Scenario: Finding three fractions that adds up to one whole
Country: Norway
Grade (student age): Grade 7 (age 12-13)
Contributed by: Ove Gunnar Drageset, University of Tromsoe, Norway
Context
The teacher has five years teaching experience. The lesson is halfway through grade seven. The competence goals in the Norwegian curriculum are not formulated for each year, but are given after the completion of grade 4, 7 and 10. The students have only met fractions like ½ and ¼ used in everyday situations prior to fifth grade. The competence goals regarding fractions after grade seven focuses on calculations using positive and negative fractions, and placing fractions on a number line.
In this case the teacher asks the students in plenary of suggestions to a task that many found difficult. The task was to find three fractions that added together was equal to one whole. One student suggested 6/12, 1/3 and 1/6, and the teacher wrote 6/12+1/3+1/6 on the blackboard. Then this followed:
SM: Yes (confirming that the teacher had written the correct fractions).
T: Yes. And how did you do now?
SM: (impossible to hear the start)… with four.
T: You expanded with four, because you wanted to have a common denominator (writes x4 behind numerator and denominator of the fraction 1/3). Yes.
SM: And then we get four twelfths.
T: Yes, if we now just write below… six twelfths (writes 6/12 on a new line below 6/12+1/3+1/6) plus four twelfth (writes +4/12 below 1/3). Yes.
SM: And then one sixth multiplied by two
T: Yes. Above and below when we expand. Yes. (writes x2 behind numerator and denominator of the fraction 1/6).
SM: And then I got two twelfths.
T: Yes… (writes +2/12 below 1/6). And then we add these (pointing at 6/12+4/12+2/12), six plus four plus two twelfths equals twelve twelfths (writes 6+4+2 as numerator and 12 as denominator on a new fraction, then =12/12=1)… and that equals one whole. Great.
The same pattern was repeated four times with different student suggestions.
Knowledge Quartet Coding Commentary
Contributed by: Ove Gunnar Drageset, University of Tromsoe
Knowledge Quartet Dimension: Foundation
Knowledge Quartet Code: Awareness of purpose (AP) and possibly Concentrates on procedure (COP)
Scenario: Finding three fractions that adds up to one whole
The task is difficult because the students do not know, or haven’t seen examples of, how to approach such a task. If the teacher had tried to enlighten how the students that found an answer managed to find this that would probably have been helpful for other students. Instead, the teacher chooses to check the correctness of the answers. This is done by adding fractions, which the students already had plenty of practice doing. It is an example of lack of awareness of purpose because the teacher does not go into the purpose of the task (finding three fractions) but instead chooses to check the answers. This example is coded as Awareness of Purpose (AP) because it is an example where the teacher lacks awareness of what the students needs to know or practice in order to solve similar tasks. Instead of trying to highlight how the answer was reached (using problem solving), the teacher focuses on a procedure of checking the answer.
It might also be an example of teachers that concentrates on procedures (COP) as the teacher in this problem-solving task uses a familiar procedure of checking correctness instead of enlightening how the students thought when searching for an answer.