.) You can't really see in the video, but the "SLOW" side takes in input more slowly than the other side spits out output. There are evenly spaced tick marks on the string to help with visualizing this. My hubby recommended putting the pieces of tape on the string to help with visualizing. If the ratio was 1, you'd expect the output length (from hole to ending position of tape) to be the same as the input length (from starting position of tape to hole).

It's magic! Or, just plain mathematics...

Wednesday, June 25, 2014

One Resource a (Week)Day #4: Calculus Applications and Sustainability

I picked up several Calculus books from school to take home to skim through over the summer. I am actually pretty happy with my Calculus curriculum from this year for its exploration-based approach, but I am (of course) looking for supplemental resources to further enrich the curriculum next year.

One book that I picked up is called Calculus and Its Applications, authored by Marvin L. Bittinger. The Post-It that I had slapped on it says "student-friendly", which was just my initial impression after flipping through it for 30 seconds. As a teacher, I try to be creative in my own presentation of material in the classroom, but I always appreciate a textbook that is straight-forward and easy for students to reference in hindsight, with a "classic" organization of ideas and skills. For example, our previous textbook had a weird content layout where they tried to present everything at once. Although some of it is innovative, that's not really a student-friendly text, in my opinion, because without a teacher's guidance, that textbook is daunting and likely to remain on the shelf untouched.

Anyway, this textbook by Bittinger is heavy on applications, which I like. I tried to interleave my class with economic, physics, and basic max/min problems as much as possible this year, but this book goes quite a bit further to research real mathematical models generated from real sets of data. The examples are (dense and therefore) more suitable for college students, perhaps, but I appreciate how they regularly cite their sources to corroborate their use of real data, and there are also little side panels that offer more information on, for example, the process and limitations of radioactive dating. The book also includes some applications with which I am not familiar, including the elasticity of demand and some environmental sustainability problems. (I really liked the practical tie-ins, although realistically I think that some of the econ bits are just a tad bit too abstract for my Calculus students, who found even marginal cost, marginal revenue, and marginal profit to be elusive notions this year without concurrently taking an economics course.) Certainly, this would be a nice companion text for someone who's looking to apply Calculus in their other subject studies, and probably a really nice text for a university-level math prof, who is looking to expand the appreciation for Calculus as a useful tool of analysis.

The student-friendliness that struck me initially is evident in the structure of the text. The front inside cover has a "Summary of Important Formulas" for easy reference, and the index of applications is also unusually placed in the very front of the book. The book starts off with a quite comprehensive review (Ch. 1) of prerequisite algebra skills, which I imagine most Calculus teachers can assign as independent review. Ch. 2 is when they start to move into limits and other Calculus-related concepts.

As a secondary school teacher, what I like about this book (besides the no-fuss organization) is the applications and some attempt at bringing technology into the curriculum.

I liked this investigation of limits using the Table feature of the calculator, and will no doubt adapt this for my classes next year. (I liked the start of the activity, but I think they could have pushed it a bit further to include numerical analysis of limits with rational functions.)

The book had a really gentle way of transitioning from the tabular / arithmetic limit approach to the graphical approach. They said that you should draw an "X" on the endpoint that the function is approaching from either side (+ and -), and if the two X's overlap, then the limit exists. It's nothing fancy, but the language is really student-friendly and I liked the tactile approach.

Further into the book, they also used movie stats and technology to model a logistics curve, and then used that to motivate the finding and analysis of derivatives. I think this is pretty cool, because you can modify the structure slightly to have students pick their own movie / album sale / etc. and to do their own analysis based on their own personal interest, and it introduces them to a useful functional form that isn't always studied in earlier courses.

Also really cool: There are problems in this book that have to do with sustainability, which I am always looking to bring into my classroom. (I am convinced that we all need to be teaching sustainability, the same way that we all teach literacy and we all teach technology. It's an essential 21st century value that they must have when they leave our classes.) The problems pictured below have to do with an application of integral Calculus in predicting how much longer certain ores or mines will last before they are depleted.

Here's another problem/discussion having to do with how to hunt animals sustainably and to maximize our return without damaging the ideal animal population. The problem is quite involved, but mostly because the ecological concepts within it are not ones that we are accustomed to thinking about, from a mathematical standpoint.

I hope you have enjoyed this installment of One Resource a (Week)Day! I feel pretty great so far about the few resources that I have looked at. I know that there are lots of invaluable digital resources out there as well, but I plan to delve into them when I am down in New Orleans. This way, I won't be carrying any textbooks with me to and fro.

Tuesday, June 24, 2014

One Resource a (Week)Day #3: Math and Simple Machines

So far, summer has been pretty great but pretty busy. School only ended last week, but it sure feels like I have been on summer break for a long time. In no particular order, I have:

* Gone on a glorious evening bike ride with a local bike group
* Checked out the annual Solstice parade, which included naked bike riders, costumed dogs, marching bands, and everything in between!
* Shopped, ate funnel cake, and all around enjoyed the Fremont Solstice fair -- on a side note,  I fell in love with a local project to help low-income and homeless youth get some real job training. trains these young people through design internships and empowers them by giving them real ownership and a sense of accomplishment in designing, printing, and selling their own t-shirts. I bought two of Sydney's awesome blue shirts after talking to her and finding out that she's an intern in this program, and I only wish that there were more sustainable options like this for helping young people get on their feet!! 
* Celebrated my husband's birthday with a really fun group!
* Finished reading my first summer book (just in time for my book club meeting)
* Slept and napped and exercised and watched a good deal of TV. I feel pretty relaxed and wonderful. :)

Anyhow, I am also trying to stay on track with my goal of One Resource a (Week)Day. Today I found some interesting activities in a book called More Mathematical Activities, by Brian Bolt. I'm frankly not as enthusiastic about this book as I am about the two previous resources for straight teaching purposes, but I do think it has a lot of promise as a resource for math circles or math clubs, that do primarily recreational mathematics. The book has lots of puzzles (some of which I have seen elsewhere), and although some of the language can be a bit confusing, it also comes with a  solution key to help elucidate the instructions.

Here's a simple activity from the book that I liked, because it tied to an interactive exhibit I have seen at the Museum of Math in NYC. The activity calls for drawing parallel line segments and then cutting/translating the paper along the diagonal slit. What you observe is that one of the original parallel segments "vanishes", much like the monkey in the MoMath rotating wheel exhibit. And, similar to that exhibit, what is observable here is that each remaining line segment gained a fraction of length in order to compensate for the loss of the extra segment / monkey. For students, I think this activity is easier to wrap their heads around than the monkey display at MoMath, and would accompany the MoMath exhibit nicely.

This (below) is also quite interesting. The book has a discussion about how mechanically you can construct something that will turn circular motion into linear motion, which is useful in engine design and other such applications. The students' task is to construct such a machine. The reason why I like this is that it seems to tie proofs and the idea of "geometric validity" into an everyday application. Is the proof doable by students? That would really enrich this task to bring the learning to the forefront. It's something for me to look into/think about a bit further.

This (below) is also interesting. Here the book goes through various ways to scale something via a physical construction. I think as math teachers, most of us are familiar with the idea of gears and gear ratios, but notice the variations they have, even to include pistons and sliding ramps! Those applications are less familiar to me. Eventually, the students' task is to construct a black-box machine that accomplishes a specified ratio in input and output rates. I think the examples are great, but the task is a bit too simple. It seems like the kids can just duplicate the designs without much further thought. In order to incorporate this meaningfully into your classroom, you would have to re-think the structure of the teaching or the assessment in order to up the cognitive level.


I have always liked these mathematical paradoxes involving infinite iterations of recursive definitions. I liked the way the author paired these two similar examples together. But, again I think the author could have used a bit more variety in their tasks.

I hope you've enjoyed this installment of One Resource a (Week)Day! More to come tomorrow. :)

Monday, June 23, 2014

One Resource a (Week)Day #2: Interesting Mathematics from 1970!

Today's resource of the day is also an old publication, and one that I have seen on more than one school's shelf. I'm pretty sure I saw this book back at my school in El Salvador, but never quite looked through it then. I picked it up immediately this summer when I saw it in the to-be-donated stack at school because of this odd familiarity. The publication date is 1970, and it reads like it, because the textbook, entitled Mathematics - A Human Endeavor and authored by Harold R. Jacobs, has the tagline on the inside cover, "A Book for Those Who Think They Don't Like the Subject" and is laid out in such a way that it overviews topics from across the spectrum of algebra, geometry, statistics, etc. It's not like a modern textbook. It doesn't bother with test preparation. The author is interested in authentic mathematics that generates genuine interest, and it really shows.

The book reads more like the mathematical storybooks that my parents had bought me as a child. The table of contents looks like this (I picked out the chapters that I particularly liked, but it's most of the book):

Ch. 1. Mathematical Ways of Thinking, introduces inductive thinking through billiard logic and deductive thinking through proving number tricks.
Ch. 2. Number Sequences, covers a variety of sequences (arithmetic, geometric, cubic, etc.) all the way through the Fibonacci Sequence, each with interesting applications.
Ch. 3. Functions and Their Graphs, starts to look at the connection between equations, predictions, and graphs but doesn't go into nitty gritty details of each form.
Ch. 4. Large Numbers and Logarithms, introduces logs and exponents in historical perspective, as well as real applications (beyond earthquakes and decibels).
Ch. 6. Mathematical Curves, introduces classic geometric shapes via their construction methods. Including spirals and cycloid! (which often gets omitted in modern textbooks since they're not immediately useful to the standardized "bottom line.")
Ch. 7. Ch. 8. Ch. 9 build up an essential understanding of basic counting, probability, and statistics, tying in interesting examples as how to apply statistics to decipher secret codes!
Ch. 10 Topics in Topology, starts with a brief exploration of networks, shortest paths, and ends with questions about the Moebius Strip. 

Anyhow. This book is amazing. It's a math-lover's book as much as it is accessible for those who perhaps don't already love math. To give you just a small taste of the brilliance of this book, here are some nice bits when I was flipping through the book.

This is a sequence problem that visually represents Fibonacci Sequence. It made me think about how much trickier it would be to construct a geometric representation of f(n) = f(n - 1) + f(n - 2) + f(n - 3).

This is the classic historical view of logs. I saw this at PCMI once, and it's pretty much presented the same way here. They give log tables in the book as well, for use in the exercises.

I think this is the first time I have seen a math problem about submarine cables.  It's an awesome application of scientific notation, especially because  I think our students probably don't realize that there are fiber-optic backbones to the internet and it can potentially generate all kinds of interesting questions.

I have seen billiard problems, but I liked these. These were different from the ones I have seen. Assuming that the initial shot goes out at 45 degrees, you can ask many interesting questions if you know the length and width of the table and can assume the ball to be of point size. 

I hope you've enjoyed today's edition of One Resource a (Week)Day! Stay tuned for more tomorrow!

PS. Incidentally, my old school in the Bronx used Elementary Algebra also by Harold R. Jacobs. That's a really nice algebra 1 and 2 textbook. Not fancy but solidly usable, with great writing and some nice problems/explanations.

Friday, June 20, 2014

One Resource a (Week)Day #1: Geometry Tasks from a Wonderful Geometry Book

Today's resource du jour is an awesome Geometry book that I found while digging through a stack of old books at school. As soon as I opened the book, I immediately felt its old-publishing style magic, because the first chapter was filled with beautiful geometric pictures, tying shapes and symmetry into culture and arts. In fact, upon a closer look, the entire first chapter is also filled with activities that the students can foreseeably complete at home as enrichment, from drawing perspective drawings to creating fancy visual illusion art of their own! The book (maybe not news to many of you) is titled, Discovering Geometry, An Inductive Approach and it is written by Michael Serra. I really cannot do this book justice, but it really has a gentle and lovely approach to Geometry. For example, to teach the students geometric vocabulary, instead of providing definitions, it provides examples and non-examples. The students would look through the examples and non-examples and attempt to create their own definitions. This is what the book means by inductive approach, which ties in nicely to Common Core mathematical practices.

Also, each section in the book ends with a little fun/challenging exercise called "Improving Visual Thinking Skills" or "Improving Reasoning Skills", that is often open-ended in nature. The task/goal is very specific, but the approach is not specified, which encourages problem-solving and critical thinking, within the framework of a task that has low entry, high ceiling. I can see kids approaching them in a variety of different ways (which again, supports Common Core math practices).

Here are a few sample tasks just from opening the book randomly to a page. They are really all fabulous!

This is really a rich resource if you are going to be a Geometry teacher! I will be on a Geometry team (of 5 teachers) next year, so some of my big-picture assignments may be a bit restricted (compared to last time when I taught Geometry and had full control of my own pacing, content, etc.), but I still LOVE this book because the tasks are short and very doable as extra/fun enrichment outside of class to improve their geometric thinking.

Stay tuned for my One Resource a (Week)Day on Monday!

Thursday, June 19, 2014

One Resource a Day

It's the summer!!!!!! I am trying to figure out what I am going to do with all my free time, since Geoff doesn't approve of when I hibernate in the summer time. (It has happened. I can sleep ~20 hours a day when I channel my sleeping chi.) I'm also trying not to be a brat about my semi-permanent vacation status when around my non-teaching friends, so I'll go ahead and be a brat here. Excuse me.

One of my colleagues said that he's trying to watch one documentary a day during the summer, to help enrich his (senior elective) Media Studies class. I think that's a great idea!! I love documentaries, and I love how his goal sounds motivating all by itself. I think in general, over the summer I am going to try to delve into my various non-math interests, in hopes that they will end up energizing my teaching or feeding into my mathematical interests. For example, a while ago when I was reading some biology thing or another, I realized that the reading tied immediately back to the exponential topic I had been teaching, in terms of tracing back to our ancestors and realizing based on average generation-length estimates that at some point within the last few hundred years (between now and the Middle Ages), there must have been lots of in-breeding if you compare exponential growth of expected number of ancestors (as a function of number of generations counted back) and the estimated world population at the time of the Middle Ages. A very interesting application of real math, coming out of an unexpected leisure reading!

But, more specifically, I like my colleague's documentary-a-day idea because it's specific, quantifiable, and not directly linked with planning for classes for next year. Here are some things that I've thought about doing over the summer, linked directly, indirectly, or not at all to teaching:

* Exercising 3-4 times a week (not connected with teaching, but a definite, stated goal because I think I may get lazy while I am on vacation in New Orleans this summer and not in easy access of my building's elliptical machine.)

* Reading a book a week (I have on my shelf: Mushroom Hunters for my book club, Americanah, and I plan to pick up some leisure reading involving math, brain/learning, or just general fiction.)

* A weekly happy hour or WEEKDAY brunch.

* Learning to sew!! (This has been on my list for a while, but it hasn't been a priority since I have been busy with other things during the year. My goal is definitely to take some intro classes at the start of the summer and to investigate some patterns that I want to make during the rest of the summer. I am really inspired by our friend Adria, who makes fabulous-looking custom quilts as a side hobby!!)

* Lastly, I want to make good use of this summer to widen my lens beyond math. It's hard to be creative in math or in teaching if you're only thinking of math all the time, so my goal is to really look outside of math and teaching, and to hope that those dots will connect themselves back to the classroom. For example, I came across this article by Margo Jefferson, which is amazing and links to a lot of the race/class issues that lie just underneath the surface at independent schools such as ours. Some ideas for tangentially related subjects include: leisure reading about chemistry, biology, and astronomy. I don't know/remember much about these subjects, so it'll be fun, like starting from scratch.

* Investigating some open-source computer-science teaching curricula, such as Girls Who Code or Our school doesn't offer any programming course or club, so I don't know what I plan to do with these curricula exactly, but I have been curious for a while and now seems the perfect time to investigate. (I have a programming background from a previous life, so I am not looking to learn myself, but rather to see how they scaffold the tasks, what their pedagogy is, etc.)

*Oh, and almost forgot to mention this, but I'm going to try to do One Teaching Resource a (Week)Day. As in, each day I'll look for something new that can add to my teaching repertoire, and maybe blog about it. I'll start tomorrow since tonight has gotten quite late. 

Is that too ambitious?? Stay tuned.