Thursday, October 10, 2013

A Spectrum of Relationships with Math

Our school had our annual "Curriculum Night" tonight. The parents followed their kids' schedules to come meet teachers, each for 10 minutes at a time for "class." 10 minutes isn't really enough to do an activity, but it was enough time to convey what I think is the most important about my approach to the class. It was my first time doing something like this, and what I chose to do was...

I started with asking the parents to stand along a spectrum on the floor, to indicate whether they LOVE and use math all the time at their jobs, or whether they feel nervous when they encounter math. Somewhere in between would be if they can try to help sometimes with their kid's math homework, but they need to first look at the textbook for a little while.

After this short little exercise, I asked the parents to sit, and I said that the reason I wanted to start with this is that all of these parents are fairly successful adults, and yet they all have different levels of affinity to math. So, as a teacher I try to keep in mind that it is only natural that kids in my class are also part of this spectrum, spanning from those kids who love abstraction and are ready to think 10 examples ahead, to those kids who need a lot of assurance everytime we progress into something unfamiliar. So, I have to plan lessons that can address all parts of the spectrum.

I spoke then a little bit about my techniques for differentiation. I said that many kids who are not so mathematically comfortable, are more comfortable with words. So, often times just asking them to write about math can help them to break it down into smaller parts, to help them understand and process each part. I also said that some kids are really comfortable with technology. So, allowing them to experiment first on a graphing calculator and to pull out numbers or observe patterns on the calculator can help them ease the transition into abstract concepts. Then some more kids are intuitive about the world, and they learn best when you anchor the theories to something very concrete that they already understand. This is why we do projects. In each class, I gave an example of a project that either we did recently (ie. bungee jumping regression project in Algebra 2), or one that is coming up soon (ie. video motion analysis via Logger Pro in Precalc), or one that will come about a little bit later in the year (ie. 3-d ceramic vase modeling in Calculus). All of these projects are ways for me to reach those kids who learn concretely, and they help to make the abstract topics more accessible to the kids. On the other end of the spectrum, for the kids who are always aching to move ahead, I always try to give them a little nudge towards what is to come, to help them anticipate the development in their mind before the entire class discusses and develops that concept. That helps this type of learner to stay challenged, because they tend to enjoy figuring things out on their own and then teaching their peers.

I then showed the parents a short sample of a piece of writing recently produced by one of my Precalc kids as his final draft to the triangular numbers and stellar numbers project. I walked them through my reasoning for writing in math -- discussing how even researchers in academia need the ability to write/communicate clearly, on the level of people much less specialized than them, in order to get funding and to get published for their discoveries. I also mentioned the importance of emphasis on testing formulas, and drew a parallel to the scientific process. The parents were just amazed when looking at the level of work and the clarity that the kid was writing with! Several parents came up to me afterwards to express their gratitude that their kids have to write so extensively in my class. I was honestly so floored by their warm enthusiasm!!

It was one of the best experiences I've ever had in meeting so many parents at once. I think the approach of starting parents off in a simple move-around activity (standing along a spectrum) helped to really engage them and helped them to consider not just their own child in my class, but also the other children who have diverse needs in the same class. We're a learning community, and I hope that I was able to convey that in my 10 minutes with each group of parents!

Although it has been a rough teaching week, tonight was truly a highlight for me to get such positive feedback from my students' parents!

Tuesday, October 8, 2013

Backwards Intro to Differential Calculus

Towards the end of the summer I was brainstorming this idea of teaching Calculus backwards, starting with applications and graphing calculators, then manual Calculus skills, then finally tying those manual Calculus skills to various limits. It is now a little more than a month in, and I have to say that although I cannot compare this approach to a traditional curriculum because I've never taught Calculus the traditional way, I love the way that I am doing it!!!

After a unit on analyzing numerical and graphical derivatives, my kids have a very strong foundation of how to read/interpret graphs visually and how to sketch related derivative graphs carefully. Some of them even have a surprising ability to draw original functions given the shapes of the derivative graphs, even though we really didn't practice that as a class (not yet... I am saving it as part of our intuitive intro to integral Calculus).

And the best part of spending the first part of the year on application is that kids are now itching for algebra, because all of the visual introduction has really primed them for more specificity. They were so, SO excited today when I set them loose on an exploration regarding the power rule of differentiation. After just observing three examples that they generated via explorations on their calculator, they were fully able to generalize the pattern in partner pairs, and by the time they got to the back side of the worksheet, they thought it was just SO COOL that they could now differentiate by hand, without a calculator. It was so AWESOME to see. I've never seen kids so excited to differentiate by hand before! And, when they weren't sure whether they were doing something by hand correctly (ie. when they needed to differentiate a term already with a negative exponent on x), I would encourage them to put in an x value into their resulting f' formula and to check it against the dy/dx value that the calculator returns. Their tech skills from Unit 1 are now becoming a powerful tool for self-monitoring their process.

And, because they've already done some mixed applications with f(x) and f'(x) analyses, I can immediately set them up tomorrow to answering a question such as, "When will the instantaneous rate reach 2 along this graph?" and the skills that we have already worked on for choosing the correct tool between f and f' will come in handy then, to smoothly transition them into the new sets of manual skills (I hope!).

I am so excited! Also, in context, we are reviewing various algebra skills incrementally. On their next quiz/practice quiz, they'll be expected to show work to find standard-form equations of polynomials by hand, and then to differentiate them in order to answer follow-up questions. My plan is to loop in all of the manual algebra modeling skills alongside the new derivative and integral skills, so that the kids would always know that they need to keep up with all of the transformational concepts from Algebra 2 and Precalc, as part of the norm for our class.

So, Calculus is pretty much my favorite class to teach right now. So incredibly fun!!!! I cannot wait until we dig into more of these derivative skills, in order for us to talk about some juicy applications that require both algebraic and geometric analysis. YES!!!!!

Sunday, October 6, 2013

Three Core Values in My Class

This blog post is my contribution to Mission #1 of Exploring the MathTwitterBlogosphere.

I already write a lot about what physically happens in my classroom, so today I am going to write about three values of my class that impact my students' experience and give it a stamp of uniqueness. They are what I consider core elements of my classroom culture.

The first: Each kid learns at a different rate, but what's non-negotiable is the quality of their efforts and the fact that each student needs to be challenged.

I think that a lot of teachers out there hold the same value, so I wouldn't say that it's uniquely mine when taken alone, but it has practical implications in my interaction with the kids.

The most important implication of this value is that in my class, deadlines are not strictly held. If I tell kids that they should bring me an assignment on Monday, then on Monday I would discuss it a bit, answer last questions, and then say, "Okay. Please hand it in to me if you've already done your best on this assignment and feel that you have understood everything. If you need to still take another day or two in order to bring me your best work, then please do that instead of giving it to me now. I don't want you to turn in something that is less than your best efforts, because it's not useful to either you or me."

The same holds for quizzes and tests. Last Friday, a kid was struggling with a quiz I gave in class. After class, I gave him a full period of extended time, but as I was looking over his shoulders, I could see that he was still making the same mistake over and over again with signs, which was causing more frustration further down the line when he realized that his answers wouldn't check against the equations. Eventually, I stopped him and just said, "Alright, look, I don't want you to keep spinning in place and to feel more frustrated. Why don't you give this to me, and we'll plan for a requiz next week after we look over your errors together?" A part of me really struggled with that recommendation, because I think persistance is such an excellent trait, and allowing a kid to persist in face of difficulty is very valuable. But, a bigger part of me believes that each kid has to learn at their own rate, and forcing this quiz upon this kid then was going to do more harm than good.

Another manifestation of this value is that in class, I always try to touch upon a higher-order application of what we are learning, in order to keep the interest of those kids who just yearn for a little more depth beyond what everyone else can grasp immediately. Inevitably, those higher-order applications will come back as bonus questions on the test, not because I love to give bonus points but because I want a way to assess which students are accessing that knowledge based on our brief discussion in class. The real, core assessment is still based on the lowest common denominator, the core skills that we have thoroughly developed and practiced as a class, but on each assessment there should be room for the upper-end kids to stretch their understanding. It is one way that I make room for kids to show me that they are acquiring knowledge at a different rate.

The second: Kids should have ways of verifying and monitoring their own correctness, beyond asking me.

This is a value that a lot of math teachers hold, but one that I really invest a lot of time to teach and to develop in my classes. Depending on the topic and what makes the most sense, I either expect them to check their results via the calculator or to check their results by hand.

At the beginning of the year, it can take me more class time to teach this and to ask kids to self-verify their work, than the time that it takes me to teach new algebra skills. This is why I don't think that other teachers really teach it; a lot of people value this skill, but when it comes down to it, they don't necessarily value it enough to put in the time to force every kid to develop this skill. But as time goes on, the time spent learning this skill shrinks rapidly, and kids get very used to doing this as part of learning and assessing any new topic. In fact, my current students have said to me, "You really force us to check every problem on a quiz!" ...And yes, I do! The time they spend completing a quiz should be roughly as follows: only 2/3 to 3/4 of the time spent on completing the quiz, and the rest of the time going through and gaining full confidence in all their answers. Ideally, they should be able to tell me after every quiz, "I know I got 100%."

I try to get away from being the source of verification, because I want my students to one day exceed me in their knowledge and understanding of math, and if I don't teach them how to verify their own answers, their knowledge will always be upper-bounded by what I know.

The third: Learning to learn immediately supports learning of the content, so time should be spent in class to explicit teach, discuss, and practice various learning strategies.

Again, I think all teachers believe that learning strategies are important and incorporate them into our daily lessons. But, we don't all teach them explicitly or discuss their usefulness. In fact, some of the best content-teachers can still overlook the importance of explicitly teaching and discussing strategies for review in the classroom. Teenagers need modeling for learning strategies as much as they need modeling for how to do a math procedure. I find that once my students and I practice a certain strategy in the class, they often come back to me to request more support in learning to learn. And, in the long run, it helps them greatly in building their confidence with math.

The three most popular strategies with my students are: creating their own concept flash cards; doing practice quizzes; and doing white-board procedural practice (while they take photos of the problems). These are tried-and-true methods that the kids find to be the most useful in terms of self-diagnosing their gaps, ironing out consistent procedural errors, and increasing mental focus.

As I think about content delivery in my classroom, I tend to think that it is inseparable from the learning culture that I set up around the content. These three elements are not unique by themselves, but together they do help to form a strong culture of high expectations when it comes to self-reliance, self-monitoring, and self-knowledge. The learners cannot be viewed as helpless and passive, but in order for them to become successful and self-reliant, we need to cultivate the tools that would help them reach those expectations.