Mathematical Proficiency

The mathematics curriculum during elementary school in Sweden has many components, but there is a strong emphasis on concepts of numbers and operations with numbers. From an international perspective, mathematics knowledge is defined as something more complex than concept of numbers and operations with numbers. Kilpatrick et al. (2001) argue for five strands which together build students’ mathematical proficiency. The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs that constitute mathematical proficiency. In their report they discuss,

1. Conceptual understanding is about comprehension of mathematical concepts, operations, and relationships. Students with conceptual understanding know more than isolated facts and methods. Items measuring conceptual understanding are for instance: “Your number is 123.45. Change the hundreds and the tenths. What is your new number?

2. Procedural fluency refers to skills in carrying out procedures flexibly, accurately, efficiently, and appropriately. Students need to be efficient in performing basic computations with whole numbers (e.g., 6+7, 17–9, 8×4) without always having to refer to tables or other aids.

3. Strategic competence is the ability to formulate, represent, and solve mathematical problems. Kilpatrick et al. (2001, p.126) give the following example of item testing strategic competence: “A cycle shop has a total of 36 bicycles and tricycles in stock. Collectively there are 80 wheels. How many bikes and how many tricycles are there?”

4. Adaptive reasoning refers to the capacity for logical thought, reflection, explanation, and justification. Kilpatrick et al. (2001) gives the following example where students can use their adaptive reasoning. “Through a carefully constructed sequence of activities about adding and removing marbles from a bag containing many marbles, second graders can reason that 5+(–6)=–1. In the context of cutting short bows from a 12-meter package of ribbon and using physical models to calculate that 12 divided by 1/3 is 36, fifth graders can reason that 12 divided by 2/3 cannot be 72 because that would mean getting more bows from a package when the individual bow is larger, which does not make sense” (p.130).

5. “Productive disposition is the habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy” (Kilpatrick et al., 2001, p.5). Items measuring productive disposition are for instance: “How confident are you in the following situations? When you count 8-1=___+3 (completely confident, confident, fairly confident, not at al confident).”