Maths in the Staff Room – Short Professional Development Meetings
This section provides suggestions and resources for a professional development meeting for teachers that can be led by the maths subject leader or another person with responsibility for developing mathematics teaching and learning in the school
Developing Mental Fluency
- Establish a common understanding of what it mean to work mentally with fluency
1. Setting the scene: Experiencing what it means to work fluently and mentally
Share the aim of the professional development meeting.
You may find it helpful to have read this article from NRICH, or ask your colleagues to read it beforehand.
Ask teachers to discuss in small groups what they currently understand ‘mental fluency’ to mean? Share ideas on a flip chart. Leave the list up to return to later.
Watch this clip from the TV quiz Countdown. Is this an example of fluency? Why?
Reveal this problem and ask colleagues to solve this problem mentally:
Take examples of strategies. Some will have added the 33 and 9 and subtracted the 19. Others may notice that 19 is 10 more than 9 so the missing box will have to be 10 less than 33. The latter is a more sophisticated strategy and that we would want pupils to use to demonstrate fluency i.e. be noticing additive relationships in problems.
Reveal the problem below and ask colleagues to solve it then share their strategies. It is more likely that teachers will now look for a relationship either side of the equals sign but of course this time the relationships are multiplicative, i.e. the unknown is found by scaling up a known number:
Using additive or multiplicative relationships will enable a calculation to be performed more efficiently. In these examples noticing these relationships would mean that fewer calculations are needed.
Provide the following problem and ask the teachers to write down as many related facts as they can in one minute:
7 x 4 = 28
Now ask some of the teachers to provide a related fact and ask them to explain how they used the original fact to derive the new fact.
Explain that we want children to be able to work flexibly when solving problems mentally, and therefore encouraging them to use related facts is one way that this might be done.
Reveal this problem:
Ask colleagues to suggest two ways to work out the missing side of a right angle triangle with height 8cm and area 24cm2. One way may be to work backwards with the formula for the area of a triangle, knowing that 8 x 3 is 24 so 3 is half the length of the base. Or combine a similar triangle with the first triangle to make a rectangle with area 48cm2 and then derive that the missing side is 6 since 8 x 6 is 48. This illustrates how we might encourage pupils to work flexibly by asking for more than one strategy and then comparing these.
In all of the above the problems have been solved with mental fluency but also with deep conceptual understanding of the mathematics involved.
Share this quotation from Russell, 2000, who defines fluency as the ability to
“…provide correct answers quickly … use facts and computation strategies they know to efficiently determine answers that they do not know”
This quotation also highlights the importance of accuracy in calculating mentally and fluently.
Summarise the session by referring to the three key elements of what it means to work mentally and fluently: efficiently, flexibly and accurately.
Refer back to the list created by the teachers at the beginning and compare – are any/all of these elements included in the list?
2. Activities to foster mental fluency (45 mins)
Choose from this selection of NRICH activities to try out:
- Find the Difference
- Roll These Dice
- Twenty Divided Into Six
- Strike it Out: watch this clip from NRICH and see if the rules of the game can be worked out. Then try and play the game
- Totality: watch the video clip with the sound turned off and see if the teachers can work out the rules of the game. Play the game, adapting own rules/ parameters
- Cops and Robbers
- Other number fluency activities.
Reflect on the activities that have been tried out and discuss which of the elements of fluency (efficiency, flexibility and accuracy) the activities develop. Discuss how mental fluency might look in your school by writing collectively a statement about what you hope to achieve in mental fluency by the end of each Key Stage.
e.g. By the end of KS1 all pupils will be able to work with addition and subtraction facts to 20 and multiplication and division facts for the 2, 5 and 10 x tables in an efficient, flexible and accurate manner.
If there's time, provide this Countdown set of numbers for fun. Give the teachers the set of 6 numbers numbers first and then the target figure. Who can find the target? [a solution is (50 + 3) x (5 x 3) – 25 – 4 = 766].