Coverage or Mastery?

I have no answers this week only questions.  Being new to teaching math, the depth of this dilemma is only now really sinking in.  My students face a district-wide exam at the end of each semester that tests half the text book (six chapters).  My original plan was the obvious one.  Six chapters in eighteen weeks equal one chapter every three weeks.  

Of course, there is some time lost at the beginning for district paperwork, establishing procedures, getting to know the students.  Then, more time disappears for district mandated standardized tests.  Then, in my case, I had to be gone most of a week for a family emergency.  At this point, I will be finishing the first nine weeks with only two chapters completed.  Now there are nine weeks left before the district exam and four chapters to cover.

Even if I had not lost time at the start, I am now understanding what any experienced math teacher could have told me from the beginning.  Mastery is a very different thing from coverage.  Mastery takes time.  I can spoon feed, force feed and drill, and perhaps, just perhaps, they will have enough procedures in their bag of tricks at the end of the semester to pass the multiple choice exam, if that is the goal.  The majority, however, will not have any in depth conceptual understanding.

If mastery means understanding and retention and the ability to build upon the basis of what has come before, then the majority won’t have it.  Mastery takes time. Of course, there are a very few who grasp concepts immediately and run with them, but the rest need time, time to discover for themselves, time to make mistakes and wrong turns, time to develop a mental map and begin to fill it in.

It is an ethical question too.  What is the goal for the students?  As an educator, I believe my purpose is to facilitate appreciation and understanding for the subject.  As an individual who believes mathematics is one of the great achievements of the human mind and fundamental to a fully rounded education, I feel an obligation to offer my students the joy that I had in discovering and building mastery myself.  They, on the other hand, without my perspective, would likely say that their goal is to pass the exam.  What is my obligation to them, to their parents, to the integrity of the subject?

In my first post in this blog, I rather cavalierly decided to take as much time as needed for what I blithely believed would be mastery for most of my students.  Of course I am now discovering that I cannot guarantee mastery anyway.  No matter how much time I take, there will be some students who still need more time, and there will be some students who refuse the journey altogether.  There will be a few who find their way without me.  The road to mastery looks very different for different students.  I will be feeling my way in what is for me uncharted territory this year.  

I look forward to any comments, suggestions, or insights.

Quality Bonus--Encouraging Neat and Legible Work

Student Work with Quality Bonus

I copied the idea of the “Quality Bonus” fromTeaching Smarter by Sandy LaBelle (1999). It is only one of the many useful procedures described in detail in her book.  I received the book a couple of years ago at a workshop, and like a lot of my books, it sat on the bookshelf unopened for a long time.  In preparation for beginning this school year, I returned to my bookshelf to brush up on basics.

The “Quality Bonus is a really effective way to encourage students to put effort into turning in neat work, which in turn means I can read and grade the work easily and quickly.

How it Works:

In my class, I apply the bonus to the students' bell work.  The students complete bell work each day during the first five minutes of class while I take attendance and attend to other business.  They turn in five days of bell work on one page on Fridays.  They receive a grade for correctness and completeness just like any other assignment, but in addition, they can receive a five point extra credit "Quality Bonus" for fulfilling very stringent neatness and organization requirements.

I copied the idea of the “Quality Bonus” from Teaching Smarter by Sandy LaBelle (1999).  It is only one of the many useful procedures described in detail in her book.  I received the book a couple of years ago at a workshop, and like a lot of my books, it sat on the bookshelf unopened for a long time.  In preparation for beginning this school year, I returned to my bookshelf to brush up on basics.

The “Quality Bonus is a really effective way to encourage students to put effort into turning in neat work, which in turn means I can read and grade the work easily and quickly. 

I use it for the student’s bell work.  The students complete bell work each day during the first five minutes of class while I take attendance and attend to other business.  They turn in five days of bell work on one sheet on Fridays.  They receive a grade for correctness and completeness just like any other assignment, but in addition they can receive a five point extra 

To receive the “Quality Bonus” in my class, the heading in the upper right corner must include first and last name, the class and the period, and the dates of the work.  The title on the page must read “Early Work Week 7” (or 8 or whatever week of the school year we are in), each day of the week must be labeled and underlined (Monday,  Tuesday,…), bell work questions must be copied completely, all work shown, and answers circled.

I use a stamp that I found in the children’s section of Wal-Mart to quickly stamp all the papers that should receive the bonus.  

Some warnings:

First: If I forget to post the Quality Bonus requirements (with an example) every single week and verbally remind the students, a lot fewer of the papers turned in get the bonus.  I want most of them to get the bonus since it reinforces the good habit, so it is important to keep reminding them that the extra credit is available.   

Second: Sandy LaBelle did not include correctness in her bonus requirements.  The students received the neatness bonus whether or not all of their answers were correct.  In my class, I go over the answers each day after the students finish, so in essence, I give them the answers.  Even though they were instructed to, students were not following along and correcting their work, so I added correctness to the bonus requirements.  I also told the students that if I saw them not working on the early work during the time given that I would make a note of it and they would not receive their full credit.

The Effect on Grades:

Overall grade category weights are set by my department, so I needed to fit the early work into preset categories.

In my Algebra classes, the bell work is recorded in “Participation,” which is weighted at only ten percent of the student’s overall grade.  The bonus on their bell work is 5 points on an assignment that is normally worth 25 points, so a full 20% extra credit, but since the whole category is limited to 10% of the grade, the effect on their overall grade remains reasonable.

In my Geometry classes, the weekly bell work paper is worth 100 points (the same as a daily homework assignment) and goes into the homework category (40% of the overall grade).  They also receive the 5 point bonus, which in this case is only 5% of the assignment, so the effect of the bonus is also not that large.

The differences in the way the grade is recorded makes no difference to the effectiveness of the system.  The students are motivated by the added five points no matter what portion of their grade it represents.

Grading, Grading, Grading...

A few days ago, I had a monster pile of grading that looked insurmountable.  Not only had the first few weeks of student work gotten the better of me, I had had a family emergency that meant I was absent for four days in the middle of the very first chapter.

I started out with the best of intentions.  Students would grade their own work, receiving that immediate feedback that would be so useful to their learning.  Nothing would pile up on my desk.  Nothing would come home with me.

Self-grading worked like a charm, but even simply spot checking a few examples and recording the grades piled up on me.  Onto that I added bell work that I intended to grade weekly and a quiz and a test that turned out to be miserable grading experiences because I hadn’t sufficiently planned ahead of time what I would and would not accept as answers.  I thought with math I would be leaving behind those kinds of judgment calls!

After two days of mandatory state testing (allowing me grading time) and a marathon weekend, the giant pile disappeared and I have made new resolutions to keep it that way. 

I don’t remember where I first heard of having students grade their own work.  I never previously thought I could make it work, but my students this year are so wonderful, that they manage it like a charm.  My version of grading works like this:

Right after their bell work, the students pull out their homework and a red pen.  I insist on the red pen and no other writing instrument being available on the desk.  I announce the numbers of five representative problems from the previous night’s assignment. 

First, the students circle those numbers.  If they are missing any of those problems, they mark the spot where it should have been and write -4 in the margin.  At the top of their paper, they write “Missing 4 x ___ = _____” multiplying the total number missing by 4. 

Second, I use an Elmo to show the answers for the given problems in the teacher edition.  The students mark an x and write -1 next to any wrong answers.  They are encouraged to write the right answer in the red pen as well so that they can study from the corrected answer.  At the top of the page, they write “Wrong = _____” and write the number wrong.

The two totals are added together and subtracted from 20 to give them their score out of 20.  I spent a lot of time the first few times reminding them how big a difference skipping problems made to their assignments.  If they do all the problems, their lowest possible score is 75%.  Given that, they are encouraged to do the work, and are not much tempted to cheat on the grading. 

Before introducing the grading the first time, I made a speech reminding students that cheating is not only dishonorable and dishonest, but downright stupid when the cost would be getting a zero and having me publicly announce that I am going to be grading their work in future, while the lowest score they will get if they grade honestly and do all the work is a 75%.  I have not seen any evidence of cheating. 

There are students that continue to misunderstand or incorrectly mark some problems, but these errors really don’t matter much.  Homework, by department policy, is set at 40% of their grade.  Even if they get full credit for all homework, their grade will be determined by tests and quizzes.  The grading system is meant to convince them that trying every problem is worth their effort.  I do have to watch for the occasional student who gives himself or herself too low a score.

After the first few times, the grading became a very quick operation.  I can quickly gauge how well students understand the material by their scores.  Any grades less than 15 indicate that the student did not do the whole assignment.  If I picked the five problems to be representative of the types of problems in the set, scores between 15 and 20 give me a pretty good idea of how well the students mastered the problems given in this set.

That was the idea anyway.  I am only beginning to be able to breathe enough to actually do more than collect papers, record them, and move on.  

Here are some examples:

Student struggling--missed 4 of 5 representative problems

Student not completing assignment.

Student has a pretty good grasp of the material and took notes on the one problem missed.

Student does not understand the grading instructions.  Notice my correction in green.

Notice I have to use green for my corrections, since I told the students to use red.  I remembered too late why the workshop leader where I first learned a version of this grading technique recommended the students use green!

Rather than let the papers pile up in my keeping, I have tried lately to record the grades on my seating chart while the students are still seated and copying notes from the board.  That way, the students keep the work the whole time, and I can transfer the grades very efficiently to the computer.

A very wise person reminded me in the middle of all this that the point is the learning not the grade, so I am trying not to fixate on my process.  Sometimes there just isn't time to grade in class.  There is too much else to do, and for some assignments, I just gave myself permission to give full credit for completion and move on.  Just for the moment, the folders are all empty, and I have that rare and wonderful feeling of being temporarily caught up. 

The Tour Bus has Left Without Me!

I have just finished five weeks of teaching math for the first time—three periods of freshman Algebra 1 and three periods of sophomore Geometry.  I got the notion of chronicling my teaching in blog form as a way to share my experience, seek input from others, and in hopes that the act of reflection would lead to improvements in my teaching.  I almost immediately was discouraged by two things. 

First, all the good names are taken!  Mathematically Speaking, Continuous Everywhere but Differentiable Nowhere, dy/dan, f(t), Math Mama, and so on and so on…

Second, these people all actually have useful information to impart based on years of experience.  I, on the other hand, am a tour guide without a map who hasn’t memorized the script yet.

Naturally, I decided I had no excuse for imposing my ramblings on an unsuspecting public. 

Having made that decision, my ego wouldn’t let it go.

It recently occurred to me that something that I haven’t seen is a blog focusing on reviewing and sharing information from all the great math books (NOT textbooks) out there.  So that is my excuse for going ahead with this experiment. 

The plan for this blog is to share a chronicle of my personal and probably idiosyncratic responses to the math books I have loved and enjoyed, as well as to the books I expect to discover and hope to have recommended to me by readers.  Off the top of my head, I expect to revisit Marilyn Burns’ great Brown Paper Bag books (Math for Smarty Pants and The I Hate Mathematics! Book)  and Kenn Amdahl’s Algebra Unplugged.

In the meantime, I can’t resist sharing a bit of my actual teaching experience, which at this point, has left me no time for reading anyway!

I am at the end of week 5 as a new math teacher.  I’ve stumbled along in a fairly predictable way as a beginning math teacher for these first weeks—lesson planning by textbook section.  For the students, it is like a quick compulsory trip through a strange country.  “It’s Tuesday, this must be Belgium” style, only for them, “It’s Tuesday, it must be Vertical Angles”  is more like it.  It didn’t help that during the first chapter, I was called away for a family emergency for four days.  At the end of five weeks, we have completed chapter 1 (scheduled to be finished in three weeks).

Worried about falling behind schedule and fearful of not “covering” all the landmarks, I went barreling straight through.  The first test showed the result.  The kids are lost in a strange land without a map.

Time for a deep breath and a “do over”.  So, I’ve let the tour bus leave without us and we’re heading out on foot to revisit the best of the country-side, and this time we’ll be practicing the language!