1. Practising mathematics
Dave Hewitt

2. Different types of practice….
Mindless Practice
Unconnected practice
Open Problems
The kind of mimicking of the method explained by a teacher where learners have not understood the mathematics behind a process but appear to be able to get correct answers as long as the questions do not deviate too much. Some examples of the downfall of this type of practice might be where learners find the hypotenuse of twenty or so right-angled triangles with a method along the lines of ‘you square this, you square that, add them together and square root it’ then proceed to do the same for questions such as this:
Find the missing side:
Or pupils may correctly solve many equations of the form: and then answer questions like with .

3. Different types of practice….
Mindless Practice
Unconnected practice
Open Problems
The kind of mimicking of the method explained by a teacher where learners have not understood the mathematics behind a process but appear to be able to get correct answers as long as the questions do not deviate too much. Some examples of the downfall of this type of practice might be where learners find the hypotenuse of twenty or so right-angled triangles with a method along the lines of ‘you square this, you square that, add them together and square root it’ then proceed to do the same for questions such as this:
Find the missing side:
Or pupils may correctly solve many equations of the form: and then answer questions like with .

4. Different types of practice….
Mindless Practice
Unconnected practice
Open Problems
The kind of mimicking of the method explained by a teacher where learners have not understood the mathematics behind a process but appear to be able to get correct answers as long as the questions do not deviate too much. Some examples of the downfall of this type of practice might be where learners find the hypotenuse of twenty or so right-angled triangles with a method along the lines of ‘you square this, you square that, add them together and square root it’ then proceed to do the same for questions such as this:
Find the missing side:
Or pupils may correctly solve many equations of the form: and then answer questions like with .

5. Different types of practice….
Mindless Practice
Unconnected practice
Open Problems
How long does it take to overtake?
The kind of mimicking of the method explained by a teacher where learners have not understood the mathematics behind a process but appear to be able to get correct answers as long as the questions do not deviate too much. Some examples of the downfall of this type of practice might be where learners find the hypotenuse of twenty or so right-angled triangles with a method along the lines of ‘you square this, you square that, add them together and square root it’ then proceed to do the same for questions such as this:
Find the missing side:
Or pupils may correctly solve many equations of the form: and then answer questions like with .

6. Practice Through Progress

7. Practice Through Progress
Progressing as a mathematician…
Practice used as a tool to explore, notice, generalise and justify; developing the mathematician as well as the mathematics.
Progressing through the curriculum
Continue practising previous topics whilst progressing on new topics.

8. Practice Through Progress
Developing the mathematician as well as the mathematics

9. 5 principles:
Exploration of this task offers the opportunity to gain further insights in related areas of mathematics.
Learners have an element of choice.
Opportunities to notice mathematical patterns and relationships, and make conjectures.
Opportunities to justify and prove.
Mathematical situations that can be adapted and extended1.
Firstly, exploration of this task offers the opportunity to gain further insights in related areas of mathematics. Awareness that where the five digits are placed makes a difference to the answer offers a further appreciation of place value and its role in this activity. It also can lead a careful examination of the process of multiplying multi-digit numbers, such as realising that 200×3 and 20×30 give the same result.
Secondly, learners have an element of choice in their work. In this case they are choosing the five numbers and then have to decide where to place those numbers within the boxes. With a traditional exercise learners are rarely asked to make choices about the mathematics. Here the choices made are central to the challenge offered by the task. Initially choices may be relatively random but after time they are likely to become increasingly informed choices which are based upon the mathematical awarenesses they have gained.
The third aspect concerns the desire for learners to notice mathematical patterns and relationships. So they may note that some arrangements of digits produce larger answers than others but it is relating that to the particular positions of these digits and beginning to notice connections between those cases where the answers are relatively large, or indeed relatively small. Noticing such relationships can lead to making conjectures, such as “If I put the two biggest digits in the units boxes, then the answer will be small”. This can be tested through considered choices of where to put the digits for the next example. This can lead to the realisation that other things need to be considered as well and gradually conjectures are tested and refined. Conjectures often involve an element of generalisation; they apply to situations beyond the particular case being considered at the time. Mathematics is full of generalisations. For example, the statement 3+2=5 is a generalisation which applies to pens, chairs, people, etc. It is saying that it does not matter what things you are talking about: three of one thing plus two of the same thing will give you five of that thing. This is not a trivial statement but a profound one as this also means that and is just as true as 3 chairs plus 2 chairs gives 5 chairs. A learner does not have to know anything about fractions or algebra in order to know that these statements must be true as a direct result of knowing that 2+3=5. The ability to generalise from particular situations and also to see the general within the particular (Mason, 1987) can be developed through such tasks. So progress is being made in the ability to notice patterns and relationships, generalise from particular cases, and make and test conjectures.
Fourthly, as conjectures begin to appear robust through testing, the question of why a particular conjecture might be true can arise. This can lead to the desire to try to justify and prove the conjecture. It is our opinion that justification and proof are not always given the emphasis they deserve in the school curriculum; yet they are vital for higher level mathematics. We feel it is important to develop the ability to justify or refute conjectures and this can be done at all levels of education and all topics within the mathematics curriculum. This can be done as part of engaging tasks which also offer the desired practice as well.
Lastly, mathematical situations can be adapted and extended (Prestage and Perks, 2001) and this idea is an important one in mathematics. For example, if a learner has difficulty with the five-box scenario above then they could start by simplifying it to a three-box situation with a two-digit number being multiplied by a single-digit number. Awareness gained from this situation can then lead to having ideas which can be applied to the original five-box situation. It is an important skill to know when to simplify a situation in order to gain initial insights which then can be applied to the original case. Even when someone has developed a conjecture about the five-box case and proved how to make the biggest product whatever the given five digits, this can be extended into considering the product of two numbers, one being an n-digit number and the other an m-digit number (this can, of course, also be extended again to having three numbers, etc.). Adapting and extending situations is an important skill when working on mathematics.
1 (Prestage and Perks, 2001)

10. Tasks: Watch yourself whilst working on the tasks:
What are you practising?
How do the tasks relate to the 5 principles?
Exploration of this task offers the opportunity to gain further insights in related areas of mathematics.
Learners have an element of choice.
Opportunities to notice mathematical patterns and relationships and make conjectures.
Opportunities to justify and prove.
Mathematical situations that can be adapted and extended.
Image from

11. Areas and perimeters

12. Area and perimeter task
Pie chart adding
Juniper green
Squares game
Function routes
Do we meet?

13. Practice Through Progress
Progressing through the curriculum

14. Practice through progress: progressing through the curriculum
Schemes of work often move on too quickly.
It takes time and a lot of practice for learners to become fluent and confident.
Practice through progress allows teachers to move to the next topic and learners to continue practising the previous topic.
When working on finding areas, there are numbers which will be multiplied.
The pedagogic decision not to allow calculators means that there is practice of multiplication as well as working on area.
If more considered thought is given to the dimensions of the areas, then multiplication of fractions or decimals can also be practised.
(Francome & Hewitt, 2017)

16. x y Complete the table of values for the equation :
The four lines below form the boundary of a parallelogram. Draw the lines and work out the area of the parallelogram:
𝑦=2𝑥 , 𝑦=4, 𝑦=2𝑥−1, 𝑦=1 4 7
Find the equations of four lines which bound a parallelogram with area of eleven and five-eighths. And another… x −2
− 4 5
3 4
1 1 2 2 y

17. Teaching: straight line graphs
Progress
Teaching: straight line graphs
Teaching: area of parallelograms
Practising: area of parallelograms
Teaching: multiplication of fractions
Practising: multiplication of fractions
Practising: multiplication of fractions
Practice

18. Practice through progress: progressing through the curriculum
The nature of this type of practice has three key properties:
a learner is learning a new topic and as such might feel a sense of ‘progressing’ through the curriculum;
a learner is gaining further practice of a previous topic taught but in a new context;
the nature of that practice is such that attention has been shifted onto a something new and as such the practising begins to make a journey from being the prime focus of attention to becoming an unconsciously fluent act.
(Francome & Hewitt, 2017)

19. The Mastery Arrow - from early learning to ‘mastery’
Conscious effortful acts Unconscious effortless acts
Focus is on what is to be learned Focus is on a challenge which requires that learning to be used

20. Task:
Choose a topic
In groups create a chain of topics where each new topic can also continue practising the previous topic
Write two example questions for each new topic which shows how the previous one or two topic(s) are still being practised at the same time.
How long a chain can you make?
30 mins

21. Random Topics Choose two topics at random
Challenge is to devise a task so that the first topic is practised whilst working on the second.
15 mins

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