Day Off

I'm taking a "day off" from my One Resource a (Week)Day series today. I have actually had a pretty rough few weeks and it culminated in a very difficult decision yesterday, a nearly sleepless night, and a challenging conversation I had to have with someone whom I care deeply about. (It's not Geoff, don't worry.)

It made me decide to take a day off and to just read a bit, to settle my mind. I leave you with a couple of old posts recommending good reads surrounding the type of nuanced work that we do: how to foster creativity (in the workplace, in education, and in yourself); what leadership looks like and how that applies to the classroom. I really like these books that straddle self-help and slippery topics. What Should I Do With My Life had really shaped me when I read it back in college, ultimately shaping the choices that led me to become the person that I am today.

I am taking a break from mathy things to read Get Out of Your Own Way today, in hopes that it will be fantastic and that it will be something I can recommend to someone else. (Although that sounds pretty bad to imply that someone else needs to get out of their own way. I really dislike that book title!)

So, anyway, be back soon with more peace of mind, I hope.

[Addendum 7/11/2014: Get Out of Your Own Way made for good dinner conversation with the hubby, but unfortunately I think it's a bit too encyclopedic / laundry-listy to strike a real chord with most readers, which I think is what necessarily characterizes a good self-help book. I'll have to keep looking around. I mentioned to the hubby that I'll keep trying to look into good self-help books. Next on the list: some book about motivation. Hoping to glean something out of it both for personal sphere and for my students!]

One Resource a (Week)Day #8: Radical Math

[Note: It's summer, and I am extra rambly. If you want just the math bits, skip ahead to Paragraph 3. "In thinking about some random social issues..."]

I write this entry from New Orleans. Geoff and I are staying here for a month to leisurely get to know the city, and on Day 1 I already fell in love with the unbelievable charm of the city. Even though we are staying in an "okay" part of town (Bywater), the houses here have such an old-school charm to them, with warm-colored trims, southern-style porches, and some with painted rocking chairs outside. The summer air here is dense with moisture and heat in between the thunderstorms. On our first night, we walked to a popular soul food restaurant called Praline Connection, which is supposed to have the best fried chicken in town according to Yelp and some locals who were waiting around in line with us. I ate some delicious fried chicken livers, but didn't dare to over-indulge because they were fried in the same deep fryer as shellfish, to which I am "severely" allergic. I survived ok in the end without getting an allergic reaction, and I rejoiced over the fried goodness! While we were waiting for our table at Praline Connection, we had walked around Frenchmen Street and heard beautiful jazz music seeping out from a bar called the Spotted Cat Music Club. We stopped in there, originally just for a drink while we waited for the restaurant, but we were so charmed by the old-timey jazz sounds and by the spontaneous dancing we saw, that we couldn't help ourselves but to also swing dance a little. (We only danced one song before we were overwhelmed by an avalanche of sweat. Afterwards, a mom kindly offered us baby wipes to attempt to dry our faces.)

Today, I spent most of the day reading up about New Orleans. Now that I have an idea where things are located relatively, I am both surprised and heart-broken to read that the Lower 9th Ward, or the area hardest-hit by Hurricane Katrina and that had never come close to recovering, is just east of where we are staying. In his morning jog, Geoff had run all the way up to the levee bordering the Lower 9th Ward and almost crossed over on the bridge. (Thank goodness he didn't, because as it turns out, it may not be safe to cross over there even during the day. Many of the houses are still vacant, and the many dogs left behind after Katrina are now feral and carrying unknown diseases borne out of the abandoned houses. Besides that, locals tell us that New Orleans, like many big cities, suffers from a lot of gang-related violence, which has proliferated in some of the areas hardest hit by the hurricane, including the area just north of the first major street from us, St. Claude Ave.) It's different when you read about it in the news from a distance than when you are here. It breaks my heart that such a beautiful city has had such a tough time, for so long now, and is still struggling to rebuild. I also don't know what to think of their nearly all-charter "public school" education.

In thinking about some random social issues, I did a bit of digging into resources to help me think about ways of linking math with social justice issues. Here is a great PDF guide for incorporating social justice into math from Jonathan Osler, which is only a starting point to digging into various specific issues at his website Radical Math.

I love that:

1. His resources are free.
2. He's straight up. "Good math doesn't mean good politics. [...] Talking about a jar of Jelly Beans can be a fun way to study Probability. But studying probability in the context of a unit on how the Lottery increases the economic divide between the rich and the poor will allow the class to cover the same mathematical content while simultaneously investigating an important issue of economic inequality. [Likewise, good] politics doesn't mean good math. [...] It is an act of social in-justice to deny young people the opportunity to master the math that they are in your class to learn." (Pg. 5 of the PDF guide)
3. His website is very usable, searchable both by math topics and by social issues.
4. He has sample lessons in his PDF guide, and they are authentically interesting.
5. The "Math Skills and Social Justice Topics Chart" at the end of his PDF is great as a starting point / dashboard of ideas. It's easy to read and could inspire you to think about social justice more regularly as an accessible lesson-planning focal point.

That is it! I hope you've found today's resource to be useful (although probably not cheerful). Do you know of other great resources on social justice math?

One Resource a (Week)Day #7: Tips for Meaningful Group Work

In case you have missed @cheesemonkeysf's recent blogpost linking to various great resources on group work, I wanted to particularly highlight the link to Malcolm Swan's summarized recommendations for designing instruction strategically. It is fabulous! I think that most of the time, we are doing some vague form of group work, as in kids randomly sit in groups (maybe they sit with the same friends they always work with), and we may feel inspired occasionally to make some sorting/activity cards or maybe we just give out worksheets. The collaboration is pretty adhoc, and students are unclear about when to ask for help from their group and when to ask the teacher. Some always prefer to ask the group, while others more eagerly turn to the teacher. I'll confess that this is typically how it looks in my class, even though students are generally pretty good about collaborating with each other. The issue with this (and why I am so invested in fixing this for next year) is that when kids work with the same kids all the time, it is almost always the same one or two kids per group who explain the concept to the rest of the group. In order to break through that, you need structured discussions. I've noticed that some students, when placed into groups with students that they don't know well, automatically facilitate a democratic sharing of ideas that includes everyone, and afterwards I always hear really positive feedback from those particular groups. But it's not just enough for that to occur as a fluke; it should be occurring in all groups all the time, thereby empowering every student to have a voice.

Malcolm Swan's recommendations are clear and easy to read, because they are bulleted lists with clear language connecting all the sections. Although most, if not all, of the activities suggested are actually ones that I've already done, he goes quite a bit further to talk about the teacher's role in making these activities more meaningful, which is the piece that I know I am lacking. For example, recently at a presentation about Complex Instruction, we (participants) were asked in groups of 3 to sort numbers on a number line. That activity, I myself have used in a Grade 7 class. But the way that it was structured under Complex Instruction guidelines ensured equal participation: Each person can only touch / place the numbers that they are individually assigned to sort. After all the numbers have been sorted, as a group we needed to try to generate 4 different ways of comparing numbers along the number line. That last bit completely elevated the level of complexity of this task and brought our discussions to a deeper level, drawing out connections that were previously rushed through or overlooked. Malcolm Swan recommend similar tactics (he would categorize this as a multiple representation task, I think, since the sorting involved fractions, decimals, and well-known irrational numbers):

The teacher’s role is to ensure that learners:
* take their time and do not rush through the task;
* take turns at matching cards, so that everyone participates;
* explain their reasoning and write reasons down;
* challenge each other when they disagree;
* find alternative ways to check answers (e.g. using calculators, finding areas in different ways, manipulating the functions);
* create further cards to show what they have learned. (Pg. 21 of the summary)

He goes on and provides similar analysis of roles for a variety of learning tasks, which makes this a nice "cheat sheet" to have right before running a particular activity, in order to get the most bang for your buck.

In fact, this summary document is fabulous as a resource, because it also gives group work guidelines that you can hand out to students to help them understand what positive group work looks like. (See Pg. 31 of the summary.) I won't be able to make it to the Twitter Math Camp this year, but what a fabulous set of resources @cheesemonkeysf has already laid out! Thanks again, MathTwitterBlogosphere! I can't wait to dig into the rest of what you guys come up with!

Core Elements of Authentic Learning

Out of pure randomness, my work desk is located in a science classroom. (It was a free desk when I arrived, and the classroom is full of windows and light, so I was more than happy to claim the desk.) I have seen some fabulous hands-on science teaching as a result of this fortunate vantage point. During the first part of the school year, the 9th-grade physical science teachers do a long unit on designing and building catapults. The goal is for the students to research various designs of catapults (or anything that can propel a paperclip) for a certain distance (~20 feet?), and to land inside a target the size of a frisbee disc consistently. The students research, prototype, and refine their designs and it culminates in a grade-wide competition during the Science Fest. In the context of the project, they learn about accuracy (landing in a specified spot) and precision (calibrating so that they can aim consistently), research, working in groups, basic woodworking (their catapults are built from wood and pipes and it involves some sawing and some drilling), and sustainability (they re-use all materials from previous years and they break down their catapults after the competition).

Towards the end of the year, those same students design and build a planetary model that would help to explain all of the celestial phenomena that we experience. The teachers don't tell them what to build. They go around and conference with each group, rotating planets and flashlights and explaining, "According to your model, we would see a solar eclipse every 6 months. Is that true?" The students then are engrossed in deep conversations about their models and attempt to fix them. Eventually, when all the groups have finished building their models correctly, they use their models to answer questions such as, "It is 3pm and the moon is in the eastern sky. What time of the year is it?" 

Seeing this level of amazing teaching and learning has made me question what we can do in math teaching to bring our kids to be curious, to model, and to experiment the same way that they do in these amazing science classes. Here are some elements that I think were critical to the success of these projects, and in many ways should be transferable to a math classroom:

Core element 1: A meaningful physical experience

Core element 2: Clear objectives and a way to self-assess against them

Core element 3: Time for an iterative process, conference, and reflection

Core element 4: Shared vision within the department

A meaningful physical experience: One complaint that I have received from past students is that we do too many worksheets in class. As much as I try to make learning exploratory and scaffolded, I do find it challenging to move away from worksheets. One thing that I really liked about our Precalculus experience this year was that about once per unit (as in, once per major topic), we had some activity that involved modeling and analyzing a certain motion through Logger Pro. This often came randomly, but should be far more thoughtful. Each unit should start with an investigation that links the math to a physical experience, to establish the goals of analysis, and this investigation should generate interesting questions (otherwise it's probably not a good investigation to use). Towards the end of the unit, there should be some form of follow-up where the students can create their own example and create a physical product to represent what they have learned. (For example, an end-of-unit quadratic product could be for students to create a quadratic sequence that grows visually, and to analyze its growth using a variety of algebra methods.)

Clear objectives and a way to self-assess against them: The book I previously read by Jo Boaler had underscored the importance of this. The better students understand what they should be achieving, the better they can achieve it. In the case of the science projects, the goals and objectives are clear and simply stated. The students are left questioning how to achieve that, and the methods are left open. In a math classroom, we are often eager to show kids the way, but the objectives are actually obscured. At the start of a unit, the objectives should be clearly stated. The students could, for example, keep learning journals with which they generate their own examples and explain connections in their own words. Whenever they feel that they have achieved the learning objectives, they can then submit those learning journals for review by the teacher. This is a mixture of self-assessment and teacher-supported (non-quiz) assessment. What it would provide is a clear expectation that the students are self-assessing their learning on an on-going basis. 

Time for an iterative process, conference, and reflection: One thing that strikes me about these science classes is the time that those teachers take in letting their students struggle. They believe so strongly in what they do, that they will spend four or so weeks on one open-ended project. It's not all roses; some of my students claimed after the fact that they "learned nothing" from Grade 9 science, but what those students don't realize (even after the fact) is that that class was trying to teach them to think, and to think deeply. The teachers often spent the whole period just checking in on half of the class and involving in deep conferences, rather than bouncing from group to group as I have the tendency to do. Quite often, the teachers would send some of the students out of the room to work unsupervised, so that they could focus on the remaining groups. I think that this is something I will experiment with -- having longer conferences with fewer students, while providing the other students with clear expectations to struggle for a bit more on their own before asking for help.

Shared vision within the department: One thing that is extremely powerful is that the science department at our school has a similar vision. Not only do those two physical science teachers believe in teaching the way that they teach (and taking long weeks to focus on deep, meaningful assignments), but their entire department believes the same. The bio teacher teaches biology as a series of mysteries to be unfolded based on your existing knowledge. The chem teachers bring out an unknown substance, and ask the students to do whatever they can to decipher its identity. The physics teacher does exciting projects like rocket design (complete with parachute deployment), building circuits for a house, and processing sound signals. That is so important in the continuum of development of a child. If we can achieve the same coherence in our math department, it won't matter which content strands we've only skimmed over, because what we will get in the end is a confident, thinking student. I feel that holding this line is especially important in the high school age range, where the pressure of content is strong, with or without standardized exams. Believing that we can do right by the kids by re-focusing on the cognitive aspects of math is more critical in high school than ever.

Sorry, maybe you were hoping for a new resource today. I hope you enjoyed this rant nonetheless. 

PS. Hello from NOLA, where I will be for about a month. What an amazing city it has been already! 

Decibels vs. Distance

This is a random mathematical musing. 

Since sound intensity drops as a function of (the square of) distance away from sound source, I was curious as I fell asleep last night how it would look to graph Decibels (which is a logarithmic quantity) as a function of distance away from source. 

Intutiively, you'd expect Decibels to still be negatively correlated to distance. As distance increases, Decibels should drop. But, would it drop linearly? Rapidly? It's basically wrapping a logarithmic curve on top of a rational curve, or in other words, a real-life function composition!

I did a bit of digging.  This website has a clear visual for why sound intensity drops as a square of the distance. (Basically, for non-physics readers, you're spreading the loudness over a greater surface area, if you consider the sound source as sending vibrations in all spatial directions, spherically outwards.) So, if you wrap the functional definition of Decibels around that, you get this function that maps x, the distance away from source, to d(x), the Decibels measured at that distance:

d(x) = 10 log(a^2/x^2) + b, where (a, b) is a point of known Decibel value. For example, if at 4 feet away from the source, the loudness is 20 decibels, then (a, b) = (4, 20).

The "common" (ie. Googlable) observation you get about this graph is that every time you double the distance away from the source (for example, if you go from x = 4 to x = 8, or x = 32 to x = 64), the decibel value drops linearly about 6 decibels. The graph looks like an upside-down log graph, which makes sense if you apply log rules to the function d(x) to decompose it into constant parts and variable parts. 

Another interesting feature of this graph that raised questions for me was the end behavior. On the right side, the function dips below the x-axis, which indicates that after 40 feet, this particular sound source can no longer be heard (0 dB is the threshold of hearing). On the left side, however, it wigs me out that the Decibel should approach infinity as you get closer and closer to the sound source. Is that because I never really understood Decibels before? Mathematically, I could see how as the surface area approaches zero, the intensity of the sound approaches infinity, but somehow, it's hard for me to wrap my mind around the fact that the measurable decibels would also go towards infinity as x --> 0. 

Now, can we corroborate this with some kind of experiment? 

One Resource a (Week)Day #6 (Continued): Amazing Jo Boaler Book!

I wanted to follow up on my previous post after finishing Jo Boaler's What's Math Got to Do with it? because I have never read something so deeply moving to me as a math educator. The book was published in 2007, but it is every bit as current today as it was then. The same issue of math wars -- teachers who hold on to their traditional ways of teaching and are scared to reform -- is as real today as it was in 2007. I came from a software engineering background, where truly, as Jo says in her introduction to the book, you use new math that you develop along the way. None of the data-processing algorithms used by any of the big software companies now existed before they developed them. Those textbook-perfected math formulas? You'll likely never see most of them again after formal schooling.

And this is why it frustrates me so much that I still hear discussions about which content strands need to be covered before the end of the year, and why we don't have the room to put in an extra project or two. YES, I know that projects and explorations take time. I know that it takes even more time for kids to navigate their own way through the projects to find their own solutions. And maybe that means you'll have to cut a unit at the end of the year, if you do enough of these interesting tasks. But, isn't it totally worth the time? You're teaching kids how math ties to real life, how to think on their own, how to persevere and (yes, sometimes) to start over. That is OK, for me, especially considering that most kids don't retain a lot of the math facts that they learn in a hurry anyhow. More depth, less breadth should be the name of the game, always!!!!

Jo's book is deeply moving, and also practical for me in thinking about some changes I was already planning to make for next year. Specifically:

• Giving group tests (which will have NO re-test option allowed) before individual tests, to force kids to communicate/explain the concepts to each other.
• Peer evaluation of work against learning standard, using the system "2 stars and a wish." Jo cites research that when you spend class time to do this, you are clarifying learning goals and you can cover the same material in half the time.
• Written math journal, intentional grouping, and structured discussions (as per Complex Instruction) to actively engage all students in a heterogeneous learning environment. (Jo cites anecdotal evidence that mixed-ability grouping yields better learning result for all.)
• Recognize kids who ask good questions by putting their questions on a poster to keep up.

Her book also talks about why girls struggle particularly in the math classroom, but in a way that praises girls for wishing to think more deeply than the boys (attributing the failure, instead, to the way math is taught in schools). It also has some nice math puzzles to break up the stories... And, although not targeting me, there are some nice sections about how to positively impact your own child's math development at home. 

In short, if you're an educator OR a parent and you have not yet read this book, I highly, highly recommend it! It's a must-read and will help provide some context and insight into the public debates on math standards and math curriculum that will surely (unfortunately) continue for the next few decades. 

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!