Compared to what? The effects of different comparisons on conceptual knowledge and procedural flexibility for equation solving

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 7th- and 8th-grade students learned to solve equations by comparing equivalent problems solved with the same solution method, by comparing different problem types solved with the same solution method, or by comparing different solution methods to the same problem. Students' conceptual knowledge and procedural flexibility were best supported by comparing solution methods, and to a lesser extent by comparing problem types. The benefits of comparison are augmented when examples differ on relevant features, and contrasting methods may be particularly useful in mathematics learning

It pays to compare: An experimental study on computational estimation

Comparing and contrasting examples is a core cognitive process that supports learning in children and adults across a variety of topics. In this experimental study, we evaluated the benefits of supporting comparison in a classroom context for children learning about computational estimation. Fifth- and sixth-grade students (n = 157) learned about estimation either by comparing alternative solution strategies or by reflecting on the strategies one at a time. At posttest and retention test, students who compared were more flexible problem solvers on a variety of measures. Comparison also supported greater conceptual knowledge, but only for students who already knew some estimation strategies. These findings indicate that comparison is an effective learning and instructional practice in a domain with multiple acceptable answers

Wait for it . . . Delaying Instruction Improves Mathematics Problem Solving: A Classroom Study

Engaging learners in exploratory problem-solving activities prior to receiving instruction (i.e., explore-instruct approach) has been endorsed as an effective learning approach. However, it remains unclear whether this approach is feasible for elementary-school children in a classroom context. In two experiments, second-graders solved mathematical equivalence problems either before or after receiving brief conceptual instruction. In Experiment 1 (n = 41), the explore-instruct approach was less effective at supporting learning than an instruct-solve approach. However, it did not include a common, but often overlooked feature of an explore-instruct approach, which is provision of a knowledge-application activity after instruction. In Experiment 2 (n = 47), we included a knowledge-application activity by having all children check their answers on previously solved problems. The explore-instruct approach led to superior learning than an instruct-solve approach. Findings suggest promise for an explore-instruct approach, provided learners have the opportunity to apply knowledge from instruction

Assessing conceptual knowledge in three online engineering courses: theory of computation and compiler construction, operating systems, and signal and systems

In the current decade understanding conceptual knowledge should be an important area of engineering science. However, it is not as widespread in this field as it is in the areas of education and psychology. Learning conceptual knowledge in engineering science could help instructors to adapt their lectures in order to overcome student misconceptions, to reinforce the learning process, and to check whether students are able to identify key features of a problem. Different methods are provided by authors to assess conceptual knowledge. One is to design and develop a concept inventory with the objective of identifying possible student misconceptions through multiple-choice questions. Another method consists of asking each student to answer a question by submitting a written explanation. This study provides two procedures for the assessment of conceptual knowledge based on the latter method. The first procedure is applied to online computer science students enrolled on an Operating Systems course. The second procedure is applied to online communication students enrolled on a Signals and Systems course. Both procedures are focused not only on assessing but also on searching for causes of potential student misconceptions. These procedures could help other instructors to assess conceptual knowledge on other engineering courses.Universidad a Distancia de Madrid, UDIMA2018-1

Comparison helps students learn to be better estimators

This article describes a recent research study that examined the effectiveness of comparison on students’ learning of strategies and concepts for computational estimation

The Role of Prior Knowledge in the Development of Strategy Flexibility: The Case of Computational Estimation

The ability to estimate is a fundamental real-world skill; it allows students to check the reasonableness of answers found through other means, and it can help students develop a better understanding of place value, mathematical operations, and general number sense. Flexibility in the use of strategies is particularly critical in computational estimation. The ability to perform complex calculations mentally is cognitively challenging for many students; thus it is important to have a broad repertoire of estimation strategies and to select the most appropriate strategy for a given problem. In this paper, we consider the role of students' prior knowledge of estimation strategies in the effectiveness of interventions designed to promote strategy flexibility across two recent studies. In the first, 65 fifth graders began the study as fluent users of one strategy for computing mental estimates to multi-digit multiplication problems such as 17 x 41. In the second, 157 fifth and sixth graders began the study with moderate to low prior knowledge of strategies for computing mental estimates. Results indicated that students' fluency with estimation strategies had an impact on which strategies they adopted. Students who exhibited high fluency at pretest were more likely to increase use of estimation strategies that led to more accurate estimates, while students with less fluency adopted strategies that were easiest to implement. Our results suggest that both the ease and accuracy of strategies as well as students’ fluency with strategies are all important factors in the development of strategy flexibility

Learning from Comparison in Algebra

Mastery of algebra is an important yet difficult milestone for students, suggesting the need for more effective teaching strategies in the algebra classroom. Learning by comparing worked-out examples of algebra problems may be one such strategy. Comparison is a powerful learning tool from cognitive science that has shown promising results in prior small-scale studies in mathematics classrooms. This study reports on a yearlong randomized controlled trial testing the effect of an Algebra I supplemental comparison curriculum on students’ mathematical knowledge. 141 Algebra I teachers were randomly assigned to either implement the comparison curriculum as a supplement to their regular curriculum or to be a ‘business as usual’ control. Use of the supplemental curriculum was much less frequent than requested for many teachers, and there was no main effect of condition on student achievement. However, greater use of the supplemental curriculum was associated with greater procedural student knowledge. These findings suggest a role for comparison in the algebra classroom but also the challenges of supporting teacher integration of new materials into the curriculum