Jon R. Star

Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations.

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventhgrade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom.

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.

Teaching Mathematics for Understanding: An Analysis of Lessons Submitted by Teachers Seeking NBPTS Certification

The authors present an analysis of portfolio entries submitted by candidates seeking certification by the National Board for Professional Teaching Standards in the area of Early Adolescence/Mathematics. Analyses of mathematical features revealed that the tasks used in instruction included a range of mathematics topics but were not consistently intellectually challenging. Analyses of key pedagogical features of the lesson materials showed that tasks involved hands-on activities or real-world contexts and technology but rarely required students to provide explanations or demonstrate mathematical reasoning. The findings suggest that, even in lessons that teachers selected for display as best practice examples of teaching for understanding, innovative pedagogical approaches were not systematically used in ways that supported students’ engagement with cognitively demanding mathematical tasks.

Relations Among Conceptual Knowledge, Procedural Knowledge, and Procedural Flexibility in Two Samples Differing in Prior Knowledge

Competence in many domains rests on children developing conceptual and procedural knowledge, as well as procedural flexibility. However, research on the developmental relations between these different types of knowledge has yielded unclear results, in part because little attention has been paid to the validity of the measures or to the effects of prior knowledge on the relations. To overcome these problems, we modeled the three constructs in the domain of equation solving as latent factors and tested (a) whether the predictive relations between conceptual and procedural knowledge were bidirectional, (b) whether these interrelations were moderated by prior knowledge, and (c) how both constructs contributed to procedural flexibility. We analyzed data from 2 measurement points each from two samples (Ns = 228 and 304) of middle school students who differed in prior knowledge. Conceptual and procedural knowledge had stable bidirectional relations that were not moderated by prior knowledge. Both kinds of knowledge contributed independently to procedural flexibility. The results demonstrate how changes in complex knowledge structures contribute to competence development.

Spurious Correlations in Mathematical Thinking.

How does detection of correlational structure affect mathematical thinking and learning? When does correlational information lead to erroneous problem solving? Are experienced students susceptible to misleading correlations? This work attempts to answer these questions by examining a source of systematic errors termed the spurious-correlation effect. This effect is hypothesized to occur when a student perceives a correlation between an irrelevant feature in a problem and the algorithm used for solving that problem and then proceeds to execute the algorithm when detecting the feature in a different problem. In this research, we investigated whether students encode spurious correlations in memory and exhibit them during the learning process leading to ineffectual problem solving. Findings suggest that even experienced students rely on surface-structural feature-algorithm correlations for solving new problems. Implications for teaching are discussed.

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.

Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality

Following Star (2005, 2007), we continue to problematize the entangling of type and quality in the use of conceptual knowledge and procedural knowledge. Although those whose work is guided by types of knowledge and those whose work is guided by qualities of knowledge seem to be referring to the same phenomena, actually they are not. This lack of mutual understanding of both the nature of the questions being asked and the results being generated causes difficulties for the continued exploration of questions of interest in mathematics teaching and learning, such as issues of teachers’ knowledge.RésuméDans la lignée de Star (2005, 2007), nous continuons de problématiser l’absence de distinction entre type et qualité lorsqu’il est question de connaissance des concepts et des procédures. Bien que ceux dont les travaux se fondent sur les types of connaissances et ceux dont les travaux se fondent sur les qualités des connaissances semblent faire référence aux mêmes phénomènes, ce n’est pas le cas en réalité. Le manque de compréhension réciproque, aussi bien de la nature des questions posées que des résultats obtenus, cause des difficultés pour l’exploration de questions importantes en enseignement et en apprentissage des mathématiques, par exemple la question des connaissances des enseignants.

The power of comparison in learning and instruction: Learning outcomes supported by different types of comparisons.

Abstract Comparison is a powerful learning process that has been leveraged to improve learning in a variety of domains. We identify five different types of comparisons that have been used in past research and develop a framework for describing them and the learning outcomes they support. For example, comparing multiple methods for solving the same problem, with a focus on which method is better for solving a particular problem, can improve procedural flexibility. We include a review of our own efforts to design and evaluate educational materials that leverage different types of comparisons to support mathematics learning in classrooms, including our ongoing effort to encourage use of comparison throughout the Algebra I curriculum. In the context of this classroom work, two new comparison types emerged. Overall, we illustrate how cognitive science research helped guide the design of effective educational materials and how educational practice revealed new ideas to test and incorporate into theories of learning.

The role of language in the construction of kinds

Publisher Summary This chapter explores several potential ways that language may affect the construction of inference-promoting kinds. Human categories are distinctive in their diversity, ranging from simple to complex, from concrete to abstract, or from arbitrary groupings to those deeply rooted in theories. To understand the role of language in categorization, it is first necessary to make some distinctions. It introduces some terminologies: “kinds” and “essentialism.” The chapter gives an overview of some of the findings demonstrating essentialist beliefs even in young children. Four distinct linguistic devices are discussed and the role of each in conveying essentialism is evaluated. Two of these forms convey membership in a richly structured category (the word “kind;” lexicalization) and two of the forms express scope of a proposition (logical quantifiers; generic noun phrases). The chapter also discusses the nature of the effects of language and potential areas for future research.

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.