# In search of mastery- lesson study in multiplying fractions

The focus on planning for students to have ‘mastery’ in their learning is much discussed at present. The term ‘mastery’ is somewhat vague and I have read many contrasting attempts by others at defining it. I will not attempt a definition; personally, I am still on the ascent in my understanding of this idea and cannot survey the landscape fully yet (if ever!).

Nonetheless, two components that I know are integral to the concept of mastery are that students need to be able to * retain their knowledge over time* and also to be able to

**.**

*transfer and apply it to different contexts*I have written a great deal about the former on this blog in recent years and my understanding of the benefits of spaced learning etc have been well documented. For example, Numeracy Ninjas was designed around research findings on optimum spacing intervals to maximise retention. In this post, I would like to concentrate on the * transfer* element of mastery and share some professional learning from some INSET delivered by two dept colleagues that I have attempted to put into practice.

In our dept meeting this week we learned about the concept of ‘lesson study’- teachers collaborating to design lessons. The process is cyclical and after the lessons are delivered the teachers come back together and collaborate on improving them. The idea is that the in-depth thinking going into the planning is then renewable rather than a one-off, evaporating into the ether. In addition to the lesson study concept, we discussed the importance of focusing planning on the key concepts you want students to learn, what the misconceptions are and how you plan your lessons to help students get around them.

**The resource follows from the planning, the planning doesn’t follow from the resource…**

The practicalities of how to make lesson study a regular feature of a dept’s planning are still something I am reflecting on. Despite the challenges, I am convinced it is a concept worth exploring as a simple, small-scale application of it this week has moved my own practice forward.

I was planning a lesson for my year 10 class on multiplying fractions. I thought about ways of communicating the concept- concrete examples and visual models etc- then thought about all the different permutations that could arise whereby the skill would be needed. From it, I produced these exercises:

First of all, I make no apologies for it being hand-written; I couldn’t find an equivalent resource quickly with a web search, it was faster to write than word processing, it will change as I improve the lesson in the future and it served its purpose.

A few years ago I would have given students 12 questions all in the format of Q1-4. I feel somewhat ashamed to say that now, but we’re all on a journey! In each question in the worksheet above I am trying to vary a single thing relating to how a multiplying fractions question can present itself. For example:

- Q5- squaring a fraction
- Q6- cubing a fraction
- Q7- multiplying three fractions
- Q8- a negative fraction multiplied by a positive fraction
- Q9- negative multiplied by negative
- Q10- algebraic
- Q11- integer times fraction
- Q12- integer times fraction where the answer simplifies to an integer

Undoubtedly I will have missed some possible permutations and the next step should certainly include some questions relating multiplying fractions in other contexts e.g. probability ‘AND’ rules etc.

What really pleased me about the way the lesson went in practice was ** the amount of time students stayed deeply engaged in thinking about the concepts** involved in multiplying fractions.

You retain what you have to think deeply about; as Daniel Willingham puts it:

Memory is the residue of thought– Daniel Willingham, Cognitive Scientist and author of ‘Why Don’t Students Like School?’

If students had only been given questions in the format of Q1-4 they would have quickly reached a level of fluency at which they were no longer thinking about the concepts. They would then simply only be practising becoming faster human calculators rather than mathematicians. By varying the format and types of numbers involved (Q5-12), students spent much longer thinking about the concepts involved in multiplying fractions and less time executing a simple algorithmic process without thinking. It is only a small component in the ‘mastery picture’, but I am going to take this forward in my practice as a technique to improve my students’ contextual transferability of their learning.

I did not work collaboratively with other teachers on the design of the lesson, but did informally seek other’s thoughts after the lesson. This has already been beneficial and I know what I’ll do to improve the questions next time. For example, a colleague pointed out the link to fraction of an amount: 2/3 of 9 and 2/3 × 9 could be discussed… You can think of 2/3 of 9 and 2/3 ‘lots of’ 9 and thus calculate 2×9 / 3×1 rather than 9 ÷ 3 × 2 etc. Had other teachers have delivered the lesson and then we have collaborated on improving it, I’m sure we could think of other ways of making it better too.

Suggested questions to discuss in the comments section:

- Do you use lesson study in your school? If so, how do you make time for it and how do you run it for it to be most effective?
- Do you vary the format/ context of the questions you give students to ensure they remain engaged in thinking about the main concepts you are trying to get them to learn?

These are exaclty the type of questions I would write, sometimes even on the spot during the lesson depending on how the students are getting on. The use of algebra really helps you figure out if they understand the methods. I always worry that I don’t give my students enough repetitive practice on the same type of question so it is great to read this and feel like I am doing something right.

An interesting read. Thank you.

Excellent resource and excellent ideas about collaboration.

I would also mention about cancelling down before you do the multiplication.

In terms of collaboration I might suggest that the lessons are planned across the whole year group and then different levels of questions are asked for each set.

I found this post inspiring and did the same for my next lesson on expanding quadratics. It worked a treat – really engaged students who were thinking about the links between questions.