## Heather Bahlmann and Professor Janet G. Walter, Mathematics Education Department

### Purpose and Framework

Inferring student understanding is at the heart of improvement in mathematics learning and teaching. Assessment provides valuable information which can be used to “promote growth, modify programs, recognize student accomplishments, and improve instruction” (NCTM, 1995, p.27). It is imperative that the methods for assessing student understanding be constantly explored and evaluated. Ou decision to research the open-response task as an assessment tool results from our perspective that students make sense of mathematics by exercising personal agency. Personal agency is the freedom and responsibility to choose to act (Walter & Gerson, 2006). Open-response tasks allow students to exercise personal agency, hence eliciting not only what students know, but also how students explore concepts. Open-response tasks require “students to explain their thinking and thus allow teachers to gain insights into…the ‘holes’ in their understanding” (MOON &Schulman, 1995, p.30). How can “holes” in student understanding be inferred from analyses of student strategies in solving open-response tasks? Does a student’s strategy choice indicate a lack of proficiency or conceptual understanding that may be evidenced in alternative strategies?

### Method

Pre- and post-tests consisting of 12 multiple-choice and 12 short-answer questions were administered to 774 elementary students. We used a mixed method of qualitative and quantitative research procedures to infer student understanding of probability and fractions and to analyze students’ strategies on open-response tasks. A detailed analysis was conducted on 2 openresponse and a combination of 13 conceptually related multiple choice and additional shortanswer questions on 172 post-tests given to fifth graders at two elementary schools. Each openresponse task was graded individually by two research team members. Discrepancies were discussed and resolved before a final score was given. Qualitative analysis was performed by assigning strategy codes to all student solutions on open-response tasks. Related questions answered incorrectly by each student were quantified. We noted relationships between incorrectly answered related questions and alternative student strategies.

### Data and Analysis

The 172 fifth graders’ exam scores improved significantly (p<.01) from their pretest to their post-test performance. We discuss one open-response question here which asks students to identify the larger of two fractions (2/3 or 3/2), and use words or pictures to explain their answer. Sixty-four students received full credit for this problem. Three strategies received full points. Strategy 1 compares 2/3 and 3/2 to one whole by using a picture or explanation (Figure 1), strategy 2 compares 2/3 and 3/2 to one-whole and represents 3/2 as a mixed number (Figure 2), and strategy 3 finds a common denominator (Figures 3 and 4).

Thirty-five of the 64 students employed an intuitive strategy (strategy 1). Of these 35 students, 57% missed two or more related questions involving finding a common denominator and 29% of the students missed multiple-choice question number 5 dealing with representing a fraction as a mixed number (see Table 1). Twenty-two students (35%) used strategy 2. These students performed better on question number 5 than those who used strategy 1.

Seven students (11%) used a procedural strategy (strategy 3) to compare 2/3 and 3/2. However, only two of these students provided evidence that finding a common denominator is accomplished by re-unitizing each fraction so that each unit is divided into the same size of pieces (see Figure 4). Four of the 7 students who found a common denominator without interpreting its meaning missed 2 or more questions dealing with finding a common denominator. These four students might lack the intuition of knowing when to use this strategy. For example, analysis of Joshua’s exam suggests that when comparing fractions he found a common denominator, but when using a common denominator would be most useful for solving certain addition or subtraction problems he added or subtracted across the numberators and denominators. Five of the seven students missed an intuitive problem. A key finding of this study is that the two of the seven students who did not miss the intuitive problem and who performed well on problems involving finding a common denominator were those two students who demonstrated their intuitive and procedural understanding as shown in Figure 4.

### Conclusions

A careful review of student strategies on open-response tasks warrants a new interpretation of the open-response task as an assessment tool. These fifth-grade students who worked procedurally on open-response problems to find common denominators did not provide evidence of intuitive understanding through drawings or explanations. Students who did not employ the strategy of finding a common denominator might not be proficient with that procedure. Students may be able to solve open-response problems, such as those involving finding common denominators, but when conceptually similar problems are placed in another context, students may be unsuccessful in demonstrating their content knowledge or mathematical skills. Implications for teaching include assessing whether or not students can flexibly use their understanding of fractions in multiple contexts. Open-response tasks provide opportunities for teachers to consider how students do not solve problems and encourage their students to develop understanding of those concepts addressed by alternative strategies. Through the use of openresponse tasks and analysis of alternative strategies, teachers can gain greater insight into student thinking and provide opportunities for their students to learn as they exercise personal agency.

### References

- Moon J., &Shulman, L. (1995). Finding the connections: Linking assessment, instruction and curriculum in elementary mathematics. Portsmouth, NH: Heinemann.
- National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: NCTM.
- Walter, J. &Gerson, H. (In press). Teachers’ personal agency: Making sense of slope through additive structures. Educational Studies in Mathematics.