Scientific constructivism is a well-developed scientific theory based on more than two decades of scientific research into gaining greater understanding of students’ learning of mathematics (Battista, 1999:6). The researcher focuses on mathematics learning because of its unique features in the learning process and because of the importance of mathematics within the SciMathUS programme.
All current major scientific theories describing students’ mathematics learning agree that mathematical ideas must be personally constructed by students as they try to make sense of situations and actively seek to interpret it (ibid.:5). However, since subjects, such as mathematics, are more ‘bounded’ than others by rules, formulae, and procedures they are more likely to be regarded by teachers as producing problems and tasks to which there are ‘correct’ answers. Individual interpretations and construction of ideas and concepts are therefore less likely to be encouraged by teachers than in subjects such as literature and writing for instance (Ismat, 1998:1-4). According to Battista (1999:9) traditional mathematical instruction which ignores students’ personal construction of mathematical meaning, results in the development of students’ mathematical thought not being properly nurtured, resulting in stunted growth. To develop powerful mathematical thinking in students, instruction must therefore focus on, guide, and support their personal construction of ideas. Such instruction encourages students to invent, test, and refine their own ideas rather than blindly follow procedures given to them by others. Research clearly shows that such ‘construction-focused’ mathematics instruction produces more powerful mathematical thinkers.
To illustrate the depth of scientific constructivism, Battista (1999:6) discusses its description of fundamental learning mechanisms namely abstraction, reflection, and learning and offers an example of the type of insight that can result from constructivist research. In scientific
constructivist accounts of learning, abstraction is the fundamental mental mechanism by which new mathematical knowledge is generated. Abstraction is the process by which the mind selects, coordinates, combines, and registers in memory a collection of mental items or acts that appear in the attention field. Abstraction is the critical mechanism that enables the mind to construct the mental entities that individuals use to reason about their ‘mathematical’ realities. But understanding mathematics requires more than abstraction. It also requires
reflection, which is the conscious process of mentally replaying experiences, actions, or
mental processes and considering their results or how they are composed. As these acts of reflection are themselves abstracted, they can become the content – what is acted upon – in future acts of reflection and abstraction.
What emerges from this theory is a picture of meaningful mathematics learning, which arises as individuals recursively cycle through phases of action (physical and mental), reflection, and abstraction in a way that enables them to integrate related abstractions into ever more sophisticated mental models of phenomena. In fact, students’ ability to understand and effectively use the formal mathematical systems of our culture to make sense of their quantitative and spatial surroundings depends on their construction of elaborated sequences of mental models. Initial models in these sequences enable students working with real-world objects to reason about their physical manipulations. Later models permit them to reason using mental images of real-world objects. Finally, symbolic models enable them to reason by meaningfully manipulating mathematical symbols that represent real-world situations. Without this recursively developed sequence of mental models, students’ learning about mathematical symbol systems is strictly syntactic, and their use of symbolic procedures is totally disconnected from real-world situations. Research has shown repeatedly that rote learning of syntactic rules for manipulating symbols is exactly what results for most students in traditional mathematics curricula (ibid.:7).
Instead of presenting students with problems from the outset, traditional education seems to be more preoccupied with giving content. The content is often content which teachers themselves are most knowledgeable about or comfortable with, or content they think will be useful for solving some problems without also dealing explicitly with the process of acquiring this content (Bolhuis, 2003:339). According to Moursund (2006:1) much of the weaknesses in the traditional education system can be discerned by carefully thinking about the following
diagram: a simplified 4-step model explaining the use of mathematics to solve a problem (see Figure 2.2).
Figure 2.2: Simplified 4-step model of using mathematics to solve a problem (Moursund, 2006:1).
Standard estimates are that 80% of mathematics education at grade 12 level is focused on part 2 of the diagram, thus helping students to learn to carry out a number of different types of ‘step 2’ combinations. When the teacher starts the instructional process from part 2 of the diagram the teacher usually shows students several examples of how to solve a certain type of problem and then has them practise this method in class and in homework leading to “mindless mimicry mathematics” (Battista, 1999:4) or as O’Brien (1999) (in Moursund, 2006:4) terms “parrot math”.
Within this traditional mathematics teaching approach, school mathematics is an endless sequence of memorizing and forgetting facts and procedures that are superficial and that make little sense to students. Doing mathematics then becomes an academic ritual that has no real- world usefulness (Battista, 1999:4-5). Because students in traditional curricula learn ideas and procedures by rote rather than meaningfully, they quickly forget them, so the ideas must be re-taught year after year which, according to Battista (1999:2) handicaps our nation in a competitive and increasingly technological global marketplace. Teachers, who teach students rote procedures for doing these novel items, do not test understanding, but mere memorization (ibid.:12). Real world problem 1 Pure mathematics problem Solved mathematics problem Statement about real world 2 3 4
In sense-making curricula, because students retain learned ideas for long periods of time, and because a natural part of sense making is to relate ideas, students accumulate an ever- increasing store of well-integrated knowledge (ibid.:11). Students therefore need enough opportunities to construct through experience the appropriate mental models to serve as the foundation for such abstract learning. If this does not occur students either drop out of the study of mathematics or resort to mindless mimicry (ibid.:12). In part 2 of the diagram mathematics topics are thus learned in a self-contained environment where what is being learned has little immediate use in the lives of students. Students therefore develop little skill at transferring their mathematics knowledge and skills into non-mathematics disciplines or into problems that they encounter outside of school (Moursund, 2006:2).
All current major scientific theories describing students’ mathematics learning therefore agree that mathematical ideas must be personally constructed by students as they try to make sense of situations and actively seek to interpret it (Battista, 1999:5-6). A new classroom environment thus envisions teachers providing students with numerous opportunities to solve complex and interesting problems; to read, write and discuss mathematics; and to formulate and test the validity of personally constructed mathematical ideas so that they can draw their own conclusions (ibid.:4). According to Battista (1999:5) obtaining the facts in the information age is therefore not the problem anymore; it is the analyzing and making sense of these facts by solving problems, reasoning, justifying ideas, making sense of complex situations, and learning new ideas independently, that are of importance. Students should therefore not just follow rules invented by others, but instead make personal sense of the ideas. Developing powerful conceptual structures and patterns of reasoning will enable students to apply their mathematical knowledge and understanding to numerous real-world situations, thus giving them intellectual autonomy in their mathematical reasoning.
The theoretical links between PBL and scientific constructivism
PBL instruction begins at part 1 of Moursund’s (2006:1) diagram which supports the students’ personal construction of ideas set in a real-world context which can produce more powerful mathematical thinkers by increasing understanding and providing opportunities to apply knowledge in novel situations. This type of practice is consistent with the latest scientific research regarding learning and specifically the learning of mathematics (Battista, 1999:12-13).
In PBL students are further encouraged to apply their existing knowledge and to identify their further learning needs. Learning is student-centred and cooperative, with students encountering real-world problem-solving situations in small groups (as seen on part 1 of Moursund’s) diagram that are guided by a tutor whose role it is to facilitate the learning process. This is quite different from most university teaching approaches which concentrate on the transmission of factual knowledge (Yeung; Au-Yeung; Chiu; Mok, & Lai, 2003:237). Students are thus actively engaged in the learning process in constructing personal meaning within their context (Prince, 2004:223). It is however important to note that PBL does not deny the importance of ‘content’ but it does deny that content is best acquired in the abstract, in vast quantities, and in a purely prepositional form, to be brought out and ‘applied’ much later to problems (as seen on part 2 of Moursund’s diagram). Problem-based learning therefore requires a much greater integration of ‘knowing that’ with ‘knowing how’ (White, 2001:69).