Research from different theoretical approaches has provided information regarding processes that are important for young children to solve mathematical problems adequately (Donlan, 1998; Koriat, 1995; Lucangeli & Cornoldi, 1997; Metcalfe, 1998; Montague, 1998; Schunn, Reder, Nhouyvanisvong, Richards, & Stroffolino, 1997; Schwartz & Metcalfe, 1994). Our model of mathematical problem-solving integrates nine cognitive processes and two metacognitive parameters. To clarify our conceptual framework, we describe the cognitive processes included in mathematical problem-solving (see NR, S, K, P, L, C, V, R, N in Table 1).
Cognitive processes' enable the translation of numerical (NR-processes), symbolic (S-processes), simple linguistic (L-processes) or more complex contextual (C-processes) information into mental representations or visualizations (V-processes) of the problem or task. Furthermore, dealing with number system knowledge (K-processes), eliminating irrelevant information (R-processes) and estimating based on number sense (N-processes) typify mathematical problem solving and precede procedural calculation processes (P-processes), leading to the computing of the solution (Desoete, Roeyers, & Buysse, 2001).
In addition 'metacognition' seems to be involved in successful mathematical problem solving (see Pr and Ev in Table 1) (Desoete, Roeyers, & Buysse, 2000b; Lucangeli & Cornoldi, 1997; Montague, 1998; Tobias & Everson, 1996). Flavell (1976) defined metacognition as '...one's knowledge concerning one's own cognitive processes and products or anything related to them' (1976, p. 232). Studies concerned with problem- solving strategies in mathematically average-performing children have shown that inetacognition is instrumental during the initial stage ('Prediction', Pr) of mathematical problem solving, when subjects build an appropriate representation of the problem, as well as in the final stage ('Evaluation', Ev) of interpretation and checking the outcome of the calculations (Verschaffel, 1999). Prediction guarantees working slowly when exercises are new or complex and working fast with easy or familiar tasks. Evaluation refers to the retrospective verbalizations after the event has transpired (Brown, 1987), wher e children look at what strategies were used and whether they led to a desired result or not.
Children with mathematics learning disabilities show some typical shortcomings in different 'cognitive' processes (NR, S, K, P, L, C, V, R, N) of mathematical problem solving (e.g. Geary, 1993; McCloskey & Macaruso, 1995; Rourke & Conway, 1997; Verschaffel, 1999). Some of these children have problems in number (NR) and symbol (S) comprehension and production. They confuse 6 with 9, 'drie' (three in Dutch) with 'vier' (four in Dutch) or x with +. Other children with mathematics learning disabilities lack the needed number system knowledge (K) or make mistakes of a procedural (P) type. These children confuse digits and tens or forget, for example, a multidigit addition, to start in the right column. Language-dependent (L) and mental representation (V) related mistakes or problems dealing with linguistic or contextual (C) information as well as a lack of number sense (N) are also typical for some children with mathematics learning disabilities (Desoete et al., 2000; Desoete, Roeyers, Buysse, & De Clercq, 2000). Furthermore, children with mathematics learning disabilities often show below-average performances on the different metacognitive (Pr, Ev) parameters included in mathematical problem-solving (Desoete et al., 2001).
To focus on the problems of students with mathematics learning disabilities and to tailor a relevant instructional program, it is necessary to assess the 'cognitive' and 'metacognitive' strengths and weaknesses of these children. No test is currently available for a combined assessment of cognitive and metacognitive skills in grade 3 of the elementary school. …