(1) Earlier this year I had a conversation with an amazing elementary school teacher who was starting her 2nd year. We were sharing first-year teacher stories when she said something to the effect of: “You know, they always say ‘It’s okay – it’ll be so much easier your second year. It gets better’. But they never tell you how it gets better, so I always wondered what it was – what’s this secret thing that happens between your first year and second year that somehow makes it easier. And I still don’t quite know what it is, but I’m in my second year and they’re right – it somehow is easier, but I can’t explain why”.

I’ve been thinking about that because I’m in my second semester and it’s going incredibly smooth and I’ve been trying to figure out why. I’ve pinpointed several little things that really just have to do with personal comfort – I now know how my school and administration works (especially discipline procedures); I know how my own work and sleep schedule works; I know approximately how long it will take to grade papers or make worksheets; I know how my organization works (what things need to be kept within arms reach at all times and what things can be stored in a filing cabinet across the room) – essentially, I know more about myself as a teacher than I did last semester.

Actually, I think that last sentence really describes all parts of this, because I’m also a lot more comfortable and confident with the activities and problems I give in the classroom. I remember experimenting a lot last semester and a year ago during student teaching – am I a teacher who has students come up to the board and do problems? Am I a teacher who uses slates? Am I a teacher who assigns homework every night (note: I’m still working on this one)? Am I a teacher who tolerates electronics in the classroom – and if so, during what times is that okay and during what times is that not okay? The hardest ones to figure out were the little housekeeping procedures – how do I keep track of tardies? Of attendence? Of late work? I was still figuring all of this out last year, but now I have my set list of activities and procedures during class. You could say I have an idea of what my ‘default classroom’ looks like. If we need to do guided practice, I have my 1-2 stock routines to use. If we do an inquiry-based lesson, I have my 1-2 group structures for them to use. I’m no longer experimenting with group roles, participation quizzes, jigsaw activities, scavenger hunts, solve-crumple-toss, or other cool things I could be doing that work well for other teachers but bombed for me (for example: have students come up with their own definitions of geometric objects ala Dan Meyer and MissCalcul8? Totally Bombed). Instead, I’ve found the ones I like and have started just focusing on improving them. Which is a nice feeling. Maybe next year I’ll branch out and try these other strategies again.

By the way – I don’t think I’d be this comfortable now if I didn’t take so many opportunities to experiment when I was in my education classes at the U of A or during Student Teaching (or, honestly, during my first semester – sorry first-semester students, you were the guinea pigs for many of those lessons). I was never the preservice teacher who planned a straightforward lesson to present to my peers – I always experimented with some questioning strategy or group structure. And most of the time it bombed, but at least it bombed in a safe environment instead of in front of 30 grumbling teenagers. And because of that, I learned a lot and made a ton of improvements, so when I tried it the second time, it worked well enough that I could call it a moderate success. Then usually the 3rd time I’ve worked out all the procedural kinks (bad/missing directions, misjudged the amount of time it would take, how explicit to be with modeling, prerequisite knowledge, etc). Anyway, the point of this paragraph is: if you are a preservice teacher, I highly highly highly advise you to take as many risks as possible when you’re still preservice – all it takes is one run-through to get some very concrete data about directions, modeling, and how much time it’ll take. And it’ll make you better by the time you get to inservice.

(2) I mentioned earlier: this semester seems to be going incredibly smooth. I have almost no discipline/management problems. I’m comfortable with the pace I’m going at. My students’ test averages keep going up. On paper and in my head, there’s a lot of good stuff happening…

This is freaking me out. I’m a first year teacher working in a high-needs school. I keep thinking to myself that there is no way my classes should be going this well. I’m quietly waiting for everything to come crashing down somehow – for me to discover that my students have been cheating on every one of my tests, or that they’ve discovered how the ‘play the game of my classroom’, which means getting good grades without actually needed to learn the material. Actually, my biggest fear as a teacher is that my perception of my classroom stops being the reality of my classroom. I’ve seen so many experienced teachers fall into this illusion and I’ll always be scared that it’ll happen to me – I don’t want to be that teacher who stands at the front of the room and asks for a choral response, the same 3 students answer, and I move on because I assume the entire class is paying attention (when really they’re asleep or texting on their cell phones). And then when I do my formative assessments, half the class copies off each other without me noticing so I think the data represents comprehension, when really it represents a lack of attention on my part. I don’t think I’ve become that, but the lightness I feel towards this semester is something I’ve only ever experienced with teachers who have built up this illusion around their classrooms (A colleague of mine has dubbed this illusion Fantasy Teaching, which I think is a good description for it).

I guess another way to express this fear is the following: I know this semester is better than last semester, but I don’t know if it’s good or great or adequate or terrible – I can’t get an objective picture of my classroom. And I fear that if I become complacent with better,  then I might also be becoming complacent with adequate  or mediocre, which isn’t what I want. And it scares me that since I’m always at the center of my classroom, I will never be the person who sees it objectively. I’ll always need an administrator or peppy student teacher to come along and point out the things I’ve started to overlook. Or I guess to be videotaped every so often and watch my classroom like a coach watches his team – nitpicking all the details that you can only see once you’re removed from the game.

(3) I’m still closely watching my students and their grades and their understanding with my switch to exit tickets and the removal of homework. If there’s one thing that I’m concerned about, it’s that they might not be getting as much practice as they need. I need to be very careful about when I do assign homework so I make sure they get enough concrete practice outside of class. So far, everything’s been going pretty smooth – I feel like I’ve established an understanding of ‘You may not have to do much work outside of class, but you will give me your  best effort while you are physically in my room and you will work the entire class period”. What I’m trying to establish now is “When I do give you work outside of class, you will take it seriously and complete it correctly on time”, which is a battle I’m fighting not only with the default state of any teenager (‘I’m lazy and just don’t want to do it’), but also the homework culture of my class from last semester (where I didn’t really care about homework so they started not to care too). Homework turnin still isn’t great, but the students who do turn it in on time have been my A/B students or my hardworking B/C students – which means the data I get from the homework is important. These are problems every student should be able to answer, so if this subset of students gets them incorrect, I can assume my slightly-lost, still figuring it out subgroup of students will miss these too. Which means I should go over them the next day and make sure everyone does them correctly, which benefits both my A students and my struggling students. This is the first time for me as a teacher that homework has had a direct impact on how I teach, and I like that.

Anyway, my current assessment cycle is: Bellwork & Exit tickets pretty much every day (formative assessments), Homework once every 1.5 weeks (evaluative assessment with written feedback and rapid turnaround), Exams at end of unit (summative), and retake packets to allow students to retake any test any time. My underlying philosophy regarding assessments is: I don’t care how long it takes or how many times it takes – just learn the material. This is something I’ve sorta gleaned from the philosophy of Standards Based Grading and their focus on mastery and reassessment. I’m liking it so far, but we’ll see what happens.

So… that’s pretty much where I am. I’m about 2/3 of the way through my first year – hope you’ve been enjoying the journey.

“Okay – doing translations tomorrow. Got the practice problems and notes made. Need them to know about coordinate notation and be able to actually draw and re-draw a shape given a translation. Just gotta put together this presentation. Let’s see… start by introducing translations as functions in 2-D. Do the whole function machine thing. Or wait – maybe some sort of game thing? You’re a mouse and you need to get to the cheese. Or you have to shoot something but can only move in x and y directions. Like we’re programming a platforming game or something! Yah! And you have to describe it using x and y directions! So it’s like you’re translating the mouse! Geez I’m brilliant. Okay – let’s open up Geogebra and see what I can do…”

(10 minutes pass)

“Okay – don’t want any graphics. But whatever – I can have a red dot and a blue dot. Need some buttons to randomly generate their positions… okay, now a way to tell the red dot where to move… Oh man, wouldn’t it be great if the red dot actually moved to the blue dot? Yah – I can do that – just gotta use sliders. Okay – so first I need these extra points to keep track of everything…”

(20 minutes pass)

“Wait, crap – I want the red dot to move in the x direction, then the y direction. That’s hard to do with just one slider. Also, how do I get a button to animate a slider? Ugh – time to google search Geogebra scripting commands. And figure out how to separate out this equation to make the red dot move using only one slider variable…”

(10 minutes later)

“So if I halve the value of the overall slider for moving in the x-direction, then try and pick that up for moving in the y-direction, I’ll be good to go. Then it will seem like the dot moved as far as it needed in the x-direction and just picked up in the y-direction. Okay – just need to mess with how I parameterized this slider variable…”

(15 minutes pass. Editors Note: If you don’t know what I’m talking about, that’s okay – you’re not supposed to. Instead, notice how little of this has to do with my lesson for tomorrow)

“I CAN’T FIGURE THIS OUT HELP HELP HELP”

(The following becomes my facebook status: “I have a tangible situation where I need a function that maps [.5, 1] onto [0, 1] and I’m struggling. HELP ME!”)

(2 minutes pass. One of my math friends responds with “2x – 1″. I feel a little silly)

“HA! IT WORKS! BRILLIANT! Okay – so now when I press Go, it’ll fire the red dot along the path I set up so it’ll reach the blue dot. Awesome!”

(5 second pause)

“Okay… so… I introduce translations as 2-D functions… uhhm…”

(10 seconds pass before I realize that the last hour of making this awesome cool thing doesn’t fit in my lesson anywhere, so if I have time at the end, my kids get to play the ‘shoot the red dot at the blue dot game’)

The moral of the story: This is not the first time I’ve spent over an hour focused on something I think will be amazing in my lesson tomorrow, only to realize once it’s done that it doesn’t fit or it only engages me without engaging the kids (in other words, the only reason my students will think it’s cool is because I think it’s REALLY cool) or it’s actually a great way to talk about this other thing that I’m not talking about today (ie: this game is great for vector notation, which I don’t plan on talking about tomorrow).

Anyway – Here’s what I made. I still think it’s cool and maybe you can use it. And I’ll try to use it tomorrow, but this is just one of many mini-projects which, once it’s completed, ends up taking only a few minutes of my lesson with minimal results.

This is probably the most informal blog post I’ve ever done. I hope it’s completely shattered any illusions of me being some sort of professional.

The reason I’ve stopped assigning homework every night is because I’ve turned it into an evaluative assessment rather than a formative assessment. I still assign it, but when I do, I want to make sure that every problem is (1) doable the day it’s assigned, and (2) meaningful- none of these problems where, if a student were to review their homework to study for a test, I’d have to say ‘oh – don’t worry about that one, we just did those because they were in that section’.

Last semester, students started class with bellwork which I checked and, at times, added problems to until I was sure students would be able to understand today’s lesson. Or, if I knew we needed a lot more practice, the lesson would be entirely guided practice. Also, I tend to teach with a lot of practice problems intermixed in the lesson – one of the way I chunk out lessons – I usually mix them in with a structured notetaker of some kind: “fill in this definition, let’s do this problem together, now you do these 3 on your own”. I also tended to have mini exit-tickets throughout the lesson which I called ‘objective checks’ – a practice problem at the end of the lesson that I felt all students should be able to do. But, with all this practice and in-class working… I never found time to talk about the homework. Plus, I found that most of the misconceptions from the homework had been cleared up when we went over the bellwork. Well, actually, sometimes it was worse than this – some students would go home and do the homework but do all of it incorrectly, then need to be untaught the incorrect method as I went over the bellwork. This is mostly my fault with the way I would design homework assignments (I made every one from scratch – I haven’t used a textbook all year). I also hated to be the teacher who went over a homework assignment that only 5 or 6 students completed. Homework wasn’t worth very much in my grading scale and wasn’t collected until the day of the unit test, so I never felt right about spending 5-10 minutes talking about problems when the majority of the class didn’t have a reason to listen. It also meant that it wasn’t a very effective formative assessment if the majority of the class wasn’t doing it.

I don’t know. I was doing a lot wrong with homework last year. I honestly started to resent it – it didn’t tell me anything about my students, my students usually did it incorrectly but we fixed it during the bellwork the next day, and it took up class time that I would have rather used for other things. I hated the feeling of sending my students off with a homework assignment knowing they wouldn’t be able to do most of it, and I hated knowing that I would have to spend class time tomorrow talking about it – what was the point of calling it homework if I was going to make them do it in class tomorrow anyway? (This, by the way, is not a horrible thing – I just think it undermines the idea of me calling it ‘homework’ – students will learn they don’t need to do it at home because they’ll get time in class the next day to do it anyway). It was nice to have the problem sets made because if a student came in for tutoring, we could do the homework together – but otherwise, most students tended to get everything they needed from my in-class activities (which, as said above, sometimes meant just doing the homework in class and me hating myself for reinforcing that they didn’t need to do it at home). Part of this homework frustration comes from the low skills of my students – it took me a while to realize just how little algebra my students knew or how bad their integer operation intuition was. I didn’t feel comfortable sending them home with any assignments because I started to fear they would do it incorrectly purely because they were trying to master so many skills at once – geometry, algebra, and arithmetic.

Anyway anyway – the bottom line is: it didn’t really have a place in my classroom. The only place it could have fit was as a formative assessment, but I already spent so much time on bellwork that I couldn’t find time to fit homework in too. One of the few ways homework served a purpose was when I ran out of time at the end of the class and gave it to the students to start before they left, which meant I could answer questions on the first few problems (this, by the way, was partly a classroom management/lesson planning crutch – ‘crap! Lesson didn’t fill the entire period! Good thing I have homework…’). However, since it was homework, half the students would just put it away and say they’d work on it at home, knowing full well I wouldn’t check it tomorrow and wouldn’t care about it until the unit test. Over the break, I realized that these few homework problems were really the only thing I valued about the day-to-day implementation of homework in my classroom – so why wasn’t I doing it every day and why was I letting it be optional? This was one of my motivations for switching to exit tickets – take those first few problems from a homework assignment, make it an exit ticket, and make every student do it in class in front of me. Then, if students struggle with it, put problems like this on the bellwork tomorrow. Either way, I’m forcing every student to try these problems and learn how to do them.

I realized that my assessment cycle was heavy on formative assessments and light on any other kind of assessment. I had bellwork, class activities, and homework. I didn’t usually give quizzes, which means the first time students got a chance to really show me what they could do independently was on a test – which, sometimes, was a disaster. I also really hated the message I was sending my students re: independent work and homework. I was saying to my students that homework isn’t important by (1) not checking it, (2) not collecting it until the unit test, (3) weighing it very low in my grading scale, and (4) not making a connection between homework success and test-score success. To my students credit, they also realized that I sometimes assigned homework that they weren’t completely prepared for, which is definitely my fault and definitely not good for creating a positive perception of homework. I’ve always wanted homework to be doable and meaningful for my students. So, after I decided I wanted to use exit tickets instead of homework every day, I realized I could be much more careful about my homework assignments and make them much more meaningful than before. I decided to assign homework once or twice a unit and have it include problems that I felt every student should be able to do by this point in time – for example, after a 4-day introduction to a unit with bellwork and exit tickets every day, assign homework with several similar problems for them to complete independently. They would need very minimal guidance except on a few stretch problems. It should contain at least 10-15 problems – something meaningful and worth their time. I decided to make the due-dates a few days long and, if possible, include a weekend.

I’ve done 4 homework assignments like this so far this semester and I like it better than before. Students are taking it seriously. They have time to work on it. They are meaningful problems and they care about getting every single one correct (partly because I grade it for correctness). I also feel more comfortable going over the assignment with the class – since I’m not assigning homework as often, I’m okay giving up that class time once a week or so and it reinforces the idea that ‘this must be important since we’re talking about it in class’. They also serve the same function that a low-stakes quiz might serve – I get a better picture of what students know by looking at their independent work. I didn’t have anything like this last semester and it’s been helpful this year.

The one thing I don’t like about the switch from homework to exit ticket (that I did not expect) is that an exit ticket has a time limit – you have 5 minutes to complete it before the bell rings. For some students, this is too short – or, on some days, I’ve planned the exit ticket poorly such that it would take longer to complete than I anticipated. Or, in the rush to finish in the time limit, they make rushed mistakes, which is one of the major areas I’m trying to fix with my students (No! Watch your negative signs! Stop making rushed mistakes!) On days like this, I wish students could take it home and finish it – but, again, I don’t really want to devote time to checking it the next day and I can’t guarantee every student will do it. Instead, I just throw it on the bellwork the next day.

However, what I do like is: I can assign more ‘explain… describe… justify…’ questions as they leave, which I really really like, and leave the ‘solve… define…’ questions for bellwork. I like that my homework now feels like home work - an independent assessment that my students are responsible for without it becoming a crutch for my class. I (as a teacher) will never have a day that is ruined by my students not doing their homework, but they (as students) will have days that are benefited by them doing their homework.

Anyway. I think that’s it. What I want to write about next: a day in the classroom of two teachers I work with who, in my opinion, use homework incredibly effectively and compare their assessment cycle to my assessment cycle.

Some of my studnets knocked this out of the park - It was really fun to see what my students knew about these shapes and how they connected it to their real life.

In grading them, I’ve come to realize that the whole point of assigning this project in the first place is to (1) allow students to creatively demonstrate they understand the properties of different quadrilaterals and their relationships, (2) make connections to the real-world, and (3) lucidly explain their answers and justifications in a manner that is convincing. I’ve been thinking of ways to adjust my rubric in light of these three points

Here is a list of A+ explanations that some of my students submitted. Some favorites:

“Mr Rhombus has an interest in law because a rhombus has all equal sides so he believes all people should have equal rights”

“Mr Parallelograms favorite movie is Master of Disguise because he can disguise himself as other shapes, like a rectangle or a square”

“Mr Square’s favorite movie is the Breakfast Club because when it was made, they called people ‘squares’ when they were losers”

“Mr Parallelogram likes any books about trains because trains run on parallel tracks and parallelograms have two parallel sides”

Most students who put in the effort received excellent grades – those that didn’t either didn’t follow directions (which my rubric penalized them heavily for), didn’t fully explain their answers, or left parts incomplete.

When I assign this next year, some things I would change…

• Make clear in my directions and rubric that students should use real movies/books/music and not make things up. More students than I can count came up with very punny bands or movies for their shapes to like – which is clever, but ignores the whole ‘make real-world connections’ aspect of the project
• Make clear that ‘having parallel lines’ is not a hobby, nor is ‘the geometry textbook’ an acceptable answer for a favorite book
• Make clear that the activities and entertainment must be related to the shape, not to the fake person that the student is creating – ‘hanging out with friends at the mall’, while a valid hobby to have for a person, is not something particularly special for a shape
• I think I’d still have them complete 4 pages, but one page would be due much earlier than the rest so I could give feedback. This is definitely something I wish I had done this year – had them turn in one as sort of a ‘rough draft’ so I can tell them if their explanations are acceptable, etc. A lot of students fell prey to the mistakes listed above, which is something that was unclear in my rubric and directions but could have been caught early on if I had essentially had them do a rough draft

So… that’s about it.

Update 2/12: If you’re reading this, there’s a chance you’re looking for ideas on how to teach a unit focusing on quadrilaterals. If that’s the case, you might want to check out how @crstn85 does it over at Drawing on Math: http://drawingonmath.blogspot.com/2012/02/sorting-quadrilaterals.html

I’ve tried to write about this for a few weeks now but it’s been hard to compress everything I want to say into one post. It took a while to realize that writing about this change to my classroom means (1) describing this new procedure I’m doing and how it fits in with the rest of my classroom procedures, (2) comparing the procedure to what I did last semester, which involves some serious reflection, (3) discussing something I casually refer to with friends as ‘The Homework Problem’, and (4) an opinion/reflection on what I consider effective use of formative assessment in my classroom. I can’t do all of this in one post, even though it’s all related. So, this post is attempting to address (1): A day in my classroom, culminating in a description of what a daily exit ticket looks like (if you just care about the exit ticket part, skip to the end)

So: I don’t assign homework every night anymore (which I did do last semester). I also don’t collect bellwork every day (which I did do last semester). Instead, my new classroom structure looks something like this: students come in and work on bellwork. I walk around and stamp the bellwork of students who have begun working diligently as soon as the bell rings. This stamp is worth 1 EC point. I do not let students make up bellwork if they are absent, so these EC points (in the long run) offset sporadic absences. The stamp is also an excellent nonverbal ‘You need to focus and begin your work’ signal with a tangible incentive (more points!) – plus, if I have a cool stamp, the kids like it.

After 5 minutes or so, we check the bellwork as a class. Students correct and score their own bellwork with a colored pen. I give partial credit if a student has corrected their mistakes on their bellwork. At the end of the week, I collect the bellwork and grade it. Bellwork is my ‘sometimes you need to learn something by doing problems, seeing it modeled by the teacher, and learning from your mistakes’ aspect of my classroom. I check bellwork by asking my students to tell me everything I need to do, which is a form of formative assessment – I get into their heads and, if 2 or 3 students can’t tell me the right answer, I know I need to use this ‘correct the bellwork’ moment as a ‘reteach this concept’ moment, then assign an extra bellwork on the fly to see if they pick it up the second time around. On a good day, a student wants to write their answer on the board instead of me, so I get to sit down with the other students and be annoying. Bellwork is also a ‘you may think you’re not good at math but you really are and I’m going to make you more confident’ moment when I cold call a student who I know got the right answer but would normally be too shy to share it with the group – they don’t like me when I call their name out of the blue, but hopefully they like how excited I get when they tell me the right answer. In my opinion, doing this enough times helps create a ‘I’m willing to take a risk and be okay if I’m wrong’ classroom, which is what I want.

After bellwork, we start the lesson, which can be very multifaceted depending on what I’m trying to do – are we focusing on practice? Are we learning something new? Is the ‘something new’ procedural or conceptual? Do I need to allot time to passing back papers and other administrative classroom interruptions? All of this happens. As it does, my kids do a lot of practice and structured notes – “Write this down. Do this problem. Refer to your notes. Check with your neighbor. Try this one now. How many people beat me to the answer? Good – let’s keep going”. Trying to paint a generic picture of my day-to-day lessons is difficult because my classroom isn’t generic.

Anyway – lesson is coming to an end. Students have the last 5-10 minutes to complete 3-4 problems that are an exit ticket. I didn’t mention this before, but these 3-4 problems are up on the board the entire class period. Students can scan the problems as they come in and see what they are expected to do by the end of the period. These problems can be purely procedural practice problems – “evaluate… solve…”, ‘were you taking notes?’ problems – “write the definition… list the properties…”, or more challenging ‘get inside your head’ questions: ‘compare… describe… Mr S claims that ____ – is he right or wrong? explain why”. I make my students complete these on a half-sheet of paper. I’ve done them too, so I know the answers. When they finish, they raise their hands and I collect it. If they got 100%, I give them a thumbs up / high five / yell ‘Boom goes the dynamite’ (inside joke, which I will probably share one day). If they get it wrong, I tell them they got a problem wrong and tell them to fix it. There’s a mastery component to this – you’re not done until you have 100%. When the bell rings, students turn in their tickets to a basket I have in the center of the room. I grade each exit ticket every night and give detailed feedback. I return them to students (hopefully) the next day and only to students where the feedback is useful. Misscalcul8 asked on twitter one day ‘What do you give feedback on and how often?’ – there’s my answer: every day there’s an exit ticket.

Part of the reason for this switch is I make my kids work the entire period, which means we do a lot of practice problems as we go through the material and they are very good at checking with their neighbor – I don’t need them to do 15 problems at home after we’ve done 10 problems in class. What I do need, though, is for them to do 1-2 rapid-fire practice problems and to see if they get them correct the first time through. For these types of exit tickets, I think to myself ‘what do I want my students to be able to do by the end of the lesson without assistance?’. They are usually procedural, based on example problems we completed, and straightforward. I can gauge how well my student’s understood by gauging how many get it correct on the first try and how they  interact with each other – are they asking each other ‘how do I do this?’, or are they asking ‘did you get this too?’ – I want the second one, and I want the other student to say ‘yes’ and then smile. Maybe they high-five too. And there’s a rainbow in the background.

Or, if the lesson is geared more towards a conceptual understanding, then my exit ticket becomes more open-ended – more explanatory. These questions are guided by me asking ‘What do I want my students to understand by the end of the lesson? What new intuition should they have?’ I don’t walk around and collect it – I let them turn it in to the basket and I check it later. Some examples of this distinction: today we did a lesson on area where I wanted them to understand that area is answering the question ‘how many squares can I fit inside a shape?’. My exit ticket: “Explain why the units to area are called square units“, “Mr S uses a grid to find the area of a rectangle. He then switches to a grid where the grid squares are smaller. Will the area of the shape increase or decrease? Explain why”. Tomorrow I want them to be able to use the formulas for area to solve area problems. Tomorrow’s exit ticket will be along the lines of “A rectangle has a base of __ and height of ___ – what’s it’s area?”.

Okay. That’s my classroom. I write all of this so it can be a base for some future thoughts on homework in a math class, some reflections on my last semester, and so I can tell you how this system is working out. But, that’s enough for now. Now you have an idea of what it’s like when I start and end my classes.

I’m preparing for a unit on area and, based on an informal exit ticket I gave a few days ago, my students don’t know very much about area even on a conceptual level. Which is sort of exciting to me, since it means I get to teach them from scratch. One thing I want to impress on them is how the choice of units effects the type of answer you get. I want to impress this on them visually – to show them a 1 cm x 1 cm square, then a 1 in x 1 in square, then a 1 ft x 1 ft square, then… etc etc etc. If area is determining the number of squares that can fit inside a shape, then I want them to know that the size of the square is a big deal. 

To do this, I thought of xkcd. They have several incredible charts of different objects that are to scale. The most recent one, and one of my favorites, is the comparative cost and distribution of wealth:

They also have the height of the observable universe and depth of a computer circuit to the neuron level. The important thing is that all three of these diagrams are to scale, which paints a powerful visual picture for comparisons.

What I just spent 30 minutes fruitlessly Google searching for: A chart/image/visual that has a to scale representation of several different area measurements – all side-by-side – so I can show it to my students and we can compare them. I imagine it would look something like this (only prettier and more accurate and better in every way because I made this in 30 seconds):

Finally he gets to the point: Do you, dear reader, know of any such chart? Any place on the internet that sounds like that I’m looking for? Something visual that I can show to my students and we can explore this idea that the bigger the unit, the larger the square, so the smaller the area – and the smaller the unit, the smaller the square, so the larger the area. The most important thing to me is that it is to scale - the accuracy matters and my kids deserve it and I’m a little OCD like that. If you know of any such resource, please please please tell me in the comments.

Tonight was the Student of the Quarter ceremony – each teacher nominates a student who deserves to be recognized for the work over the last quarter. We packed our little theater – parents and friends and students who may have never been recognized in their entire lives were suddenly the center of attention. This is a great night.

The student I nominated was there to accept her certificate. She earned a D in my course last semester, but damn did she earn her grade. She came in every day for two months, even Saturdays – no exaggeration, I have the tutoring logs to prove it – to raise her grade. But ‘raise your grade’ doesn’t mean ‘help with tonights homework’. It means: learn how integer operations work. Learn how algebra works. Learn how the coordinate system works. Learn how slope and graphing lines and order of operations and long division – learn how it all works. Those blog posts I wrote about fixing algebra mistakes? They’re based on my experiences with this student. The feedback I got right here, on this blog, helped me be a better teacher for her.

Slowly and painfully and carefully, she learned every little thing she had missed over the last two years of her life. Slowly and carefully, she was able to participate in class – to answer questions without much help – to have a discussion with the people next to her because she knew what she was talking about. Her grades started to improve. She started to pass my class. She still came in for tutoring. She patiently tried and failed and tried again and kept with it. And she was successful.

She took her final for my class early, after school one day. I watched out of the corner of my eye and realized every single thing she writes on that exam is something I have taught her. She uses the number line for every single problem, just like we practiced. She sets up her congruency problems using the ____ = _____ framework that I showed her. She calculates slope and does her algebra just like we practiced. She draws arrows like I do – underlines like I do. This is surreal. Every line on her final is something she worked hard to learn this semester, putting in more hours than any other student in any of my classes.

And she passed. Both my class and the final.

Anyway – back to tonight. Student of the quarter ceremony. I get to stand on stage and read what I wrote about her. About how I’ve never seen someone work so hard. Be so dedicated. How I think she can accomplish anything. And I mean it. I can’t be sure, but I get the feeling this is one of the first times in her life that she’s had this kind of honor. That she’s been recognized like this. That she’s been able to persevere and be successful and be rewarded.

Minutes before the ceremony, I discover that her English teacher also nominated her for Student of the Quarter. Her written statement reads almost identical to mine – she’s never seen a student turn her education around with such determination and perseverance and force as this student has. The English teacher isn’t there, so I get to read both statements – to tell her, twice, that she’s turned her whole education around and we’re all damn proud of her.

Reading that certificate was a good feeling. Shaking her hand was a good feeling. Watching her walk back to her smiling, excited family and give them all a hug was a great feeling. Fostering such tremendous growth in someone  is a great feeling, especially someone who had slipped through the cracks for years - who is rarely recognized but tonight was the only person to be recognized twice. I had a strange sense of pride that I can’t really describe.

Sometimes it’s hard for me to talk about teaching because parts of it are incredibly personal to me. When I do talk about it, I usually tell people that teaching is a continuum – there are moments of great highs and moments of tremendous lows. I like my low-income high school because I believe it has the widest continuums – when it’s bad, it’s bad - slip into a depressive coma bad, don’t want to go to work bad, question your self-worth bad. Talking to people, I get the impression that they think this is only what my school is.

This student used to call herself stupid and say nothing mattered because she’d never amount to anything. And now she doesn’t. And that’s worth it to me.

That’s what the state thinks. I, on the other hand, knew at the beginning of the unit that I wanted my kids to be able to answer the following problem: “A = (3, -1), B = (1, -3), C = (7, 3) and D = (4, 3). What kind of quadrilateral is it? Justify your answer – be sure to give evidence about why you believe you shape is what it is, as well as why your shape isn’t something else”. As I was teaching, I also realized part of this unit is teaching how to understand a hierarchical relationship used to classify things – ie the Kingdom/Phyllum/…/Species classifications in biology, or the organization of the countries of the world (The 5 continents, then the many countries, then within each country there are territories/provinces, and then individual cities/towns), or object-oriented computer programming (which they won’t care about and I can’t use as examples in class, but man is this a useful skill to have if any of my kids become Java programmers). The point is: there is a method of organizing and categorizing objects such that they split into defining categories, all of which have a common property or defining characteristic – and, anything underneath these categories have all the properties of the objects above them and continue to become specialized. It is useful, in my opinion, to make this method of organization explicit and teach my students how to understand this method of classification.

The Coordinate Stuff: I just love the problem of being able to give students 4 points and ask them what the shape is. I actually just taught this stuff today and it went great – we had finished talking about the definitions and properties of the quadrilaterals we care about and I told them “You are now an architect designing a building. All the rooms in the floorplan must be rectangular – otherwise the rooms just feel weird. The drafting program you’re using only lets you create shapes if you give it the vertices of the polygon. You’ve type in (…..). Is this a rectangular room?”. I ask for opinions, but no one really has any – only the kids who like attention speak up but they can’t justify their answer. Eventually someone asks if we can graph it (sometimes this ‘someone’ is me), so I show it on Geogebra. I’ve cleverly picked a parallelogram that is tilted and looks like a rectangle, but really isn’t. Now more people enter the debate – everyone has an opinion, but it starts off based on ‘how it looks’. Finally someone starts to say it’s because it ‘looks’ like the sides are congruent or that the sides are parallel or the angles are 90 degrees – but again, they’re not sure. Next comes the guided connection between parallel/perpendicular and slope, congruence and distance, and bisect with midpoint. At this point, we’re ready to collect some data:

Update: I can’t figure out how to embed box.net stuff into WordPress, and Scribd completely messes up the formatting of my worksheets. So, sorry that you have to add a few extra clicks in order to see what I’m walking about. Anyway – here’s the ‘Here are 4 points, what shape is it?’ worksheet I made

(Here’s a link to a template that doesn’t have any of the coordinates filled in – you can make your own!)

So now they’re finding distances and slopes and midpoints… then turning that data into geometric properties… then turning those geometric properties into arguments. I really dig it. By the way, there is an ‘efficient’ method that doesn’t require so many calculations - it’s outlined in this Flowchart (right-branches mean ‘yes’, left-branches mean ‘no’), which I may or may not give to my students at some point later. My honors kids are already thinking about ‘the most efficient method’, while my regular kids are still focusing on making 100% sure that they know what (-1 – 3) equals.

The Hierarchy/Relationship Stuff: I told my students that this unit was really about relationships - things these shapes have in common, ways in which these shapes are different, how we go from a general definitions to specific definitions, etc. I then told them that they’re already intricately involved in something that involves relationships: Facebook. Then I assigned them a project: to make 4 fake facebook pages for 4 of the quadrilaterals we’ve been learning about (minor reflective note: next year, I think I’ll only assign 3). Below are the resources for this project

Facebook Template (for making the physical pages)

Facebook Project Explanation

The Grading Rubric (because every project should have a rubric, although I don’t think this one is completely stellar. I suspect I’ll revise it after I see what kind of work my students turn in this year)

Rubric Explanation

There’s also an example Facebook page that I made, but it is hand written and I don’t have a scanner yet – that may come later.

My students have that ‘This is different and sorta neat and I’m sorta excited by it but I don’t wanna show it so I’m gonna pretend it’s dumb’ attitude. But that hasn’t stopped several students from creating real Facebook accounts for this project (like this one). I also have several English Language Learner students who have come to talk to me about the project (‘Is this okay? How do I say this?’) – and man are they creating quality work – it’s really stretching their vocabulary and forcing them to really understand these definitions and create personal connections. In fact, that’s really the whole goal of the project – to create some sort of personal connection with these shapes and to creatively show that you understand the relationships between them or other shapes. I was over-reading a twitter conversation (I do that a lot) where someone commented that their daughter in elementary school spent an entire unit learning about trees and never stepped outside to actually look at one (I forget where I saw this though – sorry) – at times, I feel like this is a problem for me in geometry simply because I haven’t found the time or structure or priorities yet to have them create meaningful constructions and actually draw these things. So, I figured the next best thing is for them to find them in pop culture and the world around them, which I hope this project does.

Now that I have some distance from my first semester, I realize that I was drawn to Khan Academy as a solution to two problems. (1) I have a weird schedule which unequally allots 25 extra minutes to certain classes throughout the day – you can read about that here. Essentially, I needed some sort of consistent enrichment activity for 25 minutes every week. (2) My students came to me in need of serious remediation. I’m teaching sophomores and the majority of them needed to be retaught basic algebra and integer arithmetic skills. There were other holes – coordinate plane, exponents, graphing lines, distributive property, etc – and my perception is that they had a bunch of rules and procedures in their heads that a previous teacher had forced them to memorize, but they got these rules all jumbled up or just plain wrong and kept making mistakes as a result. What I needed was a way for them to do lots and lots of practice with lots and lots of guidance until they got those procedures correct. I also needed this practice to be in such a way that students could do a ton of problems over and over again – something that randomized these problems, but also narrowed down the skill set I was looking for (in other words – solve liner equations vs solve linear equations with distributive property vs solve linear equations with variable on both sides of the equal sign vs solve linear equations with fractions). I was also felt like I was in a very peculiar situation since I’m not teaching an algebra class yet so many students desperately needed to be retaught algebra – how do I reteach without explicitly reteaching? Something that engages both the students who need the reteaching, but also the students who managed to master the concepts from last year?

I started using Khan Academy in that extra time and we went through a lot of the remedial topics so we could build up our mathematical understandings. I didn’t touch the videos at all – we only worked through the practice problems with the badges and points and other incentives which are meaningless to me as an adult, but sometimes I forget how attractive these things are to students (related: the enthusiasm a student has when they get a sticker on their paper). My kids were instantly engaged and motivated – they wanted to work through the problems and see how far they could progress – they were on a race against the other classes. My honors kids even wanted to work on problems that they knew they struggled with – “Hey, lets do a fractions one because we’re not good at that”. I was happy because I found an activity that was mathematical and filled up time at the end of a lesson.

Positive Experiences: This quote from David Cox, via twitter, sums up a lot of the positive experiences I had: “Give these kids 5 problems on paper and they socialize. Give them the same problems in an applet and they collaborate”. By giving these drill-and-skill problems in a fun, technological format, their motivation changed completely and we were able to do remedial work without much complaint. I also went through the Geometry strand on the day before Thanksgiving as a review-esque sort of game, which was a great use of that time. As a teacher, I loved that I had a classroom structure I could employ whenever I needed extra time – it was something for me to fall back on (which, as a first-year teacher, is something I desperately needed). I also loved that it was a way for me to do drill & skill without saying ‘here are 20 worksheet problems – work on them right now”.

How I would recommend playing the game as a full-group activity: I had a hard time finding the right classroom structure for my students – a way to hold everyone accountable but also let them collaborate with their peers. By far, the best classroom experience I had was when we used group whiteboards to answer problems. My students sit in groups of 4 – I made one person the scribe: this person was the only person allowed to write, but this person was not allowed to speak and could only write down things that were said out loud by their group members. This made the problems a little bit like a game of charades, marginalized the ‘smartest’ person in the group so they weren’t always doing all the work, and making the groups speak ‘out loud’ made sure no one was just doing the problem silently while the rest of the group watched. As students answered problems, I walked around the room and monitored – if I saw a group break the rules (writing without anyone talking, scribe speaks, etc), then that group was disqualified. I entered the correct answer once I saw that at least 6 of the 8 groups had the correct answer on their whiteboards – if more than 2 groups were disqualified, we had to pick a different question. For harder problems, I would pick one group to explain how they got their answer – for this ‘explain your answer to the class’ dynamic, the rule was that only the scribe could talk. In this way, the scribe is also accountable for knowing how the problem works because if I pick that group to explain, only the scribe can speak to explain their answer – so if they’re not listening, they can’t explain the answer and their group is disqualified. I only got a chance to try out this dynamic twice over the course of the year, but both times it worked out really well.

Edit 1/9: After over-reading a Twitter conversation (get it? like overhearing?), I realized the strategy above is an adaptation of the Sage & Scribe Kagan structure.

What I would probably change for next year (if I do it again next year): The Khan Academy questions and game framework are wonderful and fantastic and a godsend… when they fit perfectly with the skill level of my kids and the goals of my curriculum. And when they don’t, the activity isn’t really worth it. This is what I experienced about halfway through the semester when we had covered arithmetic skills, integer operations, the geometry sections, and solving linear equations. After that… there wasn’t much else in the game that I wanted to cover with my students – I didn’t care about conic sections, decimal division, line-graph intuition, graphing lines in standard form, etc (although these could be important topics if I wasn’t teaching high-school geometry). I also began to experience problems where the Khan exercises didn’t properly narrow down the skills I wanted to assess (ie: please don’t ask my answer to be in standard form when I’ve only taught my students slope-intercept form). Remember, one reason I was drawn to Khan is that it was a fun way to answer a bunch of drill-and-skill problems, and it auto-generated each question – this last bit was really important for me as a new teacher without any stock resources yet. About halfway through the semester, Khan stopped filling this niche as well as it could have. Instead, it was up to me to find more remedial materials for me to use with my students that could auto-generate tons and tons of problems and where I could narrow in on the skills being assessed. I found those resources via the Worksheet Works website (which is awesome at generating practice algebra problems) and free trials of the Kuta problem-generation software. I also developed better classroom structures for answering drill and skill problems (see group whiteboard paragraph above), so it still felt less like ‘Do these worksheets because I said so’. By the end of the year, I was using Khan academy less and less because it stopped addressing the specific holes we had, but I definitely valued the fact that it ‘prepped’ my students for working remedial drill-and-skill problems and they got used to the fun classroom structures that I continued to use.

What I Wish Khan Academy was instead: Skip to the bottom of this post and read this letter from Justin Lanier to Sal Khan: http://ichoosemath.wordpress.com/2011/09/26/a-letter-to-sal-khan/. In particular, I wholeheartedly agree with Justin’s desire for:

• Teachers could create and share their own instructional videos.
• Teachers could devise and share their own exercise sets.
• Teachers could design and share their own customized “knowledge maps”.

What I really want from the Khan Academy is the game-esque framework, the auto-generated problem sets, and the mastery component (keep practicing until you get a certain number correct). I don’t need Khan Academy exclusively for this though – in fact, David Cox created an amazing leveled Geogebra applet that does almost exactly this for Linear Functions: http://www.geogebratube.org/student/m2506.  If I could, I would have an applet like this up my sleeve for just about every key concept I planned to teach in a semester. Maybe that’ll be a summer project of mine, or maybe Khan Academy will let Justin and I create our own problem sets that fit our own students with their own individual needs. In any case, the geogebra applet fills the exact same need that Khan Academy does – and if I were the one to create it, it has the added benefit of fitting exactly within my curriculum and assessing exactly the things I want to assess. If I had several of those applets, automatically generating problems and keeping track of correct answers, I wouldn’t need Khan Academy at all.

Concluding Thoughts: I like to think of the origin of the whole Khan Academy thing – it was Sal Khan tutoring his nephews long-distance. From there, it gained traction because lots of other people used it as their own private tutor too. However, as someone who has worked as a tutor and now works as a teacher, there is a profound difference between tutoring someone and teaching something from scratch. To teach is to give intuition – to give a foundation – to plant an idea and foster its growth. To tutor is to clarify – to help assimilate knowledge, to help with the procedural steps, to model how to solve problems, to answer questions rather than ask them. Teaching is big-picture, overarching, connected – tutoring is detail-oriented, in the moment, specific. Teaching involves a conceptual component built from scratch – tutoring involves clarifying a foundation that has already been established. I think Khan Academy is an excellent tutor – an excellent model of a pre-established intuition, of clarifying those established rules and procedures that so many of my students found jumbled in their heads, of pinpointing specific problems that students should know how to solve. To Sal Khan’s credit, he usually does an excellent job of thinking out loud, making mistakes, and highlighting common mistakes.

But I don’t think I would ever use Khan academy as more than a tutor. And I would certainly feel uncomfortable if my tutor started to usurp my role as teacher unless I knew for sure that my tutor was presenting the material in a way that was aligned and consistent with my curriculum (but I can’t exactly ask Sal Khan if he has my student’s best interests in mind, or if he’s aware that the problems in Unit 2 will scaffold nicely into Unit 3). And I, as the duly-appointed teacher, would definitely feel uncomfortable if the Khan Academy took the role of teacher – if my students went there for the intuition and to me for the clarification – because how do I know how all the Khan academy units tie together? Do they tie together at all? Is Khan Academy even a curriculum? Or is it thousands of individual lessons that are loosely connected? Or, to ask a more pointed question, what’s the difference between the Khan Academy curriculum and a badly written textbook except that one talks at you with videos instead of through a static page? When second semester rolls around, I don’t intend to use Khan Academy like I did this past semester – we’re beyond that. But if I were to continue using Khan academy at all, it would be as a reference when students are confused and as a place to attempt practice problems – much like the way I use my textbook in my class.

So, goal for next year: Try to create a bunch of Geogebra Apps that do the same thing as Khan Academy, but are tailored specifically to my curriculum, and only use Khan Academy as a reference or as a resource for students who need serious remediation (as in: still need to memorize their multiplication tables, of which I have a few students).

PS – if you’re interested in more readings about the whole Khan Academy in the Classroom Discussion, I recommend:

• Matthew McCrea’s A Day in the Life of a Khan Academy Student: http://www.matthewmccrea.com/a-day-in-the-life-of-a-ka-student/
• A selection of Frank Noschese’s criticism of Khan Academy: http://fnoschese.wordpress.com/tag/khan-academy/

So… once again, thanks for reading another long and epic post. I’ve been avoiding ‘tips to good blogging’ articles because I’m scared by how many rules I probably break…

In another example of ‘The blogotwittersphere works in mysterious ways’, Kate Nowak just shared her Formula for a Math Lesson, which is incredibly similar to my own version of an ‘ideal lesson’. I highly recommend reading it.

So: I want my classroom to be one where learning is framed around solving a problem. This means introducing students to new scenarios they may not have seen before, as well as introducing them to a new way of thinking they may not have seen (or been conscious of) before. My job is to guide them through the process of applying previous knowledge to a new situation, or making a discovered pattern into something explicit. This means modeling my thinking, which is a big part of how I plan certain lessons and scaffold certain activities.

But here’s something you should know about me: I think in questions. Which means when I’ve posed a new type of problem to the class and am modeling how I would try and solve it once they’ve struggled with it a little bit, I’m asking questions. But I’m not directing them at the class – I’m asking them of myself, using them to guide my own internal dialogue. I’m asking rhetorical questions (which, I’ve been told, are the worst kind of questions – and, based on the reflection to follow, I somewhat agree). They’re big general questions that are sometimes hard to answer, like “where should I start?”, “do I know any theorems or definitions that can help me out?”, “have I solved a similar problem to this?”, “what was I given and how can I use it?”, “What happens if I…”, etc etc. I’m essentially walking through Polya’s process for problem solving right at the front of the room for all my students to see. And as I answer them, I step through how I break down a complex problem into manageable pieces, or how I relate a new situation to familiar content. I’m modeling my own problem-solving strategies (which I also have listed on a poster in my room) and showing my students its okay to not know the answer right away, but at least we have some strategies on where to start.

But I have a problem: I’ve accidentally blurred the lines the questions intended to model my own thinking, and the questions that come later meant to peek into what my students are thinking.

These same types of questions – “Where would you start?”, “Have you solved a problem like this before?”, “What do you think about the information you were given?” – are meant to help my students start problems, so it’s important that they be comfortable answering them. From there, I can get a little peek into any misconceptions that may be brewing or where there may have been a gap in my explanation. I find that when I ask my students these questions, meant to formally assess how they’re thinking, I’m usually met with silence. It takes a while for them to take a risk with this, if they take that risk at all. For a long time I thought this was just an issue with my students comfort level with mathematics and taking risks in the classroom, but I overheard a twitter conversation between Dan Meyer and David Cox that changed my opinion. I now think what I’ve actually been doing is blurring the line between the closed questions I pose to myself and the open questions I pose to the class. This terminology comes from Dan Meyer’s post on Pretending Closed Questions are Open - in fact, here’s a quote from that same post describing how the general question ‘What question do you have?’ eventually became a single specific question:

“At that point, Avery just asked the question that interested him: How many squares does the diagonal pass through? His session ended on that problem but I’m extremely curious what would have happened had he presented a new image and asked his participants for new questions. I can’t be sure but I suspect they would have held out. They’d know from their last experience that Avery had a question in mind and everyone but the apple-polishers would have waited him out.”

Here’s what I think happened: sometimes when I model my own thinking and am asking myself those rhetorical questions at the beginning of class – “What should I do next?”, “How should I start this problem?” – an excited student will raise their hand to answer the question. And then I get excited because I’m hoping that this student is really thinking the same thing I am and will take over this conversation (because, remember, I never want to say anything a kid can say). So I call on this student to answer my rhetorical question that I’m using to model my own thinking… and the student won’t be thinking the same thing as me at all. They’ll confidently announce an incorrect answer or misconception, or want to go off in a direction that proves irrelevant and I, in the interest of time and not creating any other misconceptions, will need to curtail that discussion. But really what I’m doing is: I allowed one of my closed questions, intended to have exactly one right answer that I was thinking of, become an open question – and in doing so, I allowed a student to take a risk and get shut down. As a result, when I go to ask this question later – “what should you do next?” – after we’ve done several problems together hopefully building up their confidence with these types of problems, I’m met with blank stares. Or, they sit an try to wait me out, since earlier when I asked this same question, I gave the answer. So maybe they’re thinking that if they wait long enough, I’ll give them the answer again. Only the ‘apple polishers’ are willing to volunteer an answer even though I’ve checked in with many of my students and know they now know how to do these problems.

I guess what I’m realizing is I need to be acutely aware of the points of my lesson when I’m modeling my own thinking with questions, and I need to make sure my students are aware of it too – “Let me tell you how I would handle this problem if I were in your shoes and you can see if your thinking is similar to mine”, or maybe “If you’re not sure what to do, you would probably ask me for help. And if you did, the first thing I would ask you is…”. I need to not give in to that temptation of calling on a student when I’ve asked a vague, general question but have a specific answer in mind. I need to prepare the class for when I’m expecting an answer out of them: “Alright – we’ve talked through a few as a class – let’s see if we can start this next one on our own”. Maybe at this point they turn to their partner and talk through it with them before I poll the class full-group (which is always a good idea – check with a neighbor before sharing full-group). I need to be careful about when I choose to poll the class with one of these Polya-esque questions (“Anyone have any ideas of where to start?”) – if I’m doing it as I’m introducing a new type of problem to my students, this is probably a mistake – I usually have a specific way to start this problem in mind and it’s not fair to make my students ‘guess’ what that specific strategy is. Doing it as we’re going over homework seems like a better idea – this is where I want to see what my students are thinking and, hopefully, some students will be able to explain the answer to the class.

I guess this idea of blurring the lines between an Open and Closed question has really manifested itself in my classroom with how I model problem-solving and my enthusiasm and desire to have my students tell me the answer, rather than me tell them it. If I want to keep modeling my thinking by asking questions, I need to accept that these are just very general closed questions and it’s only fair that I’m the one who answers them – otherwise, I’m setting my students up for failure by giving them too broad of a question with too specific of an answer. And if they get enough of those wrong, they stop wanting to take risks, which is the opposite of my ideal classroom.

Update 1/2: I was looking through some old teaching resources and found this – Developing Mathematical Thinking with Effective Questions. There are some goooood question starters in there.