ANCHORED INSTRUCTION IN MATHEMATICS

Anchored instruction is a teaching approach that situates, or anchors, problems in authentic-like contexts that people can explore to find plausible solutions. Anchored instruction in education is closely related to problem-based and case-based learning in other fields (e.g., business, medicine), but it differs somewhat because all the information for solving anchored problems is available whereas it may not be in actual problem solving situations. Anchors are typically shown in a short video (8-to-12 minutes), which students search to find information they need for solving the embedded problems. In a typical classroom using anchored instruction, students work together to formulate strategies for solving the subproblems embedded in the anchor. The problems are of high interest, and most students work for several days to help the main characters in the video solve the problems.

Presenting anchored problems in video format has several advantages. One important quality of an anchored problem is its ability to directly immerse students in a rich array of problem contexts, which helps to eliminate the barriers many students with low achievement in both math and reading confront when attempting typical text-based problems. Second, the dynamic nature of video enables students to notice subtleties in the mix of auditory and visual cues, which are missing in text-based problems. Finally, multimedia scaffolds enable students to access help stations as they work on generating solutions they think are plausible.

Theoretical underpinnings of anchored instruction are derived from well-known theorists such as John Dewey (1933) who stressed the importance of viewing knowledge as tools. When people (students) acquire new knowledge that they understand can help them solve problems in particular contexts, they view knowledge more as a tool than asdisconnected facts and procedures. The role that context playsinhelping students recognize how and when to use these tools (i.e., transfer) is oneofthe key components of anchored instruction. The importance of context on cognition has been termed “situated cognition” (e.g., Brown, Collins, & Duguid, 1989) and “cognitive apprenticeship” (e.g., Collins, Brown, & Newman, 1989). Contextual factors in everyday problem solving have been shown to affect learning situation-specific practices and their transfer among people across cultures (e.g., Lave, Smith, & Butler, 1988).

A primary goal of anchored instruction is to engage students in problem-solving activities that can help reduce the “inert knowledge” problem that Alfred North White-head (1929) identified decades ago. Knowledge presented as isolated disconnected facts remains inert and thus fails to transfer. In contrast, when knowledge and skills are contextualized as they are in anchored instruction, students are more apt to recognize when to appropriately apply them and use their prior knowledge to solve similar problems they encounter in the future. Research in educational settings suggests expertise is developed through problem-solving activities that involve active construction of knowledge results (Bransford, Brown, & Cocking, 2000). Thus, anchored learning environments are generative because they motivate students to actively search for relevant information, use the information to plan strategies for solving the problem, and test their solutions.

## ANCHORED INSTRUCTION IN MATHEMATICS

A well-known series of anchors for improving students' problem-solving skills in mathematics is called *The Adventures of Jasper Woodbury* (CTGV, 1997). The entire set of Jasper Adventures consists of twelve episodes, each one including a video-based problem and related extension problems. Three episodes were developed for each of four mathematical areas of study: Distance/Rate/Time, Statistics, Geometry, and Algebra. The early versions of Jasper were presented on random access videodisc technology, which enabled students to keep track of the frame numbers and to access information almost instantly with the videodisc controller or barcode reader. This feature was far superior to access methods in linear-based videotape and VCR technology.

*Kim's Komet* is one of the Jasper Adventures that helps students develop their informal understanding of pre-alge-braic concepts, such as variable, linear function, rate of change (slope), line of best fit, and reliability and measurement error. The video-based anchor involves two girls who compete in a car competition called the Grand Pentathalon in which competitors have to predict where on a ramp they should release their cars to navigate five tricks attached to the end of ramp straightaway. *Kim's Komet* first helps show students how to calculate speeds when times and distances are known. They do this watching the time trials in the video held the day before the Grand Pentathalon. The challenge is to identify the three fastest qualifiers in three regional races, where times and distances are known but the distances vary. For example, students explain whether a car that travels 15 feet in 0.9 seconds is faster or slower than a car that travels 20 feet in 1.3 seconds.

The next challenge is more difficult: It asks students to use their own stopwatches to time Kim's car in time trials prior to the Grand Pentathalon. The software allows students to pick various release points (i.e., heights) on the ramp so they can compute the speeds of Kim's car along the length of straightaway. Eventually the students realize that they should time Kim's car from the beginning to the end of the straightaway, where the car's speed is relatively constant, rather than on the ramp, where the car is accelerating. After computing these speeds, students plot on their graph the speed of Kim's car for each of the release points and then draw a line of best fit. Students use this line to predict speeds for all possible release points.

On the day of the Grand Pentathalon, the teachers reveal to students the critical speed range of Kim's car for each of five tricks, which are attached to the end of the straightaway. Students earn points for helping Kim successfully accomplishing each trick. The software enables students to enter the height of the release point for each event and to watch Kim as she releases her car from that height. If students provide Kim with the correct release point, they can watch as Kim's car successfully navigates the trick. However, when speeds and release points have been incorrectly computed, they watch as Kim's flies off the trick and crashes. Foundation skills needed to solve this problem include computation with whole numbers and decimals.

## ENHANCED ANCHORED INSTRUCTION

An instructional method based on the concept of anchored instruction is called Enhanced Anchored Instruction (EAI; Bottge, 2001). EAI has two main components. Like the Jasper series, problems are presented in video format that students navigate to find solutions to an overarching problem. Multimedia-based learning opportunities built into newer technology enable students to access technology-mediated scaffolds, which are of particular benefit to many low-achieving students. In addition, EAI extends AI by affording students additional opportunities to practice their skills as they solve new but analogous problems in applied, motivating, and challenging hands-on contexts.

One example of EAI is a multimedia-based problem called *Fraction of the Cost* and its hands-on related problem called the *Hovercraft Challenge. Fraction of the Cost* stars three middle school students who wonder if they can afford materials to build a skateboard ramp. The main instructional purpose of the problem is to help students improve their math skills in the areas of rational numbers and measurement. After students solve the problems posed in *Fraction of the Cost*, they work on solving a related problem, the *Hovercraft Challenge*, in which students have to plan and construct a rollover cage for a hovercraft out of PVC pipe. The teacher divides the class into groups of three students, and each group plans how they can make the cage in the most economical way. When students have constructed their cages, they lift them onto a 4-by-4-foot plywood platform (i.e., hovercraft). A leaf blower inserted into a hole in the plywood powers the hovercraft, which inflates the plastic attached to its underside and elevates it slightly above the floor. The last day of the project students ride on their hovercrafts in relay races up and down the halls of the school.

## RESEARCH FINDINGS OF AI AND EAI

Results of formative evaluations with Jasper materials were reported in CTGV (1997). In these studies, teachers were provided extensive training on the content and use of the anchored materials. Findings suggested students in the anchored instruction groups developed more sophisticated math skills and positive attitudes compared to students in comparison groups. In a later study conducted in fifth grade classrooms (Hickey, Moore, & Pellegrino, 2001), two pairs of closely matched schools were randomly assigned to assess the effects of several Jasper series adventures on students' math achievement and motivational responses. Results showed that the anchored instruction materials had positive effects on math achievement and motivation of students in both high-SES and low-SES schools. This latter finding was especially important because it suggests that low-achieving students can profit from complex problem-solving activities without negative or motivational consequences if they are designed appropriately.

A series of quasi-experimental studies comparing EAI to traditional modes of instruction with students at several achievement levels (i.e., low, average, high) have confirmed and expanded the earlier findings with AI. These studies have yielded medium-to-large effect sizes (Z^{2}) on curriculum-aligned problem solving tests (.31 to .79) and transfer tasks among students without disabilities (.14 to .38) (e.g., Bottge, 1999; Bottge, Heinrichs, Mehta, & Hung, 2002). Similar results have been found in studies involving students with learning disabilities (Bottge et al., 2007) and emotional/behavioral disabilities (Bottge et al., 2006). An important finding generated from these studies suggests that teachers need substantial amounts of training to teach EAI effectively. This training should include both pedagogical methods for teaching EAI and deep understanding of the math principles embedded in the EAI problems.

Although studies suggest that AI and EAI can have positive effects on students' problem-solving skills, they have also identified some points for consideration. First, teachers need considerable training in methods and math content to teach with anchored instruction. Second, low-achieving students require explicit just-in-time instruction to help them perform difficult mathematics computation procedures (e.g., adding fractions) and understand mathematical concepts (e.g., variables). Third, EAI uses instructional technology (i.e., computer and hands-on applications), which some schools may not have or cannot afford to buy. Finally, anchored instruction may not fit neatly within traditional curricula.

Anchored instruction has the potential for helping students across a range of abilities improve their problem-solving skills. Results of several studies suggest that the contextualized nature of the anchors interest students and engage them for extended periods in problem solving activities. With low-achieving students, the related hands-on problems and learning scaffolds are especially important for solidifying the concepts embedded in the video-based anchors. One of the most important implications of these findings is that success or failure on academic tasks should not be predicted on students' prior achievement but rather on the design quality of the instructional material.

See also:Constructivism

### BIBLIOGRAPHY

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Bottge, B. A., Heinrichs, M., Chan, S., & Serlin, R. (2001). Anchoring adolescents' understanding of math concepts in rich problem solving environments. *Remedial and Special Education, 22*, 299–314.

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