Background Objectives Overview The Four Tiered Problems Reading and Grading the Letters Stella Letter Feedback
Writing is a powerful way for math students to demonstrate their understanding of math concepts while developing technical writing skills. Most math teachers would agree with that simple premise, yet incorporating writing into a math curriculum in a meaningful way is a challenge. First of all, assessing a writing piece tends to be a subjective matter, whereas math teachers, by nature, are objective creatures. Isn't writing the job of the Language Arts department anyway? Add to that the lack of time most teachers have even for the basics, like preparing lesson plans, grading homework, quizzes, and tests, communicating with parents, attending meetings, oh yeah, and teaching, when would there be time to read, grade, and provide meaningful feedback for each student's writing piece? Writing in math class just doesn't 'add up'!
Well I've been there, done that, and want to say that writing in math class is not only a viable option, but a necessary one. Writing in math has been the most rewarding and powerful piece of my math curriculum for everyone involved - students, parents, and most of all for me - the teacher. And furthermore, I've worked to improve the process to the point where it is not a painful and time consuming endeavor. I believe in writing in math, but more importantly, I believe in enjoying my life outside of school. I am here to say that these two ideals do not have to be mutually exclusive. Allow me to walk you through a writing process that has evolved for me through my twelve years of teaching middle school mathematics.
The writing program is designed as a pen-pal correspondence between students and a fictitious character, Stella. Stella is actually you, the teacher, whose responsibilities include giving the four-tiered math problems to students, reading their letters of response to the questions, determining their grade, and then writing the students back regarding their solutions and problem-solving process. This process enables students to create a meaningful portfolio of written work demonstrating their understanding of various math concepts.
The problems that are provided all have the same structure. First, an overview of the problem is presented in a few short sentences. Then Question A is posed to the student. This question basically checks to see if the student read the given information. Therefore, every student should be able to correctly answer this question, as the answer is already given in the overview. Question B is a question for concrete thinkers. This question is generally answered by drawing a diagram, generating a chart, or performing simple computations. It also acts as a springboard for Question C, which is the proficient level question. This question determines whether a student is performing at grade-level, the level of most existing textbook questions. Finally, there is an advanced Question D that tests the student's abstract thinking. This question challenges the student to either generalize the applicable concept or apply it to a related situation.
These tiers have several benefits. First of all, they provide different entry points for the varying ability levels of the diverse set of students in a classroom. As the questions increase in difficulty, students are able to stretch themselves as learners while attempting each new question. Consequently, each student may reach a different level of understanding of the concept as evidenced by the tier they successfully complete. The tiers also help to make the grading more objective, by holding each student to the same standard for writing, explaining, and defending their solution(s). For example, a student is given credit for reaching a tier if and only if a complete and accurate written solution process is provided for that question, backed by a clear and unambiguous explanation. An accurate solution with no written explanation receives no credit. Therefore, a letter grade or level rating is easily assigned to each student by simply determining the number of questions answered correctly with detailed explanations.
The letters that the student writes to Stella all have the same format. The student works through as many questions as possible, starting with Question A and working up to Question D. The student must provide a written explanation of the thought process used to build a solution for each question. They are encouraged to provide charts, diagrams, and graphs, if necessary and must always include mathematical calculations within the written work. Because these questions are tiered, the letters can be read and graded rather quickly. Furthermore, because the correct solution is already known, the teacher can immediately check the accuracy of the solution and spend more of the grading time reading the explanation and checking that the student is able to provide a logically sound argument defending how they got to their solution. As previously stated, the letter grade is simply assigned by correlating the level at which the student ended with an accurate, thoroughly supported solution. The four questions then map to four letter grades (A - D), while an F is reserved for those students choosing not to produce a letter to Stella at all.
Consider the following grade equivalents or level ratings:
| Question # | Grade | Level |
|---|---|---|
| Question D | Grade A | Advanced |
| Question C | Grade B | Proficient |
| Question B | Grade C | Novice |
| Question A | Grade D | Beginner |
Pluses and minuses can also be used to further differentiate the levels of understanding. For example, a student may answer Questions A , B, and C accurately with explanations, but employ a guess and check method for Question C. While guessing and checking is a legitimate process, it does not demonstrate a proficient level of understanding of a concept. Therefore, the student receives credit for Questions A and B and a plus + for the answer to Question C. The grade then becomes a C +. This type of student is different from the student who answers Questions A and B accurately with explanations but misses Question C due to a careless mathematical error in computation, despite a logical explanation. Consequently, this student earns a B -.
The beauty of this step for the teacher is that the bulk of the work is already completed! A written letter from Stella is provided explaining her thought process for each question. Of course, there is always more than one way to solve a problem and teachers can provide additional explanations as needed, as well as any 'personal' details they may want to add for an individual student. Each student receives a personalized letter from Stella through the technology of mail-merge. The letter from Stella is stapled to the back of their letter. Furthermore, any student who earns an A or an Advanced rating on their Stella letter also receives an Advanced Stella Certificate praising their work.
At the end of the school year, students have an impressive portfolio of written work demonstrating their understanding of various math concepts. The Stella letters are divided into four different topics of math meeting the National Math Standards: Numbers and Number Relationships, Geometry and Measurement, Probability and Statistics, and Patterns, Functions and Algebra. The Questions can be copied on to colored paper by topic so that students can easily organize their portfolios.
There are seven different curriculum packets: one per grades 4 - 8 and an algebra and a geometry set for high school. In addition, there is a practice problem that can be utilized for each grade at the beginning of the year to help students learn the standards and expectations. I use this problem as a learning tool for groups of students to work on in class. Students turn in one letter per group to minimize the grading and then the letters are returned the next day so that expectations and grading can be further discussed. This practice letter has proven to be extremely beneficial for students so that they are fully prepared for the actual Stella problems.
There are four problems per proficiency area: Number and Number Relationships, Geometry and Measurement, Probability, Statistics and Data, and Patterns and Algebra. I suggest that there be some type of in-class assessment to hold students accountable for their own work. Often, when students take these problems home to work on, we never know whose work we are really grading. I have dealt with this accountability issue in a variety of ways. First of all, Stella problems are handed out on Fridays and I require a hand-written rough draft from each student by Tuesday. At this point, I just spot check the student work and put my initials on their letters to verify their authenticity. I require that these rough drafts be stapled to their final (often typed) letters so that I can see their progress. Furthermore, after each proficiency is completed, I have an in-class Stella assessment. This gives me a more honest picture of each student's abilities. I always encourage my students to seek extra help from me, their parents, and peers. With this in mind, my hope is that by the time they take the in-class assessment, they have attained the adequate skills and critical thinking that is required to successfully attack a Stella problem on their own. I weight the in-class assessment the same weight as the sum of the take home Stellas. Since there are four Stellas provided to you for each proficiency, you might want to keep one for an in-class assessment, or make up your own assessment that reflects the skills that you believe are most important for your class.
I hope that you enjoy working with Stella as much as I have over the years. Play with it, tweak it, and utilize it in any way that works for you. I am always curious as to how other teachers incorporate writing in their classrooms. Drop me a line and let me know how you're doing. I'm forever interested in your feedback and suggestions so that I can help Stella evolve into a product that best suits your needs. Thank you for your interest.
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