Scenario: Hans teaching fractions greater than one
Country: Norway
Grade (student age): Year 5 (age 11-12)
Contributed by: Bodil Kleve, Oslo and Akershus University College of Applied Sciences
Context – national, curricular, professional, other
In our present curriculum LK 06, competence aims for the subject are presented after year 2, year 4, year 7 and year 10. This lesson is from year five. With regard to fractions, competence aims after year 4 do not include any. After year 7 fractions are included in Competence aims for numbers and algebra: “The aim for the education is that the pupil shall be able to
- describe the place value system for decimal numbers, calculate with positive and negative whole numbers, decimal numbers, fractions and percentages, and place them on the real number line
- find common denominators and carry out addition, subtraction and multiplication of fractions”
In grades 5-7 (age 11-13) in Norway, it is quite common when introducing work with fractions to emphasise fractions as part of a whole. Also in most textbooks used in primary schools in Norway, fractions are introduced and emphasised as part of a whole. This was the case for the book used in the classes where the research referred to took place.
Scenario
The lesson objective was written on the smart board: ‘To be able to calculate with fractions which are bigger than one whole’. There were 4 circles, 2 on each side of an equal sign on the smart board. The circles were each divided into eight pieces. The teacher, Hans, was a young man who had finished his teacher education (specialised in mathematics) 4-5 years ago. He had shaded five pieces of the first circle and four pieces in the second on the left side of the equal sign (see below) and asked the class: How much is this all together?
Hans: Jens?
Jens: Eleven ninth, no, wait a minute
[13 seconds]
Hans: I can do it on the other side too so you can see it?
Hans then shaded five pieces in the first and four in the second circle on the right side, exactly the same as on the left side (see below),
Again he addressed Jens:
Hans: Jens?
Jens: Ten ninth
Hans: Is it? How many, how big part of the whole is this one? (pointed to the circle to the very left)
Jens: Five,…. Five ninth
Hans: Are there nine? Is it ninth? Petra?
Petra: Eight
Hans: Eight yes.
Then the lesson proceeded. The calculation 5/8+4/8=9/8 was written under the circles, and 9/8 was converted to 1 1/8. To illustrate the answer, on the right side of the equal sign, Hans erased three of the four shaded pieces in the second circle and filled up the first (see below).
Knowledge Quartet Coding Commentary
Contributed by: Bodil Kleve, Oslo and Akershus University College of Applied Sciences
Knowledge Quartet Dimension: Transformation
Knowledge Quartet Code: Choice and Use of Representations (CUR)
Scenario: Hans teaching fractions greater than one
Transformation as a dimension in the KQ is about “the capacity of a teacher to transform the content knowledge he or she possesses into forms that are pedagogically powerful” (Shulman, 1987, p. 15). In order to make one’s own knowledge accessible to others, and to assist concept formation, teachers often use examples, illustrations and representations (Rowland, Turner, Thwaites, & Huckstep, 2009).
In this episode, Hans chose to represent fractions with circles divided into eight segments. The task was to add 5/8 and 4/8 represented by shaded pieces of circles. Hans had made two “wholes” each divided into eights as representations. He shaded 5 pieces in one circle to represent 5/8 and 4 pieces in the next representing 4/8. He had drawn four circles, two on each side of an equal sign. What makes this episode interesting is that based on the representations of fractions Hans chose, there was lack of consensus between Hans and Jens what the whole was. Having divided the circles in eighths, it was evident for Hans that the whole was eights. However, Jens kept answering ninths. Also, after having shaded exactly the same on the right side (it is unclear why Hans did this), Jens kept answering ninth. Hans then broke the task down and asked how many were shaded in the first. When Jens kept answering ninth, Hans gave in and asked another child, who answered eight, which Hans confirmed.
Hans’ question, ‘are there nine’, indicates that Hans thought Jens had counted wrongly (this was confirmed in a post lesson interview afterwards). However, I suggest that Jens’ answer (five ninths) indicates that Jens looked upon the shaded pieces as the whole. Hans’ use of shaded pieces of circles as representations of fractions did not turn out (at least not for Jens) the way he had intended. Jens’ conception of the shaded pieces put a demand on Hans’ knowledge. First, to see why Jens perceived the shaded pieces as the whole, then to justify for the class why that might or might not be correct. Finally were the representations of fractions greater than one he used helpful, or did they constrain pupils’ conceptions of improper fractions?
In his illustrations of fractions, Hans emphasised fractions as part of a whole. However, confusion occurred with regard to what was the whole. The goal for the lesson was to ‘calculate with fractions which are bigger than one whole’. According to Dickson, Brown and Gibson (1984), representing fractions as sub areas of a unit area and part of a whole is inconsistent with the existence of improper fractions. Anghileri (2000) warned against emphasising fractions as part of a whole, claiming that success in working with fractions depends on the ability to see the fraction both as representing a number and as a ratio which reflects the procedure for finding the number. According to Askew (2000) possibilities for obtaining a well-developed concept are limited if one focuses on fractions as part of a whole.
The teacher’s response to Jens’ ninths suggests that he perceived eights as the only possible whole. The way Hans illustrated fractions as part of a whole in this lesson also indicates lack of knowledge about the above referred research which suggests that illustrating fractions as part of a whole constrains and limits pupils’ conceptions of fractions. A number line marked in eights might have been an alternative way of representing the fractions in this case.
References
Anghileri, J. (2000). Teaching number sense. London: Continuum.
Askew, M. (2000). What does it mean to learn? What is effective teaching? In J. Anghileri (Ed.), Principles and Practices in Arithmetic teaching (pp. 134-146). Buckingham Open University Press.
Dickson, L., Gibson, O., & Brown, M. (1984). Children learning mathematics : a teacher's guide to recent research. Eastbourne, East Sussex: Holt, Rinehart and Wilson.
Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing Primary Mathematics Teaching, Reflecting on Practice with the Knowledge Quartet. London: SAGE publications Ltd.
Shulman, L. S. (1987). Knowledge and Teaching: Foundations of the New Reform. Harvard Educational Review, 57(1), 1-22.