Albert, Lillie R.: “OutsideIn – InsideOut: SeventhGrade Students’ Mathematical Thought Processes” in Educational Studies in Mathematics 41: 10914, 2000.
In this paper, Dr. Lillie Albert presents findings from research completed as part of her doctoral dissertation at the University of Illinois at UrbanaChampaign (1995). Albert currently teaches in the Department of Teacher Education at Boston College. The paper builds on Vygotsky’s theory of scaffolding, which says that student learning takes place in the “Zone of Proximal Development (ZPD)” – a developmental stage in which students advance to new levels of competence only with the help of more experienced members of society (teachers, parents, or more advanced peers). In the ZPD, students engage in learning activities they could not complete on their own; however, with the help of their teachers, students eventually gain the ability to complete the tasks independently. Albert’s study analyzes the various activities that occur in the ZPD when middlegrade students engage in mathematical problem solving. Specifically, the research looks at seventh grade mathematics students to address the relationship between their oral and written thought processes when solving complex mathematical problems. This interpretive case study – which is a subset of a larger dissertation study  focuses on the extent to which writing in mathematics helps scaffold students to a more comprehensive level of mathematical understanding.
To analyze connection between oral thought and written thought, and to place it into the larger context of mathematical problem solving, Albert studies seven seventhgrade math students from the MidWest. The population represents a mix of Caucasian, AfricanAmerican, Hispanic, Asian or Pacific Islander, and Native American students. Of the sixty students that took part in the larger study for her dissertation, Albert focused on these seven for this interpretive case study for the following three reasons: their membership to the same class in the school, their perfect attendance over the course of the study, and their active class participation in problem solving. To collect data on these students, Albert relied on both interviews and writing samples. The interviews consisted of a series of questions that helped Albert scaffold students through the problem solving process. Not only did the interviews help students complete the problems, but they also allowed them to express their thinking orally. Students’ written work consisted of notes, diagrams, and other ideas about the strategies they employed. Albert subsequently coded these assessments by grouping their results into the following categories: approaches and strategies to problem solving, relationship, communication, examination of solution, mathematical learning. The researcher then analyzed the contents of these categories to evaluate the students’ conceptual understanding of mathematical problem solving.
The findings of this research indicate that using writing to solve mathematical problems greatly helped students conceptualize their understanding of questions, because it provided a framework for the organization of thought. Albert notes that “writing strengthened students’ individual problemsolving performance because it provided a context that allowed them to engage in ‘selftalk.’” Stemming from this notion of selftalk, Albert creates the Zone of Proximal Practice (ZPP)  a new learning zone, similar to the ZPD. This new zone allows students to think independently about the mathematics at hand, and come to their own conclusions. Just as teachers scaffold students’ learning through social interaction in the ZPD, writing similarly helps students scaffold themselves in this ZPP. Albert calls the learning that occurs in the ZPD “learning OutsideIn,” whereas she calls the learning that occurs in the ZPP “learning InsideOut.” Writing allowed students to solve problems through the “selfregulation” of internalized thought. Albert argues that interpersonal talk (oral thought) dominates the ZPD. This interpersonal talk becomes intrapersonal in the ZPP, as a direct result of student writing. Writing allows students to internalize the scaffolding process and become independent in their thinking; consequently, teachers must learn to utilize both the ZPD and the ZPP if the want their students to become proficient in mathematical problemsolving.
As stated previously, the researcher divided the data into categories that represented the way in which students used writing. Albert outlines the significance of the results by describing data that reflects three emerging themes that developed out of the coding of information. First, she discusses the processes that govern student thinking while they solve problems. This section yielded that writing was useful in identifying the question, deciding on a problem solving strategy, and implementing this plan. Second, Albert discusses the students’ attitudes about writing, noting that most students found the writing process helpful when they could not simply do the problem in their head. Interpretive analysis of the final category shows that writing prompted students to think about their answer, as opposed to simply accepting the answer as true because “the math was right.” With these three sections, Albert addresses the results of her coding and in doing so establishes a method for assessing students’ understanding of problem solving that all teachers can emulate.
The strength of this paper stems from the theory in which Albert grounds her research. Vygotsky’s scaffolding theory and the ZPD provide a strong theoretical framework that lends one to question the possibility of other learning zones. Albert shows the relevance of researching writing in mathematics by explaining how writing helps students internalize and organize their thought processes. One might think the small sample of 7 students hinders the reliability of the study; however, because this research was qualitative, not quantitative, the small sample is actually an asset. The seven students shared many common characteristics, which Albert studied in depth through interview. She basically views the seven students a one single case that speaks towards the importance of writing in mathematics problem solving. Had she used a larger sample – or looked at students individually  such accurate and indepth analysis would have been impossible to construct. Additionally, one of the largest strengths of this research is the amount of time Albert spent with the students. She follows them for fourteen weeks, the equivalent of a school semester. Consequently, her research does not describe a few unrelated observations connecting writing and math; rather, it documents the progress of these students’ mathematical thought processes over the course of four months. The length of this study complements the theoretical framework it proposes, because it shows the results of having students work repeatedly in the ZPP. An overarching theme that emerged from this study is the notion that students can use writing not only as a means of expressing their answers to problems, but also as a new way of learning to think about problem solving itself. In short, the strengths of this research culminate in the double purpose of writing – writing becomes an end, as well as a means to achieve that end.
While the strengths of this article outnumber the weaknesses in terms of relevance, one must nonetheless address the aspects of this paper that detract from its significance. First, the paper gives little background information about the students it studied, other than their ethnicity. Albert says she chose to use the seven students that were present for all fourteen weeks; however, she fails to acknowledge why these students had such perfect attendance. Perhaps the fact that these students were never absent speaks towards their academic character. It is important to consider if they were the type of students who would never miss class; this notion has far reaching implications, with respect to their need for assistance in mathematical reasoning. The description Albert gives of these seven students leads on to wonder how much help they really needed with respect to problem solving. Second, the research automatically assumes the need for a new learning zone that centers on writing. In doing so, it fails to acknowledge the fact that many students can solve, visualize, conceptualize, and truly understand the solutions to mathematical problems without writing. Third, and most important, the research makes a critical assumption about the ability of all mathematics students – that they can effectively express their ideas through their writing. Albert’s conclusions and the ZPP mean nothing to math teachers if their students do not have an ageappropriate understanding of the written language. While these weaknesses do not negate Albert’s findings, one must consider them before acting on this paper.
Albert’s research has many farreaching implications for teachers’ personal practice in their classrooms. Teachers should find ways to incorporate writing into their math assignments, in as many ways as possible. Not only does doing so give students the chance to work in the ZPP and understand the concepts more thoroughly, but doing so also provides teachers with a new means of assessing where students go wrong when they make mistakes. For example, when assigning homework problems, students could fold their paper in half to create a “writing column” and an “example column.” The example column would contain the answers just as one would complete normally; however, the writing column would leave space for a paragraph explaining the student’s logic behind why he or she performed certain operations. Another way to incorporate writing into the mathematics classroom is for teachers to consider a writing component on all assessments. Assuming students have the ability to clearly express themselves through writing, such a section would give teachers an extremely valid assessment. It allows teachers to know whether students guessed at an answer, or honestly understood the underlying concepts behind the answer at which they arrived. Similarly, including a writing section on every assessment emphasizes to students the fact that that process is just as important – if not more important – than the answer itself. As a third possibility, teachers can have students keep a “math journal” in which they work daily on certain problems that their teachers chose. In this journal, students can make a list of their difficulties and achievements, as well as brainstorm ideas for problems as they work in the ZPP. This notion of a journal provides teachers with an excellent resource to understand the intrapersonal talk taking place as students work independently. In short, allowing students to write just for the sake of writing has no useful purpose. Teachers must analyze the mathematical writing that students complete in order for that writing to influence student learning; consequently, positive feedback from teachers holds the key to success when it comes to writing in math.
For too long, writing and high school math classes have been seen as unimportant, except maybe during wordproblem activities. Albert’s research shows that writing does hold an important place in math classrooms, specifically because of the effect it has on organization and internal thought. She thoroughly articulates this conclusion by substantiating it with evidence from her research. Mathematics departments everywhere should work with their students to help them write at a level that compliments their mathematical learning.
