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.
Friday, December 21, 2012
Sunday, December 16, 2012
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.
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.
Saturday, December 15, 2012
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...
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...
Thursday, December 13, 2012
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.
loga(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, loga(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.
3x =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.
logx(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(7k) = ?? --> "log asks the question, what exponent is required to go from a base of 7 to reach a value of 7k? The answer is, well, k!"
log7(72n-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*5x =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.
-4x =-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*6x - 7 = 20
102x-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:
6x = 36x-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 36x-3 ?" And they decided that the answer to that question has already been provided, as x.
So, log6(36x-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(6m^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 2log2(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.
This is the definition I taught them:
Log is just a way to ask a specific question.
loga(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, loga(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.
3x =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.
logx(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(7k) = ?? --> "log asks the question, what exponent is required to go from a base of 7 to reach a value of 7k? The answer is, well, k!"
log7(72n-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*5x =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.
-4x =-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*6x - 7 = 20
102x-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:
6x = 36x-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 36x-3 ?" And they decided that the answer to that question has already been provided, as x.
So, log6(36x-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(6m^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 2log2(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.
Wednesday, December 12, 2012
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?
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?
Tuesday, December 11, 2012
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?
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).
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).
Saturday, December 8, 2012
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!
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.
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.
Friday, December 7, 2012
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*
"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*
Thursday, December 6, 2012
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.
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.
Saturday, December 1, 2012
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:
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.
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