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

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

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

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

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

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

## Sunday, January 29, 2012

## Saturday, January 28, 2012

### Japanese Geometry Problem Set #2

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

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

I hope you are enjoying these!

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

I hope you are enjoying these!

## Friday, January 27, 2012

### Accountability and Changing Classroom Dynamics

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

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

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

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

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

In Grade 7, the game starts to change a bit. The problems become multi-stepped, and new (algebra) skills come down the pipeline that require repeated practice in order to reach a point of effortless, automatic application. I am certainly not saying that

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

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

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

I have made a homework schedule for my IB students in grades 11 and 12. I have never been a believer in mandatory studying outside of class, until now, because part of me hopes that the kids will see the importance of pacing themselves and setting their own goals. I can see, however, that my Grade 11's and Grade 12's are too comfortable; they come to class, do what assignments I have designed for the day, feel great about their understanding of the concepts, and then most of them walk away without practicing more on their own or struggling through practice problems in the textbook. It's simply not working, because I am working way harder than them for this class, and that cannot be the case when it is

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

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

In short, as I think more about what makes a successful student, I want to think about ways of extricating myself out of that picture. I truly think that this will empower my students to feel like their learning is in

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

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

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

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

In Grade 7, the game starts to change a bit. The problems become multi-stepped, and new (algebra) skills come down the pipeline that require repeated practice in order to reach a point of effortless, automatic application. I am certainly not saying that

*understanding*is not important in Grade 7 -- in fact, we do a lot of conceptual development in class before introducing any algebra methods -- but I find many of my intuitive students struggling with procedural issues in algebra, even though they understand in their heads what they need to do. They simply have not developed the work ethic to practice and practice again until their procedural issues are ironed out and they can*consistently*solve something without effort. In Grade 7, more than anything I emphasize work habit, because I find it so dangerous that those "intuitive" kids get passed on from teacher to teacher believing that AS SOON AS they would start working, everything would be dandy.Because truly, I don't believe that. I have two amazingly intuitive 11th-graders, who lack the basic arithmetic and algebra skills to complete problems. If you throw them into a completely new type of problem, they can make headway better than their classmates. But then, put them in front of a common/simple skills application, they would have no idea what to do and invent crazy algebra rules. I can only imagine that those two kids have been praised by teachers all along for their intuition, without an equal emphasis on how much damage they are doing themselves by leaving such major gaps in their basic skills. At the same time, I've seen other kids go from struggling to mastering skills, and then tougher concepts. Those kids persevered until they mastered the basic skills, and now they can save their mental energy for the trickier/problem-solving parts of the task.

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

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

I have made a homework schedule for my IB students in grades 11 and 12. I have never been a believer in mandatory studying outside of class, until now, because part of me hopes that the kids will see the importance of pacing themselves and setting their own goals. I can see, however, that my Grade 11's and Grade 12's are too comfortable; they come to class, do what assignments I have designed for the day, feel great about their understanding of the concepts, and then most of them walk away without practicing more on their own or struggling through practice problems in the textbook. It's simply not working, because I am working way harder than them for this class, and that cannot be the case when it is

*their*learning we are talking about. So, in my new homework schedule, I assigned a Chapter Review from the textbook every week or two weeks, for a topic they should be familiar with. If there are questions they cannot answer, I urged them to look through the chapter at home to resolve their questions, and only bring to class the most complicated questions that remain unanswered.And you know what? It has been wonderful. It has completely changed the dynamic of my classroom. Instead of feeling like I am on the hook for giving kids work and setting the pace for my class and making sure they get sufficient practice during class, our roles have somewhat flipped. Kids are in more control, and they are asking me questions that they want to know the answers to. It has changed the feel of the relationship between me and the students, for the better.

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

In short, as I think more about what makes a successful student, I want to think about ways of extricating myself out of that picture. I truly think that this will empower my students to feel like their learning is in

*their*hands and to build their confidence outside of class, without changing the way I currently structure lessons inside the classroom. I think this is how I am going to help my students move towards becoming life-long learners.## Sunday, January 22, 2012

### Japanese Geometry Problem Set

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

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

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

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

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

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

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

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

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

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

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

## Saturday, January 21, 2012

### Some Resources on Patterns and Systems of Equations

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

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

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

So, let me try and do this:

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

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

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

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

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

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

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

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

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

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

So, let me try and do this:

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

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

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

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

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

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

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

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

## Friday, January 20, 2012

### A "Backwards" Approach to Completing the Square

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

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

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

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

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

## Thursday, January 19, 2012

### Departmental Ruminations

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

## Friday, January 13, 2012

### Focusing on Process and Learning from Our Mistakes

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

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.## 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.)

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!

*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

(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.

*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.

## 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,

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.

* 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.

* 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:

*

*

*

*

...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?

* 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

* "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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