The Colorado Mathematics Teacher – What are Students Really Thinking? Portfolios as Assessment

What is the purpose of assessment? According to Colorado Model Content Standards: _Students use a variety of tools and techniques to measure, apply the results in problem-solving situations and communicate the reasoning used in solving these problems._ However, the communication aspect is not often a focus in assessment.

The NCTM Assessment Standards for School Mathematics (1995) describe the critical aspects of assessment:

Standard 1: Assessment should enhance mathematics learning.

Standard 2: Assessment should promote equity.

Standard 3: Assessment should promote openness.

Standard 4: Assessment should promote valid inferences about mathematics learning.

Standard 5: Assessment should be a coherent process.

Assessment is a way of determining what students understand. I have used tests, quizzes, various methods of collecting homework, and I have found that the best method for peering into the students_ minds is through the use of portfolios. Traditionally, methods of assessment lack a focus on the communication aspect of learning in math.

I have students do portfolios where they are required to write reflective pieces defending their learning. The portfolio is broken down into chapters to align with what is happening in class. Portfolios such as these are an effective form of assessment because they require students to communicate what they understand. In additional to allowing student sot demonstrate their understanding of the math content and problem solving skills, portfolios help students develop the skills for communicating mathematical ideas.

In this sense, portfolios achieve what I understand to be the goal of assessment: providing insight as to what students understand, and what they do not understand.In a recent portfolio, a student wrote that the Pythagorean Theorem works for all triangles.If the student was just applying the Pythagorean Theorem to a series of problems, I might have been unaware of this misunderstanding.By asking the student to write about his understanding, I can develop an awareness of the gaps in understanding and then address those gaps.

Using portfolios as a means of assessment is not a new idea. However, it is my experience that portfolios are usually implemented in ways that are not necessarily valuable or insightful. I have seen many institutions that have students take the jumbled mess of their work from their backpack, hole punch it, throw it in a three-ring binder, and call it a portfolio. Looking through such portfolios has been a painful process for me, and I am often left with a vague idea of what a student understands. In my experience, the portfolio process has been most effective when it asks students to reflect on their own learning.

Following is the portfolio system that has worked at my school (please share any of your ideas with me).This system offers a clear view of what students understand and helps students form a coherent picture of the concepts they are learning.The portfolio consists of five sections: Table of Contents, Skill Chapters, Problem Solving Chapters, Assessments, and the Self-Evaluation.All of these sections except the Table of Contents require thoughtful writing.The Skill Chapters and the Problem Solving Chapters are the heart of the portfolio.The Table of Contents (I have them insert dividers) makes the portfolio user-friendly for the instructor and gives the portfolio a sense of professionalism that helps students take more pride in their work.

In the Skill Chapters students describe their learning of math content.After learning a specific math concept, such as the use of the basic trigonometric functions, students compile a Skill Chapter to document their learning of that topic.The Skill Chapter consists of two parts: the cover letter and the evidence.Students choose the evidence (usually homework and class work) that best shows their learning in the chapter.Then they write a cover letter explaining what they have learned, how they use what they have learned, and how the evidence demonstrates their learning.In the example of the trigonometric functions, students would include definitions of sine, cosine, and tangent, explain how to use those functions to solve problems, and discuss how their evidence demonstrates their learning of those concepts.The cover contains student_s reflection and communication.

In Problem Solving Chapters, students take on complex, open-ended problems and formally document their progress on those problems.These are often Problems of the Week.The write-up, which is somewhat standard, includes four sections: Problem Statement, Process, Solution, and Reflection.The key learning occurs when students are asked to translate their mental processes into writing, thereby making those processes more concrete. In addition, students are sharpening their communication and logic skills when defending these elaborate solutions.I have observed the students are more motivated to do the _writing_ in a math class when it is an essential part of the course as opposed to a seemingly superfluous activity.

In the Assessments section students apply those skills addressed in the portfolio and help me to see students_ ability to take the skills they have learned and apply them to new situations. This section distinguishes students_ ability to transfer the skill to different situations from merely being able to _parrot_ back what happened in class.

The Self-Evaluations section is designed for students to reflect on their performance as a learner _ the metacognition on the class.What did they do well in this class?What could they do better in future math classes?Students often use this as a forum to inform me of what worked well in the class and to suggest changes that may better facilitate effective learning.

This portfolio system gives me a clear understanding of what students have learned. The portfolios are assessed on the level of communication and understanding presented as well as given a level for the portfolio and narrative feedback. Students also have an opportunity for revision, allowing them to fill in their own gaps in learning.

I have encountered some hurdles in implementing portfolios. One hurdle for students is that portfolios are different from what they have seen before and often students may be inclined to avoid them. One way to combat this is to provide students with models of exemplary portfolios. It is also useful to set up a support structure of deadlines so students are compiling the portfolio in manageable segments.

A hurdle for the instructor can be time. The initial implementation of portfolios is the most time consuming because the instructor has to learn how to most efficiently review all the portfolios. However, after assessing a few portfolios, I find the review goes surprisingly quickly. Also, when students are compiling portfolios, I do not give many tests or quizzes and only check homework assignments for completion. This frees up my time for portfolio evaluation. However, portfolios may require greater time and thought commitment than other alternative forms of assessment.

Summary: Portfolios are not just a compilation of work and applied problems.My students often say they did not really understand what was happening with the math until they compiled their portfolios.Portfolios are an opportunity for students to go beyond problems and reach the meta-level where they are explaining and reflecting on the process they use to deal with mathematical situations.

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