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!
Thursday, October 10, 2013
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!!!!!
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.
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.
Saturday, September 28, 2013
Color-Worthy Handouts
My German teacher had a simple but very effective trick for organizing papers: make copies on colored paper, whenever it is a handout that you want the students to be able to quickly retrieve and to reference over time. I have started to do this this year. Regular worksheets are always on white paper, but quizzes and important reference handouts are always on colored paper. This way, after a unit is over, I can basically tell the kids to leave all the white handouts filed away at home and to just keep their colored handouts handy for future classes.
Here are some recent handouts that I think are useful enough to be on colored paper:
Here are some recent handouts that I think are useful enough to be on colored paper:
- Addressing common calculator entry errors, because seeing my kids enter them wrong repeatedly is making me want to tear my hair out. It's an indication that they are not getting enough exposure to math technology (on the computer or on their TIs) over the course of their education, because the way to enter these algebraic expressions is pretty standard across all platforms.
- Good writing transitions to help out when kids are communicating their problem-solving process via lengthy writing. I am super excited to share an example of student work, verrrry soon!
Using Logger Pro in Quadratic Modeling!
One of the wonderful things of teaching in different schools is that you get to learn from different teachers. My current school has a site-wide license for Logger Pro, which (I know, unfortunately) is a proprietary program that allows you to import and analyze videos. It pulls the scaling information based on your definition of what 1 meter looks like in the video, and it uses the timestamps built into the video to retrieve timing info. From that, this program is able to pull both position information over time, and estimated velocity information over time. (The velocity bit is not that precise, however.)
I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,
and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.
I was playing around with this piece of software this morning because our Precalculus course team wishes to incorporate it into our Quadratics unit. I imported a video from David Cox, which can be found at http://vimeo.com/16506894,
and I got this screenshot in Logger Pro. The red is the horizontal position of the yellow ball over time, for the frames that I chose. The blue is the vertical position of the yellow ball over time, for the frames that I chose.
I love this! I can see letting my kids do the same, picking out points from a video that includes both dimensions of movement, and then discussing why height is always quadratic and the horizontal distance is not. And then, they will do quadratic modeling both by hand (by setting up a system of equations) and on the calculator (via regression) in order to find the curve that fits this graph. LOVE IT!
PS. If you are lucky enough to work at a school that would agree to get a site-wide license, the really nice thing is that you get to install it at home completely legally, which is great for both you and the students. So, keep that in mind when you are talking to your admin!
Week 3 Teaching - Setbacks and Triumphs
We are in the thick of it now, the part of the semester when I see how kids handle setbacks and challenges. This is one of the ways I really get to know a kid, because I truly believe that how you handle setbacks defines your character. I tell the kids that they can keep reviewing and re-quizzing, or re-submitting drafts of a writing assignment, until they decide that their score is good enough to stop. No one is going to disallow them to keep working to get better, because I think that training kids to keep tackling something long after the class has "moved on" is how we can teach them to develop a persevering character.
For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.
Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually none of the problems, including the bonus ones which we had only briefly seen during class) had previously said to me that she always felt a little behind in other people's math classes, and 2. our school has been having some conversations surrounding issues of diversity (mainly ethnic and socioeconomic), which has been making me wonder a bit about the role of "male privilege" in the math classroom, similar to the issues of "white privilege." Anyhow, the fraction of kids who didn't do so well on their quizzes are seeing me on Tuesday for a re-quiz, so as of now it is still too early for me to say whether they just had a bad day, or they are still learning how to study effectively, or they really need some serious intervention with the topics that we have covered. In this class, unlike the other classes, I have not been collecting/grading homework past discussing the answers as a group, since thus far we have been building up introductory concepts and there are not a lot of nitty-gritty skills checkpoints for me to look at and respond to. I think that contributed to the lack of individual feedback before the first quiz, which I will remedy in the coming unit by being more hands-on with grading their homework assignments, once we step into the realm of manual differentiation techniques. Overall, I have been very happy with the way that my kids have built their conceptual understanding around derivatives and instantaneous and average rates. We take every second Friday to review algebra skills from the past, and then I assign a review homework assignment for the weekend following. So far, we have reviewed: 1. factorization techniques, 2. how to solve for a parameter within an equation, and 3. when to use their calculators to solve complex equations; and already I can see their independence growing inside the classroom from these mini skills reviews. We just wrapped up a great worksheet (if I may say so myself), because every problem in this worksheet is anchored in something very real. Problem 1 was about investment, and I had talked to the kids about how my husband does real-estate investment and how he uses this type of math to calculate mortgage rates and monthly expenses on a rental property, in order to compare those fees with his rental income to make sure that he will clear a profit every month from his investment. (Related to this I talked to the kids about why they should invest, and why investment does not mean that they cannot be contributing productively to the society.) Problem 2 was about carbon-dating, but the kids needed to read the initial C14 levels out of a graph of atmospheric carbon levels to use in their carbon decay model. This is very realistic, and we got to talk a little bit about the science behind your body equalizing with the atmospheric carbon levels while you are breathing/alive, as well as about how cow-farming is causing historic carbon levels to rise (connecting this to what they see in the graph of historical atmospheric levels). Problem 3 on the worksheet is an ad that I actually found on the web for Gap Inc's credit card offers, so although it is another review problem for interest rates, it is steeped in real world context. I am trying to make my Calculus class as inter-curicular as possible on a day-to-day basis, so that kids can see the reason/motivation behind studying what we study. Interestingly enough, a couple of the faster-moving kids have already started on our end-of-unit economic mini-project (Part 1 and Part 2), and the first question they asked me before even starting the math was, "Why would anybody care about marginal costs?" --Aren't my kids fabulous? I want them to ask me questions like this, so that Calculus can come alive for them.
Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to prove that they are doing the tests of their formulas. Some of the kids, in fact, didn't even know how to construct tables or write subscripts, so there was lots of tech education there, besides helping them out with the mathematical language. The kids thought that typing up their revisions to the rough drafts was going to be easy, but it did in fact take them two full (45-minute) class periods, and many still had to go home to take some time this weekend to re-read through it to make sure that they have hit every part of the project rubric. Anyhow, I prepared a graphical organizer template so that sometime next week, we can discuss how this idea of approaching and analyzing math sequences is going to be the big idea through the entire first Quint. (We have 5 Quints a year, as opposed to 4 Quarters.) Also, something quite cool that I tried recently was to put kids into groups and let them do mixed analysis of linear and quadratic sequences, and instead of me telling them whether they were correct, they got to check using the web interface of visualpatterns.org! The kids were super into it, and I think seeing the two types side by side really helped them to clarify mentally the different strategies for each type. Alongside the writeups, the kids have just about finished reviewing all the algebra skills for lines, so I will give another quiz next week before moving on to reviewing quadratic and transformational skills.
My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!
For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....
But, I cannot complain. I love this time of the year!
For me personally, I've always viewed myself as a second-try kind of gal. If I weren't, I would have given up the first time a teaching program told me that I wasn't their ideal candidate, and I would never have ended up doing what I do, and loving it. Too many adults give up on their goals too easily, and let other people decide for them what they can and cannot achieve. I don't want that to happen to my students.
Anyhow, enough philosophizing. More about teaching. My Calculus kids are entering a very interesting phase of the course. We had our first quiz, which was quite tricky and conceptual, even though it did not involve many numbers. To my delight, about two-thirds of the class did quite well on this quiz, and the top three or so scores were all girls!!! I cannot help myself but feel gleeful about that, especially because 1. the girl who did the best on the quiz (missed actually none of the problems, including the bonus ones which we had only briefly seen during class) had previously said to me that she always felt a little behind in other people's math classes, and 2. our school has been having some conversations surrounding issues of diversity (mainly ethnic and socioeconomic), which has been making me wonder a bit about the role of "male privilege" in the math classroom, similar to the issues of "white privilege." Anyhow, the fraction of kids who didn't do so well on their quizzes are seeing me on Tuesday for a re-quiz, so as of now it is still too early for me to say whether they just had a bad day, or they are still learning how to study effectively, or they really need some serious intervention with the topics that we have covered. In this class, unlike the other classes, I have not been collecting/grading homework past discussing the answers as a group, since thus far we have been building up introductory concepts and there are not a lot of nitty-gritty skills checkpoints for me to look at and respond to. I think that contributed to the lack of individual feedback before the first quiz, which I will remedy in the coming unit by being more hands-on with grading their homework assignments, once we step into the realm of manual differentiation techniques. Overall, I have been very happy with the way that my kids have built their conceptual understanding around derivatives and instantaneous and average rates. We take every second Friday to review algebra skills from the past, and then I assign a review homework assignment for the weekend following. So far, we have reviewed: 1. factorization techniques, 2. how to solve for a parameter within an equation, and 3. when to use their calculators to solve complex equations; and already I can see their independence growing inside the classroom from these mini skills reviews. We just wrapped up a great worksheet (if I may say so myself), because every problem in this worksheet is anchored in something very real. Problem 1 was about investment, and I had talked to the kids about how my husband does real-estate investment and how he uses this type of math to calculate mortgage rates and monthly expenses on a rental property, in order to compare those fees with his rental income to make sure that he will clear a profit every month from his investment. (Related to this I talked to the kids about why they should invest, and why investment does not mean that they cannot be contributing productively to the society.) Problem 2 was about carbon-dating, but the kids needed to read the initial C14 levels out of a graph of atmospheric carbon levels to use in their carbon decay model. This is very realistic, and we got to talk a little bit about the science behind your body equalizing with the atmospheric carbon levels while you are breathing/alive, as well as about how cow-farming is causing historic carbon levels to rise (connecting this to what they see in the graph of historical atmospheric levels). Problem 3 on the worksheet is an ad that I actually found on the web for Gap Inc's credit card offers, so although it is another review problem for interest rates, it is steeped in real world context. I am trying to make my Calculus class as inter-curicular as possible on a day-to-day basis, so that kids can see the reason/motivation behind studying what we study. Interestingly enough, a couple of the faster-moving kids have already started on our end-of-unit economic mini-project (Part 1 and Part 2), and the first question they asked me before even starting the math was, "Why would anybody care about marginal costs?" --Aren't my kids fabulous? I want them to ask me questions like this, so that Calculus can come alive for them.
Anyhow, Precalc is also going swimmingly. I heard feedback from one 11th-grade advisor that her advisee loves my class, and thinks that all the math we do thus far is very clear and very understandable. The students are in the process of finishing up their first big lab writeup, which was very exciting because when I took them down to the computer lab, I got to show them how to 1. enter and format equations properly into MS Word or Google Docs, and 2. how to connect their TI-84s to the computer via TI-ScreenConnect to prove that they are doing the tests of their formulas. Some of the kids, in fact, didn't even know how to construct tables or write subscripts, so there was lots of tech education there, besides helping them out with the mathematical language. The kids thought that typing up their revisions to the rough drafts was going to be easy, but it did in fact take them two full (45-minute) class periods, and many still had to go home to take some time this weekend to re-read through it to make sure that they have hit every part of the project rubric. Anyhow, I prepared a graphical organizer template so that sometime next week, we can discuss how this idea of approaching and analyzing math sequences is going to be the big idea through the entire first Quint. (We have 5 Quints a year, as opposed to 4 Quarters.) Also, something quite cool that I tried recently was to put kids into groups and let them do mixed analysis of linear and quadratic sequences, and instead of me telling them whether they were correct, they got to check using the web interface of visualpatterns.org! The kids were super into it, and I think seeing the two types side by side really helped them to clarify mentally the different strategies for each type. Alongside the writeups, the kids have just about finished reviewing all the algebra skills for lines, so I will give another quiz next week before moving on to reviewing quadratic and transformational skills.
My Algebra 2 classes are moving pretty slowly through their regression project, because I have discovered that they have some holes in their Algebra 1 knowledge and am taking some daily class time to discuss homework problems before assigning new review assignments. In class, we are doing white-boarding practice about once a week, because it is a great time for me to make sure that everyone is doing some algebra practice together and getting instantaneous feedback/help as needed. Following our fairly difficult first quiz, which I had written about last week, lots of kids came to see me to do re-quizzes, which I loved. Although they didn't all get 100% on their re-quizzes, it started a very productive dialogue with kids about how they are studying, what study tips I can recommend, and why things always seem easier when you do a re-quiz. One international student in particular had an 180-degree shift of attitude towards me after the re-quiz. I think she's the kind of student who thrives particularly on positive feedback, so the fact that she had failed the first time but got 100% on the second try, really boosted her confidence and her feelings towards math (and I guess, me). So, although it had been a challenging/"discouraging" week last week, I think it was a necessary reality check for many kids and now they are much more focused and strategic in their learning. My 10th-graders, for example, took a lot of photos yesterday during our white-boarding practice, because the one student who had done that last time and who had practiced with those problems later at home, had done very well on the quiz and had offered that up as a study strategy to her peers. So, we are a growing community of learners, moving in the right direction, slowly but surely!
For both Algebra 2 classes, on Monday we will go and test out the kids' predictions for the bungee drop. They will build the cords, name their rubber chickens, take a photo with their chickens, and then we will go out to the balcony. It'll be a very fun day, but the hard work that is yet to come is to write the lab reports coherently. I am a little nervous about getting their rough drafts on Tuesday, and what those will look like, especially for my international kids....
But, I cannot complain. I love this time of the year!
Saturday, September 21, 2013
Week 2 Teaching - the Gentle Push Back
The second full week of school has been a very meaty one. The kids seemed very eager to learn after the first few unstructured socializing/cohort retreat days. And I am starting to see the various personalities starting to emerge, which is both wonderful and more challenging because now it is real teaching and real learning.
In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.
By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...
In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.
In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.
After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.
Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!
In my Algebra 2 classes, we had our first quiz, which challenged all the students in different ways. My vision for the start of Algebra 2 had been to lay down a solid tech foundation alongside review or re-teaching of linearity skills, so that kids realize that a corner stone of Algebra 2 has to be using technology flexibly in order to self-monitor accuracy. I told the kids that I don't want them to ask me, "Is this right?" but I would be very happy to hear them ask, "How can I check this using the calculator?" So, on the first quiz they needed to demonstrate this skill throughout the algebra problems, in order to earn full points on reflection. (I gave separate points for: Communication, Approach, Accuracy, and Reflection.) In the end, my two groups were challenged differently; the Grade 10 native-speaker class had holes in their algebra skills and couldn't complete all problems, and the Grade 9 largely non-native speaker class generally showed stronger algebra skills (even with less in-class whiteboarding practice), but struggled with using the calculator flexibly. But, some of the kids shared their effective studying strategies in class, and some of the other kids are planning to see me on Monday for a requiz, so I feel quite hopeful that this first quiz is just the start of a learning dialogue.
By the way, my relationship with the international kids is developing in an interesting way. Because I speak Chinese, I am able to help the kids in my class without watering down the level of tasks I am asking them to complete. But, at the same time I can carry some weight when I see them being off-task and I offer to call their parents in China to have the dialogue directly about their efforts, in Chinese. What an interesting situation for me and them. Interestingly, they are better-behaved for me, and they try hard to speak English in my class, except when they need to help each other translate something. I am curious how they are going to do on their first big writing assignment, which we will be doing next week...
In my Precalculus class, kids are wrapping up the rough drafts of their first big project on special (triangular and stellar) numbers. I had to be very explicit in helping them to format their writeup. (I had to say at one point, "Take out a sheet of paper. At the top, write, 'In this task, I was asked to...' Now, complete that thought in your own words. I give you 30 seconds to do that. ...Now, write down, 'In order to accomplish that, first we had to...' and go ahead and complete that thought, make sure you insert a diagram here." But, after about 5 minutes of modeling, I think they all got the idea and were able to continue the rest at home, because all the drafts that they brought back to me the next day looked pretty coherent. So, it has been a tough project for them, for sure, but I still think it had tremendous learning value. We also had a quiz, and the kids are doing fine with function identification, interpretation of f(3)=7, and writing both recursive and explicit equations for arithmetic sequences. Some of the more clever ones were able to write formulas for quadratic sequences already, based on their learning from the Special Numbers project. So, I am pretty happy so far. As we wrap up our project (meaning, as I read over their drafts), the kids are doing mixed lines review and checking all answers via their calculator. They use either the Table or Trace to check all equations that they write from given info, and they use [2nd][Math] to verify equivalent expressions after simplifying. So far, so good, because kids in this class seem to be quite independent.
In my Calculus class, I had one student come forward to say that he really enjoys the exploratory nature of our class, and two others who came to ask me to do more examples followed by practice. I thought over this carefully and decided that although I think it is awesome that kids are being advocates for their own learning, and I really wanted to acknowledge that and to encourage that, the issue is really much more complex than their individual learning styles. I ended up describing to the class two contrasting learning models, direct instruction and inquiry-based learning. I said that in most math classes they have had, they probably experienced the former (intro, example, guided example, individual practice, closure, and eval), and that that is fine. It is comfortable, you know what to expect when you come to class. But, that way only reaches the top half of the class, the half that is fortunate enough to maintain focused attention and to comprehend at the speed of material presentation. Then I showed them a diagram of inquiry-based learning, which is a cycle of asking questions, investigation, creation of model or new knowledge, discussion, reflection, and back to asking questions. I explained to them that what we do in class is NOT true inquiry, because true inquiry would be like me saying, "Go. Find out how much universal health care is going to cost our country, both in the short run and in the long run." The problem would be entirely open-ended, complex, and vast, and we would learn all the necessary math skills as we move along. I explained to them that what we do typically in our class is a smaller version of this; within the individual topics of Calculus, I try to think about ways to structure our class so that they can create their own understanding. I ended the class with showing a little clip from Sir Ken Robinson's tedX talk (the animated one), and saying that teaching creativity is hard, and that our traditional schools have been doing a good job killing creativity. I told the kids that, yes, I think even in math there is room for creativity in the classroom, and unfortunately we don't get that by me doing an example and then handing out 25 problems that look the same. So, although I will try to find a balance between direct instruction and exploratory learning, I want the kids to keep an open mind and to appreciate opportunities for creativity in any discipline.
After this talk, I heard from another student in this class who said that she really enjoys math this year, and that our chat helps her to understand and appreciate my philosophy even more. So, one point for being authentic with kids and treating them as intellectual equals.
Good second week. Our school is awesome, by the way. I love that kids clean the school three times a week, and I adore my colleagues!!!!!
Subscribe to:
Posts (Atom)