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.

## November 21, 2011

## 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?

## 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?

## November 7, 2011

### How to Study for Math

Last week I gave a test on proofs. It went poorly. As I graded them I felt badly for rushing the test to get it in before the end of the quarter. When I returned them, I apologized for giving the test before everyone was ready but made sure to say that I was sharing the blame, they needed to take responsibility too and tell me when they didn't understand what we were doing. I've had a better sense of what everyone knows from regular quizzes this year, but I try to make those 3 short questions while the test has multi-step and cumulative problems, which is where everyone got stuck.

All this was fine, until I started reading their test corrections. The first question on the page asks "How did you study for the test?" Page after page had answers such as "I didn't" or "I read my notes" or "I flipped through notes right before the test." Now I know that most high schoolers don't know how to study for math, so from the beginning of the year I talk about how to organize notes into two columns with vocab on one side and definitions on the other so they can easily skim and quiz themselves. We make a study guide together the class before the test (which gives them 2 nights to study thanks to block scheduling). I make them write out the study guide even if they have nice notes because I know (and share) that the act of writing helps implant information in the brain. I talk to them about active vs. passive methods of studying. I specifically assign the practice test in the book. But, after all of this I get "I looked over my notes" as the sole method of studying. I no longer felt guilty for rushing the test, but frustrated with my students for not taking responsibility by preparing for the test.

So, I decided I must not be enough of an expert- I'd need something more official or more flashy to convince them. So today I provided just that. First I took this article (direct link to pdf: How to Study Math by Paul Dawkins) and broke it into 4 sections. We did a jigsaw where each kid was assigned a page to read and annotate (underline things you currently do, circle things you could do for the next test) and compared notes with people who read the same page. Then, they got into groups of four and shared out. This activity made me want to be an English teacher - they read, made notes and talked to each other! All of English must be so easy! Then I came back to my senses, I don't envy English teachers at all, but it was fun to read and discuss something.

How to Study Math

After students shared a few of the most interesting parts of their page with the whole class I showed them this diagram:

I hope that the quote and the percentages really hit home. Maybe now they'll start practicing vocabulary words as soon as they get them? And do actual practice problems since the best way to learn is by doing? Maybe?

At the very least it was a productive 30 minutes of students reflecting on how they study and being exposed to some other options from sources other than me (who they have to listen to every day). I'll let you know how the next test goes!

All this was fine, until I started reading their test corrections. The first question on the page asks "How did you study for the test?" Page after page had answers such as "I didn't" or "I read my notes" or "I flipped through notes right before the test." Now I know that most high schoolers don't know how to study for math, so from the beginning of the year I talk about how to organize notes into two columns with vocab on one side and definitions on the other so they can easily skim and quiz themselves. We make a study guide together the class before the test (which gives them 2 nights to study thanks to block scheduling). I make them write out the study guide even if they have nice notes because I know (and share) that the act of writing helps implant information in the brain. I talk to them about active vs. passive methods of studying. I specifically assign the practice test in the book. But, after all of this I get "I looked over my notes" as the sole method of studying. I no longer felt guilty for rushing the test, but frustrated with my students for not taking responsibility by preparing for the test.

So, I decided I must not be enough of an expert- I'd need something more official or more flashy to convince them. So today I provided just that. First I took this article (direct link to pdf: How to Study Math by Paul Dawkins) and broke it into 4 sections. We did a jigsaw where each kid was assigned a page to read and annotate (underline things you currently do, circle things you could do for the next test) and compared notes with people who read the same page. Then, they got into groups of four and shared out. This activity made me want to be an English teacher - they read, made notes and talked to each other! All of English must be so easy! Then I came back to my senses, I don't envy English teachers at all, but it was fun to read and discuss something.

How to Study Math

After students shared a few of the most interesting parts of their page with the whole class I showed them this diagram:

**Edit (8/6/12 7:30 pm) I just found out the Cone of Learning has no basis in research. Debating if I should white out the percents or toss it entirely. |

I hope that the quote and the percentages really hit home. Maybe now they'll start practicing vocabulary words as soon as they get them? And do actual practice problems since the best way to learn is by doing? Maybe?

At the very least it was a productive 30 minutes of students reflecting on how they study and being exposed to some other options from sources other than me (who they have to listen to every day). I'll let you know how the next test goes!

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