Mathematical proficiency

The above section reveals disagreement among mathematics educators about how to distinguish between procedural and conceptual knowledge and learning, however, what is important is to find out what constitutes mathematical competence or proficiency.

Both Niss and Højgaard Jensen (2002) in Denmark and Kilpatrick, Swafford, and Findell (2001) in the USA have worked out criteria for

what mathematical proficiency or competence means. According to the KOM-project mathematical competence:

means the ability to understand, judge, do, and use mathematics in a variety of in- tra- and extra-mathematical contexts and situations in which mathematics plays or could play a role. Necessary, but certainly not sufficient, prerequisites for mathe- matical competence are lots of factual knowledge and technical skills, in the same way as vocabulary, orthography, and grammar are necessary but not sufficient pre- requisites for literacy (Niss, 2003, p. 7).

In addition, eight distinct competencies were listed:

 Thinking mathematically (mastering mathematical modes of thought)  Posing and solving mathematical problems

 Modelling mathematically (i.e. analysing and building models)  Reasoning mathematically

 Representing mathematical entities

 Handling mathematical symbols and formalisms  Communicating in, with, and about mathematics  Making use of aids and tools (IT included)

The KOM-project has had an influence on the last curriculum in Norway, and has also made an impact on the design and analysis of the PISA stud- ies. However, in this thesis limiting the focus to algebra and mostly to the transformational activity of the topic, led to the choice of mathematical proficiency as the applied framework. This framework was worked out by Kilpatrick et al. (2001). Five intertwined strands constitute mathematical proficiency. The strands are:

 Conceptual understanding  Procedural fluency

 Strategic competence  Adaptive reasoning

 Productive disposition (ibid p. 116-117).

The strands are interwoven and interdependent and to be proficient means that all strands have to be present. A prerequisite developing proficiency is that thinking strategies are emphasised.

In the next section the different strands will be outlined. 5.4.1 Conceptual understanding

Conceptual understanding correlates with what is outlined above about conceptual knowledge and is defined to be “comprehension of mathemati- cal concepts, operations, and relations” (ibid, p. 118). It means knowing more than facts and rules, knowing both how and why. This knowledge may be tacit, in that students can understand before they are able to ex- press their understanding.

One indicator of conceptual understanding is that students are able to use different representations for mathematical situations and choose the most useful representation.

The advantage of conceptual understanding for students is that they need not have to ‘learn’ so much because they can see connections. What is already learned is encapsulated; they have made compressions and knowledge is stored in clusters. What is learned with understanding is less likely to be forgotten, and what might be forgotten can be reconstructed on the basis of the connections made to what is known from before.

What is learned with understanding creates the basis for new learning. It enables students to check themselves, to solve unfamiliar problems, to analyse similarities and differences between different mathematical situa- tions, and to give arguments for why procedures work. Conceptual under- standing makes students confident, which again helps them to reach new levels of understanding.

5.4.2 Procedural fluency

Procedural fluency correlates to what is written above in section 5.3 about procedural knowledge, and it is defined to be “skill in carrying out proce- dures flexibly, accurately, efficiently, and appropriately” (ibid, p.116). Procedural fluency supports conceptual understanding in analysing differ- ences and similarities between methods and procedures for mathematical work. One advantage of procedural fluency such as skills in performing operations and working out procedures efficiently and accurately, is that the focus can shift from a concentration on, and a struggle to follow known procedures step by step. Instead the students can use energy on developing connections and gain an overview of the topics at hand. In that way procedural fluency supports conceptual understanding. Students who learn procedures with understanding can adapt and modify procedures making them easier to use, and transfer them to new contexts.

Procedures learned without understanding are stored as isolated bits of knowledge, and then students often think that there are different proce- dures for different contexts. They see no connection.

However, students have often learned procedures and methods without understanding. To engage students in activities later, from which an un- derstanding could be developed, is experienced to be a problem. It is thus important to promote teaching for understanding from the beginning.

It is important to practice in order to gain procedural fluency, which can promote knowledge of ways to estimate and to control the results of procedures. This control is also part of procedural fluency.

Kilpatrick et. al. (ibid) claim that procedures can be developed to be general procedures, which can solve classes of problems. Examples are the general rules for arithmetic operations. These rules or procedures can be concepts in their own right. Hiebert and Lefevre (1986) call them su- per-procedures (see section 5.3).

5.4.3 Strategic competence

Strategic competence is defined to be the ability to formulate, represent, and solve mathematical problems. This strand is connected to problem solving, and problem formulation. For this thesis, this is related to solving word problems. Having strategic competence implies that the students have grasped the problem situation and the relations between the known and the unknown quantities. Then they must represent the problem situa- tion in one way or another and then reformulate it in mathematical sym- bols and operations before the problem is solved.

It is important that students can represent individual problems, but it is also necessary that they can see that some problems have the same math- ematical structure.

In becoming proficient problem solvers, students learn how to form mental repre- sentations of problems, detect mathematical relationships, and devise novel solu- tion methods when needed (ibid, p. 126).

When students have not learned procedures to follow, they have to seek for new approaches. This promotes development of flexibility. One indi- cation of a well-developed strategic competence is that a person is able to choose flexibly between different approaches. Related to this study a stra- tegic competent person could choose between reasoning, use of tables, guess and check, or equations depending on what is most suitable in the situation. It is of great importance that students are offered non-routine problems in order to develop strategic competence.

Related to strategic competence is also the competence in choosing the most efficient procedure in computations; which is related to a well-

developed procedural fluency.

Strategic competence is intertwined with the other strands. Procedural fluency develops as students use their strategic competence to solve tasks efficiently. In order to solve cognitively challenging problems students are depending on procedural fluency. Also experiences in dealing with com- plicated problems promote conceptual understanding.

5.4.4 Adaptive reasoning

Adaptive reasoning is defined as thinking logically about the relationships between concepts and situations. The authors claim this strand to be the ‘glue’ that holds everything together. Students with adaptive reasoning can think logically about what is going on in the mathematical activity. They can explain and justify their work and their arguments. The word justify is used in the sense of ‘provide sufficient reason for’ (ibid, p. 130). To promote adaptive reasoning, it is said to be important to establish classroom norms which make it an expectation that students shall provide arguments and reasons for their solutions. This is a way to improve con- ceptual understanding. For students in this thesis it could be to explain

why the rules for fraction operations work. Adaptive reasoning interacts with the other strands especially when working with non-routine prob- lems. When determining if a procedure is appropriate for the actual prob- lem, adaptive reasoning is required and interacts with the strand of strate- gic competence.

5.4.5 Productive disposition

This strand is defined to be the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.

This strand has to do with students’ motivation and attitude to mathe- matics and learning (Marton & Säljö, 1976; Novak, 2002) which was a theme in section 5.3. The point in mathematical proficiency is that this productive disposition can be promoted by conceptual understanding and confidence in mathematics. The more concepts a student understands and the more fluent and flexible he or she is in working out procedures, the more sensible mathematics is for him or her.

All the strands are of interest for this study, however, the emphases will be on the first three of them.