Using a PISA Task to Focus on Algebraic Thinking:

A Case Study

Description

We argue that PISA mathematics tasks could be used as resources for enhancing mathematics teaching and learning.  Next we report preliminary results from our work in which we closely examined students' algebraic thinking in solving a PISA task and engaged teachers in a discussion of our analysis.

Purpose and Rationale

The mathematics assessment within the Program for International Student Assessment (PISA) has received little attention in the U.S., beyond noting that performance of U.S. students is mediocre compared to students in Asia and many European countries.  In a project currently supported by the National Science Foundation, we view the mathematics tasks used in PISA as resources for the improvement of mathematics teaching and learning in the U.S. by provoking teachers of mathematics in the middle grades and high school to look at the mathematics they teach in new ways.

Just as large-scale assessments such as NAEP have been used to create resources to improve mathematics teaching and learning, we hypothesize that PISA could be an equally useful source. For example, numerous interpretive reports of findings, secondary analyses of data, and analyses of student responses from the NAEP mathematics assessments have been written for and made accessible to teachers and teacher educators via books (e.g., Kloosterman & Lester, 2004, 2007; Silver & Kenney, 2000) and articles in teacher-oriented journals (e.g., Blume, et al., 1998; Kenney & Silver, 1997; Kenney, Zawojewski & Silver, 1998; Stylianou et al., 2000).

One facet of our work with PISA focuses on algebra, a curricular topic of central concern to teachers of mathematics. There is a substantial body of research on student and teacher learning related to algebra that focuses two core ideas: the notion of algebra as an abstract structure representing generalized arithmetic, and the concept of function as a unifying idea that transcends and connects ideas within algebra (Kaput, 2000; Wu, 2001). Regarding the former, students and teachers alike struggle with connections between arithmetic and algebra, the meaning and concept of variable, and the links between symbolic, tabular, and graphical representations of algebraic concepts (e.g., Knuth, 2000; Pitts, 2003; Schoenfeld & Arcavi, 1988). With respect to the concept of function, research has focused on students' and teachers' understandings of this unifying theme and how the teaching of function has been dominated by symbolic representations of linear and quadratic families, often leading to a limited conception of function and its role in mathematics (e.g., Knuth, 2000; Leinhardt, Zaslavsky, & Stein, 1990).

Several of the 50 publicly released PISA mathematics tasks relate to knowledge and skills associated with school algebra.   We present results from a detailed analysis of a PISA algebra-related task and student work obtained from a convenience sample of middle and high school students.  We also discuss teacher reactions to our findings and how their comments can not only contribute to our work on professional development materials but also reveal what teachers might learn from encounters with such tasks and analyses.

Methods

From the 50 PISA mathematics tasks, we selected M136 (see Figure 1), hereafter called the "Apples" task, as one that appeared to be sufficiently interesting to engage middle and high school teachers with core issues of algebraic thinking. The "Apples" subtasks encompass a wide range of algebraic thinking, as the solver analyzes, generalizes, and compares two different patterns, one linear and one quadratic.  At the request of the professional developers with whom we collaborated, we revised the second of the three subtasks to make it similar to those found in middle school mathematics curricula (see Figure 2).

---Insert Figures 1 and 2 about here---

Because the original PISA test booklets were not available as a source for U.S. students' work, we solicited assistance from mathematics teacher participants in a professional development project aimed at improving the teaching and learning of algebra.  These high school and middle school teachers collected over 900 student responses to the modified version of the "Apples" task (see Figure 2). The students were in grades 5 through 12 and were enrolled in a variety of mathematics courses (e.g., algebra I, geometry, algebra II, pre-calculus, etc.). Our goal was not to produce findings from a nationally representative sample of student responses – PISA already provides that – but rather to analyze a sample of responses to identify facets of students' algebraic thinking that could provoke a fruitful conversation among teachers.

We developed a analytic coding rubric based on a merger between the original PISA scoring guide and a classification scheme adapted from Lannin (2005).   We looked specifically at students' use of recursive or explicit approaches to characterizing the generalization and the extent to which their generalization was captured using equations or verbal descriptions.

Results

Among the findings from our analysis of the responses were these that specifically related to students' algebraic thinking:

á       Students in upper grades (e.g., high school) and students in more advanced mathematics classes (algebra 2; precalculus) tended to use mathematical symbolism and equations while their counterparts in middle school and in lower level mathematics classes relied on verbal descriptions.

á       Students at all grade levels used both explicit and recursive strategies to solve subtasks 3.2.1 and 3.2.2, with more using recursion for 3.2.2. Students using recursion used only verbal description to express their generalizations.

When we presented our analytic findings and graphs such as those in Figure 3 to teachers in a professional development session, they were very interested in and surprised by the findings related to recursive reasoning and verbal descriptions, both of which were viewed by the secondary school teachers as "less sophisticated" forms of algebraic thinking.  In particular, they were surprised that so many students in algebra 1 and subsequent courses used recursive reasoning (see Figure 3).

---Insert Figure 3 about here---

Our research effort is ongoing to understand how teachers' ideas about students' algebraic thinking are affected by their experience with the "Apples" task and the corresponding analysis of student work. The professional development session mentioned above was video-recorded, and we are beginning an analysis of the session to identify indicators of teacher attention and engagement.   Based on our experience with the teachers, we are designing a workshop that can be used with other teachers to examine the "Apples" task with particular attention to students' use of recursive reasoning. 

References

Blume, G. W., Zawojewski, J. S., Silver, E. A., & Kenney, P. A.  (1998).  Focusing on worthwhile mathematical tasks in professional development: Using a task from the National Assessment of Educational Progress.  Mathematics Teacher, 91, 156-170.

Kaput, J. J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by "algebrafying" the K-12 curriculum. Washington, DC: Office of Educational Research and Improvement.

Kenney, P. A., & Silver, E. A. (1997).  Probing the foundations of algebra:  Grade-4 pattern items in NAEP.  Teaching Children Mathematics, 3, 268-274.

Kenney, P. A., Zawojewski, J. S., & Silver, E. A.  (1998). Marcy's dot pattern.  Mathematics Teaching in the Middle School, 3, 474-477.

Kloosterman, P., & Lester, F. K. (Eds.). (2004). Results and interpretations of the 1990-2000 Mathematics Assessments of the National Assessment of Educational Progress.  Reston, VA:  National Council of Teacher of Mathematics.

Kloosterman, P., & Lester, F. K. (Eds.). (2007). Results and interpretations of the 2003 Mathematics Assessment of the National Assessment of Educational Progress.  Reston, VA:  National Council of Teacher of Mathematics.

Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31, 500–508.

Leinhardt, G., Zaslavsky, O., & Stein M.K., (1990) Functions, graphs and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64.

Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231–258.

OECD (2006). PISA released items: Mathematics. Retrieved January 10, 2010 from http://www.oecd.org/dataoecd/14/10/38709418.pdf.

Pitts, V. R. (2003). Representations of functions: An examination of pre-service mathematics teachers' knowledge of translations between algebraic and graphical representations. Unpublished doctoral dissertation, University of Pittsburgh.

Schoenfeld, A. H., & Arcavi, A. (1988). On the meaning of variable. Mathematics Teacher, 81(6), 420-427.

Silver, E. A., & Kenney, P. A. (Eds.)  (2000). Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress.  Reston, VA: National Council of Teacher of Mathematics.

Stylianou, D.A., Kenney, P.A., Silver, E.A., & Alacaci, C. (2000).  Gaining insight into students' thinking through assessment tasks.  Mathematics Teaching in the Middle School, 6, 136-144.

Wu, H. (2001). How to prepare students for algebra. American Educator, 25(3), 1-15.

Figure 1. Apples task

(source: OECD (2006). PISA released items: Mathematics)

Figure 2. Modified apples task

Figure 3. Results for subtasks 3.2.1 (top) and 3.2.2 (buttom)