Numbers play an important role in our everyday lives. From the carpenter measuring lengths of wood to the physician checking a patient’s blood pressure measurements on a chart, we constantly use numbers and perform calculations to guide our actions and decisions. It has been shown that school-entry numerical skills are a more important predictor of subsequent academic achievement than early reading and socio-emotional skills.1 Furthermore, there are many reports linking numerical skills to economic outcomes, such as evidence showing that early math skills predict adult socio-economic status.2
In view of the above, it is critical that education systems seek to find the best ways to teach math and thereby equip learners with the skills necessary to succeed not only in school but also in life more generally. So what are the best ways to teach math? Are we currently using all the available evidence from fields such as educational research, cognitive psychology, developmental psychology and neuroscience to guide math pedagogy? Unfortunately, for decades there have been ongoing, fierce, partisan debates over how to teach math which have, for the most part, not been informed by the wealth of knowledge about how children learn math. This article critically discusses one of the central debates in math education and draws attention to the importance of taking an evidence-based and developmental perspective on how to teach math.
The math wars
Perhaps the most prominent debate that has been raging in math education for decades is whether children should be taught to calculate by rehearsing (also referred to as “drilling” or “rote learning”) arithmetic facts, such as learning the times tables, or whether students should be taught to learn arithmetic and other math skills by discovering the principles that underlie it, through being encouraged to use hands-on materials, invent their own strategies, solve open-ended problems and describe their problem-solving strategies without having to memorize answers (often referred to as “discovery-based learning” or “problem solving”). Another characterization of the debate is that one side advocates for greater attention to teaching children procedural knowledge for mathematical problem solving (such as explicit teaching of strategies) and encouraging them to memorize facts, while the other emphasizes the students’ construction of rich conceptual knowledge, allowing them insights into how they solve problems.
These two approaches to math instruction are frequently painted as being completely distinct and diametrically opposed to one another, creating the perception that there is a need to side with one particular view of best practice in math education. Indeed, math education curricula align themselves with one “side” or the other. History suggests that the pendulum swings between the two supposed extremes rather than settling for a balance in the middle.
Proponents of either instructional approach tend to have very strong negative views of the alternative. The debate over which of the two (supposedly completely opposing) instructional approaches should be adopted resembles a political debate with candidates on either side of the debate painting their opponents’ perspective as hurting student learning. This has frequently been referred to as the “math wars.”
The math wars in Canada
Strongly held views about instructional approaches tend to flare up whenever there is evidence suggestive of falling student achievement in mathematics. Take, for example, the most recent (2012) results from the Program for International Student Assessment (PISA) published by the Organization for Economic Co-operation and Development (OECD).3 The PISA study assesses math, reading and science skills among 15-year-old students in over 50 countries every three years (PISA started in 2000). In 2012, the latest math results from the PISA study revealed that Canadian 15-year-olds ranked 13th among students tested in 65 countries. Critically, these scores were lower than in previous years, when Canadian students had ranked in the top ten of the countries participating in PISA. Thus the conclusion from these data was that math education standards in Canada were falling. This evidence of falling achievement levels among 15-year-old Canadian students garnered significant media attention and strong reactions. For example, John Manley, CEO and president of the Canadian Council of Chief Executives, proclaimed in The Globe and Mail that the results were “on the scale of a national emergency.”4
Soon after the dust of the media frenzy surrounding the publication of the PISA results had settled, the search for the fall guy began. Quickly this search turned to discovery-based math instruction curriculum, used (with variations) in most provinces. A parent group in Alberta even initiated a petition against the discovery-based curriculum and in support of a back-to-basics approach focusing more on rehearsal of facts and procedural learning. To date this petition has over 17,000 signatories.5
The need for an evidence base
What is sorely lacking from this highly politicized and emotional public debate over math instruction and the analysis of falling student achievement (such as that uncovered by the PISA study) is the use of a solid evidence base that, without biased opinions and beliefs about what works, seeks to inform decision making.
While the math wars have been raging here in Canada and abroad, scientists from developmental and educational psychology as well as cognitive neuroscience have been busy accumulating evidence regarding the ways in which children learn math and what factors influence their learning trajectories and achievement success. This evidence suggests that the dichotomy between discovery-based or conceptual learning, on the one hand, and procedural or rote learning, on the other, is false and inconsistent with the way in which children build an understanding of mathematics. Indeed, there is a long line of research showing that children learn best when procedural and conceptual approaches are combined.6 Moreover, children’s procedural and conceptual knowledge are highly correlated with one another, speaking against creating a dichotomy between them through instructional approaches. Researchers, such as Bethany Rittle-Johnson at Vanderbilt, have demonstrated that an effective use of instructional time in math education involves the alternation of lessons focused on concepts with those concentrated on instructing students on procedures.7 While there is still some debate in the literature over the precise sequencing of procedural and conceptual instruction (i.e. which one should come first – with several studies indicating optimal learning when some conceptual instruction precedes procedural learning in elementary school mathematics), all of the literature clearly suggests that both instructional approaches are tightly related to one another and are mutual determinants of successful math learning over time.8
Discovery math proponents argue strongly against the use of setting time limits for students to complete mathematical tasks, such as calculation. Here again, the empirical evidence speaks against the notion that speeded instruction necessarily has negative consequences. For example, research I conducted in collaboration with Gavin Price (Vanderbilt University) and Michelle Mazzocco (University of Minnesota)9 demonstrated that young adults who performed well on a test of high-school math achievement (the Preliminary Scholastic Achievement Test, PSAT) activated brain regions associated with arithmetic fact retrieval in the left hemisphere more while solving simple, single-digit arithmetic problems (such as 3+4) compared to their lower-achieving peers, who recruited brain regions associated with less efficient strategies, such as counting and decomposition in areas of the right parietal cortex. These data suggest that arithmetic fluency and its neural correlates contribute to higher-level math abilities. Moreover, recent research by Lynne Fuchs and colleagues at Vanderbilt University10 has demonstrated that speeded practice can lead to larger student gains in arithmetic compared to nonspeeded practice, and that such practice can be particularly useful for low-achieving students in overcoming their math reasoning difficulties. Thus speeded practice is beneficial, when combined with other approaches.
Such evidence clearly shows that the current debate over what kind of math curriculum to adopt is ill-informed and focuses on false extreme dichotomies that paint students as either learning one way or the other – when the evidence demonstrates that both conceptual and procedural knowledge are required for successful math learning. Do we really want to create, on the one hand, students who can solve arithmetic problems quickly but who lack conceptual knowledge and are not able to be flexible mathematical thinkers; or, on the other hand, students who are able to reflect on their mathematical problem solving, but are unable to quickly retrieve the answers to intermediate solutions in the context of complex calculation problems, because they lack mathematical fluency?
When considering the evidence base for guiding math instruction it is also critical to think developmentally and to ask what sequence of learning and content of learning is most appropriate at which age/level of the student. Learning math is a cumulative process – early skills build the foundations for later abilities. For example, when we ask students to reflect on their mathematical problem-solving strategies, we need to consider whether they have the metacognitive skills (the ability to reflect on one’s thinking) necessary to articulate how they are thinking; when we train students to solve arithmetic problems under speeded conditions, we need to ascertain that they understand the meaning of the numbers that they are performing arithmetic operations with.
Finally, it is important to carefully evaluate the evidence that is drawn upon to guide educational decision-making. Take for example the PISA study that is now being used to motivate the back-to-basics approach. In this context, it is also important to note that the results of the PISA study do not directly show that the curriculum is to blame. The PISA study generates complex, multi-layered data that do not allow for straightforward conclusions about the factors that cause student performance. Furthermore, the PISA study only tells us about the math performance of 15-year-old children and can therefore not be generalized to learners of all ages. Critically, the focus of the PISA study is designed to measure the extent to which students can apply their knowledge to real-life situations and therefore the test is not directly linked to the school curriculum. Hence, to take the PISA ranking as an indication that a particular curriculum is the causal factor ignores the complexity of this international comparison study.
Enabling students to be mathematically competent is a major challenge for our education systems. For too long math education has been characterized by emotional debates that falsely dichotomize instructional approaches without consulting evidence about how students learn math. It is time to heed the empirical evidence coming from multiple scientific disciplines that clearly shows that math instruction is effective when different approaches are combined in developmentally appropriate ways. It is time for mathematics educators, educational policy makers and textbook publishers to take this evidence seriously, to move beyond opinions towards a level-headed, unemotional and evidence-based approach in order to to improve student learning in mathematics.
En Bref – Les compétences en mathématiques sont essentielles tant pour réussir ses études que pour mener une vie saine et prospère. Alors, comment peut-on le mieux enseigner les mathématiques? Dans cet article, j’examine la tendance des enseignants de mathématiques à créer de fausses dichotomies entre les approches pédagogiques visant à développer les connaissances conceptuelles des mathématiques des élèves et les méthodes d’enseignement axées sur les habiletés procédurales des apprenants. Je présente des constats tirés de la psychopédagogie, de la psychologie du développement et de la neuroscience pour démontrer que les élèves apprennent mieux lorsque ces deux approches pédagogiques sont combinées et que les éducateurs réfléchissent attentivement à la séquenced’enseignement adaptée au développement des approches conceptuelles et procédurales des mathématiques. Je préconise d’abandonner les fausses dichotomies et de tenir compte des constats existants pour élaborer des programmes d’études équilibrés améliorant l’apprentissage des élèves.
Photo: Dave Donald
First published in Education Canada, September 2015
1 G. J. Duncan, C. J. Dowsett, A. Claessens, K. Magnuson, A. C. Huston, P. Klebanov, L. S. Pagani, L. Feinstein, M. Engel, J. Brooks-Gunn, H. Sexton, K. Duckworth, and C. Japel, “School Readiness and Later Achievement,” Developmental Psychology 43 (2007): 1428-46.
2 S. J. Ritchie and T. C. Bates, “Enduring Links from Childhood Mathematics and Reading Achievement to Adult Socioeconomic Status,” Psychological Science 24 (2013): 1301-8.
3 Organization for Economic Co-operation and Development, PISA 2012 Results in Focus: What 15-year-olds know and what they can do with what they know (2014).
6 B. Rittle-Johnson, R. S. Siegler, and M. W. Alibali, “Developing Conceptual Understanding and Procedural Skill in Mathematics: An iterative process,” Journal of Educational Psychology 93 (2001): 346-362.
7 B. Rittle-Johnson and K. R. Koedinger, “Iterating Between Lessons on Concepts and Procedures Can Improve Mathematics Knowledge,” British Journal of Educational Psychology 79 (2009): 483 – 500.
8 M. Schneider, B. Rittle-Johnson, and J. Star, “Relations Between Conceptual Knowledge, Procedural Knowledge, and Procedural Flexibility in Two Samples Differing in Prior Knowledge,” Developmental Psychology 47 (2011): 1525-1538.
9 G. R. Price, M. M. Mazzocco and D. Ansari, “Why Mental Arithmetic Counts: Brain activation during single digit arithmetic predicts high-school math scores,” Journal of Neuroscience 33 (2013): 156-63.
10 L. S. Fuchs, D. C. Geary, D. L. Compton, D. Fuchs, C. Schatschneider, C. L. Hamlett, P. M. Seethaler, J. Wilson, C. F. Craddock, J. D. Bryant, K. Luther, and P. Changas, “Effects of First-grade Number Knowledge Tutoring with Contrasting Forms of Practice, Developmental Psychology 105 (2013): 58-77.