Graphing quadratic functions foldable

Nothing too fancy or mind-shattering here, but perhaps someone may find this useful. Just a quick, little graphic organizer for the steps I want students to follow to make a nice graph of a quadratic function.

Graphing quadratic function foldable (updated) by Anna

You need to print both pages and copy from one-sided to two-sided so that you can fold over the top and make it look like this:

Students should cut each numbered flap and then fill in each step with an example problem. Here is what the final copy will look like:

Outside:

Inside:

Group success!!

Today had one of those class periods. You know the ones. Where everyone is working and discussing and arguing and engaged and I'm? I'm just there. Totally and completely unnecessary to the workings of my class. Amaze.

Today was the first day of learning to solve systems by substitution. We went through an example together using @cheesemonkeysf's patented "substitution by stars" method. Then, I passed out some problems for them to work through, telling them to work with their group members and only go to the next problem when everyone in the group was totally good with the previous one. There were some where students had to apply the distributive property and some where they had to isolate a variable first. We had not talked about these. I did not stop class to discuss and fix and give hints, just circulated the room and looked on. Not one kid asked me for help for the 20 minutes that they worked - they puzzled through and argued and made mistakes and erased them and made some more and eventually figured it out all on their own. Holy wow. You guys get it. We all want group work to go exactly like this, but it so rarely does.

But finally!! All of the working on norms and redirecting questions with, "Have you asked your group yet?" or "Is this a group question?" and biting my lip to avoid butting in have paid off and there was just this amazing energy and focus and I was so, so proud. Cause there aren't that many glorious classroom moments like this, and it's so easy to focus on all the ways that we aren't perfect and they aren't either. So I want to stop and acknowledge that today was exactly right and keep it in my memory bank for when I'm feeling frustrated and disgruntled. Hooray!!!!

Article on "Math People"

Soo.... I wrote an article for Quartz magazine on why so many people identify themselves as "math people" or "non math people" and what we can do about it.  It's the first time I've ever been officially published and I'm both terrified and excited.  The magazine editors came up with the article title and the subheadings but the rest is mostly my work.  Here it is.

I don't know how many people will read it, but so far it's been well worth the two weeks I stressed over it because of some of the wonderful responses I've gotten from former students.  I have to share them.  If the article was terrible, it was worth writing just to get these wonderful words of encouragement from my students.

Student 1: Congratulations on having your article published! I miss having you as my teacher so much! You were the best teacher I have ever had <3

Student 2: You are so brilliant! I am so lucky I had you as my teacher. Not only were you a good teacher, but a super-cool one. I miss you!

Student 3 (this is the one that made me cry): I want to thank you for writing that article. I have been so scared to take another math class because of the last one I took. It was Math 110; basic college Algebra. I failed the class. I went to the math lab regularly, I participated in study groups, office hours, the works. I tried hard, but the teacher just could not explain things in a way that I could understand well and remember. After that class, I decided that I could not ever have a career in the math/science field despite my love for them because I just was not a "math person." A few weeks ago I was reading a Biology/Science textbook and realized that those were the only textbooks I had ever read for fun. I always have. But right then I nearly started crying thinking about how I just did not have the math skills to ever pursue it. Your article has given me hope... I now have the courage to try again. Thank you so much. I am so grateful for the time I had as your student. I know that all of your former students feel the same way. We love you. You are the best. Don't believe anyone who tells you otherwise

Axis of symmetry for a quadratic function

My two-year Algebra course is just starting our study of quadratics. So far, we've looked at graphs and tables for quadratic functions and compared them to linear functions and talked about the symmetry of the shape and finding the axis of symmetry from the x-intercepts and finding the reflections of points across the axis of symmetry. I would like students to derive the formula for the axis of symmetry: for the quadratic function . The Exeter Math 1 curriculum does this by having students graph functions in the form , then generalizing the idea that one of the x-intercepts is zero and the other one is . I created a worksheet that extends on this idea using desmos to graph and having students also derive the formula for the axis of symmetry by averaging the two x-intercepts to get . Then, just like in the Exeter packet, I will have students compare graphs of  to graphs of  to realize that the axis of symmetry is not changed by adding a c value to the equation. I'm hoping that the logical progression of the graphs will make sense and that the questions are open enough for this to be genuinely based on student thinking and discovery, but with enough structure that students who are not necessarily used to deriving formulas will have something concrete to think about and answer. Any and all feedback welcome!!

Axis of symmetry by Anna

Why teach Algebra

I've been thinking about a response to the article published in the New York Times, "Is Algebra Necessary?"  for over a year.  I've started maybe 10 different draft posts and scrapped them.  I've been following the follow-up debates in blogs (see Wiggins' post on algebra 1 as a poorly designed course and Honner's response to Wiggins) and thinking about related articles like "Wrong Answer: The Case Against Algebra II" and "The Mathematician's Lament".

I've been so torn about how to respond because as a math teacher, of course I believe teaching math is vital.  I became a teacher because I wanted to save the world and martyr myself with 80 hour work weeks and panicked sweats every Sunday night.  And reading these articles seems to trivialize what I have poured sweat, tears and many many gallons of coffee into.  Yet I see their side of things.  I hate the idea of algebra 1 being the barrier between a talented artist and a career in art.  I am now teaching students in algebra 1 who have failed it two or three times before and it broke my heart yesterday when I handed back a  homework assignment I had given a 7/10 to a student and her face lit up as she said that she'd never gotten a passing score on a math assignment.

Also, I've never used math in "the real world".  I'm not an engineer, an economist, a physicist or a banker.  I don't know how math gets used out there so who am I to tell students year after year that they're going to need these skills when I don't know that they will.  The argument that math sharpens general cognitive skills and teaches students problem solving strategies that will be useful later in life, especially as it's backed up by research, holds water but that doesn't help us algebra teachers argue for teaching algebra.  Why not teach statistics?  Or a formal logic class?  Should we defend the traditional math sequence, or should we branch out and give students who are failing at algebra alternative math options?

But the other day I was talking with my husband and I realized why I love math, not why I teach it or how I use it, but why I love it.  And I think the reason for my love is also the reason it needs to be taught.  I am decidedly introverted, perhaps the queen of introverts.  I can't handle phones- it's very very difficult for me to talk on the phone with those I love and even harder with those I don't know.  I need to see eyes, to gauge reactions, to be able to comment on surroundings or engage my conversation partner in a task that removes the focus of conversation off of me.  I've found the adult world intimidating and overwhelming and need frequent breaks from it.  I like playing board games to escape because they have defined protocols.  I know exactly what the object of the game is and how to get there.  I can enjoy socializing while playing because of the game's comforting structure.

The world is overwhelming for anyone- even those not so introverted as I am.  There are complex political systems to understand, the natural world can be scary and confusing, bad things happen to good people inexplicably, we are born with deficiencies and insecurities that make socializing difficult or awkward.  School is for this- to help our young students learn that knowledge will conquer their confusions and difficulties.   When they understand how something works, they aren't as afraid of it and they know how to navigate it.   Or when they understand how something works, they won't make a mess of it because of overconfidence or arrogance.  Understanding our history and political systems is vital but impossible.  We give our students the best analysis tools we can and hope that time and a love of learning will help guide them in making wise decisions for themselves.  Learning science is fascinating and practical, but requires lots of field trips, labs, props and math to even begin understanding the basics of how our world works.  Math is the only field where understanding can be created by the student with nothing more than a pencil, a paper and a system of logical rules- just like a board game.  Yet unlike a board game, math helps us untangle the mysteries of how the world around us works.  It gives us a sense of order and control over our own minds and our own environments.  Isn't our job as educators to help students make sense of the world around them and to help them feel in control of their own lives?  Math is instrumental in accomplishing these two goals but especially for helping students realize what their minds are capable of and that they don't have to go outside to conquer a small piece of their universe.

So this is why we need algebra and not just statistics or logic.  Algebra is about finding the unknowns.  It's about looking at how the complex variables in our lives that affect each other and us. It has the further advantage of being the bedrock of higher level math so that if a student chose to pursue advanced math, she could.  It's got an easily understood framework of logic so that when the basic properties of algebra are mastered, all the other results are easily provable by a 14 year old with a pencil.  But most importantly, mastering algebra - especially because it can be such a difficult transition for many students- makes a student feel powerful and in control of her mind and world.  Isn't this how we want students to feel when they go out to help shape our society?

Baby steps

There was a good discussion recently on Twitter about complex tasks and why many teachers and students shy away from engaging with them or give up in frustration and return to low-level tasks.

Thoughts/critiques of this framework? pic.twitter.com/vsTdBKbOhC
— Geoff Krall (@emergentmath) November 20, 2013

I think that we can all come up with reasons why it's difficult for many teachers (including myself) to move out of their comfort zones and implement rich tasks in their classrooms. I am also interested in figuring out why students would resist complex tasks. @MathEdnet blogged about the various reasons that complex tasks can empower students by giving them more control and a voice as mathematicians and doers. The idea is that working with rich problems allows students to see their knowledge as valuable and themselves as active users of such knowledge. In implementing such tasks in the classroom, however, I have often seen student frustration and discomfort with the change in expectations from previous classes or from how the class had been functioning. This is sometimes especially true for students who care about their progress the most and who have certain ways of doing mathematics that have worked for them in the past that no longer work in a framework of complex problem solving. For these students, complex tasks appear confusing, unfamiliar, and an obstacle to their goal of doing well in the class. It can feel very frustrating to the teacher, especially if she hopes that implementing a complex task will increase student buy-in and engagement. Everybody is unhappy.

There are many ways of working on this issue, I think, and each is unique to the particular confluence of school, teacher, and group of students. Some teachers have big enough personalities that they can persuade students to trust them and step out of their comfort zones through sheer awesomeness.

Not a teacher, but would probably be an awesome one.

Teachers like me who have a hard time not being liked by our students and are not inspiring enough to get everyone to drink the Kool Aid come up with more gentle approaches. Baby steps, if you will. I have been working on a mix of traditional and complex instruction that takes students from the type of work that they're used to doing in math classes and gradually, inserts some open problems, starting with smaller tasks that are worked on in class and give students plenty of supports to hopefully build on more and more rich problems as students' comfort level increases.

I am, by no means, amazing at this. I definitely give tasks that are too open for students to handle and they freak out. Or alternatively, too many low-level tasks, which undo some of the work I've put into pushing them past that point. But this is the type of thing that is really, really hard to learn to do. Or, at least, it is for me. It's not something that is part of a graduate course or can be picked up by watching a lesson or two taught by a master teacher. And I have certainly never seen a pre-made curriculum that does this type of nuanced dance between what this particular group of students is comfortable doing and something that's just a bit outside of their comfort and ability zone so that they feel challenged and interested, but not overwhelmed and frustrated or bored and disengaged. So. My point. I did have one. I feel like lots of us on Twitter are stabbing away at this teaching thing, but with different tools, personalities, and kids. And it's easy to feel frustrated that I'm not doing amazing open tasks every day with my students or month-long cross-curricular projects that empower and engage them to the utmost.

Wait, this isn't what your classroom looks like every day?

But, I'm working just outside of my comfort zone and pushing my students to do the same. Baby steps. But progress, nonetheless. And I'm confident that y'all are doing the same, in your own way.

So coming back to the original question - perhaps what I'm hoping for is more recognition of baby steps and meeting people where they are, both teachers and students, to help them make small, but noticeable progress, as a way out of the cycle that @emergentmath described.

Congruent Triangles Review Game

I played a review game with my geometry class a few weeks ago that they loved so I thought I'd share it.  I think I stole this idea from a blog, but I can't for the life of me remember which blog, so if it's yours let me know.

I stole the problems from the Pearson Geometry Common Core Edition. 

I printed the document below double sided but didn't staple it.  Then I shuffled the pages and made 8 or so copies of all of them for the 8 groups in my class.  The groups needed to start with the page that has "RP" at the top and do the proof.  Then they hunt for the answer in the sheaf of papers.  When they find the answer, they grade their proof against the answer key, turn the answer key over and work the problem on the back of the answer key.  Then they hunt out the answer key to the new problem.

If they keep track of the order in which they did the problems, they can write down all the letters from the upper right hand corners of the problems, unscramble them and a message appears.

I had students for the first time actually paying close attention to every step of the proof, asking great questions about why different steps appeared, if they were necessary, and if/how order in the proof matters.  They also really loved working out the code.

I know it's just drill and kill two-column proofing, but it did a nice job of getting my students to compare different proving methods and getting them to analyze their own work.

Here it is:

Triangle Congruence Proofs Review Game

Awesome Article

Here's an AWESOME article on that oft heard phase that crushes teachers' souls, "I'm not a math person."  It's titled "Miles Kimball and Noah Smith  on the fallacy of inborn math ability."

Getting students to dig deeper into rich problems

So I was going to participate in the #MTBoS Challenges, but then, life happened. I did write a blog post responding to the first challenge, and even though I'm not participating in the full scope of challenges, I'd like to post what I can. So here is what I wrote in response to the question "What is one of your favorite open-ended/rich problems?  How do you use it in your classroom?"

Even though we were directed to write about a favorite rich problem, I’m going to write about a problem that is the most recent one that I’ve done with a class because I’m having some issues with the way it’s worked out and would love some feedback on how to make it better.

The problem that I recently gave my 2-year Algebra students (these are 8^th graders in the second year of a 2-year course that covers a standard Algebra 1 curriculum) was the first one in the Integrated Math Program, Year 1 book, called Broken Eggs. This was students’ first problem of the week, in which they are to write up their problem-solving process and justify their thinking and solution, if they find one. In this problem, you are told that a number of eggs when put into groups of 2, 3, 4, 5, or 6 always had one egg left over, but fit perfectly into groups of 7. Students were asked to determine whether there was only one unique solution or whether there were many possible solutions, and if so, how they were connected to each other.

Full problem:

Broken Eggs POW by Anna

I gave students 20 minutes in class to start working on the problem (5 minutes alone, then 15 minutes with their group members) and a week and a half to complete their write-ups. I also met with students individually who were struggling.

I thought that the problem was a good one with which to start as it could be approached from a variety of angles and would encourage for the looking of patterns. However, the results were pretty disappointing. Most kids were only able to find the first solution and did so using brute force (writing out the multiples of seven and testing each one to see how it divided by 2, 3, 4, 5, and 6). Quite a few kids just looked at numbers that weren’t evenly divisible rather than looking for a specific remainder. Almost no one found any other solutions and not a single student found a pattern between the solutions. Almost no students even attempted to find one. So the problem just turned into one that required some organization to keep track of things, but almost no algebraic thinking. So basically, the result was a lot of annoying calculations with little payoff.

I am trying to think about what I could have done differently to encourage students to keep going and to notice patterns that would make their work easier. Having students share strategies maybe would have helped to disseminate some of the shortcuts that a few of the students discovered, but not ones that no one figured out. I think that part of the tension for me is that I want open problems to really be about students’ thinking and approaches, but also be a learning opportunity that stretches them past their current abilities and into something more advanced, and I don’t know how to do that without giving hints or telling kids to change their approach. Basically, I want them to learn and be stretched mathematically, but have it be organic and come as an extension of their own thinking rather than a top-down approach where I direct them.

Part of the issue also is that kids are mostly to work on problems of the week outside of class so I’m not getting much of a chance to see their thinking before they turn them in to me. So another change that I’m thinking of doing is having students turn in a “rough draft” that I can give them feedback on or that we can confer about in person before they complete their write-up. There is a thin line between pushing a student’s thinking and directing it onto a predetermined path that takes away from the problem’s openness and richness, and I am still navigating how to do this in an optimal way. Suggestions welcome!

I'm a real teacher again!

My life has kind of been in turmoil for a while.  After one cross country move last year, we've just driven back across the country.  By car: Oregon-> New York-> California.  By airplane: DC, Japan, Indiana, Ohio, and Arizona (and Oregon and New York a bunch).  This is all in a 12 month time period.  I wonder how many miles I've traveled.  I think it's probably close to a world record.

I'm also undergoing the teacher licensing process in my 4th state!  Woot!  Soon I'll collect all 50 :).  I would totally go for national board certification, but I'm not wise enough or anywhere consistent enough to go for that yet.

We moved back to the west coast in September because of a family illness.  I left my job at the one-to-one private school abruptly and was actually looking forward to some time off.  The private school was year-round so I was getting pretty tired.  But of course, obsessive compulsive me couldn't stop checking craigslist and within 24 hours of arriving in San Diego I had a job interview at a charter school.  I was really really nervous about not having a job plan when we decided to come spend some time in California so I was delighted at the prospect of a job so quickly.  But it's in a classroom again with a lot of kids and I start tomorrow and I'm terrified.

It's been a year since I worked in a classroom with more than one student and my pitiful classroom management skills have completely atrophied plus my obsessive compulsive work ethic already has me fruitlessly planning lesson after lesson even though I have no idea where the kids are and will have to scrap all this work and start over.

Do the nerves ever go away?  I wish they would.  Everyone says I'm super lucky to have landed a job in late September/early October but right now I wish I could go back two weeks and kick my over eager, initiative grabbing, hopeful self and tell her to just CHILL!

But I am happy to be teaching in a classroom again, I am.  And I will try to post more because now I'm legitimate.

Exponent Rules GAME

I've posted on this topic a bunch of times here, here and here, but I'm not tired yet of hammering more nails into this coffin.  I think that correctly mastering exponent rules is a gateway skill.  Maybe one of the most important gateway skills in algebra.  Exponent rules:

• Formalize the meaning of multiplication and division for algebra.
• Provide the first forum for students to effectively use reducing in an algebraic context.
• Introduces students for the first time to how simple algebra definitions (i.e. the definition of an exponent) can be used to prove a multitude of other cool rules that make doing math easier.  In other words exponent rules formalize the structure of mathematical logic and proof for students.
• Is often the first time students see and manipulate algebraic rules represented purely with variables.  If students can understand and use exponent rules, it prepares them for using and understanding other rules represented with abstract mathematical language. 
• Set the foundation for a student's understanding of polynomial functions, radical functions, exponential functions, and logarithmic functions.  Without a solid understanding of exponents and their properties students will struggle with all of these types of functions later.

And I think we can teach the exponent rules well because they're just not that hard to derive, but the level of abstraction is what makes it difficult for students.  So in teaching exponent rules, I believe we should focus on teaching students the abstraction, and the rules get learned along the way.  

That being said, I don't know how to do it but I keep trying.  I've updated my lesson on developing the exponent rules.  You can find that here and also, I've developed a simple game that I hope helps students cement the rules and learn to play with them.  The game involves both strategy, luck and understanding of exponents so I think it's pretty good but it's only had a few trial runs.

Materials: You'll need a set of blue cards and a set of green cards.  You can download the cards and the rules here.  Lay the cards out like so:

Object: Combine your starting expression with green cards to create the target expression.

Rules:

(1) You may use as many green cards as you wish.

(2) Cards that look like this: (   )^2 must be applied to your whole expression so far.  So if you start with the card “ab" and you grab the green card (   )^2  you will end up with a^2 b^2

(3) If neither player can find the right cards to create the target expression, three more green cards can be put down. 

(4) Once the target expression is reached by a player, that player gets the blue card and all the green cards they used to make the winning expression.  A new blue card is then put down and green cards are added until there are 9 green cards again.

(5) Once all the green cards are gone, the game is ended and the player with the most green and blue cards wins.

Examples:

Here are several examples of how the players in the set-up above could reach the target expression. 

Not Ratios again!

Students either seem to "get" ratios or they don't.  I don't know what to do about it.  I made a really detailed ratio and proportion lesson based on beats per minute and the fastest guitar player in the world for my algebra 1 students.  3 students that I used it on loved it and understood everything just fine, 1 student could not get it no matter what I tried.

I drew a picture of a person and said that he was 6 ft tall but a shrink-ray shrunk him to 2 ft.  If his legs were 3 ft long originally how long are they now?  My student thought for a second and then said "-1 feet?"  I tried pictures of triangles, I tried explaining that scale factors worked with multiplication and division, not addition and subtraction, I tried just showing him the math steps based on fractions in a last ditch attempt to get him to walk out of the class with something.  But none of it worked because he didn't have an internal sense for proportion.  He could do the mechanism of cross-multiplication, but a "sense" for proportion just eluded him.

I'm now working on a lesson for geometry introducing ratio and proportion and I'm getting a little cold and clammy because I have nothing.  My experience is just kids see it or they don't and if they don't, I don't know what to do.  Sheer perseverance and drill have helped these students eventually reach an "aha" moment, but it just seems to be based on time, not on cleverness of the activity (or I haven't found or thought of a sufficiently clever activity.)  I've been sitting in a coffee shop for an hour now and so far, I just have a warm-up:

1. Which pair of numbers is out of place?  Explain why you chose that pair.
1.    3 and 4
2.  5 and 6
3.  9 and 12
4.   27 and 36
2. Which pair of numbers is out of place?  Explain why you chose that pair.
1.    9 and 12
2.   12 and 15
3.   20 and 25
4.   32 and 40
3. You got a part time job at The Pizza Hub.  You just found out that your co-worker makes more money.  Which statement would make you angrier?  Why?
1.    Your coworker makes $10 more than you.
2. Your coworker makes double what you make. 

 If there's anything out there to follow this warm up with, I'd love to hear about it.

Update [8/4/2013]  Here's the lesson I eventually came up with.  I think it does a decent job.  

Ratio and Proportion Lesson for Geometry by ezmoreldo

And here's an "I notice, I wonder" activity that could be used to get students thinking about ratio and proportion.  Both of these are PDFs to preserve formatting, but if you go to my scribd profile you can find the .docx versions.

Ratio and Proportion- I Notice I Wonder PDF by ezmoreldo

Parallel Lines and Transversals game

Here's a simple game that helps students cement all the different vocabulary words for the angles formed by parallel lines and transversals.  I teach at school specializing in one-to-one instruction, so unfortunately it's not very much fun, but it does work!  I basically just made a geometry version of my parallel and perpendicular lines game.

Parallel Lines and Transversals Game

Sharing and Laziness

My 5 year old nephew this morning had a bowl of blueberries.  I had a devilish headache and was lying on my grandparent's porch swing.  So when I wanted a blueberry, instead of going to get one for myself  from the kitchen, I asked him if I could have one of his blueberries.  He gave me three.  And he fed them to me himself.  Isn't sharing nice?

A few weeks ago a new teacher was hired at my school.  She's certified in English, but she'll need to teach some chemistry this summer (my school is kind of crazy) so I told her I'd share all my lessons with her so that she had somewhere to start.  She was surprised and thanked me and this led to a larger discussion about sharing resources.  She said, and I'm quoting almost verbatim, "I'm one of those teachers who gets up Saturday morning excited to lesson plan all weekend.  I'm happy to share my resources with coworkers I like, but I won't share with teachers I don't know or don't like because I don't want to encourage them to be lazy!"

A week later our school's curriculum writing team got together and our bosses told us not to make the lesson plans for the curriculum they're distributing across their various campuses too detailed.  They don't want more than a page per lesson.  Most of the teachers they hire aren't certified teachers because it's a private school so they told us they need a unified, regimented curriculum to support people who have never taught before.  At the same time though, they said we could trust the teachers to know their subject matter and that they only needed an outline of content to be covered with no recommended instructional strategies or resources because "we don't want to stifle our teachers' creativity by giving them anything that's too detailed."
Is this a common view point held by teachers and administrators that providing rich, creative, and detailed lesson plans, activities and games promotes laziness and stifles creativity?  It's hogwash!  I am inspired by innovative and engaging ideas that I find in other people's lessons and the more detailed they are, the easier they are for me to adapt, to understand, or to use to help my students learn.  The richer my materials, the more freedom I have to experiment.  The more time I have to think about how to engage particular students.  When I share my lessons I get feedback on what worked and what didn't and I become a better teacher.  I can't believe how entrenched this miserly attitude is about sharing materials.  This curriculum meeting was composed of dozens of teachers and administrators from all over California, New York and New Jersey and they accepted this rationale without question.  That we should keep the good ideas secret to try to force other people to come up with their own good ideas.  Don't they know that good ideas mate with other good ideas to produce litters of new, bouncing, energetic good ideas?

And so what if you give a lazy teacher your lesson plan.  Isn't that a good thing?  Those kids in the lazy teacher's classroom maybe haven't encountered innovative teaching strategies before and if a lazy teacher uses someone else's lesson plan to good effect with their students, isn't that what we all want?  That even kids who have "burned out" teachers have access to rich educational experiences?  Why save the good teaching just for the students who happen to have been placed in your classroom?  It's so silly to hoard!  I guess I can understand not wanting someone to get credit for your idea, but in the end aren't we in this business to help students learn, not to accumulate "credit" for ourselves?  I'm probably way too idealistic for this world.  I guess I feel so strongly about this because I wouldn't be a sixth of the teacher I am today if I hadn't ripped, stolen, copied, and adapted every idea I could skim off of other teachers' blogs and websites.  So I guess I am one of the lazy teachers but boy am I a better person and a better teacher for being willing to learn from all these amazing experts surrounding me.

The Logic of Geometry

I've noticed a distinct void in the Common Core where geometric logic used to have a home.  This makes me really sad for three reasons:

•  I think it's a really great way to show students how mathematical thinking can be applied to real life 
•  I was skipped past geometry in high school and when I got to college, the lack of logic training was a real handicap
• And I've developed a tiny bit of skepticism about Common Core.  Not a lot, but I read this article about how the Common Core was developed and it creeped me out a little.  The fact that only one teacher helped develop the Common Core seems kind of terrible.  I recently made friends with an economics blogger and he told me that every profession engages in "turf defending" where those in the profession reject outsiders' perspectives.  That immediately made me not want to be a turf defender. I won't be the person who stomps on the school house floor and screams "MINE!"  Of course I want outside input, but can I also add that it's maybe a little out of hand in education?  I want to teach logic damn it! 
• Oh, also, I've found logic very challenging to teach and to give up on trying to find the perfect way to introduce it now just because Common Core gives me permission feels like a cop out. (so I am pretty selfish after all.) 

I've made a point of always teaching it thoroughly up until now.  This year at least I'm stubbornly sticking to my guns and teaching a bit of logic (although I nixed conjunctions and disjunctions, mostly because I personally never used them in college)

Below are my three logic lessons if you'd like to take a look.  I'm writing lessons for the other math teachers in my school so the notes are kind of overly detailed.  Also my school gives us only 50 hours a year (as opposed to the usual 150-180) to get through a year's worth of material, so it's all super condensed.  I don't know if it's usable outside my school, but feel free to steal and I'd love feedback.  I've uploaded the pdfs below to preserve formatting but the .docx are also available on Scribd.

Geometry Lesson 9 V2-Introduction to Logic by ezmoreldo

Geometry Lesson 10 V2- Converse, Inverse, Contra Positive and Biconditional by ezmoreldo

Geometry Lesson 11 V2-Proofs Using Logic by ezmoreldo

Note: I borrowed some of the homework problems from Harold R. Jacob's Geometry: Seeing, doing and understanding 2nd ed. and from AMSCO's Geometry.  I also totally stole the formatting from Dan Wekselgreene.

One-to-one

My school teaches students in a one-to-one classroom; one teacher, one student.  Doesn't this sound awesome?  From the teaching perspective, you can customize every lesson to the student and be sure they're really learning.  From the student perspective, you won't have to sit while the teacher's instruction outpaces your focus or interest.

We had an IEP meeting for one of our students and we were trying to convince the local school board that this student was making dramatic progress with us and thus his needs were being met.  The moderator said "who wouldn't make progress in a one-to-one classroom!"  And this got me thinking.  I wouldn't.  I would have HATED a one-to-one environment as a student.  I didn't like to be the object of too much intense teacher concentration (and really, we can be overly intense sometimes).  I liked to sit back and evaluate what the teacher said before deciding to accept or reject it.  I liked to have a little freedom and control over how I took my notes and whether or not my mind drifted during class.  I liked the opportunity to work with other students- our collective energy was so much more powerful than mine alone.

As a teacher, I'm not a huge fan of one-to-one either.  First of all, while "classroom management" is easier (this is a pretty silly word to use for one-to-one) it can also be more difficult because there's no escaping personality conflicts.  It's easier to get locked into battles of will (I avoid these like I would avoid Mexican food in NY- sorry guys, it's terrible)  but I've noticed these types of intense, pride posturing battles in classrooms near mine.  There are no other students near by to say "hey dude, chill out" and there's no one the teacher can turn to to exchange a sympathetic look with or to share a joke with to relieve the tension.  This leads to my biggest complaint against one-to-one.  It assumes that the teacher can teach all the student needs to learn.  THIS ISN'T TRUE!  Students learn so much more from each other.  The teacher can present content and establish a respectful and fun classroom culture, but the students sustain the culture, teach each other how to act, give each other support and encouragement, joke to relieve tension, boredom or frustration, and reinterpret the content in different ways so that their classmates can see it from multiple perspectives.  Learning is both a solitary and group endeavor.

So I know not many people have one-to-one classroom environments so this doesn't really matter to many besides me BUT I've been reading a little bit about the advent of personalized learning software.  It seems that since the whole constructivist approach to teaching math, where everyone had to learn everything in groups, hasn't really shown huge gains, there's a movement to go in the opposite direction.  Personalized learning software seems like it's going somewhere.  Knewton, a personalized learning software developer based in NY, and Pearson are teaming up to bring customized, competency-based learning to as many students as possible.  I think their vision of the future will be students sitting at home, in coffee shops, in libraries or even in classrooms, logging in to their program and getting a wholly customized learning experience.  The Knewton software amasses student data and has an algorithm that decides when a student is ready to move on to new material, and what that new material will be.  This is exactly like what I'm doing right now, except without the teacher.

Here's what I see happening:

1. computers probably aren't smart enough to do this yet.  I don't know if they ever will be.  I have an Algebra student who asked me if numbers go on forever.  We had a nice side discussion on the nature of number and infinity which he understood.  His abstract reasoning capabilities have far outpaced his computational competence so he asks really good, deep math questions and understands the answers even when he struggles with adding fractions.  I can satisfy his deeper curiosity while still drilling him on basic arithmetic.  A program would decide that he's not ready for anything beyond basic computation. 
2. I do believe that school is about more than learning content.  I think that learning to work cooperatively is important.  Learning to share a joke to relieve stress is important.  Learning how to speak up for yourself to an authority figure is important.  Socialization is important.  Maybe this just makes me old fashioned.  But having people by your side makes learning more fun too.  
3. Peers push you to accomplish more than you can by yourself.  This is a problem I've seen in one-to-one teaching- that it's so much harder to motivate a student to study independently or to work on projects.  When there's no one to share it with, no peer audience, teenagers shut down and disengage.  Duh!  Even teenagers with special needs (we teach a lot of students with severe social anxiety) need peers to support them.  Even if you place kids in the same classrooms to use these educational technologies, they won't be learning the same content so won't be able to support or motivate each other effectively.  
4. Finally, if you allow students to do the work at home, I think cheating will become the norm.  I think it already is.  A coworker of mine (a teacher!) admitted to me that she takes all of her sisters' online tests for her because her sister is an athlete and doesn't need school.  

Can't we find a middle ground?  There is never going to be one model that works for everyone, but we all deserve to experience different types of learning.  I may not like one-to-one learning because it makes me uncomfortable, but I went in to see my professors to get help on papers and it was good for me.  We all need a mix of different experiences.  We need technology and classrooms and individualized instruction and games and textbooks and teachers.  Why can't technology make our classrooms richer and more full of different kinds of learning?  Why is it a choice between classrooms or personalized learning?  Can't we recognized that there are good things about classrooms?  Why are we so obsessed with "or"?  Why not and?  Knewton and Pearson might say though that they are offering an "and" option.  That this is meant to enrich teachers' curricula not replace them, but I'm suspicious that this is just another way to devalue teachers' professionalism.  I read an article in Newsweek (I think but of course I can't find it now) where the head of one of these personalized software companies admitted that he didn't employ any educators on staff.  And Pearson, based on what I've seen of their textbooks, doesn't know that much about teaching either.  

Whiteboarding wins and fails

Now that whiteboarding (my posts on it here, here, and here) has become a full-fledged part of my classroom, I wanted to reflect on how it's been going. So first, the successes...

Whiteboarding wins:

• Student cooperation and engagement in group activities has been kicked up a notch. I thought that I was doing fine with structuring group work and encouraging student engagement and participation, but whiteboarding has really increased cooperation and discourse between students. There's something about having one person write for everyone that creates more investment in the process and final product. A white board also encourages larger and clearer writing, making it easier for everyone in the group to see and participate.
• All students have to demonstrate understanding and it's harder to hide or have one person do all of the work. When groups present or when I go around the room and talk to groups, I pick students at random to explain their group's thinking. Rotating the marker between group members also help keep everyone involved. 
• Kids love to get picked to show their work and work harder to make their thinking more explicit and clear.
• It results in greater perseverance and willingness to try crazy things - something about the ease of erasing makes kids more willing to take risks and get messy with a problem.
• Kids get used to asking questions of others' work and disagreeing with their process politely, and I am seeing this carry over into non-whiteboarding work.
• Students really like it, and it lends itself to other activities & structures that allow it to continue feeling fresh and exciting.

... and now for some failures...

Whiteboarding fails:

• Annoyingly, kids are still uptight about making mistakes and having errors publicly attributed to them. Even after playing the Mistake Game, kids were still getting upset and unhappy with each other when unintentional mistakes would get exposed in front of the class. I had one incident where a group of boys exploded with accusations about whose fault the mistake was and had to talk to them out in the hallway about their behavior. Kids still find it embarrassing to be wrong, which I am finding frustrating. Like, how many times do I have to say, "This is the awesomest, most interesting mistake ever! We are learning so much more from it than we would from the "correct" approach!" before it sinks in? When I would ask groups to present, kids would still anxiously ask "but is it right or wrong?" Right now, I'm working around this by putting up boards with mistakes and having the class give feedback without attributing from which group the mistake came, but I'd like to get to a place where students are actually comfortable with their errors and aren't embarrassed by having them publicly discussed.
• Kids are still kind of bad at asking questions rather than just pointing out the mistakes. Or their questions are really just statements with a lilt at the end: "Should you get 8 when you square 4?" 
• The boards are sometimes hard to display if the group wrote too small, and the whiteboard is too big to stick under the document camera. My workaround has been to take a photo of the whiteboard and send it to the projector where I can zoom in on their work. Hopefully, their writing will get better as we continue to use whiteboarding regularly.
• Students would like to have notes they can take home, and whiteboarding does not lend itself to that. I thought Bowman's post on giving students time at the end of a period spent whiteboarding to reflect and write down some notes was really great, but it's hard to find the time for this what with going over homework, whiteboarding, and student explanations of their work, all of which must fit into a 45 minute period, 4 days per week. I've thought about taking photos of their whiteboards and putting them on our class webpage and perhaps asking them to do some reflecting and summarizing for homework, but haven't actually followed up on this.

Overall, I am really happy with how whiteboarding has gone this year!!

Thoughts on homework

Julie put out a call for posts about homework differentiation, which came at a perfect time for me since this is something I've been thinking about recently. I'll get to differentiation in a sec, but first, some general issues that I'm having with homework:

It has sort of taken over my teaching life. We spend a lot of time on it in class, and I spend a lot of time on it outside of class. I am not sure if this is good or bad. The way that I do homework, students have all of the answers and must check their assignment and identify problems they have done incorrectly or that they don't know how to do. In class, students discuss their questions with each other in groups for about 5 minutes, then we make a list of questions to go over as a class. Students volunteer to present their solutions to the identified questions and answer any follow up questions from classmates. Those who got those questions wrong or didn't complete them are expected to make corrections to their work. The whole thing takes about 20 minutes, on average, which is almost half of the 45 minute class period. For my end, I look at each problem for each kid and write feedback on each paper. If there are errors or incomplete work or misunderstandings demonstrated, credit is reduced and students are expected to correct those assignments and turn them back in to me. Once the assignment meets my criteria for "goodness," full credit is given. Students are also required to write up corrections and explanations of their errors for tests and complete a journal entry reflecting on their progress each chapter. For my four classes, which have four homework assignments each week, the whole grading process takes me on average about 2 hours every day. That's a lot of time!!!

On the plus side, I think doing problems independently, discussing their thinking with other students, presenting their work and having to make sure that it adequately communicates their process, and getting regular, frequent feedback from me are all really helpful for their learning of both the math content and how to be a better student. On the minus side, it means that there isn't much time left for group explorations or as thorough grounding of each lesson as I would like, both in terms of how much class time is available and how much time I have to plan lessons outside of class.

I use this system not because I think it's super duper awesome, but because I haven't seen any other way that accomplishes the student learning piece better. A few things that have helped me be slightly more efficient:

1. If I see a bunch of errors, say three or more, I stop grading the paper and write a note to the student reminding them that they need to check their answers and make corrections, either on their own or with help from me. The assignment doesn't get any credit until that happens. I don't want them relying on me to check their answers for them, and I want them to come in and get help.
2. As the year continues, I start restricting the amount of time given before corrections can be turned back in to me. Eventually, they are not allowed to correct any more and must come in and get help before turning in the work in the first place. This helps them transition to their future math classes, none of which allow corrections on homework assignments. 
3. I try not to assign too many problems and assign ones that I think are meaningful and helpful to students' learning. Kids do see the connection between putting effort into their homework and their learning and progress in the class.

But really, this is a crazy inefficient system, especially in terms of my time use. I don't know about everyone else, but grading homework for me is somewhere up there between having hot pokers shoved into my eyes and being trapped in an airless mine shaft. It's seriously obnoxious. But I hate the idea of just grading based on completion because then there's no feedback to the student and much less of a connection between homework and learning. So, I'm stuck with my current system unless someone has any suggestions.

Okay, on to differentiation!!

So I don't currently actively differentiate a lot of homework, other than test corrections and journal entries. There is a good deal of differentiation that happens as a natural outcome of how I structure homework though because students who didn't get full credit on assignments have to correct them, which means that they are addressing their own specific misunderstandings. Also, students who score less than 70% on a test are required to complete extra work as well as meet with me.

Here are some other ideas of things I'd like to try:

1. After a quiz, breaking up the class into groups based on the types of mistakes they made. Each group will have all the kids that need to work on a particular concept together. Then, I can work with each group to reteach/address misunderstandings on that topic and their homework will be problems targeting that specific concept. The kids that rocked the quiz will work together on a challenge problem. Their homework will be to complete it and do a write-up of their process.
2. In my regular-paced class, which is completing Algebra 1 over two years, we often spend two days on a learning target. For the first day, I'd like to assign everyone the same basic homework assignment. Then, for the second night's homework assignment, students will have the option of either doing more basic practice OR to do a more challenging, shorter set.
3. I loved this post by Bob Lochel that I recently read about differentiating homework assignments in which different problems are worth different point amounts (depending on difficulty) and the students complete enough problems of their choice to earn the assigned total point value.

Looking forward to reading everyone else's differentiation strategies and ideas.