why polynomial long division is awesome

A week ago I had plans for Precalculus. I liked them alright, but I felt they could be better so I did what I always do when I need my brain to percolate over ideas: I start catching up on my reader. I ran across Jason Buell’s post about the IMPROVE model (you should go over there to read the details of that model if you’ve never heard of it). I then scrapped my plans and started re-writing them. That was Sunday night. *shakes fist at Jason and his inspirational posts* 

Before we get into the week, let me be the first to say that I have no grandeous dreams of students going forth into the world and using their ability to long divide polynomials or sketch them from factored form by hand. Polynomial Long Division (henceforth referred to as PLD) is a way for me to do the following:

1. Check on and improve student ability to add/subtract positive and negative numbers as well as distribute them
2. Improve mathematical endurance (why yes, those last two problems may take up a whole page each)
3. Improve student understanding of how the equation, the factors and the graph work together with higher order polynomials
4. Create an opportunity for my studens to feel like mathematical Rock Stars

Let me explain that last one. It goes back to my drive to help students see themselves as being ‘good at math’. PLD is impressive looking.  I mean, really impressive. Especially when you start with a 5th degree, have students figure out the zeros, create the factors and then do multiple rounds of long division in order to come up with the factored form which they then must graph by hand labeling all intercepts. Then they can check their work with the calculator. Yay for instant feedback.

At the start of last week the kids had only worked a day or two with PLD. By then end of the week, with some group assistance, they were all working through these massive problems. They’ll whine that the problem’s huge and takes up too much space, but they’ll do the whole thing and then just kind of sit back and stare at it with this look of ‘did i actually just do all that!?’ on their faces. There were lots of “got it!” mumblings while they worked. It’s not the crowing cheers you get with, say, WCYDWT problems. I liken it to the satisfaction of a job well done. Because what is the true hold up for a lot of students in math? The concepts, or the grammar? Is it that the kid can’t understand what’s going on, or that they are so hung up on the symbols and the difference between coefficients and exponents and plus’ and minus’ that their brain is spinning? I think it’s the latter much of the time that shuts off the student mind.

So here’s the worksheet given out to each table group. I put them in a plastic slip so the four students in each group would have to share. Maybe even, I dunno, read a problem to the group. Work together.

Continue reading

You are not bad at math, you just don’t read the language

How many of us have had this happen:

————————

Student: reads a problem. Attempts to solve. Completely bombs. Ignores order of operations. “I suck at math!”

Teacher: sits down with student. Breaks apart the problem, giving it verbally to the student.

Student: does perfect, exclaiming “That was so easy! Why didn’t I get it the first time? I must suck at math!”

Teacher: *facepalm*

————————–

On facebook there are a variety of little polls that go around.  Right now there is one that looks like this:

Now how many people who got the ‘1’ are thinking “my, goodness, look at all the idiots who don’t know simple mathematics!”

And how many who got the ‘9’ are thinking the same thing about the people that picked ‘1’?

Me: *facepalm*

I feel like that’s the same as making fun of a lady for not knowing the sign said ‘men’s room’ when it’s in Japanese and the lady in question only reads English. If some helpful soul had read it out loud and translated, of course that lady would have chosen the other door. It’s not willful ignorance, here, it’s that smart, hardworking individuals never figured out how to read math. Or if they did, it was short term omg-there-is-a-test-tomorrow learning which does not stick. Especially if sleep deprivation is invovled. But that’s another post another time.

I suspect I could grab just about any individual off the street and walk them through it and they would get the right answer. Possibly with some finger counting.

So does math illiteracy = not able to do math? No. Not any more than reading illiteracy = not being able to discuss complex ideas. It’s useful to write math down because when large problems are being worked through, it’s a rare individual that can keep all the number straight. The visual form of mathematics (equations and graphs and tables) give a clear why to communicate mathematical ideas just as words do for language.

Now, teaching a language that pulls from multiple alphabets, where position is important, and multiple symbols are in play? Including three different ways to indicate multiplication? That’s a part of my job. Not the biggest part by far, but one that can cause an avalanche of problems.

Kids can do math just fine. But how much of the lack of mathematical self efficacy is from young kids deciding they suck at math because of a click on Facebook?

And which did you pick? the 1 or the 9?

how my dog learned fear, and how i’m driving it back

I have a dog.  His name is Doppler.  I can actually see him right now chilling on the back lawn just looking around his domain and keeping an eye on the painters working on the house.  The Dop took to training well and loves to go hiking and sleep in tents (or hammocks!).  I started jogging last winter and his joy on our outings has been a large factor in my sticking with it through the less enjoyable ‘I have never done land-based exercising and now I remember why’ phase.  Now I’m jogging for sheer enjoyment, but between last winter and now my dog has become a very different beast on our outings.

Doppler has always been fearless in his approach to life.  He bounds over tall fields of grass, climbs up mountains, and approaches all people as though they will pet him with a wagging tail.  Jogging with him was easy even though he seemed to think I was going very slowly.  And then when leaving my grandma’s house one day he hit a car.

No, the car did not hit him, he hit it.  I was letting him walk off leash (normal, though stupid on my part as he was riled up by my grandma’s dog) and called him to get into the back of the car when he runs past me toward the street and hits the back end of a red sedan that was driving past at 25mph and bounces off with a loud thud before immediately running back my direction.  I had to grab him mid-stride as he was attempting to shoot past me and just hold on until he calmed down.  He was moving alright and looked spooked so I brought him home.  By the time we were home he was acting completely normal and greeting the cats and my husband the same way as usual (attempting to body-hug/slam the first, bringing a tug toy to the latter).  I chided myself to be more mindful of his moods and that he is still a puppy (only a year old at the time), but didn’t think too much more about it.

A few days later the Dop and I headed out for a jog and he kept doing a weird front-to-back-to-front movement around me.  I thought he was just being a nut, but then I realized that whenever he did this his tail was down between his legs (something I had NEVER seen him do) and that he was moving that way to keep me between passing cars and himself.  He was terrified, but trying his best to keep jogging with me.  I almost broke down crying on the side of the street seeing him afraid for the first time.  I immediately altered my route to get away from cars and took a less-traveled path home through a park and fed him treats every time a car would come by.

Fast forward to yesterday evening went my husband and I took the Dop out on a long jog (almost three miles, woo!).  Given our schedules, the spouse and I rarely get a chance to do this together and it was nice to tag-team phrase/treat Dop’s good jogging behavior.  I realized that even though it has been over 5 months, Doppler is still carefully placing his people in between himself and the cars that go by.  He’s a bit sneaky about it; suddenly a plant or patch of ground becomes fascinating and he’ll slow down and then speed up again once the car is gone.  His tail no longer lowers to the ground, but it does stop wagging.  Five months and he still flinches.  I know that if we keep working at it and are consistent he’ll eventually get over the fear (though I can’t say I mind my dog being wary about cars), but for such a small incident the impact on his psyche has been huge.  One momentary slip on my part as his person was all it took.

I was left wondering during last night’s jog how many of my students have hit cars.  You know the ones.  They say they’ll come in, but never do even though you see them in English class in the morning (tripped over a motorcycle).  Some of them have parents who talk about how much they struggled in math and that their child has the same ‘math gene’ deficiency while the kid is in the room (bumped a VW Bug).  Others talk about how math used to make sense but then they had a teacher that treated them like idiots and now they just don’t enjoy it (SUV hit-and-run).

I know many different ways to teach specific mathematical skills, but sometimes I get so into the math (something I love and no longer have any fear of), that I miss the signs telling me this kid has their tail between their legs.  Other kids are just really good at deflecting, e.g., being socialites, class clowns, or class delinquents instead of doing the math.  How much of a student acting out in class is from a fear of the subject they picked up somewhere else along the way?  How do I make sure I remember this when I go to work with a student?  How do I remember to not be the car with off-hand remarks that I don’t think all the way through?

All of this goes back to my drive toward building mathematical self-efficacy in my students.  I see SBG in the same way I view treats for my dog–as immediate, positive feedback.  Car coming close, Dop?  Have something small and tasty to focus on and take your brain off the OMGCARRUNAWAY! reflex I know you are currently feeling.  Trigonometry making your head spin, Student-of-mine?  Let’s work on this little skill together for a whi–oh, you know how to do that?  Awesome!  How about you show me so I can update your skill-score in the gradebook and then we can move onto the next skill-n-bits for this topic?

Sure, sure, SBG helps teachers get a clearer view of what their students know (and in the process of making the skills lists a clearer view of what they need to know according to curriculum), but it’s also a way to re-acclimate students to mathematical proficiency and remembering that math isn’t out to run them over or block their way.  That they were not always afraid of cars.

So what do you do to push back the fear?

My Adventures in SBG — An Overview

I’m going to have to start this with a massive, neon disclaimer that I’ve been reading and re-reading edu-blogs for over a year and sometimes I am not sure which thoughts are my own and which ones are things I’ve lifted and adopted from other sources.  My apologies in advance if you read this and feel like you wrote it not that long ago–no plagiarism is intended.

the background
A central tenant I hold is that humans are inherently capable in mathematics.  All of them.  I don’t think we got to be the most stupendous badasses on the planet without knowing a bit about the basics of if-then and creative minds.  My students are capable of great mathematical thinking, but years of numbers circled at the top of their papers have taught many of them to believe they are failures.  By the time I get them as teenagers I am one more person to put down one more number outlining their inadequacies.

I hate being that person.  I hate seeing that ‘yeah, whatever’ look as I tried to go over a test with a student and point out the things they did know.  I also hated looking through tests and trying to understand the specific pieces that were missing from student understanding–a 65% meant about as much to me as it did to them in regards to useful information.

For the sake of my students and my self I needed to change the conversations happening in the classroom surrounding grades and what students could do.  Their beliefs regarding their mathematical proficiency were so low, that I couldn’t have pried their mathematical self-efficacy off the floor with one of those spatulas the custodians use to get gum of the tables.  In addition, my spatula was pretty weak to begin with because I didn’t have a good grasp on what students could really do skill-wise.  Sure, sure, I had homework (mmm, copying) and class/individual discussions and some projects, but students don’t seem to regard those things as evidence that they are capable.  They want numbers.  They want a percent.  They want to be able to come in, get some help, work hard, and see their effort improve their grade.  Go-go-gadget SBG.

the what i did
Back in late 2008 I sat down with my state standards for Algebra 1 and made a spreadsheet with all of them in one column.  I then started breaking down the standards and identifying the skills that a students would have to know and be able to do in order to achieve that standard.  I’ll note that some standards cannot be broken down this way due to the amorphous nature of words like ‘synthesize’ and ‘discuss’, so I put them to the side as the types of things that class projects/discussions were invented for.

Some standards have only one skill attached while others have a small handfull.  My favorite is the one about writing quadratic equations that breaks out into about five separate skills.  The skills were then broken down into beginning level and advanced level.  At this point, there were three columns: skill, levels, and state standards.  The last column to get added was the chapter of the textbooks my district uses that the specific skill could be found in.  Final step: sort by chapter.  The document gave a chapter-by-chapter summary of all the things the state curriculum was expecting from Algebra 1 students.  It also demonstrated quite clearly that the first two chapters were all 8th grade skills and that a number of sections could be ignored if time was pressing.

I was pretty psyched.  Here was something tangible I could use to assess the basic skills my students needed to have in order to be successful.  My admin liked it as well and green-lighted me trying out the new assessment system along with a colleague of mine in a total of 3 pilot classes.  I’m not going to go into the grading specifics here as my entire assessment system is based on Dan Meyer will very little changes.  I figured I would try something tested before getting my own ride and adding a roof rack and spoiler.

Does this list address thinking skills and habits of mind?  No.  But that’s not what I was trying to do.  I think trying to get students into a mathematical frame of thinking when the merest hint of using a variable sends them into a downward spiral of despair is a bit silly.  I needed to un-stick their mathematical self-efficacy from the floor and get them to trust my sincerity when I tell them “you can do this” before going onto the big stuff.  Though that’s not to say it was all skill assessments all the time until they were bleeding roses with golden proportions.  While implementing the new assessment system my teaching stayed relatively the same.

what happened
The conversations started to change (slowly).  My spatula got stronger as the kids started to understand the system.  That part took the longest.  At the end of the year I still had kids looking at me like I was crazy when I showed them how that skill assessment they just retook improved their grade.  Need to work on that whole ‘giving instructions in a way students remember’ bit.

In the 2009-2010 school year all 11 of my schools Algebra 1 classes were shifted over to the SBG system piloted the year before.  I only had an algebra 1 support class last year so I was a bit out of the picture, but the main algebra 1 crew expanded on the original skill list, hauled kids in to skill retakes, and expressed happiness at how nice it was to look at the online grade book and be able to pinpoint exactly where a student dropped off the bandwagon and plan remediation accordingly.  Parent conversations were also improved during phone calls and conferences.  So nice to tell a students’ parent exactly why their beloved progeny is having trouble and what the student needed to do to improve their grade and that yes, so long as your progeny can demonstrate that they are back on the bandwagon and grokking these concepts their grade will improve.

But all of that stuff is secondary to the student reactions.  The “wow, I can do this stuff!” moments.  The joy and victory dances when they turn in a skill assessment for me to grade and I plunk in full credit to the gradebook and they watch their grade go up.  It’s in those moments of weakness I sneak in with a “that was awesome!  Now that you have that down, I’d like you to take a look at this skill, which is similar to the one you just pwned.”  Lips were bitten and noses would scrunch after I would say such things, but then I would see something amazing.  The student would take a breath, nod, and say “lay it on me!”  After a while, they would come in without my poking them about it with a level of determination that I hadn’t seen in the past.  A belief in their eyes that if they worked at it, they would master the skill.  A belief in their own mathematical capability.

Anecdotes
As I was mostly off the Algebra 1 train this past year, I turned towards the classes that I was teaching (precalculus and a precalculus alternative) and implemented the skill-based grading as my quiz structure.  Just going through those classes and thinking about how to break down the topics skill-by-skill was a wonderful way to start the school year.  There are no state standards for these classes, so I used the college readiness standards as a springboard supplemented with requests of ‘critical items’ by the AP calc and stats teachers at my school.

SBG with upper level kids was much different that with Algebra 1 students.  They caught onto the system quicker and needed far less poking about skills.  Their victory dances were also a bit nerdier.  And while a lot of them didn’t need any help in the mathematical self-efficacy department, a surprising number did.  It felt really good to haul in C/D students and give them focused help on exactly what they need.  Example: “Hi <insert student name here>.  I can see that you are having some problems with the exponential chapter.  Your skill quiz score for translating between exponential and logarithmic form isn’t a perfect yet, so let’s start by going over that skill.  Could you show me how you would approach this problem and talk me through your thoughts?”

my last thoughts (for now)
While I don’t feel like the system is perfect RE: scoring system, I love the power it gives students over their grade.  I love how many more positive interactions I have with my students.  I love being able to glance at a chart in my room of my different class’ skill percents and know that I’ll need to spend a bit more time on skill X or that skill Y had a fantastic percent right away so I need to make sure to look back at how I introduced that skill and do it again.

It’s not a panacea for all your ills, but I believe that by turning grade conversations toward the positive I am able to improve my classroom atmosphere from a defeatist mentality to one of determination and persistence.  Students will tell me they can’t do a skill and I respond with “yet!”  After a while, they believe in the ‘yet’.  They believe in themselves.  Because they are stupendous badasses, and I live for putting the glint of determination back into their eyes.