A short while back, Kate linked to an awesome video about learning from mistakes. Well, following my 7th-graders doing an awesome little project writeup for me this week, I thought I'd wrap up the week reviewing some of their common mistakes from last semester's big exam.
This is how I structured it. First (since it had been a while... we hadn't seen equations since December's big exam), I gave them one problem on the board with an answer written at the bottom of the board. I asked for them to figure out the process for showing how to get that answer, and the first ones to show me the clearest work can put them on the board, and I'll choose another person with the correct work to explain what has been written on the board.
Here was my first problem (not an easy one!):
-4(x – 3) + 1 = 5(3 – 2x) + 70
.
.
.
.
.
.
x = 12
The kids were instantly into it. (They were engaged by the competition aspect.) After two kids had put up two different ways of solving, I chose a normally very insecure kid to go up and explain their work, and she did great!
Then, we did another problem similarly:
4x – 13x = 2(-x + 8) + 19
.
.
.
.
.
.
x = -5
This time, a lot more kids were able to successfully complete the problem in a short amount of time. Of them, I picked two kids whose work didn't look exactly the same to put their process up on the board. Another normally unconfident kid agreed to go up and explain the work already put up on the board.
After that, we switched gears and I took the board markers and put up three problems, one at a time. I challenged the kids to quietly put up their hands when they can see where a classic mistake exists, and I waited until over half of the class had their hands up to pick a relatively weak student to tell me the answer.
Here was the first one, which many of them got right away:
-2(3x – 5) = 20
-6x - 10 = 20
-6x = 30
x = -5
They really enjoyed it, so I went ahead and put up:
5x – 3 = 3x + 11
8x = 8
x = 1
This time, a juicy discussion ensued. One of my students thought that the mistake was that 3x + 11 doesn't equal 8 but equals 14. Another student said that the second line should be 8x = 14, because the -3 "should become +3 when it goes across the equal sign." (I put it in quotes because it bothers me when kids say that, but if they've already been taught some basic algebra at home, that tends to be their phrasing.) Finally, some kids correctly identified/explained that the second line should have been 2x = 14.
Then, I put up a third problem, this time with two separate mistakes in it. Again, I challenged the class to find both mistakes.
(1/2)(x – 8) = 50
1/2*x - 8 = 50
1/2*x = 58
x = 29
It was so great! They were very excited that they could find so many mistakes.
It was perfect time to transition into Kate's suggested "My Favorite No" activity. We went through three algebra problems, increasingly more difficult each time, and I had kids submit their solutions on little scraps of paper. I wrote down my favorite incorrect problem on the board, and we started by pointing out all the things that person had done correctly, before discussing where they had gone wrong and why. In doing so, we caught: arithmetic error (some student thought -29 - 27 = 56) because they thought that you apply "the integer rules." We also caught the mistake of subtracting 2x from the same side of the equation twice. (I was so happy when the kids said, "You can't do that, because that would throw the equation off-balance!" They are talking like pros.) We also caught the mistake of going from 2.5x = 10 to x = 2.5/10 = 0.25.
It was brilliant! I think the kids had fun, AND I was able to get them to think hard about some common procedural issues ON A FRIDAY AFTERNOON.
When the class ended, I had just gotten them started on a Row Game involving some more basic algebra. It's their (my) first time doing a Row Game, so the concept of comparing answers even though the problems are not the same was a bit confusing to them. We'll have to continue with this Row Game next week, because it's supposed to address some more common procedural problems that I saw on the December exam. The worksheet I made for that is here if you want it. I am excited to continue it next week! Kids were talking to each other about math and trying to figure it out before turning to me for help (even though they were convinced that they could not have made a mistake and the problems could NOT have the same answers). It was really lovely.
So, yay to Kate, and yay for a day of trying new things and working with our conceptual mistakes instead of pretending that they don't exist.
Friday, January 13, 2012
Thursday, January 12, 2012
A Plug for PCMI
Hey, are you looking for a great mathy thing to do this summer? Try applying to PCMI. It's awesome, and Park City, Utah, is a fantastic place to be for three weeks of the summer. When you go there, you feel like the sky is bigger/cleaner and the days are way longer somehow. And there are some great math teachers who are passionate about teaching and doing math. Although you can find last year's problem sets here, it's hard to imagine the level of energy and camaraderie unless you've been to PCMI.
It's magic for three weeks, and you'll miss it when it's gone. There is also a generous stipend that covers most of your expenses.
Apply today! Deadline is the end of January, so you had better hurry.
PS. If you do go to Park City, bring your yoga mat if you've got one. It's utterly beautiful there to do yoga outdoors. Also bring your hiking shoes, your best karaoke persona, your fine dining belly, and your thinking cap. Just sayin'.
It's magic for three weeks, and you'll miss it when it's gone. There is also a generous stipend that covers most of your expenses.
Apply today! Deadline is the end of January, so you had better hurry.
PS. If you do go to Park City, bring your yoga mat if you've got one. It's utterly beautiful there to do yoga outdoors. Also bring your hiking shoes, your best karaoke persona, your fine dining belly, and your thinking cap. Just sayin'.
Wednesday, January 11, 2012
Baby Steps in Learning German
Today, I learned about a funny German category of verbs. (I just started private tutoring last week. It's amazing. I get to move at my own pace, which is pretty miraculous. I really feel that in two classes, I've already covered the equivalent of two or three weeks in a regular course, because I don't have to wait for other people to finish an exercise, and everything is chop-chop fast.)
Anrufen means to call someone up. When you conjugate it, the prefix comes off of the front of the verb and moves to the back. So, for example, to say "Are you going to call (me) on the weekend?" you say, "Rufst du (mich) am Wochenende an?" with the two parts of the verb separated by the entire rest of the sentence! Or, "Are you going to call him about it?" becomes "Rufst du ihn deshalb an?" Totally crazy cool. (By the way, Germans capitalize all nouns, which is funny and cool to me. Everything is important! Morgen, for example, is capitalized when it's a noun meaning "morning", but not capitalized when it is used as an adverb as in "tomorrow"... All sorts of very interesting, very particular grammatical rules!)
I think if I really work at it, I can cover a lot of ground in a calendar year. I am curious what that ground will look like without anyone else to set the pace, so I have decided to motivate myself to do some extra work every week in review of the last lesson and in preparation for the next, in order to maximize this year. It's so exciting! Ich will bald besser sein! (Is that right? "I want to be better soon!")
Anrufen means to call someone up. When you conjugate it, the prefix comes off of the front of the verb and moves to the back. So, for example, to say "Are you going to call (me) on the weekend?" you say, "Rufst du (mich) am Wochenende an?" with the two parts of the verb separated by the entire rest of the sentence! Or, "Are you going to call him about it?" becomes "Rufst du ihn deshalb an?" Totally crazy cool. (By the way, Germans capitalize all nouns, which is funny and cool to me. Everything is important! Morgen, for example, is capitalized when it's a noun meaning "morning", but not capitalized when it is used as an adverb as in "tomorrow"... All sorts of very interesting, very particular grammatical rules!)
I think if I really work at it, I can cover a lot of ground in a calendar year. I am curious what that ground will look like without anyone else to set the pace, so I have decided to motivate myself to do some extra work every week in review of the last lesson and in preparation for the next, in order to maximize this year. It's so exciting! Ich will bald besser sein! (Is that right? "I want to be better soon!")
Tuesday, January 10, 2012
Sin Nombre
Have you seen the movie Sin Nombre? It's about some illegal immigrants trying to get to America, and getting entangled with mareros from the infamous MS13 gang. Having lived in El Salvador, the movie was all too real and very depressing to me. After watching it, Geoff and I said to each other that our lives are truly privileged.
(Warning, major spoiler to follow)
I am sure there are worse ways to die, but plunging face forward off of/into a moving train while trying to cross the border illegally has got to be one of the worst ways. You're dying like an animal. It made me think about all the people who do die trying to cross the various borders. So utterly unjust. The only difference between them and us is their desperation; they were born into the wrong place, at the wrong time.
(Warning, major spoiler to follow)
I am sure there are worse ways to die, but plunging face forward off of/into a moving train while trying to cross the border illegally has got to be one of the worst ways. You're dying like an animal. It made me think about all the people who do die trying to cross the various borders. So utterly unjust. The only difference between them and us is their desperation; they were born into the wrong place, at the wrong time.
Saturday, January 7, 2012
Reflections Based on Types of Mistakes
I wanted to come back to talk a bit about my follow up to bucketing kid mistakes. I had my students write a detailed reflection of their exam. They had to identify which type of mistakes they had made the most frequently, to list the math concepts they had missed, and to thoroughly evaluate their study strategies in order to seek further improvement.
As the kids were looking carefully at their mistakes (I said I'd collect their reflections and compare it against their tests to make sure they were doing a thoughtful job), they changed some categorizations down from procedural to simply careless, if they are sure that they knew what to do but just didn't apply the skill carefully. They also noted to me if they had made various mistakes due to the same essential misunderstanding (ie. not looking to distribute the negative sign).
The kids' reflections that have been completed so far have been very impressively detailed and honest! In response, I corresponded with them in writing to add my assessment or recommendation for improvement during the second semester, and I am going to return the tests and their reflections next week to take home to review with their parents. In my comments to them, I wrote down things like if I think they should be checking their answers regularly against the back of the textbook, or if I think it was quite commendable that they persisted for a long time during the exam to try to get through even the hardest questions, regardless of whether they had finally succeeded. On my Grade 8 exam in particular, I commended the whole class for doing well and persisting when challenged on certain problems. For Grade 7, I noted to the kids that many of their exam scores did not accurately correspond to their normal performance, which showed me that they still have ways to go in working on their test-preparation strategies. (One of them, for example, did tons of practice problems but never checked her answers against the back of the book, so many of her practice problems were in fact incorrect when I looked at them! Another student signed up for some random math website and did random problems before the exam, instead of the problems I assigned for practice. While other students did 50 problems of the same type, and then ignored the 9 types of other problems that were going to be on the exam. These are all weird things that I am glad now I know they need to fix...)
Overall, I think this idea was a success! Instead of me saying to the kids that they are still making careless mistakes, they were pointing it out to me that they're not reading instructions, or not answering the questions fully, or making careless procedural errors -- all of which, they say, could have been avoided. I was very pleased because they were drawing the same conclusions that I wish they could have drawn, without my input.
I wouldn't do this level of reflection after every test, because it's lengthy and I don't want kids to start treating reflections as a "let's-just-get-through-this" thing of routine. But, I think I am going to stick to doing careful reflections twice a year to help them grow as students.
PS. On a totally different note, have you seen this? It's beautiful, and amazingly makes me feel (again) like the world is small. The guy who made this is my college friend's friend from high school!
As the kids were looking carefully at their mistakes (I said I'd collect their reflections and compare it against their tests to make sure they were doing a thoughtful job), they changed some categorizations down from procedural to simply careless, if they are sure that they knew what to do but just didn't apply the skill carefully. They also noted to me if they had made various mistakes due to the same essential misunderstanding (ie. not looking to distribute the negative sign).
The kids' reflections that have been completed so far have been very impressively detailed and honest! In response, I corresponded with them in writing to add my assessment or recommendation for improvement during the second semester, and I am going to return the tests and their reflections next week to take home to review with their parents. In my comments to them, I wrote down things like if I think they should be checking their answers regularly against the back of the textbook, or if I think it was quite commendable that they persisted for a long time during the exam to try to get through even the hardest questions, regardless of whether they had finally succeeded. On my Grade 8 exam in particular, I commended the whole class for doing well and persisting when challenged on certain problems. For Grade 7, I noted to the kids that many of their exam scores did not accurately correspond to their normal performance, which showed me that they still have ways to go in working on their test-preparation strategies. (One of them, for example, did tons of practice problems but never checked her answers against the back of the book, so many of her practice problems were in fact incorrect when I looked at them! Another student signed up for some random math website and did random problems before the exam, instead of the problems I assigned for practice. While other students did 50 problems of the same type, and then ignored the 9 types of other problems that were going to be on the exam. These are all weird things that I am glad now I know they need to fix...)
Overall, I think this idea was a success! Instead of me saying to the kids that they are still making careless mistakes, they were pointing it out to me that they're not reading instructions, or not answering the questions fully, or making careless procedural errors -- all of which, they say, could have been avoided. I was very pleased because they were drawing the same conclusions that I wish they could have drawn, without my input.
I wouldn't do this level of reflection after every test, because it's lengthy and I don't want kids to start treating reflections as a "let's-just-get-through-this" thing of routine. But, I think I am going to stick to doing careful reflections twice a year to help them grow as students.
PS. On a totally different note, have you seen this? It's beautiful, and amazingly makes me feel (again) like the world is small. The guy who made this is my college friend's friend from high school!
Thursday, January 5, 2012
Private School Salary Dilemma
I never gave it much thought until today (I'm not very good with money things), but recently there was a discussion about the tradeoffs between various systems of salaries and raises in private schools. It occurs to me as a very real, and fairly tricky, math problem.
First off, some brief description for you non-teachers: "salary step" is basically a grid of salaries, where for every year of experience you accumulate, you move along vertically to another area of the grid, and therefore get assigned a higher salary. Alternately, you can also move to a higher-pay area of the grid by accumulating additional training (and thereby moving horizontally along the grid). Teachers' unions typically negotiate salary increases across the entire grid, for example, to request increased benefits for every teacher in the system, and I am pretty sure they use a similar system for all public employees in general.
Obvious advantages of this salary-step system:
* It makes sure people are paid based on experience and training. (Loosely speaking, it's a logical idea that more senior teachers and better trained teachers will translate to better productivity.)
* It ensures equity among staffers hired earlier and later. ie. If you started at the school 10 years ago when you were a 3rd year teacher, you are now paid a higher salary than someone hired this year, with 6 years of previous experience elsewhere.
* It encourages retention of existing staffers, as they will continuously be rewarded for additional years accumulated on the job. Staffers who do leave, then, tend to leave for personal reasons as opposed to leaving for reasons of financial stagnation.
Disadvantages:
* The biggest disadvantage is that the overall school staffing budget will grow linearly every year, assuming that there is little attrition. At the same time, most schools will not be able to increase their tuition linearly every year, or increase their student enrollment linearly to compensate for the constantly growing staffing budget. As I see it, a private school nearing its max enrollment simply cannot afford to use salary steps (and one wonders how our government can afford to do so either).
* Some may argue that the salary-step system does not take into account teacher's actual productivity/merit. That's not a discussion I'd like to go into at this point, given all the controversy surrounding merit pay in general.
* Because staffers are continuously being rewarded for staying, it provides little incentive within the international school environment for healthy mobility and change/influx of new ideas.
Alternative systems and their tradeoffs:
* No salary step system. What salary you enter at is what you stay at, no matter how long you stay at the school. It creates weird situations like if you entered the school 10 years ago, with 5 years of previous experience, you could now (and forever) be paid significantly less than another person who now freshly enters the system with a prior experience of, say, 8 years. Even though overall you are way more senior than that person (15 years of work experience, versus their 8 years), and you have also shown that you are committed to this school, you end up forever being paid less. Needless to say, this affects morale negatively.
* Fixed annual percent increase of pay. This solves the problem of inequity due to time of hire, since by the time other new staffers have been hired, you would have already experienced various raises that put you ahead of them permanently. I think this is probably a strategy that non-mathy people would naturally come to, except that it creates the problem of an exponentially growing school budget over time, so it isn't really feasible. --Plus, in order to avoid any such "weird situations" of inequity, you actually would need to pick an annual percentage that grows FASTER than the linear increases in the salary step! No good!!
* Cost-of-living adjustments and project-based stipends. I think most schools do this, but it's still not addressing the issue of the inequities due to time of hire.
* Merit-based increases. I hate to say it, but this seems like an obvious option despite research that says otherwise. But, what do you judge merit based on? Hopefully not test scores or student opinions. Is it too much to move towards a business model of stacking employees based on peer and supervisor evaluations?
...I think this is sensitive to people because every time we talk about pay, it always gets sensitive. But truly, I see it as a mathematical/business dilemma that is objectively interesting. What do you think is a viable solution? Does one exist?
Other issues to consider:
* Your school's entry salary has to be internationally competitive for a person of that level of experience/training.
* Salary steps are not truly linear (I don't think), nor should they be. The productivity difference between a teacher in their 18th and 20th years is not at all comparable to the productivity difference between their first and third years.
* Every time natural attrition happens, depending on how you replace the lost staffer, your overall staffing budget will either shrink or expand. Therefore, the school admin still has significant control over their staffing budget regardless of the salary system in place.
Anyway, I'm throwing it out there because I have not made up my mind about it, but I am curious if there are clear-cut solutions that I am just not seeing. Y'all able to help me think this one through?
First off, some brief description for you non-teachers: "salary step" is basically a grid of salaries, where for every year of experience you accumulate, you move along vertically to another area of the grid, and therefore get assigned a higher salary. Alternately, you can also move to a higher-pay area of the grid by accumulating additional training (and thereby moving horizontally along the grid). Teachers' unions typically negotiate salary increases across the entire grid, for example, to request increased benefits for every teacher in the system, and I am pretty sure they use a similar system for all public employees in general.
Obvious advantages of this salary-step system:
* It makes sure people are paid based on experience and training. (Loosely speaking, it's a logical idea that more senior teachers and better trained teachers will translate to better productivity.)
* It ensures equity among staffers hired earlier and later. ie. If you started at the school 10 years ago when you were a 3rd year teacher, you are now paid a higher salary than someone hired this year, with 6 years of previous experience elsewhere.
* It encourages retention of existing staffers, as they will continuously be rewarded for additional years accumulated on the job. Staffers who do leave, then, tend to leave for personal reasons as opposed to leaving for reasons of financial stagnation.
Disadvantages:
* The biggest disadvantage is that the overall school staffing budget will grow linearly every year, assuming that there is little attrition. At the same time, most schools will not be able to increase their tuition linearly every year, or increase their student enrollment linearly to compensate for the constantly growing staffing budget. As I see it, a private school nearing its max enrollment simply cannot afford to use salary steps (and one wonders how our government can afford to do so either).
* Some may argue that the salary-step system does not take into account teacher's actual productivity/merit. That's not a discussion I'd like to go into at this point, given all the controversy surrounding merit pay in general.
* Because staffers are continuously being rewarded for staying, it provides little incentive within the international school environment for healthy mobility and change/influx of new ideas.
Alternative systems and their tradeoffs:
* No salary step system. What salary you enter at is what you stay at, no matter how long you stay at the school. It creates weird situations like if you entered the school 10 years ago, with 5 years of previous experience, you could now (and forever) be paid significantly less than another person who now freshly enters the system with a prior experience of, say, 8 years. Even though overall you are way more senior than that person (15 years of work experience, versus their 8 years), and you have also shown that you are committed to this school, you end up forever being paid less. Needless to say, this affects morale negatively.
* Fixed annual percent increase of pay. This solves the problem of inequity due to time of hire, since by the time other new staffers have been hired, you would have already experienced various raises that put you ahead of them permanently. I think this is probably a strategy that non-mathy people would naturally come to, except that it creates the problem of an exponentially growing school budget over time, so it isn't really feasible. --Plus, in order to avoid any such "weird situations" of inequity, you actually would need to pick an annual percentage that grows FASTER than the linear increases in the salary step! No good!!
* Cost-of-living adjustments and project-based stipends. I think most schools do this, but it's still not addressing the issue of the inequities due to time of hire.
* Merit-based increases. I hate to say it, but this seems like an obvious option despite research that says otherwise. But, what do you judge merit based on? Hopefully not test scores or student opinions. Is it too much to move towards a business model of stacking employees based on peer and supervisor evaluations?
...I think this is sensitive to people because every time we talk about pay, it always gets sensitive. But truly, I see it as a mathematical/business dilemma that is objectively interesting. What do you think is a viable solution? Does one exist?
Other issues to consider:
* Your school's entry salary has to be internationally competitive for a person of that level of experience/training.
* Salary steps are not truly linear (I don't think), nor should they be. The productivity difference between a teacher in their 18th and 20th years is not at all comparable to the productivity difference between their first and third years.
* Every time natural attrition happens, depending on how you replace the lost staffer, your overall staffing budget will either shrink or expand. Therefore, the school admin still has significant control over their staffing budget regardless of the salary system in place.
Anyway, I'm throwing it out there because I have not made up my mind about it, but I am curious if there are clear-cut solutions that I am just not seeing. Y'all able to help me think this one through?
Tuesday, January 3, 2012
Feeling Inspired After PD on Differentiation
We had an all-staff professional development session today that was actually great! The speaker was from London Gifted and Talented, but he spoke more generally about differentiation for all kids (in the context of G&T education). First off, I have to say that I am a skeptic of the whole G&T education thing and what it does for kids; anyway, I went into the PD with quite a bit of doubt.
That said, I was really glad to hear the speaker say that the best way to nurture G&T kids is to provide opportunities for enrichment for all of your students via effective differentiation. He talked a lot and went through a lot of slides, but here were my favorite points:
* "High challenge/low threshold learning" is what we should be aiming for. A truly differentiated task should be limitless on the upper bound of complexity and be truly open-ended, genuinely investigative, and to allow student choices of medium/depth/topic, but still be accessible to everyone in the class.
* Differentiation cannot/should not be an end in itself. It should be linked to a purpose, and your method should reflect your purpose. --> This was a particularly good point for me, because I realized while he was talking that I have not clarified the end goals of differentiation for myself. What am I trying to achieve? Do I want different kids to be able to approach problems using different methods? Do I want kids to be able to demonstrate their knowledge using different media/application? Do I want kids to achieve similar abstract knowledge or am I comfortable with different kids understanding the concept differently? Lots of things for me to think about!!
* "It's not that [differentiation] is not happening. Rather, it's that we don't have the shared language to talk about what is already happening [in our classrooms]."
* Provide variable-credit assignments. A complex, rich task that is done well should replace several smaller, more basic tasks. A talented student should not be punished for their talents by being assigned extra work.
* Student floundering is good. Teacher needs to create environment for kids to think independently and to allow students to struggle. (I know I'm preaching to the choir here, but I also know how much I enjoy PDs that emphasize this point, because so many educators still do not believe that themselves.)
All in all, the session was great because it reminded me of the things I had committed to doing in the beginning of the year that I am still not doing. I don't believe in new year's resolutions (since I think goal-setting should be an on-going process and allow the opportunities of failures and re-attempts), but the second half of the school year seems like as good a time as any to be more self-critical and to hold myself accountable to some of those promises!
That said, I was really glad to hear the speaker say that the best way to nurture G&T kids is to provide opportunities for enrichment for all of your students via effective differentiation. He talked a lot and went through a lot of slides, but here were my favorite points:
* "High challenge/low threshold learning" is what we should be aiming for. A truly differentiated task should be limitless on the upper bound of complexity and be truly open-ended, genuinely investigative, and to allow student choices of medium/depth/topic, but still be accessible to everyone in the class.
* Differentiation cannot/should not be an end in itself. It should be linked to a purpose, and your method should reflect your purpose. --> This was a particularly good point for me, because I realized while he was talking that I have not clarified the end goals of differentiation for myself. What am I trying to achieve? Do I want different kids to be able to approach problems using different methods? Do I want kids to be able to demonstrate their knowledge using different media/application? Do I want kids to achieve similar abstract knowledge or am I comfortable with different kids understanding the concept differently? Lots of things for me to think about!!
* "It's not that [differentiation] is not happening. Rather, it's that we don't have the shared language to talk about what is already happening [in our classrooms]."
* Provide variable-credit assignments. A complex, rich task that is done well should replace several smaller, more basic tasks. A talented student should not be punished for their talents by being assigned extra work.
* Student floundering is good. Teacher needs to create environment for kids to think independently and to allow students to struggle. (I know I'm preaching to the choir here, but I also know how much I enjoy PDs that emphasize this point, because so many educators still do not believe that themselves.)
All in all, the session was great because it reminded me of the things I had committed to doing in the beginning of the year that I am still not doing. I don't believe in new year's resolutions (since I think goal-setting should be an on-going process and allow the opportunities of failures and re-attempts), but the second half of the school year seems like as good a time as any to be more self-critical and to hold myself accountable to some of those promises!
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