Tackling misconceptions

Helen Hindle, Hugh Salter and Andrew Blair, three teachers at Longhill High School (Brighton, UK), collaborated on a lesson study in November 2015. They chose the prompt to challenge students' misconceptions about adding fractions.

The inquiries that developed from the prompt featured rich discussions in which students reconstructed their understanding of fractions.

Andrew Blair reports on the lesson study:

In the lesson study cycle, which involved year 7 classes, I went first. I decided to use a number line as a tool with which to approach the concept of a fraction. Before showing the class the prompt, we started by locating fractions on a number line.

This led immediately to our first misconception about representing 1/6, which one student argued should be placed half way along the number line as six is half of twelve.

Speculation about patterns

The students' questions and comments about the prompt provided a strong foundation for inquiry. In particular, the speculation around the solution to 1/4 + 1/5 motivated the students to request instruction in how to add fractions.

Should we continue the sequence 5/6, 7/12 by adding two to the numerator and six to the denominator, giving 9/18?

Or should we apply the 'rules' - the sum of the denominators is the numerator, and their product is the denominator - giving 9/20 ?

Inquiry for all attainment levels

One class with lower prior attainment that was part of the lesson study posed meaningful questions and made insightful observations (see picture). The students' responses show the potential of the prompt to promote questioning and noticing in all classes.

As the inquiries developed, students were taught to link the number of intervals on the number line with the product of the denominators. The students then showed the sum of any two fractions on a number line by using equivalent fractions. So, typically, a student went on to show 1/4 + 1/5 on a number line of length 20, explain why it is equivalent to 5/20 + 4/20, and give the solution 9/20.

Misconceptions identified during the lesson study

1/6 should be placed half way along a line of 12 units because the 'number' six stands for the length along the line. The student who said this had no problem marking a quarter. Thus, while students have a sense of 1/4, they might not have developed a conceptual understanding.
Having started with number lines of length 12 units, students refused to use a line of six units to show the first calculation in the prompt. This revealed an inability to conceive of a fraction as part of any whole. Once a third was represented by an arrow four units along a line of length 12, students would not accept it could also be shown as two units along a shorter line of length six.
To add two fractions, students claimed, you add the numerators and then the denominators separately. Thus, 1/2 + 1/3 = 2/5. This shows a misconception of fractions as two unrelated 'numbers'.

To add two fractions, you add the denominators to get the numerator in the answer and multiply them to get the denominator. (As students realise during their inquiry, this works for unit fractions, but not when the numerator is greater than one.)

When showing the solution to 1/2 + 1/3 on a number line, students start both fractions at zero, rather than place one fraction after the other (see illustration). The idea that the fraction of a line can only be shown from zero was surprisingly common.

Misconceptions and inquiry

Exposing misconceptions

Students form generalisations as they try to make sense of the world around them. As they transfer one general rule to another context, the rule might no longer work. For example, a method to find the sum of two integers will not work to find the sum of two fractions (or, at least, not in the same way).

The inquiry teacher aims to expose the misconceptions that students hold, examine them through an inquiry process and ensure students are given the opportunity to revise their thinking.

Agata Glonek, a year 6 teacher at Canon Barnett Primary School in Tower Hamlets (London, UK), makes this point. She contributed to a professional development project in 2020. The project focused on how teachers can close gaps caused by missed learning during the Covid-19 pandemic. (Read the project report.)

Agata highlighted the potential of Inquiry Maths prompts to identify gaps in learning and expose misconceptions. She writes that children's initial ideas are "not necessarily correct, but allow for formative assessment." Teachers can adapt their planning to meet the needs of the pupils:

"Constant assessment for learning is key if we as teachers want children to gain a greater conceptual understanding which is more meaningful and deeper. As teachers we need to be alert to any misconceptions and adapt our teaching to support our children."

In the report, Agata was concerned that inquiry learning had suffered during the pandemic and concluded, "This is unfortunate as Tower Hamlets schools were investing a lot of energy developing this area and it will be an area schools must reactivate."

July 2022

Sowing misconceptions

In an Inquiry Maths workshop at Ecolint in Geneva (Switzerland) in May 2022, Catherine and Xavier Dore, both teachers of mathematics, were concerned that students' responses to a prompt could sow misconceptions.

In the workshop, the participants looked at classroom examples of the intersecting sequences prompt. One teacher had written a student's comment on the board: "If you multiply 3n by 2n, you get 6n." Although this is wrong algebraically (and the teacher should correct the mistake as the inquiry develops), it does advance the inquiry because there is a link between the coefficients of n.

Rather than seeing the potential for launching a line of inquiry, Catherine and Xavier argued that the comment could give rise to a misconception and, in writing it on the board, the teacher had sanctioned and possibly spread the error.

When it came to the adding fractions inquiry, they argued that it was the prompt itself that could give rise to a misconception (see 'the prompt' section above). They proposed this alternative prompt to avoid the danger:

While the new prompt might encourage a correct generalisation, there is also the possibility that students could create a different erroneous rule for adding fractions - that is, the sum of the squares of the numerators equals the numerator and the product of the denominators equals the denominator.

Back to the Adding fractions inquiry.

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