Scenario: Find 5/3 when 9 is the whole
Grade (student age): Year 5 (age 10-11)
Contributed by: Ove Gunnar Drageset, University of Tromsoe, Norway
The teacher has 16 years of teaching experience. The lesson is from fifth grade (students aged 10-11) in the middle of the year. The students have only met fractions like ½ and ¼ used in everyday situations prior to fifth grade. 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 competence goals regarding fractions after grade seven focuses on calculations using positive and negative fractions, and placing fractions on a number line. The following excerpt is from one of the first lessons on fractions, and the students were not used to fractions larger than one. Also, fractions as set models were not well known.
The task is to find 5/3 when nine chips is one whole. Several students struggle with the task, and the teacher has to help almost every pair of students in the classroom. Some students try to divide the nine chips in five groups. With one pair of student the teacher sits for a while helping them through the entire task. This is an excerpt from their dialogue:
T: You have to divide them into thirds. That is what is written. Five thirds.
S1: (organises the chips in three rows of three chips)
T: Yes. Then this is one third (moves one column of three chips away from the others), one more third (moves the second column) and one more third (moves the third column), then we have three thirds. How can we have four thirds?
S2: Divide them up a little more
T: You shall not divide them more, no. That is, that… what is it that we lack now? We have three thirds, what do we lack to have four thirds?
S2: Six chips (no reason provided for the answer, might be guessing)
T: Five thirds.
S2: Oh yes.
T: Yes, than you have to fetch the chips that we lack.
S2: But why, but how can we do that?
T: They are there (points at the chips that are not in use)
Later in the dialogue the cause of the problem seems to be revealed:
S2: But the task is to build nine, isn’t it?
T: No, it was nine at the beginning.
T: Yes, but then you have to fetch more.
S2: But if nine is one whole?
Knowledge Quartet Coding Commentary
Contributed by: Ove Gunnar Drageset, University of Tromsoe
Knowledge Quartet Dimension: Connection
Knowledge Quartet Code: Anticipation of complexity (AC)
Scenario: Two examples from the teaching of fractions
First, the teacher tells the students that the nine chips have to be divided into three thirds. This is done to help them to see how much one third and three thirds are. Finding one third is necessary to be able to find five thirds, and how much one third and three thirds are is emphasised by the teacher. This is an example of some anticipation of complexity, even though it can be argued that it is the least one could expect in this situation.
Second, the teacher does not anticipate that the student has difficulty understanding why it is possible to add more chips to the nine that are one whole. This is connected to the problem of understanding that fractions can be more than one whole, which might be especially difficult using the set model of fractions. It appears that the student does not understand how it is possible to pick more chips when nine chips is the whole. This is further exemplified through questions as ‘So you are allowed to fetch more?’ and ‘How can you fetch them then?’ This seems to be connected to this teacher’s emphasis on the whole as everything. It is obviously confusing that one can suddenly take more chips when one whole is thought about as everything. The problem seems to be too difficult for these students as they are not familiar with fractions larger than one or fractions as set models. The choice of task demonstrates a lack of AC. Also, the teacher does not anticipate nor understand the complexity of adding more chips to nine when nine is defined as one whole.
Lack of AC further seems evident by presenting a “naked number problem” but not giving it a context—as in, putting it into a word problem, which would help students conceptualize what it means to have 5/3 of 9.