Formative Assessment 88

Assessment Problem: 

What can you say about c if c + d = 10 and c is less than d?

Common Responses: 

c < 5 (11%)
c = 1, 2, 3, 4 (systematic list) (19%)
c = 10 - d  (4%)
Unsystematic list (1%)
One value only (usually c=4) (39%) I: Why 4?
S: Because four is less than five

Mathematical Issues: 

In contrast with the understanding of a letter as a specific unknown value, a generalised number is able to take more than one value. A distinction can be made between the idea of a letter taking on several values in turn and a letter representing a set of values simultaneously. The problem has got to be especially confusing for students who struggle with the idea of a letter representing something “unknown”--therefore they treat it as having a specific value.

Variables & Expressions
Research References: 
Hart, K.M.; Kerslake, D; Brown, ML; Ruddock, G; Kuchemann, DE; McCartney, M. (1981). “Algebra” in Children's Understanding of Mathematics: 11-16. London: John Murray Pub.