Technology in the MS Classroom #msSunFun

See all the other #msSunFun submissions for this week

iPad apps (I have an iPad, my students do not):

• Doceri: I just started using this last week after reading about it on @danbowdoin's blog. So far, I've made a few video examples for students to watch as review and for a sub to show when I was away on a field trip one day. It is incredibly easy to use, which is a huge selling point for me. You hit the record button and write with a stylus or finger while commenting on what you're writing. Hit stop when you're done and upload the whole thing to YouTube. Done and done. We are slowly, but surely, moving to 1-to-1 for students, so this would be a cool app for students to use to create videos for each other.
• AirPlay: This allows me to mirror my iPad screen on my projector via an Apple TV device. I'm actually still waiting for the Apple TV to be installed in my room (any day now, IS...), but I've played around with this before asking for my own and it's awesome. You have the ability to project stuff wirelessly from anywhere in the room instead of being tethered to where the computer is plugged in. Although, I just saw that you can do this type of iPad mirroring without the Apple TV. If someone does this, let me know in the comments because I'm thinking of trying it also while I wait for that Apple TV to get installed.

Classroom Management Technology:

• Edmodo: I'm using it for the first time this year, and it's so much better than my old class webpage for providing kids with easy access to class materials and encouraging interaction outside of class. Students often post questions (sometimes, they take a picture of their work using their phone or iPad and post that too) and answer each other's questions. They have arguments about homework problems (whose answer is right??) and post interesting math questions they are wondering about. So, so cool.
• Google docs: I use these to survey students anonymously through Google Forms, get input on what music they want to listen to in class, and have them write and share reflections and conference preparation documents with me. I also use it to share documents with other teachers (for example, the other Algebra 1 teacher and I have a shared spreadsheet we use to plan out the unit), such as notes from meetings and committee reports.

Organization Technology:

• Evernote: Started using this over the summer, and it's helped me be much more organized this year. Because I have it on my phone, iPad, and laptop and all the accounts are synced, I can track cool teaching ideas from twitter or blogs, information about students (including the missing homework form idea I stole from @approx_normal - I take a picture of each one and stick that in the note for that student), textbook sign-out sheets, and notes from meetings/conferences, not to mention personal stuff, like receipts, restaurants, books/articles I want to read, and vacation planning materials.
• Hackpad: I blogged about this app earlier this year. I use it similarly to google docs, but it has the added feature that the name of the person writing shows up next to their text so it's really handy when you want to know which person wrote which part of a document.
• Google calendar: Our school recently moved to Google Apps for Education, so everyone has a school google account. We use shared google calendars to keep track of shared events (field trips, meetings, and advisory activities) and large tests and projects to help make sure that we're not over scheduling kids or putting too many assessments on one day. I also need this for my own personal organization to keep from imploding. All events in my life go here and my husband and I can see & edit each other's calendars to keep from double-booking ourselves or our kid.

Default mode

This week was a little disappointing. I had been very excited about how the year had started - I put in a lot of work this summer to revise the first big chapter (exponent properties and polynomial operations) to make it more constructivist and engaging. There was lots of group work (including using my new big whiteboards!!), writing and reflection questions, and classes that built from exploration --> sharing out --> a teeny bit of summary from me to pull it all together. It felt organic and like the ideas were coming from the students themselves. Almost all of the students did a really great job pulling everything together for the chapter test - the average was much higher than last year.

Working polynomial problems in their groups

I had students give me anonymous feedback via a google form towards the end of the chapter (yay for actually doing one of my goals for the year!) and was happy to see how positive students were feeling about the class and their understanding of the material. My favorite quotes:

"I think that we should keep spending a lot of time going over homework and learning by making mistakes in the notes."

"I like how you have us interacting with others with our groups to solve problems."

"I liked how last year we consistently did notes, like a pattern, and at the beginning of this year, it was hard to adjust to what we do now. But now I realize that I like figuring things out in my own ways, not just following the repetitive steps on the notes."

Given how well things were going last chapter, I was surprised with how meh this week has gone. Classes felt boring and I noticed that I was doing lots and lots of the talking and that students were passively completing problems with little engagement and interaction. Then I gave a short & sweet quiz on Friday on some early factoring concepts (factoring by GCF and factoring by grouping). Holy cow. So awful. So so so so awful. It's like they learned almost nothing this week. I can count on one hand the number of students who did even remotely well on this quiz. Thinking things through, I've realized that I've completely reverted to my default mode of teaching - here students, I'm going to work through some example problems, and you just follow along. Now, you try some and I will help you if you get stuck. Oh look, class is over. Let's do that again tomorrow. Ughhhhh!! 

This quiz was a good wake-up call for me that old habits die hard and that I need to be vigilant and keep the big picture in mind for how I want class to go and what I need to do to make sure that it's student-centered and engaging and that students are actually learning and not just following along mindlessly. My plan for Monday is to provide one actual incorrect approach for each problem from the quiz (one per group) and have students analyze the error and explain what went wrong and how to do it correctly to the class. I also want to talk about study strategies because I definitely got the sense that few students actually reviewed for this quiz or did so in an effective manner. I made this handout to help them think through studying for math quizzes and tests:

Study for Math

Then, it's back to the drawing board for me to plan the rest of the week's lessons with what I've learned this past week in mind. It's nice knowing that every day, I get a fresh start and a chance to get things right.

Writing and reflection

One of the goals that I set for myself this year was to make writing a more regular part of my class, rather than the add-on journal entries I've had students write the past few years. Kids had been resistant to these (a few refused to do them at all) and I felt like I wasn't seeing much improvement as the year went on. Those who were reflective and took the assignments seriously got something out of it, but lots of kids did a crappy job, took a 1/3 or 2/3 score and moved on with their lives.

So this year, I started the first real unit (after reviewing last year's Algebra 1A material) with worksheets that kids started on in class and that had more problems and a reflection piece at the end for them to complete at home. Here's one (adapted very closely from CME Project Algebra):

Ch. 7 Day 1 12-13

I used the same writing prompt each day:

How well did you understand today’s lesson? Use one or more of the following prompts to help you answer this question (write at least a few sentences, include at least one example).

a. One thing that I understand really well from this lesson is…

b. One thing that I didn’t understand at first from this lesson, but now do understand is...

c. One thing that is still confusing to me from this lesson is...

d. Something that I’m wondering about that is related to this lesson is…

Some positives: 

• Every kid is responding to these. Maybe because it's the last question on the assignment and they've already put in all the rest of the work, or because it is an almost daily component of their work and thus normalized, but I'm having much less resistance to writing this year.
• I feel like I'm getting a better understanding of kids' misconceptions and questions. Yes, there are some who always say "I understand everything. Here's a trivial example." But lots of kids are taking the time to write about a problem type they don't understand or a question they have about the topic that wasn't addressed in class. 

Things that still need to be worked out:

• Getting kids to use the example as evidence for what they are saying in words. I want the response to be a coherent piece of writing with the math embedded in the words, not as an add-on because it's a requirement to include an example. 
• Having kids go deeper in their explanations, rather than just stating a procedure they used (or not explaining what they did at all). I want them to explain why their approach worked (if they're using prompts a and b) or where they got stuck (for prompt c). I would also like them to write more. I think that if I require at least a paragraph minimum, fewer kids (ahem, boys) would be tempted to just pick the easiest example from the notes and try to regurgitate it back to me in the fewest number of words possible.

Here's one of the better ones from last week because this student actually explained their example in detail. Again, I'd like them to go a bit deeper into the "why," but at this point in the year, I'll take it.

So, a few things that I know I need to do to promote better writing:

• Give specific feedback. I've been saying things like, "needs to be longer" or "explain your example," but I should really talk to kids and tell them more specifically what I want them to change.
• Show examples of strong math writing and have kids point out what the person has done well, in addition to things that they can still improve on. 
• Tell kids why it is that I'm having them do math writing. Perhaps it would be helpful to (in general terms) talk about the research on metacognition and learning. 
• Change up the writing prompts and have more writing responses to actual math problems. I just had kids do an investigation in class with the homework assignment being a writeup of the problem, their process, and solution, if any. Once I grade these, I will have a better sense of where they are at with their writing about math and where to go from here.

Any other suggestions??

Making Math Class Easier

One of the things that I've been thinking about lately is this idea of "making math class easier." @ddmeyer asked about people's opinions on foldables earlier today and linked to a blog post that stated, "I really wish I had math notes like these growing up…math would have been so much easier!" Now, I'm obviously not ripping on this woman. I'm sure that she's an awesome and caring teacher. It just really made me think about this idea of "easier" and whether this is what we should be aiming for in math education. I have seen lots of teachers teach "tricks" or "mnemonic devices" that are supposed to help students remember concepts and procedures. A colleague of mine tells her students about colored socks - she pulls two socks out of her drawer and if they are the same color, that's good, but if they are different colors, that's bad. This is supposed to help students remember that when signs are the same, the product is positive, but when signs are different, the product is negative. Most of us have probably seen the little mnemonic device for absolute value inequalities: greatOR and less thAND to "help" kids convert absolute value inequalities to and/or compound inequalities. There are a lot of these hanging around.

To me, foldables are in the same category of "cute" device that will help you remember something. Yes, kids love them. They are easy. All these devices are colorful or cute or make a little rhyme or whatever. But like it or not, I feel that there's a tension between this aspect of math and the side where kids are grappling with rich problems, constructing meaning, and having ownership of ideas. I understand that many teachers use these "shortcuts" as ways to summarize a concept or wrap up a discovery lesson. And hardly anyone gets away from teaching any shortcuts whatsoever. Maybe that's not the point. But I do think that we need to think critically about what we're doing when we emphasize these shortcuts. Even if it comes at the end of a deep, rich lesson, there's going to be a bunch of kids that are going to remember the "trick" superficially, and it's going to be what the lesson was all about in their mind. It's going to train them, to some extent, to expect tricks like that in the future and avoid the harder work of understanding and internalizing the concepts underlying the shortcut.

Yes, a hook can be powerful for motivating student engagement. But I think it can also be junk food that distracts us from the substantive meal, which is not as shiny or easy to digest at first glance. Let's look instead for ways to make the mathematics more profound, more apparent, and more rich for kids. It really doesn't need to be dressed up and tricked out because it's pretty darn awesome on its own.

#msSunFun - Favorite Ways to Practice

Go here to submit yours

This week, we're blogging about ways to get kids to do practice in our classrooms. To be honest, this is so not one of my strengths. I am pretty terrible at coming up with creative ways to hide the fact that they just need to do a bunch of problems right now. I would much rather plan a discovery activity or an application lab, so I tend to treat practice with some annoyance, even though I know it's important. Here are a few classroom structures that I've liked for making practice a bit less dull:

• Speed dating from @k8nowak (students rotate through, pairing up with a different partner each time)
• Matching puzzles from @sqrt_1 (answers to problems are along the edge so students match up a piece with a problem to a piece with an answer)
• Solve Crumple Toss from @k8nowak again (students complete a problem, bring it up to you to check, and if correct, student crumples the sheet and tries to make a basket using the recycling bin or garbage can for points)

My own, much, much less creative go-to structure for practicing problems is the following:

Teacher puts up a problem. Everyone works on it - students may work with anyone else in the room that they want to until everyone is done. A random student is chosen who puts their work under the document camera and explains what they did. If they are correct, the class gets a point. For every one/two/three (depends on how generous I'm feeling) points the class earns, a homework problem is removed from that night's assignment. 

When presenting this activity, I make it very clear that if a student has the wrong solution, it's an issue for the class, not for that student - the class wins or loses as a group. Therefore, everyone has the responsibility of making sure they check what they have with others and get help if they are confused and everyone has the responsibility of checking in with others to make sure that no one is sitting alone and confused. Kids seem to take the idea of group responsibility very seriously when we do this. Maybe it's a middle school thing, but kids are running around the class, talking with each other, arguing about whether their answer is correct or not, and reaching out to kids they see sitting by themselves. The group responsibility piece also makes kids that would rather just sit on the sidelines or be quietly confused work harder and engage more fully since they don't want to let their peers down. When I first tried this, I worried that kids would feel the pressure in a bad way, but instead, it seems to result in an increase in support and encouragement, which makes me feel all warm and fuzzy.

And that's exactly what math class should feel like.

My Favorite Friday - Hackpad to organize meetings & discussions

Submit yours here

This is my first time doing one of these, but I wanted to share a site that I've been using to help organize meetings and discussion groups at school. The site is hackpad.com, and it's basically a wiki (or document that is editable by multiple people), but with some nice, extra features that make it super useful for organization. To use the site, you can either create an account or use an existing Google or Facebook account. The benefit of using one of those (I prefer using a Google account) is that you will stay logged in to Hackpad for as long as you're logged in to the other account. And if there's one thing that I hate the most in the world, it's logging into accounts. 

Uhhh... my password?????

So once you're in, you can start making "pads," which are basically blank documents that you can share with other people. The awesomestestest thing about Hackpad is that once the people you're sharing them with make accounts, their name automatically appears next to whatever text they've entered. Here's a screen grab from a pad I'm using in a professional development group on young adult literature:

A few things to note:

• There are several different privacy levels, from public to those with link only, to those on a pre-approved list only. The default setting is private (only you can see it).
• Bolding an entire line of text automatically makes it an entry in a table of contents, which is super useful for organizing long pads. (See the table of contents on the right-hand side above)
• It's super easy to embed links, images, tables, and videos - the site recognizes the format and everything is embedded in a single document.
• You can create to-do lists with check-boxes.
• When changes to the pad are made, all of the people signed-up for that pad get an email with what the changes are. They can edit by going directly to the pad or by replying to the email.
• You can group pads into collections (basically, like folders - the pad above is in my "School stuff" collection).
• You can call up specific people using the @ symbol - if they have an account, their name will pop up from a drop-down menu and they will get a notification email that says they were called up in a pad (basically like tagging in Facebook or Twitter). This is super useful if you have a question or comment for a specific person and want to make sure they see it.
• People can edit at the same time and the changes are recorded in real time. There is no need to save - it is all automatic.
• You can link to other pads so one can be one initial pad and when it gets too big, you can cut chunks off and make them into individual pads that can still be navigated to from the main one.
• You can view the document as a cohesive whole or as a series of changes, ordered from most recent to least recent. This can be helpful if you just want to see what's been changed.

Yes, there are some similarities to google docs in that pads are shared and editable by many people. The main benefits that I see with Hackpad are:

• You don't need a Google account so this can be used with lots of different groups of people with varying levels of tech-savviness. People can access it using whatever means is easiest for them (Google account, Facebook account, or with an email address & password).
• The name of the writer is automatically shown next to the text they added so it's super easy to see who is saying what in a discussion so no more typing in third person, or trying to figure out who "I" is in a document.
• You can "tag" (or call up) specific people.
• Ability to view recent changes.

Some drawbacks that I've found are that if you have a lot of pads (which I do), you can get inundated with emails for updates being made to each one (to fix this, you can set a specific pad to not get notification emails) and the site does not play well with Android mobile devices. I'm hoping that they will come out with an Android app sooner rather than later, since that would make it way more convenient for me to use on the go.

I wrote a blog post for them recently specifically about how I use Hackpad in an education setting: https://hackpad.com/7sD02CpiqYk#Using-hackpad-in-education.

Reality vs. Ideal Classroom

This has been a challenging year so far. Exhibit A: I have not been on Twitter much since school started almost 2 weeks ago. Exhibit B: I have not blogged in almost 4 weeks. Exhibit C: I haven't responded to any comments made on my blog from the past 3 weeks. Apologies, all around. And I promise to get to those comments really soon.

What I'm struggling with the most this year is that my expectations and goals are much higher than they have been since my first few years of teaching and they're running head first into the reality of school life. I blogged about some of my big plans here, here, and here. The basic gist is that I wanted my students to process at a deeper level this year, involving more writing and more problem solving, and I reworked some of the curriculum to reflect these changes. The issue that I'm having is that these goals are very difficult to achieve in the constraints of the current system. One of my courses is an accelerated Algebra 1 class taught to 7th graders, who I see for 45 minutes, 4 times each week (which doesn't include time lost due to conferences, field trips, assemblies, etc). Although I am not "teaching to the test," per se, my students do need to be able to do well in their following (accelerated) math classes and gain a reasonably strong foundation in Algebra 1 content, as well as study skills, organization, and ability to show work clearly and using standard notation. And I cannot seem to find the time to both teach a high level of content understanding and skill development while incorporating actual problem solving, writing, discussion, group work, labs, and all of the other components that I think are crucial to a rich, exciting middle school math class. 45 minutes is just not enough time to go over homework, use a problem-solving or groupwork based approach to teaching a concept, and assess students' understanding of said concept before I assign problems to be completed independently at home. Every day, I am rushing through things that just need more time to stew and develop in students' minds, telling them the conclusion because class actually ended 2 minutes ago, and they need to be able to do their assigned homework for the night, and I can't possibly write them all a late pass yet again, just because I desperately want more time for them to discover the conclusion themselves and own it on their own terms.

what I feel like this year

I know one possible solution: cut stuff out of the curriculum. Are rational functions and equations really that important for Algebra 1 students to master? What other content is really Algebra 2 material that's been pushed down into Algebra 1? What things do you cut or wish you could cut? Any other suggestions out there? (I've already petitioned for a change in the schedule that would allow for more time for math, and have been told "soon" for about 8 years.)

MS SunFun - Advisory

Our Middle School has a fairly developed advisory program in the 6th - 8th grades. This will be my seventh year as an advisor, and I really, really love it. Which is very surprising to me because it was the thing I was most nervous about when starting at this school - I know math, I don't necessarily know adolescents and their crazy thinking and feelings and struggles. But it's turned out much, much better than I had feared, and I would not like to go back to just being a classroom teacher like I was before. I really enjoy the close relationships that I develop with advisees and the community that advisory becomes as the year progresses.

Some aspects of the program:

• Advisors welcome students to a new year with either a phone call or a letter sent home before school starts. Here is my letter that I'm sending out in a few days, but I hand wrote each one on a cute notecard. Kids love getting mail!
• Advisory meets first thing every morning for 10 minutes to go through announcements and check in with students and for 45 minutes twice a week ("extended advisory").
• Advisors meet with students and parents twice a year for Parent-Advisor-Student conferences, led by the student. They discuss the student's progress and other issues that are affecting them. 8th grade advisors (that's me!) also help students register for high school classes if they are continuing into our high school.
• The advisor is basically the touch point between the student and family and the school. The advisor keeps tabs on how the student is doing academically (via other teachers), behaviorally (via the assistant principal), and emotionally (via the counselor). Concerns about the student are supposed to go to the advisor first, either from other teachers or from parents.
• During extended advisory, we do activities that are related to the advisory curriculum (more on that below), play games, play outside, or meet one-on-one with advisees. Two big things that the 8th graders also do during advisory is participate in a Little Buddies program with a younger class and do community service projects, like helping out at a food pantry. One of the extended advisories takes place on a Friday morning and students take turns bringing in breakfast so that the advisory can sit down and eat breakfast together.
• The curriculum is pretty loose, but tries to hit the following topics: 
• Learning/study strategies, goal setting, other academic type skills, including preparing to lead conferences
• Executive functioning, organization (a lot of our students struggle with this)
• Risky behaviors (sex, drugs, and rock & roll)
• Relationships (navigating friendships & dating, cliques and excluding others)
• Bullying & aggressive behavior
• Online stuff (navigating social media, safety, civility in a digital world)
• Media literacy, including being a smart consumer
• Body image & eating disorders
• Diversity & inclusion
• Leadership & communication skills
• 8th grade advisors select a book for each of their advisees as a graduation gift (the school pays for this). This is one of my fave traditions, but it takes me forever to come up with the perfect book for each kid.

I'm currently organizing all of the 8th grade advisory resources into digital form since we've had physical binders & folders for a long time, and will post an update once they have been migrated to the web in case anyone would like to use them.

NBI Post #2: Something That I'm Proud Of

Seems like the New Blogger Initiative has gotten started with a bang... my Google Reader is bursting at the seams and I'm seeing lots of new faces on Twitter. So here we go with entry #2. I chose the first prompt:

Find one worksheet or activity or test or unit or question or powerpoint slide or syllabus or anything that you are proud of. Share it.

I cheated because I couldn't pick just one, and had to settle for two that are very connected. So my favorite sequence of lessons to teach are on the topic of slope-intercept form of a linear equation. I feel like there's so much richness there, in terms of patterns, real-world applications, and connections to previous and future topics that I've always enjoyed teaching it. My main problem has been time constraints hitting against my desire to do a million different activities with this topic. Last year, this was the sequence that I used:

1) Introduce patterns that grow in a linear fashion. Students are in groups and need to predict the previous and the next figures in the pattern. Then, they need to explain the pattern - what changes? what stays the same? Then, they describe the 100th figure in the pattern and generalize to the xth figure. Repeat for a few more patterns that are still linear, but either grow faster or slower or start with a different number of tiles. We make a table showing the data (figure # versus # of tiles), graph it, and then all the awesomeness gets even more so when we start connecting and comparing all of the different representations and finally discuss the equation for each pattern and how it shows this information.

Intro to Slope-Intercept Form

I really like this activity because it is so group-focused - all I need to do is moderate the discussions, and all of the discovery and thinking comes from the students. The tasks are also low-entry and kids that maybe typically don't participate much seem to enjoy the visual patterns and predictions. I love days when I feel like the students are running the classroom and I see intrinsic engagement.

2) The next day, students complete a lab-type activity in groups, called "Linear Walks." They use motion detectors to visualize the relationship between time and distance and better understand why the graph of an equation in slope-intercept form looks the way that it does. This was adapted from the Discovering Algebra textbook, but I've seen versions of it in lots of places.

Linear Walks Lab

This is also a super fun day for me because there's such a clear connection for students between the algebraic reality (variables and equations and such) and what's actually going on in front of them. It's so clear why the graph of y = 0.5x + 2 looks the way that it does since it represents someone standing 2 meters away from the motion detector and increasing their distance by 0.5 meters every second. It also connects nicely to when we discuss point-slope form of an equation a few lessons later. An equation like y = 0.5(x – 1) + 2 now means that someone standing 2 meters away from the motion detector waited 1 second (so they lost 1 second of time, hence we subtract 1 from x) and then started increasing their distance by 0.5 meters every second.

I love that these two lessons make sense of an abstract concept like y = mx + b without memorization or "tricks," but rather through understanding of patterns and physical concepts like movement over time. It gives me a nice contextual handle to refer back to throughout the chapter: "If your graph represented someone walking, would their distance be increasing or decreasing over time?" "If your equation represented a pattern, how many tiles would it have started with?"

I'd love to hear how others teach this topic and if you have any feedback or criticism of these lessons.

New Blogger Initiative - Post #1 on First Week Goals

Super excited for the New Blogger Initiative that @samjshah has started up! I'm fairly new to blogging (started about 3 months ago), and it's wonderful to be initiated into the mathtwitterblogosphere and to be harangued & threatened with whacking if I don't keep up with my blog! Umm, I think it's with love?

Anyway, without further ado, here is my big goal for the first week of school, which is in about two and a half weeks:

Create a positive classroom culture where students feel comfortable, confident, and cared for by me and each other.

Yup, that's a raccoon group hug. 

Many of my students have struggled with math in the past or have learned that it is a weird, arbitrary set of rules that they have to memorize and regurgitate as best as they can and that their creativity, passion, and intellect don't have much of a place. Yes, it's a bit of a tall order for the first week, but I want students to have a sense of our classroom as a place where things make sense, where they are smart and capable, and where people care about each other. Since the first unit for all of my sections will focus on review, it gives me lots of opportunities for activities that emphasize collaboration, creativity, and engaging thinking. I also want to be sure to create a sense of order and safety in how the class is run, both in terms of procedures that simplify our day-to-day structures and in terms of how mistakes are received and feedback is given. Obviously, as the year goes on, I'm going to be looking at students' learning and ability to communicate mathematically, and all of the big goals that I outlined for myself earlier, but for the start of the year, I would love to just see students feeling positive.

msSunFun #3: Goals for the School Year

I'm so glad that the theme for this week was changed to goal-setting for the new school year because this is something that I've needed to sit down and write for a while now, and this was the perfect kick-in-the-butt to get myself to do it. I have set goals for a few years now, but this year, I'd like to go back and you know, actually see how I'm doing. So maybe there will be a prompt later in the year to check in on our goals?

I have two overarching goals this year:

1. Richer Mathematics
I would like to deepen the curriculum, to push for understanding that is more abiding and less surface-level or focused on discrete skills. The specific ways that I hope to achieve this are by having students do more:
• writing, processing, reflecting, and explaining
  
  We already do a lot of this in class and I've required students to do journal writing for two years now, but I want to make this part of daily homework assignments and incorporate into assessments. I don't want writing and reflecting to be an add-on that happens every week or two, but incorporated into the fabric of the class. To that end, I will be asking students to respond orally and in writing to prompts at the end of most class periods and as part of most homework assignments. I will be asking students to make videos where they explain their approach to a problem. I would also like to put more "explain this" type questions on tests.
  
  
• problem-solving
  
  In my previous post, I wrote about the various different approaches that I've tried to incorporate rich problems and tasks into my classrooms, and how I plan to use them this year. The basic gist is that I want to use more problems that are content-related in the classroom, pose more problems for kids to think about outside of the classroom, and continue to provide extra, "fun" problems to interested kids. I think that the group-sized whiteboards I made this year will help encourage better groupwork and communication about problems between students. I'm still thinking about how to assess students' work when assigning more difficult, open-ended problems, both in terms of giving good feedback and in terms of coming up with a grade of some sort at the end.
  
  
2. Communication
I would like for there to be more dialog between myself and students, more opportunities for them to give feedback on how they are doing and what they need and for me to communicate more clearly and more often back to them how they are doing in the class and what they should be working on to improve. 

Last year, I had time to meet with students in the two-year Algebra sequence about once a week to discuss how they were doing and what I wanted to them to work on, but it wasn't until the end of the year that I realized that I was doing a lot of the work for them (keeping track of missing assignments & assignments that should be corrected, as well as assessments that needed to be retested) and that they were depending on me to tell them what to do. Last year, I started making them keep track of this themselves and even gave points for having a pretty clear picture of where they were at when I checked in with them. I want to start this much earlier this year. 

I was also very unsystematic about reassessing - there wasn't a clear schedule and I didn't always follow up with students who blew it off. I would like to be more organized this year - I will have a calendar where students who miss assessments or those who are reassessing will sign up, and keep better track of students who need to reassess but avoid doing so. 

I would also like to encourage students to communicate with me about their needs. I'll be using Edmodo for the first time this year, which will allow me to periodically post surveys or questionnaires to get more feedback from students. I'm planning on taking more pictures and notes during class and sharing my observations with students throughout the year rather than just at report card time. I'm also toying with the idea of involving parents more, either through Edmodo (which allows for parent accounts) or by using Evernote to keep track of the student photos and notes and emailing them to families. I need to think about this a bit more - I'd love to hear how others choose to involve (or not involve) parents and why.

Integrating problem solving into the curriculum

Like many others (@fawnpnguyen posted recently about her approach and there were some great discussions in the comments), I have wrestled with the question of how to integrate problem solving into my teaching. The master's program through which I was trained as a teacher heavily emphasized students engaging with rich, multi-entry tasks that promoted collaboration, writing, and connections between different approaches and ideas. I strongly believe this type of work should be a vital part of every math class. At some point soon, I hope that the Global Math Department will have a presentation on how to lead/organize problem solving in the classroom. Here are the different ways that I've used rich problems in the past:

1. Found problems that connected directly with the content material that was already part of the course.
   There are many problems that lend themselves to the content found in traditional MS and HS classes. For example, many of the problems in the Interactive Mathematics Program, Years 1 and 2, lead to students creating rules for specific scenarios or functions, including linear, exponential, and inverse ones. The Mathematics in Context and Connected Mathematics series have some great problems that can be integrated into traditional Pre-Algebra and Algebra 1 classes. The drawback with trying to connect everything back to the traditional content is that there's lots of material for which I have not found good problems, such as factoring, operations with rational expressions, and radical functions and expressions. Back when I taught Algebra 2 and Pre-Calculus, I had similar difficulties finding rich problems for much of the content. There's also the issue of time - I'd like to ideally have at least one rich problem every week or two, which eats up a lot of my class time if done well. Finally, using only problems that have a clear connection to the traditional curriculum leaves out a lot of rich, awesome problems that I still want to include.
   
   
2. Assigned problems to be completed outside of class. Some were connected to the traditional content, some were not.
   This gave me a lot more flexibility in terms of good problems to use and took up much less class time. But I never found a good way to support struggling students, develop the writing and problem-solving skills that are at the core of this type of work, and make explicit the connections between the assigned problems and the rest of the curriculum. The problems gradually petered out as both I and the students lost steam and assigning the problems became stressful and unproductive. If I do this again, I will need to spend some class time teaching students how to wrestle productively with open problems and will probably need to do some ramping, with easier problems at the start of the year.
   
   
3. Provided problems to interested students outside of class. Not required, problems were usually unconnected to the content.
   This was definitely the approach that involved the least amount of work. I had a pretty straightforward system: a folder with copies of the current "Problem of the Week" stapled to the wall outside of my classroom and another folder stapled just below that where students put their completed write-ups. At the end of the week, I would read through the submitted work, write feedback, and award candy to those students who demonstrated good work on the problem. I had a spreadsheet where I kept track of students who completed these. Some positives were that I got kids who weren't even my students to participate, just because they thought it might be interesting, and because it was not required, it was very stress-free and emphasized the "fun" aspect of figuring out math problems. The cons were that there was little connection to the curriculum and the students who participated were those who already enjoyed math and the students who could stand the most to gain from this type of experience avoided it altogether.
   

So, my thoughts for this school year are that I would like to do all three of these options (hooray for overachievers!). A mix of #1 and #2 make the most sense for my class - doing those problems that have a clear content connection in class & spending more time on them, while reserving those awesome, random problems for the times when I can't find anything good that connects to what we're studying. Option #3 can co-exist as optional, more challenging or more "fun" type problems for students to do just because they want more. My biggest enemy right now is time: time in class for students to discuss and time outside of school for students to think and do math and write up their thinking and mathing. Oh, and did I mention that my students only have math for 45 minutes four days a week??? Clearly, I can't just add on more stuff without cutting anything, so I'm wondering how others have found time to do this - what do you cut? 

MS SunFun - Math Class Binders

The theme this week is Student Math Class Notebooks. Instead of notebooks, however, I like for my students to keep a 1 inch 3-ring binder. My reasons for this rather than an Interactive Notebook is that there is no cutting or gluing necessary, which cuts down on supplies needed for class as well as time to cut & glue stuff into the notebook. Instead, all handouts are hole-punched and students have blank hole-punched lined and graph paper to use. The other benefit is that the order can be changed and new pages inserted at any time. If a student is absent, they can just continue with their class work and if they later work on an assignment that happened while they were gone, they can just insert it into the right place. Homework or classwork can be turned in to me and then easily returned to the binder. Students' binders go back and forth between home and school.

The binder is organized into three sections with dividers:

1. Notes/In-class projects (basically, everything that happens in class, but isn't a quiz or test)
2. Homework/Journaling (all assignments that get taken home)
3. Quizzes/Tests, along with corrections and retakes

This year, I will be using Left Hand Page and a Right Hand Page designations for notes, as described by Megan on her blog. Students will use the RHP to write down and work out problems and take notes, and they will use the LHP to process the material, mostly through reflective journaling questions that I will ask at the end of class, as well as a general summary of the day's lesson and a list of any questions that they still have.

Another change for this year is that I will ask students to number the pages in each section and make a table of contents at the front of the In-class section. Since I give them an assignment sheet that lists all of the homework assignments for the unit, that page can be their table of contents for the homework section. I do a binder check every so often (more if the kids seem especially disorganized) where I look to see that they have the three sections organized and that they have blank lined and graph paper, as well as the required supplies for class. During the binder check, I also check in with students to see if they know what assignments, if any, they are missing, and what assignments they have not received full credit on and that they need to correct before the end of the unit. It's part of the grade for the binder check that students have a pretty accurate view of any outstanding work that they need to complete and know what concepts/topics they need to review or correct. My hope is that this helps them see the benefit of having an organized binder and puts more of the responsibility of knowing what they are supposed to do on them.

My other little tip for keeping a binder is that at the end of each unit, students clean out each section, staple them together, and put them in a file folder that I keep for each student in a crate in my classroom. This year, I may ask them to reflect on the unit and create a summary sheet of the most important concepts and skills, which they will put at the front of the packet. At the end of the year, students have a nice folder of review materials that is organized by chapter. I'm not sure yet how to effectively help them use it to review for final exams, so if you have any good ideas about that, I'd love to hear them.

(This is not from my class, but since all of my classroom stuff is still put away for the summer, it will have to do)

Counseling conference thoughts

I've been super busy the last few days attending a counseling conference for teachers and advisors in Colorado. It's been amazingly powerful. We have been working on the skills that will help me be more than just an advisor ("Let's see how we're going to fix this problem..." "Have you tried...?" "When I was a student...") and moving towards real listening and building deeper relationships that will allow students to feel truly connected and understood. This will sometimes result in them processing through their feelings and coming to a solution of some sort. Sometimes, it will mean that "the relationship is the solution," which is a new idea for me. The conference is run by the Stanley H. King Counseling Institute, and I have a few more days in which to practice these newfound skills.

On the first day, we learned about "real listening," which basically involves me talking as little as possible, only saying a few words or a question here or there to continue encouraging the speaker to go deeper and talk more. The next day, we learned about specific skills that would help us do this type of listening. I am actually thinking of making a small handout to post for myself listing these types of responses until they become more internalized:

• Summary: a broadbrush overview of what was said, used to convey that you've got the main idea
• Paraphrase: rephrases what the speaker has said into your own words, this allows the speaker to correct or clarify the listener's misunderstanding (basically, a more detailed version of summarizing)
• Feeling and source: identifies the feeling underlying the speaker's words and the perceived cause of this feeling (can be helpful in pushing the speaker to dig deeper, but have to be careful not to assume or jump too far)
• Clarifying question or statement: helps the speaker better understand what he or she is feeling. This is NOT to satisfy the listener's curiosity - the focus is on the speaker and what he or she needs
• Joining: a statement that shows empathy or shared connection with the speaker's feelings without moving attention away from his or her story (so don't say, "I had a similar experience too," but instead say, "It's really tough when x happens.")

We've done a few role plays channeling students that we struggled to advise over the years, and it was amazing how helpful these techniques were in understanding where the student was coming from and in deepening the listener's relationship with them. I was struck by the difference between this type of relationship building and the type that I usually engage in: discussing common interests, asking kids about their hobbies and athletic pursuits, sharing music or funny videos, etc. These are also good, but they don't promote deep processing and working through issues, which quite a few of my students would benefit from. I also really appreciated the importance of not placating the student or denying their feelings ("I'm sure it's not that bad." "It's okay." "Don't be sad."), which is something I'm certainly guilty of doing. I thought that I was doing a weepy student a kindness by releasing them to go to the bathroom and come back "when they're feeling better and ready for class" (I let them bring a friend! What am I - some kind of monster?), but now I see that I just couldn't handle sitting with their pain and uncomfortable with processing it with them together. This conference is helping me realize that much of what I was doing with my advising and relationship building before was about me, not about the student.

One other thing that I wanted to add that wasn't part of this conference, but that I've been working on for the past few years, is the ability to hear beneath the words that a student or parent says to me rather than interpreting them as objective representations of reality. For example, a student or parent might be upset and angry about a test grade and might start to argue. In the past, I would answer their points logically and maybe even get defensive and upset myself. I've been learning to hear under their anger to the worry and fear underneath (of failure? of weakness? of lack of control) and ignoring their words and dealing with what I'm sensing as their true reality: "I can tell that it was very important for you to do well on this test and that you're worried about your progress in the class. Let's talk about ways that we can work together to help you with that." It was amazing how quickly those types of stressful situations deescalated once we started working on the underlying issues and not the details on the surface. I have to remember that almost all of the time, it's really not about me, but about fear or worry that the student or parent is feeling and doesn't know how to resolve. 

Homework Folders - Made 4 Math #5

I think I skipped a few weeks there, but hey, that's just how I roll. Anyways, today's Made 4 Math post is basically the only original thing that I've ever thought of for teaching. I mean, literally ever. I take the "beg, borrow, and steal" mentality of teachers everywhere to a new level, which is why I am so proud of this one idea that is my very own. It came about because my biggest difficulty in teaching (and life) is organization. Loose papers are my kryptonite. So this is my system for keeping kids' homework organized. I buy 4 sets of colored folders ($0.10 each at Office Depot), one set for each of my classes, 7 folders per set.

Each folder will hold the work of one group. I generally have between 4 and 6 groups of 4 students per class, but I buy a few extra folders in case one rips or a kid loses one (more on that later). The groups are based on the kids' seats - the desks are usually arranged in groups of 4. 

The folder is labeled with the kids' names and the class and period number. When we change seats, I just stick a new label on top of the old one and write the names of the new group on it.

The folder has two parts: "In" on the left side is where kids turn in their homework and "Out" on the right side is where I place graded homework. One of my group roles is a Materials Manager. This student's role every day is to pick up his or her group folder from the magazine stands that hold all of the folders.

He or she passes out the graded homework from the right side of the folder to each member of the group and after the group or class goes over homework questions, the Materials Manager inserts everyone's homework papers into the left side in numerical order (each kid in the group is assigned the numbers 1 through 4). My gradebook is set up so that the kids are listed in the order of their groups and their assigned numbers so entering grades is a snap, straight from the group folder. It also means that I never ever ever lose homework - it just moves from the left side of the folder to the right side. I don't have to use any class time to collect or pass out assignments, and if I want to know if a kid turned in their homework that day, I can find out very quickly. I also like that it turns over responsibility to the students.

This also allows me to easily keep track of work for kids who are absent. When I pass out any worksheets or problem sets in class, I pass out a copy for the absent kid too and the Materials Manager puts his or her name on the sheet and places it in the right side of the folder so that it's ready for the student when they return to school. 

The best part is that almost all of the other MS math teachers at my school have adopted this system so now I don't even need to take the 5 minutes I used to explain how it works or remind students to not put any papers on my desk or try to hand them to me (I literally tell students that if they hand me a piece of paper, there is about a 90% chance that this paper will never be seen again by human eyes).

One more logistic thing - I added a responsibility for the class leader (a job that rotates each week) to check at the end of class that all group folders have been returned to their container so that no one walks off with one. Before I added that, I had a group folder disappear for good about once every two years, which is still probably a better statistic than how often I lost homework before I started using this system. Hopefully, this is useful to some new teachers out there, and I'd love to hear how you handle homework organization in your class.

Whiteboarding activities in the MS/Algebra 1 classroom: Part 2

So my last post got a bit long, so I decided to stop at the half-way point of Algebra 1. The second half of the Algebra 1 curriculum, for me, feels way more dry and theoretical and with fewer opportunities for rich, problem-solving tasks that would really make good use of my whiteboards. There are some content-specific ideas, but before I start with those, I also had some general ideas of how to use the large whiteboards in my classroom:

General Ideas:

1. Each student in the group gets a different colored marker and the group's product has to include each color.
2. Each group gets one marker and at least three or four problems. The marker has to rotate through each member of the group.
3. As I circulate, I need to remember to ask random group members questions about the group's work to make sure that everyone is involved.
4. Combine whiteboarding with a "participation quiz," as described on Sam Shah's blog.
5. Provide opportunities for students to give feedback to their groups as to how well they are working together, including everyone, and promoting learning.

(I would love more ideas on how to encourage/require individual accountability in group situations, since this is something that I've struggled with in the past.)

Content-Specific Ideas 

(from the second half of Algebra 1):

1. Investigate possible exponent properties; use numbers to determine which are and which aren't true (adapted from CME Project, Algebra 1).
2. Given a trinomial of the form , students will generate several factorable quadratic polynomials and several non-factorable quadratic polynomials (whenever I say factorable, I am talking about being factorable over ), and a conclusion describing how those that are factorable differ from those that are not. The same can be done with a trinomial of the form .
3. Writing a quadratic equation given its rational solutions. Groups can be challenged to come up with at least 3 or 4 possible equations and an explanation of how they are related to each other and how to describe the set of all possible correct equations.
4. Working on the Exeter problems that can be solved using quadratic functions, such as
   
   or "Is it possible for a rectangle to have a perimeter of 100 feet and an area of 100 square feet? Justify your response."
5. Investigation that leads up to exponential functions (Adapted from IMP, Year 2):
   Phone Tree
   
6. This is a question that I have found to be very helpful in uncovering misunderstandings about percent change and exponential functions, but I think that it would be even more powerful for students to work on it in groups using a whiteboard (adapted from Exeter, Math 1 problem set): 

The population of a small town increased 25% two years ago and then decreased by 25% last year. The population is now 4500 persons. What was the population before the two changes? Show and annotate your work clearly - explain what you are doing and why.

I don't have many interesting or "whiteboard-worthy" tasks involving rational or radical functions and equations, and I'm always looking for more rich content-specific problems that stimulate group discussion and where different approaches or solutions are possible, so please share if you have some that you think might fit that description.

Whiteboarding activities in the MS/Algebra 1 classroom: Part 1

In my most recent post, I discussed how to get cheap whiteboards for student use in the classroom. I made large whiteboards (24" by 32") so that I could use them for group activities and investigations. The logical next question is what would be some activities and investigations that would make good use of this resource? Since my most recent experience is with MS Math classes, specifically Algebra 1, I'm digging through my files and seeing what would work well as a whiteboarding activity.

There are some awesome ideas at http://bowmandickson.com/2011/11/05/experiments-with-math-whiteboarding/. If you know of others, please link to them in the comments or ping me on twitter (@Borschtwithanna).

So, from my files from last year, here are some activities that I think can be adapted to whiteboarding in groups.

1. "1, 2, 3, 4 competition": Groups compete to see how many different ways they can generate the integers -24 through 24 using the numbers 1, 2, 3, and 4 exactly once each and any of the standard operations (addition, subtraction, multiplication, division), as well as parentheses. Sometimes, I let groups use square roots and exponents, as an extra challenge. So for example, doing 1 + 3 • (2 – 4) generates the integer -5. Each different expression is worth one point, even if it generates the same integer. Bonus points if they can generate all of the integers in the set. After time is up, all of the whiteboards will be revealed and groups have a chance to challenge an expression they think is incorrect, getting a point for each error they find. (Adapted from an IMP, Year 1 task)
2. "Number tricks with expressions": students can work on the following problem in their groups:
   
   A magician said to a volunteer from the audience, "Pick a number, but don't tell me what it is. Add 15 to it. Multiply your answer by 3. Subtract 9. Divide by 3. Subtract 8. Now tell me your answer."
   
   "Thirty-two," replied the volunteer.
   
   Then the magician immediately guessed the number that the volunteer had originally chosen.
   
   a. What was the volunteer's number?
   b. How did the magician know so quickly? (The magician couldn't possibly have worked backwards that fast.)
   c. Create your own impressive number trick. Write down the directions that you would give an audience member and explain what you would do to figure out the number that was picked. (Adapted from Discovering Algebra)
3. "Mystery Bags" investigation, which builds towards formal equation solving using the balance scale model. (Adapted from IMP, Year 2)
Mystery Bags
4. "Pauline's Run": fractions, rates, and equation solving. Students could use whiteboards to compare and contrast different methods. (Adapted from IMP, Year 1)
   
   One day, Pauline was walking through a train tunnel on her way to town. Suddenly, she heard the whistle of a train approaching from behind her! Pauline knew that the train always traveled at an even 60 miles per hour. She also knew that she was exactly three-eighths of the way through the tunnel, and she could tell from the train whistle how far the train was from the tunnel. Pauline wasn't sure if she should run forward as fast as she could, or run back to the near end of the tunnel.
   
   Well, she did some lightning-fast calculations, based on how fast she could run and the length of the tunnel. She figured out that whichever way she ran, she would just barely make it out of the tunnel before the train reached her. Whew!!
   
   How fast could Pauline run? Carefully explain how you found your answer.
5. Groups could use whiteboards to solve different versions of one problem so that the class can then discuss how they are related to one another. For example, one group could solve and graph the solution set for each of the following:
   a. 2x – 3 = 13
   b. 2x – 3 > 13
   c. 2x – 3 < 13
   d. |2x – 3| = 13
   e. |2x – 3| > 13
   f. |2x – 3| < 13
   
6. Sketching graphs to model situations and vice versa. For example, having students sketch a variety of graphs representing motion (walking away at a constant speed, standing still, walking away at an accelerating speed, walking toward at various speeds, etc) or having students create stories from a series of graphs representing different people's walks (there are lots of these types of activities in the book A Visual Approach to Functions). Here is a similar set of questions from the Exeter Math 1 packet:
   Graphing Functions
   
7. Introducing slope-intercept form of a line through changing patterns (adapted from The Pattern and Function Connection).
Intro to Slope-Intercept Form
8. Group investigations that can be presented and summarized. One idea is that groups can use graphing calculators and sketch a variety of functions and what happens when you take their absolute value in order to reach a conclusion about the nature of the absolute value function and how it operates on other functions. For example, one group would graph and , another group would graph and , a third group would graph and , and so on. Many types of graphic transformations can be examined by having each group present an example to the class and then synthesizing the investigation into a cohesive conclusion.
   

Since this blog entry is getting a bit long, I'll put the activities from the second half of Algebra 1 into a new post.

Whiteboards for groups & student roles - Made 4 Math Monday #2

Recently, @fnoschese linked to a blog post he wrote just about two years ago, called "The $2 Interactive Whiteboard," where he described the advantages of a large dry-erase board that students can use to work collaboratively and then easily share their findings with the class. You can buy these large whiteboards from various places on the interwebs, but I thought I would try the cheaper option explained in the blog and went to Lowe's, where a 48" by 96" piece of white tile board cost me $11.50.

They cut it for me into 6 pieces that are 24" by 32" each (for free!!), which are just about perfect for two or three students to use for a partner or group activity. The edges weren't sharp, but I put duct tape around them so that the cardboard backing wouldn't get fuzzy and disintegrate.

I'm thinking that students can just stand them up on the marker tray at the blackboard at the front of the room, but if that doesn't work, I can get all fancy and attach two metal hoops to the back of each whiteboard so that it can hang on the hooks at the top of my blackboard.

There are more whiteboarding resources at Whiteboards USA and Whiteboarding in the Classroom, as well as at Frank Noschese's blog, linked above.

The second thing that I worked on today was cleaning up classroom roles and expectations. I've had an assigned class leader for the past few years, who rotates every week, and it's been helpful both in making the class run more smoothly and in fostering students' independence (it's really nice to leave sub directions like, "1. Take attendance 2. Give all handouts to class leader and let her direct class for the rest of the period."). I've gradually moved away from group roles, which has made the class roles more meaningful as well. This year, I've decided to split up some of the jobs into two roles: a Class Leader, who sort of runs many of the starting and ending class routines, and a Materials Manager, who takes care of supplies and handouts/class organization.

Class Leader

Materials Manager

Three Templates - Made 4 Math Monday #1

A few fellow math teachers on Twitter (yay @druinok and @pamjwilson!) came up with the idea of making something for your math class, at least once per week, and sharing it on Monday and attaching the tag #made4math. I decided to put together two templates that have been bouncing around in my head for some time now. The first one is inspired by @mgolding, who has written a lot about interactive notebooks (for example, here and here). I will still have students use a binder system next year rather than interactive notebooks in order to save time and avoid cutting and gluing, but I wanted to move to a more interactive system for note-taking so I created a note-taking template that uses the "input" and "output" notions of interactive notebooks to help kids move from passive note-taking to processing and making sense of what they did in class.

Note Taking Template

The idea is that students would write down what's happening in class in the "input" (right) side, then write down reflections, comments, and questions in the "output" (left) side. I will also have them write an annotated example that either shows what they understand or what is still confusing to them from that day's lesson and to write a summary at the bottom of the last page in their own words of what they learned that day. My plan is to have them do this during the last few minutes of the class period since my experience has been that kids tend to blow these types of assignments off if they are asked to do them for homework.

The second template that I created for today was inspired by @samjshah's post on participation quizzes, where he talked about giving feedback to groups on the quality of their communication and ability to work together on a given mathematical task. The idea is that while students are working in their groups, the teacher is circulating and making notes on an overhead or SmartBoard of the things that are going on in each group, both positive and negative, as well as positive phrases or quotes that are promoting good groupwork. This gives groups instantaneous feedback and an opportunity to learn about group norms and what types of behaviors and ways of talking promote or hinder their ability to work together. I'm hoping to be able to use the template below to organize my feedback to each group and to be able to do this on my iPad while circulating around the room. I've done it before on an overhead and it can get a little hectic to both circulate and keep running to the front of the room to write down comments and observations.

Participation Quiz Template

The final template I made is inspired by stolen from @approx_normal's Homework SWHHW (See, what had happened was...) sheet, which she collects from students who don't have the homework due that day. I thought that this was a great way to keep a clear record for students who habitually neglect their homework beyond what I record in my grade book. This just seems so much more objective and will be in the students' own handwriting too, so no squirreling out of that one during a conference with parents. My optimistic plan, since I am terrible at keeping track of papers, is to take a photo of the form the student fills out in class and then upload it somewhere - possibly Evernote - where I can tag it with the student's name. Then, I can call up all of the missing homework forms for a particular student by searching. In a future world where every student has some sort of digital device in my classroom, I can just have them fill it out electronically, but I'm getting ahead of myself. So, here's my version of the form, just slightly altered from @approx_normal's document.

Missing HW Form

I would love any comments or ideas on how to improve these templates since this is just my first attempt at putting them together.