a) 8 + 9 z
-I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 1 1 1 1 1 ^
0
10
20b) 11 - 6 =
j i i i I i i i i I i i i i I i i i i Lo
10
20
The n um ber line is an established model for teaching num ber that is said to b e a schem atic representation o f a string o f beads. T he num ber line can b e show n
‘co m plete’ (w ith ones), ‘h a lf com plete’ (w ith tens only m arked) o r ‘em p ty ’ (w ithout num bers (V an den H euvel-Panhuizen, and Senior, 2001, p. 82). T he em pty num ber line is now used internationally. It is closely connected w ith RM E. T his item, borrow ed from th e D elPH E project, invited students to u se a com plete n um ber line. W hile th e tw o tasks (a & b) involve small num bers, they call for different operations w ith subtraction arguably m ore difficult than addition, depending on the problem structure represented and the strategy used. F o r exam ple, at a basic counting level, the hops on the nu m ber lines m ight m ove in different directions. In T ask 2(a) (addition) the hops m ight start at 8 and m ove 9 steps to th e right. In T ask 2(b) (subtraction) the hops m ight start at 11 and m ove 6 steps to th e left. It m ight then b e argued that task 2(b) has m ore cognitive dem and than 2(a) so I am here presenting a table show ing how participants perform ed on T ask 2(b) only.
T a b l e 4 .6 : P a r t i c i p a n t s ’ r e s p o n s e s t o T a s k 2 ( b )
TASK 2b Total
Not Done Incorrect Incom plete C orrect
cose
R esult Fail0
4 17 114 135 P a s s1
0
10
5061
C redit0
0
115
16
Total1
4
28179
212
T able 4.6 show s that m o st participants (n = 179) succeeded in obtaining the correct answ er for T ask 2(b). 84% o f participants w ere able to associate th e n u m b er line w ith a satisfactory representation o f a subtraction task. Further investigation o f th e 33 responses that w ere coded either ‘n ot done’ (n = 1), ‘incorrect’ (n=4), or ‘incom plete’
(n= 28) provides interesting evidence o f different structures o f attention paid to the task b y different candidates.
S n a p s h o t s o f R e s p o n s e s t o C a t e g o r y 1 T a s k s
O ne young student, M ary aged 20, correctly filled in the digits lto 19 on the num ber line for item 2(b), but although she successfully com pleted 2(a), did nothing further w ith either th e iconic or the sym bolic representation. M ary m ight be said to h av e discerned the detailed calibration o f the nu m ber line b u t did not proceed further, so her response w as coded ‘not do ne’.
O f the four participants w hose answ ers to item 2(b) w ere coded ‘incorrect5 one treated the item as i f it were addition ( 1 1 - 6 = 17) w ith appropriate ju m p s forw ard on th e n um ber line. She appears to have read the m inus sign as a plus. T w o other
students correctly indicated six jum ps back from eleven on the n um ber line but w rote the answ er as 11 - 6 = 4. One o f these two w as m ore sophisticated. She placed a directional arrow to the left labelled it m inus ( - ) but appeared to have counted the rem aining spaces as 4 instead o f 5 (see Figure 4,4 below ). I suggest that she, like the other m entioned respondent, started counting th e ‘d ifference5 either to or from the next ‘fre e’ num eral, i.e., 4. It is tem pting to suggest th a t inviting participants to m ake an ico nic rep resentation confused these two students instead o f helping them .
F ig u r e 4 .4 : S t u d e n t ’s r e s p o n s e to T a s k 2 ( b )
b) n - 6 = 4-
i , , ■ , ... ... ..
q , -a t i * ‘i i o » 2 0
A n oth er o f the four students that were coded incorrect, B ella provides
interesting insight into levels o f fam iliarity w ith the use o f th e n u m ber line. H er iconic representation show s a jum p, starting (incorrectly) at 1 to 11. A second arrow goes tow ards the left from 11 to 6. This ju m p is labelled w ith a m inus ( - ) sign. H ow ever, the individual calibrations are not labelled as num erals. Instead, eleven little boxes are draw n. T h e difference betw een 11 and 6 is labelled “ equal 5” . F o r th e sym bolic representation, B ella w rote 11 - 6 =15, w hich suggests that she w as relying on the n u m ber line to find the answ er to the task and m isread 5 as 15.