3.7
STUDENTS’ REPRESENTATIONS OF NUMBER OPERATIONS
Research has shown that young children can solve a range of basic computational
problems presented in context by modelling the action or relationship
described in the problem (Carpenter, et al., 1999). However, when students
in the current study were presented with the cards
4 – 3
and 3 –
4 without contextual cues, it was apparent that some were
confused as to what to model and how to demonstrate the problem with the
cubes. In order to justify their conjecture six percent of the sample,
all Year 4, consistently modelled all numbers on the cards for each of
the three operationsæaddition, subtraction, and multiplication –
in the following manner:
| Q: | Is 4 plus 3 the same as 3 plus 4? | |
| qqqq qqq | qqq qqqq | |
| Q: | What about 4 minus 3 and 3 minus 4? Are they the same? | |
| qqqq qqq | qqq qqqq | |
| Q: | Does 2 times 5 give the same answer as 5 times 2? | |
| qq qqqqq | qqqqq qq | |
Students who started with the above representation of the problem used a range of strategies. One student was not able to offer any further explanationæthe cubes were expected to demonstrate the validity of the conjecture.
| Case A: Year 4 student |
| Q: | Is 4 plus 3 the same as 3 plus 4? |
| [no explanation offered] | |
| Q: | What about 4 minus 3 and 3 minus 4? Are they the same? |
| [no explanation offered] | |
| Q: | Does 2 times 5 give the same answer as 5 times 2? |
| [no explanation offered] |
Others appeared to use the modelling of each respective number to reason that the commutative law held for all three operationsæbased on the logic that in all cases the same number of cubes were present for each pair of cards (e.g., Case B). or indicating that the action (e.g., take away) would be the same, but in reverse for each representation (Case C).
| Case B: Year 8 student |
| Q: | Is 4 plus 3 the same as 3 plus 4? |
| It’s just the same but instead of the 4 going over here you put it on that side. | |
| Q: | What about 4 minus 3 and 3 minus 4? Are they the same? |
| Yes. Four take away 3 is the same as, instead of 4 being over here it’s just changed around. | |
| Q: | Does 2 times 5 give the same answer as 5 times 2? |
| You’ve got the same numbers, changed around. |
| Case C: Year 8 student |
| Q: | Is 4 plus 3 the same as 3 plus 4? |
| Well, that
is seven but the way of working it out, you are adding terms together. 4 plus 3 is 7 but they’re just changed around to 3 plus 4. |
|
| Q: | What about 4 minus 3 and 3 minus 4? Are they the same? |
| They are both the same but swapped around and equal to one because, like, if you had 4 minus 3 it would be 4 blocks and you take away 3 and you’ve only got one left. [3-4] It’s just the same as 4 minus 3 but it’s round the other way. You’ve got 4 and you just take away 3 again. | |
| Q: | Does 2 times 5 give the same answer as 5 times 2? |
| Yes, it will, the order is just changed around. |
Others students appeared to be constrained by their representation and forced into unreasonable answers (raising serious concerns about the development of their number sense) such as 3 – 4 = 3 due to their reliance on their representation of each of the numbers on the cards in the first instance:
| Case D: Year 4student |
| Q: | Is 4 plus 3 the same as 3 plus 4? |
| Yes. It shows that you know how to do your pluses . | |
| Q: | What about 4 minus 3 and 3 minus 4? Are they the same? |
| You’ve
got 4 then you take away 3 and that just leaves 4. And then you have 3 and you take away 4. You’d leave yourself 3. |
|
| Q: | Does 2 times 5 give the same answer as 5 times 2? |
| When you do five times 2 you just change it around. |
Several other students
at both year levels also used this strategy for either the subtraction
or multiplication question with modification. For example, they removed
the set of 3 cubes from both the set of 4 and the set of 3 to obtain the
correct answer of 3 for 4 – 3 = 3. Likewise, several students who
represented the multiplication problem 5 x 2 by qqqqq qq then
gave the answer 10 and counted out ten blocks, effectively representing
the equation 5 x 2 = 10 with block, rather than demonstrating the multiplicative
action in the problem.
In all cases the inability of students to provide a demonstration of addition,
subtraction or multiplication with cubes is of concern and raises serious
issues concerning the links between the students informal knowledge and
their formal mathematics.
Students who experience feedback and correction which signal low-ability may learn to behave accordingly. Continual querying or negative messages undermine the student’s self-efficacy to the extent that the student comes to believe that a poor performance will result from an attempt on a task. This is not to suggest that a student’s response should not be challenged and questioned; indeed challenging in a supportive environment is fundamental to an inquiry classroom. What is also important in these classrooms is that the authority does not reside entirely with the teacher. On-going authoritative confrontation is not conducive to enriching mathematical experiences.
Our impression from viewing the tapes was that interviewers created a supportive and encouraging context for the students. For the most part, students appeared to enjoy the experience and responded well to the challenge of the research situation. Many students received positive messages that they were capable of the mathematics involved and had articulated clearly (e.g., you are a very good teacher; I like the way you explained that; that was a very good explanation). Some interviewers were very successful at exploring and extending students’ thinking by probing. However, in a small number of instances, probing generated confusion in the students’ thinking.
It was also noted that in questions concerning the identity element (Questions 4 & 5) that several students were not provided with the opportunity to explain their reasoning.
