Rick's First Draft

"Algebra for everyone" is the policy at the large suburban San Francisco high school where I student-teach in the Mathematics department.  This means that Algebra I is not only a required course, but it is also the lowest level mathematics course offered at the school. No alternative courses are available for those who might not be fully prepared for Algebra when they enter as freshmen, and no senior can graduate without a passing grade in Algebra.

One practical result of this policy is that my fall term class of 22 students included eleven freshmen taking Algebra for the first time, and eleven upperclassmen who had failed it one, two, or even three times in prior years.  I quickly learned that some freshmen were indeed not ready for this course, in that they lacked basic skills such as computations with fractions and had conceptual difficulty in multiplying with negative numbers.  The most mathematically talented members of the freshmen class were unrepresented here, because they had completed Algebra I in middle school, and thus bypassed this course completely. Several of the upperclassmen had obviously not been ready on earlier attempts, and repeated efforts to master the Algebra course content had not addressed their pre-existing gaps in skills and concepts.  Ready or not, Algebra was what they were to learn in my class.

My first impression of this group was that they were remarkably quiet at the beginning of the year.  Some of this was probably due to the 7:30 a.m. starting time of the class, made more onerous by the hour or longer bus rides that some took to get to school. The block schedule adopted by the school put them in my class for 90 minutes each morning, five days a week, as the start to a very long day.  There was much material to master each morning, because the entire Algebra course was to be completed between September and January.

The class was an ethnically diverse group, in which girls outnumbered boys sixteen to six. I was relieved to find that all were competent in spoken English; as a beginning teacher, my tasks seemed formidable enough without the added complications of language issues.  One junior boy was a mainstreamed special education student, and a resource person was frequently in the room to help out without occasional coaching.  The class was well behaved, presenting few management challenges; indeed, I wished they would be more talkative and active, so that I could exploit that energy for class discussions.

A departmental decision had been made to emphasize linear equations as the critical course content this term.  Minimal mastery of this topic requires understanding the concept of a variable, the ability to produce and interpret coordinate graphs, and skill in operations to solve simple equations.  These were the main topics of concern for the first weeks of the course.

Initially, I would begin each day with a set of four to six warm-up problems posted on the board.  Usually, these were chosen to explore or illustrate a single theme or property, drawn from the prior day's material.  The intent was to get the kids thinking about math, wake them up, review the prior content, and add a bit of insight to what had been done the previous day. Students worked on these problems individually, and then volunteers would put solutions on the board for class discussion.  After the warm-up, a new lesson would be started.

For the lessons, I often placed the class in groups of four.  Initial group assignments were made by me on the basis of early quiz results, to ensure a mix of ability in each.  Later, I asked the students to list in confidence their preferences for the others they would most like to have assigned to their group. Apparently, the students had gotten to know and like each other in the groups I had initially created, because honoring the requested assignments required only minor rearrangement., and mixed ability grouping was preserved.

Early on, I began to notice a pattern in the performance on warm-up problems and quizzes. When I assigned a purely computational problem, or one requiring symbolic manipulation to solve an equation, most students would usually attempt the problem, and some would correctly answer it.  However, when I devised a problem written in English, perhaps asking for an application of a mathematical principle or technique, it seemed to stop the whole class dead in their tracks. When faced with a "word" problem, they would freeze up, with stunned facial expressions, reminding me of a surprised deer caught in the beam of my car headlights.

I was puzzled at first as to why these problems appeared to be so hard for the class. A problem already written in mathematical notation, or an equation already written, seemed well within their grasp.  But a word problem that could be represented by the same notation or equation brought them to a screeching halt.

I thought hard about what the possible causes might be.  The most obvious possibility was difficulty with basic reading skills. I soon decided that some sort of experiment would be necessary to explore or eliminate that possibility. I gave the class moderately complex written instructions for problems written in equation form.  They seemed to have no trouble understanding the instructions, and took the equations in stride.  I also tried a few more word problems in class, asking selected students to read them aloud from a workbook.  Reading did not seem to be the issue.

I also considered the possibility that these students were simply unaccustomed to being asked to think in a math class, and simply refused to do so.  As I continued to sprinkle word problems into daily warm-ups, I watched their faces and listened carefully to the few questions that this quiet group would ask.  However, I saw no defiance, only bewilderment.

I concluded that the difficulty must lie in recognizing the mathematical content in a problem statement, and then generating a translation into mathematical notation.  In effect, they did not know how to get started.  Once an equation was in hand, they could solve it, but they couldn't see how to begin. I realized that this would cripple them for any higher mathematics, and relegate any math they already knew to an academic curiosity, with no hope of practical application.  I decided to start a personal quest to cure the word problem phobia for these kids.

As a first step, I resolved to put at least one word problem into each daily warm-up. It was rare to get a full solution from a student, but I would always be sure to provide one if the class could not. When we reviewed the problems, I would explicitly talk about and thoroughly question the solution strategy that applied, whether the source of the solution was a student or me.  I thought that practice would help, and that explicit discussion of strategy might demystify the whole subject.  This did not produce immediate improvement, so I continued to work on ways to help the class over this hurdle.

To prevent students from simply "shutting down" when they saw a problem written in English, I modified the procedure for the warm-ups.  I created a new rule that nobody was allowed to give up on a problem. If they were stumped, they were required to get up from their seat and find some other student to talk to about it. If both students were stumped, then they could come to me for a hint.

I also decided to give the class some training in writing equations from English statements.  I showed them some fundamental ideas, such as converting "is" to "=", and "and" to "+". We also considered more complicated phrases such as "four more than a number" (x+4) and "reduced by 25%" (-.25x). A typical exercise in the early stages was to re-write "the width is four more than the length" as "W=L+4". We used a variety of common formulas, such as distance equals rate times time (d=rt), age equals grade level plus six (a=g+6), and so forth.  These exercises seemed worthwhile to include, regardless of their potential value in my quest, because they were valuable for comprehending the concept of a variable.  Most of the problems were simple linear relationships, so they fit well with the departmental objectives. More students seemed to be attempting these each day, and correct answers became more and more frequent.

I also introduced a new form of problem.  I would frequently ask the class to write an appropriate equation from a written statement, without then asking them to solve the equation they had just written. This seemed sensible, because the difficulties all seemed to lie in the initial translation steps.  It also allowed me to use a wider variety of problems. For example, I could pose area computation problems, which sometimes led to quadratic equations.  The class had not yet been taught to solve quadratics, so they could not carry through the later steps, but they had the requisite knowledge to read and write such equations.  I thought that this writing-without-solving effort might send the message that mathematics lies not only in getting numerical answers but also in modeling situations, without regard to whether one can solve the problem thus described.

I worked a bit of this into each day's activities.  Lessons continued on the normal curriculum, as I tried to fold this content in where I could.  Eventually, I also shifted to starting the class in groups, so that the warm-up problems could be done as a collaborative efforts.

I did no formal assessment on word problem mastery, instead relying on my impression that participation in solving the warm-up problems was increasing, and judging from the warm-up discussions, comprehension appeared to be increasing as well.  To demystify the subject as much as possible, I prepared a short handout with tips for getting started, that was discussed in class and applied to some sample problems.  It looked like this:

HOW TO GET STARTED WITH WORD PROBLEMS

1. Figure out what you're looking for (a price, a length, etc.).

2. Give it a name (a variable, such as X or L or P or ....)

3. See if you can write the other quantities in the problem in terms of the variable.          (L+2, or ½ X, or .07P, etc.)

4. Look at the problem to see what the overall relationship is.  Do several lengths add up to a total?  Do several prices add up to a cost?  Do two sides multiply to make an area?

5. Write out the relationship from step 4 using the expressions you wrote at step 3.

My remaining concern was that all this had perhaps gone from terrifying to boring without having passed through understanding along the way.  I decided that I would create a personalized set of problems that might have more interesting content for them.  I based this on what I knew of the hobbies and interests of many students in the class.  I knew that we had several junior varsity athletes in girl's volleyball and tennis, several devoted fans of a popular teen-idol band, and one girl who aspired to a singing career.  I prepared one custom problem for each group, based upon the individuals in it. I put considerable effort into including the kids' names in the problems, and researching a good basis for each problem setting (for example, one dealing with concert tickets reflected the actual ticket prices for the next major concert booked by the band in question, that I had determined by online research).

Each problem required writing and then solving a linear equation.  Two samples are reproduced here:

Kristin buys two tickets for the upcoming Reno concert of N-Sync, by calling the B.A.S.S. telephone service.  B.A.S.S. adds $6 per ticket service charge.  Michelle buys two tickets through Ticketmaster, and they add 20% as a service charge.  They spend a grand total of $188.  What is the ticket price?

 Brandee gets a recording contract, so Angela and Laurie decide to buy her first cd. Angela picks up one for Cathy, too. Laurie shops online with amazon.com and they charge 9% for shipping and handling.  Angela goes to Tower Records and has to pay 8% tax.  If they spend a total of $32.50, what's the price of the cd?

After three weeks of practice, explaining strategies, showing translations tips, providing handouts, and collaborative work on problems of this sort, I expected all to go well when I presented this set of problems to the groups.  I thought that this would be a good measure of the success of my quest.

When class began the next day, I handed out the problems with high hopes.  I could see the students initially reacting with surprise to see their names in the problems, and could overhear comments such as "How did he know I play tennis?".  The problems were not difficult, in my estimation.  I anticipated that allowing for a bit of inefficiency in group communication, some groups would have solutions fully worked out in under ten minutes, and all would have answers within fifteen.

However, as the clock ticked on past ten minutes, then fifteen, and then twenty, I became more and more worried.  Most groups were stumped.  I reminded everyone that they could use their "how to get started" handouts, and some renewed their efforts.  After twenty-five minutes, one group had reached a solution.  I waited another five, but no other group had meaningful progress to report.  I asked the successful group to present their solution, and explain to the class how they obtained it.  I suggested to the others that they take the problem home and consider it overnight, given whatever added insight they may have obtained from their peers' presentation. The next day, there were no additional solutions to present. 

At this point, I had a much stronger sense of failure than any of the students.  They, at least, were accustomed to being stumped by problems of the sort they were facing.  I was not accustomed to being stumped by the problem I faced.  I had tried everything I knew, and it obviously wasn't enough.  I reluctantly decided that my personal quest was sapping too much time and energy, and that I would have to abandon it for the foreseeable future.  It was time to move on to new content.

As I look back on this experience, I think that my choice of a personal quest on this topic was a good one.  I suspect that many years of prior bad experience in math classes had left these kids ill-equipped to face mathematical problems that required thinking and application rather than just mechanical manipulation and computation. If I had been able to cure this problem, it would have been one of the greatest mathematical successes imaginable in an Algebra class.  I would even be content to have my students forget most of the specific techniques taught in the course if they could only master this one difficulty. Unfortunately, my diagnostic skills and nascent repertoire of teaching techniques were not up to the job. There were too many years of conditioning for me to overcome.

I wonder how a more experienced teacher would have viewed the situation.  Had I properly diagnosed the problem to begin with? Were my attempts at curing it appropriate, or did they miss the mark?  Did I give up too easily?  What should I have done?

1.                 Questions:

1.                 What did I do wrong?

2.                 What did I fail to do?

3.                 What did I overlook, or falsely assume?