Kelley Durkin

Developing Procedural Flexibility: Are Novices Prepared to Learn from Comparing Procedures?.

BACKGROUND A key learning outcome in problem-solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. AIMS We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. SAMPLE Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi-step equation solving, which was the topic of our lessons. METHOD Students learned about multi-step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre-test, post-test, and 1-month-retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. RESULTS Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1-month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. CONCLUSIONS Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason.

Using erroneous examples to improve mathematics learning with a web-based tutoring system

Middle school students learned to solve decimal problems with a web-based tutoring system.ExErr group received erroneous examples to correct and explain.PS group received problems to solve and explain.ExErr group outperformed PS group on a delayed test and on judging answer correctness.PS group reported liking the lessons better than the ExErr group. This study examines whether asking students to critique incorrect solutions to decimal problems based on common misconceptions can help them learn about decimals better than asking them to solve the same problems and receive feedback. In a web-based tutoring system, 208 middle school students either had to identify, explain, and correct errors made by a fictional student (erroneous examples group) or solve isomorphic versions of the problems with feedback (problem-solving group). Although the two groups did not differ significantly on an immediate posttest, students in the erroneous examples group performed significantly better on a delayed posttest administered one week later (d=.62). Students in the erroneous examples group also were more accurate at judging whether their posttest answers were correct (d=.49). Students in the problem-solving group reported higher satisfaction with the materials than those in the erroneous examples group, indicating that liking instructional materials does not equate to learning from them. Overall, practice in identifying, explaining, and correcting errors may help students process decimal problems at a deeper level, and thereby help them overcome misconceptions and build a lasting understanding of decimals.

Is self‐explanation worth the time? A comparison to additional practice

BACKGROUND Self-explanation, or generating explanations to oneself in an attempt to make sense of new information, can promote learning. However, self-explaining takes time, and the learning benefits of this activity need to be rigorously evaluated against alternative uses of this time. AIMS In the current study, we compared the effectiveness of self-explanation prompts to the effectiveness of solving additional practice problems (to equate for time on task) and to solving the same number of problems (to equate for problem-solving experience). SAMPLE Participants were 69 children in grades 2-4. METHODS Students completed a pre-test, brief intervention session, and a post- and retention test. The intervention focused on solving mathematical equivalence problems such as 3 + 4 + 8 = _ + 8. Students were randomly assigned to one of three intervention conditions: self-explain, additional-practice, or control. RESULTS Compared to the control condition, self-explanation prompts promoted conceptual and procedural knowledge. Compared to the additional-practice condition, the benefits of self-explanation were more modest and only apparent on some subscales. CONCLUSIONS The findings suggest that self-explanation prompts have some small unique learning benefits, but that greater attention needs to be paid to how much self-explanation offers advantages over alternative uses of time.

Studying technology-based strategies for enhancing motivation in mathematics

BackgroundDuring the middle school years, students frequently show significant declines in motivation toward school in general and mathematics in particular. One way in which researchers have sought to spark students’ interests and build their sense of competence in mathematics and in STEM more generally is through the use of technology. Yet evidence regarding the motivational effectiveness of this approach is mixed. Here we evaluate the impact of three brief technology-based activities on students’ short-term motivation in math. 16,789 5th to 8th grade students and their teachers in one large school district were randomly assigned to three different technology-based activities, each representing a different framework for motivation and engagement and all designed around an exemplary lesson related to algebraic reasoning. We investigated the relationship between specific technology-based activities that embody various motivational constructs and students’ engagement in mathematics and perceived competence in pursuing STEM careers.ResultsResults indicate that the effect of each technology activity on students’ motivation was quite modest. No gains were found in self-efficacy; for implicit theory of ability, a lower incremental view of ability was found; we found modest declines in value beliefs. With respect to math learning, students in all three inductions had modest improvements in their scores on the math learning measure. However, these effects were modified by students’ grade level and not by their demographic variables. In addition, teacher-level variables did not have an effect on student outcomes.ConclusionsThe present findings highlight the importance of tailoring motivational experiences to students’ developmental level. Our results are also encouraging about developers’ ability to create instructional interventions and professional development that can be effective when experienced by a wide range of students and teachers. Further research is needed to determine the degree, duration of, and type of instructional intervention necessary to substantially impact multi-dimensional, deep-rooted motivational constructs, such as self-efficacy.

To err is human, to explain and correct is divine: a study of interactive erroneous examples with middle school math students

Erroneous examples are an instructional technique that hold promise to help children learn. In the study reported in this paper, sixth and seventh grade math students were presented with erroneous examples of decimal problems and were asked to explain and correct those examples. The problems were presented as interactive exercises on the Internet, with feedback provided on correctness of the student explanations and corrections. A second (control) group of students were given problems to solve, also with feedback on correctness. With over 100 students per condition, an erroneous example effect was found: students who worked with the interactive erroneous examples did significantly better than the problem solving students on a delayed posttest. While this finding is highly encouraging, our ultimate research question is this: how can erroneous examples be adaptively presented to students, targeted at their most deeply held misconceptions, to best leverage their effectiveness? This paper discusses how the results of the present study will lead us to an adaptive version of the erroneous examples material.

Comparison and Explanation of Multiple Strategies One Example of a Small Step Forward for Improving Mathematics Education

Education policy should aim to promote instructional methods that are easy for teachers to implement and have demonstrable, positive impact on student learning. Our research on comparison and explanation of multiple strategies illustrates the promise of this approach. In several short-term experimental, classroom-based studies, comparing different strategies for solving the same problem was particularly effective for promoting student learning. Thus, we developed a supplemental Algebra 1 curriculum to foster comparison in combination with explanation of multiple strategies. In a randomized control trial, teachers used our materials as intended, but much less often than expected, and student learning was not greater in experimental classrooms. Yet greater use of our comparison materials was associated with greater student learning, suggesting the approach has promise when used sufficiently often. These studies provide some evidence that easy-to-implement reforms can change teacher practice and improve student learning.

Evaluating Game-Based Learning Environments for Enhancing Motivation in Mathematics

During the middle school years, students frequently show significant declines in motivation toward school in general and mathematics in particular. One way in which researchers have sought to spark students’ interests and build their sense of competence in mathematics and in STEM more generally is through the use of game-based learning environments. Yet evidence regarding the motivational effectiveness of this approach is mixed. Here, we evaluate the impact of three brief game-based technology activities on students’ short-term motivation in math. A total number of 16,789 fifth to eighth grade students and their teachers in one large school district were randomly assigned to three different game-based technology activities, each representing a different framework for motivation and engagement and all designed around an exemplary lesson related to algebraic reasoning. We investigated the relationship between specific game-based technology activities that embody various motivational constructs and students’ engagement in mathematics and perceived competence in pursuing STEM careers. Results indicate that the effect of each game-based technology activities on students’ motivation was quite modest. However, these effects were modified by students’ grade level and not by their demographic variables. In addition, teacher-level variables did not have an effect on student outcomes.

The Power of Comparison in Mathematics Instruction: Experimental Evidence From Classrooms

Abstract Comparison is a fundamental cognitive process that supports learning in a variety of domains. To leverage comparison in mathematics instruction, evidence-based guidelines are needed for how to use comparison effectively. In this chapter, we review our classroom-based research on using comparison to help students learn mathematics. In five short-term experimental, classroom-based studies, we evaluated two types of comparison for supporting the acquisition of mathematics knowledge and tested whether prior knowledge moderated their effectiveness. Comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods. We next developed a supplemental Algebra 1 curriculum to foster comparison and evaluated its effectiveness in a randomized-control trial. Teachers used our supplemental materials much less often than expected, and student learning was not greater in classrooms that had been assigned to use our materials. Students’ procedural knowledge was positively related to greater implementation of the intervention, suggesting the approach has promise when used sufficiently often. This study suggests that teachers may need additional support in deciding what to compare and when to use comparison.