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."

I'm a real teacher again!

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

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

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

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

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

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

Exponent Rules GAME

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

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

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

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

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

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

Rules:

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

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

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

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

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

Examples:

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

Not Ratios again!

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

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

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

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

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

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

Ratio and Proportion Lesson for Geometry by ezmoreldo

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

Ratio and Proportion- I Notice I Wonder PDF by ezmoreldo

Parallel Lines and Transversals game

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

Parallel Lines and Transversals Game