A Few Minutes of Reflection

It’s Sunday and I’m sitting in my favorite coffee shop sketching out my next Geometry unit. On paper and in the textbook, the heading would be ‘Similar Triangles’. In my mind, the heading is ‘Finally getting over a fear of fractions, learning to solving proportional algebra problems, and solving really cool indirect measurement problems’. Anyway – I thought I’d take a minute to put down on paper (or, I guess, on ‘text box’, since that’s how the WordPress post interface works) some things that have been floating around in my head.

Something a First Year Teacher Does

Let me tell you how the last 2 hours of my night have gone.

“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.

Why I Switched to Exit Tickets

I posted a bit ago about a day in my classroom, which can be summarized very briefly as: bellwork, lesson, exit ticket. Part of this structure is a reaction to how I did homework last semester, as well as addressing my needs as a teacher for a balance between formative and evaluative assessment. I firmly believe that any conversation about homework is really a conversation about assessment, whether you’re aware of it or not. Formative assessments have very specific connotations in the education community – they are consistent, low-risk, informational, feedback-worthy, used to inform teaching. Evaluative assessments are medium-risk and meant to be done correctly the first time through with minimal stretching by the student. I should say that this term, evaluative assessment, is something that a colleague threw around one day during a Professional Development seminar at my school one day and I latched onto it because it perfectly described one of the problems in my classroom from last semester – I had too many formative assessments and not enough evaluative assessments. Also note that this means that if you throw around the term ‘evaluative assessment’ and expect people to know what you mean, you may find yourself facing blank looks.

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’.

Facebook Project Followup

A quick followup re: my Quadrilateral Facebook Project

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

A Day In My Classroom

I’m doing something new in my class: I’ve stopped assigning homework every night and instead assign an exit ticket every day. This is one of several posts describing this.

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.

Help: Chart of Area Units To Scale

This is me abusing the fact that there are many talented, resourceful math teachers who read this. This is me asking for your help. It’s similar to a question I would have posed on Twitter, but the question is much too big, so I’m also abusing the fact that I have a blog. Sidenote: this is another wonderful benefit to being a member of the blogotwittersphere – that instead of asking my department for resources, I get to ask the whole world!

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.

Why I Teach…

I’ve got several blog posts brewing, but I’ve got something heavy on my heart instead. I’m a first-year teacher, but I’ve wanted to be a first-year teacher for several years and I have a lot of complicated feelings and experiences about going through life wanting to be a teacher within a culture that doesn’t always understand the draw to teaching, especially when you choose to work in a low-income high school among the’ unmotivated hooligans and troubled youth’. This is for you.

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.

Quadrilaterals: Coordinate Proofs & Facebook Projects

I’ve been teaching a unit on ‘quadrilaterals’, which I would describe as investigating the relationships between the different types of quadrilaterals – parallelograms, rectangles, squares, trapezoids, kites, etc. One of the AZ state standards is to ‘use the hierarchy of quadrilaterals in deductive reasoning’, which means being able to answer questions like ‘A rectangle is (always, sometimes, never) a square’.

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.

Khan Academy Roundup

Earlier in the year I posted about how I was trying to use Khan Academy in my classroom in a nontrivial fashion. You can read about it here. Now that my first semester is over, I’ve had time to see how all this panned out and reflect on it. I was also contacted by an amazing teacher who graduated with me from the U of A who told me that her school was thinking about adopting Khan Academy and she wanted to know my opinion. So, I guess what follows is half reflection, half opinion about Khan Academy and if I would ever use it exclusively in my classroom.

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?

Questions in my Classroom

I’ve been reflecting on-and-off on the way I use questions in my classroom. If I could, I would use questions all the time - I would try to ‘tell’ the students as little as I could. In fact, if there’s one pedagogy article that I have completely and wholeheartedly internalized, it’s Never Say Anything a Kid Can Say (If you haven’t read this, I highly recommend it). My whole strategy for instruction/lesson design/unit design is asking the question ‘How can I get my students to come to this conclusion rather than have me tell it to them?’. Part of this, in my opinion, is providing the right motivation – the right problem to solve or question to answer. After they’ve started to come up with ideas and stumble onto the correct procedures or answers, my job then becomes to help them organize the information and give them the vocabulary that they’re probably missing. Jason Buell explained it so much better than I think I could in his post about Layering - essentially, in my own ideal classroom, my lessons follow a similar pattern as his: pose the question, investigate, summarize/synthesize/organize/give new vocabulary (which is a big part of teaching Geometry), practice, extend.

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.