The Mastery session took a look at the following four big ideas:
- A comprehensive, detailed conceptual journey through the mathematics
- A focus on mathematical relationships and making connections
- Supporting students with learning (small steps, appropriate scaffolding)
- Careful planning –Key concepts, representations, difficult points, variation to enable depth
Use of a “Concrete Pictorial Abstract” approach (CPA) in the teaching and learning of mathematical ideas and concepts and to support depth. This included a look at Part Whole models and the use of bar modelling.
3. Variation Theory:
A look at variation theory including considering Conceptual variation as well as Procedural variation and this in turn included a look at Intelligent practice and how connections can be made through arithmetic practise to deepen understanding of number sense and fluency.
Practice is most effective when it is intelligent practice, i.e. where the teacher is advised to avoid mechanical repetition and to create an appropriate path for practising the thinking process with increasing creativity. (Gu 2004)
4. Mathematical thinking:
This involved a look at some rich tasks and open-ended problems. The Nrich website was publicised and some of the activities were tried out. An article published on the Nrich website identifies the following five steps in the ‘chain of reasoning’ involved in mathematical thinking.
Step one: Describing: simply tells what they did.
Step two: Explaining: offers some reasons for what they did. These may or may not be correct. The argument may yet not hang together coherently. This is the beginning of inductive reasoning.
Step three: Convincing: confident that their chain of reasoning is right and may use words such as, ‘I reckon’ or ‘without doubt’. The underlying mathematical argument may or may not be accurate yet is likely to have more coherence and completeness than the explaining stage. (Inductive reasoning).
Step four: Justifying: a correct logical argument that has a complete chain of reasoning to it and uses words such as ‘because’, ‘therefore’, ‘and so’, ‘that leads to’ ...
Step five: Proving: a watertight argument that is mathematically sound, often based on generalisations and underlying structure. (Deductive reasoning).
Delegates engaged with the following problem and modelled it using multi-link cubes.
The connections with square numbers and patterns was made and was greeted with lots of comments like “Oh that’s why square numbers are called square numbers!”
We investigated other problems too, such as the Deca tree and Mystery Matrix.
Summary of points from the Mastery session included:
- Represent the mathematics in ways that are accessible
- Use whole class Ping Pong style
- Avoid cognitive overload
- Repetition and stem sentences are valuable
- Learn facts to automaticity
- Encouragement of variation and intelligent practice
- Dong Nao Jin (Use your head!)
- Mistakes are valuable
- Depth over speed
- Ability is not fixed
- Success through hard work
Fluency Session - Seven things to try!
What a lot of fun we had! We tried and evaluated so many activities to see how they could promote fluency in the essential maths skills. Here are seven things that you could bring to your classroom.
5. Rapid Recall Whiteboards
It was great to use the Rapid Recall Whiteboards and to learn how they could be used in the classroom.
Also, the other Propeller games that help to encourage repetition, collaboration, conversation and competition while ensuring that learning can take place!
6. How to practice multiplication facts
We watched a teacher practising multiplication facts with children in upper key stage 2 using a video from the NCETM website:
Link to the video and other resources is here: https://www.ncetm.org.uk/resources/40533
This provided an idea for an activity involving triangles and applying the inverse calculations for tables facts
7. How Close to 100?
We played “How Close to 100?” This is a game produced by Jo Boaler and can be found on her Youcubed website (https://www.youcubed.org/task/how-to-close-100/). It’s a great one for multiplication facts and promoting the use of arrays.
8. 'Strike it out'
The Nrich game “Strike it out” also provided lots of opportunity for fluency with addition and subtraction. More can be found here http://nrich.maths.org/6589
9. Place value game
We played a couple of games that encouraged understanding of mathematical Place value. Including this one where players generated a 2-digit number by rolling die and deciding whether to place the digit in the Tens or Ones place. Repeat so that there are two 2-digit numbers. The closest to 100 wins.
10. Four digit targets
Another great Nrich game https://nrich.maths.org/6342
The idea is to arrange two sets of 0-9 digits in the five boxes to make four-digit numbers as close to the target number as possible. You may use each digit once only. E.g. 3457 is a four digit odd number.
11. Treasure Hunt
This idea provides the vehicle for any maths area of work and is great for encouraging collaboration and team work.
Set up 10 or 12 cards (any number can be used) to start with. Put the question on the main part of the card and it’s answer on the top of the next card. Repeat until you have put the answer to the last question on the first card. Make sure all the answers are unique!
As always, we thoroughly enjoyed the course with @RuthBull! The teachers who attended had a great time and took away many ideas to use in their schools.