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Monthly Archives: August 2012

But then… “that” student

In my last post, I talked about one student who had a successful year and grew quite a lot in his mathematical ability and engagement.  I also thought about another student for whom things didn’t go so well.  Those are the ones who pull at your heartstrings, aren’t they?  You try, but you don’t reach them, and you spend a long time wondering why and what you could have done differently.

This young man (let’s call him John) was always a mystery to me.  He was a delightful kid every morning when he came into the classroom.  He greeted me cheerfully, told me a story about his weekend, even brought me little trinkets like a cool rock he found in the yard.  As soon as a lesson started, though, he would talk to those around him and shred his papers and supplies into little piles of trash in his desk.  I try very hard to have many more positive interactions with kids than negative ones.  I really believe in this.  However, I rarely had positive interactions with John outside of our morning hellos.  I was almost always asking him to get a pencil, find his supplies, move to this seat and that so he wouldn’t talk to his neighbors, or write something down so I could see what he knew.  I would fill out his point sheet at the end of the day and really try to think of something positive that happened, but on so many days there just wasn’t much that went well.  He did almost no mathematical thinking, the whole time he was in my class.  I still don’t know if he has any sense of fractions or percents or negative numbers.  He never showed me.  He wrote what seemed to be random numbers on most parts of his assessments, or argued with the paraprofessional and didn’t finish them.

There were factors that made it harder to work with him… the class was a needy one, and many kids were needy but also hard workers – I can look back and tell I avoided helping John on many days, because I could help another kid who was eager and willing and not spend ten minutes with John to try and get him to put pencil to paper.  The class was big, and I had chatty kids – I couldn’t turn my teacher eyes off of them long enough to make any headway with John.  It took a very long time to get anything done with him even when he was in the mood to work, which was rare.

We had a couple of meetings with his parents, and they ended with a commitment from him to try harder, but nothing ever changed for long.  I remember one conversation I had with John’s case manager in the hall.  She was almost in tears.  She had poured time and emotion and so much hard work into his education, and he fought and fought her.  He demanded attention, acted out, and made it hard for her to work with kids who were engaged in their education.  She had tried really hard to build a relationship with him and she was exhausted.

John did not make any measurable academic growth, and I still think and think about it.
What could I have done differently?  What would you have done?

One failing on my part, I think, was that I leaned on his case manager and paraprofessional too much.  I really should have scheduled some one-on-one time with him after school or during my planning time, to work with him on math when I wasn’t distracted by having a class in the room.  With kids in Special Education, I always know they have a case manager who will go to bat for them, so I sometimes do less to advocate for them – I pass off the responsibility.  John’s case manager was as devoted as they come, but there’s no substitute for just sitting down with a kid yourself and talking about math.

What do you do with “that” student – the one you aren’t reaching?  How do you turn things around?

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Posted by on August 25, 2012 in Lesson Reflections, Student Stories

 

Peeling back the onion

For teachers in Colorado, our most important metric is how much growth a student makes in a year’s time.  How do you help students grow?  There are so many pieces to that puzzle.  I’m fortunate in that I work in a terrific, high-functioning team.  We are very goal-oriented in our planning.  We reflect often on how our lessons went and what we need to change.

I remind myself often that I don’t need to get all of my students to be proficient in every skill we teach; I am responsible for making sure my students grow from wherever they started to wherever they end up.  To that end, I need to know from where they are starting.  We do a lot of pre-assessing on my team.  We group the kids every Friday into flexible math groups, and many Thursdays, they can expect a little quiz to find out what they know and what they need to know.

And yet there’s more to all that pre-assessing and data gathering.  Sometimes, it helps me if I think of a particular student that I felt had a real turning point during the year.  Last year, I had a student who I will call Aaron.  Aaron was a very quiet, bespectacled kid who rarely made eye contact, never turned in homework, and only did minimal work at his desk if you hovered over him.  His test scores had dropped steadily over the years, crossing into Unsatisfactory territory when he was in sixth grade.  He was new to the school.  In the fall, I decided a good first step would be to figure out if getting him to complete his homework would turn things around.  I called his parents and set up a meeting to alert them to what was going on.  I learned from them that he had a really difficult fifth-grade year in math and things started to go downhill then.  I suggested he stay after school for homework help for a while until he started to find his feet.

The homework time was valuable for both of us.  I used the time to gather information about his strengths and weaknesses.  One afternoon, while he was working problems on proportions, he puzzled over the problem:  2/9 = x/27.  I knew then that Aaron didn’t understand proportions, but I wasn’t about to teach him proportions unless he was ready to learn it.  So I asked some questions to find out what he knew.
Me: “What if you think of it as equivalent fractions? Two-ninths equals how many twenty-sevenths?”
Aaron: “I don’t know.”
Me: “Maybe an estimate then. Is two-ninths greater than half or less than half?”
Aaron: “Less than half.”
Aha.  Aaron did know some things about fractions as a portion of a whole. This was really helpful.
Me: “What would be less than half of 27?”
Aaron: “10?”
Me: “What made you say that?”
Aaron: “Well, 10 is half of 20.  And 27 is more than 20 so 10 is less than half.”
It’s not a terrible estimate.  Aaron’s numerical reasoning skills really weren’t bad at all, I realized at this point.  He has decent number sense and this is a strength.
Me: “Okay. If we scale up the original fraction, we can find the value of that x.  What’s 27 divided by 9?”
Aaron started counting on his fingers.  But I wasn’t discouraged one bit.  When he started counting on his fingers, I knew that Aaron, while not knowing his basic facts, did know a meaning of division.  He knew it meant how many times a number goes into another.
I started the homework session with a student who had an unsatisfactory standardized test score and a string of failed tests.  But I ended the homework session realizing I had a student that understood division and knew that a fraction represented a portion of a whole.  He was ready to learn proportions, with the help of diagrams, manipulatives, grids, and some flash cards to help him with his facts.  He wasn’t ready to abstract the proportions yet, but he was ready to learn about equivalent fractions and what they represented.  We had a starting point.  We worked from there.

When I gave feedback to Aaron, I told him about just what I’ve written here.  I didn’t give him meaningless praise, but I told him I noticed his good number sense and his understanding of division.  I said I thought he would make great strides this year and that he was ready to make a leap.

And he did.  He grew a ton.  Aaron’s standardized test score moved up a performance band, from Unsatisfactory to Partially Proficient. He failed tests less often.  He talked in class and engaged in group work.  He smiled more often.  He may make another leap this year – I certainly hope he does.

I think of Aaron often and how in my assessments, I need to dig deeper to find out what the kids know – not just what they don’t know.  From there you can grow.

 
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Posted by on August 23, 2012 in Lesson Reflections, Student Stories

 

Entrepreneurial Qualities

This was popped up on a slide during our beginning-of-the-year staff development meeting:

ENTREPRENEURIAL QUALITIES
Confidence
Passion
Creativity
Friends
Risk Taking
Motivation
– Yong Zhao

The “Friends” bullet item was interesting.  How do we purposefully build all of these skills, and the “Friends” skill in particular, through school?

Another tidbit – source not cited.  It was mentioned in our grade-level team meeting that if you choose an at-risk student, and spend two minutes a day, every day for ten days, checking in and having a conversation with the kid, it improves his or her outcomes significantly.  I didn’t write down what outcomes were measured, or perhaps they weren’t cited.  But still.

It reminded me of the time a not-at-risk kid said to me, “Ms. DuPriest, I see you more than I see my mom.”

Students arrive in the classroom tomorrow!  We will do our usual norming and pre-assessing and then dive into integers and order of operations.

 
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Posted by on August 21, 2012 in Uncategorized

 

Is Algebra Necessary?

I got sucked into the online buzz generated by this July 28th New York Times Op-Ed by Andrew Hacker, titled “Is Algebra Necessary?”  It struck a nerve.  Many nerves.  Hacker’s main premises were: 1) Algebra is too hard, and is responsible for many a talented student’s academic failure – backed up by statistics, and 2) Algebra is not necessary for most careers, backed up by flimsier statistics.

Of course Algebra is necessary, we cried!  Algebra teaches you to make real-life situations abstract, to simplify them to their core.  It teaches you to play with the variables involved and to quantify cause-and-effect.  It teaches you to argue logically and reasonably.  It teaches you the life skills of discipline and hard work. It teaches you organized thinking and linearity.  It’s good exercise for your brain.

These are noble goals.  Everyone should learn these skills.  These are not taught in high school mathematics.  Some students learn them.  But look at a high school Algebra 1 or Algebra 2 exit exam and tell me where the teacher is assessing the student’s ability to abstract real-life situations and argue logically.  You’ll find a couple of word problems in the problem set on systems of equations, and I can almost guarantee that will be the extent of the assessment on abstracting real-life situations.  There’s an assessment here, and also here, and here’s a standardized practice exam for Algebra II here.  The students are required to do very little other than notice the structure of a problem and follow the right algorithm.  If you’re required to show work, you can show evidence of organized, linear thinking… or, you might just show evidence of being very very good at memorizing a sequence of steps.  Students who are already good at pattern-finding and generalizing may excel at this, as they’ll recognize the structure of a problem and know right away it’s, say, a trinomial with integer coefficients they can easily factor.  Will this teach a student to become good at pattern-finding?  Students for whom this does not come easily will sweat over the same trinomial, trying to recall the steps needed to get the problem into a form the teacher will like.  They’ll probably mix up one or two of the steps and arrive at an answer that would make no sense in context, but there is no context around the problem, so one answer looks just as meaningful as another.  Students learn a set of rules to apply for a certain structure of problem.  Then another set of rules to apply for another structure of a problem.  The noble, higher purposes of Algebra – the abstract reasoning, the beauty of a logical argument, the modeling of a complex situation into a symbolic form that can be manipulated to apply to other situations or prove other truths – are not actually ever taught.

There’s more.  All of this algorithm-crunching and polymonial-factoring and imaginary-number-figuring by hand would make a lot of sense to keep in the high school curriculum if this were how anybody actually used mathematics. But here is the dirty little secret:  nobody does math this way in the real world.  Who solves a quadratic or a double-integral by hand?  Who shades in those cartesian graphs of systems of inequalities?  Who converts a cartesian graph to a polar graph with a pencil, or finds the inverse of a radical expression on paper?  NOBODY.  It’s time-consuming and error-prone.  Computers are better at it, and we have better things to do with our time.  You know how long we’ve been teaching algebra by hand, with paper and pencil?  The earliest evidence I could find was from 1923.  The application of math in the real world has changed a little since the 1920’s.

Inevitably, you’ll be teaching the simplification of expressions using complex numbers, and a student will raise her hand and ask when we’re ever going to use this. You’ll be grumpy about this, because they might use it someday, if the student becomes a theoretical physicist or a mathematician, and anyway, that isn’t the point.  The point is that when a solution can’t be found in the set of real numbers, look, there is a whole other infinite set of numbers that are imaginary that have limitless possibilities.  It’s beautiful and expands the horizons of what your brain can do.

It’s too bad students don’t actually ever discover that. They learn a set of rules for simplifying expressions if they contain the letter i.

So then you’ll be having coffee with some parents, and a parent will say Johnny has been having trouble with complex numbers, and will ask you honestly when their child might ever be expected to use this.  And you’ll say the kid might, if they become an engineer, and that math is important if a kid wants to get into a STEM career.  Algebra II success is a great predictor of college success.  Then you’ll change the subject.

You know you’ve done this.  I have!

I have a story about the way you use math in engineering.  I would like to share it so you understand why we so badly do need to completely restructure mathematics education.

I worked as an engineer in an electronic test and measurement company, during the late 1990’s and early 2000’s.  I was a software engineer and I had to study a lot of math to get there, because engineers use a lot of math.  Once, I was on a project where my job was to write the drivers that took measurements on the power output of a fiber-optic amplifier.  I remember there being a function that described the power output in terms of another variable, and although I didn’t need to solve it because it was already done, I solved it anyway to prove to myself that the equation worked.  I remember the equation contained a log function which I struggled to remember how to “undo” (is it a base 10 if there’s no base specified, or is that a natural log??), but once I refreshed my memory using the information superhighway, I was good to go.  I balanced the equation carefully and tested it with a couple of pairs of variables, and indeed the equation was derived correctly just as I received it.  I was satisfied.  I spent about an hour on this task, if you also include the time I spent searching online for cool information and simulations of logarithmic and exponential functions.  And watching a funny cat video.

So I spent an hour using all that precious knowledge I had gained in the many, many years of math I took in high school and college.  Next I had to write the program that retrieved the output from a spectrum analyzer and calculated the power from it.  The spectrum analyzer produces a curved trace, which you receive as an array of numbers.  I actually had to calculate the area under the trace.  I didn’t integrate it by hand… I used a computer program to calculate the area of a lot of tiny rectangles under the trace and add the areas together.  With a few lines of code, this took a very short time and really just used seventh-grade math.  I was going back to the very basics of what calculus is all about, and it was humbling to see I spent so much time studying integrals and here I was, a professional engineer, adding rectangles together.  But in the digital world, this is what you do.  Almost nothing is really curved in the computer age.

I finished the software and then had some decisions to make.  The more little rectangles I measured, the longer the program took.  Did the customer need the result in milliseconds, or did they want a more precise answer that took several seconds to get?  How precise was the spectrum analyzer and could I make it more precise?  Was one measurement enough?  Should I take three traces and average all my answers? Should I take ten traces, drop any outliers, and average the rest?  How would I know if a trace were an outlier?   What result should I be expecting?  How could I independently verify the results I got?  Under what extreme conditions did I need to test this?  In what format did the customer want the data saved?  What was a pass versus a fail?  These kinds of questions filled the rest of the development cycle.

The project lasted several months.  I spent one hour of that time doing algebra.  The rest of the time I spent puzzling over what information I needed, what my results meant, and how to make them presentable and understandable.  Since that project, it has never been lost on me that I spent years and years of my life manipulating equations by hand, and I used that knowledge in my career once.  And I was an engineer in a bona fide STEM career.

Think about that, and think about how we teach algebra.  A STEM career involves making lots of decisions about messy data with unknown sources of error.  Understanding how to abstract and idealize a real-life situation is certainly an important part of math.  This is what algebra is supposed to teach us to do, and it doesn’t even do that well.  We teach symbolic manipulation via a bunch of algorithms instead.  We rarely get to having students truly abstract a real-life situation, and we completely neglect teaching communcation, challenging the reasonableness of answers, sources of error, estimation, precision, tradeoffs, and general problem-solving.

I have some half-baked ideas, of course, of what a 21st century math education actually looks like.  But I’d like to hear from you:  If you were king, how would you structure secondary math?

For some great inspiration, enjoy this TED talk by Conrad Wolfram.  You’ll start to get the idea on where I stand.

http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html

For more inspiration, read a piece that has challenged me from my early days of becoming a teacher:
http://www.maa.org/devlin/lockhartslament.pdf

 
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Posted by on August 15, 2012 in Trends in Math Education