Developing mathematical fluency: comparing exercises and rich tasks

Abstract

Achieving fluency in important mathematical procedures is fundamental to students' mathematical development. The usual way to develop procedural fluency is to practise repetitive exercises, but is this the only effective way? This paper reports three quasiexperimental studies carried out in a total of 11 secondary schools involving altogether 528 students aged 12-15. In each study, parallel classes were taught the same mathematical procedure before one class undertook traditional exercises while the other worked on a Bmathematical etude^(Foster International Journal of Mathematical Education in Science and Technology, 44(5), 765-774, 2013b), designed to be a richer task involving extensive opportunities for practice of the relevant procedure. Bayesian t tests on the gain scores between pre-and post-tests in each study provided evidence of no difference between the two conditions. A Bayesian meta-analysis of the three studies gave a combined Bayes factor of 5.83, constituting Bsubstantial^evidence (Jeffreys, 1961) in favour of the null hypothesis that etudes and exercises were equally effective, relative to the alternative hypothesis that they were not. These data support the conclusion that the mathematical etudes trialled are comparable to traditional exercises in their effects on procedural fluency. This could make etudes a viable alternative to exercises, since they offer the possibility of richer, more creative problem-solving activity, with comparable effectiveness in developing procedural fluency.

An analysis of motivation strategies used within the small‐group Accelerating Mathematics Performance through Practice Strategies (AMPPS‐SG) program

“…However, after combining the Condition 2 motivational strategies with group‐based reinforcement for effort, these large effects were commensurate with evidence suggesting the overall importance of motivational strategies, especially strategies to raise students’ attention and effort (Fuchs et al, ; Schunk & Cox, ). Motivation strategies are particularly needed for fluency interventions because (a) fluency‐building requires extensive practice, and (b) students with low motivation—such as those who previously experienced a lot of difficulty in the academic area—are likely to lose interest during instruction and thus benefit less from it (Foster, ; Haring & Eaton, ). From this, it is sensible that including motivators (e.g., to help students regulate their attention and behaviors to work hard) is considered as one of the seven principles of effective interventions for students with math disabilities (Fuchs et al, ).…”

Exploring Grade Nine Geometry Spatial Mathematical Reasoning in the South African Annual National Assessment

“…First, the learner used corresponding sides, a concept for proving congruency which Fujita et al (2017) regarded as relevant SV. Second, the learner provided three matching sides, in which one was incorrect and an erroneously matched pair of angles that made the SR inappropriate (Foster, 2018). Third, the use of erroneous SR to make an invalid judgment justifies the lack of SO (Pittalis & Christou, 2010).…”

“…The first category focuses on proofs that are deductive, inductive or analogical (Amir-Mofidi et al, 2012). The second category is on comprehension of mathematical meaning (Foster, 2018). The third category requires learners to validate blended mathematical statements (Lee, 2016).…”

Menumbuhkan Kemampuan Pemecahan Masalah Matematika Siswa Melalui Model Belajar PBL Berbasis Rich Task Matematika

“…Rich Tasks dianggap sebagai komponen penting dalam praktik pengajaran guru (Aubusson, Burke, Schuck, Kearney & Frischknecht, 2014). Dalam literatur terdapat beberapa studi yang melaporkan konsep rich tasks (Aubusson at al., 2014;Foster, 2017;Glover, 2016). Definisi rich tasks yang paling komprehensif sebagaimana yang dinyatakan oleh Darling Harmond (2012:320) in her paper.…”