After a semester of use I've decided that I like blogging. More than that, I want to share my blog with my department and make it a little more than just for me. However, I've been using this blog since college and the title/url is the username I've had since I was 10. I'd like my school blog to be independent of all of that. So, I moved to DrawingOnMath.blogspot.com I like the name well enough but I was much more excited about playing with layouts than coming up with a name so I may regret rushing that later. It has multiple levels of interpretation though and since I teach almost all geometry currently it seems quite appropriate.

To those of you in a reader: I'm really sorry that you'll have to click out of your reader, go to the new site and re-subscribe. I totally understand if you "keep unread" and save that arduous task for later. Since next week is midterms for us I do plan to do quite a bit of writing though, so subscribing sooner rather than later is advisable ;)

Thanks for reading!

# crstn85

The math portion of this blog has moved to Drawing On Math Head over there for all the math posts from here and more!

## Friday, January 20, 2012

## Wednesday, January 18, 2012

### My Student Self

Last week I started an online course in Geometry and Measurement run by the EDC (love them!) for my district and a neighboring one. Being a student in a different environment has me reflecting on how I would react to being a student in my own class.

We started the course with a face-to-face meeting. I'm not going to lie, I was a little resentful that I needed to attend since I was looking forward to taking the course from my couch at home on my own schedule. The resentment only grew as I was instructed on how to sign in (which we'd been asked to do ahead of time), use a forum (something I've been doing for years) and use GeoGebra (I teach my students to use it).

When I'm teaching I need to make sure not to rehash ideas that students already know. I hope that when we discuss homework students view it as an opportunity to check their answers, not just the class re-doing the work they already did. I try not to go over every little step but I need to make sure that when the class gets it, we get on to the challenging problems. Also, I need to explicitly ask students to be resources for each other when we study a topic some of them are familiar with. I would have been happy to share what I knew about GeoGebra- but being asked to draw a triangle was just boring.

Each session of the course starts on Wednesday. Last week I'd finished all the readings by Thursday night. This week? I finished all the readings and activities by 8 pm, Wednesday. I'm a nerd and am happy to own that label. But I'm also really efficient; I'd rather just take a couple hours to go through all the material with complete focus and no distractions than drag it out over a week. This is one of the reasons I signed up for an online course- it's very self-paced and allows me to be independent.

While there is opportunity for students to do their homework whenever they like within the 2-4 day gap between classes (block scheduling), there is very little room for self-pacing in the classes I teach. Each problem set I assign in class is low-threshold/high-ceiling, but there is no room to move on to the next unit until the whole class is ready. At this point it would be logistically a nightmare to allow students in geometry to move forward since we do so much experimentation and compiling of data to form conclusions. My basic algebra class does spent part of their time on fact practice and that portion of the course is entirely self-paced. I'd be interested to see if there was anything similar in geometry to allow students more control over at least part of the curriculum.

Finally, we are required to post in the discussion forums throughout the week. That's the one piece I haven't done yet, and may no do until Friday. I own my nerd status, but that doesn't mean I'm not hesitant to look like an overachiever. This is especially true since I don't know half of the people in the course (they're from another district). I will be an overachiever in the end (last week I posted more than twice as many comments as were required) but I won't totally stick my neck out there... yet.

It's really important that students feel comfortable in the classroom. By this point in the year I forget that this means not just trusting the teacher (me) but also their peers. I am shocked when a student is helping to return papers and doesn't know a classmate's name. All of them have a few people they are okay working with (I've shuffled pairs until everyone has a few partners they gravitate to) but participating and asking questions in front of the whole class means they need to know that the entire class will be respectful and supportive of them. And that includes not getting called a brown-noser (I really hate that term, it's just so gross).

I'm excited for the rest of this course as I continue to analyze my learning style, as well as the materials presented. So far we've looked at essential knowledge for students to arrive to geometry with and methods for examining student work. The technology components are coming up soon. They include: an iPad to use for the semester (to be grouped into a set teachers can reserve next year), a new desktop to use at school and my new smart board! I'd love to hear your suggestions for iPad apps and the smart board (applets, features, uses, anything- I've never used one before!).

We started the course with a face-to-face meeting. I'm not going to lie, I was a little resentful that I needed to attend since I was looking forward to taking the course from my couch at home on my own schedule. The resentment only grew as I was instructed on how to sign in (which we'd been asked to do ahead of time), use a forum (something I've been doing for years) and use GeoGebra (I teach my students to use it).

When I'm teaching I need to make sure not to rehash ideas that students already know. I hope that when we discuss homework students view it as an opportunity to check their answers, not just the class re-doing the work they already did. I try not to go over every little step but I need to make sure that when the class gets it, we get on to the challenging problems. Also, I need to explicitly ask students to be resources for each other when we study a topic some of them are familiar with. I would have been happy to share what I knew about GeoGebra- but being asked to draw a triangle was just boring.

Each session of the course starts on Wednesday. Last week I'd finished all the readings by Thursday night. This week? I finished all the readings and activities by 8 pm, Wednesday. I'm a nerd and am happy to own that label. But I'm also really efficient; I'd rather just take a couple hours to go through all the material with complete focus and no distractions than drag it out over a week. This is one of the reasons I signed up for an online course- it's very self-paced and allows me to be independent.

While there is opportunity for students to do their homework whenever they like within the 2-4 day gap between classes (block scheduling), there is very little room for self-pacing in the classes I teach. Each problem set I assign in class is low-threshold/high-ceiling, but there is no room to move on to the next unit until the whole class is ready. At this point it would be logistically a nightmare to allow students in geometry to move forward since we do so much experimentation and compiling of data to form conclusions. My basic algebra class does spent part of their time on fact practice and that portion of the course is entirely self-paced. I'd be interested to see if there was anything similar in geometry to allow students more control over at least part of the curriculum.

Finally, we are required to post in the discussion forums throughout the week. That's the one piece I haven't done yet, and may no do until Friday. I own my nerd status, but that doesn't mean I'm not hesitant to look like an overachiever. This is especially true since I don't know half of the people in the course (they're from another district). I will be an overachiever in the end (last week I posted more than twice as many comments as were required) but I won't totally stick my neck out there... yet.

It's really important that students feel comfortable in the classroom. By this point in the year I forget that this means not just trusting the teacher (me) but also their peers. I am shocked when a student is helping to return papers and doesn't know a classmate's name. All of them have a few people they are okay working with (I've shuffled pairs until everyone has a few partners they gravitate to) but participating and asking questions in front of the whole class means they need to know that the entire class will be respectful and supportive of them. And that includes not getting called a brown-noser (I really hate that term, it's just so gross).

I'm excited for the rest of this course as I continue to analyze my learning style, as well as the materials presented. So far we've looked at essential knowledge for students to arrive to geometry with and methods for examining student work. The technology components are coming up soon. They include: an iPad to use for the semester (to be grouped into a set teachers can reserve next year), a new desktop to use at school and my new smart board! I'd love to hear your suggestions for iPad apps and the smart board (applets, features, uses, anything- I've never used one before!).

## Thursday, December 08, 2011

### Triangle Quilt

Last year I realized that even though most students claimed familiarity with the types of triangles from middle school, they still didn't really know them (especially isosceles and scalene). Plus, many were not adept at accurately constructing examples. As a week before winter break activity I had all of my geometry classes fill a square piece of paper with examples of all the triangle vocabulary we had studied, then I took all those squares and filled a section of the hallway, making a 'quilt.' This year I assigned the activity again, but instead of a review I used it as an introduction. It was a great way to make sure up front that everyone had a solid experience with the vocabulary, not to mention I always enjoy the down time of a coloring activity (especially with my CP class who did this after they finished a test).

The assignment:

Triangle Quilt

The finished product:

Marginally related:

I love putting student artwork up in my classroom. Whenever I see a student doodling something cool or showing off a drawing I ask them to contribute to my art gallery. It all started last year with a centroid sailboat and has grown to take over a corner of my classroom. I enjoy pretty things on the wall to look at, and it makes the kids feel appreciated when their work is on display. I didn't remember to take a photo of it yet, but I do have some photos of the pieces my homeroom created after a discussion on bullying. (These are in a separate corner.)

The assignment:

Triangle Quilt

The finished product:

The first round of submissions. |

Final Result (Progress reports were due today so nearly everyone has theirs in now) |

Marginally related:

I love putting student artwork up in my classroom. Whenever I see a student doodling something cool or showing off a drawing I ask them to contribute to my art gallery. It all started last year with a centroid sailboat and has grown to take over a corner of my classroom. I enjoy pretty things on the wall to look at, and it makes the kids feel appreciated when their work is on display. I didn't remember to take a photo of it yet, but I do have some photos of the pieces my homeroom created after a discussion on bullying. (These are in a separate corner.)

Aren't the birds beautiful? |

The top left one says "Don't be an angry bird!" Lots of birds because it was just before Thanksgiving and I suggested drawing hand turkeys. |

## Sunday, December 04, 2011

### Know Your Limits

By the time Friday rolled around I was lagging, I guess it's a post-Thanksgiving phenomenon but it seemed like an endless week. I'm not proud to admit that I yelled at one class on Friday when they were floundering. I tried to do some open ended experimentation (which I will share once I've tried again with my other class tomorrow) and they weren't having it. Whether I should have given more structure, support or time is yet to be determined, but getting mad is never the answer. At least I recognized that I was getting grumpy and spent my prep block coloring bubble letters and hanging more squares in our triangle quilt (I'll share that once it's done as well). I also forewarned my last class that the well of patience had run dry and we spent the beginning of class brainstorming what an A student looks like so they could be sure to be on their best behavior (which enabled me to be on mine).

As soon as the last student left after school (he stayed an hour on Friday, and he was one of the ones I was mad at earlier - can't stay mad at dedication like that!) I called my friend who I knew would still be in his classroom and announced it was time to leave the building. Several hours at a coffee shop with a good friend, mint hot chocolate, crepes and parcheesi (that's a game, not a food) and I was on my way to recovery.

The rest of the weekend I did nothing. I don't just mean no school work, but all day yesterday and today I did absolutely nothing productive. I lounged, played games, napped and read. I might break out a bit of grading this evening, but no guarantees. I'm not worried, I know all the work will get done. And I'd much rather have an hour of productive work tomorrow than 3 hours of not getting much done but feeling like I should be today.

Why am I telling you about my failures Friday and totally boring weekend? Because I worry that everyone is running themselves into the ground. We're not yet halfway through the year and we shouldn't be sprinting toward June with the knowledge that we can recover in the summer. Cold and flu season is coming, for many the holiday season means extra stresses and December break is always too busy and too short. Stop, take a break and give yourself some time off. I always declare Saturday my school sabbath, but this weekend that needed to extend into a longer break. You know you could teach all of your classes without any prep on any given day- sure it wouldn't be a great lesson, but your students would learn something. So, cut yourself a break, do whatever it is that you find rejuvenating and know your limits.

As soon as the last student left after school (he stayed an hour on Friday, and he was one of the ones I was mad at earlier - can't stay mad at dedication like that!) I called my friend who I knew would still be in his classroom and announced it was time to leave the building. Several hours at a coffee shop with a good friend, mint hot chocolate, crepes and parcheesi (that's a game, not a food) and I was on my way to recovery.

The rest of the weekend I did nothing. I don't just mean no school work, but all day yesterday and today I did absolutely nothing productive. I lounged, played games, napped and read. I might break out a bit of grading this evening, but no guarantees. I'm not worried, I know all the work will get done. And I'd much rather have an hour of productive work tomorrow than 3 hours of not getting much done but feeling like I should be today.

Why am I telling you about my failures Friday and totally boring weekend? Because I worry that everyone is running themselves into the ground. We're not yet halfway through the year and we shouldn't be sprinting toward June with the knowledge that we can recover in the summer. Cold and flu season is coming, for many the holiday season means extra stresses and December break is always too busy and too short. Stop, take a break and give yourself some time off. I always declare Saturday my school sabbath, but this weekend that needed to extend into a longer break. You know you could teach all of your classes without any prep on any given day- sure it wouldn't be a great lesson, but your students would learn something. So, cut yourself a break, do whatever it is that you find rejuvenating and know your limits.

## Monday, November 21, 2011

### Taboo Review

At the end of each chapter we spend a day making a study guide and then playing some sort of review game. In the past this games have included BINGO (fill in laminated cards with answers, then if you solve the problem you get to cross out the square), "the points game" (jar full of cards that say +5, +10, -5, -10 and x2, correct answer means you get to pull a card to determine your points) and a simple game of solving problems in teams to see which team can solve the most before the end of class. This year I have played these, and added in a few more thanks to the wealth of ideas on twitter. Last week we played basketball in my fundamentals class (solve some problems on a half sheet, if they're correct crumple and take a shot at the recycling bin, bonus point if you make it in! However- if you get one wrong you have to shoot from the far line, so there's an incentive to check your work).

Today, we finally played Taboo. I'd been hearing about the great reasons to play Taboo from lots of people, but hadn't wanted to make the cards. Last night I sat down to do it and it wasn't actually as hard as I'd expected. In fact, the worst part was fighting with Word to get the table to stay the way I wanted. I started with this set (google docs link) and then added my own to get this:

Taboo Ch 1-3

It's roughly in order of how I teach them so it should be easy to add more pages as the year goes on and play again. And I definitely will play again! This kids were really engaged, they said they learned from it and the most telling moment was when a couple kids were hesitant to take the talking role since they knew they didn't know the words well enough. Those two will definitely be doing some studying! (And they did eventually take turns in the role of describer.)

These are the rules we played by:

Taboo Rules

I assume I'm not the only high school teacher whose students have selective hearing, so you do need to go around and 'buzz' kids until they start monitoring each other. One group was just reading the words on the card! I was impressed that they knew all the vocabulary words, but it wasn't very challenging for the describer. They had a good laugh once they learned they were doing the exact opposite of the rules.

We didn't do a great job of forming teams but rather just grouped into clusters. That actually worked out fine since everyone was playing an active role (describing, checking or guessing), but I think next time if we want to have opposing teams they should sit A, B, A, B, A, B in a circle so there's no need to have kids switching seats between rounds. Finally, I didn't have enough timers for each group to have their own, so I just yelled out "Start" then when the time on my phone went off announced "Stop! Tally up your score and switch." And repeated until there was just enough time left in class to fix the desks. Overall I think it was a great activity to get students talking about math, using vocabulary and stretching themselves to do something other than recite a definition they memorized. I would highly recommend playing Taboo with your class. Especially if you want to make cards for the next few chapters in Geometry! Kidding, although I'd love to hear feedback on the taboo words I chose and other words to add to our deck.

Today, we finally played Taboo. I'd been hearing about the great reasons to play Taboo from lots of people, but hadn't wanted to make the cards. Last night I sat down to do it and it wasn't actually as hard as I'd expected. In fact, the worst part was fighting with Word to get the table to stay the way I wanted. I started with this set (google docs link) and then added my own to get this:

Taboo Ch 1-3

It's roughly in order of how I teach them so it should be easy to add more pages as the year goes on and play again. And I definitely will play again! This kids were really engaged, they said they learned from it and the most telling moment was when a couple kids were hesitant to take the talking role since they knew they didn't know the words well enough. Those two will definitely be doing some studying! (And they did eventually take turns in the role of describer.)

These are the rules we played by:

Taboo Rules

I assume I'm not the only high school teacher whose students have selective hearing, so you do need to go around and 'buzz' kids until they start monitoring each other. One group was just reading the words on the card! I was impressed that they knew all the vocabulary words, but it wasn't very challenging for the describer. They had a good laugh once they learned they were doing the exact opposite of the rules.

We didn't do a great job of forming teams but rather just grouped into clusters. That actually worked out fine since everyone was playing an active role (describing, checking or guessing), but I think next time if we want to have opposing teams they should sit A, B, A, B, A, B in a circle so there's no need to have kids switching seats between rounds. Finally, I didn't have enough timers for each group to have their own, so I just yelled out "Start" then when the time on my phone went off announced "Stop! Tally up your score and switch." And repeated until there was just enough time left in class to fix the desks. Overall I think it was a great activity to get students talking about math, using vocabulary and stretching themselves to do something other than recite a definition they memorized. I would highly recommend playing Taboo with your class. Especially if you want to make cards for the next few chapters in Geometry! Kidding, although I'd love to hear feedback on the taboo words I chose and other words to add to our deck.

## Monday, November 14, 2011

### I don't share well

A post on how I fail at co-teaching.

My school does an awesome job of supporting students with learning disabilities by offering courses co-taught by a content teacher and a special education teacher. This year I have 2 co-taught Geometry courses and I have the same co-teacher for both of those. I also have an "Algebra 1" (in quotes because we're not exactly at that level) course for students with moderate to severe learning disabilities. That course has 2 special education teachers so we've split it into 2 groups with 1 teacher (plus several paraprofessionals) in each class and me jumping between them.

The title of this post is actually true in a far more general sense than just co-teaching. I like to do things my way, I'm quite stubborn when I get my mind set on something and most of the time I'm rather independent. I do enjoy collaborating and I love the support of having another teacher in the room, but I'm still stuck in the mindset of considering my co-teachers support as opposed to equal level players.

I'd love to hear some ideas on how people have found a good balance in a shared classroom.

Some issues that have come up recently:

Grades were due last week so my Geometry co-teacher offered to do some of the grading. Problem was, I like grading those classes better than my others since they go fast (smaller class size and my other class is working on proofs- so glad I'm not an English teacher!) so I'd already graded almost everything. She later shared that she'd really like to do some of the grading since she wants to have a better sense of how everyone is doing. This struck me as totally obvious yet I'd never realized it - I'd been hoarding the grading since I want to see how the kids are doing, but she should get to share those insights! So now we're going to split it so that one of us does tests and the other does test corrections, then we both get to see. (I know, poor me, I have to give away some of my grading.)

I talked to one teacher for "Algebra 1" and shared what I thought the kids were ready to do next. She said that she was happy to put together a worksheet since I would be working with the other group the following class. When I talked to some people who had been in that class later she had gone and done something totally different than what we discussed. I was really frustrated that she didn't follow the plan that we had made, but I realized that most of the time I leave her in the dark and just show up with something to do that day. In my head I have an arc that I'm following and it all flows, but this probably isn't clear to her (especially since math isn't her area of expertise). I'm going to really try to work on communicating my goals and hope that she will offer me the courtesy of doing the same so when changes need to be made they aren't a surprise to either of us.

A success:

I've been using dropbox with the other teacher for "Algebra 1" so she gets real time updates as I create worksheets, write to-do lists and formulate plans. She did her practicum in middle school math and studied computer science, so she can see the underlying structures I'm putting in place when random files pop up on her computer. (I'm also using dropbox with some of the teachers in the math department to share everything I'm doing in my current courses and some projects I've used in the past that may apply to the courses we don't have in common. I do at least share resources well!)

I know that part of the problem I'm having is a lack of time to sit down and discuss everything with my co-teachers. We need to make that a priority in the future. Otherwise, I'm having a hard time not making all the decisions and monopolizing the small amount of teacher-centered time there is in ~~my~~ our classroom. Advice? Personal anecdotes? Articles I can share to get the conversation rolling?

## Thursday, November 10, 2011

### A Variety of Variables

This year I am teaching a course for students with moderate to severe learning disabilities. We are supposed to be studying Algebra and so we are working on the concept of a variable. I've found that many students have a really hard time understanding variables and their purpose.

I can think of three different ways to interpret variables so far, and so I'm trying to provide situations that promote comprehension of variables in each context.

1. A variable can be used to generalize, in this case it is a representation of any and all numbers. For this situation we did number tricks:

Pick a number. Add 6. Multiply by two. Subtract 4. Divide by two. Subtract your original number.

Students quickly realize that they keep getting 4, but in order to know it always works, they need something to hold the place of their original number. I talk about using a variable instead of spending the rest of your life checking numbers since you can put any number in the place of the variable and it will still work.

(Side note: this is a great way to introduce proof in Geometry since they actually see why they would want to prove something- it seems clear but they don't know why it works.)

2. A variable can be a number that changes. It could be something that varies over time, or that is different for different people. I came across this example rather circuitously. I found a worksheet translating verbal expressions into algebraic ones, but I also wanted students to substitute and evaluate the expressions. Problem was, the original author did a really awesome job of choosing different letters, so much so I didn't feel like writing in values for every variable. Then it dawned on me, there's an easy way to assign a number to each letter- a cipher! My non-math major friends in college all took cryptology which meant that I got to learn along with them and I've been surprised how often I've used ideas from that class in new situations. Using a cipher to decide the numbers to substitute did a few things- first it was a cool mini history lesson on codes, second it allowed me to easily change the values and show that we could make the same expression simplify to different things depending on the "key of the day."

**Edit- read the awesome comments below, I'm leaving #3 in its original form so you know what the comments are in reference to, but I'm no longer counting this as a valid category.

3. A variable can represent a specific number that we don't know. This is the case for most equations that we have students solve. We know the value of the variable, their goal is to find it. To introduce this concept we started by solving really simple word problems (Chris has 5 apples, Josh has 3, how many do they have together?) by writing an expression equal to a variable (5+3=A). The word problems have increased in difficulty but the idea is the same, that letter represents some specific value we are trying to determine.

I have no idea if this is a standard way of dividing up the roles variables can play, it's definitely something I'm still trying to figure out. But my goal is for students to see many different ways to approach solving problems using variables. And then, somehow, we need to merge all of these ideas into one concept of symbol represents number(s).

Finally, I'm hoping they will understand that all of these methods apply in any situation. Just because you have a number to substitute for your variable doesn't mean that substituting is the best first step. Frequently simplifying and solving before substituting can show structure (just like delayed evaluation when you only have numbers). Conversely, even if a variable is representing a particular number you need to find, guessing random numbers isn't a bad way to start out. For students who have no idea how to approach a problem having them try their favorite number will usually give some insight on the steps to solve a problem (which they can eventually generalize to an equation using a variable).

What misconceptions do you see when students are using variables? What other situations can I introduce that use variables in a different way?

"Just a darn minute! Yesterday you said x equals two!"

I can think of three different ways to interpret variables so far, and so I'm trying to provide situations that promote comprehension of variables in each context.

1. A variable can be used to generalize, in this case it is a representation of any and all numbers. For this situation we did number tricks:

Pick a number. Add 6. Multiply by two. Subtract 4. Divide by two. Subtract your original number.

Students quickly realize that they keep getting 4, but in order to know it always works, they need something to hold the place of their original number. I talk about using a variable instead of spending the rest of your life checking numbers since you can put any number in the place of the variable and it will still work.

(Side note: this is a great way to introduce proof in Geometry since they actually see why they would want to prove something- it seems clear but they don't know why it works.)

2. A variable can be a number that changes. It could be something that varies over time, or that is different for different people. I came across this example rather circuitously. I found a worksheet translating verbal expressions into algebraic ones, but I also wanted students to substitute and evaluate the expressions. Problem was, the original author did a really awesome job of choosing different letters, so much so I didn't feel like writing in values for every variable. Then it dawned on me, there's an easy way to assign a number to each letter- a cipher! My non-math major friends in college all took cryptology which meant that I got to learn along with them and I've been surprised how often I've used ideas from that class in new situations. Using a cipher to decide the numbers to substitute did a few things- first it was a cool mini history lesson on codes, second it allowed me to easily change the values and show that we could make the same expression simplify to different things depending on the "key of the day."

**Edit- read the awesome comments below, I'm leaving #3 in its original form so you know what the comments are in reference to, but I'm no longer counting this as a valid category.

3. A variable can represent a specific number that we don't know. This is the case for most equations that we have students solve. We know the value of the variable, their goal is to find it. To introduce this concept we started by solving really simple word problems (Chris has 5 apples, Josh has 3, how many do they have together?) by writing an expression equal to a variable (5+3=A). The word problems have increased in difficulty but the idea is the same, that letter represents some specific value we are trying to determine.

I have no idea if this is a standard way of dividing up the roles variables can play, it's definitely something I'm still trying to figure out. But my goal is for students to see many different ways to approach solving problems using variables. And then, somehow, we need to merge all of these ideas into one concept of symbol represents number(s).

Finally, I'm hoping they will understand that all of these methods apply in any situation. Just because you have a number to substitute for your variable doesn't mean that substituting is the best first step. Frequently simplifying and solving before substituting can show structure (just like delayed evaluation when you only have numbers). Conversely, even if a variable is representing a particular number you need to find, guessing random numbers isn't a bad way to start out. For students who have no idea how to approach a problem having them try their favorite number will usually give some insight on the steps to solve a problem (which they can eventually generalize to an equation using a variable).

What misconceptions do you see when students are using variables? What other situations can I introduce that use variables in a different way?

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