No More Math Wars: een illusie? Straks meer oorlog in Vlaanderen?

Wiskunde is inspiratie (inzicht) én transpiratie (automatiseren en memoriseren) = klassieke visie op degelijk wiskunde-onderwijs die we zelf al 50 jaar propageren 
en die voor de meeste praktijkmensen al eeuwen een evidentie is.

Jammer genoeg werd die evidentie in vraag gesteld door propagandisten van ontdekkend/constructivistisch en contextueel leren van het Freudenthal Instituut, opstellers eindtermen en leerplannen eerste graad s.o..

De visie die prof. Daniel Ansari beschrijft staat ook haaks op deze in de ZILL-bijdrage over wiskundeonderwijs.
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An evidence-based, developmental perspective on math educationby: Daniel Ansari
Vooraf: in de klassieke Vlaamse visie op degelijk wiskundeonderwijs werd steeds gesteld dat het zowel gaat om inspiratie (inzicht in concepten e.d.) als om transpiratie (uatomatiseren en memoriseren dril, inoefenen. Dit is ook de centrale idee in mijn boek ‘Rekenen tot honderd( Plantyn, 200 pagina’s)

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.

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. 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.

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 University 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 (in Canada) 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.