and portfolios: alternative assessment for preservice teachers
Teaching Children Mathematics; Reston; Nov 1999; Cathy S Liebars;
Copyright National Council of Teachers of Mathematics Nov 1999
Traditionally, assessment in mathematics courses has consisted of tests, quizzes. and textbook exercises. Thus, preservice elementary school teachers enter their mathematics-methods courses with expectations for similar assessment. The current reform movement in mathematics education recommends that student assessment be integral to instruction and that multiple means of assessment should be used (NCTM1989). The Professional Standards for Teaching Mathematics (NCTM 1991) highlights the need for teachers to reflect on their practices and to use alternative-assessnient methods. Mathematics-teacher educators must intern ene and model these practices in (lie methods and content courses for elementary education majors. In the words of the Curricu and Evaltiation Standards (NCTM 1989, 190):
[m]erely adding scores on written tests will not give a full picture of what students know. The challenge for teachers is to try different ways of grading, scoring, and reporting to determine the best ways to describe students' knowledge of mathematics as indicated in these Standards.
It is not enough to preach about alternative assessment. If preservice teachers are expected to adopt multiple assessment methods, then they must experience them. By using multiple methods of assessment, such as portfolios, writing, presentations, and projects, the teacher educator not only models behavior for the preservice teachers but also assesses their learning and understanding.
In this article I share what I have learned about using alternative forms of assessment with preservice teachers. Two assessment methods that I have incorporated in my elementary-mathematics-methods course are journals and portfolios.
Students take my course during the junior professional semester, in which they spend either one full day or two half-days each week and two full weeks at the end of the semester practice teaching in an elementary classroom. Students have a wonderful opportunity to try the ideas that they learn in class and to reflect on themselves as classroom teachers. A useful outlet for this reflection is a journal.
I use various formats for the journals, including open-ended reflections and responses to prompts. Open-ended reflections consist of weekly essays in which the students write about what they have learned in class and about their reactions to the activities that they encounter during the week. Some students simply summarize what we do in class, whereas others ask questions about topics covered in class or expressed concerns about the class, teaching, or mathematics in general. I also ask them to include reflections about their observations and mathematical experiences in the elementary classroom in which they are placed for the semester.
When I give the students prompts to which they should respond, I choose topics that could be used with, or adapted for, elementary students. Some prompts are related to course content, for example, explain to someone how to model 23 x 5 with baseten blocks. Others explore student attitude, for example, what mathematics means to me. No matter what the format, the journals are well received by the students and are an important part of the course.
During my course, each student keeps a working portfolio, which consists of homework assignments, journal entries, in-class writing assignments, professional articles, and other relevant student-selected materials. I review the portfolio briefly four times during the semester. Examples of homework assignments include a paper about how mathematics is used in different professions, reaction papers to a journal article, and homemade attribute materials.
At the end of the course, the students develop a final portfolio to showcase their learning and accomplishments throughout the semester. They select items on the basis of a given list of criteria, such as that the item demonstrates positive mathematical disposition, problem-solving skills, and knowledge of using tools and technology. Students also choose to include assignments, journal entries, and professional articles, as well as lesson plans and assessment instruments that they have created. For each item selected, students write a rationale for that particular selection. They conclude with an essay in which they summarize their experiences and describe significant insights that they have gained in the course.
I use a holistic rubric to grade each assignment in the working portfolio (see fig. 1) and an analytic rubric to evaluate the final portfolio (see fig. 2). The Assessment Standards for School Mathematics (NCTM 1995) supports this process of assessment. By predetermining the criteria for evaluation, I make the task easier and the analysis more complete. Because students are given copies of these rubrics at the beginning of the semester, they know what is expected of them and how they will be evaluated.
Lax (1989) addresses concerns about the time needed to grade these types of assessment: "The time spent in helping students explain clearly what they mean and [in examining] the stated ideas will lead to major savings in instruction time later on" (p. 253). I believe that devoting time to evaluating portfolios is worthwhile. To manage the time, I review the working portfolios while students take quizzes.
Benefits of writing
My students are required to do a significant amount of writing in their journals. "Mathematics as communication" is one of the four themes stressed at every grade level in NCTM's Curriculum and Evaluation Standards for School Mathematics (1989). Writing is one way in which teachers can help students meet this standard. Oral communication does not require the elaboration that is necessary for written work (Dougherty 1996).
Students' writings provide me with feedback about the course and students' understanding. Reading what the students write about the class gives me some insight into which topics the students most enjoy and which ones are more difficult to understand.
I found tree diagrams to be more confusing than helpful. It just got me more confused to find all the different paths to take. I know it is a method to use instead of writing down every possibility. but I think if I was confused, I could only imagine what the students might think.
This information enables me to adjust future lessons. Students' reflections indicate which concepts require more time to comprehend.
I also find that by having students write about what they learned, I can discover and correct their misunderstandings.
The relationship is that a parallelogram is a type of square. A trapezoid is a type of parallelogram. Thus a square is a type of trapezoid.
This student's confusion about these relationships tells me that the lesson has not been meaningful to her. Traditional assessments, such as multiple-choice tests or having students write definitions, might not have revealed the depth of this student's misconception. Once I identify such misconceptions through students' writing, I can make corrections and alter future lesson plans accordingly.
Writing is a means of communication between the teacher and the student. The first topic that I give my students for their journals is "What math means to me," which gives me insight into their mathematical backgrounds and their feelings about the subject. They often reveal anxiety and negative feelings, as in the following example:
Math as a subject in school has always meant feelings of anxiety. It has always been the one subject that made me feel stupid and inadequate. Outside of school I use math successfully on a daily basis. I do not experience the same negative feelings about math when I am cooking or shopping or balancing my checkbook.
Such feelings would most likely not be expressed in a full class setting. Writing allows the students to express concerns and gives the teacher a chance to recognize, and possibly address, these concerns.
Writing forces students to explain concepts and thought processes. I give the students several writing topics and assignments for which they have to explain a concept in their own words or explain how they solved a problem. Not only am I able to assess their understanding of the concepts, but the students realize how this activity aided their learning process:
I was surprised by how much the journal writing has helped me to understand the math concepts. By forcing me to write down the math in words, it helped to slow down my thought processes and realize how I mentally solve the problems.
Another student wrote the following:
The experience of journal writing has taught me that to truly understand something, you must be able to explain it in words.
Writing fosters creativity and confidence. Many students enjoy the assignments and the writing topics in which they can use their creativity to write a story or a poem about mathematics. One such topic was adapted from a PBS video series, The Eddie Files:
My favorite writing assignment was "Welcome to Palindrome City, U.S.A." This was a great way to assess my knowledge of palindromes without giving a formal test. The story had to be original and creative while at the same time conveying the concept of palindromes. I enjoy creative writing, so this was a perfect way to get me interested in this mathematical idea. Students who are more linguistically inclined may prefer this type of assignment over a routine math test.
Writing down their thoughts about a mathematical concept or creating a story about a mathematical topic can give students confidence in their abilities:
Now that it is the end of the semester and I have had the opportunity to do several journals, my attitude has changed dramatically. My writings have not only helped me to better understand the materials, but they have also given me more confidence.
Writing forces students to reflect on their own learning and teaching. Through writing in the journals, students discover things about themselves as learners of mathematics that perhaps they may never have otherwise realized:
[W]hen I couldn't figure out a problem, working it outactually writing out, word for word, every step of the problem-forced me to slow down and look at and analyze every step. It was then easier to see the trouble spots.
A crucial component of teaching is reflection. Often students use the journals to reflect on lessons that they teach in their elementary-classroom placement or on a presentation that they did in the course:
At first, [my lesson] worked well. However, I noticed I had caused some confusion ... This experience taught me to be extra careful when considering my explanations of new concepts.
Another student wrote the following about her prsentation:
I realized how much harder it was to actually explain it when I was up there than it was in planning the entire thing. I'd like to go back and change certain things I've said because my clarity of explanations may not have been sufficient.
Keeping journals changes students' opinions about writing and mathematics. When I ask students how the experience of writing in this course affects their opinions of integrating writing and mathematics, the responses are overwhelmingly positive. One student wrote the following:
To be perfectly honest, before this class I didn't see a place for writing in a mathematics curriculum. I didn't really understand how writing could be used to help children learn math. However, now that I have experienced the use of writing in mathematics myself, I feel I have a decent understanding of how I could use it in my own classroom someday. It is now my opinion that writing is essential in helping students learn mathematics ....
Another student wrote this entry:
I will definitely use journal writing with my students, because for the past three months I have been playing the role of the student, and I have been witness to the benefits that this type of an activity possesses.
Benefits of Portfolios
The benefits of using portfolios are many. Several states and individual school systems use portfolios as part of the teacher-hiring process, a trend that seems to be growing. Therefore, practice given in the college classroom on how to develop such a selfassessment instrument enhances the students' chances in the hiring process should they need to use a portfolio (Campbell et al. 1997).
Portfolios present a way for students to self-assess their learning. Student self-assessment can improve students' confidence in their ability to do mathematics and allow them to become more independent in their learning of mathematics (NCTM 1995). In choosing items to place in the final portfolio, students must reexamine and assess their work in the course:
I made another observation about my work over the semester. I found that either I was doing really well or just really bad. I think I either did what I was supposed to and did it well or I just did the job halfway and didn't really put much effort into it. I realize this now, and I am trying to get out of that pattern.
Portfolios give students an opportunity to pull together what they have learned in the course and to document their growth. Students do not always simply include their best work but instead choose examples of work that shows their growth:
I think I have grown as a learner and a teacher. I think this is evident by the lesson plans I have included in this portfolio. The lesson plans in the beginning of the semester were not as creative or involved as the ones toward the end of the semester.
Portfolios give preservice teachers a chance to experience a process that they may want to use in their future classrooms. My students enjoy putting together their portfolios. They recognize the benefits of looking back at what they have learned:
The portfolio was another major feature of the college class experience. I like the idea of the portfolio because it provides a concise synopsis of my work in the class [to] which I can refer ... in the future.
I hope that they will choose to use this means of assessment when they teach mathematics.
Although challenges arise in using alternative forms of assessment in mathematics classes, the benefits achieved override the negative aspects. The Assessment Standards for School Mathematics (NCTM 1995, 15) states that "assessment should promote equity." Using a variety of methods to assess students' understanding gives them ample opportunity to demonstrate their knowledge.
Mathematics teacher educators should model these practices in the mathematics methods and content courses. If preservice teachers are exposed to alternative assessments as students, they are more likely to adopt them as teachers.