On Being Me

For some reason, 2012 has been a year of much personal growth for me. Now that I'm finally on Christmas break, I thought I'd take a moment to just reflect on this a bit. I don't know that I had set out at the beginning of the year hoping to be a better -- or even different -- person, but I feel distinctly different from the person that I was at the start of this year, and definitely more comfortable in my own skin.

Looking back, traveling by myself over the summer was one of the best decisions I had ever made. It really gave me clarity and room to think about who I was and (very importantly) to learn to listen to other people's stories and concerns with no distractions on my mind. I learned to stay in the moment and to really value my travel companions for the stories that they had to tell about who they were and what brought them to that place, at that time. Most of those people I had met, I have no doubt that I'll never see again, but it made me feel very fortunate to have made the short connection with them in that small slice of space-time continuum. There's something very zen about meeting strangers, caring about them, and wishing the best for them without future expectations.

When Geoff and I started planning our wedding this year, there were a few dear friends who I had always envisioned would be at my wedding, who responded and said that for one reason or another, they would not be able to make it. At first, I was feeling pretty hurt about this, but soon I came around and realized that the last thing that I would want was for our wedding to be an imposition upon our friends. I think it is very easy when you plan for such a big day in your life, to forget that other people have other priorities -- families, school, etc. I was glad to have caught myself feeling that way. Instead of dwelling upon who cannot make it to our wedding, Geoff and I will simply cherish the fact that so many of our friends and family will be able to make it all the way to Belize and to spend several days with us during this joyous time in our lives!! :)

Speaking of friends, 2012 was a good year for taking some risks and hanging out with new people, for me anyway. It always takes me a bit of time to feel comfortable in a new place, before I can reach out and hang out with people one-on-one. But, once I get to that point, that's how I know that I'm really settled in to a place. This year has been a very busy one with getting to know people better and spending quality time with them outside of work. Once I had developed that personal relationship with some of my coworkers, it made working at the school a totally different experience (as it always does).

Then, towards the end of 2012 I had two separate people in my professional relationships (one a colleague and one a student parent) send me a series of very accusatory, very blatantly condescending emails. Both times, their emails really got under my skin, and I had to expend hours of energy in restraining my own responses, so that what came out was extremely polite but still firm. In both cases, the people on the other side ended up realizing eventually that they were being unreasonable, and actually became very thankful for my even-keeled handling of the whole situation. One person, following a series of making-it-up-to-me gestures, actually sat down next to me at a party recently and chatted to me for 30 minutes about their baby. Afterwards, when I walked out of that party, I felt so euphoric not because I really care or not care about this person, but because it feels so great to really forgive someone and to let what had happened to be water under the bridge -- something that would not have been possible, had I decided to reciprocate with the same tones that those emails had been written or if I could not let it go on my mind and had continued to act awkwardly around this person. It reminded me of the saying that, on your deathbed, you will never wish to have forgiven fewer people in your life. It sounds super corny, but I think it's so true that when you forgive someone else for their offenses, you're releasing yourself from the anger. (And, tied to this is the issue of general integrity. If you don't have integrity and faith in what you do, then it is extremely difficult to hold your temper when others are coming at you with extremely ill-intentioned accusations.)

Lastly, Geoff and I have been trying to make some big decisions for 2013, and it's gotten us thinking about all kinds of things. In the end, I think my priorities are clear, and we are just waiting for a bit more information before we make our decisions. This is the first time we're really making decisions as a family, and thinking ahead about what things will be like in a few years when we do have a kid. So, my whole frame of mind is different, and that's an interesting -- and nice -- place to be. It also got me thinking about all the reasons why I love Geoff. In the end, when all the other things fall away, his character deep down is so aligned with my own, that we can understand each other even without saying anything. And that is amazing.

How to Keep the Kids on Your Side (While Addressing Misbehavior)

Sometimes I see teachers get into confrontational conversations with students, where the tension escalates very fast for no reason because the child is off-task first and the teacher reacts in a way that makes the child feel "picked on." In my experience, you can avoid this kind of thing by just reframing your questions to be about the work. I often teach very active, social kids, and I find that the way I phrase my questions to them often keeps them calm, focused during the class instead of letting things spiral out of control.

For example, when I am helping a kid in class and I can see a kid all the way across the room to start to get distracted, I immediately say, "[So and so], are you talking because you are completely done with this task?" I haven't yet told the child off about the fact that they're misbehaving, but I am probing into why they're doing this. I think almost always, the child responds with one of these:

1. "No... but almost!" and turns back around to resume the task.

2. "Yes! I'm finished!" in which case I quickly walk over, give them another task, and tell them either that I'll very soon come back to check on their finished work, or I point them at a person whose work I've already checked to check their answers against. The latter usually encourages peer discussion of differences and errors.

3. "No, but I need your help!" and depending on the nature of the task and how many kids are in queue waiting for my help, I'd either direct them to asking another kid for help (again encouraging peer interaction), or I say that I'll be right there with them.

(4. Very occasionally, I get a smart-alecky response like, "Yes!" when they're clearly not finished. When that happens, I always immediately ask, "So you're saying that I should collect it from you now, grade it, and move you along to the next assignment?" to which the cheeky child always mumbles, "No...")

So, I find that asking the right question when you notice a kid is talking / off-task in class is helpful in putting the kid back on track by addressing their need or reminding them gently that our time in class is purposeful.

Similarly, after a few warnings, when I decide that I do need to move a kid's seat or to put them outside the room altogether, when I approach them, I say, "I need you to move now so that I can help you focus on this task." This way, even if they feel somewhat punished, they can hear in the back of their minds what I am saying about using it as a tool to help them learn. (At the point when I do decide to move a child though, I don't allow negotiations. They cannot still try to haggle that they'll be on task "starting now.") If a child is moved outside of the room, I tell them that they can come in to ask me a question, and if I get a chance I'd pop my head out to ask if they're still doing OK. And then, after class I always tell them that I wish I didn't have to move them, that I wish they could just learn to focus on their own -- so that the child can see that I am really on their side. I make the conversation always about their learning, not about how many times I had to ask them to get on task.

When a kid exhibits continuous, repeating instances of disruptive behavior in one class (or over the period of a few classes), then when I communicate with the kid and the parents afterwards about this (which has to be immediate and firm), I still make the conversation to be about their learning. Either the child is not keeping up with the content, and I say that "I am concerned that this type of behavior is actually really damaging their learning, [with specific examples]..." or if the child is advanced compared to their peers, then I say that "this type of behavior is damaging our learning community and therefore is not acceptable." Frame your observations in a way that is impersonal. It's not about you or the child. It's about how their actions impact the task or the learning -- either their own or that of their peers. When you communicate this way, you're helping parents and children see why they need to improve, instead of just saying, "You cannot and should not be disruptive or disrespectful."

For some kids, the change is very gradual and it can take a whole year for them to learn to control themselves. But in the mean time, VERY importantly, your relationship with the kid will not be damaged by this type of confrontation. If the kid likes your lessons and likes the way you run the class, they will slowly develop respect for you, and the confrontations will thereby decrease in frequency. But, in the mean time, do not create additional obstacles for the kid and for yourself by being overly confrontational without also being explanatory... If each time the child walks into your classroom they are already antagonizing you and your approach to discipline, then no matter how great your lessons are, you're going to have a hard time in trying to win the child over.

Just some quick thoughts about discipline. Many teachers in private schools, I find, don't discipline as much as they need to, and that's a problem as well. A kid always needs structure, and as a teacher it is our duty to help them learn to be more focused, or to point out when they are not focused by assigning specific consequences like giving them gentle cues during class, talking to them after class, moving their seats, putting them outside, or contacting their parents. If you don't follow up immediately with some action, even if your lessons are terrific, kids cannot really respect your authority or they're being distracted by their less well-behaving peers, and their learning will therefore be affected. 

Ken Ken and Classroom Problem-Solving

I randomly copied two 7-by-7 KenKen puzzles yesterday from my book of KenKen puzzles edited by Will Shortz. (I ordered it as a teaching resource at the end of last year, and I'm thinking about using it during the last classes in Middle School before Christmas break.) And, I have to say, I had forgotten just how addicting these are! I solved the two puzzles on my way home, and by the time I got home, my hands were freezing from walking outside without gloves and I was nauseating from reading and thinking on a stop-and-go bus. But, I felt exhilarated.

I find that the process of doing KenKen puzzles is quite similar to the process of solving multi-step math problems, because you have to constantly switch back and forth between processing logical relationships / self-monitoring reasonableness of an answer, and doing arithmetic calculations (such as factoring 120 into 3 numbers that are all in the correct range of 1 to 7).

I am too lazy to copy the puzzles I did yesterday, but for those of you not familiar with KenKen puzzles, I highly recommend checking out these from NY Times . Even the small-sized grids are fun because they are not nearly as repetitive as Sudokus are. So, if you're thinking of ideas of stocking stuffers for your favorite math geeks...

What just MIGHT be (for me) the Secret of Teaching Logarithms

I have been teaching logarithm for a few years now. Each year, no matter how I approach it and how exploratory I make the whole thing to be, I find that my students are fairly unsuccessful at putting everything together, and they always get confused at some point. Last year, I finally had the idea of going back to basic definitions. The whole problem, I think, with kids getting confused with logs all the time is because they simply cannot remember, in the end, what the hell log even means after I make them derive all those rules. So, this year, I started with the definition very firmly, and every time the kids are doing a new problem, I repeat the hell out of that definition until they want to rip me into pieces. And, guess what! I don't care if they want to rip me up. It has worked like a charm. NO ONE is getting confused yet this year by the notion of logs. (I've skipped the exploratory stuff this year, in order to really keep their focus on what's important.)

This is the definition I taught them:

Log is just a way to ask a specific question.
log_a(b) asks the question: "What exponent is required to go from a base of a in order to reach a value of b?"

That's IT! We go over that with an example.

For example,
log₂(8) means "What exponent is required to go from a base of 2 to reach a value of 8?"
So, log₂(8) = ??

The kids said, "3!" (...OK, maybe first they said 4. I cannot remember now. But anyhow, they understood why it would be 3. Either they self-corrected or I corrected them.)

Then, we did some more simple numerical examples, as you always would do before kids start to get confused with logs. In each case, instead of just letting them be robots and following the previous numerical pattern mindlessly, I slowed them down and hammered into them the meaning of log. They had to say it OUT LOUD for every example:

log₃(81) means "What exponent is required to go from a base of 3 to a value of 81?" and that's why it's 4.
log₅(5)  means "What exponent is required to go from a base of 5 to a value of 5?" and that's why it's 1.
log₄(16) means "What exponent is required to go from a base of 4 to a value of 16?" and that's why it's 2.

etc. And then we went over the change of base formula, log_a(b) = log(b)/log(a). I am sorry, but I didn't try to make them discover it this year. Derivation is nice if the kids are already getting the basic concept, but else it obfuscates what's already a fairly tricky topic for a majority of kids. We practiced finding some decimal log results using the calculator, and testing them (as exponents) to make sure that they did give approximately the correct values that we desired, starting from the base.


And then we jumped right into solving equations! And the kids did brilliantly. I didn't even make a worksheet, I just started writing things on the board, a couple of simple problems at a time. Each time they got stuck, I just said, "Go back to your definition. What question does log help us ask? How can we use that?"

Each time they worked on a new type of problem and they needed help, they had to laboriously say out loud what the question is that log is asking. "What exponent is required to go from base of ___ to reach a value of ___?" and they then had to identify, based on the equation given, whether that question being posed had already been answered or not. Once they said all of this out loud, they were able to figure out on their own what x was fairly easily, without any help from me.

3^x =10  --> "What exponent is required to go from base 3 to reach a value of 10? That hasn't been answered yet." so, log is going to help us ask that of the calculator: log₃(10) = x

log₄(x) = 3  --> "What exponent is required to go from base 4 to reach a value of x? That has been answered already, 3." So, 4³ = x.

log_x(36) = 2  --> "What exponent is required to go from base x to reach a value of 36? That has been answered already, 2." So, x² = 36. For this one, it led us into a brief discussion of why x could not be -6, and of limitations on log inputs.

I was really shocked by how well the kids received this. I even tried after a few problems to introduce to them the memory trick from Amy Gruen, and they looked at me like, "Why would we need this?" (which I can assure you, was not the response I had gotten in the previous year.) I really, truly believe that going back to the definition of logs is the way to teach this often confusing concept.

Shortly after, they were able to do problems such as:

log₅(1/5) = ??

log₇(7^k) = ??  --> "log asks the question, what exponent is required to go from a base of 7 to reach a value of 7^k? The answer is, well, k!"

log₇(7^2n-3) = ??

So, being very pleased by their ability to recite and apply log definition, I started to put up some questions of multiple-step equations on the board, again just to let the kids try them first. (They needed a bit of hints only in the beginning, but for the most part they were pretty OK doing them by themselves.)

2*5^x =80  --> here was my hint. "Well, log does NOT ask the question, what exponent is required so that when I raise the base of 5 to it AND THEN MULTIPLY BY 2, the final value is 80. So, clearly the 2 here is a bit problematic..." and therefore the kids figured out that it needs to go away first.

-4^x =-73 --> here I helped them visualize order of operations by circling the x with the 4, and then circling the negative sign on an outside layer. I use this 7th-grade trick now even with my 11th and 12th graders to help them visualize how to peel away layers of the onion when solving for something.

3*6^x - 7 = 20

10^2x-9 =1098

So, this was all things that happened during our previous class. Today, after they returned, they were still very successful at transferring the log definition onto more complicated equations such as:

 6^x = 36^x-3     (which I realize, yes, they can easily solve in the future as a "change of base" problem, but since we're on the topic of introducing logs, I just wanted them to see how to apply the log definition to this problem.)

So, this is the question they decided to ask: "What exponent is required in order to go from a base of 6 to a value of  36^x-3 ?"  And they decided that the answer to that question has already been provided, as x.

So, log₆(36^x-3 ) = x

Now they apply a simple log rule of dropping the exponents in the front, which makes:

(x - 3) log₆(36) = x

And clearly since they know what log means, they can immediately simplify it now as:
(x - 3)(2) = x

and then just solve the rest as a linear equation. Tada!

Easy breezy. I'm going to always teach logs using definitions from now on. My little logarithm ninjas can even solve exponential equations for x in terms of other variables, and they can also tell me that log₆(6^{m^3}) should equal m³. YEAH. Not bad for being only two days into logs, I'd say.

If kids understand the definition of logs as something that asks a certain question, then down the road they won't be so confused when we discuss that 2^log2(k) = k, because the log part simply asks the right question, and the rest of the expression actually CARRIES OUT the instruction implied by that question. I find that when the situation looks complicated, I always go back to thinking about the definition of log in my own head. So, I have every reason to be hopeful that my kids, with consistent reinforcement from me, will create the same frame of reference in their little heads.

Zero and Negative Zero

Today a random 11th-grader asked me what the difference is between zero and negative zero. He sounded so sure that there was a difference, that for a fraction of a second I had to double-check my entire logical understanding of numbers before answering him. I asked him what the difference is between -2 and 2, and he said that they're "...of course different. They are opposites!" And I asked him what the '2' means. He said, "Well, they are both 2 units from the 'center'." So, "OK," I said. "And -0 and 0 are both how many units away from the 'center'?" He said, "Zero units, but in opposite directions!" We went back and forth like this for a bit until I convinced him that -0 and 0 are the same using the number line as a reference frame.

Then, my colleague comes in and this student asks the same question to him. I didn't say anything because I was a bit curious what my colleague would say. My colleague invoked reflection on the number line to explain geometrically why 0 and -0 are the same, which is exactly the same explanation that I had given! Afterwards, we were both amused. My colleague says, "You see, there are some things that all math teachers can agree upon."

But, we are both people who are comfortable reasoning through number lines and relationships of numbers. In the end, we are able to provide an answer to the kid that is logically sound and coherent with other concepts that the kid knows and understands. The same question could easily have come up in a classroom (for example, of younger children perhaps) where the teacher is multiple subject-certified and perhaps not quite as comfortable with mathematics as they are with other subjects. In that case, what systems can we put in place in order to support those teachers to answering conceptual questions such as this? (I realize that elementary-school age children do not typically learn about negative numbers, but the same types of innocent questions can still very easily arise, with other math topics that they do learn about.) We want to encourage questioning and robust reasoning in mathematics, and that mode of thinking should be instilled starting at a young age. What can I do as a math department head of a K-12 school, in order to ensure of this and to help all teachers feel equipped to answer conceptual questions from curious learners?*

*For example, the art department head of our school regularly models lessons in the elementary school, in order to show the teachers how to deliver art lessons using the same general approach as in the middle- and high- school. But, I don't feel confident that I can manage young children or that I would be equipped to explain concepts at their young comprehension level. So, if that is not an option, then what is??

What does your school have in place in order to support vertical alignment and conceptual development at different ages, not just on paper but in tangible terms?

An "Understanding" Rubric for the Semester

I am reading parts of John Hattie's Visible Learning for Teachers, which has some real gems. Since this book reads dense like a textbook, I find that it is the most enticing when I flip through and just randomly stumble upon sections that are appealing to my wandering mind.

One part of the book talks about what successful feedback looks like.

The criteria for evaluating any learning achievements must be made transparent to students to enable them to have a clear overview of the aims of their work and what it means to complete it successfully.

Students should be encouraged to bear in mind the aims of their work and to assess their own progress to meet these aims as they proceed. They will then be able to guide their own work and so become independent learners. 

I think this is something that I can improve on. I give a lot of verbal feedback daily, and then a lot of individual written feedback when looking at a kid's work (ie. quizzes or homework or projects, depending on the class and the time of year). But, I am not so sure that the kids always have the big picture in mind. So, to that end, to help my 8th-graders frame their minds around what I will be looking for on their semester exam next week in terms of levels of mastery of the learned topics, I've pulled together this hopefully kid-friendly rubric/skills list which I'll go over with the kids prior to the exam. What do you think?? It's a skills list like many of you might have for SBG, but it's sort of hierarchical in terms of ordering what I think are more basic skills and what I think are more complex skills layered on top. Mimicking the MYP scale for "Knowledge and Understanding", the rubric goes up to 8 points in most topics, where a 5 is basically solidly at-grade-level "average" performance, and 7 and 8 are only if you are beginning to approach more advanced topics or skills that are somewhat beyond your grade level.

My goal is that I will go over this rubric with the students this week, and they will self assess which skills they still need to work on over the weekend, so that their efforts are not so randomly scattered during precious review time. In January, after the exams have been graded, they will again self-assess in order to figure out where their skills holes are from the first semester, if any. This type of self-analysis will help them gain independence as a more self-driven learner over time, instead of me always telling them what to work on next and feeling like I fall into the rather unappealing nagging mode.

Sounds like a plan?

Review Week

It's so nice to just slow down before the semester test and to review everything we have learned. It's like taking in a deep breath, because finally we are no longer trying to cover as much ground as possible. Finally the kids have come to a reasonable stopping point before THE TEST next week.

So, this week we will "just" do some review in Grades 7 and 8. For that, up my sleeve I have:
this awesome stations review format from Amy Gruen; my normal speed game format (which I may vary up this time to have 3 kids go up at a time to the board, working individually and not keeping team score), and of course mini whiteboards.

Looking forward to the rest of this week! :) There is nothing like fun with review time (coupled with the kids doing extra review problems at home, obviously).

11 Recommendations to Middle-ish Grades Math Teachers

Maybe some of these are "radical" and "offensive", or maybe they're not. I'm just throwing them out there. Please don't hate me if you do some of these (I know a couple of them are quite commonly done by many math teachers). These are just my personal opinions, but I feel quite strongly about them. I figured I'd say it here, because some of these things drive me nuts, and I need an outlet.

11. Do not tell a kid to "move things across the equal sign and just change the signs". Preferably you don't ever say this phrase to a kid ever, but if a kid comes to you and says that someone else somewhere (ie. a parent or another teacher) has taught that to them, you need to drill into them the reason why this shortcut works. Grill them until they can articulate why the term necessarily shows up on the other side with the opposite sign.

Reason: If you teach shortcuts like this without the correct foundation of understanding, then down the road the kid cannot approach a complex situation intuitively, because they cannot visualize the equation as a balance.

10. Do not allow your students to do simple equations only in their heads / showing little or no work at the early stages.

Reason: Even if the kid is bright and they can do the equation correctly every time, you are helping them to develop a bad habit that is hard to break. Down the road, when the equations get more complicated and the kid starts to make procedural mistakes, they'll have such a hard time finding their mistakes because they've habitually skipped so many steps.

9. Do not allow your students to "open parentheses" without knowing why they do this and where it is the most useful.

Reason: Normally, it just irks me to see kids do 2(3 - 5 + 1) = 6 - 10 + 2 = -2 instead of following PEMDAS to do 2(3 - 5 + 1) = 2(-1) = -2, but it's really feckin' scary when I see students do
2 (4·5 + 7) = 8 + 10 + 14 = 32 in their second year of algebra. When all the values are known, they don't need to be distributing!!

8. Do not introduce integer operations without explaining the meaning of addition and subtraction of signed numbers! You can use number lines or you can use the idea of counting and neutralizing terms, but in the end, the kids need a framework of understanding.

Reason: I recently tutored a kid on a one-time basis (I was doing a favor for her parent, who's my friend, even though the kid doesn't go to our school), and the kid only knows memorized rules (she rattled them off like a poem), with no conceptual understanding whatsoever when I probed further. Unfortunately, that kind of problem is not really fixable in a day. It can take a teacher weeks to instill the correct habit of thinking. If you teach a kid to memorize a rule for adding or subtracting integers without understanding, then I can almost guarantee you that after the summer they won't remember it. And, if you first teach the rules by memorization, even if you try to explain afterwards the conceptual reasons for the rules, there ain't no one listening.

7. Do not teach "rise over run".

Reason: In my experience, a majority of the concrete-thinking kids out there do not know what the slope means. When you ask them, they just enthusiastically say "rise over run", but they cannot identify it in a table, they cannot find it in a word problem, they certainly cannot say that it's change of y over change of x, and half of them cannot even see it in a graph.

You have other alternatives. I teach slope as "what happens (with the quantity we care about), over how long it takes." For example, the speed of a car is "miles, per hour"; the rate of growth of a plant is "inches, per number of days"; the rate of decrease of savings can be "dollars, per number of weeks". You have alternatives to "rise over run". Learn to teach kids to visualize the coordinate plane as a sort of timeline, where the y-axis describes the interesting values we are observing, and x-axis describes the stages at which those values occur. Once they understand this, then they can correctly pick out the slope given any representation of the function.

6. Do not let any kid in your class get away with saying "A linear function is something that is
y = mx + b."

Reason: Again, in my experience, I can habitually meet a full class of kids who had spent a good part of a previous year learning about lines and linear equations and linear operations, and in the following year when they move to my class and I ask them to tell me simply what is a linear function, the only thing they can say is: Y = mx + b. That is really, really bad. The first thing they should be able to say is that it is a straight line, or that it is a pattern with a constant rate.

Please, please, do teach lines in context. Make sure that kids know what y, m, x, and b all mean in a simple linear situation.

5. Do not introduce sine, cosine, and tangent without explaining their relationship to similar triangles.

Reason: The concept is too abstract, too much of a jump from anything else the kids have done. Give the kids some shadow problems to allow them to see what the ancient mathematicians saw, and why they recorded trigonometric ratios in a chart. Let the 3 ratios arise as an outcome of a natural discovery, a natural need. And then, when you introduce the terms sine, cosine, and tangent, the kids will at least already understand them as sweet nicknames for comparisons between sides.

From then on, whenever you say sin(x) = 0.1283... , remind kids what that means simply by immediately saying, "So the opposite side is about 12.8% as long as the hypotenuse." If you do this consistently, kids will never grow afraid of those ratios.

4. Do not teach right-triangle trigonometry from inside the classroom!

Reason: Kids need to be able to visualize geometry relationships, especially in word problems. Some kids have trouble doing this on a flat sheet of paper. Don't disadvantage those learners. Take them outside, make them build an inclinometer, make them experience angles of elevation, angles of depression, and the idea of measuring heights through triangular ratios. This really only takes a day, or at most two. In the end, when you take them back to the classroom, you will be amazed at how well they can now visualize even complicated word problems involving multiple right triangles.

3. Do distinguish between conceptual and procedural errors. Your modes of intervention should look very different for those two types of mistakes, and what you communicate to the kids as recommended "next steps" should look fairly different as well.

2. Do incorporate writing into your classroom. Writing is an extremely valuable way of learning mathematics. Every project should have a significant writing component. When kids write, they are forced to engage with the subject on a personal level, and so they learn much more.

1. Do make learning fun. From a biological perspective (of primitive survival), our brain makes us remember things permanently when our emotions are triggered. Fun learning isn't a waste of time. It's necessary to help the students build long-term memory!

Ich versuche in Deutsch schreiben...

Meine Deutscheskurse A2 ist gerade fertig! Die naechste Stufe beginnt im Februar. Ich glaube dass, ich meisten einfache Sachen verstehe, aber natuerlich ich muss mehr ueben. Es gibt etwas Worten an den ich kann nie mich erinnere. Und die Regeln und die Praepositionen sind auch (mehr als) ein Bisschen schwer... Aber Deutsch ist immer viel Spass zu lernen. :) Im Dezember und im Januar will ich selbstaendig studieren, um besser im Februar zu sein. Damit ich mehr angenehm mit den Regeln bin, ich moechte jede Woche ein Bisschen in Deutsch schreiben. So! Hier ist meine erste Ubung. Wenn Sie gut Deutsch sprechen, koennen Sie bitte mich korrigieren? Dankeschoen!

Geoff ist weg in den USA, um an einer Messe in Florida teil zu nehmen. Am Sontag fliegt er nach New Jersey um seinen Elterns Haus zu bleiben.  Meine Freundin Gabby ist gerade nach San Francisco zurueckgeflogen. (Wir hatten viel Spass, obwohl ich arbeiten musste.) Fur eine Woche (bis Geoffs zurueckflug nach Berlin) habe ich mehr Zeit zu mehr arbeiten... Und ich hoffe dass, wenn Geoff zurueck kommt, alle Arbeit ist schon fertig!! Naechstes Wochenende sind zwei verschiedenen Christmas-Parties, deshalb ich muss einige Rezepten vorbereiten... Ich will einige Deutsche Brataepfel kochen und zu den Parties mitbringen, und vielleicht ich koche auch etwas Anderen... 

I find that the German teacher I have is really fantastic, because (in my opinion) she's truly a teacher's teacher. I'll just say that three minutes before the class is scheduled to leave for the evening or to stop for a pause, she'd hand out a worksheet and say, "OK! Let's do one last exercise!" instead of letting us off the hook a few minutes early. And she has some really creative activities to get us to talk more in German class, and I think the class is quite dynamic (over the course of the 3 hours each night, twice a week). And sometimes when I take the train home with her, she still makes me speak German to her the entire time, and she corrects me on the train, too. Really a teacher's teacher! I am so happy that she'll be our teacher all the way through the courses (unless I drop out of the rotation at some point because I get too busy). My German still sucks, but it is through no fault of hers. I know that I just need to sit down and get more serious about memorizing and practicing rules, and then I need to find a tandem partner to practice with, in order to get my thinking-while-speaking more up to speed.

Addendum: I find it pretty fun to read back on how my German progress is coming along (however slowly) over time, so I uploaded a sample of my reading of the first paragraph so I can come back and hear myself talking in German at some other point.  Here it is if you're curious! It's a little deceptive because obviously it's easier to read something that's been written than to speak off the cuff. Hopefully at some point I can just talk fluidly without having written it down.

Weeks Before Vacation

So, it's nearing Christmas at the end of a long term of school (the last time we had any day off was in early October). In my "weak" Grade 9 group, kids are starting to lose some of their focus. In my mind, I know that they probably cannot help it, because even I am looking forward to vacation. But well, this is the kind of teacher I am. I said this very calmly and firmly to the 9th-graders in the last 10 minutes of class (after they had been idly, overly casually doing work for about 30 minutes):

"Alright, listen. You guys are dismissed today to go to lunch when you have shown me that your worksheet is complete and that everything is correct. ...No, don't complain. You know, I used to do this a lot at my last school, and those kids used to complain to me all the time about being dismissed late, but then the next year those same kids would write me emails to THANK me for teaching them so much math. So, I'll just tell you in advance: 'You are welcome!!'"

The half of the class that had been on task, laughed at this and packed up on time to go. The other half that had to stay late, mostly finished the worksheets within 5 or so minutes of regular dismissal. Only one kid was mumbling angry comments the whole time (but he gets angry pretty easily and I'm pretty used to it and so I don't take it personally), and then everyone else that had to stay late, actually thanked me on their way out. What sweet kids.

Fun fact: Some of my former students in the Bronx used to call me Yangsta or Yangdizzle. No relation to this story, of course. But these kids definitely don't want to be wasting time in my class, lest the real Yangdizzle come out. *wink*

Proportional Reasoning with Percents

I am a big fan of estimation, probably because I am lazy. I think that most percent problems (obviously not all) can be done with just basic numerical / proportional reasoning.

For example, my Grade 7s are pretty darned good by now at doing something like "find 15% of 6.4". They can articulate that 10% of 6.4 is 0.64, and then half of that is 5% = 0.32, so 15% is 0.64 + 0.32 = 0.96 . They can also do something like "find 12% of 88" by reasoning that 10% is 8.8, and then 1% is 0.88, so 12% is 8.8 + 0.88 + 0.88 = 8.8 + 1.76 = 10.56 . Recently I looked over the big semester test that we had given last year in December, and realized that we won't have time this year before the semester exam to cover enough of proportional reasoning to do a problem like "9 is 15% of what number?" using a setup of proportions and of solving by cross multiplication. (We've only just started proportional reasoning this week.) So, today I just decided to throw up some of those types of questions on the board as the Do Now, and the kids were surprisingly clever at figuring it out! One of them said, "It's not that hard. It's like playing Math Detective."

"3 is 10% of what number?" --> This was the first "backwards" type of question I had put on the board, and the kids thought it was quite easy. You're just multipling by 10 times to get to 100%, which is 30.

"12 is 15% of what number?" --> Some kids naturally, intuitively, broke it down to 5% being 4, which is a "nice" percentage to have/know because then you know that 10% is 8, and therefore 100% is 80.

"6 is 2% of what number?" --> To do this, they were also clever. Some did it as 1% is 3, so 100% must be 300 if you know that one out of the 100 parts is 3. One kid immediately thought that you can scale both values up by 50 times in order to reach 100%, to save us from doing two steps of work.

I really liked these discussions, because I think -- in lack of a formal proportional setup -- this is the most intuitive way of approaching percentages, in my mind. It reinforces the meaning of percents and frees kids from fear of these fairly basic everyday concepts.

We discussed briefly why this is proportional reasoning. It tied very nicely to the word problems we did today on proportions, but I am not going to touch cross multiplication until January. (It's just too much, too soon for many of them, and they have already worked very hard to fill in the gaps this semester.)

So, go seventh-graders! I feel very hopeful that, despite my concerns about many of them at the start of the year, the majority of the class is nearly caught up to where they ought to be at this time of the year.

Thinking About Elementary Math

Recently I've had some questions come my way about what the younger kids need to be able to do in order to be successful in the future years. I am no expert, but I am very interested in getting a discussion started about this from a Middle School perspective.

Specifically:

• I think in grades 1 - 4, the most important skills to develop are obviously basic (up to two-digit) addition, subtraction, and "nice" multiplication and division using the times table. For the young kids, manipulatives are very important in order for them to understand the meaning of these operations. 
• In Germany (and probably other places as well), they use a triangular diagram to teach the idea of inverse operations in elementary school. For example, the diagram below reinforces that 60, the total, can be divided by 5 to get 12, or divided by 12 to get 5. The two bottom operations multiply to form 60. The neat thing about this triangle is that it extends into algebraic relationships such as D = rt, D/r = t, or D/t = r. My lawyer friend who grew up in Israel told me that this is how they learned basic operations in school.
  
• I think that by grade 5, kids should be able to do addition from left to right, in order to build up their estimation skills. For example, adding 638 + 290, yo can look from left to right to get 800... And then when you look just one digit ahead, you can already estimate that 9+3 is bigger than 10, so the result is actually 900 something. 928, to be exact, if you keep adding from left to right, peeking just one digit ahead each time. The nice about this is that even if kids only quickly looked at two numbers being added, they can already estimate the sum reasonably.
• I think that by grade 5, kids should be able to multiplication of two-digit by one-digit numbers in their heads. Teaching kids to break down (82 times 6) into 80 times 6, plus 2 times 6, reinforces two things: placement values and the idea of distributive property within arithmetic. It makes introducing the algebraic distributive property in middle-school a breeze, if kids already have seen it in action.
• Sometime in grade 5 or grade 6, when kids start to learn conversion from decimals into fractions, this should be done using their proper naming of numbers. 5.6 is read properly as "five and six-tenths", and the way we write that in fractions is immediately 5   6/10. Going backwards, they should be able to do the same, at least for base-ten fractions. 3   9/100, is read as "three and nine-hundredths", which writes as 3.09 in decimals.
• There are a lot of resources out there for fractions already, but I think that the most important representation is the number line and the comparison of numbers. To find a fraction of any number, the kids need to know that 1/n is one out of n equal pieces, so k/n just means that size, multiplied by k. I think the concept behind fractions is so so so SO important, so it should always be done in context.
• Dividing by simple fractions can be done similarly using reasoning. I teach my middle-schoolers how to intuitively divide 5 by 1/3 by first asking them what is 1 divided by 1/3. We draw diagrams until everyone can see why it is 3 (and I use language like, "how many times does 1/3 fit into 1?"). And then I ask them what is 5 divided by 1/3. ("How many times does 1/3 fit into 5?") The language that you use with fractions, I think, has an immediate impact on the children's understanding of the operations. Of course, this does not bypass the need to show them the manipulation of fractions in division, but it helps to add meaning to the otherwise rote/abstract operations.
• By the way, the triangle (shown above) can be used to reinforce why 5 divided by 1/3 is 15, and why 5 divided by 15 is 1/3. One of Geoff's friends has a good analogy to cutting potatoes in order to illustrate this. (You can cut 5 potatoes into 15 groups of 1/3 potatoes each, or if you already knew that you wanted to make 1/3 potatoes the size of each group, you can make 15 groups.)
• I recently wrote a short email outlining my recommendations for decimal division in Grade 6, so I'll just paste it here. "I think for decimal division, kids should be able to reason through step-by-step, starting with normal division. For example, to teach 7.2/6, I’d start first with 72/6 = 12, and then ask kids what they think 7.2/6 will be, and then ask them what 7.2/0.6 will be, and then 7.2/0.06, etc. You can use it to introduce this idea of ratios between numbers. Have them practice this on other decimal pairs instead of teaching the rote “moving it over this many times” trick.
  
  
  Also, this is a useful trick: 700000/35000 = 700/35 = 20. Or 840/120 = 84/12 = 42/6 = 7. They should always reduce before division, if they can. It’ll make their lives much, much easier down the road to not have to divide with so many digits involved.
  
  
  Another thing is that, I don’t know how familiar the Grade 6 kids already are with fractions, but I think that if the division is very messy, the kids should stop after the unit digit and just write the rest as a fraction. The most important thing from Grade 7 on is that they can estimate decimals, such as to know that 11/7 is between 1 and 2, just past 1.5 because 11/7 = 1   4/7. They don’t need to really get that it is 1.57142857142857142857142857142857 since we all have calculators…."
   
  What do you think? Disagreements? Something I missed? I would love to hear what all the MS teachers have to say about what makes a child successful coming into MS.

A Pause in the Moment

I really like this time of the year, because the kids are very motivated and I know all of them well enough to pinpoint their strengths and weaknesses, and I have worked with them long enough to see a pattern in their learning and to see if they are growing in their efforts, responsibility, and habits of mind.

One thing I am trying this year (sort of organically) is to have all assignments be a dialogue with the students. They turn the assignments in, I look at them, give them back with written comments, and if the student's work isn't up to par, then the student knows that they need to re-do it (because I say it to them, leaving no assumptions). I think this back-and-forth dialogue is a more natural way of learning, and as a result I am a bit more lax on the deadlines. Unless a kid's work is more than a couple of weeks late, it matters more to me that it's done well than it is done absolutely on time. And it also helps to instill a culture of quality over quantity of work.

---------

We have an intern in our department this semester, and talking to her is bringing back all kinds of memories of my own first year of teaching! (All the hectic schedule, the stress, that feeling of momentary panic when you're standing in front of the room and kids won't listen, etc.) She's great though, definitely much better/firmer than I was as a first-year teacher. She's also multi-talented, totally certified to teach math, science, and German all at once. I'm still working on my one semi-professional qualification of teaching math...

Besides that, I think our department is finally, finally at a smooth-sailing pace. We've finally recovered from the shock of the start of the year, when everything was constantly backlogged. It only took us 4 months to get to this place. I can only say that the next year will be better, because I'll already know my way through the start of the year, and I can prepare for it better. Being a department chair is definitely not easy, but I think people have gradually warmed up to me in the last 4 months (and I have gradually gotten used to the responsibilities). By that, I mean now they don't bolt out the door at 4:30pm at the end of the meeting. They actually linger until we're done with business... which is a definite sign of something good, however small. I think I have a good relationship with everyone in the department, which really, really helps to smooth things over when issues arise. I cannot wait to see what the second half of the year will bring!

Late Autumn in Berlin

Time for a bit of general Berlin updates! (Math teachers who are not interested in this: Sorry, go ahead and move along in your GReader.)

October and November have been fairly busy months for us! First, the Jersey boys came to visit during Oktoberfest. It was a very rowdy good time (we had rented a house in the heart of Munich), and Geoff got all kinds of nostalgic thinking about how this might be the last time -- minus our wedding -- that they could have crazy, random experiences like this, since all the boys are nearing that age when they are thinking about settling down for good...

By the way, I don't know if you have ever been to Oktoberfest, but there they build these huge tents (like the one you see below) for only the month of that festival. It's incredibly hard to get in on Fridays and Saturday nights to one of the big tents, so we had to bribe the bouncers. Once you get in, the tent gets more and more crowded and crazier and crazier as the day rolls on...

While the boys were still here in Germany, we rented a Bier Bike, which is a mobile bar that you can pedal around Berlin while singing out loud and waving at people. We blasted Queen's Bicycle Race several times during the course of two hours on the bike, and we kept going afterwards to a hipster restaurant called White Trash, followed by karaoke until about 3am. It's very Berlin to have a bunch of random experiences all in one day, because this city is just organic and crazy!

Soon after the boys left, I heard via the grapevine that the Berlin Light Show was about to come to an end, and that it was cool to see the light show from the top of the Berlin Reichstag (Congress Hall). I didn't even know that you could make an appointment to visit the Reichstag at night! We had only been there once during the day. So, on a Sunday night in late October, we walked around Berlin to enjoy the light festivities on the famous monuments, and then went up the Reichstag.

Halloween was a bit quiet this year. On the actual day of Halloween itself, we had German class, so I went to class as normal and only wore my cat ears to be festive. My non-American classmates thought it was funny that I would celebrate this type of holiday still in costume. On the way back, I was impressed to see a holiday graffiti at the subway station, since it was quite a production and you know graffiti artists are seriously prosecuted by the subway officials. Later during the weekend, we went to a party where Geoff dressed up as the recycling goddess (he made a hula hoop skirt out of toilet paper rolls that we had collected for over 6 months, and also he made a busty bikini top out of cut-out juice containers), and I was dressed as eine Katz im Sack zu verkaufen, which means I was a cat inside a trash bag, with a for-sale sign taped to the bag. In German, "buying a cat in a bag" means to purchase something (such as a used car or an old building) without having seen it, so you don't really know how it's going to behave afterwards. In other words, I was playing off a German pun, but Germans don't really get why Americans would dress up as non-scary things for Halloween, so it was a fairly obscure costume...

Then, there was the Sparkle Army party this year! The Sparkle Army is an annual party at our favorite karaoke place, where if you dress up in sparkles then you can get in for free. We have some friends who have been going every year for about 5 or 6 years (since the inception of this idea). The slogan of the party is, amazingly, "More is more!" They really want to spread the word of sparkle. Last year we went to the party, but we were pretty last-minute about the preparations. This year, since I had coincidentally run into the Sparkle Army girls while they were shopping for supplies a weekend in advance, I went ahead and made advance preparations. We ended up bringing 20 of our people to the party. It was a blast!!

To give you just a small taste of why this party is awesome, here was actually some random guy (not one of our friends) with an absolutely awesome outfit. He even had slippers with glitter bulbs glued on, and there were stuffed animals sewn onto his pink tutu. Faaaaan-freakin-tastic!

whew. That's all the fall updates for now. December will get its own story, when it's time... My friend comes to visit this Friday, to enjoy Christmas markets in Berlin with us. I have planned already a Christmas market crawl -- in Santa Claus outfits (a la Santa Convention style, like you can find in some major cities in the States). I can't wait! Our cheap Santa costumes will get delivered on Wednesday, and we bought the most hilarious-looking ones with a funny-looking shoulder cape... Let the Christmas season begin!

Visualizing Complex Operations

I frequently feel sad that complex numbers are not part of the IB SL curriculum. During the time when I taught Algebra 2, it was always my favorite topic to connect algebra, geometry, and the history of numbers all at once.

Here is one cute activity I used to give to my students to illustrate the relationship between complex number operations and coordinate transformations. I recently gave it off-handedly to a student at our school, who was very intrigued by this, so I thought I'd share with those of you who still teach this topic. I think this activity is very eye-opening for the students and very visual, and it gives the various algebra operations a more concrete meaning / some motivation.

I vaguely remember that I had written about this a long time ago, but here I am posting it again since I cannot find the old post. (oops. Lost in the WWW, I guess.)

A Presentation with Optional Paths

I have been piecewise putting together a presentation I will be doing in the early spring at a conference called AGIS (for German international schools). I'm pretty nervous since it'll be the first time that I will be doing a presentation at a conference! My only previous presentation experiences were when I did a PD to my own department back in El Salvador, on utilizing web resources, and when I quickly presented something at PCMI 2011 for what I did with GeoGebra and unit circles.  Both of those were quite informal, I think, compared to speaking at a conference... So, I've given myself plenty of head start to think carefully about the content and presentation format for the talk in the spring. As a presentation newbie, I am nervous about not knowing who my audience will be. As a teacher who tends to lecture minimally in the classroom, I am also concerned about audience engagement as a whole when I speak to them one-sidedly for a stretch of time.

This presentation that I am preparing will be on math projects as a tool for self-differentiation. One thing I decided to do for this presentation is to not assume that all teachers are very experienced with projects, OR that they're completely inexperienced. So, my presentation will start by laying the foundation for why I think math projects are beneficial for students, just to establish a common ground with everyone. And then we go into pivot points, where I'll poll the audience quickly to find out which topics they would like me to spend the most amount of time speaking about. I was inspired by the idea that within powerpoint you can have hyperlinks to other parts of the powerpoint (eg. those Jeopardy-format powerpoints), so depending on their interest, we can click through only certain math projects to discuss them in more details and to spend some more time discussing general project structures, or to click through all of the projects and discuss actual content rather than format and framework of projects.

Here is the powerpoint I've pulled together so far. Please check it out -- using actual slideshow mode -- and give me feedback! Once you see the rounded boxes, you need to click inside the boxes in order to navigate forwards and backwards... (I used the boxes in order to avoid having ugly underlined hyperlinks everywhere.) Not all of the projects are mine, and I haven't filled all the slides in with pictures yet, but I am excited about the overall idea of a presentation that self-adjusts in real-time to the responses and vibes from the audience.

Addendum: I have revised my powerpoint to make it less verbose... Now I think it's 90 or 95% finished! Check it out and keep those suggestions coming! Thanks!!

Teaching Number Definitions Meaningfully

As a kid, I was never good with mathematical terms. I was always doodling in math class and only picked up vocabulary words osmotically (which also meant not so effectively). As a teacher, I have tried to make vocab instruction more meaningful for kids.

Recently, in Grade 8, we started talking about inequalities for the first time. I started off the discussion by asking kids to give me some examples of equalities, and we wrote them on the board. After a few minutes, we started to list examples of inequalities and I went over why in math, saying -5 is less than -2 is a bit more precise/less confusing than saying -5 is smaller than -2.

Then, I asked the kids to come up with some example numbers that can satisfy the inequality
x + 2 < 15 . The kids started to list numbers, and after a few minutes I asked them for what they wrote down, and in the context of this we discussed number types. I explained to them that I think generally, in life, when you want to brainstorm options in your head, you don't want to keep listing the same types of objects over and over again, because in doing so you are limiting your vision of what is possible. The examples that the first kid gave me were {1, 2, 3, ..... , 12} and the examples that the second kid gave me were {-1, -2, -3, -100}

I drew on the board this diagram and said that "a long time ago, when cave people started counting sticks on the ground, they came up with numbers like 1, 2, 3, 4... These were called natural numbers.* Then eventually they came up with one more number to add to that set, and they called it whole numbers. Can you guess what that new number was?"

After kids enthusiastically guessed zero, they were starting to understand this diagram representation of a subset and were beginning to appreciate the historical development of numbers. Then, I added another layer of negative numbers and asked them what that is called when we started at some point to include/consider negative numbers as well.

They figured out that it was called integers. Great!

Then, I asked them what types of numbers they still know / have learned that we haven't named yet. They gave examples of decimals and fractions, and we added them to this picture.

In the end, we went back to the original topic at hand of finding example numbers that satisfy the inequality x + 2 < 15, and this time they were much better equipped to list a variety of examples and to discuss the full range of (non-discrete) solutions, which then led to the discussion of shading the solutions on the number line, and why we need the open circle rather than the closed circle sometimes.

I find this to be a more natural way to teach number types. Another time when I have done this was in teaching 9th-graders how to think of possible counterexamples that might disprove a math statement. If in their heads they are only considering a single type of example, then they're not being effective and thorough in their consideration of possibilities. As a kid, I would have appreciated this type of instruction of categorical types, followed immediately by application of its usefulness, and it would have probably helped me remember the names better. So, for me as a teacher, I always think it is important that I don't introduce the number definitions purely in isolation just because it's part of section 1.1 in the textbook. In the end, our teaching of these numerical categories should be explicitly supporting the kids' thinking, rather than just to add to the volume of disconnected rote knowledge in their heads...

*Note: By the way, I prefer this definition of natural numbers, even though I know that mathematicians don't all agree. Some refer to zero as part of the natural number set. 

A Day in the Life: Berlin Edition

 
Today was Friday. On Fridays, unofficially speaking, I have a full day with no free periods.

I got up around 6am and got ready. I knew it was going to be cold today, so I dressed extra warm with thermals and boots. Normally I eat breakfast, but today I was running late after responding to a Facebook message, and so I scrambled out the door at 6:32am with some dry cereal in a napkin to eat on the way to the train.

After a train and a bus, I arrived at school at 7:45am. Scrambled up to my classroom (on Floor 4), dropped off my stuff, and then hurried off to my morning Homeroom which began at 8am in another building.

After morning attendance with my Homeroom, I went to my Grade 9 class. Some of the kids who had opted to take the test today were pulled out of the room by the Student Support specialist. The rest of the kids were taking advantage of my extra review day for them. This was a double-period that lasted until 9:30 and seemed to fly. In the beginning of the period, I lost patience with one kid who was expecting me to hand him all of the answers, and I spoke to him impatiently and basically said that he was being lazy. I felt a little bad, but not too much. I told him that his effort was unacceptable and that I'm not one of those teachers who's going to say that it is OK what he's doing. The class was pretty quiet and really trying the problems after that. By the end, the kids were in good spirits because they felt like they have a grasp of the word problems dealing with midpoint and distance, even though they definitely still need to review quadratic factorization before next Wednesday's test... 

Then came morning break (a 15-minute recess), during which a boarding student came to speak with me about another new boarding student whom he is helping to tutor in math. We spoke about what the new student should be working on.

I grabbed my box of supplies (10 graphing calculators + board markers + lesson materials + 2 versions of textbooks) to head over to the other building where my next class was going to be. It was Grade 12. We went over homework from the previous day, which tied into the new Calculus stuff we were going to be doing -- area between two curves. I explained how this ties to the middle-school idea of finding an irregular shaded area using subtraction of total area minus smaller part/hole. The kids then practiced some skills in class and then copied down another practice quiz, due at the start of the next class. This double-period also went by fast. When class was over at 11:05, I looked up and was surprised that half the day was already over.

During lunch, 3 eleventh-graders came to see me for a re-quiz, and so did two seventh-graders. Before they left, I graded their quizzes on the spot, gave them feedback, and then pep-talked one girl who is really persistent despite her difficulties in Standard Level IB. I told her that as long as she keeps trying, I won't remove her from the class because I think it is possible for her to catch up, even though I know that her old math teacher didn't recommend her to go into Standard Level math. Another two Grade 9 twins tried to come get some help with their American curriculum (I have been teaching them the curriculum in pieces during lunch, since they're going back to the States after this year), but I told them that today's not a good day because I needed to prep for a class after lunch still. As usual, I just ate some bread which I had bought in the morning, for lunch. I always stay in my classroom during lunch, because kids tend to drop by at this time for re-quizzes, extra help, etc.

At the end of lunch, I headed over to teach PSHE, which is basically character education for my Homeroom kids. Today we were doing some bonding activities, because another Homeroom teacher was out, and some of her students were joining us. We did a get-to-know-you-better game involving asking provoking questions. The kids really enjoyed it, and they were sad when the period ended.

My own Homeroom kids then followed me to a different room for our Grade 8 math class. (I teach in many different rooms.) They turned in their lab report final drafts and we continued our discussion from yesterday of inequalities. They asked me interesting questions like, "Did it take people a long time to discover negative infinity?" And they also made up hand signals for infinity. I told them that they can start an infinity gang with their infinity gang signs, and after that every 5 minutes I'd see them flash the infinity sign.

After the Grade 8 math class, technically I should have two periods off at the end of the day, but those are the periods when I go in to support someone else's math class (unofficially). I work with a kid who moved to our school from a war-torn country, who is several years behind in math. I sit next to her and normally she works on solving simple one-variable linear equations while the rest of the class works on trigonometry. Today was the start of a new topic in Statistics, so she was following along the rest of the class and doing tallies. While she did tallies, since she said she didn't need any help, I just graded some recent trig projects for my Grade 11 students.

My colleague needed a bit of pep-talk after class, so I sat around to wait for her after school. We chatted a bit before I headed out to catch the bus. Today I left school early (at 3:30pm) because it's a Friday and also because I had a wedding dress fitting at 5pm in the opposite corner of Berlin.

The dress fitting went well! Afterwards, I grabbed dinner alone at my favorite Thai restaurant near home. Extra spicy. (It's my guilty pleasure to dine out alone. I do it every Friday so that I don't have to deal with talking to people at the end of the week, and plus it works out well that Geoff goes to play ball on Fridays.) I came home, updated the blog a bit, and then now I am thinking about how I should probably nap before my friend's Housewarmer tonight...

yawn.

Three-Variable Investigation... in Grade 8!

Even after teaching all of grades 7 through 12, I still LOVE teaching 8th-graders the best. (This is my 5th year teaching 8th-graders.) They have just accumulated enough algebra under their belts to be able to do rich explorations, and they are still so naturally curious about the world. They are like math ninjas, always ready to pull out their math skills to apply to the world at a short notice, and never intimidated by the look of a problem.

In the past couple of years, I realized that it's a real shame that we don't do modeling with three variables in middle-school math. It's a shame because it would be such a terrific tie in to the scientific process, to show how in mathematics you can also hold one variable constant while examining the effect of another variable on the output, and vice versa, and then in the end generalize the results into one grand conclusion. (I realized this because, looking at the past IB portfolios, this is a required skill for 11th- and 12th- graders. This was news to me as a person coming from the American curriculum.)

This year, I decided that I will try to remedy this gaping hole in our curriculum by exposing my 8th-graders to a new assignment. Take a look! They will be completing parts of this at home, then bringing it to class for discussions as a group. And then they will take more of it home to do. Eventually, when all kids feel comfortable with the process and the results, they will write it up like a modeling report. (We have already written one lab report this year based on Dan's awesome activity, and they found it very challenging / a great learning experience. In that lab, they had to learn how to define variables, collect data, determine the type of regression, perform regression, interpret results, make mathematical predictions, test their prediction, and then do error analysis. We followed it up with a very rigorous write-up process that included carefully critiqued rough drafts and a day spent on discussing how to create / insert graphs using GeoGebra and how to structure their writeup in a logical sequence.) Since I am a firm believer that kids learn more through writing about their understanding, the gears in my head are already turning to think about this next modeling assignment.

Anyway, I am VERRRRY excited about this three-variable assignment. Since the topic is already abstract, I kept the patterns linear to make it more accessible to all kids. But, I am very hopeful that it could turn out to be an awesome learning experience.

Addendum: For you new readers, this analogy is what I am going to use to kick off the introduction of 3-variable relationships in the real world.

Visualizing Order of Operations

This year my Grade 7s came in to my class with some missing prerequisite skills. Half of them started the year not knowing integers or order of operations or how to calculate simple fractions and percents, which are all supposed to be prerequisite knowledge for Grade 7 at our school. So, (although I did go back and fill in those gaps more or less,) teaching them algebra skills on top of this shaky foundation has been a new challenge.

At some point, I realized that it was difficult for these kids to look at a formula (in simple algebraic evaluation, for example) and to visualize what operations need to happen first, second, third, etc. They know the rules for PEMDAS, but it is just hard for them to always do it consistently. So, I came up with a trick of teaching kids to circle the operations in the given exercises in a layered manner, so that they can train their eyes to look for the operations, rather than to look at the equation from left to right.

Something like:

My special ed helper agrees that this is really helpful for them to visualize the rules. My only frustration is that they don't do this consistently because they still think they can just see it all in their heads, and in doing so they end up missing an extra negative sign here and there and throwing off the entire answer.

Anyway, I think the same trick can be used with certain types of equations to help kids see the layering and the process of peeling away the onion.

So, this is what I'll focus on for the next couple of weeks, to see if it can help solidify their foundation with this type of layered equation. (They're pretty OK with the ones with x's on both sides, since we had started off with doing balance visualization and crossing out shapes.) Basically, anything that can help them sink their teeth into abstract representation is worth a try for me. Any other ideas on how I can help these kids?

Ant on a Wheel

The hook for circular functions I had envisioned turned out to be pretty great, even better than I had imagined. The problem about the ant on the wheel was a hit, and really brought out some nice misconceptions from the kids. When I passed out the worksheet, I told them that they needed to estimate intelligently (not randomly guess) for the values in the table in #1, and that it should take a bit of time to complete if they were doing it correctly. They took their guesses, and most of them wrote things like

t = 0 --> h = 60
t = 0.5 --> h = 40
t = 1 --> h = 20
t = 1.5 --> h = 0
t = 2 --> h = 20
t = 2.5 --> h = 40
t = 3 --> h = 60
t = 3.5 --> h = 40
t = 4 --> h = 20
t = 4.5 --> h = 0


Then, when we discussed as a class, I drew a big wheel on the board and a vertical scale next to it to show height up to 60. I asked the kids where the ant was at 0 seconds, and everyone pointed to the top of the wheel, so I labeled it 0 sec. And then I asked them where the ant was at 3 seconds, and at 1.5 seconds, and we labeled those quickly as well since those were "obvious" to the kids. Then, less obviously, I asked them where on the wheel the ant was at 1 second and 2 seconds. To do this, they figured out that you have to divide the wheel up into thirds. And then we can further divide this wheel up to see where the ant is at 0.5, 2.5, 3.5, ... seconds.

By the end of our discussion, we got a diagram that looks something like this (on the board and also on their papers):

Each time, we referred horizontally over to the height scale that I had sketched on the board, and we estimated how high up the ant actually is. In doing so, the students noticed that they had to change the heights in their table. See below:

t = 0 --> h = 60
t = 0.5 --> h = 45
t = 1 --> h =15
t = 1.5 --> h = 0
t = 2 --> h =15
t = 2.5 --> h = 45
t = 3 --> h = 60
t = 3.5 --> h = 45
t = 4 --> h =15
t = 4.5 --> h = 0

One kid said, "But that doesn't make sense! I had divided it into equal parts before and that made sense." So, we had to discuss as a class that the ant is moving mostly vertically between certain parts of the wheel (ie. between 0.5 and 1 second, or between 2 and 2.5 seconds), and mostly horizontally between other parts of the wheel (ie. between 1 and 1.5 second, or from 1.5 to 2 seconds). Finally this kid is convinced that the height change is not a steady rate at all times.

So, with this consensus, I asked the kids to take out their graphing calcs and to create a scatterplot with this data (continuing the pattern all the way to 9 seconds, by 0.5 second increments, in order to reinforce their understanding of this circular pattern around the wheel). We looked at the graphs, and tada! It looks like a wave. Some kids were able to see it immediately, while others needed me to draw the scatterplot on the board and to connect it for them in order for them to see it.

Then, I asked them what type of function this was, and they were able to vaguely say sine or cosine (but they weren't sure which). We didn't get too far, but we started finding the equation of this function by hand for both sine and cosine functions and discussing the meaning of each part of the wave equation, which we will verify afterwards using the calculator's sine regression functionality. (Good time for the kids to practice all kinds of tech skills on the calculator, which they may or may not need for their internal assessment this year, depending on their internal assessment topics.)

Anyway, just thought I'd throw it out there. I didn't do super exploratory/introductory stuff this year in introducing waves, since the kids already have seen these equations the year before (in Grade 10), but I thought this lesson hook was really nice for re-introducing waves to them.

Addendum: I have decided that I'll be starting the next class by asking the kids where on the wheel the ant will likely raise two of his legs and say, "whee!" and then we draw it on the board on the wheel as well as on the wave graph, in order to connect this to rollercoasters and an itty bitty physics. On the other side, when the ant is rising rapidly, we can draw the ant hanging on barely, with two of his legs swinging in the wind.

Challenging Exponents/Log Problem

Fun/challenging problem from my colleague's test for his HL students (ouch!):

2(5^(x+1)) = 1 + 3/(5^x), solve for x in exact form "a + log(b)/log(5)", where a and b are both integers.

A student asked me after school for help with this, but I couldn't quickly figure it out on the spot. Afterwards, I thought over this and found it really fun to think of a variety of ways to try approaching this problem. (Too bad they mostly didn't work.) Over dinner, I thought of one way and it worked!

I am curious how you'd solve this. Please share!

Addendum: In hindsight, it's pretty foolish of me that I never looked at the graph even though I had suggested the graphical approach to that student. It's quite a simpler problem if you consider graphical approach as aid to the algebra. I was over-complicating things by doing it completely manually. (Still possible, but takes more work, obviously.) Silly silly me. These are the kinds of automatic-habit things I'd get better at, I guess, if I started coaching ISMTF competition teams....

Solo Performance and Remembering a Formula

I was doing a bit of sorting through old posts today, and I remembered that a couple of summers ago Sam Shah had asked me to blog about the cute activity I use to teach Quadratic Formula to my kids, since he says he cannot remember it otherwise.

Here it is: I teach them how to sing the Quadratic Formula song (to the tune of Pop! Goes the Weasel ), and then of course we practice in class how to apply the formula, and then for homework they need to go to a non-math teacher on campus and sing their song from memory. (To hold them accountable, I give them a half-sheet that says something like, "Dear Ms. Yang, I hereby certify that your student ___________ indeed came to me and sang from memory the following song to the tune of Pop! Goes the Weasel: X equals negative B, plus or minus square root.... [blah blah you know the rest], Signed, ____________")

It is very cute, because henceforth they sing every time they practice the formula, and they really do become experts at it before singing solo to a teacher who checks them off! And plus, some cool teachers even make them dance while singing. How very silly! I have kids who come back to me years after and still credit this assignment/song for them never forgetting that formula.

So, here you go, one of my sillier teaching "moves". I give you fair warning that after a few years you get really, really sick of hearing that tune... 

Intro to Waves...?

I am just going to try introducing waves like this this year in Grade 11. I hope it doesn't suck.

PS. They've "seen" sine and cosine functions before in Grade 10. Else I don't think you can introduce it this way and expect them to associate it with sine and cosine automagically. So I guess the proper title of this post would have been "Re-Intro to Waves...?"

100% Accountability

We had yesterday off from school. I had an epiphany during my day off that I am not instituting enough accountability and mental math drills in my Grade 7 class. So, today I remedied it with 80 minutes of a mini whiteboard lesson, and the result was really lovely.

We started off doing a laborious example: 2(x + 3), and I called on every single day-dreamer kid in the class to explain over and over again why this equals 2x + 6 (and as predicted, many of them could not say why even after their classmates had explained it 5 times). Once I was satisfied with our 100% accountability for listening in class, we proceeded on to do a very similar example, something like 4(x + 5). Everyone needed to do it on their whiteboards, and I asked them to all hold it up. 100% participation, no one can pass. We then practiced something like 5(x + 2) again just to make sure we all could do it. And then I started to call on one row at a time. When I called on the row, everyone in that row would stand up, and then I'd put up a question on the board. They'd be "put on the spot" to do it (individually on their boards without collaborating), and when they were all ready, they'd hold up their boards and their classmates would say whether they were correct.

This is a huge change from my normal mode of classroom learning, where I think I am too tolerant of mistakes, and so kids think that careless mistakes are totally ok even if you keep making them. This way, they're put under a bit of friendly peer pressure and they need to perform. --AND PLUS IT IS FUN!

We progressed our way to things like -(x + 3) or -8(3x + 5), and then -2(4x - 3) type of things, basic building blocks of algebra, until every group was consistently doing them correctly while standing for their turn.

Then, we moved on to doing two-step equations like 3x - 11 = 16. It turned out to be an excellent practice of their integer skills, and by the end of class, kids were comfortable finding fractional answers to two-step equations such as -4x + 5 = -2, and some can do it in their heads. Not bad for being week 2 of algebra for some of them (and having only recently learned about integers)!

So, I am liking this. Obviously it's not an everyday thing, but I think it adds a nice twist to my normal partnered work, and it really adds accountability for each student to be on the hook to do problems, correctly and consistently and independently. (I was very strict today about kids being silent and not talking during the exercises, and not asking their neighbors for help.) Truly, even my one student who normally gets no work done in class was really shining today and proudly displaying his work and volunteering to explain his answers to class. It was marvelous to see, and the special ed expert in the room was so impressed by the change in him.

So, go mini whiteboards on instituting 100% accountability!

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.