Girls in Math

 Hello!

It's been awhile.  I was just reading "Math with Bad Drawings" and realized that the author graduated from college around the same time I did.  Yet he's here, publishing AMAZING books about math.  And I'm... still teaching, but not contributing to classrooms beyond my own the way I so desperately wanted to at the start of my career.  I'm reflecting on what happened.  Why I stopped blogging, why I felt scared to go to professional development conferences and keep shying away from opportunities to grow beyond my school.  

It's a long sad story with two prongs.  I've never had confidence, in math or anything else.  Any acclaim felt gifted to me as a fluke and thinking about striving for more felt like tempting fate.  Pitting myself against.. well.. mostly men in math my whole life has left me feeling very vulnerable to ridicule (the very fact that I saw it this way is telling).  The result has been a contented career as a teacher, but even in teaching circles I feel like I have little to contribute.  That's prong one.  Prong two is personal devastation.  I was blogging and teaching and humaning pretty well I think from 2009-2014.  Then I fell into the oldest of traps: marriage and parenthood.  I loved my marriage and I loved being a parent.  I guess I still love being a parent (I guess... but it's so much harder than teaching!).  I have been shocked however to discover how patriarchal these two frameworks still are.  I mean, I knew they were, intellectually, but I didn't think it would happen to me!  Right?  I mean... it totally hadn't happened to me at all before... heh.  

My spouse retreated from the responsibilities of adulthood.  Collapsed in on themselves when the work got heavy.  I, in my wisdom, was determined not to be one of those wives who nags.  Because that's demeaning to both of us.  I didn't want to demand too much help, I wanted modern adulthood to be fun.  Who cares if the house is a mess.  We should do math instead!  Sure, go work on the game you're designing, I'll take the kids to the park.  Slowly, I realized that housework, the work work, and the childrearing work had all fallen to me while my spouse was playing.  Then I wasn't fun, I was exhausted.  My spouse got bored and left.  There was a little more to it, as in all stories, and I don't want to fall into a blame trap. I was the one who insisted on doing the work because I was so scared that asking for help put too much of a burden on others.  I played into the system by enabling inaction.  I know also the dynamics can be reversed.  Yet I was so shocked and confused to find myself in this 1950s predicament.  It destroyed what confidence I had left in my career.

So why don't girls do math?  For me, decades of subtle sexism combined with one big sexist torpedo undercut any contribution I felt I could make.  I still teach but watching bright girls year after year choose other disciplines and bright boys thrive STILL in 2022 has me bewildered.  Our math department is chaired by two women.  Yet it's our male part-time colleague who is still pursuing studies at a university.  He has much more confidence than either of us "heads".  When he shares videos on youtube with me of math he's playing with, the majority of those videos are made by men.  My favorite bloggers and writers are men.  

I know women are out there, but just like in math class, I can't hear them.  I took a graduate level math class at my local university the year my son was born (8 years ago) and there were about 30 students, 7 or 8 of them women.  I kept tallies all year of how often women spoke in class compared to men.  I'd add a tic to the top of my paper, one column for men, one for women.   It wasn't until the first semester was finished that my cumulative count for women reached 5 tallies.  The few women in math teaching that I know, respect, and crave to hear more from are entrenched in elementary school math.  This high school math world is bleak.  The teaching of it still feels dominated by patriarchal systems of intellectual one-up-manship.  I feel the touch of patriarchy in our grading systems, in our sequencing, in who talks in class and who doesn't, in how I ask my questions.  Yet again I'm facing a calculus class all boys, and two girls, fighting to be heard or drifting out of the space.  As I did and as I still do.  I have many non-binary and trans students as well, but from my anecdotal observations, the way that students had been forced to identify earlier in their life still often paves their pathways through math. 

After reflection, these thoughts made me want to put words on my blog again.  Hesitant words and clumsy words.  Unfortunately, I think I drifted towards math because my skills in disciplines more welcoming to women are less than shiny including my articulateness... and spelling... and general ability to communicate.  Still thought I feel compelled to try.  Math communication is so tricky, a puzzle I keep wanting to solve.  

I know blogging is old fashioned now and teachers don't seem to be sharing free resources anymore.  That's still what I want to do though.  I want to share what I'm doing in the sliver of an improbable chance that it's of value to someone.  

I'm teaching Calculus, Geometry, Graph Theory and Japanese this year.  I'm very interested in sharing my Graph Theory materials as it's not a topic I've been able to find resources on at the high school level BUT IT'S SO MUCH FUN with high schoolers.  

I'd also like to share the systems I'm playing with, including a new grading system I developed for my school and my ever changing methods for organizing curricula and lesson plans.  Wish me luck!

To be good at math

Falsehoods I have internalized.  I'm thinking about asking my students to make a list too.

(1) To be good at math, you have to get concepts quickly

(2) To be good at math, you have to get things the first time

(3) To be good at math, you shouldn't have to struggle with a topic

(4) To be good at math, you have to feel certain

(5) To be good at math, you should never have had to repeat a class

(6) To be good at math, you should be vocal

(7) To be good at math, you must be so innately smart that you're above other people

(8) To be good at math, you can follow other people when they talk about math, and if you can't, you're bad at it.

(9) To be good at math, you should be younger than other people in your class.  Mastering it younger=being innately good at it.

Average Rate of Change for Pre-Calc

Here's an average rate of change activity I cooked up.  I thought I'd share it just because I couldn't find anything like it when I did my own internet dive. 

I'm teaching pre-calc in a semi flipped classroom model this year.  We're going to spent 2-3 days per section of the textbook and rather than teaching any of the textbook content, we're opening each section with a thoughtful activity, their homework is to read the textbook and do check in problems, then the next day in class they're working problems from the exercise section of the book.  So the below activity is to introduce the idea of average rate of change and how it's used without having formally talked about the concept before.  

I'm going to start with Sam Shah's "What does it mean to be going 58 miles per hour at 2:03pm" worksheet.

Then here's the worksheet I made:

Average Rate of Change Activity by ezmoreldo on Scribd

A mathematical beginning of the school year activity with low bar and a high ceiling

I'm teaching a really broad range of classes and subjects this year (6th grade math, middle school geometry, algebra 2, calculus, Japanese 1 and Japanese 3) so for my math intro activity I wanted to explore fraction concepts in a way that both 6th graders and algebra 2 students could enjoy and would reveal the struggles and confusions of both groups.

Voila: my summer in fractions.

My Summer in Fractions by ezmoreldo on Scribd

This activity went so much better than I expected that I wanted to share it somewhere.  It took one and a half class periods to complete (really just one- half of the first day after other intro activities and half of the next day).  I thought it might be funish in that the kids got to share their summers with each other and color a little, but I expected to hear some grumbling about fraction hate.  Instead, it generated some of the most interesting conversations I've had with students to date.  First I'll talk about how I ran it, then at the bottom of the post I'll list the cool conversations/insights we had.

On day 1, I did some basic introduction stuff and with the 6th graders, my favorite intro to math class game where I put cards on their backs with different numbers on them and then they have to go around asking each other yes or no questions about their number and try to guess what it is.  With Algebra 2 we did an intro activity where they wrote on a note card a number that represented their summer (45 hrs playing zelda, 17 for the age they turned, 5 am for the latest they stayed up, etc.) and their general attitude about math on the back.  Then we guessed who was associated with each card.

Then after those intro activities I handed them the above worksheet and did one or two of the categories on the top (sleep and vacation) for myself on the document camera so they could see some of the thinking involved then I let them work the rest of the period.  Finishing it was homework.

Then the next day I showed their pie charts on the document camera anonymously and they tried to figure out who had made which pie chart.  I would then ask a mixture of math questions about the charts and personal questions (if 1/7 of the pie chart represented reading, how many hours a day on average is that?  What did you read?  Oh yeah? me too.  Did you binge watch the show also?  How on earth did you spend a quarter of your summer staring off into space?  What did you think about?)

Finally, I taped all their charts to the wall.  It was a beautiful sight:

Here are some of the cool conversations/insights I had with the 6th graders

• How many days are in the summer?  How many weeks is that?  How many hours is that?  So if you spend one day a week on an activity does that represent 1/7 of your summer?  If you sleep while on vacation, how do we account for that in a pie chart?
• If you spent 8 hrs a day sleeping and 8 hrs is 1/3 of a day, how much of a week do you spend sleeping?  How much of the whole summer do you spend sleeping?
• Lots of arguments about units.  "I'm confused because the fraction that represents my sleep is in hours but the fraction that represents my vacation time is in weeks?  How can I put both of those on the same pie chart?  Do I use 84 days or 12 weeks for this fraction?"
• How do we add up all the fractions and what do they equal?  What does it mean if they don't equal one or are more than one?
• How do we break the pie chart up into enough sections that we can represent every fraction we generated?
• So much reducing!  If they were on vacation for 16/84 days what's the simper fraction.  

Here are some of the cool conversations/insights I had with the high schoolers.

• I showed them how to use their graphing calculators to add up all the fractions and output a fractional answer since their fractions were much more complicated than the 6th graders and they were more attuned to precision.  They fell in love with the calculators IMMEDIATELY because they can avoid doing fraction work with them.  But we still reviewed how to find the LCM of more complicated fractions.
• We went over how to turn the fractions into degree measures.  How to use a protractor, and how to find the center of the circle.
• When showing the pie charts on the screen I had them estimate how big different sections were (is the amount of time he spent playing video games a tenth or a twelfth) and that generated an amazing conversation when we decided one of the sections was between 1/4 and 1/5.  What fraction IS between 1/4 and 1/5?  One student suggest 1/4.5.  What IS that?  Is it halfway between?  Can we find a rule that gets us the half way point between any two fractions?

And all students loved the activity.  The lowest ability students could do it, and the higher ability students came up with lots of interesting questions.  The ones who liked geometry got to play with compasses and protractors and colors.  We all got to learn cool things about each other and tease each other a little. And I got to see everyone's comfort level with fractions and decimals and some long division.  A little geometry and algebra snuck in too.  WOOT!  Success.

With my algebra 2 and calc students I also did this answer key ethics worksheet that generated some interesting discussions.

Answer Key Ethics by ezmoreldo on Scribd

Solving systems of linear equations review for algebra 2

I whipped up this worksheet out of a balance problem worksheet I found online years ago and some desmos activities on systems.  I really wanted to do some of the desmos activites themselves, but I tried one and my students couldn't stay on-task.  They kept cutting and pasting quotes from the communist manifesto and advertisements for bitcoin into the short response boxes.  Sigh.  This activity that I mashed together actually went really well.  We did most of the exercises as think-pair-shares and they remembered enough from algebra 1 for it to be a quick, easy and intuitive review of systems solving.  I did steal everything so I take no credit except in terms of the presentation.

Solving Systems Review by ezmoreldo on Scribd

NWMC 2017

Holy Cosines!  A lot has been happening in the math world in the three years since I had my first child and my life was consumed by this entropy machine.  I just attended my first professional conference in 4 or 5 years and my mind is abuzz.  I need to write it all down before I forget.

First I attended Tom Reardon's "Problem Solving: All-Time Favorite Mathematically Rich Precalculus Activities, Individualized- with Complete Solutions".  Here are some things I want to remember:

• We started with "the great applied problem" which involved a cylindrical tank lying on it's side partially filled with water.  The goal is to figure out how much water is in the tank, and how much water is needed to finish filling up the tank.  He asked us first to ask him for the information we'd need to solve the problem.  I never do this in class because I'm always in such a hurry.  What a super important first step when tackling any problem.  This was consistent through all the activities he showed us.
• Then we did a bunch of other fun problems and we played with graphing calculators a lot.  My classroom has a set of donated calculators I scrounged up from the app Next Door this summer, so I don't know how much of this fun my kids will get to have, but it reminded me of how powerful they are, and how intimidating they are.  He had us doing stuff with them I never new was possible. But remembering how to do all the little steps and where all the different buttons were was hard, and I use TI-84s every day.  I almost asked him if it was worth it teaching students this way or if we should just switch over to Desmos, screw the standardized tests, but I was too chicken.  

Then I did Dan Meyer's "Charge Up Your Classes with Free Desmos Technology."  I have to insert another Holy Cosines here!  Dan sat next to me at one point.  I was too star dazzled to say a single word to him, but I was 2 feet away!  I actually haven't always been a big three act fan.  I've followed Dan's blog for a long time, but I was too overworked to implement any of his ideas.  I'm not tech saavy and when he was first posting his three act videos I thought they were super cool, but I couldn't see how I could make any, didn't have the technology in my classroom to even show them, and I also didn't think I could spare the class time to really do them justice.  Also I was teaching 14 preps so really didn't have the planning time either (yes, it really was 14 preps.  Isn't that insane?)  But in the few years I've been away he's made Desmos into a lean, mean three act machine.  Some things I don't want to forget.

• He did this super cool thing where we made a list of values from -5 to 5, then he defined points like (L, L) or (L, -L) or (L, L^2) and had us predict what the graph would look like.  What a super super super cool way to finally cement the idea that when we graph a function like y=x^2, the ordered pairs will be: (x, x^2).  I've had such a hard time getting my students to understand that (x,y) means the same thing as (x, f(x)) which means the same thing as (x, whatever f(x) is defined as, like x^2, or x^2 -2x+7)
• I MUST go back and finish the desmos scavenger hunt then use desmos all the time.
• Dan said something during the presentation that rubbed me the wrong way and I'm still trying to make sense of it.  He had this shtick of playing super dead-pan and skeptical of anyone's answer.  He said he worked hard to make students doubt any answer they presented.  His point that we want students to justify their thinking, to be open to alternate solutions, and to really own their answers even in the face of doubt is well taken, and maybe if I'd been taught with someone like him as a teacher, someone I trusted, I would have learned more confidence.  But I have too many experiences of being ignored or doubted when I was right in less trusting environments to be comfortable with this teaching style.  

Then I was tired and the next day we had staff development for my school so I played hookie and didn't go to any of the Friday sessions.  Which was a huge bummer.

Then I did a 7:30 AM (ON A SATURDAY!) breakfast keynote with Fawn Nguyen titled, "What if We've Been Teaching Mathematics All Wrong."  WOW.  There is so much from this that I want to remember that I wish I'd video taped it.  But I've already forgotten so much of it.

• First, I want a poster that has three rules on it.  Rule #1: Never give up.  Rule #2: Never give someone else an answer.  Rule #3: Love being stuck.
• She used visual patterns in ways I'd never thought about before.  Of course I can't remember them anymore!  She did a cool paper folding activity where you take a strip of paper and fold it in half.  Then unfold and count the creases.  Then fold it back in half and then in half again.  Count the creases.  Repeat.  I've seen this activity before, but I'd forgotten about it so hopefully this blog post will make me remember.  
• I need to play WAAAAAY more with visual patterns and between 2 numbers and everything she's ever done ever.  

Then I did Andrew Stadel's "Lessons that Make Math Stick."  Again, he may have made me a groupie for life.  Things I want to remember:

• Give every activity the 3-C test; each one should be conceptual, spark curiosity, and should connect students to real experiences.  He had a great video of a girl on a see-saw.  She put a milk crate on the other side of the see-saw and started filling it with bricks.  Each brick weighed 5 lbs.  The obvious questions was how many bricks it took to balance with the girl.  What a really cool way to model division and multiplication.  He had so many of these great videos that made me see how truly important it is to spark curiosity.
• He talked about how baseball players who practice three kinds of hitting in blocks learn less than those who do mixed practice.  Then he advocated for Steve Leinwand's 2-4-2 homework model.  Here's a link to a blog post talking about it.  I really really really want to try this with my pre-calc class.
• Finally he talked about meaningful feedback- whether feedback should be immediate or delayed.  I post all my answer keys online for students to look at.  I need to give this question a LOT more thought.  

Finally I did Jeff Crawford's "Visual Algebra: Current Research and Practical Applications."  Oh my gcf!  He had us looking at visual patterns in such cool ways.  Blocks are sooooo cool.

• I want to investigate proofs without words, Jo Boaler videos and youcubed (which I'd never heard of before!) and open-up resources which I'd already started playing around with for my 6th grade math class.
• He talked about Finger gnosia which is crazy and I want to experiment on my toddler with.  
• He showed us all the different ways our brains see patterns and how while all those different ways can be distilled into the same algebra expressions, they are beautiful in their uniqueness and the different ways we see them can lead to some understandings of the function's that are more useful than others.  (Particularly in tracing the patterns backwards)
• The key to functions is identifying what stays the same and what changes.  Then when it changes, how it changes.  
• I need to have students prove why sqrt(a^2+b^2) IS NOT a+b with blocks.  Because they KEEP making this mistake.  
• I need to read PEAK.

Whew.  I'm pooped.  Maybe later I'll come back and fill out more details but I've got the heart of the sessions distilled for future me to enjoy with tea and biscuits.  

Inverse Function Activity

Here's a quick activity to develop the idea of inverse functions for my algebra 2 students.  We just learned composition and domain and range.  I hope it'll be fun.

Inverse Functions Activity by ezmoreldo on Scribd

Here's a word version: inverse functions game

Update: We just did this activity.  The game was GREAT.  The worksheet was terrible.  I need to tweak it.  I'll upload a new version once I think it'll work better.

Update update: Here's a new version of the worksheet.  I'm actually kinda proud of this one.  I want to go back in time and kick myself.  Or at least hand this over to past me and make me teach it this way instead.

Inverse Functions Activity2 by ezmoreldo on Scribd

  And here's a word version

Sunday Funday: Classroom Organization

I guess I'm behind the times as the Sunday Funday Blog challenge prompt I want to respond to is already a little old, but it inspired me to write a post, so here I am.

I want to write about classroom organization, specifically what to do if you don't have a classroom.  There are some of us nomadic teachers out there, shlepping stuff from classroom to classroom, trying to figure out how to connect to the digital projector with the wrong cords when our laptops only have hdmi ports and the projector only has vga.  There are some interesting and unique challenges you face when you don't have your own space.

Challenge #1: Collecting homework.  If you have to move from one classroom to another, to another without much of a break in between, and you need a teeny bit of time for set-up, you don't have time to go stash homework somewhere.  So if you collect homework in one class, you have to shlep it to another classroom, then if you collect homework in that classroom, your homework shlepping multiplies.  It's impossible to do things like notebook or binder checks, unless you try to check it during class while the kids are occupied, but our periods are only 50 minutes which makes that strategy tricky too.

Solution: I don't collect homework.  I've come up with two ways of assigning homework without needing to collect it.

• Way 1: I assign the homework, show them the answer key in class and while they're grading their own work, I go around and give them a stamp for having it done on time.  I use alphabet stamps and go through the alphabet so at the end of the unit, it's easy for me to see when an assignment is missing.  Pros: the kids grade their own work so are more cognizant of their mistakes.  They can get their questions addressed way more quickly than if I collected work and they ask deeper questions because they can see where their work deviated from mine and catch misconceptions they didn't even know they had.  Cons: Takes time away from instruction.
• Way 2: I post the answer keys to the homework on my class website and the kids check their work on their own and come to class with questions.  Then in class, I choose a problem from the homework and display it.  Then I hand out blank note cards and the kids solve the problem from the homework without their homework or notes in front of them.  If they did the homework and checked their work thoughtfully, recreating the solution should be a breeze.  Pros: They have access to the answer key as they're working through their homework so can be more thoughtful and can come to class with really specific questions.  Having to then reproduce the work in class the next day really reveals if they understood it or not.  Cons: cheating is a possibility but as only the note cards are graded it won't help their score.  I also see a smaller sample of their work so I have less info on their understanding.  

  Challenge #2: Navigating different rooms and layouts is hard.  Some have chalk boards, some have whiteboards all are missing writing implements as teachers hoard them.  Also, all the rooms are laid out differently and in some, it's easy to have students come up and use the board, in others the tables and chairs and bodies are too tightly packed to do much moving around.

Solution: Our school does have a digital projector in each room so I bought myself a document camera and with that, my laptop, and a vga to hdmi converter I can reliably use the projectors.  I write a notes template for the lesson that day and have students take notes from that.  Then I can scan in the template each night and they'll always have access to the notes!!!  I don't have to ever argue with students anymore about notes.  If they didn't take them- go check the website!  If they're absent, go check the website!  Teaching this way also makes displaying student work a breeze.  We all had a great time when I had my algebra 2 students solve a quadratic formula problem on note cards.  One by one I showed the cards via document camera and every answer was different!  There was some laughter and a lot of sheepish "I guess you were right in telling us to be more careful"s.

Challenge #3: Classroom management has always been a struggle for me, and I thought I'd finally come to grips with it a few years ago when I was teaching in California.  Teaching without a classroom makes me feel less valid.  I can't control my space, I'm always puffing from one place to another frantically trying to set things up.  I lose authority this way.  I don't have a solution for this one.  It's just an interesting observation.  I do my best, I try to pretend, but without feeling like I own my space I also don't feel as in command of my students (not that I want to command them- gently trick them into doing what I want without them noticing?)

I'm definitely not as good a teacher when I have to teach this way.  But I'm pleased with the grading system and I'm really enjoying the ethical conversations we're having about how to use answer keys responsibly.  Students have so many resources at their fingers, but don't know how to properly use them.  Even with the answer key sitting in front of him a student today couldn't scan through his paper and compare it to the key.  He kept skipping around or missing details.  There are so many hidden executive functioning skills/deficiencies that are revealed when kids have to grade their own work.

I'm Back! Again?

So two years ago I posted an "I'm back" post.  But I hadn't actually started teaching again so I had nothing to say.  Now I'm really back.  Back to teaching and hopefully, back to blogging.  I want to spend this first post reflecting on why I stopped blogging in the first place because I think it speaks to some of the issues our students struggle with.

First, I stopped following teacher blogs.  The good ideas have been so helpful and so inspiring, but every good idea I didn't have the chance to use made me feel bad, every boring lesson that I didn't spice up made me feel bad, and then all those good ideas made me feel bad about my own paltry ones.  So then I stopped posting my own blogs.

It's so silly to fall into that comparison trap.  To think that because there are so many amazing things happening out there, so many things that I could never have dreamed of, that means my ideas are worthless.  That I have nothing to contribute.  But that's also not what blogging is about.  It's not an arena where the best ideas have to pin the good or mediocre ideas down and hold them down for a count of 10.  I did feel like I was "losing" at some game and I'd never be smart enough to win.

I've felt this way about math too.  I often didn't have the insight fast enough, or wasn't able to chug the numbers competently enough to shine in class.  I always thought my contributions were worth less than other peoples'.  It took me until grad school to realize that the people speaking up in class often were wrong, or were bsing, or were questioning.  They weren't better at math than me, they were better at participating in a mathematical community than I was.  My math partner who talked a lot of jargon and had a deeper pool of knowledge than I did often missed the key insights our proofs needed and I usually saw them.  I was quieter about it, and more tentative.  But I could see them.  I fight this crippling insecurity every day.  I know where it came from- a string of sexist math teachers and an older brother incredibly gifted at math- I don't know how to conquer it other than being very aware of it and fighting against it.

So I'm fighting now by resuming this blog.  And I'm going to spend some time thinking about how to get my students to fight too because I know that a lot of them also feel like they're losing the game.  They can't collect enough points, or see the ideas fast enough, or be smart enough.  There are so many ways to help all students feel valued, but at the same time they're constantly inundated with messages about achievement that are divorced from real learning and from the real contributions they can make to their mathematical community.  It's not about winning, it's about participation and I want to figure out how to weave this message into every aspect of my classroom culture.  That's my mission for this return to teaching and this return to blogging.  Wish me luck!

Teaching Again

I've taken the last year off to have my son.
It's been quite a ride.
People have been asking me which is easier, parenting or teaching.  Since he's only 11 months old, I don't really know yet what parenting consists of but I can say that this has been the most relaxing year of my adult life.  What does that say about being a teacher?
I was able to take some classes for myself (abstract algebra, graph theory 1 and graph theory 2) and I reconnected with what I love and hate about being a student.  It was really helpful for me to remember what not knowing math feels like and what a different persona I adopt as a student (super quiet, shy and uncertain) vs. who I am as a teacher (gregarious, adventurous, unashamed of making mistakes.)  I spent a lot of time observing the other women in the classes (only about a quarter of the graduate students were women) and how much they participated compared to the men (about 90% of the comments made in class were by men.)  None of the students were black or Latino. I'm still processing how these observations should influence my teaching but for now, it's clear that I need to do more for my female and minority students.  Why don't women participate?  Why don't I participate?  My personal reasons are related to fear that at some point, I will hit a wall mathematically and just won't be able to understand something (even though I've overcome every wall so far), inherent shyness and introvertedness, fear of being wrong, math being so tied to my identity that I don't want to be revealed as a fraud (which I do feel like sometimes.  What right do I have to be telling other people how to do math when I'm unsure I could have pursued math seriously.)  I did have some sexist math teachers.  I never felt encouraged in math.  But these are my reasons.  Does every woman in math share these misgivings?  Or do we all have our own individual insecurities reinforced by our cultural context?  My sample size was really tiny.  And my shyness prevented me from sharing my observations with other women in the class.
Anyway.  I'm going back to teaching.  Algebra 1 and Japanese for next year.  I'm excited and scared to go back but I'm looking forward to catching up with what everyone's been doing on the MTBoS while I've been away.  I hope I can start contributing again and I'm so grateful I have this community to lean on when I'm in need of inspiration, which I always am!  I hope someday I can contribute something useful in exchange for all this community has given me.

Intro to Proofs in Geometry

I wanted to blog about this a looooong loooong time ago but the school year got in the way along with moving across the country twice because of family health dramas (NY to California in the fall, now California to Oregon.  I know Cali to Oregon doesn't seem that far, but it is over 1,000 miles from San Diego to Portland.  California is freekishly big.)  So though I know posting lesson plans in the summer is kind of silly, I want to get it out of my system before I forget what I did.

The school in which I taught this past year was a high poverty school where 30% of our students had IEPs.  It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills.  In fact, most of my geometry students this past year hadn't even passed algebra 1 yet.  The previous algebra teacher found them so lacking in basic skills that she gave the entire algebra 1 class "incompletes" because they didn't finish the algebra 1 curriculum.  My principal decided to enroll all these students in geometry anyway because she figured (rightly I think) that they needed a bit of a break from algebra and if they saw some algebra in a geometrical context it might make going back to algebra more meaningful (which it did.  Every time algebra popped up in geometry the students were actually excited because it was familiar and wasn't too difficult.  They really mastered equation solving, writing expressions and equations of lines by studying these topics through geometry.)  All of this meant that when proofs came up I was super freaked out.  I've always struggled with teaching them and I feel like I've done a very poor job in the past.  I put a lot of thought into how to build proofs into our curriculum this past year and I feel like what I did was relatively successful.  My students weren't scared of proofs for the first time in my teaching career.  When they came up, the students knew at least where to start and always attempted them.  So I want to lay down what I did just so I don't forget.

So here is the description of how the unit flowed.  It's definitely a more traditional approach to proofs and I stuck to two-column proofs.  I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs.  First, I didn't nix the logic unit.  Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs.  

Logic Unit Lesson 1: Intro to conditional statements.

• First I did Sam Shah's lesson introducing conditional statements.  I did the drawing activity and posted all their pictures on the wall.  It went really well- I was surprised at how much trouble some students had following the directions precisely.  A lot of them didn't know the geometric vocabulary (like what an isosceles triangle is) or were hesitant drawing so it was a great activity to do towards the beginning of the class.
• Then I just did a mini-lecture on the notation of conditional statements, Euler diagrams and what a negation is.
• Then I gave them this assignment: Logic Unit Lesson 1: Conditional Statements Intro 

Logic Unit Lesson 2: Manipulations of Conditional statements

• I did the second part of Sam Shah's lesson on conditional statements
• I did a mini lecture on the vocab: converse, inverse, contrapositive and negatin
• Finally I gave them this assignment: Logic Unit Lesson 2: Manipulations of Conditional Statements.  

Logic Unit Lesson 3: Word Proofs

• First we did a syllogism activity  where I just cut the cards apart and had them put the syllogism in the correct order and a mini lecture on syllogisms
• Then we did word proofs.  This is one of the most successful lessons I've ever taught on proofs.  I totally stole it from another blogger, and of course forgot to save their name in the name of the file I downloaded like I normally do.  When I figure out who made it I'll update this post.  I reformatted the file I stole from that other blogger and did the lesson in kind of a workshop style.  I did one or two of the word proofs on the board to demonstrate how to do it, then I gave them time to work on their own and then we compared answers.  I had them come up and show different solutions they'd discovered and we talked about the fact that there's more than one way to do a proof correctly.  Like I mentioned before, this was actually a pretty bright class but one lacking in discipline and both basic math and study skills.  I had to hold the kids back.  They were chomping at the bit to do more and more and more puzzles.  I made the last two pages of the lesson optional and almost all students did them anyway.  Whoever designed this lesson was brilliant because it really hooked students who are usually disengaged with math.  You can't not want to solve one of these puzzles when they're presented to you.  
• There are two pages in the lesson to have the students make their own puzzles and switch papers with each other.  We had to skip this part because we ran out of time.
• Here's the lesson: Logic Unit Lesson 3: Word Proofs

Logic Unit Lesson 4: Angle Proofs

• This lesson was a little less fun than the last one, but it was very effective.  First we went over basic angle terminology: complementary and supplementary angles, vertical angles and linear pairs (they'd learned these before), and we also talked about the algebra properties of equality, transitivity and the substitution property.  
• Then I had them do the lesson below in pairs.  It's structured exactly like the word proofs from the last lesson with 4 or so "rules" and space for them to use the rules to go from the given to the prove.  The students were able to stumble their way through these proofs without me doing any examples on the board based on what they did in the last lesson.  
• Here's the lesson: Logic Unit Lesson 4: Angle Proofs
• Finally we concluded with a Scrambled Proofs Activity

Here's where I made a mistake.  I did parallel lines and transversals as the next unit because I liked flowing from points, to lines, to parallel lines to triangles to polygons.  It seemed like the logical way to structure the course.  Also, congruent triangle proofs are so much richer and more interesting if the students already know their parallel lines and transversals angle relationships.  But if I were to do it again, I would do congruent triangles after the logic unit and then do parallel lines and transversals.  The proofs for parallel lines and transversals are a little more abstract and involve more vocabulary than congruent triangle proofs so trying to launch from the intro to proofs unit straight into parallel lines and transversals was too big a jump.  So here is how I built up proof using congruent triangles after I failed at teaching them proofs through parallel lines and transversal relationships.  

Congruent Triangles Lesson 1: Intro to Congruence

• For this lesson I just did a standard lecture over what congruence is, the notation for congruence and examples of using congruence to find missing parts.  Nothing really exciting.  The only thing about this lesson that I like is my warm-up.  We define congruence as "identical in every way" and then I ask students if the two identical twins are congruent which leads to a great discussion of what "corresponding parts" means: 
  
• Also, I always get a laugh out of my student by choosing another teacher who's the same height as me (at this school I used the principal) and I use that teacher to discuss the difference between the "equal" symbol and the "congruence" symbol.  Our heights can be equated, but if you accidentally use the congruence symbol you're saying I and this other teacher are identical in every way.  
• Here's the assignment for that lesson: Congruent Triangle Lesson 1: Intro to Congruence

Congruent Triangles Lesson 2: Triangle Congruence Theorems

• I used this cool illuminations app on congruent triangle theorems along with this worksheet that I wrote to introduce SSS, SAS, ASA and AAS.  I've done this activity twice, the first time I let the students pair up and use their own computers.  A lot of the students wouldn't or couldn't follow the directions so I spent the whole period frantically running around trouble shooting.  The more motivated students in the class were able to make the connections I wanted but the other students left class mostly confused.  So the second time, we did each activity first as a whole class with computers closed, then I let them open their laptops and play with the simulation to confirm the results for themselves. This worked much better.  All the students were successful on the homework without need for more instruction.  
• Then I totally stole a few worksheets here and here from Math Teacher Mambo for the students to work on independently.  Here's my mash-up of her brilliance: Congruent Triangle Lesson 2: Congruent Triangle Theorems

Congruent Triangles Lesson 3: Using Theorems in Proofs

• This is a lesson I created that I'm exceptionally proud of.  It was super boring though.  I realized that students weren't really getting what theorems were for or how to use them.  They were still struggling with problems like: if angle A and angle B are a linear pair and angle A measures 40 degrees, what is the measure of angle B.  They weren't thinking about what "linear pair" means and how to connect that meaning to the problem.  Even if they got this far they didn't understand that in a "proving" situation they needed to state how they know that angle A and angle B add to 180 (that they need to say: by def of linear pair or state the theorem).
• So I made this worksheet: Congruent Triangles Lesson 3: Using Theorems to Make Deductions.  
• In pairs I had them go through their notes and fill in the blanks for all the theorems we've covered.  This was the boring part.  I gave them a time limit though which helped keep them focused and this activity also reinforced the importance of taking notes.  I refused to tell them any answers.  If they didn't have it in their notes they needed to find someone who did.  
• The "classwork" part of the lesson is where it all really paid off.  I did the first few examples with them- how to cite the correct theorem that had been used in each situation.  As soon as I started doing these problems on the board, a bunch of "ohhhh so that's why we needed those stupid theorems" exclamations went through the room.  It was especially rewarding to watch them do the last page where they have to think backwards- find the theorem that applies to the situation then figure out what deduction can be made.  
• Students guarded the list of theorems they made as the first part of the lesson fiercely and insisted that I do a similar "fill in the blank theorem review" at the end of every unit.  

Congruent Triangles Unit Lesson 4: Proving Triangles Congruent.

• This lesson went quite well although it's a very traditional lesson.  I just did a few example proofs with them and talked about how to set up a two-column proof table (I know... but these students liked structure.  I tried to show them a paragraph proof and their eyes all crossed and they started throwing paper.)  
• I gave them this packet of problems from letspracticegeometry.com (there are a lot of typos in this worksheet though.  I would like to rewrite it) but without the first two pages.  
• Then I gave them this proof template worksheet thing that I created.  It has the students choose which proofs they want to try, the harder proofs being worth more points.  They have to reach a certain number of points to get full credit.  
• This small spin on a worksheet created a night and day difference in students' attitudes about proofs.  Every other time I've taught proofs students have been super whinny about them and would give up quickly.  But when I handed out the above assignment I saw at least half the students immediately turning to the last page to do the harder proofs.  A lot of them struggled on the proofs through a good chunk of the period without finishing and I kept suggesting they just go do more of the easier proofs, or build up to the harder ones but they said that they wanted to do hardest ones.  The fact that there was a choice between easy and hard involved made them want to prove to themselves that they could do the hard.  Students with less confidence started with the easy ones and were able to advance to the harder ones pretty smoothly.  Everyone was engaged and no one was complaining that I was making them do proofs.  

Congruent Triangles Unit Lesson 5: CPCTC theorem proofs

• I taught this lesson the same as the last one.  Examples then a "choose your own problems" proof worksheet.  Here's the packet from  letspracticegeometry.com and here's the proof template worksheet I gave them.
• I've had trouble in the past with students using CPCTC inappropriately so I put the following message up on the projector in giant letters and made them recite it in unison a few times.  I kept it up as they worked on the proofs and I didn't have students misusing CPCTC!  

To use CPCTC you MUST

FIRST: Prove triangles congruent

THEN: Say parts are congruent with CPCTC

CPCTC says that:

IF two triangles are congruent THEN their corresponding parts are congruent.

Prove the "IF" first, Only then can you use the "THEN"

• Again, through this class period students were working on proofs without complaint and without giving up.  If they started to have trouble they could persevere or choose a new problem and this flexibility eliminated a lot of the griping I've experienced in the past with proof practice.  Boring but effective.  

WHEW.  That's all.  I just wanted to catalog what I'd done because this was my most successful proof teaching experience so far.  It still needs a lot of work though.  And I know that under Common Core, it may not even be relevant anymore because I didn't work in any proving congruence with transformations.  

Pedagogy vs. Compassion

This year has been the most tumultuous of my life which is why my posts this year have been so infrequent but this summer, my first summer off in my 5 years of full time teaching, I hope to spend some time reflecting on my teaching career so far.  I also want to record my experiences in the school I most recently taught at in San Diego California before they grow cobwebs.  I have a lot of lesson plans I want to share but first I want to think about how teaching at a disadvantaged, high poverty, high IEP percentage school was different from teaching at my relatively privileged charter school in Oregon and VERY privileged private school in New York.  It wasn't that different.

I did have to change how I taught.  I used inquiry based approaches at the other two schools in which I've worked.  The students had good study skills, were well organized and cared about their education so getting them engaged in self-discovery lessons wasn't that difficult.  They knew how to accept challenges and persevere even if they didn't know how to do something.  Boy did this bomb in my school in San Diego.  In geometry class, if I gave the students rulers they were immediately put to use as either projectiles or weapons.  If I asked them to spend 10 minutes completing an activity on their own, all the cell phones came out or hands went up asking for help.  No one had the initiative to even attempt an activity on their own.  Games descended into chaos.  I quickly learned that these students needed a very firm hand and they would only behave under direct instruction.  Maybe I should have persevered with inquiry based approaches and over time they would have gotten better but the standards hanging over my head made me too nervous to spend too much time on this classroom chaos.

Their study skills were so weak that most didn't take notes, bring paper or pencils to class, and many didn't know their multiplication tables.  I spent a lot of my time teaching them how to listen in class, how to take notes, how to use their notes effectively, how to show work and how to care.  I did use a lot of questioning in my direct instruction lessons- I never actually completed a problem myself on the board, always asking for student input- but it was still direct instruction.  At the end of the year though, as I was grading their final project and their final tests, I was astonished to realize that they'd mastered as much content as the students in my relatively privileged Oregon school and also exhibited the same enthusiasm for math that my Oregon students exhibited.  Here's an excerpt from an e-mail a student sent me at the end of this school year- it's almost identical to letters I received from my Oregon students:

I'm not sure why, but it just recently dawned upon me that you will be leaving after this year and I'll probably never see you again, so I decided to write you a farewell letter. I've never really been compelled to write one to a teacher before so you'll have to bear with me here. I wanted to start off by thanking you for everything you've done, I can honestly say you're the best teacher I've ever had in my entire life. That being said, the support you've given me and the mentality of perseverance you have instilled in the classroom has really inspired me to work even harder and I wanted you to know you have made a big impact on my life. I want you to know that you'll always have a special place in my heart, even years from now, I'm sure I'll look back and be able to confidently say you helped me achieve my goals.

Without inquiry based learning, you-tube videos, gimmicks, games or technology my students in San Diego reached a similar mastery of content and a similar changing of attitudes about math that my students in Oregon attained.

My pedagogy didn't matter.  Or rather, I used the methods that I thought would work for my students.  Method mattered much less than I would have thought.

I don't want this post to sound boastful- I had the same number of failures and frustrations as other teachers but I did feel successful at the end of the year.  I am left questioning the amount of time I've poured into thinking about my method- feeling guilty over not using more inquiry based approaches or not doing enough projects or relying too much on direct instruction or not letting learners of different styles shine since direct instruction caters to auditory and visual learners.  Certainly method was important but it wasn't a question of "is direct instruction or inquiry instruction the correct way to teach," it was a question of "is direct instruction or inquiry instruction the correct way to teach for my students."  Would my Oregon students have learned as well as they did had I used direct instruction on them?  I don't know.  Or was the method of instruction really not that important at all?

I got numerous notes from students at the end of this school year and all of them cited my ability to listen to their difficulties, to work with them after school, and my stubborn refusal to let them give up that helped them succeed.  (There were a few students that I failed though, don't get me wrong.  And I felt like giving up on a lot of them sometimes.)  None of them mentioned my lectures as being boring and a lot of them thanked me for teaching them how to listen and take notes because it helped them in their other classes.  Where does this leave me in the pedagogy wars? I don't know... but maybe it's time I directed my guilt away from my methods of instruction and try to hone what does make me feel successful- treating each student with compassion and trying to be flexible in finding what works for them, regardless of what methods are fashionable in the larger ed-community.

Absenteeism

Right now, I'm making lesson plans for my first period algebra 1 class.  Here is the first slide I decided to add to my presentation for tomorrow.

"Right now...

• 4 of you have As
• 2 of you have Cs
• 9 of you have Fs

It's really easy for me to tell though without looking at my gradebook who is passing and who is not: those who are here everyday are passing.  Those of you who are absent two or more times a week are failing."

What do I do?  I can't teach students who don't show up.  When I make this announcement, most likely at least half my class will be absent and won't even hear the message I'm trying to convey.

Update: 3/8
The Monday after writing the above post I decided to try a new grading system in my class to see if that could help with attendance. I had been assigning homework and calling it homework, but I had been giving the students time to complete it in class.  Only if they didn't finish it in class would they need to do it at home.  I did this because I didn't want students to feel pressured to get the work done quickly- when I assign only classwork the slower and more careful students tend to get stressed out.  The problem though was that students were not using class time well.  When I asked them to work they said they would finish it at home, and then of course it (and the student) never came back.

I thought that maybe if I made their grade entirely based on them showing up and using class time well, then I would have more luck with both attendance and with comprehension.  Miraculously all the students did show up on Monday and I told my students that attendance was our biggest problem.  That those who were failing were failing because they weren't here.  I explained that I was going to make their grade based entirely on if they came to class, took notes and if they completed the work asked of them during class.  Immediately, I saw relief wash through the classroom.  I think because for the first time all year, they realized that they could pass.  That they could do what I was asking them to do.  The late homework, missed lessons and poor classwork completion had been weighing on them and had been causing them to avoid class.  It was easy for them to not show up because this class is first period and at our school, only freshmen and sophomores have to come to first period.  So my freshmen were hanging out with their Junior and Senior friends instead of coming to class.

They want to do well and only their guilt and lack of confidence had been keeping them away from class.  They constantly tell me that they like me as a teacher which is why I was so baffled by their poor attendance.  Maybe the fact that they do seem to like me contributed to them not wanting to face me when they thought they'd let me down.

Since changing my grading system two weeks ago, my attendance has sky rocketed.  They're all completing class work, asking questions and performing well on quizzes.  They still definitely lack initiative.  Since I require them to turn in an exit ticket to receive credit for the day's work, the end of class has gotten awfully chaotic as students frantically try to get my help because they don't trust their own abilities.  But they're trying and showing up now.  We can work on initiative later.  

I am torn about this no- homework system.  I have been following the homework vs. no homework debate and I'm more on the side of assigning homework because I've seen students grow so much from wresting with problems when they have no one around to help them.  They take better notes, ask better questions, and demonstrate much more mastery over the material than when I don't assign homework.  This experiment has reinforced my belief that homework does significantly contribute to learning because my other algebra 1 class to whom I still assign homework are demonstrating much more confidence with the material and are growing more rapidly.  Both my first and fifth period Algebra 1 classes are composed of low-income students who have failed algebra at least once before.  But my fifth period class has time earlier in the day (usually during lunch) to complete their homework so their homework turn in rate is good, their attendance is good and their learning is evident.  But clearly when students can't do homework and the not doing it wears down their self confidence and causes them to avoid class, the homework needs to be nixed because it's doing much more harm than good.

I guess this just reinforces my belief that there are no absolutes in education.  Every thing about teaching needs to be modified depending on the composition of students sitting in your classroom.  When students do homework it's good for them, but when they can't do it and are still expected to do it, it's bad for them.

Asking for help

I seek help on-line constantly when it comes to lesson planning.  I've grown used to the idea that anything I can think of, someone out there in the blogosphere has probably already perfected and I love that I can see kernels of lessons I've just dreamed come alive in others' hands.  This doesn't even include the gazillions of ideas I've never thought of that are about a hundred times better than anything I can dream.

But when it comes to actually teaching- implementing the lessons, getting my kids excited, supporting their growth, encouraging them to persevere, I've never received much help (administrators never pop in.  I've been formally observed only once and that was by a coworker) and I feel like at this stage, I don't need much help.  I have a thriving community of students coming during lunch to do math because they enjoy it, and I've watched the most recalcitrant math students slowly gain confidence and enthusiasm and I feel like this is what I'm good at.  I'm good at patiently coaxing students into learning that they can learn math and over time, that they enjoy learning it.

But this semester I have the most stubbornly anti math student I've ever taught.  For three weeks, she was an angel in my advisory and a demon in my math classroom.  She refuses to accept help saying that she doesn't need it, she'll do it at home.  Then she proceeds to do nothing at all through the whole 80 minute block.  When I try to help her she slides under her desk, covers her paper, refuses to look at the problem, gets up and walks away, or starts ranting about the uselessness of math.  She's a wonderful student in advisory so I know she's bright and capable, but she refuses to cooperate in math (especially whenever division becomes involved.  She says she never learned it and she never wants to learn it.)  She slept through all of my math classes two weeks ago and refused to stir when I tried to rouse her.  She got a 30% on her first test and even though I discussed with her the consequences of her actions through all of advisory that day she slept through math again the next day. I asked her if she wants to fail? It means she'll have to do it all again next year.  She replied she doesn't but she can BS her way through the other tests.  I said that learning to read is tedious, but once you do learn, it's magical what you can discover and that math is the same way.  She replied that reading is vital but math is superfluous.  I said that everyone needs help to learn math because it's several thousand years of accumulated knowledge that we're trying to impart in a few short years and that all I would like is for her to let me help her.  Right now I don't even care about notes or homework or tests.  I would just like her to allow me to talk to her about math without arguing.  She wouldn't budge.

I thought that I'd have to just wait her out.  I'd need to stop nagging her and let her come around on her own.  Maybe over time she'd start to feel left out.  Or she'd realize that she couldn't BS her way on her own and she didn't want to fail.  She was so obstinate that maybe just the fact that I was pushing was making her push against me and if I stopped pushing she'd stop fighting.  I was worried though that she would get so far behind by the time she came round that it would be too late to learn what she needed to learn since she was already so far behind.

So I turned to our vice principle, explained what was going on and what I'd tried and he said he'd talk to her.  The next day she took notes, completed her homework and asked for help.  I asked him what he said and he told me he'd talked about how many thousands of years of knowledge we were trying to teach her in a tiny span of time and that she could not learn without my help.  He said that this will be maybe the only time in her life where she had a teacher who was willing to give her extra time, extra help and who really cared about her and if she waited, she would never get the help she needed.  It was almost exactly the same logic I'd tried on her.  Her efforts have continued through the week.

I guess this just reinforces my belief that if I ever get to a place where I think I've figured it out- that means I've grown too complacent.  Teaching will always and forever be something I'll need help with and that's the way it's supposed to be because it's a collaborative endeavor.  I hope I'm always humble enough to ask for the help I need.

Angleatron Failure and Distance Formula Game Success

I taught the lesson on angleatrons that I previously posted about and it was not very successful at all.  I'm reluctant to write about my failures because I'm already the type of person who doubts everything I do and even my most successful lessons leave me feeling like I'm not the teacher I wish I was.  This is also why I'm a terrible blogger.  In my most insecure moments, I can't help but compare my teaching to these fantastic teachers I so admire and aspire to me more like.  I hope the fact that I am constantly striving to be better makes me a better teacher, but it also makes me very uncomfortable in my own skin much of the time.

The lesson was unsuccessful for several reasons beyond my control.  My speakers broke partway through showing the video so the students couldn't hear Vi Hart's narrations.  I then tried to paraphrase what she was doing with paper folding but students grew bored watching a soundless video.  This made me rush through the video to move on to the activity, but then students were confused about how to do the paper folding.  Their confusion reinforced my reasoning behind doing the activity because if students couldn't grasp the idea that the corner of their paper can be used as a 90 degree angle, then they really did need to practice basic angle drawings.  About half the class did take off doing drawings and folding angles.  A couple of them produced really beautiful designs and I think all of them did grasp what 90 degree and 45 degree angles are supposed to look like.  The other half of the class adamantly refused to draw, or refused to draw precisely (sloppily drawing 90 degree angles that looked more like they were 100 degrees because they refused to use the corner of their papers to guide their drawings.)  Their reluctance and difficulty only convinced me that they did need to practice, but the activity didn't work for the students who needed the practice.

I did try some other games this past week and they were much more successful.  For me, the simpler the game, the easier it is for me to pull off because I have a very minimalist classroom (I have to buy all my own supplies, the students have tiny desks and we don't have a white board, only a smart board which allows only one student to write on it at a time.)  I came up with a game to practice the distance formula which worked beautifully mostly because it was so simple.  First, I had to bribe the students to play because playing games involves more thinking than taking notes and they actually wanted me to keep lecturing so that they could passively copy/ sleep.  Then I asked them to group into threes and told them they were competing against their group members to convince them to work with people other than their best friends.  Finally, I just displayed four numbers on the smart board.  The students could rearrange the numbers into two ordered pairs however they wanted and could add negatives if they wanted.  The person in their group that was able to organize the ordered pairs in such a way as to maximize distance won and earned a candy.  I started with 0,0, 4, 12.  Then gave them 2,3,4,5.  Then started giving them bigger numbers.  At first the students just paired the first two digits and the last two digits and used the distance formula.  But after a round or two they started figuring out how to add negatives and rearrange the bigger numbers with smaller numbers to get larger distances.  They also were doing a good job of checking each other's work because they only earned candy if they did the calculations correctly.  By the end of the game, every student had figured out how to maximize distance and they were all tying and I was going bankrupt on Jolly Ranchers.  My favorite part was when one person in a group announced their largest distance was 13.2 and students from a different group came over and clustered around asking the person from the first group how they'd gotten such a big distance. I think the game worked very nicely because it was simple, strategic, competitive but not so competitive that students who were "losing" became disheartened.  By the end everyone was winning.

I didn't like the game because I don't like the distance formula.  I would much rather students use the Pythagorean theorem enough that they could then extrapolate the distance formula by picturing triangles on the coordinate plane without needing to graph.  Unfortunately I just didn't have the time to reinforce this method of calculating distance so I caved and taught them the distance formula (but at least I did show them how it came from the Pythagorean theorem, though half my class fell asleep or glazed over when I tried to show the derivation to them.  I've tried having them do the derivation themselves but their algebra skills are too weak.)  At least though, they did do some critical thinking in terms of figuring out how to maximize distance.  That was the saving grace of this game.

Angleatrons

I inherited a geometry class last semester that was already two months into the curriculum and it was very frustrating that their basic sense of shape hadn't been strengthened.  I was supposed to start with congruent triangles, but many of them didn't even know what a right angle was supposed to look like.  It was too late to go back and work on basic drawing skills but I've been thinking about how to help students with little practical drawing experience succeed in geometry.  Gone are the days when all students had formal art classes and without these classes, their visualization and drawing skills are so weak that geometry can be really challenging and frustrating.

With the new semester, I'm starting over with a new class and I'm working on building more drawing and visualization into my curriculum.  I've just written up a lesson tied to Vi Hart's angleatron video. I want my students to be able to do rough sketches of all the basic angles so that their drawings, when we get to triangles and polygons can be at least a little bit accurate.

First I'll show my students the video and have them try to explain how the different angleatrons were formed.

Then I'll have the students make the different angleatrons, name their vertices, sides and the angles themselves.

Then I want my students to try making 3 different geometric patterns using their angleatrons like Vi Hart did.
Finally, they'll each pick the pattern they like the best and we'll make a class quilt out of their different patterns.

I'm a little nervous because a lot of my students really hate drawing, but I hope the structure of this activity and Vi Hart's beautiful examples will help.

Here's the lesson sheet I'm planning to use:

Angleatrons Lab

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

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?

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."