Tuesday, January 26, 2021

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.

Monday, September 16, 2019

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:

Sunday, September 9, 2018

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.

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.

Monday, November 13, 2017

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.

Sunday, October 15, 2017

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.  

Tuesday, October 3, 2017

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.

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.
  And here's a word version

Thursday, September 14, 2017

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.