EXEMPLARS from Community Works Institute
Teaching Math Without an Answer Key
by Matt Mayberry, Math Teacher
Spaulding High School, Barre, VT
Matt Mayberry is a second-year math teacher at Spaulding High School in Barre, Vermont. Matt also works as a business consultant, specializing in executive training for Fortune 500 companies. Prior to consulting, he worked as an experimental physicist in fusion energy research. Matt also shares a Geometry service-learning project involving students in school design, along with an insightful reflection on his experiences
Most high school math classes closely follow a textbook. The pathway through the material is laid out clearly for the teacher in fine detail, complete with explanations, homework problems, quizzes, tests and all the correct answers. Of course, the real world (that place that our students always ask us about in class) with its real problems is not so neatly packaged for us. It certainly doesnt come with an answer key.
When I began my first full year of teaching high school math last year, I wanted to incorporate a service-learning project into my algebra and geometry classes. Having come from a career of business consulting, I wanted to expose my students to more in-depth problem-solving. I also wanted to motivate students in a way that I felt a textbook couldnt, by having them work as consultants on a project, for a client, with a purpose. Beyond that, I really had no particular projects in mind, no client under contract, and certainly no plan for integrating a CSL project into my classes. My professional experience told me to expect many surprises ahead. Its pretty safe to say that I wasnt disappointed.
Analyzing Why Freshmen Fail
Early in the school year I approached one of our co-Principals Cindy Donlan and told her that I had a class of nineteen consultants in my algebra class ready to spend about 10% of our class time on a project. When I asked her if she had any ideas, she immediately began sharing her concerns about the lack of student engagement and the high level of freshman failure in recent years. She shared some of her ideas about the possible causes of the problem and suggested that we start by analyzing student attendance data, since she believed that poor attendance contributed to poor academic performance. Just like that we had our client and we had a project! The topic seemed perfectly suited to a math class since it involved analysis of data.
Soon thereafter we kicked off the project with a client meeting. At this meeting Mrs. Donlon visited our class and shared her perspectives about freshmen failure to the students. After a brief discussion and negotiating a contract for a free lunch in return for their services, the students agreed to do the analysis.
It can be tricky to stay balanced on the knife-edge between order and chaos, where authentic learning (and teaching) resides. Doing so requires putting aside the neatly packaged teachers guides and answer keys and taking some personal risks.
The students working as consulting teams of three or four began their analysis by looking for trends in the attendance data. Students sorted the attendance data along several dimensions (class, school year, month) and they compared attendance for students who failed classes in their freshmen year and those who didnt. Along the way, they learned how to use Microsoft Excel to perform calculations and create tables and charts. Each team analyzed the same data using more or less the same approach.
The first surprise came when the students turned in their tables and graphs for me to grade. I found a multitude of math errors and numerical inconsistencies between the groups. Charts were improperly labeled or not labeled at all. Most of the conclusions reached by the teams were erroneous. I had taken for granted that the students would check their results and take the care needed to ensure their validity.
When I shared my dismay with the students they seemed surprised. After discussing as a class why such an analysis would be useless to our client, I asked them to reanalyze the data. It was hard to enforce this discipline upon the students, but this was an essential step in their learning process. It was critical that they discover their own mistakes and correct them. There was no answer key to guide; they had to rely on each other.
Real Data Compells Attention to Detail
What I learned from this experience was the difference between solving book problems and analyzing real data is quite a leap for many students. In the pre-packaged math curriculum, the notion of sanity checking results was absent. In fact, the project helped to fill a void in the curriculum. The second time we reviewed the results, there were fewer errors and inconsistencies, but many of the conclusions that students reached were still not justified. Finally, after a third round of analysis, checking and discussion, it appeared that we could reach some tentative conclusions based on the data.
What the students found was contrary to the principals initial hypothesis. On average there was no significant difference in attendance between the students who had failed during their freshman year and those who hadnt. There were a significantly higher number of absences for a few of the failing students, but these were the exceptions rather than the rule. In fact, attendance (tardiness) seemed to be worse for seniors than freshmen, and yet the seniors werent failing their classes.
The students put together posters and presented their results to the principal and the CSL coordinator. An example of a poster is shown on the previous page.
Immediately after the students presented their findings to the client, they described to me their sense of failure at not having given the principal the answer she was looking for. They felt a negative result was not useful or significant. This reaction reminded me of my experiences as a physicist (before I was a consultant) where my positive experimental results received lots of attention but my negative results were discounted or ignored. I tried to assure my students that negative results are often the most valuable because they can help to rule out false theories. Im not sure they bought my argument. When I asked them whether they were surprised about their results, they responded with a resounding NO. Then they told me that they never believed that attendance was an important factor to begin with.
Why Do Students Really Fail?
So if attendance wasnt the root cause of freshmen failure, what was? I posed this as a journal question to my students. I asked them what their life would be like over the next ten years if they decided to fully engage in their high school education. I also asked them to describe their life if they decided to disengage from their high school education. I asked them whether they thought freshman that failed their classes understood the long-term consequences of their choices. And if they did understand the consequences, what near-term rewards were they choosing over long-term consequences? All of my students (even the failing ones) responded with thoughtful analyses of issues related to low student engagement. There is room here only for a small sampling:
[Failing freshmen] are having fun now, but they will regret it if they dont shape up. I say this because if you get a habit of not doing work, it will follow into following years and make it harder. I heard this from classmates.
Some freshmen just dont care about school because of their attitude that they get from their parents. They would rather live in the moment and be happy now instead of later.
High school is a big jump from elementary school. Teachers dont hunt you down anymore. If you dont do work, you might as well give up and stop wasting other peoples time.
A lot of freshmen dont understand how their grades will affect their life. They think that freshman year doesnt matter and so they just blow it off.
[Freshman are] probably thinking they have three more years in front of them because thats what I thought when I was a freshman. Plus I got told before I got to high school that the only years that mattered were junior and senior. So thats why I kind of slacked off. But they will have to find out the hard way like I did. . . . I dont think freshmen are given enough info. . . How can you go into high school not really knowing really about classes and credits?
Some are failing because of pressure of their friends . . . I dont know maybe spending time with their friends is more important than anything.
I think that the freshmen consider the near-term rewards because they want or enjoy being young and they are influenced by their friends. I think that they are being influenced by everything around that tells them to be enjoying life and having a good time. They do not have to worry about being in school. They do not have to worry about their future.
I know that my grades will help me a lot. Other people that I know have succeeded in life because they worked hard in school. . . A lot of freshmen dont understand how their grades will affect their life. They think that freshman year doesnt matter and so they just blow it off.
Freshmen who do succeed probably do realize that they have better chances of succeeding later on. Most probably do care about college or careers, and that may be their motivation. But for some people, maybe they do it because of their expectations from other people and of themselves.
Some freshmen dont understand the long-term consequences because they have too much on their mind. For example, sports because they have practice every day and a job they have every day and when they do these activities they forget about homework or maybe forget the time thats gone by and tomorrows Monday and they havent done anything.
The next ten years will be harsh. It will be hard to find a job with my grades.
No, I dont think the freshmen know the long-term consequences. The reason I think this is because in Jr. High they would just blow off classes and their homework. They didnt care about school. They saw no point in homework. I know thats what they did and because of that (everyone was doing it and I didnt want to be the only one not doing it) I did what other people did, I became one of them. A person who is lazy, one that did nothing.
These were many of the same issues that I had read about in recent books about school reform (see, for example, Beyond the Classroom: Why School Reform has Failed and What Parents Need to Do, by Laurence Steinberg).
One issue that I hadnt expected to surface as a theme from my students was the difficulty of making the transition from middle school to high school. Students wrote about or shared in class some of the false rumors that were spread in middle school about high school. These included stories of freshmen paddlings in high school, or about freshmen grades not counting towards college applications. They talked openly about the difficulties of adjusting to high school life, both socially and academically.
It became increasingly apparent to all of us that this could be one area where more experienced students could help incoming freshmen. When we asked Mrs. Donlon for suggestions about what we might do about this, she suggested that the students help out with Step-up Night. This is an orientation held in the spring where incoming freshmen attend an abbreviated class schedule and meet their future teachers. The class agreed to do this, bargaining with me this time for some bonus quiz points (sometimes even community service needs just a little extra motivating).
On Step-up Night, my students paired themselves up with English teachers. They helped with presentations, answered questions from the incoming freshmen, and helped the new students find their way around the high school. The level of preparation and participation varied depending on which teacher the students worked with. As I made the rounds that night, I couldnt help but enjoy hearing several of my students give impassioned pleas to their younger audience to do their homework and take school seriously. The most compelling dont do what I did messages came from those students who were currently struggling to pass my class.
Did Our Work Matter?
I wondered whether my students felt that they had made any difference that evening, so I asked them to write a journal entry about it. While a few students felt it was a waste of time, most felt they had made some difference. Here is one of the more positive responses, this one from a sophomore who had failed algebra the previous year:
On Step-up Night, Tim and I worked with Mrs. Spencer. She is an English teacher. We thought working with her would be good because we each had her and we know her teaching style. So when we arrived she had planned for us to talk a little about the class and what key things you should do to pass the class. So the students arrived and we talked, but the main thing we kept [reinforcing to] to kids was doing homework and you will pass the class. And after we talked to the two classes we talked to her and she was very excited and thinks it would help tremendously and that we did a great job. She also said it went very well which I think it did too.
So on to the real point, do I think it helped? Yes I do. I think we did a good job and I think it will help them a lot hearing it from the people that know best (the students). I think it should be turned into a yearly thing with students speaking with a teacher. And from my point of view I think most teachers liked it and think everyone did a good job and that it helps. I even got complements from three parents out of school that said I did a great job and that their kids liked it. They also think it should be something that should happen at every Step-up Night.
I would like to thank my fellow geometry teachers, John Pandolfo and Kevin Beard, for their valuable contributions to the planning and execution of the architecture project. Id also like to thank the following administrators for their continuing support and encouragement: Cindy Donlon, Co-Principal at Spaulding High School); Jeff Mahr, Curriculum Director at the Barre Supervisory Union; and Bev Scofield, CSL Director at Spaulding High School.