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 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.
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?
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