I Hope This Old Train Breaks Down...

Graphical Organizer Charts

noreply@blogger.com (Mimi) — Sun, 29 Jan 2012 14:06:00 +0000

One of the teachers I work with closely at my school has sheets and sheets of handouts on the theories behind various concepts. Over the years, she has polished her theory handouts and they seem to work very well for her classes. Although it's not my preferred method of teaching, observing her handouts has gotten me to think about how I can use graphical organizers to help elucidate certain confusing concepts to the students.

For example, my students and I developed some notes together in December on how to look at graph of f(x) and use it to generate rough estimates of the graphs of f'(x) and f"(x). And we also started with a graph of f'(x) and worked our way to developing f(x) and f"(x) graphs. Keeping those notes in a 3-column format helps the kids see side-by-side the correspondence between the graphs.

As another example, one thing that I was nervous about was teaching integral Calculus concepts for the first time this year. I read up on various resources from Sam's Virtual Filing Cabinet, and decided that I liked the suggestion of first introducing integration techniques before discussing the meaning of integrals. So, I created this worksheet. The kids started with the middle column and first differentiated to get the left column answers. This was a mini-review because it had been a few weeks since I had seen the kids (Christmas vacation). And then, I urged them to observe the pattern and to work backwards to find the indefinite integral column.

It was great! The kids were certainly able to figure out some of the simple ones on their own, but in general, they found it helpful to think about what dy/dx is, before thinking about how to "work backwards" to find the indefinite integral. They carried that technique with them later on even when they weren't given a grid to work with. We still needed a couple more days of practice before they felt comfortable with the idea of "working backwards" to un-do differentiation, but this was a good way to introduce it without introducing much fear.

Another attempt at teaching schematically is this: a framework for taking notes on wave transformations. I haven't actually used this yet, but I think it's definitely an improvement over whatever I did last year. We will start by seeing/comparing how sine and cosine waves are similar under each simple transformation, and then work our way to seeing how they transform under a series of steps and under complicated IB language. Then, the kids will work backwards, going from waves to equations to solidify their understanding.

Do you teach with graphical organizers? If so, for which concepts?

Japanese Geometry Problem Set #2

noreply@blogger.com (Mimi) — Sat, 28 Jan 2012 15:20:00 +0000

Here are these week's Geometry problems from Japan, again loosely scaffolded. They are getting juicier.

From doing/translating these problems, I have noticed that the Japanese expect a lot more from their students in terms of rigorously applying algebra to geometric settings. Check out the problem set and try out some of the non-guided problems, and you'll see what I mean. The problems are fun for me to do and to think about, because they're just one step up in terms of complexity from our normal Geometry problems in the States, even though the concepts involved are relatively few. Technically, all of the problems I've linked to thus far are just Pythagorean Theorem problems, with a couple of circle theorems and special right triangles tossed in here and there, but the way the problems are structured is significantly (far??) more complex than our typical American applications of the Pythagorean Theorem, requiring a fairly sophisticated understanding of algebra. It's amazing to think that this is what they expect from their public school children in middle school.

I hope you are enjoying these!

Accountability and Changing Classroom Dynamics

noreply@blogger.com (Mimi) — Fri, 27 Jan 2012 17:29:00 +0000

I believe that I work hard to teach developmentally. I consider what my students need in order to understand/create the next concepts from scratch and to integrate those concepts into their existing worldview, and I constantly reflect on how their age affects their motivation and their understanding of the learning process. I constantly revise, in my mind, what successful learning looks like, in order to better help my students.

But recently, I have been thinking that I have it all wrong. I am working way too hard, and ironically enough, my students are working way too little. Let me explain.

My latest belief is that what makes a student successful in math is a combination of sophisticated skills, raw intuition/understanding, and confidence in their own ability to attempt new problems. The more I think about this, the more sense it all makes to me. We often over-praise children for their "potential" or "understanding" in mathematics, and those same children walk away from that praise believing that being good at math means having an innate ability to reason. The more I think about it and observe children and think about it some more, the more I disagree with that belief and think it's bogus that we praise kids for their "potential" without making an immediate, stern emphasis on their lack of effort.

Here is a simple analogy I draw for myself: A musician or an athlete would never prepare for a gig / game by ONLY thinking in their heads that they need to "dribble the ball past their opponents and then shoot it into the basket." In order to do it successfully and consistently, they need to put in hours of practice to bridge the gap between theory and technical expertise. So, why do our children think that they can get better at math simply by thinking about it abstractly and passively looking at examples??

In Grades 3 through 6, we more or less teach kids the same skills and concepts over and over again, with a bit more depth each time. The kids who are intuitive and/or clever (and yes, they do exist) cannot help but be "good at math" by the end of Grades 5 or 6, because they've already seen every skill at least twice. Does it mean that they've "mastered" those skills? I think not, based on the fact that some of my most intuitive/clever incoming 7th-graders still could not recall basic fractional skills until we had reviewed them, even after years of learning the concepts. But, at this point, whether kids commit to practicing the concepts repeatedly does not immediately determine their test performance.

In Grade 7, the game starts to change a bit. The problems become multi-stepped, and new (algebra) skills come down the pipeline that require repeated practice in order to reach a point of effortless, automatic application. I am certainly not saying that understanding is not important in Grade 7 -- in fact, we do a lot of conceptual development in class before introducing any algebra methods -- but I find many of my intuitive students struggling with procedural issues in algebra, even though they understand in their heads what they need to do. They simply have not developed the work ethic to practice and practice again until their procedural issues are ironed out and they can consistently solve something without effort. In Grade 7, more than anything I emphasize work habit, because I find it so dangerous that those "intuitive" kids get passed on from teacher to teacher believing that AS SOON AS they would start working, everything would be dandy.

Because truly, I don't believe that. I have two amazingly intuitive 11th-graders, who lack the basic arithmetic and algebra skills to complete problems. If you throw them into a completely new type of problem, they can make headway better than their classmates. But then, put them in front of a common/simple skills application, they would have no idea what to do and invent crazy algebra rules. I can only imagine that those two kids have been praised by teachers all along for their intuition, without an equal emphasis on how much damage they are doing themselves by leaving such major gaps in their basic skills. At the same time, I've seen other kids go from struggling to mastering skills, and then tougher concepts. Those kids persevered until they mastered the basic skills, and now they can save their mental energy for the trickier/problem-solving parts of the task.

So, practically, what does that mean in my classroom? Besides praising effort, what else have I started doing in support of this belief that dedicated practice is important?

Well, one of the things I have been thinking about is that, despite my efforts to make learning exploratory and constructivist, my students are still far too dependent on me. They expect me to set the pace of my classroom, and when they are absent, for example, their learning suffers tremendously because they do minimal work at home. And this simply cannot be the case for the Grade 11's and Grade 12's.

I have made a homework schedule for my IB students in grades 11 and 12. I have never been a believer in mandatory studying outside of class, until now, because part of me hopes that the kids will see the importance of pacing themselves and setting their own goals. I can see, however, that my Grade 11's and Grade 12's are too comfortable; they come to class, do what assignments I have designed for the day, feel great about their understanding of the concepts, and then most of them walk away without practicing more on their own or struggling through practice problems in the textbook. It's simply not working, because I am working way harder than them for this class, and that cannot be the case when it is their learning we are talking about. So, in my new homework schedule, I assigned a Chapter Review from the textbook every week or two weeks, for a topic they should be familiar with. If there are questions they cannot answer, I urged them to look through the chapter at home to resolve their questions, and only bring to class the most complicated questions that remain unanswered.

And you know what? It has been wonderful. It has completely changed the dynamic of my classroom. Instead of feeling like I am on the hook for giving kids work and setting the pace for my class and making sure they get sufficient practice during class, our roles have somewhat flipped. Kids are in more control, and they are asking me questions that they want to know the answers to. It has changed the feel of the relationship between me and the students, for the better.

I am also experimenting the same with my younger students, except I am going to give them a quantity of problems (ie. 15) to bring me each week from the textbook, and they get to decide which problems they wish to practice. In doing so, the kids get to decide if they need more practice with current material (ie. weaker students) or if they want to use the "mandatory" practice as an opportunity to spiral review.

In short, as I think more about what makes a successful student, I want to think about ways of extricating myself out of that picture. I truly think that this will empower my students to feel like their learning is in their hands and to build their confidence outside of class, without changing the way I currently structure lessons inside the classroom. I think this is how I am going to help my students move towards becoming life-long learners.

Japanese Geometry Problem Set

noreply@blogger.com (Mimi) — Sun, 22 Jan 2012 15:30:00 +0000

I am doing a foray into trying to differentiate for my very advanced Japanese student! He's awesomely hard-working, but I recently had a chat with him because I am concerned that he has already learned all of our Grade 8 topics in Japan, and he's concerned that he'll fall behind the curriculum in Japan. I actually noticed this much earlier this year, but he has just now gained enough basic English fluency to communicate academically, so I decided that now is a good time to start his individual math program.

Given that he's planning on moving back in two years to finish high school there, I think it's important to try to help him keep up with the Japanese curriculum. So, I asked him to bring me some Japanese math textbooks to give me a sense of what goes on in Grade 8 in Japan. I sat down today to take a look, and wow! It was tough stuff considering he's only in Grade 8! Since I obviously cannot read Japanese (I can read some Kanji, since they use Chinese characters, but it doesn't always mean the same thing), I did my best to cross check his geometry diagrams and the solution guide he gave me, to get a feel for what was given in each problem, what was expected, and what prior knowledge he must already have.

This is going to be my new pet project for Grade 8. I've got at least one other very bright kid in similar shoes, actually, who is transitioning back to a different curriculum after this year and wishes to be studying Geometry to supplement the algebra we are doing in class. So, for at least the two of them (and anyone else who wishes for the challenge), I am going to do my best to offer Japanese math problems as enrichment in our class. I think it'll be a great way to force the bright kids to work together and to support each other, and it will help our class appreciate math from other cultures!!! (We are an international school, after all.)

As for the Japanese kid in my class, the advantages are obvious -- I can help him bridge the gap between the curricula, and because I've translated the problems to English, he can receive my support in English, as well as develop a bilingual vocabulary, hopefully to be able to read the questions further on down the book on his own and translate for me what they're giving him and asking him to do!

And for me, this is also an exciting opportunity to take a look at math problems from another culture, to see what they consider "basic" and "difficult" and how they scaffold. I am very excited about this pet project! It's a win-win-win!

Here is the first Japanese problem set I translated/loosely scaffolded, if you're curious. I don't know what the day-to-day math pacing is like, but they do all this and MORE in one lesson in the textbook!!

Some Resources on Patterns and Systems of Equations

noreply@blogger.com (Mimi) — Sat, 21 Jan 2012 09:46:00 +0000

Here are some resources I'd like to share (yes, I'm blogging a lot, but that's because various things have been on my mind and I have been too busy doing semester grades to post anything).

First: Some fun visual patterns from NCTM. You can use this for your elementary students simply as pattern recognition exercises, OR expand them into algebraic exercises / modeling writeups as I am going to do for my middle-schoolers.

Secondly, I realized that when I shared my shapes puzzles for teaching systems of equations a while ago, I didn't share the surrounding lessons that then reinforced their conceptual understanding and eased the student transition to symbolic manipulations. I'm reusing these lessons this year and they are simply working magically for me. The kids are doing substitutions in their heads for entire expressions. When they look at 2n + 5p = 44 and n + p = 10, they can quickly say to me: "You can replace the smaller equation into the larger one twice, with 3p left over, so that means 10 + 10 + 3p = 44, so 3p = 24 and p = 8." And they completely understand why algebraic "elimination" is simply a shortcut that comes out of the substitution concept. (Some of them are even mad at me for giving it another name, because in their heads elimination and substitution are exactly same methods with a couple of steps skipped.)

So, let me try and do this:
Lesson 0: Savings Race was the first lesson I used after getting back from vacation, to get the kids thinking about linear functions and to briefly preview the idea of break-even points. It primes the kids for some of the concepts that will come along soon.

Lesson 1: Shapes puzzles introduces the idea of solving for unknowns with multiple requirements and visualizing variables/equations as composed of visual shapes. Also, the kids start to concretely develop an understanding of substitution and "scaling down" a value. No mini-lesson teaching necessary. Just go around and facilitate if they get stuck with the puzzles, but there was that eerie silence for much of the period when the kids were just thinking and they didn't want my help. I introduced the term "substitution" at the end of class using a pair of shapes equations.

Lesson 2: Line segments reinforces the same concepts from Lesson 1, plus it asks the kids to write algebraic descriptions of the relationships so that I can go around and facilitate how they could have substituted using symbols instead of using pictures. Again, no mini-lesson teaching necessary. I re-introduced the term "substitution" at the end of class using a pair of equations they can visualize easily in their heads.

Lesson 3: Transition to Algebra asked the kids to solve various pairs of equations. I asked the kids to do mostly algebraic manipulation at this point, or if they need to, draw out a few shapes but still write equivalent algebraic symbols next to them to show a transition to symbolic thinking. I went around and facilitated, but there was no mini lesson. THE KIDS TALKED A LOT TO EACH OTHER DURING THIS CLASS WHILE MAKING JOINT DISCOVERIES! At the end of this assignment, I went over the different terminology of "substitution" versus "elimination" methods using examples from this worksheet. Since most of them had been doing elimination in their heads, I asked them to start writing down -( SMALL EQUATION ) to show they are mentally subtracting the smaller equation from the larger one.

Lesson 4: Drilling elimination method asked the kids to do everything using elimination. I did not tell the kids how to decide if it's addition or subtraction between the two equations, but I told them that the ones where they don't want to subtract in their heads are going to "feel a little different" when they get to them. More often than not, kids grabbed me when they thought subtraction would not work for that system, and we discussed how addition would cancel out opposite terms. This day, I also gave the kids computers and asked them to check their answers using Wolfram Alpha instead of turning to me or checking with their partners.

Lesson 5: Graphical solution and meaning of systems helps the kids see one type of situation where systems are used and gets them to practice some basic linear skills and graph-reading skills. No mini lesson necessary, although I did chat with individual kids to tie this to the graphs they saw the day before on Wolfram Alpha.

Lesson 6: Group project lets the kids practice analyzing break-even points. Both lessons 5 and 6 are easy transitions, because they followed from Lesson 0 above.

...We're not done with the unit yet (I intend on going all the way through quadratic-linear systems), but I thought instead of sharing the whole unit, I'd share just the bits above on how I developed the most fundamental concepts of systems. I hope this is useful to you! Like I said, these are lesson I dug up from the dusty digital filing cabinet, but they're working like magic for me with no rote teaching, so I am linking to them here in hopes that parts of them can be used in more than just my classroom!

A "Backwards" Approach to Completing the Square

noreply@blogger.com (Mimi) — Sat, 21 Jan 2012 00:48:00 +0000

One of my students is an intuitive math student, even though she's not so good with memorizing algorithms and often her intuition isn't enough to complete the problem all the way through. She recently encountered a completing-the-square problem on a test, and couldn't remember how to do it. The question said to go from f(x) = 2x^2 - 12x + 5 to the form f(x) = 2(x - k)^2 + h, in order to do further transformational analysis.

Being an intuitive math student, she took the unusual approach of expanding f(x) = 2(x - k)^2 + h into f(x) = 2x^2 - 4kx + 2k^2 + h. And then she set this equal to 2x^2 - 12x + 5 and then just got stuck. I looked at her work and thought it was an interesting alternative to completing the square. Based on her approach, you can simply observe that if the two equations are equal, then it must be true that their x terms are equal: -4kx = -12x, and similarly, their constant terms are equal: 2k^2 + h = 5. So, it follows that k = 3 and h = -13, leading us to the vertex form of the equation as f(x) = 2(x - 3)^2 - 13.

Just thought I'd share an interesting alternative to completing the square "forwards". I see this as an alternative in working "backwards". Funny what the kids can help you see, even when they're not skilled/experienced enough to make it all the way through a problem.

Departmental Ruminations

noreply@blogger.com (Mimi) — Thu, 19 Jan 2012 18:52:00 +0000

My school's math department runs from the elementary school all the way through the high school. That's very interesting, because it's gotten me thinking beyond my own classroom about the end-to-end process of math education. For example, recently we discussed the issue of what we want math teaching to look like in the elementary school, in order to boost student understanding in the middle and high schools. That discussion revolved around not just what topics should be taught, but also what elementary school assessments should look like, which rote methods (such as the lattice method of multiplication) should be avoided, which traditional concepts/skills emphasized, etc. For example, I feel quite strongly that kids need to be able to multiply 1 digit by 2 digit numbers in their heads by the end of Grade 5. They should also be able to add two multi-digit numbers in their heads from left to right. If they cannot do those things, their estimation skills suffer and it can affect their overall number sense, or at least reflect a lack thereof. But, anyway, our departmental discussion got me thinking about other cross-grade improvement/coordination possibilities (which could apply at any school, not just ours).

* We could develop a clear calculator policy. When (at what grade) do we transition over to using calculators instead of calculating manually? What calculator skills should be taught in what grades, in order to ensure a comprehensive exposure?

* We could develop and maintain writing samples and rubrics for analytical/applied mathematics at all grades. We have MYP and IB rubrics that we have adopted, but I think the writing samples can be a nice addition. Ideally, each kid would carry a math writing portfolio around with them from grade to grade, to showcase their growth on the rubric over time. As part of their math portfolios, they'd reflect on their own growth in understanding the flexible use of mathematics.

* This idea came up about making a Celebrating Math Week that spans the entire school. We could all coordinate projects to happen around the same time in our math classes, and the kids would present their results on an evening when their parents are invited.

* I would love for all the teachers to sit down and have a thoughtful discussion about scope and sequence, across the grades.

* We can encourage more interaction between grades. For example, we can get 8th-graders to conference with 7th-graders about their projects, to offer objective feedback on how clear the written explanations are to someone who's not already familiar with the mathematical task. Similar cooperation can happen at the high-school level, with Grade 12's giving Grade 11's advice about preparing for the IB exams.

These are some ideas I have so far. I'd love to tackle one or more of these things next year, if I could drum up some support...

Have you had success doing department-wide improvement projects? What other ideas do you have?

Focusing on Process and Learning from Our Mistakes

noreply@blogger.com (Mimi) — Fri, 13 Jan 2012 13:19:00 +0000

A short while back, Kate linked to an awesome video about learning from mistakes. Well, following my 7th-graders doing an awesome little project writeup for me this week, I thought I'd wrap up the week reviewing some of their common mistakes from last semester's big exam.

This is how I structured it. First (since it had been a while... we hadn't seen equations since December's big exam), I gave them one problem on the board with an answer written at the bottom of the board. I asked for them to figure out the process for showing how to get that answer, and the first ones to show me the clearest work can put them on the board, and I'll choose another person with the correct work to explain what has been written on the board.

Here was my first problem (not an easy one!):

-4(x – 3) + 1 = 5(3 – 2x) + 70
.
.
.
.
.
.
x = 12

The kids were instantly into it. (They were engaged by the competition aspect.) After two kids had put up two different ways of solving, I chose a normally very insecure kid to go up and explain their work, and she did great!

Then, we did another problem similarly:
4x – 13x = 2(-x + 8) + 19
.
.
.
.
.
.
x = -5

This time, a lot more kids were able to successfully complete the problem in a short amount of time. Of them, I picked two kids whose work didn't look exactly the same to put their process up on the board. Another normally unconfident kid agreed to go up and explain the work already put up on the board.

After that, we switched gears and I took the board markers and put up three problems, one at a time. I challenged the kids to quietly put up their hands when they can see where a classic mistake exists, and I waited until over half of the class had their hands up to pick a relatively weak student to tell me the answer.

Here was the first one, which many of them got right away:
-2(3x – 5) = 20
-6x - 10 = 20
-6x = 30
x = -5

They really enjoyed it, so I went ahead and put up:
5x – 3 = 3x + 11
8x = 8
x = 1

This time, a juicy discussion ensued. One of my students thought that the mistake was that 3x + 11 doesn't equal 8 but equals 14. Another student said that the second line should be 8x = 14, because the -3 "should become +3 when it goes across the equal sign." (I put it in quotes because it bothers me when kids say that, but if they've already been taught some basic algebra at home, that tends to be their phrasing.) Finally, some kids correctly identified/explained that the second line should have been 2x = 14.

Then, I put up a third problem, this time with two separate mistakes in it. Again, I challenged the class to find both mistakes.
(1/2)(x – 8) = 50
1/2*x - 8 = 50
1/2*x = 58
x = 29

It was so great! They were very excited that they could find so many mistakes.

It was perfect time to transition into Kate's suggested "My Favorite No" activity. We went through three algebra problems, increasingly more difficult each time, and I had kids submit their solutions on little scraps of paper. I wrote down my favorite incorrect problem on the board, and we started by pointing out all the things that person had done correctly, before discussing where they had gone wrong and why. In doing so, we caught: arithmetic error (some student thought -29 - 27 = 56) because they thought that you apply "the integer rules." We also caught the mistake of subtracting 2x from the same side of the equation twice. (I was so happy when the kids said, "You can't do that, because that would throw the equation off-balance!" They are talking like pros.) We also caught the mistake of going from 2.5x = 10 to x = 2.5/10 = 0.25.

It was brilliant! I think the kids had fun, AND I was able to get them to think hard about some common procedural issues ON A FRIDAY AFTERNOON.

When the class ended, I had just gotten them started on a Row Game involving some more basic algebra. It's their (my) first time doing a Row Game, so the concept of comparing answers even though the problems are not the same was a bit confusing to them. We'll have to continue with this Row Game next week, because it's supposed to address some more common procedural problems that I saw on the December exam. The worksheet I made for that is here if you want it. I am excited to continue it next week! Kids were talking to each other about math and trying to figure it out before turning to me for help (even though they were convinced that they could not have made a mistake and the problems could NOT have the same answers). It was really lovely.

So, yay to Kate, and yay for a day of trying new things and working with our conceptual mistakes instead of pretending that they don't exist.

A Plug for PCMI

noreply@blogger.com (Mimi) — Thu, 12 Jan 2012 21:25:00 +0000

Hey, are you looking for a great mathy thing to do this summer? Try applying to PCMI. It's awesome, and Park City, Utah, is a fantastic place to be for three weeks of the summer. When you go there, you feel like the sky is bigger/cleaner and the days are way longer somehow. And there are some great math teachers who are passionate about teaching and doing math. Although you can find last year's problem sets here, it's hard to imagine the level of energy and camaraderie unless you've been to PCMI.

It's magic for three weeks, and you'll miss it when it's gone. There is also a generous stipend that covers most of your expenses.

Apply today! Deadline is the end of January, so you had better hurry.

PS. If you do go to Park City, bring your yoga mat if you've got one. It's utterly beautiful there to do yoga outdoors. Also bring your hiking shoes, your best karaoke persona, your fine dining belly, and your thinking cap. Just sayin'.

Baby Steps in Learning German

noreply@blogger.com (Mimi) — Wed, 11 Jan 2012 21:19:00 +0000

Today, I learned about a funny German category of verbs. (I just started private tutoring last week. It's amazing. I get to move at my own pace, which is pretty miraculous. I really feel that in two classes, I've already covered the equivalent of two or three weeks in a regular course, because I don't have to wait for other people to finish an exercise, and everything is chop-chop fast.)

Anrufen means to call someone up. When you conjugate it, the prefix comes off of the front of the verb and moves to the back. So, for example, to say "Are you going to call (me) on the weekend?" you say, "Rufst du (mich) am Wochenende an?" with the two parts of the verb separated by the entire rest of the sentence! Or, "Are you going to call him about it?" becomes "Rufst du ihn deshalb an?" Totally crazy cool. (By the way, Germans capitalize all nouns, which is funny and cool to me. Everything is important! Morgen, for example, is capitalized when it's a noun meaning "morning", but not capitalized when it is used as an adverb as in "tomorrow"... All sorts of very interesting, very particular grammatical rules!)

I think if I really work at it, I can cover a lot of ground in a calendar year. I am curious what that ground will look like without anyone else to set the pace, so I have decided to motivate myself to do some extra work every week in review of the last lesson and in preparation for the next, in order to maximize this year. It's so exciting! Ich will bald besser sein! (Is that right? "I want to be better soon!")

Sin Nombre

noreply@blogger.com (Mimi) — Tue, 10 Jan 2012 18:15:00 +0000

Have you seen the movie Sin Nombre? It's about some illegal immigrants trying to get to America, and getting entangled with mareros from the infamous MS13 gang. Having lived in El Salvador, the movie was all too real and very depressing to me. After watching it, Geoff and I said to each other that our lives are truly privileged.

(Warning, major spoiler to follow)

I am sure there are worse ways to die, but plunging face forward off of/into a moving train while trying to cross the border illegally has got to be one of the worst ways. You're dying like an animal. It made me think about all the people who do die trying to cross the various borders. So utterly unjust. The only difference between them and us is their desperation; they were born into the wrong place, at the wrong time.

Reflections Based on Types of Mistakes

noreply@blogger.com (Mimi) — Sat, 07 Jan 2012 08:16:00 +0000

I wanted to come back to talk a bit about my follow up to bucketing kid mistakes. I had my students write a detailed reflection of their exam. They had to identify which type of mistakes they had made the most frequently, to list the math concepts they had missed, and to thoroughly evaluate their study strategies in order to seek further improvement.

As the kids were looking carefully at their mistakes (I said I'd collect their reflections and compare it against their tests to make sure they were doing a thoughtful job), they changed some categorizations down from procedural to simply careless, if they are sure that they knew what to do but just didn't apply the skill carefully. They also noted to me if they had made various mistakes due to the same essential misunderstanding (ie. not looking to distribute the negative sign).

The kids' reflections that have been completed so far have been very impressively detailed and honest! In response, I corresponded with them in writing to add my assessment or recommendation for improvement during the second semester, and I am going to return the tests and their reflections next week to take home to review with their parents. In my comments to them, I wrote down things like if I think they should be checking their answers regularly against the back of the textbook, or if I think it was quite commendable that they persisted for a long time during the exam to try to get through even the hardest questions, regardless of whether they had finally succeeded. On my Grade 8 exam in particular, I commended the whole class for doing well and persisting when challenged on certain problems. For Grade 7, I noted to the kids that many of their exam scores did not accurately correspond to their normal performance, which showed me that they still have ways to go in working on their test-preparation strategies. (One of them, for example, did tons of practice problems but never checked her answers against the back of the book, so many of her practice problems were in fact incorrect when I looked at them! Another student signed up for some random math website and did random problems before the exam, instead of the problems I assigned for practice. While other students did 50 problems of the same type, and then ignored the 9 types of other problems that were going to be on the exam. These are all weird things that I am glad now I know they need to fix...)

Overall, I think this idea was a success! Instead of me saying to the kids that they are still making careless mistakes, they were pointing it out to me that they're not reading instructions, or not answering the questions fully, or making careless procedural errors -- all of which, they say, could have been avoided. I was very pleased because they were drawing the same conclusions that I wish they could have drawn, without my input.

I wouldn't do this level of reflection after every test, because it's lengthy and I don't want kids to start treating reflections as a "let's-just-get-through-this" thing of routine. But, I think I am going to stick to doing careful reflections twice a year to help them grow as students.

PS. On a totally different note, have you seen this? It's beautiful, and amazingly makes me feel (again) like the world is small. The guy who made this is my college friend's friend from high school!

Private School Salary Dilemma

noreply@blogger.com (Mimi) — Thu, 05 Jan 2012 20:35:00 +0000

I never gave it much thought until today (I'm not very good with money things), but recently there was a discussion about the tradeoffs between various systems of salaries and raises in private schools. It occurs to me as a very real, and fairly tricky, math problem.

First off, some brief description for you non-teachers: "salary step" is basically a grid of salaries, where for every year of experience you accumulate, you move along vertically to another area of the grid, and therefore get assigned a higher salary. Alternately, you can also move to a higher-pay area of the grid by accumulating additional training (and thereby moving horizontally along the grid). Teachers' unions typically negotiate salary increases across the entire grid, for example, to request increased benefits for every teacher in the system, and I am pretty sure they use a similar system for all public employees in general.

Obvious advantages of this salary-step system:

* It makes sure people are paid based on experience and training. (Loosely speaking, it's a logical idea that more senior teachers and better trained teachers will translate to better productivity.)

* It ensures equity among staffers hired earlier and later. ie. If you started at the school 10 years ago when you were a 3rd year teacher, you are now paid a higher salary than someone hired this year, with 6 years of previous experience elsewhere.

* It encourages retention of existing staffers, as they will continuously be rewarded for additional years accumulated on the job. Staffers who do leave, then, tend to leave for personal reasons as opposed to leaving for reasons of financial stagnation.

Disadvantages:

* The biggest disadvantage is that the overall school staffing budget will grow linearly every year, assuming that there is little attrition. At the same time, most schools will not be able to increase their tuition linearly every year, or increase their student enrollment linearly to compensate for the constantly growing staffing budget. As I see it, a private school nearing its max enrollment simply cannot afford to use salary steps (and one wonders how our government can afford to do so either).

* Some may argue that the salary-step system does not take into account teacher's actual productivity/merit. That's not a discussion I'd like to go into at this point, given all the controversy surrounding merit pay in general.

* Because staffers are continuously being rewarded for staying, it provides little incentive within the international school environment for healthy mobility and change/influx of new ideas.

Alternative systems and their tradeoffs:

* No salary step system. What salary you enter at is what you stay at, no matter how long you stay at the school. It creates weird situations like if you entered the school 10 years ago, with 5 years of previous experience, you could now (and forever) be paid significantly less than another person who now freshly enters the system with a prior experience of, say, 8 years. Even though overall you are way more senior than that person (15 years of work experience, versus their 8 years), and you have also shown that you are committed to this school, you end up forever being paid less. Needless to say, this affects morale negatively.

* Fixed annual percent increase of pay. This solves the problem of inequity due to time of hire, since by the time other new staffers have been hired, you would have already experienced various raises that put you ahead of them permanently. I think this is probably a strategy that non-mathy people would naturally come to, except that it creates the problem of an exponentially growing school budget over time, so it isn't really feasible. --Plus, in order to avoid any such "weird situations" of inequity, you actually would need to pick an annual percentage that grows FASTER than the linear increases in the salary step! No good!!

* Cost-of-living adjustments and project-based stipends. I think most schools do this, but it's still not addressing the issue of the inequities due to time of hire.

* Merit-based increases. I hate to say it, but this seems like an obvious option despite research that says otherwise. But, what do you judge merit based on? Hopefully not test scores or student opinions. Is it too much to move towards a business model of stacking employees based on peer and supervisor evaluations?

...I think this is sensitive to people because every time we talk about pay, it always gets sensitive. But truly, I see it as a mathematical/business dilemma that is objectively interesting. What do you think is a viable solution? Does one exist?

Other issues to consider:

* Your school's entry salary has to be internationally competitive for a person of that level of experience/training.

* Salary steps are not truly linear (I don't think), nor should they be. The productivity difference between a teacher in their 18th and 20th years is not at all comparable to the productivity difference between their first and third years.

* Every time natural attrition happens, depending on how you replace the lost staffer, your overall staffing budget will either shrink or expand. Therefore, the school admin still has significant control over their staffing budget regardless of the salary system in place.

Anyway, I'm throwing it out there because I have not made up my mind about it, but I am curious if there are clear-cut solutions that I am just not seeing. Y'all able to help me think this one through?

Feeling Inspired After PD on Differentiation

noreply@blogger.com (Mimi) — Tue, 03 Jan 2012 16:51:00 +0000

We had an all-staff professional development session today that was actually great! The speaker was from London Gifted and Talented, but he spoke more generally about differentiation for all kids (in the context of G&T education). First off, I have to say that I am a skeptic of the whole G&T education thing and what it does for kids; anyway, I went into the PD with quite a bit of doubt.

That said, I was really glad to hear the speaker say that the best way to nurture G&T kids is to provide opportunities for enrichment for all of your students via effective differentiation. He talked a lot and went through a lot of slides, but here were my favorite points:

* "High challenge/low threshold learning" is what we should be aiming for. A truly differentiated task should be limitless on the upper bound of complexity and be truly open-ended, genuinely investigative, and to allow student choices of medium/depth/topic, but still be accessible to everyone in the class.

* Differentiation cannot/should not be an end in itself. It should be linked to a purpose, and your method should reflect your purpose. --> This was a particularly good point for me, because I realized while he was talking that I have not clarified the end goals of differentiation for myself. What am I trying to achieve? Do I want different kids to be able to approach problems using different methods? Do I want kids to be able to demonstrate their knowledge using different media/application? Do I want kids to achieve similar abstract knowledge or am I comfortable with different kids understanding the concept differently? Lots of things for me to think about!!

* "It's not that [differentiation] is not happening. Rather, it's that we don't have the shared language to talk about what is already happening [in our classrooms]."

* Provide variable-credit assignments. A complex, rich task that is done well should replace several smaller, more basic tasks. A talented student should not be punished for their talents by being assigned extra work.

* Student floundering is good. Teacher needs to create environment for kids to think independently and to allow students to struggle. (I know I'm preaching to the choir here, but I also know how much I enjoy PDs that emphasize this point, because so many educators still do not believe that themselves.)

All in all, the session was great because it reminded me of the things I had committed to doing in the beginning of the year that I am still not doing. I don't believe in new year's resolutions (since I think goal-setting should be an on-going process and allow the opportunities of failures and re-attempts), but the second half of the school year seems like as good a time as any to be more self-critical and to hold myself accountable to some of those promises!

Christmas in Turkey

noreply@blogger.com (Mimi) — Sat, 31 Dec 2011 13:02:00 +0000

We just got back from a week-long trip to beautiful Turkey! Even though we had not much time to look around, from what we could see, it was truly an amazing mix of East and West, of modern and ancient cultures. Geoff and I both agreed that it is incredible that you can hear the songs calling Muslims to prayer 6 times a day in the backdrop of Istanbul, yet the city is vibrant with modernly (and risquely) dressed women. The city, at least, seems to embrace people's choice of lifestyles, more so perhaps than most parts of the West. (I think that in the States, as much as we claim to embrace liberal ideas, if you choose to pray 6 times a day towards Mecca, people at your work place would definitely look at you funny.)

Istanbul itself is also an amazing mix of cultures -- there are bar-lined streets galore, such as in the Taksim Square area, terraces overlooking the hilly city, trams that run from tourist point to tourist point, and yet the guitar and drums music seeping out from the bars are traditional-sounding, somewhere between Arabic and Indian.

The Turkish food is incredible; there are elegant restaurants to match the best of Europe. Three restaurants in our hostel area that we can immediately recommend are De L'Artiste, Morro, and Solera. Of these, Solera was my favorite, because they serve up local Turkish wine, coupled with delicious cold appetizers that are of a local variety, but beautifully done with elements of savory surprise. The city also has countless bazaars -- the only one we visited was the Grand Bazaar, but the price and the quality of the goods there were fantastic. Between Geoff and me, we bought: a silver necklace, an exquisite mirror for my sister, a lamp set, a rug, a leather-made silhouette puppet, and a beautifully woven pillow case. :)

While in Turkey, we did the typical touristy thing. We flew into Istanbul, spent a few days there, took a flight out to Izmir, spent a couple of days there on an all-inclusive tour, and then flew back to Istanbul for a few more days. Istanbul was amazing because it was a party spot, but backdropped by the ancient buildings. It's incredible to think about the unique culture exported from Istanbul to the rest of the world over the centuries. It is of little wonder that it prides itself as the Cultural Capital of the World.

This is me fake-crying because of the weather. :) It was snowing our first day in Istanbul! Colder than in Berlin!!!

While we were in Izmir, we got to visit Ephesus, the third largest ancient city. A good amount of the stuff has been rebuilt from the excavated material, and the excavation is still on-going, but it was still impressive to see the ruins left after thousands of years.

Besides Ephesus, we also got to visit Pammukkale, which is a beautiful calcium bicarbonate deposit formed by centuries of active hot springs. Some of the springs were cooled greatly during the winter, but others were still bath-water warm! You can only walk through the labyrinth of springs with bare feet, but we braved the cold anyway....

As an added bonus of going to Turkey during the off-season, we got to stay at a five-star hotel for a night as part of our all-inclusive trip to Izmir. The food was delicious and our room had bubble jet stream bath, and a view of the ocean. It had been a long time since we had fallen asleep to the sound of ocean waves outside of our window, so it was a real treat. (Especially because we had anticipated staying at a hostel.)

Anyway, here were some other random things we did:

Geoff was fed some stuffed clams by a cute-looking Einstein man! Clams stuffed with rice is apparently a local specialty.

We visited a local (Geoff's favorite) shisha spot. Surrounded by colorful Turkish lamps and lots of locals, it was the perfect spot to enjoy some apple tea and some shisha.

Here is Geoff inside the Circumcision Room at the Topkapi Palace.

We couldn't take any pictures at the Whirling Dervish religious ceremony, but it was pretty cool and inside a fixed up bath house.

Speaking of bath houses, we had our first experience with a Turkish bath house. For the equivalent of about 30 to 35 Euros, you can strip down naked, lie inside a sauna, and then have a person scrub you clean. (It's not coed. Geoff and I were in separate parts of the bath house; he had an overweight man leaning over him asking him, "Is this GOOD? IS THIS VERY GOOD?" while scrubbing him down. I had an old lady and she also asked me if it was very good while I lay completely naked and she scrubbed me down. It was very unique -- definitely an experience I would recommend.) The bath house we went to was a traditional one -- it had been running since the 1400s.

And, as a last note (harhar, no pun intended), I wanted to share with you my excitement to see the Sigma notation on a Turkish bill. I am pretty sure that in Turkey, they put random accomplished people (not just politicians) on bills, so they also put down some visual representation of why that person's famous. You know I had to share this:

Categorizing Student Mistakes

noreply@blogger.com (Mimi) — Thu, 22 Dec 2011 14:05:00 +0000

I am experimenting with a new method of grading exams, in which I look through the problems to determine what type of mistake the student made, and instead of writing a lengthy explanation of the mistake (which is what I typically used to do), I just circle the place where they messed up and then write down the category of the mistake.

So far, these are the labels I've come up with:

* Wrong approach - ie. the student was completely not on the right track.

* Conceptual mistake - the student had an inkling of what to do, but they made some severe error in the initial setup of the problem.

* Procedural mistake - the student understands at a high level what the question is asking and what procedures are required, but made some fundamental mechanical error in the procedure.

* Arithmetic mistake - mistakes involving combining decimals, fractions, or integers.

* Careless mistake - mistakes involving miscarried signs or wrongly recorded results, when the student exhibits overall competence in the process.

* Incomplete operation mistake - the student failed to completely answer the question or completely simplify their answers. Or, they were off to a good start and then bailed halfway...

* Mistake in interpreting the instructions - the student did not carefully follow the written directions and therefore did irrelevant calculations.

My hopes are that in this way, I can help kids to focus first on the bigger conceptual issues, and next on the other types of issues. I'd say that if a high level kid sees that they're consistently making the same types of careless or incomplete-operation mistake, it is valuable feedback for them to keep in mind for the future. Versus if a lower-performing kid sees that they're at least not missing the major concepts, then that is a good feedback for them as well, so that they know they would just have to focus on the procedural issues.

Thoughts? I'm grading semester exams as we speak. (sigh.) It's the last road block between me and a real vacation...

Doodle #3 and Rotating Calipers

noreply@blogger.com (Mimi) — Tue, 20 Dec 2011 15:31:00 +0000

I got a sketchpad and am fooling around on it, practicing sketching motions and emotions. Motions are easy, emotions are hard. So far, this is my favorite from the motions sketches:

If you're looking for some mathy updates, perhaps you should consider reading Geoff's very technical blog. He recently put up an implementation of the Rotating Calipers algorithm for finding the minimum bounding rectangle around any polygon. (The algorithm finds the best rotated rectangle, not just the best right-side-up rectangle, which would have been too easy.) His tech blog is super dry like the stock market books that he reads in his leisure time, but he likes it that way. :) He says he just wants to dump information on the internet to facilitate other people who might come across the same issues, so reader-friendliness isn't one of his main concerns.

Anyway, if you read through his implementation, you'd see that there's a lot of vectors math in there. It's neat... He's a computer programmer who actually uses high-school math on a regular basis to solve problems!

Doodle #2

noreply@blogger.com (Mimi) — Sun, 18 Dec 2011 23:03:00 +0000

Here's a second doodle I did today (you can click on it to see the full-sized version). My pen ran out of ink near the end, so I think this'll be the last sketch in a while and I'll go back to reading during my spare time.

I threw out the previous one (of the dancer) after I took a photo, since I had made it on scrap paper and there were other things on the back side. I think this one I'll keep. I didn't do such a good job on their faces, but I still liked the overall feel of the piece.

Ah, winter break...

noreply@blogger.com (Mimi) — Sun, 18 Dec 2011 14:44:00 +0000

I miss drawing. I think this Christmas, I'm going to buy Geoff and myself some charcoal and drawing pad, so that we can be drawing hippies on Saturday mornings. I made this today because I was bored and the art store was closed. It's based loosely on this picture, but of course I messed up on the arms since I was drawing with an ink pen (one I normally use for grading) and I hadn't made anything in years.

Hello, winter break. :) It's only Day 2, and we have already: partied a good bit, finished hanging all kinds of things up in our apartment (Geoff measured/built three art frames from scratch and stapled the canvasses to them! It was awesome watching him sawing and banging things together while I laid back, ate chocolate, and watched TV... but I'm extra happy that finally all of our Salvadoran art is hung up), and we rode our bikes today to eat yummy breakfast in the park. In about 5 days we will be off to Turkey. I can't wait!!!

PS. Geoff's parents sent us chilled champagne in the mail. Two bottles. I managed to convince Geoff to immediately crack open one bottle upon receiving them, because hey -- how often in your life would you get chilled champagne in the mail?! That seems like as good an occasion as any to enjoy them. :)

PPS. Our Christmas tree/bush is coming along. It's crooked and small (that's what she said?), but it's filled with holiday spirit! :) I am so excited about our first jointly owned Christmas tree ever!! (Last two years we lived in the tropics, and before that we each lived separately in NYC.)

Anyway, I hope your holiday spirits are bright. Setting up the Christmas tree made me all sorts of sentimental. It was the first time I had actually set one up without my sister around (even though it has been 7 or so years since we've spent Christmas together). Made me miss her extra much.

PPPS. You know that Australian claymation movie Mary and Max? Please watch it. It's phenomenal (although not really suitable for children) and made me both laugh and cry.

Holiday Geometry Activities

noreply@blogger.com (Mimi) — Thu, 15 Dec 2011 21:38:00 +0000

Today was the last full day before Christmas break, and another teacher and I had talked about gathering up both of our classes to do some fun holiday geometry. In the end, the other teacher was absent this week and then very busy when they returned, so they trusted me to just plan the session by myself.

This is how I structured it.

First, we did this snowflake prediction activity in partners. Everybody folded the papers together for the first snowflake, drew out their predictions, and cut it out. Then, I monitored that each pair of partners finished predicting the next two before I gave them each one piece of paper to have them cut out a snowflake to verify their predictions. (They split up what they would cut up, both to save time and save paper.) Then they proceeded to make more predictions, followed by more testing by cutting out snowflakes. This took about 40 minutes. Meanwhile, both the other teacher and I circulated to make sure that kids were understanding how to apply the idea of symmetry to making appropriate predictions.

Then, with the remaining 40 minutes, the kids got to choose between either doing a tetrahedron origami (mostly unassisted; the exercise was in reading and deciphering diagrammed instructions... the hardest part is reading the earlier instructions on how to create the regular hexagons out of a rectangular sheet of paper), or making a geometric sequence/recursive pattern (see below).

This second activity, by the way, was one that I learned at PCMI. :) You keep cutting each segment into smaller thirds (or any fixed fraction), and folding up the middle part. In the end, you end up with a very intricate design. I'll post a photo when I get a chance!

It was awesome! We wrapped up the class by talking a bit about the rotational symmetry of the tetrahedron and also about why we could fold hexagons up into tetrahedrons (same base shape, the equilateral triangle). It was a lovely way to inject some last-minute holiday cheers after all the heavy-duty algebra we had been doing.

Happy holidays!

Being the New Kid On the Block

noreply@blogger.com (Mimi) — Wed, 14 Dec 2011 18:04:00 +0000

I have to admit that my professional transition from El Salvador to Berlin has not been a warm-and-fuzzy one. At my old school, I had built a reputation with my students and their families, and their parents would regularly come up to tell me how much they appreciated their kids being in my class and how they wished that I could have stayed to teach their younger sons and daughters. At the end of my first year, an entire group of kids went to talk to the administration to request for me to move up with them to the next grade. Former students would come back to visit me, and even now they are sending me emails to let me know where they are off to next year for college.

When you move to a new school, for better or worse, you start off anew. You leave whatever reputation you have built up behind you -- the respect that you have gained from your colleagues and the administration, the affection from kids whom you've known over multiple years. Most of all, you leave behind the trust of your students and their families. When you move schools, you start again at the bottom of the totem pole and have to prove yourself every step of the way, to everyone who might be watching.

In my case, I took on the slower-paced classes in two grades this year, because 1. I didn't really mind, I enjoy teaching things at a manageable speed for the kids, 2. I wanted to make sure the kids at the bottom would get the extra attention/support that they needed. Well, in those two grades, I have had various resistance from a few kids who feel that they should belong to a faster-paced group. Those kids care not for fun learning or meaningful tasks; they just want to move along faster through the topics. How do you convince these kids that conceptual development is worth taking the time to get right?? I refuse to short-change their conceptual foundation in order to speed through the topics, and I don't believe that it is good for their mathematical growth in the long run, or good for their problem-solving abilities. (Case in point, one of those kids moved up to a faster-paced group on a trial basis, and went from getting 100% to getting 25% on assessments.)

ARGH! It is frustrating to feel like the new kid on the block. ...I know that being new doesn't mean I am less qualified to teach these kids, or that I'm making bad decisions for their learning. But it does mean that what I value carries a lot less currency around here, as far as my kids are concerned. sigh.

Christmas break cannot come soon enough. The last week has been fairly demoralizing, and I don't have a lot of umph left in me before the holidays. :(

Updates from Deutschland

noreply@blogger.com (Mimi) — Sat, 10 Dec 2011 13:46:00 +0000

I thought I'd take a minute and do some life updates. Time is flying by!

Things are going smoothly over in Berlin. I had put in a lot of work (ie. over 10-hour work days daily) from August to November to learn the new curricula and to earn the trust of my colleagues and student parents, and finally I was ready to re-focus on what I needed for myself. So, in a non-trivial gesture, I'm back doing yoga on a weekly basis and am LOVING it more than ever. (My new yoga teacher is really tough, but I love him!) I've also arranged for a private German teacher to work with me starting in the new year, since I feel like I cannot commit to 6 hours of classes like Geoff does during the work week. Socially, things are good and we've met a lot of fun people. :) Overall, despite the weather getting colder and the days getting darker, we are enjoying our first winter in Germany!

Recently, we got some free tickets to check out a Christmas market outside of Berlin, so we took the local train there last Sunday. It turned out to be a fairly small market, but the town was very charming!

The town had a lot of old people on bikes. I did not see a single young person riding a bike that day. I told Geoff that some day, we're going to retire to towns with old people on bikes. :) The town also had statues that looked like they were from old Grimmes' tales.

There were gondolas ferrying people back and forth between the two sides of the Christmas market. The ride was long and a bit chilly, but we had blankets and people were drinking hot mulled wine ("gluhwein"), which is common at Christmas markets and really all around Germany at this time of year.

We also saw a for-rent sign next to this cute little barrel of a room. It's even smaller than our apartments in NYC!!

Geoff took a photo of some locals moving a tractor via two gondolas.

All in all, it was a lovely day away from the city, and a much needed break from all of the stress I had been feeling from nearing the big semester exams. :)

I am looking forward to checking out some of Berlin's very own Christmas markets this weekend. Gluhwein, here we come!

Math in Psychology

noreply@blogger.com (Mimi) — Sat, 03 Dec 2011 12:55:00 +0000

Since my Kindle got fixed mid last week, I've been reading a rather delightful book called Thinking, Fast and Slow. I've read books like this one before, on the clever psychological experiments that people have managed to design over the years and what they reveal about the human mind. But this book, in particular, pulls together a lot of interesting bits that I've either read or heard over the years and organizes them into one cohesive and surprisingly elegant theory. (I won't spoil the book for you here, but it's worth looking into. The theory is elegant and seemingly simple, but the details are very interesting and not always so predictable or obvious.)

I am also surprised by how mathematical the author is and how easily he ties abstract math ideas into concrete experiments. For this, I highly recommend the book to math teachers. For example, the author talks about how if you get one person to look at a jar of pennies to guess at how much money is inside, that person might over- or under- estimate by a lot. But then, if you repeat the experiment with a large number of people, their average errors will actually approach zero (in the absence of a systematic bias), and therefore if you average all of their guesses, that average is actually going to be quite accurate. This is a logical idea that kids can grasp, and it's a nice extension of the absolute-value error concept. By the same token, he ties this in general to public opinions. If you survey a large enough sample population on a certain issue, in the absence of a systematic bias, the average of their answers will represent the truth.

Another issue that the author addresses is basic numeracy when reading current event reports or statistics in the media. He illustrates with a simple picking-colored-balls-out-of-a-box example why, with small sample groups, we end up with more extreme values more often. And then he extends this to why when you poll different counties for health information, it's easy to see rural counties with more extreme health statistics. Again, it's not impressive math, but the ease with which he ties math to something real is delightful.

And, as an aside, try to answer this question:

"How many animals of each type did Moses bring into the ark?"*

If you're like me and (the author so says) most people, you let your mental image of the ark prevent you from noticing that Moses is the wrong biblical character here in this context. Our mind has the tendency to smooth over the little inconsistencies using preconstructed expectations, in order to make its job easier. And that's both advantageous and troublesome, depending on the context.

Anyway, so far, I thoroughly enjoy the book! :)

Addendum 12/07/11: I've reached a part of the book where the author talks about how we let our stereotypes affect our judgment of the likelihood of certain combined events. For example, after being exposed to a description to a liberal woman, people -- even those who are mathematically inclined -- would rank the probability of her being a "feminist banker" to be more likely than her being a banker, even though any added details should diminish the overall probability! --What an interesting intersection of math and psychology!!

My Favorite Student

noreply@blogger.com (Mimi) — Tue, 29 Nov 2011 18:36:00 +0000

My favorite student (this year) is someone who is not particularly quick at picking up math concepts. After she got the first test back, she cried. She came to see me everyday during lunch to go over a section of the test at a time. We did the corrections together, and then when she went home, she re-did the problems at home that we had worked on together earlier that day. It took us several days just to go through the entire test. And then, because she had worked so hard, I offered to give her a re-test not for grade, but just to see if she now knew the material. On the re-test, she made no conceptual mistakes, and only some arithmetic errors. When she went home and did corrections again (this time, unprompted), she thought the corrections were very easy.

This kid works so hard that when I announced that we were having a big exam in December, she came to me within a couple of days to start working on the topics that will be included. We sat down and made flashcards for strategies while approaching problems, and we did some practice problems and put them also on the flashcards. She has been reviewing the flashcards since at home, and she says that they are very helpful as a starting point to solving problems.

This favorite student of mine came to me yesterday after school to ask me to teach her long division, because all of her friends are able to divide 3 digit numbers by 2 digit numbers and she couldn't remember how to divide. I was running a fever when she came by after school yesterday and my body ached all over and I longed to go home, but hearing her question warmed my heart. Big time. She is my favorite kid because she isn't embarrassed to get help, and I know that some day she will master all of the skills that she lacks, if she can keep up that amazing spirit of hers.

That's my favorite type of student. Over the years, I have seen other students like her grow into excellent math students, who can connect the dots faster than anyone else and to work quickly through complex scenarios of problems. As a teacher, may I always remember to appreciate when a kid has the willingness for hard work, regardless of what they currently achieve.

Introducing Limits and Derivatives

noreply@blogger.com (Mimi) — Sat, 26 Nov 2011 08:17:00 +0000

I know you non-IB people probably cannot fathom this, but Calculus is only one of many units in the IB curriculum. I am introducing it to my kids now, and it's the last big topic I plan on teaching before we start to do mixed review in the spring time.

The reason why Calculus is condensed into a single unit in the IB makes sense to me, actually. Although there are so many applications of Calculus, I would be just as happy if a kid can walk away from Calculus knowing the big ideas of limits, differentiation, and integration, and to be able to do basic differentiation/integration of polynomials by hand without a formula sheet. Everything else, I'm OK with them relying on a formula sheet in order to remember the mechanics. So, it's very do-able as a single unit and to still develop the relevant concepts as a class.

To that end, I have already introduced instantaneous rates vs. average rates. Then, after that I wanted to pull in the idea of limits, which the kids had already seen a little bit of in the context of geometric series. So, I gave the kids this worksheet, and as they worked through it, we set up a huge grid of comparison charts on the board to go over when a "forbidden x value" inside a rational function will be a vertical asymptote, versus a "hole".... and to highlight the idea that a limit is what happens to the theoretical output as you approach those "impossible" x values. We are not done with the worksheet yet, but my hope is that by the start of the next class, the kids will fully grasp the idea that a function with a "hole" somewhere (as opposed to a vertical asymptote) can still have a limit at that breaking point, and many functions also have limits at extreme values of x. That (along with the previous instantaneous rate intro) will prime us for going into talking about the mechanics of differentiation in this next intro to differentiation lesson. One of the things I want to immediately tie into differentiation is that you can check your sensibility of your answers using a graph. I will also right away tie differentiation to the algebraic meaning of turning points. This way, they are immediately exposed to the key concepts in an integrated manner before we do any further practice.

Thoughts?? It's my first time teaching Calculus, so I'm still muddling my way through it while doing my best to sequence the concepts clearly. Your feedback is welcome!