Overview

We conducted user, domain, and product research in four different areas. We interviewed teachers to better understand their goals and intents in teaching Algebra and using calculators as instructional technology. We interviewed students to better understand their lifestyles and their goals and motivations for learning Algebra. We reviewed handheld technology products to generate design ideas that benefit from the work companies have already done in this area. Finally, we observed Algebra 1 classrooms to learn how teachers teach, reviewed education research literature to discover teaching best practices, and reviewed Algebra textbooks and curriculum to develop a comprehensive list of topics taught in Algebra 1.

Algebra 1 Education Research

To better understand teaching best practices and issues in Algebra 1 education, we reviewed 21 papers on Algebra and Pre-Algebra education found off of educational researchers' web pages and academic databases like ERIC. Word problems and symbol manipulation were the most mentioned topics in our literature search. Teachers and students alike find them the most difficult part of Algebra I. Though there is no clear consensus, several possible reasons include: the terminology and wording used in textbooks is arcane to the typical Algebra I student, students are unable to form a good mental model of the problem, students have adequate symbol manipulation skills, or that students see the problems as boring, contrived and irrelevant. Symbol manipulation is closely tied to word problems (many textbooks teach students to form a mental model of the problem, translate it to symbols, then solve it symbolically), and has its own set of difficulties. Often there is little mapping to real world concepts, mathematical notation is introduced too early, students are unable to apply the rules of combining monomials, and students fail to understand the generality of variables and become fixated on using x and y.

Several articles described students' problem solving skills. Students often employ guess and check tactics to answer questions, rely on plugging in numbers instead of manipulating algebraic expressions, and apply procedures indiscriminately when they don't understand the math.

The literature also discusses more specific problems. For example, the equals sign is often seen as meaning "compute the answer" (instead of defining an equality relationship) and students have problems with addition that requires aligning decimal places.

Our literature search gave us a general understanding of typical Algebra 1 student problems. This influenced the foci of our initial research with teachers, and directly lead us to investigate designs for symbol manipulation and word problem tools.

See our bibliography for the titles and authors of the papers we reviewed.

User Research – Algebra Teachers

We interviewed 31 Algebra 1 teachers from Pittsburgh, New York, Baltimore, D.C., and Toronto. Our initial interviews had two foci: 1) what techniques do teachers employ to teach the concepts of Algebra 1, and 2) what kind of environments do teachers work in and how does this affect their teaching. The interviews were similar in style to contextual inquiries; we began with a list of questions, then narrowed or expanded focus accordingly. We primarily searched for new behaviors, though in many cases we were able to validate previously observed behaviors by confirming them with multiple teachers.

Many of the teachers we spoke with were under significant time pressure to get through a pre-defined set of lesson plans to prepare students for standardized tests. Some were frustrated with the educational system and felt they didn't have much freedom to teach in the style they felt was best for their students. Some were also frustrated with their students, who they perceived as being uninterested in learning math. As a result, most teachers fell back on teaching scripted procedures for transforming a problem into an answer and drilled these procedures into students through in-class and homework exercises.

Based on these findings, we decided that our calculator must support procedural learning (despite the misgivings of academics) by making it easier for students to focus on the important Algebra 1 concepts while working through exercises. As a result, many of our designs attempt to automate the mechanical portions of problems while requiring students to understand the underlying concepts in order to use the calculator effectively.

Unfortunately, we were unable to speak with as many teachers who were unfamiliar with graphing calculators as we would have liked. Thus, for our second round of studies we shifted focus to teachers who already use graphing calculators for teaching Algebra 1, with the caveat that these were not the types of people we were directly designing for. We aimed to get ideas we could adapt to our target market by discovering 1) how teachers currently use graphing calculators to teach Algebra 1 topics and 2) what portions of the Algebra 1 curriculum were particularly difficult to teach to students.

This research led to many of the specific usage patterns that appear in the motivation sections for several of our designs. We took some of the things sophisticated teachers currently use graphing calculators for (e.g., plotting statistics data and finding a best-fit line) and simplified the interface to make it more accessible to less tech-savvy teachers.

To get a better sense for how teachers and students really interact, we observed 9 Algebra 1 classrooms. During these observations we noted how students used calculators in the context of a class in session, how teachers employed the calculator to teach a concept, and how teams of students collaborated on assignments. We found that students frequently viewed each other's calculators to compare answers and procedures even though each student had his or her own model. We also saw that the most common use of calculators was to check answers, an exercise we decided to explicitly support in the Formula Library.