A lot of students are taught traditional algorithms for fraction operations without a solid foundation of conceptual understanding. By the time they finish the middle school years of their education, many students have the rules all mixed up – and no sense of the reasonableness of their answers. Division of fractions, in particular, is one of the most rote procedures (“Ours is not to reason why-just invert and multiply”) in arithmetic instruction. Students easily confuse the series of steps for correct calculations, and even if they can divide fractions accurately, they do not know why the procedures they used make sense.

 

Apart from wanting to teach fraction division, I was trying in this lesson to continue building on a classroom culture where mistakes were seen as useful. I was beginning to shift the emphasis in classroom discussions from whether an answer was correct to whether an explanation was clear and convincing. By use of the mini whiteboards for students to write answers in their table groups, it also gave students some anonymity if they did feel uncomfortable in making a mistake, but also the opportunity to discuss an idea presented without giving away the fact they were the author. The mini whiteboards also made it easier for students to share their ideas in their table groups.

 

The main body of the lesson starts with two very simple questions – 6÷2 = 3 and 6÷3 = 2. These were given for three reasons. Firstly, so I could lay down my expectation as to how I wanted students to lay out their mini whiteboard work. Secondly, the next question (6 ÷½) students often give an answer of 2 or 3. By considering these questions first, I had hoped to alleviate these common errors. I felt I was successful in doing this, as you will note none of the groups gave an answer other than 12 to the 6÷½ question posed, although some students not visible to the camera required some further convincing. Finally, these simple questions allowed me the opportunity to guide students to how I wanted them to present their fraction diagrams without actually giving them a diagram for the first fraction division question presented.

 

The video ends when the class was just about to move out for lunchtime break. To reinforce learning, I decided students had to earn a ‘ticket out’ of the class by being able to answer fraction division questions similar to the ones studied in the lesson. Apart from that, I had Jack (the boy who arrived late) and Eden (the boy drawing pictures on the mini whiteboard) to stay for a few minutes to chat about my concerns.

 

The follow on lesson started with some quick fraction division warm up questions on the whiteboard followed by a discussion and write up on the whiteboard (for students to copy down in their notebooks) about methods for dividing fractions, being;

1. draw a diagram (as shown in the lesson video)

2. convert all fractions into fractions with common denominators

3. the ‘invert and multiply’ method

 

Points for improvement

Some parts of the instruction seemed to drag. Upon reflection it was at these times I was out of view of the camera discussing ideas with student table groups towards the middle and back of the classroom. Enquiry based lessons can at times go slower than what the traditional approach offers, but this is because instruction is aiming for depth rather than ‘getting the right answer and moving on’.

 

At times the students in view of the camera became fidgety while waiting for the other groups to record on mini whiteboard their thinking. The Brain Break in the second half of the lesson was designed to give such students some ‘wiggle time’ to shake off any excess energy.

 

If I were to do this lesson again, I would give students three mini whiteboards to write on rather than just one. Often students had written work which would have made a good discussion point, only to have them wipe it off. With three mini whiteboards and the instruction not to wipe any board clean until given direction by the teacher, I felt we would have had some additional discussion that was curtailed in this lesson video.