Wednesday, October 31, 2012

Strategies for Making Math Manageable for Kids

The first part of this school year has been pretty gratifying for me. My students who moved on to new teachers and new classes are all faring well, much better than they were doing at the same time last year. For me it is like seeing the light at the end of the tunnel. Although there are completely new students this year with whom I need to repeat the struggle, I can see that it does take a full year to bring a kid up to speed to become the student that I want them to be (something that my old Math Coach used to always say to me.... words of the wise!). Sometimes, as an impatient person, I want to see immediate results. But, it is just as well to see it a full year later, that the kids are more confident, more independent, more responsible, and that they feel that they can actually tackle mathematics.

More and more I am thinking about the utter importance of confidence in a student's mathematical success. I have always tried to think of the magical formula for teaching, and more and more I believe that confidence is it. There is a high correlation, I find, between a student's general personality and their energy in learning math. If the student is generally confident (and especially in math), they can take risks, challenge solutions, see a problem through different angles, and if they fall behind they can do extra work on their own in hopes of catching up. If they are not confident, then they often try to do the minimum, cannot persist through difficult problems, and cannot begin to enjoy the process of thinking about math. Of course, I am not saying that this erases the need to bring in interesting ways of teaching kids, but confidence makes such a huge difference, and if you can find ways to increase a student's confidence in math, the payoff is in fact doubled in the long run.

Concretely speaking, I increase kids' confidence levels in three ways: I explicitly teach study strategies for mathematics (and we spend class time making a few flash cards, for example, with problems on one side and solutions on the other side, or problem type on one side and strategies on the other); I help kids identify/fix basic skills they are still missing from years prior (such as integers or multiplication) that are causing them frustration in current material; I offer re-quizzes and re-tests as much as the kids want, so that they can feel successful about having gone back and mastered an old topic.

Enough rambling and philosophizing. This year, thus far, I have been fairly successful with teaching linear functions and quadratic factorization to my "low" Grade 9 students. Again, I find that the confidence building is a huge element of my classroom (as many of my students come from backgrounds where for years they have felt largely unsuccessful with math, for one reason or another), but I also tried to layer the concepts this year in a way that eliminates confusion as much as possible. Some specific feedback I have had from parents of students in this class is that 1. Their child is really feeling much better about math this year, 2. They can now actually enjoy math. Some parents are actually less concerned about the learning results, and more grateful that math is not the class keeping their kids from wanting to go to school anymore!! From my side, obviously confidence is necessary, but more gratifying is to see the increasing independence in their work, and in their ability / enthusiasm to discuss and to help each other -- something that they simply were not able to do at the start of the year.

Since I have worked on these units so carefully, I wanted to put them on here and discuss the extra scaffolding that went into them. They are not ideal for your regular lessons, probably, but if you find yourself in the peculiar situation of teaching math to kids who have no prior retention AND who dread mathematics, maybe this would help to see how I scaffold things for my kids.

The first day of lines (see above, but actual Dropbox link to file here), I didn't want to make the assumption that kids knew what linear functions looked like. Actually, I thought that since our kids were coming from all over the place, maybe I shouldn't assume that they knew how to graph points either. So, we started the class with going over some key words (see the box) and copying definitions -- this was a strategy for reaching my EAL learners, as well as the highlighting of key words in each question. Sure enough, on this first day I discovered that we had to review how to graph points (x, y). Problem #8 and #9 introduces the idea of line equations, and ask the kids to look for a visual connection between an equation and an already graphed line, and to begin writing their own equations based on their observation. Problem #10 was a bridge between two different representations (list of points and a table), which again I didn't want to assume was obvious. By the end of the first day, kids were expected to be able to graph lines and to write equations based on a table or a graphical line.

The second day of lines (see above, but actual file is here), we again started the class with introducing some key terms (again for our EAL learners). Then, the worksheet started off with review problems very similar to the problems from the previous day -- this is a confidence builder. I wanted my students to feel successful, like they have by now mastered something from a previous lesson, or at least that they had the resources immediately available to recall those skills. Then, Day 2 is the reversal of Day 1 -- they needed to now go from equations to tables, in order to really solidify their understanding of the different parts of an equation. (By the way, I don't teach the phrase rise over run because I don't think that has any real meaning to a normal child. I always say that the rate is what is happening over how long it takes, kind of like miles per hour, so a slope of 1/3 means "the value goes up 1 every 3 stages"). Anyway, problem #4 and #3 are related, because when they start graphing from line equations, I still recommend that they always make a table first. (And this has shown to work like a charm. All my kids can graph consistently.) #5 introduces the idea of parallel lines (which we come back to at the start of next class).

The third and really fourth day of lines (see above, but actual file is here), we quickly took notes on some new key terms and the worksheet started off again with review problems. At this point, the kids were fairly confident in their acquired skills, so I pushed them a bit further by introducing lines in non-slope intercept form, and also asking them to find equations repeatedly in a diagram. We also went over why vertical lines are x=... (because they "hit the x axis at ...") and why horizontal lines are y=...

Then, finally, to pave way towards more abstract ideas like "finding b", I used this worksheet (see below, and Dropbox link here). This worksheet starts off with a very "simple" idea of points fitting through lines, and then hooks it up to the idea that "b", the y-intercept, must have one specific value in order to allow the point to still fall on that line. Then, we apply the idea of parallel lines and gradients (I took out the perpendicular lines skill for now, but I'll put it back in later this year when we do more integrated practice, once the kids have more confidence in their general abilities).

To help the kids study for the linear functions quiz and test, we did one day of team games (Powerpoint here), one day of mini-whiteboard practice exercises, a couple of days of textbook practice (problems here), and one day of move-around review where I put up problems on index cards around the classroom with answers on the back.

I hope that helps to give you ideas for teaching a low-confidence population! For my quadratics teaching, we focused a lot on identifying which stage a quadratic equation already is in, using an introduction/visual organization chart that looked like this. (I am a big proponent of the box method factorisation, as I think it covers all cases of factorisation and eliminates the need for a student to memorize 10 different methods for essentially the same situation.) Since the kids were motivated after the linear functions unit, they also did an amazing amount of work on their own during the October break, which greatly sharpened their factorisation skills. Right now I am very happy with where my Grade 9 students stand, and I hope that it gives you hope that all students can be reached, if we keep trying different methods and keep trying to build their confidence.

Tuesday, October 23, 2012

IB Internal Assessment 2014: Choosing a Topic

My students and I have been brainstorming topics appropriate for the new 2014 IB internal assessments. Our department created a timeline that would allow kids to submit topic choices by early November, write an outline and then some drafts in the winter / spring, followed by final submission in May. One of my colleagues recently went to an IB Category 3 training, which focused on the new internal assessments. After we sat down as a grade-level team to discuss the issues that came up at this conference, I now feel fairly confident that I understand what is expected (as much as is possible at this early stage of implementation of the new format), so here is what I gather. I hope it is helpful to you and/or your students.

The new format entails two parts: research and your own application. The math topic selected should be one that goes beyond the IB curriculum (quadratics is too easy, for example). The student is supposed to research and read literature on the math content beyond the course related to their topic, and then break down the math literature in their own words. Then, they need to basically show "engagement" by applying the learned knowledge to a situation of their own creation.

For example, one student of mine wishes to study a certain card game. I said that is OK, and that the research would entail this student explaining step-by-step how to calculate all of the probabilities in the standard rules of the game. And then, as part of their project, they could create a second game similar to the one that they have studied, but with varied rules, and then they would re-do all calculations to show that they had made the understanding uniquely their own. This kid can also run actual game simulations to compare with their theoretical results, in order to further reflect upon their understanding.


If your kids are lost (as mine were) about exploration topics to pursue, I find that the Numbers Guy from WSJ is a good place to start to think about the connection between math and real-world issues. If they are familiar with GeoGebra, there are also possibilities of studying various GeoGebra animation-based problems that are comfortably within the realm of the SL topics. Or, any extension from 2-D to 3-D is always interesting. Besides that, Vi Hart has some unusual connections between math and various art forms, that might also jog some ideas. The kids can basically study anything, or even model data, so long as the data is sufficiently complicated. For me, a good rule of thumb is that if I can already imagine without any research what the data / result will likely look like, then the topic chosen is already too simple and not sufficiently "juicy" to get all the upper marks. In that case, the kid would need to add a twist to it to make it more interesting / complex, or abandon that idea and look for a new one.

Happy hunting for good topics! What an exciting time! Let me know if you find other sources of inspiration for you and your students. I find in my own class that if a kid picks a topic as a starting point, and they just take the time to start talking to me about it, usually we can brainstorm together until we find something that is sufficiently interesting/complex. The resulting direction may have nothing to do with what we started with, but just the process of talking a kid through a topic can jog my mind through related ideas that I have come across/read before.

Saturday, October 20, 2012

It Takes a Village

I have been very gratified by what I consider significant progress this week in some of my students.

Before the October break, I had two students in my "low" Grade 9 class who were refusing to take a test. One of them has Dyscalculia and was simply afraid to try and fail. The other came from a different school, where they had a very difficult time with math, and was afraid of showing the same weakness at the new school. Eventually, after a lot of coaxing and some stern-talking, the kids each sat down to take the test, and they did great!! What an amazing feeling it was for them and me, to get that positive feedback that despite their own tremendous self-doubts, understanding is possible for them. I wish I could take some credit for their success, but the truth is that all the credit goes to them being brave and really, really putting themselves out there by studying for hours for a short test. There is nothing scarier than giving something 100%, and both of them did that and earned the most well-deserved marks in return!!

Then, in Grade 7 I had one child who had done nothing since the beginning of the year, and was falling very behind. Despite my many chats with her, her efforts were minimal and in class she would spend 80 minutes sorting papers or fixing her pen. Outside of class, when I would work with her one-on-one, she would be great, but if there was even a single other kid there, she would stop working entirely. When I set her up with an older student as a tutor, she flaked out twice and the tutor simply gave up on her. Because I was extremely concerned, I had been chatting with the mom over email and in person, and eventually the mom had recommended me to reach out to her elementary school teacher, who had taught her two years ago with some level of success. I reached out to the teacher, talked to the kid, talked to other colleagues to get both the kid and myself out of class in order to meet the elementary teacher (who was only free during a time that we were not free), and we sat down for a three-way conference with the kid. After that, the kid has been a totally different person. She is motivated and progressing rapidly, quickly catching up to the rest of her peers (because she is in fact extremely bright, well above average in terms of her ability to grasp complex material). To encourage her to stay focused, the elementary teacher (who had taught her for 2 years in ES) has offered to support her on the more basic skills for a couple of months until she is completely caught up. I am amazed by my colleague's generosity, and so thankful by the difference it has made in this child to know that she is being looked after and cared for by all the adults in her life.

In the same Grade 7 class I have another child who cannot add or multiply. I reached out to his parents immediately when I discovered that he was having problems, and they said that because he is at a difficult age, we should get him an older boy to act as his mentor / tutor instead of expecting him to always come see me. I found an 11th-grader who has been working with him since, and in my 11th-grader I can see the future of a great math teacher. He is so patient with this kid. The first time he was supposed to meet the 7th-grader, the 7th-grader purposely bailed out on him and stood him up for 30 minutes. Instead of giving up, the 11th-grader dragged me around campus looking for the kid until we found him. This 11th-grade tutor called the kid's parents after the first session and said that the kid needs support at home to work on basic multiplication, and the parents received it well and have been actually doing timed drills with the kid at home. The boy is making slow but sure progress, and my amazing 11th-grader has the goal in mind that by the end of the year, this boy will be on grade level!

One small step at a time. I am trying to reach all kids, and sometimes it is just not possible to do it all myself. It is so gratifying to see when we all pitch in and it makes a difference for one kid. This week, I've been very humbled and thankful and thinking about the phrase that it takes a village to educate a single child.

Tuesday, October 9, 2012

Mini Wedding Updates

So, now that I am on October break and work is temporarily put into the back of my mind, I thought I'd give a little bit of wedding prep updates. Brace yourselves!

On a semi-happy note, I think I found a dress. It's short, classy, and suits who I am without being "loud." The complication is that I had let 3 or 4 weeks pass in between the first time I had tried on the dress, and going back to purchase it.* So, during that time the dress I had tried on (which was in my size) had been sold, and so I decided to buy a dress that is 6 sizes too big to have them size it down. No problem, except when I went back today to try on the dress again, it's now too small and the dress bunches up in weird places instead of falling smoothly like it's supposed to. The store seems to still think that they can fix it OK, but I really needed their help to zip it up today and it looked very funny anyway -- definitely not the fitting experience I had hoped for. To me, it's all a bit nerve-racking, because the alternative is that I'd have to start over looking for a new dress, with less time left.**

On top of this, there are some not-so-straight-forward stylistic alterations that I want them to make to the existing dress (as recommended by my dress jury and my own gut instincts). It requires them to crinkle some fabric in the same style that the rest of the dress is already crinkled. The store people told me that they're not sure that they can do it, but that they'll try (with a mixture of chemicals, getting the fabric wet, and repeated ironing) and then let me know if it's even possible. So, at this point, the whole dress is a bit of an unknown variable still.

On another note, I am really looking forward to the wedding itself. My boss granted me 2 extra days off (attached to a two-week-long spring break, so it's no trivial matter), and that means that we're heading over to Puerto Rico for a mini pre-wedding honeymoon en route to Belize! Geoff, the ever responsible one, researched to find out that there is some bio-luminescent area of PR that is a must-see. So, we're going to brave Mosquito Bay merely days before our big day. :)

This is not wedding related, but for you non-Facebookers, here is a recent photo of us from Oktoberfest in Munich. Lovely time all around, with a great group of friends from the States!

*Problems in paradise: I went to the Berlin Music Festival, which took up one whole weekend in September. Then, the next weekend I was in Leipzig for a Coldplay concert on Friday night, and couldn't make it back in time on Saturday to go to the store. The following weekend my girlfriends were not around to give me the thumbs up. So, it was 4 weeks later when I finally made it back to the store. In the interim, I had been trying to convince myself that I always get paranoid about things for no reason. Little did I know that my paranoia was totally real, and the one correct-sized wedding dress did sell out within those 4 weeks! 

**I googled this problem, as googling is my way of self-therapy whenever I feel anxious about something. As you would expect, many brides-to-be purchase dresses that are several sizes too small -- on purpose! And then, weeks before their wedding they panic if they still cannot fit into the dress that never fit them in the first place. OMG. As a chronic stresser, I cannot even imagine how panicked I would be to still have a dress that I cannot fit into, two or so weeks prior to the actual wedding. I can only hope that people who do this have allotted extra money to buy a second dress!

Wednesday, October 3, 2012

Benefits of Shared Assessments

One of the things I really wanted to implement in the MYP (grades 6 through 10) this year is shared assessments across different classes. I cannot emphasize the importance of this, if you are are a department chair reading this. Our school is fortunate to have very strong, very experienced MS and HS math teachers with varying areas of expertise, but the danger of that is that you end up closing the door and teaching in isolation, because you don't need to collaborate with others. Before I came to the school, in most MYP grades there were already twice a year shared assessments (in December or January, and then again in May or June). This is pretty good for making sure that half-way and full-way through the year, we are more or less synchronized in terms of topics covered. But, what I personally found last year as a first-year teacher at the school, was that there was too much anxiety placed upon me and my students the month before those "big tests", to adjust our pacing and content to match the other classes. The truth is that no matter how thorough your curriculum documents are, they are still up to the teachers' interpretations; the only way to ensure that students are not missing any part of pertinent information is by looking at what is assessed, at the end of the unit, in each other's classes, and then adjusting instruction to fill in gaps in real time.

So, this year I recommended that at least 4 times a year (including the midyear and end-of-year exams), we try to create shared assessments. And thus far, this has been really great, because it has allowed/forced us as a department to dialogue on an on-going basis about what we are teaching, how fast we are teaching it, and whether we think our students are moving faster or slower than other groups in the same grade. This makes adjustments possible as an on-going process, instead of only twice a year when the crunch time rolls around. Also, because these smaller shared assessments are based on a single topic/unit only and each teacher can give it when they feel that the class is ready, we can really take the time to fine-tune their content to match the MYP expectations. (In the MYP framework, for example, each assessment must contain "new, unfamiliar" situations for applying the learned topics in order to encourage students to think flexibly beyond normal applications. When we each create assessments for our classes, it's time-consuming to think of good problems like this to use, but when we all collaborate, it's really not so bad.) Another hidden benefit is that shared assessments helps us to provide a shared meaning of grades (ie. a "5" means more or less the same in all classes in the same grade), as well as helps us to identify those kids who are truly behind or truly ahead in their classes, compared to peers across the entire grade. In short, good things are already coming out of this change, and I am feeling quite encouraged by the small changes that are, in the long run, making our students' learning experiences much better!

Another thing we have agreed to try to do this year is to build in a common schedule for MYP projects, which would be a couple of weeks (surrounding vacations) during which time each teacher will come up with their own favorite projects to use for their classes. We have not actually gotten around to doing this yet this year, but I am very interested in getting the conversation started very soon within our department about the particulars and logistics of projects. It will hopefully be another opportunity for us to share ideas on projects and to have a dialogue on what makes a good MYP project.

Fractions Project

My fractions project in Grade 7 is almost done! We had taken a break from it to work on some introductory algebra stuff (and to allow kids to turn in additional rough drafts to me outside of class), but on Tuesday, I gave the kids a full 80-minute period to work on their final drafts, and the polished final drafts will be due on Thursday when they walk into class. I was very pleased with the upper-end products I have seen thus far (complete with explanatory diagrams), and I think they showed me that all kids had to learn some new skills and concepts in order to complete this rather involved project. The kids towards the bottom half of the class required a lot of hands-on support on this project, and some of them will be turning it in late because they are still struggling with basic things like responsibility... (The transition from Grade 6 to my Grade 7 class has been a pretty tough reality check for some of them in terms of my expectations for what they need to complete, by what time, and how well. For three of them, I have extended the deadline and made explicit to them that they need to come conference with me later this week during lunch, and then bring me an awesome final draft following the week-long October break.)

So, I think the specs are a good starting point for a decent fractions project. They are poorly formatted though, in hindsight. For itty-bitty beginning-Grade 7 students, there was too much information all on one page, and as a result it was difficult for the kids to find on the page the culminating big-idea questions (at the bottom of the page) that they needed to answer; if I am to repeat this project in the future, I'd have to re-think how I would do this, maybe provide them with a layout or outline to help them structure their write-up, since it is the first time they have had to do a math writing of this length and complexity.

Here it is on Dropbox. If you do use it as a basis for a fractions project, please share with me your modifications so I can make mine better for next year. Thanks! If you check back on this post, I'll also update it next week to include some fragments of student explanations (and/or maybe photos of finished dartboard designs).