Grote niveaudaling wiskunde als gevolg van invoering van discovery-based instruction – also called problem-based, inquiry, experiential, and constructivist learning in enkele Canadese provincies

Vooraf: een recente studie van een Canadees onderzoeks-instituut bevestigt de kritiek op de ontdekkende & contextuele, constructivistische aanpak van het wiskundeonderwijs die we al meer dan 25 jaar formuleerden in Onderwijskrant e.d.

We zorgden er destijds voor dat die aanpak – à la Freudenthal Instituut - niet doordrong in het leerplan wiskunde (katholiek) basisonderwijs 1998. Merkwaardig genoeg pleiten de ZILL-leerplanarchitecten nu voor die nefaste aanpak die ook in een aantal Canadese provincies tot een sterke niveaudaling leidde. En jammer genoeg drong die nefaste aanpak ook al door in het leerplan wiskunde 1ste graad secundair onderwijs van 1997 & 2009.

Studie van Institut C.D. HOWE Institute commentary NO. 427 What to Do about Canada’s Declining Math Scores

Conclusies

The correlation between early math achievement and later academic success and the decline in Canadian students’ mathematics scores on international tests in recent years suggest that provincial governments would be wise to improve the way mathematics is taught in Canadian schools.
Best teaching practices in math have been at the forefront of discussions regarding declining math scores in Canada. Discovery-based instruction – also called problem-based, inquiry, experiential, and constructivist learning – has become popular in North America in recent years, pushing aside direct instruction techniques, like times table memorization, explicit teacher instruction, pencil-and-paper practice, and mastery of standard mathematical procedures.

Based on international and domestic evidence, this Commentary finds that studies consistently show direct instruction is much more effective than discovery-based instruction, which leads to straightforward recommendations on how to tilt the balance toward best instructional techniques. Student fluency with particular math concepts, such as fraction arithmetic, in early and middle years has been shown to predict future math success.

This Commentary recommends that provincial math curricula be rewritten to remove ineffective pedagogical directives and to stress specific topics, at appropriate grade levels, that are known to lead to later success in math. Evidence shows that teachers who are most comfortable and knowledgeable with the content they are required to teach tend to transmit that knowledge best to students. This Commentary suggests that future early and middle-years teachers be required to pass a math-content licensure exam prior to receiving certification to teach mathematics.

Recent shifts in math teaching practices coupled with radical, discovery-based math curricula are seriously hampering math learning by Canadian students. Evidence shows that direct instructional techniques work better than discovery-based techniques, so teachers should follow an 80/20 rule, devoting at least 80 percent of their math instructional time to direct instructional techniques.

Curricula also should be revised to remove ineffective instructional directives, and streamlined to focus on explicit topics and concepts that have been shown to predict later success in math. As well, the need to improve the math content knowledge of future elementary and middle-years math teachers should be addressed through course requirements and licensure exams. Adopting these recommendations should help solve some of the root problems behind the falling math scores of Canadian students, and result in improvements in the years to come.

Discovery-based Instruction: Changes in Canadian Classrooms Discovery-based instruction has become popular in North America in recent years, making its way into Canadian curricula, teachers’ professional development sessions and textbooks. A discoverybased learning environment often uses a top-down approach in which students are taught through problem solving

Discovery-based learning environments typically have some of the following characteristics:
• minimal guidance from the teacher and few explicit teacher explanations;
• use of multiple, preferably student-invented, strategies;
• open-ended problems with multiple solutions
• frequent use of hands-on materials such as blocks, fraction strips and algebra tiles or drawing pictures to solve problems; • minimal worksheet practice or written symbolic work; • memorization of math facts is deprioritized;
• standard methods such as column addition or long division are downplayed;
• a top-down approach in which students work on complex problems, even though foundational skills might not be present.

Expliciete instructie

The more conventional instructional approach is often called direct or explicit instruction. In this setting, students are directly taught concepts and given explicit explanations, followed by plenty of student practice, often with paper and pencil, feedback from the teacher and conventional assessment.

Standard methods like column addition and long division are emphasized and students are encouraged to memorize basic facts like times tables. Direct instruction often follows a bottom-up approach in which students are taught foundational skills that are practiced to mastery, gradually preparing them for complex problem solving. Note, however, that the teaching of understanding – or why mathematical procedures and rules hold – can be encouraged in any instructional setting, including during direct instruction.
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Discovery methods ignore the limitations of working memory
by eschewing conventional techniques such as times table memorization and by encouraging multiple, convoluted strategies instead of efficient, standard methods. Teaching through problem solving without providing the foundational skills necessary to solve problems overburdens working memory (Sweller 1988), and might not alter long-term memory, thereby inhibiting learning of mathematical concepts. Controlled studies have shown that direct instruction is a much more effective teaching method: when learners are presented with new information, it should be explicitly taught by a teacher.

Discovery-based learning environments often result in students becoming confused and frustrated and it is an inefficient style of instruction characterized by frequent false starts (Kirschner, Sweller, and Clark 2006). Numerous studies have found that, in contrast, direct instruction techniques such as worked examples, scaffolding, explicit explanations and consistent feedback are extremely beneficial for learning (Alfieri et al. 2011; Hattie and Yates 2014).

In a review of 200 research studies, Sutton Trust identifies key characteristics of effective instruction: teachers’ use of assessment, reviewing previous learning, working through examples for students, giving adequate time for practice to embed skills securely in long-term memory, and introducing topics incrementally. The review also finds that less effective teachers often teach math using handson materials, delay the introduction of formal methods until they feel pupils are ready to move on, and encourage students to work things out for themselves using any method with which they feel comfortable (Sutton Trust 2014).

As well, studies consistently find that students who have difficulty with mathematics by the end of their primary school years have not memorized basic number facts, making further math learning difficult and resulting in feelings of helplessness and a lack of confidence and enjoyment (Hattie and Yates 2014).

A great deal of time and effort is required to commit basic number facts to long-term memory, but the ability to recall them instantly frees up working memory, making it easier to learn new concepts. Overemphasis on hands-on materials and pictures also presents problems. Although some materials, such as base-ten blocks, might assist initial learning, overuse can prevent the transfer of information to long-term memory, because working memory is assaulted with extraneous information.

Transfer is more likely to occur if mathematical symbols are stressed over concrete materials (Kaminski, Sloutsky, and Heckler 2009). A survey of eighth grade Canadian students who participated in the nationally administered 2010 PCAP, which tested mathematics as a major domain, shows a pattern similar to the international studies cited above. Students were asked to report the frequency with which their teachers used both indirect instruction methods and direct instructional techniques.

The use of direct instruction was positively correlated with better math performance for most students, except the highest achievers, who seemed to succeed regardless of the instructional method used. Furthermore, greater use of indirect instruction was found to be strongly associated with lower scores (CMEC 2012).

Best Methods to Teach Problem Solving and Understanding
Equipping students with strong problem-solving skills is an important goal of math education, but Canadian curricula and prominent resources ignore what research in cognitive science reveals about how problem-solving skills are acquired. Experienced, effective problem solvers store organized techniques in long-term memory, which allows them to categorize new problems and implement effective strategies to solve them (Sweller and Cooper 1985).

The best way to ensure that students are well positioned to solve new problems is to provide them a library of knowledge and techniques and to teach thinking skills through direct instruction (Hattie and Yates 2014). A large body of evidence shows that direct instruction through worked examples followed by practice with problems similar to the worked examples respects working-memory limitations and improves problem-solving performance (Paas and van Gog 2006; Sweller and Cooper 1985). Gradually increasing the difficulty level of worked examples and practice problems results in the ability of students to transfer problem-solving skills to new situations. However, when students are presented with problems that they do not have the techniques to solve without reference to worked examples,

The PCAP survey characterized indirect instruction techniques by teachers’ use of hands-on materials (base-ten blocks, colour tiles, geometric solids), use of computer software, working in groups on investigations or problems, sharing solutions with other students in the class, and having opportunities to reflect on what was learned. Direct instruction techniques were characterized by conventional teaching methods: watching the teacher do examples, listening to the teacher give explanations, copying notes given by the teacher, practising new skills, undertaking teacher-guided investigations, reviewing skills learned, solving problems and working individually on investigations or problems.

As well, discovery-based learning does not lead to a better understanding of concepts or a higher quality of learning than direct instruction. On the contrary, Klahr and Nigam (2004) find that direct instruction results in much more learning than discovery-based instruction, and that students who learn in a direct instruction environment are no less proficient at translating learning to new situations. A particularly disturbing finding, from a number of studies, is that low-aptitude students perform worse on post-test measures after receiving discoverybased instruction than they do on pre-test measures. In other words, discovery-based instruction might result in learning losses and widen the gap between low- and high-performing students (Clark 1989)

Benchmarks for the Critical Foundations of Mathematics (lager onderwijs)

Fluency With Whole Numbers

1) By the end of Grade 3 (eerste leerjaar), students should be proficient with the addition and subtraction of whole numbers.
2) By the end of Grade 5, students should be proficient with multiplication and division of whole numbers.

Fluency With Fractions

1) By the end of Grade 4 (tweede leerjaar) , students should be able to identify and represent fractions and decimals, and compare them on a number line or with other common representations of fractions and decimals.
2) By the end of Grade 5, students should be proficient with comparing fractions and decimals and common percent, and with the addition and subtraction of fractions and decimals.
3) By the end of Grade 6, students should be proficient with multiplication and division of fractions and decimals.
4) By the end of Grade 6, students should be proficient with all operations involving positive and negative integers.
5) By the end of Grade 7, students should be proficient with all operations involving positive and negative fractions.
6) By the end of Grade 7, students should be able to solve problems involving percent, ratio, and rate and extend this work to proportionality.

Geometry and Measurement

1) By the end of Grade 5 (3de leerjaar), students should be able to solve problems involving perimeter and area of triangles and all quadrilaterals having at least one pair of parallel sides (i.e., trapezoids).
2) By the end of Grade 6, students should be able to analyze the properties of two-dimensional shapes and solve problems involving perimeter and area, and analyze the properties of three-dimensional shapes and solve problems involving surface area and volume.
3) By the end of Grade 7, students should be familiar with the relationship between similar triangles and the concept of the slope of a line.