Numeracy – our everyday play with numbers – is essential to the rhythm of life and our adaptive success as a species. There is a wonderful beauty in numbers that infants and toddlers intuitively appreciate as those around them play with numbers in song, story, dance, and life drama. It doesn’t take long before toddlers are counting, adding, judging the basic fairness of quantities, and playing with numbers in every new form they appear. Much like language and the learning of new words, there is a basic curiosity in relation to numbers. The fundamental value of numbers is no different than words – every new idea is interesting and potentially valuable. So how do we explain the unique trajectory of skill development? Why is it that we so often hear an emerging negative attitude in relation to numeracy skill development as time goes on? — ‘I can’t do maths’, ‘I don’t like maths’, ‘Maths is for nerds’, ‘I don’t want to be a scientist, so I don’t need maths’, and so on. Why does the playful activity cease in relation to a certain class of ideas and progress in relation to others? How can we continue to reinforce increasingly complex behavioral fluency and flexibility in the sphere of numeracy skills? This is a problem worth considering.
The importance of numeracy and mathematical knowledge has been likened to that of basic literacy in that it is essential for independent functioning in society (Geary, Hoard, Nugent, & Bailey, 2012). The development of mathematical expertise facilitates advancement in numerous areas including business, medicine, engineering and science; and how students perform mathematically also impacts labour quality and national growth (Lin & Kubina, 2005).
The development of numeracy skills is an ongoing concern internationally. The findings from the 2009 Program for International Assessment found that students from the U.S. performed significantly below the OECD average in mathematics, with students from Ireland similarly performing below average (OECD, 2010). Since 2009, there has been a relative improvement in the performance of Irish students to just above the OECD average while students in the U.S. continued to perform below average (OECD, 2014). Low performance in mathematics drives continued analysis of instructional programs and interventions which could enhance students’ development of mathematical skills (Codding, Archer, & Connell, 2010; Poncy, Fontenelle, & Skinner, 2013).
There are many ways to approach the training of mathematical skills, but we believe that a foundation stone for all skilled performance is fluency in the operation of sub-skills. Notably, research findings indicate that levels of fluency with component mathematics skills predict overall mathematical ability (Carr, Steiner, Kyser & Biddlecomb, 2008) and growth trajectories in mathematical ability (Carr & Alexeev, 2011). These findings suggest that instructional programs might do well to focus on promoting fluency in the early school years (Carr & Alexeev, 2011).
Fluency is a reflection of skill. A fluent performance has been described as combining both accuracy and speed of responding (Binder, 1996), is often described and observed as effortless and without error, and is dependent on deliberate and well-designed practice (Johnson & Street, 2013). Fluency has also been described in terms of behaviours that reflect the endurance, stability, and application of skills. For example, when skills can be performed fluently, individuals can engage in that skill for prolonged periods without fatiguing (Endurance; Brady & Kubina, 2010; McDowell & Keenan, 2001); they can continue to perform amidst distractions (Stability; Johnson & Street, 2013); and they can improve their ability to apply skills to more complex problems (Application; Bucklin et al., 2000; Cavallini & Perini, 2009; Chiesa & Roberston, 2000; Kubina et al., 2004). Despite the importance of fluent performances, a gap in the research exists with respect to empirical validation of fluency-based instruction (Codding et al., 2009; Poncy et al., 2013) and the impact of fluency instruction on overall mathematical ability as measured by standardised assessments.
We employed a randomised controlled trial (RCT) to evaluate the effects of fluency training with component mathematical skills in comparison to a treatment as usual control (TAU) condition. We used an instructional approach and mathematics curriculum derived from the Morningside Mathematics Fluency method (Johnson, 2008). Participants included 28 males aged 9-11 years all of whom demonstrated difficulties in performing at an age equivalent level on the mainstream mathematics curriculum and were receiving daily remedial support in mathematics. Pre- and post-test measures of fluency with targeted mathematics skills were recorded for all participants to evaluate the impact of our instructional approach on their performances. As a link between fluency with component mathematics skills and overall mathematical ability has been identified in previous research (Carr & Alexeev, 2011; Carr et al., 2008), we also measured standardised scores of mathematical ability using the Wechsler Individual Achievement Test (WIAT-II; Wechsler, 2005). Importantly, we also tracked measures of endurance, stability, and application of mathematical skills. Controlling for baseline differences in mathematical ability, it was hypothesised that the fluency training group would show significantly greater performance than the control group for all post-intervention outcome measures. Participants in the control group received typical classroom instruction and additional learning support classes. Participants in the experimental group received additional fluency-based instruction for twenty minute sessions, one day per week. This continued until they had achieved fluency with 26 different mathematics worksheets pertaining to 12 math fact families outlined in the curriculum.
Our instructional approach combined Explicit Timing (ET) and performance feedback (Hartnedy, Mozzoni, & Fahoum, 2005; Poncy, Duhon, Lee, & Key, 2010). Explicit Timing (ET) is used to allow timed practice of skills and involves the presentation of a task and a specific amount of time allocated to complete it (Gross et al., 2013). One-minute timings were conducted during which students completed as many repetitions of the target skills as they could before the timing ended (e.g., 2 × 2 = ?, 4 ÷ 2 = ?; 2 × 4 = ?). Positive reinforcement for accurate and fluent responding was provided and corrective feedback for incorrect responding. A measure of social validity was conducted with each participant post-intervention demonstrating that the majority of participants enjoyed this instructional approach and 100% of the participants responded that they learned their math facts well.
Fluency-based instruction conducted once per week across 32 weeks resulted in a number of positive outcomes for children. The majority of children completed the intervention curriculum, demonstrated enhanced performance on critical fluency outcomes and also demonstrated transfer of benefits to standardised tests of mathematical ability.
First, in relation to measures of basic fluency, controlling for pre-test rate of accurate responding, the experimental group’s rate of accurate responding at post-test was significantly higher than that of the control group. Second, controlling for baseline ability, there were significant differences between groups on post-test measures of endurance and stability – important dimensions of more generalised fluent performances. Finally, controlling for baseline ability, there were significant differences between groups on post-test measures of mathematical reasoning assessed using the WIAT-II (Wechsler, 2005).
Researchers suggest that promoting accuracy and fluency with mathematics skills should be a focus of instruction during the early school years (Carr & Alexeev, 2011); however, a gap in the research exists with respect to empirical validation of fluency-based instruction (Codding et al., 2009; Poncy et al., 2013). Our research findings provide evidence for the efficacy of the Morningside Math Facts: Multiplication and Division curriculum (Johnson, 2008) in conjunction with explicit timing (ET) and performance feedback and provides a strong rationale for its incorporation into educational settings to increase fluency with component mathematics skills. Codding, Hilt-Panahon, Panahon and Benson (2009), in the context of a meta-analysis investigating fluency-based instruction with mathematics skills, note that very few studies examine generalised or transfer effects of fluency instruction. Our study contributes to the literature by incorporating measures of generalised effects (i.e., measures of critical learning outcomes and overall mathematical ability) and demonstrates the value of building fluent component repertoires to improve overall mathematical performance.
Mathematics is a fundamental part of the educational curriculum and students’ mathematical proficiency can significantly impact their independent functioning in society (Geary et al., 2012) and future success across many academic and applied domains. It is suggested that promoting fluency with mathematics skills should be a focus of instruction in the early school years (Carr & Alexeev, 2011); however, opportunities to practice component skills and maximize fluency are rarely provided within educational settings (Codding et al, 2010). Since a gap in the research exists with respect to empirical validation of fluency-based instruction (Codding et al., 2009; Poncy et al., 2013), our research findings contribute to the existing literature and have important implications for educational practice. While continued research should be conducted to further validate fluency-based instructional approaches, our research findings suggest that fluency-based instruction can drive mathematics skills development. Our hope is to not only advance skill development but also to change emerging attitudes in relation to numeracy skill development. We want to hear students say ‘I can do maths’, ‘I like maths’, ‘Maths is for everyone’, and ‘Maths is incredibly useful in so many ways!’. We want to continue to reinforce increasingly complex behavioral fluency and flexibility in the sphere of numeracy skills. This is something we believe we can do.
Originally published May 12, 2015 in ‘In One Lifespan’ @ PsychologyToday.com
Some links contained within this post are external
A full list of references can be found in our paper:
McTiernan, A., Holloway, J., Healy, O., & Hogan, M. (in press). A Randomized Controlled Trial of the Morningside Math Facts Curriculum on Fluency, Stability, Endurance and Application Outcomes. Journal of Behavioral Education.
Pre-print is available to read here